LMFDB - Embedded newform 475.2.b.b.324.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [475,2,Mod(324,475)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(475, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("475.324"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$475 = 5^{2} \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	475.b (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,-8,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	no
Analytic conductor:	$3.79289409601$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.1827904.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} + 9x^{4} + 14x^{2} + 1$ x^6 + 9x^4 + 14x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			324.1
Root			$0.273891i$ of defining polynomial
Character	$\chi$	$=$	475.324
Dual form			475.2.b.b.324.6

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.37720i q^{2} +1.27389i q^{3} -3.65109 q^{4} +3.02830 q^{6} -0.726109i q^{7} +3.92498i q^{8} +1.37720 q^{9} -0.273891 q^{11} -4.65109i q^{12} -5.95328i q^{13} -1.72611 q^{14} +2.02830 q^{16} -5.27389i q^{17} -3.27389i q^{18} -1.00000 q^{19} +0.924984 q^{21} +0.651093i q^{22} -3.67939i q^{23} -5.00000 q^{24} -14.1522 q^{26} +5.57608i q^{27} +2.65109i q^{28} +2.27389 q^{29} +3.19887 q^{31} +3.02830i q^{32} -0.348907i q^{33} -12.5371 q^{34} -5.02830 q^{36} -8.12386i q^{37} +2.37720i q^{38} +7.58383 q^{39} -9.43380 q^{41} -2.19887i q^{42} +9.81100i q^{43} +1.00000 q^{44} -8.74666 q^{46} -12.1599i q^{47} +2.58383i q^{48} +6.47277 q^{49} +6.71836 q^{51} +21.7360i q^{52} +5.69781i q^{53} +13.2555 q^{54} +2.84997 q^{56} -1.27389i q^{57} -5.40550i q^{58} +4.20662 q^{59} -0.103312 q^{61} -7.60437i q^{62} -1.00000i q^{63} +11.2555 q^{64} -0.829422 q^{66} +11.7827i q^{67} +19.2555i q^{68} +4.68714 q^{69} +5.75441 q^{71} +5.40550i q^{72} +6.67939i q^{73} -19.3121 q^{74} +3.65109 q^{76} +0.198875i q^{77} -18.0283i q^{78} -3.87826 q^{79} -2.97170 q^{81} +22.4260i q^{82} -0.488265i q^{83} -3.37720 q^{84} +23.3227 q^{86} +2.89669i q^{87} -1.07502i q^{88} +16.4338 q^{89} -4.32273 q^{91} +13.4338i q^{92} +4.07502i q^{93} -28.9066 q^{94} -3.85772 q^{96} +4.44447i q^{97} -15.3871i q^{98} -0.377203 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q - 8 q^{4} - 6 q^{6} - 2 q^{9} + 2 q^{11} - 14 q^{14} - 12 q^{16} - 6 q^{19} - 12 q^{21} - 30 q^{24} - 22 q^{26} + 10 q^{29} - 2 q^{31} - 10 q^{34} - 6 q^{36} + 22 q^{39} + 2 q^{41} + 6 q^{44} - 24 q^{46}+ \cdots + 8 q^{99}+O(q^{100})$ 6 * q - 8 * q^4 - 6 * q^6 - 2 * q^9 + 2 * q^11 - 14 * q^14 - 12 * q^16 - 6 * q^19 - 12 * q^21 - 30 * q^24 - 22 * q^26 + 10 * q^29 - 2 * q^31 - 10 * q^34 - 6 * q^36 + 22 * q^39 + 2 * q^41 + 6 * q^44 - 24 * q^46 + 14 * q^49 + 36 * q^51 + 10 * q^54 - 18 * q^56 + 12 * q^59 + 6 * q^61 - 2 * q^64 - 2 * q^66 - 2 * q^69 + 14 * q^71 + 2 * q^74 + 8 * q^76 + 36 * q^79 - 42 * q^81 - 10 * q^84 + 80 * q^86 + 40 * q^89 + 34 * q^91 - 90 * q^94 + 4 * q^96 + 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$ .

$n$	$77$	$401$
$\chi(n)$	$-1$	$1$

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	− 2.37720i	− 1.68094i	−0.541861	−	0.840468i	$-0.682280\pi$
$2$	− 2.37720i	− 1.68094i	0.541861	−	0.840468i	$-0.317720\pi$
$3$	1.27389i	0.735481i	0.929928	+	0.367741i	$0.119869\pi$
$3$	1.27389i	0.735481i	−0.929928	+	0.367741i	$0.880131\pi$
$4$	−3.65109	−1.82555
$5$	0	0
$5$	0	0
$6$	3.02830	1.23630
$7$	− 0.726109i	− 0.274444i	−0.990540	−	0.137222i	$-0.956183\pi$
$7$	− 0.726109i	− 0.274444i	0.990540	−	0.137222i	$-0.0438173\pi$
$8$	3.92498i	1.38769i
$9$	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	$0.513147\pi$
$12$	− 4.65109i	− 1.34266i
$13$	− 5.95328i	− 1.65114i	−0.564298	−	0.825571i	$-0.690853\pi$
$13$	− 5.95328i	− 1.65114i	0.564298	−	0.825571i	$-0.309147\pi$
$14$	−1.72611	−0.461322
$15$	0	0
$16$	2.02830	0.507074
$17$	− 5.27389i	− 1.27911i	−0.768747	−	0.639553i	$-0.779119\pi$
$17$	− 5.27389i	− 1.27911i	0.768747	−	0.639553i	$-0.220881\pi$
$18$	− 3.27389i	− 0.771663i
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.924984	0.201848
$22$	0.651093i	0.138814i
$23$	− 3.67939i	− 0.767206i	−0.923498	−	0.383603i	$-0.874683\pi$
$23$	− 3.67939i	− 0.767206i	0.923498	−	0.383603i	$-0.125317\pi$
$24$	−5.00000	−1.02062
$25$	0	0
$26$	−14.1522	−2.77547
$27$	5.57608i	1.07312i
$28$	2.65109i	0.501010i
$29$	2.27389	0.422251	0.211125	−	0.977459i	$-0.432287\pi$
$29$	2.27389	0.422251	0.211125	+	0.977459i	$0.432287\pi$
$30$	0	0
$31$	3.19887	0.574535	0.287267	−	0.957850i	$-0.407253\pi$
$31$	3.19887	0.574535	0.287267	+	0.957850i	$0.407253\pi$
$32$	3.02830i	0.535332i
$33$	− 0.348907i	− 0.0607368i
$34$	−12.5371	−2.15010
$35$	0	0
$36$	−5.02830	−0.838049
$37$	− 8.12386i	− 1.33555i	−0.744361	−	0.667777i	$-0.767246\pi$
$37$	− 8.12386i	− 1.33555i	0.744361	−	0.667777i	$-0.232754\pi$
$38$	2.37720i	0.385633i
$39$	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	$0.763596\pi$
$42$	− 2.19887i	− 0.339294i
$43$	9.81100i	1.49616i	0.663607	+	0.748082i	$0.269025\pi$
$43$	9.81100i	1.49616i	−0.663607	+	0.748082i	$0.730975\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−8.74666	−1.28962
$47$	− 12.1599i	− 1.77370i	−0.462053	−	0.886852i	$-0.652887\pi$
$47$	− 12.1599i	− 1.77370i	0.462053	−	0.886852i	$-0.347113\pi$
$48$	2.58383i	0.372943i
$49$	6.47277	0.924681
$50$	0	0
$51$	6.71836	0.940758
$52$	21.7360i	3.01424i
$53$	5.69781i	0.782655i	0.920252	+	0.391327i	$0.127984\pi$
$53$	5.69781i	0.782655i	−0.920252	+	0.391327i	$0.872016\pi$
$54$	13.2555	1.80384
$55$	0	0
$56$	2.84997	0.380843
$57$	− 1.27389i	− 0.168731i
$58$	− 5.40550i	− 0.709777i
$59$	4.20662	0.547656	0.273828	−	0.961779i	$-0.411710\pi$
$59$	4.20662	0.547656	0.273828	+	0.961779i	$0.411710\pi$
$60$	0	0
$61$	−0.103312	−0.0132278	−0.00661389	−	0.999978i	$-0.502105\pi$
$61$	−0.103312	−0.0132278	−0.00661389	+	0.999978i	$0.502105\pi$
$62$	− 7.60437i	− 0.965756i
$63$	− 1.00000i	− 0.125988i
$64$	11.2555	1.40693
$65$	0	0
$66$	−0.829422	−0.102095
$67$	11.7827i	1.43949i	0.694241	+	0.719743i	$0.255740\pi$
$67$	11.7827i	1.43949i	−0.694241	+	0.719743i	$0.744260\pi$
$68$	19.2555i	2.33507i
$69$	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	$0.389078\pi$
$72$	5.40550i	0.637044i
$73$	6.67939i	0.781763i	0.920441	+	0.390882i	$0.127830\pi$
$73$	6.67939i	0.781763i	−0.920441	+	0.390882i	$0.872170\pi$
$74$	−19.3121	−2.24498
$75$	0	0
$76$	3.65109	0.418809
$77$	0.198875i	0.0226639i
$78$	− 18.0283i	− 2.04130i
$79$	−3.87826	−0.436339	−0.218169	−	0.975911i	$-0.570009\pi$
$79$	−3.87826	−0.436339	−0.218169	+	0.975911i	$0.570009\pi$
$80$	0	0
$81$	−2.97170	−0.330189
$82$	22.4260i	2.47654i
$83$	− 0.488265i	− 0.0535941i	−0.999641	−	0.0267970i	$-0.991469\pi$
$83$	− 0.488265i	− 0.0535941i	0.999641	−	0.0267970i	$-0.00853078\pi$
$84$	−3.37720	−0.368483
$85$	0	0
$86$	23.3227	2.51495
$87$	2.89669i	0.310558i
$88$	− 1.07502i	− 0.114597i
$89$	16.4338	1.74198	0.870989	−	0.491302i	$-0.163479\pi$
$89$	16.4338	1.74198	0.870989	+	0.491302i	$0.163479\pi$
$90$	0	0
$91$	−4.32273	−0.453146
$92$	13.4338i	1.40057i
$93$	4.07502i	0.422559i
$94$	−28.9066	−2.98148
$95$	0	0
$96$	−3.85772	−0.393727
$97$	4.44447i	0.451267i	0.974212	+	0.225634i	$0.0724453\pi$
$97$	4.44447i	0.451267i	−0.974212	+	0.225634i	$0.927555\pi$
$98$	− 15.3871i	− 1.55433i
$99$	−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	$0.429995\pi$
$102$	− 15.9709i	− 1.58136i
$103$	3.33048i	0.328162i	0.986447	+	0.164081i	$0.0524659\pi$
$103$	3.33048i	0.328162i	−0.986447	+	0.164081i	$0.947534\pi$
$104$	23.3665	2.29128
$105$	0	0
$106$	13.5449	1.31559
$107$	− 16.4904i	− 1.59419i	−0.603857	−	0.797093i	$-0.706370\pi$
$107$	− 16.4904i	− 1.59419i	0.603857	−	0.797093i	$-0.293630\pi$
$108$	− 20.3588i	− 1.95902i
$109$	−7.79045	−0.746190	−0.373095	−	0.927793i	$-0.621703\pi$
$109$	−7.79045	−0.746190	−0.373095	+	0.927793i	$0.621703\pi$
$110$	0	0
$111$	10.3489	0.982275
$112$	− 1.47277i	− 0.139163i
$113$	− 0.142282i	− 0.0133848i	−0.999978	−	0.00669238i	$-0.997870\pi$
$113$	− 0.142282i	− 0.0133848i	0.999978	−	0.00669238i	$-0.00213027\pi$
$114$	−3.02830	−0.283626
$115$	0	0
$116$	−8.30219	−0.770839
$117$	− 8.19887i	− 0.757986i
$118$	− 10.0000i	− 0.920575i
$119$	−3.82942	−0.351043
$120$	0	0
$121$	−10.9250	−0.993180
$122$	0.245594i	0.0222351i
$123$	− 12.0176i	− 1.08359i
$124$	−11.6794	−1.04884
$125$	0	0
$126$	−2.37720	−0.211778
$127$	15.1316i	1.34271i	0.741135	+	0.671357i	$0.234288\pi$
$127$	15.1316i	1.34271i	−0.741135	+	0.671357i	$0.765712\pi$
$128$	− 20.6999i	− 1.82963i
$129$	−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	$0.421563\pi$
$132$	1.27389i	0.110878i
$133$	0.726109i	0.0629617i
$134$	28.0099	2.41968
$135$	0	0
$136$	20.6999	1.77500
$137$	− 12.8294i	− 1.09609i	−0.836448	−	0.548046i	$-0.815372\pi$
$137$	− 12.8294i	− 1.09609i	0.836448	−	0.548046i	$-0.184628\pi$
$138$	− 11.1423i	− 0.948494i
$139$	15.2477	1.29329	0.646647	−	0.762789i	$-0.276171\pi$
$139$	15.2477	1.29329	0.646647	+	0.762789i	$0.276171\pi$
$140$	0	0
$141$	15.4904	1.30453
$142$	− 13.6794i	− 1.14795i
$143$	1.63055i	0.136353i
$144$	2.79338	0.232781
$145$	0	0
$146$	15.8783	1.31409
$147$	8.24559i	0.680085i
$148$	29.6610i	2.43812i
$149$	−13.8315	−1.13312	−0.566562	−	0.824019i	$-0.691727\pi$
$149$	−13.8315	−1.13312	−0.566562	+	0.824019i	$0.691727\pi$
$150$	0	0
$151$	−11.7077	−0.952758	−0.476379	−	0.879240i	$-0.658051\pi$
$151$	−11.7077	−0.952758	−0.476379	+	0.879240i	$0.658051\pi$
$152$	− 3.92498i	− 0.318358i
$153$	− 7.26322i	− 0.587196i
$154$	0.472765	0.0380965
$155$	0	0
$156$	−27.6893	−2.21692
$157$	4.79045i	0.382320i	0.981559	+	0.191160i	$0.0612249\pi$
$157$	4.79045i	0.382320i	−0.981559	+	0.191160i	$0.938775\pi$
$158$	9.21942i	0.733458i
$159$	−7.25839	−0.575628
$160$	0	0
$161$	−2.67164	−0.210555
$162$	7.06434i	0.555027i
$163$	12.8011i	1.00266i	0.865256	+	0.501331i	$0.167156\pi$
$163$	12.8011i	1.00266i	−0.865256	+	0.501331i	$0.832844\pi$
$164$	34.4437	2.68960
$165$	0	0
$166$	−1.16071	−0.0900882
$167$	20.9426i	1.62059i	0.586024	+	0.810294i	$0.300692\pi$
$167$	20.9426i	1.62059i	−0.586024	+	0.810294i	$0.699308\pi$
$168$	3.63055i	0.280103i
$169$	−22.4415	−1.72627
$170$	0	0
$171$	−1.37720	−0.105317
$172$	− 35.8209i	− 2.73132i
$173$	15.7282i	1.19580i	0.801572	+	0.597898i	$0.203997\pi$
$173$	15.7282i	1.19580i	−0.801572	+	0.597898i	$0.796003\pi$
$174$	6.88601	0.522027
$175$	0	0
$176$	−0.555531	−0.0418747
$177$	5.35878i	0.402791i
$178$	− 39.0665i	− 2.92816i
$179$	−3.41325	−0.255118	−0.127559	−	0.991831i	$-0.540714\pi$
$179$	−3.41325	−0.255118	−0.127559	+	0.991831i	$0.540714\pi$
$180$	0	0
$181$	23.5109	1.74755	0.873777	−	0.486327i	$-0.161664\pi$
$181$	23.5109	1.74755	0.873777	+	0.486327i	$0.161664\pi$
$182$	10.2760i	0.761709i
$183$	− 0.131609i	− 0.00972878i
$184$	14.4415	1.06464
$185$	0	0
$186$	9.68714	0.710296
$187$	1.44447i	0.105630i
$188$	44.3969i	3.23798i
$189$	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	$0.351079\pi$
$192$	14.3382i	1.03477i
$193$	19.2993i	1.38919i	0.719400	+	0.694596i	$0.244417\pi$
$193$	19.2993i	1.38919i	−0.719400	+	0.694596i	$0.755583\pi$
$194$	10.5654	0.758552
$195$	0	0
$196$	−23.6327	−1.68805
$197$	− 6.63055i	− 0.472407i	−0.971704	−	0.236203i	$-0.924097\pi$
$197$	− 6.63055i	− 0.472407i	0.971704	−	0.236203i	$-0.0759032\pi$
$198$	0.896688i	0.0637248i
$199$	23.0849	1.63644	0.818222	−	0.574902i	$-0.194960\pi$
$199$	23.0849	1.63644	0.818222	+	0.574902i	$0.194960\pi$
$200$	0	0
$201$	−15.0099	−1.05871
$202$	− 10.4239i	− 0.733425i
$203$	− 1.65109i	− 0.115884i
$204$	−24.5294	−1.71740
$205$	0	0
$206$	7.91723	0.551620
$207$	− 5.06727i	− 0.352199i
$208$	− 12.0750i	− 0.837252i
$209$	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	$0.583658\pi$
$212$	− 20.8032i	− 1.42877i
$213$	7.33048i	0.502276i
$214$	−39.2010	−2.67973
$215$	0	0
$216$	−21.8860	−1.48915
$217$	− 2.32273i	− 0.157677i
$218$	18.5195i	1.25430i
$219$	−8.50881	−0.574972
$220$	0	0
$221$	−31.3969	−2.11199
$222$	− 24.6015i	− 1.65114i
$223$	− 1.09344i	− 0.0732221i	−0.999330	−	0.0366111i	$-0.988344\pi$
$223$	− 1.09344i	− 0.0732221i	0.999330	−	0.0366111i	$-0.0116563\pi$
$224$	2.19887	0.146918
$225$	0	0
$226$	−0.338233	−0.0224989
$227$	20.1316i	1.33618i	0.744080	+	0.668091i	$0.232888\pi$
$227$	20.1316i	1.33618i	−0.744080	+	0.668091i	$0.767112\pi$
$228$	4.65109i	0.308026i
$229$	5.51656	0.364545	0.182272	−	0.983248i	$-0.441655\pi$
$229$	5.51656	0.364545	0.182272	+	0.983248i	$0.441655\pi$
$230$	0	0
$231$	−0.253344	−0.0166688
$232$	8.92498i	0.585954i
$233$	− 18.1805i	− 1.19104i	−0.803340	−	0.595520i	$-0.796946\pi$
$233$	− 18.1805i	− 1.19104i	0.803340	−	0.595520i	$-0.203054\pi$
$234$	−19.4904	−1.27413
$235$	0	0
$236$	−15.3588	−0.999771
$237$	− 4.94048i	− 0.320919i
$238$	9.10331i	0.590080i
$239$	−21.9164	−1.41766	−0.708828	−	0.705381i	$-0.750776\pi$
$239$	−21.9164	−1.41766	−0.708828	+	0.705381i	$0.750776\pi$
$240$	0	0
$241$	−28.1882	−1.81576	−0.907881	−	0.419228i	$-0.862301\pi$
$241$	−28.1882	−1.81576	−0.907881	+	0.419228i	$0.862301\pi$
$242$	25.9709i	1.66947i
$243$	12.9426i	0.830269i
$244$	0.377203	0.0241479
$245$	0	0
$246$	−28.5683	−1.82145
$247$	5.95328i	0.378798i
$248$	12.5555i	0.797277i
$249$	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	$0.408223\pi$
$252$	3.65109i	0.229997i
$253$	1.00775i	0.0633567i
$254$	35.9709	2.25702
$255$	0	0
$256$	−26.6970	−1.66856
$257$	− 6.86064i	− 0.427955i	−0.976839	−	0.213978i	$-0.931358\pi$
$257$	− 6.86064i	− 0.427955i	0.976839	−	0.213978i	$-0.0686419\pi$
$258$	29.7106i	1.84970i
$259$	−5.89881	−0.366534
$260$	0	0
$261$	3.13161	0.193842
$262$	− 13.2739i	− 0.820064i
$263$	− 9.25547i	− 0.570717i	−0.958421	−	0.285358i	$-0.907887\pi$
$263$	− 9.25547i	− 0.570717i	0.958421	−	0.285358i	$-0.0921126\pi$
$264$	1.36945	0.0842840
$265$	0	0
$266$	1.72611	0.105835
$267$	20.9349i	1.28119i
$268$	− 43.0197i	− 2.62785i
$269$	−0.498939	−0.0304208	−0.0152104	−	0.999884i	$-0.504842\pi$
$269$	−0.498939	−0.0304208	−0.0152104	+	0.999884i	$0.504842\pi$
$270$	0	0
$271$	3.71061	0.225403	0.112702	−	0.993629i	$-0.464050\pi$
$271$	3.71061	0.225403	0.112702	+	0.993629i	$0.464050\pi$
$272$	− 10.6970i	− 0.648602i
$273$	− 5.50669i	− 0.333280i
$274$	−30.4981	−1.84246
$275$	0	0
$276$	−17.1132	−1.03009
$277$	− 4.58675i	− 0.275591i	−0.990461	−	0.137796i	$-0.955998\pi$
$277$	− 4.58675i	− 0.275591i	0.990461	−	0.137796i	$-0.0440017\pi$
$278$	− 36.2469i	− 2.17395i
$279$	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	$0.197692\pi$
$282$	− 36.8238i	− 2.19283i
$283$	− 10.2661i	− 0.610259i	−0.952311	−	0.305129i	$-0.901300\pi$
$283$	− 10.2661i	− 0.610259i	0.952311	−	0.305129i	$-0.0986997\pi$
$284$	−21.0099	−1.24671
$285$	0	0
$286$	3.87614	0.229201
$287$	6.84997i	0.404341i
$288$	4.17058i	0.245754i
$289$	−10.8139	−0.636113
$290$	0	0
$291$	−5.66177	−0.331899
$292$	− 24.3871i	− 1.42715i
$293$	− 1.87051i	− 0.109277i	−0.998506	−	0.0546383i	$-0.982599\pi$
$293$	− 1.87051i	− 0.109277i	0.998506	−	0.0546383i	$-0.0174006\pi$
$294$	19.6015	1.14318
$295$	0	0
$296$	31.8860	1.85334
$297$	− 1.52723i	− 0.0886192i
$298$	32.8804i	1.90471i
$299$	−21.9044	−1.26677
$300$	0	0
$301$	7.12386	0.410612
$302$	27.8315i	1.60153i
$303$	5.58595i	0.320904i
$304$	−2.02830	−0.116331
$305$	0	0
$306$	−17.2661	−0.987040
$307$	− 0.227171i	− 0.0129653i	−0.999979	−	0.00648266i	$-0.997936\pi$
$307$	− 0.227171i	− 0.0129653i	0.999979	−	0.00648266i	$-0.00206351\pi$
$308$	− 0.726109i	− 0.0413739i
$309$	−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	$0.297494\pi$
$312$	29.7664i	1.68519i
$313$	− 11.2349i	− 0.635035i	−0.948252	−	0.317518i	$-0.897151\pi$
$313$	− 11.2349i	− 0.635035i	0.948252	−	0.317518i	$-0.102849\pi$
$314$	11.3879	0.642655
$315$	0	0
$316$	14.1599	0.796557
$317$	− 18.6228i	− 1.04596i	−0.852345	−	0.522980i	$-0.824820\pi$
$317$	− 18.6228i	− 1.04596i	0.852345	−	0.522980i	$-0.175180\pi$
$318$	17.2547i	0.967594i
$319$	−0.622797	−0.0348699
$320$	0	0
$321$	21.0069	1.17249
$322$	6.35103i	0.353929i
$323$	5.27389i	0.293447i
$324$	10.8500	0.602776
$325$	0	0
$326$	30.4309	1.68541
$327$	− 9.92418i	− 0.548809i
$328$	− 37.0275i	− 2.04450i
$329$	−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	$0.626714\pi$
$332$	1.78270i	0.0978385i
$333$	− 11.1882i	− 0.613110i
$334$	49.7848	2.72410
$335$	0	0
$336$	1.87614	0.102352
$337$	− 22.9709i	− 1.25130i	−0.780102	−	0.625652i	$-0.784833\pi$
$337$	− 22.9709i	− 1.25130i	0.780102	−	0.625652i	$-0.215167\pi$
$338$	53.3481i	2.90175i
$339$	0.181252	0.00984424
$340$	0	0
$341$	−0.876142	−0.0474457
$342$	3.27389i	0.177032i
$343$	− 9.78270i	− 0.528216i
$344$	−38.5080	−2.07621
$345$	0	0
$346$	37.3892	2.01006
$347$	3.93273i	0.211120i	0.994413	+	0.105560i	$0.0336635\pi$
$347$	3.93273i	0.211120i	−0.994413	+	0.105560i	$0.966336\pi$
$348$	− 10.5761i	− 0.566937i
$349$	34.4252	1.84274	0.921371	−	0.388685i	$-0.127071\pi$
$349$	34.4252	1.84274	0.921371	+	0.388685i	$0.127071\pi$
$350$	0	0
$351$	33.1960	1.77187
$352$	− 0.829422i	− 0.0442083i
$353$	− 4.25547i	− 0.226496i	−0.993567	−	0.113248i	$-0.963875\pi$
$353$	− 4.25547i	− 0.226496i	0.993567	−	0.113248i	$-0.0361254\pi$
$354$	12.7389	0.677065
$355$	0	0
$356$	−60.0013	−3.18006
$357$	− 4.87826i	− 0.258185i
$358$	8.11399i	0.428837i
$359$	20.2944	1.07110	0.535550	−	0.844504i	$-0.320104\pi$
$359$	20.2944	1.07110	0.535550	+	0.844504i	$0.320104\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	− 55.8903i	− 2.93753i
$363$	− 13.9172i	− 0.730465i
$364$	15.7827	0.827238
$365$	0	0
$366$	−0.312860	−0.0163535
$367$	− 3.85289i	− 0.201119i	−0.994931	−	0.100560i	$-0.967937\pi$
$367$	− 3.85289i	− 0.201119i	0.994931	−	0.100560i	$-0.0320633\pi$
$368$	− 7.46289i	− 0.389030i
$369$	−12.9922	−0.676350
$370$	0	0
$371$	4.13724	0.214795
$372$	− 14.8783i	− 0.771402i
$373$	14.6356i	0.757802i	0.925437	+	0.378901i	$0.123698\pi$
$373$	14.6356i	0.757802i	−0.925437	+	0.378901i	$0.876302\pi$
$374$	3.43380	0.177557
$375$	0	0
$376$	47.7274	2.46135
$377$	− 13.5371i	− 0.697197i
$378$	− 9.62492i	− 0.495052i
$379$	22.0099	1.13057	0.565286	−	0.824895i	$-0.308766\pi$
$379$	22.0099	1.13057	0.565286	+	0.824895i	$0.308766\pi$
$380$	0	0
$381$	−19.2760	−0.987540
$382$	− 29.6319i	− 1.51610i
$383$	− 3.08569i	− 0.157671i	−0.996888	−	0.0788357i	$-0.974880\pi$
$383$	− 3.08569i	− 0.157671i	0.996888	−	0.0788357i	$-0.0251202\pi$
$384$	26.3695	1.34566
$385$	0	0
$386$	45.8783	2.33514
$387$	13.5117i	0.686840i
$388$	− 16.2272i	− 0.823810i
$389$	−8.77203	−0.444760	−0.222380	−	0.974960i	$-0.571382\pi$
$389$	−8.77203	−0.444760	−0.222380	+	0.974960i	$0.571382\pi$
$390$	0	0
$391$	−19.4047	−0.981338
$392$	25.4055i	1.28317i
$393$	7.11319i	0.358813i
$394$	−15.7622	−0.794086
$395$	0	0
$396$	1.37720	0.0692070
$397$	1.59450i	0.0800257i	0.999199	+	0.0400129i	$0.0127399\pi$
$397$	1.59450i	0.0800257i	−0.999199	+	0.0400129i	$0.987260\pi$
$398$	− 54.8775i	− 2.75076i
$399$	−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	$0.644408\pi$
$402$	35.6815i	1.77963i
$403$	− 19.0438i	− 0.948639i
$404$	−16.0099	−0.796521
$405$	0	0
$406$	−3.92498	−0.194794
$407$	2.22505i	0.110292i
$408$	26.3695i	1.30548i
$409$	−36.6815	−1.81378	−0.906892	−	0.421363i	$-0.861552\pi$
$409$	−36.6815	−1.81378	−0.906892	+	0.421363i	$0.861552\pi$
$410$	0	0
$411$	16.3433	0.806155
$412$	− 12.1599i	− 0.599076i
$413$	− 3.05447i	− 0.150301i
$414$	−12.0459	−0.592025
$415$	0	0
$416$	18.0283	0.883910
$417$	19.4239i	0.951194i
$418$	− 0.651093i	− 0.0318460i
$419$	18.8187	0.919356	0.459678	−	0.888086i	$-0.347965\pi$
$419$	18.8187	0.919356	0.459678	+	0.888086i	$0.347965\pi$
$420$	0	0
$421$	−33.7819	−1.64643	−0.823215	−	0.567730i	$-0.807822\pi$
$421$	−33.7819	−1.64643	−0.823215	+	0.567730i	$0.807822\pi$
$422$	17.9426i	0.873432i
$423$	− 16.7467i	− 0.814250i
$424$	−22.3638	−1.08608
$425$	0	0
$426$	17.4260	0.844295
$427$	0.0750160i	0.00363028i
$428$	60.2079i	2.91026i
$429$	−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	$0.599471\pi$
$432$	11.3099i	0.544150i
$433$	− 16.0771i	− 0.772618i	−0.922369	−	0.386309i	$-0.873750\pi$
$433$	− 16.0771i	− 0.772618i	0.922369	−	0.386309i	$-0.126250\pi$
$434$	−5.52161	−0.265046
$435$	0	0
$436$	28.4437	1.36220
$437$	3.67939i	0.176009i
$438$	20.2272i	0.966492i
$439$	−1.36945	−0.0653604	−0.0326802	−	0.999466i	$-0.510404\pi$
$439$	−1.36945	−0.0653604	−0.0326802	+	0.999466i	$0.510404\pi$
$440$	0	0
$441$	8.91431	0.424491
$442$	74.6369i	3.55012i
$443$	4.62280i	0.219636i	0.993952	+	0.109818i	$0.0350268\pi$
$443$	4.62280i	0.219636i	−0.993952	+	0.109818i	$0.964973\pi$
$444$	−37.7848	−1.79319
$445$	0	0
$446$	−2.59933	−0.123082
$447$	− 17.6199i	− 0.833391i
$448$	− 8.17270i	− 0.386124i
$449$	−23.2555	−1.09749	−0.548747	−	0.835989i	$-0.684895\pi$
$449$	−23.2555	−1.09749	−0.548747	+	0.835989i	$0.684895\pi$
$450$	0	0
$451$	2.58383	0.121668
$452$	0.519485i	0.0244345i
$453$	− 14.9143i	− 0.700735i
$454$	47.8569	2.24604
$455$	0	0
$456$	5.00000	0.234146
$457$	35.8443i	1.67673i	0.545111	+	0.838364i	$0.316487\pi$
$457$	35.8443i	1.67673i	−0.545111	+	0.838364i	$0.683513\pi$
$458$	− 13.1140i	− 0.612776i
$459$	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	$0.612645\pi$
$462$	0.602251i	0.0280193i
$463$	− 29.9554i	− 1.39215i	−0.717971	−	0.696073i	$-0.754929\pi$
$463$	− 29.9554i	− 1.39215i	0.717971	−	0.696073i	$-0.245071\pi$
$464$	4.61212	0.214112
$465$	0	0
$466$	−43.2186	−2.00206
$467$	6.73598i	0.311704i	0.987780	+	0.155852i	$0.0498123\pi$
$467$	6.73598i	0.311704i	−0.987780	+	0.155852i	$0.950188\pi$
$468$	29.9349i	1.38374i
$469$	8.55553	0.395058
$470$	0	0
$471$	−6.10251	−0.281189
$472$	16.5109i	0.759977i
$473$	− 2.68714i	− 0.123555i
$474$	−11.7445	−0.539444
$475$	0	0
$476$	13.9816	0.640845
$477$	7.84704i	0.359291i
$478$	52.0998i	2.38299i
$479$	−16.6978	−0.762943	−0.381471	−	0.924381i	$-0.624582\pi$
$479$	−16.6978	−0.762943	−0.381471	+	0.924381i	$0.624582\pi$
$480$	0	0
$481$	−48.3636	−2.20519
$482$	67.0091i	3.05218i
$483$	− 3.40338i	− 0.154859i
$484$	39.8881	1.81310
$485$	0	0
$486$	30.7672	1.39563
$487$	− 3.64042i	− 0.164963i	−0.996593	−	0.0824816i	$-0.973715\pi$
$487$	− 3.64042i	− 0.164963i	0.996593	−	0.0824816i	$-0.0262846\pi$
$488$	− 0.405499i	− 0.0183561i
$489$	−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	$0.770894\pi$
$492$	43.8775i	1.97815i
$493$	− 11.9922i	− 0.540104i
$494$	14.1522	0.636736
$495$	0	0
$496$	6.48827	0.291332
$497$	− 4.17833i	− 0.187424i
$498$	− 1.47861i	− 0.0662582i
$499$	37.9914	1.70073	0.850365	−	0.526193i	$-0.176381\pi$
$499$	37.9914	1.70073	0.850365	+	0.526193i	$0.176381\pi$
$500$	0	0
$501$	−26.6786	−1.19191
$502$	− 21.4183i	− 0.955945i
$503$	− 42.1826i	− 1.88083i	−0.340032	−	0.940414i	$-0.610438\pi$
$503$	− 42.1826i	− 1.88083i	0.340032	−	0.940414i	$-0.389562\pi$
$504$	3.92498	0.174833
$505$	0	0
$506$	2.39563	0.106499
$507$	− 28.5881i	− 1.26964i
$508$	− 55.2469i	− 2.45119i
$509$	21.9971	0.975003	0.487502	−	0.873122i	$-0.337908\pi$
$509$	21.9971	0.975003	0.487502	+	0.873122i	$0.337908\pi$
$510$	0	0
$511$	4.84997	0.214550
$512$	22.0643i	0.975115i
$513$	− 5.57608i	− 0.246190i
$514$	−16.3091	−0.719365
$515$	0	0
$516$	45.6319	2.00883
$517$	3.33048i	0.146474i
$518$	14.0227i	0.616121i
$519$	−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	$0.354891\pi$
$522$	− 7.44447i	− 0.325836i
$523$	− 4.64817i	− 0.203250i	−0.994823	−	0.101625i	$-0.967596\pi$
$523$	− 4.64817i	− 0.203250i	0.994823	−	0.101625i	$-0.0324042\pi$
$524$	−20.3871	−0.890614
$525$	0	0
$526$	−22.0021	−0.959338
$527$	− 16.8705i	− 0.734891i
$528$	− 0.707686i	− 0.0307981i
$529$	9.46209	0.411395
$530$	0	0
$531$	5.79338	0.251411
$532$	− 2.65109i	− 0.114939i
$533$	56.1620i	2.43265i
$534$	49.7664	2.15360
$535$	0	0
$536$	−46.2469	−1.99756
$537$	− 4.34811i	− 0.187635i
$538$	1.18608i	0.0511355i
$539$	−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	$0.358027\pi$
$542$	− 8.82087i	− 0.378889i
$543$	29.9504i	1.28529i
$544$	15.9709	0.684747
$545$	0	0
$546$	−13.0905	−0.560222
$547$	37.2010i	1.59060i	0.606216	+	0.795300i	$0.292687\pi$
$547$	37.2010i	1.59060i	−0.606216	+	0.795300i	$0.707313\pi$
$548$	46.8414i	2.00097i
$549$	−0.142282	−0.00607245
$550$	0	0
$551$	−2.27389	−0.0968710
$552$	18.3969i	0.783026i
$553$	2.81604i	0.119750i
$554$	−10.9036	−0.463251
$555$	0	0
$556$	−55.6708	−2.36097
$557$	− 44.8393i	− 1.89990i	−0.312399	−	0.949951i	$-0.601133\pi$
$557$	− 44.8393i	− 1.89990i	0.312399	−	0.949951i	$-0.398867\pi$
$558$	− 10.4728i	− 0.443347i
$559$	58.4076	2.47038
$560$	0	0
$561$	−1.84010	−0.0776889
$562$	− 64.8152i	− 2.73407i
$563$	− 21.9172i	− 0.923701i	−0.886958	−	0.461851i	$-0.847186\pi$
$563$	− 21.9172i	− 0.923701i	0.886958	−	0.461851i	$-0.152814\pi$
$564$	−56.5569	−2.38147
$565$	0	0
$566$	−24.4047	−1.02581
$567$	2.15778i	0.0906183i
$568$	22.5860i	0.947685i
$569$	−9.90656	−0.415305	−0.207652	−	0.978203i	$-0.566582\pi$
$569$	−9.90656	−0.415305	−0.207652	+	0.978203i	$0.566582\pi$
$570$	0	0
$571$	17.6404	0.738229	0.369114	−	0.929384i	$-0.379661\pi$
$571$	17.6404	0.738229	0.369114	+	0.929384i	$0.379661\pi$
$572$	− 5.95328i	− 0.248919i
$573$	15.8791i	0.663357i
$574$	16.2838	0.679671
$575$	0	0
$576$	15.5011	0.645878
$577$	12.7048i	0.528906i	0.964399	+	0.264453i	$0.0851914\pi$
$577$	12.7048i	0.528906i	−0.964399	+	0.264453i	$0.914809\pi$
$578$	25.7069i	1.06927i
$579$	−24.5851	−1.02172
$580$	0	0
$581$	−0.354534	−0.0147085
$582$	13.4592i	0.557900i
$583$	− 1.56058i	− 0.0646325i
$584$	−26.2165	−1.08485
$585$	0	0
$586$	−4.44659	−0.183687
$587$	− 15.0438i	− 0.620924i	−0.950586	−	0.310462i	$-0.899516\pi$
$587$	− 15.0438i	− 0.620924i	0.950586	−	0.310462i	$-0.100484\pi$
$588$	− 30.1054i	− 1.24153i
$589$	−3.19887	−0.131807
$590$	0	0
$591$	8.44659	0.347446
$592$	− 16.4776i	− 0.677225i
$593$	− 16.4231i	− 0.674417i	−0.941430	−	0.337208i	$-0.890517\pi$
$593$	− 16.4231i	− 0.674417i	0.941430	−	0.337208i	$-0.109483\pi$
$594$	−3.63055	−0.148963
$595$	0	0
$596$	50.5003	2.06857
$597$	29.4076i	1.20357i
$598$	52.0713i	2.12935i
$599$	19.1260	0.781466	0.390733	−	0.920504i	$-0.372222\pi$
$599$	19.1260	0.781466	0.390733	+	0.920504i	$0.372222\pi$
$600$	0	0
$601$	31.4124	1.28134	0.640670	−	0.767816i	$-0.278657\pi$
$601$	31.4124	1.28134	0.640670	+	0.767816i	$0.278657\pi$
$602$	− 16.9349i	− 0.690213i
$603$	16.2272i	0.660821i
$604$	42.7459	1.73930
$605$	0	0
$606$	13.2789	0.539420
$607$	41.5315i	1.68571i	0.538140	+	0.842855i	$0.319127\pi$
$607$	41.5315i	1.68571i	−0.538140	+	0.842855i	$0.680873\pi$
$608$	− 3.02830i	− 0.122814i
$609$	2.10331	0.0852305
$610$	0	0
$611$	−72.3913	−2.92864
$612$	26.5187i	1.07195i
$613$	21.7274i	0.877563i	0.898594	+	0.438781i	$0.144590\pi$
$613$	21.7274i	0.877563i	−0.898594	+	0.438781i	$0.855410\pi$
$614$	−0.540031	−0.0217939
$615$	0	0
$616$	−0.780579	−0.0314504
$617$	− 33.6065i	− 1.35295i	−0.736467	−	0.676473i	$-0.763507\pi$
$617$	− 33.6065i	− 1.35295i	0.736467	−	0.676473i	$-0.236493\pi$
$618$	10.0857i	0.405706i
$619$	27.6036	1.10948	0.554741	−	0.832023i	$-0.312817\pi$
$619$	27.6036	1.10948	0.554741	+	0.832023i	$0.312817\pi$
$620$	0	0
$621$	20.5166	0.823301
$622$	− 49.8152i	− 1.99741i
$623$	− 11.9327i	− 0.478075i
$624$	15.3822	0.615783
$625$	0	0
$626$	−26.7077	−1.06745
$627$	0.348907i	0.0139340i
$628$	− 17.4904i	− 0.697942i
$629$	−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	$0.487702\pi$
$632$	− 15.2221i	− 0.605504i
$633$	− 9.61505i	− 0.382164i
$634$	−44.2702	−1.75819
$635$	0	0
$636$	26.5011	1.05084
$637$	− 38.5342i	− 1.52678i
$638$	1.48052i	0.0586142i
$639$	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	$0.493589\pi$
$642$	− 49.9378i	− 1.97089i
$643$	36.9866i	1.45861i	0.684189	+	0.729305i	$0.260156\pi$
$643$	36.9866i	1.45861i	−0.684189	+	0.729305i	$0.739844\pi$
$644$	9.75441	0.384377
$645$	0	0
$646$	12.5371	0.493266
$647$	− 24.1182i	− 0.948186i	−0.880475	−	0.474093i	$-0.842776\pi$
$647$	− 24.1182i	− 0.948186i	0.880475	−	0.474093i	$-0.157224\pi$
$648$	− 11.6639i	− 0.458201i
$649$	−1.15215	−0.0452260
$650$	0	0
$651$	2.95891	0.115969
$652$	− 46.7381i	− 1.83041i
$653$	37.2603i	1.45811i	0.684456	+	0.729054i	$0.260040\pi$
$653$	37.2603i	1.45811i	−0.684456	+	0.729054i	$0.739960\pi$
$654$	−23.5918	−0.922512
$655$	0	0
$656$	−19.1345	−0.747078
$657$	9.19887i	0.358882i
$658$	20.9893i	0.818249i
$659$	−21.4386	−0.835130	−0.417565	−	0.908647i	$-0.637116\pi$
$659$	−21.4386	−0.835130	−0.417565	+	0.908647i	$0.637116\pi$
$660$	0	0
$661$	0.783503	0.0304747	0.0152374	−	0.999884i	$-0.495150\pi$
$661$	0.783503	0.0304747	0.0152374	+	0.999884i	$0.495150\pi$
$662$	33.5315i	1.30324i
$663$	− 39.9963i	− 1.55333i
$664$	1.91643	0.0743720
$665$	0	0
$666$	−26.5966	−1.03060
$667$	− 8.36653i	− 0.323953i
$668$	− 76.4634i	− 2.95846i
$669$	1.39292	0.0538535
$670$	0	0
$671$	0.0282963	0.00109237
$672$	2.80113i	0.108056i
$673$	− 50.1903i	− 1.93469i	−0.253454	−	0.967347i	$-0.581567\pi$
$673$	− 50.1903i	− 1.93469i	0.253454	−	0.967347i	$-0.418433\pi$
$674$	−54.6065	−2.10336
$675$	0	0
$676$	81.9362	3.15139
$677$	29.8804i	1.14840i	0.818716	+	0.574198i	$0.194686\pi$
$677$	29.8804i	1.14840i	−0.818716	+	0.574198i	$0.805314\pi$
$678$	− 0.430872i	− 0.0165475i
$679$	3.22717	0.123847
$680$	0	0
$681$	−25.6455	−0.982736
$682$	2.08277i	0.0797532i
$683$	12.3326i	0.471894i	0.971766	+	0.235947i	$0.0758192\pi$
$683$	12.3326i	0.471894i	−0.971766	+	0.235947i	$0.924181\pi$
$684$	5.02830	0.192262
$685$	0	0
$686$	−23.2555	−0.887898
$687$	7.02750i	0.268116i
$688$	19.8996i	0.758666i
$689$	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	$0.478053\pi$
$692$	− 57.4252i	− 2.18298i
$693$	0.273891i	0.0104042i
$694$	9.34891	0.354880
$695$	0	0
$696$	−11.3695	−0.430958
$697$	49.7528i	1.88452i
$698$	− 81.8358i	− 3.09753i
$699$	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	$0.277122\pi$
$702$	− 78.9135i	− 2.97840i
$703$	8.12386i	0.306397i
$704$	−3.08277	−0.116186
$705$	0	0
$706$	−10.1161	−0.380725
$707$	− 3.18396i	− 0.119745i
$708$	− 19.5654i	− 0.735313i
$709$	17.1209	0.642990	0.321495	−	0.946911i	$-0.395815\pi$
$709$	17.1209	0.642990	0.321495	+	0.946911i	$0.395815\pi$
$710$	0	0
$711$	−5.34116	−0.200309
$712$	64.5024i	2.41733i
$713$	− 11.7699i	− 0.440786i
$714$	−11.5966	−0.433993
$715$	0	0
$716$	12.4621	0.465730
$717$	− 27.9191i	− 1.04266i
$718$	− 48.2440i	− 1.80045i
$719$	7.02750	0.262081	0.131041	−	0.991377i	$-0.458168\pi$
$719$	7.02750	0.262081	0.131041	+	0.991377i	$0.458168\pi$
$720$	0	0
$721$	2.41830	0.0900620
$722$	− 2.37720i	− 0.0884703i
$723$	− 35.9087i	− 1.33546i
$724$	−85.8406	−3.19024
$725$	0	0
$726$	−33.0841	−1.22787
$727$	11.8938i	0.441115i	0.975374	+	0.220558i	$0.0707877\pi$
$727$	11.8938i	0.441115i	−0.975374	+	0.220558i	$0.929212\pi$
$728$	− 16.9667i	− 0.628826i
$729$	−25.4026	−0.940836
$730$	0	0
$731$	51.7421	1.91375
$732$	0.480515i	0.0177604i
$733$	20.7154i	0.765142i	0.923926	+	0.382571i	$0.124961\pi$
$733$	20.7154i	0.765142i	−0.923926	+	0.382571i	$0.875039\pi$
$734$	−9.15910	−0.338069
$735$	0	0
$736$	11.1423	0.410710
$737$	− 3.22717i	− 0.118874i
$738$	30.8852i	1.13690i
$739$	−33.8620	−1.24563	−0.622816	−	0.782368i	$-0.714012\pi$
$739$	−33.8620	−1.24563	−0.622816	+	0.782368i	$0.714012\pi$
$740$	0	0
$741$	−7.58383	−0.278599
$742$	− 9.83505i	− 0.361056i
$743$	42.7381i	1.56791i	0.620818	+	0.783955i	$0.286800\pi$
$743$	42.7381i	1.56791i	−0.620818	+	0.783955i	$0.713200\pi$
$744$	−15.9944	−0.586382
$745$	0	0
$746$	34.7918	1.27382
$747$	− 0.672440i	− 0.0246033i
$748$	− 5.27389i	− 0.192833i
$749$	−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	$0.429988\pi$
$752$	− 24.6639i	− 0.899400i
$753$	11.4776i	0.418267i
$754$	−32.1805	−1.17194
$755$	0	0
$756$	−14.7827	−0.537642
$757$	− 29.1103i	− 1.05803i	−0.848612	−	0.529015i	$-0.822561\pi$
$757$	− 29.1103i	− 1.05803i	0.848612	−	0.529015i	$-0.177439\pi$
$758$	− 52.3219i	− 1.90042i
$759$	−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	$0.632454\pi$
$762$	45.8230i	1.65999i
$763$	5.65672i	0.204787i
$764$	−45.5109	−1.64653
$765$	0	0
$766$	−7.33531	−0.265036
$767$	− 25.0432i	− 0.904258i
$768$	− 34.0091i	− 1.22720i
$769$	−3.95891	−0.142762	−0.0713809	−	0.997449i	$-0.522741\pi$
$769$	−3.95891	−0.142762	−0.0713809	+	0.997449i	$0.522741\pi$
$770$	0	0
$771$	8.73971	0.314753
$772$	− 70.4634i	− 2.53603i
$773$	27.8139i	1.00040i	0.865911	+	0.500199i	$0.166740\pi$
$773$	27.8139i	1.00040i	−0.865911	+	0.500199i	$0.833260\pi$
$774$	32.1201	1.15453
$775$	0	0
$776$	−17.4445	−0.626220
$777$	− 7.51444i	− 0.269579i
$778$	20.8529i	0.747612i
$779$	9.43380	0.338001
$780$	0	0
$781$	−1.57608	−0.0563965
$782$	46.1289i	1.64957i
$783$	12.6794i	0.453124i
$784$	13.1287	0.468882
$785$	0	0
$786$	16.9095	0.603141
$787$	1.82460i	0.0650398i	0.999471	+	0.0325199i	$0.0103532\pi$
$787$	1.82460i	0.0650398i	−0.999471	+	0.0325199i	$0.989647\pi$
$788$	24.2087i	0.862401i
$789$	11.7905	0.419751
$790$	0	0
$791$	−0.103312	−0.00367336
$792$	− 1.48052i	− 0.0526078i
$793$	0.615047i	0.0218410i
$794$	3.79045	0.134518
$795$	0	0
$796$	−84.2851	−2.98741
$797$	21.0360i	0.745135i	0.928005	+	0.372567i	$0.121522\pi$
$797$	21.0360i	0.745135i	−0.928005	+	0.372567i	$0.878478\pi$
$798$	2.19887i	0.0778393i
$799$	−64.1300	−2.26876
$800$	0	0
$801$	22.6327	0.799686
$802$	41.7261i	1.47340i
$803$	− 1.82942i	− 0.0645589i
$804$	54.8024	1.93273
$805$	0	0
$806$	−45.2710	−1.59460
$807$	− 0.635593i	− 0.0223739i
$808$	17.2109i	0.605476i
$809$	0.0819654	0.00288175	0.00144088	−	0.999999i	$-0.499541\pi$
$809$	0.0819654	0.00288175	0.00144088	+	0.999999i	$0.499541\pi$
$810$	0	0
$811$	−6.72531	−0.236158	−0.118079	−	0.993004i	$-0.537674\pi$
$811$	−6.72531	−0.236158	−0.118079	+	0.993004i	$0.537674\pi$
$812$	6.02830i	0.211552i
$813$	4.72691i	0.165780i
$814$	5.28939	0.185393
$815$	0	0
$816$	13.6268	0.477034
$817$	− 9.81100i	− 0.343243i
$818$	87.1994i	3.04886i
$819$	−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	$0.318442\pi$
$822$	− 38.8513i	− 1.35509i
$823$	26.5908i	0.926896i	0.886124	+	0.463448i	$0.153388\pi$
$823$	26.5908i	0.926896i	−0.886124	+	0.463448i	$0.846612\pi$
$824$	−13.0721	−0.455388
$825$	0	0
$826$	−7.26109	−0.252646
$827$	5.57900i	0.194001i	0.995284	+	0.0970004i	$0.0309248\pi$
$827$	5.57900i	0.194001i	−0.995284	+	0.0970004i	$0.969075\pi$
$828$	18.5011i	0.642956i
$829$	−18.9765	−0.659082	−0.329541	−	0.944141i	$-0.606894\pi$
$829$	−18.9765	−0.659082	−0.329541	+	0.944141i	$0.606894\pi$
$830$	0	0
$831$	5.84302	0.202692
$832$	− 67.0069i	− 2.32305i
$833$	− 34.1367i	− 1.18276i
$834$	46.1746	1.59890
$835$	0	0
$836$	−1.00000	−0.0345857
$837$	17.8372i	0.616543i
$838$	− 44.7360i	− 1.54538i
$839$	−12.9143	−0.445852	−0.222926	−	0.974835i	$-0.571561\pi$
$839$	−12.9143	−0.445852	−0.222926	+	0.974835i	$0.571561\pi$
$840$	0	0
$841$	−23.8294	−0.821704
$842$	80.3064i	2.76754i
$843$	34.7331i	1.19627i
$844$	27.5577	0.948574
$845$	0	0
$846$	−39.8102	−1.36870
$847$	7.93273i	0.272572i
$848$	11.5569i	0.396864i
$849$	13.0779	0.448834
$850$	0	0
$851$	−29.8908	−1.02464
$852$	− 26.7643i	− 0.916929i
$853$	6.46077i	0.221213i	0.993864	+	0.110606i	$0.0352793\pi$
$853$	6.46077i	0.221213i	−0.993864	+	0.110606i	$0.964721\pi$
$854$	0.178328	0.00610227
$855$	0	0
$856$	64.7245	2.21224
$857$	12.4055i	0.423764i	0.977295	+	0.211882i	$0.0679592\pi$
$857$	12.4055i	0.423764i	−0.977295	+	0.211882i	$0.932041\pi$
$858$	4.93778i	0.168573i
$859$	−40.3425	−1.37647	−0.688234	−	0.725489i	$-0.741614\pi$
$859$	−40.3425	−1.37647	−0.688234	+	0.725489i	$0.741614\pi$
$860$	0	0
$861$	−8.72611	−0.297385
$862$	30.3452i	1.03356i
$863$	1.83235i	0.0623738i	0.999514	+	0.0311869i	$0.00992870\pi$
$863$	1.83235i	0.0623738i	−0.999514	+	0.0311869i	$0.990071\pi$
$864$	−16.8860	−0.574474
$865$	0	0
$866$	−38.2186	−1.29872
$867$	− 13.7758i	− 0.467849i
$868$	8.48052i	0.287847i
$869$	1.06222	0.0360333
$870$	0	0
$871$	70.1457	2.37680
$872$	− 30.5774i	− 1.03548i
$873$	6.12094i	0.207162i
$874$	8.74666	0.295860
$875$	0	0
$876$	31.0665	1.04964
$877$	8.14419i	0.275010i	0.990501	+	0.137505i	$0.0439083\pi$
$877$	8.14419i	0.275010i	−0.990501	+	0.137505i	$0.956092\pi$
$878$	3.25547i	0.109867i
$879$	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	$0.434275\pi$
$882$	− 21.1911i	− 0.713542i
$883$	46.7614i	1.57364i	0.617179	+	0.786822i	$0.288275\pi$
$883$	46.7614i	1.57364i	−0.617179	+	0.786822i	$0.711725\pi$
$884$	114.633	3.85553
$885$	0	0
$886$	10.9893	0.369194
$887$	34.8804i	1.17117i	0.810611	+	0.585584i	$0.199135\pi$
$887$	34.8804i	1.17117i	−0.810611	+	0.585584i	$0.800865\pi$
$888$	40.6193i	1.36309i
$889$	10.9872	0.368499
$890$	0	0
$891$	0.813922	0.0272674
$892$	3.99225i	0.133670i
$893$	12.1599i	0.406916i
$894$	−41.8860	−1.40088
$895$	0	0
$896$	−15.0304	−0.502131
$897$	− 27.9039i	− 0.931683i
$898$	55.2830i	1.84482i
$899$	7.27389	0.242598
$900$	0	0
$901$	30.0496	1.00110
$902$	− 6.14228i	− 0.204516i
$903$	9.07502i	0.301998i
$904$	0.558455	0.0185739
$905$	0	0
$906$	−35.4543	−1.17789
$907$	25.5080i	0.846980i	0.905901	+	0.423490i	$0.139195\pi$
$907$	25.5080i	0.846980i	−0.905901	+	0.423490i	$0.860805\pi$
$908$	− 73.5024i	− 2.43926i
$909$	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	$0.615933\pi$
$912$	− 2.58383i	− 0.0855591i
$913$	0.133731i	0.00442586i
$914$	85.2093	2.81847
$915$	0	0
$916$	−20.1415	−0.665493
$917$	− 4.05447i	− 0.133890i
$918$	− 69.9079i	− 2.30730i
$919$	−37.1386	−1.22509	−0.612544	−	0.790436i	$-0.709854\pi$
$919$	−37.1386	−1.22509	−0.612544	+	0.790436i	$0.709854\pi$
$920$	0	0
$921$	0.289391	0.00953575
$922$	35.3756i	1.16503i
$923$	− 34.2576i	− 1.12760i
$924$	0.924984	0.0304297
$925$	0	0
$926$	−71.2101	−2.34011
$927$	4.58675i	0.150649i
$928$	6.88601i	0.226044i
$929$	3.36170	0.110294	0.0551469	−	0.998478i	$-0.482437\pi$
$929$	3.36170	0.110294	0.0551469	+	0.998478i	$0.482437\pi$
$930$	0	0
$931$	−6.47277	−0.212136
$932$	66.3785i	2.17430i
$933$	26.6949i	0.873951i
$934$	16.0128	0.523955
$935$	0	0
$936$	32.1805	1.05185
$937$	− 10.0694i	− 0.328953i	−0.986381	−	0.164476i	$-0.947407\pi$
$937$	− 10.0694i	− 0.328953i	0.986381	−	0.164476i	$-0.0525934\pi$
$938$	− 20.3382i	− 0.664067i
$939$	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	$0.389824\pi$
$942$	14.5069i	0.472661i
$943$	34.7106i	1.13033i
$944$	8.53228	0.277702
$945$	0	0
$946$	−6.38788	−0.207688
$947$	30.4904i	0.990804i	0.868664	+	0.495402i	$0.164979\pi$
$947$	30.4904i	0.990804i	−0.868664	+	0.495402i	$0.835021\pi$
$948$	18.0382i	0.585852i
$949$	39.7643	1.29080
$950$	0	0
$951$	23.7234	0.769284
$952$	− 15.0304i	− 0.487139i
$953$	7.58383i	0.245664i	0.992427	+	0.122832i	$0.0391977\pi$
$953$	7.58383i	0.245664i	−0.992427	+	0.122832i	$0.960802\pi$
$954$	18.6540	0.603946
$955$	0	0
$956$	80.0189	2.58800
$957$	− 0.793375i	− 0.0256462i
$958$	39.6941i	1.28246i
$959$	−9.31556	−0.300815
$960$	0	0
$961$	−20.7672	−0.669910
$962$	114.970i	3.70678i
$963$	− 22.7106i	− 0.731839i
$964$	102.918	3.31476
$965$	0	0
$966$	−8.09052	−0.260308
$967$	− 54.6687i	− 1.75803i	−0.476797	−	0.879014i	$-0.658202\pi$
$967$	− 54.6687i	− 1.75803i	0.476797	−	0.879014i	$-0.341798\pi$
$968$	− 42.8804i	− 1.37823i
$969$	−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	$0.721579\pi$
$972$	− 47.2547i	− 1.51569i
$973$	− 11.0715i	− 0.354936i
$974$	−8.65402	−0.277293
$975$	0	0
$976$	−0.209548	−0.00670747
$977$	− 15.2400i	− 0.487570i	−0.969829	−	0.243785i	$-0.921611\pi$
$977$	− 15.2400i	− 0.487570i	0.969829	−	0.243785i	$-0.0783891\pi$
$978$	38.7656i	1.23959i
$979$	−4.50106	−0.143855
$980$	0	0
$981$	−10.7290	−0.342552
$982$	79.2199i	2.52801i
$983$	− 45.3609i	− 1.44679i	−0.690435	−	0.723394i	$-0.742581\pi$
$983$	− 45.3609i	− 1.44679i	0.690435	−	0.723394i	$-0.257419\pi$
$984$	47.1690	1.50369
$985$	0	0
$986$	−28.5080	−0.907880
$987$	− 11.2477i	− 0.358019i
$988$	− 21.7360i	− 0.691514i
$989$	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	$0.583639\pi$
$992$	9.68714i	0.307567i
$993$	− 17.9688i	− 0.570222i
$994$	−9.93273	−0.315047
$995$	0	0
$996$	−2.27097	−0.0719583
$997$	33.2037i	1.05157i	0.850617	+	0.525786i	$0.176229\pi$
$997$	33.2037i	1.05157i	−0.850617	+	0.525786i	$0.823771\pi$
$998$	− 90.3134i	− 2.85882i
$999$	45.2993	1.43321

Display

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p

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a_n

instead) (See

a_n

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a_n

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.b.b.324.1		6
5.2	odd	4		475.2.a.g.1.3	yes	3
5.3	odd	4		475.2.a.e.1.1	✓	3
5.4	even	2	inner	475.2.b.b.324.6		6
15.2	even	4		4275.2.a.ba.1.1		3
15.8	even	4		4275.2.a.bm.1.3		3
20.3	even	4		7600.2.a.cc.1.2		3
20.7	even	4		7600.2.a.bh.1.2		3
95.18	even	4		9025.2.a.bc.1.3		3
95.37	even	4		9025.2.a.y.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.a.e.1.1	✓	3	5.3	odd	4
475.2.a.g.1.3	yes	3	5.2	odd	4
475.2.b.b.324.1		6	1.1	even	1	trivial
475.2.b.b.324.6		6	5.4	even	2	inner
4275.2.a.ba.1.1		3	15.2	even	4
4275.2.a.bm.1.3		3	15.8	even	4
7600.2.a.bh.1.2		3	20.7	even	4
7600.2.a.cc.1.2		3	20.3	even	4
9025.2.a.y.1.1		3	95.37	even	4
9025.2.a.bc.1.3		3	95.18	even	4

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	475.2.b.b.324.1

Level	$475$
Weight	$2$
Character	475.324
Analytic conductor	$3.793$
Analytic rank	$0$
Dimension	$6$
Inner twists	$2$