LMFDB - Embedded newform 704.4.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [704,4,Mod(1,704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(704, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

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

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

chi := DirichletCharacter("704.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 704 = 2^{6} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	704.a (trivial)

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:	yes
Analytic conductor:	$41.5373446440$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 11)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	704.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.92820 q^{3} +12.8564 q^{5} -16.9282 q^{7} +8.14359 q^{9} +O(q^{10})$	$q+5.92820 q^{3} +12.8564 q^{5} -16.9282 q^{7} +8.14359 q^{9} -11.0000 q^{11} -74.6410 q^{13} +76.2154 q^{15} -82.7846 q^{17} -67.9230 q^{19} -100.354 q^{21} -13.3538 q^{23} +40.2872 q^{25} -111.785 q^{27} -168.995 q^{29} +65.4974 q^{31} -65.2102 q^{33} -217.636 q^{35} -40.8564 q^{37} -442.487 q^{39} +274.928 q^{41} -2.28719 q^{43} +104.697 q^{45} -71.8461 q^{47} -56.4359 q^{49} -490.764 q^{51} +149.005 q^{53} -141.420 q^{55} -402.662 q^{57} +545.631 q^{59} -101.303 q^{61} -137.856 q^{63} -959.615 q^{65} +411.641 q^{67} -79.1642 q^{69} +470.636 q^{71} +610.600 q^{73} +238.831 q^{75} +186.210 q^{77} +978.225 q^{79} -882.559 q^{81} +26.1539 q^{83} -1064.31 q^{85} -1001.84 q^{87} -352.887 q^{89} +1263.54 q^{91} +388.282 q^{93} -873.246 q^{95} +847.585 q^{97} -89.5795 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} - 20 q^{7} + 44 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 - 20 * q^7 + 44 * q^9	$ 2 q - 2 q^{3} - 2 q^{5} - 20 q^{7} + 44 q^{9} - 22 q^{11} - 80 q^{13} + 194 q^{15} - 124 q^{17} + 72 q^{19} - 76 q^{21} + 98 q^{23} + 136 q^{25} - 182 q^{27} - 144 q^{29} + 34 q^{31} + 22 q^{33} - 172 q^{35} - 54 q^{37} - 400 q^{39} + 536 q^{41} - 60 q^{43} - 428 q^{45} + 272 q^{47} - 390 q^{49} - 164 q^{51} + 492 q^{53} + 22 q^{55} - 1512 q^{57} + 634 q^{59} - 840 q^{61} - 248 q^{63} - 880 q^{65} + 754 q^{67} - 962 q^{69} + 678 q^{71} - 400 q^{73} - 520 q^{75} + 220 q^{77} - 316 q^{79} - 1294 q^{81} + 468 q^{83} - 452 q^{85} - 1200 q^{87} - 1842 q^{89} + 1280 q^{91} + 638 q^{93} - 2952 q^{95} + 2194 q^{97} - 484 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 - 20 * q^7 + 44 * q^9 - 22 * q^11 - 80 * q^13 + 194 * q^15 - 124 * q^17 + 72 * q^19 - 76 * q^21 + 98 * q^23 + 136 * q^25 - 182 * q^27 - 144 * q^29 + 34 * q^31 + 22 * q^33 - 172 * q^35 - 54 * q^37 - 400 * q^39 + 536 * q^41 - 60 * q^43 - 428 * q^45 + 272 * q^47 - 390 * q^49 - 164 * q^51 + 492 * q^53 + 22 * q^55 - 1512 * q^57 + 634 * q^59 - 840 * q^61 - 248 * q^63 - 880 * q^65 + 754 * q^67 - 962 * q^69 + 678 * q^71 - 400 * q^73 - 520 * q^75 + 220 * q^77 - 316 * q^79 - 1294 * q^81 + 468 * q^83 - 452 * q^85 - 1200 * q^87 - 1842 * q^89 + 1280 * q^91 + 638 * q^93 - 2952 * q^95 + 2194 * q^97 - 484 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	5.92820	1.14088	0.570442	−	0.821338i	$-0.306772\pi$
$3$	5.92820	1.14088	0.570442	+	0.821338i	$0.306772\pi$
$4$	0	0
$5$	12.8564	1.14991	0.574956	−	0.818184i	$-0.305019\pi$
$5$	12.8564	1.14991	0.574956	+	0.818184i	$0.305019\pi$
$6$	0	0
$7$	−16.9282	−0.914037	−0.457019	−	0.889457i	$-0.651083\pi$
$7$	−16.9282	−0.914037	−0.457019	+	0.889457i	$0.651083\pi$
$8$	0	0
$9$	8.14359	0.301615
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−74.6410	−1.59244	−0.796219	−	0.605009i	$-0.793170\pi$
$13$	−74.6410	−1.59244	−0.796219	+	0.605009i	$0.793170\pi$
$14$	0	0
$15$	76.2154	1.31192
$16$	0	0
$17$	−82.7846	−1.18107	−0.590536	−	0.807011i	$-0.701084\pi$
$17$	−82.7846	−1.18107	−0.590536	+	0.807011i	$0.701084\pi$
$18$	0	0
$19$	−67.9230	−0.820138	−0.410069	−	0.912055i	$-0.634495\pi$
$19$	−67.9230	−0.820138	−0.410069	+	0.912055i	$0.634495\pi$
$20$	0	0
$21$	−100.354	−1.04281
$22$	0	0
$23$	−13.3538	−0.121064	−0.0605319	−	0.998166i	$-0.519280\pi$
$23$	−13.3538	−0.121064	−0.0605319	+	0.998166i	$0.519280\pi$
$24$	0	0
$25$	40.2872	0.322297
$26$	0	0
$27$	−111.785	−0.796776
$28$	0	0
$29$	−168.995	−1.08212	−0.541061	−	0.840983i	$-0.681977\pi$
$29$	−168.995	−1.08212	−0.541061	+	0.840983i	$0.681977\pi$
$30$	0	0
$31$	65.4974	0.379474	0.189737	−	0.981835i	$-0.439237\pi$
$31$	65.4974	0.379474	0.189737	+	0.981835i	$0.439237\pi$
$32$	0	0
$33$	−65.2102	−0.343989
$34$	0	0
$35$	−217.636	−1.05106
$36$	0	0
$37$	−40.8564	−0.181534	−0.0907669	−	0.995872i	$-0.528932\pi$
$37$	−40.8564	−0.181534	−0.0907669	+	0.995872i	$0.528932\pi$
$38$	0	0
$39$	−442.487	−1.81679
$40$	0	0
$41$	274.928	1.04723	0.523617	−	0.851954i	$-0.324582\pi$
$41$	274.928	1.04723	0.523617	+	0.851954i	$0.324582\pi$
$42$	0	0
$43$	−2.28719	−0.00811146	−0.00405573	−	0.999992i	$-0.501291\pi$
$43$	−2.28719	−0.00811146	−0.00405573	+	0.999992i	$0.501291\pi$
$44$	0	0
$45$	104.697	0.346830
$46$	0	0
$47$	−71.8461	−0.222975	−0.111488	−	0.993766i	$-0.535562\pi$
$47$	−71.8461	−0.222975	−0.111488	+	0.993766i	$0.535562\pi$
$48$	0	0
$49$	−56.4359	−0.164536
$50$	0	0
$51$	−490.764	−1.34746
$52$	0	0
$53$	149.005	0.386178	0.193089	−	0.981181i	$-0.438149\pi$
$53$	149.005	0.386178	0.193089	+	0.981181i	$0.438149\pi$
$54$	0	0
$55$	−141.420	−0.346711
$56$	0	0
$57$	−402.662	−0.935681
$58$	0	0
$59$	545.631	1.20398	0.601992	−	0.798502i	$-0.294374\pi$
$59$	545.631	1.20398	0.601992	+	0.798502i	$0.294374\pi$
$60$	0	0
$61$	−101.303	−0.212631	−0.106315	−	0.994332i	$-0.533905\pi$
$61$	−101.303	−0.212631	−0.106315	+	0.994332i	$0.533905\pi$
$62$	0	0
$63$	−137.856	−0.275687
$64$	0	0
$65$	−959.615	−1.83116
$66$	0	0
$67$	411.641	0.750596	0.375298	−	0.926904i	$-0.377540\pi$
$67$	411.641	0.750596	0.375298	+	0.926904i	$0.377540\pi$
$68$	0	0
$69$	−79.1642	−0.138120
$70$	0	0
$71$	470.636	0.786679	0.393339	−	0.919393i	$-0.371320\pi$
$71$	470.636	0.786679	0.393339	+	0.919393i	$0.371320\pi$
$72$	0	0
$73$	610.600	0.978977	0.489488	−	0.872010i	$-0.337184\pi$
$73$	610.600	0.978977	0.489488	+	0.872010i	$0.337184\pi$
$74$	0	0
$75$	238.831	0.367704
$76$	0	0
$77$	186.210	0.275593
$78$	0	0
$79$	978.225	1.39315	0.696576	−	0.717483i	$-0.254706\pi$
$79$	978.225	1.39315	0.696576	+	0.717483i	$0.254706\pi$
$80$	0	0
$81$	−882.559	−1.21064
$82$	0	0
$83$	26.1539	0.0345875	0.0172938	−	0.999850i	$-0.494495\pi$
$83$	26.1539	0.0345875	0.0172938	+	0.999850i	$0.494495\pi$
$84$	0	0
$85$	−1064.31	−1.35813
$86$	0	0
$87$	−1001.84	−1.23458
$88$	0	0
$89$	−352.887	−0.420292	−0.210146	−	0.977670i	$-0.567394\pi$
$89$	−352.887	−0.420292	−0.210146	+	0.977670i	$0.567394\pi$
$90$	0	0
$91$	1263.54	1.45555
$92$	0	0
$93$	388.282	0.432935
$94$	0	0
$95$	−873.246	−0.943086
$96$	0	0
$97$	847.585	0.887208	0.443604	−	0.896223i	$-0.353700\pi$
$97$	847.585	0.887208	0.443604	+	0.896223i	$0.353700\pi$
$98$	0	0
$99$	−89.5795	−0.0909402
$100$	0	0
$101$	−1293.46	−1.27430	−0.637150	−	0.770740i	$-0.719887\pi$
$101$	−1293.46	−1.27430	−0.637150	+	0.770740i	$0.719887\pi$
$102$	0	0
$103$	1725.24	1.65042	0.825209	−	0.564828i	$-0.191057\pi$
$103$	1725.24	1.65042	0.825209	+	0.564828i	$0.191057\pi$
$104$	0	0
$105$	−1290.19	−1.19914
$106$	0	0
$107$	−484.179	−0.437452	−0.218726	−	0.975786i	$-0.570190\pi$
$107$	−484.179	−0.437452	−0.218726	+	0.975786i	$0.570190\pi$
$108$	0	0
$109$	64.2563	0.0564645	0.0282323	−	0.999601i	$-0.491012\pi$
$109$	64.2563	0.0564645	0.0282323	+	0.999601i	$0.491012\pi$
$110$	0	0
$111$	−242.205	−0.207109
$112$	0	0
$113$	−2005.08	−1.66922	−0.834612	−	0.550839i	$-0.814308\pi$
$113$	−2005.08	−1.66922	−0.834612	+	0.550839i	$0.814308\pi$
$114$	0	0
$115$	−171.682	−0.139213
$116$	0	0
$117$	−607.846	−0.480302
$118$	0	0
$119$	1401.39	1.07954
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	1629.83	1.19477
$124$	0	0
$125$	−1089.10	−0.779298
$126$	0	0
$127$	−109.605	−0.0765816	−0.0382908	−	0.999267i	$-0.512191\pi$
$127$	−109.605	−0.0765816	−0.0382908	+	0.999267i	$0.512191\pi$
$128$	0	0
$129$	−13.5589	−0.00925423
$130$	0	0
$131$	1156.71	0.771469	0.385734	−	0.922610i	$-0.373948\pi$
$131$	1156.71	0.771469	0.385734	+	0.922610i	$0.373948\pi$
$132$	0	0
$133$	1149.82	0.749636
$134$	0	0
$135$	−1437.15	−0.916223
$136$	0	0
$137$	198.323	0.123678	0.0618391	−	0.998086i	$-0.480303\pi$
$137$	198.323	0.123678	0.0618391	+	0.998086i	$0.480303\pi$
$138$	0	0
$139$	−2900.14	−1.76969	−0.884844	−	0.465888i	$-0.845735\pi$
$139$	−2900.14	−1.76969	−0.884844	+	0.465888i	$0.845735\pi$
$140$	0	0
$141$	−425.918	−0.254389
$142$	0	0
$143$	821.051	0.480138
$144$	0	0
$145$	−2172.67	−1.24435
$146$	0	0
$147$	−334.564	−0.187717
$148$	0	0
$149$	−3488.34	−1.91796	−0.958980	−	0.283472i	$-0.908514\pi$
$149$	−3488.34	−1.91796	−0.958980	+	0.283472i	$0.908514\pi$
$150$	0	0
$151$	1163.32	0.626953	0.313477	−	0.949596i	$-0.398506\pi$
$151$	1163.32	0.626953	0.313477	+	0.949596i	$0.398506\pi$
$152$	0	0
$153$	−674.164	−0.356228
$154$	0	0
$155$	842.061	0.436361
$156$	0	0
$157$	−342.057	−0.173880	−0.0869398	−	0.996214i	$-0.527709\pi$
$157$	−342.057	−0.173880	−0.0869398	+	0.996214i	$0.527709\pi$
$158$	0	0
$159$	883.333	0.440584
$160$	0	0
$161$	226.056	0.110657
$162$	0	0
$163$	−1394.89	−0.670285	−0.335142	−	0.942167i	$-0.608784\pi$
$163$	−1394.89	−0.670285	−0.335142	+	0.942167i	$0.608784\pi$
$164$	0	0
$165$	−838.369	−0.395557
$166$	0	0
$167$	−478.703	−0.221815	−0.110908	−	0.993831i	$-0.535376\pi$
$167$	−478.703	−0.221815	−0.110908	+	0.993831i	$0.535376\pi$
$168$	0	0
$169$	3374.28	1.53586
$170$	0	0
$171$	−553.138	−0.247365
$172$	0	0
$173$	−1808.58	−0.794822	−0.397411	−	0.917641i	$-0.630091\pi$
$173$	−1808.58	−0.794822	−0.397411	+	0.917641i	$0.630091\pi$
$174$	0	0
$175$	−681.990	−0.294592
$176$	0	0
$177$	3234.61	1.37361
$178$	0	0
$179$	−4429.85	−1.84973	−0.924867	−	0.380292i	$-0.875824\pi$
$179$	−4429.85	−1.84973	−0.924867	+	0.380292i	$0.875824\pi$
$180$	0	0
$181$	−3409.17	−1.40001	−0.700005	−	0.714138i	$-0.746819\pi$
$181$	−3409.17	−1.40001	−0.700005	+	0.714138i	$0.746819\pi$
$182$	0	0
$183$	−600.543	−0.242587
$184$	0	0
$185$	−525.267	−0.208748
$186$	0	0
$187$	910.631	0.356106
$188$	0	0
$189$	1892.31	0.728283
$190$	0	0
$191$	−2923.75	−1.10762	−0.553810	−	0.832643i	$-0.686827\pi$
$191$	−2923.75	−1.10762	−0.553810	+	0.832643i	$0.686827\pi$
$192$	0	0
$193$	−2484.18	−0.926505	−0.463253	−	0.886226i	$-0.653318\pi$
$193$	−2484.18	−0.926505	−0.463253	+	0.886226i	$0.653318\pi$
$194$	0	0
$195$	−5688.79	−2.08914
$196$	0	0
$197$	5125.67	1.85375	0.926876	−	0.375369i	$-0.122484\pi$
$197$	5125.67	1.85375	0.926876	+	0.375369i	$0.122484\pi$
$198$	0	0
$199$	7.69219	0.00274013	0.00137006	−	0.999999i	$-0.499564\pi$
$199$	7.69219	0.00274013	0.00137006	+	0.999999i	$0.499564\pi$
$200$	0	0
$201$	2440.29	0.856343
$202$	0	0
$203$	2860.78	0.989100
$204$	0	0
$205$	3534.59	1.20423
$206$	0	0
$207$	−108.748	−0.0365146
$208$	0	0
$209$	747.154	0.247281
$210$	0	0
$211$	3107.34	1.01383	0.506915	−	0.861996i	$-0.330786\pi$
$211$	3107.34	1.01383	0.506915	+	0.861996i	$0.330786\pi$
$212$	0	0
$213$	2790.03	0.897509
$214$	0	0
$215$	−29.4050	−0.00932746
$216$	0	0
$217$	−1108.75	−0.346853
$218$	0	0
$219$	3619.76	1.11690
$220$	0	0
$221$	6179.13	1.88078
$222$	0	0
$223$	12.3185	0.00369913	0.00184957	−	0.999998i	$-0.499411\pi$
$223$	12.3185	0.00369913	0.00184957	+	0.999998i	$0.499411\pi$
$224$	0	0
$225$	328.082	0.0972096
$226$	0	0
$227$	4615.90	1.34964	0.674820	−	0.737983i	$-0.264221\pi$
$227$	4615.90	1.34964	0.674820	+	0.737983i	$0.264221\pi$
$228$	0	0
$229$	−5074.63	−1.46437	−0.732186	−	0.681105i	$-0.761500\pi$
$229$	−5074.63	−1.46437	−0.732186	+	0.681105i	$0.761500\pi$
$230$	0	0
$231$	1103.89	0.314419
$232$	0	0
$233$	211.683	0.0595184	0.0297592	−	0.999557i	$-0.490526\pi$
$233$	211.683	0.0595184	0.0297592	+	0.999557i	$0.490526\pi$
$234$	0	0
$235$	−923.683	−0.256402
$236$	0	0
$237$	5799.12	1.58942
$238$	0	0
$239$	−4312.49	−1.16716	−0.583581	−	0.812055i	$-0.698349\pi$
$239$	−4312.49	−1.16716	−0.583581	+	0.812055i	$0.698349\pi$
$240$	0	0
$241$	−996.584	−0.266372	−0.133186	−	0.991091i	$-0.542521\pi$
$241$	−996.584	−0.266372	−0.133186	+	0.991091i	$0.542521\pi$
$242$	0	0
$243$	−2213.80	−0.584426
$244$	0	0
$245$	−725.563	−0.189202
$246$	0	0
$247$	5069.85	1.30602
$248$	0	0
$249$	155.046	0.0394603
$250$	0	0
$251$	−276.892	−0.0696306	−0.0348153	−	0.999394i	$-0.511084\pi$
$251$	−276.892	−0.0696306	−0.0348153	+	0.999394i	$0.511084\pi$
$252$	0	0
$253$	146.892	0.0365021
$254$	0	0
$255$	−6309.46	−1.54947
$256$	0	0
$257$	−3235.18	−0.785233	−0.392617	−	0.919702i	$-0.628430\pi$
$257$	−3235.18	−0.785233	−0.392617	+	0.919702i	$0.628430\pi$
$258$	0	0
$259$	691.626	0.165929
$260$	0	0
$261$	−1376.23	−0.326384
$262$	0	0
$263$	−207.944	−0.0487544	−0.0243772	−	0.999703i	$-0.507760\pi$
$263$	−207.944	−0.0487544	−0.0243772	+	0.999703i	$0.507760\pi$
$264$	0	0
$265$	1915.67	0.444071
$266$	0	0
$267$	−2091.99	−0.479504
$268$	0	0
$269$	−5033.04	−1.14078	−0.570390	−	0.821374i	$-0.693208\pi$
$269$	−5033.04	−1.14078	−0.570390	+	0.821374i	$0.693208\pi$
$270$	0	0
$271$	−1487.01	−0.333319	−0.166660	−	0.986015i	$-0.553298\pi$
$271$	−1487.01	−0.333319	−0.166660	+	0.986015i	$0.553298\pi$
$272$	0	0
$273$	7490.51	1.66061
$274$	0	0
$275$	−443.159	−0.0971764
$276$	0	0
$277$	235.836	0.0511552	0.0255776	−	0.999673i	$-0.491858\pi$
$277$	235.836	0.0511552	0.0255776	+	0.999673i	$0.491858\pi$
$278$	0	0
$279$	533.384	0.114455
$280$	0	0
$281$	−4915.01	−1.04343	−0.521717	−	0.853118i	$-0.674708\pi$
$281$	−4915.01	−1.04343	−0.521717	+	0.853118i	$0.674708\pi$
$282$	0	0
$283$	−5199.56	−1.09216	−0.546081	−	0.837733i	$-0.683881\pi$
$283$	−5199.56	−1.09216	−0.546081	+	0.837733i	$0.683881\pi$
$284$	0	0
$285$	−5176.78	−1.07595
$286$	0	0
$287$	−4654.04	−0.957210
$288$	0	0
$289$	1940.29	0.394930
$290$	0	0
$291$	5024.65	1.01220
$292$	0	0
$293$	8880.92	1.77075	0.885373	−	0.464881i	$-0.153903\pi$
$293$	8880.92	1.77075	0.885373	+	0.464881i	$0.153903\pi$
$294$	0	0
$295$	7014.85	1.38448
$296$	0	0
$297$	1229.63	0.240237
$298$	0	0
$299$	996.743	0.192786
$300$	0	0
$301$	38.7180	0.00741417
$302$	0	0
$303$	−7667.90	−1.45383
$304$	0	0
$305$	−1302.39	−0.244507
$306$	0	0
$307$	−1497.93	−0.278474	−0.139237	−	0.990259i	$-0.544465\pi$
$307$	−1497.93	−0.278474	−0.139237	+	0.990259i	$0.544465\pi$
$308$	0	0
$309$	10227.6	1.88293
$310$	0	0
$311$	7484.71	1.36469	0.682345	−	0.731030i	$-0.260960\pi$
$311$	7484.71	1.36469	0.682345	+	0.731030i	$0.260960\pi$
$312$	0	0
$313$	−658.363	−0.118891	−0.0594455	−	0.998232i	$-0.518933\pi$
$313$	−658.363	−0.118891	−0.0594455	+	0.998232i	$0.518933\pi$
$314$	0	0
$315$	−1772.34	−0.317016
$316$	0	0
$317$	−233.708	−0.0414080	−0.0207040	−	0.999786i	$-0.506591\pi$
$317$	−233.708	−0.0414080	−0.0207040	+	0.999786i	$0.506591\pi$
$318$	0	0
$319$	1858.94	0.326272
$320$	0	0
$321$	−2870.31	−0.499082
$322$	0	0
$323$	5622.98	0.968641
$324$	0	0
$325$	−3007.08	−0.513239
$326$	0	0
$327$	380.924	0.0644194
$328$	0	0
$329$	1216.23	0.203808
$330$	0	0
$331$	8532.95	1.41696	0.708480	−	0.705731i	$-0.249381\pi$
$331$	8532.95	1.41696	0.708480	+	0.705731i	$0.249381\pi$
$332$	0	0
$333$	−332.718	−0.0547533
$334$	0	0
$335$	5292.22	0.863120
$336$	0	0
$337$	11691.2	1.88979	0.944895	−	0.327373i	$-0.106163\pi$
$337$	11691.2	1.88979	0.944895	+	0.327373i	$0.106163\pi$
$338$	0	0
$339$	−11886.5	−1.90439
$340$	0	0
$341$	−720.472	−0.114416
$342$	0	0
$343$	6761.73	1.06443
$344$	0	0
$345$	−1017.77	−0.158825
$346$	0	0
$347$	4598.79	0.711459	0.355729	−	0.934589i	$-0.384232\pi$
$347$	4598.79	0.711459	0.355729	+	0.934589i	$0.384232\pi$
$348$	0	0
$349$	−6720.27	−1.03074	−0.515369	−	0.856968i	$-0.672345\pi$
$349$	−6720.27	−1.03074	−0.515369	+	0.856968i	$0.672345\pi$
$350$	0	0
$351$	8343.72	1.26882
$352$	0	0
$353$	5738.70	0.865270	0.432635	−	0.901569i	$-0.357584\pi$
$353$	5738.70	0.865270	0.432635	+	0.901569i	$0.357584\pi$
$354$	0	0
$355$	6050.69	0.904611
$356$	0	0
$357$	8307.75	1.23163
$358$	0	0
$359$	4115.27	0.605001	0.302501	−	0.953149i	$-0.402179\pi$
$359$	4115.27	0.605001	0.302501	+	0.953149i	$0.402179\pi$
$360$	0	0
$361$	−2245.46	−0.327374
$362$	0	0
$363$	717.313	0.103717
$364$	0	0
$365$	7850.12	1.12574
$366$	0	0
$367$	−9662.99	−1.37440	−0.687199	−	0.726469i	$-0.741160\pi$
$367$	−9662.99	−1.37440	−0.687199	+	0.726469i	$0.741160\pi$
$368$	0	0
$369$	2238.90	0.315861
$370$	0	0
$371$	−2522.39	−0.352981
$372$	0	0
$373$	141.780	0.0196812	0.00984062	−	0.999952i	$-0.496868\pi$
$373$	141.780	0.0196812	0.00984062	+	0.999952i	$0.496868\pi$
$374$	0	0
$375$	−6456.42	−0.889088
$376$	0	0
$377$	12613.9	1.72321
$378$	0	0
$379$	−2819.73	−0.382163	−0.191082	−	0.981574i	$-0.561200\pi$
$379$	−2819.73	−0.382163	−0.191082	+	0.981574i	$0.561200\pi$
$380$	0	0
$381$	−649.760	−0.0873707
$382$	0	0
$383$	6337.84	0.845557	0.422778	−	0.906233i	$-0.361055\pi$
$383$	6337.84	0.845557	0.422778	+	0.906233i	$0.361055\pi$
$384$	0	0
$385$	2393.99	0.316907
$386$	0	0
$387$	−18.6259	−0.00244653
$388$	0	0
$389$	8805.25	1.14767	0.573836	−	0.818970i	$-0.305455\pi$
$389$	8805.25	1.14767	0.573836	+	0.818970i	$0.305455\pi$
$390$	0	0
$391$	1105.49	0.142985
$392$	0	0
$393$	6857.23	0.880156
$394$	0	0
$395$	12576.5	1.60200
$396$	0	0
$397$	−4315.26	−0.545534	−0.272767	−	0.962080i	$-0.587939\pi$
$397$	−4315.26	−0.545534	−0.272767	+	0.962080i	$0.587939\pi$
$398$	0	0
$399$	6816.34	0.855247
$400$	0	0
$401$	361.681	0.0450411	0.0225206	−	0.999746i	$-0.492831\pi$
$401$	361.681	0.0450411	0.0225206	+	0.999746i	$0.492831\pi$
$402$	0	0
$403$	−4888.79	−0.604288
$404$	0	0
$405$	−11346.5	−1.39213
$406$	0	0
$407$	449.420	0.0547345
$408$	0	0
$409$	9220.50	1.11473	0.557365	−	0.830268i	$-0.311812\pi$
$409$	9220.50	1.11473	0.557365	+	0.830268i	$0.311812\pi$
$410$	0	0
$411$	1175.70	0.141102
$412$	0	0
$413$	−9236.55	−1.10049
$414$	0	0
$415$	336.245	0.0397726
$416$	0	0
$417$	−17192.6	−2.01901
$418$	0	0
$419$	−14912.9	−1.73876	−0.869380	−	0.494144i	$-0.835481\pi$
$419$	−14912.9	−1.73876	−0.869380	+	0.494144i	$0.835481\pi$
$420$	0	0
$421$	13486.0	1.56121	0.780603	−	0.625027i	$-0.214912\pi$
$421$	13486.0	1.56121	0.780603	+	0.625027i	$0.214912\pi$
$422$	0	0
$423$	−585.085	−0.0672525
$424$	0	0
$425$	−3335.16	−0.380656
$426$	0	0
$427$	1714.87	0.194352
$428$	0	0
$429$	4867.36	0.547782
$430$	0	0
$431$	−406.334	−0.0454116	−0.0227058	−	0.999742i	$-0.507228\pi$
$431$	−406.334	−0.0454116	−0.0227058	+	0.999742i	$0.507228\pi$
$432$	0	0
$433$	−1766.69	−0.196078	−0.0980391	−	0.995183i	$-0.531257\pi$
$433$	−1766.69	−0.196078	−0.0980391	+	0.995183i	$0.531257\pi$
$434$	0	0
$435$	−12880.0	−1.41965
$436$	0	0
$437$	907.033	0.0992889
$438$	0	0
$439$	−7824.19	−0.850634	−0.425317	−	0.905044i	$-0.639837\pi$
$439$	−7824.19	−0.850634	−0.425317	+	0.905044i	$0.639837\pi$
$440$	0	0
$441$	−459.591	−0.0496265
$442$	0	0
$443$	11667.9	1.25137	0.625686	−	0.780075i	$-0.284819\pi$
$443$	11667.9	1.25137	0.625686	+	0.780075i	$0.284819\pi$
$444$	0	0
$445$	−4536.86	−0.483299
$446$	0	0
$447$	−20679.6	−2.18817
$448$	0	0
$449$	16975.3	1.78421	0.892107	−	0.451825i	$-0.149227\pi$
$449$	16975.3	1.78421	0.892107	+	0.451825i	$0.149227\pi$
$450$	0	0
$451$	−3024.21	−0.315753
$452$	0	0
$453$	6896.42	0.715280
$454$	0	0
$455$	16244.6	1.67375
$456$	0	0
$457$	−16192.9	−1.65748	−0.828741	−	0.559632i	$-0.810943\pi$
$457$	−16192.9	−1.65748	−0.828741	+	0.559632i	$0.810943\pi$
$458$	0	0
$459$	9254.05	0.941050
$460$	0	0
$461$	−8586.04	−0.867444	−0.433722	−	0.901047i	$-0.642800\pi$
$461$	−8586.04	−0.867444	−0.433722	+	0.901047i	$0.642800\pi$
$462$	0	0
$463$	7917.20	0.794694	0.397347	−	0.917668i	$-0.369931\pi$
$463$	7917.20	0.794694	0.397347	+	0.917668i	$0.369931\pi$
$464$	0	0
$465$	4991.91	0.497837
$466$	0	0
$467$	−15155.0	−1.50169	−0.750844	−	0.660480i	$-0.770353\pi$
$467$	−15155.0	−1.50169	−0.750844	+	0.660480i	$0.770353\pi$
$468$	0	0
$469$	−6968.34	−0.686073
$470$	0	0
$471$	−2027.78	−0.198376
$472$	0	0
$473$	25.1591	0.00244570
$474$	0	0
$475$	−2736.43	−0.264328
$476$	0	0
$477$	1213.44	0.116477
$478$	0	0
$479$	−10001.1	−0.953993	−0.476996	−	0.878905i	$-0.658275\pi$
$479$	−10001.1	−0.953993	−0.476996	+	0.878905i	$0.658275\pi$
$480$	0	0
$481$	3049.56	0.289081
$482$	0	0
$483$	1340.11	0.126246
$484$	0	0
$485$	10896.9	1.02021
$486$	0	0
$487$	−7044.54	−0.655480	−0.327740	−	0.944768i	$-0.606287\pi$
$487$	−7044.54	−0.655480	−0.327740	+	0.944768i	$0.606287\pi$
$488$	0	0
$489$	−8269.21	−0.764717
$490$	0	0
$491$	−13326.4	−1.22487	−0.612437	−	0.790520i	$-0.709811\pi$
$491$	−13326.4	−1.22487	−0.612437	+	0.790520i	$0.709811\pi$
$492$	0	0
$493$	13990.2	1.27806
$494$	0	0
$495$	−1151.67	−0.104573
$496$	0	0
$497$	−7967.02	−0.719054
$498$	0	0
$499$	−20069.1	−1.80044	−0.900218	−	0.435440i	$-0.856593\pi$
$499$	−20069.1	−1.80044	−0.900218	+	0.435440i	$0.856593\pi$
$500$	0	0
$501$	−2837.85	−0.253065
$502$	0	0
$503$	−7782.35	−0.689856	−0.344928	−	0.938629i	$-0.612097\pi$
$503$	−7782.35	−0.689856	−0.344928	+	0.938629i	$0.612097\pi$
$504$	0	0
$505$	−16629.3	−1.46533
$506$	0	0
$507$	20003.4	1.75224
$508$	0	0
$509$	1475.93	0.128526	0.0642628	−	0.997933i	$-0.479530\pi$
$509$	1475.93	0.128526	0.0642628	+	0.997933i	$0.479530\pi$
$510$	0	0
$511$	−10336.4	−0.894821
$512$	0	0
$513$	7592.75	0.653466
$514$	0	0
$515$	22180.4	1.89784
$516$	0	0
$517$	790.307	0.0672295
$518$	0	0
$519$	−10721.7	−0.906799
$520$	0	0
$521$	7609.43	0.639875	0.319938	−	0.947439i	$-0.396338\pi$
$521$	7609.43	0.639875	0.319938	+	0.947439i	$0.396338\pi$
$522$	0	0
$523$	12452.9	1.04116	0.520581	−	0.853812i	$-0.325715\pi$
$523$	12452.9	1.04116	0.520581	+	0.853812i	$0.325715\pi$
$524$	0	0
$525$	−4042.97	−0.336095
$526$	0	0
$527$	−5422.18	−0.448186
$528$	0	0
$529$	−11988.7	−0.985344
$530$	0	0
$531$	4443.39	0.363139
$532$	0	0
$533$	−20520.9	−1.66765
$534$	0	0
$535$	−6224.81	−0.503031
$536$	0	0
$537$	−26261.0	−2.11033
$538$	0	0
$539$	620.795	0.0496095
$540$	0	0
$541$	−9312.17	−0.740039	−0.370020	−	0.929024i	$-0.620649\pi$
$541$	−9312.17	−0.740039	−0.370020	+	0.929024i	$0.620649\pi$
$542$	0	0
$543$	−20210.3	−1.59725
$544$	0	0
$545$	826.105	0.0649292
$546$	0	0
$547$	−11018.6	−0.861278	−0.430639	−	0.902524i	$-0.641712\pi$
$547$	−11018.6	−0.861278	−0.430639	+	0.902524i	$0.641712\pi$
$548$	0	0
$549$	−824.968	−0.0641325
$550$	0	0
$551$	11478.6	0.887490
$552$	0	0
$553$	−16559.6	−1.27339
$554$	0	0
$555$	−3113.89	−0.238157
$556$	0	0
$557$	12018.4	0.914250	0.457125	−	0.889403i	$-0.348879\pi$
$557$	12018.4	0.914250	0.457125	+	0.889403i	$0.348879\pi$
$558$	0	0
$559$	170.718	0.0129170
$560$	0	0
$561$	5398.40	0.406276
$562$	0	0
$563$	−8763.89	−0.656046	−0.328023	−	0.944670i	$-0.606382\pi$
$563$	−8763.89	−0.656046	−0.328023	+	0.944670i	$0.606382\pi$
$564$	0	0
$565$	−25778.1	−1.91946
$566$	0	0
$567$	14940.1	1.10657
$568$	0	0
$569$	−10273.2	−0.756895	−0.378447	−	0.925623i	$-0.623542\pi$
$569$	−10273.2	−0.756895	−0.378447	+	0.925623i	$0.623542\pi$
$570$	0	0
$571$	2602.62	0.190747	0.0953734	−	0.995442i	$-0.469596\pi$
$571$	2602.62	0.190747	0.0953734	+	0.995442i	$0.469596\pi$
$572$	0	0
$573$	−17332.6	−1.26366
$574$	0	0
$575$	−537.988	−0.0390185
$576$	0	0
$577$	−19727.0	−1.42331	−0.711653	−	0.702532i	$-0.752053\pi$
$577$	−19727.0	−1.42331	−0.711653	+	0.702532i	$0.752053\pi$
$578$	0	0
$579$	−14726.7	−1.05703
$580$	0	0
$581$	−442.739	−0.0316143
$582$	0	0
$583$	−1639.06	−0.116437
$584$	0	0
$585$	−7814.72	−0.552306
$586$	0	0
$587$	−10116.2	−0.711309	−0.355654	−	0.934618i	$-0.615742\pi$
$587$	−10116.2	−0.711309	−0.355654	+	0.934618i	$0.615742\pi$
$588$	0	0
$589$	−4448.78	−0.311221
$590$	0	0
$591$	30386.0	2.11491
$592$	0	0
$593$	3130.32	0.216774	0.108387	−	0.994109i	$-0.465431\pi$
$593$	3130.32	0.216774	0.108387	+	0.994109i	$0.465431\pi$
$594$	0	0
$595$	18016.9	1.24138
$596$	0	0
$597$	45.6009	0.00312616
$598$	0	0
$599$	−10080.1	−0.687581	−0.343790	−	0.939046i	$-0.611711\pi$
$599$	−10080.1	−0.687581	−0.343790	+	0.939046i	$0.611711\pi$
$600$	0	0
$601$	4777.02	0.324224	0.162112	−	0.986772i	$-0.448169\pi$
$601$	4777.02	0.324224	0.162112	+	0.986772i	$0.448169\pi$
$602$	0	0
$603$	3352.24	0.226391
$604$	0	0
$605$	1555.63	0.104537
$606$	0	0
$607$	2571.35	0.171941	0.0859703	−	0.996298i	$-0.472601\pi$
$607$	2571.35	0.171941	0.0859703	+	0.996298i	$0.472601\pi$
$608$	0	0
$609$	16959.3	1.12845
$610$	0	0
$611$	5362.67	0.355074
$612$	0	0
$613$	−12711.9	−0.837564	−0.418782	−	0.908087i	$-0.637543\pi$
$613$	−12711.9	−0.837564	−0.418782	+	0.908087i	$0.637543\pi$
$614$	0	0
$615$	20953.8	1.37388
$616$	0	0
$617$	16236.1	1.05939	0.529693	−	0.848189i	$-0.322307\pi$
$617$	16236.1	1.05939	0.529693	+	0.848189i	$0.322307\pi$
$618$	0	0
$619$	12657.3	0.821874	0.410937	−	0.911664i	$-0.365202\pi$
$619$	12657.3	0.821874	0.410937	+	0.911664i	$0.365202\pi$
$620$	0	0
$621$	1492.75	0.0964607
$622$	0	0
$623$	5973.75	0.384162
$624$	0	0
$625$	−19037.8	−1.21842
$626$	0	0
$627$	4429.28	0.282119
$628$	0	0
$629$	3382.28	0.214404
$630$	0	0
$631$	3949.97	0.249201	0.124600	−	0.992207i	$-0.460235\pi$
$631$	3949.97	0.249201	0.124600	+	0.992207i	$0.460235\pi$
$632$	0	0
$633$	18421.0	1.15666
$634$	0	0
$635$	−1409.13	−0.0880621
$636$	0	0
$637$	4212.44	0.262014
$638$	0	0
$639$	3832.67	0.237274
$640$	0	0
$641$	−7398.27	−0.455872	−0.227936	−	0.973676i	$-0.573198\pi$
$641$	−7398.27	−0.455872	−0.227936	+	0.973676i	$0.573198\pi$
$642$	0	0
$643$	−12491.7	−0.766134	−0.383067	−	0.923721i	$-0.625132\pi$
$643$	−12491.7	−0.766134	−0.383067	+	0.923721i	$0.625132\pi$
$644$	0	0
$645$	−174.319	−0.0106415
$646$	0	0
$647$	10472.0	0.636315	0.318158	−	0.948038i	$-0.396936\pi$
$647$	10472.0	0.636315	0.318158	+	0.948038i	$0.396936\pi$
$648$	0	0
$649$	−6001.94	−0.363015
$650$	0	0
$651$	−6572.92	−0.395719
$652$	0	0
$653$	−6337.94	−0.379820	−0.189910	−	0.981801i	$-0.560820\pi$
$653$	−6337.94	−0.379820	−0.189910	+	0.981801i	$0.560820\pi$
$654$	0	0
$655$	14871.2	0.887121
$656$	0	0
$657$	4972.48	0.295274
$658$	0	0
$659$	15196.7	0.898302	0.449151	−	0.893456i	$-0.351726\pi$
$659$	15196.7	0.898302	0.449151	+	0.893456i	$0.351726\pi$
$660$	0	0
$661$	−2298.17	−0.135232	−0.0676161	−	0.997711i	$-0.521539\pi$
$661$	−2298.17	−0.135232	−0.0676161	+	0.997711i	$0.521539\pi$
$662$	0	0
$663$	36631.1	2.14575
$664$	0	0
$665$	14782.5	0.862016
$666$	0	0
$667$	2256.73	0.131006
$668$	0	0
$669$	73.0265	0.00422028
$670$	0	0
$671$	1114.33	0.0641106
$672$	0	0
$673$	23199.6	1.32880	0.664398	−	0.747379i	$-0.268688\pi$
$673$	23199.6	1.32880	0.664398	+	0.747379i	$0.268688\pi$
$674$	0	0
$675$	−4503.49	−0.256799
$676$	0	0
$677$	2145.38	0.121793	0.0608963	−	0.998144i	$-0.480604\pi$
$677$	2145.38	0.121793	0.0608963	+	0.998144i	$0.480604\pi$
$678$	0	0
$679$	−14348.1	−0.810941
$680$	0	0
$681$	27364.0	1.53978
$682$	0	0
$683$	−29544.6	−1.65519	−0.827593	−	0.561329i	$-0.810290\pi$
$683$	−29544.6	−1.65519	−0.827593	+	0.561329i	$0.810290\pi$
$684$	0	0
$685$	2549.72	0.142219
$686$	0	0
$687$	−30083.4	−1.67068
$688$	0	0
$689$	−11121.9	−0.614964
$690$	0	0
$691$	27803.1	1.53065	0.765325	−	0.643644i	$-0.222578\pi$
$691$	27803.1	1.53065	0.765325	+	0.643644i	$0.222578\pi$
$692$	0	0
$693$	1516.42	0.0831227
$694$	0	0
$695$	−37285.4	−2.03498
$696$	0	0
$697$	−22759.8	−1.23686
$698$	0	0
$699$	1254.90	0.0679036
$700$	0	0
$701$	19697.8	1.06130	0.530652	−	0.847590i	$-0.321947\pi$
$701$	19697.8	1.06130	0.530652	+	0.847590i	$0.321947\pi$
$702$	0	0
$703$	2775.09	0.148883
$704$	0	0
$705$	−5475.78	−0.292524
$706$	0	0
$707$	21896.0	1.16476
$708$	0	0
$709$	−19122.5	−1.01292	−0.506460	−	0.862263i	$-0.669046\pi$
$709$	−19122.5	−1.01292	−0.506460	+	0.862263i	$0.669046\pi$
$710$	0	0
$711$	7966.27	0.420195
$712$	0	0
$713$	−874.641	−0.0459405
$714$	0	0
$715$	10555.8	0.552117
$716$	0	0
$717$	−25565.3	−1.33160
$718$	0	0
$719$	−1837.44	−0.0953060	−0.0476530	−	0.998864i	$-0.515174\pi$
$719$	−1837.44	−0.0953060	−0.0476530	+	0.998864i	$0.515174\pi$
$720$	0	0
$721$	−29205.2	−1.50854
$722$	0	0
$723$	−5907.95	−0.303899
$724$	0	0
$725$	−6808.33	−0.348765
$726$	0	0
$727$	7555.46	0.385442	0.192721	−	0.981254i	$-0.438269\pi$
$727$	7555.46	0.385442	0.192721	+	0.981254i	$0.438269\pi$
$728$	0	0
$729$	10705.2	0.543881
$730$	0	0
$731$	189.344	0.00958021
$732$	0	0
$733$	−11984.6	−0.603905	−0.301952	−	0.953323i	$-0.597638\pi$
$733$	−11984.6	−0.603905	−0.301952	+	0.953323i	$0.597638\pi$
$734$	0	0
$735$	−4301.29	−0.215858
$736$	0	0
$737$	−4528.05	−0.226313
$738$	0	0
$739$	−27142.5	−1.35109	−0.675543	−	0.737321i	$-0.736091\pi$
$739$	−27142.5	−1.35109	−0.675543	+	0.737321i	$0.736091\pi$
$740$	0	0
$741$	30055.1	1.49001
$742$	0	0
$743$	29222.6	1.44290	0.721450	−	0.692467i	$-0.243476\pi$
$743$	29222.6	1.44290	0.721450	+	0.692467i	$0.243476\pi$
$744$	0	0
$745$	−44847.6	−2.20549
$746$	0	0
$747$	212.987	0.0104321
$748$	0	0
$749$	8196.29	0.399847
$750$	0	0
$751$	8859.39	0.430471	0.215236	−	0.976562i	$-0.430948\pi$
$751$	8859.39	0.430471	0.215236	+	0.976562i	$0.430948\pi$
$752$	0	0
$753$	−1641.47	−0.0794404
$754$	0	0
$755$	14956.2	0.720941
$756$	0	0
$757$	−35734.4	−1.71571	−0.857853	−	0.513896i	$-0.828202\pi$
$757$	−35734.4	−1.71571	−0.857853	+	0.513896i	$0.828202\pi$
$758$	0	0
$759$	870.806	0.0416446
$760$	0	0
$761$	34394.7	1.63838	0.819189	−	0.573524i	$-0.194424\pi$
$761$	34394.7	1.63838	0.819189	+	0.573524i	$0.194424\pi$
$762$	0	0
$763$	−1087.74	−0.0516107
$764$	0	0
$765$	−8667.33	−0.409631
$766$	0	0
$767$	−40726.4	−1.91727
$768$	0	0
$769$	−11602.7	−0.544091	−0.272045	−	0.962284i	$-0.587700\pi$
$769$	−11602.7	−0.544091	−0.272045	+	0.962284i	$0.587700\pi$
$770$	0	0
$771$	−19178.8	−0.895859
$772$	0	0
$773$	12680.6	0.590026	0.295013	−	0.955493i	$-0.404676\pi$
$773$	12680.6	0.590026	0.295013	+	0.955493i	$0.404676\pi$
$774$	0	0
$775$	2638.71	0.122303
$776$	0	0
$777$	4100.10	0.189305
$778$	0	0
$779$	−18674.0	−0.858876
$780$	0	0
$781$	−5176.99	−0.237193
$782$	0	0
$783$	18891.0	0.862210
$784$	0	0
$785$	−4397.62	−0.199946
$786$	0	0
$787$	−4417.61	−0.200090	−0.100045	−	0.994983i	$-0.531899\pi$
$787$	−4417.61	−0.200090	−0.100045	+	0.994983i	$0.531899\pi$
$788$	0	0
$789$	−1232.74	−0.0556231
$790$	0	0
$791$	33942.4	1.52573
$792$	0	0
$793$	7561.33	0.338601
$794$	0	0
$795$	11356.5	0.506633
$796$	0	0
$797$	27030.1	1.20132	0.600661	−	0.799504i	$-0.294904\pi$
$797$	27030.1	1.20132	0.600661	+	0.799504i	$0.294904\pi$
$798$	0	0
$799$	5947.75	0.263350
$800$	0	0
$801$	−2873.77	−0.126766
$802$	0	0
$803$	−6716.60	−0.295173
$804$	0	0
$805$	2906.27	0.127246
$806$	0	0
$807$	−29836.9	−1.30150
$808$	0	0
$809$	23647.0	1.02767	0.513835	−	0.857889i	$-0.328224\pi$
$809$	23647.0	1.02767	0.513835	+	0.857889i	$0.328224\pi$
$810$	0	0
$811$	33486.1	1.44988	0.724941	−	0.688811i	$-0.241867\pi$
$811$	33486.1	1.44988	0.724941	+	0.688811i	$0.241867\pi$
$812$	0	0
$813$	−8815.30	−0.380278
$814$	0	0
$815$	−17933.3	−0.770768
$816$	0	0
$817$	155.353	0.00665251
$818$	0	0
$819$	10289.7	0.439014
$820$	0	0
$821$	−2605.69	−0.110766	−0.0553832	−	0.998465i	$-0.517638\pi$
$821$	−2605.69	−0.110766	−0.0553832	+	0.998465i	$0.517638\pi$
$822$	0	0
$823$	−31976.2	−1.35434	−0.677169	−	0.735828i	$-0.736793\pi$
$823$	−31976.2	−1.35434	−0.677169	+	0.735828i	$0.736793\pi$
$824$	0	0
$825$	−2627.14	−0.110867
$826$	0	0
$827$	−37759.0	−1.58768	−0.793839	−	0.608128i	$-0.791921\pi$
$827$	−37759.0	−1.58768	−0.793839	+	0.608128i	$0.791921\pi$
$828$	0	0
$829$	1137.55	0.0476584	0.0238292	−	0.999716i	$-0.492414\pi$
$829$	1137.55	0.0476584	0.0238292	+	0.999716i	$0.492414\pi$
$830$	0	0
$831$	1398.08	0.0583621
$832$	0	0
$833$	4672.03	0.194329
$834$	0	0
$835$	−6154.40	−0.255068
$836$	0	0
$837$	−7321.60	−0.302356
$838$	0	0
$839$	37372.2	1.53782	0.768911	−	0.639356i	$-0.220799\pi$
$839$	37372.2	1.53782	0.768911	+	0.639356i	$0.220799\pi$
$840$	0	0
$841$	4170.26	0.170989
$842$	0	0
$843$	−29137.2	−1.19044
$844$	0	0
$845$	43381.1	1.76610
$846$	0	0
$847$	−2048.31	−0.0830943
$848$	0	0
$849$	−30824.0	−1.24603
$850$	0	0
$851$	545.589	0.0219772
$852$	0	0
$853$	22490.8	0.902780	0.451390	−	0.892327i	$-0.350928\pi$
$853$	22490.8	0.902780	0.451390	+	0.892327i	$0.350928\pi$
$854$	0	0
$855$	−7111.36	−0.284449
$856$	0	0
$857$	43409.5	1.73027	0.865135	−	0.501539i	$-0.167233\pi$
$857$	43409.5	1.73027	0.865135	+	0.501539i	$0.167233\pi$
$858$	0	0
$859$	29533.2	1.17306	0.586532	−	0.809926i	$-0.300493\pi$
$859$	29533.2	1.17306	0.586532	+	0.809926i	$0.300493\pi$
$860$	0	0
$861$	−27590.1	−1.09207
$862$	0	0
$863$	−14351.6	−0.566090	−0.283045	−	0.959107i	$-0.591345\pi$
$863$	−14351.6	−0.566090	−0.283045	+	0.959107i	$0.591345\pi$
$864$	0	0
$865$	−23251.9	−0.913975
$866$	0	0
$867$	11502.4	0.450569
$868$	0	0
$869$	−10760.5	−0.420051
$870$	0	0
$871$	−30725.3	−1.19528
$872$	0	0
$873$	6902.39	0.267595
$874$	0	0
$875$	18436.5	0.712307
$876$	0	0
$877$	43248.7	1.66523	0.832614	−	0.553854i	$-0.186844\pi$
$877$	43248.7	1.66523	0.832614	+	0.553854i	$0.186844\pi$
$878$	0	0
$879$	52647.9	2.02021
$880$	0	0
$881$	3816.13	0.145935	0.0729675	−	0.997334i	$-0.476753\pi$
$881$	3816.13	0.145935	0.0729675	+	0.997334i	$0.476753\pi$
$882$	0	0
$883$	48787.6	1.85938	0.929690	−	0.368343i	$-0.120075\pi$
$883$	48787.6	1.85938	0.929690	+	0.368343i	$0.120075\pi$
$884$	0	0
$885$	41585.5	1.57953
$886$	0	0
$887$	−41495.1	−1.57077	−0.785384	−	0.619009i	$-0.787534\pi$
$887$	−41495.1	−1.57077	−0.785384	+	0.619009i	$0.787534\pi$
$888$	0	0
$889$	1855.41	0.0699984
$890$	0	0
$891$	9708.15	0.365023
$892$	0	0
$893$	4880.01	0.182870
$894$	0	0
$895$	−56951.9	−2.12703
$896$	0	0
$897$	5908.90	0.219947
$898$	0	0
$899$	−11068.7	−0.410637
$900$	0	0
$901$	−12335.3	−0.456104
$902$	0	0
$903$	229.528	0.00845871
$904$	0	0
$905$	−43829.7	−1.60989
$906$	0	0
$907$	21615.3	0.791316	0.395658	−	0.918398i	$-0.370517\pi$
$907$	21615.3	0.791316	0.395658	+	0.918398i	$0.370517\pi$
$908$	0	0
$909$	−10533.4	−0.384347
$910$	0	0
$911$	−3646.35	−0.132611	−0.0663057	−	0.997799i	$-0.521121\pi$
$911$	−3646.35	−0.132611	−0.0663057	+	0.997799i	$0.521121\pi$
$912$	0	0
$913$	−287.693	−0.0104285
$914$	0	0
$915$	−7720.82	−0.278954
$916$	0	0
$917$	−19581.1	−0.705151
$918$	0	0
$919$	−31280.0	−1.12278	−0.561388	−	0.827553i	$-0.689733\pi$
$919$	−31280.0	−1.12278	−0.561388	+	0.827553i	$0.689733\pi$
$920$	0	0
$921$	−8880.05	−0.317707
$922$	0	0
$923$	−35128.7	−1.25274
$924$	0	0
$925$	−1645.99	−0.0585079
$926$	0	0
$927$	14049.7	0.497790
$928$	0	0
$929$	6557.92	0.231602	0.115801	−	0.993272i	$-0.463056\pi$
$929$	6557.92	0.231602	0.115801	+	0.993272i	$0.463056\pi$
$930$	0	0
$931$	3833.30	0.134942
$932$	0	0
$933$	44370.9	1.55695
$934$	0	0
$935$	11707.4	0.409491
$936$	0	0
$937$	−24473.3	−0.853265	−0.426632	−	0.904425i	$-0.640300\pi$
$937$	−24473.3	−0.853265	−0.426632	+	0.904425i	$0.640300\pi$
$938$	0	0
$939$	−3902.91	−0.135641
$940$	0	0
$941$	−15420.8	−0.534224	−0.267112	−	0.963665i	$-0.586069\pi$
$941$	−15420.8	−0.534224	−0.267112	+	0.963665i	$0.586069\pi$
$942$	0	0
$943$	−3671.34	−0.126782
$944$	0	0
$945$	24328.3	0.837461
$946$	0	0
$947$	−33141.2	−1.13722	−0.568608	−	0.822609i	$-0.692518\pi$
$947$	−33141.2	−1.13722	−0.568608	+	0.822609i	$0.692518\pi$
$948$	0	0
$949$	−45575.8	−1.55896
$950$	0	0
$951$	−1385.47	−0.0472417
$952$	0	0
$953$	−20735.4	−0.704813	−0.352406	−	0.935847i	$-0.614637\pi$
$953$	−20735.4	−0.704813	−0.352406	+	0.935847i	$0.614637\pi$
$954$	0	0
$955$	−37589.0	−1.27367
$956$	0	0
$957$	11020.2	0.372239
$958$	0	0
$959$	−3357.26	−0.113046
$960$	0	0
$961$	−25501.1	−0.856000
$962$	0	0
$963$	−3942.96	−0.131942
$964$	0	0
$965$	−31937.7	−1.06540
$966$	0	0
$967$	8178.87	0.271990	0.135995	−	0.990710i	$-0.456577\pi$
$967$	8178.87	0.271990	0.135995	+	0.990710i	$0.456577\pi$
$968$	0	0
$969$	33334.2	1.10511
$970$	0	0
$971$	−20576.1	−0.680039	−0.340020	−	0.940418i	$-0.610434\pi$
$971$	−20576.1	−0.680039	−0.340020	+	0.940418i	$0.610434\pi$
$972$	0	0
$973$	49094.1	1.61756
$974$	0	0
$975$	−17826.6	−0.585546
$976$	0	0
$977$	14541.9	0.476188	0.238094	−	0.971242i	$-0.423477\pi$
$977$	14541.9	0.476188	0.238094	+	0.971242i	$0.423477\pi$
$978$	0	0
$979$	3881.76	0.126723
$980$	0	0
$981$	523.277	0.0170305
$982$	0	0
$983$	29285.7	0.950223	0.475111	−	0.879926i	$-0.342408\pi$
$983$	29285.7	0.950223	0.475111	+	0.879926i	$0.342408\pi$
$984$	0	0
$985$	65897.7	2.13165
$986$	0	0
$987$	7210.03	0.232521
$988$	0	0
$989$	30.5427	0.000982004 0
$990$	0	0
$991$	38085.9	1.22083	0.610413	−	0.792083i	$-0.291003\pi$
$991$	38085.9	1.22083	0.610413	+	0.792083i	$0.291003\pi$
$992$	0	0
$993$	50585.1	1.61658
$994$	0	0
$995$	98.8940	0.00315090
$996$	0	0
$997$	26803.6	0.851434	0.425717	−	0.904856i	$-0.360022\pi$
$997$	26803.6	0.851434	0.425717	+	0.904856i	$0.360022\pi$
$998$	0	0
$999$	4567.12	0.144642

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	704.4.a.n.1.2		2
4.3	odd	2		704.4.a.p.1.1		2
8.3	odd	2		11.4.a.a.1.1	✓	2
8.5	even	2		176.4.a.i.1.1		2
24.5	odd	2		1584.4.a.bc.1.2		2
24.11	even	2		99.4.a.c.1.2		2
40.3	even	4		275.4.b.c.199.3		4
40.19	odd	2		275.4.a.b.1.2		2
40.27	even	4		275.4.b.c.199.2		4
56.27	even	2		539.4.a.e.1.1		2
88.3	odd	10		121.4.c.c.9.1		8
88.19	even	10		121.4.c.f.9.2		8
88.21	odd	2		1936.4.a.w.1.1		2
88.27	odd	10		121.4.c.c.3.2		8
88.35	even	10		121.4.c.f.81.1		8
88.43	even	2		121.4.a.c.1.2		2
88.51	even	10		121.4.c.f.27.2		8
88.59	odd	10		121.4.c.c.27.1		8
88.75	odd	10		121.4.c.c.81.2		8
88.83	even	10		121.4.c.f.3.1		8
104.51	odd	2		1859.4.a.a.1.2		2
120.59	even	2		2475.4.a.q.1.1		2
264.131	odd	2		1089.4.a.v.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.4.a.a.1.1	✓	2	8.3	odd	2
99.4.a.c.1.2		2	24.11	even	2
121.4.a.c.1.2		2	88.43	even	2
121.4.c.c.3.2		8	88.27	odd	10
121.4.c.c.9.1		8	88.3	odd	10
121.4.c.c.27.1		8	88.59	odd	10
121.4.c.c.81.2		8	88.75	odd	10
121.4.c.f.3.1		8	88.83	even	10
121.4.c.f.9.2		8	88.19	even	10
121.4.c.f.27.2		8	88.51	even	10
121.4.c.f.81.1		8	88.35	even	10
176.4.a.i.1.1		2	8.5	even	2
275.4.a.b.1.2		2	40.19	odd	2
275.4.b.c.199.2		4	40.27	even	4
275.4.b.c.199.3		4	40.3	even	4
539.4.a.e.1.1		2	56.27	even	2
704.4.a.n.1.2		2	1.1	even	1	trivial
704.4.a.p.1.1		2	4.3	odd	2
1089.4.a.v.1.1		2	264.131	odd	2
1584.4.a.bc.1.2		2	24.5	odd	2
1859.4.a.a.1.2		2	104.51	odd	2
1936.4.a.w.1.1		2	88.21	odd	2
2475.4.a.q.1.1		2	120.59	even	2

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:	\( N \)	\(=\)	\( 704 = 2^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	704.a (trivial)

\(f(q)\)	\(=\)	\(q+5.92820 q^{3} +12.8564 q^{5} -16.9282 q^{7} +8.14359 q^{9} +O(q^{10})\)	\(q+5.92820 q^{3} +12.8564 q^{5} -16.9282 q^{7} +8.14359 q^{9} -11.0000 q^{11} -74.6410 q^{13} +76.2154 q^{15} -82.7846 q^{17} -67.9230 q^{19} -100.354 q^{21} -13.3538 q^{23} +40.2872 q^{25} -111.785 q^{27} -168.995 q^{29} +65.4974 q^{31} -65.2102 q^{33} -217.636 q^{35} -40.8564 q^{37} -442.487 q^{39} +274.928 q^{41} -2.28719 q^{43} +104.697 q^{45} -71.8461 q^{47} -56.4359 q^{49} -490.764 q^{51} +149.005 q^{53} -141.420 q^{55} -402.662 q^{57} +545.631 q^{59} -101.303 q^{61} -137.856 q^{63} -959.615 q^{65} +411.641 q^{67} -79.1642 q^{69} +470.636 q^{71} +610.600 q^{73} +238.831 q^{75} +186.210 q^{77} +978.225 q^{79} -882.559 q^{81} +26.1539 q^{83} -1064.31 q^{85} -1001.84 q^{87} -352.887 q^{89} +1263.54 q^{91} +388.282 q^{93} -873.246 q^{95} +847.585 q^{97} -89.5795 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} - 20 q^{7} + 44 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 - 20 * q^7 + 44 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} - 20 q^{7} + 44 q^{9} - 22 q^{11} - 80 q^{13} + 194 q^{15} - 124 q^{17} + 72 q^{19} - 76 q^{21} + 98 q^{23} + 136 q^{25} - 182 q^{27} - 144 q^{29} + 34 q^{31} + 22 q^{33} - 172 q^{35} - 54 q^{37} - 400 q^{39} + 536 q^{41} - 60 q^{43} - 428 q^{45} + 272 q^{47} - 390 q^{49} - 164 q^{51} + 492 q^{53} + 22 q^{55} - 1512 q^{57} + 634 q^{59} - 840 q^{61} - 248 q^{63} - 880 q^{65} + 754 q^{67} - 962 q^{69} + 678 q^{71} - 400 q^{73} - 520 q^{75} + 220 q^{77} - 316 q^{79} - 1294 q^{81} + 468 q^{83} - 452 q^{85} - 1200 q^{87} - 1842 q^{89} + 1280 q^{91} + 638 q^{93} - 2952 q^{95} + 2194 q^{97} - 484 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 - 20 * q^7 + 44 * q^9 - 22 * q^11 - 80 * q^13 + 194 * q^15 - 124 * q^17 + 72 * q^19 - 76 * q^21 + 98 * q^23 + 136 * q^25 - 182 * q^27 - 144 * q^29 + 34 * q^31 + 22 * q^33 - 172 * q^35 - 54 * q^37 - 400 * q^39 + 536 * q^41 - 60 * q^43 - 428 * q^45 + 272 * q^47 - 390 * q^49 - 164 * q^51 + 492 * q^53 + 22 * q^55 - 1512 * q^57 + 634 * q^59 - 840 * q^61 - 248 * q^63 - 880 * q^65 + 754 * q^67 - 962 * q^69 + 678 * q^71 - 400 * q^73 - 520 * q^75 + 220 * q^77 - 316 * q^79 - 1294 * q^81 + 468 * q^83 - 452 * q^85 - 1200 * q^87 - 1842 * q^89 + 1280 * q^91 + 638 * q^93 - 2952 * q^95 + 2194 * q^97 - 484 * q^99

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