LMFDB - Embedded newform 7569.2.a.bl.1.9

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:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 14x^{7} + 9x^{6} + 70x^{5} - 23x^{4} - 141x^{3} + 14x^{2} + 84x - 7 $ x^9 - x^8 - 14x^7 + 9x^6 + 70x^5 - 23x^4 - 141x^3 + 14x^2 + 84*x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.9
Root			$2.68096$ 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)

$f(q)$	$=$	$q+2.68096 q^{2} +5.18756 q^{4} -3.11062 q^{5} -2.37023 q^{7} +8.54571 q^{8} +O(q^{10})$	$q+2.68096 q^{2} +5.18756 q^{4} -3.11062 q^{5} -2.37023 q^{7} +8.54571 q^{8} -8.33945 q^{10} -2.60272 q^{11} -2.30853 q^{13} -6.35450 q^{14} +12.5356 q^{16} +4.83107 q^{17} +1.63818 q^{19} -16.1365 q^{20} -6.97779 q^{22} +1.18327 q^{23} +4.67594 q^{25} -6.18908 q^{26} -12.2957 q^{28} +5.78321 q^{31} +16.5161 q^{32} +12.9519 q^{34} +7.37288 q^{35} +6.89817 q^{37} +4.39190 q^{38} -26.5824 q^{40} +7.66791 q^{41} +1.88741 q^{43} -13.5017 q^{44} +3.17231 q^{46} +9.30167 q^{47} -1.38200 q^{49} +12.5360 q^{50} -11.9756 q^{52} +5.17313 q^{53} +8.09606 q^{55} -20.2553 q^{56} +4.90494 q^{59} -11.5358 q^{61} +15.5046 q^{62} +19.2078 q^{64} +7.18096 q^{65} -6.97413 q^{67} +25.0615 q^{68} +19.7664 q^{70} -3.73571 q^{71} +2.39344 q^{73} +18.4937 q^{74} +8.49816 q^{76} +6.16905 q^{77} +3.75941 q^{79} -38.9935 q^{80} +20.5574 q^{82} +10.5949 q^{83} -15.0276 q^{85} +5.06009 q^{86} -22.2421 q^{88} +5.70261 q^{89} +5.47175 q^{91} +6.13830 q^{92} +24.9374 q^{94} -5.09576 q^{95} +16.7931 q^{97} -3.70510 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + q^{2} + 11 q^{4} + 5 q^{7} + 12 q^{8}+O(q^{10}) $ 9 * q + q^2 + 11 * q^4 + 5 * q^7 + 12 * q^8	$ 9 q + q^{2} + 11 q^{4} + 5 q^{7} + 12 q^{8} - 4 q^{10} + 3 q^{11} + 5 q^{13} + 15 q^{14} - 5 q^{16} + 16 q^{17} + q^{19} - 8 q^{20} + 24 q^{22} + 10 q^{23} + 9 q^{25} + 12 q^{26} - 24 q^{28} + 4 q^{31} + 25 q^{32} + 24 q^{34} + 44 q^{35} - 25 q^{37} + 10 q^{38} - 5 q^{40} + 34 q^{41} - 12 q^{43} - 23 q^{44} - 6 q^{46} + 8 q^{47} + 26 q^{49} + 27 q^{50} - 23 q^{52} + 32 q^{53} + 5 q^{55} - 14 q^{56} - 10 q^{59} - 51 q^{61} - 8 q^{62} - 8 q^{64} + 11 q^{65} + 7 q^{67} + 11 q^{68} + 14 q^{70} - 7 q^{71} - 17 q^{73} + 62 q^{74} + 6 q^{76} + 64 q^{77} - 13 q^{79} - 54 q^{80} + 37 q^{82} + 31 q^{83} - 42 q^{85} + 70 q^{86} - 29 q^{88} - 32 q^{89} + 45 q^{91} - 9 q^{92} + 38 q^{94} - 20 q^{95} - 16 q^{97} - 12 q^{98}+O(q^{100}) $ 9 * q + q^2 + 11 * q^4 + 5 * q^7 + 12 * q^8 - 4 * q^10 + 3 * q^11 + 5 * q^13 + 15 * q^14 - 5 * q^16 + 16 * q^17 + q^19 - 8 * q^20 + 24 * q^22 + 10 * q^23 + 9 * q^25 + 12 * q^26 - 24 * q^28 + 4 * q^31 + 25 * q^32 + 24 * q^34 + 44 * q^35 - 25 * q^37 + 10 * q^38 - 5 * q^40 + 34 * q^41 - 12 * q^43 - 23 * q^44 - 6 * q^46 + 8 * q^47 + 26 * q^49 + 27 * q^50 - 23 * q^52 + 32 * q^53 + 5 * q^55 - 14 * q^56 - 10 * q^59 - 51 * q^61 - 8 * q^62 - 8 * q^64 + 11 * q^65 + 7 * q^67 + 11 * q^68 + 14 * q^70 - 7 * q^71 - 17 * q^73 + 62 * q^74 + 6 * q^76 + 64 * q^77 - 13 * q^79 - 54 * q^80 + 37 * q^82 + 31 * q^83 - 42 * q^85 + 70 * q^86 - 29 * q^88 - 32 * q^89 + 45 * q^91 - 9 * q^92 + 38 * q^94 - 20 * q^95 - 16 * q^97 - 12 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	2.68096	1.89573	0.947863	−	0.318678i	$-0.103239\pi$
$2$	2.68096	1.89573	0.947863	+	0.318678i	$0.103239\pi$
$3$	0	0
$3$	0	0
$4$	5.18756	2.59378
$5$	−3.11062	−1.39111	−0.695555	−	0.718473i	$-0.744842\pi$
$5$	−3.11062	−1.39111	−0.695555	+	0.718473i	$0.744842\pi$
$6$	0	0
$7$	−2.37023	−0.895863	−0.447932	−	0.894068i	$-0.647839\pi$
$7$	−2.37023	−0.895863	−0.447932	+	0.894068i	$0.647839\pi$
$8$	8.54571	3.02137
$9$	0	0
$10$	−8.33945	−2.63716
$11$	−2.60272	−0.784749	−0.392375	−	0.919805i	$-0.628346\pi$
$11$	−2.60272	−0.784749	−0.392375	+	0.919805i	$0.628346\pi$
$12$	0	0
$13$	−2.30853	−0.640271	−0.320136	−	0.947372i	$-0.603729\pi$
$13$	−2.30853	−0.640271	−0.320136	+	0.947372i	$0.603729\pi$
$14$	−6.35450	−1.69831
$15$	0	0
$16$	12.5356	3.13391
$17$	4.83107	1.17171	0.585854	−	0.810417i	$-0.300759\pi$
$17$	4.83107	1.17171	0.585854	+	0.810417i	$0.300759\pi$
$18$	0	0
$19$	1.63818	0.375825	0.187912	−	0.982186i	$-0.439828\pi$
$19$	1.63818	0.375825	0.187912	+	0.982186i	$0.439828\pi$
$20$	−16.1365	−3.60823
$21$	0	0
$22$	−6.97779	−1.48767
$23$	1.18327	0.246730	0.123365	−	0.992361i	$-0.460631\pi$
$23$	1.18327	0.246730	0.123365	+	0.992361i	$0.460631\pi$
$24$	0	0
$25$	4.67594	0.935188
$26$	−6.18908	−1.21378
$27$	0	0
$28$	−12.2957	−2.32367
$29$	0	0
$29$	0	0
$30$	0	0
$31$	5.78321	1.03869	0.519347	−	0.854563i	$-0.326175\pi$
$31$	5.78321	1.03869	0.519347	+	0.854563i	$0.326175\pi$
$32$	16.5161	2.91966
$33$	0	0
$34$	12.9519	2.22124
$35$	7.37288	1.24624
$36$	0	0
$37$	6.89817	1.13405	0.567026	−	0.823700i	$-0.308094\pi$
$37$	6.89817	1.13405	0.567026	+	0.823700i	$0.308094\pi$
$38$	4.39190	0.712461
$39$	0	0
$40$	−26.5824	−4.20305
$41$	7.66791	1.19753	0.598763	−	0.800926i	$-0.295659\pi$
$41$	7.66791	1.19753	0.598763	+	0.800926i	$0.295659\pi$
$42$	0	0
$43$	1.88741	0.287828	0.143914	−	0.989590i	$-0.454031\pi$
$43$	1.88741	0.287828	0.143914	+	0.989590i	$0.454031\pi$
$44$	−13.5017	−2.03547
$45$	0	0
$46$	3.17231	0.467732
$47$	9.30167	1.35679	0.678394	−	0.734698i	$-0.262676\pi$
$47$	9.30167	1.35679	0.678394	+	0.734698i	$0.262676\pi$
$48$	0	0
$49$	−1.38200	−0.197429
$50$	12.5360	1.77286
$51$	0	0
$52$	−11.9756	−1.66072
$53$	5.17313	0.710584	0.355292	−	0.934755i	$-0.384381\pi$
$53$	5.17313	0.710584	0.355292	+	0.934755i	$0.384381\pi$
$54$	0	0
$55$	8.09606	1.09167
$56$	−20.2553	−2.70673
$57$	0	0
$58$	0	0
$59$	4.90494	0.638569	0.319285	−	0.947659i	$-0.396557\pi$
$59$	4.90494	0.638569	0.319285	+	0.947659i	$0.396557\pi$
$60$	0	0
$61$	−11.5358	−1.47701	−0.738507	−	0.674246i	$-0.764469\pi$
$61$	−11.5358	−1.47701	−0.738507	+	0.674246i	$0.764469\pi$
$62$	15.5046	1.96908
$63$	0	0
$64$	19.2078	2.40097
$65$	7.18096	0.890688
$66$	0	0
$67$	−6.97413	−0.852026	−0.426013	−	0.904717i	$-0.640082\pi$
$67$	−6.97413	−0.852026	−0.426013	+	0.904717i	$0.640082\pi$
$68$	25.0615	3.03915
$69$	0	0
$70$	19.7664	2.36254
$71$	−3.73571	−0.443347	−0.221674	−	0.975121i	$-0.571152\pi$
$71$	−3.73571	−0.443347	−0.221674	+	0.975121i	$0.571152\pi$
$72$	0	0
$73$	2.39344	0.280132	0.140066	−	0.990142i	$-0.455269\pi$
$73$	2.39344	0.280132	0.140066	+	0.990142i	$0.455269\pi$
$74$	18.4937	2.14985
$75$	0	0
$76$	8.49816	0.974806
$77$	6.16905	0.703028
$78$	0	0
$79$	3.75941	0.422967	0.211483	−	0.977382i	$-0.432171\pi$
$79$	3.75941	0.422967	0.211483	+	0.977382i	$0.432171\pi$
$80$	−38.9935	−4.35961
$81$	0	0
$82$	20.5574	2.27018
$83$	10.5949	1.16294	0.581472	−	0.813566i	$-0.302477\pi$
$83$	10.5949	1.16294	0.581472	+	0.813566i	$0.302477\pi$
$84$	0	0
$85$	−15.0276	−1.62997
$86$	5.06009	0.545643
$87$	0	0
$88$	−22.2421	−2.37101
$89$	5.70261	0.604475	0.302238	−	0.953233i	$-0.402266\pi$
$89$	5.70261	0.604475	0.302238	+	0.953233i	$0.402266\pi$
$90$	0	0
$91$	5.47175	0.573596
$92$	6.13830	0.639962
$93$	0	0
$94$	24.9374	2.57210
$95$	−5.09576	−0.522814
$96$	0	0
$97$	16.7931	1.70508	0.852539	−	0.522663i	$-0.175061\pi$
$97$	16.7931	1.70508	0.852539	+	0.522663i	$0.175061\pi$
$98$	−3.70510	−0.374271
$99$	0	0
$100$	24.2567	2.42567
$101$	−4.58223	−0.455949	−0.227974	−	0.973667i	$-0.573210\pi$
$101$	−4.58223	−0.455949	−0.227974	+	0.973667i	$0.573210\pi$
$102$	0	0
$103$	10.5312	1.03767	0.518834	−	0.854875i	$-0.326366\pi$
$103$	10.5312	1.03767	0.518834	+	0.854875i	$0.326366\pi$
$104$	−19.7281	−1.93449
$105$	0	0
$106$	13.8690	1.34707
$107$	−13.7534	−1.32959	−0.664794	−	0.747026i	$-0.731481\pi$
$107$	−13.7534	−1.32959	−0.664794	+	0.747026i	$0.731481\pi$
$108$	0	0
$109$	−5.35230	−0.512657	−0.256329	−	0.966590i	$-0.582513\pi$
$109$	−5.35230	−0.512657	−0.256329	+	0.966590i	$0.582513\pi$
$110$	21.7052	2.06951
$111$	0	0
$112$	−29.7123	−2.80755
$113$	10.7979	1.01578	0.507892	−	0.861421i	$-0.330425\pi$
$113$	10.7979	1.01578	0.507892	+	0.861421i	$0.330425\pi$
$114$	0	0
$115$	−3.68071	−0.343228
$116$	0	0
$117$	0	0
$118$	13.1500	1.21055
$119$	−11.4508	−1.04969
$120$	0	0
$121$	−4.22585	−0.384169
$122$	−30.9272	−2.80001
$123$	0	0
$124$	30.0007	2.69414
$125$	1.00803	0.0901609
$126$	0	0
$127$	16.8611	1.49618	0.748088	−	0.663599i	$-0.230972\pi$
$127$	16.8611	1.49618	0.748088	+	0.663599i	$0.230972\pi$
$128$	18.4631	1.63192
$129$	0	0
$130$	19.2519	1.68850
$131$	−4.55312	−0.397808	−0.198904	−	0.980019i	$-0.563738\pi$
$131$	−4.55312	−0.397808	−0.198904	+	0.980019i	$0.563738\pi$
$132$	0	0
$133$	−3.88287	−0.336688
$134$	−18.6974	−1.61521
$135$	0	0
$136$	41.2850	3.54016
$137$	22.0092	1.88038	0.940188	−	0.340657i	$-0.110650\pi$
$137$	22.0092	1.88038	0.940188	+	0.340657i	$0.110650\pi$
$138$	0	0
$139$	−8.09367	−0.686496	−0.343248	−	0.939245i	$-0.611527\pi$
$139$	−8.09367	−0.686496	−0.343248	+	0.939245i	$0.611527\pi$
$140$	38.2472	3.23248
$141$	0	0
$142$	−10.0153	−0.840465
$143$	6.00846	0.502453
$144$	0	0
$145$	0	0
$146$	6.41673	0.531053
$147$	0	0
$148$	35.7846	2.94148
$149$	−0.743685	−0.0609251	−0.0304625	−	0.999536i	$-0.509698\pi$
$149$	−0.743685	−0.0609251	−0.0304625	+	0.999536i	$0.509698\pi$
$150$	0	0
$151$	−17.2880	−1.40688	−0.703440	−	0.710755i	$-0.748354\pi$
$151$	−17.2880	−1.40688	−0.703440	+	0.710755i	$0.748354\pi$
$152$	13.9994	1.13550
$153$	0	0
$154$	16.5390	1.33275
$155$	−17.9893	−1.44494
$156$	0	0
$157$	−0.399286	−0.0318665	−0.0159332	−	0.999873i	$-0.505072\pi$
$157$	−0.399286	−0.0318665	−0.0159332	+	0.999873i	$0.505072\pi$
$158$	10.0788	0.801829
$159$	0	0
$160$	−51.3752	−4.06157
$161$	−2.80463	−0.221036
$162$	0	0
$163$	9.91343	0.776479	0.388240	−	0.921558i	$-0.373083\pi$
$163$	9.91343	0.776479	0.388240	+	0.921558i	$0.373083\pi$
$164$	39.7777	3.10612
$165$	0	0
$166$	28.4046	2.20462
$167$	11.7190	0.906842	0.453421	−	0.891297i	$-0.350203\pi$
$167$	11.7190	0.906842	0.453421	+	0.891297i	$0.350203\pi$
$168$	0	0
$169$	−7.67068	−0.590052
$170$	−40.2885	−3.08999
$171$	0	0
$172$	9.79107	0.746562
$173$	−8.14339	−0.619131	−0.309565	−	0.950878i	$-0.600184\pi$
$173$	−8.14339	−0.619131	−0.309565	+	0.950878i	$0.600184\pi$
$174$	0	0
$175$	−11.0831	−0.837800
$176$	−32.6267	−2.45933
$177$	0	0
$178$	15.2885	1.14592
$179$	−10.7785	−0.805625	−0.402812	−	0.915283i	$-0.631967\pi$
$179$	−10.7785	−0.805625	−0.402812	+	0.915283i	$0.631967\pi$
$180$	0	0
$181$	20.6125	1.53211	0.766057	−	0.642773i	$-0.222216\pi$
$181$	20.6125	1.53211	0.766057	+	0.642773i	$0.222216\pi$
$182$	14.6696	1.08738
$183$	0	0
$184$	10.1119	0.745461
$185$	−21.4576	−1.57759
$186$	0	0
$187$	−12.5739	−0.919497
$188$	48.2529	3.51921
$189$	0	0
$190$	−13.6615	−0.991112
$191$	24.4869	1.77181	0.885906	−	0.463865i	$-0.153538\pi$
$191$	24.4869	1.77181	0.885906	+	0.463865i	$0.153538\pi$
$192$	0	0
$193$	−15.8432	−1.14042	−0.570208	−	0.821501i	$-0.693137\pi$
$193$	−15.8432	−1.14042	−0.570208	+	0.821501i	$0.693137\pi$
$194$	45.0216	3.23236
$195$	0	0
$196$	−7.16922	−0.512087
$197$	−12.7505	−0.908436	−0.454218	−	0.890891i	$-0.650081\pi$
$197$	−12.7505	−0.908436	−0.454218	+	0.890891i	$0.650081\pi$
$198$	0	0
$199$	−8.12481	−0.575952	−0.287976	−	0.957638i	$-0.592982\pi$
$199$	−8.12481	−0.575952	−0.287976	+	0.957638i	$0.592982\pi$
$200$	39.9592	2.82555
$201$	0	0
$202$	−12.2848	−0.864354
$203$	0	0
$204$	0	0
$205$	−23.8519	−1.66589
$206$	28.2337	1.96713
$207$	0	0
$208$	−28.9389	−2.00655
$209$	−4.26373	−0.294928
$210$	0	0
$211$	−20.9954	−1.44538	−0.722692	−	0.691170i	$-0.757096\pi$
$211$	−20.9954	−1.44538	−0.722692	+	0.691170i	$0.757096\pi$
$212$	26.8359	1.84310
$213$	0	0
$214$	−36.8723	−2.52054
$215$	−5.87102	−0.400401
$216$	0	0
$217$	−13.7075	−0.930528
$218$	−14.3493	−0.971857
$219$	0	0
$220$	41.9988	2.83156
$221$	−11.1527	−0.750211
$222$	0	0
$223$	21.2070	1.42013	0.710064	−	0.704137i	$-0.248666\pi$
$223$	21.2070	1.42013	0.710064	+	0.704137i	$0.248666\pi$
$224$	−39.1470	−2.61562
$225$	0	0
$226$	28.9488	1.92565
$227$	−10.7043	−0.710469	−0.355234	−	0.934777i	$-0.615599\pi$
$227$	−10.7043	−0.710469	−0.355234	+	0.934777i	$0.615599\pi$
$228$	0	0
$229$	−14.0058	−0.925531	−0.462765	−	0.886481i	$-0.653143\pi$
$229$	−14.0058	−0.925531	−0.462765	+	0.886481i	$0.653143\pi$
$230$	−9.86785	−0.650667
$231$	0	0
$232$	0	0
$233$	12.3851	0.811377	0.405688	−	0.914011i	$-0.367032\pi$
$233$	12.3851	0.811377	0.405688	+	0.914011i	$0.367032\pi$
$234$	0	0
$235$	−28.9339	−1.88744
$236$	25.4447	1.65631
$237$	0	0
$238$	−30.6991	−1.98992
$239$	−8.90340	−0.575913	−0.287956	−	0.957644i	$-0.592976\pi$
$239$	−8.90340	−0.575913	−0.287956	+	0.957644i	$0.592976\pi$
$240$	0	0
$241$	−26.1498	−1.68446	−0.842228	−	0.539121i	$-0.818757\pi$
$241$	−26.1498	−1.68446	−0.842228	+	0.539121i	$0.818757\pi$
$242$	−11.3294	−0.728278
$243$	0	0
$244$	−59.8429	−3.83105
$245$	4.29888	0.274645
$246$	0	0
$247$	−3.78180	−0.240630
$248$	49.4216	3.13828
$249$	0	0
$250$	2.70249	0.170920
$251$	−3.87652	−0.244683	−0.122342	−	0.992488i	$-0.539040\pi$
$251$	−3.87652	−0.244683	−0.122342	+	0.992488i	$0.539040\pi$
$252$	0	0
$253$	−3.07973	−0.193621
$254$	45.2038	2.83634
$255$	0	0
$256$	11.0833	0.692709
$257$	1.61631	0.100823	0.0504114	−	0.998729i	$-0.483947\pi$
$257$	1.61631	0.100823	0.0504114	+	0.998729i	$0.483947\pi$
$258$	0	0
$259$	−16.3503	−1.01596
$260$	37.2516	2.31025
$261$	0	0
$262$	−12.2067	−0.754135
$263$	22.6623	1.39741	0.698707	−	0.715408i	$-0.253759\pi$
$263$	22.6623	1.39741	0.698707	+	0.715408i	$0.253759\pi$
$264$	0	0
$265$	−16.0916	−0.988500
$266$	−10.4098	−0.638267
$267$	0	0
$268$	−36.1787	−2.20997
$269$	5.22022	0.318283	0.159141	−	0.987256i	$-0.449127\pi$
$269$	5.22022	0.318283	0.159141	+	0.987256i	$0.449127\pi$
$270$	0	0
$271$	1.49938	0.0910808	0.0455404	−	0.998962i	$-0.485499\pi$
$271$	1.49938	0.0910808	0.0455404	+	0.998962i	$0.485499\pi$
$272$	60.5605	3.67202
$273$	0	0
$274$	59.0059	3.56468
$275$	−12.1702	−0.733888
$276$	0	0
$277$	2.18118	0.131055	0.0655273	−	0.997851i	$-0.479127\pi$
$277$	2.18118	0.131055	0.0655273	+	0.997851i	$0.479127\pi$
$278$	−21.6988	−1.30141
$279$	0	0
$280$	63.0066	3.76536
$281$	17.2274	1.02770	0.513850	−	0.857880i	$-0.328219\pi$
$281$	17.2274	1.02770	0.513850	+	0.857880i	$0.328219\pi$
$282$	0	0
$283$	−11.4305	−0.679470	−0.339735	−	0.940521i	$-0.610337\pi$
$283$	−11.4305	−0.679470	−0.339735	+	0.940521i	$0.610337\pi$
$284$	−19.3792	−1.14994
$285$	0	0
$286$	16.1084	0.952512
$287$	−18.1747	−1.07282
$288$	0	0
$289$	6.33927	0.372898
$290$	0	0
$291$	0	0
$292$	12.4161	0.726599
$293$	5.12453	0.299378	0.149689	−	0.988733i	$-0.452173\pi$
$293$	5.12453	0.299378	0.149689	+	0.988733i	$0.452173\pi$
$294$	0	0
$295$	−15.2574	−0.888320
$296$	58.9498	3.42639
$297$	0	0
$298$	−1.99379	−0.115497
$299$	−2.73163	−0.157974
$300$	0	0
$301$	−4.47361	−0.257855
$302$	−46.3486	−2.66706
$303$	0	0
$304$	20.5356	1.17780
$305$	35.8836	2.05469
$306$	0	0
$307$	3.39033	0.193496	0.0967482	−	0.995309i	$-0.469156\pi$
$307$	3.39033	0.193496	0.0967482	+	0.995309i	$0.469156\pi$
$308$	32.0023	1.82350
$309$	0	0
$310$	−48.2287	−2.73921
$311$	−17.1182	−0.970687	−0.485343	−	0.874324i	$-0.661305\pi$
$311$	−17.1182	−0.970687	−0.485343	+	0.874324i	$0.661305\pi$
$312$	0	0
$313$	−1.44818	−0.0818559	−0.0409280	−	0.999162i	$-0.513031\pi$
$313$	−1.44818	−0.0818559	−0.0409280	+	0.999162i	$0.513031\pi$
$314$	−1.07047	−0.0604101
$315$	0	0
$316$	19.5022	1.09708
$317$	17.5169	0.983849	0.491925	−	0.870638i	$-0.336294\pi$
$317$	17.5169	0.983849	0.491925	+	0.870638i	$0.336294\pi$
$318$	0	0
$319$	0	0
$320$	−59.7480	−3.34002
$321$	0	0
$322$	−7.51912	−0.419024
$323$	7.91418	0.440357
$324$	0	0
$325$	−10.7946	−0.598774
$326$	26.5775	1.47199
$327$	0	0
$328$	65.5278	3.61817
$329$	−22.0471	−1.21550
$330$	0	0
$331$	10.4276	0.573155	0.286577	−	0.958057i	$-0.407482\pi$
$331$	10.4276	0.573155	0.286577	+	0.958057i	$0.407482\pi$
$332$	54.9618	3.01642
$333$	0	0
$334$	31.4181	1.71912
$335$	21.6939	1.18526
$336$	0	0
$337$	−32.2012	−1.75411	−0.877056	−	0.480387i	$-0.840496\pi$
$337$	−32.2012	−1.75411	−0.877056	+	0.480387i	$0.840496\pi$
$338$	−20.5648	−1.11858
$339$	0	0
$340$	−77.9566	−4.22779
$341$	−15.0521	−0.815115
$342$	0	0
$343$	19.8673	1.07273
$344$	16.1293	0.869634
$345$	0	0
$346$	−21.8321	−1.17370
$347$	−6.96578	−0.373942	−0.186971	−	0.982365i	$-0.559867\pi$
$347$	−6.96578	−0.373942	−0.186971	+	0.982365i	$0.559867\pi$
$348$	0	0
$349$	13.9714	0.747871	0.373935	−	0.927455i	$-0.378008\pi$
$349$	13.9714	0.747871	0.373935	+	0.927455i	$0.378008\pi$
$350$	−29.7133	−1.58824
$351$	0	0
$352$	−42.9867	−2.29120
$353$	−25.8422	−1.37544	−0.687722	−	0.725974i	$-0.741389\pi$
$353$	−25.8422	−1.37544	−0.687722	+	0.725974i	$0.741389\pi$
$354$	0	0
$355$	11.6204	0.616745
$356$	29.5826	1.56787
$357$	0	0
$358$	−28.8968	−1.52724
$359$	17.5435	0.925909	0.462955	−	0.886382i	$-0.346789\pi$
$359$	17.5435	0.925909	0.462955	+	0.886382i	$0.346789\pi$
$360$	0	0
$361$	−16.3164	−0.858756
$362$	55.2613	2.90447
$363$	0	0
$364$	28.3850	1.48778
$365$	−7.44509	−0.389694
$366$	0	0
$367$	29.7629	1.55361	0.776805	−	0.629741i	$-0.216839\pi$
$367$	29.7629	1.55361	0.776805	+	0.629741i	$0.216839\pi$
$368$	14.8331	0.773228
$369$	0	0
$370$	−57.5269	−2.99068
$371$	−12.2615	−0.636586
$372$	0	0
$373$	19.2955	0.999082	0.499541	−	0.866290i	$-0.333502\pi$
$373$	19.2955	0.999082	0.499541	+	0.866290i	$0.333502\pi$
$374$	−33.7102	−1.74311
$375$	0	0
$376$	79.4894	4.09936
$377$	0	0
$378$	0	0
$379$	−8.58926	−0.441200	−0.220600	−	0.975364i	$-0.570802\pi$
$379$	−8.58926	−0.441200	−0.220600	+	0.975364i	$0.570802\pi$
$380$	−26.4345	−1.35606
$381$	0	0
$382$	65.6485	3.35887
$383$	28.2918	1.44564	0.722822	−	0.691034i	$-0.242845\pi$
$383$	28.2918	1.44564	0.722822	+	0.691034i	$0.242845\pi$
$384$	0	0
$385$	−19.1895	−0.977990
$386$	−42.4749	−2.16191
$387$	0	0
$388$	87.1150	4.42260
$389$	27.1843	1.37830	0.689149	−	0.724619i	$-0.257984\pi$
$389$	27.1843	1.37830	0.689149	+	0.724619i	$0.257984\pi$
$390$	0	0
$391$	5.71649	0.289095
$392$	−11.8102	−0.596505
$393$	0	0
$394$	−34.1836	−1.72215
$395$	−11.6941	−0.588394
$396$	0	0
$397$	4.95624	0.248747	0.124373	−	0.992235i	$-0.460308\pi$
$397$	4.95624	0.248747	0.124373	+	0.992235i	$0.460308\pi$
$398$	−21.7823	−1.09185
$399$	0	0
$400$	58.6158	2.93079
$401$	−6.21952	−0.310588	−0.155294	−	0.987868i	$-0.549632\pi$
$401$	−6.21952	−0.310588	−0.155294	+	0.987868i	$0.549632\pi$
$402$	0	0
$403$	−13.3507	−0.665046
$404$	−23.7706	−1.18263
$405$	0	0
$406$	0	0
$407$	−17.9540	−0.889946
$408$	0	0
$409$	4.63871	0.229369	0.114685	−	0.993402i	$-0.463414\pi$
$409$	4.63871	0.229369	0.114685	+	0.993402i	$0.463414\pi$
$410$	−63.9461	−3.15807
$411$	0	0
$412$	54.6311	2.69148
$413$	−11.6259	−0.572071
$414$	0	0
$415$	−32.9568	−1.61778
$416$	−38.1279	−1.86938
$417$	0	0
$418$	−11.4309	−0.559103
$419$	−13.6679	−0.667722	−0.333861	−	0.942622i	$-0.608352\pi$
$419$	−13.6679	−0.667722	−0.333861	+	0.942622i	$0.608352\pi$
$420$	0	0
$421$	−11.9030	−0.580118	−0.290059	−	0.957009i	$-0.593675\pi$
$421$	−11.9030	−0.580118	−0.290059	+	0.957009i	$0.593675\pi$
$422$	−56.2879	−2.74005
$423$	0	0
$424$	44.2081	2.14693
$425$	22.5898	1.09577
$426$	0	0
$427$	27.3426	1.32320
$428$	−71.3464	−3.44866
$429$	0	0
$430$	−15.7400	−0.759050
$431$	22.7320	1.09496	0.547480	−	0.836819i	$-0.315587\pi$
$431$	22.7320	1.09496	0.547480	+	0.836819i	$0.315587\pi$
$432$	0	0
$433$	−5.51061	−0.264823	−0.132412	−	0.991195i	$-0.542272\pi$
$433$	−5.51061	−0.264823	−0.132412	+	0.991195i	$0.542272\pi$
$434$	−36.7494	−1.76403
$435$	0	0
$436$	−27.7653	−1.32972
$437$	1.93842	0.0927271
$438$	0	0
$439$	20.2638	0.967140	0.483570	−	0.875306i	$-0.339340\pi$
$439$	20.2638	0.967140	0.483570	+	0.875306i	$0.339340\pi$
$440$	69.1866	3.29834
$441$	0	0
$442$	−29.8999	−1.42219
$443$	−33.8979	−1.61054	−0.805269	−	0.592910i	$-0.797979\pi$
$443$	−33.8979	−1.61054	−0.805269	+	0.592910i	$0.797979\pi$
$444$	0	0
$445$	−17.7386	−0.840892
$446$	56.8553	2.69217
$447$	0	0
$448$	−45.5269	−2.15094
$449$	−32.2546	−1.52219	−0.761095	−	0.648640i	$-0.775338\pi$
$449$	−32.2546	−1.52219	−0.761095	+	0.648640i	$0.775338\pi$
$450$	0	0
$451$	−19.9574	−0.939758
$452$	56.0148	2.63472
$453$	0	0
$454$	−28.6978	−1.34685
$455$	−17.0205	−0.797935
$456$	0	0
$457$	−26.1317	−1.22239	−0.611194	−	0.791480i	$-0.709311\pi$
$457$	−26.1317	−1.22239	−0.611194	+	0.791480i	$0.709311\pi$
$458$	−37.5491	−1.75455
$459$	0	0
$460$	−19.0939	−0.890258
$461$	12.5362	0.583867	0.291933	−	0.956439i	$-0.405701\pi$
$461$	12.5362	0.583867	0.291933	+	0.956439i	$0.405701\pi$
$462$	0	0
$463$	−38.0989	−1.77061	−0.885303	−	0.465014i	$-0.846049\pi$
$463$	−38.0989	−1.77061	−0.885303	+	0.465014i	$0.846049\pi$
$464$	0	0
$465$	0	0
$466$	33.2041	1.53815
$467$	6.90918	0.319719	0.159859	−	0.987140i	$-0.448896\pi$
$467$	6.90918	0.319719	0.159859	+	0.987140i	$0.448896\pi$
$468$	0	0
$469$	16.5303	0.763299
$470$	−77.5708	−3.57807
$471$	0	0
$472$	41.9163	1.92935
$473$	−4.91241	−0.225873
$474$	0	0
$475$	7.66004	0.351467
$476$	−59.4015	−2.72266
$477$	0	0
$478$	−23.8697	−1.09177
$479$	15.2304	0.695894	0.347947	−	0.937514i	$-0.386879\pi$
$479$	15.2304	0.695894	0.347947	+	0.937514i	$0.386879\pi$
$480$	0	0
$481$	−15.9246	−0.726101
$482$	−70.1066	−3.19327
$483$	0	0
$484$	−21.9219	−0.996448
$485$	−52.2368	−2.37195
$486$	0	0
$487$	−31.6610	−1.43470	−0.717350	−	0.696713i	$-0.754645\pi$
$487$	−31.6610	−1.43470	−0.717350	+	0.696713i	$0.754645\pi$
$488$	−98.5821	−4.46260
$489$	0	0
$490$	11.5251	0.520653
$491$	−10.7278	−0.484137	−0.242068	−	0.970259i	$-0.577826\pi$
$491$	−10.7278	−0.484137	−0.242068	+	0.970259i	$0.577826\pi$
$492$	0	0
$493$	0	0
$494$	−10.1388	−0.456168
$495$	0	0
$496$	72.4961	3.25517
$497$	8.85450	0.397179
$498$	0	0
$499$	−9.76966	−0.437350	−0.218675	−	0.975798i	$-0.570173\pi$
$499$	−9.76966	−0.437350	−0.218675	+	0.975798i	$0.570173\pi$
$500$	5.22921	0.233857
$501$	0	0
$502$	−10.3928	−0.463853
$503$	24.2897	1.08302	0.541512	−	0.840693i	$-0.317852\pi$
$503$	24.2897	1.08302	0.541512	+	0.840693i	$0.317852\pi$
$504$	0	0
$505$	14.2536	0.634275
$506$	−8.25664	−0.367052
$507$	0	0
$508$	87.4677	3.88075
$509$	4.04253	0.179182	0.0895910	−	0.995979i	$-0.471444\pi$
$509$	4.04253	0.179182	0.0895910	+	0.995979i	$0.471444\pi$
$510$	0	0
$511$	−5.67302	−0.250960
$512$	−7.21219	−0.318737
$513$	0	0
$514$	4.33327	0.191133
$515$	−32.7585	−1.44351
$516$	0	0
$517$	−24.2096	−1.06474
$518$	−43.8344	−1.92597
$519$	0	0
$520$	61.3664	2.69110
$521$	20.3240	0.890412	0.445206	−	0.895428i	$-0.353130\pi$
$521$	20.3240	0.890412	0.445206	+	0.895428i	$0.353130\pi$
$522$	0	0
$523$	0.181906	0.00795418	0.00397709	−	0.999992i	$-0.498734\pi$
$523$	0.181906	0.00795418	0.00397709	+	0.999992i	$0.498734\pi$
$524$	−23.6196	−1.03183
$525$	0	0
$526$	60.7566	2.64911
$527$	27.9391	1.21705
$528$	0	0
$529$	−21.5999	−0.939124
$530$	−43.1410	−1.87393
$531$	0	0
$532$	−20.1426	−0.873293
$533$	−17.7016	−0.766742
$534$	0	0
$535$	42.7815	1.84960
$536$	−59.5989	−2.57428
$537$	0	0
$538$	13.9952	0.603377
$539$	3.59696	0.154932
$540$	0	0
$541$	−26.9847	−1.16016	−0.580082	−	0.814558i	$-0.696979\pi$
$541$	−26.9847	−1.16016	−0.580082	+	0.814558i	$0.696979\pi$
$542$	4.01978	0.172664
$543$	0	0
$544$	79.7905	3.42099
$545$	16.6489	0.713162
$546$	0	0
$547$	8.74053	0.373718	0.186859	−	0.982387i	$-0.440169\pi$
$547$	8.74053	0.373718	0.186859	+	0.982387i	$0.440169\pi$
$548$	114.174	4.87728
$549$	0	0
$550$	−32.6277	−1.39125
$551$	0	0
$552$	0	0
$553$	−8.91068	−0.378920
$554$	5.84767	0.248444
$555$	0	0
$556$	−41.9864	−1.78062
$557$	28.3288	1.20033	0.600166	−	0.799876i	$-0.295101\pi$
$557$	28.3288	1.20033	0.600166	+	0.799876i	$0.295101\pi$
$558$	0	0
$559$	−4.35716	−0.184288
$560$	92.4237	3.90561
$561$	0	0
$562$	46.1860	1.94824
$563$	1.24481	0.0524623	0.0262312	−	0.999656i	$-0.491649\pi$
$563$	1.24481	0.0524623	0.0262312	+	0.999656i	$0.491649\pi$
$564$	0	0
$565$	−33.5882	−1.41307
$566$	−30.6446	−1.28809
$567$	0	0
$568$	−31.9243	−1.33951
$569$	8.25260	0.345967	0.172983	−	0.984925i	$-0.444659\pi$
$569$	8.25260	0.345967	0.172983	+	0.984925i	$0.444659\pi$
$570$	0	0
$571$	27.8020	1.16348	0.581738	−	0.813376i	$-0.302373\pi$
$571$	27.8020	1.16348	0.581738	+	0.813376i	$0.302373\pi$
$572$	31.1692	1.30325
$573$	0	0
$574$	−48.7258	−2.03377
$575$	5.53292	0.230739
$576$	0	0
$577$	7.19670	0.299602	0.149801	−	0.988716i	$-0.452137\pi$
$577$	7.19670	0.299602	0.149801	+	0.988716i	$0.452137\pi$
$578$	16.9954	0.706913
$579$	0	0
$580$	0	0
$581$	−25.1124	−1.04184
$582$	0	0
$583$	−13.4642	−0.557630
$584$	20.4537	0.846380
$585$	0	0
$586$	13.7387	0.567539
$587$	−8.46040	−0.349198	−0.174599	−	0.984640i	$-0.555863\pi$
$587$	−8.46040	−0.349198	−0.174599	+	0.984640i	$0.555863\pi$
$588$	0	0
$589$	9.47394	0.390367
$590$	−40.9045	−1.68401
$591$	0	0
$592$	86.4728	3.55401
$593$	28.7932	1.18239	0.591197	−	0.806527i	$-0.298656\pi$
$593$	28.7932	1.18239	0.591197	+	0.806527i	$0.298656\pi$
$594$	0	0
$595$	35.6189	1.46023
$596$	−3.85791	−0.158026
$597$	0	0
$598$	−7.32338	−0.299475
$599$	2.47991	0.101327	0.0506633	−	0.998716i	$-0.483866\pi$
$599$	2.47991	0.101327	0.0506633	+	0.998716i	$0.483866\pi$
$600$	0	0
$601$	−39.7004	−1.61941	−0.809706	−	0.586836i	$-0.800373\pi$
$601$	−39.7004	−1.61941	−0.809706	+	0.586836i	$0.800373\pi$
$602$	−11.9936	−0.488822
$603$	0	0
$604$	−89.6826	−3.64913
$605$	13.1450	0.534421
$606$	0	0
$607$	35.3535	1.43495	0.717476	−	0.696583i	$-0.245297\pi$
$607$	35.3535	1.43495	0.717476	+	0.696583i	$0.245297\pi$
$608$	27.0564	1.09728
$609$	0	0
$610$	96.2026	3.89513
$611$	−21.4732	−0.868713
$612$	0	0
$613$	8.09395	0.326912	0.163456	−	0.986551i	$-0.447736\pi$
$613$	8.09395	0.326912	0.163456	+	0.986551i	$0.447736\pi$
$614$	9.08935	0.366816
$615$	0	0
$616$	52.7189	2.12411
$617$	−27.3619	−1.10155	−0.550775	−	0.834654i	$-0.685668\pi$
$617$	−27.3619	−1.10155	−0.550775	+	0.834654i	$0.685668\pi$
$618$	0	0
$619$	27.1831	1.09258	0.546291	−	0.837596i	$-0.316039\pi$
$619$	27.1831	1.09258	0.546291	+	0.837596i	$0.316039\pi$
$620$	−93.3207	−3.74785
$621$	0	0
$622$	−45.8934	−1.84016
$623$	−13.5165	−0.541527
$624$	0	0
$625$	−26.5153	−1.06061
$626$	−3.88251	−0.155176
$627$	0	0
$628$	−2.07132	−0.0826545
$629$	33.3256	1.32878
$630$	0	0
$631$	19.6997	0.784231	0.392116	−	0.919916i	$-0.371743\pi$
$631$	19.6997	0.784231	0.392116	+	0.919916i	$0.371743\pi$
$632$	32.1269	1.27794
$633$	0	0
$634$	46.9622	1.86511
$635$	−52.4483	−2.08135
$636$	0	0
$637$	3.19040	0.126408
$638$	0	0
$639$	0	0
$640$	−57.4317	−2.27019
$641$	−0.280270	−0.0110700	−0.00553501	−	0.999985i	$-0.501762\pi$
$641$	−0.280270	−0.0110700	−0.00553501	+	0.999985i	$0.501762\pi$
$642$	0	0
$643$	−47.1001	−1.85745	−0.928724	−	0.370773i	$-0.879093\pi$
$643$	−47.1001	−1.85745	−0.928724	+	0.370773i	$0.879093\pi$
$644$	−14.5492	−0.573319
$645$	0	0
$646$	21.2176	0.834796
$647$	1.09365	0.0429956	0.0214978	−	0.999769i	$-0.493157\pi$
$647$	1.09365	0.0429956	0.0214978	+	0.999769i	$0.493157\pi$
$648$	0	0
$649$	−12.7662	−0.501117
$650$	−28.9398	−1.13511
$651$	0	0
$652$	51.4264	2.01402
$653$	29.8076	1.16646	0.583230	−	0.812307i	$-0.301788\pi$
$653$	29.8076	1.16646	0.583230	+	0.812307i	$0.301788\pi$
$654$	0	0
$655$	14.1630	0.553395
$656$	96.1221	3.75294
$657$	0	0
$658$	−59.1075	−2.30425
$659$	33.3371	1.29863	0.649315	−	0.760519i	$-0.275056\pi$
$659$	33.3371	1.29863	0.649315	+	0.760519i	$0.275056\pi$
$660$	0	0
$661$	2.44877	0.0952462	0.0476231	−	0.998865i	$-0.484835\pi$
$661$	2.44877	0.0952462	0.0476231	+	0.998865i	$0.484835\pi$
$662$	27.9561	1.08654
$663$	0	0
$664$	90.5412	3.51368
$665$	12.0781	0.468370
$666$	0	0
$667$	0	0
$668$	60.7929	2.35215
$669$	0	0
$670$	58.1604	2.24693
$671$	30.0246	1.15909
$672$	0	0
$673$	11.2261	0.432734	0.216367	−	0.976312i	$-0.430579\pi$
$673$	11.2261	0.432734	0.216367	+	0.976312i	$0.430579\pi$
$674$	−86.3303	−3.32532
$675$	0	0
$676$	−39.7921	−1.53046
$677$	5.12465	0.196956	0.0984781	−	0.995139i	$-0.468603\pi$
$677$	5.12465	0.196956	0.0984781	+	0.995139i	$0.468603\pi$
$678$	0	0
$679$	−39.8035	−1.52752
$680$	−128.422	−4.92475
$681$	0	0
$682$	−40.3540	−1.54523
$683$	−29.5618	−1.13115	−0.565575	−	0.824697i	$-0.691346\pi$
$683$	−29.5618	−1.13115	−0.565575	+	0.824697i	$0.691346\pi$
$684$	0	0
$685$	−68.4623	−2.61581
$686$	53.2634	2.03361
$687$	0	0
$688$	23.6599	0.902026
$689$	−11.9423	−0.454966
$690$	0	0
$691$	44.2064	1.68169	0.840845	−	0.541275i	$-0.182058\pi$
$691$	44.2064	1.68169	0.840845	+	0.541275i	$0.182058\pi$
$692$	−42.2443	−1.60589
$693$	0	0
$694$	−18.6750	−0.708892
$695$	25.1763	0.954992
$696$	0	0
$697$	37.0443	1.40315
$698$	37.4567	1.41776
$699$	0	0
$700$	−57.4940	−2.17307
$701$	25.6911	0.970341	0.485171	−	0.874420i	$-0.338758\pi$
$701$	25.6911	0.970341	0.485171	+	0.874420i	$0.338758\pi$
$702$	0	0
$703$	11.3005	0.426205
$704$	−49.9924	−1.88416
$705$	0	0
$706$	−69.2821	−2.60747
$707$	10.8609	0.408468
$708$	0	0
$709$	18.1323	0.680972	0.340486	−	0.940250i	$-0.389408\pi$
$709$	18.1323	0.680972	0.340486	+	0.940250i	$0.389408\pi$
$710$	31.1538	1.16918
$711$	0	0
$712$	48.7329	1.82634
$713$	6.84312	0.256277
$714$	0	0
$715$	−18.6900	−0.698967
$716$	−55.9142	−2.08961
$717$	0	0
$718$	47.0334	1.75527
$719$	−5.50638	−0.205353	−0.102677	−	0.994715i	$-0.532741\pi$
$719$	−5.50638	−0.205353	−0.102677	+	0.994715i	$0.532741\pi$
$720$	0	0
$721$	−24.9613	−0.929608
$722$	−43.7435	−1.62797
$723$	0	0
$724$	106.928	3.97396
$725$	0	0
$726$	0	0
$727$	−27.5811	−1.02293	−0.511463	−	0.859305i	$-0.670896\pi$
$727$	−27.5811	−1.02293	−0.511463	+	0.859305i	$0.670896\pi$
$728$	46.7601	1.73304
$729$	0	0
$730$	−19.9600	−0.738753
$731$	9.11824	0.337250
$732$	0	0
$733$	−42.7886	−1.58043	−0.790217	−	0.612828i	$-0.790032\pi$
$733$	−42.7886	−1.58043	−0.790217	+	0.612828i	$0.790032\pi$
$734$	79.7932	2.94522
$735$	0	0
$736$	19.5431	0.720367
$737$	18.1517	0.668627
$738$	0	0
$739$	29.9595	1.10208	0.551040	−	0.834479i	$-0.314231\pi$
$739$	29.9595	1.10208	0.551040	+	0.834479i	$0.314231\pi$
$740$	−111.312	−4.09192
$741$	0	0
$742$	−32.8726	−1.20679
$743$	−1.55470	−0.0570363	−0.0285181	−	0.999593i	$-0.509079\pi$
$743$	−1.55470	−0.0570363	−0.0285181	+	0.999593i	$0.509079\pi$
$744$	0	0
$745$	2.31332	0.0847535
$746$	51.7304	1.89399
$747$	0	0
$748$	−65.2279	−2.38497
$749$	32.5987	1.19113
$750$	0	0
$751$	−42.0480	−1.53435	−0.767177	−	0.641436i	$-0.778339\pi$
$751$	−42.0480	−1.53435	−0.767177	+	0.641436i	$0.778339\pi$
$752$	116.602	4.25205
$753$	0	0
$754$	0	0
$755$	53.7765	1.95713
$756$	0	0
$757$	−27.1592	−0.987117	−0.493559	−	0.869713i	$-0.664304\pi$
$757$	−27.1592	−0.987117	−0.493559	+	0.869713i	$0.664304\pi$
$758$	−23.0275	−0.836395
$759$	0	0
$760$	−43.5469	−1.57961
$761$	35.0279	1.26976	0.634881	−	0.772610i	$-0.281049\pi$
$761$	35.0279	1.26976	0.634881	+	0.772610i	$0.281049\pi$
$762$	0	0
$763$	12.6862	0.459271
$764$	127.027	4.59569
$765$	0	0
$766$	75.8493	2.74055
$767$	−11.3232	−0.408858
$768$	0	0
$769$	21.5891	0.778523	0.389261	−	0.921127i	$-0.372730\pi$
$769$	21.5891	0.778523	0.389261	+	0.921127i	$0.372730\pi$
$770$	−51.4464	−1.85400
$771$	0	0
$772$	−82.1872	−2.95798
$773$	−44.3882	−1.59653	−0.798265	−	0.602306i	$-0.794249\pi$
$773$	−44.3882	−1.59653	−0.798265	+	0.602306i	$0.794249\pi$
$774$	0	0
$775$	27.0419	0.971374
$776$	143.509	5.15167
$777$	0	0
$778$	72.8801	2.61288
$779$	12.5614	0.450060
$780$	0	0
$781$	9.72301	0.347917
$782$	15.3257	0.548045
$783$	0	0
$784$	−17.3243	−0.618724
$785$	1.24202	0.0443298
$786$	0	0
$787$	20.7242	0.738739	0.369369	−	0.929283i	$-0.379574\pi$
$787$	20.7242	0.738739	0.369369	+	0.929283i	$0.379574\pi$
$788$	−66.1440	−2.35628
$789$	0	0
$790$	−31.3514	−1.11543
$791$	−25.5936	−0.910003
$792$	0	0
$793$	26.6309	0.945690
$794$	13.2875	0.471556
$795$	0	0
$796$	−42.1479	−1.49389
$797$	4.93609	0.174845	0.0874227	−	0.996171i	$-0.472137\pi$
$797$	4.93609	0.174845	0.0874227	+	0.996171i	$0.472137\pi$
$798$	0	0
$799$	44.9371	1.58976
$800$	77.2282	2.73043
$801$	0	0
$802$	−16.6743	−0.588790
$803$	−6.22946	−0.219833
$804$	0	0
$805$	8.72414	0.307486
$806$	−35.7928	−1.26075
$807$	0	0
$808$	−39.1584	−1.37759
$809$	−1.19173	−0.0418990	−0.0209495	−	0.999781i	$-0.506669\pi$
$809$	−1.19173	−0.0418990	−0.0209495	+	0.999781i	$0.506669\pi$
$810$	0	0
$811$	46.6639	1.63859	0.819296	−	0.573370i	$-0.194364\pi$
$811$	46.6639	1.63859	0.819296	+	0.573370i	$0.194364\pi$
$812$	0	0
$813$	0	0
$814$	−48.1340	−1.68709
$815$	−30.8369	−1.08017
$816$	0	0
$817$	3.09193	0.108173
$818$	12.4362	0.434822
$819$	0	0
$820$	−123.733	−4.32095
$821$	−31.7367	−1.10762	−0.553808	−	0.832644i	$-0.686826\pi$
$821$	−31.7367	−1.10762	−0.553808	+	0.832644i	$0.686826\pi$
$822$	0	0
$823$	−38.3538	−1.33693	−0.668464	−	0.743744i	$-0.733048\pi$
$823$	−38.3538	−1.33693	−0.668464	+	0.743744i	$0.733048\pi$
$824$	89.9964	3.13517
$825$	0	0
$826$	−31.1685	−1.08449
$827$	17.1249	0.595491	0.297745	−	0.954645i	$-0.403765\pi$
$827$	17.1249	0.595491	0.297745	+	0.954645i	$0.403765\pi$
$828$	0	0
$829$	50.9294	1.76885	0.884425	−	0.466682i	$-0.154551\pi$
$829$	50.9294	1.76885	0.884425	+	0.466682i	$0.154551\pi$
$830$	−88.3558	−3.06688
$831$	0	0
$832$	−44.3417	−1.53727
$833$	−6.67656	−0.231329
$834$	0	0
$835$	−36.4533	−1.26152
$836$	−22.1183	−0.764978
$837$	0	0
$838$	−36.6432	−1.26582
$839$	−38.4991	−1.32914	−0.664568	−	0.747228i	$-0.731384\pi$
$839$	−38.4991	−1.32914	−0.664568	+	0.747228i	$0.731384\pi$
$840$	0	0
$841$	0	0
$842$	−31.9116	−1.09974
$843$	0	0
$844$	−108.915	−3.74901
$845$	23.8606	0.820828
$846$	0	0
$847$	10.0163	0.344163
$848$	64.8484	2.22690
$849$	0	0
$850$	60.5624	2.07727
$851$	8.16242	0.279804
$852$	0	0
$853$	−19.5496	−0.669364	−0.334682	−	0.942331i	$-0.608629\pi$
$853$	−19.5496	−0.669364	−0.334682	+	0.942331i	$0.608629\pi$
$854$	73.3045	2.50843
$855$	0	0
$856$	−117.532	−4.01717
$857$	6.02847	0.205929	0.102964	−	0.994685i	$-0.467167\pi$
$857$	6.02847	0.205929	0.102964	+	0.994685i	$0.467167\pi$
$858$	0	0
$859$	15.2968	0.521921	0.260961	−	0.965349i	$-0.415961\pi$
$859$	15.2968	0.521921	0.260961	+	0.965349i	$0.415961\pi$
$860$	−30.4563	−1.03855
$861$	0	0
$862$	60.9435	2.07574
$863$	5.09773	0.173529	0.0867643	−	0.996229i	$-0.472347\pi$
$863$	5.09773	0.173529	0.0867643	+	0.996229i	$0.472347\pi$
$864$	0	0
$865$	25.3310	0.861279
$866$	−14.7737	−0.502032
$867$	0	0
$868$	−71.1086	−2.41358
$869$	−9.78469	−0.331923
$870$	0	0
$871$	16.1000	0.545528
$872$	−45.7392	−1.54892
$873$	0	0
$874$	5.19683	0.175785
$875$	−2.38926	−0.0807718
$876$	0	0
$877$	56.3739	1.90361	0.951805	−	0.306704i	$-0.0992262\pi$
$877$	56.3739	1.90361	0.951805	+	0.306704i	$0.0992262\pi$
$878$	54.3266	1.83343
$879$	0	0
$880$	101.489	3.42120
$881$	−8.76452	−0.295284	−0.147642	−	0.989041i	$-0.547168\pi$
$881$	−8.76452	−0.295284	−0.147642	+	0.989041i	$0.547168\pi$
$882$	0	0
$883$	−44.5671	−1.49980	−0.749902	−	0.661549i	$-0.769899\pi$
$883$	−44.5671	−1.49980	−0.749902	+	0.661549i	$0.769899\pi$
$884$	−57.8552	−1.94588
$885$	0	0
$886$	−90.8790	−3.05314
$887$	−50.8916	−1.70877	−0.854386	−	0.519639i	$-0.826067\pi$
$887$	−50.8916	−1.70877	−0.854386	+	0.519639i	$0.826067\pi$
$888$	0	0
$889$	−39.9646	−1.34037
$890$	−47.5566	−1.59410
$891$	0	0
$892$	110.013	3.68350
$893$	15.2378	0.509915
$894$	0	0
$895$	33.5279	1.12071
$896$	−43.7618	−1.46198
$897$	0	0
$898$	−86.4734	−2.88566
$899$	0	0
$900$	0	0
$901$	24.9918	0.832596
$902$	−53.5051	−1.78152
$903$	0	0
$904$	92.2760	3.06905
$905$	−64.1176	−2.13134
$906$	0	0
$907$	−22.1563	−0.735689	−0.367844	−	0.929887i	$-0.619904\pi$
$907$	−22.1563	−0.735689	−0.367844	+	0.929887i	$0.619904\pi$
$908$	−55.5291	−1.84280
$909$	0	0
$910$	−45.6314	−1.51267
$911$	−2.98463	−0.0988851	−0.0494425	−	0.998777i	$-0.515744\pi$
$911$	−2.98463	−0.0988851	−0.0494425	+	0.998777i	$0.515744\pi$
$912$	0	0
$913$	−27.5756	−0.912620
$914$	−70.0581	−2.31731
$915$	0	0
$916$	−72.6560	−2.40062
$917$	10.7919	0.356382
$918$	0	0
$919$	−4.29559	−0.141698	−0.0708492	−	0.997487i	$-0.522571\pi$
$919$	−4.29559	−0.141698	−0.0708492	+	0.997487i	$0.522571\pi$
$920$	−31.4543	−1.03702
$921$	0	0
$922$	33.6089	1.10685
$923$	8.62401	0.283863
$924$	0	0
$925$	32.2554	1.06055
$926$	−102.142	−3.35659
$927$	0	0
$928$	0	0
$929$	−33.6586	−1.10430	−0.552152	−	0.833743i	$-0.686193\pi$
$929$	−33.6586	−1.10430	−0.552152	+	0.833743i	$0.686193\pi$
$930$	0	0
$931$	−2.26397	−0.0741987
$932$	64.2485	2.10453
$933$	0	0
$934$	18.5232	0.606099
$935$	39.1127	1.27912
$936$	0	0
$937$	24.1673	0.789511	0.394755	−	0.918786i	$-0.370829\pi$
$937$	24.1673	0.789511	0.394755	+	0.918786i	$0.370829\pi$
$938$	44.3171	1.44701
$939$	0	0
$940$	−150.096	−4.89561
$941$	25.2276	0.822395	0.411197	−	0.911546i	$-0.365111\pi$
$941$	25.2276	0.822395	0.411197	+	0.911546i	$0.365111\pi$
$942$	0	0
$943$	9.07324	0.295465
$944$	61.4865	2.00122
$945$	0	0
$946$	−13.1700	−0.428193
$947$	−48.1298	−1.56401	−0.782004	−	0.623274i	$-0.785802\pi$
$947$	−48.1298	−1.56401	−0.782004	+	0.623274i	$0.785802\pi$
$948$	0	0
$949$	−5.52534	−0.179360
$950$	20.5363	0.666285
$951$	0	0
$952$	−97.8550	−3.17150
$953$	−10.3433	−0.335052	−0.167526	−	0.985868i	$-0.553578\pi$
$953$	−10.3433	−0.335052	−0.167526	+	0.985868i	$0.553578\pi$
$954$	0	0
$955$	−76.1694	−2.46479
$956$	−46.1869	−1.49379
$957$	0	0
$958$	40.8321	1.31922
$959$	−52.1670	−1.68456
$960$	0	0
$961$	2.44547	0.0788862
$962$	−42.6933	−1.37649
$963$	0	0
$964$	−135.654	−4.36911
$965$	49.2820	1.58644
$966$	0	0
$967$	−19.7278	−0.634404	−0.317202	−	0.948358i	$-0.602743\pi$
$967$	−19.7278	−0.634404	−0.317202	+	0.948358i	$0.602743\pi$
$968$	−36.1129	−1.16071
$969$	0	0
$970$	−140.045	−4.49657
$971$	8.37868	0.268885	0.134442	−	0.990921i	$-0.457076\pi$
$971$	8.37868	0.268885	0.134442	+	0.990921i	$0.457076\pi$
$972$	0	0
$973$	19.1839	0.615007
$974$	−84.8821	−2.71980
$975$	0	0
$976$	−144.609	−4.62882
$977$	−27.6767	−0.885457	−0.442728	−	0.896656i	$-0.645989\pi$
$977$	−27.6767	−0.885457	−0.442728	+	0.896656i	$0.645989\pi$
$978$	0	0
$979$	−14.8423	−0.474362
$980$	22.3007	0.712369
$981$	0	0
$982$	−28.7607	−0.917791
$983$	1.80073	0.0574344	0.0287172	−	0.999588i	$-0.490858\pi$
$983$	1.80073	0.0574344	0.0287172	+	0.999588i	$0.490858\pi$
$984$	0	0
$985$	39.6619	1.26373
$986$	0	0
$987$	0	0
$988$	−19.6183	−0.624140
$989$	2.23333	0.0710157
$990$	0	0
$991$	16.7954	0.533525	0.266762	−	0.963762i	$-0.414046\pi$
$991$	16.7954	0.533525	0.266762	+	0.963762i	$0.414046\pi$
$992$	95.5160	3.03264
$993$	0	0
$994$	23.7386	0.752942
$995$	25.2732	0.801213
$996$	0	0
$997$	26.6523	0.844087	0.422043	−	0.906576i	$-0.361313\pi$
$997$	26.6523	0.844087	0.422043	+	0.906576i	$0.361313\pi$
$998$	−26.1921	−0.829096
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.bl.1.9		9
3.2	odd	2		2523.2.a.p.1.1		9
29.23	even	7		261.2.k.b.181.1		18
29.24	even	7		261.2.k.b.199.1		18
29.28	even	2		7569.2.a.bk.1.1		9
87.23	odd	14		87.2.g.b.7.3	✓	18
87.53	odd	14		87.2.g.b.25.3	yes	18
87.86	odd	2		2523.2.a.q.1.9		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.g.b.7.3	✓	18	87.23	odd	14
87.2.g.b.25.3	yes	18	87.53	odd	14
261.2.k.b.181.1		18	29.23	even	7
261.2.k.b.199.1		18	29.24	even	7
2523.2.a.p.1.1		9	3.2	odd	2
2523.2.a.q.1.9		9	87.86	odd	2
7569.2.a.bk.1.1		9	29.28	even	2
7569.2.a.bl.1.9		9	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)

\(f(q)\)	\(=\)	\(q+2.68096 q^{2} +5.18756 q^{4} -3.11062 q^{5} -2.37023 q^{7} +8.54571 q^{8} +O(q^{10})\)	\(q+2.68096 q^{2} +5.18756 q^{4} -3.11062 q^{5} -2.37023 q^{7} +8.54571 q^{8} -8.33945 q^{10} -2.60272 q^{11} -2.30853 q^{13} -6.35450 q^{14} +12.5356 q^{16} +4.83107 q^{17} +1.63818 q^{19} -16.1365 q^{20} -6.97779 q^{22} +1.18327 q^{23} +4.67594 q^{25} -6.18908 q^{26} -12.2957 q^{28} +5.78321 q^{31} +16.5161 q^{32} +12.9519 q^{34} +7.37288 q^{35} +6.89817 q^{37} +4.39190 q^{38} -26.5824 q^{40} +7.66791 q^{41} +1.88741 q^{43} -13.5017 q^{44} +3.17231 q^{46} +9.30167 q^{47} -1.38200 q^{49} +12.5360 q^{50} -11.9756 q^{52} +5.17313 q^{53} +8.09606 q^{55} -20.2553 q^{56} +4.90494 q^{59} -11.5358 q^{61} +15.5046 q^{62} +19.2078 q^{64} +7.18096 q^{65} -6.97413 q^{67} +25.0615 q^{68} +19.7664 q^{70} -3.73571 q^{71} +2.39344 q^{73} +18.4937 q^{74} +8.49816 q^{76} +6.16905 q^{77} +3.75941 q^{79} -38.9935 q^{80} +20.5574 q^{82} +10.5949 q^{83} -15.0276 q^{85} +5.06009 q^{86} -22.2421 q^{88} +5.70261 q^{89} +5.47175 q^{91} +6.13830 q^{92} +24.9374 q^{94} -5.09576 q^{95} +16.7931 q^{97} -3.70510 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{2} + 11 q^{4} + 5 q^{7} + 12 q^{8}+O(q^{10}) \) 9 * q + q^2 + 11 * q^4 + 5 * q^7 + 12 * q^8	\( 9 q + q^{2} + 11 q^{4} + 5 q^{7} + 12 q^{8} - 4 q^{10} + 3 q^{11} + 5 q^{13} + 15 q^{14} - 5 q^{16} + 16 q^{17} + q^{19} - 8 q^{20} + 24 q^{22} + 10 q^{23} + 9 q^{25} + 12 q^{26} - 24 q^{28} + 4 q^{31} + 25 q^{32} + 24 q^{34} + 44 q^{35} - 25 q^{37} + 10 q^{38} - 5 q^{40} + 34 q^{41} - 12 q^{43} - 23 q^{44} - 6 q^{46} + 8 q^{47} + 26 q^{49} + 27 q^{50} - 23 q^{52} + 32 q^{53} + 5 q^{55} - 14 q^{56} - 10 q^{59} - 51 q^{61} - 8 q^{62} - 8 q^{64} + 11 q^{65} + 7 q^{67} + 11 q^{68} + 14 q^{70} - 7 q^{71} - 17 q^{73} + 62 q^{74} + 6 q^{76} + 64 q^{77} - 13 q^{79} - 54 q^{80} + 37 q^{82} + 31 q^{83} - 42 q^{85} + 70 q^{86} - 29 q^{88} - 32 q^{89} + 45 q^{91} - 9 q^{92} + 38 q^{94} - 20 q^{95} - 16 q^{97} - 12 q^{98}+O(q^{100}) \) 9 * q + q^2 + 11 * q^4 + 5 * q^7 + 12 * q^8 - 4 * q^10 + 3 * q^11 + 5 * q^13 + 15 * q^14 - 5 * q^16 + 16 * q^17 + q^19 - 8 * q^20 + 24 * q^22 + 10 * q^23 + 9 * q^25 + 12 * q^26 - 24 * q^28 + 4 * q^31 + 25 * q^32 + 24 * q^34 + 44 * q^35 - 25 * q^37 + 10 * q^38 - 5 * q^40 + 34 * q^41 - 12 * q^43 - 23 * q^44 - 6 * q^46 + 8 * q^47 + 26 * q^49 + 27 * q^50 - 23 * q^52 + 32 * q^53 + 5 * q^55 - 14 * q^56 - 10 * q^59 - 51 * q^61 - 8 * q^62 - 8 * q^64 + 11 * q^65 + 7 * q^67 + 11 * q^68 + 14 * q^70 - 7 * q^71 - 17 * q^73 + 62 * q^74 + 6 * q^76 + 64 * q^77 - 13 * q^79 - 54 * q^80 + 37 * q^82 + 31 * q^83 - 42 * q^85 + 70 * q^86 - 29 * q^88 - 32 * q^89 + 45 * q^91 - 9 * q^92 + 38 * q^94 - 20 * q^95 - 16 * q^97 - 12 * q^98

Introduction

L-functions

Modular forms

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Database

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