LMFDB - Embedded newform 7569.2.a.bm.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:	$0$
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} - 4x^{8} - 6x^{7} + 33x^{6} + 6x^{5} - 90x^{4} + 21x^{3} + 84x^{2} - 36x - 8 $ x^9 - 4x^8 - 6x^7 + 33x^6 + 6x^5 - 90x^4 + 21x^3 + 84x^2 - 36x - 8
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			1.5
Root			$0.822927$ 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)

$f(q)$	$=$	\(q+0.177073 q^{2} -1.96865 q^{4} +3.45143 q^{5} -3.79521 q^{7} -0.702739 q^{8} +0.611154 q^{10} -4.26673 q^{11} -4.64311 q^{13} -0.672028 q^{14} +3.81285 q^{16} -3.07208 q^{17} +3.59269 q^{19} -6.79464 q^{20} -0.755521 q^{22} +0.433603 q^{23} +6.91237 q^{25} -0.822167 q^{26} +7.47142 q^{28} +4.52230 q^{31} +2.08063 q^{32} -0.543981 q^{34} -13.0989 q^{35} -6.32794 q^{37} +0.636166 q^{38} -2.42545 q^{40} -1.97128 q^{41} +0.251178 q^{43} +8.39968 q^{44} +0.0767793 q^{46} -4.81629 q^{47} +7.40363 q^{49} +1.22399 q^{50} +9.14063 q^{52} -10.9694 q^{53} -14.7263 q^{55} +2.66704 q^{56} +6.06991 q^{59} -3.79174 q^{61} +0.800776 q^{62} -7.25729 q^{64} -16.0254 q^{65} -5.26714 q^{67} +6.04783 q^{68} -2.31946 q^{70} +5.83609 q^{71} +10.5495 q^{73} -1.12051 q^{74} -7.07272 q^{76} +16.1931 q^{77} -10.0367 q^{79} +13.1598 q^{80} -0.349059 q^{82} +7.09816 q^{83} -10.6031 q^{85} +0.0444768 q^{86} +2.99840 q^{88} -13.9762 q^{89} +17.6216 q^{91} -0.853611 q^{92} -0.852834 q^{94} +12.3999 q^{95} +14.3224 q^{97} +1.31098 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 5 q^{2} + 11 q^{4} + 4 q^{5} + 5 q^{7} + 24 q^{8} - q^{11} + q^{13} + 9 q^{14} + 35 q^{16} + 2 q^{17} - 9 q^{19} + 18 q^{20} - 4 q^{22} + 4 q^{23} + q^{25} - 8 q^{26} + 40 q^{28} - 8 q^{31} + 43 q^{32}+ \cdots - 30 q^{98}+O(q^{100}) $ 9 * q + 5 * q^2 + 11 * q^4 + 4 * q^5 + 5 * q^7 + 24 * q^8 - q^11 + q^13 + 9 * q^14 + 35 * q^16 + 2 * q^17 - 9 * q^19 + 18 * q^20 - 4 * q^22 + 4 * q^23 + q^25 - 8 * q^26 + 40 * q^28 - 8 * q^31 + 43 * q^32 - 4 * q^34 - 22 * q^35 - 27 * q^37 + 30 * q^38 + 29 * q^40 + 12 * q^41 - 16 * q^43 + 37 * q^44 + 22 * q^46 - 8 * q^47 - 6 * q^49 - 7 * q^50 + 33 * q^52 + 8 * q^53 - 9 * q^55 + 40 * q^56 + 16 * q^59 - 21 * q^61 + 32 * q^62 + 36 * q^64 + 31 * q^65 + 3 * q^67 + 33 * q^68 + 6 * q^70 + 33 * q^71 - 3 * q^73 - 28 * q^74 + 26 * q^76 + 24 * q^77 - 3 * q^79 + 64 * q^80 + 13 * q^82 - 13 * q^83 - 6 * q^85 - 58 * q^86 + 27 * q^88 + 6 * q^89 + q^91 + 29 * q^92 - 18 * q^94 + 48 * q^95 - 4 * q^97 - 30 * 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 $

$2$	0.177073	0.125209	0.0626046	−	0.998038i	$-0.480059\pi$
$2$	0.177073	0.125209	0.0626046	+	0.998038i	$0.480059\pi$
$3$	0	0
$3$	0	0
$4$	−1.96865	−0.984323
$5$	3.45143	1.54353	0.771763	−	0.635910i	$-0.219375\pi$
$5$	3.45143	1.54353	0.771763	+	0.635910i	$0.219375\pi$
$6$	0	0
$7$	−3.79521	−1.43445	−0.717227	−	0.696839i	$-0.754589\pi$
$7$	−3.79521	−1.43445	−0.717227	+	0.696839i	$0.754589\pi$
$8$	−0.702739	−0.248456
$9$	0	0
$10$	0.611154	0.193264
$11$	−4.26673	−1.28647	−0.643234	−	0.765670i	$-0.722408\pi$
$11$	−4.26673	−1.28647	−0.643234	+	0.765670i	$0.722408\pi$
$12$	0	0
$13$	−4.64311	−1.28777	−0.643883	−	0.765124i	$-0.722678\pi$
$13$	−4.64311	−1.28777	−0.643883	+	0.765124i	$0.722678\pi$
$14$	−0.672028	−0.179607
$15$	0	0
$16$	3.81285	0.953214
$17$	−3.07208	−0.745088	−0.372544	−	0.928014i	$-0.621514\pi$
$17$	−3.07208	−0.745088	−0.372544	+	0.928014i	$0.621514\pi$
$18$	0	0
$19$	3.59269	0.824219	0.412109	−	0.911134i	$-0.364792\pi$
$19$	3.59269	0.824219	0.412109	+	0.911134i	$0.364792\pi$
$20$	−6.79464	−1.51933
$21$	0	0
$22$	−0.755521	−0.161078
$23$	0.433603	0.0904125	0.0452063	−	0.998978i	$-0.485605\pi$
$23$	0.433603	0.0904125	0.0452063	+	0.998978i	$0.485605\pi$
$24$	0	0
$25$	6.91237	1.38247
$26$	−0.822167	−0.161240
$27$	0	0
$28$	7.47142	1.41197
$29$	0	0
$29$	0	0
$30$	0	0
$31$	4.52230	0.812230	0.406115	−	0.913822i	$-0.366883\pi$
$31$	4.52230	0.812230	0.406115	+	0.913822i	$0.366883\pi$
$32$	2.08063	0.367807
$33$	0	0
$34$	−0.543981	−0.0932920
$35$	−13.0989	−2.21412
$36$	0	0
$37$	−6.32794	−1.04031	−0.520153	−	0.854073i	$-0.674125\pi$
$37$	−6.32794	−1.04031	−0.520153	+	0.854073i	$0.674125\pi$
$38$	0.636166	0.103200
$39$	0	0
$40$	−2.42545	−0.383498
$41$	−1.97128	−0.307862	−0.153931	−	0.988082i	$-0.549193\pi$
$41$	−1.97128	−0.307862	−0.153931	+	0.988082i	$0.549193\pi$
$42$	0	0
$43$	0.251178	0.0383043	0.0191522	−	0.999817i	$-0.493903\pi$
$43$	0.251178	0.0383043	0.0191522	+	0.999817i	$0.493903\pi$
$44$	8.39968	1.26630
$45$	0	0
$46$	0.0767793	0.0113205
$47$	−4.81629	−0.702529	−0.351264	−	0.936276i	$-0.614248\pi$
$47$	−4.81629	−0.702529	−0.351264	+	0.936276i	$0.614248\pi$
$48$	0	0
$49$	7.40363	1.05766
$50$	1.22399	0.173099
$51$	0	0
$52$	9.14063	1.26758
$53$	−10.9694	−1.50676	−0.753382	−	0.657583i	$-0.771579\pi$
$53$	−10.9694	−1.50676	−0.753382	+	0.657583i	$0.771579\pi$
$54$	0	0
$55$	−14.7263	−1.98570
$56$	2.66704	0.356398
$57$	0	0
$58$	0	0
$59$	6.06991	0.790234	0.395117	−	0.918631i	$-0.370704\pi$
$59$	6.06991	0.790234	0.395117	+	0.918631i	$0.370704\pi$
$60$	0	0
$61$	−3.79174	−0.485483	−0.242741	−	0.970091i	$-0.578047\pi$
$61$	−3.79174	−0.485483	−0.242741	+	0.970091i	$0.578047\pi$
$62$	0.800776	0.101699
$63$	0	0
$64$	−7.25729	−0.907161
$65$	−16.0254	−1.98770
$66$	0	0
$67$	−5.26714	−0.643484	−0.321742	−	0.946827i	$-0.604268\pi$
$67$	−5.26714	−0.643484	−0.321742	+	0.946827i	$0.604268\pi$
$68$	6.04783	0.733407
$69$	0	0
$70$	−2.31946	−0.277228
$71$	5.83609	0.692616	0.346308	−	0.938121i	$-0.387435\pi$
$71$	5.83609	0.692616	0.346308	+	0.938121i	$0.387435\pi$
$72$	0	0
$73$	10.5495	1.23473	0.617365	−	0.786677i	$-0.288200\pi$
$73$	10.5495	1.23473	0.617365	+	0.786677i	$0.288200\pi$
$74$	−1.12051	−0.130256
$75$	0	0
$76$	−7.07272	−0.811297
$77$	16.1931	1.84538
$78$	0	0
$79$	−10.0367	−1.12921	−0.564607	−	0.825360i	$-0.690972\pi$
$79$	−10.0367	−1.12921	−0.564607	+	0.825360i	$0.690972\pi$
$80$	13.1598	1.47131
$81$	0	0
$82$	−0.349059	−0.0385471
$83$	7.09816	0.779124	0.389562	−	0.921000i	$-0.372626\pi$
$83$	7.09816	0.779124	0.389562	+	0.921000i	$0.372626\pi$
$84$	0	0
$85$	−10.6031	−1.15006
$86$	0.0444768	0.00479606
$87$	0	0
$88$	2.99840	0.319630
$89$	−13.9762	−1.48147	−0.740737	−	0.671795i	$-0.765524\pi$
$89$	−13.9762	−1.48147	−0.740737	+	0.671795i	$0.765524\pi$
$90$	0	0
$91$	17.6216	1.84724
$92$	−0.853611	−0.0889951
$93$	0	0
$94$	−0.852834	−0.0879631
$95$	12.3999	1.27220
$96$	0	0
$97$	14.3224	1.45422	0.727111	−	0.686520i	$-0.240863\pi$
$97$	14.3224	1.45422	0.727111	+	0.686520i	$0.240863\pi$
$98$	1.31098	0.132429
$99$	0	0
$100$	−13.6080	−1.36080
$101$	0.816445	0.0812393	0.0406196	−	0.999175i	$-0.487067\pi$
$101$	0.816445	0.0812393	0.0406196	+	0.999175i	$0.487067\pi$
$102$	0	0
$103$	−4.95795	−0.488521	−0.244260	−	0.969710i	$-0.578545\pi$
$103$	−4.95795	−0.488521	−0.244260	+	0.969710i	$0.578545\pi$
$104$	3.26289	0.319953
$105$	0	0
$106$	−1.94238	−0.188661
$107$	11.5359	1.11521	0.557607	−	0.830105i	$-0.311720\pi$
$107$	11.5359	1.11521	0.557607	+	0.830105i	$0.311720\pi$
$108$	0	0
$109$	8.64718	0.828250	0.414125	−	0.910220i	$-0.364088\pi$
$109$	8.64718	0.828250	0.414125	+	0.910220i	$0.364088\pi$
$110$	−2.60763	−0.248628
$111$	0	0
$112$	−14.4706	−1.36734
$113$	7.58551	0.713584	0.356792	−	0.934184i	$-0.383870\pi$
$113$	7.58551	0.713584	0.356792	+	0.934184i	$0.383870\pi$
$114$	0	0
$115$	1.49655	0.139554
$116$	0	0
$117$	0	0
$118$	1.07481	0.0989447
$119$	11.6592	1.06880
$120$	0	0
$121$	7.20498	0.654998
$122$	−0.671414	−0.0607869
$123$	0	0
$124$	−8.90281	−0.799496
$125$	6.60043	0.590360
$126$	0	0
$127$	−1.23469	−0.109561	−0.0547807	−	0.998498i	$-0.517446\pi$
$127$	−1.23469	−0.109561	−0.0547807	+	0.998498i	$0.517446\pi$
$128$	−5.44633	−0.481392
$129$	0	0
$130$	−2.83765	−0.248879
$131$	−3.99564	−0.349101	−0.174550	−	0.984648i	$-0.555847\pi$
$131$	−3.99564	−0.349101	−0.174550	+	0.984648i	$0.555847\pi$
$132$	0	0
$133$	−13.6350	−1.18230
$134$	−0.932667	−0.0805702
$135$	0	0
$136$	2.15887	0.185121
$137$	16.6539	1.42284	0.711418	−	0.702769i	$-0.248053\pi$
$137$	16.6539	1.42284	0.711418	+	0.702769i	$0.248053\pi$
$138$	0	0
$139$	8.27106	0.701543	0.350771	−	0.936461i	$-0.385919\pi$
$139$	8.27106	0.701543	0.350771	+	0.936461i	$0.385919\pi$
$140$	25.7871	2.17941
$141$	0	0
$142$	1.03341	0.0867219
$143$	19.8109	1.65667
$144$	0	0
$145$	0	0
$146$	1.86803	0.154600
$147$	0	0
$148$	12.4575	1.02400
$149$	9.38848	0.769135	0.384567	−	0.923097i	$-0.374351\pi$
$149$	9.38848	0.769135	0.384567	+	0.923097i	$0.374351\pi$
$150$	0	0
$151$	19.6633	1.60018	0.800088	−	0.599883i	$-0.204786\pi$
$151$	19.6633	1.60018	0.800088	+	0.599883i	$0.204786\pi$
$152$	−2.52472	−0.204782
$153$	0	0
$154$	2.86736	0.231059
$155$	15.6084	1.25370
$156$	0	0
$157$	2.61457	0.208665	0.104333	−	0.994542i	$-0.466729\pi$
$157$	2.61457	0.208665	0.104333	+	0.994542i	$0.466729\pi$
$158$	−1.77722	−0.141388
$159$	0	0
$160$	7.18115	0.567720
$161$	−1.64562	−0.129693
$162$	0	0
$163$	0.527800	0.0413405	0.0206702	−	0.999786i	$-0.493420\pi$
$163$	0.527800	0.0413405	0.0206702	+	0.999786i	$0.493420\pi$
$164$	3.88074	0.303035
$165$	0	0
$166$	1.25689	0.0975536
$167$	15.0489	1.16452	0.582261	−	0.813002i	$-0.302168\pi$
$167$	15.0489	1.16452	0.582261	+	0.813002i	$0.302168\pi$
$168$	0	0
$169$	8.55844	0.658342
$170$	−1.87751	−0.143999
$171$	0	0
$172$	−0.494481	−0.0377038
$173$	−0.103233	−0.00784865	−0.00392433	−	0.999992i	$-0.501249\pi$
$173$	−0.103233	−0.00784865	−0.00392433	+	0.999992i	$0.501249\pi$
$174$	0	0
$175$	−26.2339	−1.98310
$176$	−16.2684	−1.22628
$177$	0	0
$178$	−2.47480	−0.185494
$179$	4.04916	0.302649	0.151324	−	0.988484i	$-0.451646\pi$
$179$	4.04916	0.302649	0.151324	+	0.988484i	$0.451646\pi$
$180$	0	0
$181$	−11.5222	−0.856442	−0.428221	−	0.903674i	$-0.640860\pi$
$181$	−11.5222	−0.856442	−0.428221	+	0.903674i	$0.640860\pi$
$182$	3.12030	0.231292
$183$	0	0
$184$	−0.304710	−0.0224635
$185$	−21.8404	−1.60574
$186$	0	0
$187$	13.1077	0.958532
$188$	9.48157	0.691515
$189$	0	0
$190$	2.19568	0.159292
$191$	1.55942	0.112836	0.0564179	−	0.998407i	$-0.482032\pi$
$191$	1.55942	0.112836	0.0564179	+	0.998407i	$0.482032\pi$
$192$	0	0
$193$	−13.3775	−0.962932	−0.481466	−	0.876465i	$-0.659896\pi$
$193$	−13.3775	−0.962932	−0.481466	+	0.876465i	$0.659896\pi$
$194$	2.53611	0.182082
$195$	0	0
$196$	−14.5751	−1.04108
$197$	12.7756	0.910226	0.455113	−	0.890434i	$-0.349599\pi$
$197$	12.7756	0.910226	0.455113	+	0.890434i	$0.349599\pi$
$198$	0	0
$199$	8.82206	0.625379	0.312690	−	0.949855i	$-0.398770\pi$
$199$	8.82206	0.625379	0.312690	+	0.949855i	$0.398770\pi$
$200$	−4.85759	−0.343484
$201$	0	0
$202$	0.144570	0.0101719
$203$	0	0
$204$	0	0
$205$	−6.80372	−0.475193
$206$	−0.877917	−0.0611674
$207$	0	0
$208$	−17.7035	−1.22752
$209$	−15.3290	−1.06033
$210$	0	0
$211$	15.0234	1.03425	0.517127	−	0.855909i	$-0.327002\pi$
$211$	15.0234	1.03425	0.517127	+	0.855909i	$0.327002\pi$
$212$	21.5949	1.48314
$213$	0	0
$214$	2.04269	0.139635
$215$	0.866924	0.0591237
$216$	0	0
$217$	−17.1631	−1.16511
$218$	1.53118	0.103705
$219$	0	0
$220$	28.9909	1.95457
$221$	14.2640	0.959500
$222$	0	0
$223$	23.9456	1.60352	0.801759	−	0.597648i	$-0.203898\pi$
$223$	23.9456	1.60352	0.801759	+	0.597648i	$0.203898\pi$
$224$	−7.89643	−0.527602
$225$	0	0
$226$	1.34319	0.0893474
$227$	5.15851	0.342382	0.171191	−	0.985238i	$-0.445238\pi$
$227$	5.15851	0.342382	0.171191	+	0.985238i	$0.445238\pi$
$228$	0	0
$229$	8.10170	0.535375	0.267688	−	0.963506i	$-0.413740\pi$
$229$	8.10170	0.535375	0.267688	+	0.963506i	$0.413740\pi$
$230$	0.264998	0.0174735
$231$	0	0
$232$	0	0
$233$	−24.5153	−1.60605	−0.803027	−	0.595943i	$-0.796778\pi$
$233$	−24.5153	−1.60605	−0.803027	+	0.595943i	$0.796778\pi$
$234$	0	0
$235$	−16.6231	−1.08437
$236$	−11.9495	−0.777846
$237$	0	0
$238$	2.06452	0.133823
$239$	27.9408	1.80734	0.903669	−	0.428232i	$-0.140863\pi$
$239$	27.9408	1.80734	0.903669	+	0.428232i	$0.140863\pi$
$240$	0	0
$241$	3.96836	0.255625	0.127812	−	0.991798i	$-0.459204\pi$
$241$	3.96836	0.255625	0.127812	+	0.991798i	$0.459204\pi$
$242$	1.27581	0.0820119
$243$	0	0
$244$	7.46459	0.477871
$245$	25.5531	1.63253
$246$	0	0
$247$	−16.6812	−1.06140
$248$	−3.17800	−0.201803
$249$	0	0
$250$	1.16876	0.0739186
$251$	−31.6093	−1.99516	−0.997581	−	0.0695077i	$-0.977857\pi$
$251$	−31.6093	−1.99516	−0.997581	+	0.0695077i	$0.977857\pi$
$252$	0	0
$253$	−1.85007	−0.116313
$254$	−0.218631	−0.0137181
$255$	0	0
$256$	13.5502	0.846886
$257$	4.05027	0.252649	0.126324	−	0.991989i	$-0.459682\pi$
$257$	4.05027	0.252649	0.126324	+	0.991989i	$0.459682\pi$
$258$	0	0
$259$	24.0159	1.49227
$260$	31.5483	1.95654
$261$	0	0
$262$	−0.707518	−0.0437106
$263$	−14.3001	−0.881782	−0.440891	−	0.897561i	$-0.645338\pi$
$263$	−14.3001	−0.881782	−0.440891	+	0.897561i	$0.645338\pi$
$264$	0	0
$265$	−37.8602	−2.32573
$266$	−2.41439	−0.148035
$267$	0	0
$268$	10.3691	0.633396
$269$	21.5174	1.31194	0.655970	−	0.754787i	$-0.272260\pi$
$269$	21.5174	1.31194	0.655970	+	0.754787i	$0.272260\pi$
$270$	0	0
$271$	19.7901	1.20216	0.601080	−	0.799189i	$-0.294737\pi$
$271$	19.7901	1.20216	0.601080	+	0.799189i	$0.294737\pi$
$272$	−11.7134	−0.710228
$273$	0	0
$274$	2.94895	0.178152
$275$	−29.4932	−1.77851
$276$	0	0
$277$	5.60355	0.336685	0.168342	−	0.985729i	$-0.446159\pi$
$277$	5.60355	0.336685	0.168342	+	0.985729i	$0.446159\pi$
$278$	1.46458	0.0878397
$279$	0	0
$280$	9.20511	0.550110
$281$	10.8186	0.645382	0.322691	−	0.946504i	$-0.395413\pi$
$281$	10.8186	0.645382	0.322691	+	0.946504i	$0.395413\pi$
$282$	0	0
$283$	−7.43490	−0.441959	−0.220980	−	0.975278i	$-0.570925\pi$
$283$	−7.43490	−0.441959	−0.220980	+	0.975278i	$0.570925\pi$
$284$	−11.4892	−0.681757
$285$	0	0
$286$	3.50797	0.207430
$287$	7.48141	0.441614
$288$	0	0
$289$	−7.56234	−0.444843
$290$	0	0
$291$	0	0
$292$	−20.7683	−1.21537
$293$	12.4164	0.725373	0.362686	−	0.931911i	$-0.381860\pi$
$293$	12.4164	0.725373	0.362686	+	0.931911i	$0.381860\pi$
$294$	0	0
$295$	20.9499	1.21975
$296$	4.44689	0.258470
$297$	0	0
$298$	1.66244	0.0963028
$299$	−2.01327	−0.116430
$300$	0	0
$301$	−0.953274	−0.0549458
$302$	3.48183	0.200357
$303$	0	0
$304$	13.6984	0.785656
$305$	−13.0869	−0.749355
$306$	0	0
$307$	−24.0387	−1.37196	−0.685981	−	0.727620i	$-0.740627\pi$
$307$	−24.0387	−1.37196	−0.685981	+	0.727620i	$0.740627\pi$
$308$	−31.8785	−1.81645
$309$	0	0
$310$	2.76382	0.156975
$311$	−31.2545	−1.77228	−0.886140	−	0.463418i	$-0.846623\pi$
$311$	−31.2545	−1.77228	−0.886140	+	0.463418i	$0.846623\pi$
$312$	0	0
$313$	−21.0508	−1.18986	−0.594931	−	0.803777i	$-0.702821\pi$
$313$	−21.0508	−1.18986	−0.594931	+	0.803777i	$0.702821\pi$
$314$	0.462969	0.0261268
$315$	0	0
$316$	19.7586	1.11151
$317$	−15.6515	−0.879073	−0.439537	−	0.898225i	$-0.644857\pi$
$317$	−15.6515	−0.879073	−0.439537	+	0.898225i	$0.644857\pi$
$318$	0	0
$319$	0	0
$320$	−25.0480	−1.40023
$321$	0	0
$322$	−0.291394	−0.0162387
$323$	−11.0370	−0.614116
$324$	0	0
$325$	−32.0949	−1.78030
$326$	0.0934589	0.00517621
$327$	0	0
$328$	1.38529	0.0764900
$329$	18.2788	1.00775
$330$	0	0
$331$	19.4595	1.06959	0.534796	−	0.844981i	$-0.320388\pi$
$331$	19.4595	1.06959	0.534796	+	0.844981i	$0.320388\pi$
$332$	−13.9738	−0.766910
$333$	0	0
$334$	2.66476	0.145809
$335$	−18.1792	−0.993235
$336$	0	0
$337$	−20.1780	−1.09916	−0.549582	−	0.835440i	$-0.685213\pi$
$337$	−20.1780	−1.09916	−0.549582	+	0.835440i	$0.685213\pi$
$338$	1.51547	0.0824305
$339$	0	0
$340$	20.8737	1.13203
$341$	−19.2954	−1.04491
$342$	0	0
$343$	−1.53184	−0.0827119
$344$	−0.176513	−0.00951692
$345$	0	0
$346$	−0.0182797	−0.000982724 0
$347$	−0.788354	−0.0423211	−0.0211605	−	0.999776i	$-0.506736\pi$
$347$	−0.788354	−0.0423211	−0.0211605	+	0.999776i	$0.506736\pi$
$348$	0	0
$349$	20.9417	1.12098	0.560491	−	0.828161i	$-0.310613\pi$
$349$	20.9417	1.12098	0.560491	+	0.828161i	$0.310613\pi$
$350$	−4.64531	−0.248302
$351$	0	0
$352$	−8.87748	−0.473172
$353$	36.0882	1.92078	0.960390	−	0.278659i	$-0.0898898\pi$
$353$	36.0882	1.92078	0.960390	+	0.278659i	$0.0898898\pi$
$354$	0	0
$355$	20.1428	1.06907
$356$	27.5142	1.45825
$357$	0	0
$358$	0.716996	0.0378944
$359$	18.6229	0.982881	0.491441	−	0.870911i	$-0.336471\pi$
$359$	18.6229	0.982881	0.491441	+	0.870911i	$0.336471\pi$
$360$	0	0
$361$	−6.09261	−0.320664
$362$	−2.04028	−0.107235
$363$	0	0
$364$	−34.6906	−1.81828
$365$	36.4110	1.90584
$366$	0	0
$367$	−16.1322	−0.842094	−0.421047	−	0.907039i	$-0.638337\pi$
$367$	−16.1322	−0.842094	−0.421047	+	0.907039i	$0.638337\pi$
$368$	1.65327	0.0861824
$369$	0	0
$370$	−3.86735	−0.201054
$371$	41.6312	2.16139
$372$	0	0
$373$	1.13592	0.0588156	0.0294078	−	0.999567i	$-0.490638\pi$
$373$	1.13592	0.0588156	0.0294078	+	0.999567i	$0.490638\pi$
$374$	2.32102	0.120017
$375$	0	0
$376$	3.38460	0.174547
$377$	0	0
$378$	0	0
$379$	−30.2475	−1.55371	−0.776856	−	0.629679i	$-0.783187\pi$
$379$	−30.2475	−1.55371	−0.776856	+	0.629679i	$0.783187\pi$
$380$	−24.4110	−1.25226
$381$	0	0
$382$	0.276131	0.0141281
$383$	17.5797	0.898280	0.449140	−	0.893461i	$-0.351730\pi$
$383$	17.5797	0.898280	0.449140	+	0.893461i	$0.351730\pi$
$384$	0	0
$385$	55.8895	2.84839
$386$	−2.36879	−0.120568
$387$	0	0
$388$	−28.1958	−1.43142
$389$	38.5442	1.95427	0.977134	−	0.212625i	$-0.0682014\pi$
$389$	38.5442	1.95427	0.977134	+	0.212625i	$0.0682014\pi$
$390$	0	0
$391$	−1.33206	−0.0673653
$392$	−5.20281	−0.262782
$393$	0	0
$394$	2.26222	0.113969
$395$	−34.6409	−1.74297
$396$	0	0
$397$	−21.0742	−1.05768	−0.528841	−	0.848721i	$-0.677373\pi$
$397$	−21.0742	−1.05768	−0.528841	+	0.848721i	$0.677373\pi$
$398$	1.56215	0.0783033
$399$	0	0
$400$	26.3559	1.31779
$401$	34.7249	1.73408	0.867039	−	0.498240i	$-0.166020\pi$
$401$	34.7249	1.73408	0.867039	+	0.498240i	$0.166020\pi$
$402$	0	0
$403$	−20.9975	−1.04596
$404$	−1.60729	−0.0799657
$405$	0	0
$406$	0	0
$407$	26.9996	1.33832
$408$	0	0
$409$	3.13466	0.154999	0.0774995	−	0.996992i	$-0.475306\pi$
$409$	3.13466	0.154999	0.0774995	+	0.996992i	$0.475306\pi$
$410$	−1.20475	−0.0594985
$411$	0	0
$412$	9.76044	0.480862
$413$	−23.0366	−1.13356
$414$	0	0
$415$	24.4988	1.20260
$416$	−9.66059	−0.473649
$417$	0	0
$418$	−2.71435	−0.132763
$419$	8.68305	0.424195	0.212097	−	0.977249i	$-0.431971\pi$
$419$	8.68305	0.424195	0.212097	+	0.977249i	$0.431971\pi$
$420$	0	0
$421$	−28.3830	−1.38330	−0.691651	−	0.722232i	$-0.743116\pi$
$421$	−28.3830	−1.38330	−0.691651	+	0.722232i	$0.743116\pi$
$422$	2.66023	0.129498
$423$	0	0
$424$	7.70863	0.374364
$425$	−21.2354	−1.03007
$426$	0	0
$427$	14.3905	0.696403
$428$	−22.7100	−1.09773
$429$	0	0
$430$	0.153509	0.00740284
$431$	−0.0823506	−0.00396669	−0.00198334	−	0.999998i	$-0.500631\pi$
$431$	−0.0823506	−0.00396669	−0.00198334	+	0.999998i	$0.500631\pi$
$432$	0	0
$433$	−1.26450	−0.0607678	−0.0303839	−	0.999538i	$-0.509673\pi$
$433$	−1.26450	−0.0607678	−0.0303839	+	0.999538i	$0.509673\pi$
$434$	−3.03912	−0.145882
$435$	0	0
$436$	−17.0232	−0.815265
$437$	1.55780	0.0745197
$438$	0	0
$439$	29.1767	1.39253	0.696263	−	0.717787i	$-0.254845\pi$
$439$	29.1767	1.39253	0.696263	+	0.717787i	$0.254845\pi$
$440$	10.3488	0.493358
$441$	0	0
$442$	2.52576	0.120138
$443$	−18.2966	−0.869298	−0.434649	−	0.900600i	$-0.643128\pi$
$443$	−18.2966	−0.869298	−0.434649	+	0.900600i	$0.643128\pi$
$444$	0	0
$445$	−48.2379	−2.28669
$446$	4.24012	0.200775
$447$	0	0
$448$	27.5429	1.30128
$449$	−10.6432	−0.502283	−0.251142	−	0.967950i	$-0.580806\pi$
$449$	−10.6432	−0.502283	−0.251142	+	0.967950i	$0.580806\pi$
$450$	0	0
$451$	8.41090	0.396054
$452$	−14.9332	−0.702397
$453$	0	0
$454$	0.913431	0.0428694
$455$	60.8196	2.85127
$456$	0	0
$457$	30.2026	1.41282	0.706408	−	0.707804i	$-0.250314\pi$
$457$	30.2026	1.41282	0.706408	+	0.707804i	$0.250314\pi$
$458$	1.43459	0.0670340
$459$	0	0
$460$	−2.94618	−0.137366
$461$	−36.4593	−1.69808	−0.849041	−	0.528328i	$-0.822819\pi$
$461$	−36.4593	−1.69808	−0.849041	+	0.528328i	$0.822819\pi$
$462$	0	0
$463$	18.1919	0.845450	0.422725	−	0.906258i	$-0.361074\pi$
$463$	18.1919	0.845450	0.422725	+	0.906258i	$0.361074\pi$
$464$	0	0
$465$	0	0
$466$	−4.34100	−0.201093
$467$	9.91290	0.458714	0.229357	−	0.973342i	$-0.426338\pi$
$467$	9.91290	0.458714	0.229357	+	0.973342i	$0.426338\pi$
$468$	0	0
$469$	19.9899	0.923049
$470$	−2.94350	−0.135773
$471$	0	0
$472$	−4.26556	−0.196338
$473$	−1.07171	−0.0492772
$474$	0	0
$475$	24.8340	1.13946
$476$	−22.9528	−1.05204
$477$	0	0
$478$	4.94755	0.226296
$479$	42.7297	1.95237	0.976184	−	0.216945i	$-0.0696091\pi$
$479$	42.7297	1.95237	0.976184	+	0.216945i	$0.0696091\pi$
$480$	0	0
$481$	29.3813	1.33967
$482$	0.702689	0.0320066
$483$	0	0
$484$	−14.1841	−0.644730
$485$	49.4328	2.24463
$486$	0	0
$487$	−25.4364	−1.15263	−0.576316	−	0.817227i	$-0.695510\pi$
$487$	−25.4364	−1.15263	−0.576316	+	0.817227i	$0.695510\pi$
$488$	2.66460	0.120621
$489$	0	0
$490$	4.52476	0.204408
$491$	31.4216	1.41804	0.709018	−	0.705190i	$-0.249138\pi$
$491$	31.4216	1.41804	0.709018	+	0.705190i	$0.249138\pi$
$492$	0	0
$493$	0	0
$494$	−2.95379	−0.132897
$495$	0	0
$496$	17.2429	0.774228
$497$	−22.1492	−0.993526
$498$	0	0
$499$	−6.94660	−0.310973	−0.155486	−	0.987838i	$-0.549694\pi$
$499$	−6.94660	−0.310973	−0.155486	+	0.987838i	$0.549694\pi$
$500$	−12.9939	−0.581105
$501$	0	0
$502$	−5.59715	−0.249813
$503$	26.5513	1.18386	0.591932	−	0.805988i	$-0.298365\pi$
$503$	26.5513	1.18386	0.591932	+	0.805988i	$0.298365\pi$
$504$	0	0
$505$	2.81790	0.125395
$506$	−0.327596	−0.0145634
$507$	0	0
$508$	2.43067	0.107844
$509$	1.38050	0.0611896	0.0305948	−	0.999532i	$-0.490260\pi$
$509$	1.38050	0.0611896	0.0305948	+	0.999532i	$0.490260\pi$
$510$	0	0
$511$	−40.0377	−1.77116
$512$	13.2920	0.587430
$513$	0	0
$514$	0.717192	0.0316340
$515$	−17.1120	−0.754045
$516$	0	0
$517$	20.5498	0.903780
$518$	4.25255	0.186846
$519$	0	0
$520$	11.2616	0.493856
$521$	−8.46722	−0.370956	−0.185478	−	0.982648i	$-0.559383\pi$
$521$	−8.46722	−0.370956	−0.185478	+	0.982648i	$0.559383\pi$
$522$	0	0
$523$	4.74716	0.207579	0.103789	−	0.994599i	$-0.466903\pi$
$523$	4.74716	0.207579	0.103789	+	0.994599i	$0.466903\pi$
$524$	7.86599	0.343628
$525$	0	0
$526$	−2.53216	−0.110407
$527$	−13.8929	−0.605183
$528$	0	0
$529$	−22.8120	−0.991826
$530$	−6.70400	−0.291203
$531$	0	0
$532$	26.8425	1.16377
$533$	9.15285	0.396454
$534$	0	0
$535$	39.8152	1.72136
$536$	3.70143	0.159877
$537$	0	0
$538$	3.81015	0.164267
$539$	−31.5893	−1.36065
$540$	0	0
$541$	−2.44407	−0.105079	−0.0525394	−	0.998619i	$-0.516732\pi$
$541$	−2.44407	−0.105079	−0.0525394	+	0.998619i	$0.516732\pi$
$542$	3.50428	0.150522
$543$	0	0
$544$	−6.39186	−0.274049
$545$	29.8451	1.27843
$546$	0	0
$547$	6.17333	0.263952	0.131976	−	0.991253i	$-0.457868\pi$
$547$	6.17333	0.263952	0.131976	+	0.991253i	$0.457868\pi$
$548$	−32.7856	−1.40053
$549$	0	0
$550$	−5.22245	−0.222686
$551$	0	0
$552$	0	0
$553$	38.0913	1.61981
$554$	0.992235	0.0421560
$555$	0	0
$556$	−16.2828	−0.690544
$557$	12.4564	0.527794	0.263897	−	0.964551i	$-0.414992\pi$
$557$	12.4564	0.527794	0.263897	+	0.964551i	$0.414992\pi$
$558$	0	0
$559$	−1.16625	−0.0493270
$560$	−49.9442	−2.11053
$561$	0	0
$562$	1.91567	0.0808078
$563$	−3.58213	−0.150969	−0.0754843	−	0.997147i	$-0.524050\pi$
$563$	−3.58213	−0.150969	−0.0754843	+	0.997147i	$0.524050\pi$
$564$	0	0
$565$	26.1809	1.10144
$566$	−1.31652	−0.0553374
$567$	0	0
$568$	−4.10124	−0.172084
$569$	−26.8881	−1.12721	−0.563603	−	0.826046i	$-0.690585\pi$
$569$	−26.8881	−1.12721	−0.563603	+	0.826046i	$0.690585\pi$
$570$	0	0
$571$	−30.0109	−1.25592	−0.627959	−	0.778247i	$-0.716109\pi$
$571$	−30.0109	−1.25592	−0.627959	+	0.778247i	$0.716109\pi$
$572$	−39.0006	−1.63070
$573$	0	0
$574$	1.32475	0.0552941
$575$	2.99723	0.124993
$576$	0	0
$577$	2.86591	0.119309	0.0596546	−	0.998219i	$-0.481000\pi$
$577$	2.86591	0.119309	0.0596546	+	0.998219i	$0.481000\pi$
$578$	−1.33908	−0.0556985
$579$	0	0
$580$	0	0
$581$	−26.9390	−1.11762
$582$	0	0
$583$	46.8035	1.93840
$584$	−7.41357	−0.306776
$585$	0	0
$586$	2.19860	0.0908234
$587$	−21.8403	−0.901446	−0.450723	−	0.892664i	$-0.648834\pi$
$587$	−21.8403	−0.901446	−0.450723	+	0.892664i	$0.648834\pi$
$588$	0	0
$589$	16.2472	0.669455
$590$	3.70965	0.152724
$591$	0	0
$592$	−24.1275	−0.991635
$593$	−19.9669	−0.819941	−0.409971	−	0.912099i	$-0.634461\pi$
$593$	−19.9669	−0.819941	−0.409971	+	0.912099i	$0.634461\pi$
$594$	0	0
$595$	40.2409	1.64971
$596$	−18.4826	−0.757077
$597$	0	0
$598$	−0.356494	−0.0145781
$599$	−25.8925	−1.05794	−0.528969	−	0.848641i	$-0.677421\pi$
$599$	−25.8925	−1.05794	−0.528969	+	0.848641i	$0.677421\pi$
$600$	0	0
$601$	−24.1209	−0.983911	−0.491956	−	0.870620i	$-0.663718\pi$
$601$	−24.1209	−0.983911	−0.491956	+	0.870620i	$0.663718\pi$
$602$	−0.168799	−0.00687973
$603$	0	0
$604$	−38.7100	−1.57509
$605$	24.8675	1.01101
$606$	0	0
$607$	5.67771	0.230451	0.115226	−	0.993339i	$-0.463241\pi$
$607$	5.67771	0.230451	0.115226	+	0.993339i	$0.463241\pi$
$608$	7.47505	0.303153
$609$	0	0
$610$	−2.31734	−0.0938262
$611$	22.3626	0.904693
$612$	0	0
$613$	−44.8240	−1.81042	−0.905212	−	0.424961i	$-0.860288\pi$
$613$	−44.8240	−1.81042	−0.905212	+	0.424961i	$0.860288\pi$
$614$	−4.25660	−0.171782
$615$	0	0
$616$	−11.3795	−0.458495
$617$	−29.0434	−1.16924	−0.584622	−	0.811306i	$-0.698757\pi$
$617$	−29.0434	−1.16924	−0.584622	+	0.811306i	$0.698757\pi$
$618$	0	0
$619$	7.61305	0.305994	0.152997	−	0.988227i	$-0.451107\pi$
$619$	7.61305	0.305994	0.152997	+	0.988227i	$0.451107\pi$
$620$	−30.7274	−1.23404
$621$	0	0
$622$	−5.53432	−0.221906
$623$	53.0426	2.12511
$624$	0	0
$625$	−11.7810	−0.471238
$626$	−3.72753	−0.148982
$627$	0	0
$628$	−5.14716	−0.205394
$629$	19.4399	0.775121
$630$	0	0
$631$	43.7635	1.74220	0.871099	−	0.491107i	$-0.163408\pi$
$631$	43.7635	1.74220	0.871099	+	0.491107i	$0.163408\pi$
$632$	7.05316	0.280560
$633$	0	0
$634$	−2.77144	−0.110068
$635$	−4.26146	−0.169111
$636$	0	0
$637$	−34.3758	−1.36202
$638$	0	0
$639$	0	0
$640$	−18.7976	−0.743041
$641$	15.3364	0.605750	0.302875	−	0.953030i	$-0.402053\pi$
$641$	15.3364	0.605750	0.302875	+	0.953030i	$0.402053\pi$
$642$	0	0
$643$	27.2674	1.07532	0.537661	−	0.843161i	$-0.319308\pi$
$643$	27.2674	1.07532	0.537661	+	0.843161i	$0.319308\pi$
$644$	3.23963	0.127659
$645$	0	0
$646$	−1.95435	−0.0768930
$647$	24.1819	0.950690	0.475345	−	0.879800i	$-0.342323\pi$
$647$	24.1819	0.950690	0.475345	+	0.879800i	$0.342323\pi$
$648$	0	0
$649$	−25.8986	−1.01661
$650$	−5.68313	−0.222911
$651$	0	0
$652$	−1.03905	−0.0406923
$653$	−25.1321	−0.983497	−0.491748	−	0.870737i	$-0.663642\pi$
$653$	−25.1321	−0.983497	−0.491748	+	0.870737i	$0.663642\pi$
$654$	0	0
$655$	−13.7907	−0.538846
$656$	−7.51619	−0.293458
$657$	0	0
$658$	3.23668	0.126179
$659$	9.64762	0.375818	0.187909	−	0.982186i	$-0.439829\pi$
$659$	9.64762	0.375818	0.187909	+	0.982186i	$0.439829\pi$
$660$	0	0
$661$	15.7907	0.614188	0.307094	−	0.951679i	$-0.400643\pi$
$661$	15.7907	0.614188	0.307094	+	0.951679i	$0.400643\pi$
$662$	3.44575	0.133923
$663$	0	0
$664$	−4.98815	−0.193578
$665$	−47.0603	−1.82492
$666$	0	0
$667$	0	0
$668$	−29.6260	−1.14627
$669$	0	0
$670$	−3.21904	−0.124362
$671$	16.1783	0.624557
$672$	0	0
$673$	−16.3293	−0.629450	−0.314725	−	0.949183i	$-0.601912\pi$
$673$	−16.3293	−0.629450	−0.314725	+	0.949183i	$0.601912\pi$
$674$	−3.57297	−0.137625
$675$	0	0
$676$	−16.8485	−0.648021
$677$	11.3525	0.436314	0.218157	−	0.975914i	$-0.429996\pi$
$677$	11.3525	0.436314	0.218157	+	0.975914i	$0.429996\pi$
$678$	0	0
$679$	−54.3566	−2.08602
$680$	7.45118	0.285740
$681$	0	0
$682$	−3.41670	−0.130832
$683$	−21.0754	−0.806426	−0.403213	−	0.915106i	$-0.632107\pi$
$683$	−21.0754	−0.806426	−0.403213	+	0.915106i	$0.632107\pi$
$684$	0	0
$685$	57.4797	2.19619
$686$	−0.271248	−0.0103563
$687$	0	0
$688$	0.957706	0.0365122
$689$	50.9321	1.94036
$690$	0	0
$691$	5.59507	0.212846	0.106423	−	0.994321i	$-0.466060\pi$
$691$	5.59507	0.212846	0.106423	+	0.994321i	$0.466060\pi$
$692$	0.203229	0.00772561
$693$	0	0
$694$	−0.139596	−0.00529899
$695$	28.5470	1.08285
$696$	0	0
$697$	6.05592	0.229384
$698$	3.70820	0.140357
$699$	0	0
$700$	51.6453	1.95201
$701$	17.3045	0.653583	0.326791	−	0.945097i	$-0.394033\pi$
$701$	17.3045	0.653583	0.326791	+	0.945097i	$0.394033\pi$
$702$	0	0
$703$	−22.7343	−0.857440
$704$	30.9649	1.16703
$705$	0	0
$706$	6.39023	0.240500
$707$	−3.09858	−0.116534
$708$	0	0
$709$	18.6967	0.702171	0.351085	−	0.936343i	$-0.385813\pi$
$709$	18.6967	0.702171	0.351085	+	0.936343i	$0.385813\pi$
$710$	3.56675	0.133858
$711$	0	0
$712$	9.82162	0.368081
$713$	1.96089	0.0734357
$714$	0	0
$715$	68.3759	2.55711
$716$	−7.97137	−0.297904
$717$	0	0
$718$	3.29761	0.123066
$719$	17.2847	0.644610	0.322305	−	0.946636i	$-0.395542\pi$
$719$	17.2847	0.644610	0.322305	+	0.946636i	$0.395542\pi$
$720$	0	0
$721$	18.8164	0.700761
$722$	−1.07884	−0.0401501
$723$	0	0
$724$	22.6832	0.843015
$725$	0	0
$726$	0	0
$727$	−2.87185	−0.106511	−0.0532556	−	0.998581i	$-0.516960\pi$
$727$	−2.87185	−0.106511	−0.0532556	+	0.998581i	$0.516960\pi$
$728$	−12.3834	−0.458958
$729$	0	0
$730$	6.44739	0.238629
$731$	−0.771639	−0.0285401
$732$	0	0
$733$	−46.8195	−1.72932	−0.864659	−	0.502359i	$-0.832466\pi$
$733$	−46.8195	−1.72932	−0.864659	+	0.502359i	$0.832466\pi$
$734$	−2.85657	−0.105438
$735$	0	0
$736$	0.902168	0.0332543
$737$	22.4735	0.827821
$738$	0	0
$739$	−25.0754	−0.922415	−0.461208	−	0.887292i	$-0.652584\pi$
$739$	−25.0754	−0.922415	−0.461208	+	0.887292i	$0.652584\pi$
$740$	42.9961	1.58057
$741$	0	0
$742$	7.37175	0.270626
$743$	28.1550	1.03291	0.516453	−	0.856316i	$-0.327252\pi$
$743$	28.1550	1.03291	0.516453	+	0.856316i	$0.327252\pi$
$744$	0	0
$745$	32.4037	1.18718
$746$	0.201140	0.00736426
$747$	0	0
$748$	−25.8045	−0.943505
$749$	−43.7810	−1.59972
$750$	0	0
$751$	40.8050	1.48900	0.744498	−	0.667624i	$-0.232689\pi$
$751$	40.8050	1.48900	0.744498	+	0.667624i	$0.232689\pi$
$752$	−18.3638	−0.669660
$753$	0	0
$754$	0	0
$755$	67.8664	2.46991
$756$	0	0
$757$	−40.8015	−1.48296	−0.741478	−	0.670977i	$-0.765875\pi$
$757$	−40.8015	−1.48296	−0.741478	+	0.670977i	$0.765875\pi$
$758$	−5.35601	−0.194539
$759$	0	0
$760$	−8.71389	−0.316086
$761$	41.6230	1.50883	0.754417	−	0.656395i	$-0.227920\pi$
$761$	41.6230	1.50883	0.754417	+	0.656395i	$0.227920\pi$
$762$	0	0
$763$	−32.8179	−1.18809
$764$	−3.06995	−0.111067
$765$	0	0
$766$	3.11288	0.112473
$767$	−28.1832	−1.01764
$768$	0	0
$769$	9.30333	0.335487	0.167743	−	0.985831i	$-0.446352\pi$
$769$	9.30333	0.335487	0.167743	+	0.985831i	$0.446352\pi$
$770$	9.89650	0.356645
$771$	0	0
$772$	26.3355	0.947836
$773$	14.4098	0.518284	0.259142	−	0.965839i	$-0.416560\pi$
$773$	14.4098	0.518284	0.259142	+	0.965839i	$0.416560\pi$
$774$	0	0
$775$	31.2599	1.12289
$776$	−10.0649	−0.361310
$777$	0	0
$778$	6.82512	0.244692
$779$	−7.08218	−0.253745
$780$	0	0
$781$	−24.9010	−0.891028
$782$	−0.235872	−0.00843476
$783$	0	0
$784$	28.2289	1.00818
$785$	9.02400	0.322080
$786$	0	0
$787$	31.0529	1.10692	0.553459	−	0.832877i	$-0.313308\pi$
$787$	31.0529	1.10692	0.553459	+	0.832877i	$0.313308\pi$
$788$	−25.1507	−0.895957
$789$	0	0
$790$	−6.13395	−0.218236
$791$	−28.7886	−1.02360
$792$	0	0
$793$	17.6055	0.625188
$794$	−3.73166	−0.132432
$795$	0	0
$796$	−17.3675	−0.615575
$797$	−33.4931	−1.18639	−0.593193	−	0.805060i	$-0.702133\pi$
$797$	−33.4931	−1.18639	−0.593193	+	0.805060i	$0.702133\pi$
$798$	0	0
$799$	14.7960	0.523446
$800$	14.3821	0.508484
$801$	0	0
$802$	6.14883	0.217123
$803$	−45.0120	−1.58844
$804$	0	0
$805$	−5.67973	−0.200184
$806$	−3.71809	−0.130964
$807$	0	0
$808$	−0.573747	−0.0201844
$809$	17.9475	0.631000	0.315500	−	0.948926i	$-0.397828\pi$
$809$	17.9475	0.631000	0.315500	+	0.948926i	$0.397828\pi$
$810$	0	0
$811$	6.11810	0.214836	0.107418	−	0.994214i	$-0.465742\pi$
$811$	6.11810	0.214836	0.107418	+	0.994214i	$0.465742\pi$
$812$	0	0
$813$	0	0
$814$	4.78089	0.167570
$815$	1.82166	0.0638101
$816$	0	0
$817$	0.902404	0.0315711
$818$	0.555063	0.0194073
$819$	0	0
$820$	13.3941	0.467743
$821$	38.5521	1.34548	0.672738	−	0.739881i	$-0.265118\pi$
$821$	38.5521	1.34548	0.672738	+	0.739881i	$0.265118\pi$
$822$	0	0
$823$	15.0217	0.523625	0.261813	−	0.965119i	$-0.415680\pi$
$823$	15.0217	0.523625	0.261813	+	0.965119i	$0.415680\pi$
$824$	3.48414	0.121376
$825$	0	0
$826$	−4.07915	−0.141932
$827$	17.5838	0.611448	0.305724	−	0.952120i	$-0.401101\pi$
$827$	17.5838	0.611448	0.305724	+	0.952120i	$0.401101\pi$
$828$	0	0
$829$	27.9182	0.969640	0.484820	−	0.874614i	$-0.338885\pi$
$829$	27.9182	0.969640	0.484820	+	0.874614i	$0.338885\pi$
$830$	4.33807	0.150577
$831$	0	0
$832$	33.6964	1.16821
$833$	−22.7445	−0.788051
$834$	0	0
$835$	51.9404	1.79747
$836$	30.1774	1.04371
$837$	0	0
$838$	1.53753	0.0531131
$839$	−22.1987	−0.766384	−0.383192	−	0.923669i	$-0.625175\pi$
$839$	−22.1987	−0.766384	−0.383192	+	0.923669i	$0.625175\pi$
$840$	0	0
$841$	0	0
$842$	−5.02585	−0.173202
$843$	0	0
$844$	−29.5757	−1.01804
$845$	29.5389	1.01617
$846$	0	0
$847$	−27.3444	−0.939566
$848$	−41.8248	−1.43627
$849$	0	0
$850$	−3.76020	−0.128974
$851$	−2.74381	−0.0940568
$852$	0	0
$853$	−26.4548	−0.905794	−0.452897	−	0.891563i	$-0.649609\pi$
$853$	−26.4548	−0.905794	−0.452897	+	0.891563i	$0.649609\pi$
$854$	2.54816	0.0871961
$855$	0	0
$856$	−8.10670	−0.277081
$857$	45.8418	1.56592	0.782962	−	0.622069i	$-0.213708\pi$
$857$	45.8418	1.56592	0.782962	+	0.622069i	$0.213708\pi$
$858$	0	0
$859$	−8.29444	−0.283003	−0.141501	−	0.989938i	$-0.545193\pi$
$859$	−8.29444	−0.283003	−0.141501	+	0.989938i	$0.545193\pi$
$860$	−1.70667	−0.0581968
$861$	0	0
$862$	−0.0145820	−0.000496666 0
$863$	7.74485	0.263638	0.131819	−	0.991274i	$-0.457918\pi$
$863$	7.74485	0.263638	0.131819	+	0.991274i	$0.457918\pi$
$864$	0	0
$865$	−0.356301	−0.0121146
$866$	−0.223908	−0.00760869
$867$	0	0
$868$	33.7880	1.14684
$869$	42.8238	1.45270
$870$	0	0
$871$	24.4559	0.828657
$872$	−6.07671	−0.205783
$873$	0	0
$874$	0.275844	0.00933055
$875$	−25.0500	−0.846845
$876$	0	0
$877$	37.2741	1.25866	0.629328	−	0.777139i	$-0.283330\pi$
$877$	37.2741	1.25866	0.629328	+	0.777139i	$0.283330\pi$
$878$	5.16639	0.174357
$879$	0	0
$880$	−56.1493	−1.89279
$881$	−15.2380	−0.513382	−0.256691	−	0.966493i	$-0.582632\pi$
$881$	−15.2380	−0.513382	−0.256691	+	0.966493i	$0.582632\pi$
$882$	0	0
$883$	−48.7085	−1.63917	−0.819586	−	0.572956i	$-0.805797\pi$
$883$	−48.7085	−1.63917	−0.819586	+	0.572956i	$0.805797\pi$
$884$	−28.0807	−0.944457
$885$	0	0
$886$	−3.23983	−0.108844
$887$	−8.21801	−0.275934	−0.137967	−	0.990437i	$-0.544057\pi$
$887$	−8.21801	−0.275934	−0.137967	+	0.990437i	$0.544057\pi$
$888$	0	0
$889$	4.68592	0.157161
$890$	−8.54161	−0.286315
$891$	0	0
$892$	−47.1404	−1.57838
$893$	−17.3034	−0.579037
$894$	0	0
$895$	13.9754	0.467146
$896$	20.6700	0.690535
$897$	0	0
$898$	−1.88462	−0.0628906
$899$	0	0
$900$	0	0
$901$	33.6989	1.12267
$902$	1.48934	0.0495896
$903$	0	0
$904$	−5.33063	−0.177294
$905$	−39.7682	−1.32194
$906$	0	0
$907$	4.20133	0.139503	0.0697514	−	0.997564i	$-0.477779\pi$
$907$	4.20133	0.139503	0.0697514	+	0.997564i	$0.477779\pi$
$908$	−10.1553	−0.337014
$909$	0	0
$910$	10.7695	0.357005
$911$	−13.0361	−0.431906	−0.215953	−	0.976404i	$-0.569286\pi$
$911$	−13.0361	−0.431906	−0.215953	+	0.976404i	$0.569286\pi$
$912$	0	0
$913$	−30.2859	−1.00232
$914$	5.34805	0.176898
$915$	0	0
$916$	−15.9494	−0.526982
$917$	15.1643	0.500769
$918$	0	0
$919$	2.15993	0.0712497	0.0356248	−	0.999365i	$-0.488658\pi$
$919$	2.15993	0.0712497	0.0356248	+	0.999365i	$0.488658\pi$
$920$	−1.05168	−0.0346730
$921$	0	0
$922$	−6.45595	−0.212616
$923$	−27.0976	−0.891927
$924$	0	0
$925$	−43.7411	−1.43820
$926$	3.22129	0.105858
$927$	0	0
$928$	0	0
$929$	−5.52224	−0.181179	−0.0905894	−	0.995888i	$-0.528875\pi$
$929$	−5.52224	−0.181179	−0.0905894	+	0.995888i	$0.528875\pi$
$930$	0	0
$931$	26.5989	0.871744
$932$	48.2620	1.58087
$933$	0	0
$934$	1.75530	0.0574353
$935$	45.2404	1.47952
$936$	0	0
$937$	55.7255	1.82047	0.910237	−	0.414089i	$-0.135900\pi$
$937$	55.7255	1.82047	0.910237	+	0.414089i	$0.135900\pi$
$938$	3.53967	0.115574
$939$	0	0
$940$	32.7250	1.06737
$941$	−52.7691	−1.72022	−0.860112	−	0.510105i	$-0.829606\pi$
$941$	−52.7691	−1.72022	−0.860112	+	0.510105i	$0.829606\pi$
$942$	0	0
$943$	−0.854752	−0.0278345
$944$	23.1437	0.753262
$945$	0	0
$946$	−0.189770	−0.00616997
$947$	42.3685	1.37679	0.688396	−	0.725335i	$-0.258315\pi$
$947$	42.3685	1.37679	0.688396	+	0.725335i	$0.258315\pi$
$948$	0	0
$949$	−48.9826	−1.59004
$950$	4.39742	0.142671
$951$	0	0
$952$	−8.19336	−0.265548
$953$	−1.43433	−0.0464623	−0.0232312	−	0.999730i	$-0.507395\pi$
$953$	−1.43433	−0.0464623	−0.0232312	+	0.999730i	$0.507395\pi$
$954$	0	0
$955$	5.38224	0.174165
$956$	−55.0055	−1.77900
$957$	0	0
$958$	7.56626	0.244455
$959$	−63.2050	−2.04100
$960$	0	0
$961$	−10.5488	−0.340283
$962$	5.20263	0.167739
$963$	0	0
$964$	−7.81230	−0.251617
$965$	−46.1715	−1.48631
$966$	0	0
$967$	10.4202	0.335092	0.167546	−	0.985864i	$-0.446416\pi$
$967$	10.4202	0.335092	0.167546	+	0.985864i	$0.446416\pi$
$968$	−5.06322	−0.162738
$969$	0	0
$970$	8.75321	0.281049
$971$	−39.4639	−1.26646	−0.633228	−	0.773965i	$-0.718271\pi$
$971$	−39.4639	−1.26646	−0.633228	+	0.773965i	$0.718271\pi$
$972$	0	0
$973$	−31.3904	−1.00633
$974$	−4.50408	−0.144320
$975$	0	0
$976$	−14.4574	−0.462769
$977$	−7.31111	−0.233903	−0.116952	−	0.993138i	$-0.537312\pi$
$977$	−7.31111	−0.233903	−0.116952	+	0.993138i	$0.537312\pi$
$978$	0	0
$979$	59.6327	1.90587
$980$	−50.3050	−1.60693
$981$	0	0
$982$	5.56390	0.177551
$983$	−39.1740	−1.24946	−0.624728	−	0.780842i	$-0.714790\pi$
$983$	−39.1740	−1.24946	−0.624728	+	0.780842i	$0.714790\pi$
$984$	0	0
$985$	44.0942	1.40496
$986$	0	0
$987$	0	0
$988$	32.8394	1.04476
$989$	0.108912	0.00346319
$990$	0	0
$991$	−47.8488	−1.51997	−0.759983	−	0.649942i	$-0.774793\pi$
$991$	−47.8488	−1.51997	−0.759983	+	0.649942i	$0.774793\pi$
$992$	9.40924	0.298744
$993$	0	0
$994$	−3.92201	−0.124399
$995$	30.4487	0.965290
$996$	0	0
$997$	−51.3829	−1.62731	−0.813656	−	0.581347i	$-0.802526\pi$
$997$	−51.3829	−1.62731	−0.813656	+	0.581347i	$0.802526\pi$
$998$	−1.23005	−0.0389367
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.bm.1.5		9
3.2	odd	2		2523.2.a.o.1.5		9
29.9	even	14		261.2.k.c.226.2		18
29.13	even	14		261.2.k.c.82.2		18
29.28	even	2		7569.2.a.bj.1.5		9
87.38	odd	14		87.2.g.a.52.2	✓	18
87.71	odd	14		87.2.g.a.82.2	yes	18
87.86	odd	2		2523.2.a.r.1.5		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.g.a.52.2	✓	18	87.38	odd	14
87.2.g.a.82.2	yes	18	87.71	odd	14
261.2.k.c.82.2		18	29.13	even	14
261.2.k.c.226.2		18	29.9	even	14
2523.2.a.o.1.5		9	3.2	odd	2
2523.2.a.r.1.5		9	87.86	odd	2
7569.2.a.bj.1.5		9	29.28	even	2
7569.2.a.bm.1.5		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)

\(f(q)\)	\(=\)	\(q+0.177073 q^{2} -1.96865 q^{4} +3.45143 q^{5} -3.79521 q^{7} -0.702739 q^{8} +0.611154 q^{10} -4.26673 q^{11} -4.64311 q^{13} -0.672028 q^{14} +3.81285 q^{16} -3.07208 q^{17} +3.59269 q^{19} -6.79464 q^{20} -0.755521 q^{22} +0.433603 q^{23} +6.91237 q^{25} -0.822167 q^{26} +7.47142 q^{28} +4.52230 q^{31} +2.08063 q^{32} -0.543981 q^{34} -13.0989 q^{35} -6.32794 q^{37} +0.636166 q^{38} -2.42545 q^{40} -1.97128 q^{41} +0.251178 q^{43} +8.39968 q^{44} +0.0767793 q^{46} -4.81629 q^{47} +7.40363 q^{49} +1.22399 q^{50} +9.14063 q^{52} -10.9694 q^{53} -14.7263 q^{55} +2.66704 q^{56} +6.06991 q^{59} -3.79174 q^{61} +0.800776 q^{62} -7.25729 q^{64} -16.0254 q^{65} -5.26714 q^{67} +6.04783 q^{68} -2.31946 q^{70} +5.83609 q^{71} +10.5495 q^{73} -1.12051 q^{74} -7.07272 q^{76} +16.1931 q^{77} -10.0367 q^{79} +13.1598 q^{80} -0.349059 q^{82} +7.09816 q^{83} -10.6031 q^{85} +0.0444768 q^{86} +2.99840 q^{88} -13.9762 q^{89} +17.6216 q^{91} -0.853611 q^{92} -0.852834 q^{94} +12.3999 q^{95} +14.3224 q^{97} +1.31098 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 5 q^{2} + 11 q^{4} + 4 q^{5} + 5 q^{7} + 24 q^{8} - q^{11} + q^{13} + 9 q^{14} + 35 q^{16} + 2 q^{17} - 9 q^{19} + 18 q^{20} - 4 q^{22} + 4 q^{23} + q^{25} - 8 q^{26} + 40 q^{28} - 8 q^{31} + 43 q^{32}+ \cdots - 30 q^{98}+O(q^{100}) \) 9 * q + 5 * q^2 + 11 * q^4 + 4 * q^5 + 5 * q^7 + 24 * q^8 - q^11 + q^13 + 9 * q^14 + 35 * q^16 + 2 * q^17 - 9 * q^19 + 18 * q^20 - 4 * q^22 + 4 * q^23 + q^25 - 8 * q^26 + 40 * q^28 - 8 * q^31 + 43 * q^32 - 4 * q^34 - 22 * q^35 - 27 * q^37 + 30 * q^38 + 29 * q^40 + 12 * q^41 - 16 * q^43 + 37 * q^44 + 22 * q^46 - 8 * q^47 - 6 * q^49 - 7 * q^50 + 33 * q^52 + 8 * q^53 - 9 * q^55 + 40 * q^56 + 16 * q^59 - 21 * q^61 + 32 * q^62 + 36 * q^64 + 31 * q^65 + 3 * q^67 + 33 * q^68 + 6 * q^70 + 33 * q^71 - 3 * q^73 - 28 * q^74 + 26 * q^76 + 24 * q^77 - 3 * q^79 + 64 * q^80 + 13 * q^82 - 13 * q^83 - 6 * q^85 - 58 * q^86 + 27 * q^88 + 6 * q^89 + q^91 + 29 * q^92 - 18 * q^94 + 48 * q^95 - 4 * q^97 - 30 * q^98

Introduction

L-functions

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