LMFDB - Embedded newform 7600.2.a.bh.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7600,2,Mod(1,7600)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [3,0,-2,0,0,0,-4,0,1,0,-1,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$60.6863055362$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.169.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x - 1 $ x^3 - x^2 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 475)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.37720$ 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)

$f(q)$	$=$	$q-2.37720 q^{3} +0.377203 q^{7} +2.65109 q^{9} +1.37720 q^{11} +2.82167 q^{13} +6.37720 q^{17} -1.00000 q^{19} -0.896688 q^{21} -6.19887 q^{23} +0.829422 q^{27} -3.37720 q^{29} -2.48052 q^{31} -3.27389 q^{33} +5.58383 q^{37} -6.70769 q^{39} +8.50106 q^{41} +12.1522 q^{43} +6.87826 q^{47} -6.85772 q^{49} -15.1599 q^{51} +11.5478 q^{53} +2.37720 q^{57} -6.05659 q^{59} +5.02830 q^{61} +1.00000 q^{63} -3.22717 q^{67} +14.7360 q^{69} +2.30219 q^{71} -3.19887 q^{73} +0.519485 q^{77} +6.71836 q^{79} -9.92498 q^{81} -18.2165 q^{83} +8.02830 q^{87} +1.50106 q^{89} +1.06434 q^{91} +5.89669 q^{93} -11.7827 q^{97} +3.65109 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} - 4 q^{7} + q^{9} - q^{11} + 3 q^{13} + 14 q^{17} - 3 q^{19} - 6 q^{21} - 8 q^{23} + q^{27} - 5 q^{29} + q^{31} - 8 q^{33} + 5 q^{37} + 11 q^{39} + q^{41} + 5 q^{43} - 9 q^{47} - 7 q^{49}+ \cdots + 4 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 - 4 * q^7 + q^9 - q^11 + 3 * q^13 + 14 * q^17 - 3 * q^19 - 6 * q^21 - 8 * q^23 + q^27 - 5 * q^29 + q^31 - 8 * q^33 + 5 * q^37 + 11 * q^39 + q^41 + 5 * q^43 - 9 * q^47 - 7 * q^49 - 18 * q^51 + 31 * q^53 + 2 * q^57 + 6 * q^59 + 3 * q^61 + 3 * q^63 + 13 * q^67 + q^69 - 7 * q^71 + q^73 + 10 * q^77 + 18 * q^79 - 21 * q^81 - 3 * q^83 + 12 * q^87 - 20 * q^89 - 17 * q^91 + 21 * q^93 - 13 * q^97 + 4 * q^99

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
$3$	−2.37720	−1.37248	−0.686239	−	0.727376i	$-0.740740\pi$
$3$	−2.37720	−1.37248	−0.686239	+	0.727376i	$0.740740\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.377203	0.142569	0.0712846	−	0.997456i	$-0.477290\pi$
$7$	0.377203	0.142569	0.0712846	+	0.997456i	$0.477290\pi$
$8$	0	0
$9$	2.65109	0.883698
$10$	0	0
$11$	1.37720	0.415242	0.207621	−	0.978209i	$-0.433428\pi$
$11$	1.37720	0.415242	0.207621	+	0.978209i	$0.433428\pi$
$12$	0	0
$13$	2.82167	0.782591	0.391295	−	0.920265i	$-0.372027\pi$
$13$	2.82167	0.782591	0.391295	+	0.920265i	$0.372027\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.37720	1.54670	0.773349	−	0.633980i	$-0.218580\pi$
$17$	6.37720	1.54670	0.773349	+	0.633980i	$0.218580\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−0.896688	−0.195673
$22$	0	0
$23$	−6.19887	−1.29255	−0.646277	−	0.763103i	$-0.723675\pi$
$23$	−6.19887	−1.29255	−0.646277	+	0.763103i	$0.723675\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0.829422	0.159622
$28$	0	0
$29$	−3.37720	−0.627131	−0.313565	−	0.949567i	$-0.601524\pi$
$29$	−3.37720	−0.627131	−0.313565	+	0.949567i	$0.601524\pi$
$30$	0	0
$31$	−2.48052	−0.445514	−0.222757	−	0.974874i	$-0.571506\pi$
$31$	−2.48052	−0.445514	−0.222757	+	0.974874i	$0.571506\pi$
$32$	0	0
$33$	−3.27389	−0.569911
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.58383	0.917976	0.458988	−	0.888443i	$-0.348212\pi$
$37$	5.58383	0.917976	0.458988	+	0.888443i	$0.348212\pi$
$38$	0	0
$39$	−6.70769	−1.07409
$40$	0	0
$41$	8.50106	1.32764	0.663821	−	0.747891i	$-0.268934\pi$
$41$	8.50106	1.32764	0.663821	+	0.747891i	$0.268934\pi$
$42$	0	0
$43$	12.1522	1.85319	0.926593	−	0.376065i	$-0.122723\pi$
$43$	12.1522	1.85319	0.926593	+	0.376065i	$0.122723\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.87826	1.00330	0.501649	−	0.865071i	$-0.332727\pi$
$47$	6.87826	1.00330	0.501649	+	0.865071i	$0.332727\pi$
$48$	0	0
$49$	−6.85772	−0.979674
$50$	0	0
$51$	−15.1599	−2.12281
$52$	0	0
$53$	11.5478	1.58621	0.793105	−	0.609085i	$-0.208463\pi$
$53$	11.5478	1.58621	0.793105	+	0.609085i	$0.208463\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.37720	0.314868
$58$	0	0
$59$	−6.05659	−0.788501	−0.394251	−	0.919003i	$-0.628996\pi$
$59$	−6.05659	−0.788501	−0.394251	+	0.919003i	$0.628996\pi$
$60$	0	0
$61$	5.02830	0.643807	0.321904	−	0.946772i	$-0.395677\pi$
$61$	5.02830	0.643807	0.321904	+	0.946772i	$0.395677\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−3.22717	−0.394262	−0.197131	−	0.980377i	$-0.563162\pi$
$67$	−3.22717	−0.394262	−0.197131	+	0.980377i	$0.563162\pi$
$68$	0	0
$69$	14.7360	1.77400
$70$	0	0
$71$	2.30219	0.273219	0.136610	−	0.990625i	$-0.456379\pi$
$71$	2.30219	0.273219	0.136610	+	0.990625i	$0.456379\pi$
$72$	0	0
$73$	−3.19887	−0.374400	−0.187200	−	0.982322i	$-0.559941\pi$
$73$	−3.19887	−0.374400	−0.187200	+	0.982322i	$0.559941\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.519485	0.0592008
$78$	0	0
$79$	6.71836	0.755874	0.377937	−	0.925831i	$-0.376634\pi$
$79$	6.71836	0.755874	0.377937	+	0.925831i	$0.376634\pi$
$80$	0	0
$81$	−9.92498	−1.10278
$82$	0	0
$83$	−18.2165	−1.99952	−0.999760	−	0.0218996i	$-0.993029\pi$
$83$	−18.2165	−1.99952	−0.999760	+	0.0218996i	$0.993029\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.02830	0.860724
$88$	0	0
$89$	1.50106	0.159112	0.0795561	−	0.996830i	$-0.474650\pi$
$89$	1.50106	0.159112	0.0795561	+	0.996830i	$0.474650\pi$
$90$	0	0
$91$	1.06434	0.111573
$92$	0	0
$93$	5.89669	0.611458
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−11.7827	−1.19635	−0.598176	−	0.801365i	$-0.704108\pi$
$97$	−11.7827	−1.19635	−0.598176	+	0.801365i	$0.704108\pi$
$98$	0	0
$99$	3.65109	0.366949
$100$	0	0
$101$	−9.18820	−0.914260	−0.457130	−	0.889400i	$-0.651123\pi$
$101$	−9.18820	−0.914260	−0.457130	+	0.889400i	$0.651123\pi$
$102$	0	0
$103$	9.47277	0.933379	0.466690	−	0.884421i	$-0.345447\pi$
$103$	9.47277	0.933379	0.466690	+	0.884421i	$0.345447\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.3510	1.48404	0.742020	−	0.670378i	$-0.233868\pi$
$107$	15.3510	1.48404	0.742020	+	0.670378i	$0.233868\pi$
$108$	0	0
$109$	−16.7643	−1.60573	−0.802863	−	0.596163i	$-0.796691\pi$
$109$	−16.7643	−1.60573	−0.802863	+	0.596163i	$0.796691\pi$
$110$	0	0
$111$	−13.2739	−1.25990
$112$	0	0
$113$	−13.3305	−1.25403	−0.627013	−	0.779009i	$-0.715723\pi$
$113$	−13.3305	−1.25403	−0.627013	+	0.779009i	$0.715723\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	7.48052	0.691574
$118$	0	0
$119$	2.40550	0.220512
$120$	0	0
$121$	−9.10331	−0.827574
$122$	0	0
$123$	−20.2087	−1.82216
$124$	0	0
$125$	0	0
$126$	0	0
$127$	3.04672	0.270353	0.135176	−	0.990822i	$-0.456840\pi$
$127$	3.04672	0.270353	0.135176	+	0.990822i	$0.456840\pi$
$128$	0	0
$129$	−28.8881	−2.54346
$130$	0	0
$131$	8.70769	0.760794	0.380397	−	0.924823i	$-0.375787\pi$
$131$	8.70769	0.760794	0.380397	+	0.924823i	$0.375787\pi$
$132$	0	0
$133$	−0.377203	−0.0327076
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.59450	0.563406	0.281703	−	0.959502i	$-0.409101\pi$
$137$	6.59450	0.563406	0.281703	+	0.959502i	$0.409101\pi$
$138$	0	0
$139$	10.1677	0.862409	0.431205	−	0.902254i	$-0.358089\pi$
$139$	10.1677	0.862409	0.431205	+	0.902254i	$0.358089\pi$
$140$	0	0
$141$	−16.3510	−1.37701
$142$	0	0
$143$	3.88601	0.324965
$144$	0	0
$145$	0	0
$146$	0	0
$147$	16.3022	1.34458
$148$	0	0
$149$	−5.54003	−0.453857	−0.226929	−	0.973911i	$-0.572868\pi$
$149$	−5.54003	−0.453857	−0.226929	+	0.973911i	$0.572868\pi$
$150$	0	0
$151$	−5.12386	−0.416974	−0.208487	−	0.978025i	$-0.566854\pi$
$151$	−5.12386	−0.416974	−0.208487	+	0.978025i	$0.566854\pi$
$152$	0	0
$153$	16.9066	1.36681
$154$	0	0
$155$	0	0
$156$	0	0
$157$	19.7643	1.57736	0.788681	−	0.614803i	$-0.210765\pi$
$157$	19.7643	1.57736	0.788681	+	0.614803i	$0.210765\pi$
$158$	0	0
$159$	−27.4514	−2.17704
$160$	0	0
$161$	−2.33823	−0.184279
$162$	0	0
$163$	−13.5195	−1.05893	−0.529464	−	0.848332i	$-0.677607\pi$
$163$	−13.5195	−1.05893	−0.529464	+	0.848332i	$0.677607\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−13.1054	−1.01413	−0.507065	−	0.861908i	$-0.669269\pi$
$167$	−13.1054	−1.01413	−0.507065	+	0.861908i	$0.669269\pi$
$168$	0	0
$169$	−5.03817	−0.387551
$170$	0	0
$171$	−2.65109	−0.202734
$172$	0	0
$173$	1.48827	0.113151	0.0565754	−	0.998398i	$-0.481982\pi$
$173$	1.48827	0.113151	0.0565754	+	0.998398i	$0.481982\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	14.3977	1.08220
$178$	0	0
$179$	17.1132	1.27910	0.639550	−	0.768750i	$-0.279121\pi$
$179$	17.1132	1.27910	0.639550	+	0.768750i	$0.279121\pi$
$180$	0	0
$181$	−5.73891	−0.426569	−0.213285	−	0.976990i	$-0.568416\pi$
$181$	−5.73891	−0.426569	−0.213285	+	0.976990i	$0.568416\pi$
$182$	0	0
$183$	−11.9533	−0.883612
$184$	0	0
$185$	0	0
$186$	0	0
$187$	8.78270	0.642255
$188$	0	0
$189$	0.312860	0.0227572
$190$	0	0
$191$	−22.3948	−1.62043	−0.810216	−	0.586131i	$-0.800650\pi$
$191$	−22.3948	−1.62043	−0.810216	+	0.586131i	$0.800650\pi$
$192$	0	0
$193$	−21.3687	−1.53815	−0.769075	−	0.639159i	$-0.779283\pi$
$193$	−21.3687	−1.53815	−0.769075	+	0.639159i	$0.779283\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.11399	0.0793682	0.0396841	−	0.999212i	$-0.487365\pi$
$197$	1.11399	0.0793682	0.0396841	+	0.999212i	$0.487365\pi$
$198$	0	0
$199$	2.22505	0.157729	0.0788647	−	0.996885i	$-0.474870\pi$
$199$	2.22505	0.157729	0.0788647	+	0.996885i	$0.474870\pi$
$200$	0	0
$201$	7.67164	0.541116
$202$	0	0
$203$	−1.27389	−0.0894096
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−16.4338	−1.14223
$208$	0	0
$209$	−1.37720	−0.0952631
$210$	0	0
$211$	9.75441	0.671521	0.335760	−	0.941947i	$-0.391007\pi$
$211$	9.75441	0.671521	0.335760	+	0.941947i	$0.391007\pi$
$212$	0	0
$213$	−5.47277	−0.374988
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−0.935657	−0.0635166
$218$	0	0
$219$	7.60437	0.513856
$220$	0	0
$221$	17.9944	1.21043
$222$	0	0
$223$	18.6433	1.24845	0.624225	−	0.781244i	$-0.285415\pi$
$223$	18.6433	1.24845	0.624225	+	0.781244i	$0.285415\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.04672	0.534080	0.267040	−	0.963686i	$-0.413954\pi$
$227$	8.04672	0.534080	0.267040	+	0.963686i	$0.413954\pi$
$228$	0	0
$229$	20.1415	1.33099	0.665493	−	0.746404i	$-0.268221\pi$
$229$	20.1415	1.33099	0.665493	+	0.746404i	$0.268221\pi$
$230$	0	0
$231$	−1.23492	−0.0812518
$232$	0	0
$233$	−1.73386	−0.113589	−0.0567945	−	0.998386i	$-0.518088\pi$
$233$	−1.73386	−0.113589	−0.0567945	+	0.998386i	$0.518088\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−15.9709	−1.03742
$238$	0	0
$239$	18.3150	1.18470	0.592349	−	0.805682i	$-0.298201\pi$
$239$	18.3150	1.18470	0.592349	+	0.805682i	$0.298201\pi$
$240$	0	0
$241$	−2.19675	−0.141505	−0.0707526	−	0.997494i	$-0.522540\pi$
$241$	−2.19675	−0.141505	−0.0707526	+	0.997494i	$0.522540\pi$
$242$	0	0
$243$	21.1054	1.35391
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.82167	−0.179539
$248$	0	0
$249$	43.3043	2.74430
$250$	0	0
$251$	13.6716	0.862946	0.431473	−	0.902126i	$-0.357994\pi$
$251$	13.6716	0.862946	0.431473	+	0.902126i	$0.357994\pi$
$252$	0	0
$253$	−8.53711	−0.536723
$254$	0	0
$255$	0	0
$256$	0	0
$257$	28.4904	1.77718	0.888591	−	0.458701i	$-0.151685\pi$
$257$	28.4904	1.77718	0.888591	+	0.458701i	$0.151685\pi$
$258$	0	0
$259$	2.10624	0.130875
$260$	0	0
$261$	−8.95328	−0.554194
$262$	0	0
$263$	−5.36945	−0.331095	−0.165547	−	0.986202i	$-0.552939\pi$
$263$	−5.36945	−0.331095	−0.165547	+	0.986202i	$0.552939\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−3.56833	−0.218378
$268$	0	0
$269$	7.06727	0.430899	0.215449	−	0.976515i	$-0.430878\pi$
$269$	7.06727	0.430899	0.215449	+	0.976515i	$0.430878\pi$
$270$	0	0
$271$	−21.6970	−1.31800	−0.659000	−	0.752143i	$-0.729020\pi$
$271$	−21.6970	−1.31800	−0.659000	+	0.752143i	$0.729020\pi$
$272$	0	0
$273$	−2.53016	−0.153132
$274$	0	0
$275$	0	0
$276$	0	0
$277$	25.1132	1.50891	0.754453	−	0.656355i	$-0.227902\pi$
$277$	25.1132	1.50891	0.754453	+	0.656355i	$0.227902\pi$
$278$	0	0
$279$	−6.57608	−0.393699
$280$	0	0
$281$	−10.0411	−0.599001	−0.299501	−	0.954096i	$-0.596820\pi$
$281$	−10.0411	−0.599001	−0.299501	+	0.954096i	$0.596820\pi$
$282$	0	0
$283$	20.9143	1.24323	0.621613	−	0.783324i	$-0.286478\pi$
$283$	20.9143	1.24323	0.621613	+	0.783324i	$0.286478\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.20662	0.189281
$288$	0	0
$289$	23.6687	1.39228
$290$	0	0
$291$	28.0099	1.64197
$292$	0	0
$293$	−0.818748	−0.0478318	−0.0239159	−	0.999714i	$-0.507613\pi$
$293$	−0.818748	−0.0478318	−0.0239159	+	0.999714i	$0.507613\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.14228	0.0662819
$298$	0	0
$299$	−17.4912	−1.01154
$300$	0	0
$301$	4.58383	0.264207
$302$	0	0
$303$	21.8422	1.25480
$304$	0	0
$305$	0	0
$306$	0	0
$307$	7.44447	0.424878	0.212439	−	0.977174i	$-0.431859\pi$
$307$	7.44447	0.424878	0.212439	+	0.977174i	$0.431859\pi$
$308$	0	0
$309$	−22.5187	−1.28104
$310$	0	0
$311$	0.956204	0.0542213	0.0271107	−	0.999632i	$-0.491369\pi$
$311$	0.956204	0.0542213	0.0271107	+	0.999632i	$0.491369\pi$
$312$	0	0
$313$	5.98158	0.338099	0.169049	−	0.985608i	$-0.445930\pi$
$313$	5.98158	0.338099	0.169049	+	0.985608i	$0.445930\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.6511	1.27221	0.636106	−	0.771602i	$-0.280544\pi$
$317$	22.6511	1.27221	0.636106	+	0.771602i	$0.280544\pi$
$318$	0	0
$319$	−4.65109	−0.260411
$320$	0	0
$321$	−36.4925	−2.03681
$322$	0	0
$323$	−6.37720	−0.354837
$324$	0	0
$325$	0	0
$326$	0	0
$327$	39.8521	2.20383
$328$	0	0
$329$	2.59450	0.143039
$330$	0	0
$331$	−4.16283	−0.228810	−0.114405	−	0.993434i	$-0.536496\pi$
$331$	−4.16283	−0.228810	−0.114405	+	0.993434i	$0.536496\pi$
$332$	0	0
$333$	14.8032	0.811213
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−18.0304	−0.982180	−0.491090	−	0.871109i	$-0.663401\pi$
$337$	−18.0304	−0.982180	−0.491090	+	0.871109i	$0.663401\pi$
$338$	0	0
$339$	31.6893	1.72112
$340$	0	0
$341$	−3.41617	−0.184996
$342$	0	0
$343$	−5.22717	−0.282241
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7.43380	−0.399067	−0.199534	−	0.979891i	$-0.563943\pi$
$347$	−7.43380	−0.399067	−0.199534	+	0.979891i	$0.563943\pi$
$348$	0	0
$349$	21.9194	1.17332	0.586658	−	0.809835i	$-0.300443\pi$
$349$	21.9194	1.17332	0.586658	+	0.809835i	$0.300443\pi$
$350$	0	0
$351$	2.34036	0.124919
$352$	0	0
$353$	10.3695	0.551910	0.275955	−	0.961171i	$-0.411006\pi$
$353$	10.3695	0.551910	0.275955	+	0.961171i	$0.411006\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−5.71836	−0.302648
$358$	0	0
$359$	23.9893	1.26611	0.633054	−	0.774108i	$-0.281801\pi$
$359$	23.9893	1.26611	0.633054	+	0.774108i	$0.281801\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	21.6404	1.13583
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−35.0275	−1.82842	−0.914210	−	0.405240i	$-0.867188\pi$
$367$	−35.0275	−1.82842	−0.914210	+	0.405240i	$0.867188\pi$
$368$	0	0
$369$	22.5371	1.17323
$370$	0	0
$371$	4.35586	0.226145
$372$	0	0
$373$	30.8003	1.59478	0.797390	−	0.603464i	$-0.206213\pi$
$373$	30.8003	1.59478	0.797390	+	0.603464i	$0.206213\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−9.52936	−0.490787
$378$	0	0
$379$	−0.671640	−0.0344998	−0.0172499	−	0.999851i	$-0.505491\pi$
$379$	−0.671640	−0.0344998	−0.0172499	+	0.999851i	$0.505491\pi$
$380$	0	0
$381$	−7.24267	−0.371053
$382$	0	0
$383$	30.1805	1.54215	0.771075	−	0.636745i	$-0.219720\pi$
$383$	30.1805	1.54215	0.771075	+	0.636745i	$0.219720\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	32.2165	1.63766
$388$	0	0
$389$	−31.5109	−1.59767	−0.798834	−	0.601552i	$-0.794549\pi$
$389$	−31.5109	−1.59767	−0.798834	+	0.601552i	$0.794549\pi$
$390$	0	0
$391$	−39.5315	−1.99919
$392$	0	0
$393$	−20.6999	−1.04417
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−12.5761	−0.631175	−0.315588	−	0.948896i	$-0.602202\pi$
$397$	−12.5761	−0.631175	−0.315588	+	0.948896i	$0.602202\pi$
$398$	0	0
$399$	0.896688	0.0448905
$400$	0	0
$401$	24.6036	1.22864	0.614322	−	0.789056i	$-0.289430\pi$
$401$	24.6036	1.22864	0.614322	+	0.789056i	$0.289430\pi$
$402$	0	0
$403$	−6.99920	−0.348655
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.69006	0.381182
$408$	0	0
$409$	13.6666	0.675770	0.337885	−	0.941187i	$-0.390289\pi$
$409$	13.6666	0.675770	0.337885	+	0.941187i	$0.390289\pi$
$410$	0	0
$411$	−15.6765	−0.773263
$412$	0	0
$413$	−2.28456	−0.112416
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−24.1706	−1.18364
$418$	0	0
$419$	−12.6893	−0.619911	−0.309956	−	0.950751i	$-0.600314\pi$
$419$	−12.6893	−0.619911	−0.309956	+	0.950751i	$0.600314\pi$
$420$	0	0
$421$	29.1826	1.42227	0.711136	−	0.703055i	$-0.248181\pi$
$421$	29.1826	1.42227	0.711136	+	0.703055i	$0.248181\pi$
$422$	0	0
$423$	18.2349	0.886612
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.89669	0.0917872
$428$	0	0
$429$	−9.23784	−0.446007
$430$	0	0
$431$	29.9816	1.44416	0.722081	−	0.691809i	$-0.243186\pi$
$431$	29.9816	1.44416	0.722081	+	0.691809i	$0.243186\pi$
$432$	0	0
$433$	−4.76216	−0.228855	−0.114427	−	0.993432i	$-0.536503\pi$
$433$	−4.76216	−0.228855	−0.114427	+	0.993432i	$0.536503\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.19887	0.296532
$438$	0	0
$439$	−6.88601	−0.328652	−0.164326	−	0.986406i	$-0.552545\pi$
$439$	−6.88601	−0.328652	−0.164326	+	0.986406i	$0.552545\pi$
$440$	0	0
$441$	−18.1805	−0.865736
$442$	0	0
$443$	−8.65109	−0.411026	−0.205513	−	0.978654i	$-0.565886\pi$
$443$	−8.65109	−0.411026	−0.205513	+	0.978654i	$0.565886\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	13.1698	0.622909
$448$	0	0
$449$	8.63055	0.407301	0.203650	−	0.979044i	$-0.434719\pi$
$449$	8.63055	0.407301	0.203650	+	0.979044i	$0.434719\pi$
$450$	0	0
$451$	11.7077	0.551293
$452$	0	0
$453$	12.1805	0.572288
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−28.6092	−1.33828	−0.669141	−	0.743135i	$-0.733338\pi$
$457$	−28.6092	−1.33828	−0.669141	+	0.743135i	$0.733338\pi$
$458$	0	0
$459$	5.28939	0.246888
$460$	0	0
$461$	−39.1025	−1.82119	−0.910593	−	0.413305i	$-0.864374\pi$
$461$	−39.1025	−1.82119	−0.910593	+	0.413305i	$0.864374\pi$
$462$	0	0
$463$	8.04380	0.373827	0.186913	−	0.982376i	$-0.440152\pi$
$463$	8.04380	0.373827	0.186913	+	0.982376i	$0.440152\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−17.0488	−0.788926	−0.394463	−	0.918912i	$-0.629069\pi$
$467$	−17.0488	−0.788926	−0.394463	+	0.918912i	$0.629069\pi$
$468$	0	0
$469$	−1.21730	−0.0562096
$470$	0	0
$471$	−46.9837	−2.16489
$472$	0	0
$473$	16.7360	0.769521
$474$	0	0
$475$	0	0
$476$	0	0
$477$	30.6142	1.40173
$478$	0	0
$479$	−22.5478	−1.03023	−0.515117	−	0.857120i	$-0.672252\pi$
$479$	−22.5478	−1.03023	−0.515117	+	0.857120i	$0.672252\pi$
$480$	0	0
$481$	15.7557	0.718399
$482$	0	0
$483$	5.55845	0.252918
$484$	0	0
$485$	0	0
$486$	0	0
$487$	24.5577	1.11281	0.556407	−	0.830910i	$-0.312180\pi$
$487$	24.5577	1.11281	0.556407	+	0.830910i	$0.312180\pi$
$488$	0	0
$489$	32.1386	1.45336
$490$	0	0
$491$	16.9298	0.764032	0.382016	−	0.924156i	$-0.375230\pi$
$491$	16.9298	0.764032	0.382016	+	0.924156i	$0.375230\pi$
$492$	0	0
$493$	−21.5371	−0.969983
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.868391	0.0389527
$498$	0	0
$499$	−0.418295	−0.0187255	−0.00936273	−	0.999956i	$-0.502980\pi$
$499$	−0.418295	−0.0187255	−0.00936273	+	0.999956i	$0.502980\pi$
$500$	0	0
$501$	31.1543	1.39187
$502$	0	0
$503$	12.5993	0.561776	0.280888	−	0.959741i	$-0.409371\pi$
$503$	12.5993	0.561776	0.280888	+	0.959741i	$0.409371\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	11.9767	0.531906
$508$	0	0
$509$	12.8209	0.568275	0.284138	−	0.958784i	$-0.408293\pi$
$509$	12.8209	0.568275	0.284138	+	0.958784i	$0.408293\pi$
$510$	0	0
$511$	−1.20662	−0.0533779
$512$	0	0
$513$	−0.829422	−0.0366199
$514$	0	0
$515$	0	0
$516$	0	0
$517$	9.47277	0.416612
$518$	0	0
$519$	−3.53791	−0.155297
$520$	0	0
$521$	11.3743	0.498316	0.249158	−	0.968463i	$-0.419846\pi$
$521$	11.3743	0.498316	0.249158	+	0.968463i	$0.419846\pi$
$522$	0	0
$523$	−33.0948	−1.44713	−0.723566	−	0.690255i	$-0.757498\pi$
$523$	−33.0948	−1.44713	−0.723566	+	0.690255i	$0.757498\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−15.8187	−0.689075
$528$	0	0
$529$	15.4260	0.670698
$530$	0	0
$531$	−16.0566	−0.696797
$532$	0	0
$533$	23.9872	1.03900
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−40.6815	−1.75554
$538$	0	0
$539$	−9.44447	−0.406802
$540$	0	0
$541$	31.4338	1.35144	0.675722	−	0.737156i	$-0.263832\pi$
$541$	31.4338	1.35144	0.675722	+	0.737156i	$0.263832\pi$
$542$	0	0
$543$	13.6425	0.585458
$544$	0	0
$545$	0	0
$546$	0	0
$547$	23.3460	0.998202	0.499101	−	0.866544i	$-0.333664\pi$
$547$	23.3460	0.998202	0.499101	+	0.866544i	$0.333664\pi$
$548$	0	0
$549$	13.3305	0.568931
$550$	0	0
$551$	3.37720	0.143874
$552$	0	0
$553$	2.53418	0.107764
$554$	0	0
$555$	0	0
$556$	0	0
$557$	15.9229	0.674673	0.337337	−	0.941384i	$-0.390474\pi$
$557$	15.9229	0.674673	0.337337	+	0.941384i	$0.390474\pi$
$558$	0	0
$559$	34.2894	1.45029
$560$	0	0
$561$	−20.8783	−0.881481
$562$	0	0
$563$	29.6404	1.24919	0.624597	−	0.780947i	$-0.285263\pi$
$563$	29.6404	1.24919	0.624597	+	0.780947i	$0.285263\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−3.74373	−0.157222
$568$	0	0
$569$	−7.64334	−0.320426	−0.160213	−	0.987082i	$-0.551218\pi$
$569$	−7.64334	−0.320426	−0.160213	+	0.987082i	$0.551218\pi$
$570$	0	0
$571$	10.5577	0.441824	0.220912	−	0.975294i	$-0.429097\pi$
$571$	10.5577	0.441824	0.220912	+	0.975294i	$0.429097\pi$
$572$	0	0
$573$	53.2370	2.22401
$574$	0	0
$575$	0	0
$576$	0	0
$577$	38.9447	1.62129	0.810645	−	0.585538i	$-0.199117\pi$
$577$	38.9447	1.62129	0.810645	+	0.585538i	$0.199117\pi$
$578$	0	0
$579$	50.7976	2.11108
$580$	0	0
$581$	−6.87131	−0.285070
$582$	0	0
$583$	15.9036	0.658661
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10.9992	0.453986	0.226993	−	0.973896i	$-0.427111\pi$
$587$	10.9992	0.453986	0.226993	+	0.973896i	$0.427111\pi$
$588$	0	0
$589$	2.48052	0.102208
$590$	0	0
$591$	−2.64817	−0.108931
$592$	0	0
$593$	26.7848	1.09992	0.549960	−	0.835191i	$-0.314643\pi$
$593$	26.7848	1.09992	0.549960	+	0.835191i	$0.314643\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−5.28939	−0.216480
$598$	0	0
$599$	3.44930	0.140934	0.0704672	−	0.997514i	$-0.477551\pi$
$599$	3.44930	0.140934	0.0704672	+	0.997514i	$0.477551\pi$
$600$	0	0
$601$	−37.0686	−1.51206	−0.756030	−	0.654537i	$-0.772863\pi$
$601$	−37.0686	−1.51206	−0.756030	+	0.654537i	$0.772863\pi$
$602$	0	0
$603$	−8.55553	−0.348408
$604$	0	0
$605$	0	0
$606$	0	0
$607$	14.8732	0.603685	0.301843	−	0.953358i	$-0.402398\pi$
$607$	14.8732	0.603685	0.301843	+	0.953358i	$0.402398\pi$
$608$	0	0
$609$	3.02830	0.122713
$610$	0	0
$611$	19.4082	0.785172
$612$	0	0
$613$	−40.4671	−1.63445	−0.817226	−	0.576317i	$-0.804489\pi$
$613$	−40.4671	−1.63445	−0.817226	+	0.576317i	$0.804489\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8.76991	0.353063	0.176532	−	0.984295i	$-0.443512\pi$
$617$	8.76991	0.353063	0.176532	+	0.984295i	$0.443512\pi$
$618$	0	0
$619$	−32.0510	−1.28824	−0.644119	−	0.764926i	$-0.722776\pi$
$619$	−32.0510	−1.28824	−0.644119	+	0.764926i	$0.722776\pi$
$620$	0	0
$621$	−5.14148	−0.206321
$622$	0	0
$623$	0.566205	0.0226845
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.27389	0.130747
$628$	0	0
$629$	35.6092	1.41983
$630$	0	0
$631$	18.9709	0.755220	0.377610	−	0.925965i	$-0.376746\pi$
$631$	18.9709	0.755220	0.377610	+	0.925965i	$0.376746\pi$
$632$	0	0
$633$	−23.1882	−0.921648
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−19.3502	−0.766684
$638$	0	0
$639$	6.10331	0.241443
$640$	0	0
$641$	−44.3433	−1.75145	−0.875727	−	0.482806i	$-0.839617\pi$
$641$	−44.3433	−1.75145	−0.875727	+	0.482806i	$0.839617\pi$
$642$	0	0
$643$	−42.9397	−1.69338	−0.846688	−	0.532090i	$-0.821407\pi$
$643$	−42.9397	−1.69338	−0.846688	+	0.532090i	$0.821407\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−17.9864	−0.707118	−0.353559	−	0.935412i	$-0.615029\pi$
$647$	−17.9864	−0.707118	−0.353559	+	0.935412i	$0.615029\pi$
$648$	0	0
$649$	−8.34116	−0.327419
$650$	0	0
$651$	2.22425	0.0871751
$652$	0	0
$653$	−21.7274	−0.850260	−0.425130	−	0.905132i	$-0.639772\pi$
$653$	−21.7274	−0.850260	−0.425130	+	0.905132i	$0.639772\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−8.48052	−0.330856
$658$	0	0
$659$	40.8590	1.59164	0.795821	−	0.605532i	$-0.207040\pi$
$659$	40.8590	1.59164	0.795821	+	0.605532i	$0.207040\pi$
$660$	0	0
$661$	33.7282	1.31188	0.655938	−	0.754815i	$-0.272273\pi$
$661$	33.7282	1.31188	0.655938	+	0.754815i	$0.272273\pi$
$662$	0	0
$663$	−42.7763	−1.66129
$664$	0	0
$665$	0	0
$666$	0	0
$667$	20.9349	0.810601
$668$	0	0
$669$	−44.3190	−1.71347
$670$	0	0
$671$	6.92498	0.267336
$672$	0	0
$673$	−11.0622	−0.426417	−0.213209	−	0.977007i	$-0.568391\pi$
$673$	−11.0622	−0.426417	−0.213209	+	0.977007i	$0.568391\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6.14711	−0.236253	−0.118126	−	0.992999i	$-0.537689\pi$
$677$	−6.14711	−0.236253	−0.118126	+	0.992999i	$0.537689\pi$
$678$	0	0
$679$	−4.44447	−0.170563
$680$	0	0
$681$	−19.1287	−0.733013
$682$	0	0
$683$	13.6073	0.520669	0.260334	−	0.965519i	$-0.416167\pi$
$683$	13.6073	0.520669	0.260334	+	0.965519i	$0.416167\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−47.8804	−1.82675
$688$	0	0
$689$	32.5840	1.24135
$690$	0	0
$691$	40.3043	1.53325	0.766624	−	0.642096i	$-0.221935\pi$
$691$	40.3043	1.53325	0.766624	+	0.642096i	$0.221935\pi$
$692$	0	0
$693$	1.37720	0.0523156
$694$	0	0
$695$	0	0
$696$	0	0
$697$	54.2130	2.05346
$698$	0	0
$699$	4.12174	0.155898
$700$	0	0
$701$	−3.23704	−0.122261	−0.0611307	−	0.998130i	$-0.519471\pi$
$701$	−3.23704	−0.122261	−0.0611307	+	0.998130i	$0.519471\pi$
$702$	0	0
$703$	−5.58383	−0.210598
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.46582	−0.130345
$708$	0	0
$709$	20.2370	0.760018	0.380009	−	0.924983i	$-0.375921\pi$
$709$	20.2370	0.760018	0.380009	+	0.924983i	$0.375921\pi$
$710$	0	0
$711$	17.8110	0.667965
$712$	0	0
$713$	15.3764	0.575851
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−43.5384	−1.62597
$718$	0	0
$719$	−47.8804	−1.78564	−0.892819	−	0.450416i	$-0.851276\pi$
$719$	−47.8804	−1.78564	−0.892819	+	0.450416i	$0.851276\pi$
$720$	0	0
$721$	3.57315	0.133071
$722$	0	0
$723$	5.22212	0.194213
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−17.7926	−0.659890	−0.329945	−	0.944000i	$-0.607030\pi$
$727$	−17.7926	−0.659890	−0.329945	+	0.944000i	$0.607030\pi$
$728$	0	0
$729$	−20.3969	−0.755443
$730$	0	0
$731$	77.4968	2.86632
$732$	0	0
$733$	−5.66097	−0.209093	−0.104546	−	0.994520i	$-0.533339\pi$
$733$	−5.66097	−0.209093	−0.104546	+	0.994520i	$0.533339\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.44447	−0.163714
$738$	0	0
$739$	5.59955	0.205983	0.102991	−	0.994682i	$-0.467159\pi$
$739$	5.59955	0.205983	0.102991	+	0.994682i	$0.467159\pi$
$740$	0	0
$741$	6.70769	0.246413
$742$	0	0
$743$	−5.81663	−0.213391	−0.106696	−	0.994292i	$-0.534027\pi$
$743$	−5.81663	−0.213391	−0.106696	+	0.994292i	$0.534027\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−48.2936	−1.76697
$748$	0	0
$749$	5.79045	0.211579
$750$	0	0
$751$	41.1797	1.50267	0.751333	−	0.659923i	$-0.229411\pi$
$751$	41.1797	1.50267	0.751333	+	0.659923i	$0.229411\pi$
$752$	0	0
$753$	−32.5003	−1.18438
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−33.5208	−1.21833	−0.609167	−	0.793042i	$-0.708496\pi$
$757$	−33.5208	−1.21833	−0.609167	+	0.793042i	$0.708496\pi$
$758$	0	0
$759$	20.2944	0.736641
$760$	0	0
$761$	31.5032	1.14199	0.570995	−	0.820954i	$-0.306558\pi$
$761$	31.5032	1.14199	0.570995	+	0.820954i	$0.306558\pi$
$762$	0	0
$763$	−6.32353	−0.228927
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−17.0897	−0.617074
$768$	0	0
$769$	−1.22425	−0.0441475	−0.0220737	−	0.999756i	$-0.507027\pi$
$769$	−1.22425	−0.0441475	−0.0220737	+	0.999756i	$0.507027\pi$
$770$	0	0
$771$	−67.7274	−2.43914
$772$	0	0
$773$	40.6687	1.46275	0.731376	−	0.681974i	$-0.238878\pi$
$773$	40.6687	1.46275	0.731376	+	0.681974i	$0.238878\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−5.00695	−0.179623
$778$	0	0
$779$	−8.50106	−0.304582
$780$	0	0
$781$	3.17058	0.113452
$782$	0	0
$783$	−2.80113	−0.100104
$784$	0	0
$785$	0	0
$786$	0	0
$787$	39.9525	1.42415	0.712076	−	0.702102i	$-0.247755\pi$
$787$	39.9525	1.42415	0.712076	+	0.702102i	$0.247755\pi$
$788$	0	0
$789$	12.7643	0.454420
$790$	0	0
$791$	−5.02830	−0.178786
$792$	0	0
$793$	14.1882	0.503838
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−4.53791	−0.160741	−0.0803705	−	0.996765i	$-0.525610\pi$
$797$	−4.53791	−0.160741	−0.0803705	+	0.996765i	$0.525610\pi$
$798$	0	0
$799$	43.8641	1.55180
$800$	0	0
$801$	3.97945	0.140607
$802$	0	0
$803$	−4.40550	−0.155467
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−16.8003	−0.591399
$808$	0	0
$809$	55.5958	1.95465	0.977323	−	0.211756i	$-0.0679183\pi$
$809$	55.5958	1.95465	0.977323	+	0.211756i	$0.0679183\pi$
$810$	0	0
$811$	−42.3326	−1.48650	−0.743249	−	0.669014i	$-0.766716\pi$
$811$	−42.3326	−1.48650	−0.743249	+	0.669014i	$0.766716\pi$
$812$	0	0
$813$	51.5782	1.80893
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−12.1522	−0.425150
$818$	0	0
$819$	2.82167	0.0985972
$820$	0	0
$821$	−3.10543	−0.108380	−0.0541902	−	0.998531i	$-0.517258\pi$
$821$	−3.10543	−0.108380	−0.0541902	+	0.998531i	$0.517258\pi$
$822$	0	0
$823$	45.2002	1.57558	0.787790	−	0.615944i	$-0.211225\pi$
$823$	45.2002	1.57558	0.787790	+	0.615944i	$0.211225\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	35.6503	1.23968	0.619841	−	0.784727i	$-0.287197\pi$
$827$	35.6503	1.23968	0.619841	+	0.784727i	$0.287197\pi$
$828$	0	0
$829$	−18.4330	−0.640204	−0.320102	−	0.947383i	$-0.603717\pi$
$829$	−18.4330	−0.640204	−0.320102	+	0.947383i	$0.603717\pi$
$830$	0	0
$831$	−59.6991	−2.07094
$832$	0	0
$833$	−43.7331	−1.51526
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2.05739	−0.0711139
$838$	0	0
$839$	14.1805	0.489564	0.244782	−	0.969578i	$-0.421284\pi$
$839$	14.1805	0.489564	0.244782	+	0.969578i	$0.421284\pi$
$840$	0	0
$841$	−17.5945	−0.606707
$842$	0	0
$843$	23.8697	0.822117
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−3.43380	−0.117987
$848$	0	0
$849$	−49.7176	−1.70630
$850$	0	0
$851$	−34.6134	−1.18653
$852$	0	0
$853$	42.6639	1.46078	0.730392	−	0.683028i	$-0.239337\pi$
$853$	42.6639	1.46078	0.730392	+	0.683028i	$0.239337\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.42392	−0.0486403	−0.0243201	−	0.999704i	$-0.507742\pi$
$857$	−1.42392	−0.0486403	−0.0243201	+	0.999704i	$0.507742\pi$
$858$	0	0
$859$	8.27894	0.282474	0.141237	−	0.989976i	$-0.454892\pi$
$859$	8.27894	0.282474	0.141237	+	0.989976i	$0.454892\pi$
$860$	0	0
$861$	−7.62280	−0.259784
$862$	0	0
$863$	−30.4154	−1.03535	−0.517676	−	0.855577i	$-0.673203\pi$
$863$	−30.4154	−1.03535	−0.517676	+	0.855577i	$0.673203\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−56.2653	−1.91087
$868$	0	0
$869$	9.25254	0.313871
$870$	0	0
$871$	−9.10602	−0.308546
$872$	0	0
$873$	−31.2370	−1.05721
$874$	0	0
$875$	0	0
$876$	0	0
$877$	57.8484	1.95340	0.976700	−	0.214608i	$-0.0688474\pi$
$877$	57.8484	1.95340	0.976700	+	0.214608i	$0.0688474\pi$
$878$	0	0
$879$	1.94633	0.0656481
$880$	0	0
$881$	18.4055	0.620097	0.310049	−	0.950721i	$-0.399655\pi$
$881$	18.4055	0.620097	0.310049	+	0.950721i	$0.399655\pi$
$882$	0	0
$883$	18.7947	0.632492	0.316246	−	0.948677i	$-0.397578\pi$
$883$	18.7947	0.632492	0.316246	+	0.948677i	$0.397578\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11.1471	0.374283	0.187142	−	0.982333i	$-0.440078\pi$
$887$	11.1471	0.374283	0.187142	+	0.982333i	$0.440078\pi$
$888$	0	0
$889$	1.14923	0.0385440
$890$	0	0
$891$	−13.6687	−0.457919
$892$	0	0
$893$	−6.87826	−0.230172
$894$	0	0
$895$	0	0
$896$	0	0
$897$	41.5801	1.38832
$898$	0	0
$899$	8.37720	0.279395
$900$	0	0
$901$	73.6425	2.45339
$902$	0	0
$903$	−10.8967	−0.362619
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−38.5598	−1.28036	−0.640178	−	0.768226i	$-0.721139\pi$
$907$	−38.5598	−1.28036	−0.640178	+	0.768226i	$0.721139\pi$
$908$	0	0
$909$	−24.3588	−0.807930
$910$	0	0
$911$	1.79820	0.0595771	0.0297885	−	0.999556i	$-0.490517\pi$
$911$	1.79820	0.0595771	0.0297885	+	0.999556i	$0.490517\pi$
$912$	0	0
$913$	−25.0878	−0.830285
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.28456	0.108466
$918$	0	0
$919$	32.4458	1.07029	0.535144	−	0.844761i	$-0.320257\pi$
$919$	32.4458	1.07029	0.535144	+	0.844761i	$0.320257\pi$
$920$	0	0
$921$	−17.6970	−0.583136
$922$	0	0
$923$	6.49602	0.213819
$924$	0	0
$925$	0	0
$926$	0	0
$927$	25.1132	0.824825
$928$	0	0
$929$	−18.4231	−0.604443	−0.302222	−	0.953238i	$-0.597728\pi$
$929$	−18.4231	−0.604443	−0.302222	+	0.953238i	$0.597728\pi$
$930$	0	0
$931$	6.85772	0.224753
$932$	0	0
$933$	−2.27309	−0.0744176
$934$	0	0
$935$	0	0
$936$	0	0
$937$	8.29926	0.271125	0.135563	−	0.990769i	$-0.456716\pi$
$937$	8.29926	0.271125	0.135563	+	0.990769i	$0.456716\pi$
$938$	0	0
$939$	−14.2194	−0.464033
$940$	0	0
$941$	33.6687	1.09757	0.548784	−	0.835964i	$-0.315091\pi$
$941$	33.6687	1.09757	0.548784	+	0.835964i	$0.315091\pi$
$942$	0	0
$943$	−52.6970	−1.71605
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.35103	−0.0439026	−0.0219513	−	0.999759i	$-0.506988\pi$
$947$	−1.35103	−0.0439026	−0.0219513	+	0.999759i	$0.506988\pi$
$948$	0	0
$949$	−9.02617	−0.293002
$950$	0	0
$951$	−53.8462	−1.74608
$952$	0	0
$953$	−6.70769	−0.217283	−0.108642	−	0.994081i	$-0.534650\pi$
$953$	−6.70769	−0.217283	−0.108642	+	0.994081i	$0.534650\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	11.0566	0.357409
$958$	0	0
$959$	2.48746	0.0803244
$960$	0	0
$961$	−24.8470	−0.801518
$962$	0	0
$963$	40.6970	1.31144
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−19.5174	−0.627636	−0.313818	−	0.949483i	$-0.601608\pi$
$967$	−19.5174	−0.627636	−0.313818	+	0.949483i	$0.601608\pi$
$968$	0	0
$969$	15.1599	0.487006
$970$	0	0
$971$	8.50669	0.272993	0.136496	−	0.990641i	$-0.456416\pi$
$971$	8.50669	0.272993	0.136496	+	0.990641i	$0.456416\pi$
$972$	0	0
$973$	3.83527	0.122953
$974$	0	0
$975$	0	0
$976$	0	0
$977$	19.7048	0.630411	0.315206	−	0.949023i	$-0.397927\pi$
$977$	19.7048	0.630411	0.315206	+	0.949023i	$0.397927\pi$
$978$	0	0
$979$	2.06727	0.0660701
$980$	0	0
$981$	−44.4437	−1.41898
$982$	0	0
$983$	12.4677	0.397658	0.198829	−	0.980034i	$-0.436286\pi$
$983$	12.4677	0.397658	0.198829	+	0.980034i	$0.436286\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−6.16765	−0.196319
$988$	0	0
$989$	−75.3297	−2.39534
$990$	0	0
$991$	−25.0841	−0.796822	−0.398411	−	0.917207i	$-0.630438\pi$
$991$	−25.0841	−0.796822	−0.398411	+	0.917207i	$0.630438\pi$
$992$	0	0
$993$	9.89589	0.314036
$994$	0	0
$995$	0	0
$996$	0	0
$997$	11.8775	0.376163	0.188082	−	0.982153i	$-0.439773\pi$
$997$	11.8775	0.376163	0.188082	+	0.982153i	$0.439773\pi$
$998$	0	0
$999$	4.63135	0.146529

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.bh.1.1		3
4.3	odd	2		475.2.a.g.1.1	yes	3
5.4	even	2		7600.2.a.cc.1.3		3
12.11	even	2		4275.2.a.ba.1.3		3
20.3	even	4		475.2.b.b.324.5		6
20.7	even	4		475.2.b.b.324.2		6
20.19	odd	2		475.2.a.e.1.3	✓	3
60.59	even	2		4275.2.a.bm.1.1		3
76.75	even	2		9025.2.a.y.1.3		3
380.379	even	2		9025.2.a.bc.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.a.e.1.3	✓	3	20.19	odd	2
475.2.a.g.1.1	yes	3	4.3	odd	2
475.2.b.b.324.2		6	20.7	even	4
475.2.b.b.324.5		6	20.3	even	4
4275.2.a.ba.1.3		3	12.11	even	2
4275.2.a.bm.1.1		3	60.59	even	2
7600.2.a.bh.1.1		3	1.1	even	1	trivial
7600.2.a.cc.1.3		3	5.4	even	2
9025.2.a.y.1.3		3	76.75	even	2
9025.2.a.bc.1.1		3	380.379	even	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-2.37720 q^{3} +0.377203 q^{7} +2.65109 q^{9} +1.37720 q^{11} +2.82167 q^{13} +6.37720 q^{17} -1.00000 q^{19} -0.896688 q^{21} -6.19887 q^{23} +0.829422 q^{27} -3.37720 q^{29} -2.48052 q^{31} -3.27389 q^{33} +5.58383 q^{37} -6.70769 q^{39} +8.50106 q^{41} +12.1522 q^{43} +6.87826 q^{47} -6.85772 q^{49} -15.1599 q^{51} +11.5478 q^{53} +2.37720 q^{57} -6.05659 q^{59} +5.02830 q^{61} +1.00000 q^{63} -3.22717 q^{67} +14.7360 q^{69} +2.30219 q^{71} -3.19887 q^{73} +0.519485 q^{77} +6.71836 q^{79} -9.92498 q^{81} -18.2165 q^{83} +8.02830 q^{87} +1.50106 q^{89} +1.06434 q^{91} +5.89669 q^{93} -11.7827 q^{97} +3.65109 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} - 4 q^{7} + q^{9} - q^{11} + 3 q^{13} + 14 q^{17} - 3 q^{19} - 6 q^{21} - 8 q^{23} + q^{27} - 5 q^{29} + q^{31} - 8 q^{33} + 5 q^{37} + 11 q^{39} + q^{41} + 5 q^{43} - 9 q^{47} - 7 q^{49}+ \cdots + 4 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 - 4 * q^7 + q^9 - q^11 + 3 * q^13 + 14 * q^17 - 3 * q^19 - 6 * q^21 - 8 * q^23 + q^27 - 5 * q^29 + q^31 - 8 * q^33 + 5 * q^37 + 11 * q^39 + q^41 + 5 * q^43 - 9 * q^47 - 7 * q^49 - 18 * q^51 + 31 * q^53 + 2 * q^57 + 6 * q^59 + 3 * q^61 + 3 * q^63 + 13 * q^67 + q^69 - 7 * q^71 + q^73 + 10 * q^77 + 18 * q^79 - 21 * q^81 - 3 * q^83 + 12 * q^87 - 20 * q^89 - 17 * q^91 + 21 * q^93 - 13 * q^97 + 4 * q^99

Introduction

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