LMFDB - Embedded newform 7500.2.a.n.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7500,2,Mod(1,7500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7500, 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("7500.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7500 = 2^{2} \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7500.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:	$59.8878015160$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} - 11 x^{10} + 94 x^{9} + 27 x^{8} - 460 x^{7} + 55 x^{6} + 812 x^{5} - 127 x^{4} + \cdots - 4 $ x^12 - 6x^11 - 11x^10 + 94x^9 + 27x^8 - 460x^7 + 55x^6 + 812x^5 - 127x^4 - 512x^3 + 40x^2 + 72*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 5^{3} $
Twist minimal:	no (minimal twist has level 300)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.20648$ of defining polynomial
Character	$\chi$	$=$	7500.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.57893 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.57893 q^{7} +1.00000 q^{9} +3.88059 q^{11} +0.343277 q^{13} +6.07054 q^{17} +3.69189 q^{19} -1.57893 q^{21} -1.39041 q^{23} +1.00000 q^{27} -3.32990 q^{29} -5.25496 q^{31} +3.88059 q^{33} +8.56667 q^{37} +0.343277 q^{39} -1.27815 q^{41} -1.42438 q^{43} +0.375462 q^{47} -4.50698 q^{49} +6.07054 q^{51} +11.2992 q^{53} +3.69189 q^{57} +11.5818 q^{59} -10.9673 q^{61} -1.57893 q^{63} +10.4591 q^{67} -1.39041 q^{69} -10.1261 q^{71} -13.1900 q^{73} -6.12718 q^{77} +13.7995 q^{79} +1.00000 q^{81} +4.62458 q^{83} -3.32990 q^{87} -7.26904 q^{89} -0.542010 q^{91} -5.25496 q^{93} +6.05557 q^{97} +3.88059 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{3} + 8 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q + 12 * q^3 + 8 * q^7 + 12 * q^9	$ 12 q + 12 q^{3} + 8 q^{7} + 12 q^{9} + 2 q^{11} + 8 q^{17} + 10 q^{19} + 8 q^{21} + 18 q^{23} + 12 q^{27} + 8 q^{29} - 2 q^{31} + 2 q^{33} + 4 q^{37} + 10 q^{41} + 28 q^{43} + 22 q^{47} + 28 q^{49} + 8 q^{51} + 16 q^{53} + 10 q^{57} - 2 q^{59} + 34 q^{61} + 8 q^{63} + 32 q^{67} + 18 q^{69} + 24 q^{73} + 18 q^{77} + 6 q^{79} + 12 q^{81} + 28 q^{83} + 8 q^{87} + 10 q^{89} + 20 q^{91} - 2 q^{93} + 16 q^{97} + 2 q^{99}+O(q^{100}) $ 12 * q + 12 * q^3 + 8 * q^7 + 12 * q^9 + 2 * q^11 + 8 * q^17 + 10 * q^19 + 8 * q^21 + 18 * q^23 + 12 * q^27 + 8 * q^29 - 2 * q^31 + 2 * q^33 + 4 * q^37 + 10 * q^41 + 28 * q^43 + 22 * q^47 + 28 * q^49 + 8 * q^51 + 16 * q^53 + 10 * q^57 - 2 * q^59 + 34 * q^61 + 8 * q^63 + 32 * q^67 + 18 * q^69 + 24 * q^73 + 18 * q^77 + 6 * q^79 + 12 * q^81 + 28 * q^83 + 8 * q^87 + 10 * q^89 + 20 * q^91 - 2 * q^93 + 16 * q^97 + 2 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.57893	−0.596780	−0.298390	−	0.954444i	$-0.596450\pi$
$7$	−1.57893	−0.596780	−0.298390	+	0.954444i	$0.596450\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.88059	1.17004	0.585021	−	0.811018i	$-0.301086\pi$
$11$	3.88059	1.17004	0.585021	+	0.811018i	$0.301086\pi$
$12$	0	0
$13$	0.343277	0.0952078	0.0476039	−	0.998866i	$-0.484841\pi$
$13$	0.343277	0.0952078	0.0476039	+	0.998866i	$0.484841\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.07054	1.47232	0.736161	−	0.676807i	$-0.236637\pi$
$17$	6.07054	1.47232	0.736161	+	0.676807i	$0.236637\pi$
$18$	0	0
$19$	3.69189	0.846977	0.423489	−	0.905901i	$-0.360805\pi$
$19$	3.69189	0.846977	0.423489	+	0.905901i	$0.360805\pi$
$20$	0	0
$21$	−1.57893	−0.344551
$22$	0	0
$23$	−1.39041	−0.289921	−0.144961	−	0.989437i	$-0.546306\pi$
$23$	−1.39041	−0.289921	−0.144961	+	0.989437i	$0.546306\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.32990	−0.618347	−0.309174	−	0.951006i	$-0.600052\pi$
$29$	−3.32990	−0.618347	−0.309174	+	0.951006i	$0.600052\pi$
$30$	0	0
$31$	−5.25496	−0.943818	−0.471909	−	0.881647i	$-0.656435\pi$
$31$	−5.25496	−0.943818	−0.471909	+	0.881647i	$0.656435\pi$
$32$	0	0
$33$	3.88059	0.675524
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.56667	1.40835	0.704176	−	0.710025i	$-0.251317\pi$
$37$	8.56667	1.40835	0.704176	+	0.710025i	$0.251317\pi$
$38$	0	0
$39$	0.343277	0.0549682
$40$	0	0
$41$	−1.27815	−0.199613	−0.0998067	−	0.995007i	$-0.531822\pi$
$41$	−1.27815	−0.199613	−0.0998067	+	0.995007i	$0.531822\pi$
$42$	0	0
$43$	−1.42438	−0.217216	−0.108608	−	0.994085i	$-0.534639\pi$
$43$	−1.42438	−0.217216	−0.108608	+	0.994085i	$0.534639\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.375462	0.0547667	0.0273834	−	0.999625i	$-0.491283\pi$
$47$	0.375462	0.0547667	0.0273834	+	0.999625i	$0.491283\pi$
$48$	0	0
$49$	−4.50698	−0.643854
$50$	0	0
$51$	6.07054	0.850045
$52$	0	0
$53$	11.2992	1.55207	0.776033	−	0.630692i	$-0.217229\pi$
$53$	11.2992	1.55207	0.776033	+	0.630692i	$0.217229\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.69189	0.489002
$58$	0	0
$59$	11.5818	1.50783	0.753914	−	0.656973i	$-0.228163\pi$
$59$	11.5818	1.50783	0.753914	+	0.656973i	$0.228163\pi$
$60$	0	0
$61$	−10.9673	−1.40422	−0.702111	−	0.712068i	$-0.747759\pi$
$61$	−10.9673	−1.40422	−0.702111	+	0.712068i	$0.747759\pi$
$62$	0	0
$63$	−1.57893	−0.198927
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.4591	1.27778	0.638892	−	0.769296i	$-0.279393\pi$
$67$	10.4591	1.27778	0.638892	+	0.769296i	$0.279393\pi$
$68$	0	0
$69$	−1.39041	−0.167386
$70$	0	0
$71$	−10.1261	−1.20175	−0.600874	−	0.799344i	$-0.705181\pi$
$71$	−10.1261	−1.20175	−0.600874	+	0.799344i	$0.705181\pi$
$72$	0	0
$73$	−13.1900	−1.54377	−0.771887	−	0.635760i	$-0.780687\pi$
$73$	−13.1900	−1.54377	−0.771887	+	0.635760i	$0.780687\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.12718	−0.698257
$78$	0	0
$79$	13.7995	1.55256	0.776282	−	0.630386i	$-0.217103\pi$
$79$	13.7995	1.55256	0.776282	+	0.630386i	$0.217103\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.62458	0.507613	0.253807	−	0.967255i	$-0.418317\pi$
$83$	4.62458	0.507613	0.253807	+	0.967255i	$0.418317\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−3.32990	−0.357003
$88$	0	0
$89$	−7.26904	−0.770516	−0.385258	−	0.922809i	$-0.625888\pi$
$89$	−7.26904	−0.770516	−0.385258	+	0.922809i	$0.625888\pi$
$90$	0	0
$91$	−0.542010	−0.0568181
$92$	0	0
$93$	−5.25496	−0.544914
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.05557	0.614850	0.307425	−	0.951572i	$-0.400533\pi$
$97$	6.05557	0.614850	0.307425	+	0.951572i	$0.400533\pi$
$98$	0	0
$99$	3.88059	0.390014
$100$	0	0
$101$	3.23036	0.321432	0.160716	−	0.987001i	$-0.448620\pi$
$101$	3.23036	0.321432	0.160716	+	0.987001i	$0.448620\pi$
$102$	0	0
$103$	−7.69725	−0.758433	−0.379216	−	0.925308i	$-0.623806\pi$
$103$	−7.69725	−0.758433	−0.379216	+	0.925308i	$0.623806\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.0376	1.74376	0.871878	−	0.489722i	$-0.162902\pi$
$107$	18.0376	1.74376	0.871878	+	0.489722i	$0.162902\pi$
$108$	0	0
$109$	16.7259	1.60205	0.801027	−	0.598629i	$-0.204288\pi$
$109$	16.7259	1.60205	0.801027	+	0.598629i	$0.204288\pi$
$110$	0	0
$111$	8.56667	0.813113
$112$	0	0
$113$	2.26898	0.213448	0.106724	−	0.994289i	$-0.465964\pi$
$113$	2.26898	0.213448	0.106724	+	0.994289i	$0.465964\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.343277	0.0317359
$118$	0	0
$119$	−9.58496	−0.878652
$120$	0	0
$121$	4.05896	0.368996
$122$	0	0
$123$	−1.27815	−0.115247
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1.50725	−0.133747	−0.0668735	−	0.997761i	$-0.521302\pi$
$127$	−1.50725	−0.133747	−0.0668735	+	0.997761i	$0.521302\pi$
$128$	0	0
$129$	−1.42438	−0.125410
$130$	0	0
$131$	−15.9388	−1.39258	−0.696289	−	0.717761i	$-0.745167\pi$
$131$	−15.9388	−1.39258	−0.696289	+	0.717761i	$0.745167\pi$
$132$	0	0
$133$	−5.82924	−0.505459
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.44654	−0.465329	−0.232665	−	0.972557i	$-0.574744\pi$
$137$	−5.44654	−0.465329	−0.232665	+	0.972557i	$0.574744\pi$
$138$	0	0
$139$	−9.88325	−0.838286	−0.419143	−	0.907920i	$-0.637669\pi$
$139$	−9.88325	−0.838286	−0.419143	+	0.907920i	$0.637669\pi$
$140$	0	0
$141$	0.375462	0.0316196
$142$	0	0
$143$	1.33211	0.111397
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−4.50698	−0.371729
$148$	0	0
$149$	0.0649364	0.00531979	0.00265990	−	0.999996i	$-0.499153\pi$
$149$	0.0649364	0.00531979	0.00265990	+	0.999996i	$0.499153\pi$
$150$	0	0
$151$	−12.1221	−0.986481	−0.493240	−	0.869893i	$-0.664188\pi$
$151$	−12.1221	−0.986481	−0.493240	+	0.869893i	$0.664188\pi$
$152$	0	0
$153$	6.07054	0.490774
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−23.4721	−1.87328	−0.936638	−	0.350300i	$-0.886080\pi$
$157$	−23.4721	−1.87328	−0.936638	+	0.350300i	$0.886080\pi$
$158$	0	0
$159$	11.2992	0.896086
$160$	0	0
$161$	2.19537	0.173019
$162$	0	0
$163$	5.99585	0.469632	0.234816	−	0.972040i	$-0.424551\pi$
$163$	5.99585	0.469632	0.234816	+	0.972040i	$0.424551\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.39850	−0.495130	−0.247565	−	0.968871i	$-0.579630\pi$
$167$	−6.39850	−0.495130	−0.247565	+	0.968871i	$0.579630\pi$
$168$	0	0
$169$	−12.8822	−0.990935
$170$	0	0
$171$	3.69189	0.282326
$172$	0	0
$173$	−5.22138	−0.396974	−0.198487	−	0.980104i	$-0.563603\pi$
$173$	−5.22138	−0.396974	−0.198487	+	0.980104i	$0.563603\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	11.5818	0.870545
$178$	0	0
$179$	23.3040	1.74183	0.870913	−	0.491437i	$-0.163528\pi$
$179$	23.3040	1.74183	0.870913	+	0.491437i	$0.163528\pi$
$180$	0	0
$181$	25.1739	1.87116	0.935582	−	0.353108i	$-0.114875\pi$
$181$	25.1739	1.87116	0.935582	+	0.353108i	$0.114875\pi$
$182$	0	0
$183$	−10.9673	−0.810728
$184$	0	0
$185$	0	0
$186$	0	0
$187$	23.5573	1.72268
$188$	0	0
$189$	−1.57893	−0.114850
$190$	0	0
$191$	8.03152	0.581140	0.290570	−	0.956854i	$-0.406155\pi$
$191$	8.03152	0.581140	0.290570	+	0.956854i	$0.406155\pi$
$192$	0	0
$193$	20.2575	1.45817	0.729084	−	0.684424i	$-0.239946\pi$
$193$	20.2575	1.45817	0.729084	+	0.684424i	$0.239946\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.0158	1.42607	0.713034	−	0.701130i	$-0.247321\pi$
$197$	20.0158	1.42607	0.713034	+	0.701130i	$0.247321\pi$
$198$	0	0
$199$	−22.4180	−1.58917	−0.794585	−	0.607153i	$-0.792311\pi$
$199$	−22.4180	−1.58917	−0.794585	+	0.607153i	$0.792311\pi$
$200$	0	0
$201$	10.4591	0.737729
$202$	0	0
$203$	5.25769	0.369017
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.39041	−0.0966404
$208$	0	0
$209$	14.3267	0.990998
$210$	0	0
$211$	8.66554	0.596560	0.298280	−	0.954478i	$-0.403587\pi$
$211$	8.66554	0.596560	0.298280	+	0.954478i	$0.403587\pi$
$212$	0	0
$213$	−10.1261	−0.693830
$214$	0	0
$215$	0	0
$216$	0	0
$217$	8.29722	0.563252
$218$	0	0
$219$	−13.1900	−0.891298
$220$	0	0
$221$	2.08387	0.140177
$222$	0	0
$223$	12.3840	0.829293	0.414647	−	0.909983i	$-0.363905\pi$
$223$	12.3840	0.829293	0.414647	+	0.909983i	$0.363905\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	18.2228	1.20949	0.604745	−	0.796419i	$-0.293275\pi$
$227$	18.2228	1.20949	0.604745	+	0.796419i	$0.293275\pi$
$228$	0	0
$229$	16.4164	1.08483	0.542413	−	0.840112i	$-0.317511\pi$
$229$	16.4164	1.08483	0.542413	+	0.840112i	$0.317511\pi$
$230$	0	0
$231$	−6.12718	−0.403139
$232$	0	0
$233$	5.13782	0.336590	0.168295	−	0.985737i	$-0.446174\pi$
$233$	5.13782	0.336590	0.168295	+	0.985737i	$0.446174\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	13.7995	0.896373
$238$	0	0
$239$	24.5593	1.58861	0.794304	−	0.607520i	$-0.207835\pi$
$239$	24.5593	1.58861	0.794304	+	0.607520i	$0.207835\pi$
$240$	0	0
$241$	4.83511	0.311457	0.155728	−	0.987800i	$-0.450228\pi$
$241$	4.83511	0.311457	0.155728	+	0.987800i	$0.450228\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.26734	0.0806388
$248$	0	0
$249$	4.62458	0.293071
$250$	0	0
$251$	−15.1395	−0.955594	−0.477797	−	0.878470i	$-0.658565\pi$
$251$	−15.1395	−0.955594	−0.477797	+	0.878470i	$0.658565\pi$
$252$	0	0
$253$	−5.39562	−0.339220
$254$	0	0
$255$	0	0
$256$	0	0
$257$	22.7976	1.42207	0.711036	−	0.703155i	$-0.248226\pi$
$257$	22.7976	1.42207	0.711036	+	0.703155i	$0.248226\pi$
$258$	0	0
$259$	−13.5262	−0.840476
$260$	0	0
$261$	−3.32990	−0.206116
$262$	0	0
$263$	−9.44456	−0.582376	−0.291188	−	0.956666i	$-0.594051\pi$
$263$	−9.44456	−0.582376	−0.291188	+	0.956666i	$0.594051\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−7.26904	−0.444858
$268$	0	0
$269$	−29.3838	−1.79156	−0.895780	−	0.444497i	$-0.853382\pi$
$269$	−29.3838	−1.79156	−0.895780	+	0.444497i	$0.853382\pi$
$270$	0	0
$271$	15.4883	0.940848	0.470424	−	0.882441i	$-0.344101\pi$
$271$	15.4883	0.940848	0.470424	+	0.882441i	$0.344101\pi$
$272$	0	0
$273$	−0.542010	−0.0328039
$274$	0	0
$275$	0	0
$276$	0	0
$277$	15.3786	0.924010	0.462005	−	0.886877i	$-0.347130\pi$
$277$	15.3786	0.924010	0.462005	+	0.886877i	$0.347130\pi$
$278$	0	0
$279$	−5.25496	−0.314606
$280$	0	0
$281$	3.87277	0.231030	0.115515	−	0.993306i	$-0.463148\pi$
$281$	3.87277	0.231030	0.115515	+	0.993306i	$0.463148\pi$
$282$	0	0
$283$	−11.8869	−0.706605	−0.353303	−	0.935509i	$-0.614941\pi$
$283$	−11.8869	−0.706605	−0.353303	+	0.935509i	$0.614941\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.01811	0.119125
$288$	0	0
$289$	19.8514	1.16773
$290$	0	0
$291$	6.05557	0.354984
$292$	0	0
$293$	−15.4596	−0.903161	−0.451581	−	0.892230i	$-0.649140\pi$
$293$	−15.4596	−0.903161	−0.451581	+	0.892230i	$0.649140\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.88059	0.225175
$298$	0	0
$299$	−0.477297	−0.0276028
$300$	0	0
$301$	2.24900	0.129630
$302$	0	0
$303$	3.23036	0.185579
$304$	0	0
$305$	0	0
$306$	0	0
$307$	26.6092	1.51867	0.759334	−	0.650702i	$-0.225525\pi$
$307$	26.6092	1.51867	0.759334	+	0.650702i	$0.225525\pi$
$308$	0	0
$309$	−7.69725	−0.437881
$310$	0	0
$311$	−5.48062	−0.310778	−0.155389	−	0.987853i	$-0.549663\pi$
$311$	−5.48062	−0.310778	−0.155389	+	0.987853i	$0.549663\pi$
$312$	0	0
$313$	7.45099	0.421155	0.210578	−	0.977577i	$-0.432466\pi$
$313$	7.45099	0.421155	0.210578	+	0.977577i	$0.432466\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.96844	−0.166724	−0.0833622	−	0.996519i	$-0.526566\pi$
$317$	−2.96844	−0.166724	−0.0833622	+	0.996519i	$0.526566\pi$
$318$	0	0
$319$	−12.9220	−0.723492
$320$	0	0
$321$	18.0376	1.00676
$322$	0	0
$323$	22.4117	1.24702
$324$	0	0
$325$	0	0
$326$	0	0
$327$	16.7259	0.924946
$328$	0	0
$329$	−0.592828	−0.0326837
$330$	0	0
$331$	27.4282	1.50759	0.753795	−	0.657109i	$-0.228221\pi$
$331$	27.4282	1.50759	0.753795	+	0.657109i	$0.228221\pi$
$332$	0	0
$333$	8.56667	0.469451
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−7.51695	−0.409474	−0.204737	−	0.978817i	$-0.565634\pi$
$337$	−7.51695	−0.409474	−0.204737	+	0.978817i	$0.565634\pi$
$338$	0	0
$339$	2.26898	0.123234
$340$	0	0
$341$	−20.3923	−1.10431
$342$	0	0
$343$	18.1687	0.981019
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11.9946	0.643902	0.321951	−	0.946756i	$-0.395661\pi$
$347$	11.9946	0.643902	0.321951	+	0.946756i	$0.395661\pi$
$348$	0	0
$349$	−3.50169	−0.187441	−0.0937207	−	0.995599i	$-0.529876\pi$
$349$	−3.50169	−0.187441	−0.0937207	+	0.995599i	$0.529876\pi$
$350$	0	0
$351$	0.343277	0.0183227
$352$	0	0
$353$	−24.9521	−1.32807	−0.664033	−	0.747704i	$-0.731156\pi$
$353$	−24.9521	−1.32807	−0.664033	+	0.747704i	$0.731156\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−9.58496	−0.507290
$358$	0	0
$359$	0.802395	0.0423488	0.0211744	−	0.999776i	$-0.493259\pi$
$359$	0.802395	0.0423488	0.0211744	+	0.999776i	$0.493259\pi$
$360$	0	0
$361$	−5.36997	−0.282630
$362$	0	0
$363$	4.05896	0.213040
$364$	0	0
$365$	0	0
$366$	0	0
$367$	15.8049	0.825007	0.412504	−	0.910956i	$-0.364654\pi$
$367$	15.8049	0.825007	0.412504	+	0.910956i	$0.364654\pi$
$368$	0	0
$369$	−1.27815	−0.0665378
$370$	0	0
$371$	−17.8407	−0.926242
$372$	0	0
$373$	3.60958	0.186897	0.0934484	−	0.995624i	$-0.470211\pi$
$373$	3.60958	0.186897	0.0934484	+	0.995624i	$0.470211\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.14308	−0.0588715
$378$	0	0
$379$	27.3130	1.40297	0.701486	−	0.712683i	$-0.252520\pi$
$379$	27.3130	1.40297	0.701486	+	0.712683i	$0.252520\pi$
$380$	0	0
$381$	−1.50725	−0.0772189
$382$	0	0
$383$	−19.2816	−0.985244	−0.492622	−	0.870243i	$-0.663961\pi$
$383$	−19.2816	−0.985244	−0.492622	+	0.870243i	$0.663961\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.42438	−0.0724054
$388$	0	0
$389$	−17.0594	−0.864946	−0.432473	−	0.901647i	$-0.642359\pi$
$389$	−17.0594	−0.864946	−0.432473	+	0.901647i	$0.642359\pi$
$390$	0	0
$391$	−8.44056	−0.426857
$392$	0	0
$393$	−15.9388	−0.804006
$394$	0	0
$395$	0	0
$396$	0	0
$397$	27.3641	1.37336	0.686682	−	0.726958i	$-0.259066\pi$
$397$	27.3641	1.37336	0.686682	+	0.726958i	$0.259066\pi$
$398$	0	0
$399$	−5.82924	−0.291827
$400$	0	0
$401$	−14.7793	−0.738042	−0.369021	−	0.929421i	$-0.620307\pi$
$401$	−14.7793	−0.738042	−0.369021	+	0.929421i	$0.620307\pi$
$402$	0	0
$403$	−1.80390	−0.0898589
$404$	0	0
$405$	0	0
$406$	0	0
$407$	33.2437	1.64783
$408$	0	0
$409$	21.7633	1.07613	0.538063	−	0.842905i	$-0.319156\pi$
$409$	21.7633	1.07613	0.538063	+	0.842905i	$0.319156\pi$
$410$	0	0
$411$	−5.44654	−0.268658
$412$	0	0
$413$	−18.2869	−0.899842
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−9.88325	−0.483985
$418$	0	0
$419$	−4.53705	−0.221649	−0.110825	−	0.993840i	$-0.535349\pi$
$419$	−4.53705	−0.221649	−0.110825	+	0.993840i	$0.535349\pi$
$420$	0	0
$421$	3.05548	0.148915	0.0744574	−	0.997224i	$-0.476278\pi$
$421$	3.05548	0.148915	0.0744574	+	0.997224i	$0.476278\pi$
$422$	0	0
$423$	0.375462	0.0182556
$424$	0	0
$425$	0	0
$426$	0	0
$427$	17.3167	0.838011
$428$	0	0
$429$	1.33211	0.0643151
$430$	0	0
$431$	−18.8971	−0.910241	−0.455120	−	0.890430i	$-0.650404\pi$
$431$	−18.8971	−0.910241	−0.455120	+	0.890430i	$0.650404\pi$
$432$	0	0
$433$	−3.41899	−0.164306	−0.0821532	−	0.996620i	$-0.526180\pi$
$433$	−3.41899	−0.164306	−0.0821532	+	0.996620i	$0.526180\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.13325	−0.245557
$438$	0	0
$439$	−5.95625	−0.284276	−0.142138	−	0.989847i	$-0.545398\pi$
$439$	−5.95625	−0.284276	−0.142138	+	0.989847i	$0.545398\pi$
$440$	0	0
$441$	−4.50698	−0.214618
$442$	0	0
$443$	33.0705	1.57122	0.785612	−	0.618719i	$-0.212348\pi$
$443$	33.0705	1.57122	0.785612	+	0.618719i	$0.212348\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0.0649364	0.00307138
$448$	0	0
$449$	−8.68077	−0.409671	−0.204835	−	0.978796i	$-0.565666\pi$
$449$	−8.68077	−0.409671	−0.204835	+	0.978796i	$0.565666\pi$
$450$	0	0
$451$	−4.95997	−0.233556
$452$	0	0
$453$	−12.1221	−0.569545
$454$	0	0
$455$	0	0
$456$	0	0
$457$	13.3667	0.625269	0.312635	−	0.949873i	$-0.398788\pi$
$457$	13.3667	0.625269	0.312635	+	0.949873i	$0.398788\pi$
$458$	0	0
$459$	6.07054	0.283348
$460$	0	0
$461$	−40.9583	−1.90762	−0.953809	−	0.300413i	$-0.902876\pi$
$461$	−40.9583	−1.90762	−0.953809	+	0.300413i	$0.902876\pi$
$462$	0	0
$463$	23.1280	1.07485	0.537425	−	0.843311i	$-0.319397\pi$
$463$	23.1280	1.07485	0.537425	+	0.843311i	$0.319397\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	30.2890	1.40161	0.700804	−	0.713354i	$-0.252825\pi$
$467$	30.2890	1.40161	0.700804	+	0.713354i	$0.252825\pi$
$468$	0	0
$469$	−16.5142	−0.762556
$470$	0	0
$471$	−23.4721	−1.08154
$472$	0	0
$473$	−5.52744	−0.254152
$474$	0	0
$475$	0	0
$476$	0	0
$477$	11.2992	0.517355
$478$	0	0
$479$	−24.0798	−1.10023	−0.550116	−	0.835088i	$-0.685417\pi$
$479$	−24.0798	−1.10023	−0.550116	+	0.835088i	$0.685417\pi$
$480$	0	0
$481$	2.94074	0.134086
$482$	0	0
$483$	2.19537	0.0998927
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.25482	0.102176	0.0510878	−	0.998694i	$-0.483731\pi$
$487$	2.25482	0.102176	0.0510878	+	0.998694i	$0.483731\pi$
$488$	0	0
$489$	5.99585	0.271142
$490$	0	0
$491$	−28.6098	−1.29114	−0.645572	−	0.763699i	$-0.723381\pi$
$491$	−28.6098	−1.29114	−0.645572	+	0.763699i	$0.723381\pi$
$492$	0	0
$493$	−20.2143	−0.910406
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15.9884	0.717179
$498$	0	0
$499$	−26.9489	−1.20640	−0.603199	−	0.797590i	$-0.706108\pi$
$499$	−26.9489	−1.20640	−0.603199	+	0.797590i	$0.706108\pi$
$500$	0	0
$501$	−6.39850	−0.285864
$502$	0	0
$503$	0.416998	0.0185930	0.00929651	−	0.999957i	$-0.497041\pi$
$503$	0.416998	0.0185930	0.00929651	+	0.999957i	$0.497041\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−12.8822	−0.572117
$508$	0	0
$509$	13.4030	0.594079	0.297039	−	0.954865i	$-0.404001\pi$
$509$	13.4030	0.594079	0.297039	+	0.954865i	$0.404001\pi$
$510$	0	0
$511$	20.8261	0.921293
$512$	0	0
$513$	3.69189	0.163001
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.45701	0.0640793
$518$	0	0
$519$	−5.22138	−0.229193
$520$	0	0
$521$	21.0744	0.923288	0.461644	−	0.887065i	$-0.347260\pi$
$521$	21.0744	0.923288	0.461644	+	0.887065i	$0.347260\pi$
$522$	0	0
$523$	11.8755	0.519281	0.259640	−	0.965705i	$-0.416396\pi$
$523$	11.8755	0.519281	0.259640	+	0.965705i	$0.416396\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−31.9004	−1.38960
$528$	0	0
$529$	−21.0667	−0.915946
$530$	0	0
$531$	11.5818	0.502609
$532$	0	0
$533$	−0.438759	−0.0190047
$534$	0	0
$535$	0	0
$536$	0	0
$537$	23.3040	1.00564
$538$	0	0
$539$	−17.4897	−0.753335
$540$	0	0
$541$	36.5654	1.57207	0.786034	−	0.618183i	$-0.212131\pi$
$541$	36.5654	1.57207	0.786034	+	0.618183i	$0.212131\pi$
$542$	0	0
$543$	25.1739	1.08032
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−39.5812	−1.69237	−0.846185	−	0.532890i	$-0.821106\pi$
$547$	−39.5812	−1.69237	−0.846185	+	0.532890i	$0.821106\pi$
$548$	0	0
$549$	−10.9673	−0.468074
$550$	0	0
$551$	−12.2936	−0.523726
$552$	0	0
$553$	−21.7885	−0.926539
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12.1804	0.516102	0.258051	−	0.966131i	$-0.416920\pi$
$557$	12.1804	0.516102	0.258051	+	0.966131i	$0.416920\pi$
$558$	0	0
$559$	−0.488957	−0.0206807
$560$	0	0
$561$	23.5573	0.994588
$562$	0	0
$563$	36.5886	1.54203	0.771014	−	0.636819i	$-0.219750\pi$
$563$	36.5886	1.54203	0.771014	+	0.636819i	$0.219750\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.57893	−0.0663089
$568$	0	0
$569$	−33.8355	−1.41846	−0.709230	−	0.704978i	$-0.750957\pi$
$569$	−33.8355	−1.41846	−0.709230	+	0.704978i	$0.750957\pi$
$570$	0	0
$571$	−1.20423	−0.0503953	−0.0251976	−	0.999682i	$-0.508022\pi$
$571$	−1.20423	−0.0503953	−0.0251976	+	0.999682i	$0.508022\pi$
$572$	0	0
$573$	8.03152	0.335522
$574$	0	0
$575$	0	0
$576$	0	0
$577$	16.4040	0.682909	0.341455	−	0.939898i	$-0.389080\pi$
$577$	16.4040	0.682909	0.341455	+	0.939898i	$0.389080\pi$
$578$	0	0
$579$	20.2575	0.841874
$580$	0	0
$581$	−7.30189	−0.302933
$582$	0	0
$583$	43.8476	1.81598
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−33.6919	−1.39062	−0.695308	−	0.718712i	$-0.744732\pi$
$587$	−33.6919	−1.39062	−0.695308	+	0.718712i	$0.744732\pi$
$588$	0	0
$589$	−19.4007	−0.799392
$590$	0	0
$591$	20.0158	0.823340
$592$	0	0
$593$	−32.2208	−1.32315	−0.661576	−	0.749878i	$-0.730112\pi$
$593$	−32.2208	−1.32315	−0.661576	+	0.749878i	$0.730112\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−22.4180	−0.917507
$598$	0	0
$599$	−14.7284	−0.601784	−0.300892	−	0.953658i	$-0.597284\pi$
$599$	−14.7284	−0.601784	−0.300892	+	0.953658i	$0.597284\pi$
$600$	0	0
$601$	35.5643	1.45070	0.725348	−	0.688382i	$-0.241679\pi$
$601$	35.5643	1.45070	0.725348	+	0.688382i	$0.241679\pi$
$602$	0	0
$603$	10.4591	0.425928
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−16.0986	−0.653421	−0.326710	−	0.945124i	$-0.605940\pi$
$607$	−16.0986	−0.653421	−0.326710	+	0.945124i	$0.605940\pi$
$608$	0	0
$609$	5.25769	0.213052
$610$	0	0
$611$	0.128887	0.00521422
$612$	0	0
$613$	−43.2663	−1.74751	−0.873754	−	0.486368i	$-0.838321\pi$
$613$	−43.2663	−1.74751	−0.873754	+	0.486368i	$0.838321\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.58485	0.0638036	0.0319018	−	0.999491i	$-0.489844\pi$
$617$	1.58485	0.0638036	0.0319018	+	0.999491i	$0.489844\pi$
$618$	0	0
$619$	−13.4791	−0.541770	−0.270885	−	0.962612i	$-0.587316\pi$
$619$	−13.4791	−0.541770	−0.270885	+	0.962612i	$0.587316\pi$
$620$	0	0
$621$	−1.39041	−0.0557954
$622$	0	0
$623$	11.4773	0.459829
$624$	0	0
$625$	0	0
$626$	0	0
$627$	14.3267	0.572153
$628$	0	0
$629$	52.0043	2.07355
$630$	0	0
$631$	−29.2879	−1.16593	−0.582967	−	0.812496i	$-0.698108\pi$
$631$	−29.2879	−1.16593	−0.582967	+	0.812496i	$0.698108\pi$
$632$	0	0
$633$	8.66554	0.344424
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−1.54714	−0.0612999
$638$	0	0
$639$	−10.1261	−0.400583
$640$	0	0
$641$	−17.2911	−0.682958	−0.341479	−	0.939889i	$-0.610928\pi$
$641$	−17.2911	−0.682958	−0.341479	+	0.939889i	$0.610928\pi$
$642$	0	0
$643$	−46.8857	−1.84899	−0.924497	−	0.381190i	$-0.875514\pi$
$643$	−46.8857	−1.84899	−0.924497	+	0.381190i	$0.875514\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14.3970	0.566005	0.283003	−	0.959119i	$-0.408669\pi$
$647$	14.3970	0.566005	0.283003	+	0.959119i	$0.408669\pi$
$648$	0	0
$649$	44.9444	1.76422
$650$	0	0
$651$	8.29722	0.325194
$652$	0	0
$653$	31.9828	1.25158	0.625791	−	0.779991i	$-0.284776\pi$
$653$	31.9828	1.25158	0.625791	+	0.779991i	$0.284776\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−13.1900	−0.514591
$658$	0	0
$659$	18.6889	0.728014	0.364007	−	0.931396i	$-0.381408\pi$
$659$	18.6889	0.728014	0.364007	+	0.931396i	$0.381408\pi$
$660$	0	0
$661$	−0.679140	−0.0264155	−0.0132078	−	0.999913i	$-0.504204\pi$
$661$	−0.679140	−0.0264155	−0.0132078	+	0.999913i	$0.504204\pi$
$662$	0	0
$663$	2.08387	0.0809309
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.62994	0.179272
$668$	0	0
$669$	12.3840	0.478793
$670$	0	0
$671$	−42.5597	−1.64300
$672$	0	0
$673$	−7.25680	−0.279729	−0.139864	−	0.990171i	$-0.544667\pi$
$673$	−7.25680	−0.279729	−0.139864	+	0.990171i	$0.544667\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−33.9536	−1.30494	−0.652471	−	0.757814i	$-0.726267\pi$
$677$	−33.9536	−1.30494	−0.652471	+	0.757814i	$0.726267\pi$
$678$	0	0
$679$	−9.56133	−0.366930
$680$	0	0
$681$	18.2228	0.698299
$682$	0	0
$683$	28.1579	1.07743	0.538716	−	0.842488i	$-0.318910\pi$
$683$	28.1579	1.07743	0.538716	+	0.842488i	$0.318910\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	16.4164	0.626325
$688$	0	0
$689$	3.87876	0.147769
$690$	0	0
$691$	22.1805	0.843787	0.421893	−	0.906645i	$-0.361366\pi$
$691$	22.1805	0.843787	0.421893	+	0.906645i	$0.361366\pi$
$692$	0	0
$693$	−6.12718	−0.232752
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−7.75905	−0.293895
$698$	0	0
$699$	5.13782	0.194330
$700$	0	0
$701$	16.9652	0.640768	0.320384	−	0.947288i	$-0.396188\pi$
$701$	16.9652	0.640768	0.320384	+	0.947288i	$0.396188\pi$
$702$	0	0
$703$	31.6272	1.19284
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.10051	−0.191824
$708$	0	0
$709$	14.3275	0.538081	0.269041	−	0.963129i	$-0.413293\pi$
$709$	14.3275	0.538081	0.269041	+	0.963129i	$0.413293\pi$
$710$	0	0
$711$	13.7995	0.517521
$712$	0	0
$713$	7.30657	0.273633
$714$	0	0
$715$	0	0
$716$	0	0
$717$	24.5593	0.917184
$718$	0	0
$719$	11.8215	0.440869	0.220434	−	0.975402i	$-0.429253\pi$
$719$	11.8215	0.440869	0.220434	+	0.975402i	$0.429253\pi$
$720$	0	0
$721$	12.1534	0.452618
$722$	0	0
$723$	4.83511	0.179820
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.23634	0.0458534	0.0229267	−	0.999737i	$-0.492702\pi$
$727$	1.23634	0.0458534	0.0229267	+	0.999737i	$0.492702\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−8.64677	−0.319812
$732$	0	0
$733$	−16.5080	−0.609739	−0.304869	−	0.952394i	$-0.598613\pi$
$733$	−16.5080	−0.609739	−0.304869	+	0.952394i	$0.598613\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	40.5875	1.49506
$738$	0	0
$739$	−16.8461	−0.619693	−0.309847	−	0.950787i	$-0.600278\pi$
$739$	−16.8461	−0.619693	−0.309847	+	0.950787i	$0.600278\pi$
$740$	0	0
$741$	1.26734	0.0465568
$742$	0	0
$743$	−11.8940	−0.436347	−0.218174	−	0.975910i	$-0.570010\pi$
$743$	−11.8940	−0.436347	−0.218174	+	0.975910i	$0.570010\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	4.62458	0.169204
$748$	0	0
$749$	−28.4801	−1.04064
$750$	0	0
$751$	0.821377	0.0299725	0.0149862	−	0.999888i	$-0.495230\pi$
$751$	0.821377	0.0299725	0.0149862	+	0.999888i	$0.495230\pi$
$752$	0	0
$753$	−15.1395	−0.551713
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−32.5591	−1.18338	−0.591690	−	0.806166i	$-0.701539\pi$
$757$	−32.5591	−1.18338	−0.591690	+	0.806166i	$0.701539\pi$
$758$	0	0
$759$	−5.39562	−0.195849
$760$	0	0
$761$	−10.8675	−0.393946	−0.196973	−	0.980409i	$-0.563111\pi$
$761$	−10.8675	−0.393946	−0.196973	+	0.980409i	$0.563111\pi$
$762$	0	0
$763$	−26.4091	−0.956073
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.97578	0.143557
$768$	0	0
$769$	−2.38400	−0.0859692	−0.0429846	−	0.999076i	$-0.513687\pi$
$769$	−2.38400	−0.0859692	−0.0429846	+	0.999076i	$0.513687\pi$
$770$	0	0
$771$	22.7976	0.821034
$772$	0	0
$773$	−37.2341	−1.33922	−0.669609	−	0.742714i	$-0.733538\pi$
$773$	−37.2341	−1.33922	−0.669609	+	0.742714i	$0.733538\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−13.5262	−0.485249
$778$	0	0
$779$	−4.71878	−0.169068
$780$	0	0
$781$	−39.2953	−1.40609
$782$	0	0
$783$	−3.32990	−0.119001
$784$	0	0
$785$	0	0
$786$	0	0
$787$	22.8974	0.816206	0.408103	−	0.912936i	$-0.366191\pi$
$787$	22.8974	0.816206	0.408103	+	0.912936i	$0.366191\pi$
$788$	0	0
$789$	−9.44456	−0.336235
$790$	0	0
$791$	−3.58257	−0.127382
$792$	0	0
$793$	−3.76483	−0.133693
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−8.87469	−0.314358	−0.157179	−	0.987570i	$-0.550240\pi$
$797$	−8.87469	−0.314358	−0.157179	+	0.987570i	$0.550240\pi$
$798$	0	0
$799$	2.27925	0.0806342
$800$	0	0
$801$	−7.26904	−0.256839
$802$	0	0
$803$	−51.1850	−1.80628
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−29.3838	−1.03436
$808$	0	0
$809$	−40.7380	−1.43227	−0.716136	−	0.697961i	$-0.754091\pi$
$809$	−40.7380	−1.43227	−0.716136	+	0.697961i	$0.754091\pi$
$810$	0	0
$811$	44.7586	1.57169	0.785844	−	0.618424i	$-0.212229\pi$
$811$	44.7586	1.57169	0.785844	+	0.618424i	$0.212229\pi$
$812$	0	0
$813$	15.4883	0.543199
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−5.25866	−0.183977
$818$	0	0
$819$	−0.542010	−0.0189394
$820$	0	0
$821$	−49.9466	−1.74315	−0.871574	−	0.490264i	$-0.836901\pi$
$821$	−49.9466	−1.74315	−0.871574	+	0.490264i	$0.836901\pi$
$822$	0	0
$823$	−15.9822	−0.557104	−0.278552	−	0.960421i	$-0.589854\pi$
$823$	−15.9822	−0.557104	−0.278552	+	0.960421i	$0.589854\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−45.1514	−1.57007	−0.785034	−	0.619453i	$-0.787354\pi$
$827$	−45.1514	−1.57007	−0.785034	+	0.619453i	$0.787354\pi$
$828$	0	0
$829$	−15.4474	−0.536510	−0.268255	−	0.963348i	$-0.586447\pi$
$829$	−15.4474	−0.536510	−0.268255	+	0.963348i	$0.586447\pi$
$830$	0	0
$831$	15.3786	0.533477
$832$	0	0
$833$	−27.3598	−0.947960
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−5.25496	−0.181638
$838$	0	0
$839$	−23.1609	−0.799603	−0.399802	−	0.916602i	$-0.630921\pi$
$839$	−23.1609	−0.799603	−0.399802	+	0.916602i	$0.630921\pi$
$840$	0	0
$841$	−17.9118	−0.617647
$842$	0	0
$843$	3.87277	0.133385
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−6.40882	−0.220210
$848$	0	0
$849$	−11.8869	−0.407959
$850$	0	0
$851$	−11.9112	−0.408311
$852$	0	0
$853$	−0.817929	−0.0280054	−0.0140027	−	0.999902i	$-0.504457\pi$
$853$	−0.817929	−0.0280054	−0.0140027	+	0.999902i	$0.504457\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	38.5882	1.31815	0.659074	−	0.752078i	$-0.270948\pi$
$857$	38.5882	1.31815	0.659074	+	0.752078i	$0.270948\pi$
$858$	0	0
$859$	40.3774	1.37766	0.688829	−	0.724924i	$-0.258125\pi$
$859$	40.3774	1.37766	0.688829	+	0.724924i	$0.258125\pi$
$860$	0	0
$861$	2.01811	0.0687770
$862$	0	0
$863$	−26.8839	−0.915139	−0.457569	−	0.889174i	$-0.651280\pi$
$863$	−26.8839	−0.915139	−0.457569	+	0.889174i	$0.651280\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	19.8514	0.674190
$868$	0	0
$869$	53.5501	1.81656
$870$	0	0
$871$	3.59037	0.121655
$872$	0	0
$873$	6.05557	0.204950
$874$	0	0
$875$	0	0
$876$	0	0
$877$	22.6946	0.766344	0.383172	−	0.923677i	$-0.374832\pi$
$877$	22.6946	0.766344	0.383172	+	0.923677i	$0.374832\pi$
$878$	0	0
$879$	−15.4596	−0.521440
$880$	0	0
$881$	−30.0485	−1.01236	−0.506180	−	0.862428i	$-0.668943\pi$
$881$	−30.0485	−1.01236	−0.506180	+	0.862428i	$0.668943\pi$
$882$	0	0
$883$	34.8902	1.17415	0.587074	−	0.809533i	$-0.300280\pi$
$883$	34.8902	1.17415	0.587074	+	0.809533i	$0.300280\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−38.8006	−1.30280	−0.651398	−	0.758736i	$-0.725817\pi$
$887$	−38.8006	−1.30280	−0.651398	+	0.758736i	$0.725817\pi$
$888$	0	0
$889$	2.37985	0.0798175
$890$	0	0
$891$	3.88059	0.130005
$892$	0	0
$893$	1.38616	0.0463861
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−0.477297	−0.0159365
$898$	0	0
$899$	17.4985	0.583608
$900$	0	0
$901$	68.5923	2.28514
$902$	0	0
$903$	2.24900	0.0748421
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−10.0886	−0.334988	−0.167494	−	0.985873i	$-0.553567\pi$
$907$	−10.0886	−0.334988	−0.167494	+	0.985873i	$0.553567\pi$
$908$	0	0
$909$	3.23036	0.107144
$910$	0	0
$911$	−28.7150	−0.951372	−0.475686	−	0.879615i	$-0.657800\pi$
$911$	−28.7150	−0.951372	−0.475686	+	0.879615i	$0.657800\pi$
$912$	0	0
$913$	17.9461	0.593928
$914$	0	0
$915$	0	0
$916$	0	0
$917$	25.1663	0.831063
$918$	0	0
$919$	1.47549	0.0486719	0.0243360	−	0.999704i	$-0.492253\pi$
$919$	1.47549	0.0486719	0.0243360	+	0.999704i	$0.492253\pi$
$920$	0	0
$921$	26.6092	0.876803
$922$	0	0
$923$	−3.47606	−0.114416
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−7.69725	−0.252811
$928$	0	0
$929$	−40.8836	−1.34135	−0.670674	−	0.741752i	$-0.733995\pi$
$929$	−40.8836	−1.34135	−0.670674	+	0.741752i	$0.733995\pi$
$930$	0	0
$931$	−16.6392	−0.545329
$932$	0	0
$933$	−5.48062	−0.179428
$934$	0	0
$935$	0	0
$936$	0	0
$937$	32.4829	1.06117	0.530584	−	0.847632i	$-0.321972\pi$
$937$	32.4829	1.06117	0.530584	+	0.847632i	$0.321972\pi$
$938$	0	0
$939$	7.45099	0.243154
$940$	0	0
$941$	−7.64940	−0.249363	−0.124682	−	0.992197i	$-0.539791\pi$
$941$	−7.64940	−0.249363	−0.124682	+	0.992197i	$0.539791\pi$
$942$	0	0
$943$	1.77716	0.0578722
$944$	0	0
$945$	0	0
$946$	0	0
$947$	52.1075	1.69326	0.846632	−	0.532178i	$-0.178626\pi$
$947$	52.1075	1.69326	0.846632	+	0.532178i	$0.178626\pi$
$948$	0	0
$949$	−4.52782	−0.146979
$950$	0	0
$951$	−2.96844	−0.0962584
$952$	0	0
$953$	4.20767	0.136300	0.0681500	−	0.997675i	$-0.478290\pi$
$953$	4.20767	0.136300	0.0681500	+	0.997675i	$0.478290\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−12.9220	−0.417708
$958$	0	0
$959$	8.59971	0.277699
$960$	0	0
$961$	−3.38541	−0.109207
$962$	0	0
$963$	18.0376	0.581252
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−50.5324	−1.62501	−0.812506	−	0.582952i	$-0.801897\pi$
$967$	−50.5324	−1.62501	−0.812506	+	0.582952i	$0.801897\pi$
$968$	0	0
$969$	22.4117	0.719969
$970$	0	0
$971$	15.8566	0.508863	0.254432	−	0.967091i	$-0.418112\pi$
$971$	15.8566	0.508863	0.254432	+	0.967091i	$0.418112\pi$
$972$	0	0
$973$	15.6050	0.500272
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−43.3101	−1.38561	−0.692806	−	0.721124i	$-0.743626\pi$
$977$	−43.3101	−1.38561	−0.692806	+	0.721124i	$0.743626\pi$
$978$	0	0
$979$	−28.2081	−0.901536
$980$	0	0
$981$	16.7259	0.534018
$982$	0	0
$983$	34.9708	1.11540	0.557698	−	0.830044i	$-0.311685\pi$
$983$	34.9708	1.11540	0.557698	+	0.830044i	$0.311685\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−0.592828	−0.0188699
$988$	0	0
$989$	1.98048	0.0629757
$990$	0	0
$991$	−15.4006	−0.489216	−0.244608	−	0.969622i	$-0.578659\pi$
$991$	−15.4006	−0.489216	−0.244608	+	0.969622i	$0.578659\pi$
$992$	0	0
$993$	27.4282	0.870408
$994$	0	0
$995$	0	0
$996$	0	0
$997$	30.7935	0.975239	0.487620	−	0.873056i	$-0.337865\pi$
$997$	30.7935	0.975239	0.487620	+	0.873056i	$0.337865\pi$
$998$	0	0
$999$	8.56667	0.271038

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7500.2.a.n.1.3		12
5.2	odd	4		7500.2.d.g.1249.3		24
5.3	odd	4		7500.2.d.g.1249.22		24
5.4	even	2		7500.2.a.m.1.10		12
25.2	odd	20		300.2.o.a.229.1	yes	24
25.9	even	10		1500.2.m.d.901.5		24
25.11	even	5		1500.2.m.c.601.2		24
25.12	odd	20		1500.2.o.c.349.6		24
25.13	odd	20		300.2.o.a.169.1	✓	24
25.14	even	10		1500.2.m.d.601.5		24
25.16	even	5		1500.2.m.c.901.2		24
25.23	odd	20		1500.2.o.c.649.6		24
75.2	even	20		900.2.w.c.829.6		24
75.38	even	20		900.2.w.c.469.6		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.o.a.169.1	✓	24	25.13	odd	20
300.2.o.a.229.1	yes	24	25.2	odd	20
900.2.w.c.469.6		24	75.38	even	20
900.2.w.c.829.6		24	75.2	even	20
1500.2.m.c.601.2		24	25.11	even	5
1500.2.m.c.901.2		24	25.16	even	5
1500.2.m.d.601.5		24	25.14	even	10
1500.2.m.d.901.5		24	25.9	even	10
1500.2.o.c.349.6		24	25.12	odd	20
1500.2.o.c.649.6		24	25.23	odd	20
7500.2.a.m.1.10		12	5.4	even	2
7500.2.a.n.1.3		12	1.1	even	1	trivial
7500.2.d.g.1249.3		24	5.2	odd	4
7500.2.d.g.1249.22		24	5.3	odd	4

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:	\( N \)	\(=\)	\( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7500.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.57893 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.57893 q^{7} +1.00000 q^{9} +3.88059 q^{11} +0.343277 q^{13} +6.07054 q^{17} +3.69189 q^{19} -1.57893 q^{21} -1.39041 q^{23} +1.00000 q^{27} -3.32990 q^{29} -5.25496 q^{31} +3.88059 q^{33} +8.56667 q^{37} +0.343277 q^{39} -1.27815 q^{41} -1.42438 q^{43} +0.375462 q^{47} -4.50698 q^{49} +6.07054 q^{51} +11.2992 q^{53} +3.69189 q^{57} +11.5818 q^{59} -10.9673 q^{61} -1.57893 q^{63} +10.4591 q^{67} -1.39041 q^{69} -10.1261 q^{71} -13.1900 q^{73} -6.12718 q^{77} +13.7995 q^{79} +1.00000 q^{81} +4.62458 q^{83} -3.32990 q^{87} -7.26904 q^{89} -0.542010 q^{91} -5.25496 q^{93} +6.05557 q^{97} +3.88059 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{3} + 8 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q + 12 * q^3 + 8 * q^7 + 12 * q^9	\( 12 q + 12 q^{3} + 8 q^{7} + 12 q^{9} + 2 q^{11} + 8 q^{17} + 10 q^{19} + 8 q^{21} + 18 q^{23} + 12 q^{27} + 8 q^{29} - 2 q^{31} + 2 q^{33} + 4 q^{37} + 10 q^{41} + 28 q^{43} + 22 q^{47} + 28 q^{49} + 8 q^{51} + 16 q^{53} + 10 q^{57} - 2 q^{59} + 34 q^{61} + 8 q^{63} + 32 q^{67} + 18 q^{69} + 24 q^{73} + 18 q^{77} + 6 q^{79} + 12 q^{81} + 28 q^{83} + 8 q^{87} + 10 q^{89} + 20 q^{91} - 2 q^{93} + 16 q^{97} + 2 q^{99}+O(q^{100}) \) 12 * q + 12 * q^3 + 8 * q^7 + 12 * q^9 + 2 * q^11 + 8 * q^17 + 10 * q^19 + 8 * q^21 + 18 * q^23 + 12 * q^27 + 8 * q^29 - 2 * q^31 + 2 * q^33 + 4 * q^37 + 10 * q^41 + 28 * q^43 + 22 * q^47 + 28 * q^49 + 8 * q^51 + 16 * q^53 + 10 * q^57 - 2 * q^59 + 34 * q^61 + 8 * q^63 + 32 * q^67 + 18 * q^69 + 24 * q^73 + 18 * q^77 + 6 * q^79 + 12 * q^81 + 28 * q^83 + 8 * q^87 + 10 * q^89 + 20 * q^91 - 2 * q^93 + 16 * q^97 + 2 * q^99

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