LMFDB - Embedded newform 7350.2.a.dq.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7350,2,Mod(1,7350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7350, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("7350.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7350.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:	$58.6900454856$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.2700.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 15x - 20 $ x^3 - 15*x - 20
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 210)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$4.41883$ of defining polynomial
Character	$\chi$	$=$	7350.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +6.10725 q^{11} +1.00000 q^{12} +1.68842 q^{13} +1.00000 q^{16} -6.83767 q^{17} +1.00000 q^{18} +1.68842 q^{19} +6.10725 q^{22} +2.31158 q^{23} +1.00000 q^{24} +1.68842 q^{26} +1.00000 q^{27} +8.41883 q^{29} -0.688417 q^{31} +1.00000 q^{32} +6.10725 q^{33} -6.83767 q^{34} +1.00000 q^{36} +2.31158 q^{37} +1.68842 q^{38} +1.68842 q^{39} +5.14925 q^{41} -2.00000 q^{43} +6.10725 q^{44} +2.31158 q^{46} -3.06525 q^{47} +1.00000 q^{48} -6.83767 q^{51} +1.68842 q^{52} -10.4608 q^{53} +1.00000 q^{54} +1.68842 q^{57} +8.41883 q^{58} -7.04200 q^{59} +9.46083 q^{61} -0.688417 q^{62} +1.00000 q^{64} +6.10725 q^{66} -12.2145 q^{67} -6.83767 q^{68} +2.31158 q^{69} +6.00000 q^{71} +1.00000 q^{72} +4.00000 q^{73} +2.31158 q^{74} +1.68842 q^{76} +1.68842 q^{78} +7.31158 q^{79} +1.00000 q^{81} +5.14925 q^{82} +13.5261 q^{83} -2.00000 q^{86} +8.41883 q^{87} +6.10725 q^{88} -12.2145 q^{89} +2.31158 q^{92} -0.688417 q^{93} -3.06525 q^{94} +1.00000 q^{96} -4.95800 q^{97} +6.10725 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^6 + 3 * q^8 + 3 * q^9	$ 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{6} + 3 q^{8} + 3 q^{9} + 3 q^{11} + 3 q^{12} + 3 q^{13} + 3 q^{16} + 6 q^{17} + 3 q^{18} + 3 q^{19} + 3 q^{22} + 9 q^{23} + 3 q^{24} + 3 q^{26} + 3 q^{27} + 12 q^{29} + 3 q^{32} + 3 q^{33} + 6 q^{34} + 3 q^{36} + 9 q^{37} + 3 q^{38} + 3 q^{39} - 9 q^{41} - 6 q^{43} + 3 q^{44} + 9 q^{46} - 3 q^{47} + 3 q^{48} + 6 q^{51} + 3 q^{52} - 9 q^{53} + 3 q^{54} + 3 q^{57} + 12 q^{58} - 12 q^{59} + 6 q^{61} + 3 q^{64} + 3 q^{66} - 6 q^{67} + 6 q^{68} + 9 q^{69} + 18 q^{71} + 3 q^{72} + 12 q^{73} + 9 q^{74} + 3 q^{76} + 3 q^{78} + 24 q^{79} + 3 q^{81} - 9 q^{82} + 12 q^{83} - 6 q^{86} + 12 q^{87} + 3 q^{88} - 6 q^{89} + 9 q^{92} - 3 q^{94} + 3 q^{96} - 24 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^6 + 3 * q^8 + 3 * q^9 + 3 * q^11 + 3 * q^12 + 3 * q^13 + 3 * q^16 + 6 * q^17 + 3 * q^18 + 3 * q^19 + 3 * q^22 + 9 * q^23 + 3 * q^24 + 3 * q^26 + 3 * q^27 + 12 * q^29 + 3 * q^32 + 3 * q^33 + 6 * q^34 + 3 * q^36 + 9 * q^37 + 3 * q^38 + 3 * q^39 - 9 * q^41 - 6 * q^43 + 3 * q^44 + 9 * q^46 - 3 * q^47 + 3 * q^48 + 6 * q^51 + 3 * q^52 - 9 * q^53 + 3 * q^54 + 3 * q^57 + 12 * q^58 - 12 * q^59 + 6 * q^61 + 3 * q^64 + 3 * q^66 - 6 * q^67 + 6 * q^68 + 9 * q^69 + 18 * q^71 + 3 * q^72 + 12 * q^73 + 9 * q^74 + 3 * q^76 + 3 * q^78 + 24 * q^79 + 3 * q^81 - 9 * q^82 + 12 * q^83 - 6 * q^86 + 12 * q^87 + 3 * q^88 - 6 * q^89 + 9 * q^92 - 3 * q^94 + 3 * q^96 - 24 * q^97 + 3 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	6.10725	1.84141	0.920703	−	0.390265i	$-0.127617\pi$
$11$	6.10725	1.84141	0.920703	+	0.390265i	$0.127617\pi$
$12$	1.00000	0.288675
$13$	1.68842	0.468283	0.234141	−	0.972203i	$-0.424772\pi$
$13$	1.68842	0.468283	0.234141	+	0.972203i	$0.424772\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.83767	−1.65838	−0.829189	−	0.558969i	$-0.811197\pi$
$17$	−6.83767	−1.65838	−0.829189	+	0.558969i	$0.811197\pi$
$18$	1.00000	0.235702
$19$	1.68842	0.387349	0.193675	−	0.981066i	$-0.437959\pi$
$19$	1.68842	0.387349	0.193675	+	0.981066i	$0.437959\pi$
$20$	0	0
$21$	0	0
$22$	6.10725	1.30207
$23$	2.31158	0.481998	0.240999	−	0.970525i	$-0.422525\pi$
$23$	2.31158	0.481998	0.240999	+	0.970525i	$0.422525\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	1.68842	0.331126
$27$	1.00000	0.192450
$28$	0	0
$29$	8.41883	1.56334	0.781669	−	0.623694i	$-0.214369\pi$
$29$	8.41883	1.56334	0.781669	+	0.623694i	$0.214369\pi$
$30$	0	0
$31$	−0.688417	−0.123643	−0.0618217	−	0.998087i	$-0.519691\pi$
$31$	−0.688417	−0.123643	−0.0618217	+	0.998087i	$0.519691\pi$
$32$	1.00000	0.176777
$33$	6.10725	1.06314
$34$	−6.83767	−1.17265
$35$	0	0
$36$	1.00000	0.166667
$37$	2.31158	0.380022	0.190011	−	0.981782i	$-0.439148\pi$
$37$	2.31158	0.380022	0.190011	+	0.981782i	$0.439148\pi$
$38$	1.68842	0.273897
$39$	1.68842	0.270363
$40$	0	0
$41$	5.14925	0.804178	0.402089	−	0.915601i	$-0.368284\pi$
$41$	5.14925	0.804178	0.402089	+	0.915601i	$0.368284\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	6.10725	0.920703
$45$	0	0
$46$	2.31158	0.340824
$47$	−3.06525	−0.447113	−0.223556	−	0.974691i	$-0.571767\pi$
$47$	−3.06525	−0.447113	−0.223556	+	0.974691i	$0.571767\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	0	0
$51$	−6.83767	−0.957465
$52$	1.68842	0.234141
$53$	−10.4608	−1.43691	−0.718453	−	0.695576i	$-0.755149\pi$
$53$	−10.4608	−1.43691	−0.718453	+	0.695576i	$0.755149\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	0	0
$57$	1.68842	0.223636
$58$	8.41883	1.10545
$59$	−7.04200	−0.916790	−0.458395	−	0.888749i	$-0.651576\pi$
$59$	−7.04200	−0.916790	−0.458395	+	0.888749i	$0.651576\pi$
$60$	0	0
$61$	9.46083	1.21134	0.605668	−	0.795718i	$-0.292906\pi$
$61$	9.46083	1.21134	0.605668	+	0.795718i	$0.292906\pi$
$62$	−0.688417	−0.0874290
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	6.10725	0.751750
$67$	−12.2145	−1.49224	−0.746119	−	0.665812i	$-0.768085\pi$
$67$	−12.2145	−1.49224	−0.746119	+	0.665812i	$0.768085\pi$
$68$	−6.83767	−0.829189
$69$	2.31158	0.278282
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	1.00000	0.117851
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	2.31158	0.268716
$75$	0	0
$76$	1.68842	0.193675
$77$	0	0
$78$	1.68842	0.191176
$79$	7.31158	0.822617	0.411309	−	0.911496i	$-0.365072\pi$
$79$	7.31158	0.822617	0.411309	+	0.911496i	$0.365072\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	5.14925	0.568639
$83$	13.5261	1.48468	0.742340	−	0.670023i	$-0.233716\pi$
$83$	13.5261	1.48468	0.742340	+	0.670023i	$0.233716\pi$
$84$	0	0
$85$	0	0
$86$	−2.00000	−0.215666
$87$	8.41883	0.902594
$88$	6.10725	0.651035
$89$	−12.2145	−1.29473	−0.647367	−	0.762178i	$-0.724130\pi$
$89$	−12.2145	−1.29473	−0.647367	+	0.762178i	$0.724130\pi$
$90$	0	0
$91$	0	0
$92$	2.31158	0.240999
$93$	−0.688417	−0.0713855
$94$	−3.06525	−0.316156
$95$	0	0
$96$	1.00000	0.102062
$97$	−4.95800	−0.503409	−0.251704	−	0.967804i	$-0.580991\pi$
$97$	−4.95800	−0.503409	−0.251704	+	0.967804i	$0.580991\pi$
$98$	0	0
$99$	6.10725	0.613802
$100$	0	0
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	−6.83767	−0.677030
$103$	0.837665	0.0825376	0.0412688	−	0.999148i	$-0.486860\pi$
$103$	0.837665	0.0825376	0.0412688	+	0.999148i	$0.486860\pi$
$104$	1.68842	0.165563
$105$	0	0
$106$	−10.4608	−1.01605
$107$	15.5261	1.50096	0.750482	−	0.660891i	$-0.229822\pi$
$107$	15.5261	1.50096	0.750482	+	0.660891i	$0.229822\pi$
$108$	1.00000	0.0962250
$109$	−11.6753	−1.11829	−0.559147	−	0.829069i	$-0.688871\pi$
$109$	−11.6753	−1.11829	−0.559147	+	0.829069i	$0.688871\pi$
$110$	0	0
$111$	2.31158	0.219406
$112$	0	0
$113$	−3.37683	−0.317666	−0.158833	−	0.987305i	$-0.550773\pi$
$113$	−3.37683	−0.317666	−0.158833	+	0.987305i	$0.550773\pi$
$114$	1.68842	0.158135
$115$	0	0
$116$	8.41883	0.781669
$117$	1.68842	0.156094
$118$	−7.04200	−0.648269
$119$	0	0
$120$	0	0
$121$	26.2985	2.39077
$122$	9.46083	0.856543
$123$	5.14925	0.464292
$124$	−0.688417	−0.0618217
$125$	0	0
$126$	0	0
$127$	−4.94491	−0.438790	−0.219395	−	0.975636i	$-0.570408\pi$
$127$	−4.94491	−0.438790	−0.219395	+	0.975636i	$0.570408\pi$
$128$	1.00000	0.0883883
$129$	−2.00000	−0.176090
$130$	0	0
$131$	−6.73042	−0.588039	−0.294020	−	0.955799i	$-0.594993\pi$
$131$	−6.73042	−0.588039	−0.294020	+	0.955799i	$0.594993\pi$
$132$	6.10725	0.531568
$133$	0	0
$134$	−12.2145	−1.05517
$135$	0	0
$136$	−6.83767	−0.586325
$137$	19.0522	1.62774	0.813868	−	0.581050i	$-0.197358\pi$
$137$	19.0522	1.62774	0.813868	+	0.581050i	$0.197358\pi$
$138$	2.31158	0.196775
$139$	13.3768	1.13461	0.567304	−	0.823508i	$-0.307986\pi$
$139$	13.3768	1.13461	0.567304	+	0.823508i	$0.307986\pi$
$140$	0	0
$141$	−3.06525	−0.258141
$142$	6.00000	0.503509
$143$	10.3116	0.862298
$144$	1.00000	0.0833333
$145$	0	0
$146$	4.00000	0.331042
$147$	0	0
$148$	2.31158	0.190011
$149$	12.6232	1.03413	0.517065	−	0.855946i	$-0.327025\pi$
$149$	12.6232	1.03413	0.517065	+	0.855946i	$0.327025\pi$
$150$	0	0
$151$	1.93475	0.157448	0.0787238	−	0.996896i	$-0.474915\pi$
$151$	1.93475	0.157448	0.0787238	+	0.996896i	$0.474915\pi$
$152$	1.68842	0.136949
$153$	−6.83767	−0.552792
$154$	0	0
$155$	0	0
$156$	1.68842	0.135182
$157$	17.1492	1.36866	0.684330	−	0.729173i	$-0.260095\pi$
$157$	17.1492	1.36866	0.684330	+	0.729173i	$0.260095\pi$
$158$	7.31158	0.581678
$159$	−10.4608	−0.829598
$160$	0	0
$161$	0	0
$162$	1.00000	0.0785674
$163$	0.539168	0.0422309	0.0211155	−	0.999777i	$-0.493278\pi$
$163$	0.539168	0.0422309	0.0211155	+	0.999777i	$0.493278\pi$
$164$	5.14925	0.402089
$165$	0	0
$166$	13.5261	1.04983
$167$	−3.14925	−0.243696	−0.121848	−	0.992549i	$-0.538882\pi$
$167$	−3.14925	−0.243696	−0.121848	+	0.992549i	$0.538882\pi$
$168$	0	0
$169$	−10.1492	−0.780711
$170$	0	0
$171$	1.68842	0.129116
$172$	−2.00000	−0.152499
$173$	−10.5261	−0.800283	−0.400142	−	0.916453i	$-0.631039\pi$
$173$	−10.5261	−0.800283	−0.400142	+	0.916453i	$0.631039\pi$
$174$	8.41883	0.638230
$175$	0	0
$176$	6.10725	0.460351
$177$	−7.04200	−0.529309
$178$	−12.2145	−0.915515
$179$	4.93475	0.368840	0.184420	−	0.982847i	$-0.440959\pi$
$179$	4.93475	0.368840	0.184420	+	0.982847i	$0.440959\pi$
$180$	0	0
$181$	6.13050	0.455677	0.227838	−	0.973699i	$-0.426834\pi$
$181$	6.13050	0.455677	0.227838	+	0.973699i	$0.426834\pi$
$182$	0	0
$183$	9.46083	0.699365
$184$	2.31158	0.170412
$185$	0	0
$186$	−0.688417	−0.0504772
$187$	−41.7593	−3.05374
$188$	−3.06525	−0.223556
$189$	0	0
$190$	0	0
$191$	−14.9217	−1.07969	−0.539847	−	0.841763i	$-0.681518\pi$
$191$	−14.9217	−1.07969	−0.539847	+	0.841763i	$0.681518\pi$
$192$	1.00000	0.0721688
$193$	−5.58117	−0.401741	−0.200871	−	0.979618i	$-0.564377\pi$
$193$	−5.58117	−0.401741	−0.200871	+	0.979618i	$0.564377\pi$
$194$	−4.95800	−0.355964
$195$	0	0
$196$	0	0
$197$	14.5261	1.03494	0.517470	−	0.855701i	$-0.326874\pi$
$197$	14.5261	1.03494	0.517470	+	0.855701i	$0.326874\pi$
$198$	6.10725	0.434023
$199$	−8.62317	−0.611280	−0.305640	−	0.952147i	$-0.598870\pi$
$199$	−8.62317	−0.611280	−0.305640	+	0.952147i	$0.598870\pi$
$200$	0	0
$201$	−12.2145	−0.861544
$202$	−8.00000	−0.562878
$203$	0	0
$204$	−6.83767	−0.478732
$205$	0	0
$206$	0.837665	0.0583629
$207$	2.31158	0.160666
$208$	1.68842	0.117071
$209$	10.3116	0.713267
$210$	0	0
$211$	−7.68842	−0.529292	−0.264646	−	0.964346i	$-0.585255\pi$
$211$	−7.68842	−0.529292	−0.264646	+	0.964346i	$0.585255\pi$
$212$	−10.4608	−0.718453
$213$	6.00000	0.411113
$214$	15.5261	1.06134
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	−11.6753	−0.790753
$219$	4.00000	0.270295
$220$	0	0
$221$	−11.5448	−0.776589
$222$	2.31158	0.155143
$223$	−11.1725	−0.748166	−0.374083	−	0.927395i	$-0.622042\pi$
$223$	−11.1725	−0.748166	−0.374083	+	0.927395i	$0.622042\pi$
$224$	0	0
$225$	0	0
$226$	−3.37683	−0.224624
$227$	3.85075	0.255583	0.127792	−	0.991801i	$-0.459211\pi$
$227$	3.85075	0.255583	0.127792	+	0.991801i	$0.459211\pi$
$228$	1.68842	0.111818
$229$	5.67533	0.375036	0.187518	−	0.982261i	$-0.439956\pi$
$229$	5.67533	0.375036	0.187518	+	0.982261i	$0.439956\pi$
$230$	0	0
$231$	0	0
$232$	8.41883	0.552723
$233$	−11.4608	−0.750824	−0.375412	−	0.926858i	$-0.622499\pi$
$233$	−11.4608	−0.750824	−0.375412	+	0.926858i	$0.622499\pi$
$234$	1.68842	0.110375
$235$	0	0
$236$	−7.04200	−0.458395
$237$	7.31158	0.474938
$238$	0	0
$239$	15.4608	1.00008	0.500039	−	0.866003i	$-0.333319\pi$
$239$	15.4608	1.00008	0.500039	+	0.866003i	$0.333319\pi$
$240$	0	0
$241$	4.37683	0.281937	0.140968	−	0.990014i	$-0.454978\pi$
$241$	4.37683	0.281937	0.140968	+	0.990014i	$0.454978\pi$
$242$	26.2985	1.69053
$243$	1.00000	0.0641500
$244$	9.46083	0.605668
$245$	0	0
$246$	5.14925	0.328304
$247$	2.85075	0.181389
$248$	−0.688417	−0.0437145
$249$	13.5261	0.857181
$250$	0	0
$251$	−26.5362	−1.67495	−0.837477	−	0.546473i	$-0.815970\pi$
$251$	−26.5362	−1.67495	−0.837477	+	0.546473i	$0.815970\pi$
$252$	0	0
$253$	14.1174	0.887554
$254$	−4.94491	−0.310272
$255$	0	0
$256$	1.00000	0.0625000
$257$	22.4290	1.39908	0.699541	−	0.714592i	$-0.253388\pi$
$257$	22.4290	1.39908	0.699541	+	0.714592i	$0.253388\pi$
$258$	−2.00000	−0.124515
$259$	0	0
$260$	0	0
$261$	8.41883	0.521113
$262$	−6.73042	−0.415806
$263$	−19.5913	−1.20805	−0.604027	−	0.796964i	$-0.706438\pi$
$263$	−19.5913	−1.20805	−0.604027	+	0.796964i	$0.706438\pi$
$264$	6.10725	0.375875
$265$	0	0
$266$	0	0
$267$	−12.2145	−0.747515
$268$	−12.2145	−0.746119
$269$	−6.95800	−0.424237	−0.212118	−	0.977244i	$-0.568036\pi$
$269$	−6.95800	−0.424237	−0.212118	+	0.977244i	$0.568036\pi$
$270$	0	0
$271$	−28.3637	−1.72297	−0.861487	−	0.507779i	$-0.830467\pi$
$271$	−28.3637	−1.72297	−0.861487	+	0.507779i	$0.830467\pi$
$272$	−6.83767	−0.414594
$273$	0	0
$274$	19.0522	1.15098
$275$	0	0
$276$	2.31158	0.139141
$277$	5.05216	0.303555	0.151778	−	0.988415i	$-0.451500\pi$
$277$	5.05216	0.303555	0.151778	+	0.988415i	$0.451500\pi$
$278$	13.3768	0.802289
$279$	−0.688417	−0.0412144
$280$	0	0
$281$	7.68842	0.458652	0.229326	−	0.973350i	$-0.426348\pi$
$281$	7.68842	0.458652	0.229326	+	0.973350i	$0.426348\pi$
$282$	−3.06525	−0.182533
$283$	18.4290	1.09549	0.547745	−	0.836645i	$-0.315486\pi$
$283$	18.4290	1.09549	0.547745	+	0.836645i	$0.315486\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	10.3116	0.609737
$287$	0	0
$288$	1.00000	0.0589256
$289$	29.7537	1.75022
$290$	0	0
$291$	−4.95800	−0.290643
$292$	4.00000	0.234082
$293$	−1.83767	−0.107358	−0.0536788	−	0.998558i	$-0.517095\pi$
$293$	−1.83767	−0.107358	−0.0536788	+	0.998558i	$0.517095\pi$
$294$	0	0
$295$	0	0
$296$	2.31158	0.134358
$297$	6.10725	0.354379
$298$	12.6232	0.731240
$299$	3.90292	0.225711
$300$	0	0
$301$	0	0
$302$	1.93475	0.111332
$303$	−8.00000	−0.459588
$304$	1.68842	0.0968373
$305$	0	0
$306$	−6.83767	−0.390883
$307$	−16.0840	−0.917962	−0.458981	−	0.888446i	$-0.651785\pi$
$307$	−16.0840	−0.917962	−0.458981	+	0.888446i	$0.651785\pi$
$308$	0	0
$309$	0.837665	0.0476531
$310$	0	0
$311$	−15.3768	−0.871940	−0.435970	−	0.899961i	$-0.643595\pi$
$311$	−15.3768	−0.871940	−0.435970	+	0.899961i	$0.643595\pi$
$312$	1.68842	0.0955878
$313$	−22.4188	−1.26719	−0.633594	−	0.773666i	$-0.718421\pi$
$313$	−22.4188	−1.26719	−0.633594	+	0.773666i	$0.718421\pi$
$314$	17.1492	0.967788
$315$	0	0
$316$	7.31158	0.411309
$317$	−10.0653	−0.565321	−0.282660	−	0.959220i	$-0.591217\pi$
$317$	−10.0653	−0.565321	−0.282660	+	0.959220i	$0.591217\pi$
$318$	−10.4608	−0.586614
$319$	51.4159	2.87874
$320$	0	0
$321$	15.5261	0.866581
$322$	0	0
$323$	−11.5448	−0.642371
$324$	1.00000	0.0555556
$325$	0	0
$326$	0.539168	0.0298618
$327$	−11.6753	−0.645647
$328$	5.14925	0.284320
$329$	0	0
$330$	0	0
$331$	23.2797	1.27957	0.639785	−	0.768554i	$-0.279023\pi$
$331$	23.2797	1.27957	0.639785	+	0.768554i	$0.279023\pi$
$332$	13.5261	0.742340
$333$	2.31158	0.126674
$334$	−3.14925	−0.172319
$335$	0	0
$336$	0	0
$337$	−5.17250	−0.281764	−0.140882	−	0.990026i	$-0.544994\pi$
$337$	−5.17250	−0.281764	−0.140882	+	0.990026i	$0.544994\pi$
$338$	−10.1492	−0.552046
$339$	−3.37683	−0.183404
$340$	0	0
$341$	−4.20433	−0.227677
$342$	1.68842	0.0912991
$343$	0	0
$344$	−2.00000	−0.107833
$345$	0	0
$346$	−10.5261	−0.565886
$347$	−9.67533	−0.519399	−0.259699	−	0.965689i	$-0.583624\pi$
$347$	−9.67533	−0.519399	−0.259699	+	0.965689i	$0.583624\pi$
$348$	8.41883	0.451297
$349$	28.2985	1.51478	0.757392	−	0.652961i	$-0.226473\pi$
$349$	28.2985	1.51478	0.757392	+	0.652961i	$0.226473\pi$
$350$	0	0
$351$	1.68842	0.0901210
$352$	6.10725	0.325517
$353$	17.6753	0.940763	0.470381	−	0.882463i	$-0.344116\pi$
$353$	17.6753	0.940763	0.470381	+	0.882463i	$0.344116\pi$
$354$	−7.04200	−0.374278
$355$	0	0
$356$	−12.2145	−0.647367
$357$	0	0
$358$	4.93475	0.260810
$359$	−4.83767	−0.255322	−0.127661	−	0.991818i	$-0.540747\pi$
$359$	−4.83767	−0.255322	−0.127661	+	0.991818i	$0.540747\pi$
$360$	0	0
$361$	−16.1492	−0.849960
$362$	6.13050	0.322212
$363$	26.2985	1.38031
$364$	0	0
$365$	0	0
$366$	9.46083	0.494526
$367$	−10.6464	−0.555738	−0.277869	−	0.960619i	$-0.589628\pi$
$367$	−10.6464	−0.555738	−0.277869	+	0.960619i	$0.589628\pi$
$368$	2.31158	0.120500
$369$	5.14925	0.268059
$370$	0	0
$371$	0	0
$372$	−0.688417	−0.0356928
$373$	34.7275	1.79812	0.899061	−	0.437824i	$-0.144251\pi$
$373$	34.7275	1.79812	0.899061	+	0.437824i	$0.144251\pi$
$374$	−41.7593	−2.15932
$375$	0	0
$376$	−3.06525	−0.158078
$377$	14.2145	0.732084
$378$	0	0
$379$	−19.4477	−0.998964	−0.499482	−	0.866324i	$-0.666476\pi$
$379$	−19.4477	−0.998964	−0.499482	+	0.866324i	$0.666476\pi$
$380$	0	0
$381$	−4.94491	−0.253336
$382$	−14.9217	−0.763459
$383$	−0.850752	−0.0434714	−0.0217357	−	0.999764i	$-0.506919\pi$
$383$	−0.850752	−0.0434714	−0.0217357	+	0.999764i	$0.506919\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−5.58117	−0.284074
$387$	−2.00000	−0.101666
$388$	−4.95800	−0.251704
$389$	26.2985	1.33339	0.666693	−	0.745332i	$-0.267709\pi$
$389$	26.2985	1.33339	0.666693	+	0.745332i	$0.267709\pi$
$390$	0	0
$391$	−15.8058	−0.799335
$392$	0	0
$393$	−6.73042	−0.339505
$394$	14.5261	0.731813
$395$	0	0
$396$	6.10725	0.306901
$397$	−1.70150	−0.0853960	−0.0426980	−	0.999088i	$-0.513595\pi$
$397$	−1.70150	−0.0853960	−0.0426980	+	0.999088i	$0.513595\pi$
$398$	−8.62317	−0.432240
$399$	0	0
$400$	0	0
$401$	3.01875	0.150749	0.0753745	−	0.997155i	$-0.475985\pi$
$401$	3.01875	0.150749	0.0753745	+	0.997155i	$0.475985\pi$
$402$	−12.2145	−0.609204
$403$	−1.16233	−0.0579000
$404$	−8.00000	−0.398015
$405$	0	0
$406$	0	0
$407$	14.1174	0.699774
$408$	−6.83767	−0.338515
$409$	−4.14925	−0.205167	−0.102584	−	0.994724i	$-0.532711\pi$
$409$	−4.14925	−0.205167	−0.102584	+	0.994724i	$0.532711\pi$
$410$	0	0
$411$	19.0522	0.939774
$412$	0.837665	0.0412688
$413$	0	0
$414$	2.31158	0.113608
$415$	0	0
$416$	1.68842	0.0827814
$417$	13.3768	0.655066
$418$	10.3116	0.504356
$419$	13.8189	0.675098	0.337549	−	0.941308i	$-0.390402\pi$
$419$	13.8189	0.675098	0.337549	+	0.941308i	$0.390402\pi$
$420$	0	0
$421$	25.2667	1.23142	0.615711	−	0.787972i	$-0.288869\pi$
$421$	25.2667	1.23142	0.615711	+	0.787972i	$0.288869\pi$
$422$	−7.68842	−0.374266
$423$	−3.06525	−0.149038
$424$	−10.4608	−0.508023
$425$	0	0
$426$	6.00000	0.290701
$427$	0	0
$428$	15.5261	0.750482
$429$	10.3116	0.497848
$430$	0	0
$431$	19.5448	0.941441	0.470721	−	0.882282i	$-0.343994\pi$
$431$	19.5448	0.941441	0.470721	+	0.882282i	$0.343994\pi$
$432$	1.00000	0.0481125
$433$	−14.7537	−0.709016	−0.354508	−	0.935053i	$-0.615352\pi$
$433$	−14.7537	−0.709016	−0.354508	+	0.935053i	$0.615352\pi$
$434$	0	0
$435$	0	0
$436$	−11.6753	−0.559147
$437$	3.90292	0.186702
$438$	4.00000	0.191127
$439$	−15.9347	−0.760524	−0.380262	−	0.924879i	$-0.624166\pi$
$439$	−15.9347	−0.760524	−0.380262	+	0.924879i	$0.624166\pi$
$440$	0	0
$441$	0	0
$442$	−11.5448	−0.549132
$443$	21.8246	1.03692	0.518459	−	0.855103i	$-0.326506\pi$
$443$	21.8246	1.03692	0.518459	+	0.855103i	$0.326506\pi$
$444$	2.31158	0.109703
$445$	0	0
$446$	−11.1725	−0.529033
$447$	12.6232	0.597055
$448$	0	0
$449$	−11.3637	−0.536288	−0.268144	−	0.963379i	$-0.586410\pi$
$449$	−11.3637	−0.536288	−0.268144	+	0.963379i	$0.586410\pi$
$450$	0	0
$451$	31.4477	1.48082
$452$	−3.37683	−0.158833
$453$	1.93475	0.0909025
$454$	3.85075	0.180725
$455$	0	0
$456$	1.68842	0.0790674
$457$	6.41883	0.300260	0.150130	−	0.988666i	$-0.452031\pi$
$457$	6.41883	0.300260	0.150130	+	0.988666i	$0.452031\pi$
$458$	5.67533	0.265191
$459$	−6.83767	−0.319155
$460$	0	0
$461$	1.50733	0.0702036	0.0351018	−	0.999384i	$-0.488824\pi$
$461$	1.50733	0.0702036	0.0351018	+	0.999384i	$0.488824\pi$
$462$	0	0
$463$	−24.1174	−1.12083	−0.560416	−	0.828211i	$-0.689359\pi$
$463$	−24.1174	−1.12083	−0.560416	+	0.828211i	$0.689359\pi$
$464$	8.41883	0.390834
$465$	0	0
$466$	−11.4608	−0.530913
$467$	23.5448	1.08952	0.544762	−	0.838590i	$-0.316620\pi$
$467$	23.5448	1.08952	0.544762	+	0.838590i	$0.316620\pi$
$468$	1.68842	0.0780471
$469$	0	0
$470$	0	0
$471$	17.1492	0.790196
$472$	−7.04200	−0.324134
$473$	−12.2145	−0.561623
$474$	7.31158	0.335832
$475$	0	0
$476$	0	0
$477$	−10.4608	−0.478969
$478$	15.4608	0.707162
$479$	29.9738	1.36954	0.684770	−	0.728760i	$-0.259903\pi$
$479$	29.9738	1.36954	0.684770	+	0.728760i	$0.259903\pi$
$480$	0	0
$481$	3.90292	0.177958
$482$	4.37683	0.199359
$483$	0	0
$484$	26.2985	1.19539
$485$	0	0
$486$	1.00000	0.0453609
$487$	−30.5028	−1.38222	−0.691108	−	0.722752i	$-0.742877\pi$
$487$	−30.5028	−1.38222	−0.691108	+	0.722752i	$0.742877\pi$
$488$	9.46083	0.428272
$489$	0.539168	0.0243820
$490$	0	0
$491$	−3.87966	−0.175087	−0.0875434	−	0.996161i	$-0.527902\pi$
$491$	−3.87966	−0.175087	−0.0875434	+	0.996161i	$0.527902\pi$
$492$	5.14925	0.232146
$493$	−57.5652	−2.59260
$494$	2.85075	0.128261
$495$	0	0
$496$	−0.688417	−0.0309108
$497$	0	0
$498$	13.5261	0.606118
$499$	9.37683	0.419765	0.209882	−	0.977727i	$-0.432692\pi$
$499$	9.37683	0.419765	0.209882	+	0.977727i	$0.432692\pi$
$500$	0	0
$501$	−3.14925	−0.140698
$502$	−26.5362	−1.18437
$503$	14.2145	0.633793	0.316897	−	0.948460i	$-0.397359\pi$
$503$	14.2145	0.633793	0.316897	+	0.948460i	$0.397359\pi$
$504$	0	0
$505$	0	0
$506$	14.1174	0.627596
$507$	−10.1492	−0.450744
$508$	−4.94491	−0.219395
$509$	31.9637	1.41676	0.708382	−	0.705829i	$-0.249425\pi$
$509$	31.9637	1.41676	0.708382	+	0.705829i	$0.249425\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	1.68842	0.0745454
$514$	22.4290	0.989301
$515$	0	0
$516$	−2.00000	−0.0880451
$517$	−18.7203	−0.823316
$518$	0	0
$519$	−10.5261	−0.462044
$520$	0	0
$521$	−36.2014	−1.58601	−0.793006	−	0.609213i	$-0.791485\pi$
$521$	−36.2014	−1.58601	−0.793006	+	0.609213i	$0.791485\pi$
$522$	8.41883	0.368482
$523$	−36.9682	−1.61651	−0.808253	−	0.588836i	$-0.799586\pi$
$523$	−36.9682	−1.61651	−0.808253	+	0.588836i	$0.799586\pi$
$524$	−6.73042	−0.294020
$525$	0	0
$526$	−19.5913	−0.854223
$527$	4.70716	0.205047
$528$	6.10725	0.265784
$529$	−17.6566	−0.767678
$530$	0	0
$531$	−7.04200	−0.305597
$532$	0	0
$533$	8.69408	0.376582
$534$	−12.2145	−0.528573
$535$	0	0
$536$	−12.2145	−0.527586
$537$	4.93475	0.212950
$538$	−6.95800	−0.299981
$539$	0	0
$540$	0	0
$541$	−17.2667	−0.742352	−0.371176	−	0.928563i	$-0.621045\pi$
$541$	−17.2667	−0.742352	−0.371176	+	0.928563i	$0.621045\pi$
$542$	−28.3637	−1.21833
$543$	6.13050	0.263085
$544$	−6.83767	−0.293162
$545$	0	0
$546$	0	0
$547$	39.4347	1.68610	0.843052	−	0.537832i	$-0.180756\pi$
$547$	39.4347	1.68610	0.843052	+	0.537832i	$0.180756\pi$
$548$	19.0522	0.813868
$549$	9.46083	0.403778
$550$	0	0
$551$	14.2145	0.605558
$552$	2.31158	0.0983875
$553$	0	0
$554$	5.05216	0.214646
$555$	0	0
$556$	13.3768	0.567304
$557$	−32.8898	−1.39359	−0.696793	−	0.717272i	$-0.745391\pi$
$557$	−32.8898	−1.39359	−0.696793	+	0.717272i	$0.745391\pi$
$558$	−0.688417	−0.0291430
$559$	−3.37683	−0.142825
$560$	0	0
$561$	−41.7593	−1.76308
$562$	7.68842	0.324316
$563$	−35.2014	−1.48356	−0.741781	−	0.670642i	$-0.766019\pi$
$563$	−35.2014	−1.48356	−0.741781	+	0.670642i	$0.766019\pi$
$564$	−3.06525	−0.129070
$565$	0	0
$566$	18.4290	0.774629
$567$	0	0
$568$	6.00000	0.251754
$569$	17.4942	0.733397	0.366699	−	0.930340i	$-0.380488\pi$
$569$	17.4942	0.733397	0.366699	+	0.930340i	$0.380488\pi$
$570$	0	0
$571$	8.83767	0.369845	0.184922	−	0.982753i	$-0.440797\pi$
$571$	8.83767	0.369845	0.184922	+	0.982753i	$0.440797\pi$
$572$	10.3116	0.431149
$573$	−14.9217	−0.623361
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	−40.3087	−1.67807	−0.839036	−	0.544076i	$-0.816880\pi$
$577$	−40.3087	−1.67807	−0.839036	+	0.544076i	$0.816880\pi$
$578$	29.7537	1.23759
$579$	−5.58117	−0.231945
$580$	0	0
$581$	0	0
$582$	−4.95800	−0.205516
$583$	−63.8869	−2.64593
$584$	4.00000	0.165521
$585$	0	0
$586$	−1.83767	−0.0759133
$587$	36.1492	1.49204	0.746020	−	0.665924i	$-0.231963\pi$
$587$	36.1492	1.49204	0.746020	+	0.665924i	$0.231963\pi$
$588$	0	0
$589$	−1.16233	−0.0478932
$590$	0	0
$591$	14.5261	0.597523
$592$	2.31158	0.0950055
$593$	−4.08400	−0.167710	−0.0838548	−	0.996478i	$-0.526723\pi$
$593$	−4.08400	−0.167710	−0.0838548	+	0.996478i	$0.526723\pi$
$594$	6.10725	0.250583
$595$	0	0
$596$	12.6232	0.517065
$597$	−8.62317	−0.352923
$598$	3.90292	0.159602
$599$	−19.7593	−0.807344	−0.403672	−	0.914904i	$-0.632266\pi$
$599$	−19.7593	−0.807344	−0.403672	+	0.914904i	$0.632266\pi$
$600$	0	0
$601$	3.72025	0.151752	0.0758761	−	0.997117i	$-0.475825\pi$
$601$	3.72025	0.151752	0.0758761	+	0.997117i	$0.475825\pi$
$602$	0	0
$603$	−12.2145	−0.497413
$604$	1.93475	0.0787238
$605$	0	0
$606$	−8.00000	−0.324978
$607$	11.2434	0.456356	0.228178	−	0.973619i	$-0.426723\pi$
$607$	11.2434	0.456356	0.228178	+	0.973619i	$0.426723\pi$
$608$	1.68842	0.0684743
$609$	0	0
$610$	0	0
$611$	−5.17542	−0.209375
$612$	−6.83767	−0.276396
$613$	−26.9347	−1.08788	−0.543942	−	0.839123i	$-0.683069\pi$
$613$	−26.9347	−1.08788	−0.543942	+	0.839123i	$0.683069\pi$
$614$	−16.0840	−0.649097
$615$	0	0
$616$	0	0
$617$	−33.4608	−1.34708	−0.673541	−	0.739150i	$-0.735228\pi$
$617$	−33.4608	−1.34708	−0.673541	+	0.739150i	$0.735228\pi$
$618$	0.837665	0.0336958
$619$	−34.8711	−1.40159	−0.700794	−	0.713364i	$-0.747171\pi$
$619$	−34.8711	−1.40159	−0.700794	+	0.713364i	$0.747171\pi$
$620$	0	0
$621$	2.31158	0.0927606
$622$	−15.3768	−0.616555
$623$	0	0
$624$	1.68842	0.0675908
$625$	0	0
$626$	−22.4188	−0.896037
$627$	10.3116	0.411805
$628$	17.1492	0.684330
$629$	−15.8058	−0.630220
$630$	0	0
$631$	−10.4942	−0.417769	−0.208885	−	0.977940i	$-0.566983\pi$
$631$	−10.4942	−0.417769	−0.208885	+	0.977940i	$0.566983\pi$
$632$	7.31158	0.290839
$633$	−7.68842	−0.305587
$634$	−10.0653	−0.399742
$635$	0	0
$636$	−10.4608	−0.414799
$637$	0	0
$638$	51.4159	2.03558
$639$	6.00000	0.237356
$640$	0	0
$641$	7.77241	0.306992	0.153496	−	0.988149i	$-0.450947\pi$
$641$	7.77241	0.306992	0.153496	+	0.988149i	$0.450947\pi$
$642$	15.5261	0.612766
$643$	21.7218	0.856626	0.428313	−	0.903631i	$-0.359108\pi$
$643$	21.7218	0.856626	0.428313	+	0.903631i	$0.359108\pi$
$644$	0	0
$645$	0	0
$646$	−11.5448	−0.454225
$647$	−9.55792	−0.375760	−0.187880	−	0.982192i	$-0.560162\pi$
$647$	−9.55792	−0.375760	−0.187880	+	0.982192i	$0.560162\pi$
$648$	1.00000	0.0392837
$649$	−43.0072	−1.68818
$650$	0	0
$651$	0	0
$652$	0.539168	0.0211155
$653$	−50.2667	−1.96709	−0.983543	−	0.180673i	$-0.942173\pi$
$653$	−50.2667	−1.96709	−0.983543	+	0.180673i	$0.942173\pi$
$654$	−11.6753	−0.456542
$655$	0	0
$656$	5.14925	0.201044
$657$	4.00000	0.156055
$658$	0	0
$659$	11.1623	0.434823	0.217411	−	0.976080i	$-0.430239\pi$
$659$	11.1623	0.434823	0.217411	+	0.976080i	$0.430239\pi$
$660$	0	0
$661$	14.3450	0.557956	0.278978	−	0.960298i	$-0.410004\pi$
$661$	14.3450	0.557956	0.278978	+	0.960298i	$0.410004\pi$
$662$	23.2797	0.904793
$663$	−11.5448	−0.448364
$664$	13.5261	0.524914
$665$	0	0
$666$	2.31158	0.0895720
$667$	19.4608	0.753526
$668$	−3.14925	−0.121848
$669$	−11.1725	−0.431954
$670$	0	0
$671$	57.7797	2.23056
$672$	0	0
$673$	−30.8013	−1.18730	−0.593652	−	0.804722i	$-0.702314\pi$
$673$	−30.8013	−1.18730	−0.593652	+	0.804722i	$0.702314\pi$
$674$	−5.17250	−0.199237
$675$	0	0
$676$	−10.1492	−0.390356
$677$	14.1623	0.544303	0.272151	−	0.962254i	$-0.412265\pi$
$677$	14.1623	0.544303	0.272151	+	0.962254i	$0.412265\pi$
$678$	−3.37683	−0.129687
$679$	0	0
$680$	0	0
$681$	3.85075	0.147561
$682$	−4.20433	−0.160992
$683$	30.7087	1.17504	0.587519	−	0.809211i	$-0.300105\pi$
$683$	30.7087	1.17504	0.587519	+	0.809211i	$0.300105\pi$
$684$	1.68842	0.0645582
$685$	0	0
$686$	0	0
$687$	5.67533	0.216527
$688$	−2.00000	−0.0762493
$689$	−17.6622	−0.672878
$690$	0	0
$691$	−39.3507	−1.49697	−0.748485	−	0.663152i	$-0.769218\pi$
$691$	−39.3507	−1.49697	−0.748485	+	0.663152i	$0.769218\pi$
$692$	−10.5261	−0.400142
$693$	0	0
$694$	−9.67533	−0.367271
$695$	0	0
$696$	8.41883	0.319115
$697$	−35.2088	−1.33363
$698$	28.2985	1.07111
$699$	−11.4608	−0.433488
$700$	0	0
$701$	−3.79567	−0.143360	−0.0716802	−	0.997428i	$-0.522836\pi$
$701$	−3.79567	−0.143360	−0.0716802	+	0.997428i	$0.522836\pi$
$702$	1.68842	0.0637252
$703$	3.90292	0.147201
$704$	6.10725	0.230176
$705$	0	0
$706$	17.6753	0.665220
$707$	0	0
$708$	−7.04200	−0.264655
$709$	−20.5595	−0.772128	−0.386064	−	0.922472i	$-0.626166\pi$
$709$	−20.5595	−0.772128	−0.386064	+	0.922472i	$0.626166\pi$
$710$	0	0
$711$	7.31158	0.274206
$712$	−12.2145	−0.457758
$713$	−1.59133	−0.0595959
$714$	0	0
$715$	0	0
$716$	4.93475	0.184420
$717$	15.4608	0.577395
$718$	−4.83767	−0.180540
$719$	−24.8115	−0.925313	−0.462656	−	0.886538i	$-0.653104\pi$
$719$	−24.8115	−0.925313	−0.462656	+	0.886538i	$0.653104\pi$
$720$	0	0
$721$	0	0
$722$	−16.1492	−0.601013
$723$	4.37683	0.162776
$724$	6.13050	0.227838
$725$	0	0
$726$	26.2985	0.976029
$727$	−39.2899	−1.45718	−0.728591	−	0.684949i	$-0.759825\pi$
$727$	−39.2899	−1.45718	−0.728591	+	0.684949i	$0.759825\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	13.6753	0.505800
$732$	9.46083	0.349682
$733$	−9.01875	−0.333115	−0.166558	−	0.986032i	$-0.553265\pi$
$733$	−9.01875	−0.333115	−0.166558	+	0.986032i	$0.553265\pi$
$734$	−10.6464	−0.392966
$735$	0	0
$736$	2.31158	0.0852061
$737$	−74.5970	−2.74782
$738$	5.14925	0.189546
$739$	−14.0709	−0.517607	−0.258803	−	0.965930i	$-0.583328\pi$
$739$	−14.0709	−0.517607	−0.258803	+	0.965930i	$0.583328\pi$
$740$	0	0
$741$	2.85075	0.104725
$742$	0	0
$743$	−39.1492	−1.43625	−0.718123	−	0.695916i	$-0.754999\pi$
$743$	−39.1492	−1.43625	−0.718123	+	0.695916i	$0.754999\pi$
$744$	−0.688417	−0.0252386
$745$	0	0
$746$	34.7275	1.27146
$747$	13.5261	0.494893
$748$	−41.7593	−1.52687
$749$	0	0
$750$	0	0
$751$	−39.2854	−1.43355	−0.716773	−	0.697307i	$-0.754381\pi$
$751$	−39.2854	−1.43355	−0.716773	+	0.697307i	$0.754381\pi$
$752$	−3.06525	−0.111778
$753$	−26.5362	−0.967035
$754$	14.2145	0.517662
$755$	0	0
$756$	0	0
$757$	−28.9217	−1.05118	−0.525588	−	0.850739i	$-0.676155\pi$
$757$	−28.9217	−1.05118	−0.525588	+	0.850739i	$0.676155\pi$
$758$	−19.4477	−0.706374
$759$	14.1174	0.512430
$760$	0	0
$761$	15.5579	0.563974	0.281987	−	0.959418i	$-0.409006\pi$
$761$	15.5579	0.563974	0.281987	+	0.959418i	$0.409006\pi$
$762$	−4.94491	−0.179135
$763$	0	0
$764$	−14.9217	−0.539847
$765$	0	0
$766$	−0.850752	−0.0307389
$767$	−11.8898	−0.429317
$768$	1.00000	0.0360844
$769$	−4.05216	−0.146125	−0.0730624	−	0.997327i	$-0.523277\pi$
$769$	−4.05216	−0.146125	−0.0730624	+	0.997327i	$0.523277\pi$
$770$	0	0
$771$	22.4290	0.807761
$772$	−5.58117	−0.200871
$773$	−21.4477	−0.771422	−0.385711	−	0.922620i	$-0.626044\pi$
$773$	−21.4477	−0.771422	−0.385711	+	0.922620i	$0.626044\pi$
$774$	−2.00000	−0.0718885
$775$	0	0
$776$	−4.95800	−0.177982
$777$	0	0
$778$	26.2985	0.942847
$779$	8.69408	0.311498
$780$	0	0
$781$	36.6435	1.31121
$782$	−15.8058	−0.565215
$783$	8.41883	0.300865
$784$	0	0
$785$	0	0
$786$	−6.73042	−0.240066
$787$	−4.32467	−0.154158	−0.0770789	−	0.997025i	$-0.524559\pi$
$787$	−4.32467	−0.154158	−0.0770789	+	0.997025i	$0.524559\pi$
$788$	14.5261	0.517470
$789$	−19.5913	−0.697470
$790$	0	0
$791$	0	0
$792$	6.10725	0.217012
$793$	15.9738	0.567247
$794$	−1.70150	−0.0603841
$795$	0	0
$796$	−8.62317	−0.305640
$797$	41.1549	1.45778	0.728891	−	0.684630i	$-0.240036\pi$
$797$	41.1549	1.45778	0.728891	+	0.684630i	$0.240036\pi$
$798$	0	0
$799$	20.9592	0.741482
$800$	0	0
$801$	−12.2145	−0.431578
$802$	3.01875	0.106596
$803$	24.4290	0.862081
$804$	−12.2145	−0.430772
$805$	0	0
$806$	−1.16233	−0.0409415
$807$	−6.95800	−0.244933
$808$	−8.00000	−0.281439
$809$	−17.7724	−0.624845	−0.312422	−	0.949943i	$-0.601140\pi$
$809$	−17.7724	−0.624845	−0.312422	+	0.949943i	$0.601140\pi$
$810$	0	0
$811$	47.1231	1.65472	0.827358	−	0.561676i	$-0.189843\pi$
$811$	47.1231	1.65472	0.827358	+	0.561676i	$0.189843\pi$
$812$	0	0
$813$	−28.3637	−0.994760
$814$	14.1174	0.494815
$815$	0	0
$816$	−6.83767	−0.239366
$817$	−3.37683	−0.118140
$818$	−4.14925	−0.145075
$819$	0	0
$820$	0	0
$821$	5.71167	0.199339	0.0996693	−	0.995021i	$-0.468222\pi$
$821$	5.71167	0.199339	0.0996693	+	0.995021i	$0.468222\pi$
$822$	19.0522	0.664521
$823$	−30.3450	−1.05776	−0.528880	−	0.848697i	$-0.677388\pi$
$823$	−30.3450	−1.05776	−0.528880	+	0.848697i	$0.677388\pi$
$824$	0.837665	0.0291815
$825$	0	0
$826$	0	0
$827$	−27.2014	−0.945886	−0.472943	−	0.881093i	$-0.656808\pi$
$827$	−27.2014	−0.945886	−0.472943	+	0.881093i	$0.656808\pi$
$828$	2.31158	0.0803331
$829$	10.8377	0.376408	0.188204	−	0.982130i	$-0.439733\pi$
$829$	10.8377	0.376408	0.188204	+	0.982130i	$0.439733\pi$
$830$	0	0
$831$	5.05216	0.175258
$832$	1.68842	0.0585353
$833$	0	0
$834$	13.3768	0.463202
$835$	0	0
$836$	10.3116	0.356634
$837$	−0.688417	−0.0237952
$838$	13.8189	0.477367
$839$	−1.59133	−0.0549389	−0.0274695	−	0.999623i	$-0.508745\pi$
$839$	−1.59133	−0.0549389	−0.0274695	+	0.999623i	$0.508745\pi$
$840$	0	0
$841$	41.8767	1.44403
$842$	25.2667	0.870747
$843$	7.68842	0.264803
$844$	−7.68842	−0.264646
$845$	0	0
$846$	−3.06525	−0.105385
$847$	0	0
$848$	−10.4608	−0.359226
$849$	18.4290	0.632482
$850$	0	0
$851$	5.34342	0.183170
$852$	6.00000	0.205557
$853$	−29.9869	−1.02673	−0.513366	−	0.858170i	$-0.671602\pi$
$853$	−29.9869	−1.02673	−0.513366	+	0.858170i	$0.671602\pi$
$854$	0	0
$855$	0	0
$856$	15.5261	0.530671
$857$	37.0057	1.26409	0.632045	−	0.774932i	$-0.282216\pi$
$857$	37.0057	1.26409	0.632045	+	0.774932i	$0.282216\pi$
$858$	10.3116	0.352032
$859$	27.7593	0.947136	0.473568	−	0.880757i	$-0.342966\pi$
$859$	27.7593	0.947136	0.473568	+	0.880757i	$0.342966\pi$
$860$	0	0
$861$	0	0
$862$	19.5448	0.665700
$863$	−3.60442	−0.122696	−0.0613479	−	0.998116i	$-0.519540\pi$
$863$	−3.60442	−0.122696	−0.0613479	+	0.998116i	$0.519540\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−14.7537	−0.501350
$867$	29.7537	1.01049
$868$	0	0
$869$	44.6537	1.51477
$870$	0	0
$871$	−20.6232	−0.698789
$872$	−11.6753	−0.395377
$873$	−4.95800	−0.167803
$874$	3.90292	0.132018
$875$	0	0
$876$	4.00000	0.135147
$877$	−12.0971	−0.408490	−0.204245	−	0.978920i	$-0.565474\pi$
$877$	−12.0971	−0.408490	−0.204245	+	0.978920i	$0.565474\pi$
$878$	−15.9347	−0.537772
$879$	−1.83767	−0.0619829
$880$	0	0
$881$	−44.7406	−1.50735	−0.753674	−	0.657248i	$-0.771720\pi$
$881$	−44.7406	−1.50735	−0.753674	+	0.657248i	$0.771720\pi$
$882$	0	0
$883$	36.5970	1.23159	0.615793	−	0.787908i	$-0.288836\pi$
$883$	36.5970	1.23159	0.615793	+	0.787908i	$0.288836\pi$
$884$	−11.5448	−0.388295
$885$	0	0
$886$	21.8246	0.733211
$887$	−56.0578	−1.88224	−0.941119	−	0.338076i	$-0.890224\pi$
$887$	−56.0578	−1.88224	−0.941119	+	0.338076i	$0.890224\pi$
$888$	2.31158	0.0775717
$889$	0	0
$890$	0	0
$891$	6.10725	0.204601
$892$	−11.1725	−0.374083
$893$	−5.17542	−0.173189
$894$	12.6232	0.422182
$895$	0	0
$896$	0	0
$897$	3.90292	0.130315
$898$	−11.3637	−0.379213
$899$	−5.79567	−0.193296
$900$	0	0
$901$	71.5277	2.38293
$902$	31.4477	1.04710
$903$	0	0
$904$	−3.37683	−0.112312
$905$	0	0
$906$	1.93475	0.0642777
$907$	18.6435	0.619047	0.309524	−	0.950892i	$-0.399830\pi$
$907$	18.6435	0.619047	0.309524	+	0.950892i	$0.399830\pi$
$908$	3.85075	0.127792
$909$	−8.00000	−0.265343
$910$	0	0
$911$	−6.00000	−0.198789	−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000	−0.198789	−0.0993944	+	0.995048i	$0.531691\pi$
$912$	1.68842	0.0559091
$913$	82.6072	2.73390
$914$	6.41883	0.212316
$915$	0	0
$916$	5.67533	0.187518
$917$	0	0
$918$	−6.83767	−0.225677
$919$	−38.9217	−1.28391	−0.641954	−	0.766743i	$-0.721876\pi$
$919$	−38.9217	−1.28391	−0.641954	+	0.766743i	$0.721876\pi$
$920$	0	0
$921$	−16.0840	−0.529986
$922$	1.50733	0.0496414
$923$	10.1305	0.333450
$924$	0	0
$925$	0	0
$926$	−24.1174	−0.792547
$927$	0.837665	0.0275125
$928$	8.41883	0.276362
$929$	13.7349	0.450628	0.225314	−	0.974286i	$-0.427659\pi$
$929$	13.7349	0.450628	0.225314	+	0.974286i	$0.427659\pi$
$930$	0	0
$931$	0	0
$932$	−11.4608	−0.375412
$933$	−15.3768	−0.503415
$934$	23.5448	0.770410
$935$	0	0
$936$	1.68842	0.0551876
$937$	−20.7435	−0.677661	−0.338830	−	0.940847i	$-0.610031\pi$
$937$	−20.7435	−0.677661	−0.338830	+	0.940847i	$0.610031\pi$
$938$	0	0
$939$	−22.4188	−0.731611
$940$	0	0
$941$	13.0885	0.426673	0.213336	−	0.976979i	$-0.431567\pi$
$941$	13.0885	0.426673	0.213336	+	0.976979i	$0.431567\pi$
$942$	17.1492	0.558753
$943$	11.9029	0.387612
$944$	−7.04200	−0.229198
$945$	0	0
$946$	−12.2145	−0.397128
$947$	20.2610	0.658394	0.329197	−	0.944261i	$-0.393222\pi$
$947$	20.2610	0.658394	0.329197	+	0.944261i	$0.393222\pi$
$948$	7.31158	0.237469
$949$	6.75367	0.219233
$950$	0	0
$951$	−10.0653	−0.326388
$952$	0	0
$953$	1.65500	0.0536107	0.0268053	−	0.999641i	$-0.491467\pi$
$953$	1.65500	0.0536107	0.0268053	+	0.999641i	$0.491467\pi$
$954$	−10.4608	−0.338682
$955$	0	0
$956$	15.4608	0.500039
$957$	51.4159	1.66204
$958$	29.9738	0.968410
$959$	0	0
$960$	0	0
$961$	−30.5261	−0.984712
$962$	3.90292	0.125835
$963$	15.5261	0.500321
$964$	4.37683	0.140968
$965$	0	0
$966$	0	0
$967$	−15.3405	−0.493317	−0.246659	−	0.969102i	$-0.579333\pi$
$967$	−15.3405	−0.493317	−0.246659	+	0.969102i	$0.579333\pi$
$968$	26.2985	0.845266
$969$	−11.5448	−0.370873
$970$	0	0
$971$	47.1129	1.51193	0.755963	−	0.654615i	$-0.227169\pi$
$971$	47.1129	1.51193	0.755963	+	0.654615i	$0.227169\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−30.5028	−0.977374
$975$	0	0
$976$	9.46083	0.302834
$977$	29.8898	0.956260	0.478130	−	0.878289i	$-0.341315\pi$
$977$	29.8898	0.956260	0.478130	+	0.878289i	$0.341315\pi$
$978$	0.539168	0.0172407
$979$	−74.5970	−2.38413
$980$	0	0
$981$	−11.6753	−0.372765
$982$	−3.87966	−0.123805
$983$	5.14925	0.164236	0.0821178	−	0.996623i	$-0.473832\pi$
$983$	5.14925	0.164236	0.0821178	+	0.996623i	$0.473832\pi$
$984$	5.14925	0.164152
$985$	0	0
$986$	−57.5652	−1.83325
$987$	0	0
$988$	2.85075	0.0906945
$989$	−4.62317	−0.147008
$990$	0	0
$991$	6.81892	0.216610	0.108305	−	0.994118i	$-0.465458\pi$
$991$	6.81892	0.216610	0.108305	+	0.994118i	$0.465458\pi$
$992$	−0.688417	−0.0218573
$993$	23.2797	0.738761
$994$	0	0
$995$	0	0
$996$	13.5261	0.428590
$997$	51.5652	1.63309	0.816543	−	0.577285i	$-0.195888\pi$
$997$	51.5652	1.63309	0.816543	+	0.577285i	$0.195888\pi$
$998$	9.37683	0.296818
$999$	2.31158	0.0731353

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7350.2.a.dq.1.3		3
5.2	odd	4		1470.2.g.i.589.6		6
5.3	odd	4		1470.2.g.i.589.3		6
5.4	even	2		7350.2.a.dn.1.3		3
7.2	even	3		1050.2.i.u.151.1		6
7.4	even	3		1050.2.i.u.751.1		6
7.6	odd	2		7350.2.a.dp.1.3		3
35.2	odd	12		210.2.n.b.109.5	yes	12
35.3	even	12		1470.2.n.j.79.5		12
35.4	even	6		1050.2.i.v.751.3		6
35.9	even	6		1050.2.i.v.151.3		6
35.12	even	12		1470.2.n.j.949.5		12
35.13	even	4		1470.2.g.h.589.1		6
35.17	even	12		1470.2.n.j.79.3		12
35.18	odd	12		210.2.n.b.79.5	yes	12
35.23	odd	12		210.2.n.b.109.1	yes	12
35.27	even	4		1470.2.g.h.589.4		6
35.32	odd	12		210.2.n.b.79.1	✓	12
35.33	even	12		1470.2.n.j.949.3		12
35.34	odd	2		7350.2.a.do.1.3		3
105.2	even	12		630.2.u.f.109.2		12
105.23	even	12		630.2.u.f.109.6		12
105.32	even	12		630.2.u.f.289.6		12
105.53	even	12		630.2.u.f.289.2		12
140.23	even	12		1680.2.di.c.529.4		12
140.67	even	12		1680.2.di.c.289.4		12
140.107	even	12		1680.2.di.c.529.2		12
140.123	even	12		1680.2.di.c.289.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.n.b.79.1	✓	12	35.32	odd	12
210.2.n.b.79.5	yes	12	35.18	odd	12
210.2.n.b.109.1	yes	12	35.23	odd	12
210.2.n.b.109.5	yes	12	35.2	odd	12
630.2.u.f.109.2		12	105.2	even	12
630.2.u.f.109.6		12	105.23	even	12
630.2.u.f.289.2		12	105.53	even	12
630.2.u.f.289.6		12	105.32	even	12
1050.2.i.u.151.1		6	7.2	even	3
1050.2.i.u.751.1		6	7.4	even	3
1050.2.i.v.151.3		6	35.9	even	6
1050.2.i.v.751.3		6	35.4	even	6
1470.2.g.h.589.1		6	35.13	even	4
1470.2.g.h.589.4		6	35.27	even	4
1470.2.g.i.589.3		6	5.3	odd	4
1470.2.g.i.589.6		6	5.2	odd	4
1470.2.n.j.79.3		12	35.17	even	12
1470.2.n.j.79.5		12	35.3	even	12
1470.2.n.j.949.3		12	35.33	even	12
1470.2.n.j.949.5		12	35.12	even	12
1680.2.di.c.289.2		12	140.123	even	12
1680.2.di.c.289.4		12	140.67	even	12
1680.2.di.c.529.2		12	140.107	even	12
1680.2.di.c.529.4		12	140.23	even	12
7350.2.a.dn.1.3		3	5.4	even	2
7350.2.a.do.1.3		3	35.34	odd	2
7350.2.a.dp.1.3		3	7.6	odd	2
7350.2.a.dq.1.3		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7350.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +6.10725 q^{11} +1.00000 q^{12} +1.68842 q^{13} +1.00000 q^{16} -6.83767 q^{17} +1.00000 q^{18} +1.68842 q^{19} +6.10725 q^{22} +2.31158 q^{23} +1.00000 q^{24} +1.68842 q^{26} +1.00000 q^{27} +8.41883 q^{29} -0.688417 q^{31} +1.00000 q^{32} +6.10725 q^{33} -6.83767 q^{34} +1.00000 q^{36} +2.31158 q^{37} +1.68842 q^{38} +1.68842 q^{39} +5.14925 q^{41} -2.00000 q^{43} +6.10725 q^{44} +2.31158 q^{46} -3.06525 q^{47} +1.00000 q^{48} -6.83767 q^{51} +1.68842 q^{52} -10.4608 q^{53} +1.00000 q^{54} +1.68842 q^{57} +8.41883 q^{58} -7.04200 q^{59} +9.46083 q^{61} -0.688417 q^{62} +1.00000 q^{64} +6.10725 q^{66} -12.2145 q^{67} -6.83767 q^{68} +2.31158 q^{69} +6.00000 q^{71} +1.00000 q^{72} +4.00000 q^{73} +2.31158 q^{74} +1.68842 q^{76} +1.68842 q^{78} +7.31158 q^{79} +1.00000 q^{81} +5.14925 q^{82} +13.5261 q^{83} -2.00000 q^{86} +8.41883 q^{87} +6.10725 q^{88} -12.2145 q^{89} +2.31158 q^{92} -0.688417 q^{93} -3.06525 q^{94} +1.00000 q^{96} -4.95800 q^{97} +6.10725 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^6 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{6} + 3 q^{8} + 3 q^{9} + 3 q^{11} + 3 q^{12} + 3 q^{13} + 3 q^{16} + 6 q^{17} + 3 q^{18} + 3 q^{19} + 3 q^{22} + 9 q^{23} + 3 q^{24} + 3 q^{26} + 3 q^{27} + 12 q^{29} + 3 q^{32} + 3 q^{33} + 6 q^{34} + 3 q^{36} + 9 q^{37} + 3 q^{38} + 3 q^{39} - 9 q^{41} - 6 q^{43} + 3 q^{44} + 9 q^{46} - 3 q^{47} + 3 q^{48} + 6 q^{51} + 3 q^{52} - 9 q^{53} + 3 q^{54} + 3 q^{57} + 12 q^{58} - 12 q^{59} + 6 q^{61} + 3 q^{64} + 3 q^{66} - 6 q^{67} + 6 q^{68} + 9 q^{69} + 18 q^{71} + 3 q^{72} + 12 q^{73} + 9 q^{74} + 3 q^{76} + 3 q^{78} + 24 q^{79} + 3 q^{81} - 9 q^{82} + 12 q^{83} - 6 q^{86} + 12 q^{87} + 3 q^{88} - 6 q^{89} + 9 q^{92} - 3 q^{94} + 3 q^{96} - 24 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^6 + 3 * q^8 + 3 * q^9 + 3 * q^11 + 3 * q^12 + 3 * q^13 + 3 * q^16 + 6 * q^17 + 3 * q^18 + 3 * q^19 + 3 * q^22 + 9 * q^23 + 3 * q^24 + 3 * q^26 + 3 * q^27 + 12 * q^29 + 3 * q^32 + 3 * q^33 + 6 * q^34 + 3 * q^36 + 9 * q^37 + 3 * q^38 + 3 * q^39 - 9 * q^41 - 6 * q^43 + 3 * q^44 + 9 * q^46 - 3 * q^47 + 3 * q^48 + 6 * q^51 + 3 * q^52 - 9 * q^53 + 3 * q^54 + 3 * q^57 + 12 * q^58 - 12 * q^59 + 6 * q^61 + 3 * q^64 + 3 * q^66 - 6 * q^67 + 6 * q^68 + 9 * q^69 + 18 * q^71 + 3 * q^72 + 12 * q^73 + 9 * q^74 + 3 * q^76 + 3 * q^78 + 24 * q^79 + 3 * q^81 - 9 * q^82 + 12 * q^83 - 6 * q^86 + 12 * q^87 + 3 * q^88 - 6 * q^89 + 9 * q^92 - 3 * q^94 + 3 * q^96 - 24 * q^97 + 3 * q^99

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