LMFDB - Embedded newform 475.2.a.e.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [475,2,Mod(1,475)] 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("475.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(475, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	475.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$3.79289409601$
Analytic rank:	$1$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.65109$ of defining polynomial
Character	$\chi$	$=$	475.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.65109 q^{2} -2.37720 q^{3} +0.726109 q^{4} -3.92498 q^{6} +0.377203 q^{7} -2.10331 q^{8} +2.65109 q^{9} -1.37720 q^{11} -1.72611 q^{12} -2.82167 q^{13} +0.622797 q^{14} -4.92498 q^{16} -6.37720 q^{17} +4.37720 q^{18} +1.00000 q^{19} -0.896688 q^{21} -2.27389 q^{22} -6.19887 q^{23} +5.00000 q^{24} -4.65884 q^{26} +0.829422 q^{27} +0.273891 q^{28} -3.37720 q^{29} +2.48052 q^{31} -3.92498 q^{32} +3.27389 q^{33} -10.5294 q^{34} +1.92498 q^{36} -5.58383 q^{37} +1.65109 q^{38} +6.70769 q^{39} +8.50106 q^{41} -1.48052 q^{42} +12.1522 q^{43} -1.00000 q^{44} -10.2349 q^{46} +6.87826 q^{47} +11.7077 q^{48} -6.85772 q^{49} +15.1599 q^{51} -2.04884 q^{52} -11.5478 q^{53} +1.36945 q^{54} -0.793375 q^{56} -2.37720 q^{57} -5.57608 q^{58} +6.05659 q^{59} +5.02830 q^{61} +4.09556 q^{62} +1.00000 q^{63} +3.36945 q^{64} +5.40550 q^{66} -3.22717 q^{67} -4.63055 q^{68} +14.7360 q^{69} -2.30219 q^{71} -5.57608 q^{72} +3.19887 q^{73} -9.21942 q^{74} +0.726109 q^{76} -0.519485 q^{77} +11.0750 q^{78} -6.71836 q^{79} -9.92498 q^{81} +14.0360 q^{82} -18.2165 q^{83} -0.651093 q^{84} +20.0643 q^{86} +8.02830 q^{87} +2.89669 q^{88} +1.50106 q^{89} -1.06434 q^{91} -4.50106 q^{92} -5.89669 q^{93} +11.3567 q^{94} +9.33048 q^{96} +11.7827 q^{97} -11.3227 q^{98} -3.65109 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 3 q^{6} - 4 q^{7} - 3 q^{8} + q^{9} + q^{11} - 7 q^{12} - 3 q^{13} + 7 q^{14} - 6 q^{16} - 14 q^{17} + 8 q^{18} + 3 q^{19} - 6 q^{21} - 5 q^{22} - 8 q^{23} + 15 q^{24}+ \cdots - 4 q^{99}+O(q^{100}) $ 3 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 - 3 * q^6 - 4 * q^7 - 3 * q^8 + q^9 + q^11 - 7 * q^12 - 3 * q^13 + 7 * q^14 - 6 * q^16 - 14 * q^17 + 8 * q^18 + 3 * q^19 - 6 * q^21 - 5 * q^22 - 8 * q^23 + 15 * q^24 - 11 * q^26 + q^27 - q^28 - 5 * q^29 - q^31 - 3 * q^32 + 8 * q^33 + 5 * q^34 - 3 * q^36 - 5 * q^37 - 2 * q^38 - 11 * q^39 + q^41 + 4 * q^42 + 5 * q^43 - 3 * q^44 - 12 * q^46 - 9 * q^47 + 4 * q^48 - 7 * q^49 + 18 * q^51 + 22 * q^52 - 31 * q^53 - 5 * q^54 - 9 * q^56 - 2 * q^57 - q^58 - 6 * q^59 + 3 * q^61 + 5 * q^62 + 3 * q^63 + q^64 - q^66 + 13 * q^67 - 23 * q^68 + q^69 + 7 * q^71 - q^72 - q^73 - q^74 + 4 * q^76 - 10 * q^77 + 42 * q^78 - 18 * q^79 - 21 * q^81 + 34 * q^82 - 3 * q^83 + 5 * q^84 + 40 * q^86 + 12 * q^87 + 12 * q^88 - 20 * q^89 + 17 * q^91 + 11 * q^92 - 21 * q^93 + 45 * q^94 + 2 * q^96 + 13 * q^97 - 4 * q^98 - 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 $

$2$	1.65109	1.16750	0.583750	−	0.811934i	$-0.301585\pi$
$2$	1.65109	1.16750	0.583750	+	0.811934i	$0.301585\pi$
$3$	−2.37720	−1.37248	−0.686239	−	0.727376i	$-0.740740\pi$
$3$	−2.37720	−1.37248	−0.686239	+	0.727376i	$0.740740\pi$
$4$	0.726109	0.363055
$5$	0	0
$5$	0	0
$6$	−3.92498	−1.60237
$7$	0.377203	0.142569	0.0712846	−	0.997456i	$-0.477290\pi$
$7$	0.377203	0.142569	0.0712846	+	0.997456i	$0.477290\pi$
$8$	−2.10331	−0.743633
$9$	2.65109	0.883698
$10$	0	0
$11$	−1.37720	−0.415242	−0.207621	−	0.978209i	$-0.566572\pi$
$11$	−1.37720	−0.415242	−0.207621	+	0.978209i	$0.566572\pi$
$12$	−1.72611	−0.498285
$13$	−2.82167	−0.782591	−0.391295	−	0.920265i	$-0.627973\pi$
$13$	−2.82167	−0.782591	−0.391295	+	0.920265i	$0.627973\pi$
$14$	0.622797	0.166450
$15$	0	0
$16$	−4.92498	−1.23125
$17$	−6.37720	−1.54670	−0.773349	−	0.633980i	$-0.781420\pi$
$17$	−6.37720	−1.54670	−0.773349	+	0.633980i	$0.781420\pi$
$18$	4.37720	1.03172
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−0.896688	−0.195673
$22$	−2.27389	−0.484795
$23$	−6.19887	−1.29255	−0.646277	−	0.763103i	$-0.723675\pi$
$23$	−6.19887	−1.29255	−0.646277	+	0.763103i	$0.723675\pi$
$24$	5.00000	1.02062
$25$	0	0
$26$	−4.65884	−0.913674
$27$	0.829422	0.159622
$28$	0.273891	0.0517604
$29$	−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	$-0.428494\pi$
$31$	2.48052	0.445514	0.222757	+	0.974874i	$0.428494\pi$
$32$	−3.92498	−0.693846
$33$	3.27389	0.569911
$34$	−10.5294	−1.80577
$35$	0	0
$36$	1.92498	0.320831
$37$	−5.58383	−0.917976	−0.458988	−	0.888443i	$-0.651788\pi$
$37$	−5.58383	−0.917976	−0.458988	+	0.888443i	$0.651788\pi$
$38$	1.65109	0.267843
$39$	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	$0.268934\pi$
$42$	−1.48052	−0.228448
$43$	12.1522	1.85319	0.926593	−	0.376065i	$-0.122723\pi$
$43$	12.1522	1.85319	0.926593	+	0.376065i	$0.122723\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−10.2349	−1.50906
$47$	6.87826	1.00330	0.501649	−	0.865071i	$-0.332727\pi$
$47$	6.87826	1.00330	0.501649	+	0.865071i	$0.332727\pi$
$48$	11.7077	1.68986
$49$	−6.85772	−0.979674
$50$	0	0
$51$	15.1599	2.12281
$52$	−2.04884	−0.284123
$53$	−11.5478	−1.58621	−0.793105	−	0.609085i	$-0.791537\pi$
$53$	−11.5478	−1.58621	−0.793105	+	0.609085i	$0.791537\pi$
$54$	1.36945	0.186359
$55$	0	0
$56$	−0.793375	−0.106019
$57$	−2.37720	−0.314868
$58$	−5.57608	−0.732175
$59$	6.05659	0.788501	0.394251	−	0.919003i	$-0.371004\pi$
$59$	6.05659	0.788501	0.394251	+	0.919003i	$0.371004\pi$
$60$	0	0
$61$	5.02830	0.643807	0.321904	−	0.946772i	$-0.395677\pi$
$61$	5.02830	0.643807	0.321904	+	0.946772i	$0.395677\pi$
$62$	4.09556	0.520137
$63$	1.00000	0.125988
$64$	3.36945	0.421182
$65$	0	0
$66$	5.40550	0.665371
$67$	−3.22717	−0.394262	−0.197131	−	0.980377i	$-0.563162\pi$
$67$	−3.22717	−0.394262	−0.197131	+	0.980377i	$0.563162\pi$
$68$	−4.63055	−0.561536
$69$	14.7360	1.77400
$70$	0	0
$71$	−2.30219	−0.273219	−0.136610	−	0.990625i	$-0.543621\pi$
$71$	−2.30219	−0.273219	−0.136610	+	0.990625i	$0.543621\pi$
$72$	−5.57608	−0.657147
$73$	3.19887	0.374400	0.187200	−	0.982322i	$-0.440059\pi$
$73$	3.19887	0.374400	0.187200	+	0.982322i	$0.440059\pi$
$74$	−9.21942	−1.07174
$75$	0	0
$76$	0.726109	0.0832905
$77$	−0.519485	−0.0592008
$78$	11.0750	1.25400
$79$	−6.71836	−0.755874	−0.377937	−	0.925831i	$-0.623366\pi$
$79$	−6.71836	−0.755874	−0.377937	+	0.925831i	$0.623366\pi$
$80$	0	0
$81$	−9.92498	−1.10278
$82$	14.0360	1.55002
$83$	−18.2165	−1.99952	−0.999760	−	0.0218996i	$-0.993029\pi$
$83$	−18.2165	−1.99952	−0.999760	+	0.0218996i	$0.993029\pi$
$84$	−0.651093	−0.0710401
$85$	0	0
$86$	20.0643	2.16359
$87$	8.02830	0.860724
$88$	2.89669	0.308788
$89$	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
$92$	−4.50106	−0.469268
$93$	−5.89669	−0.611458
$94$	11.3567	1.17135
$95$	0	0
$96$	9.33048	0.952288
$97$	11.7827	1.19635	0.598176	−	0.801365i	$-0.295892\pi$
$97$	11.7827	1.19635	0.598176	+	0.801365i	$0.295892\pi$
$98$	−11.3227	−1.14377
$99$	−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	$0.651123\pi$
$102$	25.0304	2.47838
$103$	9.47277	0.933379	0.466690	−	0.884421i	$-0.345447\pi$
$103$	9.47277	0.933379	0.466690	+	0.884421i	$0.345447\pi$
$104$	5.93486	0.581961
$105$	0	0
$106$	−19.0665	−1.85190
$107$	15.3510	1.48404	0.742020	−	0.670378i	$-0.233868\pi$
$107$	15.3510	1.48404	0.742020	+	0.670378i	$0.233868\pi$
$108$	0.602251	0.0579516
$109$	−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
$112$	−1.85772	−0.175538
$113$	13.3305	1.25403	0.627013	−	0.779009i	$-0.284277\pi$
$113$	13.3305	1.25403	0.627013	+	0.779009i	$0.284277\pi$
$114$	−3.92498	−0.367608
$115$	0	0
$116$	−2.45222	−0.227683
$117$	−7.48052	−0.691574
$118$	10.0000	0.920575
$119$	−2.40550	−0.220512
$120$	0	0
$121$	−9.10331	−0.827574
$122$	8.30219	0.751645
$123$	−20.2087	−1.82216
$124$	1.80113	0.161746
$125$	0	0
$126$	1.65109	0.147091
$127$	3.04672	0.270353	0.135176	−	0.990822i	$-0.456840\pi$
$127$	3.04672	0.270353	0.135176	+	0.990822i	$0.456840\pi$
$128$	13.4132	1.18557
$129$	−28.8881	−2.54346
$130$	0	0
$131$	−8.70769	−0.760794	−0.380397	−	0.924823i	$-0.624213\pi$
$131$	−8.70769	−0.760794	−0.380397	+	0.924823i	$0.624213\pi$
$132$	2.37720	0.206909
$133$	0.377203	0.0327076
$134$	−5.32836	−0.460300
$135$	0	0
$136$	13.4132	1.15018
$137$	−6.59450	−0.563406	−0.281703	−	0.959502i	$-0.590899\pi$
$137$	−6.59450	−0.563406	−0.281703	+	0.959502i	$0.590899\pi$
$138$	24.3305	2.07115
$139$	−10.1677	−0.862409	−0.431205	−	0.902254i	$-0.641911\pi$
$139$	−10.1677	−0.862409	−0.431205	+	0.902254i	$0.641911\pi$
$140$	0	0
$141$	−16.3510	−1.37701
$142$	−3.80113	−0.318983
$143$	3.88601	0.324965
$144$	−13.0566	−1.08805
$145$	0	0
$146$	5.28164	0.437112
$147$	16.3022	1.34458
$148$	−4.05447	−0.333275
$149$	−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	$-0.433146\pi$
$151$	5.12386	0.416974	0.208487	+	0.978025i	$0.433146\pi$
$152$	−2.10331	−0.170601
$153$	−16.9066	−1.36681
$154$	−0.857718	−0.0691169
$155$	0	0
$156$	4.87051	0.389953
$157$	−19.7643	−1.57736	−0.788681	−	0.614803i	$-0.789235\pi$
$157$	−19.7643	−1.57736	−0.788681	+	0.614803i	$0.789235\pi$
$158$	−11.0926	−0.882483
$159$	27.4514	2.17704
$160$	0	0
$161$	−2.33823	−0.184279
$162$	−16.3871	−1.28749
$163$	−13.5195	−1.05893	−0.529464	−	0.848332i	$-0.677607\pi$
$163$	−13.5195	−1.05893	−0.529464	+	0.848332i	$0.677607\pi$
$164$	6.17270	0.482007
$165$	0	0
$166$	−30.0771	−2.33444
$167$	−13.1054	−1.01413	−0.507065	−	0.861908i	$-0.669269\pi$
$167$	−13.1054	−1.01413	−0.507065	+	0.861908i	$0.669269\pi$
$168$	1.88601	0.145509
$169$	−5.03817	−0.387551
$170$	0	0
$171$	2.65109	0.202734
$172$	8.82379	0.672808
$173$	−1.48827	−0.113151	−0.0565754	−	0.998398i	$-0.518018\pi$
$173$	−1.48827	−0.113151	−0.0565754	+	0.998398i	$0.518018\pi$
$174$	13.2555	1.00489
$175$	0	0
$176$	6.78270	0.511265
$177$	−14.3977	−1.08220
$178$	2.47839	0.185763
$179$	−17.1132	−1.27910	−0.639550	−	0.768750i	$-0.720879\pi$
$179$	−17.1132	−1.27910	−0.639550	+	0.768750i	$0.720879\pi$
$180$	0	0
$181$	−5.73891	−0.426569	−0.213285	−	0.976990i	$-0.568416\pi$
$181$	−5.73891	−0.426569	−0.213285	+	0.976990i	$0.568416\pi$
$182$	−1.75733	−0.130262
$183$	−11.9533	−0.883612
$184$	13.0382	0.961187
$185$	0	0
$186$	−9.73598	−0.713877
$187$	8.78270	0.642255
$188$	4.99437	0.364252
$189$	0.312860	0.0227572
$190$	0	0
$191$	22.3948	1.62043	0.810216	−	0.586131i	$-0.199350\pi$
$191$	22.3948	1.62043	0.810216	+	0.586131i	$0.199350\pi$
$192$	−8.00987	−0.578063
$193$	21.3687	1.53815	0.769075	−	0.639159i	$-0.220717\pi$
$193$	21.3687	1.53815	0.769075	+	0.639159i	$0.220717\pi$
$194$	19.4543	1.39674
$195$	0	0
$196$	−4.97945	−0.355675
$197$	−1.11399	−0.0793682	−0.0396841	−	0.999212i	$-0.512635\pi$
$197$	−1.11399	−0.0793682	−0.0396841	+	0.999212i	$0.512635\pi$
$198$	−6.02830	−0.428412
$199$	−2.22505	−0.157729	−0.0788647	−	0.996885i	$-0.525130\pi$
$199$	−2.22505	−0.157729	−0.0788647	+	0.996885i	$0.525130\pi$
$200$	0	0
$201$	7.67164	0.541116
$202$	−15.1706	−1.06740
$203$	−1.27389	−0.0894096
$204$	11.0078	0.770697
$205$	0	0
$206$	15.6404	1.08972
$207$	−16.4338	−1.14223
$208$	13.8967	0.963562
$209$	−1.37720	−0.0952631
$210$	0	0
$211$	−9.75441	−0.671521	−0.335760	−	0.941947i	$-0.608993\pi$
$211$	−9.75441	−0.671521	−0.335760	+	0.941947i	$0.608993\pi$
$212$	−8.38495	−0.575881
$213$	5.47277	0.374988
$214$	25.3460	1.73262
$215$	0	0
$216$	−1.74453	−0.118700
$217$	0.935657	0.0635166
$218$	−27.6794	−1.87468
$219$	−7.60437	−0.513856
$220$	0	0
$221$	17.9944	1.21043
$222$	21.9164	1.47093
$223$	18.6433	1.24845	0.624225	−	0.781244i	$-0.285415\pi$
$223$	18.6433	1.24845	0.624225	+	0.781244i	$0.285415\pi$
$224$	−1.48052	−0.0989211
$225$	0	0
$226$	22.0099	1.46407
$227$	8.04672	0.534080	0.267040	−	0.963686i	$-0.413954\pi$
$227$	8.04672	0.534080	0.267040	+	0.963686i	$0.413954\pi$
$228$	−1.72611	−0.114314
$229$	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
$232$	7.10331	0.466355
$233$	1.73386	0.113589	0.0567945	−	0.998386i	$-0.481912\pi$
$233$	1.73386	0.113589	0.0567945	+	0.998386i	$0.481912\pi$
$234$	−12.3510	−0.807412
$235$	0	0
$236$	4.39775	0.286269
$237$	15.9709	1.03742
$238$	−3.97170	−0.257447
$239$	−18.3150	−1.18470	−0.592349	−	0.805682i	$-0.701799\pi$
$239$	−18.3150	−1.18470	−0.592349	+	0.805682i	$0.701799\pi$
$240$	0	0
$241$	−2.19675	−0.141505	−0.0707526	−	0.997494i	$-0.522540\pi$
$241$	−2.19675	−0.141505	−0.0707526	+	0.997494i	$0.522540\pi$
$242$	−15.0304	−0.966192
$243$	21.1054	1.35391
$244$	3.65109	0.233737
$245$	0	0
$246$	−33.3665	−2.12737
$247$	−2.82167	−0.179539
$248$	−5.21730	−0.331299
$249$	43.3043	2.74430
$250$	0	0
$251$	−13.6716	−0.862946	−0.431473	−	0.902126i	$-0.642006\pi$
$251$	−13.6716	−0.862946	−0.431473	+	0.902126i	$0.642006\pi$
$252$	0.726109	0.0457406
$253$	8.53711	0.536723
$254$	5.03042	0.315637
$255$	0	0
$256$	15.4076	0.962976
$257$	−28.4904	−1.77718	−0.888591	−	0.458701i	$-0.848315\pi$
$257$	−28.4904	−1.77718	−0.888591	+	0.458701i	$0.848315\pi$
$258$	−47.6970	−2.96949
$259$	−2.10624	−0.130875
$260$	0	0
$261$	−8.95328	−0.554194
$262$	−14.3772	−0.888227
$263$	−5.36945	−0.331095	−0.165547	−	0.986202i	$-0.552939\pi$
$263$	−5.36945	−0.331095	−0.165547	+	0.986202i	$0.552939\pi$
$264$	−6.88601	−0.423805
$265$	0	0
$266$	0.622797	0.0381861
$267$	−3.56833	−0.218378
$268$	−2.34328	−0.143139
$269$	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	$-0.270980\pi$
$271$	21.6970	1.31800	0.659000	+	0.752143i	$0.270980\pi$
$272$	31.4076	1.90437
$273$	2.53016	0.153132
$274$	−10.8881	−0.657776
$275$	0	0
$276$	10.6999	0.644060
$277$	−25.1132	−1.50891	−0.754453	−	0.656355i	$-0.772098\pi$
$277$	−25.1132	−1.50891	−0.754453	+	0.656355i	$0.772098\pi$
$278$	−16.7877	−1.00686
$279$	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	$0.596820\pi$
$282$	−26.9971	−1.60765
$283$	20.9143	1.24323	0.621613	−	0.783324i	$-0.286478\pi$
$283$	20.9143	1.24323	0.621613	+	0.783324i	$0.286478\pi$
$284$	−1.67164	−0.0991936
$285$	0	0
$286$	6.41617	0.379396
$287$	3.20662	0.189281
$288$	−10.4055	−0.613150
$289$	23.6687	1.39228
$290$	0	0
$291$	−28.0099	−1.64197
$292$	2.32273	0.135928
$293$	0.818748	0.0478318	0.0239159	−	0.999714i	$-0.492387\pi$
$293$	0.818748	0.0478318	0.0239159	+	0.999714i	$0.492387\pi$
$294$	26.9164	1.56980
$295$	0	0
$296$	11.7445	0.682637
$297$	−1.14228	−0.0662819
$298$	−9.14711	−0.529878
$299$	17.4912	1.01154
$300$	0	0
$301$	4.58383	0.264207
$302$	8.45997	0.486817
$303$	21.8422	1.25480
$304$	−4.92498	−0.282467
$305$	0	0
$306$	−27.9143	−1.59575
$307$	7.44447	0.424878	0.212439	−	0.977174i	$-0.431859\pi$
$307$	7.44447	0.424878	0.212439	+	0.977174i	$0.431859\pi$
$308$	−0.377203	−0.0214931
$309$	−22.5187	−1.28104
$310$	0	0
$311$	−0.956204	−0.0542213	−0.0271107	−	0.999632i	$-0.508631\pi$
$311$	−0.956204	−0.0542213	−0.0271107	+	0.999632i	$0.508631\pi$
$312$	−14.1084	−0.798729
$313$	−5.98158	−0.338099	−0.169049	−	0.985608i	$-0.554070\pi$
$313$	−5.98158	−0.338099	−0.169049	+	0.985608i	$0.554070\pi$
$314$	−32.6327	−1.84157
$315$	0	0
$316$	−4.87826	−0.274424
$317$	−22.6511	−1.27221	−0.636106	−	0.771602i	$-0.719456\pi$
$317$	−22.6511	−1.27221	−0.636106	+	0.771602i	$0.719456\pi$
$318$	45.3249	2.54169
$319$	4.65109	0.260411
$320$	0	0
$321$	−36.4925	−2.03681
$322$	−3.86064	−0.215145
$323$	−6.37720	−0.354837
$324$	−7.20662	−0.400368
$325$	0	0
$326$	−22.3219	−1.23630
$327$	39.8521	2.20383
$328$	−17.8804	−0.987279
$329$	2.59450	0.143039
$330$	0	0
$331$	4.16283	0.228810	0.114405	−	0.993434i	$-0.463504\pi$
$331$	4.16283	0.228810	0.114405	+	0.993434i	$0.463504\pi$
$332$	−13.2272	−0.725935
$333$	−14.8032	−0.811213
$334$	−21.6383	−1.18399
$335$	0	0
$336$	4.41617	0.240922
$337$	18.0304	0.982180	0.491090	−	0.871109i	$-0.336599\pi$
$337$	18.0304	0.982180	0.491090	+	0.871109i	$0.336599\pi$
$338$	−8.31849	−0.452466
$339$	−31.6893	−1.72112
$340$	0	0
$341$	−3.41617	−0.184996
$342$	4.37720	0.236692
$343$	−5.22717	−0.282241
$344$	−25.5598	−1.37809
$345$	0	0
$346$	−2.45726	−0.132103
$347$	−7.43380	−0.399067	−0.199534	−	0.979891i	$-0.563943\pi$
$347$	−7.43380	−0.399067	−0.199534	+	0.979891i	$0.563943\pi$
$348$	5.82942	0.312490
$349$	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
$352$	5.40550	0.288114
$353$	−10.3695	−0.551910	−0.275955	−	0.961171i	$-0.588994\pi$
$353$	−10.3695	−0.551910	−0.275955	+	0.961171i	$0.588994\pi$
$354$	−23.7720	−1.26347
$355$	0	0
$356$	1.08993	0.0577664
$357$	5.71836	0.302648
$358$	−28.2555	−1.49335
$359$	−23.9893	−1.26611	−0.633054	−	0.774108i	$-0.718199\pi$
$359$	−23.9893	−1.26611	−0.633054	+	0.774108i	$0.718199\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−9.47547	−0.498020
$363$	21.6404	1.13583
$364$	−0.772829	−0.0405073
$365$	0	0
$366$	−19.7360	−1.03162
$367$	−35.0275	−1.82842	−0.914210	−	0.405240i	$-0.867188\pi$
$367$	−35.0275	−1.82842	−0.914210	+	0.405240i	$0.867188\pi$
$368$	30.5294	1.59145
$369$	22.5371	1.17323
$370$	0	0
$371$	−4.35586	−0.226145
$372$	−4.28164	−0.221993
$373$	−30.8003	−1.59478	−0.797390	−	0.603464i	$-0.793787\pi$
$373$	−30.8003	−1.59478	−0.797390	+	0.603464i	$0.793787\pi$
$374$	14.5011	0.749832
$375$	0	0
$376$	−14.4671	−0.746086
$377$	9.52936	0.490787
$378$	0.516561	0.0265691
$379$	0.671640	0.0344998	0.0172499	−	0.999851i	$-0.494509\pi$
$379$	0.671640	0.0344998	0.0172499	+	0.999851i	$0.494509\pi$
$380$	0	0
$381$	−7.24267	−0.371053
$382$	36.9759	1.89185
$383$	30.1805	1.54215	0.771075	−	0.636745i	$-0.219720\pi$
$383$	30.1805	1.54215	0.771075	+	0.636745i	$0.219720\pi$
$384$	−31.8860	−1.62718
$385$	0	0
$386$	35.2816	1.79579
$387$	32.2165	1.63766
$388$	8.55553	0.434341
$389$	−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
$392$	14.4239	0.728518
$393$	20.6999	1.04417
$394$	−1.83929	−0.0926623
$395$	0	0
$396$	−2.65109	−0.133222
$397$	12.5761	0.631175	0.315588	−	0.948896i	$-0.397798\pi$
$397$	12.5761	0.631175	0.315588	+	0.948896i	$0.397798\pi$
$398$	−3.67376	−0.184149
$399$	−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	$0.289430\pi$
$402$	12.6666	0.631752
$403$	−6.99920	−0.348655
$404$	−6.67164	−0.331926
$405$	0	0
$406$	−2.10331	−0.104386
$407$	7.69006	0.381182
$408$	−31.8860	−1.57859
$409$	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
$412$	6.87826	0.338868
$413$	2.28456	0.112416
$414$	−27.1337	−1.33355
$415$	0	0
$416$	11.0750	0.542997
$417$	24.1706	1.18364
$418$	−2.27389	−0.111220
$419$	12.6893	0.619911	0.309956	−	0.950751i	$-0.399686\pi$
$419$	12.6893	0.619911	0.309956	+	0.950751i	$0.399686\pi$
$420$	0	0
$421$	29.1826	1.42227	0.711136	−	0.703055i	$-0.248181\pi$
$421$	29.1826	1.42227	0.711136	+	0.703055i	$0.248181\pi$
$422$	−16.1054	−0.784000
$423$	18.2349	0.886612
$424$	24.2886	1.17956
$425$	0	0
$426$	9.03605	0.437798
$427$	1.89669	0.0917872
$428$	11.1465	0.538788
$429$	−9.23784	−0.446007
$430$	0	0
$431$	−29.9816	−1.44416	−0.722081	−	0.691809i	$-0.756814\pi$
$431$	−29.9816	−1.44416	−0.722081	+	0.691809i	$0.756814\pi$
$432$	−4.08489	−0.196534
$433$	4.76216	0.228855	0.114427	−	0.993432i	$-0.463497\pi$
$433$	4.76216	0.228855	0.114427	+	0.993432i	$0.463497\pi$
$434$	1.54486	0.0741555
$435$	0	0
$436$	−12.1727	−0.582967
$437$	−6.19887	−0.296532
$438$	−12.5555	−0.599926
$439$	6.88601	0.328652	0.164326	−	0.986406i	$-0.447455\pi$
$439$	6.88601	0.328652	0.164326	+	0.986406i	$0.447455\pi$
$440$	0	0
$441$	−18.1805	−0.865736
$442$	29.7104	1.41318
$443$	−8.65109	−0.411026	−0.205513	−	0.978654i	$-0.565886\pi$
$443$	−8.65109	−0.411026	−0.205513	+	0.978654i	$0.565886\pi$
$444$	9.63830	0.457413
$445$	0	0
$446$	30.7819	1.45757
$447$	13.1698	0.622909
$448$	1.27097	0.0600476
$449$	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
$452$	9.67939	0.455280
$453$	−12.1805	−0.572288
$454$	13.2859	0.623538
$455$	0	0
$456$	5.00000	0.234146
$457$	28.6092	1.33828	0.669141	−	0.743135i	$-0.266662\pi$
$457$	28.6092	1.33828	0.669141	+	0.743135i	$0.266662\pi$
$458$	33.2555	1.55393
$459$	−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	$0.864374\pi$
$462$	2.03897	0.0948615
$463$	8.04380	0.373827	0.186913	−	0.982376i	$-0.440152\pi$
$463$	8.04380	0.373827	0.186913	+	0.982376i	$0.440152\pi$
$464$	16.6327	0.772152
$465$	0	0
$466$	2.86276	0.132615
$467$	−17.0488	−0.788926	−0.394463	−	0.918912i	$-0.629069\pi$
$467$	−17.0488	−0.788926	−0.394463	+	0.918912i	$0.629069\pi$
$468$	−5.43167	−0.251079
$469$	−1.21730	−0.0562096
$470$	0	0
$471$	46.9837	2.16489
$472$	−12.7389	−0.586356
$473$	−16.7360	−0.769521
$474$	26.3695	1.21119
$475$	0	0
$476$	−1.74666	−0.0800578
$477$	−30.6142	−1.40173
$478$	−30.2397	−1.38313
$479$	22.5478	1.03023	0.515117	−	0.857120i	$-0.327748\pi$
$479$	22.5478	1.03023	0.515117	+	0.857120i	$0.327748\pi$
$480$	0	0
$481$	15.7557	0.718399
$482$	−3.62704	−0.165207
$483$	5.55845	0.252918
$484$	−6.61000	−0.300455
$485$	0	0
$486$	34.8470	1.58069
$487$	24.5577	1.11281	0.556407	−	0.830910i	$-0.312180\pi$
$487$	24.5577	1.11281	0.556407	+	0.830910i	$0.312180\pi$
$488$	−10.5761	−0.478757
$489$	32.1386	1.45336
$490$	0	0
$491$	−16.9298	−0.764032	−0.382016	−	0.924156i	$-0.624770\pi$
$491$	−16.9298	−0.764032	−0.382016	+	0.924156i	$0.624770\pi$
$492$	−14.6738	−0.661544
$493$	21.5371	0.969983
$494$	−4.65884	−0.209611
$495$	0	0
$496$	−12.2165	−0.548537
$497$	−0.868391	−0.0389527
$498$	71.4995	3.20397
$499$	0.418295	0.0187255	0.00936273	−	0.999956i	$-0.497020\pi$
$499$	0.418295	0.0187255	0.00936273	+	0.999956i	$0.497020\pi$
$500$	0	0
$501$	31.1543	1.39187
$502$	−22.5732	−1.00749
$503$	12.5993	0.561776	0.280888	−	0.959741i	$-0.409371\pi$
$503$	12.5993	0.561776	0.280888	+	0.959741i	$0.409371\pi$
$504$	−2.10331	−0.0936890
$505$	0	0
$506$	14.0956	0.626624
$507$	11.9767	0.531906
$508$	2.21225	0.0981528
$509$	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
$512$	−1.38708	−0.0613007
$513$	0.829422	0.0366199
$514$	−47.0403	−2.07486
$515$	0	0
$516$	−20.9759	−0.923415
$517$	−9.47277	−0.416612
$518$	−3.47759	−0.152797
$519$	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	$0.419846\pi$
$522$	−14.7827	−0.647021
$523$	−33.0948	−1.44713	−0.723566	−	0.690255i	$-0.757498\pi$
$523$	−33.0948	−1.44713	−0.723566	+	0.690255i	$0.757498\pi$
$524$	−6.32273	−0.276210
$525$	0	0
$526$	−8.86547	−0.386553
$527$	−15.8187	−0.689075
$528$	−16.1239	−0.701701
$529$	15.4260	0.670698
$530$	0	0
$531$	16.0566	0.696797
$532$	0.273891	0.0118747
$533$	−23.9872	−1.03900
$534$	−5.89164	−0.254956
$535$	0	0
$536$	6.78775	0.293186
$537$	40.6815	1.75554
$538$	11.6687	0.503074
$539$	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	$0.263832\pi$
$542$	35.8238	1.53876
$543$	13.6425	0.585458
$544$	25.0304	1.07317
$545$	0	0
$546$	4.17753	0.178782
$547$	23.3460	0.998202	0.499101	−	0.866544i	$-0.333664\pi$
$547$	23.3460	0.998202	0.499101	+	0.866544i	$0.333664\pi$
$548$	−4.78833	−0.204547
$549$	13.3305	0.568931
$550$	0	0
$551$	−3.37720	−0.143874
$552$	−30.9944	−1.31921
$553$	−2.53418	−0.107764
$554$	−41.4642	−1.76165
$555$	0	0
$556$	−7.38283	−0.313102
$557$	−15.9229	−0.674673	−0.337337	−	0.941384i	$-0.609526\pi$
$557$	−15.9229	−0.674673	−0.337337	+	0.941384i	$0.609526\pi$
$558$	10.8577	0.459644
$559$	−34.2894	−1.45029
$560$	0	0
$561$	−20.8783	−0.881481
$562$	−16.5788	−0.699334
$563$	29.6404	1.24919	0.624597	−	0.780947i	$-0.285263\pi$
$563$	29.6404	1.24919	0.624597	+	0.780947i	$0.285263\pi$
$564$	−11.8726	−0.499928
$565$	0	0
$566$	34.5315	1.45147
$567$	−3.74373	−0.157222
$568$	4.84222	0.203175
$569$	−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	$-0.570903\pi$
$571$	−10.5577	−0.441824	−0.220912	+	0.975294i	$0.570903\pi$
$572$	2.82167	0.117980
$573$	−53.2370	−2.22401
$574$	5.29444	0.220986
$575$	0	0
$576$	8.93273	0.372197
$577$	−38.9447	−1.62129	−0.810645	−	0.585538i	$-0.800883\pi$
$577$	−38.9447	−1.62129	−0.810645	+	0.585538i	$0.800883\pi$
$578$	39.0793	1.62548
$579$	−50.7976	−2.11108
$580$	0	0
$581$	−6.87131	−0.285070
$582$	−46.2469	−1.91700
$583$	15.9036	0.658661
$584$	−6.72823	−0.278416
$585$	0	0
$586$	1.35183	0.0558436
$587$	10.9992	0.453986	0.226993	−	0.973896i	$-0.427111\pi$
$587$	10.9992	0.453986	0.226993	+	0.973896i	$0.427111\pi$
$588$	11.8372	0.488157
$589$	2.48052	0.102208
$590$	0	0
$591$	2.64817	0.108931
$592$	27.5003	1.13025
$593$	−26.7848	−1.09992	−0.549960	−	0.835191i	$-0.685357\pi$
$593$	−26.7848	−1.09992	−0.549960	+	0.835191i	$0.685357\pi$
$594$	−1.88601	−0.0773841
$595$	0	0
$596$	−4.02267	−0.164775
$597$	5.28939	0.216480
$598$	28.8796	1.18097
$599$	−3.44930	−0.140934	−0.0704672	−	0.997514i	$-0.522449\pi$
$599$	−3.44930	−0.140934	−0.0704672	+	0.997514i	$0.522449\pi$
$600$	0	0
$601$	−37.0686	−1.51206	−0.756030	−	0.654537i	$-0.772863\pi$
$601$	−37.0686	−1.51206	−0.756030	+	0.654537i	$0.772863\pi$
$602$	7.56833	0.308462
$603$	−8.55553	−0.348408
$604$	3.72048	0.151384
$605$	0	0
$606$	36.0635	1.46498
$607$	14.8732	0.603685	0.301843	−	0.953358i	$-0.402398\pi$
$607$	14.8732	0.603685	0.301843	+	0.953358i	$0.402398\pi$
$608$	−3.92498	−0.159179
$609$	3.02830	0.122713
$610$	0	0
$611$	−19.4082	−0.785172
$612$	−12.2760	−0.496228
$613$	40.4671	1.63445	0.817226	−	0.576317i	$-0.195511\pi$
$613$	40.4671	1.63445	0.817226	+	0.576317i	$0.195511\pi$
$614$	12.2915	0.496045
$615$	0	0
$616$	1.09264	0.0440237
$617$	−8.76991	−0.353063	−0.176532	−	0.984295i	$-0.556488\pi$
$617$	−8.76991	−0.353063	−0.176532	+	0.984295i	$0.556488\pi$
$618$	−37.1805	−1.49562
$619$	32.0510	1.28824	0.644119	−	0.764926i	$-0.277224\pi$
$619$	32.0510	1.28824	0.644119	+	0.764926i	$0.277224\pi$
$620$	0	0
$621$	−5.14148	−0.206321
$622$	−1.57878	−0.0633034
$623$	0.566205	0.0226845
$624$	−33.0352	−1.32247
$625$	0	0
$626$	−9.87614	−0.394730
$627$	3.27389	0.130747
$628$	−14.3510	−0.572668
$629$	35.6092	1.41983
$630$	0	0
$631$	−18.9709	−0.755220	−0.377610	−	0.925965i	$-0.623254\pi$
$631$	−18.9709	−0.755220	−0.377610	+	0.925965i	$0.623254\pi$
$632$	14.1308	0.562093
$633$	23.1882	0.921648
$634$	−37.3991	−1.48531
$635$	0	0
$636$	19.9327	0.790384
$637$	19.3502	0.766684
$638$	7.67939	0.304030
$639$	−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	$0.839617\pi$
$642$	−60.2525	−2.37798
$643$	−42.9397	−1.69338	−0.846688	−	0.532090i	$-0.821407\pi$
$643$	−42.9397	−1.69338	−0.846688	+	0.532090i	$0.821407\pi$
$644$	−1.69781	−0.0669032
$645$	0	0
$646$	−10.5294	−0.414272
$647$	−17.9864	−0.707118	−0.353559	−	0.935412i	$-0.615029\pi$
$647$	−17.9864	−0.707118	−0.353559	+	0.935412i	$0.615029\pi$
$648$	20.8753	0.820061
$649$	−8.34116	−0.327419
$650$	0	0
$651$	−2.22425	−0.0871751
$652$	−9.81663	−0.384449
$653$	21.7274	0.850260	0.425130	−	0.905132i	$-0.360228\pi$
$653$	21.7274	0.850260	0.425130	+	0.905132i	$0.360228\pi$
$654$	65.7995	2.57297
$655$	0	0
$656$	−41.8676	−1.63465
$657$	8.48052	0.330856
$658$	4.28376	0.166998
$659$	−40.8590	−1.59164	−0.795821	−	0.605532i	$-0.792960\pi$
$659$	−40.8590	−1.59164	−0.795821	+	0.605532i	$0.792960\pi$
$660$	0	0
$661$	33.7282	1.31188	0.655938	−	0.754815i	$-0.272273\pi$
$661$	33.7282	1.31188	0.655938	+	0.754815i	$0.272273\pi$
$662$	6.87322	0.267135
$663$	−42.7763	−1.66129
$664$	38.3150	1.48691
$665$	0	0
$666$	−24.4415	−0.947091
$667$	20.9349	0.810601
$668$	−9.51598	−0.368184
$669$	−44.3190	−1.71347
$670$	0	0
$671$	−6.92498	−0.267336
$672$	3.51948	0.135767
$673$	11.0622	0.426417	0.213209	−	0.977007i	$-0.431609\pi$
$673$	11.0622	0.426417	0.213209	+	0.977007i	$0.431609\pi$
$674$	29.7699	1.14669
$675$	0	0
$676$	−3.65826	−0.140702
$677$	6.14711	0.236253	0.118126	−	0.992999i	$-0.462311\pi$
$677$	6.14711	0.236253	0.118126	+	0.992999i	$0.462311\pi$
$678$	−52.3219	−2.00941
$679$	4.44447	0.170563
$680$	0	0
$681$	−19.1287	−0.733013
$682$	−5.64042	−0.215983
$683$	13.6073	0.520669	0.260334	−	0.965519i	$-0.416167\pi$
$683$	13.6073	0.520669	0.260334	+	0.965519i	$0.416167\pi$
$684$	1.92498	0.0736036
$685$	0	0
$686$	−8.63055	−0.329516
$687$	−47.8804	−1.82675
$688$	−59.8492	−2.28173
$689$	32.5840	1.24135
$690$	0	0
$691$	−40.3043	−1.53325	−0.766624	−	0.642096i	$-0.778065\pi$
$691$	−40.3043	−1.53325	−0.766624	+	0.642096i	$0.778065\pi$
$692$	−1.08064	−0.0410799
$693$	−1.37720	−0.0523156
$694$	−12.2739	−0.465911
$695$	0	0
$696$	−16.8860	−0.640063
$697$	−54.2130	−2.05346
$698$	36.1909	1.36985
$699$	−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	$0.519471\pi$
$702$	−3.86415	−0.145843
$703$	−5.58383	−0.210598
$704$	−4.64042	−0.174892
$705$	0	0
$706$	−17.1209	−0.644355
$707$	−3.46582	−0.130345
$708$	−10.4543	−0.392898
$709$	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
$712$	−3.15720	−0.118321
$713$	−15.3764	−0.575851
$714$	9.44155	0.353341
$715$	0	0
$716$	−12.4260	−0.464383
$717$	43.5384	1.62597
$718$	−39.6086	−1.47818
$719$	47.8804	1.78564	0.892819	−	0.450416i	$-0.148724\pi$
$719$	47.8804	1.78564	0.892819	+	0.450416i	$0.148724\pi$
$720$	0	0
$721$	3.57315	0.133071
$722$	1.65109	0.0614473
$723$	5.22212	0.194213
$724$	−4.16707	−0.154868
$725$	0	0
$726$	35.7304	1.32608
$727$	−17.7926	−0.659890	−0.329945	−	0.944000i	$-0.607030\pi$
$727$	−17.7926	−0.659890	−0.329945	+	0.944000i	$0.607030\pi$
$728$	2.23864	0.0829697
$729$	−20.3969	−0.755443
$730$	0	0
$731$	−77.4968	−2.86632
$732$	−8.67939	−0.320799
$733$	5.66097	0.209093	0.104546	−	0.994520i	$-0.466661\pi$
$733$	5.66097	0.209093	0.104546	+	0.994520i	$0.466661\pi$
$734$	−57.8337	−2.13468
$735$	0	0
$736$	24.3305	0.896834
$737$	4.44447	0.163714
$738$	37.2109	1.36975
$739$	−5.59955	−0.205983	−0.102991	−	0.994682i	$-0.532841\pi$
$739$	−5.59955	−0.205983	−0.102991	+	0.994682i	$0.532841\pi$
$740$	0	0
$741$	6.70769	0.246413
$742$	−7.19193	−0.264024
$743$	−5.81663	−0.213391	−0.106696	−	0.994292i	$-0.534027\pi$
$743$	−5.81663	−0.213391	−0.106696	+	0.994292i	$0.534027\pi$
$744$	12.4026	0.454700
$745$	0	0
$746$	−50.8542	−1.86191
$747$	−48.2936	−1.76697
$748$	6.37720	0.233174
$749$	5.79045	0.211579
$750$	0	0
$751$	−41.1797	−1.50267	−0.751333	−	0.659923i	$-0.770589\pi$
$751$	−41.1797	−1.50267	−0.751333	+	0.659923i	$0.770589\pi$
$752$	−33.8753	−1.23531
$753$	32.5003	1.18438
$754$	15.7339	0.572993
$755$	0	0
$756$	0.227171	0.00826212
$757$	33.5208	1.21833	0.609167	−	0.793042i	$-0.291504\pi$
$757$	33.5208	1.21833	0.609167	+	0.793042i	$0.291504\pi$
$758$	1.10894	0.0402785
$759$	−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	$0.306558\pi$
$762$	−11.9583	−0.433204
$763$	−6.32353	−0.228927
$764$	16.2611	0.588306
$765$	0	0
$766$	49.8307	1.80046
$767$	−17.0897	−0.617074
$768$	−36.6270	−1.32166
$769$	−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
$772$	15.5160	0.558432
$773$	−40.6687	−1.46275	−0.731376	−	0.681974i	$-0.761122\pi$
$773$	−40.6687	−1.46275	−0.731376	+	0.681974i	$0.761122\pi$
$774$	53.1924	1.91196
$775$	0	0
$776$	−24.7827	−0.889647
$777$	5.00695	0.179623
$778$	−52.0275	−1.86528
$779$	8.50106	0.304582
$780$	0	0
$781$	3.17058	0.113452
$782$	65.2702	2.33406
$783$	−2.80113	−0.100104
$784$	33.7742	1.20622
$785$	0	0
$786$	34.1775	1.21907
$787$	39.9525	1.42415	0.712076	−	0.702102i	$-0.247755\pi$
$787$	39.9525	1.42415	0.712076	+	0.702102i	$0.247755\pi$
$788$	−0.808876	−0.0288150
$789$	12.7643	0.454420
$790$	0	0
$791$	5.02830	0.178786
$792$	7.67939	0.272875
$793$	−14.1882	−0.503838
$794$	20.7643	0.736897
$795$	0	0
$796$	−1.61563	−0.0572644
$797$	4.53791	0.160741	0.0803705	−	0.996765i	$-0.474390\pi$
$797$	4.53791	0.160741	0.0803705	+	0.996765i	$0.474390\pi$
$798$	−1.48052	−0.0524097
$799$	−43.8641	−1.55180
$800$	0	0
$801$	3.97945	0.140607
$802$	40.6228	1.43444
$803$	−4.40550	−0.155467
$804$	5.57045	0.196455
$805$	0	0
$806$	−11.5563	−0.407054
$807$	−16.8003	−0.591399
$808$	19.3257	0.679874
$809$	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	$-0.233284\pi$
$811$	42.3326	1.48650	0.743249	+	0.669014i	$0.233284\pi$
$812$	−0.924984	−0.0324606
$813$	−51.5782	−1.80893
$814$	12.6970	0.445030
$815$	0	0
$816$	−74.6623	−2.61370
$817$	12.1522	0.425150
$818$	22.5648	0.788961
$819$	−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	$0.517258\pi$
$822$	25.8833	0.902784
$823$	45.2002	1.57558	0.787790	−	0.615944i	$-0.211225\pi$
$823$	45.2002	1.57558	0.787790	+	0.615944i	$0.211225\pi$
$824$	−19.9242	−0.694092
$825$	0	0
$826$	3.77203	0.131246
$827$	35.6503	1.23968	0.619841	−	0.784727i	$-0.287197\pi$
$827$	35.6503	1.23968	0.619841	+	0.784727i	$0.287197\pi$
$828$	−11.9327	−0.414691
$829$	−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
$832$	−9.50749	−0.329613
$833$	43.7331	1.51526
$834$	39.9079	1.38190
$835$	0	0
$836$	−1.00000	−0.0345857
$837$	2.05739	0.0711139
$838$	20.9512	0.723746
$839$	−14.1805	−0.489564	−0.244782	−	0.969578i	$-0.578716\pi$
$839$	−14.1805	−0.489564	−0.244782	+	0.969578i	$0.578716\pi$
$840$	0	0
$841$	−17.5945	−0.606707
$842$	48.1832	1.66050
$843$	23.8697	0.822117
$844$	−7.08277	−0.243799
$845$	0	0
$846$	30.1076	1.03512
$847$	−3.43380	−0.117987
$848$	56.8726	1.95301
$849$	−49.7176	−1.70630
$850$	0	0
$851$	34.6134	1.18653
$852$	3.97383	0.136141
$853$	−42.6639	−1.46078	−0.730392	−	0.683028i	$-0.760663\pi$
$853$	−42.6639	−1.46078	−0.730392	+	0.683028i	$0.760663\pi$
$854$	3.13161	0.107161
$855$	0	0
$856$	−32.2880	−1.10358
$857$	1.42392	0.0486403	0.0243201	−	0.999704i	$-0.492258\pi$
$857$	1.42392	0.0486403	0.0243201	+	0.999704i	$0.492258\pi$
$858$	−15.2525	−0.520713
$859$	−8.27894	−0.282474	−0.141237	−	0.989976i	$-0.545108\pi$
$859$	−8.27894	−0.282474	−0.141237	+	0.989976i	$0.545108\pi$
$860$	0	0
$861$	−7.62280	−0.259784
$862$	−49.5024	−1.68606
$863$	−30.4154	−1.03535	−0.517676	−	0.855577i	$-0.673203\pi$
$863$	−30.4154	−1.03535	−0.517676	+	0.855577i	$0.673203\pi$
$864$	−3.25547	−0.110753
$865$	0	0
$866$	7.86276	0.267188
$867$	−56.2653	−1.91087
$868$	0.679390	0.0230600
$869$	9.25254	0.313871
$870$	0	0
$871$	9.10602	0.308546
$872$	35.2605	1.19407
$873$	31.2370	1.05721
$874$	−10.2349	−0.346201
$875$	0	0
$876$	−5.52161	−0.186558
$877$	−57.8484	−1.95340	−0.976700	−	0.214608i	$-0.931153\pi$
$877$	−57.8484	−1.95340	−0.976700	+	0.214608i	$0.931153\pi$
$878$	11.3695	0.383700
$879$	−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	$0.399655\pi$
$882$	−30.0176	−1.01075
$883$	18.7947	0.632492	0.316246	−	0.948677i	$-0.397578\pi$
$883$	18.7947	0.632492	0.316246	+	0.948677i	$0.397578\pi$
$884$	13.0659	0.439453
$885$	0	0
$886$	−14.2838	−0.479872
$887$	11.1471	0.374283	0.187142	−	0.982333i	$-0.440078\pi$
$887$	11.1471	0.374283	0.187142	+	0.982333i	$0.440078\pi$
$888$	−27.9191	−0.936905
$889$	1.14923	0.0385440
$890$	0	0
$891$	13.6687	0.457919
$892$	13.5371	0.453256
$893$	6.87826	0.230172
$894$	21.7445	0.727246
$895$	0	0
$896$	5.05952	0.169027
$897$	−41.5801	−1.38832
$898$	14.2498	0.475523
$899$	−8.37720	−0.279395
$900$	0	0
$901$	73.6425	2.45339
$902$	−19.3305	−0.643635
$903$	−10.8967	−0.362619
$904$	−28.0382	−0.932536
$905$	0	0
$906$	−20.1111	−0.668145
$907$	−38.5598	−1.28036	−0.640178	−	0.768226i	$-0.721139\pi$
$907$	−38.5598	−1.28036	−0.640178	+	0.768226i	$0.721139\pi$
$908$	5.84280	0.193900
$909$	−24.3588	−0.807930
$910$	0	0
$911$	−1.79820	−0.0595771	−0.0297885	−	0.999556i	$-0.509483\pi$
$911$	−1.79820	−0.0595771	−0.0297885	+	0.999556i	$0.509483\pi$
$912$	11.7077	0.387680
$913$	25.0878	0.830285
$914$	47.2365	1.56244
$915$	0	0
$916$	14.6249	0.483221
$917$	−3.28456	−0.108466
$918$	−8.73328	−0.288241
$919$	−32.4458	−1.07029	−0.535144	−	0.844761i	$-0.679743\pi$
$919$	−32.4458	−1.07029	−0.535144	+	0.844761i	$0.679743\pi$
$920$	0	0
$921$	−17.6970	−0.583136
$922$	−64.5619	−2.12623
$923$	6.49602	0.213819
$924$	0.896688	0.0294989
$925$	0	0
$926$	13.2811	0.436443
$927$	25.1132	0.824825
$928$	13.2555	0.435132
$929$	−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
$932$	1.25897	0.0412390
$933$	2.27309	0.0744176
$934$	−28.1492	−0.921071
$935$	0	0
$936$	15.7339	0.514277
$937$	−8.29926	−0.271125	−0.135563	−	0.990769i	$-0.543284\pi$
$937$	−8.29926	−0.271125	−0.135563	+	0.990769i	$0.543284\pi$
$938$	−2.00987	−0.0656247
$939$	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	$0.315091\pi$
$942$	77.5745	2.52751
$943$	−52.6970	−1.71605
$944$	−29.8286	−0.970839
$945$	0	0
$946$	−27.6327	−0.898416
$947$	−1.35103	−0.0439026	−0.0219513	−	0.999759i	$-0.506988\pi$
$947$	−1.35103	−0.0439026	−0.0219513	+	0.999759i	$0.506988\pi$
$948$	11.5966	0.376641
$949$	−9.02617	−0.293002
$950$	0	0
$951$	53.8462	1.74608
$952$	5.05952	0.163980
$953$	6.70769	0.217283	0.108642	−	0.994081i	$-0.465350\pi$
$953$	6.70769	0.217283	0.108642	+	0.994081i	$0.465350\pi$
$954$	−50.5470	−1.63652
$955$	0	0
$956$	−13.2987	−0.430110
$957$	−11.0566	−0.357409
$958$	37.2285	1.20280
$959$	−2.48746	−0.0803244
$960$	0	0
$961$	−24.8470	−0.801518
$962$	26.0142	0.838731
$963$	40.6970	1.31144
$964$	−1.59508	−0.0513741
$965$	0	0
$966$	9.17753	0.295282
$967$	−19.5174	−0.627636	−0.313818	−	0.949483i	$-0.601608\pi$
$967$	−19.5174	−0.627636	−0.313818	+	0.949483i	$0.601608\pi$
$968$	19.1471	0.615411
$969$	15.1599	0.487006
$970$	0	0
$971$	−8.50669	−0.272993	−0.136496	−	0.990641i	$-0.543584\pi$
$971$	−8.50669	−0.272993	−0.136496	+	0.990641i	$0.543584\pi$
$972$	15.3249	0.491545
$973$	−3.83527	−0.122953
$974$	40.5470	1.29921
$975$	0	0
$976$	−24.7643	−0.792685
$977$	−19.7048	−0.630411	−0.315206	−	0.949023i	$-0.602073\pi$
$977$	−19.7048	−0.630411	−0.315206	+	0.949023i	$0.602073\pi$
$978$	53.0638	1.69679
$979$	−2.06727	−0.0660701
$980$	0	0
$981$	−44.4437	−1.41898
$982$	−27.9527	−0.892006
$983$	12.4677	0.397658	0.198829	−	0.980034i	$-0.436286\pi$
$983$	12.4677	0.397658	0.198829	+	0.980034i	$0.436286\pi$
$984$	42.5053	1.35502
$985$	0	0
$986$	35.5598	1.13245
$987$	−6.16765	−0.196319
$988$	−2.04884	−0.0651824
$989$	−75.3297	−2.39534
$990$	0	0
$991$	25.0841	0.796822	0.398411	−	0.917207i	$-0.369562\pi$
$991$	25.0841	0.796822	0.398411	+	0.917207i	$0.369562\pi$
$992$	−9.73598	−0.309118
$993$	−9.89589	−0.314036
$994$	−1.43380	−0.0454772
$995$	0	0
$996$	31.4437	0.996331
$997$	−11.8775	−0.376163	−0.188082	−	0.982153i	$-0.560227\pi$
$997$	−11.8775	−0.376163	−0.188082	+	0.982153i	$0.560227\pi$
$998$	0.690644	0.0218620
$999$	−4.63135	−0.146529

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.a.e.1.3	✓	3
3.2	odd	2		4275.2.a.bm.1.1		3
4.3	odd	2		7600.2.a.cc.1.3		3
5.2	odd	4		475.2.b.b.324.5		6
5.3	odd	4		475.2.b.b.324.2		6
5.4	even	2		475.2.a.g.1.1	yes	3
15.14	odd	2		4275.2.a.ba.1.3		3
19.18	odd	2		9025.2.a.bc.1.1		3
20.19	odd	2		7600.2.a.bh.1.1		3
95.94	odd	2		9025.2.a.y.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.a.e.1.3	✓	3	1.1	even	1	trivial
475.2.a.g.1.1	yes	3	5.4	even	2
475.2.b.b.324.2		6	5.3	odd	4
475.2.b.b.324.5		6	5.2	odd	4
4275.2.a.ba.1.3		3	15.14	odd	2
4275.2.a.bm.1.1		3	3.2	odd	2
7600.2.a.bh.1.1		3	20.19	odd	2
7600.2.a.cc.1.3		3	4.3	odd	2
9025.2.a.y.1.3		3	95.94	odd	2
9025.2.a.bc.1.1		3	19.18	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.a (trivial)

\(f(q)\)	\(=\)	\(q+1.65109 q^{2} -2.37720 q^{3} +0.726109 q^{4} -3.92498 q^{6} +0.377203 q^{7} -2.10331 q^{8} +2.65109 q^{9} -1.37720 q^{11} -1.72611 q^{12} -2.82167 q^{13} +0.622797 q^{14} -4.92498 q^{16} -6.37720 q^{17} +4.37720 q^{18} +1.00000 q^{19} -0.896688 q^{21} -2.27389 q^{22} -6.19887 q^{23} +5.00000 q^{24} -4.65884 q^{26} +0.829422 q^{27} +0.273891 q^{28} -3.37720 q^{29} +2.48052 q^{31} -3.92498 q^{32} +3.27389 q^{33} -10.5294 q^{34} +1.92498 q^{36} -5.58383 q^{37} +1.65109 q^{38} +6.70769 q^{39} +8.50106 q^{41} -1.48052 q^{42} +12.1522 q^{43} -1.00000 q^{44} -10.2349 q^{46} +6.87826 q^{47} +11.7077 q^{48} -6.85772 q^{49} +15.1599 q^{51} -2.04884 q^{52} -11.5478 q^{53} +1.36945 q^{54} -0.793375 q^{56} -2.37720 q^{57} -5.57608 q^{58} +6.05659 q^{59} +5.02830 q^{61} +4.09556 q^{62} +1.00000 q^{63} +3.36945 q^{64} +5.40550 q^{66} -3.22717 q^{67} -4.63055 q^{68} +14.7360 q^{69} -2.30219 q^{71} -5.57608 q^{72} +3.19887 q^{73} -9.21942 q^{74} +0.726109 q^{76} -0.519485 q^{77} +11.0750 q^{78} -6.71836 q^{79} -9.92498 q^{81} +14.0360 q^{82} -18.2165 q^{83} -0.651093 q^{84} +20.0643 q^{86} +8.02830 q^{87} +2.89669 q^{88} +1.50106 q^{89} -1.06434 q^{91} -4.50106 q^{92} -5.89669 q^{93} +11.3567 q^{94} +9.33048 q^{96} +11.7827 q^{97} -11.3227 q^{98} -3.65109 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 3 q^{6} - 4 q^{7} - 3 q^{8} + q^{9} + q^{11} - 7 q^{12} - 3 q^{13} + 7 q^{14} - 6 q^{16} - 14 q^{17} + 8 q^{18} + 3 q^{19} - 6 q^{21} - 5 q^{22} - 8 q^{23} + 15 q^{24}+ \cdots - 4 q^{99}+O(q^{100}) \) 3 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 - 3 * q^6 - 4 * q^7 - 3 * q^8 + q^9 + q^11 - 7 * q^12 - 3 * q^13 + 7 * q^14 - 6 * q^16 - 14 * q^17 + 8 * q^18 + 3 * q^19 - 6 * q^21 - 5 * q^22 - 8 * q^23 + 15 * q^24 - 11 * q^26 + q^27 - q^28 - 5 * q^29 - q^31 - 3 * q^32 + 8 * q^33 + 5 * q^34 - 3 * q^36 - 5 * q^37 - 2 * q^38 - 11 * q^39 + q^41 + 4 * q^42 + 5 * q^43 - 3 * q^44 - 12 * q^46 - 9 * q^47 + 4 * q^48 - 7 * q^49 + 18 * q^51 + 22 * q^52 - 31 * q^53 - 5 * q^54 - 9 * q^56 - 2 * q^57 - q^58 - 6 * q^59 + 3 * q^61 + 5 * q^62 + 3 * q^63 + q^64 - q^66 + 13 * q^67 - 23 * q^68 + q^69 + 7 * q^71 - q^72 - q^73 - q^74 + 4 * q^76 - 10 * q^77 + 42 * q^78 - 18 * q^79 - 21 * q^81 + 34 * q^82 - 3 * q^83 + 5 * q^84 + 40 * q^86 + 12 * q^87 + 12 * q^88 - 20 * q^89 + 17 * q^91 + 11 * q^92 - 21 * q^93 + 45 * q^94 + 2 * q^96 + 13 * q^97 - 4 * q^98 - 4 * q^99

Introduction

L-functions

Modular forms

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Database

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