LMFDB - Embedded newform 7600.2.a.bc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, 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("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7600 = 2^{4} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7600.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:	$60.6863055362$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 760)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	7600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421 q^{3} -2.41421 q^{7} +2.82843 q^{9} +O(q^{10})$	$q+2.41421 q^{3} -2.41421 q^{7} +2.82843 q^{9} -1.41421 q^{11} -1.82843 q^{13} -1.00000 q^{17} +1.00000 q^{19} -5.82843 q^{21} +5.24264 q^{23} -0.414214 q^{27} +3.82843 q^{29} -3.41421 q^{31} -3.41421 q^{33} -5.17157 q^{37} -4.41421 q^{39} -7.07107 q^{41} +2.24264 q^{43} +8.00000 q^{47} -1.17157 q^{49} -2.41421 q^{51} +1.82843 q^{53} +2.41421 q^{57} -14.4142 q^{59} +3.41421 q^{61} -6.82843 q^{63} -6.07107 q^{67} +12.6569 q^{69} -3.07107 q^{71} -13.8284 q^{73} +3.41421 q^{77} +0.828427 q^{79} -9.48528 q^{81} -2.48528 q^{83} +9.24264 q^{87} -3.75736 q^{89} +4.41421 q^{91} -8.24264 q^{93} -7.65685 q^{97} -4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^7	$ 2 q + 2 q^{3} - 2 q^{7} + 2 q^{13} - 2 q^{17} + 2 q^{19} - 6 q^{21} + 2 q^{23} + 2 q^{27} + 2 q^{29} - 4 q^{31} - 4 q^{33} - 16 q^{37} - 6 q^{39} - 4 q^{43} + 16 q^{47} - 8 q^{49} - 2 q^{51} - 2 q^{53} + 2 q^{57} - 26 q^{59} + 4 q^{61} - 8 q^{63} + 2 q^{67} + 14 q^{69} + 8 q^{71} - 22 q^{73} + 4 q^{77} - 4 q^{79} - 2 q^{81} + 12 q^{83} + 10 q^{87} - 16 q^{89} + 6 q^{91} - 8 q^{93} - 4 q^{97} - 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^13 - 2 * q^17 + 2 * q^19 - 6 * q^21 + 2 * q^23 + 2 * q^27 + 2 * q^29 - 4 * q^31 - 4 * q^33 - 16 * q^37 - 6 * q^39 - 4 * q^43 + 16 * q^47 - 8 * q^49 - 2 * q^51 - 2 * q^53 + 2 * q^57 - 26 * q^59 + 4 * q^61 - 8 * q^63 + 2 * q^67 + 14 * q^69 + 8 * q^71 - 22 * q^73 + 4 * q^77 - 4 * q^79 - 2 * q^81 + 12 * q^83 + 10 * q^87 - 16 * q^89 + 6 * q^91 - 8 * q^93 - 4 * q^97 - 8 * 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
$3$	2.41421	1.39385	0.696923	−	0.717146i	$-0.254552\pi$
$3$	2.41421	1.39385	0.696923	+	0.717146i	$0.254552\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.41421	−0.912487	−0.456243	−	0.889855i	$-0.650805\pi$
$7$	−2.41421	−0.912487	−0.456243	+	0.889855i	$0.650805\pi$
$8$	0	0
$9$	2.82843	0.942809
$10$	0	0
$11$	−1.41421	−0.426401	−0.213201	−	0.977008i	$-0.568389\pi$
$11$	−1.41421	−0.426401	−0.213201	+	0.977008i	$0.568389\pi$
$12$	0	0
$13$	−1.82843	−0.507114	−0.253557	−	0.967320i	$-0.581601\pi$
$13$	−1.82843	−0.507114	−0.253557	+	0.967320i	$0.581601\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−5.82843	−1.27187
$22$	0	0
$23$	5.24264	1.09317	0.546583	−	0.837405i	$-0.315928\pi$
$23$	5.24264	1.09317	0.546583	+	0.837405i	$0.315928\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−0.414214	−0.0797154
$28$	0	0
$29$	3.82843	0.710921	0.355461	−	0.934691i	$-0.384324\pi$
$29$	3.82843	0.710921	0.355461	+	0.934691i	$0.384324\pi$
$30$	0	0
$31$	−3.41421	−0.613211	−0.306605	−	0.951837i	$-0.599193\pi$
$31$	−3.41421	−0.613211	−0.306605	+	0.951837i	$0.599193\pi$
$32$	0	0
$33$	−3.41421	−0.594338
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.17157	−0.850201	−0.425101	−	0.905146i	$-0.639761\pi$
$37$	−5.17157	−0.850201	−0.425101	+	0.905146i	$0.639761\pi$
$38$	0	0
$39$	−4.41421	−0.706840
$40$	0	0
$41$	−7.07107	−1.10432	−0.552158	−	0.833740i	$-0.686195\pi$
$41$	−7.07107	−1.10432	−0.552158	+	0.833740i	$0.686195\pi$
$42$	0	0
$43$	2.24264	0.341999	0.171000	−	0.985271i	$-0.445300\pi$
$43$	2.24264	0.341999	0.171000	+	0.985271i	$0.445300\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	−1.17157	−0.167368
$50$	0	0
$51$	−2.41421	−0.338058
$52$	0	0
$53$	1.82843	0.251154	0.125577	−	0.992084i	$-0.459922\pi$
$53$	1.82843	0.251154	0.125577	+	0.992084i	$0.459922\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.41421	0.319770
$58$	0	0
$59$	−14.4142	−1.87657	−0.938285	−	0.345862i	$-0.887587\pi$
$59$	−14.4142	−1.87657	−0.938285	+	0.345862i	$0.887587\pi$
$60$	0	0
$61$	3.41421	0.437145	0.218573	−	0.975821i	$-0.429860\pi$
$61$	3.41421	0.437145	0.218573	+	0.975821i	$0.429860\pi$
$62$	0	0
$63$	−6.82843	−0.860301
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−6.07107	−0.741699	−0.370849	−	0.928693i	$-0.620933\pi$
$67$	−6.07107	−0.741699	−0.370849	+	0.928693i	$0.620933\pi$
$68$	0	0
$69$	12.6569	1.52371
$70$	0	0
$71$	−3.07107	−0.364469	−0.182234	−	0.983255i	$-0.558333\pi$
$71$	−3.07107	−0.364469	−0.182234	+	0.983255i	$0.558333\pi$
$72$	0	0
$73$	−13.8284	−1.61849	−0.809247	−	0.587468i	$-0.800125\pi$
$73$	−13.8284	−1.61849	−0.809247	+	0.587468i	$0.800125\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.41421	0.389086
$78$	0	0
$79$	0.828427	0.0932053	0.0466027	−	0.998914i	$-0.485161\pi$
$79$	0.828427	0.0932053	0.0466027	+	0.998914i	$0.485161\pi$
$80$	0	0
$81$	−9.48528	−1.05392
$82$	0	0
$83$	−2.48528	−0.272795	−0.136398	−	0.990654i	$-0.543552\pi$
$83$	−2.48528	−0.272795	−0.136398	+	0.990654i	$0.543552\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	9.24264	0.990915
$88$	0	0
$89$	−3.75736	−0.398279	−0.199140	−	0.979971i	$-0.563815\pi$
$89$	−3.75736	−0.398279	−0.199140	+	0.979971i	$0.563815\pi$
$90$	0	0
$91$	4.41421	0.462735
$92$	0	0
$93$	−8.24264	−0.854722
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−7.65685	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$97$	−7.65685	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	0	0
$101$	15.8995	1.58206	0.791029	−	0.611778i	$-0.209545\pi$
$101$	15.8995	1.58206	0.791029	+	0.611778i	$0.209545\pi$
$102$	0	0
$103$	−9.89949	−0.975426	−0.487713	−	0.873004i	$-0.662169\pi$
$103$	−9.89949	−0.975426	−0.487713	+	0.873004i	$0.662169\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.41421	−0.233391	−0.116695	−	0.993168i	$-0.537230\pi$
$107$	−2.41421	−0.233391	−0.116695	+	0.993168i	$0.537230\pi$
$108$	0	0
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	0	0
$111$	−12.4853	−1.18505
$112$	0	0
$113$	13.0711	1.22962	0.614811	−	0.788674i	$-0.289232\pi$
$113$	13.0711	1.22962	0.614811	+	0.788674i	$0.289232\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−5.17157	−0.478112
$118$	0	0
$119$	2.41421	0.221311
$120$	0	0
$121$	−9.00000	−0.818182
$122$	0	0
$123$	−17.0711	−1.53925
$124$	0	0
$125$	0	0
$126$	0	0
$127$	11.1716	0.991317	0.495658	−	0.868518i	$-0.334927\pi$
$127$	11.1716	0.991317	0.495658	+	0.868518i	$0.334927\pi$
$128$	0	0
$129$	5.41421	0.476695
$130$	0	0
$131$	−21.6569	−1.89217	−0.946084	−	0.323921i	$-0.894999\pi$
$131$	−21.6569	−1.89217	−0.946084	+	0.323921i	$0.894999\pi$
$132$	0	0
$133$	−2.41421	−0.209339
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−16.3137	−1.39377	−0.696887	−	0.717181i	$-0.745432\pi$
$137$	−16.3137	−1.39377	−0.696887	+	0.717181i	$0.745432\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	19.3137	1.62651
$142$	0	0
$143$	2.58579	0.216234
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−2.82843	−0.233285
$148$	0	0
$149$	−17.6569	−1.44651	−0.723253	−	0.690583i	$-0.757354\pi$
$149$	−17.6569	−1.44651	−0.723253	+	0.690583i	$0.757354\pi$
$150$	0	0
$151$	−11.1716	−0.909130	−0.454565	−	0.890714i	$-0.650205\pi$
$151$	−11.1716	−0.909130	−0.454565	+	0.890714i	$0.650205\pi$
$152$	0	0
$153$	−2.82843	−0.228665
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−0.343146	−0.0273860	−0.0136930	−	0.999906i	$-0.504359\pi$
$157$	−0.343146	−0.0273860	−0.0136930	+	0.999906i	$0.504359\pi$
$158$	0	0
$159$	4.41421	0.350070
$160$	0	0
$161$	−12.6569	−0.997500
$162$	0	0
$163$	2.92893	0.229412	0.114706	−	0.993400i	$-0.463407\pi$
$163$	2.92893	0.229412	0.114706	+	0.993400i	$0.463407\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	23.2132	1.79629	0.898146	−	0.439698i	$-0.144914\pi$
$167$	23.2132	1.79629	0.898146	+	0.439698i	$0.144914\pi$
$168$	0	0
$169$	−9.65685	−0.742835
$170$	0	0
$171$	2.82843	0.216295
$172$	0	0
$173$	−18.8284	−1.43150	−0.715749	−	0.698357i	$-0.753915\pi$
$173$	−18.8284	−1.43150	−0.715749	+	0.698357i	$0.753915\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−34.7990	−2.61565
$178$	0	0
$179$	22.9706	1.71690	0.858450	−	0.512897i	$-0.171428\pi$
$179$	22.9706	1.71690	0.858450	+	0.512897i	$0.171428\pi$
$180$	0	0
$181$	0.485281	0.0360707	0.0180353	−	0.999837i	$-0.494259\pi$
$181$	0.485281	0.0360707	0.0180353	+	0.999837i	$0.494259\pi$
$182$	0	0
$183$	8.24264	0.609314
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.41421	0.103418
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	11.2426	0.813489	0.406744	−	0.913542i	$-0.366664\pi$
$191$	11.2426	0.813489	0.406744	+	0.913542i	$0.366664\pi$
$192$	0	0
$193$	−7.65685	−0.551152	−0.275576	−	0.961279i	$-0.588869\pi$
$193$	−7.65685	−0.551152	−0.275576	+	0.961279i	$0.588869\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−20.7279	−1.47680	−0.738402	−	0.674361i	$-0.764419\pi$
$197$	−20.7279	−1.47680	−0.738402	+	0.674361i	$0.764419\pi$
$198$	0	0
$199$	−16.0711	−1.13925	−0.569624	−	0.821905i	$-0.692911\pi$
$199$	−16.0711	−1.13925	−0.569624	+	0.821905i	$0.692911\pi$
$200$	0	0
$201$	−14.6569	−1.03381
$202$	0	0
$203$	−9.24264	−0.648706
$204$	0	0
$205$	0	0
$206$	0	0
$207$	14.8284	1.03065
$208$	0	0
$209$	−1.41421	−0.0978232
$210$	0	0
$211$	24.8995	1.71415	0.857076	−	0.515190i	$-0.172279\pi$
$211$	24.8995	1.71415	0.857076	+	0.515190i	$0.172279\pi$
$212$	0	0
$213$	−7.41421	−0.508014
$214$	0	0
$215$	0	0
$216$	0	0
$217$	8.24264	0.559547
$218$	0	0
$219$	−33.3848	−2.25593
$220$	0	0
$221$	1.82843	0.122993
$222$	0	0
$223$	3.17157	0.212384	0.106192	−	0.994346i	$-0.466134\pi$
$223$	3.17157	0.212384	0.106192	+	0.994346i	$0.466134\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−17.2426	−1.14443	−0.572217	−	0.820102i	$-0.693917\pi$
$227$	−17.2426	−1.14443	−0.572217	+	0.820102i	$0.693917\pi$
$228$	0	0
$229$	−26.9706	−1.78226	−0.891132	−	0.453743i	$-0.850088\pi$
$229$	−26.9706	−1.78226	−0.891132	+	0.453743i	$0.850088\pi$
$230$	0	0
$231$	8.24264	0.542326
$232$	0	0
$233$	2.34315	0.153505	0.0767523	−	0.997050i	$-0.475545\pi$
$233$	2.34315	0.153505	0.0767523	+	0.997050i	$0.475545\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2.00000	0.129914
$238$	0	0
$239$	−2.27208	−0.146969	−0.0734843	−	0.997296i	$-0.523412\pi$
$239$	−2.27208	−0.146969	−0.0734843	+	0.997296i	$0.523412\pi$
$240$	0	0
$241$	5.65685	0.364390	0.182195	−	0.983262i	$-0.441680\pi$
$241$	5.65685	0.364390	0.182195	+	0.983262i	$0.441680\pi$
$242$	0	0
$243$	−21.6569	−1.38929
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.82843	−0.116340
$248$	0	0
$249$	−6.00000	−0.380235
$250$	0	0
$251$	27.5563	1.73934	0.869671	−	0.493632i	$-0.164331\pi$
$251$	27.5563	1.73934	0.869671	+	0.493632i	$0.164331\pi$
$252$	0	0
$253$	−7.41421	−0.466128
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.27208	−0.204107	−0.102053	−	0.994779i	$-0.532541\pi$
$257$	−3.27208	−0.204107	−0.102053	+	0.994779i	$0.532541\pi$
$258$	0	0
$259$	12.4853	0.775798
$260$	0	0
$261$	10.8284	0.670263
$262$	0	0
$263$	−4.34315	−0.267810	−0.133905	−	0.990994i	$-0.542752\pi$
$263$	−4.34315	−0.267810	−0.133905	+	0.990994i	$0.542752\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−9.07107	−0.555140
$268$	0	0
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	21.3848	1.29903	0.649516	−	0.760348i	$-0.274971\pi$
$271$	21.3848	1.29903	0.649516	+	0.760348i	$0.274971\pi$
$272$	0	0
$273$	10.6569	0.644982
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−19.3137	−1.16045	−0.580224	−	0.814457i	$-0.697035\pi$
$277$	−19.3137	−1.16045	−0.580224	+	0.814457i	$0.697035\pi$
$278$	0	0
$279$	−9.65685	−0.578141
$280$	0	0
$281$	−12.2426	−0.730335	−0.365167	−	0.930942i	$-0.618988\pi$
$281$	−12.2426	−0.730335	−0.365167	+	0.930942i	$0.618988\pi$
$282$	0	0
$283$	26.4853	1.57439	0.787193	−	0.616706i	$-0.211533\pi$
$283$	26.4853	1.57439	0.787193	+	0.616706i	$0.211533\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	17.0711	1.00767
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	−18.4853	−1.08363
$292$	0	0
$293$	28.7990	1.68245	0.841227	−	0.540681i	$-0.181834\pi$
$293$	28.7990	1.68245	0.841227	+	0.540681i	$0.181834\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0.585786	0.0339908
$298$	0	0
$299$	−9.58579	−0.554360
$300$	0	0
$301$	−5.41421	−0.312070
$302$	0	0
$303$	38.3848	2.20515
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−24.2843	−1.38598	−0.692988	−	0.720949i	$-0.743706\pi$
$307$	−24.2843	−1.38598	−0.692988	+	0.720949i	$0.743706\pi$
$308$	0	0
$309$	−23.8995	−1.35959
$310$	0	0
$311$	−14.7574	−0.836813	−0.418407	−	0.908260i	$-0.637411\pi$
$311$	−14.7574	−0.836813	−0.418407	+	0.908260i	$0.637411\pi$
$312$	0	0
$313$	4.65685	0.263221	0.131610	−	0.991302i	$-0.457985\pi$
$313$	4.65685	0.263221	0.131610	+	0.991302i	$0.457985\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	32.7990	1.84217	0.921087	−	0.389356i	$-0.127302\pi$
$317$	32.7990	1.84217	0.921087	+	0.389356i	$0.127302\pi$
$318$	0	0
$319$	−5.41421	−0.303138
$320$	0	0
$321$	−5.82843	−0.325311
$322$	0	0
$323$	−1.00000	−0.0556415
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−12.0711	−0.667532
$328$	0	0
$329$	−19.3137	−1.06480
$330$	0	0
$331$	17.3848	0.955554	0.477777	−	0.878481i	$-0.341443\pi$
$331$	17.3848	0.955554	0.477777	+	0.878481i	$0.341443\pi$
$332$	0	0
$333$	−14.6274	−0.801578
$334$	0	0
$335$	0	0
$336$	0	0
$337$	12.7279	0.693334	0.346667	−	0.937988i	$-0.387313\pi$
$337$	12.7279	0.693334	0.346667	+	0.937988i	$0.387313\pi$
$338$	0	0
$339$	31.5563	1.71391
$340$	0	0
$341$	4.82843	0.261474
$342$	0	0
$343$	19.7279	1.06521
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.8284	1.11813	0.559064	−	0.829124i	$-0.311160\pi$
$347$	20.8284	1.11813	0.559064	+	0.829124i	$0.311160\pi$
$348$	0	0
$349$	−16.6274	−0.890045	−0.445023	−	0.895519i	$-0.646804\pi$
$349$	−16.6274	−0.890045	−0.445023	+	0.895519i	$0.646804\pi$
$350$	0	0
$351$	0.757359	0.0404248
$352$	0	0
$353$	26.1127	1.38984	0.694919	−	0.719088i	$-0.255440\pi$
$353$	26.1127	1.38984	0.694919	+	0.719088i	$0.255440\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	5.82843	0.308473
$358$	0	0
$359$	−8.07107	−0.425975	−0.212987	−	0.977055i	$-0.568319\pi$
$359$	−8.07107	−0.425975	−0.212987	+	0.977055i	$0.568319\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−21.7279	−1.14042
$364$	0	0
$365$	0	0
$366$	0	0
$367$	17.4558	0.911188	0.455594	−	0.890188i	$-0.349427\pi$
$367$	17.4558	0.911188	0.455594	+	0.890188i	$0.349427\pi$
$368$	0	0
$369$	−20.0000	−1.04116
$370$	0	0
$371$	−4.41421	−0.229175
$372$	0	0
$373$	−5.97056	−0.309144	−0.154572	−	0.987982i	$-0.549400\pi$
$373$	−5.97056	−0.309144	−0.154572	+	0.987982i	$0.549400\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7.00000	−0.360518
$378$	0	0
$379$	−33.3848	−1.71486	−0.857430	−	0.514600i	$-0.827940\pi$
$379$	−33.3848	−1.71486	−0.857430	+	0.514600i	$0.827940\pi$
$380$	0	0
$381$	26.9706	1.38174
$382$	0	0
$383$	4.72792	0.241586	0.120793	−	0.992678i	$-0.461456\pi$
$383$	4.72792	0.241586	0.120793	+	0.992678i	$0.461456\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.34315	0.322440
$388$	0	0
$389$	−11.8995	−0.603328	−0.301664	−	0.953414i	$-0.597542\pi$
$389$	−11.8995	−0.603328	−0.301664	+	0.953414i	$0.597542\pi$
$390$	0	0
$391$	−5.24264	−0.265132
$392$	0	0
$393$	−52.2843	−2.63739
$394$	0	0
$395$	0	0
$396$	0	0
$397$	22.6274	1.13564	0.567819	−	0.823154i	$-0.307787\pi$
$397$	22.6274	1.13564	0.567819	+	0.823154i	$0.307787\pi$
$398$	0	0
$399$	−5.82843	−0.291786
$400$	0	0
$401$	−25.8995	−1.29336	−0.646680	−	0.762762i	$-0.723843\pi$
$401$	−25.8995	−1.29336	−0.646680	+	0.762762i	$0.723843\pi$
$402$	0	0
$403$	6.24264	0.310968
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.31371	0.362527
$408$	0	0
$409$	5.41421	0.267716	0.133858	−	0.991001i	$-0.457263\pi$
$409$	5.41421	0.267716	0.133858	+	0.991001i	$0.457263\pi$
$410$	0	0
$411$	−39.3848	−1.94271
$412$	0	0
$413$	34.7990	1.71235
$414$	0	0
$415$	0	0
$416$	0	0
$417$	9.65685	0.472898
$418$	0	0
$419$	−15.2132	−0.743214	−0.371607	−	0.928390i	$-0.621193\pi$
$419$	−15.2132	−0.743214	−0.371607	+	0.928390i	$0.621193\pi$
$420$	0	0
$421$	−15.1421	−0.737983	−0.368991	−	0.929433i	$-0.620297\pi$
$421$	−15.1421	−0.737983	−0.368991	+	0.929433i	$0.620297\pi$
$422$	0	0
$423$	22.6274	1.10018
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−8.24264	−0.398889
$428$	0	0
$429$	6.24264	0.301398
$430$	0	0
$431$	2.58579	0.124553	0.0622765	−	0.998059i	$-0.480164\pi$
$431$	2.58579	0.124553	0.0622765	+	0.998059i	$0.480164\pi$
$432$	0	0
$433$	9.07107	0.435928	0.217964	−	0.975957i	$-0.430059\pi$
$433$	9.07107	0.435928	0.217964	+	0.975957i	$0.430059\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.24264	0.250790
$438$	0	0
$439$	5.27208	0.251623	0.125811	−	0.992054i	$-0.459847\pi$
$439$	5.27208	0.251623	0.125811	+	0.992054i	$0.459847\pi$
$440$	0	0
$441$	−3.31371	−0.157796
$442$	0	0
$443$	−15.5563	−0.739104	−0.369552	−	0.929210i	$-0.620489\pi$
$443$	−15.5563	−0.739104	−0.369552	+	0.929210i	$0.620489\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−42.6274	−2.01621
$448$	0	0
$449$	31.9411	1.50739	0.753697	−	0.657222i	$-0.228268\pi$
$449$	31.9411	1.50739	0.753697	+	0.657222i	$0.228268\pi$
$450$	0	0
$451$	10.0000	0.470882
$452$	0	0
$453$	−26.9706	−1.26719
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.31371	0.295343	0.147671	−	0.989036i	$-0.452822\pi$
$457$	6.31371	0.295343	0.147671	+	0.989036i	$0.452822\pi$
$458$	0	0
$459$	0.414214	0.0193338
$460$	0	0
$461$	−32.7279	−1.52429	−0.762146	−	0.647406i	$-0.775854\pi$
$461$	−32.7279	−1.52429	−0.762146	+	0.647406i	$0.775854\pi$
$462$	0	0
$463$	−6.14214	−0.285449	−0.142725	−	0.989762i	$-0.545586\pi$
$463$	−6.14214	−0.285449	−0.142725	+	0.989762i	$0.545586\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−2.10051	−0.0971998	−0.0485999	−	0.998818i	$-0.515476\pi$
$467$	−2.10051	−0.0971998	−0.0485999	+	0.998818i	$0.515476\pi$
$468$	0	0
$469$	14.6569	0.676791
$470$	0	0
$471$	−0.828427	−0.0381719
$472$	0	0
$473$	−3.17157	−0.145829
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5.17157	0.236790
$478$	0	0
$479$	17.4558	0.797578	0.398789	−	0.917043i	$-0.369431\pi$
$479$	17.4558	0.797578	0.398789	+	0.917043i	$0.369431\pi$
$480$	0	0
$481$	9.45584	0.431149
$482$	0	0
$483$	−30.5563	−1.39036
$484$	0	0
$485$	0	0
$486$	0	0
$487$	12.8284	0.581312	0.290656	−	0.956828i	$-0.406127\pi$
$487$	12.8284	0.581312	0.290656	+	0.956828i	$0.406127\pi$
$488$	0	0
$489$	7.07107	0.319765
$490$	0	0
$491$	−26.2426	−1.18431	−0.592157	−	0.805823i	$-0.701723\pi$
$491$	−26.2426	−1.18431	−0.592157	+	0.805823i	$0.701723\pi$
$492$	0	0
$493$	−3.82843	−0.172424
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.41421	0.332573
$498$	0	0
$499$	−33.0711	−1.48046	−0.740232	−	0.672351i	$-0.765284\pi$
$499$	−33.0711	−1.48046	−0.740232	+	0.672351i	$0.765284\pi$
$500$	0	0
$501$	56.0416	2.50376
$502$	0	0
$503$	13.1005	0.584123	0.292061	−	0.956400i	$-0.405659\pi$
$503$	13.1005	0.584123	0.292061	+	0.956400i	$0.405659\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−23.3137	−1.03540
$508$	0	0
$509$	−1.65685	−0.0734388	−0.0367194	−	0.999326i	$-0.511691\pi$
$509$	−1.65685	−0.0734388	−0.0367194	+	0.999326i	$0.511691\pi$
$510$	0	0
$511$	33.3848	1.47686
$512$	0	0
$513$	−0.414214	−0.0182880
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−11.3137	−0.497576
$518$	0	0
$519$	−45.4558	−1.99529
$520$	0	0
$521$	−28.0000	−1.22670	−0.613351	−	0.789810i	$-0.710179\pi$
$521$	−28.0000	−1.22670	−0.613351	+	0.789810i	$0.710179\pi$
$522$	0	0
$523$	22.2132	0.971316	0.485658	−	0.874149i	$-0.338580\pi$
$523$	22.2132	0.971316	0.485658	+	0.874149i	$0.338580\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.41421	0.148725
$528$	0	0
$529$	4.48528	0.195012
$530$	0	0
$531$	−40.7696	−1.76925
$532$	0	0
$533$	12.9289	0.560014
$534$	0	0
$535$	0	0
$536$	0	0
$537$	55.4558	2.39310
$538$	0	0
$539$	1.65685	0.0713658
$540$	0	0
$541$	−18.0416	−0.775670	−0.387835	−	0.921729i	$-0.626777\pi$
$541$	−18.0416	−0.775670	−0.387835	+	0.921729i	$0.626777\pi$
$542$	0	0
$543$	1.17157	0.0502770
$544$	0	0
$545$	0	0
$546$	0	0
$547$	43.9411	1.87879	0.939393	−	0.342841i	$-0.111389\pi$
$547$	43.9411	1.87879	0.939393	+	0.342841i	$0.111389\pi$
$548$	0	0
$549$	9.65685	0.412144
$550$	0	0
$551$	3.82843	0.163096
$552$	0	0
$553$	−2.00000	−0.0850487
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20.6863	0.876506	0.438253	−	0.898852i	$-0.355597\pi$
$557$	20.6863	0.876506	0.438253	+	0.898852i	$0.355597\pi$
$558$	0	0
$559$	−4.10051	−0.173433
$560$	0	0
$561$	3.41421	0.144148
$562$	0	0
$563$	−11.0294	−0.464835	−0.232418	−	0.972616i	$-0.574664\pi$
$563$	−11.0294	−0.464835	−0.232418	+	0.972616i	$0.574664\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	22.8995	0.961688
$568$	0	0
$569$	24.9706	1.04682	0.523410	−	0.852081i	$-0.324660\pi$
$569$	24.9706	1.04682	0.523410	+	0.852081i	$0.324660\pi$
$570$	0	0
$571$	11.6985	0.489566	0.244783	−	0.969578i	$-0.421283\pi$
$571$	11.6985	0.489566	0.244783	+	0.969578i	$0.421283\pi$
$572$	0	0
$573$	27.1421	1.13388
$574$	0	0
$575$	0	0
$576$	0	0
$577$	25.0000	1.04076	0.520382	−	0.853934i	$-0.325790\pi$
$577$	25.0000	1.04076	0.520382	+	0.853934i	$0.325790\pi$
$578$	0	0
$579$	−18.4853	−0.768222
$580$	0	0
$581$	6.00000	0.248922
$582$	0	0
$583$	−2.58579	−0.107092
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4.38478	0.180979	0.0904895	−	0.995897i	$-0.471157\pi$
$587$	4.38478	0.180979	0.0904895	+	0.995897i	$0.471157\pi$
$588$	0	0
$589$	−3.41421	−0.140680
$590$	0	0
$591$	−50.0416	−2.05844
$592$	0	0
$593$	42.0000	1.72473	0.862367	−	0.506284i	$-0.168981\pi$
$593$	42.0000	1.72473	0.862367	+	0.506284i	$0.168981\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−38.7990	−1.58794
$598$	0	0
$599$	2.44365	0.0998449	0.0499224	−	0.998753i	$-0.484103\pi$
$599$	2.44365	0.0998449	0.0499224	+	0.998753i	$0.484103\pi$
$600$	0	0
$601$	31.5563	1.28721	0.643605	−	0.765358i	$-0.277438\pi$
$601$	31.5563	1.28721	0.643605	+	0.765358i	$0.277438\pi$
$602$	0	0
$603$	−17.1716	−0.699281
$604$	0	0
$605$	0	0
$606$	0	0
$607$	13.5147	0.548546	0.274273	−	0.961652i	$-0.411563\pi$
$607$	13.5147	0.548546	0.274273	+	0.961652i	$0.411563\pi$
$608$	0	0
$609$	−22.3137	−0.904197
$610$	0	0
$611$	−14.6274	−0.591762
$612$	0	0
$613$	−22.7279	−0.917972	−0.458986	−	0.888443i	$-0.651787\pi$
$613$	−22.7279	−0.917972	−0.458986	+	0.888443i	$0.651787\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−7.79899	−0.313976	−0.156988	−	0.987601i	$-0.550178\pi$
$617$	−7.79899	−0.313976	−0.156988	+	0.987601i	$0.550178\pi$
$618$	0	0
$619$	30.3848	1.22127	0.610634	−	0.791913i	$-0.290915\pi$
$619$	30.3848	1.22127	0.610634	+	0.791913i	$0.290915\pi$
$620$	0	0
$621$	−2.17157	−0.0871422
$622$	0	0
$623$	9.07107	0.363425
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−3.41421	−0.136351
$628$	0	0
$629$	5.17157	0.206204
$630$	0	0
$631$	38.9706	1.55139	0.775697	−	0.631106i	$-0.217399\pi$
$631$	38.9706	1.55139	0.775697	+	0.631106i	$0.217399\pi$
$632$	0	0
$633$	60.1127	2.38927
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.14214	0.0848745
$638$	0	0
$639$	−8.68629	−0.343624
$640$	0	0
$641$	−39.2132	−1.54883	−0.774414	−	0.632679i	$-0.781955\pi$
$641$	−39.2132	−1.54883	−0.774414	+	0.632679i	$0.781955\pi$
$642$	0	0
$643$	31.1716	1.22929	0.614643	−	0.788805i	$-0.289300\pi$
$643$	31.1716	1.22929	0.614643	+	0.788805i	$0.289300\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.0711	0.474563	0.237281	−	0.971441i	$-0.423744\pi$
$647$	12.0711	0.474563	0.237281	+	0.971441i	$0.423744\pi$
$648$	0	0
$649$	20.3848	0.800172
$650$	0	0
$651$	19.8995	0.779923
$652$	0	0
$653$	32.2843	1.26338	0.631691	−	0.775221i	$-0.282361\pi$
$653$	32.2843	1.26338	0.631691	+	0.775221i	$0.282361\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−39.1127	−1.52593
$658$	0	0
$659$	8.89949	0.346675	0.173338	−	0.984862i	$-0.444545\pi$
$659$	8.89949	0.346675	0.173338	+	0.984862i	$0.444545\pi$
$660$	0	0
$661$	43.4853	1.69138	0.845691	−	0.533673i	$-0.179189\pi$
$661$	43.4853	1.69138	0.845691	+	0.533673i	$0.179189\pi$
$662$	0	0
$663$	4.41421	0.171434
$664$	0	0
$665$	0	0
$666$	0	0
$667$	20.0711	0.777155
$668$	0	0
$669$	7.65685	0.296031
$670$	0	0
$671$	−4.82843	−0.186399
$672$	0	0
$673$	−28.0000	−1.07932	−0.539660	−	0.841883i	$-0.681447\pi$
$673$	−28.0000	−1.07932	−0.539660	+	0.841883i	$0.681447\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	21.3431	0.820284	0.410142	−	0.912022i	$-0.365479\pi$
$677$	21.3431	0.820284	0.410142	+	0.912022i	$0.365479\pi$
$678$	0	0
$679$	18.4853	0.709400
$680$	0	0
$681$	−41.6274	−1.59517
$682$	0	0
$683$	23.3137	0.892074	0.446037	−	0.895014i	$-0.352835\pi$
$683$	23.3137	0.892074	0.446037	+	0.895014i	$0.352835\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−65.1127	−2.48420
$688$	0	0
$689$	−3.34315	−0.127364
$690$	0	0
$691$	−7.45584	−0.283634	−0.141817	−	0.989893i	$-0.545294\pi$
$691$	−7.45584	−0.283634	−0.141817	+	0.989893i	$0.545294\pi$
$692$	0	0
$693$	9.65685	0.366834
$694$	0	0
$695$	0	0
$696$	0	0
$697$	7.07107	0.267836
$698$	0	0
$699$	5.65685	0.213962
$700$	0	0
$701$	32.9706	1.24528	0.622640	−	0.782508i	$-0.286060\pi$
$701$	32.9706	1.24528	0.622640	+	0.782508i	$0.286060\pi$
$702$	0	0
$703$	−5.17157	−0.195050
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−38.3848	−1.44361
$708$	0	0
$709$	10.6863	0.401332	0.200666	−	0.979660i	$-0.435689\pi$
$709$	10.6863	0.401332	0.200666	+	0.979660i	$0.435689\pi$
$710$	0	0
$711$	2.34315	0.0878748
$712$	0	0
$713$	−17.8995	−0.670341
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−5.48528	−0.204852
$718$	0	0
$719$	−30.4142	−1.13426	−0.567129	−	0.823629i	$-0.691946\pi$
$719$	−30.4142	−1.13426	−0.567129	+	0.823629i	$0.691946\pi$
$720$	0	0
$721$	23.8995	0.890064
$722$	0	0
$723$	13.6569	0.507904
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1.78680	−0.0662686	−0.0331343	−	0.999451i	$-0.510549\pi$
$727$	−1.78680	−0.0662686	−0.0331343	+	0.999451i	$0.510549\pi$
$728$	0	0
$729$	−23.8284	−0.882534
$730$	0	0
$731$	−2.24264	−0.0829471
$732$	0	0
$733$	10.6274	0.392533	0.196266	−	0.980551i	$-0.437118\pi$
$733$	10.6274	0.392533	0.196266	+	0.980551i	$0.437118\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.58579	0.316262
$738$	0	0
$739$	−7.41421	−0.272736	−0.136368	−	0.990658i	$-0.543543\pi$
$739$	−7.41421	−0.272736	−0.136368	+	0.990658i	$0.543543\pi$
$740$	0	0
$741$	−4.41421	−0.162160
$742$	0	0
$743$	14.2426	0.522512	0.261256	−	0.965270i	$-0.415863\pi$
$743$	14.2426	0.522512	0.261256	+	0.965270i	$0.415863\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−7.02944	−0.257194
$748$	0	0
$749$	5.82843	0.212966
$750$	0	0
$751$	−29.7574	−1.08586	−0.542931	−	0.839777i	$-0.682685\pi$
$751$	−29.7574	−1.08586	−0.542931	+	0.839777i	$0.682685\pi$
$752$	0	0
$753$	66.5269	2.42438
$754$	0	0
$755$	0	0
$756$	0	0
$757$	16.3431	0.594002	0.297001	−	0.954877i	$-0.404014\pi$
$757$	16.3431	0.594002	0.297001	+	0.954877i	$0.404014\pi$
$758$	0	0
$759$	−17.8995	−0.649711
$760$	0	0
$761$	−4.02944	−0.146067	−0.0730335	−	0.997329i	$-0.523268\pi$
$761$	−4.02944	−0.146067	−0.0730335	+	0.997329i	$0.523268\pi$
$762$	0	0
$763$	12.0711	0.437002
$764$	0	0
$765$	0	0
$766$	0	0
$767$	26.3553	0.951636
$768$	0	0
$769$	−30.7990	−1.11064	−0.555320	−	0.831637i	$-0.687404\pi$
$769$	−30.7990	−1.11064	−0.555320	+	0.831637i	$0.687404\pi$
$770$	0	0
$771$	−7.89949	−0.284493
$772$	0	0
$773$	−9.00000	−0.323708	−0.161854	−	0.986815i	$-0.551747\pi$
$773$	−9.00000	−0.323708	−0.161854	+	0.986815i	$0.551747\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	30.1421	1.08134
$778$	0	0
$779$	−7.07107	−0.253347
$780$	0	0
$781$	4.34315	0.155410
$782$	0	0
$783$	−1.58579	−0.0566714
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−22.7574	−0.811212	−0.405606	−	0.914048i	$-0.632940\pi$
$787$	−22.7574	−0.811212	−0.405606	+	0.914048i	$0.632940\pi$
$788$	0	0
$789$	−10.4853	−0.373286
$790$	0	0
$791$	−31.5563	−1.12201
$792$	0	0
$793$	−6.24264	−0.221683
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−35.1421	−1.24480	−0.622399	−	0.782700i	$-0.713842\pi$
$797$	−35.1421	−1.24480	−0.622399	+	0.782700i	$0.713842\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	0	0
$801$	−10.6274	−0.375501
$802$	0	0
$803$	19.5563	0.690129
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−24.1421	−0.849843
$808$	0	0
$809$	−8.51472	−0.299362	−0.149681	−	0.988734i	$-0.547825\pi$
$809$	−8.51472	−0.299362	−0.149681	+	0.988734i	$0.547825\pi$
$810$	0	0
$811$	42.0122	1.47525	0.737624	−	0.675212i	$-0.235948\pi$
$811$	42.0122	1.47525	0.737624	+	0.675212i	$0.235948\pi$
$812$	0	0
$813$	51.6274	1.81065
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.24264	0.0784601
$818$	0	0
$819$	12.4853	0.436271
$820$	0	0
$821$	−51.5980	−1.80078	−0.900391	−	0.435082i	$-0.856719\pi$
$821$	−51.5980	−1.80078	−0.900391	+	0.435082i	$0.856719\pi$
$822$	0	0
$823$	−47.5269	−1.65668	−0.828342	−	0.560223i	$-0.810716\pi$
$823$	−47.5269	−1.65668	−0.828342	+	0.560223i	$0.810716\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−33.8701	−1.17778	−0.588889	−	0.808214i	$-0.700434\pi$
$827$	−33.8701	−1.17778	−0.588889	+	0.808214i	$0.700434\pi$
$828$	0	0
$829$	25.7696	0.895014	0.447507	−	0.894281i	$-0.352312\pi$
$829$	25.7696	0.895014	0.447507	+	0.894281i	$0.352312\pi$
$830$	0	0
$831$	−46.6274	−1.61749
$832$	0	0
$833$	1.17157	0.0405926
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.41421	0.0488824
$838$	0	0
$839$	49.9411	1.72416	0.862080	−	0.506773i	$-0.169162\pi$
$839$	49.9411	1.72416	0.862080	+	0.506773i	$0.169162\pi$
$840$	0	0
$841$	−14.3431	−0.494591
$842$	0	0
$843$	−29.5563	−1.01797
$844$	0	0
$845$	0	0
$846$	0	0
$847$	21.7279	0.746580
$848$	0	0
$849$	63.9411	2.19445
$850$	0	0
$851$	−27.1127	−0.929411
$852$	0	0
$853$	19.9411	0.682771	0.341386	−	0.939923i	$-0.389104\pi$
$853$	19.9411	0.682771	0.341386	+	0.939923i	$0.389104\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−16.0000	−0.546550	−0.273275	−	0.961936i	$-0.588107\pi$
$857$	−16.0000	−0.546550	−0.273275	+	0.961936i	$0.588107\pi$
$858$	0	0
$859$	−5.21320	−0.177872	−0.0889361	−	0.996037i	$-0.528347\pi$
$859$	−5.21320	−0.177872	−0.0889361	+	0.996037i	$0.528347\pi$
$860$	0	0
$861$	41.2132	1.40454
$862$	0	0
$863$	−35.0711	−1.19383	−0.596917	−	0.802303i	$-0.703608\pi$
$863$	−35.0711	−1.19383	−0.596917	+	0.802303i	$0.703608\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−38.6274	−1.31186
$868$	0	0
$869$	−1.17157	−0.0397429
$870$	0	0
$871$	11.1005	0.376126
$872$	0	0
$873$	−21.6569	−0.732973
$874$	0	0
$875$	0	0
$876$	0	0
$877$	29.4853	0.995647	0.497824	−	0.867278i	$-0.334133\pi$
$877$	29.4853	0.995647	0.497824	+	0.867278i	$0.334133\pi$
$878$	0	0
$879$	69.5269	2.34508
$880$	0	0
$881$	12.2843	0.413868	0.206934	−	0.978355i	$-0.433652\pi$
$881$	12.2843	0.413868	0.206934	+	0.978355i	$0.433652\pi$
$882$	0	0
$883$	15.4558	0.520131	0.260065	−	0.965591i	$-0.416256\pi$
$883$	15.4558	0.520131	0.260065	+	0.965591i	$0.416256\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0.928932	0.0311905	0.0155952	−	0.999878i	$-0.495036\pi$
$887$	0.928932	0.0311905	0.0155952	+	0.999878i	$0.495036\pi$
$888$	0	0
$889$	−26.9706	−0.904564
$890$	0	0
$891$	13.4142	0.449393
$892$	0	0
$893$	8.00000	0.267710
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−23.1421	−0.772693
$898$	0	0
$899$	−13.0711	−0.435945
$900$	0	0
$901$	−1.82843	−0.0609137
$902$	0	0
$903$	−13.0711	−0.434978
$904$	0	0
$905$	0	0
$906$	0	0
$907$	8.55635	0.284109	0.142054	−	0.989859i	$-0.454629\pi$
$907$	8.55635	0.284109	0.142054	+	0.989859i	$0.454629\pi$
$908$	0	0
$909$	44.9706	1.49158
$910$	0	0
$911$	9.65685	0.319946	0.159973	−	0.987121i	$-0.448859\pi$
$911$	9.65685	0.319946	0.159973	+	0.987121i	$0.448859\pi$
$912$	0	0
$913$	3.51472	0.116320
$914$	0	0
$915$	0	0
$916$	0	0
$917$	52.2843	1.72658
$918$	0	0
$919$	−46.8406	−1.54513	−0.772565	−	0.634936i	$-0.781026\pi$
$919$	−46.8406	−1.54513	−0.772565	+	0.634936i	$0.781026\pi$
$920$	0	0
$921$	−58.6274	−1.93184
$922$	0	0
$923$	5.61522	0.184827
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−28.0000	−0.919641
$928$	0	0
$929$	25.4853	0.836145	0.418072	−	0.908414i	$-0.362706\pi$
$929$	25.4853	0.836145	0.418072	+	0.908414i	$0.362706\pi$
$930$	0	0
$931$	−1.17157	−0.0383968
$932$	0	0
$933$	−35.6274	−1.16639
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−22.3137	−0.728957	−0.364479	−	0.931212i	$-0.618753\pi$
$937$	−22.3137	−0.728957	−0.364479	+	0.931212i	$0.618753\pi$
$938$	0	0
$939$	11.2426	0.366890
$940$	0	0
$941$	57.1421	1.86278	0.931390	−	0.364022i	$-0.118597\pi$
$941$	57.1421	1.86278	0.931390	+	0.364022i	$0.118597\pi$
$942$	0	0
$943$	−37.0711	−1.20720
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−41.2548	−1.34060	−0.670301	−	0.742089i	$-0.733835\pi$
$947$	−41.2548	−1.34060	−0.670301	+	0.742089i	$0.733835\pi$
$948$	0	0
$949$	25.2843	0.820762
$950$	0	0
$951$	79.1838	2.56771
$952$	0	0
$953$	−19.2132	−0.622377	−0.311188	−	0.950348i	$-0.600727\pi$
$953$	−19.2132	−0.622377	−0.311188	+	0.950348i	$0.600727\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−13.0711	−0.422528
$958$	0	0
$959$	39.3848	1.27180
$960$	0	0
$961$	−19.3431	−0.623972
$962$	0	0
$963$	−6.82843	−0.220043
$964$	0	0
$965$	0	0
$966$	0	0
$967$	50.1421	1.61246	0.806231	−	0.591601i	$-0.201504\pi$
$967$	50.1421	1.61246	0.806231	+	0.591601i	$0.201504\pi$
$968$	0	0
$969$	−2.41421	−0.0775557
$970$	0	0
$971$	35.5980	1.14239	0.571197	−	0.820813i	$-0.306479\pi$
$971$	35.5980	1.14239	0.571197	+	0.820813i	$0.306479\pi$
$972$	0	0
$973$	−9.65685	−0.309585
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.61522	0.0516756	0.0258378	−	0.999666i	$-0.491775\pi$
$977$	1.61522	0.0516756	0.0258378	+	0.999666i	$0.491775\pi$
$978$	0	0
$979$	5.31371	0.169827
$980$	0	0
$981$	−14.1421	−0.451524
$982$	0	0
$983$	−53.7990	−1.71592	−0.857961	−	0.513715i	$-0.828269\pi$
$983$	−53.7990	−1.71592	−0.857961	+	0.513715i	$0.828269\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−46.6274	−1.48417
$988$	0	0
$989$	11.7574	0.373862
$990$	0	0
$991$	−10.5269	−0.334398	−0.167199	−	0.985923i	$-0.553472\pi$
$991$	−10.5269	−0.334398	−0.167199	+	0.985923i	$0.553472\pi$
$992$	0	0
$993$	41.9706	1.33190
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−27.5563	−0.872718	−0.436359	−	0.899773i	$-0.643732\pi$
$997$	−27.5563	−0.872718	−0.436359	+	0.899773i	$0.643732\pi$
$998$	0	0
$999$	2.14214	0.0677742

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.bc.1.2		2
4.3	odd	2		3800.2.a.l.1.1		2
5.2	odd	4		1520.2.d.d.609.1		4
5.3	odd	4		1520.2.d.d.609.4		4
5.4	even	2		7600.2.a.x.1.1		2
20.3	even	4		760.2.d.c.609.1	✓	4
20.7	even	4		760.2.d.c.609.4	yes	4
20.19	odd	2		3800.2.a.p.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.d.c.609.1	✓	4	20.3	even	4
760.2.d.c.609.4	yes	4	20.7	even	4
1520.2.d.d.609.1		4	5.2	odd	4
1520.2.d.d.609.4		4	5.3	odd	4
3800.2.a.l.1.1		2	4.3	odd	2
3800.2.a.p.1.2		2	20.19	odd	2
7600.2.a.x.1.1		2	5.4	even	2
7600.2.a.bc.1.2		2	1.1	even	1	trivial

Overview	Random
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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:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q+2.41421 q^{3} -2.41421 q^{7} +2.82843 q^{9} +O(q^{10})\)	\(q+2.41421 q^{3} -2.41421 q^{7} +2.82843 q^{9} -1.41421 q^{11} -1.82843 q^{13} -1.00000 q^{17} +1.00000 q^{19} -5.82843 q^{21} +5.24264 q^{23} -0.414214 q^{27} +3.82843 q^{29} -3.41421 q^{31} -3.41421 q^{33} -5.17157 q^{37} -4.41421 q^{39} -7.07107 q^{41} +2.24264 q^{43} +8.00000 q^{47} -1.17157 q^{49} -2.41421 q^{51} +1.82843 q^{53} +2.41421 q^{57} -14.4142 q^{59} +3.41421 q^{61} -6.82843 q^{63} -6.07107 q^{67} +12.6569 q^{69} -3.07107 q^{71} -13.8284 q^{73} +3.41421 q^{77} +0.828427 q^{79} -9.48528 q^{81} -2.48528 q^{83} +9.24264 q^{87} -3.75736 q^{89} +4.41421 q^{91} -8.24264 q^{93} -7.65685 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^7	\( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{13} - 2 q^{17} + 2 q^{19} - 6 q^{21} + 2 q^{23} + 2 q^{27} + 2 q^{29} - 4 q^{31} - 4 q^{33} - 16 q^{37} - 6 q^{39} - 4 q^{43} + 16 q^{47} - 8 q^{49} - 2 q^{51} - 2 q^{53} + 2 q^{57} - 26 q^{59} + 4 q^{61} - 8 q^{63} + 2 q^{67} + 14 q^{69} + 8 q^{71} - 22 q^{73} + 4 q^{77} - 4 q^{79} - 2 q^{81} + 12 q^{83} + 10 q^{87} - 16 q^{89} + 6 q^{91} - 8 q^{93} - 4 q^{97} - 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^13 - 2 * q^17 + 2 * q^19 - 6 * q^21 + 2 * q^23 + 2 * q^27 + 2 * q^29 - 4 * q^31 - 4 * q^33 - 16 * q^37 - 6 * q^39 - 4 * q^43 + 16 * q^47 - 8 * q^49 - 2 * q^51 - 2 * q^53 + 2 * q^57 - 26 * q^59 + 4 * q^61 - 8 * q^63 + 2 * q^67 + 14 * q^69 + 8 * q^71 - 22 * q^73 + 4 * q^77 - 4 * q^79 - 2 * q^81 + 12 * q^83 + 10 * q^87 - 16 * q^89 + 6 * q^91 - 8 * q^93 - 4 * q^97 - 8 * q^99

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