LMFDB - Embedded newform 2325.2.c.m.1024.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2325,2,Mod(1024,2325)] 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("2325.1024"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2325 = 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2325.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,-4,0,0,0,0,-6,0,-14,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	no
Analytic conductor:	$18.5652184699$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1024.4
Root			$-0.854638 - 0.854638i$ of defining polynomial
Character	$\chi$	$=$	2325.1024
Dual form			2325.2.c.m.1024.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.539189i q^{2} -1.00000i q^{3} +1.70928 q^{4} +0.539189 q^{6} -3.70928i q^{7} +2.00000i q^{8} -1.00000 q^{9} +1.87936 q^{11} -1.70928i q^{12} -6.58864i q^{13} +2.00000 q^{14} +2.34017 q^{16} +3.09171i q^{17} -0.539189i q^{18} +7.34017 q^{19} -3.70928 q^{21} +1.01333i q^{22} +7.29791i q^{23} +2.00000 q^{24} +3.55252 q^{26} +1.00000i q^{27} -6.34017i q^{28} -0.170086 q^{29} -1.00000 q^{31} +5.26180i q^{32} -1.87936i q^{33} -1.66701 q^{34} -1.70928 q^{36} -6.82991i q^{37} +3.95774i q^{38} -6.58864 q^{39} -7.95774 q^{41} -2.00000i q^{42} +4.18342i q^{43} +3.21235 q^{44} -3.93495 q^{46} -11.7587i q^{47} -2.34017i q^{48} -6.75872 q^{49} +3.09171 q^{51} -11.2618i q^{52} -13.9288i q^{53} -0.539189 q^{54} +7.41855 q^{56} -7.34017i q^{57} -0.0917087i q^{58} +6.72261 q^{59} +1.84324 q^{61} -0.539189i q^{62} +3.70928i q^{63} +1.84324 q^{64} +1.01333 q^{66} +1.81432i q^{67} +5.28458i q^{68} +7.29791 q^{69} -11.2618 q^{71} -2.00000i q^{72} +2.92162i q^{73} +3.68261 q^{74} +12.5464 q^{76} -6.97107i q^{77} -3.55252i q^{78} -6.24846 q^{79} +1.00000 q^{81} -4.29072i q^{82} +6.21953i q^{83} -6.34017 q^{84} -2.25565 q^{86} +0.170086i q^{87} +3.75872i q^{88} +12.4030 q^{89} -24.4391 q^{91} +12.4741i q^{92} +1.00000i q^{93} +6.34017 q^{94} +5.26180 q^{96} -3.94441i q^{97} -3.64423i q^{98} -1.87936 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{4} - 6 q^{9} - 14 q^{11} + 12 q^{14} - 8 q^{16} + 22 q^{19} - 8 q^{21} + 12 q^{24} + 20 q^{26} + 10 q^{29} - 6 q^{31} + 36 q^{34} + 4 q^{36} - 16 q^{41} + 40 q^{44} - 32 q^{46} + 10 q^{49} + 14 q^{51}+ \cdots + 14 q^{99}+O(q^{100}) $ 6 * q - 4 * q^4 - 6 * q^9 - 14 * q^11 + 12 * q^14 - 8 * q^16 + 22 * q^19 - 8 * q^21 + 12 * q^24 + 20 * q^26 + 10 * q^29 - 6 * q^31 + 36 * q^34 + 4 * q^36 - 16 * q^41 + 40 * q^44 - 32 * q^46 + 10 * q^49 + 14 * q^51 + 16 * q^56 + 28 * q^59 + 24 * q^61 + 24 * q^64 + 8 * q^66 - 10 * q^69 - 52 * q^71 + 4 * q^74 + 4 * q^76 - 20 * q^79 + 6 * q^81 - 16 * q^84 + 72 * q^86 + 18 * q^89 - 52 * q^91 + 16 * q^94 + 16 * q^96 + 14 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2325\mathbb{Z}\right)^\times$.

$n$	$652$	$776$	$1801$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	0.539189i	0.381264i	0.981662	+	0.190632i	$0.0610537\pi$
$2$	0.539189i	0.381264i	−0.981662	+	0.190632i	$0.938946\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	1.70928	0.854638
$5$	0	0
$5$	0	0
$6$	0.539189	0.220123
$7$	− 3.70928i	− 1.40197i	−0.713174	−	0.700987i	$-0.752743\pi$
$7$	− 3.70928i	− 1.40197i	0.713174	−	0.700987i	$-0.247257\pi$
$8$	2.00000i	0.707107i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	1.87936	0.566649	0.283324	−	0.959024i	$-0.408563\pi$
$11$	1.87936	0.566649	0.283324	+	0.959024i	$0.408563\pi$
$12$	− 1.70928i	− 0.493425i
$13$	− 6.58864i	− 1.82736i	−0.406435	−	0.913680i	$-0.633228\pi$
$13$	− 6.58864i	− 1.82736i	0.406435	−	0.913680i	$-0.366772\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	2.34017	0.585043
$17$	3.09171i	0.749850i	0.927055	+	0.374925i	$0.122331\pi$
$17$	3.09171i	0.749850i	−0.927055	+	0.374925i	$0.877669\pi$
$18$	− 0.539189i	− 0.127088i
$19$	7.34017	1.68395	0.841976	−	0.539516i	$-0.181393\pi$
$19$	7.34017	1.68395	0.841976	+	0.539516i	$0.181393\pi$
$20$	0	0
$21$	−3.70928	−0.809430
$22$	1.01333i	0.216043i
$23$	7.29791i	1.52172i	0.648916	+	0.760860i	$0.275223\pi$
$23$	7.29791i	1.52172i	−0.648916	+	0.760860i	$0.724777\pi$
$24$	2.00000	0.408248
$25$	0	0
$26$	3.55252	0.696706
$27$	1.00000i	0.192450i
$28$	− 6.34017i	− 1.19818i
$29$	−0.170086	−0.0315843	−0.0157921	−	0.999875i	$-0.505027\pi$
$29$	−0.170086	−0.0315843	−0.0157921	+	0.999875i	$0.505027\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	5.26180i	0.930163i
$33$	− 1.87936i	− 0.327155i
$34$	−1.66701	−0.285891
$35$	0	0
$36$	−1.70928	−0.284879
$37$	− 6.82991i	− 1.12283i	−0.827534	−	0.561415i	$-0.810257\pi$
$37$	− 6.82991i	− 1.12283i	0.827534	−	0.561415i	$-0.189743\pi$
$38$	3.95774i	0.642030i
$39$	−6.58864	−1.05503
$40$	0	0
$41$	−7.95774	−1.24279	−0.621395	−	0.783497i	$-0.713434\pi$
$41$	−7.95774	−1.24279	−0.621395	+	0.783497i	$0.713434\pi$
$42$	− 2.00000i	− 0.308607i
$43$	4.18342i	0.637965i	0.947761	+	0.318983i	$0.103341\pi$
$43$	4.18342i	0.637965i	−0.947761	+	0.318983i	$0.896659\pi$
$44$	3.21235	0.484280
$45$	0	0
$46$	−3.93495	−0.580177
$47$	− 11.7587i	− 1.71519i	−0.514329	−	0.857593i	$-0.671959\pi$
$47$	− 11.7587i	− 1.71519i	0.514329	−	0.857593i	$-0.328041\pi$
$48$	− 2.34017i	− 0.337775i
$49$	−6.75872	−0.965532
$50$	0	0
$51$	3.09171	0.432926
$52$	− 11.2618i	− 1.56173i
$53$	− 13.9288i	− 1.91327i	−0.291291	−	0.956635i	$-0.594085\pi$
$53$	− 13.9288i	− 1.91327i	0.291291	−	0.956635i	$-0.405915\pi$
$54$	−0.539189	−0.0733743
$55$	0	0
$56$	7.41855	0.991346
$57$	− 7.34017i	− 0.972230i
$58$	− 0.0917087i	− 0.0120419i
$59$	6.72261	0.875209	0.437604	−	0.899168i	$-0.355827\pi$
$59$	6.72261	0.875209	0.437604	+	0.899168i	$0.355827\pi$
$60$	0	0
$61$	1.84324	0.236003	0.118002	−	0.993013i	$-0.462351\pi$
$61$	1.84324	0.236003	0.118002	+	0.993013i	$0.462351\pi$
$62$	− 0.539189i	− 0.0684771i
$63$	3.70928i	0.467325i
$64$	1.84324	0.230406
$65$	0	0
$66$	1.01333	0.124732
$67$	1.81432i	0.221654i	0.993840	+	0.110827i	$0.0353499\pi$
$67$	1.81432i	0.221654i	−0.993840	+	0.110827i	$0.964650\pi$
$68$	5.28458i	0.640850i
$69$	7.29791	0.878565
$70$	0	0
$71$	−11.2618	−1.33653	−0.668265	−	0.743924i	$-0.732963\pi$
$71$	−11.2618	−1.33653	−0.668265	+	0.743924i	$0.732963\pi$
$72$	− 2.00000i	− 0.235702i
$73$	2.92162i	0.341950i	0.985275	+	0.170975i	$0.0546917\pi$
$73$	2.92162i	0.341950i	−0.985275	+	0.170975i	$0.945308\pi$
$74$	3.68261	0.428095
$75$	0	0
$76$	12.5464	1.43917
$77$	− 6.97107i	− 0.794427i
$78$	− 3.55252i	− 0.402244i
$79$	−6.24846	−0.703007	−0.351504	−	0.936187i	$-0.614329\pi$
$79$	−6.24846	−0.703007	−0.351504	+	0.936187i	$0.614329\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 4.29072i	− 0.473831i
$83$	6.21953i	0.682683i	0.939939	+	0.341341i	$0.110881\pi$
$83$	6.21953i	0.682683i	−0.939939	+	0.341341i	$0.889119\pi$
$84$	−6.34017	−0.691770
$85$	0	0
$86$	−2.25565	−0.243233
$87$	0.170086i	0.0182352i
$88$	3.75872i	0.400681i
$89$	12.4030	1.31471	0.657355	−	0.753581i	$-0.271675\pi$
$89$	12.4030	1.31471	0.657355	+	0.753581i	$0.271675\pi$
$90$	0	0
$91$	−24.4391	−2.56191
$92$	12.4741i	1.30052i
$93$	1.00000i	0.103695i
$94$	6.34017	0.653939
$95$	0	0
$96$	5.26180	0.537030
$97$	− 3.94441i	− 0.400494i	−0.979745	−	0.200247i	$-0.935826\pi$
$97$	− 3.94441i	− 0.400494i	0.979745	−	0.200247i	$-0.0641745\pi$
$98$	− 3.64423i	− 0.368123i
$99$	−1.87936	−0.188883
$100$	0	0
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	1.66701i	0.165059i
$103$	2.60197i	0.256380i	0.991750	+	0.128190i	$0.0409167\pi$
$103$	2.60197i	0.256380i	−0.991750	+	0.128190i	$0.959083\pi$
$104$	13.1773	1.29214
$105$	0	0
$106$	7.51026	0.729461
$107$	− 15.7431i	− 1.52195i	−0.648784	−	0.760973i	$-0.724722\pi$
$107$	− 15.7431i	− 1.52195i	0.648784	−	0.760973i	$-0.275278\pi$
$108$	1.70928i	0.164475i
$109$	4.81432	0.461128	0.230564	−	0.973057i	$-0.425943\pi$
$109$	4.81432	0.461128	0.230564	+	0.973057i	$0.425943\pi$
$110$	0	0
$111$	−6.82991	−0.648267
$112$	− 8.68035i	− 0.820216i
$113$	− 9.91548i	− 0.932770i	−0.884582	−	0.466385i	$-0.845556\pi$
$113$	− 9.91548i	− 0.932770i	0.884582	−	0.466385i	$-0.154444\pi$
$114$	3.95774	0.370676
$115$	0	0
$116$	−0.290725	−0.0269931
$117$	6.58864i	0.609120i
$118$	3.62475i	0.333686i
$119$	11.4680	1.05127
$120$	0	0
$121$	−7.46800	−0.678909
$122$	0.993857i	0.0899796i
$123$	7.95774i	0.717525i
$124$	−1.70928	−0.153497
$125$	0	0
$126$	−2.00000	−0.178174
$127$	6.68035i	0.592785i	0.955066	+	0.296392i	$0.0957836\pi$
$127$	6.68035i	0.592785i	−0.955066	+	0.296392i	$0.904216\pi$
$128$	11.5174i	1.01801i
$129$	4.18342	0.368329
$130$	0	0
$131$	3.98440	0.348119	0.174059	−	0.984735i	$-0.444312\pi$
$131$	3.98440	0.348119	0.174059	+	0.984735i	$0.444312\pi$
$132$	− 3.21235i	− 0.279599i
$133$	− 27.2267i	− 2.36086i
$134$	−0.978259	−0.0845087
$135$	0	0
$136$	−6.18342	−0.530224
$137$	23.1906i	1.98131i	0.136402	+	0.990654i	$0.456446\pi$
$137$	23.1906i	1.98131i	−0.136402	+	0.990654i	$0.543554\pi$
$138$	3.93495i	0.334965i
$139$	12.7454	1.08105	0.540525	−	0.841328i	$-0.318226\pi$
$139$	12.7454	1.08105	0.540525	+	0.841328i	$0.318226\pi$
$140$	0	0
$141$	−11.7587	−0.990263
$142$	− 6.07223i	− 0.509571i
$143$	− 12.3824i	− 1.03547i
$144$	−2.34017	−0.195014
$145$	0	0
$146$	−1.57531	−0.130373
$147$	6.75872i	0.557450i
$148$	− 11.6742i	− 0.959614i
$149$	11.0361	0.904114	0.452057	−	0.891989i	$-0.350690\pi$
$149$	11.0361	0.904114	0.452057	+	0.891989i	$0.350690\pi$
$150$	0	0
$151$	−11.1050	−0.903715	−0.451858	−	0.892090i	$-0.649239\pi$
$151$	−11.1050	−0.903715	−0.451858	+	0.892090i	$0.649239\pi$
$152$	14.6803i	1.19073i
$153$	− 3.09171i	− 0.249950i
$154$	3.75872	0.302887
$155$	0	0
$156$	−11.2618	−0.901665
$157$	14.4186i	1.15073i	0.817898	+	0.575363i	$0.195139\pi$
$157$	14.4186i	1.15073i	−0.817898	+	0.575363i	$0.804861\pi$
$158$	− 3.36910i	− 0.268031i
$159$	−13.9288	−1.10463
$160$	0	0
$161$	27.0700	2.13341
$162$	0.539189i	0.0423627i
$163$	− 16.6742i	− 1.30602i	−0.757347	−	0.653012i	$-0.773505\pi$
$163$	− 16.6742i	− 1.30602i	0.757347	−	0.653012i	$-0.226495\pi$
$164$	−13.6020	−1.06214
$165$	0	0
$166$	−3.35350	−0.260282
$167$	6.65368i	0.514878i	0.966295	+	0.257439i	$0.0828786\pi$
$167$	6.65368i	0.514878i	−0.966295	+	0.257439i	$0.917121\pi$
$168$	− 7.41855i	− 0.572354i
$169$	−30.4101	−2.33924
$170$	0	0
$171$	−7.34017	−0.561317
$172$	7.15061i	0.545229i
$173$	− 19.7165i	− 1.49901i	−0.661996	−	0.749507i	$-0.730290\pi$
$173$	− 19.7165i	− 1.49901i	0.661996	−	0.749507i	$-0.269710\pi$
$174$	−0.0917087	−0.00695242
$175$	0	0
$176$	4.39803	0.331514
$177$	− 6.72261i	− 0.505302i
$178$	6.68753i	0.501252i
$179$	3.68261	0.275251	0.137626	−	0.990484i	$-0.456053\pi$
$179$	3.68261	0.275251	0.137626	+	0.990484i	$0.456053\pi$
$180$	0	0
$181$	−1.51026	−0.112257	−0.0561284	−	0.998424i	$-0.517876\pi$
$181$	−1.51026	−0.112257	−0.0561284	+	0.998424i	$0.517876\pi$
$182$	− 13.1773i	− 0.976765i
$183$	− 1.84324i	− 0.136257i
$184$	−14.5958	−1.07602
$185$	0	0
$186$	−0.539189	−0.0395352
$187$	5.81044i	0.424901i
$188$	− 20.0989i	− 1.46586i
$189$	3.70928	0.269810
$190$	0	0
$191$	−7.43415	−0.537916	−0.268958	−	0.963152i	$-0.586679\pi$
$191$	−7.43415	−0.537916	−0.268958	+	0.963152i	$0.586679\pi$
$192$	− 1.84324i	− 0.133025i
$193$	22.9916i	1.65497i	0.561487	+	0.827485i	$0.310229\pi$
$193$	22.9916i	1.65497i	−0.561487	+	0.827485i	$0.689771\pi$
$194$	2.12678	0.152694
$195$	0	0
$196$	−11.5525	−0.825180
$197$	14.3135i	1.01980i	0.860235	+	0.509898i	$0.170317\pi$
$197$	14.3135i	1.01980i	−0.860235	+	0.509898i	$0.829683\pi$
$198$	− 1.01333i	− 0.0720143i
$199$	4.64650	0.329381	0.164691	−	0.986345i	$-0.447337\pi$
$199$	4.64650	0.329381	0.164691	+	0.986345i	$0.447337\pi$
$200$	0	0
$201$	1.81432	0.127972
$202$	− 4.31351i	− 0.303498i
$203$	0.630898i	0.0442803i
$204$	5.28458	0.369995
$205$	0	0
$206$	−1.40295	−0.0977483
$207$	− 7.29791i	− 0.507240i
$208$	− 15.4186i	− 1.06908i
$209$	13.7948	0.954209
$210$	0	0
$211$	26.6163	1.83234	0.916172	−	0.400785i	$-0.131262\pi$
$211$	26.6163	1.83234	0.916172	+	0.400785i	$0.131262\pi$
$212$	− 23.8082i	− 1.63515i
$213$	11.2618i	0.771645i
$214$	8.48852	0.580263
$215$	0	0
$216$	−2.00000	−0.136083
$217$	3.70928i	0.251802i
$218$	2.59583i	0.175811i
$219$	2.92162	0.197425
$220$	0	0
$221$	20.3701	1.37024
$222$	− 3.68261i	− 0.247161i
$223$	17.0205i	1.13978i	0.821722	+	0.569889i	$0.193014\pi$
$223$	17.0205i	1.13978i	−0.821722	+	0.569889i	$0.806986\pi$
$224$	19.5174	1.30406
$225$	0	0
$226$	5.34632	0.355632
$227$	9.78539i	0.649479i	0.945804	+	0.324739i	$0.105277\pi$
$227$	9.78539i	0.649479i	−0.945804	+	0.324739i	$0.894723\pi$
$228$	− 12.5464i	− 0.830904i
$229$	22.3474	1.47676	0.738378	−	0.674388i	$-0.235592\pi$
$229$	22.3474	1.47676	0.738378	+	0.674388i	$0.235592\pi$
$230$	0	0
$231$	−6.97107	−0.458663
$232$	− 0.340173i	− 0.0223334i
$233$	20.6803i	1.35481i	0.735608	+	0.677407i	$0.236896\pi$
$233$	20.6803i	1.35481i	−0.735608	+	0.677407i	$0.763104\pi$
$234$	−3.55252	−0.232235
$235$	0	0
$236$	11.4908	0.747986
$237$	6.24846i	0.405881i
$238$	6.18342i	0.400811i
$239$	3.43188	0.221990	0.110995	−	0.993821i	$-0.464596\pi$
$239$	3.43188	0.221990	0.110995	+	0.993821i	$0.464596\pi$
$240$	0	0
$241$	3.56812	0.229843	0.114921	−	0.993375i	$-0.463338\pi$
$241$	3.56812	0.229843	0.114921	+	0.993375i	$0.463338\pi$
$242$	− 4.02666i	− 0.258844i
$243$	− 1.00000i	− 0.0641500i
$244$	3.15061	0.201697
$245$	0	0
$246$	−4.29072	−0.273567
$247$	− 48.3617i	− 3.07718i
$248$	− 2.00000i	− 0.127000i
$249$	6.21953	0.394147
$250$	0	0
$251$	7.74539	0.488885	0.244442	−	0.969664i	$-0.421395\pi$
$251$	7.74539	0.488885	0.244442	+	0.969664i	$0.421395\pi$
$252$	6.34017i	0.399393i
$253$	13.7154i	0.862281i
$254$	−3.60197	−0.226008
$255$	0	0
$256$	−2.52359	−0.157724
$257$	20.4547i	1.27593i	0.770067	+	0.637964i	$0.220223\pi$
$257$	20.4547i	1.27593i	−0.770067	+	0.637964i	$0.779777\pi$
$258$	2.25565i	0.140431i
$259$	−25.3340	−1.57418
$260$	0	0
$261$	0.170086	0.0105281
$262$	2.14834i	0.132725i
$263$	− 24.5113i	− 1.51143i	−0.654900	−	0.755716i	$-0.727289\pi$
$263$	− 24.5113i	− 1.51143i	0.654900	−	0.755716i	$-0.272711\pi$
$264$	3.75872	0.231333
$265$	0	0
$266$	14.6803	0.900110
$267$	− 12.4030i	− 0.759048i
$268$	3.10116i	0.189434i
$269$	−27.2762	−1.66306	−0.831529	−	0.555482i	$-0.812534\pi$
$269$	−27.2762	−1.66306	−0.831529	+	0.555482i	$0.812534\pi$
$270$	0	0
$271$	−12.4897	−0.758698	−0.379349	−	0.925254i	$-0.623852\pi$
$271$	−12.4897	−0.758698	−0.379349	+	0.925254i	$0.623852\pi$
$272$	7.23513i	0.438694i
$273$	24.4391i	1.47912i
$274$	−12.5041	−0.755401
$275$	0	0
$276$	12.4741	0.750855
$277$	− 14.8638i	− 0.893077i	−0.894764	−	0.446538i	$-0.852657\pi$
$277$	− 14.8638i	− 0.893077i	0.894764	−	0.446538i	$-0.147343\pi$
$278$	6.87217i	0.412166i
$279$	1.00000	0.0598684
$280$	0	0
$281$	−8.83710	−0.527177	−0.263589	−	0.964635i	$-0.584906\pi$
$281$	−8.83710	−0.527177	−0.263589	+	0.964635i	$0.584906\pi$
$282$	− 6.34017i	− 0.377552i
$283$	− 14.7009i	− 0.873876i	−0.899492	−	0.436938i	$-0.856063\pi$
$283$	− 14.7009i	− 0.873876i	0.899492	−	0.436938i	$-0.143937\pi$
$284$	−19.2495	−1.14225
$285$	0	0
$286$	6.67647	0.394788
$287$	29.5174i	1.74236i
$288$	− 5.26180i	− 0.310054i
$289$	7.44134	0.437726
$290$	0	0
$291$	−3.94441	−0.231225
$292$	4.99386i	0.292243i
$293$	− 22.6381i	− 1.32253i	−0.750152	−	0.661266i	$-0.770020\pi$
$293$	− 22.6381i	− 1.32253i	0.750152	−	0.661266i	$-0.229980\pi$
$294$	−3.64423	−0.212536
$295$	0	0
$296$	13.6598	0.793961
$297$	1.87936i	0.109052i
$298$	5.95055i	0.344706i
$299$	48.0833	2.78073
$300$	0	0
$301$	15.5174	0.894411
$302$	− 5.98771i	− 0.344554i
$303$	8.00000i	0.459588i
$304$	17.1773	0.985184
$305$	0	0
$306$	1.66701	0.0952969
$307$	2.81432i	0.160621i	0.996770	+	0.0803107i	$0.0255913\pi$
$307$	2.81432i	0.160621i	−0.996770	+	0.0803107i	$0.974409\pi$
$308$	− 11.9155i	− 0.678947i
$309$	2.60197	0.148021
$310$	0	0
$311$	0.780465	0.0442561	0.0221281	−	0.999755i	$-0.492956\pi$
$311$	0.780465	0.0442561	0.0221281	+	0.999755i	$0.492956\pi$
$312$	− 13.1773i	− 0.746016i
$313$	33.3679i	1.88606i	0.332702	+	0.943032i	$0.392040\pi$
$313$	33.3679i	1.88606i	−0.332702	+	0.943032i	$0.607960\pi$
$314$	−7.77432	−0.438730
$315$	0	0
$316$	−10.6803	−0.600816
$317$	14.7805i	0.830154i	0.909786	+	0.415077i	$0.136245\pi$
$317$	14.7805i	0.830154i	−0.909786	+	0.415077i	$0.863755\pi$
$318$	− 7.51026i	− 0.421154i
$319$	−0.319654	−0.0178972
$320$	0	0
$321$	−15.7431	−0.878696
$322$	14.5958i	0.813394i
$323$	22.6937i	1.26271i
$324$	1.70928	0.0949597
$325$	0	0
$326$	8.99054	0.497940
$327$	− 4.81432i	− 0.266232i
$328$	− 15.9155i	− 0.878785i
$329$	−43.6163	−2.40465
$330$	0	0
$331$	2.16394	0.118941	0.0594705	−	0.998230i	$-0.481059\pi$
$331$	2.16394	0.118941	0.0594705	+	0.998230i	$0.481059\pi$
$332$	10.6309i	0.583446i
$333$	6.82991i	0.374277i
$334$	−3.58759	−0.196304
$335$	0	0
$336$	−8.68035	−0.473552
$337$	− 33.3100i	− 1.81451i	−0.420578	−	0.907256i	$-0.638173\pi$
$337$	− 33.3100i	− 1.81451i	0.420578	−	0.907256i	$-0.361827\pi$
$338$	− 16.3968i	− 0.891869i
$339$	−9.91548	−0.538535
$340$	0	0
$341$	−1.87936	−0.101773
$342$	− 3.95774i	− 0.214010i
$343$	− 0.894960i	− 0.0483233i
$344$	−8.36683	−0.451110
$345$	0	0
$346$	10.6309	0.571521
$347$	2.88655i	0.154958i	0.996994	+	0.0774791i	$0.0246871\pi$
$347$	2.88655i	0.154958i	−0.996994	+	0.0774791i	$0.975313\pi$
$348$	0.290725i	0.0155845i
$349$	−3.34017	−0.178795	−0.0893977	−	0.995996i	$-0.528494\pi$
$349$	−3.34017	−0.178795	−0.0893977	+	0.995996i	$0.528494\pi$
$350$	0	0
$351$	6.58864	0.351675
$352$	9.88882i	0.527076i
$353$	24.2111i	1.28863i	0.764761	+	0.644314i	$0.222857\pi$
$353$	24.2111i	1.28863i	−0.764761	+	0.644314i	$0.777143\pi$
$354$	3.62475	0.192654
$355$	0	0
$356$	21.2001	1.12360
$357$	− 11.4680i	− 0.606951i
$358$	1.98562i	0.104943i
$359$	−31.3751	−1.65591	−0.827956	−	0.560793i	$-0.810497\pi$
$359$	−31.3751	−1.65591	−0.827956	+	0.560793i	$0.810497\pi$
$360$	0	0
$361$	34.8781	1.83569
$362$	− 0.814315i	− 0.0427995i
$363$	7.46800i	0.391968i
$364$	−41.7731	−2.18951
$365$	0	0
$366$	0.993857	0.0519497
$367$	13.6286i	0.711409i	0.934599	+	0.355704i	$0.115759\pi$
$367$	13.6286i	0.711409i	−0.934599	+	0.355704i	$0.884241\pi$
$368$	17.0784i	0.890272i
$369$	7.95774	0.414263
$370$	0	0
$371$	−51.6658	−2.68235
$372$	1.70928i	0.0886218i
$373$	− 2.39576i	− 0.124048i	−0.998075	−	0.0620240i	$-0.980244\pi$
$373$	− 2.39576i	− 0.124048i	0.998075	−	0.0620240i	$-0.0197555\pi$
$374$	−3.13292	−0.162000
$375$	0	0
$376$	23.5174	1.21282
$377$	1.12064i	0.0577158i
$378$	2.00000i	0.102869i
$379$	11.3958	0.585361	0.292681	−	0.956210i	$-0.405453\pi$
$379$	11.3958	0.585361	0.292681	+	0.956210i	$0.405453\pi$
$380$	0	0
$381$	6.68035	0.342244
$382$	− 4.00841i	− 0.205088i
$383$	− 4.77820i	− 0.244155i	−0.992521	−	0.122077i	$-0.961044\pi$
$383$	− 4.77820i	− 0.244155i	0.992521	−	0.122077i	$-0.0389556\pi$
$384$	11.5174	0.587747
$385$	0	0
$386$	−12.3968	−0.630981
$387$	− 4.18342i	− 0.212655i
$388$	− 6.74208i	− 0.342277i
$389$	−3.84324	−0.194860	−0.0974301	−	0.995242i	$-0.531062\pi$
$389$	−3.84324	−0.194860	−0.0974301	+	0.995242i	$0.531062\pi$
$390$	0	0
$391$	−22.5630	−1.14106
$392$	− 13.5174i	− 0.682734i
$393$	− 3.98440i	− 0.200986i
$394$	−7.71769	−0.388811
$395$	0	0
$396$	−3.21235	−0.161427
$397$	12.7093i	0.637860i	0.947778	+	0.318930i	$0.103324\pi$
$397$	12.7093i	0.637860i	−0.947778	+	0.318930i	$0.896676\pi$
$398$	2.50534i	0.125581i
$399$	−27.2267	−1.36304
$400$	0	0
$401$	−21.3968	−1.06851	−0.534253	−	0.845325i	$-0.679407\pi$
$401$	−21.3968	−1.06851	−0.534253	+	0.845325i	$0.679407\pi$
$402$	0.978259i	0.0487911i
$403$	6.58864i	0.328203i
$404$	−13.6742	−0.680317
$405$	0	0
$406$	−0.340173	−0.0168825
$407$	− 12.8359i	− 0.636251i
$408$	6.18342i	0.306125i
$409$	36.1256	1.78629	0.893147	−	0.449765i	$-0.148492\pi$
$409$	36.1256	1.78629	0.893147	+	0.449765i	$0.148492\pi$
$410$	0	0
$411$	23.1906	1.14391
$412$	4.44748i	0.219112i
$413$	− 24.9360i	− 1.22702i
$414$	3.93495	0.193392
$415$	0	0
$416$	34.6681	1.69974
$417$	− 12.7454i	− 0.624145i
$418$	7.43802i	0.363806i
$419$	2.92162	0.142731	0.0713653	−	0.997450i	$-0.477264\pi$
$419$	2.92162	0.142731	0.0713653	+	0.997450i	$0.477264\pi$
$420$	0	0
$421$	−28.6309	−1.39538	−0.697692	−	0.716398i	$-0.745790\pi$
$421$	−28.6309	−1.39538	−0.697692	+	0.716398i	$0.745790\pi$
$422$	14.3512i	0.698607i
$423$	11.7587i	0.571729i
$424$	27.8576	1.35289
$425$	0	0
$426$	−6.07223	−0.294201
$427$	− 6.83710i	− 0.330871i
$428$	− 26.9093i	− 1.30071i
$429$	−12.3824	−0.597830
$430$	0	0
$431$	−2.41241	−0.116202	−0.0581008	−	0.998311i	$-0.518504\pi$
$431$	−2.41241	−0.116202	−0.0581008	+	0.998311i	$0.518504\pi$
$432$	2.34017i	0.112592i
$433$	26.2290i	1.26048i	0.776398	+	0.630242i	$0.217044\pi$
$433$	26.2290i	1.26048i	−0.776398	+	0.630242i	$0.782956\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	8.22899	0.394097
$437$	53.5679i	2.56250i
$438$	1.57531i	0.0752710i
$439$	−28.5197	−1.36117	−0.680586	−	0.732668i	$-0.738275\pi$
$439$	−28.5197	−1.36117	−0.680586	+	0.732668i	$0.738275\pi$
$440$	0	0
$441$	6.75872	0.321844
$442$	10.9834i	0.522425i
$443$	12.4235i	0.590257i	0.955458	+	0.295128i	$0.0953624\pi$
$443$	12.4235i	0.590257i	−0.955458	+	0.295128i	$0.904638\pi$
$444$	−11.6742	−0.554033
$445$	0	0
$446$	−9.17727	−0.434557
$447$	− 11.0361i	− 0.521990i
$448$	− 6.83710i	− 0.323023i
$449$	17.0833	0.806211	0.403105	−	0.915154i	$-0.367931\pi$
$449$	17.0833	0.806211	0.403105	+	0.915154i	$0.367931\pi$
$450$	0	0
$451$	−14.9555	−0.704226
$452$	− 16.9483i	− 0.797180i
$453$	11.1050i	0.521760i
$454$	−5.27617	−0.247623
$455$	0	0
$456$	14.6803	0.687470
$457$	11.0277i	0.515854i	0.966164	+	0.257927i	$0.0830395\pi$
$457$	11.0277i	0.515854i	−0.966164	+	0.257927i	$0.916961\pi$
$458$	12.0494i	0.563034i
$459$	−3.09171	−0.144309
$460$	0	0
$461$	16.7165	0.778563	0.389282	−	0.921119i	$-0.372723\pi$
$461$	16.7165	0.778563	0.389282	+	0.921119i	$0.372723\pi$
$462$	− 3.75872i	− 0.174872i
$463$	12.4124i	0.576854i	0.957502	+	0.288427i	$0.0931322\pi$
$463$	12.4124i	0.576854i	−0.957502	+	0.288427i	$0.906868\pi$
$464$	−0.398032	−0.0184782
$465$	0	0
$466$	−11.1506	−0.516542
$467$	14.7115i	0.680769i	0.940286	+	0.340385i	$0.110557\pi$
$467$	14.7115i	0.680769i	−0.940286	+	0.340385i	$0.889443\pi$
$468$	11.2618i	0.520577i
$469$	6.72979	0.310753
$470$	0	0
$471$	14.4186	0.664372
$472$	13.4452i	0.618866i
$473$	7.86216i	0.361502i
$474$	−3.36910	−0.154748
$475$	0	0
$476$	19.6020	0.898455
$477$	13.9288i	0.637756i
$478$	1.85043i	0.0846368i
$479$	16.8371	0.769307	0.384653	−	0.923061i	$-0.374321\pi$
$479$	16.8371	0.769307	0.384653	+	0.923061i	$0.374321\pi$
$480$	0	0
$481$	−44.9998	−2.05182
$482$	1.92389i	0.0876308i
$483$	− 27.0700i	− 1.23173i
$484$	−12.7649	−0.580221
$485$	0	0
$486$	0.539189	0.0244581
$487$	− 2.39084i	− 0.108339i	−0.998532	−	0.0541697i	$-0.982749\pi$
$487$	− 2.39084i	− 0.108339i	0.998532	−	0.0541697i	$-0.0172512\pi$
$488$	3.68649i	0.166880i
$489$	−16.6742	−0.754034
$490$	0	0
$491$	−9.24459	−0.417202	−0.208601	−	0.978001i	$-0.566891\pi$
$491$	−9.24459	−0.417202	−0.208601	+	0.978001i	$0.566891\pi$
$492$	13.6020i	0.613224i
$493$	− 0.525858i	− 0.0236834i
$494$	26.0761	1.17322
$495$	0	0
$496$	−2.34017	−0.105077
$497$	41.7731i	1.87378i
$498$	3.35350i	0.150274i
$499$	−7.72034	−0.345610	−0.172805	−	0.984956i	$-0.555283\pi$
$499$	−7.72034	−0.345610	−0.172805	+	0.984956i	$0.555283\pi$
$500$	0	0
$501$	6.65368	0.297265
$502$	4.17623i	0.186394i
$503$	− 20.3090i	− 0.905532i	−0.891629	−	0.452766i	$-0.850437\pi$
$503$	− 20.3090i	− 0.905532i	0.891629	−	0.452766i	$-0.149563\pi$
$504$	−7.41855	−0.330449
$505$	0	0
$506$	−7.39520	−0.328757
$507$	30.4101i	1.35056i
$508$	11.4186i	0.506616i
$509$	31.1906	1.38250	0.691250	−	0.722616i	$-0.257060\pi$
$509$	31.1906	1.38250	0.691250	+	0.722616i	$0.257060\pi$
$510$	0	0
$511$	10.8371	0.479405
$512$	21.6742i	0.957873i
$513$	7.34017i	0.324077i
$514$	−11.0289	−0.486465
$515$	0	0
$516$	7.15061	0.314788
$517$	− 22.0989i	− 0.971908i
$518$	− 13.6598i	− 0.600178i
$519$	−19.7165	−0.865457
$520$	0	0
$521$	14.6693	0.642673	0.321336	−	0.946965i	$-0.395868\pi$
$521$	14.6693	0.642673	0.321336	+	0.946965i	$0.395868\pi$
$522$	0.0917087i	0.00401398i
$523$	− 2.52813i	− 0.110547i	−0.998471	−	0.0552736i	$-0.982397\pi$
$523$	− 2.52813i	− 0.110547i	0.998471	−	0.0552736i	$-0.0176031\pi$
$524$	6.81044	0.297515
$525$	0	0
$526$	13.2162	0.576255
$527$	− 3.09171i	− 0.134677i
$528$	− 4.39803i	− 0.191400i
$529$	−30.2595	−1.31563
$530$	0	0
$531$	−6.72261	−0.291736
$532$	− 46.5380i	− 2.01768i
$533$	52.4307i	2.27102i
$534$	6.68753	0.289398
$535$	0	0
$536$	−3.62863	−0.156733
$537$	− 3.68261i	− 0.158916i
$538$	− 14.7070i	− 0.634064i
$539$	−12.7021	−0.547118
$540$	0	0
$541$	−5.08065	−0.218434	−0.109217	−	0.994018i	$-0.534834\pi$
$541$	−5.08065	−0.218434	−0.109217	+	0.994018i	$0.534834\pi$
$542$	− 6.73433i	− 0.289264i
$543$	1.51026i	0.0648114i
$544$	−16.2679	−0.697482
$545$	0	0
$546$	−13.1773	−0.563935
$547$	40.2990i	1.72306i	0.507708	+	0.861529i	$0.330493\pi$
$547$	40.2990i	1.72306i	−0.507708	+	0.861529i	$0.669507\pi$
$548$	39.6391i	1.69330i
$549$	−1.84324	−0.0786678
$550$	0	0
$551$	−1.24846	−0.0531864
$552$	14.5958i	0.621240i
$553$	23.1773i	0.985598i
$554$	8.01438	0.340498
$555$	0	0
$556$	21.7854	0.923906
$557$	− 11.5848i	− 0.490862i	−0.969414	−	0.245431i	$-0.921071\pi$
$557$	− 11.5848i	− 0.490862i	0.969414	−	0.245431i	$-0.0789295\pi$
$558$	0.539189i	0.0228257i
$559$	27.5630	1.16579
$560$	0	0
$561$	5.81044	0.245317
$562$	− 4.76487i	− 0.200994i
$563$	3.60319i	0.151856i	0.997113	+	0.0759282i	$0.0241920\pi$
$563$	3.60319i	0.151856i	−0.997113	+	0.0759282i	$0.975808\pi$
$564$	−20.0989	−0.846316
$565$	0	0
$566$	7.92654	0.333177
$567$	− 3.70928i	− 0.155775i
$568$	− 22.5236i	− 0.945069i
$569$	−19.6959	−0.825697	−0.412848	−	0.910800i	$-0.635466\pi$
$569$	−19.6959	−0.825697	−0.412848	+	0.910800i	$0.635466\pi$
$570$	0	0
$571$	38.7214	1.62044	0.810220	−	0.586126i	$-0.199348\pi$
$571$	38.7214	1.62044	0.810220	+	0.586126i	$0.199348\pi$
$572$	− 21.1650i	− 0.884953i
$573$	7.43415i	0.310566i
$574$	−15.9155	−0.664299
$575$	0	0
$576$	−1.84324	−0.0768019
$577$	8.23287i	0.342739i	0.985207	+	0.171369i	$0.0548191\pi$
$577$	8.23287i	0.342739i	−0.985207	+	0.171369i	$0.945181\pi$
$578$	4.01229i	0.166889i
$579$	22.9916	0.955498
$580$	0	0
$581$	23.0700	0.957104
$582$	− 2.12678i	− 0.0881579i
$583$	− 26.1773i	− 1.08415i
$584$	−5.84324	−0.241795
$585$	0	0
$586$	12.2062	0.504234
$587$	18.6130i	0.768242i	0.923283	+	0.384121i	$0.125495\pi$
$587$	18.6130i	0.768242i	−0.923283	+	0.384121i	$0.874505\pi$
$588$	11.5525i	0.476418i
$589$	−7.34017	−0.302447
$590$	0	0
$591$	14.3135	0.588779
$592$	− 15.9832i	− 0.656905i
$593$	9.03612i	0.371069i	0.982638	+	0.185534i	$0.0594016\pi$
$593$	9.03612i	0.371069i	−0.982638	+	0.185534i	$0.940598\pi$
$594$	−1.01333	−0.0415775
$595$	0	0
$596$	18.8638	0.772690
$597$	− 4.64650i	− 0.190168i
$598$	25.9260i	1.06019i
$599$	5.80098	0.237022	0.118511	−	0.992953i	$-0.462188\pi$
$599$	5.80098	0.237022	0.118511	+	0.992953i	$0.462188\pi$
$600$	0	0
$601$	−8.50412	−0.346890	−0.173445	−	0.984844i	$-0.555490\pi$
$601$	−8.50412	−0.346890	−0.173445	+	0.984844i	$0.555490\pi$
$602$	8.36683i	0.341007i
$603$	− 1.81432i	− 0.0738846i
$604$	−18.9816	−0.772349
$605$	0	0
$606$	−4.31351	−0.175224
$607$	− 6.90707i	− 0.280349i	−0.990127	−	0.140175i	$-0.955234\pi$
$607$	− 6.90707i	− 0.280349i	0.990127	−	0.140175i	$-0.0447664\pi$
$608$	38.6225i	1.56635i
$609$	0.630898	0.0255653
$610$	0	0
$611$	−77.4740	−3.13426
$612$	− 5.28458i	− 0.213617i
$613$	21.2618i	0.858756i	0.903125	+	0.429378i	$0.141267\pi$
$613$	21.2618i	0.858756i	−0.903125	+	0.429378i	$0.858733\pi$
$614$	−1.51745	−0.0612392
$615$	0	0
$616$	13.9421	0.561745
$617$	5.40295i	0.217515i	0.994068	+	0.108757i	$0.0346871\pi$
$617$	5.40295i	0.217515i	−0.994068	+	0.108757i	$0.965313\pi$
$618$	1.40295i	0.0564350i
$619$	−4.75258	−0.191022	−0.0955112	−	0.995428i	$-0.530449\pi$
$619$	−4.75258	−0.191022	−0.0955112	+	0.995428i	$0.530449\pi$
$620$	0	0
$621$	−7.29791	−0.292855
$622$	0.420818i	0.0168733i
$623$	− 46.0060i	− 1.84319i
$624$	−15.4186	−0.617236
$625$	0	0
$626$	−17.9916	−0.719089
$627$	− 13.7948i	− 0.550913i
$628$	24.6453i	0.983453i
$629$	21.1161	0.841954
$630$	0	0
$631$	25.5369	1.01661	0.508304	−	0.861177i	$-0.330272\pi$
$631$	25.5369	1.01661	0.508304	+	0.861177i	$0.330272\pi$
$632$	− 12.4969i	− 0.497101i
$633$	− 26.6163i	− 1.05790i
$634$	−7.96946	−0.316508
$635$	0	0
$636$	−23.8082	−0.944055
$637$	44.5308i	1.76437i
$638$	− 0.172354i	− 0.00682356i
$639$	11.2618	0.445510
$640$	0	0
$641$	8.08557	0.319361	0.159680	−	0.987169i	$-0.448954\pi$
$641$	8.08557	0.319361	0.159680	+	0.987169i	$0.448954\pi$
$642$	− 8.48852i	− 0.335015i
$643$	3.51479i	0.138610i	0.997596	+	0.0693050i	$0.0220782\pi$
$643$	3.51479i	0.138610i	−0.997596	+	0.0693050i	$0.977922\pi$
$644$	46.2700	1.82329
$645$	0	0
$646$	−12.2362	−0.481426
$647$	21.0566i	0.827822i	0.910317	+	0.413911i	$0.135837\pi$
$647$	21.0566i	0.827822i	−0.910317	+	0.413911i	$0.864163\pi$
$648$	2.00000i	0.0785674i
$649$	12.6342	0.495936
$650$	0	0
$651$	3.70928	0.145378
$652$	− 28.5008i	− 1.11618i
$653$	19.2918i	0.754945i	0.926021	+	0.377473i	$0.123207\pi$
$653$	19.2918i	0.754945i	−0.926021	+	0.377473i	$0.876793\pi$
$654$	2.59583	0.101505
$655$	0	0
$656$	−18.6225	−0.727086
$657$	− 2.92162i	− 0.113983i
$658$	− 23.5174i	− 0.916806i
$659$	26.1399	1.01827	0.509134	−	0.860687i	$-0.329966\pi$
$659$	26.1399	1.01827	0.509134	+	0.860687i	$0.329966\pi$
$660$	0	0
$661$	−15.4475	−0.600837	−0.300419	−	0.953807i	$-0.597126\pi$
$661$	−15.4475	−0.600837	−0.300419	+	0.953807i	$0.597126\pi$
$662$	1.16677i	0.0453480i
$663$	− 20.3701i	− 0.791111i
$664$	−12.4391	−0.482730
$665$	0	0
$666$	−3.68261	−0.142698
$667$	− 1.24128i	− 0.0480624i
$668$	11.3730i	0.440034i
$669$	17.0205	0.658051
$670$	0	0
$671$	3.46412	0.133731
$672$	− 19.5174i	− 0.752902i
$673$	39.0277i	1.50441i	0.658931	+	0.752204i	$0.271009\pi$
$673$	39.0277i	1.50441i	−0.658931	+	0.752204i	$0.728991\pi$
$674$	17.9604	0.691808
$675$	0	0
$676$	−51.9793	−1.99920
$677$	− 3.24354i	− 0.124660i	−0.998056	−	0.0623298i	$-0.980147\pi$
$677$	− 3.24354i	− 0.124660i	0.998056	−	0.0623298i	$-0.0198530\pi$
$678$	− 5.34632i	− 0.205324i
$679$	−14.6309	−0.561482
$680$	0	0
$681$	9.78539	0.374977
$682$	− 1.01333i	− 0.0388024i
$683$	− 47.4908i	− 1.81718i	−0.417684	−	0.908592i	$-0.637158\pi$
$683$	− 47.4908i	− 1.81718i	0.417684	−	0.908592i	$-0.362842\pi$
$684$	−12.5464	−0.479723
$685$	0	0
$686$	0.482553	0.0184239
$687$	− 22.3474i	− 0.852605i
$688$	9.78992i	0.373237i
$689$	−91.7719	−3.49623
$690$	0	0
$691$	−9.71154	−0.369444	−0.184722	−	0.982791i	$-0.559139\pi$
$691$	−9.71154	−0.369444	−0.184722	+	0.982791i	$0.559139\pi$
$692$	− 33.7009i	− 1.28111i
$693$	6.97107i	0.264809i
$694$	−1.55640	−0.0590800
$695$	0	0
$696$	−0.340173	−0.0128942
$697$	− 24.6030i	− 0.931906i
$698$	− 1.80098i	− 0.0681683i
$699$	20.6803	0.782203
$700$	0	0
$701$	9.47519	0.357873	0.178936	−	0.983861i	$-0.442734\pi$
$701$	9.47519	0.357873	0.178936	+	0.983861i	$0.442734\pi$
$702$	3.55252i	0.134081i
$703$	− 50.1327i	− 1.89079i
$704$	3.46412	0.130559
$705$	0	0
$706$	−13.0544	−0.491308
$707$	29.6742i	1.11601i
$708$	− 11.4908i	− 0.431850i
$709$	−50.0482	−1.87960	−0.939800	−	0.341724i	$-0.888989\pi$
$709$	−50.0482	−1.87960	−0.939800	+	0.341724i	$0.888989\pi$
$710$	0	0
$711$	6.24846	0.234336
$712$	24.8059i	0.929641i
$713$	− 7.29791i	− 0.273309i
$714$	6.18342	0.231409
$715$	0	0
$716$	6.29460	0.235240
$717$	− 3.43188i	− 0.128166i
$718$	− 16.9171i	− 0.631340i
$719$	−41.5753	−1.55050	−0.775249	−	0.631656i	$-0.782375\pi$
$719$	−41.5753	−1.55050	−0.775249	+	0.631656i	$0.782375\pi$
$720$	0	0
$721$	9.65142	0.359438
$722$	18.8059i	0.699883i
$723$	− 3.56812i	− 0.132700i
$724$	−2.58145	−0.0959388
$725$	0	0
$726$	−4.02666	−0.149443
$727$	2.99773i	0.111180i	0.998454	+	0.0555899i	$0.0177039\pi$
$727$	2.99773i	0.111180i	−0.998454	+	0.0555899i	$0.982296\pi$
$728$	− 48.8781i	− 1.81154i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−12.9339	−0.478378
$732$	− 3.15061i	− 0.116450i
$733$	36.6020i	1.35192i	0.736936	+	0.675962i	$0.236272\pi$
$733$	36.6020i	1.35192i	−0.736936	+	0.675962i	$0.763728\pi$
$734$	−7.34841	−0.271235
$735$	0	0
$736$	−38.4001	−1.41545
$737$	3.40975i	0.125600i
$738$	4.29072i	0.157944i
$739$	−4.16394	−0.153173	−0.0765866	−	0.997063i	$-0.524402\pi$
$739$	−4.16394	−0.153173	−0.0765866	+	0.997063i	$0.524402\pi$
$740$	0	0
$741$	−48.3617	−1.77661
$742$	− 27.8576i	− 1.02269i
$743$	− 17.3824i	− 0.637700i	−0.947805	−	0.318850i	$-0.896703\pi$
$743$	− 17.3824i	− 0.637700i	0.947805	−	0.318850i	$-0.103297\pi$
$744$	−2.00000	−0.0733236
$745$	0	0
$746$	1.29177	0.0472950
$747$	− 6.21953i	− 0.227561i
$748$	9.93164i	0.363137i
$749$	−58.3956	−2.13373
$750$	0	0
$751$	−33.4307	−1.21990	−0.609951	−	0.792439i	$-0.708811\pi$
$751$	−33.4307	−1.21990	−0.609951	+	0.792439i	$0.708811\pi$
$752$	− 27.5174i	− 1.00346i
$753$	− 7.74539i	− 0.282258i
$754$	−0.604236	−0.0220050
$755$	0	0
$756$	6.34017	0.230590
$757$	− 5.02771i	− 0.182735i	−0.995817	−	0.0913676i	$-0.970876\pi$
$757$	− 5.02771i	− 0.182735i	0.995817	−	0.0913676i	$-0.0291238\pi$
$758$	6.14447i	0.223177i
$759$	13.7154	0.497838
$760$	0	0
$761$	23.6560	0.857528	0.428764	−	0.903417i	$-0.358949\pi$
$761$	23.6560	0.857528	0.428764	+	0.903417i	$0.358949\pi$
$762$	3.60197i	0.130486i
$763$	− 17.8576i	− 0.646489i
$764$	−12.7070	−0.459723
$765$	0	0
$766$	2.57635	0.0930873
$767$	− 44.2928i	− 1.59932i
$768$	2.52359i	0.0910622i
$769$	43.5197	1.56936	0.784681	−	0.619900i	$-0.212827\pi$
$769$	43.5197	1.56936	0.784681	+	0.619900i	$0.212827\pi$
$770$	0	0
$771$	20.4547	0.736657
$772$	39.2990i	1.41440i
$773$	45.3267i	1.63029i	0.579259	+	0.815143i	$0.303342\pi$
$773$	45.3267i	1.63029i	−0.579259	+	0.815143i	$0.696658\pi$
$774$	2.25565	0.0810777
$775$	0	0
$776$	7.88882	0.283192
$777$	25.3340i	0.908853i
$778$	− 2.07223i	− 0.0742932i
$779$	−58.4112	−2.09280
$780$	0	0
$781$	−21.1650	−0.757343
$782$	− 12.1657i	− 0.435046i
$783$	− 0.170086i	− 0.00607840i
$784$	−15.8166	−0.564878
$785$	0	0
$786$	2.14834	0.0766289
$787$	− 11.2618i	− 0.401440i	−0.979649	−	0.200720i	$-0.935672\pi$
$787$	− 11.2618i	− 0.401440i	0.979649	−	0.200720i	$-0.0643281\pi$
$788$	24.4657i	0.871556i
$789$	−24.5113	−0.872626
$790$	0	0
$791$	−36.7792	−1.30772
$792$	− 3.75872i	− 0.133560i
$793$	− 12.1445i	− 0.431263i
$794$	−6.85270	−0.243193
$795$	0	0
$796$	7.94214	0.281502
$797$	− 8.00597i	− 0.283586i	−0.989896	−	0.141793i	$-0.954713\pi$
$797$	− 8.00597i	− 0.283586i	0.989896	−	0.141793i	$-0.0452867\pi$
$798$	− 14.6803i	− 0.519679i
$799$	36.3545	1.28613
$800$	0	0
$801$	−12.4030	−0.438237
$802$	− 11.5369i	− 0.407383i
$803$	5.49079i	0.193766i
$804$	3.10116	0.109370
$805$	0	0
$806$	−3.55252	−0.125132
$807$	27.2762i	0.960167i
$808$	− 16.0000i	− 0.562878i
$809$	8.70805	0.306159	0.153079	−	0.988214i	$-0.451081\pi$
$809$	8.70805	0.306159	0.153079	+	0.988214i	$0.451081\pi$
$810$	0	0
$811$	21.5357	0.756221	0.378110	−	0.925761i	$-0.376574\pi$
$811$	21.5357	0.756221	0.378110	+	0.925761i	$0.376574\pi$
$812$	1.07838i	0.0378436i
$813$	12.4897i	0.438034i
$814$	6.92096	0.242580
$815$	0	0
$816$	7.23513	0.253280
$817$	30.7070i	1.07430i
$818$	19.4785i	0.681050i
$819$	24.4391	0.853970
$820$	0	0
$821$	33.8092	1.17995	0.589975	−	0.807422i	$-0.299138\pi$
$821$	33.8092	1.17995	0.589975	+	0.807422i	$0.299138\pi$
$822$	12.5041i	0.436131i
$823$	50.2700i	1.75230i	0.482036	+	0.876152i	$0.339898\pi$
$823$	50.2700i	1.75230i	−0.482036	+	0.876152i	$0.660102\pi$
$824$	−5.20394	−0.181288
$825$	0	0
$826$	13.4452	0.467819
$827$	16.3268i	0.567740i	0.958863	+	0.283870i	$0.0916184\pi$
$827$	16.3268i	0.567740i	−0.958863	+	0.283870i	$0.908382\pi$
$828$	− 12.4741i	− 0.433506i
$829$	−12.3207	−0.427916	−0.213958	−	0.976843i	$-0.568635\pi$
$829$	−12.3207	−0.427916	−0.213958	+	0.976843i	$0.568635\pi$
$830$	0	0
$831$	−14.8638	−0.515618
$832$	− 12.1445i	− 0.421034i
$833$	− 20.8960i	− 0.724004i
$834$	6.87217	0.237964
$835$	0	0
$836$	23.5792	0.815503
$837$	− 1.00000i	− 0.0345651i
$838$	1.57531i	0.0544181i
$839$	1.47641	0.0509713	0.0254857	−	0.999675i	$-0.491887\pi$
$839$	1.47641	0.0509713	0.0254857	+	0.999675i	$0.491887\pi$
$840$	0	0
$841$	−28.9711	−0.999002
$842$	− 15.4375i	− 0.532010i
$843$	8.83710i	0.304366i
$844$	45.4947	1.56599
$845$	0	0
$846$	−6.34017	−0.217980
$847$	27.7009i	0.951813i
$848$	− 32.5958i	− 1.11935i
$849$	−14.7009	−0.504532
$850$	0	0
$851$	49.8441	1.70863
$852$	19.2495i	0.659477i
$853$	− 22.2907i	− 0.763220i	−0.924323	−	0.381610i	$-0.875370\pi$
$853$	− 22.2907i	− 0.763220i	0.924323	−	0.381610i	$-0.124630\pi$
$854$	3.68649	0.126149
$855$	0	0
$856$	31.4863	1.07618
$857$	− 54.4235i	− 1.85907i	−0.368733	−	0.929535i	$-0.620208\pi$
$857$	− 54.4235i	− 1.85907i	0.368733	−	0.929535i	$-0.379792\pi$
$858$	− 6.67647i	− 0.227931i
$859$	6.55479	0.223646	0.111823	−	0.993728i	$-0.464331\pi$
$859$	6.55479	0.223646	0.111823	+	0.993728i	$0.464331\pi$
$860$	0	0
$861$	29.5174	1.00595
$862$	− 1.30074i	− 0.0443035i
$863$	38.1399i	1.29830i	0.760661	+	0.649149i	$0.224875\pi$
$863$	38.1399i	1.29830i	−0.760661	+	0.649149i	$0.775125\pi$
$864$	−5.26180	−0.179010
$865$	0	0
$866$	−14.1424	−0.480578
$867$	− 7.44134i	− 0.252721i
$868$	6.34017i	0.215199i
$869$	−11.7431	−0.398358
$870$	0	0
$871$	11.9539	0.405041
$872$	9.62863i	0.326067i
$873$	3.94441i	0.133498i
$874$	−28.8832	−0.976990
$875$	0	0
$876$	4.99386	0.168727
$877$	47.2762i	1.59640i	0.602391	+	0.798201i	$0.294215\pi$
$877$	47.2762i	1.59640i	−0.602391	+	0.798201i	$0.705785\pi$
$878$	− 15.3775i	− 0.518966i
$879$	−22.6381	−0.763564
$880$	0	0
$881$	28.2557	0.951957	0.475979	−	0.879457i	$-0.342094\pi$
$881$	28.2557	0.951957	0.475979	+	0.879457i	$0.342094\pi$
$882$	3.64423i	0.122708i
$883$	− 7.39908i	− 0.248999i	−0.992220	−	0.124499i	$-0.960268\pi$
$883$	− 7.39908i	− 0.248999i	0.992220	−	0.124499i	$-0.0397325\pi$
$884$	34.8182	1.17106
$885$	0	0
$886$	−6.69860	−0.225044
$887$	32.1822i	1.08057i	0.841481	+	0.540286i	$0.181684\pi$
$887$	32.1822i	1.08057i	−0.841481	+	0.540286i	$0.818316\pi$
$888$	− 13.6598i	− 0.458394i
$889$	24.7792	0.831069
$890$	0	0
$891$	1.87936	0.0629610
$892$	29.0928i	0.974097i
$893$	− 86.3111i	− 2.88829i
$894$	5.95055	0.199016
$895$	0	0
$896$	42.7214	1.42722
$897$	− 48.0833i	− 1.60545i
$898$	9.21112i	0.307379i
$899$	0.170086	0.00567270
$900$	0	0
$901$	43.0638	1.43466
$902$	− 8.06382i	− 0.268496i
$903$	− 15.5174i	− 0.516388i
$904$	19.8310	0.659568
$905$	0	0
$906$	−5.98771	−0.198928
$907$	− 56.8453i	− 1.88752i	−0.330634	−	0.943759i	$-0.607262\pi$
$907$	− 56.8453i	− 1.88752i	0.330634	−	0.943759i	$-0.392738\pi$
$908$	16.7259i	0.555069i
$909$	8.00000	0.265343
$910$	0	0
$911$	17.6153	0.583621	0.291810	−	0.956476i	$-0.405742\pi$
$911$	17.6153	0.583621	0.291810	+	0.956476i	$0.405742\pi$
$912$	− 17.1773i	− 0.568796i
$913$	11.6888i	0.386841i
$914$	−5.94602	−0.196677
$915$	0	0
$916$	38.1978	1.26209
$917$	− 14.7792i	− 0.488054i
$918$	− 1.66701i	− 0.0550197i
$919$	40.1012	1.32282	0.661408	−	0.750027i	$-0.269959\pi$
$919$	40.1012	1.32282	0.661408	+	0.750027i	$0.269959\pi$
$920$	0	0
$921$	2.81432	0.0927348
$922$	9.01333i	0.296838i
$923$	74.1999i	2.44232i
$924$	−11.9155	−0.391990
$925$	0	0
$926$	−6.69263	−0.219934
$927$	− 2.60197i	− 0.0854598i
$928$	− 0.894960i	− 0.0293785i
$929$	0.326842	0.0107233	0.00536167	−	0.999986i	$-0.498293\pi$
$929$	0.326842	0.0107233	0.00536167	+	0.999986i	$0.498293\pi$
$930$	0	0
$931$	−49.6102	−1.62591
$932$	35.3484i	1.15788i
$933$	− 0.780465i	− 0.0255513i
$934$	−7.93230	−0.259553
$935$	0	0
$936$	−13.1773	−0.430713
$937$	− 35.1978i	− 1.14986i	−0.818202	−	0.574931i	$-0.805029\pi$
$937$	− 35.1978i	− 1.14986i	0.818202	−	0.574931i	$-0.194971\pi$
$938$	3.62863i	0.118479i
$939$	33.3679	1.08892
$940$	0	0
$941$	34.1038	1.11175	0.555876	−	0.831265i	$-0.312383\pi$
$941$	34.1038	1.11175	0.555876	+	0.831265i	$0.312383\pi$
$942$	7.77432i	0.253301i
$943$	− 58.0749i	− 1.89118i
$944$	15.7321	0.512035
$945$	0	0
$946$	−4.23919	−0.137828
$947$	14.6631i	0.476488i	0.971205	+	0.238244i	$0.0765718\pi$
$947$	14.6631i	0.476488i	−0.971205	+	0.238244i	$0.923428\pi$
$948$	10.6803i	0.346882i
$949$	19.2495	0.624866
$950$	0	0
$951$	14.7805	0.479289
$952$	22.9360i	0.743360i
$953$	− 7.89657i	− 0.255795i	−0.991787	−	0.127897i	$-0.959177\pi$
$953$	− 7.89657i	− 0.255795i	0.991787	−	0.127897i	$-0.0408228\pi$
$954$	−7.51026	−0.243154
$955$	0	0
$956$	5.86603	0.189721
$957$	0.319654i	0.0103329i
$958$	9.07838i	0.293309i
$959$	86.0203	2.77774
$960$	0	0
$961$	1.00000	0.0322581
$962$	− 24.2634i	− 0.782284i
$963$	15.7431i	0.507315i
$964$	6.09890	0.196432
$965$	0	0
$966$	14.5958	0.469613
$967$	11.7126i	0.376651i	0.982107	+	0.188326i	$0.0603061\pi$
$967$	11.7126i	0.376651i	−0.982107	+	0.188326i	$0.939694\pi$
$968$	− 14.9360i	− 0.480061i
$969$	22.6937	0.729026
$970$	0	0
$971$	2.24459	0.0720323	0.0360161	−	0.999351i	$-0.488533\pi$
$971$	2.24459	0.0720323	0.0360161	+	0.999351i	$0.488533\pi$
$972$	− 1.70928i	− 0.0548250i
$973$	− 47.2762i	− 1.51560i
$974$	1.28912	0.0413060
$975$	0	0
$976$	4.31351	0.138072
$977$	− 56.3701i	− 1.80344i	−0.432320	−	0.901720i	$-0.642305\pi$
$977$	− 56.3701i	− 1.80344i	0.432320	−	0.901720i	$-0.357695\pi$
$978$	− 8.99054i	− 0.287486i
$979$	23.3096	0.744979
$980$	0	0
$981$	−4.81432	−0.153709
$982$	− 4.98458i	− 0.159064i
$983$	− 9.92881i	− 0.316680i	−0.987385	−	0.158340i	$-0.949386\pi$
$983$	− 9.92881i	− 0.316680i	0.987385	−	0.158340i	$-0.0506142\pi$
$984$	−15.9155	−0.507367
$985$	0	0
$986$	0.283537	0.00902965
$987$	43.6163i	1.38832i
$988$	− 82.6635i	− 2.62988i
$989$	−30.5302	−0.970804
$990$	0	0
$991$	−4.61530	−0.146610	−0.0733049	−	0.997310i	$-0.523355\pi$
$991$	−4.61530	−0.146610	−0.0733049	+	0.997310i	$0.523355\pi$
$992$	− 5.26180i	− 0.167062i
$993$	− 2.16394i	− 0.0686707i
$994$	−22.5236	−0.714405
$995$	0	0
$996$	10.6309	0.336853
$997$	− 8.72365i	− 0.276281i	−0.990413	−	0.138140i	$-0.955887\pi$
$997$	− 8.72365i	− 0.276281i	0.990413	−	0.138140i	$-0.0441125\pi$
$998$	− 4.16272i	− 0.131769i
$999$	6.82991	0.216089

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2325.2.c.m.1024.4		6
5.2	odd	4		2325.2.a.t.1.2	✓	3
5.3	odd	4		2325.2.a.u.1.2	yes	3
5.4	even	2	inner	2325.2.c.m.1024.3		6
15.2	even	4		6975.2.a.bd.1.2		3
15.8	even	4		6975.2.a.bc.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2325.2.a.t.1.2	✓	3	5.2	odd	4
2325.2.a.u.1.2	yes	3	5.3	odd	4
2325.2.c.m.1024.3		6	5.4	even	2	inner
2325.2.c.m.1024.4		6	1.1	even	1	trivial
6975.2.a.bc.1.2		3	15.8	even	4
6975.2.a.bd.1.2		3	15.2	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

Level:	\( N \)	\(=\)	\( 2325 = 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2325.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.539189i q^{2} -1.00000i q^{3} +1.70928 q^{4} +0.539189 q^{6} -3.70928i q^{7} +2.00000i q^{8} -1.00000 q^{9} +1.87936 q^{11} -1.70928i q^{12} -6.58864i q^{13} +2.00000 q^{14} +2.34017 q^{16} +3.09171i q^{17} -0.539189i q^{18} +7.34017 q^{19} -3.70928 q^{21} +1.01333i q^{22} +7.29791i q^{23} +2.00000 q^{24} +3.55252 q^{26} +1.00000i q^{27} -6.34017i q^{28} -0.170086 q^{29} -1.00000 q^{31} +5.26180i q^{32} -1.87936i q^{33} -1.66701 q^{34} -1.70928 q^{36} -6.82991i q^{37} +3.95774i q^{38} -6.58864 q^{39} -7.95774 q^{41} -2.00000i q^{42} +4.18342i q^{43} +3.21235 q^{44} -3.93495 q^{46} -11.7587i q^{47} -2.34017i q^{48} -6.75872 q^{49} +3.09171 q^{51} -11.2618i q^{52} -13.9288i q^{53} -0.539189 q^{54} +7.41855 q^{56} -7.34017i q^{57} -0.0917087i q^{58} +6.72261 q^{59} +1.84324 q^{61} -0.539189i q^{62} +3.70928i q^{63} +1.84324 q^{64} +1.01333 q^{66} +1.81432i q^{67} +5.28458i q^{68} +7.29791 q^{69} -11.2618 q^{71} -2.00000i q^{72} +2.92162i q^{73} +3.68261 q^{74} +12.5464 q^{76} -6.97107i q^{77} -3.55252i q^{78} -6.24846 q^{79} +1.00000 q^{81} -4.29072i q^{82} +6.21953i q^{83} -6.34017 q^{84} -2.25565 q^{86} +0.170086i q^{87} +3.75872i q^{88} +12.4030 q^{89} -24.4391 q^{91} +12.4741i q^{92} +1.00000i q^{93} +6.34017 q^{94} +5.26180 q^{96} -3.94441i q^{97} -3.64423i q^{98} -1.87936 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{4} - 6 q^{9} - 14 q^{11} + 12 q^{14} - 8 q^{16} + 22 q^{19} - 8 q^{21} + 12 q^{24} + 20 q^{26} + 10 q^{29} - 6 q^{31} + 36 q^{34} + 4 q^{36} - 16 q^{41} + 40 q^{44} - 32 q^{46} + 10 q^{49} + 14 q^{51}+ \cdots + 14 q^{99}+O(q^{100}) \) 6 * q - 4 * q^4 - 6 * q^9 - 14 * q^11 + 12 * q^14 - 8 * q^16 + 22 * q^19 - 8 * q^21 + 12 * q^24 + 20 * q^26 + 10 * q^29 - 6 * q^31 + 36 * q^34 + 4 * q^36 - 16 * q^41 + 40 * q^44 - 32 * q^46 + 10 * q^49 + 14 * q^51 + 16 * q^56 + 28 * q^59 + 24 * q^61 + 24 * q^64 + 8 * q^66 - 10 * q^69 - 52 * q^71 + 4 * q^74 + 4 * q^76 - 20 * q^79 + 6 * q^81 - 16 * q^84 + 72 * q^86 + 18 * q^89 - 52 * q^91 + 16 * q^94 + 16 * q^96 + 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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