LMFDB - Embedded newform 378.4.k.d.269.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(215,378)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("378.215");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	378.k (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$22.3027219822$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} + 2 x^{18} - 32 x^{17} - 669 x^{16} + 1752 x^{15} - 1654 x^{14} + 13878 x^{13} + \cdots + 2458624 $ x^20 - 2x^19 + 2x^18 - 32x^17 - 669x^16 + 1752x^15 - 1654x^14 + 13878x^13 + 494085x^12 - 780096x^11 + 556152x^10 - 2898760x^9 - 1242580x^8 + 9104368x^7 - 4109728x^6 + 12152000x^5 + 28375312x^4 - 23774016x^3 + 7683200x^2 - 6146560*x + 2458624
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{18}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.5
Root			$-1.40021 + 0.375184i$ of defining polynomial
Character	$\chi$	$=$	378.269
Dual form			378.4.k.d.215.5

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.73205 + 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(10.6672 + 18.4761i) q^{5} +(-18.4334 + 1.79115i) q^{7} +8.00000i q^{8} +O(q^{10})$	\(q+(-1.73205 + 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(10.6672 + 18.4761i) q^{5} +(-18.4334 + 1.79115i) q^{7} +8.00000i q^{8} +(-36.9523 - 21.3344i) q^{10} +(-46.0414 - 26.5820i) q^{11} -81.4296i q^{13} +(30.1365 - 21.5358i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(46.7297 - 80.9382i) q^{17} +(-3.55276 + 2.05119i) q^{19} +85.3376 q^{20} +106.328 q^{22} +(73.0455 - 42.1728i) q^{23} +(-165.078 + 285.924i) q^{25} +(81.4296 + 141.040i) q^{26} +(-30.6622 + 67.4376i) q^{28} -87.6823i q^{29} +(-94.0150 - 54.2796i) q^{31} +(27.7128 + 16.0000i) q^{32} +186.919i q^{34} +(-229.727 - 321.472i) q^{35} +(20.7410 + 35.9245i) q^{37} +(4.10238 - 7.10553i) q^{38} +(-147.809 + 85.3376i) q^{40} +381.404 q^{41} +193.793 q^{43} +(-184.166 + 106.328i) q^{44} +(-84.3457 + 146.091i) q^{46} +(-47.8563 - 82.8895i) q^{47} +(336.584 - 66.0340i) q^{49} -660.314i q^{50} +(-282.080 - 162.859i) q^{52} +(174.922 + 100.991i) q^{53} -1134.22i q^{55} +(-14.3292 - 147.468i) q^{56} +(87.6823 + 151.870i) q^{58} +(-179.989 + 311.749i) q^{59} +(-219.018 + 126.450i) q^{61} +217.118 q^{62} -64.0000 q^{64} +(1504.50 - 868.626i) q^{65} +(63.6002 - 110.159i) q^{67} +(-186.919 - 323.753i) q^{68} +(719.371 + 327.080i) q^{70} +283.152i q^{71} +(-256.393 - 148.029i) q^{73} +(-71.8490 - 41.4820i) q^{74} +16.4095i q^{76} +(896.314 + 407.531i) q^{77} +(-627.622 - 1087.07i) q^{79} +(170.675 - 295.618i) q^{80} +(-660.611 + 381.404i) q^{82} -319.040 q^{83} +1993.90 q^{85} +(-335.659 + 193.793i) q^{86} +(212.656 - 368.331i) q^{88} +(-379.377 - 657.100i) q^{89} +(145.852 + 1501.03i) q^{91} -337.383i q^{92} +(165.779 + 95.7126i) q^{94} +(-75.7961 - 43.7609i) q^{95} -1576.58i q^{97} +(-516.946 + 450.958i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 40 q^{4} + 4 q^{7}+O(q^{10}) $ 20 * q + 40 * q^4 + 4 * q^7	$ 20 q + 40 q^{4} + 4 q^{7} - 24 q^{10} - 160 q^{16} + 408 q^{19} + 240 q^{22} - 646 q^{25} + 56 q^{28} - 102 q^{31} + 194 q^{37} - 96 q^{40} - 2332 q^{43} - 624 q^{46} + 2840 q^{49} - 648 q^{52} + 96 q^{58} + 1878 q^{61} - 1280 q^{64} - 386 q^{67} + 3672 q^{70} + 1788 q^{73} + 814 q^{79} - 672 q^{82} - 4560 q^{85} + 480 q^{88} + 2724 q^{91} - 1536 q^{94}+O(q^{100}) $ 20 * q + 40 * q^4 + 4 * q^7 - 24 * q^10 - 160 * q^16 + 408 * q^19 + 240 * q^22 - 646 * q^25 + 56 * q^28 - 102 * q^31 + 194 * q^37 - 96 * q^40 - 2332 * q^43 - 624 * q^46 + 2840 * q^49 - 648 * q^52 + 96 * q^58 + 1878 * q^61 - 1280 * q^64 - 386 * q^67 + 3672 * q^70 + 1788 * q^73 + 814 * q^79 - 672 * q^82 - 4560 * q^85 + 480 * q^88 + 2724 * q^91 - 1536 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−1.73205	+	1.00000i	−0.612372	+	0.353553i
$2$	−1.73205	+	1.00000i	−0.612372	+	0.353553i
$3$		0			0
$3$		0			0
$4$	2.00000	−	3.46410i	0.250000	−	0.433013i
$5$	10.6672	+	18.4761i	0.954104	+	1.65256i	0.736406	+	0.676540i	$0.236521\pi$
$5$	10.6672	+	18.4761i	0.954104	+	1.65256i	0.217697	+	0.976016i	$0.430145\pi$
$6$		0			0
$7$	−18.4334	+	1.79115i	−0.995312	+	0.0967128i
$7$	−18.4334	+	1.79115i	−0.995312	+	0.0967128i
$8$			8.00000i			0.353553i
$9$		0			0
$10$	−36.9523	−	21.3344i	−1.16853	−	0.674653i
$11$	−46.0414	−	26.5820i	−1.26200	−	0.728616i	−0.288539	−	0.957468i	$-0.593170\pi$
$11$	−46.0414	−	26.5820i	−1.26200	−	0.728616i	−0.973461	+	0.228852i	$0.926503\pi$
$12$		0			0
$13$		−	81.4296i		−	1.73727i	−0.495453	−	0.868635i	$-0.664998\pi$
$13$		−	81.4296i		−	1.73727i	0.495453	−	0.868635i	$-0.335002\pi$
$14$	30.1365	−	21.5358i	0.575309	−	0.411120i
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.125000	−	0.216506i
$17$	46.7297	−	80.9382i	0.666683	−	1.15473i	−0.312142	−	0.950035i	$-0.601047\pi$
$17$	46.7297	−	80.9382i	0.666683	−	1.15473i	0.978826	−	0.204694i	$-0.0656200\pi$
$18$		0			0
$19$	−3.55276	+	2.05119i	−0.0428979	+	0.0247671i	−0.521296	−	0.853376i	$-0.674551\pi$
$19$	−3.55276	+	2.05119i	−0.0428979	+	0.0247671i	0.478398	+	0.878143i	$0.341218\pi$
$20$	85.3376			0.954104
$21$		0			0
$22$	106.328			1.03042
$23$	73.0455	−	42.1728i	0.662219	−	0.382332i	−0.130903	−	0.991395i	$-0.541788\pi$
$23$	73.0455	−	42.1728i	0.662219	−	0.382332i	0.793122	+	0.609063i	$0.208454\pi$
$24$		0			0
$25$	−165.078	+	285.924i	−1.32063	+	2.28739i
$26$	81.4296	+	141.040i	0.614218	+	1.06386i
$27$		0			0
$28$	−30.6622	+	67.4376i	−0.206950	+	0.455161i
$29$		−	87.6823i		−	0.561455i	−0.959788	−	0.280728i	$-0.909424\pi$
$29$		−	87.6823i		−	0.561455i	0.959788	−	0.280728i	$-0.0905757\pi$
$30$		0			0
$31$	−94.0150	−	54.2796i	−0.544696	−	0.314481i	0.202284	−	0.979327i	$-0.435164\pi$
$31$	−94.0150	−	54.2796i	−0.544696	−	0.314481i	−0.746980	+	0.664846i	$0.768497\pi$
$32$	27.7128	+	16.0000i	0.153093	+	0.0883883i
$33$		0			0
$34$			186.919i			0.942833i
$35$	−229.727	−	321.472i	−1.10945	−	1.55254i
$36$		0			0
$37$	20.7410	+	35.9245i	0.0921568	+	0.159620i	0.908418	−	0.418062i	$-0.137291\pi$
$37$	20.7410	+	35.9245i	0.0921568	+	0.159620i	−0.816262	+	0.577682i	$0.803957\pi$
$38$	4.10238	−	7.10553i	0.0175130	−	0.0303334i
$39$		0			0
$40$	−147.809	+	85.3376i	−0.584267	+	0.337327i
$41$	381.404			1.45281			0.726406	−	0.687266i	$-0.241189\pi$
$41$	381.404			1.45281			0.726406	+	0.687266i	$0.241189\pi$
$42$		0			0
$43$	193.793			0.687282			0.343641	−	0.939101i	$-0.388340\pi$
$43$	193.793			0.687282			0.343641	+	0.939101i	$0.388340\pi$
$44$	−184.166	+	106.328i	−0.631000	+	0.364308i
$45$		0			0
$46$	−84.3457	+	146.091i	−0.270350	+	0.468260i
$47$	−47.8563	−	82.8895i	−0.148522	−	0.257248i	0.782159	−	0.623079i	$-0.214118\pi$
$47$	−47.8563	−	82.8895i	−0.148522	−	0.257248i	−0.930682	+	0.365830i	$0.880785\pi$
$48$		0			0
$49$	336.584	−	66.0340i	0.981293	−	0.192519i
$50$		−	660.314i		−	1.86765i
$51$		0			0
$52$	−282.080	−	162.859i	−0.752260	−	0.434317i
$53$	174.922	+	100.991i	0.453347	+	0.261740i	0.709243	−	0.704965i	$-0.249037\pi$
$53$	174.922	+	100.991i	0.453347	+	0.261740i	−0.255896	+	0.966704i	$0.582370\pi$
$54$		0			0
$55$		−	1134.22i		−	2.78070i
$56$	−14.3292	−	147.468i	−0.0341931	−	0.351896i
$57$		0			0
$58$	87.6823	+	151.870i	0.198504	+	0.343820i
$59$	−179.989	+	311.749i	−0.397161	+	0.687904i	−0.993374	−	0.114923i	$-0.963338\pi$
$59$	−179.989	+	311.749i	−0.397161	+	0.687904i	0.596213	+	0.802826i	$0.296671\pi$
$60$		0			0
$61$	−219.018	+	126.450i	−0.459711	+	0.265414i	−0.711923	−	0.702258i	$-0.752175\pi$
$61$	−219.018	+	126.450i	−0.459711	+	0.265414i	0.252212	+	0.967672i	$0.418842\pi$
$62$	217.118			0.444743
$63$		0			0
$64$	−64.0000			−0.125000
$65$	1504.50	−	868.626i	2.87093	−	1.65754i
$66$		0			0
$67$	63.6002	−	110.159i	0.115970	−	0.200866i	−0.802197	−	0.597059i	$-0.796336\pi$
$67$	63.6002	−	110.159i	0.115970	−	0.200866i	0.918167	+	0.396193i	$0.129669\pi$
$68$	−186.919	−	323.753i	−0.333342	−	0.577365i
$69$		0			0
$70$	719.371	+	327.080i	1.22830	+	0.558478i
$71$			283.152i			0.473296i	0.971596	+	0.236648i	$0.0760487\pi$
$71$			283.152i			0.473296i	−0.971596	+	0.236648i	$0.923951\pi$
$72$		0			0
$73$	−256.393	−	148.029i	−0.411076	−	0.237335i	0.280176	−	0.959949i	$-0.409607\pi$
$73$	−256.393	−	148.029i	−0.411076	−	0.237335i	−0.691252	+	0.722614i	$0.742940\pi$
$74$	−71.8490	−	41.4820i	−0.112869	−	0.0651647i
$75$		0			0
$76$			16.4095i			0.0247671i
$77$	896.314	+	407.531i	1.32655	+	0.603149i
$78$		0			0
$79$	−627.622	−	1087.07i	−0.893835	−	1.54817i	−0.835240	−	0.549886i	$-0.814671\pi$
$79$	−627.622	−	1087.07i	−0.893835	−	1.54817i	−0.0585954	−	0.998282i	$-0.518662\pi$
$80$	170.675	−	295.618i	0.238526	−	0.413139i
$81$		0			0
$82$	−660.611	+	381.404i	−0.889662	+	0.513647i
$83$	−319.040			−0.421918			−0.210959	−	0.977495i	$-0.567659\pi$
$83$	−319.040			−0.421918			−0.210959	+	0.977495i	$0.567659\pi$
$84$		0			0
$85$	1993.90			2.54434
$86$	−335.659	+	193.793i	−0.420872	+	0.242991i
$87$		0			0
$88$	212.656	−	368.331i	0.257605	−	0.446185i
$89$	−379.377	−	657.100i	−0.451841	−	0.782612i	0.546659	−	0.837355i	$-0.315899\pi$
$89$	−379.377	−	657.100i	−0.451841	−	0.782612i	−0.998500	+	0.0547430i	$0.982566\pi$
$90$		0			0
$91$	145.852	+	1501.03i	0.168016	+	1.72913i
$92$		−	337.383i		−	0.382332i
$93$		0			0
$94$	165.779	+	95.7126i	0.181902	+	0.105021i
$95$	−75.7961	−	43.7609i	−0.0818581	−	0.0472608i
$96$		0			0
$97$		−	1576.58i		−	1.65028i	−0.564928	−	0.825141i	$-0.691096\pi$
$97$		−	1576.58i		−	1.65028i	0.564928	−	0.825141i	$-0.308904\pi$
$98$	−516.946	+	450.958i	−0.532851	+	0.464833i
$99$		0			0
$100$	660.314	+	1143.70i	0.660314	+	1.14370i
$101$	387.236	−	670.713i	0.381500	−	0.660777i	−0.609777	−	0.792573i	$-0.708741\pi$
$101$	387.236	−	670.713i	0.381500	−	0.660777i	0.991277	+	0.131796i	$0.0420744\pi$
$102$		0			0
$103$	−649.489	+	374.983i	−0.621321	+	0.358720i	−0.777383	−	0.629027i	$-0.783453\pi$
$103$	−649.489	+	374.983i	−0.621321	+	0.358720i	0.156062	+	0.987747i	$0.450120\pi$
$104$	651.437			0.614218
$105$		0			0
$106$	−403.965			−0.370156
$107$	558.765	−	322.603i	0.504840	−	0.291469i	−0.225870	−	0.974157i	$-0.572523\pi$
$107$	558.765	−	322.603i	0.504840	−	0.291469i	0.730710	+	0.682688i	$0.239189\pi$
$108$		0			0
$109$	403.739	−	699.296i	0.354781	−	0.614499i	−0.632299	−	0.774724i	$-0.717889\pi$
$109$	403.739	−	699.296i	0.354781	−	0.614499i	0.987081	+	0.160225i	$0.0512220\pi$
$110$	1134.22	+	1964.53i	0.983127	+	1.70283i
$111$		0			0
$112$	172.286	+	241.092i	0.145353	+	0.203402i
$113$		−	154.242i		−	0.128406i	−0.997937	−	0.0642030i	$-0.979549\pi$
$113$		−	154.242i		−	0.128406i	0.997937	−	0.0642030i	$-0.0204505\pi$
$114$		0			0
$115$	1558.38	+	899.733i	1.26365	+	0.729570i
$116$	−303.740	−	175.365i	−0.243117	−	0.140364i
$117$		0			0
$118$		−	719.954i		−	0.561671i
$119$	−716.417	+	1575.67i	−0.551881	+	1.21379i
$120$		0			0
$121$	747.707	+	1295.07i	0.561764	+	0.973003i
$122$	252.900	−	438.036i	0.187676	−	0.325065i
$123$		0			0
$124$	−376.060	+	217.118i	−0.272348	+	0.157240i
$125$	−4376.90			−3.13185
$126$		0			0
$127$	−2351.54			−1.64304			−0.821518	−	0.570182i	$-0.806873\pi$
$127$	−2351.54			−1.64304			−0.821518	+	0.570182i	$0.806873\pi$
$128$	110.851	−	64.0000i	0.0765466	−	0.0441942i
$129$		0			0
$130$	−1737.25	+	3009.01i	−1.17205	+	2.03006i
$131$	−813.733	−	1409.43i	−0.542719	−	0.940017i	−0.998747	−	0.0500514i	$-0.984061\pi$
$131$	−813.733	−	1409.43i	−0.542719	−	0.940017i	0.456028	−	0.889966i	$-0.349272\pi$
$132$		0			0
$133$	61.8157	−	44.1740i	0.0403015	−	0.0287998i
$134$			254.401i			0.164007i
$135$		0			0
$136$	647.506	+	373.838i	0.408259	+	0.235708i
$137$	1001.70	+	578.332i	0.624679	+	0.360658i	0.778688	−	0.627411i	$-0.215885\pi$
$137$	1001.70	+	578.332i	0.624679	+	0.360658i	−0.154010	+	0.988069i	$0.549219\pi$
$138$		0			0
$139$		−	1863.67i		−	1.13722i	−0.822606	−	0.568612i	$-0.807481\pi$
$139$		−	1863.67i		−	1.13722i	0.822606	−	0.568612i	$-0.192519\pi$
$140$	−1573.07	+	152.852i	−0.949631	+	0.0922740i
$141$		0			0
$142$	−283.152	−	490.434i	−0.167335	−	0.289833i
$143$	−2164.56	+	3749.13i	−1.26580	+	2.19244i
$144$		0			0
$145$	1620.03	−	935.325i	0.927836	−	0.535686i
$146$	592.114			0.335642
$147$		0			0
$148$	165.928			0.0921568
$149$	383.626	−	221.487i	0.210925	−	0.121778i	−0.390816	−	0.920469i	$-0.627807\pi$
$149$	383.626	−	221.487i	0.210925	−	0.121778i	0.601741	+	0.798691i	$0.294474\pi$
$150$		0			0
$151$	917.610	−	1589.35i	0.494530	−	0.856551i	−0.505450	−	0.862856i	$-0.668674\pi$
$151$	917.610	−	1589.35i	0.494530	−	0.856551i	0.999980	+	0.00630486i	$0.00200691\pi$
$152$	−16.4095	−	28.4221i	−0.00875650	−	0.0151667i
$153$		0			0
$154$	−1959.99	+	190.449i	−1.02559	+	0.0996547i
$155$		−	2316.04i		−	1.20019i
$156$		0			0
$157$	−2742.86	−	1583.59i	−1.39429	−	0.804996i	−0.400507	−	0.916294i	$-0.631166\pi$
$157$	−2742.86	−	1583.59i	−1.39429	−	0.804996i	−0.993787	+	0.111298i	$0.964499\pi$
$158$	2174.15	+	1255.24i	1.09472	+	0.632037i
$159$		0			0
$160$			682.701i			0.337327i
$161$	−1270.94	+	908.226i	−0.622139	+	0.444585i
$162$		0			0
$163$	−869.560	−	1506.12i	−0.417848	−	0.723734i	0.577875	−	0.816125i	$-0.303882\pi$
$163$	−869.560	−	1506.12i	−0.417848	−	0.723734i	−0.995723	+	0.0923914i	$0.970549\pi$
$164$	762.808	−	1321.22i	0.363203	−	0.629086i
$165$		0			0
$166$	552.593	−	319.040i	0.258371	−	0.149170i
$167$	1811.27			0.839283			0.419642	−	0.907690i	$-0.362156\pi$
$167$	1811.27			0.839283			0.419642	+	0.907690i	$0.362156\pi$
$168$		0			0
$169$	−4433.78			−2.01811
$170$	−3453.54	+	1993.90i	−1.55808	+	0.899560i
$171$		0			0
$172$	387.585	−	671.318i	0.171820	−	0.297602i
$173$	2020.67	+	3499.90i	0.888027	+	1.53811i	0.842204	+	0.539159i	$0.181258\pi$
$173$	2020.67	+	3499.90i	0.888027	+	1.53811i	0.0458234	+	0.998950i	$0.485409\pi$
$174$		0			0
$175$	2530.83	−	5566.25i	1.09322	−	2.40439i
$176$			850.625i			0.364308i
$177$		0			0
$178$	1314.20	+	758.754i	0.553390	+	0.319500i
$179$	−2692.00	−	1554.23i	−1.12408	−	0.648986i	−0.181638	−	0.983366i	$-0.558140\pi$
$179$	−2692.00	−	1554.23i	−1.12408	−	0.648986i	−0.942438	+	0.334380i	$0.891473\pi$
$180$		0			0
$181$		−	467.849i		−	0.192127i	−0.995375	−	0.0960633i	$-0.969375\pi$
$181$		−	467.849i		−	0.192127i	0.995375	−	0.0960633i	$-0.0306251\pi$
$182$	−1753.65	−	2454.00i	−0.714227	−	0.999466i
$183$		0			0
$184$	337.383	+	584.364i	0.135175	+	0.234130i
$185$	−442.497	+	766.428i	−0.175854	+	0.304589i
$186$		0			0
$187$	−4303.00	+	2484.34i	−1.68271	+	0.971513i
$188$	−382.850			−0.148522
$189$		0			0
$190$	175.044			0.0668368
$191$	762.438	−	440.194i	0.288838	−	0.166761i	−0.348580	−	0.937279i	$-0.613336\pi$
$191$	762.438	−	440.194i	0.288838	−	0.166761i	0.637418	+	0.770518i	$0.280003\pi$
$192$		0			0
$193$	−1709.27	+	2960.54i	−0.637492	+	1.10417i	0.348489	+	0.937313i	$0.386695\pi$
$193$	−1709.27	+	2960.54i	−0.637492	+	1.10417i	−0.985981	+	0.166856i	$0.946639\pi$
$194$	1576.58	+	2730.71i	0.583462	+	1.01059i
$195$		0			0
$196$	444.419	−	1298.03i	0.161960	−	0.473042i
$197$			2403.67i			0.869310i	0.900597	+	0.434655i	$0.143130\pi$
$197$			2403.67i			0.869310i	−0.900597	+	0.434655i	$0.856870\pi$
$198$		0			0
$199$	3229.49	+	1864.55i	1.15041	+	0.664192i	0.948988	−	0.315311i	$-0.102109\pi$
$199$	3229.49	+	1864.55i	1.15041	+	0.664192i	0.201427	+	0.979504i	$0.435442\pi$
$200$	−2287.39	−	1320.63i	−0.808716	−	0.466912i
$201$		0			0
$202$			1548.95i			0.539522i
$203$	157.052	+	1616.29i	0.0542999	+	0.558823i
$204$		0			0
$205$	4068.51	+	7046.87i	1.38613	+	2.40085i
$206$	749.966	−	1298.98i	0.253653	−	0.439341i
$207$		0			0
$208$	−1128.32	+	651.437i	−0.376130	+	0.217159i
$209$	218.099			0.0721829
$210$		0			0
$211$	−1531.59			−0.499710			−0.249855	−	0.968283i	$-0.580383\pi$
$211$	−1531.59			−0.499710			−0.249855	+	0.968283i	$0.580383\pi$
$212$	699.687	−	403.965i	0.226673	−	0.130870i
$213$		0			0
$214$	−645.206	+	1117.53i	−0.206100	+	0.356976i
$215$	2067.23	+	3580.54i	0.655738	+	1.13577i
$216$		0			0
$217$	1830.24	+	832.165i	0.572557	+	0.260327i
$218$			1614.96i			0.501737i
$219$		0			0
$220$	−3929.06	−	2268.45i	−1.20408	−	0.695175i
$221$	−6590.77	−	3805.18i	−2.00608	−	1.15821i
$222$		0			0
$223$			5610.92i			1.68491i	0.538766	+	0.842456i	$0.318891\pi$
$223$			5610.92i			1.68491i	−0.538766	+	0.842456i	$0.681109\pi$
$224$	−539.501	−	245.297i	−0.160924	−	0.0731680i
$225$		0			0
$226$	154.242	+	267.155i	0.0453984	+	0.0786323i
$227$	−170.270	+	294.916i	−0.0497851	+	0.0862303i	−0.889844	−	0.456265i	$-0.849187\pi$
$227$	−170.270	+	294.916i	−0.0497851	+	0.0862303i	0.840059	+	0.542495i	$0.182520\pi$
$228$		0			0
$229$	5536.24	−	3196.35i	1.59758	−	0.922362i	0.605625	−	0.795750i	$-0.292923\pi$
$229$	5536.24	−	3196.35i	1.59758	−	0.922362i	0.991952	−	0.126612i	$-0.0404102\pi$
$230$	−3598.93			−1.03177
$231$		0			0
$232$	701.459			0.198504
$233$	−639.169	+	369.024i	−0.179714	+	0.103758i	−0.587158	−	0.809472i	$-0.699753\pi$
$233$	−639.169	+	369.024i	−0.179714	+	0.103758i	0.407444	+	0.913230i	$0.366420\pi$
$234$		0			0
$235$	1020.99	−	1768.40i	0.283412	−	0.490883i
$236$	719.954	+	1247.00i	0.198581	+	0.343952i
$237$		0			0
$238$	−334.799	−	3445.56i	−0.0911840	−	0.938413i
$239$			2529.61i			0.684632i	0.939585	+	0.342316i	$0.111211\pi$
$239$			2529.61i			0.684632i	−0.939585	+	0.342316i	$0.888789\pi$
$240$		0			0
$241$	−12.2097	−	7.04925i	−0.00326346	−	0.00188416i	0.498367	−	0.866966i	$-0.333933\pi$
$241$	−12.2097	−	7.04925i	−0.00326346	−	0.00188416i	−0.501631	+	0.865082i	$0.667266\pi$
$242$	−2590.13	−	1495.41i	−0.688017	−	0.397227i
$243$		0			0
$244$			1011.60i			0.265414i
$245$	4810.46	+	5514.37i	1.25440	+	1.43796i
$246$		0			0
$247$	167.028	+	289.300i	0.0430272	+	0.0745252i
$248$	434.237	−	752.120i	0.111186	−	0.192579i
$249$		0			0
$250$	7581.01	−	4376.90i	1.91786	−	1.10728i
$251$	3804.24			0.956660			0.478330	−	0.878180i	$-0.341242\pi$
$251$	3804.24			0.956660			0.478330	+	0.878180i	$0.341242\pi$
$252$		0			0
$253$	−4484.16			−1.11429
$254$	4072.99	−	2351.54i	1.00615	−	0.580901i
$255$		0			0
$256$	−128.000	+	221.703i	−0.0312500	+	0.0541266i
$257$	663.399	+	1149.04i	0.161018	+	0.278892i	0.935234	−	0.354030i	$-0.115189\pi$
$257$	663.399	+	1149.04i	0.161018	+	0.278892i	−0.774216	+	0.632922i	$0.781856\pi$
$258$		0			0
$259$	−446.674	−	625.062i	−0.107162	−	0.149959i
$260$		−	6949.01i		−	1.65754i
$261$		0			0
$262$	2818.85	+	1627.47i	0.664692	+	0.383760i
$263$	−2508.03	−	1448.01i	−0.588030	−	0.339499i	0.176288	−	0.984339i	$-0.443591\pi$
$263$	−2508.03	−	1448.01i	−0.588030	−	0.339499i	−0.764318	+	0.644839i	$0.776924\pi$
$264$		0			0
$265$			4309.17i			0.998907i
$266$	−62.8939	+	138.327i	−0.0144973	+	0.0318849i
$267$		0			0
$268$	−254.401	−	440.635i	−0.0579851	−	0.100433i
$269$	1610.81	−	2790.01i	0.365104	−	0.632378i	−0.623689	−	0.781673i	$-0.714367\pi$
$269$	1610.81	−	2790.01i	0.365104	−	0.632378i	0.988793	+	0.149294i	$0.0477002\pi$
$270$		0			0
$271$	3374.30	−	1948.15i	0.756363	−	0.436686i	−0.0716256	−	0.997432i	$-0.522819\pi$
$271$	3374.30	−	1948.15i	0.756363	−	0.436686i	0.827988	+	0.560745i	$0.189485\pi$
$272$	−1495.35			−0.333342
$273$		0			0
$274$	−2313.33			−0.510048
$275$	15200.9	−	8776.23i	3.33326	−	1.92446i
$276$		0			0
$277$	3104.44	−	5377.05i	0.673386	−	1.16634i	−0.303552	−	0.952815i	$-0.598173\pi$
$277$	3104.44	−	5377.05i	0.673386	−	1.16634i	0.976938	−	0.213523i	$-0.0684940\pi$
$278$	1863.67	+	3227.97i	0.402069	+	0.696405i
$279$		0			0
$280$	2571.78	−	1837.81i	0.548904	−	0.392251i
$281$		−	8367.31i		−	1.77634i	−0.459514	−	0.888170i	$-0.651976\pi$
$281$		−	8367.31i		−	1.77634i	0.459514	−	0.888170i	$-0.348024\pi$
$282$		0			0
$283$	−3150.57	−	1818.98i	−0.661774	−	0.382075i	0.131179	−	0.991359i	$-0.458124\pi$
$283$	−3150.57	−	1818.98i	−0.661774	−	0.382075i	−0.792953	+	0.609283i	$0.791457\pi$
$284$	980.868	+	566.304i	0.204943	+	0.118324i
$285$		0			0
$286$		−	8658.25i		−	1.79012i
$287$	−7030.59	+	683.150i	−1.44600	+	0.140505i
$288$		0			0
$289$	−1910.83	−	3309.66i	−0.388934	−	0.673653i
$290$	−1870.65	+	3240.06i	−0.378787	+	0.656079i
$291$		0			0
$292$	−1025.57	+	592.114i	−0.205538	+	0.118667i
$293$	−6359.39			−1.26799			−0.633993	−	0.773339i	$-0.718585\pi$
$293$	−6359.39			−1.26799			−0.633993	+	0.773339i	$0.718585\pi$
$294$		0			0
$295$	−7679.90			−1.51573
$296$	−287.396	+	165.928i	−0.0564343	+	0.0325824i
$297$		0			0
$298$	−442.973	+	767.252i	−0.0861099	+	0.149147i
$299$	−3434.12	−	5948.07i	−0.664215	−	1.15045i
$300$		0			0
$301$	−3572.27	+	347.111i	−0.684060	+	0.0664689i
$302$			3670.44i			0.699371i
$303$		0			0
$304$	56.8442	+	32.8190i	0.0107245	+	0.00619178i
$305$	−4672.61	−	2697.73i	−0.877223	−	0.506465i
$306$		0			0
$307$		−	4187.27i		−	0.778436i	−0.921146	−	0.389218i	$-0.872745\pi$
$307$		−	4187.27i		−	0.778436i	0.921146	−	0.389218i	$-0.127255\pi$
$308$	3204.36	−	2289.86i	0.592809	−	0.423626i
$309$		0			0
$310$	2316.04	+	4011.51i	0.424331	+	0.734962i
$311$	336.470	−	582.784i	0.0613488	−	0.106259i	−0.833720	−	0.552188i	$-0.813793\pi$
$311$	336.470	−	582.784i	0.0613488	−	0.106259i	0.895069	+	0.445929i	$0.147126\pi$
$312$		0			0
$313$	1196.22	−	690.637i	0.216020	−	0.124719i	−0.388086	−	0.921623i	$-0.626864\pi$
$313$	1196.22	−	690.637i	0.216020	−	0.124719i	0.604106	+	0.796904i	$0.293530\pi$
$314$	6334.37			1.13844
$315$		0			0
$316$	−5020.97			−0.893835
$317$	−719.658	+	415.495i	−0.127508	+	0.0736168i	−0.562397	−	0.826867i	$-0.690121\pi$
$317$	−719.658	+	415.495i	−0.127508	+	0.0736168i	0.434889	+	0.900484i	$0.356787\pi$
$318$		0			0
$319$	−2330.77	+	4037.02i	−0.409085	+	0.708557i
$320$	−682.701	−	1182.47i	−0.119263	−	0.206569i
$321$		0			0
$322$	1293.11	−	2844.04i	0.223796	−	0.492211i
$323$			383.406i			0.0660473i
$324$		0			0
$325$	23282.7	+	13442.3i	3.97382	+	2.29429i
$326$	3012.25	+	1739.12i	0.511757	+	0.295463i
$327$		0			0
$328$			3051.23i			0.513647i
$329$	1030.62	+	1442.22i	0.172705	+	0.241679i
$330$		0			0
$331$	−2189.22	−	3791.84i	−0.363536	−	0.629663i	0.625004	−	0.780621i	$-0.285097\pi$
$331$	−2189.22	−	3791.84i	−0.363536	−	0.629663i	−0.988540	+	0.150959i	$0.951764\pi$
$332$	−638.079	+	1105.19i	−0.105479	+	0.182696i
$333$		0			0
$334$	−3137.21	+	1811.27i	−0.513954	+	0.296731i
$335$	2713.74			0.442590
$336$		0			0
$337$	3249.47			0.525252			0.262626	−	0.964898i	$-0.415412\pi$
$337$	3249.47			0.525252			0.262626	+	0.964898i	$0.415412\pi$
$338$	7679.53	−	4433.78i	1.23583	−	0.713508i
$339$		0			0
$340$	3987.80	−	6907.08i	0.636085	−	1.10173i
$341$	2885.72	+	4998.21i	0.458271	+	0.793749i
$342$		0			0
$343$	−6086.12	+	1820.10i	−0.958074	+	0.286520i
$344$			1550.34i			0.242991i
$345$		0			0
$346$	−6999.81	−	4041.34i	−1.08761	−	0.627930i
$347$	1959.38	+	1131.25i	0.303126	+	0.175010i	0.643846	−	0.765155i	$-0.277338\pi$
$347$	1959.38	+	1131.25i	0.303126	+	0.175010i	−0.340720	+	0.940165i	$0.610671\pi$
$348$		0			0
$349$			2135.14i			0.327483i	0.986503	+	0.163741i	$0.0523562\pi$
$349$			2135.14i			0.327483i	−0.986503	+	0.163741i	$0.947644\pi$
$350$	1182.72	+	12171.9i	0.180625	+	1.85889i
$351$		0			0
$352$	−850.625	−	1473.33i	−0.128802	−	0.223092i
$353$	4430.40	−	7673.68i	0.668007	−	1.15702i	−0.310453	−	0.950589i	$-0.600481\pi$
$353$	4430.40	−	7673.68i	0.668007	−	1.15702i	0.978461	−	0.206434i	$-0.0661859\pi$
$354$		0			0
$355$	−5231.56	+	3020.44i	−0.782147	+	0.451573i
$356$	−3035.02			−0.451841
$357$		0			0
$358$	6216.91			0.917804
$359$	9259.26	−	5345.84i	1.36124	−	0.785912i	0.371451	−	0.928453i	$-0.378861\pi$
$359$	9259.26	−	5345.84i	1.36124	−	0.785912i	0.989789	+	0.142541i	$0.0455272\pi$
$360$		0			0
$361$	−3421.09	+	5925.49i	−0.498773	+	0.863900i
$362$	467.849	+	810.337i	0.0679270	+	0.117653i
$363$		0			0
$364$	5491.42	+	2496.81i	0.790737	+	0.359528i
$365$		−	6316.20i		−	0.905767i
$366$		0			0
$367$	−2755.42	−	1590.84i	−0.391912	−	0.226270i	0.291076	−	0.956700i	$-0.405987\pi$
$367$	−2755.42	−	1590.84i	−0.391912	−	0.226270i	−0.682988	+	0.730429i	$0.739320\pi$
$368$	−1168.73	−	674.766i	−0.165555	−	0.0955831i
$369$		0			0
$370$		−	1769.99i		−	0.248696i
$371$	−3405.30	−	1548.30i	−0.476535	−	0.216668i
$372$		0			0
$373$	3475.05	+	6018.96i	0.482389	+	0.835522i	0.999796	−	0.0202173i	$-0.00643582\pi$
$373$	3475.05	+	6018.96i	0.482389	+	0.835522i	−0.517407	+	0.855740i	$0.673102\pi$
$374$	4968.68	−	8606.01i	0.686963	−	1.18986i
$375$		0			0
$376$	663.116	−	382.850i	0.0909511	−	0.0525106i
$377$	−7139.93			−0.975399
$378$		0			0
$379$	−982.417			−0.133149			−0.0665744	−	0.997781i	$-0.521207\pi$
$379$	−982.417			−0.133149			−0.0665744	+	0.997781i	$0.521207\pi$
$380$	−303.184	+	175.044i	−0.0409290	+	0.0236304i
$381$		0			0
$382$	−880.387	+	1524.88i	−0.117918	+	0.204239i
$383$	5978.41	+	10354.9i	0.797604	+	1.38149i	0.921172	+	0.389155i	$0.127233\pi$
$383$	5978.41	+	10354.9i	0.797604	+	1.38149i	−0.123568	+	0.992336i	$0.539434\pi$
$384$		0			0
$385$	2031.56	+	20907.6i	0.268929	+	2.76767i
$386$		−	6837.08i		−	0.901550i
$387$		0			0
$388$	−5461.43	−	3153.16i	−0.714593	−	0.412570i
$389$	1261.63	+	728.402i	0.164440	+	0.0949394i	0.579961	−	0.814644i	$-0.303068\pi$
$389$	1261.63	+	728.402i	0.164440	+	0.0949394i	−0.415522	+	0.909583i	$0.636401\pi$
$390$		0			0
$391$		−	7882.90i		−	1.01958i
$392$	528.272	+	2692.67i	0.0680657	+	0.346940i
$393$		0			0
$394$	−2403.67	−	4163.27i	−0.307348	−	0.532342i
$395$	13389.9	−	23192.1i	1.70562	−	2.95422i
$396$		0			0
$397$	−13498.4	+	7793.28i	−1.70646	+	0.985223i	−0.767592	+	0.640939i	$0.778545\pi$
$397$	−13498.4	+	7793.28i	−1.70646	+	0.985223i	−0.938865	+	0.344284i	$0.888122\pi$
$398$	−7458.19			−0.939310
$399$		0			0
$400$	5282.51			0.660314
$401$	−6662.95	+	3846.85i	−0.829755	+	0.479059i	−0.853769	−	0.520652i	$-0.825689\pi$
$401$	−6662.95	+	3846.85i	−0.829755	+	0.479059i	0.0240138	+	0.999712i	$0.492355\pi$
$402$		0			0
$403$	−4419.96	+	7655.60i	−0.546338	+	0.946284i
$404$	−1548.95	−	2682.85i	−0.190750	−	0.330388i
$405$		0			0
$406$	−1888.31	−	2642.44i	−0.230826	−	0.323010i
$407$		−	2205.35i		−	0.268588i
$408$		0			0
$409$	2013.20	+	1162.32i	0.243390	+	0.140521i	0.616734	−	0.787172i	$-0.288456\pi$
$409$	2013.20	+	1162.32i	0.243390	+	0.140521i	−0.373344	+	0.927693i	$0.621789\pi$
$410$	−14093.7	−	8137.02i	−1.69766	−	0.980144i
$411$		0			0
$412$			2999.86i			0.358720i
$413$	2759.42	−	6069.00i	0.328770	−	0.723089i
$414$		0			0
$415$	−3403.26	−	5894.62i	−0.402553	−	0.697242i
$416$	1302.87	−	2256.64i	0.153554	−	0.265964i
$417$		0			0
$418$	−377.759	+	218.099i	−0.0442028	+	0.0255205i
$419$	8067.13			0.940585			0.470292	−	0.882511i	$-0.344148\pi$
$419$	8067.13			0.940585			0.470292	+	0.882511i	$0.344148\pi$
$420$		0			0
$421$	285.530			0.0330543			0.0165272	−	0.999863i	$-0.494739\pi$
$421$	285.530			0.0330543			0.0165272	+	0.999863i	$0.494739\pi$
$422$	2652.79	−	1531.59i	0.306009	−	0.176674i
$423$		0			0
$424$	−807.930	+	1399.37i	−0.0925390	+	0.160282i
$425$	15428.1	+	26722.3i	1.76088	+	3.04993i
$426$		0			0
$427$	3810.76	−	2723.20i	0.431887	−	0.308630i
$428$		−	2580.82i		−	0.291469i
$429$		0			0
$430$	−7161.08	−	4134.45i	−0.803111	−	0.463677i
$431$	3491.09	+	2015.58i	0.390163	+	0.225261i	0.682231	−	0.731137i	$-0.261010\pi$
$431$	3491.09	+	2015.58i	0.390163	+	0.225261i	−0.292068	+	0.956398i	$0.594343\pi$
$432$		0			0
$433$		−	6255.26i		−	0.694247i	−0.937819	−	0.347123i	$-0.887159\pi$
$433$		−	6255.26i		−	0.694247i	0.937819	−	0.347123i	$-0.112841\pi$
$434$	−4002.24	+	388.890i	−0.442658	+	0.0430123i
$435$		0			0
$436$	−1614.96	−	2797.19i	−0.177391	−	0.307250i
$437$	−173.009	+	299.660i	−0.0189385	+	0.0328025i
$438$		0			0
$439$	−12027.8	+	6944.26i	−1.30764	+	0.754969i	−0.981702	−	0.190421i	$-0.939015\pi$
$439$	−12027.8	+	6944.26i	−1.30764	+	0.754969i	−0.325942	+	0.945390i	$0.605681\pi$
$440$	9073.78			0.983127
$441$		0			0
$442$	15220.7			1.63795
$443$	−14842.1	+	8569.10i	−1.59181	+	0.919029i	−0.598808	+	0.800892i	$0.704359\pi$
$443$	−14842.1	+	8569.10i	−1.59181	+	0.919029i	−0.992997	+	0.118137i	$0.962308\pi$
$444$		0			0
$445$	8093.78	−	14018.8i	0.862207	−	1.49339i
$446$	−5610.92	−	9718.40i	−0.595706	−	1.03179i
$447$		0			0
$448$	1179.74	−	114.633i	0.124414	−	0.0120891i
$449$			3361.40i			0.353306i	0.984273	+	0.176653i	$0.0565270\pi$
$449$			3361.40i			0.353306i	−0.984273	+	0.176653i	$0.943473\pi$
$450$		0			0
$451$	−17560.4	−	10138.5i	−1.83345	−	1.05854i
$452$	−534.310	−	308.484i	−0.0556014	−	0.0321015i
$453$		0			0
$454$		−	681.080i		−	0.0704068i
$455$	−26177.4	+	18706.6i	−2.69717	+	1.92742i
$456$		0			0
$457$	−902.285	−	1562.80i	−0.0923569	−	0.159967i	0.816146	−	0.577846i	$-0.196107\pi$
$457$	−902.285	−	1562.80i	−0.0923569	−	0.159967i	−0.908502	+	0.417880i	$0.862773\pi$
$458$	−6392.70	+	11072.5i	−0.652208	+	1.12966i
$459$		0			0
$460$	6233.53	−	3598.93i	0.631826	−	0.364785i
$461$	−15471.3			−1.56305			−0.781527	−	0.623871i	$-0.785559\pi$
$461$	−15471.3			−1.56305			−0.781527	+	0.623871i	$0.785559\pi$
$462$		0			0
$463$	10252.0			1.02905			0.514526	−	0.857475i	$-0.327968\pi$
$463$	10252.0			1.02905			0.514526	+	0.857475i	$0.327968\pi$
$464$	−1214.96	+	701.459i	−0.121559	+	0.0701819i
$465$		0			0
$466$	738.048	−	1278.34i	0.0733679	−	0.127077i
$467$	6456.58	+	11183.1i	0.639775	+	1.10812i	0.985482	+	0.169780i	$0.0543058\pi$
$467$	6456.58	+	11183.1i	0.639775	+	1.10812i	−0.345707	+	0.938343i	$0.612361\pi$
$468$		0			0
$469$	−975.060	+	2144.52i	−0.0960002	+	0.211140i
$470$			4083.94i			0.400805i
$471$		0			0
$472$	−2493.99	−	1439.91i	−0.243211	−	0.140418i
$473$	−8922.49	−	5151.40i	−0.867350	−	0.500765i
$474$		0			0
$475$		−	1354.43i		−	0.130833i
$476$	4025.45	+	5633.08i	0.387618	+	0.542420i
$477$		0			0
$478$	−2529.61	−	4381.42i	−0.242054	−	0.419250i
$479$	−7793.03	+	13497.9i	−0.743367	+	1.28755i	0.207587	+	0.978216i	$0.433439\pi$
$479$	−7793.03	+	13497.9i	−0.743367	+	1.28755i	−0.950954	+	0.309332i	$0.899894\pi$
$480$		0			0
$481$	2925.32	−	1688.93i	0.277303	−	0.160101i
$482$	28.1970			0.00266460
$483$		0			0
$484$	5981.66			0.561764
$485$	29129.1	−	16817.7i	2.72718	−	1.57454i
$486$		0			0
$487$	4744.08	−	8216.98i	0.441426	−	0.764573i	−0.556369	−	0.830935i	$-0.687806\pi$
$487$	4744.08	−	8216.98i	0.441426	−	0.764573i	0.997796	+	0.0663623i	$0.0211393\pi$
$488$	−1011.60	−	1752.14i	−0.0938381	−	0.162532i
$489$		0			0
$490$	−13846.3	−	4740.71i	−1.27656	−	0.437068i
$491$		−	1921.42i		−	0.176604i	−0.996094	−	0.0883019i	$-0.971856\pi$
$491$		−	1921.42i		−	0.176604i	0.996094	−	0.0883019i	$-0.0281440\pi$
$492$		0			0
$493$	−7096.85	−	4097.37i	−0.648329	−	0.374313i
$494$	−578.600	−	334.055i	−0.0526973	−	0.0304248i
$495$		0			0
$496$			1736.95i			0.157240i
$497$	−507.167	−	5219.47i	−0.0457737	−	0.471077i
$498$		0			0
$499$	−3690.95	−	6392.91i	−0.331121	−	0.573519i	0.651611	−	0.758553i	$-0.274094\pi$
$499$	−3690.95	−	6392.91i	−0.331121	−	0.573519i	−0.982732	+	0.185035i	$0.940760\pi$
$500$	−8753.80	+	15162.0i	−0.782963	+	1.35613i
$501$		0			0
$502$	−6589.14	+	3804.24i	−0.585832	+	0.338230i
$503$	−9968.70			−0.883663			−0.441832	−	0.897098i	$-0.645671\pi$
$503$	−9968.70			−0.883663			−0.441832	+	0.897098i	$0.645671\pi$
$504$		0			0
$505$	16522.9			1.45596
$506$	7766.79	−	4484.16i	0.682363	−	0.393963i
$507$		0			0
$508$	−4703.08	+	8145.98i	−0.410759	+	0.711456i
$509$	2857.28	+	4948.95i	0.248814	+	0.430959i	0.963197	−	0.268796i	$-0.0866257\pi$
$509$	2857.28	+	4948.95i	0.248814	+	0.430959i	−0.714383	+	0.699755i	$0.753292\pi$
$510$		0			0
$511$	4991.34	+	2269.44i	0.432102	+	0.196466i
$512$		−	512.000i		−	0.0441942i
$513$		0			0
$514$	−2298.08	−	1326.80i	−0.197206	−	0.113857i
$515$	−13856.5	−	8000.04i	−1.18561	−	0.684512i
$516$		0			0
$517$			5088.47i			0.432864i
$518$	1398.72	+	635.965i	0.118642	+	0.0539434i
$519$		0			0
$520$	6949.01	+	12036.0i	0.586027	+	1.01503i
$521$	−5369.66	+	9300.52i	−0.451534	+	0.782079i	−0.998482	−	0.0550874i	$-0.982456\pi$
$521$	−5369.66	+	9300.52i	−0.451534	+	0.782079i	0.546948	+	0.837167i	$0.315790\pi$
$522$		0			0
$523$	7704.40	−	4448.14i	0.644149	−	0.371899i	−0.142062	−	0.989858i	$-0.545373\pi$
$523$	7704.40	−	4448.14i	0.644149	−	0.371899i	0.786211	+	0.617958i	$0.212040\pi$
$524$	−6509.87			−0.542719
$525$		0			0
$526$	5792.05			0.480125
$527$	−8786.58	+	5072.94i	−0.726280	+	0.419318i
$528$		0			0
$529$	−2526.40	+	4375.86i	−0.207644	+	0.359650i
$530$	−4309.17	−	7463.71i	−0.353167	−	0.611703i
$531$		0			0
$532$	−29.3918	−	302.484i	−0.00239530	−	0.0246510i
$533$		−	31057.6i		−	2.52393i
$534$		0			0
$535$	11920.9	+	6882.54i	0.963339	+	0.556184i
$536$	881.270	+	508.801i	0.0710169	+	0.0410016i
$537$		0			0
$538$			6443.25i			0.516335i
$539$	−17252.1	−	5906.78i	−1.37867	−	0.472027i
$540$		0			0
$541$	8170.75	+	14152.2i	0.649331	+	1.12467i	0.983283	+	0.182084i	$0.0582844\pi$
$541$	8170.75	+	14152.2i	0.649331	+	1.12467i	−0.333952	+	0.942590i	$0.608382\pi$
$542$	−3896.31	+	6748.61i	−0.308784	+	0.534829i
$543$		0			0
$544$	2590.02	−	1495.35i	0.204129	−	0.117854i
$545$	17227.1			1.35399
$546$		0			0
$547$	8782.16			0.686468			0.343234	−	0.939250i	$-0.388478\pi$
$547$	8782.16			0.686468			0.343234	+	0.939250i	$0.388478\pi$
$548$	4006.80	−	2313.33i	0.312339	−	0.180329i
$549$		0			0
$550$	−17552.5	+	30401.8i	−1.36080	+	2.35697i
$551$	179.853	+	311.515i	0.0139056	+	0.0240852i
$552$		0			0
$553$	13516.3	+	18914.3i	1.03937	+	1.45447i
$554$			12417.8i			0.952311i
$555$		0			0
$556$	−6455.93	−	3727.33i	−0.492432	−	0.284306i
$557$	12763.3	+	7368.88i	0.970911	+	0.560556i	0.899514	−	0.436892i	$-0.143921\pi$
$557$	12763.3	+	7368.88i	0.970911	+	0.560556i	0.0713972	+	0.997448i	$0.477254\pi$
$558$		0			0
$559$		−	15780.5i		−	1.19399i
$560$	−2616.64	+	5754.96i	−0.197452	+	0.434271i
$561$		0			0
$562$	8367.31	+	14492.6i	0.628031	+	1.08778i
$563$	−2246.20	+	3890.54i	−0.168146	+	0.291237i	−0.937768	−	0.347262i	$-0.887111\pi$
$563$	−2246.20	+	3890.54i	−0.168146	+	0.291237i	0.769622	+	0.638500i	$0.220445\pi$
$564$		0			0
$565$	2849.80	−	1645.33i	0.212198	−	0.122513i
$566$	7275.93			0.540336
$567$		0			0
$568$	−2265.22			−0.167335
$569$	−5334.21	+	3079.71i	−0.393008	+	0.226903i	−0.683463	−	0.729985i	$-0.739527\pi$
$569$	−5334.21	+	3079.71i	−0.393008	+	0.226903i	0.290455	+	0.956889i	$0.406193\pi$
$570$		0			0
$571$	12049.8	−	20870.8i	0.883131	−	1.52963i	0.0352902	−	0.999377i	$-0.488764\pi$
$571$	12049.8	−	20870.8i	0.883131	−	1.52963i	0.847841	−	0.530251i	$-0.177902\pi$
$572$	8658.25	+	14996.5i	0.632902	+	1.09622i
$573$		0			0
$574$	11494.2	−	8213.84i	0.835815	−	0.597280i
$575$			27847.3i			2.01967i
$576$		0			0
$577$	1304.05	+	752.896i	0.0940875	+	0.0543215i	0.546306	−	0.837586i	$-0.316034\pi$
$577$	1304.05	+	752.896i	0.0940875	+	0.0543215i	−0.452218	+	0.891907i	$0.649367\pi$
$578$	6619.31	+	3821.66i	0.476345	+	0.275018i
$579$		0			0
$580$		−	7482.60i		−	0.535686i
$581$	5881.00	−	571.446i	0.419940	−	0.0408048i
$582$		0			0
$583$	−5369.10	−	9299.55i	−0.381416	−	0.660631i
$584$	1184.23	−	2051.14i	0.0839105	−	0.145337i
$585$		0			0
$586$	11014.8	−	6359.39i	0.776479	−	0.448300i
$587$	−5651.33			−0.397369			−0.198684	−	0.980064i	$-0.563667\pi$
$587$	−5651.33			−0.397369			−0.198684	+	0.980064i	$0.563667\pi$
$588$		0			0
$589$	445.351			0.0311551
$590$	13302.0	−	7679.90i	0.928193	−	0.535892i
$591$		0			0
$592$	331.856	−	574.792i	0.0230392	−	0.0399051i
$593$	7479.49	+	12954.9i	0.517952	+	0.897120i	0.999783	+	0.0208551i	$0.00663887\pi$
$593$	7479.49	+	12954.9i	0.517952	+	0.897120i	−0.481830	+	0.876265i	$0.660028\pi$
$594$		0			0
$595$	−36754.5	+	3571.37i	−2.53241	+	0.246070i
$596$		−	1771.89i		−	0.121778i
$597$		0			0
$598$	11896.1	+	6868.24i	0.813493	+	0.469671i
$599$	−15145.7	−	8744.35i	−1.03311	−	0.596468i	−0.115238	−	0.993338i	$-0.536763\pi$
$599$	−15145.7	−	8744.35i	−1.03311	−	0.596468i	−0.917875	+	0.396870i	$0.870097\pi$
$600$		0			0
$601$			8969.78i			0.608794i	0.952545	+	0.304397i	$0.0984549\pi$
$601$			8969.78i			0.608794i	−0.952545	+	0.304397i	$0.901545\pi$
$602$	5840.24	−	4173.48i	0.395399	−	0.282555i
$603$		0			0
$604$	−3670.44	−	6357.39i	−0.247265	−	0.428275i
$605$	−15951.9	+	27629.5i	−1.07196	+	1.85669i
$606$		0			0
$607$	−9922.55	+	5728.79i	−0.663499	+	0.383071i	−0.793609	−	0.608428i	$-0.791800\pi$
$607$	−9922.55	+	5728.79i	−0.663499	+	0.383071i	0.130110	+	0.991500i	$0.458467\pi$
$608$	−131.276			−0.00875650
$609$		0			0
$610$	10790.9			0.716250
$611$	−6749.66	+	3896.92i	−0.446910	+	0.258024i
$612$		0			0
$613$	−1234.17	+	2137.64i	−0.0813173	+	0.140846i	−0.903816	−	0.427921i	$-0.859246\pi$
$613$	−1234.17	+	2137.64i	−0.0813173	+	0.140846i	0.822499	+	0.568767i	$0.192579\pi$
$614$	4187.27	+	7252.56i	0.275219	+	0.476693i
$615$		0			0
$616$	−3260.25	+	7170.51i	−0.213245	+	0.469007i
$617$			995.871i			0.0649794i	0.999472	+	0.0324897i	$0.0103436\pi$
$617$			995.871i			0.0649794i	−0.999472	+	0.0324897i	$0.989656\pi$
$618$		0			0
$619$	1067.72	+	616.446i	0.0693297	+	0.0400275i	0.534264	−	0.845318i	$-0.320589\pi$
$619$	1067.72	+	616.446i	0.0693297	+	0.0400275i	−0.464934	+	0.885345i	$0.653922\pi$
$620$	−8023.01	−	4632.09i	−0.519697	−	0.300047i
$621$		0			0
$622$			1345.88i			0.0867603i
$623$	8170.19	+	11433.1i	0.525412	+	0.735245i
$624$		0			0
$625$	−26054.5	−	45127.6i	−1.66749	−	2.88817i
$626$	−1381.27	+	2392.44i	−0.0881898	+	0.152749i
$627$		0			0
$628$	−10971.4	+	6334.37i	−0.697147	+	0.402498i
$629$	3876.89			0.245758
$630$		0			0
$631$	8164.30			0.515081			0.257540	−	0.966268i	$-0.417088\pi$
$631$	8164.30			0.515081			0.257540	+	0.966268i	$0.417088\pi$
$632$	8696.58	−	5020.97i	0.547360	−	0.316018i
$633$		0			0
$634$	830.990	−	1439.32i	0.0520549	−	0.0901617i
$635$	−25084.4	−	43447.4i	−1.56763	−	2.71521i
$636$		0			0
$637$	−5377.12	−	27407.9i	−0.334457	−	1.70477i
$638$		−	9323.09i		−	0.578534i
$639$		0			0
$640$	2364.95	+	1365.40i	0.146067	+	0.0843316i
$641$	−20528.8	−	11852.3i	−1.26496	−	0.730323i	−0.290927	−	0.956745i	$-0.593964\pi$
$641$	−20528.8	−	11852.3i	−1.26496	−	0.730323i	−0.974029	+	0.226422i	$0.927297\pi$
$642$		0			0
$643$			21094.9i			1.29378i	0.762582	+	0.646892i	$0.223931\pi$
$643$			21094.9i			1.29378i	−0.762582	+	0.646892i	$0.776069\pi$
$644$	604.302	+	6219.13i	0.0369764	+	0.380540i
$645$		0			0
$646$	−383.406	−	664.079i	−0.0233513	−	0.0404456i
$647$	10195.4	−	17658.9i	0.619508	−	1.07302i	−0.370068	−	0.929005i	$-0.620665\pi$
$647$	10195.4	−	17658.9i	0.619508	−	1.07302i	0.989576	−	0.144015i	$-0.0460012\pi$
$648$		0			0
$649$	16573.9	−	9568.92i	1.00244	−	0.578756i
$650$	−53769.1			−3.24461
$651$		0			0
$652$	−6956.48			−0.417848
$653$	−5839.99	+	3371.72i	−0.349979	+	0.202061i	−0.664676	−	0.747132i	$-0.731430\pi$
$653$	−5839.99	+	3371.72i	−0.349979	+	0.202061i	0.314697	+	0.949192i	$0.398097\pi$
$654$		0			0
$655$	17360.5	−	30069.3i	1.03562	−	1.79375i
$656$	−3051.23	−	5284.89i	−0.181601	−	0.314543i
$657$		0			0
$658$	−3227.31	−	1467.38i	−0.191206	−	0.0869367i
$659$			16824.4i			0.994514i	0.867603	+	0.497257i	$0.165659\pi$
$659$			16824.4i			0.994514i	−0.867603	+	0.497257i	$0.834341\pi$
$660$		0			0
$661$	16345.3	+	9436.99i	0.961816	+	0.555305i	0.896731	−	0.442575i	$-0.145935\pi$
$661$	16345.3	+	9436.99i	0.961816	+	0.555305i	0.0650845	+	0.997880i	$0.479268\pi$
$662$	7583.68	+	4378.44i	0.445239	+	0.257059i
$663$		0			0
$664$		−	2552.32i		−	0.149170i
$665$	1475.57	+	670.902i	0.0860451	+	0.0391225i
$666$		0			0
$667$	−3697.81	−	6404.80i	−0.214663	−	0.371806i
$668$	3622.54	−	6274.43i	0.209821	−	0.363420i
$669$		0			0
$670$	−4700.34	+	2713.74i	−0.271030	+	0.156479i
$671$	13445.2			0.773540
$672$		0			0
$673$	−2698.61			−0.154567			−0.0772835	−	0.997009i	$-0.524625\pi$
$673$	−2698.61			−0.154567			−0.0772835	+	0.997009i	$0.524625\pi$
$674$	−5628.24	+	3249.47i	−0.321650	+	0.185705i
$675$		0			0
$676$	−8867.56	+	15359.1i	−0.504526	+	0.873865i
$677$	10404.1	+	18020.5i	0.590641	+	1.02302i	0.994146	+	0.108042i	$0.0344582\pi$
$677$	10404.1	+	18020.5i	0.590641	+	1.02302i	−0.403506	+	0.914977i	$0.632208\pi$
$678$		0			0
$679$	2823.88	+	29061.8i	0.159603	+	1.64255i
$680$			15951.2i			0.899560i
$681$		0			0
$682$	−9996.43	−	5771.44i	−0.561266	−	0.324047i
$683$	2747.64	+	1586.35i	0.153932	+	0.0888726i	0.574987	−	0.818162i	$-0.305007\pi$
$683$	2747.64	+	1586.35i	0.153932	+	0.0888726i	−0.421056	+	0.907035i	$0.638340\pi$
$684$		0			0
$685$			24676.7i			1.37642i
$686$	8721.36	−	9238.63i	0.485398	−	0.514187i
$687$		0			0
$688$	−1550.34	−	2685.27i	−0.0859102	−	0.148801i
$689$	8223.67	−	14243.8i	0.454712	−	0.787585i
$690$		0			0
$691$	10077.6	−	5818.29i	0.554803	−	0.320316i	−0.196254	−	0.980553i	$-0.562878\pi$
$691$	10077.6	−	5818.29i	0.554803	−	0.320316i	0.751057	+	0.660238i	$0.229544\pi$
$692$	16165.4			0.888027
$693$		0			0
$694$	−4524.98			−0.247502
$695$	34433.4	−	19880.1i	1.87933	−	1.08503i
$696$		0			0
$697$	17822.9	−	30870.2i	0.968566	−	1.67760i
$698$	−2135.14	−	3698.17i	−0.115783	−	0.200541i
$699$		0			0
$700$	−14220.4	−	19899.6i	−0.767828	−	1.07447i
$701$		−	26715.2i		−	1.43940i	−0.694285	−	0.719700i	$-0.744279\pi$
$701$		−	26715.2i		−	1.43940i	0.694285	−	0.719700i	$-0.255721\pi$
$702$		0			0
$703$	−147.376	−	85.0875i	−0.00790667	−	0.00456492i
$704$	2946.65	+	1701.25i	0.157750	+	0.0910770i
$705$		0			0
$706$			17721.6i			0.944705i
$707$	−5936.76	+	13057.1i	−0.315806	+	0.694575i
$708$		0			0
$709$	−7605.75	−	13173.5i	−0.402877	−	0.697804i	0.591195	−	0.806529i	$-0.298657\pi$
$709$	−7605.75	−	13173.5i	−0.402877	−	0.697804i	−0.994072	+	0.108725i	$0.965323\pi$
$710$	6040.88	−	10463.1i	0.319310	−	0.553062i
$711$		0			0
$712$	5256.80	−	3035.02i	0.276695	−	0.159750i
$713$	−9156.50			−0.480945
$714$		0			0
$715$	−92359.3			−4.83083
$716$	−10768.0	+	6216.91i	−0.562038	+	0.324493i
$717$		0			0
$718$	−10691.7	+	18518.5i	−0.555724	+	0.962542i
$719$	−2336.90	−	4047.62i	−0.121212	−	0.209946i	0.799034	−	0.601286i	$-0.205345\pi$
$719$	−2336.90	−	4047.62i	−0.121212	−	0.209946i	−0.920246	+	0.391341i	$0.872011\pi$
$720$		0			0
$721$	11300.7	−	8075.56i	0.583716	−	0.417128i
$722$		−	13684.3i		−	0.705372i
$723$		0			0
$724$	−1620.67	−	935.697i	−0.0831932	−	0.0480316i
$725$	25070.5	+	14474.5i	1.28427	+	0.741473i
$726$		0			0
$727$			8839.42i			0.450943i	0.974250	+	0.225472i	$0.0723923\pi$
$727$			8839.42i			0.450943i	−0.974250	+	0.225472i	$0.927608\pi$
$728$	−12008.2	+	1166.82i	−0.611338	+	0.0594027i
$729$		0			0
$730$	6316.20	+	10940.0i	0.320237	+	0.554667i
$731$	9055.88	−	15685.2i	0.458199	−	0.793624i
$732$		0			0
$733$	−10708.8	+	6182.72i	−0.539615	+	0.311547i	−0.744923	−	0.667150i	$-0.767514\pi$
$733$	−10708.8	+	6182.72i	−0.539615	+	0.311547i	0.205308	+	0.978697i	$0.434180\pi$
$734$	6363.37			0.319995
$735$		0			0
$736$	2699.06			0.135175
$737$	−5856.48	+	3381.24i	−0.292709	+	0.168995i
$738$		0			0
$739$	4807.49	−	8326.82i	0.239305	−	0.414489i	−0.721210	−	0.692717i	$-0.756414\pi$
$739$	4807.49	−	8326.82i	0.239305	−	0.414489i	0.960515	+	0.278228i	$0.0897471\pi$
$740$	1769.99	+	3065.71i	0.0879272	+	0.152294i
$741$		0			0
$742$	7446.46	−	723.560i	0.368421	−	0.0357988i
$743$		−	8138.35i		−	0.401840i	−0.979608	−	0.200920i	$-0.935607\pi$
$743$		−	8138.35i		−	0.401840i	0.979608	−	0.200920i	$-0.0643931\pi$
$744$		0			0
$745$	8184.44	+	4725.29i	0.402489	+	0.232377i
$746$	−12037.9	−	6950.09i	−0.590804	−	0.341101i
$747$		0			0
$748$			19874.7i			0.971513i
$749$	−9722.13	+	6947.51i	−0.474284	+	0.338927i
$750$		0			0
$751$	3974.64	+	6884.28i	0.193125	+	0.334502i	0.946284	−	0.323336i	$-0.104804\pi$
$751$	3974.64	+	6884.28i	0.193125	+	0.334502i	−0.753159	+	0.657838i	$0.771471\pi$
$752$	−765.701	+	1326.23i	−0.0371306	+	0.0643121i
$753$		0			0
$754$	12366.7	−	7139.93i	0.597307	−	0.344856i
$755$	39153.3			1.88733
$756$		0			0
$757$	−18333.7			−0.880250			−0.440125	−	0.897936i	$-0.645066\pi$
$757$	−18333.7			−0.880250			−0.440125	+	0.897936i	$0.645066\pi$
$758$	1701.60	−	982.417i	0.0815366	−	0.0470752i
$759$		0			0
$760$	350.087	−	606.369i	0.0167092	−	0.0289412i
$761$	9101.80	+	15764.8i	0.433561	+	0.750950i	0.997177	−	0.0750873i	$-0.0239236\pi$
$761$	9101.80	+	15764.8i	0.433561	+	0.750950i	−0.563616	+	0.826037i	$0.690590\pi$
$762$		0			0
$763$	−6189.76	+	13613.6i	−0.293688	+	0.645931i
$764$		−	3521.55i		−	0.166761i
$765$		0			0
$766$	−20709.8	−	11956.8i	−0.976862	−	0.563991i
$767$	25385.6	+	14656.4i	1.19507	+	0.689976i
$768$		0			0
$769$			15817.7i			0.741743i	0.928684	+	0.370871i	$0.120941\pi$
$769$			15817.7i			0.741743i	−0.928684	+	0.370871i	$0.879059\pi$
$770$	−24426.4	−	34181.5i	−1.14320	−	1.59976i
$771$		0			0
$772$	6837.08	+	11842.2i	0.318746	+	0.552084i
$773$	−4338.21	+	7514.01i	−0.201856	+	0.349625i	−0.949126	−	0.314895i	$-0.898031\pi$
$773$	−4338.21	+	7514.01i	−0.201856	+	0.349625i	0.747270	+	0.664520i	$0.231364\pi$
$774$		0			0
$775$	31039.7	−	17920.8i	1.43868	−	0.830623i
$776$	12612.6			0.583462
$777$		0			0
$778$	−2913.61			−0.134265
$779$	−1355.04	+	782.332i	−0.0623226	+	0.0359820i
$780$		0			0
$781$	7526.76	−	13036.7i	0.344851	−	0.597299i
$782$	7882.90	+	13653.6i	0.360476	+	0.624362i
$783$		0			0
$784$	−3607.66	−	4135.57i	−0.164343	−	0.188391i
$785$		−	67570.0i		−	3.07220i
$786$		0			0
$787$	31827.5	+	18375.6i	1.44158	+	0.832299i	0.997956	−	0.0639115i	$-0.0203575\pi$
$787$	31827.5	+	18375.6i	1.44158	+	0.832299i	0.443629	+	0.896211i	$0.353691\pi$
$788$	8326.55	+	4807.33i	0.376422	+	0.217328i
$789$		0			0
$790$			53559.8i			2.41211i
$791$	276.270	+	2843.21i	0.0124185	+	0.127804i
$792$		0			0
$793$	10296.8	+	17834.5i	0.461096	+	0.798641i
$794$	15586.6	−	26996.7i	0.696658	−	1.20665i
$795$		0			0
$796$	12918.0	−	7458.19i	0.575207	−	0.332096i
$797$	−21144.6			−0.939750			−0.469875	−	0.882733i	$-0.655701\pi$
$797$	−21144.6			−0.939750			−0.469875	+	0.882733i	$0.655701\pi$
$798$		0			0
$799$	−8945.24			−0.396070
$800$	−9149.57	+	5282.51i	−0.404358	+	0.233456i
$801$		0			0
$802$	7693.71	−	13325.9i	0.338746	−	0.586725i
$803$	7869.79	+	13630.9i	0.345852	+	0.599033i
$804$		0			0
$805$	−30337.9	−	13793.9i	−1.32829	−	0.603938i
$806$		−	17679.8i		−	0.772638i
$807$		0			0
$808$	5365.71	+	3097.89i	0.233620	+	0.134880i
$809$	23879.7	+	13786.9i	1.03778	+	0.599163i	0.919204	−	0.393781i	$-0.128833\pi$
$809$	23879.7	+	13786.9i	1.03778	+	0.599163i	0.118577	+	0.992945i	$0.462167\pi$
$810$		0			0
$811$		−	29871.4i		−	1.29338i	−0.762755	−	0.646688i	$-0.776154\pi$
$811$		−	29871.4i		−	1.29338i	0.762755	−	0.646688i	$-0.223846\pi$
$812$	5913.09	+	2688.53i	0.255553	+	0.116193i
$813$		0			0
$814$	2205.35	+	3819.78i	0.0949602	+	0.164476i
$815$	18551.6	−	32132.2i	0.797341	−	1.38103i
$816$		0			0
$817$	−688.500	+	397.506i	−0.0294829	+	0.0170220i
$818$	−4649.29			−0.198727
$819$		0			0
$820$	32548.1			1.38613
$821$	−12157.3	+	7018.99i	−0.516798	+	0.298373i	−0.735624	−	0.677391i	$-0.763111\pi$
$821$	−12157.3	+	7018.99i	−0.516798	+	0.298373i	0.218826	+	0.975764i	$0.429777\pi$
$822$		0			0
$823$	9659.82	−	16731.3i	0.409137	−	0.708647i	−0.585656	−	0.810560i	$-0.699163\pi$
$823$	9659.82	−	16731.3i	0.409137	−	0.708647i	0.994793	+	0.101913i	$0.0324963\pi$
$824$	−2999.86	−	5195.92i	−0.126827	−	0.219670i
$825$		0			0
$826$	1289.54	+	13271.2i	0.0543207	+	0.559038i
$827$			33926.3i			1.42652i	0.700899	+	0.713260i	$0.252782\pi$
$827$			33926.3i			1.42652i	−0.700899	+	0.713260i	$0.747218\pi$
$828$		0			0
$829$	16380.4	+	9457.22i	0.686266	+	0.396216i	0.802212	−	0.597040i	$-0.203657\pi$
$829$	16380.4	+	9457.22i	0.686266	+	0.396216i	−0.115946	+	0.993256i	$0.536990\pi$
$830$	11789.2	+	6806.52i	0.493025	+	0.284648i
$831$		0			0
$832$			5211.49i			0.217159i
$833$	10383.8	−	30328.2i	0.431905	−	1.26148i
$834$		0			0
$835$	19321.2	+	33465.3i	0.800763	+	1.38696i
$836$	436.198	−	755.517i	0.0180457	−	0.0312561i
$837$		0			0
$838$	−13972.7	+	8067.13i	−0.575988	+	0.332547i
$839$	24415.7			1.00468			0.502338	−	0.864672i	$-0.332473\pi$
$839$	24415.7			1.00468			0.502338	+	0.864672i	$0.332473\pi$
$840$		0			0
$841$	16700.8			0.684768
$842$	−494.552	+	285.530i	−0.0202416	+	0.0116865i
$843$		0			0
$844$	−3063.17	+	5305.57i	−0.124927	+	0.216381i
$845$	−47296.0	−	81919.1i	−1.92548	−	3.33503i
$846$		0			0
$847$	−16102.5	−	22533.3i	−0.653232	−	0.914112i
$848$		−	3231.72i		−	0.130870i
$849$		0			0
$850$	−53444.6	−	30856.3i	−2.15663	−	1.24513i
$851$	3030.08	+	1749.42i	0.122056	+	0.0704691i
$852$		0			0
$853$			6934.51i			0.278351i	0.990268	+	0.139175i	$0.0444452\pi$
$853$			6934.51i			0.278351i	−0.990268	+	0.139175i	$0.955555\pi$
$854$	−3877.23	+	8527.48i	−0.155358	+	0.341691i
$855$		0			0
$856$	2580.82	+	4470.12i	0.103050	+	0.178488i
$857$	−12226.3	+	21176.5i	−0.487330	+	0.844079i	−0.999894	−	0.0145693i	$-0.995362\pi$
$857$	−12226.3	+	21176.5i	−0.487330	+	0.844079i	0.512564	+	0.858649i	$0.328696\pi$
$858$		0			0
$859$	−12143.1	+	7010.80i	−0.482324	+	0.278470i	−0.721384	−	0.692535i	$-0.756494\pi$
$859$	−12143.1	+	7010.80i	−0.482324	+	0.278470i	0.239061	+	0.971005i	$0.423161\pi$
$860$	16537.8			0.655738
$861$		0			0
$862$	−8062.34			−0.318567
$863$	5409.09	−	3122.94i	0.213357	−	0.123182i	−0.389513	−	0.921021i	$-0.627357\pi$
$863$	5409.09	−	3122.94i	0.213357	−	0.123182i	0.602871	+	0.797839i	$0.294023\pi$
$864$		0			0
$865$	−43109.8	+	74668.4i	−1.69454	+	2.93503i
$866$	6255.26	+	10834.4i	0.245453	+	0.425138i
$867$		0			0
$868$	6543.19	−	4675.81i	0.255864	−	0.182843i
$869$			66733.8i			2.60505i
$870$		0			0
$871$	−8970.18	−	5178.94i	−0.348959	−	0.201471i
$872$	5594.37	+	3229.91i	0.217258	+	0.125434i
$873$		0			0
$874$		−	692.036i		−	0.0267831i
$875$	80681.3	−	7839.66i	3.11717	−	0.302890i
$876$		0			0
$877$	−13892.1	−	24061.8i	−0.534893	−	0.926463i	−0.999168	−	0.0407715i	$-0.987018\pi$
$877$	−13892.1	−	24061.8i	−0.534893	−	0.926463i	0.464275	−	0.885691i	$-0.346315\pi$
$878$	13888.5	−	24055.6i	0.533843	−	0.924644i
$879$		0			0
$880$	−15716.3	+	9073.78i	−0.602040	+	0.347588i
$881$	−51742.0			−1.97870			−0.989349	−	0.145566i	$-0.953500\pi$
$881$	−51742.0			−1.97870			−0.989349	+	0.145566i	$0.953500\pi$
$882$		0			0
$883$	−453.488			−0.0172832			−0.00864162	−	0.999963i	$-0.502751\pi$
$883$	−453.488			−0.0172832			−0.00864162	+	0.999963i	$0.502751\pi$
$884$	−26363.1	+	15220.7i	−1.00304	+	0.579104i
$885$		0			0
$886$	17138.2	−	29684.2i	0.649852	−	1.12558i
$887$	−9730.42	−	16853.6i	−0.368338	−	0.637980i	0.620968	−	0.783836i	$-0.286740\pi$
$887$	−9730.42	−	16853.6i	−0.368338	−	0.637980i	−0.989306	+	0.145856i	$0.953406\pi$
$888$		0			0
$889$	43347.0	−	4211.95i	1.63533	−	0.158903i
$890$			32375.1i			1.21934i
$891$		0			0
$892$	19436.8	+	11221.8i	0.729588	+	0.421228i
$893$	340.044	+	196.325i	0.0127426	+	0.00735695i
$894$		0			0
$895$		−	66317.0i		−	2.47680i
$896$	−1928.74	+	1378.29i	−0.0719136	+	0.0513900i
$897$		0			0
$898$	−3361.40	−	5822.12i	−0.124912	−	0.216355i
$899$	−4759.36	+	8243.45i	−0.176567	+	0.305823i
$900$		0			0
$901$	16348.1	−	9438.58i	0.604477	−	0.348995i
$902$	40553.9			1.49701
$903$		0			0
$904$	1233.94			0.0453984
$905$	8644.03	−	4990.63i	0.317500	−	0.183309i
$906$		0			0
$907$	7482.91	−	12960.8i	0.273942	−	0.474482i	−0.695925	−	0.718114i	$-0.745006\pi$
$907$	7482.91	−	12960.8i	0.273942	−	0.474482i	0.969868	+	0.243632i	$0.0783389\pi$
$908$	681.080	+	1179.67i	0.0248926	+	0.0431152i
$909$		0			0
$910$	26634.0	−	58578.1i	0.970228	−	2.13389i
$911$		−	36481.0i		−	1.32675i	−0.748287	−	0.663375i	$-0.769123\pi$
$911$		−	36481.0i		−	1.32675i	0.748287	−	0.663375i	$-0.230877\pi$
$912$		0			0
$913$	14689.0	+	8480.72i	0.532460	+	0.307416i
$914$	3125.61	+	1804.57i	0.113114	+	0.0653062i
$915$		0			0
$916$		−	25570.8i		−	0.922362i
$917$	17524.4	+	24523.1i	0.631087	+	0.883123i
$918$		0			0
$919$	−8212.13	−	14223.8i	−0.294770	−	0.510556i	0.680162	−	0.733062i	$-0.261910\pi$
$919$	−8212.13	−	14223.8i	−0.294770	−	0.510556i	−0.974931	+	0.222506i	$0.928576\pi$
$920$	−7197.86	+	12467.1i	−0.257942	+	0.446768i
$921$		0			0
$922$	26797.0	−	15471.3i	0.957171	−	0.552623i
$923$	23057.0			0.822242
$924$		0			0
$925$	−13695.6			−0.486819
$926$	−17757.0	+	10252.0i	−0.630163	+	0.363825i
$927$		0			0
$928$	1402.92	−	2429.92i	0.0496261	−	0.0859549i
$929$	6020.37	+	10427.6i	0.212618	+	0.368265i	0.952533	−	0.304435i	$-0.0984678\pi$
$929$	6020.37	+	10427.6i	0.212618	+	0.368265i	−0.739915	+	0.672700i	$0.765134\pi$
$930$		0			0
$931$	−1060.35	+	925.000i	−0.0373273	+	0.0325625i
$932$			2952.19i			0.103758i
$933$		0			0
$934$	−22366.3	−	12913.2i	−0.783561	−	0.452389i
$935$	−91802.0	−	53001.9i	−3.21096	−	1.85385i
$936$		0			0
$937$		−	24517.1i		−	0.854791i	−0.904065	−	0.427395i	$-0.859431\pi$
$937$		−	24517.1i		−	0.854791i	0.904065	−	0.427395i	$-0.140569\pi$
$938$	−455.669	−	4689.48i	−0.0158615	−	0.163238i
$939$		0			0
$940$	−4083.94	−	7073.59i	−0.141706	−	0.245442i
$941$	16522.7	−	28618.2i	0.572397	−	0.991421i	−0.423922	−	0.905699i	$-0.639347\pi$
$941$	16522.7	−	28618.2i	0.572397	−	0.991421i	0.996319	−	0.0857221i	$-0.0273197\pi$
$942$		0			0
$943$	27859.8	−	16084.9i	0.962080	−	0.555457i
$944$	5759.63			0.198581
$945$		0			0
$946$	20605.6			0.708188
$947$	−11184.0	+	6457.09i	−0.383771	+	0.221571i	−0.679458	−	0.733715i	$-0.737785\pi$
$947$	−11184.0	+	6457.09i	−0.383771	+	0.221571i	0.295686	+	0.955285i	$0.404452\pi$
$948$		0			0
$949$	−12053.9	+	20878.0i	−0.412314	+	0.714149i
$950$	1354.43	+	2345.94i	0.0462563	+	0.0801182i
$951$		0			0
$952$	−12605.4	−	5731.34i	−0.429141	−	0.195119i
$953$			15108.2i			0.513539i	0.966473	+	0.256770i	$0.0826581\pi$
$953$			15108.2i			0.513539i	−0.966473	+	0.256770i	$0.917342\pi$
$954$		0			0
$955$	16266.2	+	9391.27i	0.551163	+	0.318214i
$956$	8762.83	+	5059.22i	0.296454	+	0.171158i
$957$		0			0
$958$		−	31172.1i		−	1.05128i
$959$	−19500.7	−	8866.45i	−0.656631	−	0.298553i
$960$		0			0
$961$	−9002.96	−	15593.6i	−0.302204	−	0.523433i
$962$	−3377.87	+	5850.63i	−0.113209	+	0.196083i
$963$		0			0
$964$	−48.8387	+	28.1970i	−0.00163173	+	0.000942079i
$965$	−72932.5			−2.43293
$966$		0			0
$967$	51533.1			1.71375			0.856873	−	0.515527i	$-0.172404\pi$
$967$	51533.1			1.71375			0.856873	+	0.515527i	$0.172404\pi$
$968$	−10360.5	+	5981.66i	−0.344009	+	0.198613i
$969$		0			0
$970$	−33635.4	+	58258.1i	−1.11337	+	1.92841i
$971$	−18870.9	−	32685.4i	−0.623684	−	1.08025i	−0.988794	−	0.149288i	$-0.952302\pi$
$971$	−18870.9	−	32685.4i	−0.623684	−	1.08025i	0.365110	−	0.930964i	$-0.381031\pi$
$972$		0			0
$973$	3338.10	+	34353.8i	0.109984	+	1.13189i
$974$			18976.3i			0.624271i
$975$		0			0
$976$	3504.28	+	2023.20i	0.114928	+	0.0663535i
$977$	−12665.0	−	7312.16i	−0.414729	−	0.239444i	0.278091	−	0.960555i	$-0.410298\pi$
$977$	−12665.0	−	7312.16i	−0.414729	−	0.239444i	−0.692820	+	0.721111i	$0.743632\pi$
$978$		0			0
$979$			40338.4i			1.31688i
$980$	28723.2	−	5635.18i	0.936255	−	0.183683i
$981$		0			0
$982$	1921.42	+	3328.00i	0.0624388	+	0.108147i
$983$	−5340.27	+	9249.62i	−0.173274	+	0.300119i	−0.939563	−	0.342377i	$-0.888768\pi$
$983$	−5340.27	+	9249.62i	−0.173274	+	0.300119i	0.766289	+	0.642496i	$0.222101\pi$
$984$		0			0
$985$	−44410.5	+	25640.4i	−1.43658	+	0.829412i
$986$	16389.5			0.529358
$987$		0			0
$988$	1336.22			0.0430272
$989$	14155.7	−	8172.79i	0.455131	−	0.262770i
$990$		0			0
$991$	6492.75	−	11245.8i	0.208122	−	0.360478i	−0.743001	−	0.669290i	$-0.766598\pi$
$991$	6492.75	−	11245.8i	0.208122	−	0.360478i	0.951123	+	0.308813i	$0.0999316\pi$
$992$	−1736.95	−	3008.48i	−0.0555928	−	0.0962896i
$993$		0			0
$994$	6097.91	+	8533.22i	0.194581	+	0.272291i
$995$			79558.0i			2.53483i
$996$		0			0
$997$	908.938	+	524.775i	0.0288730	+	0.0166698i	0.514367	−	0.857570i	$-0.328027\pi$
$997$	908.938	+	524.775i	0.0288730	+	0.0166698i	−0.485494	+	0.874240i	$0.661360\pi$
$998$	12785.8	+	7381.89i	0.405539	+	0.234138i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.4.k.d.269.5	yes	20
3.2	odd	2	inner	378.4.k.d.269.6	yes	20
7.5	odd	6	inner	378.4.k.d.215.6	yes	20
21.5	even	6	inner	378.4.k.d.215.5	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.k.d.215.5	✓	20	21.5	even	6	inner
378.4.k.d.215.6	yes	20	7.5	odd	6	inner
378.4.k.d.269.5	yes	20	1.1	even	1	trivial
378.4.k.d.269.6	yes	20	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	378.k (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.73205 + 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(10.6672 + 18.4761i) q^{5} +(-18.4334 + 1.79115i) q^{7} +8.00000i q^{8} +O(q^{10})\)	\(q+(-1.73205 + 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(10.6672 + 18.4761i) q^{5} +(-18.4334 + 1.79115i) q^{7} +8.00000i q^{8} +(-36.9523 - 21.3344i) q^{10} +(-46.0414 - 26.5820i) q^{11} -81.4296i q^{13} +(30.1365 - 21.5358i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(46.7297 - 80.9382i) q^{17} +(-3.55276 + 2.05119i) q^{19} +85.3376 q^{20} +106.328 q^{22} +(73.0455 - 42.1728i) q^{23} +(-165.078 + 285.924i) q^{25} +(81.4296 + 141.040i) q^{26} +(-30.6622 + 67.4376i) q^{28} -87.6823i q^{29} +(-94.0150 - 54.2796i) q^{31} +(27.7128 + 16.0000i) q^{32} +186.919i q^{34} +(-229.727 - 321.472i) q^{35} +(20.7410 + 35.9245i) q^{37} +(4.10238 - 7.10553i) q^{38} +(-147.809 + 85.3376i) q^{40} +381.404 q^{41} +193.793 q^{43} +(-184.166 + 106.328i) q^{44} +(-84.3457 + 146.091i) q^{46} +(-47.8563 - 82.8895i) q^{47} +(336.584 - 66.0340i) q^{49} -660.314i q^{50} +(-282.080 - 162.859i) q^{52} +(174.922 + 100.991i) q^{53} -1134.22i q^{55} +(-14.3292 - 147.468i) q^{56} +(87.6823 + 151.870i) q^{58} +(-179.989 + 311.749i) q^{59} +(-219.018 + 126.450i) q^{61} +217.118 q^{62} -64.0000 q^{64} +(1504.50 - 868.626i) q^{65} +(63.6002 - 110.159i) q^{67} +(-186.919 - 323.753i) q^{68} +(719.371 + 327.080i) q^{70} +283.152i q^{71} +(-256.393 - 148.029i) q^{73} +(-71.8490 - 41.4820i) q^{74} +16.4095i q^{76} +(896.314 + 407.531i) q^{77} +(-627.622 - 1087.07i) q^{79} +(170.675 - 295.618i) q^{80} +(-660.611 + 381.404i) q^{82} -319.040 q^{83} +1993.90 q^{85} +(-335.659 + 193.793i) q^{86} +(212.656 - 368.331i) q^{88} +(-379.377 - 657.100i) q^{89} +(145.852 + 1501.03i) q^{91} -337.383i q^{92} +(165.779 + 95.7126i) q^{94} +(-75.7961 - 43.7609i) q^{95} -1576.58i q^{97} +(-516.946 + 450.958i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 40 q^{4} + 4 q^{7}+O(q^{10}) \) 20 * q + 40 * q^4 + 4 * q^7	\( 20 q + 40 q^{4} + 4 q^{7} - 24 q^{10} - 160 q^{16} + 408 q^{19} + 240 q^{22} - 646 q^{25} + 56 q^{28} - 102 q^{31} + 194 q^{37} - 96 q^{40} - 2332 q^{43} - 624 q^{46} + 2840 q^{49} - 648 q^{52} + 96 q^{58} + 1878 q^{61} - 1280 q^{64} - 386 q^{67} + 3672 q^{70} + 1788 q^{73} + 814 q^{79} - 672 q^{82} - 4560 q^{85} + 480 q^{88} + 2724 q^{91} - 1536 q^{94}+O(q^{100}) \) 20 * q + 40 * q^4 + 4 * q^7 - 24 * q^10 - 160 * q^16 + 408 * q^19 + 240 * q^22 - 646 * q^25 + 56 * q^28 - 102 * q^31 + 194 * q^37 - 96 * q^40 - 2332 * q^43 - 624 * q^46 + 2840 * q^49 - 648 * q^52 + 96 * q^58 + 1878 * q^61 - 1280 * q^64 - 386 * q^67 + 3672 * q^70 + 1788 * q^73 + 814 * q^79 - 672 * q^82 - 4560 * q^85 + 480 * q^88 + 2724 * q^91 - 1536 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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