LMFDB - Embedded newform 4704.2.a.bz.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4704,2,Mod(1,4704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4704, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4704.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4704 = 2^{5} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4704.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:	$37.5616291108$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.2624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 3x^{2} + 2x + 1 $ x^4 - 2x^3 - 3x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.814115$ of defining polynomial
Character	$\chi$	$=$	4704.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.04244 q^{5} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.04244 q^{5} +1.00000 q^{9} -3.93089 q^{11} +4.88845 q^{13} +3.04244 q^{15} -5.34511 q^{17} -2.30266 q^{19} +7.93089 q^{23} +4.25646 q^{25} +1.00000 q^{27} +5.55912 q^{29} -0.645810 q^{31} -3.93089 q^{33} +5.65685 q^{37} +4.88845 q^{39} +10.0886 q^{41} -8.91331 q^{43} +3.04244 q^{45} -6.61065 q^{47} -5.34511 q^{51} +1.25646 q^{53} -11.9595 q^{55} -2.30266 q^{57} +3.04621 q^{59} -2.97334 q^{61} +14.8728 q^{65} +13.5186 q^{67} +7.93089 q^{69} +13.5877 q^{71} +4.67067 q^{73} +4.25646 q^{75} +1.05153 q^{79} +1.00000 q^{81} +8.60533 q^{83} -16.2622 q^{85} +5.55912 q^{87} +4.85983 q^{89} -0.645810 q^{93} -7.00572 q^{95} +18.3275 q^{97} -3.93089 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{5} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^5 + 4 * q^9	$ 4 q + 4 q^{3} + 4 q^{5} + 4 q^{9} + 4 q^{11} + 8 q^{13} + 4 q^{15} + 4 q^{17} + 8 q^{19} + 12 q^{23} + 12 q^{25} + 4 q^{27} - 8 q^{31} + 4 q^{33} + 8 q^{39} + 20 q^{41} - 8 q^{43} + 4 q^{45} - 16 q^{47} + 4 q^{51} - 8 q^{55} + 8 q^{57} + 16 q^{61} - 8 q^{65} - 8 q^{67} + 12 q^{69} + 12 q^{71} + 8 q^{73} + 12 q^{75} + 16 q^{79} + 4 q^{81} - 8 q^{85} + 28 q^{89} - 8 q^{93} + 24 q^{95} + 40 q^{97} + 4 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^5 + 4 * q^9 + 4 * q^11 + 8 * q^13 + 4 * q^15 + 4 * q^17 + 8 * q^19 + 12 * q^23 + 12 * q^25 + 4 * q^27 - 8 * q^31 + 4 * q^33 + 8 * q^39 + 20 * q^41 - 8 * q^43 + 4 * q^45 - 16 * q^47 + 4 * q^51 - 8 * q^55 + 8 * q^57 + 16 * q^61 - 8 * q^65 - 8 * q^67 + 12 * q^69 + 12 * q^71 + 8 * q^73 + 12 * q^75 + 16 * q^79 + 4 * q^81 - 8 * q^85 + 28 * q^89 - 8 * q^93 + 24 * q^95 + 40 * q^97 + 4 * 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
$5$	3.04244	1.36062	0.680311	−	0.732924i	$-0.261845\pi$
$5$	3.04244	1.36062	0.680311	+	0.732924i	$0.261845\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.93089	−1.18521	−0.592604	−	0.805494i	$-0.701900\pi$
$11$	−3.93089	−1.18521	−0.592604	+	0.805494i	$0.701900\pi$
$12$	0	0
$13$	4.88845	1.35581	0.677906	−	0.735149i	$-0.262888\pi$
$13$	4.88845	1.35581	0.677906	+	0.735149i	$0.262888\pi$
$14$	0	0
$15$	3.04244	0.785555
$16$	0	0
$17$	−5.34511	−1.29638	−0.648189	−	0.761479i	$-0.724474\pi$
$17$	−5.34511	−1.29638	−0.648189	+	0.761479i	$0.724474\pi$
$18$	0	0
$19$	−2.30266	−0.528267	−0.264134	−	0.964486i	$-0.585086\pi$
$19$	−2.30266	−0.528267	−0.264134	+	0.964486i	$0.585086\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.93089	1.65371	0.826853	−	0.562418i	$-0.190129\pi$
$23$	7.93089	1.65371	0.826853	+	0.562418i	$0.190129\pi$
$24$	0	0
$25$	4.25646	0.851292
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	5.55912	1.03230	0.516152	−	0.856497i	$-0.327364\pi$
$29$	5.55912	1.03230	0.516152	+	0.856497i	$0.327364\pi$
$30$	0	0
$31$	−0.645810	−0.115991	−0.0579954	−	0.998317i	$-0.518471\pi$
$31$	−0.645810	−0.115991	−0.0579954	+	0.998317i	$0.518471\pi$
$32$	0	0
$33$	−3.93089	−0.684281
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.65685	0.929981	0.464991	−	0.885316i	$-0.346058\pi$
$37$	5.65685	0.929981	0.464991	+	0.885316i	$0.346058\pi$
$38$	0	0
$39$	4.88845	0.782779
$40$	0	0
$41$	10.0886	1.57558	0.787791	−	0.615943i	$-0.211225\pi$
$41$	10.0886	1.57558	0.787791	+	0.615943i	$0.211225\pi$
$42$	0	0
$43$	−8.91331	−1.35927	−0.679634	−	0.733552i	$-0.737861\pi$
$43$	−8.91331	−1.35927	−0.679634	+	0.733552i	$0.737861\pi$
$44$	0	0
$45$	3.04244	0.453541
$46$	0	0
$47$	−6.61065	−0.964262	−0.482131	−	0.876099i	$-0.660137\pi$
$47$	−6.61065	−0.964262	−0.482131	+	0.876099i	$0.660137\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−5.34511	−0.748465
$52$	0	0
$53$	1.25646	0.172588	0.0862939	−	0.996270i	$-0.472498\pi$
$53$	1.25646	0.172588	0.0862939	+	0.996270i	$0.472498\pi$
$54$	0	0
$55$	−11.9595	−1.61262
$56$	0	0
$57$	−2.30266	−0.304995
$58$	0	0
$59$	3.04621	0.396582	0.198291	−	0.980143i	$-0.436461\pi$
$59$	3.04621	0.396582	0.198291	+	0.980143i	$0.436461\pi$
$60$	0	0
$61$	−2.97334	−0.380697	−0.190348	−	0.981717i	$-0.560962\pi$
$61$	−2.97334	−0.380697	−0.190348	+	0.981717i	$0.560962\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	14.8728	1.84475
$66$	0	0
$67$	13.5186	1.65156	0.825782	−	0.563989i	$-0.190734\pi$
$67$	13.5186	1.65156	0.825782	+	0.563989i	$0.190734\pi$
$68$	0	0
$69$	7.93089	0.954767
$70$	0	0
$71$	13.5877	1.61257	0.806284	−	0.591528i	$-0.201475\pi$
$71$	13.5877	1.61257	0.806284	+	0.591528i	$0.201475\pi$
$72$	0	0
$73$	4.67067	0.546661	0.273330	−	0.961920i	$-0.411875\pi$
$73$	4.67067	0.546661	0.273330	+	0.961920i	$0.411875\pi$
$74$	0	0
$75$	4.25646	0.491494
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1.05153	0.118306	0.0591530	−	0.998249i	$-0.481160\pi$
$79$	1.05153	0.118306	0.0591530	+	0.998249i	$0.481160\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.60533	0.944557	0.472279	−	0.881449i	$-0.343432\pi$
$83$	8.60533	0.944557	0.472279	+	0.881449i	$0.343432\pi$
$84$	0	0
$85$	−16.2622	−1.76388
$86$	0	0
$87$	5.55912	0.596001
$88$	0	0
$89$	4.85983	0.515140	0.257570	−	0.966260i	$-0.417078\pi$
$89$	4.85983	0.515140	0.257570	+	0.966260i	$0.417078\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−0.645810	−0.0669673
$94$	0	0
$95$	−7.00572	−0.718772
$96$	0	0
$97$	18.3275	1.86088	0.930439	−	0.366446i	$-0.119426\pi$
$97$	18.3275	1.86088	0.930439	+	0.366446i	$0.119426\pi$
$98$	0	0
$99$	−3.93089	−0.395070
$100$	0	0
$101$	5.87087	0.584173	0.292087	−	0.956392i	$-0.405650\pi$
$101$	5.87087	0.584173	0.292087	+	0.956392i	$0.405650\pi$
$102$	0	0
$103$	15.0778	1.48566	0.742828	−	0.669482i	$-0.233484\pi$
$103$	15.0778	1.48566	0.742828	+	0.669482i	$0.233484\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.238878	−0.0230932	−0.0115466	−	0.999933i	$-0.503675\pi$
$107$	−0.238878	−0.0230932	−0.0115466	+	0.999933i	$0.503675\pi$
$108$	0	0
$109$	−12.9133	−1.23687	−0.618436	−	0.785836i	$-0.712233\pi$
$109$	−12.9133	−1.23687	−0.618436	+	0.785836i	$0.712233\pi$
$110$	0	0
$111$	5.65685	0.536925
$112$	0	0
$113$	−13.3137	−1.25245	−0.626224	−	0.779643i	$-0.715401\pi$
$113$	−13.3137	−1.25245	−0.626224	+	0.779643i	$0.715401\pi$
$114$	0	0
$115$	24.1293	2.25007
$116$	0	0
$117$	4.88845	0.451937
$118$	0	0
$119$	0	0
$120$	0	0
$121$	4.45192	0.404720
$122$	0	0
$123$	10.0886	0.909663
$124$	0	0
$125$	−2.26218	−0.202336
$126$	0	0
$127$	−3.69202	−0.327613	−0.163807	−	0.986492i	$-0.552377\pi$
$127$	−3.69202	−0.327613	−0.163807	+	0.986492i	$0.552377\pi$
$128$	0	0
$129$	−8.91331	−0.784773
$130$	0	0
$131$	−18.4320	−1.61041	−0.805204	−	0.592998i	$-0.797944\pi$
$131$	−18.4320	−1.61041	−0.805204	+	0.592998i	$0.797944\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	3.04244	0.261852
$136$	0	0
$137$	−4.26750	−0.364597	−0.182299	−	0.983243i	$-0.558354\pi$
$137$	−4.26750	−0.364597	−0.182299	+	0.983243i	$0.558354\pi$
$138$	0	0
$139$	−7.31371	−0.620341	−0.310170	−	0.950681i	$-0.600386\pi$
$139$	−7.31371	−0.620341	−0.310170	+	0.950681i	$0.600386\pi$
$140$	0	0
$141$	−6.61065	−0.556717
$142$	0	0
$143$	−19.2160	−1.60692
$144$	0	0
$145$	16.9133	1.40457
$146$	0	0
$147$	0	0
$148$	0	0
$149$	14.0572	1.15161	0.575807	−	0.817585i	$-0.304688\pi$
$149$	14.0572	1.15161	0.575807	+	0.817585i	$0.304688\pi$
$150$	0	0
$151$	−22.5702	−1.83673	−0.918367	−	0.395730i	$-0.870492\pi$
$151$	−22.5702	−1.83673	−0.918367	+	0.395730i	$0.870492\pi$
$152$	0	0
$153$	−5.34511	−0.432126
$154$	0	0
$155$	−1.96484	−0.157820
$156$	0	0
$157$	−12.3147	−0.982818	−0.491409	−	0.870929i	$-0.663518\pi$
$157$	−12.3147	−0.982818	−0.491409	+	0.870929i	$0.663518\pi$
$158$	0	0
$159$	1.25646	0.0996436
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−3.45192	−0.270375	−0.135188	−	0.990820i	$-0.543164\pi$
$163$	−3.45192	−0.270375	−0.135188	+	0.990820i	$0.543164\pi$
$164$	0	0
$165$	−11.9595	−0.931047
$166$	0	0
$167$	−12.5076	−0.967867	−0.483933	−	0.875105i	$-0.660792\pi$
$167$	−12.5076	−0.967867	−0.483933	+	0.875105i	$0.660792\pi$
$168$	0	0
$169$	10.8969	0.838227
$170$	0	0
$171$	−2.30266	−0.176089
$172$	0	0
$173$	19.0997	1.45212	0.726061	−	0.687630i	$-0.241349\pi$
$173$	19.0997	1.45212	0.726061	+	0.687630i	$0.241349\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.04621	0.228967
$178$	0	0
$179$	11.6908	0.873811	0.436906	−	0.899507i	$-0.356074\pi$
$179$	11.6908	0.873811	0.436906	+	0.899507i	$0.356074\pi$
$180$	0	0
$181$	−11.3204	−0.841439	−0.420720	−	0.907191i	$-0.638222\pi$
$181$	−11.3204	−0.841439	−0.420720	+	0.907191i	$0.638222\pi$
$182$	0	0
$183$	−2.97334	−0.219795
$184$	0	0
$185$	17.2107	1.26535
$186$	0	0
$187$	21.0110	1.53648
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.9015	1.22295	0.611473	−	0.791265i	$-0.290577\pi$
$191$	16.9015	1.22295	0.611473	+	0.791265i	$0.290577\pi$
$192$	0	0
$193$	−23.6884	−1.70513	−0.852565	−	0.522622i	$-0.824954\pi$
$193$	−23.6884	−1.70513	−0.852565	+	0.522622i	$0.824954\pi$
$194$	0	0
$195$	14.8728	1.06507
$196$	0	0
$197$	−11.0831	−0.789637	−0.394819	−	0.918759i	$-0.629193\pi$
$197$	−11.0831	−0.789637	−0.394819	+	0.918759i	$0.629193\pi$
$198$	0	0
$199$	−6.51292	−0.461688	−0.230844	−	0.972991i	$-0.574149\pi$
$199$	−6.51292	−0.461688	−0.230844	+	0.972991i	$0.574149\pi$
$200$	0	0
$201$	13.5186	0.953531
$202$	0	0
$203$	0	0
$204$	0	0
$205$	30.6941	2.14377
$206$	0	0
$207$	7.93089	0.551235
$208$	0	0
$209$	9.05153	0.626107
$210$	0	0
$211$	−3.83023	−0.263684	−0.131842	−	0.991271i	$-0.542089\pi$
$211$	−3.83023	−0.263684	−0.131842	+	0.991271i	$0.542089\pi$
$212$	0	0
$213$	13.5877	0.931017
$214$	0	0
$215$	−27.1182	−1.84945
$216$	0	0
$217$	0	0
$218$	0	0
$219$	4.67067	0.315615
$220$	0	0
$221$	−26.1293	−1.75765
$222$	0	0
$223$	12.4099	0.831026	0.415513	−	0.909587i	$-0.363602\pi$
$223$	12.4099	0.831026	0.415513	+	0.909587i	$0.363602\pi$
$224$	0	0
$225$	4.25646	0.283764
$226$	0	0
$227$	−17.4782	−1.16007	−0.580033	−	0.814593i	$-0.696960\pi$
$227$	−17.4782	−1.16007	−0.580033	+	0.814593i	$0.696960\pi$
$228$	0	0
$229$	21.1783	1.39950	0.699750	−	0.714388i	$-0.253295\pi$
$229$	21.1783	1.39950	0.699750	+	0.714388i	$0.253295\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5.75459	0.376995	0.188498	−	0.982074i	$-0.439638\pi$
$233$	5.75459	0.376995	0.188498	+	0.982074i	$0.439638\pi$
$234$	0	0
$235$	−20.1125	−1.31200
$236$	0	0
$237$	1.05153	0.0683040
$238$	0	0
$239$	7.93089	0.513007	0.256503	−	0.966543i	$-0.417430\pi$
$239$	7.93089	0.513007	0.256503	+	0.966543i	$0.417430\pi$
$240$	0	0
$241$	20.9605	1.35018	0.675092	−	0.737734i	$-0.264104\pi$
$241$	20.9605	1.35018	0.675092	+	0.737734i	$0.264104\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−11.2565	−0.716231
$248$	0	0
$249$	8.60533	0.545340
$250$	0	0
$251$	−7.45607	−0.470623	−0.235311	−	0.971920i	$-0.575611\pi$
$251$	−7.45607	−0.470623	−0.235311	+	0.971920i	$0.575611\pi$
$252$	0	0
$253$	−31.1755	−1.95999
$254$	0	0
$255$	−16.2622	−1.01838
$256$	0	0
$257$	−12.3433	−0.769954	−0.384977	−	0.922926i	$-0.625791\pi$
$257$	−12.3433	−0.769954	−0.384977	+	0.922926i	$0.625791\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	5.55912	0.344101
$262$	0	0
$263$	−23.4144	−1.44379	−0.721896	−	0.692002i	$-0.756729\pi$
$263$	−23.4144	−1.44379	−0.721896	+	0.692002i	$0.756729\pi$
$264$	0	0
$265$	3.82270	0.234827
$266$	0	0
$267$	4.85983	0.297416
$268$	0	0
$269$	5.81362	0.354463	0.177231	−	0.984169i	$-0.443286\pi$
$269$	5.81362	0.354463	0.177231	+	0.984169i	$0.443286\pi$
$270$	0	0
$271$	−24.3694	−1.48033	−0.740167	−	0.672423i	$-0.765254\pi$
$271$	−24.3694	−1.48033	−0.740167	+	0.672423i	$0.765254\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−16.7317	−1.00896
$276$	0	0
$277$	15.7587	0.946851	0.473425	−	0.880834i	$-0.343017\pi$
$277$	15.7587	0.946851	0.473425	+	0.880834i	$0.343017\pi$
$278$	0	0
$279$	−0.645810	−0.0386636
$280$	0	0
$281$	−30.7698	−1.83557	−0.917786	−	0.397076i	$-0.870025\pi$
$281$	−30.7698	−1.83557	−0.917786	+	0.397076i	$0.870025\pi$
$282$	0	0
$283$	22.9080	1.36174	0.680869	−	0.732405i	$-0.261602\pi$
$283$	22.9080	1.36174	0.680869	+	0.732405i	$0.261602\pi$
$284$	0	0
$285$	−7.00572	−0.414983
$286$	0	0
$287$	0	0
$288$	0	0
$289$	11.5702	0.680598
$290$	0	0
$291$	18.3275	1.07438
$292$	0	0
$293$	19.3323	1.12940	0.564701	−	0.825295i	$-0.308991\pi$
$293$	19.3323	1.12940	0.564701	+	0.825295i	$0.308991\pi$
$294$	0	0
$295$	9.26791	0.539598
$296$	0	0
$297$	−3.93089	−0.228094
$298$	0	0
$299$	38.7698	2.24211
$300$	0	0
$301$	0	0
$302$	0	0
$303$	5.87087	0.337273
$304$	0	0
$305$	−9.04621	−0.517984
$306$	0	0
$307$	−26.8377	−1.53171	−0.765853	−	0.643015i	$-0.777683\pi$
$307$	−26.8377	−1.53171	−0.765853	+	0.643015i	$0.777683\pi$
$308$	0	0
$309$	15.0778	0.857744
$310$	0	0
$311$	−3.49240	−0.198036	−0.0990180	−	0.995086i	$-0.531570\pi$
$311$	−3.49240	−0.198036	−0.0990180	+	0.995086i	$0.531570\pi$
$312$	0	0
$313$	−2.75556	−0.155753	−0.0778767	−	0.996963i	$-0.524814\pi$
$313$	−2.75556	−0.155753	−0.0778767	+	0.996963i	$0.524814\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7.15341	−0.401775	−0.200888	−	0.979614i	$-0.564383\pi$
$317$	−7.15341	−0.401775	−0.200888	+	0.979614i	$0.564383\pi$
$318$	0	0
$319$	−21.8523	−1.22349
$320$	0	0
$321$	−0.238878	−0.0133329
$322$	0	0
$323$	12.3080	0.684835
$324$	0	0
$325$	20.8075	1.15419
$326$	0	0
$327$	−12.9133	−0.714108
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−31.7236	−1.74369	−0.871843	−	0.489786i	$-0.837075\pi$
$331$	−31.7236	−1.74369	−0.871843	+	0.489786i	$0.837075\pi$
$332$	0	0
$333$	5.65685	0.309994
$334$	0	0
$335$	41.1297	2.24716
$336$	0	0
$337$	12.6053	0.686656	0.343328	−	0.939216i	$-0.388446\pi$
$337$	12.6053	0.686656	0.343328	+	0.939216i	$0.388446\pi$
$338$	0	0
$339$	−13.3137	−0.723101
$340$	0	0
$341$	2.53861	0.137473
$342$	0	0
$343$	0	0
$344$	0	0
$345$	24.1293	1.29908
$346$	0	0
$347$	31.1064	1.66988	0.834939	−	0.550342i	$-0.185503\pi$
$347$	31.1064	1.66988	0.834939	+	0.550342i	$0.185503\pi$
$348$	0	0
$349$	15.7741	0.844370	0.422185	−	0.906510i	$-0.361263\pi$
$349$	15.7741	0.844370	0.422185	+	0.906510i	$0.361263\pi$
$350$	0	0
$351$	4.88845	0.260926
$352$	0	0
$353$	−18.3785	−0.978187	−0.489094	−	0.872231i	$-0.662672\pi$
$353$	−18.3785	−0.978187	−0.489094	+	0.872231i	$0.662672\pi$
$354$	0	0
$355$	41.3399	2.19410
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−17.7575	−0.937206	−0.468603	−	0.883409i	$-0.655243\pi$
$359$	−17.7575	−0.937206	−0.468603	+	0.883409i	$0.655243\pi$
$360$	0	0
$361$	−13.6977	−0.720934
$362$	0	0
$363$	4.45192	0.233665
$364$	0	0
$365$	14.2103	0.743799
$366$	0	0
$367$	20.9449	1.09331	0.546657	−	0.837357i	$-0.315900\pi$
$367$	20.9449	1.09331	0.546657	+	0.837357i	$0.315900\pi$
$368$	0	0
$369$	10.0886	0.525194
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−20.2365	−1.04781	−0.523903	−	0.851778i	$-0.675525\pi$
$373$	−20.2365	−1.04781	−0.523903	+	0.851778i	$0.675525\pi$
$374$	0	0
$375$	−2.26218	−0.116819
$376$	0	0
$377$	27.1755	1.39961
$378$	0	0
$379$	−36.5244	−1.87613	−0.938065	−	0.346459i	$-0.887384\pi$
$379$	−36.5244	−1.87613	−0.938065	+	0.346459i	$0.887384\pi$
$380$	0	0
$381$	−3.69202	−0.189148
$382$	0	0
$383$	−2.97417	−0.151973	−0.0759864	−	0.997109i	$-0.524211\pi$
$383$	−2.97417	−0.151973	−0.0759864	+	0.997109i	$0.524211\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−8.91331	−0.453089
$388$	0	0
$389$	−2.16445	−0.109742	−0.0548710	−	0.998493i	$-0.517475\pi$
$389$	−2.16445	−0.109742	−0.0548710	+	0.998493i	$0.517475\pi$
$390$	0	0
$391$	−42.3915	−2.14383
$392$	0	0
$393$	−18.4320	−0.929769
$394$	0	0
$395$	3.19921	0.160970
$396$	0	0
$397$	−12.0249	−0.603511	−0.301755	−	0.953385i	$-0.597573\pi$
$397$	−12.0249	−0.603511	−0.301755	+	0.953385i	$0.597573\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−21.5591	−1.07661	−0.538306	−	0.842750i	$-0.680935\pi$
$401$	−21.5591	−1.07661	−0.538306	+	0.842750i	$0.680935\pi$
$402$	0	0
$403$	−3.15701	−0.157262
$404$	0	0
$405$	3.04244	0.151180
$406$	0	0
$407$	−22.2365	−1.10222
$408$	0	0
$409$	−24.5143	−1.21215	−0.606077	−	0.795406i	$-0.707258\pi$
$409$	−24.5143	−1.21215	−0.606077	+	0.795406i	$0.707258\pi$
$410$	0	0
$411$	−4.26750	−0.210500
$412$	0	0
$413$	0	0
$414$	0	0
$415$	26.1812	1.28519
$416$	0	0
$417$	−7.31371	−0.358154
$418$	0	0
$419$	35.3048	1.72475	0.862376	−	0.506269i	$-0.168976\pi$
$419$	35.3048	1.72475	0.862376	+	0.506269i	$0.168976\pi$
$420$	0	0
$421$	2.47776	0.120758	0.0603792	−	0.998176i	$-0.480769\pi$
$421$	2.47776	0.120758	0.0603792	+	0.998176i	$0.480769\pi$
$422$	0	0
$423$	−6.61065	−0.321421
$424$	0	0
$425$	−22.7512	−1.10360
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−19.2160	−0.927756
$430$	0	0
$431$	20.3408	0.979780	0.489890	−	0.871784i	$-0.337037\pi$
$431$	20.3408	0.979780	0.489890	+	0.871784i	$0.337037\pi$
$432$	0	0
$433$	15.7964	0.759129	0.379564	−	0.925165i	$-0.376074\pi$
$433$	15.7964	0.759129	0.379564	+	0.925165i	$0.376074\pi$
$434$	0	0
$435$	16.9133	0.810931
$436$	0	0
$437$	−18.2622	−0.873599
$438$	0	0
$439$	−12.6863	−0.605484	−0.302742	−	0.953073i	$-0.597902\pi$
$439$	−12.6863	−0.605484	−0.302742	+	0.953073i	$0.597902\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	25.9273	1.23184	0.615921	−	0.787808i	$-0.288784\pi$
$443$	25.9273	1.23184	0.615921	+	0.787808i	$0.288784\pi$
$444$	0	0
$445$	14.7857	0.700911
$446$	0	0
$447$	14.0572	0.664885
$448$	0	0
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	−39.6574	−1.86739
$452$	0	0
$453$	−22.5702	−1.06044
$454$	0	0
$455$	0	0
$456$	0	0
$457$	29.7236	1.39041	0.695205	−	0.718811i	$-0.255314\pi$
$457$	29.7236	1.39041	0.695205	+	0.718811i	$0.255314\pi$
$458$	0	0
$459$	−5.34511	−0.249488
$460$	0	0
$461$	12.6314	0.588303	0.294152	−	0.955759i	$-0.404963\pi$
$461$	12.6314	0.588303	0.294152	+	0.955759i	$0.404963\pi$
$462$	0	0
$463$	−33.6884	−1.56563	−0.782817	−	0.622252i	$-0.786218\pi$
$463$	−33.6884	−1.56563	−0.782817	+	0.622252i	$0.786218\pi$
$464$	0	0
$465$	−1.96484	−0.0911172
$466$	0	0
$467$	34.0941	1.57769	0.788844	−	0.614593i	$-0.210680\pi$
$467$	34.0941	1.57769	0.788844	+	0.614593i	$0.210680\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−12.3147	−0.567431
$472$	0	0
$473$	35.0373	1.61102
$474$	0	0
$475$	−9.80119	−0.449710
$476$	0	0
$477$	1.25646	0.0575293
$478$	0	0
$479$	−0.0977317	−0.00446547	−0.00223274	−	0.999998i	$-0.500711\pi$
$479$	−0.0977317	−0.00446547	−0.00223274	+	0.999998i	$0.500711\pi$
$480$	0	0
$481$	27.6533	1.26088
$482$	0	0
$483$	0	0
$484$	0	0
$485$	55.7605	2.53195
$486$	0	0
$487$	−38.5702	−1.74778	−0.873891	−	0.486123i	$-0.838411\pi$
$487$	−38.5702	−1.74778	−0.873891	+	0.486123i	$0.838411\pi$
$488$	0	0
$489$	−3.45192	−0.156101
$490$	0	0
$491$	−19.0748	−0.860835	−0.430418	−	0.902630i	$-0.641634\pi$
$491$	−19.0748	−0.860835	−0.430418	+	0.902630i	$0.641634\pi$
$492$	0	0
$493$	−29.7141	−1.33826
$494$	0	0
$495$	−11.9595	−0.537540
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−8.05725	−0.360692	−0.180346	−	0.983603i	$-0.557722\pi$
$499$	−8.05725	−0.360692	−0.180346	+	0.983603i	$0.557722\pi$
$500$	0	0
$501$	−12.5076	−0.558798
$502$	0	0
$503$	4.94487	0.220481	0.110240	−	0.993905i	$-0.464838\pi$
$503$	4.94487	0.220481	0.110240	+	0.993905i	$0.464838\pi$
$504$	0	0
$505$	17.8618	0.794839
$506$	0	0
$507$	10.8969	0.483950
$508$	0	0
$509$	37.2161	1.64958	0.824788	−	0.565442i	$-0.191294\pi$
$509$	37.2161	1.64958	0.824788	+	0.565442i	$0.191294\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2.30266	−0.101665
$514$	0	0
$515$	45.8732	2.02142
$516$	0	0
$517$	25.9858	1.14285
$518$	0	0
$519$	19.0997	0.838383
$520$	0	0
$521$	13.1050	0.574141	0.287071	−	0.957909i	$-0.407319\pi$
$521$	13.1050	0.574141	0.287071	+	0.957909i	$0.407319\pi$
$522$	0	0
$523$	19.5795	0.856151	0.428076	−	0.903743i	$-0.359192\pi$
$523$	19.5795	0.856151	0.428076	+	0.903743i	$0.359192\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.45192	0.150368
$528$	0	0
$529$	39.8991	1.73474
$530$	0	0
$531$	3.04621	0.132194
$532$	0	0
$533$	49.3179	2.13619
$534$	0	0
$535$	−0.726773	−0.0314211
$536$	0	0
$537$	11.6908	0.504495
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−11.9648	−0.514409	−0.257204	−	0.966357i	$-0.582801\pi$
$541$	−11.9648	−0.514409	−0.257204	+	0.966357i	$0.582801\pi$
$542$	0	0
$543$	−11.3204	−0.485805
$544$	0	0
$545$	−39.2880	−1.68291
$546$	0	0
$547$	39.8827	1.70526	0.852631	−	0.522514i	$-0.175006\pi$
$547$	39.8827	1.70526	0.852631	+	0.522514i	$0.175006\pi$
$548$	0	0
$549$	−2.97334	−0.126899
$550$	0	0
$551$	−12.8008	−0.545332
$552$	0	0
$553$	0	0
$554$	0	0
$555$	17.2107	0.730552
$556$	0	0
$557$	28.9800	1.22792	0.613962	−	0.789336i	$-0.289575\pi$
$557$	28.9800	1.22792	0.613962	+	0.789336i	$0.289575\pi$
$558$	0	0
$559$	−43.5723	−1.84291
$560$	0	0
$561$	21.0110	0.887087
$562$	0	0
$563$	27.3857	1.15417	0.577086	−	0.816683i	$-0.304190\pi$
$563$	27.3857	1.15417	0.577086	+	0.816683i	$0.304190\pi$
$564$	0	0
$565$	−40.5062	−1.70411
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6.37056	−0.267068	−0.133534	−	0.991044i	$-0.542632\pi$
$569$	−6.37056	−0.267068	−0.133534	+	0.991044i	$0.542632\pi$
$570$	0	0
$571$	−14.9485	−0.625574	−0.312787	−	0.949823i	$-0.601263\pi$
$571$	−14.9485	−0.625574	−0.312787	+	0.949823i	$0.601263\pi$
$572$	0	0
$573$	16.9015	0.706068
$574$	0	0
$575$	33.7575	1.40779
$576$	0	0
$577$	−6.21500	−0.258734	−0.129367	−	0.991597i	$-0.541295\pi$
$577$	−6.21500	−0.258734	−0.129367	+	0.991597i	$0.541295\pi$
$578$	0	0
$579$	−23.6884	−0.984457
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−4.93900	−0.204553
$584$	0	0
$585$	14.8728	0.614916
$586$	0	0
$587$	27.0462	1.11632	0.558158	−	0.829735i	$-0.311508\pi$
$587$	27.0462	1.11632	0.558158	+	0.829735i	$0.311508\pi$
$588$	0	0
$589$	1.48708	0.0612742
$590$	0	0
$591$	−11.0831	−0.455897
$592$	0	0
$593$	−15.8837	−0.652266	−0.326133	−	0.945324i	$-0.605746\pi$
$593$	−15.8837	−0.652266	−0.326133	+	0.945324i	$0.605746\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6.51292	−0.266556
$598$	0	0
$599$	5.24820	0.214436	0.107218	−	0.994236i	$-0.465806\pi$
$599$	5.24820	0.214436	0.107218	+	0.994236i	$0.465806\pi$
$600$	0	0
$601$	−33.3830	−1.36172	−0.680860	−	0.732414i	$-0.738394\pi$
$601$	−33.3830	−1.36172	−0.680860	+	0.732414i	$0.738394\pi$
$602$	0	0
$603$	13.5186	0.550522
$604$	0	0
$605$	13.5447	0.550671
$606$	0	0
$607$	−18.9228	−0.768052	−0.384026	−	0.923322i	$-0.625463\pi$
$607$	−18.9228	−0.768052	−0.384026	+	0.923322i	$0.625463\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−32.3158	−1.30736
$612$	0	0
$613$	12.4928	0.504580	0.252290	−	0.967652i	$-0.418816\pi$
$613$	12.4928	0.504580	0.252290	+	0.967652i	$0.418816\pi$
$614$	0	0
$615$	30.6941	1.23771
$616$	0	0
$617$	34.4230	1.38582	0.692910	−	0.721025i	$-0.256329\pi$
$617$	34.4230	1.38582	0.692910	+	0.721025i	$0.256329\pi$
$618$	0	0
$619$	22.6381	0.909900	0.454950	−	0.890517i	$-0.349657\pi$
$619$	22.6381	0.909900	0.454950	+	0.890517i	$0.349657\pi$
$620$	0	0
$621$	7.93089	0.318256
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−28.1649	−1.12659
$626$	0	0
$627$	9.05153	0.361483
$628$	0	0
$629$	−30.2365	−1.20561
$630$	0	0
$631$	−25.3489	−1.00912	−0.504561	−	0.863376i	$-0.668346\pi$
$631$	−25.3489	−1.00912	−0.504561	+	0.863376i	$0.668346\pi$
$632$	0	0
$633$	−3.83023	−0.152238
$634$	0	0
$635$	−11.2327	−0.445758
$636$	0	0
$637$	0	0
$638$	0	0
$639$	13.5877	0.537523
$640$	0	0
$641$	−13.2196	−0.522142	−0.261071	−	0.965320i	$-0.584076\pi$
$641$	−13.2196	−0.522142	−0.261071	+	0.965320i	$0.584076\pi$
$642$	0	0
$643$	−23.7194	−0.935403	−0.467701	−	0.883887i	$-0.654918\pi$
$643$	−23.7194	−0.935403	−0.467701	+	0.883887i	$0.654918\pi$
$644$	0	0
$645$	−27.1182	−1.06778
$646$	0	0
$647$	−15.8213	−0.622000	−0.311000	−	0.950410i	$-0.600664\pi$
$647$	−15.8213	−0.622000	−0.311000	+	0.950410i	$0.600664\pi$
$648$	0	0
$649$	−11.9743	−0.470033
$650$	0	0
$651$	0	0
$652$	0	0
$653$	19.4561	0.761375	0.380687	−	0.924704i	$-0.375687\pi$
$653$	19.4561	0.761375	0.380687	+	0.924704i	$0.375687\pi$
$654$	0	0
$655$	−56.0782	−2.19116
$656$	0	0
$657$	4.67067	0.182220
$658$	0	0
$659$	10.5820	0.412217	0.206109	−	0.978529i	$-0.433920\pi$
$659$	10.5820	0.412217	0.206109	+	0.978529i	$0.433920\pi$
$660$	0	0
$661$	9.95814	0.387327	0.193663	−	0.981068i	$-0.437963\pi$
$661$	9.95814	0.387327	0.193663	+	0.981068i	$0.437963\pi$
$662$	0	0
$663$	−26.1293	−1.01478
$664$	0	0
$665$	0	0
$666$	0	0
$667$	44.0888	1.70713
$668$	0	0
$669$	12.4099	0.479793
$670$	0	0
$671$	11.6879	0.451205
$672$	0	0
$673$	−21.0724	−0.812283	−0.406141	−	0.913810i	$-0.633126\pi$
$673$	−21.0724	−0.812283	−0.406141	+	0.913810i	$0.633126\pi$
$674$	0	0
$675$	4.25646	0.163831
$676$	0	0
$677$	−4.66021	−0.179107	−0.0895533	−	0.995982i	$-0.528544\pi$
$677$	−4.66021	−0.179107	−0.0895533	+	0.995982i	$0.528544\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−17.4782	−0.669765
$682$	0	0
$683$	9.24820	0.353873	0.176936	−	0.984222i	$-0.443381\pi$
$683$	9.24820	0.353873	0.176936	+	0.984222i	$0.443381\pi$
$684$	0	0
$685$	−12.9836	−0.496079
$686$	0	0
$687$	21.1783	0.808001
$688$	0	0
$689$	6.14214	0.233997
$690$	0	0
$691$	−10.0924	−0.383933	−0.191967	−	0.981401i	$-0.561487\pi$
$691$	−10.0924	−0.383933	−0.191967	+	0.981401i	$0.561487\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−22.2515	−0.844049
$696$	0	0
$697$	−53.9249	−2.04255
$698$	0	0
$699$	5.75459	0.217658
$700$	0	0
$701$	17.0683	0.644661	0.322330	−	0.946627i	$-0.395534\pi$
$701$	17.0683	0.644661	0.322330	+	0.946627i	$0.395534\pi$
$702$	0	0
$703$	−13.0258	−0.491279
$704$	0	0
$705$	−20.1125	−0.757481
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−27.2012	−1.02156	−0.510781	−	0.859711i	$-0.670644\pi$
$709$	−27.2012	−1.02156	−0.510781	+	0.859711i	$0.670644\pi$
$710$	0	0
$711$	1.05153	0.0394353
$712$	0	0
$713$	−5.12185	−0.191815
$714$	0	0
$715$	−58.4635	−2.18641
$716$	0	0
$717$	7.93089	0.296185
$718$	0	0
$719$	24.5350	0.915001	0.457501	−	0.889209i	$-0.348745\pi$
$719$	24.5350	0.915001	0.457501	+	0.889209i	$0.348745\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	20.9605	0.779529
$724$	0	0
$725$	23.6622	0.878791
$726$	0	0
$727$	−40.7603	−1.51172	−0.755858	−	0.654735i	$-0.772780\pi$
$727$	−40.7603	−1.51172	−0.755858	+	0.654735i	$0.772780\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	47.6426	1.76213
$732$	0	0
$733$	−28.1318	−1.03907	−0.519537	−	0.854448i	$-0.673895\pi$
$733$	−28.1318	−1.03907	−0.519537	+	0.854448i	$0.673895\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−53.1403	−1.95745
$738$	0	0
$739$	12.6626	0.465800	0.232900	−	0.972501i	$-0.425178\pi$
$739$	12.6626	0.465800	0.232900	+	0.972501i	$0.425178\pi$
$740$	0	0
$741$	−11.2565	−0.413516
$742$	0	0
$743$	13.3477	0.489678	0.244839	−	0.969564i	$-0.421265\pi$
$743$	13.3477	0.489678	0.244839	+	0.969564i	$0.421265\pi$
$744$	0	0
$745$	42.7684	1.56691
$746$	0	0
$747$	8.60533	0.314852
$748$	0	0
$749$	0	0
$750$	0	0
$751$	28.2843	1.03211	0.516054	−	0.856556i	$-0.327400\pi$
$751$	28.2843	1.03211	0.516054	+	0.856556i	$0.327400\pi$
$752$	0	0
$753$	−7.45607	−0.271714
$754$	0	0
$755$	−68.6684	−2.49910
$756$	0	0
$757$	9.71410	0.353065	0.176533	−	0.984295i	$-0.443512\pi$
$757$	9.71410	0.353065	0.176533	+	0.984295i	$0.443512\pi$
$758$	0	0
$759$	−31.1755	−1.13160
$760$	0	0
$761$	−16.5569	−0.600188	−0.300094	−	0.953910i	$-0.597018\pi$
$761$	−16.5569	−0.600188	−0.300094	+	0.953910i	$0.597018\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−16.2622	−0.587960
$766$	0	0
$767$	14.8912	0.537691
$768$	0	0
$769$	−12.3884	−0.446736	−0.223368	−	0.974734i	$-0.571705\pi$
$769$	−12.3884	−0.446736	−0.223368	+	0.974734i	$0.571705\pi$
$770$	0	0
$771$	−12.3433	−0.444533
$772$	0	0
$773$	−55.4211	−1.99336	−0.996679	−	0.0814351i	$-0.974050\pi$
$773$	−55.4211	−1.99336	−0.996679	+	0.0814351i	$0.974050\pi$
$774$	0	0
$775$	−2.74886	−0.0987421
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−23.2308	−0.832329
$780$	0	0
$781$	−53.4120	−1.91123
$782$	0	0
$783$	5.55912	0.198667
$784$	0	0
$785$	−37.4667	−1.33724
$786$	0	0
$787$	13.4061	0.477877	0.238938	−	0.971035i	$-0.423201\pi$
$787$	13.4061	0.477877	0.238938	+	0.971035i	$0.423201\pi$
$788$	0	0
$789$	−23.4144	−0.833574
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−14.5350	−0.516153
$794$	0	0
$795$	3.82270	0.135577
$796$	0	0
$797$	−26.4837	−0.938102	−0.469051	−	0.883171i	$-0.655404\pi$
$797$	−26.4837	−0.938102	−0.469051	+	0.883171i	$0.655404\pi$
$798$	0	0
$799$	35.3346	1.25005
$800$	0	0
$801$	4.85983	0.171713
$802$	0	0
$803$	−18.3599	−0.647907
$804$	0	0
$805$	0	0
$806$	0	0
$807$	5.81362	0.204649
$808$	0	0
$809$	−36.5760	−1.28594	−0.642972	−	0.765889i	$-0.722299\pi$
$809$	−36.5760	−1.28594	−0.642972	+	0.765889i	$0.722299\pi$
$810$	0	0
$811$	4.67565	0.164184	0.0820921	−	0.996625i	$-0.473840\pi$
$811$	4.67565	0.164184	0.0820921	+	0.996625i	$0.473840\pi$
$812$	0	0
$813$	−24.3694	−0.854672
$814$	0	0
$815$	−10.5023	−0.367879
$816$	0	0
$817$	20.5244	0.718057
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−42.9212	−1.49796	−0.748979	−	0.662593i	$-0.769456\pi$
$821$	−42.9212	−1.49796	−0.748979	+	0.662593i	$0.769456\pi$
$822$	0	0
$823$	12.0888	0.421389	0.210695	−	0.977552i	$-0.432427\pi$
$823$	12.0888	0.421389	0.210695	+	0.977552i	$0.432427\pi$
$824$	0	0
$825$	−16.7317	−0.582522
$826$	0	0
$827$	3.23676	0.112553	0.0562765	−	0.998415i	$-0.482077\pi$
$827$	3.23676	0.112553	0.0562765	+	0.998415i	$0.482077\pi$
$828$	0	0
$829$	−2.06002	−0.0715476	−0.0357738	−	0.999360i	$-0.511390\pi$
$829$	−2.06002	−0.0715476	−0.0357738	+	0.999360i	$0.511390\pi$
$830$	0	0
$831$	15.7587	0.546664
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−38.0536	−1.31690
$836$	0	0
$837$	−0.645810	−0.0223224
$838$	0	0
$839$	−5.99468	−0.206959	−0.103480	−	0.994632i	$-0.532998\pi$
$839$	−5.99468	−0.206959	−0.103480	+	0.994632i	$0.532998\pi$
$840$	0	0
$841$	1.90384	0.0656498
$842$	0	0
$843$	−30.7698	−1.05977
$844$	0	0
$845$	33.1533	1.14051
$846$	0	0
$847$	0	0
$848$	0	0
$849$	22.9080	0.786200
$850$	0	0
$851$	44.8639	1.53791
$852$	0	0
$853$	−40.3040	−1.37998	−0.689992	−	0.723817i	$-0.742386\pi$
$853$	−40.3040	−1.37998	−0.689992	+	0.723817i	$0.742386\pi$
$854$	0	0
$855$	−7.00572	−0.239591
$856$	0	0
$857$	24.1735	0.825752	0.412876	−	0.910787i	$-0.364524\pi$
$857$	24.1735	0.825752	0.412876	+	0.910787i	$0.364524\pi$
$858$	0	0
$859$	−22.6316	−0.772179	−0.386090	−	0.922461i	$-0.626174\pi$
$859$	−22.6316	−0.772179	−0.386090	+	0.922461i	$0.626174\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	48.9015	1.66462	0.832312	−	0.554307i	$-0.187017\pi$
$863$	48.9015	1.66462	0.832312	+	0.554307i	$0.187017\pi$
$864$	0	0
$865$	58.1097	1.97579
$866$	0	0
$867$	11.5702	0.392943
$868$	0	0
$869$	−4.13344	−0.140217
$870$	0	0
$871$	66.0852	2.23921
$872$	0	0
$873$	18.3275	0.620293
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−32.1461	−1.08550	−0.542748	−	0.839896i	$-0.682616\pi$
$877$	−32.1461	−1.08550	−0.542748	+	0.839896i	$0.682616\pi$
$878$	0	0
$879$	19.3323	0.652061
$880$	0	0
$881$	1.94291	0.0654583	0.0327291	−	0.999464i	$-0.489580\pi$
$881$	1.94291	0.0654583	0.0327291	+	0.999464i	$0.489580\pi$
$882$	0	0
$883$	−48.8770	−1.64484	−0.822421	−	0.568880i	$-0.807377\pi$
$883$	−48.8770	−1.64484	−0.822421	+	0.568880i	$0.807377\pi$
$884$	0	0
$885$	9.26791	0.311537
$886$	0	0
$887$	−5.59546	−0.187877	−0.0939385	−	0.995578i	$-0.529946\pi$
$887$	−5.59546	−0.187877	−0.0939385	+	0.995578i	$0.529946\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−3.93089	−0.131690
$892$	0	0
$893$	15.2221	0.509388
$894$	0	0
$895$	35.5686	1.18893
$896$	0	0
$897$	38.7698	1.29449
$898$	0	0
$899$	−3.59014	−0.119738
$900$	0	0
$901$	−6.71591	−0.223739
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−34.4417	−1.14488
$906$	0	0
$907$	−51.0945	−1.69657	−0.848283	−	0.529543i	$-0.822363\pi$
$907$	−51.0945	−1.69657	−0.848283	+	0.529543i	$0.822363\pi$
$908$	0	0
$909$	5.87087	0.194724
$910$	0	0
$911$	−22.6965	−0.751969	−0.375985	−	0.926626i	$-0.622695\pi$
$911$	−22.6965	−0.751969	−0.375985	+	0.926626i	$0.622695\pi$
$912$	0	0
$913$	−33.8266	−1.11950
$914$	0	0
$915$	−9.04621	−0.299058
$916$	0	0
$917$	0	0
$918$	0	0
$919$	12.9836	0.428291	0.214145	−	0.976802i	$-0.431303\pi$
$919$	12.9836	0.428291	0.214145	+	0.976802i	$0.431303\pi$
$920$	0	0
$921$	−26.8377	−0.884331
$922$	0	0
$923$	66.4230	2.18634
$924$	0	0
$925$	24.0782	0.791685
$926$	0	0
$927$	15.0778	0.495219
$928$	0	0
$929$	−1.84856	−0.0606491	−0.0303246	−	0.999540i	$-0.509654\pi$
$929$	−1.84856	−0.0606491	−0.0303246	+	0.999540i	$0.509654\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−3.49240	−0.114336
$934$	0	0
$935$	63.9249	2.09057
$936$	0	0
$937$	−53.0177	−1.73201	−0.866007	−	0.500032i	$-0.833322\pi$
$937$	−53.0177	−1.73201	−0.866007	+	0.500032i	$0.833322\pi$
$938$	0	0
$939$	−2.75556	−0.0899242
$940$	0	0
$941$	−10.9615	−0.357334	−0.178667	−	0.983910i	$-0.557178\pi$
$941$	−10.9615	−0.357334	−0.178667	+	0.983910i	$0.557178\pi$
$942$	0	0
$943$	80.0120	2.60555
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.61718	0.280021	0.140010	−	0.990150i	$-0.455286\pi$
$947$	8.61718	0.280021	0.140010	+	0.990150i	$0.455286\pi$
$948$	0	0
$949$	22.8323	0.741169
$950$	0	0
$951$	−7.15341	−0.231965
$952$	0	0
$953$	6.40986	0.207636	0.103818	−	0.994596i	$-0.466894\pi$
$953$	6.40986	0.207636	0.103818	+	0.994596i	$0.466894\pi$
$954$	0	0
$955$	51.4217	1.66397
$956$	0	0
$957$	−21.8523	−0.706385
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.5829	−0.986546
$962$	0	0
$963$	−0.238878	−0.00769774
$964$	0	0
$965$	−72.0706	−2.32004
$966$	0	0
$967$	43.2749	1.39163	0.695814	−	0.718222i	$-0.255044\pi$
$967$	43.2749	1.39163	0.695814	+	0.718222i	$0.255044\pi$
$968$	0	0
$969$	12.3080	0.395389
$970$	0	0
$971$	21.8266	0.700450	0.350225	−	0.936666i	$-0.386105\pi$
$971$	21.8266	0.700450	0.350225	+	0.936666i	$0.386105\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	20.8075	0.666373
$976$	0	0
$977$	−58.5040	−1.87171	−0.935854	−	0.352387i	$-0.885370\pi$
$977$	−58.5040	−1.87171	−0.935854	+	0.352387i	$0.885370\pi$
$978$	0	0
$979$	−19.1035	−0.610549
$980$	0	0
$981$	−12.9133	−0.412290
$982$	0	0
$983$	−6.17337	−0.196900	−0.0984500	−	0.995142i	$-0.531388\pi$
$983$	−6.17337	−0.196900	−0.0984500	+	0.995142i	$0.531388\pi$
$984$	0	0
$985$	−33.7197	−1.07440
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−70.6905	−2.24783
$990$	0	0
$991$	−5.11942	−0.162624	−0.0813118	−	0.996689i	$-0.525911\pi$
$991$	−5.11942	−0.162624	−0.0813118	+	0.996689i	$0.525911\pi$
$992$	0	0
$993$	−31.7236	−1.00672
$994$	0	0
$995$	−19.8152	−0.628183
$996$	0	0
$997$	49.7448	1.57543	0.787717	−	0.616037i	$-0.211263\pi$
$997$	49.7448	1.57543	0.787717	+	0.616037i	$0.211263\pi$
$998$	0	0
$999$	5.65685	0.178975

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4704.2.a.bz.1.3	yes	4
4.3	odd	2		4704.2.a.bx.1.3	yes	4
7.6	odd	2		4704.2.a.bw.1.2	✓	4
8.3	odd	2		9408.2.a.em.1.2		4
8.5	even	2		9408.2.a.ek.1.2		4
28.27	even	2		4704.2.a.by.1.2	yes	4
56.13	odd	2		9408.2.a.en.1.3		4
56.27	even	2		9408.2.a.el.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4704.2.a.bw.1.2	✓	4	7.6	odd	2
4704.2.a.bx.1.3	yes	4	4.3	odd	2
4704.2.a.by.1.2	yes	4	28.27	even	2
4704.2.a.bz.1.3	yes	4	1.1	even	1	trivial
9408.2.a.ek.1.2		4	8.5	even	2
9408.2.a.el.1.3		4	56.27	even	2
9408.2.a.em.1.2		4	8.3	odd	2
9408.2.a.en.1.3		4	56.13	odd	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:	\( N \)	\(=\)	\( 4704 = 2^{5} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4704.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.04244 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.04244 q^{5} +1.00000 q^{9} -3.93089 q^{11} +4.88845 q^{13} +3.04244 q^{15} -5.34511 q^{17} -2.30266 q^{19} +7.93089 q^{23} +4.25646 q^{25} +1.00000 q^{27} +5.55912 q^{29} -0.645810 q^{31} -3.93089 q^{33} +5.65685 q^{37} +4.88845 q^{39} +10.0886 q^{41} -8.91331 q^{43} +3.04244 q^{45} -6.61065 q^{47} -5.34511 q^{51} +1.25646 q^{53} -11.9595 q^{55} -2.30266 q^{57} +3.04621 q^{59} -2.97334 q^{61} +14.8728 q^{65} +13.5186 q^{67} +7.93089 q^{69} +13.5877 q^{71} +4.67067 q^{73} +4.25646 q^{75} +1.05153 q^{79} +1.00000 q^{81} +8.60533 q^{83} -16.2622 q^{85} +5.55912 q^{87} +4.85983 q^{89} -0.645810 q^{93} -7.00572 q^{95} +18.3275 q^{97} -3.93089 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{5} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^5 + 4 * q^9	\( 4 q + 4 q^{3} + 4 q^{5} + 4 q^{9} + 4 q^{11} + 8 q^{13} + 4 q^{15} + 4 q^{17} + 8 q^{19} + 12 q^{23} + 12 q^{25} + 4 q^{27} - 8 q^{31} + 4 q^{33} + 8 q^{39} + 20 q^{41} - 8 q^{43} + 4 q^{45} - 16 q^{47} + 4 q^{51} - 8 q^{55} + 8 q^{57} + 16 q^{61} - 8 q^{65} - 8 q^{67} + 12 q^{69} + 12 q^{71} + 8 q^{73} + 12 q^{75} + 16 q^{79} + 4 q^{81} - 8 q^{85} + 28 q^{89} - 8 q^{93} + 24 q^{95} + 40 q^{97} + 4 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^5 + 4 * q^9 + 4 * q^11 + 8 * q^13 + 4 * q^15 + 4 * q^17 + 8 * q^19 + 12 * q^23 + 12 * q^25 + 4 * q^27 - 8 * q^31 + 4 * q^33 + 8 * q^39 + 20 * q^41 - 8 * q^43 + 4 * q^45 - 16 * q^47 + 4 * q^51 - 8 * q^55 + 8 * q^57 + 16 * q^61 - 8 * q^65 - 8 * q^67 + 12 * q^69 + 12 * q^71 + 8 * q^73 + 12 * q^75 + 16 * q^79 + 4 * q^81 - 8 * q^85 + 28 * q^89 - 8 * q^93 + 24 * q^95 + 40 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more