LMFDB - Embedded newform 7569.2.a.bt.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7569,2,Mod(1,7569)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7569, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7569.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7569 = 3^{2} \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7569.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$60.4387692899$
Analytic rank:	$1$
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} - 6 x^{11} + 2 x^{10} + 38 x^{9} - 30 x^{8} - 90 x^{7} + 55 x^{6} + 90 x^{5} - 30 x^{4} + \cdots + 1 $ x^12 - 6x^11 + 2x^10 + 38x^9 - 30x^8 - 90x^7 + 55x^6 + 90x^5 - 30x^4 - 38x^3 + 2x^2 + 6*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 87)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.757124$ of defining polynomial
Character	$\chi$	$=$	7569.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.242876 q^{2} -1.94101 q^{4} -0.975857 q^{5} +3.22234 q^{7} -0.957176 q^{8} -0.237012 q^{10} +5.81530 q^{11} -2.71844 q^{13} +0.782628 q^{14} +3.64955 q^{16} -0.0461060 q^{17} -3.13915 q^{19} +1.89415 q^{20} +1.41240 q^{22} -0.317052 q^{23} -4.04770 q^{25} -0.660243 q^{26} -6.25460 q^{28} -7.55971 q^{31} +2.80074 q^{32} -0.0111980 q^{34} -3.14454 q^{35} -3.22197 q^{37} -0.762423 q^{38} +0.934067 q^{40} -10.7403 q^{41} -0.324676 q^{43} -11.2876 q^{44} -0.0770043 q^{46} +8.90113 q^{47} +3.38347 q^{49} -0.983089 q^{50} +5.27652 q^{52} +12.6243 q^{53} -5.67490 q^{55} -3.08435 q^{56} +1.44348 q^{59} -6.78143 q^{61} -1.83607 q^{62} -6.61886 q^{64} +2.65281 q^{65} -6.15519 q^{67} +0.0894923 q^{68} -0.763733 q^{70} -11.2594 q^{71} +5.12818 q^{73} -0.782540 q^{74} +6.09312 q^{76} +18.7389 q^{77} +10.1502 q^{79} -3.56144 q^{80} -2.60855 q^{82} -8.06599 q^{83} +0.0449929 q^{85} -0.0788560 q^{86} -5.56627 q^{88} +3.21940 q^{89} -8.75973 q^{91} +0.615402 q^{92} +2.16187 q^{94} +3.06336 q^{95} -14.2107 q^{97} +0.821763 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 6 q^{2} + 8 q^{4} - 2 q^{5} - 10 q^{7} - 20 q^{10} + 14 q^{11} - 16 q^{13} - 4 q^{16} + 22 q^{17} - 16 q^{19} - 4 q^{20} + 12 q^{22} - 2 q^{23} - 2 q^{25} - 8 q^{26} - 4 q^{28} - 4 q^{31} + 16 q^{32}+ \cdots + 6 q^{98}+O(q^{100}) $ 12 * q + 6 * q^2 + 8 * q^4 - 2 * q^5 - 10 * q^7 - 20 * q^10 + 14 * q^11 - 16 * q^13 - 4 * q^16 + 22 * q^17 - 16 * q^19 - 4 * q^20 + 12 * q^22 - 2 * q^23 - 2 * q^25 - 8 * q^26 - 4 * q^28 - 4 * q^31 + 16 * q^32 - 10 * q^34 + 14 * q^35 - 28 * q^37 - 16 * q^38 - 32 * q^40 - 14 * q^41 - 36 * q^43 + 26 * q^44 - 24 * q^46 + 44 * q^47 - 18 * q^49 + 24 * q^50 - 20 * q^52 + 18 * q^53 - 4 * q^55 - 10 * q^56 - 14 * q^59 - 24 * q^61 - 4 * q^62 - 18 * q^64 - 2 * q^65 - 18 * q^67 + 52 * q^68 + 36 * q^70 - 14 * q^71 - 44 * q^73 - 12 * q^74 - 56 * q^76 + 32 * q^77 - 24 * q^79 - 52 * q^80 + 52 * q^82 - 26 * q^83 - 48 * q^85 - 64 * q^86 - 10 * q^88 + 10 * q^89 - 10 * q^91 - 64 * q^92 - 8 * q^94 - 4 * q^95 - 28 * q^97 + 6 * q^98

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 $

$2$	0.242876	0.171739	0.0858695	−	0.996306i	$-0.472633\pi$
$2$	0.242876	0.171739	0.0858695	+	0.996306i	$0.472633\pi$
$3$	0	0
$3$	0	0
$4$	−1.94101	−0.970506
$5$	−0.975857	−0.436417	−0.218208	−	0.975902i	$-0.570021\pi$
$5$	−0.975857	−0.436417	−0.218208	+	0.975902i	$0.570021\pi$
$6$	0	0
$7$	3.22234	1.21793	0.608965	−	0.793197i	$-0.291585\pi$
$7$	3.22234	1.21793	0.608965	+	0.793197i	$0.291585\pi$
$8$	−0.957176	−0.338413
$9$	0	0
$10$	−0.237012	−0.0749498
$11$	5.81530	1.75338	0.876689	−	0.481057i	$-0.159747\pi$
$11$	5.81530	1.75338	0.876689	+	0.481057i	$0.159747\pi$
$12$	0	0
$13$	−2.71844	−0.753959	−0.376979	−	0.926222i	$-0.623037\pi$
$13$	−2.71844	−0.753959	−0.376979	+	0.926222i	$0.623037\pi$
$14$	0.782628	0.209166
$15$	0	0
$16$	3.64955	0.912387
$17$	−0.0461060	−0.0111823	−0.00559117	−	0.999984i	$-0.501780\pi$
$17$	−0.0461060	−0.0111823	−0.00559117	+	0.999984i	$0.501780\pi$
$18$	0	0
$19$	−3.13915	−0.720170	−0.360085	−	0.932920i	$-0.617252\pi$
$19$	−3.13915	−0.720170	−0.360085	+	0.932920i	$0.617252\pi$
$20$	1.89415	0.423545
$21$	0	0
$22$	1.41240	0.301124
$23$	−0.317052	−0.0661100	−0.0330550	−	0.999454i	$-0.510524\pi$
$23$	−0.317052	−0.0661100	−0.0330550	+	0.999454i	$0.510524\pi$
$24$	0	0
$25$	−4.04770	−0.809540
$26$	−0.660243	−0.129484
$27$	0	0
$28$	−6.25460	−1.18201
$29$	0	0
$29$	0	0
$30$	0	0
$31$	−7.55971	−1.35776	−0.678882	−	0.734247i	$-0.737535\pi$
$31$	−7.55971	−1.35776	−0.678882	+	0.734247i	$0.737535\pi$
$32$	2.80074	0.495105
$33$	0	0
$34$	−0.0111980	−0.00192045
$35$	−3.14454	−0.531525
$36$	0	0
$37$	−3.22197	−0.529689	−0.264845	−	0.964291i	$-0.585321\pi$
$37$	−3.22197	−0.529689	−0.264845	+	0.964291i	$0.585321\pi$
$38$	−0.762423	−0.123681
$39$	0	0
$40$	0.934067	0.147689
$41$	−10.7403	−1.67735	−0.838673	−	0.544635i	$-0.816668\pi$
$41$	−10.7403	−1.67735	−0.838673	+	0.544635i	$0.816668\pi$
$42$	0	0
$43$	−0.324676	−0.0495127	−0.0247563	−	0.999694i	$-0.507881\pi$
$43$	−0.324676	−0.0495127	−0.0247563	+	0.999694i	$0.507881\pi$
$44$	−11.2876	−1.70166
$45$	0	0
$46$	−0.0770043	−0.0113537
$47$	8.90113	1.29836	0.649182	−	0.760633i	$-0.275111\pi$
$47$	8.90113	1.29836	0.649182	+	0.760633i	$0.275111\pi$
$48$	0	0
$49$	3.38347	0.483353
$50$	−0.983089	−0.139030
$51$	0	0
$52$	5.27652	0.731721
$53$	12.6243	1.73408	0.867041	−	0.498237i	$-0.166019\pi$
$53$	12.6243	1.73408	0.867041	+	0.498237i	$0.166019\pi$
$54$	0	0
$55$	−5.67490	−0.765204
$56$	−3.08435	−0.412163
$57$	0	0
$58$	0	0
$59$	1.44348	0.187925	0.0939624	−	0.995576i	$-0.470047\pi$
$59$	1.44348	0.187925	0.0939624	+	0.995576i	$0.470047\pi$
$60$	0	0
$61$	−6.78143	−0.868273	−0.434136	−	0.900847i	$-0.642946\pi$
$61$	−6.78143	−0.868273	−0.434136	+	0.900847i	$0.642946\pi$
$62$	−1.83607	−0.233181
$63$	0	0
$64$	−6.61886	−0.827358
$65$	2.65281	0.329040
$66$	0	0
$67$	−6.15519	−0.751976	−0.375988	−	0.926624i	$-0.622697\pi$
$67$	−6.15519	−0.751976	−0.375988	+	0.926624i	$0.622697\pi$
$68$	0.0894923	0.0108525
$69$	0	0
$70$	−0.763733	−0.0912836
$71$	−11.2594	−1.33625	−0.668123	−	0.744051i	$-0.732902\pi$
$71$	−11.2594	−1.33625	−0.668123	+	0.744051i	$0.732902\pi$
$72$	0	0
$73$	5.12818	0.600208	0.300104	−	0.953906i	$-0.402979\pi$
$73$	5.12818	0.600208	0.300104	+	0.953906i	$0.402979\pi$
$74$	−0.782540	−0.0909684
$75$	0	0
$76$	6.09312	0.698929
$77$	18.7389	2.13549
$78$	0	0
$79$	10.1502	1.14199	0.570993	−	0.820955i	$-0.306558\pi$
$79$	10.1502	1.14199	0.570993	+	0.820955i	$0.306558\pi$
$80$	−3.56144	−0.398181
$81$	0	0
$82$	−2.60855	−0.288066
$83$	−8.06599	−0.885358	−0.442679	−	0.896680i	$-0.645972\pi$
$83$	−8.06599	−0.885358	−0.442679	+	0.896680i	$0.645972\pi$
$84$	0	0
$85$	0.0449929	0.00488016
$86$	−0.0788560	−0.00850327
$87$	0	0
$88$	−5.56627	−0.593366
$89$	3.21940	0.341255	0.170628	−	0.985336i	$-0.445420\pi$
$89$	3.21940	0.341255	0.170628	+	0.985336i	$0.445420\pi$
$90$	0	0
$91$	−8.75973	−0.918269
$92$	0.615402	0.0641601
$93$	0	0
$94$	2.16187	0.222980
$95$	3.06336	0.314294
$96$	0	0
$97$	−14.2107	−1.44288	−0.721440	−	0.692477i	$-0.756519\pi$
$97$	−14.2107	−1.44288	−0.721440	+	0.692477i	$0.756519\pi$
$98$	0.821763	0.0830106
$99$	0	0
$100$	7.85664	0.785664
$101$	−2.48892	−0.247657	−0.123828	−	0.992304i	$-0.539517\pi$
$101$	−2.48892	−0.247657	−0.123828	+	0.992304i	$0.539517\pi$
$102$	0	0
$103$	14.2892	1.40795	0.703977	−	0.710223i	$-0.251406\pi$
$103$	14.2892	1.40795	0.703977	+	0.710223i	$0.251406\pi$
$104$	2.60202	0.255149
$105$	0	0
$106$	3.06614	0.297810
$107$	−14.5407	−1.40570	−0.702851	−	0.711337i	$-0.748090\pi$
$107$	−14.5407	−1.40570	−0.702851	+	0.711337i	$0.748090\pi$
$108$	0	0
$109$	−3.85519	−0.369260	−0.184630	−	0.982808i	$-0.559109\pi$
$109$	−3.85519	−0.369260	−0.184630	+	0.982808i	$0.559109\pi$
$110$	−1.37830	−0.131415
$111$	0	0
$112$	11.7601	1.11122
$113$	8.41550	0.791664	0.395832	−	0.918323i	$-0.370456\pi$
$113$	8.41550	0.791664	0.395832	+	0.918323i	$0.370456\pi$
$114$	0	0
$115$	0.309398	0.0288515
$116$	0	0
$117$	0	0
$118$	0.350586	0.0322740
$119$	−0.148569	−0.0136193
$120$	0	0
$121$	22.8177	2.07434
$122$	−1.64704	−0.149116
$123$	0	0
$124$	14.6735	1.31772
$125$	8.82927	0.789714
$126$	0	0
$127$	−2.19393	−0.194680	−0.0973400	−	0.995251i	$-0.531033\pi$
$127$	−2.19393	−0.194680	−0.0973400	+	0.995251i	$0.531033\pi$
$128$	−7.20904	−0.637195
$129$	0	0
$130$	0.644303	0.0565091
$131$	18.3171	1.60037	0.800184	−	0.599754i	$-0.204735\pi$
$131$	18.3171	1.60037	0.800184	+	0.599754i	$0.204735\pi$
$132$	0	0
$133$	−10.1154	−0.877116
$134$	−1.49495	−0.129144
$135$	0	0
$136$	0.0441316	0.00378425
$137$	0.362343	0.0309571	0.0154785	−	0.999880i	$-0.495073\pi$
$137$	0.362343	0.0309571	0.0154785	+	0.999880i	$0.495073\pi$
$138$	0	0
$139$	9.39836	0.797158	0.398579	−	0.917134i	$-0.369503\pi$
$139$	9.39836	0.797158	0.398579	+	0.917134i	$0.369503\pi$
$140$	6.10359	0.515848
$141$	0	0
$142$	−2.73464	−0.229486
$143$	−15.8085	−1.32198
$144$	0	0
$145$	0	0
$146$	1.24551	0.103079
$147$	0	0
$148$	6.25389	0.514067
$149$	2.89876	0.237476	0.118738	−	0.992926i	$-0.462115\pi$
$149$	2.89876	0.237476	0.118738	+	0.992926i	$0.462115\pi$
$150$	0	0
$151$	−10.8109	−0.879780	−0.439890	−	0.898052i	$-0.644982\pi$
$151$	−10.8109	−0.879780	−0.439890	+	0.898052i	$0.644982\pi$
$152$	3.00472	0.243715
$153$	0	0
$154$	4.55122	0.366747
$155$	7.37720	0.592551
$156$	0	0
$157$	−20.7432	−1.65549	−0.827743	−	0.561107i	$-0.810375\pi$
$157$	−20.7432	−1.65549	−0.827743	+	0.561107i	$0.810375\pi$
$158$	2.46523	0.196124
$159$	0	0
$160$	−2.73312	−0.216072
$161$	−1.02165	−0.0805173
$162$	0	0
$163$	−24.2502	−1.89942	−0.949712	−	0.313125i	$-0.898624\pi$
$163$	−24.2502	−1.89942	−0.949712	+	0.313125i	$0.898624\pi$
$164$	20.8470	1.62787
$165$	0	0
$166$	−1.95903	−0.152051
$167$	5.34975	0.413976	0.206988	−	0.978343i	$-0.433634\pi$
$167$	5.34975	0.413976	0.206988	+	0.978343i	$0.433634\pi$
$168$	0	0
$169$	−5.61010	−0.431546
$170$	0.0109277	0.000838115 0
$171$	0	0
$172$	0.630201	0.0480524
$173$	−9.02908	−0.686469	−0.343234	−	0.939250i	$-0.611522\pi$
$173$	−9.02908	−0.686469	−0.343234	+	0.939250i	$0.611522\pi$
$174$	0	0
$175$	−13.0431	−0.985963
$176$	21.2232	1.59976
$177$	0	0
$178$	0.781914	0.0586069
$179$	23.8270	1.78092	0.890459	−	0.455064i	$-0.150384\pi$
$179$	23.8270	1.78092	0.890459	+	0.455064i	$0.150384\pi$
$180$	0	0
$181$	−18.1286	−1.34749	−0.673743	−	0.738966i	$-0.735314\pi$
$181$	−18.1286	−1.34749	−0.673743	+	0.738966i	$0.735314\pi$
$182$	−2.12753	−0.157703
$183$	0	0
$184$	0.303475	0.0223725
$185$	3.14419	0.231165
$186$	0	0
$187$	−0.268120	−0.0196069
$188$	−17.2772	−1.26007
$189$	0	0
$190$	0.744016	0.0539766
$191$	11.5710	0.837249	0.418625	−	0.908159i	$-0.362512\pi$
$191$	11.5710	0.837249	0.418625	+	0.908159i	$0.362512\pi$
$192$	0	0
$193$	1.34897	0.0971013	0.0485507	−	0.998821i	$-0.484540\pi$
$193$	1.34897	0.0971013	0.0485507	+	0.998821i	$0.484540\pi$
$194$	−3.45144	−0.247799
$195$	0	0
$196$	−6.56736	−0.469097
$197$	−18.8288	−1.34150	−0.670748	−	0.741686i	$-0.734027\pi$
$197$	−18.8288	−1.34150	−0.670748	+	0.741686i	$0.734027\pi$
$198$	0	0
$199$	−4.97851	−0.352917	−0.176459	−	0.984308i	$-0.556464\pi$
$199$	−4.97851	−0.352917	−0.176459	+	0.984308i	$0.556464\pi$
$200$	3.87436	0.273959
$201$	0	0
$202$	−0.604498	−0.0425324
$203$	0	0
$204$	0	0
$205$	10.4810	0.732022
$206$	3.47049	0.241801
$207$	0	0
$208$	−9.92107	−0.687902
$209$	−18.2551	−1.26273
$210$	0	0
$211$	0.601335	0.0413976	0.0206988	−	0.999786i	$-0.493411\pi$
$211$	0.601335	0.0413976	0.0206988	+	0.999786i	$0.493411\pi$
$212$	−24.5039	−1.68294
$213$	0	0
$214$	−3.53158	−0.241414
$215$	0.316838	0.0216082
$216$	0	0
$217$	−24.3600	−1.65366
$218$	−0.936332	−0.0634164
$219$	0	0
$220$	11.0151	0.742635
$221$	0.125336	0.00843103
$222$	0	0
$223$	−6.11125	−0.409240	−0.204620	−	0.978842i	$-0.565596\pi$
$223$	−6.11125	−0.409240	−0.204620	+	0.978842i	$0.565596\pi$
$224$	9.02493	0.603004
$225$	0	0
$226$	2.04392	0.135960
$227$	9.40550	0.624265	0.312132	−	0.950039i	$-0.398957\pi$
$227$	9.40550	0.624265	0.312132	+	0.950039i	$0.398957\pi$
$228$	0	0
$229$	0.392271	0.0259220	0.0129610	−	0.999916i	$-0.495874\pi$
$229$	0.392271	0.0259220	0.0129610	+	0.999916i	$0.495874\pi$
$230$	0.0751452	0.00495493
$231$	0	0
$232$	0	0
$233$	−22.9385	−1.50275	−0.751376	−	0.659874i	$-0.770609\pi$
$233$	−22.9385	−1.50275	−0.751376	+	0.659874i	$0.770609\pi$
$234$	0	0
$235$	−8.68623	−0.566627
$236$	−2.80181	−0.182382
$237$	0	0
$238$	−0.0360838	−0.00233897
$239$	1.66088	0.107434	0.0537168	−	0.998556i	$-0.482893\pi$
$239$	1.66088	0.107434	0.0537168	+	0.998556i	$0.482893\pi$
$240$	0	0
$241$	−23.2131	−1.49529	−0.747645	−	0.664099i	$-0.768815\pi$
$241$	−23.2131	−1.49529	−0.747645	+	0.664099i	$0.768815\pi$
$242$	5.54187	0.356245
$243$	0	0
$244$	13.1628	0.842664
$245$	−3.30178	−0.210943
$246$	0	0
$247$	8.53358	0.542978
$248$	7.23598	0.459485
$249$	0	0
$250$	2.14441	0.135625
$251$	−5.21093	−0.328911	−0.164455	−	0.986385i	$-0.552587\pi$
$251$	−5.21093	−0.328911	−0.164455	+	0.986385i	$0.552587\pi$
$252$	0	0
$253$	−1.84375	−0.115916
$254$	−0.532853	−0.0334342
$255$	0	0
$256$	11.4868	0.717927
$257$	−5.19998	−0.324366	−0.162183	−	0.986761i	$-0.551853\pi$
$257$	−5.19998	−0.324366	−0.162183	+	0.986761i	$0.551853\pi$
$258$	0	0
$259$	−10.3823	−0.645124
$260$	−5.14913	−0.319335
$261$	0	0
$262$	4.44877	0.274846
$263$	8.96684	0.552919	0.276459	−	0.961026i	$-0.410839\pi$
$263$	8.96684	0.552919	0.276459	+	0.961026i	$0.410839\pi$
$264$	0	0
$265$	−12.3195	−0.756782
$266$	−2.45678	−0.150635
$267$	0	0
$268$	11.9473	0.729797
$269$	−4.42050	−0.269523	−0.134761	−	0.990878i	$-0.543027\pi$
$269$	−4.42050	−0.269523	−0.134761	+	0.990878i	$0.543027\pi$
$270$	0	0
$271$	26.6551	1.61918	0.809592	−	0.586993i	$-0.199688\pi$
$271$	26.6551	1.61918	0.809592	+	0.586993i	$0.199688\pi$
$272$	−0.168266	−0.0102026
$273$	0	0
$274$	0.0880043	0.00531654
$275$	−23.5386	−1.41943
$276$	0	0
$277$	−2.55670	−0.153617	−0.0768087	−	0.997046i	$-0.524473\pi$
$277$	−2.55670	−0.153617	−0.0768087	+	0.997046i	$0.524473\pi$
$278$	2.28263	0.136903
$279$	0	0
$280$	3.00988	0.179875
$281$	−21.4241	−1.27805	−0.639026	−	0.769185i	$-0.720663\pi$
$281$	−21.4241	−1.27805	−0.639026	+	0.769185i	$0.720663\pi$
$282$	0	0
$283$	16.7062	0.993079	0.496539	−	0.868014i	$-0.334604\pi$
$283$	16.7062	0.993079	0.496539	+	0.868014i	$0.334604\pi$
$284$	21.8546	1.29683
$285$	0	0
$286$	−3.83951	−0.227035
$287$	−34.6088	−2.04289
$288$	0	0
$289$	−16.9979	−0.999875
$290$	0	0
$291$	0	0
$292$	−9.95386	−0.582505
$293$	−24.0034	−1.40229	−0.701146	−	0.713018i	$-0.747328\pi$
$293$	−24.0034	−1.40229	−0.701146	+	0.713018i	$0.747328\pi$
$294$	0	0
$295$	−1.40863	−0.0820135
$296$	3.08400	0.179254
$297$	0	0
$298$	0.704038	0.0407838
$299$	0.861887	0.0498442
$300$	0	0
$301$	−1.04622	−0.0603030
$302$	−2.62571	−0.151093
$303$	0	0
$304$	−11.4565	−0.657073
$305$	6.61770	0.378929
$306$	0	0
$307$	9.02358	0.515003	0.257502	−	0.966278i	$-0.417101\pi$
$307$	9.02358	0.515003	0.257502	+	0.966278i	$0.417101\pi$
$308$	−36.3724	−2.07251
$309$	0	0
$310$	1.79174	0.101764
$311$	−26.6145	−1.50917	−0.754586	−	0.656201i	$-0.772162\pi$
$311$	−26.6145	−1.50917	−0.754586	+	0.656201i	$0.772162\pi$
$312$	0	0
$313$	−13.2311	−0.747864	−0.373932	−	0.927456i	$-0.621991\pi$
$313$	−13.2311	−0.747864	−0.373932	+	0.927456i	$0.621991\pi$
$314$	−5.03802	−0.284312
$315$	0	0
$316$	−19.7016	−1.10830
$317$	5.57496	0.313121	0.156561	−	0.987668i	$-0.449959\pi$
$317$	5.57496	0.313121	0.156561	+	0.987668i	$0.449959\pi$
$318$	0	0
$319$	0	0
$320$	6.45907	0.361073
$321$	0	0
$322$	−0.248134	−0.0138280
$323$	0.144734	0.00805319
$324$	0	0
$325$	11.0034	0.610360
$326$	−5.88979	−0.326205
$327$	0	0
$328$	10.2803	0.567636
$329$	28.6825	1.58132
$330$	0	0
$331$	9.86617	0.542294	0.271147	−	0.962538i	$-0.412597\pi$
$331$	9.86617	0.542294	0.271147	+	0.962538i	$0.412597\pi$
$332$	15.6562	0.859245
$333$	0	0
$334$	1.29933	0.0710959
$335$	6.00659	0.328175
$336$	0	0
$337$	10.7438	0.585252	0.292626	−	0.956227i	$-0.405471\pi$
$337$	10.7438	0.585252	0.292626	+	0.956227i	$0.405471\pi$
$338$	−1.36256	−0.0741133
$339$	0	0
$340$	−0.0873317	−0.00473623
$341$	−43.9620	−2.38067
$342$	0	0
$343$	−11.6537	−0.629240
$344$	0.310773	0.0167557
$345$	0	0
$346$	−2.19295	−0.117893
$347$	−29.6747	−1.59302	−0.796510	−	0.604625i	$-0.793323\pi$
$347$	−29.6747	−1.59302	−0.796510	+	0.604625i	$0.793323\pi$
$348$	0	0
$349$	15.6483	0.837632	0.418816	−	0.908071i	$-0.362445\pi$
$349$	15.6483	0.837632	0.418816	+	0.908071i	$0.362445\pi$
$350$	−3.16785	−0.169328
$351$	0	0
$352$	16.2871	0.868107
$353$	−4.26989	−0.227264	−0.113632	−	0.993523i	$-0.536248\pi$
$353$	−4.26989	−0.227264	−0.113632	+	0.993523i	$0.536248\pi$
$354$	0	0
$355$	10.9876	0.583160
$356$	−6.24889	−0.331190
$357$	0	0
$358$	5.78701	0.305853
$359$	25.9811	1.37123	0.685615	−	0.727964i	$-0.259533\pi$
$359$	25.9811	1.37123	0.685615	+	0.727964i	$0.259533\pi$
$360$	0	0
$361$	−9.14575	−0.481356
$362$	−4.40299	−0.231416
$363$	0	0
$364$	17.0027	0.891185
$365$	−5.00437	−0.261941
$366$	0	0
$367$	14.5300	0.758462	0.379231	−	0.925302i	$-0.376189\pi$
$367$	14.5300	0.758462	0.379231	+	0.925302i	$0.376189\pi$
$368$	−1.15710	−0.0603179
$369$	0	0
$370$	0.763647	0.0397001
$371$	40.6798	2.11199
$372$	0	0
$373$	−23.5680	−1.22031	−0.610153	−	0.792284i	$-0.708892\pi$
$373$	−23.5680	−1.22031	−0.610153	+	0.792284i	$0.708892\pi$
$374$	−0.0651199	−0.00336727
$375$	0	0
$376$	−8.51995	−0.439383
$377$	0	0
$378$	0	0
$379$	18.1033	0.929902	0.464951	−	0.885336i	$-0.346072\pi$
$379$	18.1033	0.929902	0.464951	+	0.885336i	$0.346072\pi$
$380$	−5.94602	−0.305024
$381$	0	0
$382$	2.81032	0.143788
$383$	−27.4106	−1.40062	−0.700309	−	0.713840i	$-0.746954\pi$
$383$	−27.4106	−1.40062	−0.700309	+	0.713840i	$0.746954\pi$
$384$	0	0
$385$	−18.2865	−0.931964
$386$	0.327633	0.0166761
$387$	0	0
$388$	27.5832	1.40032
$389$	−6.98440	−0.354123	−0.177062	−	0.984200i	$-0.556659\pi$
$389$	−6.98440	−0.354123	−0.177062	+	0.984200i	$0.556659\pi$
$390$	0	0
$391$	0.0146180	0.000739265 0
$392$	−3.23858	−0.163573
$393$	0	0
$394$	−4.57305	−0.230387
$395$	−9.90514	−0.498381
$396$	0	0
$397$	−9.45477	−0.474522	−0.237261	−	0.971446i	$-0.576250\pi$
$397$	−9.45477	−0.474522	−0.237261	+	0.971446i	$0.576250\pi$
$398$	−1.20916	−0.0606096
$399$	0	0
$400$	−14.7723	−0.738614
$401$	−29.4747	−1.47190	−0.735948	−	0.677038i	$-0.763263\pi$
$401$	−29.4747	−1.47190	−0.735948	+	0.677038i	$0.763263\pi$
$402$	0	0
$403$	20.5506	1.02370
$404$	4.83102	0.240352
$405$	0	0
$406$	0	0
$407$	−18.7367	−0.928746
$408$	0	0
$409$	−13.9350	−0.689039	−0.344520	−	0.938779i	$-0.611958\pi$
$409$	−13.9350	−0.689039	−0.344520	+	0.938779i	$0.611958\pi$
$410$	2.54557	0.125717
$411$	0	0
$412$	−27.7354	−1.36643
$413$	4.65138	0.228879
$414$	0	0
$415$	7.87126	0.386385
$416$	−7.61363	−0.373289
$417$	0	0
$418$	−4.43372	−0.216860
$419$	2.66976	0.130426	0.0652132	−	0.997871i	$-0.479227\pi$
$419$	2.66976	0.130426	0.0652132	+	0.997871i	$0.479227\pi$
$420$	0	0
$421$	−5.55692	−0.270828	−0.135414	−	0.990789i	$-0.543236\pi$
$421$	−5.55692	−0.270828	−0.135414	+	0.990789i	$0.543236\pi$
$422$	0.146050	0.00710959
$423$	0	0
$424$	−12.0837	−0.586835
$425$	0.186623	0.00905256
$426$	0	0
$427$	−21.8521	−1.05750
$428$	28.2237	1.36424
$429$	0	0
$430$	0.0769523	0.00371097
$431$	−16.6060	−0.799883	−0.399942	−	0.916541i	$-0.630970\pi$
$431$	−16.6060	−0.799883	−0.399942	+	0.916541i	$0.630970\pi$
$432$	0	0
$433$	10.2686	0.493480	0.246740	−	0.969082i	$-0.420641\pi$
$433$	10.2686	0.493480	0.246740	+	0.969082i	$0.420641\pi$
$434$	−5.91644	−0.283998
$435$	0	0
$436$	7.48297	0.358369
$437$	0.995274	0.0476104
$438$	0	0
$439$	4.26072	0.203353	0.101677	−	0.994818i	$-0.467579\pi$
$439$	4.26072	0.203353	0.101677	+	0.994818i	$0.467579\pi$
$440$	5.43188	0.258955
$441$	0	0
$442$	0.0304411	0.00144794
$443$	26.9235	1.27917	0.639587	−	0.768719i	$-0.279105\pi$
$443$	26.9235	1.27917	0.639587	+	0.768719i	$0.279105\pi$
$444$	0	0
$445$	−3.14167	−0.148930
$446$	−1.48428	−0.0702825
$447$	0	0
$448$	−21.3282	−1.00766
$449$	−27.9572	−1.31938	−0.659691	−	0.751537i	$-0.729313\pi$
$449$	−27.9572	−1.31938	−0.659691	+	0.751537i	$0.729313\pi$
$450$	0	0
$451$	−62.4578	−2.94102
$452$	−16.3346	−0.768315
$453$	0	0
$454$	2.28437	0.107211
$455$	8.54824	0.400748
$456$	0	0
$457$	−26.5876	−1.24371	−0.621857	−	0.783131i	$-0.713622\pi$
$457$	−26.5876	−1.24371	−0.621857	+	0.783131i	$0.713622\pi$
$458$	0.0952730	0.00445182
$459$	0	0
$460$	−0.600545	−0.0280005
$461$	−16.1451	−0.751951	−0.375975	−	0.926630i	$-0.622692\pi$
$461$	−16.1451	−0.751951	−0.375975	+	0.926630i	$0.622692\pi$
$462$	0	0
$463$	13.9805	0.649729	0.324865	−	0.945761i	$-0.394681\pi$
$463$	13.9805	0.649729	0.324865	+	0.945761i	$0.394681\pi$
$464$	0	0
$465$	0	0
$466$	−5.57121	−0.258081
$467$	−17.4597	−0.807937	−0.403968	−	0.914773i	$-0.632369\pi$
$467$	−17.4597	−0.807937	−0.403968	+	0.914773i	$0.632369\pi$
$468$	0	0
$469$	−19.8341	−0.915854
$470$	−2.10968	−0.0973121
$471$	0	0
$472$	−1.38166	−0.0635962
$473$	−1.88809	−0.0868145
$474$	0	0
$475$	12.7063	0.583007
$476$	0.288374	0.0132176
$477$	0	0
$478$	0.403388	0.0184505
$479$	23.7399	1.08470	0.542352	−	0.840152i	$-0.317534\pi$
$479$	23.7399	1.08470	0.542352	+	0.840152i	$0.317534\pi$
$480$	0	0
$481$	8.75874	0.399364
$482$	−5.63791	−0.256800
$483$	0	0
$484$	−44.2894	−2.01316
$485$	13.8676	0.629697
$486$	0	0
$487$	28.7264	1.30172	0.650859	−	0.759199i	$-0.274409\pi$
$487$	28.7264	1.30172	0.650859	+	0.759199i	$0.274409\pi$
$488$	6.49102	0.293835
$489$	0	0
$490$	−0.801923	−0.0362272
$491$	−12.0365	−0.543201	−0.271600	−	0.962410i	$-0.587553\pi$
$491$	−12.0365	−0.543201	−0.271600	+	0.962410i	$0.587553\pi$
$492$	0	0
$493$	0	0
$494$	2.07260	0.0932506
$495$	0	0
$496$	−27.5895	−1.23881
$497$	−36.2816	−1.62745
$498$	0	0
$499$	18.4422	0.825584	0.412792	−	0.910825i	$-0.364554\pi$
$499$	18.4422	0.825584	0.412792	+	0.910825i	$0.364554\pi$
$500$	−17.1377	−0.766422
$501$	0	0
$502$	−1.26561	−0.0564868
$503$	−26.7470	−1.19259	−0.596294	−	0.802766i	$-0.703361\pi$
$503$	−26.7470	−1.19259	−0.596294	+	0.802766i	$0.703361\pi$
$504$	0	0
$505$	2.42883	0.108082
$506$	−0.447803	−0.0199073
$507$	0	0
$508$	4.25845	0.188938
$509$	34.9643	1.54976	0.774882	−	0.632106i	$-0.217809\pi$
$509$	34.9643	1.54976	0.774882	+	0.632106i	$0.217809\pi$
$510$	0	0
$511$	16.5247	0.731011
$512$	17.2080	0.760491
$513$	0	0
$514$	−1.26295	−0.0557062
$515$	−13.9442	−0.614455
$516$	0	0
$517$	51.7627	2.27652
$518$	−2.52161	−0.110793
$519$	0	0
$520$	−2.53920	−0.111351
$521$	−19.6126	−0.859243	−0.429621	−	0.903009i	$-0.641353\pi$
$521$	−19.6126	−0.859243	−0.429621	+	0.903009i	$0.641353\pi$
$522$	0	0
$523$	−39.2690	−1.71711	−0.858556	−	0.512720i	$-0.828638\pi$
$523$	−39.2690	−1.71711	−0.858556	+	0.512720i	$0.828638\pi$
$524$	−35.5536	−1.55317
$525$	0	0
$526$	2.17783	0.0949578
$527$	0.348548	0.0151830
$528$	0	0
$529$	−22.8995	−0.995629
$530$	−2.99211	−0.129969
$531$	0	0
$532$	19.6341	0.851246
$533$	29.1967	1.26465
$534$	0	0
$535$	14.1897	0.613472
$536$	5.89160	0.254478
$537$	0	0
$538$	−1.07363	−0.0462876
$539$	19.6759	0.847501
$540$	0	0
$541$	8.46210	0.363814	0.181907	−	0.983316i	$-0.441773\pi$
$541$	8.46210	0.363814	0.181907	+	0.983316i	$0.441773\pi$
$542$	6.47389	0.278077
$543$	0	0
$544$	−0.129131	−0.00553644
$545$	3.76212	0.161151
$546$	0	0
$547$	−10.4695	−0.447646	−0.223823	−	0.974630i	$-0.571854\pi$
$547$	−10.4695	−0.447646	−0.223823	+	0.974630i	$0.571854\pi$
$548$	−0.703312	−0.0300440
$549$	0	0
$550$	−5.71696	−0.243772
$551$	0	0
$552$	0	0
$553$	32.7073	1.39086
$554$	−0.620962	−0.0263821
$555$	0	0
$556$	−18.2423	−0.773647
$557$	−8.77203	−0.371683	−0.185841	−	0.982580i	$-0.559501\pi$
$557$	−8.77203	−0.371683	−0.185841	+	0.982580i	$0.559501\pi$
$558$	0	0
$559$	0.882613	0.0373305
$560$	−11.4762	−0.484956
$561$	0	0
$562$	−5.20339	−0.219492
$563$	11.5047	0.484867	0.242433	−	0.970168i	$-0.422054\pi$
$563$	11.5047	0.484867	0.242433	+	0.970168i	$0.422054\pi$
$564$	0	0
$565$	−8.21233	−0.345495
$566$	4.05752	0.170550
$567$	0	0
$568$	10.7772	0.452203
$569$	29.3623	1.23093	0.615466	−	0.788163i	$-0.288968\pi$
$569$	29.3623	1.23093	0.615466	+	0.788163i	$0.288968\pi$
$570$	0	0
$571$	10.1743	0.425782	0.212891	−	0.977076i	$-0.431712\pi$
$571$	10.1743	0.425782	0.212891	+	0.977076i	$0.431712\pi$
$572$	30.6845	1.28298
$573$	0	0
$574$	−8.40563	−0.350844
$575$	1.28333	0.0535187
$576$	0	0
$577$	19.9508	0.830562	0.415281	−	0.909693i	$-0.363683\pi$
$577$	19.9508	0.830562	0.415281	+	0.909693i	$0.363683\pi$
$578$	−4.12837	−0.171718
$579$	0	0
$580$	0	0
$581$	−25.9914	−1.07830
$582$	0	0
$583$	73.4141	3.04050
$584$	−4.90857	−0.203118
$585$	0	0
$586$	−5.82984	−0.240828
$587$	−28.2713	−1.16688	−0.583441	−	0.812155i	$-0.698294\pi$
$587$	−28.2713	−1.16688	−0.583441	+	0.812155i	$0.698294\pi$
$588$	0	0
$589$	23.7310	0.977821
$590$	−0.342122	−0.0140849
$591$	0	0
$592$	−11.7588	−0.483282
$593$	−3.94837	−0.162140	−0.0810700	−	0.996708i	$-0.525834\pi$
$593$	−3.94837	−0.162140	−0.0810700	+	0.996708i	$0.525834\pi$
$594$	0	0
$595$	0.144982	0.00594370
$596$	−5.62652	−0.230471
$597$	0	0
$598$	0.209331	0.00856020
$599$	3.49455	0.142783	0.0713917	−	0.997448i	$-0.477256\pi$
$599$	3.49455	0.142783	0.0713917	+	0.997448i	$0.477256\pi$
$600$	0	0
$601$	−1.79919	−0.0733905	−0.0366953	−	0.999327i	$-0.511683\pi$
$601$	−1.79919	−0.0733905	−0.0366953	+	0.999327i	$0.511683\pi$
$602$	−0.254101	−0.0103564
$603$	0	0
$604$	20.9841	0.853831
$605$	−22.2668	−0.905275
$606$	0	0
$607$	1.86238	0.0755916	0.0377958	−	0.999285i	$-0.487966\pi$
$607$	1.86238	0.0755916	0.0377958	+	0.999285i	$0.487966\pi$
$608$	−8.79193	−0.356560
$609$	0	0
$610$	1.60728	0.0650769
$611$	−24.1972	−0.978913
$612$	0	0
$613$	9.37869	0.378802	0.189401	−	0.981900i	$-0.439345\pi$
$613$	9.37869	0.378802	0.189401	+	0.981900i	$0.439345\pi$
$614$	2.19161	0.0884462
$615$	0	0
$616$	−17.9364	−0.722678
$617$	10.2686	0.413398	0.206699	−	0.978405i	$-0.433728\pi$
$617$	10.2686	0.413398	0.206699	+	0.978405i	$0.433728\pi$
$618$	0	0
$619$	−20.4500	−0.821956	−0.410978	−	0.911645i	$-0.634813\pi$
$619$	−20.4500	−0.821956	−0.410978	+	0.911645i	$0.634813\pi$
$620$	−14.3192	−0.575074
$621$	0	0
$622$	−6.46402	−0.259184
$623$	10.3740	0.415625
$624$	0	0
$625$	11.6224	0.464896
$626$	−3.21351	−0.128438
$627$	0	0
$628$	40.2627	1.60666
$629$	0.148552	0.00592317
$630$	0	0
$631$	−20.2476	−0.806045	−0.403022	−	0.915190i	$-0.632040\pi$
$631$	−20.2476	−0.806045	−0.403022	+	0.915190i	$0.632040\pi$
$632$	−9.71552	−0.386463
$633$	0	0
$634$	1.35402	0.0537752
$635$	2.14097	0.0849616
$636$	0	0
$637$	−9.19775	−0.364428
$638$	0	0
$639$	0	0
$640$	7.03499	0.278083
$641$	−1.97133	−0.0778630	−0.0389315	−	0.999242i	$-0.512395\pi$
$641$	−1.97133	−0.0778630	−0.0389315	+	0.999242i	$0.512395\pi$
$642$	0	0
$643$	36.3376	1.43301	0.716507	−	0.697580i	$-0.245740\pi$
$643$	36.3376	1.43301	0.716507	+	0.697580i	$0.245740\pi$
$644$	1.98303	0.0781425
$645$	0	0
$646$	0.0351523	0.00138305
$647$	−1.77749	−0.0698803	−0.0349402	−	0.999389i	$-0.511124\pi$
$647$	−1.77749	−0.0698803	−0.0349402	+	0.999389i	$0.511124\pi$
$648$	0	0
$649$	8.39426	0.329503
$650$	2.67247	0.104823
$651$	0	0
$652$	47.0699	1.84340
$653$	21.1086	0.826043	0.413021	−	0.910721i	$-0.364474\pi$
$653$	21.1086	0.826043	0.413021	+	0.910721i	$0.364474\pi$
$654$	0	0
$655$	−17.8748	−0.698428
$656$	−39.1971	−1.53039
$657$	0	0
$658$	6.96628	0.271574
$659$	18.7017	0.728516	0.364258	−	0.931298i	$-0.381323\pi$
$659$	18.7017	0.728516	0.364258	+	0.931298i	$0.381323\pi$
$660$	0	0
$661$	−13.6054	−0.529189	−0.264594	−	0.964360i	$-0.585238\pi$
$661$	−13.6054	−0.529189	−0.264594	+	0.964360i	$0.585238\pi$
$662$	2.39625	0.0931330
$663$	0	0
$664$	7.72058	0.299617
$665$	9.87119	0.382788
$666$	0	0
$667$	0	0
$668$	−10.3839	−0.401766
$669$	0	0
$670$	1.45885	0.0563605
$671$	−39.4360	−1.52241
$672$	0	0
$673$	−15.4327	−0.594888	−0.297444	−	0.954739i	$-0.596134\pi$
$673$	−15.4327	−0.594888	−0.297444	+	0.954739i	$0.596134\pi$
$674$	2.60941	0.100511
$675$	0	0
$676$	10.8893	0.418818
$677$	7.35606	0.282716	0.141358	−	0.989959i	$-0.454853\pi$
$677$	7.35606	0.282716	0.141358	+	0.989959i	$0.454853\pi$
$678$	0	0
$679$	−45.7918	−1.75733
$680$	−0.0430661	−0.00165151
$681$	0	0
$682$	−10.6773	−0.408855
$683$	12.4365	0.475870	0.237935	−	0.971281i	$-0.423530\pi$
$683$	12.4365	0.475870	0.237935	+	0.971281i	$0.423530\pi$
$684$	0	0
$685$	−0.353595	−0.0135102
$686$	−2.83040	−0.108065
$687$	0	0
$688$	−1.18492	−0.0451747
$689$	−34.3184	−1.30743
$690$	0	0
$691$	−30.7572	−1.17006	−0.585030	−	0.811012i	$-0.698917\pi$
$691$	−30.7572	−1.17006	−0.585030	+	0.811012i	$0.698917\pi$
$692$	17.5256	0.666222
$693$	0	0
$694$	−7.20726	−0.273584
$695$	−9.17146	−0.347893
$696$	0	0
$697$	0.495190	0.0187567
$698$	3.80058	0.143854
$699$	0	0
$700$	25.3167	0.956883
$701$	−28.3853	−1.07210	−0.536050	−	0.844186i	$-0.680084\pi$
$701$	−28.3853	−1.07210	−0.536050	+	0.844186i	$0.680084\pi$
$702$	0	0
$703$	10.1143	0.381466
$704$	−38.4907	−1.45067
$705$	0	0
$706$	−1.03705	−0.0390300
$707$	−8.02014	−0.301629
$708$	0	0
$709$	21.2643	0.798597	0.399299	−	0.916821i	$-0.369254\pi$
$709$	21.2643	0.798597	0.399299	+	0.916821i	$0.369254\pi$
$710$	2.66862	0.100151
$711$	0	0
$712$	−3.08153	−0.115485
$713$	2.39682	0.0897618
$714$	0	0
$715$	15.4269	0.576932
$716$	−46.2486	−1.72839
$717$	0	0
$718$	6.31018	0.235494
$719$	24.6441	0.919070	0.459535	−	0.888160i	$-0.348016\pi$
$719$	24.6441	0.919070	0.459535	+	0.888160i	$0.348016\pi$
$720$	0	0
$721$	46.0446	1.71479
$722$	−2.22128	−0.0826676
$723$	0	0
$724$	35.1877	1.30774
$725$	0	0
$726$	0	0
$727$	−2.61086	−0.0968316	−0.0484158	−	0.998827i	$-0.515417\pi$
$727$	−2.61086	−0.0968316	−0.0484158	+	0.998827i	$0.515417\pi$
$728$	8.38460	0.310754
$729$	0	0
$730$	−1.21544	−0.0449855
$731$	0.0149695	0.000553668 0
$732$	0	0
$733$	−8.31243	−0.307027	−0.153513	−	0.988147i	$-0.549059\pi$
$733$	−8.31243	−0.307027	−0.153513	+	0.988147i	$0.549059\pi$
$734$	3.52900	0.130258
$735$	0	0
$736$	−0.887981	−0.0327314
$737$	−35.7943	−1.31850
$738$	0	0
$739$	4.87707	0.179406	0.0897029	−	0.995969i	$-0.471408\pi$
$739$	4.87707	0.179406	0.0897029	+	0.995969i	$0.471408\pi$
$740$	−6.10290	−0.224347
$741$	0	0
$742$	9.88013	0.362711
$743$	−5.87529	−0.215544	−0.107772	−	0.994176i	$-0.534372\pi$
$743$	−5.87529	−0.215544	−0.107772	+	0.994176i	$0.534372\pi$
$744$	0	0
$745$	−2.82877	−0.103638
$746$	−5.72410	−0.209574
$747$	0	0
$748$	0.520424	0.0190286
$749$	−46.8551	−1.71205
$750$	0	0
$751$	−11.0687	−0.403904	−0.201952	−	0.979395i	$-0.564728\pi$
$751$	−11.0687	−0.403904	−0.201952	+	0.979395i	$0.564728\pi$
$752$	32.4851	1.18461
$753$	0	0
$754$	0	0
$755$	10.5499	0.383950
$756$	0	0
$757$	−15.9704	−0.580455	−0.290227	−	0.956958i	$-0.593731\pi$
$757$	−15.9704	−0.580455	−0.290227	+	0.956958i	$0.593731\pi$
$758$	4.39684	0.159701
$759$	0	0
$760$	−2.93218	−0.106361
$761$	35.0081	1.26904	0.634522	−	0.772905i	$-0.281197\pi$
$761$	35.0081	1.26904	0.634522	+	0.772905i	$0.281197\pi$
$762$	0	0
$763$	−12.4227	−0.449733
$764$	−22.4595	−0.812555
$765$	0	0
$766$	−6.65738	−0.240541
$767$	−3.92401	−0.141688
$768$	0	0
$769$	−13.6731	−0.493066	−0.246533	−	0.969134i	$-0.579291\pi$
$769$	−13.6731	−0.493066	−0.246533	+	0.969134i	$0.579291\pi$
$770$	−4.44134	−0.160055
$771$	0	0
$772$	−2.61837	−0.0942374
$773$	−27.7310	−0.997414	−0.498707	−	0.866771i	$-0.666192\pi$
$773$	−27.7310	−0.997414	−0.498707	+	0.866771i	$0.666192\pi$
$774$	0	0
$775$	30.5995	1.09917
$776$	13.6022	0.488289
$777$	0	0
$778$	−1.69634	−0.0608168
$779$	33.7153	1.20797
$780$	0	0
$781$	−65.4768	−2.34295
$782$	0.00355036	0.000126961 0
$783$	0	0
$784$	12.3481	0.441005
$785$	20.2424	0.722482
$786$	0	0
$787$	−42.5533	−1.51686	−0.758431	−	0.651754i	$-0.774034\pi$
$787$	−42.5533	−1.51686	−0.758431	+	0.651754i	$0.774034\pi$
$788$	36.5469	1.30193
$789$	0	0
$790$	−2.40572	−0.0855916
$791$	27.1176	0.964191
$792$	0	0
$793$	18.4349	0.654642
$794$	−2.29634	−0.0814939
$795$	0	0
$796$	9.66334	0.342508
$797$	4.17550	0.147904	0.0739519	−	0.997262i	$-0.476439\pi$
$797$	4.17550	0.147904	0.0739519	+	0.997262i	$0.476439\pi$
$798$	0	0
$799$	−0.410396	−0.0145187
$800$	−11.3366	−0.400808
$801$	0	0
$802$	−7.15869	−0.252782
$803$	29.8219	1.05239
$804$	0	0
$805$	0.996985	0.0351391
$806$	4.99124	0.175809
$807$	0	0
$808$	2.38233	0.0838102
$809$	25.7708	0.906053	0.453027	−	0.891497i	$-0.350344\pi$
$809$	25.7708	0.906053	0.453027	+	0.891497i	$0.350344\pi$
$810$	0	0
$811$	−7.66158	−0.269035	−0.134517	−	0.990911i	$-0.542948\pi$
$811$	−7.66158	−0.269035	−0.134517	+	0.990911i	$0.542948\pi$
$812$	0	0
$813$	0	0
$814$	−4.55070	−0.159502
$815$	23.6648	0.828940
$816$	0	0
$817$	1.01921	0.0356575
$818$	−3.38446	−0.118335
$819$	0	0
$820$	−20.3437	−0.710432
$821$	7.67282	0.267783	0.133892	−	0.990996i	$-0.457253\pi$
$821$	7.67282	0.267783	0.133892	+	0.990996i	$0.457253\pi$
$822$	0	0
$823$	23.9251	0.833978	0.416989	−	0.908912i	$-0.363085\pi$
$823$	23.9251	0.833978	0.416989	+	0.908912i	$0.363085\pi$
$824$	−13.6773	−0.476470
$825$	0	0
$826$	1.12971	0.0393075
$827$	1.24451	0.0432760	0.0216380	−	0.999766i	$-0.493112\pi$
$827$	1.24451	0.0432760	0.0216380	+	0.999766i	$0.493112\pi$
$828$	0	0
$829$	1.48564	0.0515982	0.0257991	−	0.999667i	$-0.491787\pi$
$829$	1.48564	0.0515982	0.0257991	+	0.999667i	$0.491787\pi$
$830$	1.91174	0.0663574
$831$	0	0
$832$	17.9930	0.623794
$833$	−0.155998	−0.00540502
$834$	0	0
$835$	−5.22060	−0.180666
$836$	35.4333	1.22549
$837$	0	0
$838$	0.648421	0.0223993
$839$	−24.3023	−0.839007	−0.419504	−	0.907754i	$-0.637796\pi$
$839$	−24.3023	−0.839007	−0.419504	+	0.907754i	$0.637796\pi$
$840$	0	0
$841$	0	0
$842$	−1.34964	−0.0465117
$843$	0	0
$844$	−1.16720	−0.0401766
$845$	5.47466	0.188334
$846$	0	0
$847$	73.5264	2.52640
$848$	46.0730	1.58215
$849$	0	0
$850$	0.0453263	0.00155468
$851$	1.02153	0.0350178
$852$	0	0
$853$	50.0864	1.71492	0.857462	−	0.514546i	$-0.172040\pi$
$853$	50.0864	1.71492	0.857462	+	0.514546i	$0.172040\pi$
$854$	−5.30734	−0.181613
$855$	0	0
$856$	13.9180	0.475708
$857$	10.1152	0.345529	0.172764	−	0.984963i	$-0.444730\pi$
$857$	10.1152	0.345529	0.172764	+	0.984963i	$0.444730\pi$
$858$	0	0
$859$	2.06499	0.0704566	0.0352283	−	0.999379i	$-0.488784\pi$
$859$	2.06499	0.0704566	0.0352283	+	0.999379i	$0.488784\pi$
$860$	−0.614986	−0.0209708
$861$	0	0
$862$	−4.03320	−0.137371
$863$	49.7775	1.69444	0.847222	−	0.531239i	$-0.178273\pi$
$863$	49.7775	1.69444	0.847222	+	0.531239i	$0.178273\pi$
$864$	0	0
$865$	8.81110	0.299586
$866$	2.49401	0.0847498
$867$	0	0
$868$	47.2830	1.60489
$869$	59.0264	2.00233
$870$	0	0
$871$	16.7325	0.566959
$872$	3.69010	0.124962
$873$	0	0
$874$	0.241728	0.00817657
$875$	28.4509	0.961816
$876$	0	0
$877$	16.6796	0.563230	0.281615	−	0.959527i	$-0.409130\pi$
$877$	16.6796	0.563230	0.281615	+	0.959527i	$0.409130\pi$
$878$	1.03483	0.0349237
$879$	0	0
$880$	−20.7108	−0.698162
$881$	−24.6163	−0.829344	−0.414672	−	0.909971i	$-0.636104\pi$
$881$	−24.6163	−0.829344	−0.414672	+	0.909971i	$0.636104\pi$
$882$	0	0
$883$	−20.2292	−0.680767	−0.340384	−	0.940287i	$-0.610557\pi$
$883$	−20.2292	−0.680767	−0.340384	+	0.940287i	$0.610557\pi$
$884$	−0.243279	−0.00818236
$885$	0	0
$886$	6.53907	0.219684
$887$	−20.4379	−0.686238	−0.343119	−	0.939292i	$-0.611483\pi$
$887$	−20.4379	−0.686238	−0.343119	+	0.939292i	$0.611483\pi$
$888$	0	0
$889$	−7.06960	−0.237107
$890$	−0.763036	−0.0255770
$891$	0	0
$892$	11.8620	0.397169
$893$	−27.9420	−0.935042
$894$	0	0
$895$	−23.2518	−0.777222
$896$	−23.2300	−0.776059
$897$	0	0
$898$	−6.79013	−0.226590
$899$	0	0
$900$	0	0
$901$	−0.582056	−0.0193911
$902$	−15.1695	−0.505089
$903$	0	0
$904$	−8.05512	−0.267909
$905$	17.6909	0.588065
$906$	0	0
$907$	34.5604	1.14756	0.573780	−	0.819010i	$-0.305477\pi$
$907$	34.5604	1.14756	0.573780	+	0.819010i	$0.305477\pi$
$908$	−18.2562	−0.605853
$909$	0	0
$910$	2.07616	0.0688241
$911$	−15.3718	−0.509290	−0.254645	−	0.967035i	$-0.581959\pi$
$911$	−15.3718	−0.509290	−0.254645	+	0.967035i	$0.581959\pi$
$912$	0	0
$913$	−46.9062	−1.55237
$914$	−6.45748	−0.213594
$915$	0	0
$916$	−0.761402	−0.0251574
$917$	59.0238	1.94914
$918$	0	0
$919$	12.6545	0.417434	0.208717	−	0.977976i	$-0.433071\pi$
$919$	12.6545	0.417434	0.208717	+	0.977976i	$0.433071\pi$
$920$	−0.296148	−0.00976372
$921$	0	0
$922$	−3.92125	−0.129139
$923$	30.6080	1.00747
$924$	0	0
$925$	13.0416	0.428805
$926$	3.39553	0.111584
$927$	0	0
$928$	0	0
$929$	41.8885	1.37432	0.687158	−	0.726508i	$-0.258858\pi$
$929$	41.8885	1.37432	0.687158	+	0.726508i	$0.258858\pi$
$930$	0	0
$931$	−10.6212	−0.348096
$932$	44.5239	1.45843
$933$	0	0
$934$	−4.24053	−0.138754
$935$	0.261647	0.00855677
$936$	0	0
$937$	−25.5748	−0.835493	−0.417746	−	0.908564i	$-0.637180\pi$
$937$	−25.5748	−0.835493	−0.417746	+	0.908564i	$0.637180\pi$
$938$	−4.81723	−0.157288
$939$	0	0
$940$	16.8601	0.549915
$941$	−3.86852	−0.126110	−0.0630551	−	0.998010i	$-0.520084\pi$
$941$	−3.86852	−0.126110	−0.0630551	+	0.998010i	$0.520084\pi$
$942$	0	0
$943$	3.40522	0.110889
$944$	5.26804	0.171460
$945$	0	0
$946$	−0.458572	−0.0149094
$947$	−9.17000	−0.297985	−0.148992	−	0.988838i	$-0.547603\pi$
$947$	−9.17000	−0.297985	−0.148992	+	0.988838i	$0.547603\pi$
$948$	0	0
$949$	−13.9406	−0.452532
$950$	3.08606	0.100125
$951$	0	0
$952$	0.142207	0.00460895
$953$	−11.8876	−0.385078	−0.192539	−	0.981289i	$-0.561672\pi$
$953$	−11.8876	−0.385078	−0.192539	+	0.981289i	$0.561672\pi$
$954$	0	0
$955$	−11.2917	−0.365389
$956$	−3.22379	−0.104265
$957$	0	0
$958$	5.76584	0.186286
$959$	1.16759	0.0377035
$960$	0	0
$961$	26.1492	0.843524
$962$	2.12728	0.0685864
$963$	0	0
$964$	45.0570	1.45119
$965$	−1.31641	−0.0423766
$966$	0	0
$967$	52.6637	1.69355	0.846775	−	0.531951i	$-0.178541\pi$
$967$	52.6637	1.69355	0.846775	+	0.531951i	$0.178541\pi$
$968$	−21.8406	−0.701982
$969$	0	0
$970$	3.36811	0.108144
$971$	30.6487	0.983565	0.491783	−	0.870718i	$-0.336345\pi$
$971$	30.6487	0.983565	0.491783	+	0.870718i	$0.336345\pi$
$972$	0	0
$973$	30.2847	0.970883
$974$	6.97695	0.223556
$975$	0	0
$976$	−24.7491	−0.792201
$977$	32.0309	1.02476	0.512379	−	0.858759i	$-0.328764\pi$
$977$	32.0309	1.02476	0.512379	+	0.858759i	$0.328764\pi$
$978$	0	0
$979$	18.7218	0.598350
$980$	6.40880	0.204722
$981$	0	0
$982$	−2.92338	−0.0932888
$983$	−8.72525	−0.278292	−0.139146	−	0.990272i	$-0.544436\pi$
$983$	−8.72525	−0.278292	−0.139146	+	0.990272i	$0.544436\pi$
$984$	0	0
$985$	18.3742	0.585451
$986$	0	0
$987$	0	0
$988$	−16.5638	−0.526964
$989$	0.102939	0.00327328
$990$	0	0
$991$	−6.42765	−0.204181	−0.102090	−	0.994775i	$-0.532553\pi$
$991$	−6.42765	−0.204181	−0.102090	+	0.994775i	$0.532553\pi$
$992$	−21.1728	−0.672236
$993$	0	0
$994$	−8.81193	−0.279497
$995$	4.85831	0.154019
$996$	0	0
$997$	−59.4725	−1.88351	−0.941757	−	0.336294i	$-0.890826\pi$
$997$	−59.4725	−1.88351	−0.941757	+	0.336294i	$0.890826\pi$
$998$	4.47915	0.141785
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.bt.1.5		12
3.2	odd	2		2523.2.a.s.1.8		12
29.19	odd	28		261.2.o.b.100.3		24
29.26	odd	28		261.2.o.b.154.3		24
29.28	even	2		7569.2.a.bn.1.8		12
87.26	even	28		87.2.i.a.67.2	yes	24
87.77	even	28		87.2.i.a.13.2	✓	24
87.86	odd	2		2523.2.a.v.1.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.i.a.13.2	✓	24	87.77	even	28
87.2.i.a.67.2	yes	24	87.26	even	28
261.2.o.b.100.3		24	29.19	odd	28
261.2.o.b.154.3		24	29.26	odd	28
2523.2.a.s.1.8		12	3.2	odd	2
2523.2.a.v.1.5		12	87.86	odd	2
7569.2.a.bn.1.8		12	29.28	even	2
7569.2.a.bt.1.5		12	1.1	even	1	trivial

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 7569 = 3^{2} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7569.a (trivial)

\(f(q)\)	\(=\)	\(q+0.242876 q^{2} -1.94101 q^{4} -0.975857 q^{5} +3.22234 q^{7} -0.957176 q^{8} -0.237012 q^{10} +5.81530 q^{11} -2.71844 q^{13} +0.782628 q^{14} +3.64955 q^{16} -0.0461060 q^{17} -3.13915 q^{19} +1.89415 q^{20} +1.41240 q^{22} -0.317052 q^{23} -4.04770 q^{25} -0.660243 q^{26} -6.25460 q^{28} -7.55971 q^{31} +2.80074 q^{32} -0.0111980 q^{34} -3.14454 q^{35} -3.22197 q^{37} -0.762423 q^{38} +0.934067 q^{40} -10.7403 q^{41} -0.324676 q^{43} -11.2876 q^{44} -0.0770043 q^{46} +8.90113 q^{47} +3.38347 q^{49} -0.983089 q^{50} +5.27652 q^{52} +12.6243 q^{53} -5.67490 q^{55} -3.08435 q^{56} +1.44348 q^{59} -6.78143 q^{61} -1.83607 q^{62} -6.61886 q^{64} +2.65281 q^{65} -6.15519 q^{67} +0.0894923 q^{68} -0.763733 q^{70} -11.2594 q^{71} +5.12818 q^{73} -0.782540 q^{74} +6.09312 q^{76} +18.7389 q^{77} +10.1502 q^{79} -3.56144 q^{80} -2.60855 q^{82} -8.06599 q^{83} +0.0449929 q^{85} -0.0788560 q^{86} -5.56627 q^{88} +3.21940 q^{89} -8.75973 q^{91} +0.615402 q^{92} +2.16187 q^{94} +3.06336 q^{95} -14.2107 q^{97} +0.821763 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{2} + 8 q^{4} - 2 q^{5} - 10 q^{7} - 20 q^{10} + 14 q^{11} - 16 q^{13} - 4 q^{16} + 22 q^{17} - 16 q^{19} - 4 q^{20} + 12 q^{22} - 2 q^{23} - 2 q^{25} - 8 q^{26} - 4 q^{28} - 4 q^{31} + 16 q^{32}+ \cdots + 6 q^{98}+O(q^{100}) \) 12 * q + 6 * q^2 + 8 * q^4 - 2 * q^5 - 10 * q^7 - 20 * q^10 + 14 * q^11 - 16 * q^13 - 4 * q^16 + 22 * q^17 - 16 * q^19 - 4 * q^20 + 12 * q^22 - 2 * q^23 - 2 * q^25 - 8 * q^26 - 4 * q^28 - 4 * q^31 + 16 * q^32 - 10 * q^34 + 14 * q^35 - 28 * q^37 - 16 * q^38 - 32 * q^40 - 14 * q^41 - 36 * q^43 + 26 * q^44 - 24 * q^46 + 44 * q^47 - 18 * q^49 + 24 * q^50 - 20 * q^52 + 18 * q^53 - 4 * q^55 - 10 * q^56 - 14 * q^59 - 24 * q^61 - 4 * q^62 - 18 * q^64 - 2 * q^65 - 18 * q^67 + 52 * q^68 + 36 * q^70 - 14 * q^71 - 44 * q^73 - 12 * q^74 - 56 * q^76 + 32 * q^77 - 24 * q^79 - 52 * q^80 + 52 * q^82 - 26 * q^83 - 48 * q^85 - 64 * q^86 - 10 * q^88 + 10 * q^89 - 10 * q^91 - 64 * q^92 - 8 * q^94 - 4 * q^95 - 28 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more