LMFDB - Embedded newform 7600.2.a.ce.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:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 4 $ x^4 - 6*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 380)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.874032$ 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)

$f(q)$	$=$	$q-0.874032 q^{3} -2.82843 q^{7} -2.23607 q^{9} +O(q^{10})$	$q-0.874032 q^{3} -2.82843 q^{7} -2.23607 q^{9} +0.763932 q^{11} -5.45052 q^{13} -7.40492 q^{17} +1.00000 q^{19} +2.47214 q^{21} +1.08036 q^{23} +4.57649 q^{27} -4.47214 q^{29} +4.00000 q^{31} -0.667701 q^{33} -2.62210 q^{37} +4.76393 q^{39} -6.00000 q^{41} -8.48528 q^{43} -8.48528 q^{47} +1.00000 q^{49} +6.47214 q^{51} +2.62210 q^{53} -0.874032 q^{57} +1.52786 q^{59} -11.7082 q^{61} +6.32456 q^{63} -11.1074 q^{67} -0.944272 q^{69} +10.4721 q^{71} +5.24419 q^{73} -2.16073 q^{77} -15.4164 q^{79} +2.70820 q^{81} -13.7295 q^{83} +3.90879 q^{87} +2.94427 q^{89} +15.4164 q^{91} -3.49613 q^{93} +13.9358 q^{97} -1.70820 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 12 q^{11} + 4 q^{19} - 8 q^{21} + 16 q^{31} + 28 q^{39} - 24 q^{41} + 4 q^{49} + 8 q^{51} + 24 q^{59} - 20 q^{61} + 32 q^{69} + 24 q^{71} - 8 q^{79} - 16 q^{81} - 24 q^{89} + 8 q^{91} + 20 q^{99}+O(q^{100}) $ 4 * q + 12 * q^11 + 4 * q^19 - 8 * q^21 + 16 * q^31 + 28 * q^39 - 24 * q^41 + 4 * q^49 + 8 * q^51 + 24 * q^59 - 20 * q^61 + 32 * q^69 + 24 * q^71 - 8 * q^79 - 16 * q^81 - 24 * q^89 + 8 * q^91 + 20 * 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
$3$	−0.874032	−0.504623	−0.252311	−	0.967646i	$-0.581191\pi$
$3$	−0.874032	−0.504623	−0.252311	+	0.967646i	$0.581191\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	0	0
$9$	−2.23607	−0.745356
$10$	0	0
$11$	0.763932	0.230334	0.115167	−	0.993346i	$-0.463260\pi$
$11$	0.763932	0.230334	0.115167	+	0.993346i	$0.463260\pi$
$12$	0	0
$13$	−5.45052	−1.51170	−0.755852	−	0.654743i	$-0.772777\pi$
$13$	−5.45052	−1.51170	−0.755852	+	0.654743i	$0.772777\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.40492	−1.79596	−0.897978	−	0.440040i	$-0.854964\pi$
$17$	−7.40492	−1.79596	−0.897978	+	0.440040i	$0.854964\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	2.47214	0.539464
$22$	0	0
$23$	1.08036	0.225271	0.112636	−	0.993636i	$-0.464071\pi$
$23$	1.08036	0.225271	0.112636	+	0.993636i	$0.464071\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.57649	0.880746
$28$	0	0
$29$	−4.47214	−0.830455	−0.415227	−	0.909718i	$-0.636298\pi$
$29$	−4.47214	−0.830455	−0.415227	+	0.909718i	$0.636298\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	−0.667701	−0.116232
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.62210	−0.431070	−0.215535	−	0.976496i	$-0.569150\pi$
$37$	−2.62210	−0.431070	−0.215535	+	0.976496i	$0.569150\pi$
$38$	0	0
$39$	4.76393	0.762840
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−8.48528	−1.29399	−0.646997	−	0.762493i	$-0.723975\pi$
$43$	−8.48528	−1.29399	−0.646997	+	0.762493i	$0.723975\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.48528	−1.23771	−0.618853	−	0.785507i	$-0.712402\pi$
$47$	−8.48528	−1.23771	−0.618853	+	0.785507i	$0.712402\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.47214	0.906280
$52$	0	0
$53$	2.62210	0.360173	0.180086	−	0.983651i	$-0.442362\pi$
$53$	2.62210	0.360173	0.180086	+	0.983651i	$0.442362\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.874032	−0.115768
$58$	0	0
$59$	1.52786	0.198911	0.0994555	−	0.995042i	$-0.468290\pi$
$59$	1.52786	0.198911	0.0994555	+	0.995042i	$0.468290\pi$
$60$	0	0
$61$	−11.7082	−1.49908	−0.749541	−	0.661958i	$-0.769726\pi$
$61$	−11.7082	−1.49908	−0.749541	+	0.661958i	$0.769726\pi$
$62$	0	0
$63$	6.32456	0.796819
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−11.1074	−1.35698	−0.678491	−	0.734609i	$-0.737366\pi$
$67$	−11.1074	−1.35698	−0.678491	+	0.734609i	$0.737366\pi$
$68$	0	0
$69$	−0.944272	−0.113677
$70$	0	0
$71$	10.4721	1.24281	0.621407	−	0.783488i	$-0.286561\pi$
$71$	10.4721	1.24281	0.621407	+	0.783488i	$0.286561\pi$
$72$	0	0
$73$	5.24419	0.613786	0.306893	−	0.951744i	$-0.400711\pi$
$73$	5.24419	0.613786	0.306893	+	0.951744i	$0.400711\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.16073	−0.246238
$78$	0	0
$79$	−15.4164	−1.73448	−0.867241	−	0.497889i	$-0.834109\pi$
$79$	−15.4164	−1.73448	−0.867241	+	0.497889i	$0.834109\pi$
$80$	0	0
$81$	2.70820	0.300912
$82$	0	0
$83$	−13.7295	−1.50701	−0.753503	−	0.657445i	$-0.771637\pi$
$83$	−13.7295	−1.50701	−0.753503	+	0.657445i	$0.771637\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.90879	0.419066
$88$	0	0
$89$	2.94427	0.312092	0.156046	−	0.987750i	$-0.450125\pi$
$89$	2.94427	0.312092	0.156046	+	0.987750i	$0.450125\pi$
$90$	0	0
$91$	15.4164	1.61608
$92$	0	0
$93$	−3.49613	−0.362532
$94$	0	0
$95$	0	0
$96$	0	0
$97$	13.9358	1.41497	0.707483	−	0.706730i	$-0.249830\pi$
$97$	13.9358	1.41497	0.707483	+	0.706730i	$0.249830\pi$
$98$	0	0
$99$	−1.70820	−0.171681
$100$	0	0
$101$	5.23607	0.521008	0.260504	−	0.965473i	$-0.416111\pi$
$101$	5.23607	0.521008	0.260504	+	0.965473i	$0.416111\pi$
$102$	0	0
$103$	5.86319	0.577717	0.288858	−	0.957372i	$-0.406724\pi$
$103$	5.86319	0.577717	0.288858	+	0.957372i	$0.406724\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.2681	1.28268	0.641338	−	0.767258i	$-0.278380\pi$
$107$	13.2681	1.28268	0.641338	+	0.767258i	$0.278380\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	2.29180	0.217528
$112$	0	0
$113$	−16.3516	−1.53823	−0.769113	−	0.639113i	$-0.779302\pi$
$113$	−16.3516	−1.53823	−0.769113	+	0.639113i	$0.779302\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	12.1877	1.12676
$118$	0	0
$119$	20.9443	1.91996
$120$	0	0
$121$	−10.4164	−0.946946
$122$	0	0
$123$	5.24419	0.472853
$124$	0	0
$125$	0	0
$126$	0	0
$127$	19.5927	1.73857	0.869284	−	0.494314i	$-0.164581\pi$
$127$	19.5927	1.73857	0.869284	+	0.494314i	$0.164581\pi$
$128$	0	0
$129$	7.41641	0.652978
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−2.82843	−0.245256
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−7.40492	−0.632645	−0.316322	−	0.948652i	$-0.602448\pi$
$137$	−7.40492	−0.632645	−0.316322	+	0.948652i	$0.602448\pi$
$138$	0	0
$139$	−2.29180	−0.194388	−0.0971938	−	0.995265i	$-0.530987\pi$
$139$	−2.29180	−0.194388	−0.0971938	+	0.995265i	$0.530987\pi$
$140$	0	0
$141$	7.41641	0.624574
$142$	0	0
$143$	−4.16383	−0.348197
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−0.874032	−0.0720889
$148$	0	0
$149$	−12.6525	−1.03653	−0.518266	−	0.855220i	$-0.673422\pi$
$149$	−12.6525	−1.03653	−0.518266	+	0.855220i	$0.673422\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	16.5579	1.33863
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−0.412662	−0.0329340	−0.0164670	−	0.999864i	$-0.505242\pi$
$157$	−0.412662	−0.0329340	−0.0164670	+	0.999864i	$0.505242\pi$
$158$	0	0
$159$	−2.29180	−0.181751
$160$	0	0
$161$	−3.05573	−0.240825
$162$	0	0
$163$	8.48528	0.664619	0.332309	−	0.943170i	$-0.392172\pi$
$163$	8.48528	0.664619	0.332309	+	0.943170i	$0.392172\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−17.4319	−1.34892	−0.674462	−	0.738310i	$-0.735624\pi$
$167$	−17.4319	−1.34892	−0.674462	+	0.738310i	$0.735624\pi$
$168$	0	0
$169$	16.7082	1.28525
$170$	0	0
$171$	−2.23607	−0.170996
$172$	0	0
$173$	8.94665	0.680201	0.340101	−	0.940389i	$-0.389539\pi$
$173$	8.94665	0.680201	0.340101	+	0.940389i	$0.389539\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1.33540	−0.100375
$178$	0	0
$179$	22.4721	1.67965	0.839823	−	0.542860i	$-0.182659\pi$
$179$	22.4721	1.67965	0.839823	+	0.542860i	$0.182659\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	10.2333	0.756471
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−5.65685	−0.413670
$188$	0	0
$189$	−12.9443	−0.941557
$190$	0	0
$191$	3.05573	0.221105	0.110552	−	0.993870i	$-0.464738\pi$
$191$	3.05573	0.221105	0.110552	+	0.993870i	$0.464738\pi$
$192$	0	0
$193$	13.9358	1.00312	0.501561	−	0.865123i	$-0.332759\pi$
$193$	13.9358	1.00312	0.501561	+	0.865123i	$0.332759\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.6491	0.901212	0.450606	−	0.892723i	$-0.351208\pi$
$197$	12.6491	0.901212	0.450606	+	0.892723i	$0.351208\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	9.70820	0.684764
$202$	0	0
$203$	12.6491	0.887794
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2.41577	−0.167907
$208$	0	0
$209$	0.763932	0.0528423
$210$	0	0
$211$	−15.4164	−1.06131	−0.530655	−	0.847588i	$-0.678054\pi$
$211$	−15.4164	−1.06131	−0.530655	+	0.847588i	$0.678054\pi$
$212$	0	0
$213$	−9.15298	−0.627152
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−11.3137	−0.768025
$218$	0	0
$219$	−4.58359	−0.309730
$220$	0	0
$221$	40.3607	2.71495
$222$	0	0
$223$	−18.7673	−1.25675	−0.628377	−	0.777909i	$-0.716280\pi$
$223$	−18.7673	−1.25675	−0.628377	+	0.777909i	$0.716280\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5.86319	−0.389153	−0.194577	−	0.980887i	$-0.562333\pi$
$227$	−5.86319	−0.389153	−0.194577	+	0.980887i	$0.562333\pi$
$228$	0	0
$229$	7.70820	0.509372	0.254686	−	0.967024i	$-0.418028\pi$
$229$	7.70820	0.509372	0.254686	+	0.967024i	$0.418028\pi$
$230$	0	0
$231$	1.88854	0.124257
$232$	0	0
$233$	−12.6491	−0.828671	−0.414335	−	0.910124i	$-0.635986\pi$
$233$	−12.6491	−0.828671	−0.414335	+	0.910124i	$0.635986\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	13.4744	0.875259
$238$	0	0
$239$	29.8885	1.93333	0.966665	−	0.256046i	$-0.0824199\pi$
$239$	29.8885	1.93333	0.966665	+	0.256046i	$0.0824199\pi$
$240$	0	0
$241$	−28.8328	−1.85728	−0.928642	−	0.370976i	$-0.879023\pi$
$241$	−28.8328	−1.85728	−0.928642	+	0.370976i	$0.879023\pi$
$242$	0	0
$243$	−16.0965	−1.03259
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5.45052	−0.346808
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	5.88854	0.371682	0.185841	−	0.982580i	$-0.440499\pi$
$251$	5.88854	0.371682	0.185841	+	0.982580i	$0.440499\pi$
$252$	0	0
$253$	0.825324	0.0518877
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−17.4319	−1.08737	−0.543687	−	0.839288i	$-0.682972\pi$
$257$	−17.4319	−1.08737	−0.543687	+	0.839288i	$0.682972\pi$
$258$	0	0
$259$	7.41641	0.460833
$260$	0	0
$261$	10.0000	0.618984
$262$	0	0
$263$	−15.8902	−0.979832	−0.489916	−	0.871770i	$-0.662973\pi$
$263$	−15.8902	−0.979832	−0.489916	+	0.871770i	$0.662973\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−2.57339	−0.157489
$268$	0	0
$269$	14.9443	0.911168	0.455584	−	0.890193i	$-0.349430\pi$
$269$	14.9443	0.911168	0.455584	+	0.890193i	$0.349430\pi$
$270$	0	0
$271$	−21.1246	−1.28323	−0.641614	−	0.767027i	$-0.721735\pi$
$271$	−21.1246	−1.28323	−0.641614	+	0.767027i	$0.721735\pi$
$272$	0	0
$273$	−13.4744	−0.815510
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−22.2148	−1.33476	−0.667378	−	0.744719i	$-0.732583\pi$
$277$	−22.2148	−1.33476	−0.667378	+	0.744719i	$0.732583\pi$
$278$	0	0
$279$	−8.94427	−0.535480
$280$	0	0
$281$	−25.4164	−1.51622	−0.758108	−	0.652129i	$-0.773876\pi$
$281$	−25.4164	−1.51622	−0.758108	+	0.652129i	$0.773876\pi$
$282$	0	0
$283$	2.41577	0.143602	0.0718012	−	0.997419i	$-0.477125\pi$
$283$	2.41577	0.143602	0.0718012	+	0.997419i	$0.477125\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	16.9706	1.00174
$288$	0	0
$289$	37.8328	2.22546
$290$	0	0
$291$	−12.1803	−0.714024
$292$	0	0
$293$	2.62210	0.153184	0.0765922	−	0.997062i	$-0.475596\pi$
$293$	2.62210	0.153184	0.0765922	+	0.997062i	$0.475596\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.49613	0.202866
$298$	0	0
$299$	−5.88854	−0.340543
$300$	0	0
$301$	24.0000	1.38334
$302$	0	0
$303$	−4.57649	−0.262913
$304$	0	0
$305$	0	0
$306$	0	0
$307$	5.45052	0.311078	0.155539	−	0.987830i	$-0.450289\pi$
$307$	5.45052	0.311078	0.155539	+	0.987830i	$0.450289\pi$
$308$	0	0
$309$	−5.12461	−0.291529
$310$	0	0
$311$	9.70820	0.550502	0.275251	−	0.961372i	$-0.411239\pi$
$311$	9.70820	0.550502	0.275251	+	0.961372i	$0.411239\pi$
$312$	0	0
$313$	−10.4884	−0.592839	−0.296419	−	0.955058i	$-0.595793\pi$
$313$	−10.4884	−0.592839	−0.296419	+	0.955058i	$0.595793\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.94665	−0.502494	−0.251247	−	0.967923i	$-0.580841\pi$
$317$	−8.94665	−0.502494	−0.251247	+	0.967923i	$0.580841\pi$
$318$	0	0
$319$	−3.41641	−0.191282
$320$	0	0
$321$	−11.5967	−0.647267
$322$	0	0
$323$	−7.40492	−0.412021
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1.74806	−0.0966682
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	−7.41641	−0.407643	−0.203821	−	0.979008i	$-0.565336\pi$
$331$	−7.41641	−0.407643	−0.203821	+	0.979008i	$0.565336\pi$
$332$	0	0
$333$	5.86319	0.321301
$334$	0	0
$335$	0	0
$336$	0	0
$337$	16.7642	0.913206	0.456603	−	0.889671i	$-0.349066\pi$
$337$	16.7642	0.913206	0.456603	+	0.889671i	$0.349066\pi$
$338$	0	0
$339$	14.2918	0.776224
$340$	0	0
$341$	3.05573	0.165477
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9.40802	0.505049	0.252525	−	0.967590i	$-0.418739\pi$
$347$	9.40802	0.505049	0.252525	+	0.967590i	$0.418739\pi$
$348$	0	0
$349$	25.4164	1.36051	0.680255	−	0.732976i	$-0.261869\pi$
$349$	25.4164	1.36051	0.680255	+	0.732976i	$0.261869\pi$
$350$	0	0
$351$	−24.9443	−1.33143
$352$	0	0
$353$	9.56564	0.509128	0.254564	−	0.967056i	$-0.418068\pi$
$353$	9.56564	0.509128	0.254564	+	0.967056i	$0.418068\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−18.3060	−0.968854
$358$	0	0
$359$	−29.1246	−1.53714	−0.768569	−	0.639767i	$-0.779031\pi$
$359$	−29.1246	−1.53714	−0.768569	+	0.639767i	$0.779031\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	9.10427	0.477850
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−8.48528	−0.442928	−0.221464	−	0.975169i	$-0.571084\pi$
$367$	−8.48528	−0.442928	−0.221464	+	0.975169i	$0.571084\pi$
$368$	0	0
$369$	13.4164	0.698430
$370$	0	0
$371$	−7.41641	−0.385041
$372$	0	0
$373$	13.1105	0.678835	0.339417	−	0.940636i	$-0.389770\pi$
$373$	13.1105	0.678835	0.339417	+	0.940636i	$0.389770\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	24.3755	1.25540
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	−17.1246	−0.877320
$382$	0	0
$383$	36.4056	1.86024	0.930120	−	0.367257i	$-0.119703\pi$
$383$	36.4056	1.86024	0.930120	+	0.367257i	$0.119703\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	18.9737	0.964486
$388$	0	0
$389$	−14.9443	−0.757705	−0.378852	−	0.925457i	$-0.623681\pi$
$389$	−14.9443	−0.757705	−0.378852	+	0.925457i	$0.623681\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	11.7264	0.588530	0.294265	−	0.955724i	$-0.404925\pi$
$397$	11.7264	0.588530	0.294265	+	0.955724i	$0.404925\pi$
$398$	0	0
$399$	2.47214	0.123762
$400$	0	0
$401$	−4.47214	−0.223328	−0.111664	−	0.993746i	$-0.535618\pi$
$401$	−4.47214	−0.223328	−0.111664	+	0.993746i	$0.535618\pi$
$402$	0	0
$403$	−21.8021	−1.08604
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.00310	−0.0992901
$408$	0	0
$409$	17.4164	0.861186	0.430593	−	0.902546i	$-0.358304\pi$
$409$	17.4164	0.861186	0.430593	+	0.902546i	$0.358304\pi$
$410$	0	0
$411$	6.47214	0.319247
$412$	0	0
$413$	−4.32145	−0.212645
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.00310	0.0980924
$418$	0	0
$419$	20.9443	1.02319	0.511597	−	0.859225i	$-0.329054\pi$
$419$	20.9443	1.02319	0.511597	+	0.859225i	$0.329054\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	0	0
$423$	18.9737	0.922531
$424$	0	0
$425$	0	0
$426$	0	0
$427$	33.1158	1.60259
$428$	0	0
$429$	3.63932	0.175708
$430$	0	0
$431$	−1.52786	−0.0735946	−0.0367973	−	0.999323i	$-0.511716\pi$
$431$	−1.52786	−0.0735946	−0.0367973	+	0.999323i	$0.511716\pi$
$432$	0	0
$433$	−28.4906	−1.36917	−0.684585	−	0.728933i	$-0.740017\pi$
$433$	−28.4906	−1.36917	−0.684585	+	0.728933i	$0.740017\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.08036	0.0516808
$438$	0	0
$439$	18.8328	0.898841	0.449421	−	0.893320i	$-0.351630\pi$
$439$	18.8328	0.898841	0.449421	+	0.893320i	$0.351630\pi$
$440$	0	0
$441$	−2.23607	−0.106479
$442$	0	0
$443$	13.7295	0.652307	0.326153	−	0.945317i	$-0.394247\pi$
$443$	13.7295	0.652307	0.326153	+	0.945317i	$0.394247\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	11.0587	0.523057
$448$	0	0
$449$	−16.4721	−0.777368	−0.388684	−	0.921371i	$-0.627070\pi$
$449$	−16.4721	−0.777368	−0.388684	+	0.921371i	$0.627070\pi$
$450$	0	0
$451$	−4.58359	−0.215833
$452$	0	0
$453$	−6.99226	−0.328525
$454$	0	0
$455$	0	0
$456$	0	0
$457$	37.9473	1.77510	0.887551	−	0.460710i	$-0.152405\pi$
$457$	37.9473	1.77510	0.887551	+	0.460710i	$0.152405\pi$
$458$	0	0
$459$	−33.8885	−1.58178
$460$	0	0
$461$	23.8885	1.11260	0.556300	−	0.830981i	$-0.312220\pi$
$461$	23.8885	1.11260	0.556300	+	0.830981i	$0.312220\pi$
$462$	0	0
$463$	−14.5548	−0.676419	−0.338209	−	0.941071i	$-0.609821\pi$
$463$	−14.5548	−0.676419	−0.338209	+	0.941071i	$0.609821\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−30.7000	−1.42063	−0.710314	−	0.703885i	$-0.751447\pi$
$467$	−30.7000	−1.42063	−0.710314	+	0.703885i	$0.751447\pi$
$468$	0	0
$469$	31.4164	1.45067
$470$	0	0
$471$	0.360680	0.0166192
$472$	0	0
$473$	−6.48218	−0.298051
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−5.86319	−0.268457
$478$	0	0
$479$	−11.2361	−0.513389	−0.256695	−	0.966493i	$-0.582633\pi$
$479$	−11.2361	−0.513389	−0.256695	+	0.966493i	$0.582633\pi$
$480$	0	0
$481$	14.2918	0.651650
$482$	0	0
$483$	2.67080	0.121526
$484$	0	0
$485$	0	0
$486$	0	0
$487$	10.6947	0.484624	0.242312	−	0.970198i	$-0.422094\pi$
$487$	10.6947	0.484624	0.242312	+	0.970198i	$0.422094\pi$
$488$	0	0
$489$	−7.41641	−0.335382
$490$	0	0
$491$	−5.88854	−0.265746	−0.132873	−	0.991133i	$-0.542420\pi$
$491$	−5.88854	−0.265746	−0.132873	+	0.991133i	$0.542420\pi$
$492$	0	0
$493$	33.1158	1.49146
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−29.6197	−1.32862
$498$	0	0
$499$	17.1246	0.766603	0.383301	−	0.923623i	$-0.374787\pi$
$499$	17.1246	0.766603	0.383301	+	0.923623i	$0.374787\pi$
$500$	0	0
$501$	15.2361	0.680697
$502$	0	0
$503$	−20.2117	−0.901193	−0.450597	−	0.892728i	$-0.648789\pi$
$503$	−20.2117	−0.901193	−0.450597	+	0.892728i	$0.648789\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−14.6035	−0.648564
$508$	0	0
$509$	−10.5836	−0.469109	−0.234555	−	0.972103i	$-0.575363\pi$
$509$	−10.5836	−0.469109	−0.234555	+	0.972103i	$0.575363\pi$
$510$	0	0
$511$	−14.8328	−0.656165
$512$	0	0
$513$	4.57649	0.202057
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−6.48218	−0.285086
$518$	0	0
$519$	−7.81966	−0.343245
$520$	0	0
$521$	0.111456	0.00488298	0.00244149	−	0.999997i	$-0.499223\pi$
$521$	0.111456	0.00488298	0.00244149	+	0.999997i	$0.499223\pi$
$522$	0	0
$523$	24.8369	1.08604	0.543020	−	0.839720i	$-0.317281\pi$
$523$	24.8369	1.08604	0.543020	+	0.839720i	$0.317281\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−29.6197	−1.29025
$528$	0	0
$529$	−21.8328	−0.949253
$530$	0	0
$531$	−3.41641	−0.148259
$532$	0	0
$533$	32.7031	1.41653
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−19.6414	−0.847588
$538$	0	0
$539$	0.763932	0.0329049
$540$	0	0
$541$	11.7082	0.503375	0.251688	−	0.967809i	$-0.419014\pi$
$541$	11.7082	0.503375	0.251688	+	0.967809i	$0.419014\pi$
$542$	0	0
$543$	−5.24419	−0.225050
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8.27895	0.353982	0.176991	−	0.984212i	$-0.443364\pi$
$547$	8.27895	0.353982	0.176991	+	0.984212i	$0.443364\pi$
$548$	0	0
$549$	26.1803	1.11735
$550$	0	0
$551$	−4.47214	−0.190519
$552$	0	0
$553$	43.6042	1.85424
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20.0540	0.849716	0.424858	−	0.905260i	$-0.360324\pi$
$557$	20.0540	0.849716	0.424858	+	0.905260i	$0.360324\pi$
$558$	0	0
$559$	46.2492	1.95613
$560$	0	0
$561$	4.94427	0.208747
$562$	0	0
$563$	−30.0810	−1.26776	−0.633882	−	0.773429i	$-0.718540\pi$
$563$	−30.0810	−1.26776	−0.633882	+	0.773429i	$0.718540\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−7.65996	−0.321688
$568$	0	0
$569$	−38.9443	−1.63263	−0.816314	−	0.577608i	$-0.803986\pi$
$569$	−38.9443	−1.63263	−0.816314	+	0.577608i	$0.803986\pi$
$570$	0	0
$571$	−25.1246	−1.05143	−0.525716	−	0.850660i	$-0.676203\pi$
$571$	−25.1246	−1.05143	−0.525716	+	0.850660i	$0.676203\pi$
$572$	0	0
$573$	−2.67080	−0.111574
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−5.65685	−0.235498	−0.117749	−	0.993043i	$-0.537568\pi$
$577$	−5.65685	−0.235498	−0.117749	+	0.993043i	$0.537568\pi$
$578$	0	0
$579$	−12.1803	−0.506198
$580$	0	0
$581$	38.8328	1.61106
$582$	0	0
$583$	2.00310	0.0829601
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−5.40182	−0.222957	−0.111478	−	0.993767i	$-0.535559\pi$
$587$	−5.40182	−0.222957	−0.111478	+	0.993767i	$0.535559\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	−11.0557	−0.454772
$592$	0	0
$593$	−2.16073	−0.0887304	−0.0443652	−	0.999015i	$-0.514127\pi$
$593$	−2.16073	−0.0887304	−0.0443652	+	0.999015i	$0.514127\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−13.9845	−0.572348
$598$	0	0
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	0	0
$601$	16.8328	0.686625	0.343312	−	0.939221i	$-0.388451\pi$
$601$	16.8328	0.686625	0.343312	+	0.939221i	$0.388451\pi$
$602$	0	0
$603$	24.8369	1.01143
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−22.4211	−0.910044	−0.455022	−	0.890480i	$-0.650369\pi$
$607$	−22.4211	−0.910044	−0.455022	+	0.890480i	$0.650369\pi$
$608$	0	0
$609$	−11.0557	−0.448001
$610$	0	0
$611$	46.2492	1.87104
$612$	0	0
$613$	−39.1853	−1.58268	−0.791340	−	0.611376i	$-0.790616\pi$
$613$	−39.1853	−1.58268	−0.791340	+	0.611376i	$0.790616\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.08347	0.124136	0.0620678	−	0.998072i	$-0.480230\pi$
$617$	3.08347	0.124136	0.0620678	+	0.998072i	$0.480230\pi$
$618$	0	0
$619$	−41.1246	−1.65294	−0.826469	−	0.562982i	$-0.809654\pi$
$619$	−41.1246	−1.65294	−0.826469	+	0.562982i	$0.809654\pi$
$620$	0	0
$621$	4.94427	0.198407
$622$	0	0
$623$	−8.32766	−0.333641
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−0.667701	−0.0266654
$628$	0	0
$629$	19.4164	0.774183
$630$	0	0
$631$	1.70820	0.0680025	0.0340013	−	0.999422i	$-0.489175\pi$
$631$	1.70820	0.0680025	0.0340013	+	0.999422i	$0.489175\pi$
$632$	0	0
$633$	13.4744	0.535561
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−5.45052	−0.215958
$638$	0	0
$639$	−23.4164	−0.926339
$640$	0	0
$641$	−12.1115	−0.478374	−0.239187	−	0.970974i	$-0.576881\pi$
$641$	−12.1115	−0.478374	−0.239187	+	0.970974i	$0.576881\pi$
$642$	0	0
$643$	30.7000	1.21069	0.605346	−	0.795963i	$-0.293035\pi$
$643$	30.7000	1.21069	0.605346	+	0.795963i	$0.293035\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9.40802	0.369867	0.184934	−	0.982751i	$-0.440793\pi$
$647$	9.40802	0.369867	0.184934	+	0.982751i	$0.440793\pi$
$648$	0	0
$649$	1.16718	0.0458160
$650$	0	0
$651$	9.88854	0.387563
$652$	0	0
$653$	−22.2148	−0.869331	−0.434665	−	0.900592i	$-0.643133\pi$
$653$	−22.2148	−0.869331	−0.434665	+	0.900592i	$0.643133\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−11.7264	−0.457489
$658$	0	0
$659$	44.9443	1.75078	0.875390	−	0.483417i	$-0.160605\pi$
$659$	44.9443	1.75078	0.875390	+	0.483417i	$0.160605\pi$
$660$	0	0
$661$	−26.0000	−1.01128	−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000	−1.01128	−0.505641	+	0.862744i	$0.668744\pi$
$662$	0	0
$663$	−35.2765	−1.37003
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.83153	−0.187078
$668$	0	0
$669$	16.4033	0.634186
$670$	0	0
$671$	−8.94427	−0.345290
$672$	0	0
$673$	−4.62520	−0.178288	−0.0891442	−	0.996019i	$-0.528413\pi$
$673$	−4.62520	−0.178288	−0.0891442	+	0.996019i	$0.528413\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	42.7302	1.64225	0.821127	−	0.570746i	$-0.193346\pi$
$677$	42.7302	1.64225	0.821127	+	0.570746i	$0.193346\pi$
$678$	0	0
$679$	−39.4164	−1.51266
$680$	0	0
$681$	5.12461	0.196376
$682$	0	0
$683$	−19.5927	−0.749692	−0.374846	−	0.927087i	$-0.622304\pi$
$683$	−19.5927	−0.749692	−0.374846	+	0.927087i	$0.622304\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−6.73722	−0.257041
$688$	0	0
$689$	−14.2918	−0.544474
$690$	0	0
$691$	−25.1246	−0.955785	−0.477893	−	0.878418i	$-0.658599\pi$
$691$	−25.1246	−0.955785	−0.477893	+	0.878418i	$0.658599\pi$
$692$	0	0
$693$	4.83153	0.183535
$694$	0	0
$695$	0	0
$696$	0	0
$697$	44.4295	1.68289
$698$	0	0
$699$	11.0557	0.418166
$700$	0	0
$701$	0.875388	0.0330630	0.0165315	−	0.999863i	$-0.494738\pi$
$701$	0.875388	0.0330630	0.0165315	+	0.999863i	$0.494738\pi$
$702$	0	0
$703$	−2.62210	−0.0988942
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−14.8098	−0.556981
$708$	0	0
$709$	33.4164	1.25498	0.627490	−	0.778625i	$-0.284082\pi$
$709$	33.4164	1.25498	0.627490	+	0.778625i	$0.284082\pi$
$710$	0	0
$711$	34.4721	1.29281
$712$	0	0
$713$	4.32145	0.161840
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−26.1235	−0.975602
$718$	0	0
$719$	−27.5967	−1.02919	−0.514593	−	0.857435i	$-0.672057\pi$
$719$	−27.5967	−1.02919	−0.514593	+	0.857435i	$0.672057\pi$
$720$	0	0
$721$	−16.5836	−0.617605
$722$	0	0
$723$	25.2008	0.937228
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−8.48528	−0.314702	−0.157351	−	0.987543i	$-0.550295\pi$
$727$	−8.48528	−0.314702	−0.157351	+	0.987543i	$0.550295\pi$
$728$	0	0
$729$	5.94427	0.220158
$730$	0	0
$731$	62.8328	2.32396
$732$	0	0
$733$	−33.9411	−1.25364	−0.626822	−	0.779162i	$-0.715645\pi$
$733$	−33.9411	−1.25364	−0.626822	+	0.779162i	$0.715645\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.48528	−0.312559
$738$	0	0
$739$	18.8328	0.692776	0.346388	−	0.938091i	$-0.387408\pi$
$739$	18.8328	0.692776	0.346388	+	0.938091i	$0.387408\pi$
$740$	0	0
$741$	4.76393	0.175007
$742$	0	0
$743$	2.62210	0.0961954	0.0480977	−	0.998843i	$-0.484684\pi$
$743$	2.62210	0.0961954	0.0480977	+	0.998843i	$0.484684\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	30.7000	1.12326
$748$	0	0
$749$	−37.5279	−1.37124
$750$	0	0
$751$	−20.5836	−0.751106	−0.375553	−	0.926801i	$-0.622547\pi$
$751$	−20.5836	−0.751106	−0.375553	+	0.926801i	$0.622547\pi$
$752$	0	0
$753$	−5.14678	−0.187559
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−38.7727	−1.40922	−0.704608	−	0.709597i	$-0.748877\pi$
$757$	−38.7727	−1.40922	−0.704608	+	0.709597i	$0.748877\pi$
$758$	0	0
$759$	−0.721360	−0.0261837
$760$	0	0
$761$	−42.5410	−1.54211	−0.771055	−	0.636768i	$-0.780271\pi$
$761$	−42.5410	−1.54211	−0.771055	+	0.636768i	$0.780271\pi$
$762$	0	0
$763$	−5.65685	−0.204792
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.32766	−0.300694
$768$	0	0
$769$	−38.5410	−1.38982	−0.694912	−	0.719094i	$-0.744557\pi$
$769$	−38.5410	−1.38982	−0.694912	+	0.719094i	$0.744557\pi$
$770$	0	0
$771$	15.2361	0.548714
$772$	0	0
$773$	5.86319	0.210884	0.105442	−	0.994425i	$-0.466374\pi$
$773$	5.86319	0.210884	0.105442	+	0.994425i	$0.466374\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−6.48218	−0.232547
$778$	0	0
$779$	−6.00000	−0.214972
$780$	0	0
$781$	8.00000	0.286263
$782$	0	0
$783$	−20.4667	−0.731420
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−24.8369	−0.885338	−0.442669	−	0.896685i	$-0.645968\pi$
$787$	−24.8369	−0.885338	−0.442669	+	0.896685i	$0.645968\pi$
$788$	0	0
$789$	13.8885	0.494445
$790$	0	0
$791$	46.2492	1.64443
$792$	0	0
$793$	63.8158	2.26617
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−52.2958	−1.85241	−0.926206	−	0.377018i	$-0.876950\pi$
$797$	−52.2958	−1.85241	−0.926206	+	0.377018i	$0.876950\pi$
$798$	0	0
$799$	62.8328	2.22287
$800$	0	0
$801$	−6.58359	−0.232620
$802$	0	0
$803$	4.00621	0.141376
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−13.0618	−0.459796
$808$	0	0
$809$	−7.52786	−0.264666	−0.132333	−	0.991205i	$-0.542247\pi$
$809$	−7.52786	−0.264666	−0.132333	+	0.991205i	$0.542247\pi$
$810$	0	0
$811$	−8.00000	−0.280918	−0.140459	−	0.990086i	$-0.544858\pi$
$811$	−8.00000	−0.280918	−0.140459	+	0.990086i	$0.544858\pi$
$812$	0	0
$813$	18.4636	0.647546
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−8.48528	−0.296862
$818$	0	0
$819$	−34.4721	−1.20455
$820$	0	0
$821$	38.9443	1.35916	0.679582	−	0.733599i	$-0.262161\pi$
$821$	38.9443	1.35916	0.679582	+	0.733599i	$0.262161\pi$
$822$	0	0
$823$	35.9442	1.25294	0.626469	−	0.779447i	$-0.284500\pi$
$823$	35.9442	1.25294	0.626469	+	0.779447i	$0.284500\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.86629	0.273538	0.136769	−	0.990603i	$-0.456328\pi$
$827$	7.86629	0.273538	0.136769	+	0.990603i	$0.456328\pi$
$828$	0	0
$829$	18.0000	0.625166	0.312583	−	0.949890i	$-0.398806\pi$
$829$	18.0000	0.625166	0.312583	+	0.949890i	$0.398806\pi$
$830$	0	0
$831$	19.4164	0.673548
$832$	0	0
$833$	−7.40492	−0.256565
$834$	0	0
$835$	0	0
$836$	0	0
$837$	18.3060	0.632747
$838$	0	0
$839$	−49.3050	−1.70220	−0.851098	−	0.525007i	$-0.824063\pi$
$839$	−49.3050	−1.70220	−0.851098	+	0.525007i	$0.824063\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	0	0
$843$	22.2148	0.765117
$844$	0	0
$845$	0	0
$846$	0	0
$847$	29.4621	1.01233
$848$	0	0
$849$	−2.11146	−0.0724650
$850$	0	0
$851$	−2.83282	−0.0971077
$852$	0	0
$853$	−27.4589	−0.940176	−0.470088	−	0.882619i	$-0.655778\pi$
$853$	−27.4589	−0.940176	−0.470088	+	0.882619i	$0.655778\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	31.3190	1.06984	0.534919	−	0.844903i	$-0.320342\pi$
$857$	31.3190	1.06984	0.534919	+	0.844903i	$0.320342\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	0	0
$861$	−14.8328	−0.505501
$862$	0	0
$863$	−7.86629	−0.267772	−0.133886	−	0.990997i	$-0.542746\pi$
$863$	−7.86629	−0.267772	−0.133886	+	0.990997i	$0.542746\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−33.0671	−1.12302
$868$	0	0
$869$	−11.7771	−0.399510
$870$	0	0
$871$	60.5410	2.05135
$872$	0	0
$873$	−31.1614	−1.05465
$874$	0	0
$875$	0	0
$876$	0	0
$877$	13.9358	0.470579	0.235289	−	0.971925i	$-0.424396\pi$
$877$	13.9358	0.470579	0.235289	+	0.971925i	$0.424396\pi$
$878$	0	0
$879$	−2.29180	−0.0773004
$880$	0	0
$881$	0.652476	0.0219825	0.0109912	−	0.999940i	$-0.496501\pi$
$881$	0.652476	0.0219825	0.0109912	+	0.999940i	$0.496501\pi$
$882$	0	0
$883$	52.9148	1.78072	0.890362	−	0.455253i	$-0.150451\pi$
$883$	52.9148	1.78072	0.890362	+	0.455253i	$0.150451\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	49.0547	1.64710	0.823548	−	0.567247i	$-0.191991\pi$
$887$	49.0547	1.64710	0.823548	+	0.567247i	$0.191991\pi$
$888$	0	0
$889$	−55.4164	−1.85861
$890$	0	0
$891$	2.06888	0.0693102
$892$	0	0
$893$	−8.48528	−0.283949
$894$	0	0
$895$	0	0
$896$	0	0
$897$	5.14678	0.171846
$898$	0	0
$899$	−17.8885	−0.596616
$900$	0	0
$901$	−19.4164	−0.646854
$902$	0	0
$903$	−20.9768	−0.698063
$904$	0	0
$905$	0	0
$906$	0	0
$907$	38.5663	1.28057	0.640287	−	0.768136i	$-0.278815\pi$
$907$	38.5663	1.28057	0.640287	+	0.768136i	$0.278815\pi$
$908$	0	0
$909$	−11.7082	−0.388337
$910$	0	0
$911$	46.4721	1.53969	0.769845	−	0.638231i	$-0.220333\pi$
$911$	46.4721	1.53969	0.769845	+	0.638231i	$0.220333\pi$
$912$	0	0
$913$	−10.4884	−0.347115
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−26.8328	−0.885133	−0.442566	−	0.896736i	$-0.645932\pi$
$919$	−26.8328	−0.885133	−0.442566	+	0.896736i	$0.645932\pi$
$920$	0	0
$921$	−4.76393	−0.156977
$922$	0	0
$923$	−57.0786	−1.87877
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−13.1105	−0.430605
$928$	0	0
$929$	7.52786	0.246981	0.123491	−	0.992346i	$-0.460591\pi$
$929$	7.52786	0.246981	0.123491	+	0.992346i	$0.460591\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	−8.48528	−0.277796
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−43.1915	−1.41101	−0.705503	−	0.708707i	$-0.749279\pi$
$937$	−43.1915	−1.41101	−0.705503	+	0.708707i	$0.749279\pi$
$938$	0	0
$939$	9.16718	0.299160
$940$	0	0
$941$	−38.9443	−1.26955	−0.634773	−	0.772698i	$-0.718907\pi$
$941$	−38.9443	−1.26955	−0.634773	+	0.772698i	$0.718907\pi$
$942$	0	0
$943$	−6.48218	−0.211089
$944$	0	0
$945$	0	0
$946$	0	0
$947$	47.6706	1.54909	0.774543	−	0.632521i	$-0.217980\pi$
$947$	47.6706	1.54909	0.774543	+	0.632521i	$0.217980\pi$
$948$	0	0
$949$	−28.5836	−0.927863
$950$	0	0
$951$	7.81966	0.253570
$952$	0	0
$953$	−36.5632	−1.18440	−0.592199	−	0.805791i	$-0.701740\pi$
$953$	−36.5632	−1.18440	−0.592199	+	0.805791i	$0.701740\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.98605	0.0965253
$958$	0	0
$959$	20.9443	0.676326
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−29.6684	−0.956050
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.41577	0.0776858	0.0388429	−	0.999245i	$-0.487633\pi$
$967$	2.41577	0.0776858	0.0388429	+	0.999245i	$0.487633\pi$
$968$	0	0
$969$	6.47214	0.207915
$970$	0	0
$971$	40.3607	1.29524	0.647618	−	0.761965i	$-0.275765\pi$
$971$	40.3607	1.29524	0.647618	+	0.761965i	$0.275765\pi$
$972$	0	0
$973$	6.48218	0.207809
$974$	0	0
$975$	0	0
$976$	0	0
$977$	24.9945	0.799644	0.399822	−	0.916593i	$-0.369072\pi$
$977$	24.9945	0.799644	0.399822	+	0.916593i	$0.369072\pi$
$978$	0	0
$979$	2.24922	0.0718855
$980$	0	0
$981$	−4.47214	−0.142784
$982$	0	0
$983$	30.2387	0.964464	0.482232	−	0.876044i	$-0.339826\pi$
$983$	30.2387	0.964464	0.482232	+	0.876044i	$0.339826\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−20.9768	−0.667698
$988$	0	0
$989$	−9.16718	−0.291500
$990$	0	0
$991$	−38.8328	−1.23357	−0.616783	−	0.787134i	$-0.711564\pi$
$991$	−38.8328	−1.23357	−0.616783	+	0.787134i	$0.711564\pi$
$992$	0	0
$993$	6.48218	0.205706
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−11.7264	−0.371378	−0.185689	−	0.982609i	$-0.559452\pi$
$997$	−11.7264	−0.371378	−0.185689	+	0.982609i	$0.559452\pi$
$998$	0	0
$999$	−12.0000	−0.379663

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.ce.1.2		4
4.3	odd	2		1900.2.a.j.1.3		4
5.2	odd	4		1520.2.d.f.609.3		4
5.3	odd	4		1520.2.d.f.609.2		4
5.4	even	2	inner	7600.2.a.ce.1.3		4
20.3	even	4		380.2.c.a.229.3	yes	4
20.7	even	4		380.2.c.a.229.2	✓	4
20.19	odd	2		1900.2.a.j.1.2		4
60.23	odd	4		3420.2.f.a.1369.1		4
60.47	odd	4		3420.2.f.a.1369.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.c.a.229.2	✓	4	20.7	even	4
380.2.c.a.229.3	yes	4	20.3	even	4
1520.2.d.f.609.2		4	5.3	odd	4
1520.2.d.f.609.3		4	5.2	odd	4
1900.2.a.j.1.2		4	20.19	odd	2
1900.2.a.j.1.3		4	4.3	odd	2
3420.2.f.a.1369.1		4	60.23	odd	4
3420.2.f.a.1369.2		4	60.47	odd	4
7600.2.a.ce.1.2		4	1.1	even	1	trivial
7600.2.a.ce.1.3		4	5.4	even	2	inner

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

\(f(q)\)	\(=\)	\(q-0.874032 q^{3} -2.82843 q^{7} -2.23607 q^{9} +O(q^{10})\)	\(q-0.874032 q^{3} -2.82843 q^{7} -2.23607 q^{9} +0.763932 q^{11} -5.45052 q^{13} -7.40492 q^{17} +1.00000 q^{19} +2.47214 q^{21} +1.08036 q^{23} +4.57649 q^{27} -4.47214 q^{29} +4.00000 q^{31} -0.667701 q^{33} -2.62210 q^{37} +4.76393 q^{39} -6.00000 q^{41} -8.48528 q^{43} -8.48528 q^{47} +1.00000 q^{49} +6.47214 q^{51} +2.62210 q^{53} -0.874032 q^{57} +1.52786 q^{59} -11.7082 q^{61} +6.32456 q^{63} -11.1074 q^{67} -0.944272 q^{69} +10.4721 q^{71} +5.24419 q^{73} -2.16073 q^{77} -15.4164 q^{79} +2.70820 q^{81} -13.7295 q^{83} +3.90879 q^{87} +2.94427 q^{89} +15.4164 q^{91} -3.49613 q^{93} +13.9358 q^{97} -1.70820 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 12 q^{11} + 4 q^{19} - 8 q^{21} + 16 q^{31} + 28 q^{39} - 24 q^{41} + 4 q^{49} + 8 q^{51} + 24 q^{59} - 20 q^{61} + 32 q^{69} + 24 q^{71} - 8 q^{79} - 16 q^{81} - 24 q^{89} + 8 q^{91} + 20 q^{99}+O(q^{100}) \) 4 * q + 12 * q^11 + 4 * q^19 - 8 * q^21 + 16 * q^31 + 28 * q^39 - 24 * q^41 + 4 * q^49 + 8 * q^51 + 24 * q^59 - 20 * q^61 + 32 * q^69 + 24 * q^71 - 8 * q^79 - 16 * q^81 - 24 * q^89 + 8 * q^91 + 20 * q^99

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