LMFDB - Embedded newform 475.2.a.g.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:	$0$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.37720$ 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)

$f(q)$	$=$	\(q+2.37720 q^{2} +1.27389 q^{3} +3.65109 q^{4} +3.02830 q^{6} +0.726109 q^{7} +3.92498 q^{8} -1.37720 q^{9} -0.273891 q^{11} +4.65109 q^{12} -5.95328 q^{13} +1.72611 q^{14} +2.02830 q^{16} +5.27389 q^{17} -3.27389 q^{18} +1.00000 q^{19} +0.924984 q^{21} -0.651093 q^{22} -3.67939 q^{23} +5.00000 q^{24} -14.1522 q^{26} -5.57608 q^{27} +2.65109 q^{28} -2.27389 q^{29} +3.19887 q^{31} -3.02830 q^{32} -0.348907 q^{33} +12.5371 q^{34} -5.02830 q^{36} +8.12386 q^{37} +2.37720 q^{38} -7.58383 q^{39} -9.43380 q^{41} +2.19887 q^{42} +9.81100 q^{43} -1.00000 q^{44} -8.74666 q^{46} +12.1599 q^{47} +2.58383 q^{48} -6.47277 q^{49} +6.71836 q^{51} -21.7360 q^{52} +5.69781 q^{53} -13.2555 q^{54} +2.84997 q^{56} +1.27389 q^{57} -5.40550 q^{58} -4.20662 q^{59} -0.103312 q^{61} +7.60437 q^{62} -1.00000 q^{63} -11.2555 q^{64} -0.829422 q^{66} -11.7827 q^{67} +19.2555 q^{68} -4.68714 q^{69} +5.75441 q^{71} -5.40550 q^{72} +6.67939 q^{73} +19.3121 q^{74} +3.65109 q^{76} -0.198875 q^{77} -18.0283 q^{78} +3.87826 q^{79} -2.97170 q^{81} -22.4260 q^{82} -0.488265 q^{83} +3.37720 q^{84} +23.3227 q^{86} -2.89669 q^{87} -1.07502 q^{88} -16.4338 q^{89} -4.32273 q^{91} -13.4338 q^{92} +4.07502 q^{93} +28.9066 q^{94} -3.85772 q^{96} -4.44447 q^{97} -15.3871 q^{98} +0.377203 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.37720	1.68094	0.840468	−	0.541861i	$-0.182280\pi$
$2$	2.37720	1.68094	0.840468	+	0.541861i	$0.182280\pi$
$3$	1.27389	0.735481	0.367741	−	0.929928i	$-0.380131\pi$
$3$	1.27389	0.735481	0.367741	+	0.929928i	$0.380131\pi$
$4$	3.65109	1.82555
$5$	0	0
$5$	0	0
$6$	3.02830	1.23630
$7$	0.726109	0.274444	0.137222	−	0.990540i	$-0.456183\pi$
$7$	0.726109	0.274444	0.137222	+	0.990540i	$0.456183\pi$
$8$	3.92498	1.38769
$9$	−1.37720	−0.459068
$10$	0	0
$11$	−0.273891	−0.0825811	−0.0412906	−	0.999147i	$-0.513147\pi$
$11$	−0.273891	−0.0825811	−0.0412906	+	0.999147i	$0.513147\pi$
$12$	4.65109	1.34266
$13$	−5.95328	−1.65114	−0.825571	−	0.564298i	$-0.809147\pi$
$13$	−5.95328	−1.65114	−0.825571	+	0.564298i	$0.809147\pi$
$14$	1.72611	0.461322
$15$	0	0
$16$	2.02830	0.507074
$17$	5.27389	1.27911	0.639553	−	0.768747i	$-0.279119\pi$
$17$	5.27389	1.27911	0.639553	+	0.768747i	$0.279119\pi$
$18$	−3.27389	−0.771663
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0.924984	0.201848
$22$	−0.651093	−0.138814
$23$	−3.67939	−0.767206	−0.383603	−	0.923498i	$-0.625317\pi$
$23$	−3.67939	−0.767206	−0.383603	+	0.923498i	$0.625317\pi$
$24$	5.00000	1.02062
$25$	0	0
$26$	−14.1522	−2.77547
$27$	−5.57608	−1.07312
$28$	2.65109	0.501010
$29$	−2.27389	−0.422251	−0.211125	−	0.977459i	$-0.567713\pi$
$29$	−2.27389	−0.422251	−0.211125	+	0.977459i	$0.567713\pi$
$30$	0	0
$31$	3.19887	0.574535	0.287267	−	0.957850i	$-0.407253\pi$
$31$	3.19887	0.574535	0.287267	+	0.957850i	$0.407253\pi$
$32$	−3.02830	−0.535332
$33$	−0.348907	−0.0607368
$34$	12.5371	2.15010
$35$	0	0
$36$	−5.02830	−0.838049
$37$	8.12386	1.33555	0.667777	−	0.744361i	$-0.267246\pi$
$37$	8.12386	1.33555	0.667777	+	0.744361i	$0.267246\pi$
$38$	2.37720	0.385633
$39$	−7.58383	−1.21438
$40$	0	0
$41$	−9.43380	−1.47331	−0.736656	−	0.676268i	$-0.763596\pi$
$41$	−9.43380	−1.47331	−0.736656	+	0.676268i	$0.763596\pi$
$42$	2.19887	0.339294
$43$	9.81100	1.49616	0.748082	−	0.663607i	$-0.230975\pi$
$43$	9.81100	1.49616	0.748082	+	0.663607i	$0.230975\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−8.74666	−1.28962
$47$	12.1599	1.77370	0.886852	−	0.462053i	$-0.152887\pi$
$47$	12.1599	1.77370	0.886852	+	0.462053i	$0.152887\pi$
$48$	2.58383	0.372943
$49$	−6.47277	−0.924681
$50$	0	0
$51$	6.71836	0.940758
$52$	−21.7360	−3.01424
$53$	5.69781	0.782655	0.391327	−	0.920252i	$-0.372016\pi$
$53$	5.69781	0.782655	0.391327	+	0.920252i	$0.372016\pi$
$54$	−13.2555	−1.80384
$55$	0	0
$56$	2.84997	0.380843
$57$	1.27389	0.168731
$58$	−5.40550	−0.709777
$59$	−4.20662	−0.547656	−0.273828	−	0.961779i	$-0.588290\pi$
$59$	−4.20662	−0.547656	−0.273828	+	0.961779i	$0.588290\pi$
$60$	0	0
$61$	−0.103312	−0.0132278	−0.00661389	−	0.999978i	$-0.502105\pi$
$61$	−0.103312	−0.0132278	−0.00661389	+	0.999978i	$0.502105\pi$
$62$	7.60437	0.965756
$63$	−1.00000	−0.125988
$64$	−11.2555	−1.40693
$65$	0	0
$66$	−0.829422	−0.102095
$67$	−11.7827	−1.43949	−0.719743	−	0.694241i	$-0.755740\pi$
$67$	−11.7827	−1.43949	−0.719743	+	0.694241i	$0.755740\pi$
$68$	19.2555	2.33507
$69$	−4.68714	−0.564265
$70$	0	0
$71$	5.75441	0.682922	0.341461	−	0.939896i	$-0.389078\pi$
$71$	5.75441	0.682922	0.341461	+	0.939896i	$0.389078\pi$
$72$	−5.40550	−0.637044
$73$	6.67939	0.781763	0.390882	−	0.920441i	$-0.372170\pi$
$73$	6.67939	0.781763	0.390882	+	0.920441i	$0.372170\pi$
$74$	19.3121	2.24498
$75$	0	0
$76$	3.65109	0.418809
$77$	−0.198875	−0.0226639
$78$	−18.0283	−2.04130
$79$	3.87826	0.436339	0.218169	−	0.975911i	$-0.429991\pi$
$79$	3.87826	0.436339	0.218169	+	0.975911i	$0.429991\pi$
$80$	0	0
$81$	−2.97170	−0.330189
$82$	−22.4260	−2.47654
$83$	−0.488265	−0.0535941	−0.0267970	−	0.999641i	$-0.508531\pi$
$83$	−0.488265	−0.0535941	−0.0267970	+	0.999641i	$0.508531\pi$
$84$	3.37720	0.368483
$85$	0	0
$86$	23.3227	2.51495
$87$	−2.89669	−0.310558
$88$	−1.07502	−0.114597
$89$	−16.4338	−1.74198	−0.870989	−	0.491302i	$-0.836521\pi$
$89$	−16.4338	−1.74198	−0.870989	+	0.491302i	$0.836521\pi$
$90$	0	0
$91$	−4.32273	−0.453146
$92$	−13.4338	−1.40057
$93$	4.07502	0.422559
$94$	28.9066	2.98148
$95$	0	0
$96$	−3.85772	−0.393727
$97$	−4.44447	−0.451267	−0.225634	−	0.974212i	$-0.572445\pi$
$97$	−4.44447	−0.451267	−0.225634	+	0.974212i	$0.572445\pi$
$98$	−15.3871	−1.55433
$99$	0.377203	0.0379103
$100$	0	0
$101$	4.38495	0.436319	0.218160	−	0.975913i	$-0.429995\pi$
$101$	4.38495	0.436319	0.218160	+	0.975913i	$0.429995\pi$
$102$	15.9709	1.58136
$103$	3.33048	0.328162	0.164081	−	0.986447i	$-0.447534\pi$
$103$	3.33048	0.328162	0.164081	+	0.986447i	$0.447534\pi$
$104$	−23.3665	−2.29128
$105$	0	0
$106$	13.5449	1.31559
$107$	16.4904	1.59419	0.797093	−	0.603857i	$-0.206370\pi$
$107$	16.4904	1.59419	0.797093	+	0.603857i	$0.206370\pi$
$108$	−20.3588	−1.95902
$109$	7.79045	0.746190	0.373095	−	0.927793i	$-0.378297\pi$
$109$	7.79045	0.746190	0.373095	+	0.927793i	$0.378297\pi$
$110$	0	0
$111$	10.3489	0.982275
$112$	1.47277	0.139163
$113$	−0.142282	−0.0133848	−0.00669238	−	0.999978i	$-0.502130\pi$
$113$	−0.142282	−0.0133848	−0.00669238	+	0.999978i	$0.502130\pi$
$114$	3.02830	0.283626
$115$	0	0
$116$	−8.30219	−0.770839
$117$	8.19887	0.757986
$118$	−10.0000	−0.920575
$119$	3.82942	0.351043
$120$	0	0
$121$	−10.9250	−0.993180
$122$	−0.245594	−0.0222351
$123$	−12.0176	−1.08359
$124$	11.6794	1.04884
$125$	0	0
$126$	−2.37720	−0.211778
$127$	−15.1316	−1.34271	−0.671357	−	0.741135i	$-0.734288\pi$
$127$	−15.1316	−1.34271	−0.671357	+	0.741135i	$0.734288\pi$
$128$	−20.6999	−1.82963
$129$	12.4981	1.10040
$130$	0	0
$131$	5.58383	0.487861	0.243931	−	0.969793i	$-0.421563\pi$
$131$	5.58383	0.487861	0.243931	+	0.969793i	$0.421563\pi$
$132$	−1.27389	−0.110878
$133$	0.726109	0.0629617
$134$	−28.0099	−2.41968
$135$	0	0
$136$	20.6999	1.77500
$137$	12.8294	1.09609	0.548046	−	0.836448i	$-0.315372\pi$
$137$	12.8294	1.09609	0.548046	+	0.836448i	$0.315372\pi$
$138$	−11.1423	−0.948494
$139$	−15.2477	−1.29329	−0.646647	−	0.762789i	$-0.723829\pi$
$139$	−15.2477	−1.29329	−0.646647	+	0.762789i	$0.723829\pi$
$140$	0	0
$141$	15.4904	1.30453
$142$	13.6794	1.14795
$143$	1.63055	0.136353
$144$	−2.79338	−0.232781
$145$	0	0
$146$	15.8783	1.31409
$147$	−8.24559	−0.680085
$148$	29.6610	2.43812
$149$	13.8315	1.13312	0.566562	−	0.824019i	$-0.308273\pi$
$149$	13.8315	1.13312	0.566562	+	0.824019i	$0.308273\pi$
$150$	0	0
$151$	−11.7077	−0.952758	−0.476379	−	0.879240i	$-0.658051\pi$
$151$	−11.7077	−0.952758	−0.476379	+	0.879240i	$0.658051\pi$
$152$	3.92498	0.318358
$153$	−7.26322	−0.587196
$154$	−0.472765	−0.0380965
$155$	0	0
$156$	−27.6893	−2.21692
$157$	−4.79045	−0.382320	−0.191160	−	0.981559i	$-0.561225\pi$
$157$	−4.79045	−0.382320	−0.191160	+	0.981559i	$0.561225\pi$
$158$	9.21942	0.733458
$159$	7.25839	0.575628
$160$	0	0
$161$	−2.67164	−0.210555
$162$	−7.06434	−0.555027
$163$	12.8011	1.00266	0.501331	−	0.865256i	$-0.332844\pi$
$163$	12.8011	1.00266	0.501331	+	0.865256i	$0.332844\pi$
$164$	−34.4437	−2.68960
$165$	0	0
$166$	−1.16071	−0.0900882
$167$	−20.9426	−1.62059	−0.810294	−	0.586024i	$-0.800692\pi$
$167$	−20.9426	−1.62059	−0.810294	+	0.586024i	$0.800692\pi$
$168$	3.63055	0.280103
$169$	22.4415	1.72627
$170$	0	0
$171$	−1.37720	−0.105317
$172$	35.8209	2.73132
$173$	15.7282	1.19580	0.597898	−	0.801572i	$-0.296003\pi$
$173$	15.7282	1.19580	0.597898	+	0.801572i	$0.296003\pi$
$174$	−6.88601	−0.522027
$175$	0	0
$176$	−0.555531	−0.0418747
$177$	−5.35878	−0.402791
$178$	−39.0665	−2.92816
$179$	3.41325	0.255118	0.127559	−	0.991831i	$-0.459286\pi$
$179$	3.41325	0.255118	0.127559	+	0.991831i	$0.459286\pi$
$180$	0	0
$181$	23.5109	1.74755	0.873777	−	0.486327i	$-0.161664\pi$
$181$	23.5109	1.74755	0.873777	+	0.486327i	$0.161664\pi$
$182$	−10.2760	−0.761709
$183$	−0.131609	−0.00972878
$184$	−14.4415	−1.06464
$185$	0	0
$186$	9.68714	0.710296
$187$	−1.44447	−0.105630
$188$	44.3969	3.23798
$189$	−4.04884	−0.294510
$190$	0	0
$191$	12.4650	0.901937	0.450968	−	0.892540i	$-0.351079\pi$
$191$	12.4650	0.901937	0.450968	+	0.892540i	$0.351079\pi$
$192$	−14.3382	−1.03477
$193$	19.2993	1.38919	0.694596	−	0.719400i	$-0.255583\pi$
$193$	19.2993	1.38919	0.694596	+	0.719400i	$0.255583\pi$
$194$	−10.5654	−0.758552
$195$	0	0
$196$	−23.6327	−1.68805
$197$	6.63055	0.472407	0.236203	−	0.971704i	$-0.424097\pi$
$197$	6.63055	0.472407	0.236203	+	0.971704i	$0.424097\pi$
$198$	0.896688	0.0637248
$199$	−23.0849	−1.63644	−0.818222	−	0.574902i	$-0.805040\pi$
$199$	−23.0849	−1.63644	−0.818222	+	0.574902i	$0.805040\pi$
$200$	0	0
$201$	−15.0099	−1.05871
$202$	10.4239	0.733425
$203$	−1.65109	−0.115884
$204$	24.5294	1.71740
$205$	0	0
$206$	7.91723	0.551620
$207$	5.06727	0.352199
$208$	−12.0750	−0.837252
$209$	−0.273891	−0.0189454
$210$	0	0
$211$	−7.54778	−0.519611	−0.259805	−	0.965661i	$-0.583658\pi$
$211$	−7.54778	−0.519611	−0.259805	+	0.965661i	$0.583658\pi$
$212$	20.8032	1.42877
$213$	7.33048	0.502276
$214$	39.2010	2.67973
$215$	0	0
$216$	−21.8860	−1.48915
$217$	2.32273	0.157677
$218$	18.5195	1.25430
$219$	8.50881	0.574972
$220$	0	0
$221$	−31.3969	−2.11199
$222$	24.6015	1.65114
$223$	−1.09344	−0.0732221	−0.0366111	−	0.999330i	$-0.511656\pi$
$223$	−1.09344	−0.0732221	−0.0366111	+	0.999330i	$0.511656\pi$
$224$	−2.19887	−0.146918
$225$	0	0
$226$	−0.338233	−0.0224989
$227$	−20.1316	−1.33618	−0.668091	−	0.744080i	$-0.732888\pi$
$227$	−20.1316	−1.33618	−0.668091	+	0.744080i	$0.732888\pi$
$228$	4.65109	0.308026
$229$	−5.51656	−0.364545	−0.182272	−	0.983248i	$-0.558345\pi$
$229$	−5.51656	−0.364545	−0.182272	+	0.983248i	$0.558345\pi$
$230$	0	0
$231$	−0.253344	−0.0166688
$232$	−8.92498	−0.585954
$233$	−18.1805	−1.19104	−0.595520	−	0.803340i	$-0.703054\pi$
$233$	−18.1805	−1.19104	−0.595520	+	0.803340i	$0.703054\pi$
$234$	19.4904	1.27413
$235$	0	0
$236$	−15.3588	−0.999771
$237$	4.94048	0.320919
$238$	9.10331	0.590080
$239$	21.9164	1.41766	0.708828	−	0.705381i	$-0.249224\pi$
$239$	21.9164	1.41766	0.708828	+	0.705381i	$0.249224\pi$
$240$	0	0
$241$	−28.1882	−1.81576	−0.907881	−	0.419228i	$-0.862301\pi$
$241$	−28.1882	−1.81576	−0.907881	+	0.419228i	$0.862301\pi$
$242$	−25.9709	−1.66947
$243$	12.9426	0.830269
$244$	−0.377203	−0.0241479
$245$	0	0
$246$	−28.5683	−1.82145
$247$	−5.95328	−0.378798
$248$	12.5555	0.797277
$249$	−0.621996	−0.0394174
$250$	0	0
$251$	9.00987	0.568698	0.284349	−	0.958721i	$-0.408223\pi$
$251$	9.00987	0.568698	0.284349	+	0.958721i	$0.408223\pi$
$252$	−3.65109	−0.229997
$253$	1.00775	0.0633567
$254$	−35.9709	−2.25702
$255$	0	0
$256$	−26.6970	−1.66856
$257$	6.86064	0.427955	0.213978	−	0.976839i	$-0.431358\pi$
$257$	6.86064	0.427955	0.213978	+	0.976839i	$0.431358\pi$
$258$	29.7106	1.84970
$259$	5.89881	0.366534
$260$	0	0
$261$	3.13161	0.193842
$262$	13.2739	0.820064
$263$	−9.25547	−0.570717	−0.285358	−	0.958421i	$-0.592113\pi$
$263$	−9.25547	−0.570717	−0.285358	+	0.958421i	$0.592113\pi$
$264$	−1.36945	−0.0842840
$265$	0	0
$266$	1.72611	0.105835
$267$	−20.9349	−1.28119
$268$	−43.0197	−2.62785
$269$	0.498939	0.0304208	0.0152104	−	0.999884i	$-0.495158\pi$
$269$	0.498939	0.0304208	0.0152104	+	0.999884i	$0.495158\pi$
$270$	0	0
$271$	3.71061	0.225403	0.112702	−	0.993629i	$-0.464050\pi$
$271$	3.71061	0.225403	0.112702	+	0.993629i	$0.464050\pi$
$272$	10.6970	0.648602
$273$	−5.50669	−0.333280
$274$	30.4981	1.84246
$275$	0	0
$276$	−17.1132	−1.03009
$277$	4.58675	0.275591	0.137796	−	0.990461i	$-0.455998\pi$
$277$	4.58675	0.275591	0.137796	+	0.990461i	$0.455998\pi$
$278$	−36.2469	−2.17395
$279$	−4.40550	−0.263750
$280$	0	0
$281$	27.2653	1.62651	0.813257	−	0.581905i	$-0.197692\pi$
$281$	27.2653	1.62651	0.813257	+	0.581905i	$0.197692\pi$
$282$	36.8238	2.19283
$283$	−10.2661	−0.610259	−0.305129	−	0.952311i	$-0.598700\pi$
$283$	−10.2661	−0.610259	−0.305129	+	0.952311i	$0.598700\pi$
$284$	21.0099	1.24671
$285$	0	0
$286$	3.87614	0.229201
$287$	−6.84997	−0.404341
$288$	4.17058	0.245754
$289$	10.8139	0.636113
$290$	0	0
$291$	−5.66177	−0.331899
$292$	24.3871	1.42715
$293$	−1.87051	−0.109277	−0.0546383	−	0.998506i	$-0.517401\pi$
$293$	−1.87051	−0.109277	−0.0546383	+	0.998506i	$0.517401\pi$
$294$	−19.6015	−1.14318
$295$	0	0
$296$	31.8860	1.85334
$297$	1.52723	0.0886192
$298$	32.8804	1.90471
$299$	21.9044	1.26677
$300$	0	0
$301$	7.12386	0.410612
$302$	−27.8315	−1.60153
$303$	5.58595	0.320904
$304$	2.02830	0.116331
$305$	0	0
$306$	−17.2661	−0.987040
$307$	0.227171	0.0129653	0.00648266	−	0.999979i	$-0.497936\pi$
$307$	0.227171	0.0129653	0.00648266	+	0.999979i	$0.497936\pi$
$308$	−0.726109	−0.0413739
$309$	4.24267	0.241357
$310$	0	0
$311$	20.9554	1.18827	0.594136	−	0.804365i	$-0.297494\pi$
$311$	20.9554	1.18827	0.594136	+	0.804365i	$0.297494\pi$
$312$	−29.7664	−1.68519
$313$	−11.2349	−0.635035	−0.317518	−	0.948252i	$-0.602849\pi$
$313$	−11.2349	−0.635035	−0.317518	+	0.948252i	$0.602849\pi$
$314$	−11.3879	−0.642655
$315$	0	0
$316$	14.1599	0.796557
$317$	18.6228	1.04596	0.522980	−	0.852345i	$-0.324820\pi$
$317$	18.6228	1.04596	0.522980	+	0.852345i	$0.324820\pi$
$318$	17.2547	0.967594
$319$	0.622797	0.0348699
$320$	0	0
$321$	21.0069	1.17249
$322$	−6.35103	−0.353929
$323$	5.27389	0.293447
$324$	−10.8500	−0.602776
$325$	0	0
$326$	30.4309	1.68541
$327$	9.92418	0.548809
$328$	−37.0275	−2.04450
$329$	8.82942	0.486782
$330$	0	0
$331$	−14.1054	−0.775305	−0.387652	−	0.921806i	$-0.626714\pi$
$331$	−14.1054	−0.775305	−0.387652	+	0.921806i	$0.626714\pi$
$332$	−1.78270	−0.0978385
$333$	−11.1882	−0.613110
$334$	−49.7848	−2.72410
$335$	0	0
$336$	1.87614	0.102352
$337$	22.9709	1.25130	0.625652	−	0.780102i	$-0.284833\pi$
$337$	22.9709	1.25130	0.625652	+	0.780102i	$0.284833\pi$
$338$	53.3481	2.90175
$339$	−0.181252	−0.00984424
$340$	0	0
$341$	−0.876142	−0.0474457
$342$	−3.27389	−0.177032
$343$	−9.78270	−0.528216
$344$	38.5080	2.07621
$345$	0	0
$346$	37.3892	2.01006
$347$	−3.93273	−0.211120	−0.105560	−	0.994413i	$-0.533664\pi$
$347$	−3.93273	−0.211120	−0.105560	+	0.994413i	$0.533664\pi$
$348$	−10.5761	−0.566937
$349$	−34.4252	−1.84274	−0.921371	−	0.388685i	$-0.872929\pi$
$349$	−34.4252	−1.84274	−0.921371	+	0.388685i	$0.872929\pi$
$350$	0	0
$351$	33.1960	1.77187
$352$	0.829422	0.0442083
$353$	−4.25547	−0.226496	−0.113248	−	0.993567i	$-0.536125\pi$
$353$	−4.25547	−0.226496	−0.113248	+	0.993567i	$0.536125\pi$
$354$	−12.7389	−0.677065
$355$	0	0
$356$	−60.0013	−3.18006
$357$	4.87826	0.258185
$358$	8.11399	0.428837
$359$	−20.2944	−1.07110	−0.535550	−	0.844504i	$-0.679896\pi$
$359$	−20.2944	−1.07110	−0.535550	+	0.844504i	$0.679896\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	55.8903	2.93753
$363$	−13.9172	−0.730465
$364$	−15.7827	−0.827238
$365$	0	0
$366$	−0.312860	−0.0163535
$367$	3.85289	0.201119	0.100560	−	0.994931i	$-0.467937\pi$
$367$	3.85289	0.201119	0.100560	+	0.994931i	$0.467937\pi$
$368$	−7.46289	−0.389030
$369$	12.9922	0.676350
$370$	0	0
$371$	4.13724	0.214795
$372$	14.8783	0.771402
$373$	14.6356	0.757802	0.378901	−	0.925437i	$-0.376302\pi$
$373$	14.6356	0.757802	0.378901	+	0.925437i	$0.376302\pi$
$374$	−3.43380	−0.177557
$375$	0	0
$376$	47.7274	2.46135
$377$	13.5371	0.697197
$378$	−9.62492	−0.495052
$379$	−22.0099	−1.13057	−0.565286	−	0.824895i	$-0.691234\pi$
$379$	−22.0099	−1.13057	−0.565286	+	0.824895i	$0.691234\pi$
$380$	0	0
$381$	−19.2760	−0.987540
$382$	29.6319	1.51610
$383$	−3.08569	−0.157671	−0.0788357	−	0.996888i	$-0.525120\pi$
$383$	−3.08569	−0.157671	−0.0788357	+	0.996888i	$0.525120\pi$
$384$	−26.3695	−1.34566
$385$	0	0
$386$	45.8783	2.33514
$387$	−13.5117	−0.686840
$388$	−16.2272	−0.823810
$389$	8.77203	0.444760	0.222380	−	0.974960i	$-0.428618\pi$
$389$	8.77203	0.444760	0.222380	+	0.974960i	$0.428618\pi$
$390$	0	0
$391$	−19.4047	−0.981338
$392$	−25.4055	−1.28317
$393$	7.11319	0.358813
$394$	15.7622	0.794086
$395$	0	0
$396$	1.37720	0.0692070
$397$	−1.59450	−0.0800257	−0.0400129	−	0.999199i	$-0.512740\pi$
$397$	−1.59450	−0.0800257	−0.0400129	+	0.999199i	$0.512740\pi$
$398$	−54.8775	−2.75076
$399$	0.924984	0.0463071
$400$	0	0
$401$	−17.5526	−0.876535	−0.438268	−	0.898844i	$-0.644408\pi$
$401$	−17.5526	−0.876535	−0.438268	+	0.898844i	$0.644408\pi$
$402$	−35.6815	−1.77963
$403$	−19.0438	−0.948639
$404$	16.0099	0.796521
$405$	0	0
$406$	−3.92498	−0.194794
$407$	−2.22505	−0.110292
$408$	26.3695	1.30548
$409$	36.6815	1.81378	0.906892	−	0.421363i	$-0.138448\pi$
$409$	36.6815	1.81378	0.906892	+	0.421363i	$0.138448\pi$
$410$	0	0
$411$	16.3433	0.806155
$412$	12.1599	0.599076
$413$	−3.05447	−0.150301
$414$	12.0459	0.592025
$415$	0	0
$416$	18.0283	0.883910
$417$	−19.4239	−0.951194
$418$	−0.651093	−0.0318460
$419$	−18.8187	−0.919356	−0.459678	−	0.888086i	$-0.652035\pi$
$419$	−18.8187	−0.919356	−0.459678	+	0.888086i	$0.652035\pi$
$420$	0	0
$421$	−33.7819	−1.64643	−0.823215	−	0.567730i	$-0.807822\pi$
$421$	−33.7819	−1.64643	−0.823215	+	0.567730i	$0.807822\pi$
$422$	−17.9426	−0.873432
$423$	−16.7467	−0.814250
$424$	22.3638	1.08608
$425$	0	0
$426$	17.4260	0.844295
$427$	−0.0750160	−0.00363028
$428$	60.2079	2.91026
$429$	2.07714	0.100285
$430$	0	0
$431$	−12.7651	−0.614872	−0.307436	−	0.951569i	$-0.599471\pi$
$431$	−12.7651	−0.614872	−0.307436	+	0.951569i	$0.599471\pi$
$432$	−11.3099	−0.544150
$433$	−16.0771	−0.772618	−0.386309	−	0.922369i	$-0.626250\pi$
$433$	−16.0771	−0.772618	−0.386309	+	0.922369i	$0.626250\pi$
$434$	5.52161	0.265046
$435$	0	0
$436$	28.4437	1.36220
$437$	−3.67939	−0.176009
$438$	20.2272	0.966492
$439$	1.36945	0.0653604	0.0326802	−	0.999466i	$-0.489596\pi$
$439$	1.36945	0.0653604	0.0326802	+	0.999466i	$0.489596\pi$
$440$	0	0
$441$	8.91431	0.424491
$442$	−74.6369	−3.55012
$443$	4.62280	0.219636	0.109818	−	0.993952i	$-0.464973\pi$
$443$	4.62280	0.219636	0.109818	+	0.993952i	$0.464973\pi$
$444$	37.7848	1.79319
$445$	0	0
$446$	−2.59933	−0.123082
$447$	17.6199	0.833391
$448$	−8.17270	−0.386124
$449$	23.2555	1.09749	0.548747	−	0.835989i	$-0.315105\pi$
$449$	23.2555	1.09749	0.548747	+	0.835989i	$0.315105\pi$
$450$	0	0
$451$	2.58383	0.121668
$452$	−0.519485	−0.0244345
$453$	−14.9143	−0.700735
$454$	−47.8569	−2.24604
$455$	0	0
$456$	5.00000	0.234146
$457$	−35.8443	−1.67673	−0.838364	−	0.545111i	$-0.816487\pi$
$457$	−35.8443	−1.67673	−0.838364	+	0.545111i	$0.816487\pi$
$458$	−13.1140	−0.612776
$459$	−29.4076	−1.37263
$460$	0	0
$461$	−14.8812	−0.693086	−0.346543	−	0.938034i	$-0.612645\pi$
$461$	−14.8812	−0.693086	−0.346543	+	0.938034i	$0.612645\pi$
$462$	−0.602251	−0.0280193
$463$	−29.9554	−1.39215	−0.696073	−	0.717971i	$-0.745071\pi$
$463$	−29.9554	−1.39215	−0.696073	+	0.717971i	$0.745071\pi$
$464$	−4.61212	−0.214112
$465$	0	0
$466$	−43.2186	−2.00206
$467$	−6.73598	−0.311704	−0.155852	−	0.987780i	$-0.549812\pi$
$467$	−6.73598	−0.311704	−0.155852	+	0.987780i	$0.549812\pi$
$468$	29.9349	1.38374
$469$	−8.55553	−0.395058
$470$	0	0
$471$	−6.10251	−0.281189
$472$	−16.5109	−0.759977
$473$	−2.68714	−0.123555
$474$	11.7445	0.539444
$475$	0	0
$476$	13.9816	0.640845
$477$	−7.84704	−0.359291
$478$	52.0998	2.38299
$479$	16.6978	0.762943	0.381471	−	0.924381i	$-0.375418\pi$
$479$	16.6978	0.762943	0.381471	+	0.924381i	$0.375418\pi$
$480$	0	0
$481$	−48.3636	−2.20519
$482$	−67.0091	−3.05218
$483$	−3.40338	−0.154859
$484$	−39.8881	−1.81310
$485$	0	0
$486$	30.7672	1.39563
$487$	3.64042	0.164963	0.0824816	−	0.996593i	$-0.473715\pi$
$487$	3.64042	0.164963	0.0824816	+	0.996593i	$0.473715\pi$
$488$	−0.405499	−0.0183561
$489$	16.3072	0.737439
$490$	0	0
$491$	−33.3249	−1.50393	−0.751965	−	0.659203i	$-0.770894\pi$
$491$	−33.3249	−1.50393	−0.751965	+	0.659203i	$0.770894\pi$
$492$	−43.8775	−1.97815
$493$	−11.9922	−0.540104
$494$	−14.1522	−0.636736
$495$	0	0
$496$	6.48827	0.291332
$497$	4.17833	0.187424
$498$	−1.47861	−0.0662582
$499$	−37.9914	−1.70073	−0.850365	−	0.526193i	$-0.823619\pi$
$499$	−37.9914	−1.70073	−0.850365	+	0.526193i	$0.823619\pi$
$500$	0	0
$501$	−26.6786	−1.19191
$502$	21.4183	0.955945
$503$	−42.1826	−1.88083	−0.940414	−	0.340032i	$-0.889562\pi$
$503$	−42.1826	−1.88083	−0.940414	+	0.340032i	$0.889562\pi$
$504$	−3.92498	−0.174833
$505$	0	0
$506$	2.39563	0.106499
$507$	28.5881	1.26964
$508$	−55.2469	−2.45119
$509$	−21.9971	−0.975003	−0.487502	−	0.873122i	$-0.662092\pi$
$509$	−21.9971	−0.975003	−0.487502	+	0.873122i	$0.662092\pi$
$510$	0	0
$511$	4.84997	0.214550
$512$	−22.0643	−0.975115
$513$	−5.57608	−0.246190
$514$	16.3091	0.719365
$515$	0	0
$516$	45.6319	2.00883
$517$	−3.33048	−0.146474
$518$	14.0227	0.616121
$519$	20.0360	0.879485
$520$	0	0
$521$	20.0977	0.880496	0.440248	−	0.897876i	$-0.354891\pi$
$521$	20.0977	0.880496	0.440248	+	0.897876i	$0.354891\pi$
$522$	7.44447	0.325836
$523$	−4.64817	−0.203250	−0.101625	−	0.994823i	$-0.532404\pi$
$523$	−4.64817	−0.203250	−0.101625	+	0.994823i	$0.532404\pi$
$524$	20.3871	0.890614
$525$	0	0
$526$	−22.0021	−0.959338
$527$	16.8705	0.734891
$528$	−0.707686	−0.0307981
$529$	−9.46209	−0.411395
$530$	0	0
$531$	5.79338	0.251411
$532$	2.65109	0.114939
$533$	56.1620	2.43265
$534$	−49.7664	−2.15360
$535$	0	0
$536$	−46.2469	−1.99756
$537$	4.34811	0.187635
$538$	1.18608	0.0511355
$539$	1.77283	0.0763612
$540$	0	0
$541$	20.0673	0.862759	0.431380	−	0.902171i	$-0.358027\pi$
$541$	20.0673	0.862759	0.431380	+	0.902171i	$0.358027\pi$
$542$	8.82087	0.378889
$543$	29.9504	1.28529
$544$	−15.9709	−0.684747
$545$	0	0
$546$	−13.0905	−0.560222
$547$	−37.2010	−1.59060	−0.795300	−	0.606216i	$-0.792687\pi$
$547$	−37.2010	−1.59060	−0.795300	+	0.606216i	$0.792687\pi$
$548$	46.8414	2.00097
$549$	0.142282	0.00607245
$550$	0	0
$551$	−2.27389	−0.0968710
$552$	−18.3969	−0.783026
$553$	2.81604	0.119750
$554$	10.9036	0.463251
$555$	0	0
$556$	−55.6708	−2.36097
$557$	44.8393	1.89990	0.949951	−	0.312399i	$-0.101133\pi$
$557$	44.8393	1.89990	0.949951	+	0.312399i	$0.101133\pi$
$558$	−10.4728	−0.443347
$559$	−58.4076	−2.47038
$560$	0	0
$561$	−1.84010	−0.0776889
$562$	64.8152	2.73407
$563$	−21.9172	−0.923701	−0.461851	−	0.886958i	$-0.652814\pi$
$563$	−21.9172	−0.923701	−0.461851	+	0.886958i	$0.652814\pi$
$564$	56.5569	2.38147
$565$	0	0
$566$	−24.4047	−1.02581
$567$	−2.15778	−0.0906183
$568$	22.5860	0.947685
$569$	9.90656	0.415305	0.207652	−	0.978203i	$-0.433418\pi$
$569$	9.90656	0.415305	0.207652	+	0.978203i	$0.433418\pi$
$570$	0	0
$571$	17.6404	0.738229	0.369114	−	0.929384i	$-0.379661\pi$
$571$	17.6404	0.738229	0.369114	+	0.929384i	$0.379661\pi$
$572$	5.95328	0.248919
$573$	15.8791	0.663357
$574$	−16.2838	−0.679671
$575$	0	0
$576$	15.5011	0.645878
$577$	−12.7048	−0.528906	−0.264453	−	0.964399i	$-0.585191\pi$
$577$	−12.7048	−0.528906	−0.264453	+	0.964399i	$0.585191\pi$
$578$	25.7069	1.06927
$579$	24.5851	1.02172
$580$	0	0
$581$	−0.354534	−0.0147085
$582$	−13.4592	−0.557900
$583$	−1.56058	−0.0646325
$584$	26.2165	1.08485
$585$	0	0
$586$	−4.44659	−0.183687
$587$	15.0438	0.620924	0.310462	−	0.950586i	$-0.399516\pi$
$587$	15.0438	0.620924	0.310462	+	0.950586i	$0.399516\pi$
$588$	−30.1054	−1.24153
$589$	3.19887	0.131807
$590$	0	0
$591$	8.44659	0.347446
$592$	16.4776	0.677225
$593$	−16.4231	−0.674417	−0.337208	−	0.941430i	$-0.609483\pi$
$593$	−16.4231	−0.674417	−0.337208	+	0.941430i	$0.609483\pi$
$594$	3.63055	0.148963
$595$	0	0
$596$	50.5003	2.06857
$597$	−29.4076	−1.20357
$598$	52.0713	2.12935
$599$	−19.1260	−0.781466	−0.390733	−	0.920504i	$-0.627778\pi$
$599$	−19.1260	−0.781466	−0.390733	+	0.920504i	$0.627778\pi$
$600$	0	0
$601$	31.4124	1.28134	0.640670	−	0.767816i	$-0.278657\pi$
$601$	31.4124	1.28134	0.640670	+	0.767816i	$0.278657\pi$
$602$	16.9349	0.690213
$603$	16.2272	0.660821
$604$	−42.7459	−1.73930
$605$	0	0
$606$	13.2789	0.539420
$607$	−41.5315	−1.68571	−0.842855	−	0.538140i	$-0.819127\pi$
$607$	−41.5315	−1.68571	−0.842855	+	0.538140i	$0.819127\pi$
$608$	−3.02830	−0.122814
$609$	−2.10331	−0.0852305
$610$	0	0
$611$	−72.3913	−2.92864
$612$	−26.5187	−1.07195
$613$	21.7274	0.877563	0.438781	−	0.898594i	$-0.355410\pi$
$613$	21.7274	0.877563	0.438781	+	0.898594i	$0.355410\pi$
$614$	0.540031	0.0217939
$615$	0	0
$616$	−0.780579	−0.0314504
$617$	33.6065	1.35295	0.676473	−	0.736467i	$-0.263507\pi$
$617$	33.6065	1.35295	0.676473	+	0.736467i	$0.263507\pi$
$618$	10.0857	0.405706
$619$	−27.6036	−1.10948	−0.554741	−	0.832023i	$-0.687183\pi$
$619$	−27.6036	−1.10948	−0.554741	+	0.832023i	$0.687183\pi$
$620$	0	0
$621$	20.5166	0.823301
$622$	49.8152	1.99741
$623$	−11.9327	−0.478075
$624$	−15.3822	−0.615783
$625$	0	0
$626$	−26.7077	−1.06745
$627$	−0.348907	−0.0139340
$628$	−17.4904	−0.697942
$629$	42.8443	1.70832
$630$	0	0
$631$	1.94048	0.0772495	0.0386247	−	0.999254i	$-0.487702\pi$
$631$	1.94048	0.0772495	0.0386247	+	0.999254i	$0.487702\pi$
$632$	15.2221	0.605504
$633$	−9.61505	−0.382164
$634$	44.2702	1.75819
$635$	0	0
$636$	26.5011	1.05084
$637$	38.5342	1.52678
$638$	1.48052	0.0586142
$639$	−7.92498	−0.313508
$640$	0	0
$641$	1.01975	0.0402775	0.0201388	−	0.999797i	$-0.493589\pi$
$641$	1.01975	0.0402775	0.0201388	+	0.999797i	$0.493589\pi$
$642$	49.9378	1.97089
$643$	36.9866	1.45861	0.729305	−	0.684189i	$-0.239844\pi$
$643$	36.9866	1.45861	0.729305	+	0.684189i	$0.239844\pi$
$644$	−9.75441	−0.384377
$645$	0	0
$646$	12.5371	0.493266
$647$	24.1182	0.948186	0.474093	−	0.880475i	$-0.342776\pi$
$647$	24.1182	0.948186	0.474093	+	0.880475i	$0.342776\pi$
$648$	−11.6639	−0.458201
$649$	1.15215	0.0452260
$650$	0	0
$651$	2.95891	0.115969
$652$	46.7381	1.83041
$653$	37.2603	1.45811	0.729054	−	0.684456i	$-0.239960\pi$
$653$	37.2603	1.45811	0.729054	+	0.684456i	$0.239960\pi$
$654$	23.5918	0.922512
$655$	0	0
$656$	−19.1345	−0.747078
$657$	−9.19887	−0.358882
$658$	20.9893	0.818249
$659$	21.4386	0.835130	0.417565	−	0.908647i	$-0.362884\pi$
$659$	21.4386	0.835130	0.417565	+	0.908647i	$0.362884\pi$
$660$	0	0
$661$	0.783503	0.0304747	0.0152374	−	0.999884i	$-0.495150\pi$
$661$	0.783503	0.0304747	0.0152374	+	0.999884i	$0.495150\pi$
$662$	−33.5315	−1.30324
$663$	−39.9963	−1.55333
$664$	−1.91643	−0.0743720
$665$	0	0
$666$	−26.5966	−1.03060
$667$	8.36653	0.323953
$668$	−76.4634	−2.95846
$669$	−1.39292	−0.0538535
$670$	0	0
$671$	0.0282963	0.00109237
$672$	−2.80113	−0.108056
$673$	−50.1903	−1.93469	−0.967347	−	0.253454i	$-0.918433\pi$
$673$	−50.1903	−1.93469	−0.967347	+	0.253454i	$0.918433\pi$
$674$	54.6065	2.10336
$675$	0	0
$676$	81.9362	3.15139
$677$	−29.8804	−1.14840	−0.574198	−	0.818716i	$-0.694686\pi$
$677$	−29.8804	−1.14840	−0.574198	+	0.818716i	$0.694686\pi$
$678$	−0.430872	−0.0165475
$679$	−3.22717	−0.123847
$680$	0	0
$681$	−25.6455	−0.982736
$682$	−2.08277	−0.0797532
$683$	12.3326	0.471894	0.235947	−	0.971766i	$-0.424181\pi$
$683$	12.3326	0.471894	0.235947	+	0.971766i	$0.424181\pi$
$684$	−5.02830	−0.192262
$685$	0	0
$686$	−23.2555	−0.887898
$687$	−7.02750	−0.268116
$688$	19.8996	0.758666
$689$	−33.9207	−1.29227
$690$	0	0
$691$	3.62200	0.137787	0.0688936	−	0.997624i	$-0.478053\pi$
$691$	3.62200	0.137787	0.0688936	+	0.997624i	$0.478053\pi$
$692$	57.4252	2.18298
$693$	0.273891	0.0104042
$694$	−9.34891	−0.354880
$695$	0	0
$696$	−11.3695	−0.430958
$697$	−49.7528	−1.88452
$698$	−81.8358	−3.09753
$699$	−23.1599	−0.875988
$700$	0	0
$701$	34.1209	1.28873	0.644365	−	0.764718i	$-0.277122\pi$
$701$	34.1209	1.28873	0.644365	+	0.764718i	$0.277122\pi$
$702$	78.9135	2.97840
$703$	8.12386	0.306397
$704$	3.08277	0.116186
$705$	0	0
$706$	−10.1161	−0.380725
$707$	3.18396	0.119745
$708$	−19.5654	−0.735313
$709$	−17.1209	−0.642990	−0.321495	−	0.946911i	$-0.604185\pi$
$709$	−17.1209	−0.642990	−0.321495	+	0.946911i	$0.604185\pi$
$710$	0	0
$711$	−5.34116	−0.200309
$712$	−64.5024	−2.41733
$713$	−11.7699	−0.440786
$714$	11.5966	0.433993
$715$	0	0
$716$	12.4621	0.465730
$717$	27.9191	1.04266
$718$	−48.2440	−1.80045
$719$	−7.02750	−0.262081	−0.131041	−	0.991377i	$-0.541832\pi$
$719$	−7.02750	−0.262081	−0.131041	+	0.991377i	$0.541832\pi$
$720$	0	0
$721$	2.41830	0.0900620
$722$	2.37720	0.0884703
$723$	−35.9087	−1.33546
$724$	85.8406	3.19024
$725$	0	0
$726$	−33.0841	−1.22787
$727$	−11.8938	−0.441115	−0.220558	−	0.975374i	$-0.570788\pi$
$727$	−11.8938	−0.441115	−0.220558	+	0.975374i	$0.570788\pi$
$728$	−16.9667	−0.628826
$729$	25.4026	0.940836
$730$	0	0
$731$	51.7421	1.91375
$732$	−0.480515	−0.0177604
$733$	20.7154	0.765142	0.382571	−	0.923926i	$-0.375039\pi$
$733$	20.7154	0.765142	0.382571	+	0.923926i	$0.375039\pi$
$734$	9.15910	0.338069
$735$	0	0
$736$	11.1423	0.410710
$737$	3.22717	0.118874
$738$	30.8852	1.13690
$739$	33.8620	1.24563	0.622816	−	0.782368i	$-0.285988\pi$
$739$	33.8620	1.24563	0.622816	+	0.782368i	$0.285988\pi$
$740$	0	0
$741$	−7.58383	−0.278599
$742$	9.83505	0.361056
$743$	42.7381	1.56791	0.783955	−	0.620818i	$-0.213200\pi$
$743$	42.7381	1.56791	0.783955	+	0.620818i	$0.213200\pi$
$744$	15.9944	0.586382
$745$	0	0
$746$	34.7918	1.27382
$747$	0.672440	0.0246033
$748$	−5.27389	−0.192833
$749$	11.9738	0.437514
$750$	0	0
$751$	11.9581	0.436358	0.218179	−	0.975909i	$-0.429988\pi$
$751$	11.9581	0.436358	0.218179	+	0.975909i	$0.429988\pi$
$752$	24.6639	0.899400
$753$	11.4776	0.418267
$754$	32.1805	1.17194
$755$	0	0
$756$	−14.7827	−0.537642
$757$	29.1103	1.05803	0.529015	−	0.848612i	$-0.322561\pi$
$757$	29.1103	1.05803	0.529015	+	0.848612i	$0.322561\pi$
$758$	−52.3219	−1.90042
$759$	1.28376	0.0465977
$760$	0	0
$761$	−22.3014	−0.808425	−0.404212	−	0.914665i	$-0.632454\pi$
$761$	−22.3014	−0.808425	−0.404212	+	0.914665i	$0.632454\pi$
$762$	−45.8230	−1.65999
$763$	5.65672	0.204787
$764$	45.5109	1.64653
$765$	0	0
$766$	−7.33531	−0.265036
$767$	25.0432	0.904258
$768$	−34.0091	−1.22720
$769$	3.95891	0.142762	0.0713809	−	0.997449i	$-0.477259\pi$
$769$	3.95891	0.142762	0.0713809	+	0.997449i	$0.477259\pi$
$770$	0	0
$771$	8.73971	0.314753
$772$	70.4634	2.53603
$773$	27.8139	1.00040	0.500199	−	0.865911i	$-0.333260\pi$
$773$	27.8139	1.00040	0.500199	+	0.865911i	$0.333260\pi$
$774$	−32.1201	−1.15453
$775$	0	0
$776$	−17.4445	−0.626220
$777$	7.51444	0.269579
$778$	20.8529	0.747612
$779$	−9.43380	−0.338001
$780$	0	0
$781$	−1.57608	−0.0563965
$782$	−46.1289	−1.64957
$783$	12.6794	0.453124
$784$	−13.1287	−0.468882
$785$	0	0
$786$	16.9095	0.603141
$787$	−1.82460	−0.0650398	−0.0325199	−	0.999471i	$-0.510353\pi$
$787$	−1.82460	−0.0650398	−0.0325199	+	0.999471i	$0.510353\pi$
$788$	24.2087	0.862401
$789$	−11.7905	−0.419751
$790$	0	0
$791$	−0.103312	−0.00367336
$792$	1.48052	0.0526078
$793$	0.615047	0.0218410
$794$	−3.79045	−0.134518
$795$	0	0
$796$	−84.2851	−2.98741
$797$	−21.0360	−0.745135	−0.372567	−	0.928005i	$-0.621522\pi$
$797$	−21.0360	−0.745135	−0.372567	+	0.928005i	$0.621522\pi$
$798$	2.19887	0.0778393
$799$	64.1300	2.26876
$800$	0	0
$801$	22.6327	0.799686
$802$	−41.7261	−1.47340
$803$	−1.82942	−0.0645589
$804$	−54.8024	−1.93273
$805$	0	0
$806$	−45.2710	−1.59460
$807$	0.635593	0.0223739
$808$	17.2109	0.605476
$809$	−0.0819654	−0.00288175	−0.00144088	−	0.999999i	$-0.500459\pi$
$809$	−0.0819654	−0.00288175	−0.00144088	+	0.999999i	$0.500459\pi$
$810$	0	0
$811$	−6.72531	−0.236158	−0.118079	−	0.993004i	$-0.537674\pi$
$811$	−6.72531	−0.236158	−0.118079	+	0.993004i	$0.537674\pi$
$812$	−6.02830	−0.211552
$813$	4.72691	0.165780
$814$	−5.28939	−0.185393
$815$	0	0
$816$	13.6268	0.477034
$817$	9.81100	0.343243
$818$	87.1994	3.04886
$819$	5.95328	0.208024
$820$	0	0
$821$	30.9426	1.07990	0.539952	−	0.841696i	$-0.318442\pi$
$821$	30.9426	1.07990	0.539952	+	0.841696i	$0.318442\pi$
$822$	38.8513	1.35509
$823$	26.5908	0.926896	0.463448	−	0.886124i	$-0.346612\pi$
$823$	26.5908	0.926896	0.463448	+	0.886124i	$0.346612\pi$
$824$	13.0721	0.455388
$825$	0	0
$826$	−7.26109	−0.252646
$827$	−5.57900	−0.194001	−0.0970004	−	0.995284i	$-0.530925\pi$
$827$	−5.57900	−0.194001	−0.0970004	+	0.995284i	$0.530925\pi$
$828$	18.5011	0.642956
$829$	18.9765	0.659082	0.329541	−	0.944141i	$-0.393106\pi$
$829$	18.9765	0.659082	0.329541	+	0.944141i	$0.393106\pi$
$830$	0	0
$831$	5.84302	0.202692
$832$	67.0069	2.32305
$833$	−34.1367	−1.18276
$834$	−46.1746	−1.59890
$835$	0	0
$836$	−1.00000	−0.0345857
$837$	−17.8372	−0.616543
$838$	−44.7360	−1.54538
$839$	12.9143	0.445852	0.222926	−	0.974835i	$-0.428439\pi$
$839$	12.9143	0.445852	0.222926	+	0.974835i	$0.428439\pi$
$840$	0	0
$841$	−23.8294	−0.821704
$842$	−80.3064	−2.76754
$843$	34.7331	1.19627
$844$	−27.5577	−0.948574
$845$	0	0
$846$	−39.8102	−1.36870
$847$	−7.93273	−0.272572
$848$	11.5569	0.396864
$849$	−13.0779	−0.448834
$850$	0	0
$851$	−29.8908	−1.02464
$852$	26.7643	0.916929
$853$	6.46077	0.221213	0.110606	−	0.993864i	$-0.464721\pi$
$853$	6.46077	0.221213	0.110606	+	0.993864i	$0.464721\pi$
$854$	−0.178328	−0.00610227
$855$	0	0
$856$	64.7245	2.21224
$857$	−12.4055	−0.423764	−0.211882	−	0.977295i	$-0.567959\pi$
$857$	−12.4055	−0.423764	−0.211882	+	0.977295i	$0.567959\pi$
$858$	4.93778	0.168573
$859$	40.3425	1.37647	0.688234	−	0.725489i	$-0.258386\pi$
$859$	40.3425	1.37647	0.688234	+	0.725489i	$0.258386\pi$
$860$	0	0
$861$	−8.72611	−0.297385
$862$	−30.3452	−1.03356
$863$	1.83235	0.0623738	0.0311869	−	0.999514i	$-0.490071\pi$
$863$	1.83235	0.0623738	0.0311869	+	0.999514i	$0.490071\pi$
$864$	16.8860	0.574474
$865$	0	0
$866$	−38.2186	−1.29872
$867$	13.7758	0.467849
$868$	8.48052	0.287847
$869$	−1.06222	−0.0360333
$870$	0	0
$871$	70.1457	2.37680
$872$	30.5774	1.03548
$873$	6.12094	0.207162
$874$	−8.74666	−0.295860
$875$	0	0
$876$	31.0665	1.04964
$877$	−8.14419	−0.275010	−0.137505	−	0.990501i	$-0.543908\pi$
$877$	−8.14419	−0.275010	−0.137505	+	0.990501i	$0.543908\pi$
$878$	3.25547	0.109867
$879$	−2.38283	−0.0803709
$880$	0	0
$881$	12.1706	0.410037	0.205019	−	0.978758i	$-0.434275\pi$
$881$	12.1706	0.410037	0.205019	+	0.978758i	$0.434275\pi$
$882$	21.1911	0.713542
$883$	46.7614	1.57364	0.786822	−	0.617179i	$-0.211725\pi$
$883$	46.7614	1.57364	0.786822	+	0.617179i	$0.211725\pi$
$884$	−114.633	−3.85553
$885$	0	0
$886$	10.9893	0.369194
$887$	−34.8804	−1.17117	−0.585584	−	0.810611i	$-0.699135\pi$
$887$	−34.8804	−1.17117	−0.585584	+	0.810611i	$0.699135\pi$
$888$	40.6193	1.36309
$889$	−10.9872	−0.368499
$890$	0	0
$891$	0.813922	0.0272674
$892$	−3.99225	−0.133670
$893$	12.1599	0.406916
$894$	41.8860	1.40088
$895$	0	0
$896$	−15.0304	−0.502131
$897$	27.9039	0.931683
$898$	55.2830	1.84482
$899$	−7.27389	−0.242598
$900$	0	0
$901$	30.0496	1.00110
$902$	6.14228	0.204516
$903$	9.07502	0.301998
$904$	−0.558455	−0.0185739
$905$	0	0
$906$	−35.4543	−1.17789
$907$	−25.5080	−0.846980	−0.423490	−	0.905901i	$-0.639195\pi$
$907$	−25.5080	−0.846980	−0.423490	+	0.905901i	$0.639195\pi$
$908$	−73.5024	−2.43926
$909$	−6.03897	−0.200300
$910$	0	0
$911$	−21.5032	−0.712432	−0.356216	−	0.934404i	$-0.615933\pi$
$911$	−21.5032	−0.712432	−0.356216	+	0.934404i	$0.615933\pi$
$912$	2.58383	0.0855591
$913$	0.133731	0.00442586
$914$	−85.2093	−2.81847
$915$	0	0
$916$	−20.1415	−0.665493
$917$	4.05447	0.133890
$918$	−69.9079	−2.30730
$919$	37.1386	1.22509	0.612544	−	0.790436i	$-0.290146\pi$
$919$	37.1386	1.22509	0.612544	+	0.790436i	$0.290146\pi$
$920$	0	0
$921$	0.289391	0.00953575
$922$	−35.3756	−1.16503
$923$	−34.2576	−1.12760
$924$	−0.924984	−0.0304297
$925$	0	0
$926$	−71.2101	−2.34011
$927$	−4.58675	−0.150649
$928$	6.88601	0.226044
$929$	−3.36170	−0.110294	−0.0551469	−	0.998478i	$-0.517563\pi$
$929$	−3.36170	−0.110294	−0.0551469	+	0.998478i	$0.517563\pi$
$930$	0	0
$931$	−6.47277	−0.212136
$932$	−66.3785	−2.17430
$933$	26.6949	0.873951
$934$	−16.0128	−0.523955
$935$	0	0
$936$	32.1805	1.05185
$937$	10.0694	0.328953	0.164476	−	0.986381i	$-0.447407\pi$
$937$	10.0694	0.328953	0.164476	+	0.986381i	$0.447407\pi$
$938$	−20.3382	−0.664067
$939$	−14.3121	−0.467056
$940$	0	0
$941$	20.8139	0.678514	0.339257	−	0.940694i	$-0.389824\pi$
$941$	20.8139	0.678514	0.339257	+	0.940694i	$0.389824\pi$
$942$	−14.5069	−0.472661
$943$	34.7106	1.13033
$944$	−8.53228	−0.277702
$945$	0	0
$946$	−6.38788	−0.207688
$947$	−30.4904	−0.990804	−0.495402	−	0.868664i	$-0.664979\pi$
$947$	−30.4904	−0.990804	−0.495402	+	0.868664i	$0.664979\pi$
$948$	18.0382	0.585852
$949$	−39.7643	−1.29080
$950$	0	0
$951$	23.7234	0.769284
$952$	15.0304	0.487139
$953$	7.58383	0.245664	0.122832	−	0.992427i	$-0.460802\pi$
$953$	7.58383	0.245664	0.122832	+	0.992427i	$0.460802\pi$
$954$	−18.6540	−0.603946
$955$	0	0
$956$	80.0189	2.58800
$957$	0.793375	0.0256462
$958$	39.6941	1.28246
$959$	9.31556	0.300815
$960$	0	0
$961$	−20.7672	−0.669910
$962$	−114.970	−3.70678
$963$	−22.7106	−0.731839
$964$	−102.918	−3.31476
$965$	0	0
$966$	−8.09052	−0.260308
$967$	54.6687	1.75803	0.879014	−	0.476797i	$-0.158202\pi$
$967$	54.6687	1.75803	0.879014	+	0.476797i	$0.158202\pi$
$968$	−42.8804	−1.37823
$969$	6.71836	0.215825
$970$	0	0
$971$	−39.9632	−1.28248	−0.641239	−	0.767341i	$-0.721579\pi$
$971$	−39.9632	−1.28248	−0.641239	+	0.767341i	$0.721579\pi$
$972$	47.2547	1.51569
$973$	−11.0715	−0.354936
$974$	8.65402	0.277293
$975$	0	0
$976$	−0.209548	−0.00670747
$977$	15.2400	0.487570	0.243785	−	0.969829i	$-0.421611\pi$
$977$	15.2400	0.487570	0.243785	+	0.969829i	$0.421611\pi$
$978$	38.7656	1.23959
$979$	4.50106	0.143855
$980$	0	0
$981$	−10.7290	−0.342552
$982$	−79.2199	−2.52801
$983$	−45.3609	−1.44679	−0.723394	−	0.690435i	$-0.757419\pi$
$983$	−45.3609	−1.44679	−0.723394	+	0.690435i	$0.757419\pi$
$984$	−47.1690	−1.50369
$985$	0	0
$986$	−28.5080	−0.907880
$987$	11.2477	0.358019
$988$	−21.7360	−0.691514
$989$	−36.0985	−1.14787
$990$	0	0
$991$	−16.3537	−0.519493	−0.259747	−	0.965677i	$-0.583639\pi$
$991$	−16.3537	−0.519493	−0.259747	+	0.965677i	$0.583639\pi$
$992$	−9.68714	−0.307567
$993$	−17.9688	−0.570222
$994$	9.93273	0.315047
$995$	0	0
$996$	−2.27097	−0.0719583
$997$	−33.2037	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$997$	−33.2037	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$998$	−90.3134	−2.85882
$999$	−45.2993	−1.43321

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.a.g.1.3	yes	3
3.2	odd	2		4275.2.a.ba.1.1		3
4.3	odd	2		7600.2.a.bh.1.2		3
5.2	odd	4		475.2.b.b.324.6		6
5.3	odd	4		475.2.b.b.324.1		6
5.4	even	2		475.2.a.e.1.1	✓	3
15.14	odd	2		4275.2.a.bm.1.3		3
19.18	odd	2		9025.2.a.y.1.1		3
20.19	odd	2		7600.2.a.cc.1.2		3
95.94	odd	2		9025.2.a.bc.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.a.e.1.1	✓	3	5.4	even	2
475.2.a.g.1.3	yes	3	1.1	even	1	trivial
475.2.b.b.324.1		6	5.3	odd	4
475.2.b.b.324.6		6	5.2	odd	4
4275.2.a.ba.1.1		3	3.2	odd	2
4275.2.a.bm.1.3		3	15.14	odd	2
7600.2.a.bh.1.2		3	4.3	odd	2
7600.2.a.cc.1.2		3	20.19	odd	2
9025.2.a.y.1.1		3	19.18	odd	2
9025.2.a.bc.1.3		3	95.94	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)

\(f(q)\)	\(=\)	\(q+2.37720 q^{2} +1.27389 q^{3} +3.65109 q^{4} +3.02830 q^{6} +0.726109 q^{7} +3.92498 q^{8} -1.37720 q^{9} -0.273891 q^{11} +4.65109 q^{12} -5.95328 q^{13} +1.72611 q^{14} +2.02830 q^{16} +5.27389 q^{17} -3.27389 q^{18} +1.00000 q^{19} +0.924984 q^{21} -0.651093 q^{22} -3.67939 q^{23} +5.00000 q^{24} -14.1522 q^{26} -5.57608 q^{27} +2.65109 q^{28} -2.27389 q^{29} +3.19887 q^{31} -3.02830 q^{32} -0.348907 q^{33} +12.5371 q^{34} -5.02830 q^{36} +8.12386 q^{37} +2.37720 q^{38} -7.58383 q^{39} -9.43380 q^{41} +2.19887 q^{42} +9.81100 q^{43} -1.00000 q^{44} -8.74666 q^{46} +12.1599 q^{47} +2.58383 q^{48} -6.47277 q^{49} +6.71836 q^{51} -21.7360 q^{52} +5.69781 q^{53} -13.2555 q^{54} +2.84997 q^{56} +1.27389 q^{57} -5.40550 q^{58} -4.20662 q^{59} -0.103312 q^{61} +7.60437 q^{62} -1.00000 q^{63} -11.2555 q^{64} -0.829422 q^{66} -11.7827 q^{67} +19.2555 q^{68} -4.68714 q^{69} +5.75441 q^{71} -5.40550 q^{72} +6.67939 q^{73} +19.3121 q^{74} +3.65109 q^{76} -0.198875 q^{77} -18.0283 q^{78} +3.87826 q^{79} -2.97170 q^{81} -22.4260 q^{82} -0.488265 q^{83} +3.37720 q^{84} +23.3227 q^{86} -2.89669 q^{87} -1.07502 q^{88} -16.4338 q^{89} -4.32273 q^{91} -13.4338 q^{92} +4.07502 q^{93} +28.9066 q^{94} -3.85772 q^{96} -4.44447 q^{97} -15.3871 q^{98} +0.377203 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 2 q^{3} + 4 q^{4} - 3 q^{6} + 4 q^{7} + 3 q^{8} + q^{9} + q^{11} + 7 q^{12} + 3 q^{13} + 7 q^{14} - 6 q^{16} + 14 q^{17} - 8 q^{18} + 3 q^{19} - 6 q^{21} + 5 q^{22} + 8 q^{23} + 15 q^{24}+ \cdots - 4 q^{99}+O(q^{100}) \) 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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