LMFDB - Embedded newform 2325.2.c.r.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 = [12,0,0,-14,0,2,0,0,-12,0,14,0,0,-20] 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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 19x^{10} + 127x^{8} + 357x^{6} + 412x^{4} + 204x^{2} + 36 $ x^12 + 19x^10 + 127x^8 + 357x^6 + 412x^4 + 204*x^2 + 36
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1024.4
Root			$-0.864597i$ of defining polynomial
Character	$\chi$	$=$	2325.1024
Dual form			2325.2.c.r.1024.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.864597i q^{2} +1.00000i q^{3} +1.25247 q^{4} +0.864597 q^{6} +4.28308i q^{7} -2.81208i q^{8} -1.00000 q^{9} +0.353929 q^{11} +1.25247i q^{12} +3.89854i q^{13} +3.70313 q^{14} +0.0736318 q^{16} +5.22766i q^{17} +0.864597i q^{18} -8.23515 q^{19} -4.28308 q^{21} -0.306006i q^{22} -1.28131i q^{23} +2.81208 q^{24} +3.37067 q^{26} -1.00000i q^{27} +5.36443i q^{28} -7.27223 q^{29} -1.00000 q^{31} -5.68782i q^{32} +0.353929i q^{33} +4.51982 q^{34} -1.25247 q^{36} -2.30601i q^{37} +7.12009i q^{38} -3.89854 q^{39} -2.68303 q^{41} +3.70313i q^{42} +6.32275i q^{43} +0.443286 q^{44} -1.10782 q^{46} -6.68146i q^{47} +0.0736318i q^{48} -11.3447 q^{49} -5.22766 q^{51} +4.88282i q^{52} -4.33661i q^{53} -0.864597 q^{54} +12.0443 q^{56} -8.23515i q^{57} +6.28755i q^{58} -1.04616 q^{59} -0.983032 q^{61} +0.864597i q^{62} -4.28308i q^{63} -4.77040 q^{64} +0.306006 q^{66} -4.92592i q^{67} +6.54751i q^{68} +1.28131 q^{69} +10.7048 q^{71} +2.81208i q^{72} +4.56262i q^{73} -1.99377 q^{74} -10.3143 q^{76} +1.51590i q^{77} +3.37067i q^{78} +8.87953 q^{79} +1.00000 q^{81} +2.31974i q^{82} +2.37726i q^{83} -5.36443 q^{84} +5.46663 q^{86} -7.27223i q^{87} -0.995275i q^{88} -0.170731 q^{89} -16.6978 q^{91} -1.60481i q^{92} -1.00000i q^{93} -5.77676 q^{94} +5.68782 q^{96} +1.01227i q^{97} +9.80862i q^{98} -0.353929 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 14 q^{4} + 2 q^{6} - 12 q^{9} + 14 q^{11} - 20 q^{14} + 34 q^{16} - 34 q^{19} + 4 q^{21} + 6 q^{24} + 4 q^{26} + 16 q^{29} - 12 q^{31} + 26 q^{34} + 14 q^{36} - 8 q^{39} + 36 q^{41} - 8 q^{44} - 20 q^{46}+ \cdots - 14 q^{99}+O(q^{100}) $ 12 * q - 14 * q^4 + 2 * q^6 - 12 * q^9 + 14 * q^11 - 20 * q^14 + 34 * q^16 - 34 * q^19 + 4 * q^21 + 6 * q^24 + 4 * q^26 + 16 * q^29 - 12 * q^31 + 26 * q^34 + 14 * q^36 - 8 * q^39 + 36 * q^41 - 8 * q^44 - 20 * q^46 - 36 * q^49 - 2 * q^54 + 64 * q^56 - 64 * q^59 - 58 * q^64 + 4 * q^66 + 2 * q^69 + 2 * q^71 - 40 * q^74 + 84 * q^76 - 26 * q^79 + 12 * q^81 - 44 * q^84 - 22 * q^86 - 68 * q^89 - 4 * q^91 - 38 * q^94 - 70 * 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 $

$2$	− 0.864597i	− 0.611362i	−0.952134	−	0.305681i	$-0.901116\pi$
$2$	− 0.864597i	− 0.611362i	0.952134	−	0.305681i	$-0.0988842\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	1.25247	0.626236
$5$	0	0
$5$	0	0
$6$	0.864597	0.352970
$7$	4.28308i	1.61885i	0.587223	+	0.809425i	$0.300221\pi$
$7$	4.28308i	1.61885i	−0.587223	+	0.809425i	$0.699779\pi$
$8$	− 2.81208i	− 0.994219i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0.353929	0.106714	0.0533568	−	0.998576i	$-0.483008\pi$
$11$	0.353929	0.106714	0.0533568	+	0.998576i	$0.483008\pi$
$12$	1.25247i	0.361558i
$13$	3.89854i	1.08126i	0.841260	+	0.540631i	$0.181814\pi$
$13$	3.89854i	1.08126i	−0.841260	+	0.540631i	$0.818186\pi$
$14$	3.70313	0.989704
$15$	0	0
$16$	0.0736318	0.0184079
$17$	5.22766i	1.26789i	0.773376	+	0.633947i	$0.218566\pi$
$17$	5.22766i	1.26789i	−0.773376	+	0.633947i	$0.781434\pi$
$18$	0.864597i	0.203787i
$19$	−8.23515	−1.88927	−0.944637	−	0.328118i	$-0.893586\pi$
$19$	−8.23515	−1.88927	−0.944637	+	0.328118i	$0.893586\pi$
$20$	0	0
$21$	−4.28308	−0.934644
$22$	− 0.306006i	− 0.0652407i
$23$	− 1.28131i	− 0.267172i	−0.991037	−	0.133586i	$-0.957351\pi$
$23$	− 1.28131i	− 0.267172i	0.991037	−	0.133586i	$-0.0426492\pi$
$24$	2.81208	0.574013
$25$	0	0
$26$	3.37067	0.661042
$27$	− 1.00000i	− 0.192450i
$28$	5.36443i	1.01378i
$29$	−7.27223	−1.35042	−0.675209	−	0.737626i	$-0.735947\pi$
$29$	−7.27223	−1.35042	−0.675209	+	0.737626i	$0.735947\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	− 5.68782i	− 1.00547i
$33$	0.353929i	0.0616111i
$34$	4.51982	0.775143
$35$	0	0
$36$	−1.25247	−0.208745
$37$	− 2.30601i	− 0.379105i	−0.981871	−	0.189553i	$-0.939296\pi$
$37$	− 2.30601i	− 0.379105i	0.981871	−	0.189553i	$-0.0607037\pi$
$38$	7.12009i	1.15503i
$39$	−3.89854	−0.624267
$40$	0	0
$41$	−2.68303	−0.419020	−0.209510	−	0.977807i	$-0.567187\pi$
$41$	−2.68303	−0.419020	−0.209510	+	0.977807i	$0.567187\pi$
$42$	3.70313i	0.571406i
$43$	6.32275i	0.964210i	0.876114	+	0.482105i	$0.160128\pi$
$43$	6.32275i	0.964210i	−0.876114	+	0.482105i	$0.839872\pi$
$44$	0.443286	0.0668279
$45$	0	0
$46$	−1.10782	−0.163339
$47$	− 6.68146i	− 0.974590i	−0.873237	−	0.487295i	$-0.837984\pi$
$47$	− 6.68146i	− 0.974590i	0.873237	−	0.487295i	$-0.162016\pi$
$48$	0.0736318i	0.0106278i
$49$	−11.3447	−1.62068
$50$	0	0
$51$	−5.22766	−0.732019
$52$	4.88282i	0.677125i
$53$	− 4.33661i	− 0.595679i	−0.954616	−	0.297840i	$-0.903734\pi$
$53$	− 4.33661i	− 0.595679i	0.954616	−	0.297840i	$-0.0962660\pi$
$54$	−0.864597	−0.117657
$55$	0	0
$56$	12.0443	1.60949
$57$	− 8.23515i	− 1.09077i
$58$	6.28755i	0.825595i
$59$	−1.04616	−0.136198	−0.0680991	−	0.997679i	$-0.521693\pi$
$59$	−1.04616	−0.136198	−0.0680991	+	0.997679i	$0.521693\pi$
$60$	0	0
$61$	−0.983032	−0.125864	−0.0629321	−	0.998018i	$-0.520045\pi$
$61$	−0.983032	−0.125864	−0.0629321	+	0.998018i	$0.520045\pi$
$62$	0.864597i	0.109804i
$63$	− 4.28308i	− 0.539617i
$64$	−4.77040	−0.596301
$65$	0	0
$66$	0.306006	0.0376667
$67$	− 4.92592i	− 0.601796i	−0.953656	−	0.300898i	$-0.902714\pi$
$67$	− 4.92592i	− 0.601796i	0.953656	−	0.300898i	$-0.0972864\pi$
$68$	6.54751i	0.794002i
$69$	1.28131	0.154252
$70$	0	0
$71$	10.7048	1.27042	0.635212	−	0.772338i	$-0.280913\pi$
$71$	10.7048	1.27042	0.635212	+	0.772338i	$0.280913\pi$
$72$	2.81208i	0.331406i
$73$	4.56262i	0.534015i	0.963695	+	0.267007i	$0.0860348\pi$
$73$	4.56262i	0.534015i	−0.963695	+	0.267007i	$0.913965\pi$
$74$	−1.99377	−0.231771
$75$	0	0
$76$	−10.3143	−1.18313
$77$	1.51590i	0.172753i
$78$	3.37067i	0.381653i
$79$	8.87953	0.999025	0.499513	−	0.866307i	$-0.333512\pi$
$79$	8.87953	0.999025	0.499513	+	0.866307i	$0.333512\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.31974i	0.256173i
$83$	2.37726i	0.260938i	0.991452	+	0.130469i	$0.0416483\pi$
$83$	2.37726i	0.260938i	−0.991452	+	0.130469i	$0.958352\pi$
$84$	−5.36443	−0.585308
$85$	0	0
$86$	5.46663	0.589481
$87$	− 7.27223i	− 0.779665i
$88$	− 0.995275i	− 0.106097i
$89$	−0.170731	−0.0180974	−0.00904871	−	0.999959i	$-0.502880\pi$
$89$	−0.170731	−0.0180974	−0.00904871	+	0.999959i	$0.502880\pi$
$90$	0	0
$91$	−16.6978	−1.75040
$92$	− 1.60481i	− 0.167313i
$93$	− 1.00000i	− 0.103695i
$94$	−5.77676	−0.595828
$95$	0	0
$96$	5.68782	0.580510
$97$	1.01227i	0.102780i	0.998679	+	0.0513902i	$0.0163652\pi$
$97$	1.01227i	0.102780i	−0.998679	+	0.0513902i	$0.983635\pi$
$98$	9.80862i	0.990820i
$99$	−0.353929	−0.0355712
$100$	0	0
$101$	17.9045	1.78157	0.890783	−	0.454428i	$-0.150156\pi$
$101$	17.9045	1.78157	0.890783	+	0.454428i	$0.150156\pi$
$102$	4.51982i	0.447529i
$103$	8.49535i	0.837072i	0.908200	+	0.418536i	$0.137457\pi$
$103$	8.49535i	0.837072i	−0.908200	+	0.418536i	$0.862543\pi$
$104$	10.9630	1.07501
$105$	0	0
$106$	−3.74942	−0.364176
$107$	4.50652i	0.435662i	0.975986	+	0.217831i	$0.0698982\pi$
$107$	4.50652i	0.435662i	−0.975986	+	0.217831i	$0.930102\pi$
$108$	− 1.25247i	− 0.120519i
$109$	−14.3051	−1.37018	−0.685088	−	0.728460i	$-0.740236\pi$
$109$	−14.3051	−1.37018	−0.685088	+	0.728460i	$0.740236\pi$
$110$	0	0
$111$	2.30601	0.218876
$112$	0.315370i	0.0297997i
$113$	13.4229i	1.26272i	0.775489	+	0.631360i	$0.217503\pi$
$113$	13.4229i	1.26272i	−0.775489	+	0.631360i	$0.782497\pi$
$114$	−7.12009	−0.666857
$115$	0	0
$116$	−9.10827	−0.845681
$117$	− 3.89854i	− 0.360420i
$118$	0.904506i	0.0832665i
$119$	−22.3905	−2.05253
$120$	0	0
$121$	−10.8747	−0.988612
$122$	0.849926i	0.0769487i
$123$	− 2.68303i	− 0.241921i
$124$	−1.25247	−0.112475
$125$	0	0
$126$	−3.70313	−0.329901
$127$	13.7611i	1.22110i	0.791977	+	0.610550i	$0.209052\pi$
$127$	13.7611i	1.22110i	−0.791977	+	0.610550i	$0.790948\pi$
$128$	− 7.25116i	− 0.640918i
$129$	−6.32275	−0.556687
$130$	0	0
$131$	6.94358	0.606663	0.303332	−	0.952885i	$-0.401901\pi$
$131$	6.94358	0.606663	0.303332	+	0.952885i	$0.401901\pi$
$132$	0.443286i	0.0385831i
$133$	− 35.2718i	− 3.05845i
$134$	−4.25893	−0.367916
$135$	0	0
$136$	14.7006	1.26057
$137$	7.86949i	0.672336i	0.941802	+	0.336168i	$0.109131\pi$
$137$	7.86949i	0.672336i	−0.941802	+	0.336168i	$0.890869\pi$
$138$	− 1.10782i	− 0.0943037i
$139$	−7.69265	−0.652482	−0.326241	−	0.945287i	$-0.605782\pi$
$139$	−7.69265	−0.652482	−0.326241	+	0.945287i	$0.605782\pi$
$140$	0	0
$141$	6.68146	0.562680
$142$	− 9.25532i	− 0.776689i
$143$	1.37981i	0.115385i
$144$	−0.0736318	−0.00613598
$145$	0	0
$146$	3.94483	0.326476
$147$	− 11.3447i	− 0.935698i
$148$	− 2.88821i	− 0.237409i
$149$	12.6799	1.03878	0.519388	−	0.854539i	$-0.326160\pi$
$149$	12.6799	1.03878	0.519388	+	0.854539i	$0.326160\pi$
$150$	0	0
$151$	−2.94960	−0.240035	−0.120017	−	0.992772i	$-0.538295\pi$
$151$	−2.94960	−0.240035	−0.120017	+	0.992772i	$0.538295\pi$
$152$	23.1579i	1.87835i
$153$	− 5.22766i	− 0.422632i
$154$	1.31065	0.105615
$155$	0	0
$156$	−4.88282	−0.390938
$157$	3.49659i	0.279058i	0.990218	+	0.139529i	$0.0445589\pi$
$157$	3.49659i	0.279058i	−0.990218	+	0.139529i	$0.955441\pi$
$158$	− 7.67721i	− 0.610766i
$159$	4.33661	0.343915
$160$	0	0
$161$	5.48795	0.432511
$162$	− 0.864597i	− 0.0679291i
$163$	− 4.07999i	− 0.319570i	−0.987152	−	0.159785i	$-0.948920\pi$
$163$	− 4.07999i	− 0.319570i	0.987152	−	0.159785i	$-0.0510800\pi$
$164$	−3.36043	−0.262405
$165$	0	0
$166$	2.05537	0.159528
$167$	11.2784i	0.872747i	0.899766	+	0.436373i	$0.143737\pi$
$167$	11.2784i	0.872747i	−0.899766	+	0.436373i	$0.856263\pi$
$168$	12.0443i	0.929241i
$169$	−2.19864	−0.169126
$170$	0	0
$171$	8.23515	0.629758
$172$	7.91906i	0.603823i
$173$	− 22.4690i	− 1.70829i	−0.520036	−	0.854145i	$-0.674081\pi$
$173$	− 22.4690i	− 1.70829i	0.520036	−	0.854145i	$-0.325919\pi$
$174$	−6.28755	−0.476658
$175$	0	0
$176$	0.0260604	0.00196438
$177$	− 1.04616i	− 0.0786341i
$178$	0.147613i	0.0110641i
$179$	−6.71387	−0.501818	−0.250909	−	0.968011i	$-0.580729\pi$
$179$	−6.71387	−0.501818	−0.250909	+	0.968011i	$0.580729\pi$
$180$	0	0
$181$	26.4167	1.96354	0.981769	−	0.190076i	$-0.0608733\pi$
$181$	26.4167	1.96354	0.981769	+	0.190076i	$0.0608733\pi$
$182$	14.4368i	1.07013i
$183$	− 0.983032i	− 0.0726678i
$184$	−3.60315	−0.265627
$185$	0	0
$186$	−0.864597	−0.0633953
$187$	1.85022i	0.135302i
$188$	− 8.36834i	− 0.610324i
$189$	4.28308	0.311548
$190$	0	0
$191$	−19.9355	−1.44248	−0.721240	−	0.692685i	$-0.756427\pi$
$191$	−19.9355	−1.44248	−0.721240	+	0.692685i	$0.756427\pi$
$192$	− 4.77040i	− 0.344274i
$193$	3.98672i	0.286970i	0.989652	+	0.143485i	$0.0458309\pi$
$193$	3.98672i	0.286970i	−0.989652	+	0.143485i	$0.954169\pi$
$194$	0.875204	0.0628360
$195$	0	0
$196$	−14.2090	−1.01493
$197$	23.1920i	1.65236i	0.563407	+	0.826179i	$0.309490\pi$
$197$	23.1920i	1.65236i	−0.563407	+	0.826179i	$0.690510\pi$
$198$	0.306006i	0.0217469i
$199$	12.5109	0.886876	0.443438	−	0.896305i	$-0.353759\pi$
$199$	12.5109	0.886876	0.443438	+	0.896305i	$0.353759\pi$
$200$	0	0
$201$	4.92592	0.347447
$202$	− 15.4802i	− 1.08918i
$203$	− 31.1475i	− 2.18613i
$204$	−6.54751	−0.458417
$205$	0	0
$206$	7.34505	0.511754
$207$	1.28131i	0.0890573i
$208$	0.287057i	0.0199038i
$209$	−2.91466	−0.201611
$210$	0	0
$211$	23.1316	1.59245	0.796224	−	0.605002i	$-0.206828\pi$
$211$	23.1316	1.59245	0.796224	+	0.605002i	$0.206828\pi$
$212$	− 5.43148i	− 0.373036i
$213$	10.7048i	0.733480i
$214$	3.89633	0.266347
$215$	0	0
$216$	−2.81208	−0.191338
$217$	− 4.28308i	− 0.290754i
$218$	12.3681i	0.837674i
$219$	−4.56262	−0.308313
$220$	0	0
$221$	−20.3803	−1.37093
$222$	− 1.99377i	− 0.133813i
$223$	23.6450i	1.58339i	0.610920	+	0.791693i	$0.290800\pi$
$223$	23.6450i	1.58339i	−0.610920	+	0.791693i	$0.709200\pi$
$224$	24.3613	1.62771
$225$	0	0
$226$	11.6054	0.771980
$227$	26.4506i	1.75559i	0.479039	+	0.877794i	$0.340985\pi$
$227$	26.4506i	1.75559i	−0.479039	+	0.877794i	$0.659015\pi$
$228$	− 10.3143i	− 0.683081i
$229$	7.61479	0.503199	0.251600	−	0.967831i	$-0.419043\pi$
$229$	7.61479	0.503199	0.251600	+	0.967831i	$0.419043\pi$
$230$	0	0
$231$	−1.51590	−0.0997392
$232$	20.4501i	1.34261i
$233$	3.13938i	0.205668i	0.994699	+	0.102834i	$0.0327910\pi$
$233$	3.13938i	0.205668i	−0.994699	+	0.102834i	$0.967209\pi$
$234$	−3.37067	−0.220347
$235$	0	0
$236$	−1.31028	−0.0852923
$237$	8.87953i	0.576788i
$238$	19.3587i	1.25484i
$239$	−26.0939	−1.68787	−0.843936	−	0.536444i	$-0.819767\pi$
$239$	−26.0939	−1.68787	−0.843936	+	0.536444i	$0.819767\pi$
$240$	0	0
$241$	−5.22027	−0.336267	−0.168134	−	0.985764i	$-0.553774\pi$
$241$	−5.22027	−0.336267	−0.168134	+	0.985764i	$0.553774\pi$
$242$	9.40226i	0.604400i
$243$	1.00000i	0.0641500i
$244$	−1.23122	−0.0788208
$245$	0	0
$246$	−2.31974	−0.147901
$247$	− 32.1051i	− 2.04280i
$248$	2.81208i	0.178567i
$249$	−2.37726	−0.150653
$250$	0	0
$251$	29.8930	1.88683	0.943414	−	0.331617i	$-0.107594\pi$
$251$	29.8930	1.88683	0.943414	+	0.331617i	$0.107594\pi$
$252$	− 5.36443i	− 0.337928i
$253$	− 0.453493i	− 0.0285109i
$254$	11.8978	0.746535
$255$	0	0
$256$	−15.8101	−0.988133
$257$	15.6602i	0.976855i	0.872604	+	0.488427i	$0.162429\pi$
$257$	15.6602i	0.976855i	−0.872604	+	0.488427i	$0.837571\pi$
$258$	5.46663i	0.340337i
$259$	9.87680	0.613714
$260$	0	0
$261$	7.27223	0.450140
$262$	− 6.00339i	− 0.370891i
$263$	26.1422i	1.61200i	0.591918	+	0.805998i	$0.298371\pi$
$263$	26.1422i	1.61200i	−0.591918	+	0.805998i	$0.701629\pi$
$264$	0.995275	0.0612550
$265$	0	0
$266$	−30.4959	−1.86982
$267$	− 0.170731i	− 0.0104486i
$268$	− 6.16957i	− 0.376867i
$269$	32.0456	1.95386	0.976928	−	0.213567i	$-0.0685083\pi$
$269$	32.0456	1.95386	0.976928	+	0.213567i	$0.0685083\pi$
$270$	0	0
$271$	−17.3240	−1.05236	−0.526179	−	0.850374i	$-0.676376\pi$
$271$	−17.3240	−1.05236	−0.526179	+	0.850374i	$0.676376\pi$
$272$	0.384922i	0.0233393i
$273$	− 16.6978i	− 1.01059i
$274$	6.80394	0.411041
$275$	0	0
$276$	1.60481	0.0965980
$277$	− 16.6490i	− 1.00034i	−0.865926	−	0.500172i	$-0.833270\pi$
$277$	− 16.6490i	− 1.00034i	0.865926	−	0.500172i	$-0.166730\pi$
$278$	6.65104i	0.398903i
$279$	1.00000	0.0598684
$280$	0	0
$281$	13.1488	0.784392	0.392196	−	0.919882i	$-0.371716\pi$
$281$	13.1488	0.784392	0.392196	+	0.919882i	$0.371716\pi$
$282$	− 5.77676i	− 0.344001i
$283$	25.9728i	1.54392i	0.635669	+	0.771962i	$0.280724\pi$
$283$	25.9728i	1.54392i	−0.635669	+	0.771962i	$0.719276\pi$
$284$	13.4074	0.795586
$285$	0	0
$286$	1.19298	0.0705422
$287$	− 11.4916i	− 0.678330i
$288$	5.68782i	0.335158i
$289$	−10.3285	−0.607558
$290$	0	0
$291$	−1.01227	−0.0593403
$292$	5.71456i	0.334419i
$293$	27.0981i	1.58309i	0.611114	+	0.791543i	$0.290722\pi$
$293$	27.0981i	1.58309i	−0.611114	+	0.791543i	$0.709278\pi$
$294$	−9.80862	−0.572050
$295$	0	0
$296$	−6.48467	−0.376914
$297$	− 0.353929i	− 0.0205370i
$298$	− 10.9630i	− 0.635068i
$299$	4.99525	0.288883
$300$	0	0
$301$	−27.0808	−1.56091
$302$	2.55021i	0.146748i
$303$	17.9045i	1.02859i
$304$	−0.606369	−0.0347776
$305$	0	0
$306$	−4.51982	−0.258381
$307$	7.83898i	0.447394i	0.974659	+	0.223697i	$0.0718126\pi$
$307$	7.83898i	0.447394i	−0.974659	+	0.223697i	$0.928187\pi$
$308$	1.89863i	0.108184i
$309$	−8.49535	−0.483284
$310$	0	0
$311$	0.0311685	0.00176740	0.000883702	−	1.00000i	$-0.499719\pi$
$311$	0.0311685	0.00176740	0.000883702		1.00000i	$0.499719\pi$
$312$	10.9630i	0.620658i
$313$	− 2.44289i	− 0.138080i	−0.997614	−	0.0690401i	$-0.978006\pi$
$313$	− 2.44289i	− 0.138080i	0.997614	−	0.0690401i	$-0.0219936\pi$
$314$	3.02314	0.170606
$315$	0	0
$316$	11.1214	0.625626
$317$	− 4.03351i	− 0.226544i	−0.993564	−	0.113272i	$-0.963867\pi$
$317$	− 4.03351i	− 0.226544i	0.993564	−	0.113272i	$-0.0361332\pi$
$318$	− 3.74942i	− 0.210257i
$319$	−2.57385	−0.144108
$320$	0	0
$321$	−4.50652	−0.251530
$322$	− 4.74487i	− 0.264421i
$323$	− 43.0506i	− 2.39540i
$324$	1.25247	0.0695818
$325$	0	0
$326$	−3.52755	−0.195373
$327$	− 14.3051i	− 0.791072i
$328$	7.54490i	0.416597i
$329$	28.6172	1.57772
$330$	0	0
$331$	−11.0802	−0.609025	−0.304512	−	0.952508i	$-0.598493\pi$
$331$	−11.0802	−0.609025	−0.304512	+	0.952508i	$0.598493\pi$
$332$	2.97745i	0.163409i
$333$	2.30601i	0.126368i
$334$	9.75125	0.533565
$335$	0	0
$336$	−0.315370	−0.0172049
$337$	− 10.4054i	− 0.566817i	−0.958999	−	0.283409i	$-0.908535\pi$
$337$	− 10.4054i	− 0.566817i	0.958999	−	0.283409i	$-0.0914653\pi$
$338$	1.90094i	0.103397i
$339$	−13.4229	−0.729032
$340$	0	0
$341$	−0.353929	−0.0191663
$342$	− 7.12009i	− 0.385010i
$343$	− 18.6088i	− 1.00478i
$344$	17.7800	0.958636
$345$	0	0
$346$	−19.4267	−1.04438
$347$	− 23.7995i	− 1.27762i	−0.769363	−	0.638812i	$-0.779426\pi$
$347$	− 23.7995i	− 1.27762i	0.769363	−	0.638812i	$-0.220574\pi$
$348$	− 9.10827i	− 0.488254i
$349$	26.1849	1.40165	0.700824	−	0.713335i	$-0.252816\pi$
$349$	26.1849	1.40165	0.700824	+	0.713335i	$0.252816\pi$
$350$	0	0
$351$	3.89854	0.208089
$352$	− 2.01308i	− 0.107298i
$353$	− 14.3318i	− 0.762807i	−0.924409	−	0.381404i	$-0.875441\pi$
$353$	− 14.3318i	− 0.762807i	0.924409	−	0.381404i	$-0.124559\pi$
$354$	−0.904506	−0.0480739
$355$	0	0
$356$	−0.213836	−0.0113333
$357$	− 22.3905i	− 1.18503i
$358$	5.80479i	0.306793i
$359$	−29.2468	−1.54359	−0.771793	−	0.635874i	$-0.780640\pi$
$359$	−29.2468	−1.54359	−0.771793	+	0.635874i	$0.780640\pi$
$360$	0	0
$361$	48.8177	2.56935
$362$	− 22.8398i	− 1.20043i
$363$	− 10.8747i	− 0.570776i
$364$	−20.9135	−1.09616
$365$	0	0
$366$	−0.849926	−0.0444263
$367$	− 21.6111i	− 1.12809i	−0.825743	−	0.564046i	$-0.809244\pi$
$367$	− 21.6111i	− 1.12809i	0.825743	−	0.564046i	$-0.190756\pi$
$368$	− 0.0943452i	− 0.00491808i
$369$	2.68303	0.139673
$370$	0	0
$371$	18.5740	0.964315
$372$	− 1.25247i	− 0.0649377i
$373$	− 21.6784i	− 1.12247i	−0.827658	−	0.561233i	$-0.810327\pi$
$373$	− 21.6784i	− 1.12247i	0.827658	−	0.561233i	$-0.189673\pi$
$374$	1.59970	0.0827183
$375$	0	0
$376$	−18.7888	−0.968957
$377$	− 28.3511i	− 1.46016i
$378$	− 3.70313i	− 0.190469i
$379$	−35.3697	−1.81682	−0.908408	−	0.418084i	$-0.862702\pi$
$379$	−35.3697	−1.81682	−0.908408	+	0.418084i	$0.862702\pi$
$380$	0	0
$381$	−13.7611	−0.705003
$382$	17.2361i	0.881878i
$383$	− 30.1843i	− 1.54235i	−0.636626	−	0.771173i	$-0.719670\pi$
$383$	− 30.1843i	− 1.54235i	0.636626	−	0.771173i	$-0.280330\pi$
$384$	7.25116	0.370034
$385$	0	0
$386$	3.44690	0.175443
$387$	− 6.32275i	− 0.321403i
$388$	1.26784i	0.0643648i
$389$	8.07568	0.409453	0.204727	−	0.978819i	$-0.434370\pi$
$389$	8.07568	0.409453	0.204727	+	0.978819i	$0.434370\pi$
$390$	0	0
$391$	6.69826	0.338746
$392$	31.9023i	1.61131i
$393$	6.94358i	0.350257i
$394$	20.0517	1.01019
$395$	0	0
$396$	−0.443286	−0.0222760
$397$	34.0991i	1.71139i	0.517483	+	0.855693i	$0.326869\pi$
$397$	34.0991i	1.71139i	−0.517483	+	0.855693i	$0.673131\pi$
$398$	− 10.8169i	− 0.542202i
$399$	35.2718	1.76580
$400$	0	0
$401$	35.4194	1.76876	0.884380	−	0.466768i	$-0.154582\pi$
$401$	35.4194	1.76876	0.884380	+	0.466768i	$0.154582\pi$
$402$	− 4.25893i	− 0.212416i
$403$	− 3.89854i	− 0.194200i
$404$	22.4249	1.11568
$405$	0	0
$406$	−26.9300	−1.33652
$407$	− 0.816162i	− 0.0404557i
$408$	14.7006i	0.727788i
$409$	4.67021	0.230927	0.115463	−	0.993312i	$-0.463165\pi$
$409$	4.67021	0.230927	0.115463	+	0.993312i	$0.463165\pi$
$410$	0	0
$411$	−7.86949	−0.388173
$412$	10.6402i	0.524205i
$413$	− 4.48078i	− 0.220485i
$414$	1.10782	0.0544463
$415$	0	0
$416$	22.1742	1.08718
$417$	− 7.69265i	− 0.376711i
$418$	2.52000i	0.123257i
$419$	−11.4046	−0.557150	−0.278575	−	0.960414i	$-0.589862\pi$
$419$	−11.4046	−0.557150	−0.278575	+	0.960414i	$0.589862\pi$
$420$	0	0
$421$	6.33585	0.308791	0.154395	−	0.988009i	$-0.450657\pi$
$421$	6.33585	0.308791	0.154395	+	0.988009i	$0.450657\pi$
$422$	− 19.9995i	− 0.973563i
$423$	6.68146i	0.324863i
$424$	−12.1949	−0.592236
$425$	0	0
$426$	9.25532	0.448422
$427$	− 4.21040i	− 0.203755i
$428$	5.64430i	0.272827i
$429$	−1.37981	−0.0666177
$430$	0	0
$431$	−2.01105	−0.0968691	−0.0484346	−	0.998826i	$-0.515423\pi$
$431$	−2.01105	−0.0968691	−0.0484346	+	0.998826i	$0.515423\pi$
$432$	− 0.0736318i	− 0.00354261i
$433$	37.7917i	1.81615i	0.418803	+	0.908077i	$0.362450\pi$
$433$	37.7917i	1.81615i	−0.418803	+	0.908077i	$0.637550\pi$
$434$	−3.70313	−0.177756
$435$	0	0
$436$	−17.9167	−0.858054
$437$	10.5518i	0.504761i
$438$	3.94483i	0.188491i
$439$	−14.3895	−0.686771	−0.343386	−	0.939194i	$-0.611574\pi$
$439$	−14.3895	−0.686771	−0.343386	+	0.939194i	$0.611574\pi$
$440$	0	0
$441$	11.3447	0.540225
$442$	17.6207i	0.838132i
$443$	− 35.8785i	− 1.70464i	−0.523021	−	0.852320i	$-0.675195\pi$
$443$	− 35.8785i	− 1.70464i	0.523021	−	0.852320i	$-0.324805\pi$
$444$	2.88821	0.137068
$445$	0	0
$446$	20.4434	0.968022
$447$	12.6799i	0.599738i
$448$	− 20.4320i	− 0.965321i
$449$	35.4975	1.67523	0.837615	−	0.546261i	$-0.183949\pi$
$449$	35.4975	1.67523	0.837615	+	0.546261i	$0.183949\pi$
$450$	0	0
$451$	−0.949604	−0.0447151
$452$	16.8118i	0.790762i
$453$	− 2.94960i	− 0.138584i
$454$	22.8691	1.07330
$455$	0	0
$456$	−23.1579	−1.08447
$457$	− 33.8742i	− 1.58457i	−0.610153	−	0.792283i	$-0.708892\pi$
$457$	− 33.8742i	− 1.58457i	0.610153	−	0.792283i	$-0.291108\pi$
$458$	− 6.58372i	− 0.307637i
$459$	5.22766	0.244006
$460$	0	0
$461$	−19.2748	−0.897717	−0.448859	−	0.893603i	$-0.648169\pi$
$461$	−19.2748	−0.897717	−0.448859	+	0.893603i	$0.648169\pi$
$462$	1.31065i	0.0609768i
$463$	1.98874i	0.0924244i	0.998932	+	0.0462122i	$0.0147150\pi$
$463$	1.98874i	0.0924244i	−0.998932	+	0.0462122i	$0.985285\pi$
$464$	−0.535467	−0.0248584
$465$	0	0
$466$	2.71430	0.125737
$467$	− 3.18274i	− 0.147279i	−0.997285	−	0.0736397i	$-0.976539\pi$
$467$	− 3.18274i	− 0.147279i	0.997285	−	0.0736397i	$-0.0234615\pi$
$468$	− 4.88282i	− 0.225708i
$469$	21.0981	0.974218
$470$	0	0
$471$	−3.49659	−0.161114
$472$	2.94188i	0.135411i
$473$	2.23780i	0.102894i
$474$	7.67721	0.352626
$475$	0	0
$476$	−28.0435	−1.28537
$477$	4.33661i	0.198560i
$478$	22.5607i	1.03190i
$479$	−9.28348	−0.424173	−0.212086	−	0.977251i	$-0.568026\pi$
$479$	−9.28348	−0.424173	−0.212086	+	0.977251i	$0.568026\pi$
$480$	0	0
$481$	8.99006	0.409912
$482$	4.51343i	0.205581i
$483$	5.48795i	0.249710i
$484$	−13.6203	−0.619105
$485$	0	0
$486$	0.864597	0.0392189
$487$	8.58292i	0.388929i	0.980910	+	0.194465i	$0.0622969\pi$
$487$	8.58292i	0.388929i	−0.980910	+	0.194465i	$0.937703\pi$
$488$	2.76436i	0.125137i
$489$	4.07999	0.184504
$490$	0	0
$491$	−34.6474	−1.56362	−0.781808	−	0.623519i	$-0.785702\pi$
$491$	−34.6474	−1.56362	−0.781808	+	0.623519i	$0.785702\pi$
$492$	− 3.36043i	− 0.151500i
$493$	− 38.0168i	− 1.71219i
$494$	−27.7580	−1.24889
$495$	0	0
$496$	−0.0736318	−0.00330616
$497$	45.8494i	2.05663i
$498$	2.05537i	0.0921033i
$499$	11.2295	0.502703	0.251352	−	0.967896i	$-0.419125\pi$
$499$	11.2295	0.502703	0.251352	+	0.967896i	$0.419125\pi$
$500$	0	0
$501$	−11.2784	−0.503881
$502$	− 25.8454i	− 1.15354i
$503$	− 12.4460i	− 0.554941i	−0.960734	−	0.277470i	$-0.910504\pi$
$503$	− 12.4460i	− 0.554941i	0.960734	−	0.277470i	$-0.0894960\pi$
$504$	−12.0443	−0.536497
$505$	0	0
$506$	−0.392089	−0.0174305
$507$	− 2.19864i	− 0.0976451i
$508$	17.2354i	0.764697i
$509$	26.3376	1.16740	0.583698	−	0.811971i	$-0.301605\pi$
$509$	26.3376	1.16740	0.583698	+	0.811971i	$0.301605\pi$
$510$	0	0
$511$	−19.5421	−0.864490
$512$	− 0.832921i	− 0.0368102i
$513$	8.23515i	0.363591i
$514$	13.5397	0.597212
$515$	0	0
$516$	−7.91906	−0.348617
$517$	− 2.36476i	− 0.104002i
$518$	− 8.53945i	− 0.375202i
$519$	22.4690	0.986281
$520$	0	0
$521$	−8.41110	−0.368497	−0.184248	−	0.982880i	$-0.558985\pi$
$521$	−8.41110	−0.368497	−0.184248	+	0.982880i	$0.558985\pi$
$522$	− 6.28755i	− 0.275198i
$523$	8.89885i	0.389120i	0.980891	+	0.194560i	$0.0623278\pi$
$523$	8.89885i	0.389120i	−0.980891	+	0.194560i	$0.937672\pi$
$524$	8.69664	0.379914
$525$	0	0
$526$	22.6025	0.985514
$527$	− 5.22766i	− 0.227721i
$528$	0.0260604i	0.00113413i
$529$	21.3582	0.928619
$530$	0	0
$531$	1.04616	0.0453994
$532$	− 44.1769i	− 1.91531i
$533$	− 10.4599i	− 0.453070i
$534$	−0.147613	−0.00638785
$535$	0	0
$536$	−13.8521	−0.598318
$537$	− 6.71387i	− 0.289725i
$538$	− 27.7066i	− 1.19451i
$539$	−4.01523	−0.172948
$540$	0	0
$541$	−27.0487	−1.16291	−0.581457	−	0.813577i	$-0.697517\pi$
$541$	−27.0487	−1.16291	−0.581457	+	0.813577i	$0.697517\pi$
$542$	14.9783i	0.643372i
$543$	26.4167i	1.13365i
$544$	29.7340	1.27483
$545$	0	0
$546$	−14.4368	−0.617839
$547$	− 20.4371i	− 0.873829i	−0.899503	−	0.436914i	$-0.856071\pi$
$547$	− 20.4371i	− 0.873829i	0.899503	−	0.436914i	$-0.143929\pi$
$548$	9.85632i	0.421041i
$549$	0.983032	0.0419548
$550$	0	0
$551$	59.8879	2.55131
$552$	− 3.60315i	− 0.153360i
$553$	38.0317i	1.61727i
$554$	−14.3947	−0.611573
$555$	0	0
$556$	−9.63484	−0.408608
$557$	− 33.5070i	− 1.41974i	−0.704333	−	0.709869i	$-0.748754\pi$
$557$	− 33.5070i	− 1.41974i	0.704333	−	0.709869i	$-0.251246\pi$
$558$	− 0.864597i	− 0.0366013i
$559$	−24.6495	−1.04256
$560$	0	0
$561$	−1.85022	−0.0781164
$562$	− 11.3684i	− 0.479547i
$563$	− 1.03505i	− 0.0436223i	−0.999762	−	0.0218112i	$-0.993057\pi$
$563$	− 1.03505i	− 0.0436223i	0.999762	−	0.0218112i	$-0.00694326\pi$
$564$	8.36834	0.352371
$565$	0	0
$566$	22.4560	0.943897
$567$	4.28308i	0.179872i
$568$	− 30.1027i	− 1.26308i
$569$	−2.63277	−0.110371	−0.0551857	−	0.998476i	$-0.517575\pi$
$569$	−2.63277	−0.110371	−0.0551857	+	0.998476i	$0.517575\pi$
$570$	0	0
$571$	26.6263	1.11428	0.557138	−	0.830420i	$-0.311899\pi$
$571$	26.6263	1.11428	0.557138	+	0.830420i	$0.311899\pi$
$572$	1.72817i	0.0722584i
$573$	− 19.9355i	− 0.832816i
$574$	−9.93564	−0.414705
$575$	0	0
$576$	4.77040	0.198767
$577$	− 20.4641i	− 0.851933i	−0.904739	−	0.425967i	$-0.859934\pi$
$577$	− 20.4641i	− 0.851933i	0.904739	−	0.425967i	$-0.140066\pi$
$578$	8.92997i	0.371438i
$579$	−3.98672	−0.165682
$580$	0	0
$581$	−10.1820	−0.422420
$582$	0.875204i	0.0362784i
$583$	− 1.53485i	− 0.0635670i
$584$	12.8304	0.530928
$585$	0	0
$586$	23.4289	0.967839
$587$	− 21.2696i	− 0.877888i	−0.898514	−	0.438944i	$-0.855353\pi$
$587$	− 21.2696i	− 0.877888i	0.898514	−	0.438944i	$-0.144647\pi$
$588$	− 14.2090i	− 0.585968i
$589$	8.23515	0.339324
$590$	0	0
$591$	−23.1920	−0.953990
$592$	− 0.169795i	− 0.00697854i
$593$	− 39.3701i	− 1.61674i	−0.588676	−	0.808369i	$-0.700351\pi$
$593$	− 39.3701i	− 1.61674i	0.588676	−	0.808369i	$-0.299649\pi$
$594$	−0.306006	−0.0125556
$595$	0	0
$596$	15.8812	0.650519
$597$	12.5109i	0.512038i
$598$	− 4.31887i	− 0.176612i
$599$	−28.4292	−1.16158	−0.580792	−	0.814052i	$-0.697257\pi$
$599$	−28.4292	−1.16158	−0.580792	+	0.814052i	$0.697257\pi$
$600$	0	0
$601$	20.6046	0.840480	0.420240	−	0.907413i	$-0.361946\pi$
$601$	20.6046	0.840480	0.420240	+	0.907413i	$0.361946\pi$
$602$	23.4140i	0.954282i
$603$	4.92592i	0.200599i
$604$	−3.69429	−0.150318
$605$	0	0
$606$	15.4802	0.628840
$607$	18.8232i	0.764010i	0.924160	+	0.382005i	$0.124766\pi$
$607$	18.8232i	0.764010i	−0.924160	+	0.382005i	$0.875234\pi$
$608$	46.8400i	1.89961i
$609$	31.1475	1.26216
$610$	0	0
$611$	26.0479	1.05379
$612$	− 6.54751i	− 0.264667i
$613$	− 18.0049i	− 0.727213i	−0.931553	−	0.363606i	$-0.881545\pi$
$613$	− 18.0049i	− 0.727213i	0.931553	−	0.363606i	$-0.118455\pi$
$614$	6.77755	0.273520
$615$	0	0
$616$	4.26284	0.171755
$617$	− 26.2389i	− 1.05634i	−0.849139	−	0.528169i	$-0.822879\pi$
$617$	− 26.2389i	− 1.05634i	0.849139	−	0.528169i	$-0.177121\pi$
$618$	7.34505i	0.295461i
$619$	−30.1624	−1.21233	−0.606166	−	0.795339i	$-0.707293\pi$
$619$	−30.1624	−1.21233	−0.606166	+	0.795339i	$0.707293\pi$
$620$	0	0
$621$	−1.28131	−0.0514172
$622$	− 0.0269482i	− 0.00108052i
$623$	− 0.731253i	− 0.0292970i
$624$	−0.287057	−0.0114915
$625$	0	0
$626$	−2.11211	−0.0844170
$627$	− 2.91466i	− 0.116400i
$628$	4.37938i	0.174756i
$629$	12.0550	0.480665
$630$	0	0
$631$	30.7023	1.22224	0.611121	−	0.791538i	$-0.290719\pi$
$631$	30.7023	1.22224	0.611121	+	0.791538i	$0.290719\pi$
$632$	− 24.9699i	− 0.993250i
$633$	23.1316i	0.919400i
$634$	−3.48736	−0.138501
$635$	0	0
$636$	5.43148	0.215372
$637$	− 44.2279i	− 1.75237i
$638$	2.22534i	0.0881022i
$639$	−10.7048	−0.423475
$640$	0	0
$641$	4.11497	0.162531	0.0812657	−	0.996692i	$-0.474104\pi$
$641$	4.11497	0.162531	0.0812657	+	0.996692i	$0.474104\pi$
$642$	3.89633i	0.153776i
$643$	− 24.2235i	− 0.955281i	−0.878555	−	0.477641i	$-0.841492\pi$
$643$	− 24.2235i	− 0.955281i	0.878555	−	0.477641i	$-0.158508\pi$
$644$	6.87351	0.270854
$645$	0	0
$646$	−37.2214	−1.46446
$647$	− 32.4635i	− 1.27627i	−0.769924	−	0.638136i	$-0.779706\pi$
$647$	− 32.4635i	− 1.27627i	0.769924	−	0.638136i	$-0.220294\pi$
$648$	− 2.81208i	− 0.110469i
$649$	−0.370266	−0.0145342
$650$	0	0
$651$	4.28308	0.167867
$652$	− 5.11008i	− 0.200126i
$653$	− 17.3165i	− 0.677648i	−0.940850	−	0.338824i	$-0.889971\pi$
$653$	− 17.3165i	− 0.677648i	0.940850	−	0.338824i	$-0.110029\pi$
$654$	−12.3681	−0.483631
$655$	0	0
$656$	−0.197557	−0.00771329
$657$	− 4.56262i	− 0.178005i
$658$	− 24.7423i	− 0.964556i
$659$	−10.3947	−0.404922	−0.202461	−	0.979290i	$-0.564894\pi$
$659$	−10.3947	−0.404922	−0.202461	+	0.979290i	$0.564894\pi$
$660$	0	0
$661$	35.1133	1.36575	0.682874	−	0.730536i	$-0.260730\pi$
$661$	35.1133	1.36575	0.682874	+	0.730536i	$0.260730\pi$
$662$	9.57994i	0.372335i
$663$	− 20.3803i	− 0.791504i
$664$	6.68503	0.259430
$665$	0	0
$666$	1.99377	0.0772568
$667$	9.31799i	0.360794i
$668$	14.1259i	0.546546i
$669$	−23.6450	−0.914168
$670$	0	0
$671$	−0.347923	−0.0134314
$672$	24.3613i	0.939759i
$673$	16.3883i	0.631724i	0.948805	+	0.315862i	$0.102294\pi$
$673$	16.3883i	0.631724i	−0.948805	+	0.315862i	$0.897706\pi$
$674$	−8.99646	−0.346531
$675$	0	0
$676$	−2.75374	−0.105913
$677$	11.3973i	0.438036i	0.975721	+	0.219018i	$0.0702853\pi$
$677$	11.3973i	0.438036i	−0.975721	+	0.219018i	$0.929715\pi$
$678$	11.6054i	0.445703i
$679$	−4.33562	−0.166386
$680$	0	0
$681$	−26.4506	−1.01359
$682$	0.306006i	0.0117176i
$683$	− 33.7148i	− 1.29006i	−0.764157	−	0.645031i	$-0.776845\pi$
$683$	− 33.7148i	− 1.29006i	0.764157	−	0.645031i	$-0.223155\pi$
$684$	10.3143	0.394377
$685$	0	0
$686$	−16.0891	−0.614286
$687$	7.61479i	0.290522i
$688$	0.465555i	0.0177491i
$689$	16.9065	0.644085
$690$	0	0
$691$	−26.4611	−1.00663	−0.503313	−	0.864104i	$-0.667886\pi$
$691$	−26.4611	−1.00663	−0.503313	+	0.864104i	$0.667886\pi$
$692$	− 28.1418i	− 1.06979i
$693$	− 1.51590i	− 0.0575844i
$694$	−20.5770	−0.781091
$695$	0	0
$696$	−20.4501	−0.775158
$697$	− 14.0260i	− 0.531273i
$698$	− 22.6394i	− 0.856914i
$699$	−3.13938	−0.118742
$700$	0	0
$701$	−48.7391	−1.84085	−0.920425	−	0.390919i	$-0.872157\pi$
$701$	−48.7391	−1.84085	−0.920425	+	0.390919i	$0.872157\pi$
$702$	− 3.37067i	− 0.127218i
$703$	18.9903i	0.716233i
$704$	−1.68838	−0.0636334
$705$	0	0
$706$	−12.3913	−0.466352
$707$	76.6864i	2.88409i
$708$	− 1.31028i	− 0.0492435i
$709$	−13.1850	−0.495172	−0.247586	−	0.968866i	$-0.579637\pi$
$709$	−13.1850	−0.495172	−0.247586	+	0.968866i	$0.579637\pi$
$710$	0	0
$711$	−8.87953	−0.333008
$712$	0.480108i	0.0179928i
$713$	1.28131i	0.0479855i
$714$	−19.3587	−0.724483
$715$	0	0
$716$	−8.40893	−0.314257
$717$	− 26.0939i	− 0.974493i
$718$	25.2867i	0.943690i
$719$	12.3742	0.461481	0.230740	−	0.973015i	$-0.425885\pi$
$719$	12.3742	0.461481	0.230740	+	0.973015i	$0.425885\pi$
$720$	0	0
$721$	−36.3862	−1.35509
$722$	− 42.2077i	− 1.57081i
$723$	− 5.22027i	− 0.194144i
$724$	33.0862	1.22964
$725$	0	0
$726$	−9.40226	−0.348951
$727$	18.0273i	0.668597i	0.942467	+	0.334299i	$0.108499\pi$
$727$	18.0273i	0.668597i	−0.942467	+	0.334299i	$0.891501\pi$
$728$	46.9554i	1.74028i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−33.0532	−1.22252
$732$	− 1.23122i	− 0.0455072i
$733$	16.6357i	0.614453i	0.951636	+	0.307226i	$0.0994009\pi$
$733$	16.6357i	0.614453i	−0.951636	+	0.307226i	$0.900599\pi$
$734$	−18.6849	−0.689673
$735$	0	0
$736$	−7.28786	−0.268634
$737$	− 1.74342i	− 0.0642198i
$738$	− 2.31974i	− 0.0853909i
$739$	3.71551	0.136677	0.0683385	−	0.997662i	$-0.478230\pi$
$739$	3.71551	0.136677	0.0683385	+	0.997662i	$0.478230\pi$
$740$	0	0
$741$	32.1051	1.17941
$742$	− 16.0590i	− 0.589546i
$743$	18.8360i	0.691028i	0.938414	+	0.345514i	$0.112295\pi$
$743$	18.8360i	0.691028i	−0.938414	+	0.345514i	$0.887705\pi$
$744$	−2.81208	−0.103096
$745$	0	0
$746$	−18.7431	−0.686233
$747$	− 2.37726i	− 0.0869793i
$748$	2.31735i	0.0847308i
$749$	−19.3018	−0.705272
$750$	0	0
$751$	−12.6253	−0.460703	−0.230351	−	0.973108i	$-0.573988\pi$
$751$	−12.6253	−0.460703	−0.230351	+	0.973108i	$0.573988\pi$
$752$	− 0.491967i	− 0.0179402i
$753$	29.8930i	1.08936i
$754$	−24.5123	−0.892684
$755$	0	0
$756$	5.36443	0.195103
$757$	52.2392i	1.89866i	0.314274	+	0.949332i	$0.398239\pi$
$757$	52.2392i	1.89866i	−0.314274	+	0.949332i	$0.601761\pi$
$758$	30.5805i	1.11073i
$759$	0.453493	0.0164608
$760$	0	0
$761$	1.78070	0.0645502	0.0322751	−	0.999479i	$-0.489725\pi$
$761$	1.78070	0.0645502	0.0322751	+	0.999479i	$0.489725\pi$
$762$	11.8978i	0.431012i
$763$	− 61.2697i	− 2.21811i
$764$	−24.9686	−0.903333
$765$	0	0
$766$	−26.0972	−0.942932
$767$	− 4.07850i	− 0.147266i
$768$	− 15.8101i	− 0.570499i
$769$	−32.5936	−1.17536	−0.587678	−	0.809095i	$-0.699958\pi$
$769$	−32.5936	−1.17536	−0.587678	+	0.809095i	$0.699958\pi$
$770$	0	0
$771$	−15.6602	−0.563987
$772$	4.99325i	0.179711i
$773$	− 23.1130i	− 0.831317i	−0.909521	−	0.415659i	$-0.863551\pi$
$773$	− 23.1130i	− 0.831317i	0.909521	−	0.415659i	$-0.136449\pi$
$774$	−5.46663	−0.196494
$775$	0	0
$776$	2.84658	0.102186
$777$	9.87680i	0.354328i
$778$	− 6.98220i	− 0.250324i
$779$	22.0952	0.791643
$780$	0	0
$781$	3.78873	0.135572
$782$	− 5.79130i	− 0.207096i
$783$	7.27223i	0.259888i
$784$	−0.835333	−0.0298333
$785$	0	0
$786$	6.00339	0.214134
$787$	29.9984i	1.06933i	0.845065	+	0.534664i	$0.179562\pi$
$787$	29.9984i	1.06933i	−0.845065	+	0.534664i	$0.820438\pi$
$788$	29.0473i	1.03477i
$789$	−26.1422	−0.930687
$790$	0	0
$791$	−57.4913	−2.04416
$792$	0.995275i	0.0353656i
$793$	− 3.83239i	− 0.136092i
$794$	29.4820	1.04628
$795$	0	0
$796$	15.6696	0.555394
$797$	− 18.4489i	− 0.653493i	−0.945112	−	0.326747i	$-0.894048\pi$
$797$	− 18.4489i	− 0.653493i	0.945112	−	0.326747i	$-0.105952\pi$
$798$	− 30.4959i	− 1.07954i
$799$	34.9284	1.23568
$800$	0	0
$801$	0.170731	0.00603247
$802$	− 30.6235i	− 1.08135i
$803$	1.61484i	0.0569866i
$804$	6.16957	0.217584
$805$	0	0
$806$	−3.37067	−0.118727
$807$	32.0456i	1.12806i
$808$	− 50.3489i	− 1.77127i
$809$	38.1306	1.34060	0.670301	−	0.742089i	$-0.266165\pi$
$809$	38.1306	1.34060	0.670301	+	0.742089i	$0.266165\pi$
$810$	0	0
$811$	−3.36292	−0.118088	−0.0590440	−	0.998255i	$-0.518805\pi$
$811$	−3.36292	−0.118088	−0.0590440	+	0.998255i	$0.518805\pi$
$812$	− 39.0114i	− 1.36903i
$813$	− 17.3240i	− 0.607579i
$814$	−0.705651	−0.0247331
$815$	0	0
$816$	−0.384922	−0.0134750
$817$	− 52.0688i	− 1.82166i
$818$	− 4.03785i	− 0.141180i
$819$	16.6978	0.583467
$820$	0	0
$821$	−6.17661	−0.215565	−0.107783	−	0.994174i	$-0.534375\pi$
$821$	−6.17661	−0.215565	−0.107783	+	0.994174i	$0.534375\pi$
$822$	6.80394i	0.237315i
$823$	− 7.45159i	− 0.259746i	−0.991531	−	0.129873i	$-0.958543\pi$
$823$	− 7.45159i	− 0.259746i	0.991531	−	0.129873i	$-0.0414570\pi$
$824$	23.8896	0.832233
$825$	0	0
$826$	−3.87407	−0.134796
$827$	0.0604717i	0.00210281i	0.999999	+	0.00105140i	$0.000334672\pi$
$827$	0.0604717i	0.00210281i	−0.999999	+	0.00105140i	$0.999665\pi$
$828$	1.60481i	0.0557709i
$829$	−13.5457	−0.470463	−0.235231	−	0.971939i	$-0.575585\pi$
$829$	−13.5457	−0.470463	−0.235231	+	0.971939i	$0.575585\pi$
$830$	0	0
$831$	16.6490	0.577549
$832$	− 18.5976i	− 0.644757i
$833$	− 59.3065i	− 2.05485i
$834$	−6.65104	−0.230307
$835$	0	0
$836$	−3.65053	−0.126256
$837$	1.00000i	0.0345651i
$838$	9.86036i	0.340621i
$839$	16.1362	0.557085	0.278542	−	0.960424i	$-0.410149\pi$
$839$	16.1362	0.557085	0.278542	+	0.960424i	$0.410149\pi$
$840$	0	0
$841$	23.8853	0.823631
$842$	− 5.47796i	− 0.188783i
$843$	13.1488i	0.452869i
$844$	28.9718	0.997249
$845$	0	0
$846$	5.77676	0.198609
$847$	− 46.5773i	− 1.60042i
$848$	− 0.319312i	− 0.0109652i
$849$	−25.9728	−0.891385
$850$	0	0
$851$	−2.95471	−0.101286
$852$	13.4074i	0.459332i
$853$	3.88888i	0.133153i	0.997781	+	0.0665763i	$0.0212076\pi$
$853$	3.88888i	0.133153i	−0.997781	+	0.0665763i	$0.978792\pi$
$854$	−3.64030	−0.124568
$855$	0	0
$856$	12.6727	0.433144
$857$	30.7605i	1.05076i	0.850869	+	0.525379i	$0.176076\pi$
$857$	30.7605i	1.05076i	−0.850869	+	0.525379i	$0.823924\pi$
$858$	1.19298i	0.0407276i
$859$	−18.8972	−0.644765	−0.322382	−	0.946610i	$-0.604484\pi$
$859$	−18.8972	−0.644765	−0.322382	+	0.946610i	$0.604484\pi$
$860$	0	0
$861$	11.4916	0.391634
$862$	1.73875i	0.0592221i
$863$	46.3032i	1.57618i	0.615561	+	0.788089i	$0.288929\pi$
$863$	46.3032i	1.57618i	−0.615561	+	0.788089i	$0.711071\pi$
$864$	−5.68782	−0.193503
$865$	0	0
$866$	32.6746	1.11033
$867$	− 10.3285i	− 0.350774i
$868$	− 5.36443i	− 0.182081i
$869$	3.14272	0.106610
$870$	0	0
$871$	19.2039	0.650699
$872$	40.2269i	1.36226i
$873$	− 1.01227i	− 0.0342601i
$874$	9.12304	0.308592
$875$	0	0
$876$	−5.71456	−0.193077
$877$	− 14.7762i	− 0.498957i	−0.968380	−	0.249479i	$-0.919741\pi$
$877$	− 14.7762i	− 0.498957i	0.968380	−	0.249479i	$-0.0802592\pi$
$878$	12.4411i	0.419866i
$879$	−27.0981	−0.913995
$880$	0	0
$881$	−2.84895	−0.0959835	−0.0479917	−	0.998848i	$-0.515282\pi$
$881$	−2.84895	−0.0959835	−0.0479917	+	0.998848i	$0.515282\pi$
$882$	− 9.80862i	− 0.330273i
$883$	15.8536i	0.533516i	0.963764	+	0.266758i	$0.0859524\pi$
$883$	15.8536i	0.533516i	−0.963764	+	0.266758i	$0.914048\pi$
$884$	−25.5257	−0.858523
$885$	0	0
$886$	−31.0205	−1.04215
$887$	− 5.21365i	− 0.175057i	−0.996162	−	0.0875286i	$-0.972103\pi$
$887$	− 5.21365i	− 0.175057i	0.996162	−	0.0875286i	$-0.0278969\pi$
$888$	− 6.48467i	− 0.217611i
$889$	−58.9399	−1.97678
$890$	0	0
$891$	0.353929	0.0118571
$892$	29.6147i	0.991573i
$893$	55.0228i	1.84127i
$894$	10.9630	0.366657
$895$	0	0
$896$	31.0572	1.03755
$897$	4.99525i	0.166786i
$898$	− 30.6910i	− 1.02417i
$899$	7.27223	0.242542
$900$	0	0
$901$	22.6703	0.755258
$902$	0.821024i	0.0273371i
$903$	− 27.0808i	− 0.901192i
$904$	37.7463	1.25542
$905$	0	0
$906$	−2.55021	−0.0847251
$907$	13.8569i	0.460111i	0.973178	+	0.230055i	$0.0738907\pi$
$907$	13.8569i	0.460111i	−0.973178	+	0.230055i	$0.926109\pi$
$908$	33.1286i	1.09941i
$909$	−17.9045	−0.593856
$910$	0	0
$911$	−5.44928	−0.180543	−0.0902713	−	0.995917i	$-0.528773\pi$
$911$	−5.44928	−0.180543	−0.0902713	+	0.995917i	$0.528773\pi$
$912$	− 0.606369i	− 0.0200789i
$913$	0.841380i	0.0278456i
$914$	−29.2875	−0.968744
$915$	0	0
$916$	9.53731	0.315122
$917$	29.7399i	0.982097i
$918$	− 4.51982i	− 0.149176i
$919$	36.4516	1.20243	0.601214	−	0.799088i	$-0.294684\pi$
$919$	36.4516	1.20243	0.601214	+	0.799088i	$0.294684\pi$
$920$	0	0
$921$	−7.83898	−0.258303
$922$	16.6649i	0.548830i
$923$	41.7331i	1.37366i
$924$	−1.89863	−0.0624603
$925$	0	0
$926$	1.71945	0.0565048
$927$	− 8.49535i	− 0.279024i
$928$	41.3631i	1.35781i
$929$	7.33460	0.240641	0.120320	−	0.992735i	$-0.461608\pi$
$929$	7.33460	0.240641	0.120320	+	0.992735i	$0.461608\pi$
$930$	0	0
$931$	93.4256	3.06190
$932$	3.93199i	0.128797i
$933$	0.0311685i	0.00102041i
$934$	−2.75178	−0.0900411
$935$	0	0
$936$	−10.9630	−0.358337
$937$	49.1036i	1.60415i	0.597227	+	0.802073i	$0.296269\pi$
$937$	49.1036i	1.60415i	−0.597227	+	0.802073i	$0.703731\pi$
$938$	− 18.2413i	− 0.595600i
$939$	2.44289	0.0797206
$940$	0	0
$941$	−20.2169	−0.659053	−0.329527	−	0.944146i	$-0.606889\pi$
$941$	−20.2169	−0.659053	−0.329527	+	0.944146i	$0.606889\pi$
$942$	3.02314i	0.0984993i
$943$	3.43780i	0.111950i
$944$	−0.0770305	−0.00250713
$945$	0	0
$946$	1.93480	0.0629057
$947$	30.1596i	0.980055i	0.871707	+	0.490028i	$0.163013\pi$
$947$	30.1596i	0.980055i	−0.871707	+	0.490028i	$0.836987\pi$
$948$	11.1214i	0.361205i
$949$	−17.7876	−0.577409
$950$	0	0
$951$	4.03351	0.130795
$952$	62.9638i	2.04067i
$953$	14.6953i	0.476026i	0.971262	+	0.238013i	$0.0764961\pi$
$953$	14.6953i	0.476026i	−0.971262	+	0.238013i	$0.923504\pi$
$954$	3.74942	0.121392
$955$	0	0
$956$	−32.6818	−1.05701
$957$	− 2.57385i	− 0.0832008i
$958$	8.02646i	0.259323i
$959$	−33.7056	−1.08841
$960$	0	0
$961$	1.00000	0.0322581
$962$	− 7.77278i	− 0.250604i
$963$	− 4.50652i	− 0.145221i
$964$	−6.53825	−0.210583
$965$	0	0
$966$	4.74487	0.152664
$967$	50.8330i	1.63468i	0.576155	+	0.817340i	$0.304552\pi$
$967$	50.8330i	1.63468i	−0.576155	+	0.817340i	$0.695448\pi$
$968$	30.5806i	0.982897i
$969$	43.0506	1.38298
$970$	0	0
$971$	36.0990	1.15847	0.579236	−	0.815160i	$-0.303351\pi$
$971$	36.0990	1.15847	0.579236	+	0.815160i	$0.303351\pi$
$972$	1.25247i	0.0401731i
$973$	− 32.9482i	− 1.05627i
$974$	7.42077	0.237777
$975$	0	0
$976$	−0.0723824	−0.00231690
$977$	8.14233i	0.260496i	0.991481	+	0.130248i	$0.0415774\pi$
$977$	8.14233i	0.260496i	−0.991481	+	0.130248i	$0.958423\pi$
$978$	− 3.52755i	− 0.112799i
$979$	−0.0604265	−0.00193124
$980$	0	0
$981$	14.3051	0.456726
$982$	29.9561i	0.955936i
$983$	− 31.3377i	− 0.999517i	−0.866165	−	0.499759i	$-0.833422\pi$
$983$	− 31.3377i	− 0.999517i	0.866165	−	0.499759i	$-0.166578\pi$
$984$	−7.54490	−0.240523
$985$	0	0
$986$	−32.8692	−1.04677
$987$	28.6172i	0.910895i
$988$	− 40.2107i	− 1.27927i
$989$	8.10140	0.257610
$990$	0	0
$991$	−56.4797	−1.79414	−0.897068	−	0.441893i	$-0.854307\pi$
$991$	−56.4797	−1.79414	−0.897068	+	0.441893i	$0.854307\pi$
$992$	5.68782i	0.180588i
$993$	− 11.0802i	− 0.351621i
$994$	39.6412	1.25734
$995$	0	0
$996$	−2.97745	−0.0943441
$997$	− 47.4373i	− 1.50236i	−0.660100	−	0.751178i	$-0.729486\pi$
$997$	− 47.4373i	− 1.50236i	0.660100	−	0.751178i	$-0.270514\pi$
$998$	− 9.70903i	− 0.307334i
$999$	−2.30601	−0.0729588

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2325.2.c.r.1024.4		12
5.2	odd	4		2325.2.a.bb.1.4	yes	6
5.3	odd	4		2325.2.a.y.1.3	✓	6
5.4	even	2	inner	2325.2.c.r.1024.9		12
15.2	even	4		6975.2.a.ca.1.3		6
15.8	even	4		6975.2.a.cc.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2325.2.a.y.1.3	✓	6	5.3	odd	4
2325.2.a.bb.1.4	yes	6	5.2	odd	4
2325.2.c.r.1024.4		12	1.1	even	1	trivial
2325.2.c.r.1024.9		12	5.4	even	2	inner
6975.2.a.ca.1.3		6	15.2	even	4
6975.2.a.cc.1.4		6	15.8	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)

\(f(q)\)	\(=\)	\(q-0.864597i q^{2} +1.00000i q^{3} +1.25247 q^{4} +0.864597 q^{6} +4.28308i q^{7} -2.81208i q^{8} -1.00000 q^{9} +0.353929 q^{11} +1.25247i q^{12} +3.89854i q^{13} +3.70313 q^{14} +0.0736318 q^{16} +5.22766i q^{17} +0.864597i q^{18} -8.23515 q^{19} -4.28308 q^{21} -0.306006i q^{22} -1.28131i q^{23} +2.81208 q^{24} +3.37067 q^{26} -1.00000i q^{27} +5.36443i q^{28} -7.27223 q^{29} -1.00000 q^{31} -5.68782i q^{32} +0.353929i q^{33} +4.51982 q^{34} -1.25247 q^{36} -2.30601i q^{37} +7.12009i q^{38} -3.89854 q^{39} -2.68303 q^{41} +3.70313i q^{42} +6.32275i q^{43} +0.443286 q^{44} -1.10782 q^{46} -6.68146i q^{47} +0.0736318i q^{48} -11.3447 q^{49} -5.22766 q^{51} +4.88282i q^{52} -4.33661i q^{53} -0.864597 q^{54} +12.0443 q^{56} -8.23515i q^{57} +6.28755i q^{58} -1.04616 q^{59} -0.983032 q^{61} +0.864597i q^{62} -4.28308i q^{63} -4.77040 q^{64} +0.306006 q^{66} -4.92592i q^{67} +6.54751i q^{68} +1.28131 q^{69} +10.7048 q^{71} +2.81208i q^{72} +4.56262i q^{73} -1.99377 q^{74} -10.3143 q^{76} +1.51590i q^{77} +3.37067i q^{78} +8.87953 q^{79} +1.00000 q^{81} +2.31974i q^{82} +2.37726i q^{83} -5.36443 q^{84} +5.46663 q^{86} -7.27223i q^{87} -0.995275i q^{88} -0.170731 q^{89} -16.6978 q^{91} -1.60481i q^{92} -1.00000i q^{93} -5.77676 q^{94} +5.68782 q^{96} +1.01227i q^{97} +9.80862i q^{98} -0.353929 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 14 q^{4} + 2 q^{6} - 12 q^{9} + 14 q^{11} - 20 q^{14} + 34 q^{16} - 34 q^{19} + 4 q^{21} + 6 q^{24} + 4 q^{26} + 16 q^{29} - 12 q^{31} + 26 q^{34} + 14 q^{36} - 8 q^{39} + 36 q^{41} - 8 q^{44} - 20 q^{46}+ \cdots - 14 q^{99}+O(q^{100}) \) 12 * q - 14 * q^4 + 2 * q^6 - 12 * q^9 + 14 * q^11 - 20 * q^14 + 34 * q^16 - 34 * q^19 + 4 * q^21 + 6 * q^24 + 4 * q^26 + 16 * q^29 - 12 * q^31 + 26 * q^34 + 14 * q^36 - 8 * q^39 + 36 * q^41 - 8 * q^44 - 20 * q^46 - 36 * q^49 - 2 * q^54 + 64 * q^56 - 64 * q^59 - 58 * q^64 + 4 * q^66 + 2 * q^69 + 2 * q^71 - 40 * q^74 + 84 * q^76 - 26 * q^79 + 12 * q^81 - 44 * q^84 - 22 * q^86 - 68 * q^89 - 4 * q^91 - 38 * q^94 - 70 * q^96 - 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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