LMFDB - Embedded newform 6525.2.a.bs.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6525,2,Mod(1,6525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6525, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6525 = 3^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6525.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:	$52.1023873189$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.246832.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 $ x^5 - 2x^4 - 5x^3 + 6x^2 + 7x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.71250$ of defining polynomial
Character	$\chi$	$=$	6525.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.712495 q^{2} -1.49235 q^{4} -2.77986 q^{7} +2.48828 q^{8} +O(q^{10})$	$q-0.712495 q^{2} -1.49235 q^{4} -2.77986 q^{7} +2.48828 q^{8} -4.26814 q^{11} +0.779856 q^{13} +1.98063 q^{14} +1.21181 q^{16} +1.90354 q^{17} +6.72036 q^{19} +3.04103 q^{22} -2.17168 q^{23} -0.555643 q^{26} +4.14852 q^{28} -1.00000 q^{29} -8.82061 q^{31} -5.83998 q^{32} -1.35626 q^{34} +1.48402 q^{37} -4.78822 q^{38} +7.71389 q^{41} -8.19624 q^{43} +6.36956 q^{44} +1.54731 q^{46} +5.19381 q^{47} +0.727598 q^{49} -1.16382 q^{52} +11.7853 q^{53} -6.91707 q^{56} +0.712495 q^{58} +4.46028 q^{59} -5.24905 q^{61} +6.28464 q^{62} +1.73733 q^{64} -8.49375 q^{67} -2.84075 q^{68} -0.663102 q^{71} +16.5345 q^{73} -1.05736 q^{74} -10.0291 q^{76} +11.8648 q^{77} +9.54554 q^{79} -5.49611 q^{82} +0.0123998 q^{83} +5.83978 q^{86} -10.6203 q^{88} +5.46783 q^{89} -2.16789 q^{91} +3.24091 q^{92} -3.70056 q^{94} -0.952006 q^{97} -0.518410 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{2} + 5 q^{4} - 8 q^{7} + 9 q^{8}+O(q^{10}) $ 5 * q + 3 * q^2 + 5 * q^4 - 8 * q^7 + 9 * q^8	$ 5 q + 3 q^{2} + 5 q^{4} - 8 q^{7} + 9 q^{8} - 12 q^{11} - 2 q^{13} - 6 q^{14} + q^{16} - 2 q^{19} - 14 q^{22} + 8 q^{23} + 6 q^{28} - 5 q^{29} + 2 q^{31} + q^{32} + 4 q^{34} - 16 q^{37} - 14 q^{38} + 14 q^{41} - 20 q^{44} - 6 q^{46} + 2 q^{47} - 7 q^{49} - 16 q^{52} + 26 q^{53} + 2 q^{56} - 3 q^{58} - 4 q^{59} - 12 q^{61} - 9 q^{64} - 12 q^{67} - 20 q^{68} - 30 q^{71} + 12 q^{73} - 2 q^{74} - 44 q^{76} + 18 q^{77} - 18 q^{79} - 10 q^{82} + 2 q^{83} - 30 q^{86} - 42 q^{88} + 22 q^{89} - 12 q^{91} + 20 q^{92} - 50 q^{94} - 20 q^{97} - 9 q^{98}+O(q^{100}) $ 5 * q + 3 * q^2 + 5 * q^4 - 8 * q^7 + 9 * q^8 - 12 * q^11 - 2 * q^13 - 6 * q^14 + q^16 - 2 * q^19 - 14 * q^22 + 8 * q^23 + 6 * q^28 - 5 * q^29 + 2 * q^31 + q^32 + 4 * q^34 - 16 * q^37 - 14 * q^38 + 14 * q^41 - 20 * q^44 - 6 * q^46 + 2 * q^47 - 7 * q^49 - 16 * q^52 + 26 * q^53 + 2 * q^56 - 3 * q^58 - 4 * q^59 - 12 * q^61 - 9 * q^64 - 12 * q^67 - 20 * q^68 - 30 * q^71 + 12 * q^73 - 2 * q^74 - 44 * q^76 + 18 * q^77 - 18 * q^79 - 10 * q^82 + 2 * q^83 - 30 * q^86 - 42 * q^88 + 22 * q^89 - 12 * q^91 + 20 * q^92 - 50 * q^94 - 20 * q^97 - 9 * 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 $

$2$	−0.712495	−0.503810	−0.251905	−	0.967752i	$-0.581057\pi$
$2$	−0.712495	−0.503810	−0.251905	+	0.967752i	$0.581057\pi$
$3$	0	0
$3$	0	0
$4$	−1.49235	−0.746175
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.77986	−1.05069	−0.525343	−	0.850890i	$-0.676063\pi$
$7$	−2.77986	−1.05069	−0.525343	+	0.850890i	$0.676063\pi$
$8$	2.48828	0.879741
$9$	0	0
$10$	0	0
$11$	−4.26814	−1.28689	−0.643446	−	0.765491i	$-0.722496\pi$
$11$	−4.26814	−1.28689	−0.643446	+	0.765491i	$0.722496\pi$
$12$	0	0
$13$	0.779856	0.216293	0.108147	−	0.994135i	$-0.465508\pi$
$13$	0.779856	0.216293	0.108147	+	0.994135i	$0.465508\pi$
$14$	1.98063	0.529347
$15$	0	0
$16$	1.21181	0.302953
$17$	1.90354	0.461677	0.230838	−	0.972992i	$-0.425853\pi$
$17$	1.90354	0.461677	0.230838	+	0.972992i	$0.425853\pi$
$18$	0	0
$19$	6.72036	1.54176	0.770878	−	0.636983i	$-0.219818\pi$
$19$	6.72036	1.54176	0.770878	+	0.636983i	$0.219818\pi$
$20$	0	0
$21$	0	0
$22$	3.04103	0.648349
$23$	−2.17168	−0.452827	−0.226413	−	0.974031i	$-0.572700\pi$
$23$	−2.17168	−0.452827	−0.226413	+	0.974031i	$0.572700\pi$
$24$	0	0
$25$	0	0
$26$	−0.555643	−0.108971
$27$	0	0
$28$	4.14852	0.783997
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−8.82061	−1.58423	−0.792114	−	0.610373i	$-0.791020\pi$
$31$	−8.82061	−1.58423	−0.792114	+	0.610373i	$0.791020\pi$
$32$	−5.83998	−1.03237
$33$	0	0
$34$	−1.35626	−0.232597
$35$	0	0
$36$	0	0
$37$	1.48402	0.243971	0.121986	−	0.992532i	$-0.461074\pi$
$37$	1.48402	0.243971	0.121986	+	0.992532i	$0.461074\pi$
$38$	−4.78822	−0.776752
$39$	0	0
$40$	0	0
$41$	7.71389	1.20471	0.602354	−	0.798229i	$-0.294230\pi$
$41$	7.71389	1.20471	0.602354	+	0.798229i	$0.294230\pi$
$42$	0	0
$43$	−8.19624	−1.24991	−0.624957	−	0.780659i	$-0.714884\pi$
$43$	−8.19624	−1.24991	−0.624957	+	0.780659i	$0.714884\pi$
$44$	6.36956	0.960247
$45$	0	0
$46$	1.54731	0.228139
$47$	5.19381	0.757595	0.378798	−	0.925480i	$-0.376338\pi$
$47$	5.19381	0.757595	0.378798	+	0.925480i	$0.376338\pi$
$48$	0	0
$49$	0.727598	0.103943
$50$	0	0
$51$	0	0
$52$	−1.16382	−0.161393
$53$	11.7853	1.61884	0.809419	−	0.587231i	$-0.199782\pi$
$53$	11.7853	1.61884	0.809419	+	0.587231i	$0.199782\pi$
$54$	0	0
$55$	0	0
$56$	−6.91707	−0.924332
$57$	0	0
$58$	0.712495	0.0935552
$59$	4.46028	0.580679	0.290339	−	0.956924i	$-0.406232\pi$
$59$	4.46028	0.580679	0.290339	+	0.956924i	$0.406232\pi$
$60$	0	0
$61$	−5.24905	−0.672071	−0.336036	−	0.941849i	$-0.609086\pi$
$61$	−5.24905	−0.672071	−0.336036	+	0.941849i	$0.609086\pi$
$62$	6.28464	0.798150
$63$	0	0
$64$	1.73733	0.217166
$65$	0	0
$66$	0	0
$67$	−8.49375	−1.03768	−0.518838	−	0.854872i	$-0.673635\pi$
$67$	−8.49375	−1.03768	−0.518838	+	0.854872i	$0.673635\pi$
$68$	−2.84075	−0.344492
$69$	0	0
$70$	0	0
$71$	−0.663102	−0.0786957	−0.0393479	−	0.999226i	$-0.512528\pi$
$71$	−0.663102	−0.0786957	−0.0393479	+	0.999226i	$0.512528\pi$
$72$	0	0
$73$	16.5345	1.93522	0.967609	−	0.252455i	$-0.0812380\pi$
$73$	16.5345	1.93522	0.967609	+	0.252455i	$0.0812380\pi$
$74$	−1.05736	−0.122915
$75$	0	0
$76$	−10.0291	−1.15042
$77$	11.8648	1.35212
$78$	0	0
$79$	9.54554	1.07396	0.536978	−	0.843596i	$-0.319566\pi$
$79$	9.54554	1.07396	0.536978	+	0.843596i	$0.319566\pi$
$80$	0	0
$81$	0	0
$82$	−5.49611	−0.606944
$83$	0.0123998	0.00136106	0.000680528	−	1.00000i	$-0.499783\pi$
$83$	0.0123998	0.00136106	0.000680528		1.00000i	$0.499783\pi$
$84$	0	0
$85$	0	0
$86$	5.83978	0.629720
$87$	0	0
$88$	−10.6203	−1.13213
$89$	5.46783	0.579588	0.289794	−	0.957089i	$-0.406413\pi$
$89$	5.46783	0.579588	0.289794	+	0.957089i	$0.406413\pi$
$90$	0	0
$91$	−2.16789	−0.227256
$92$	3.24091	0.337888
$93$	0	0
$94$	−3.70056	−0.381684
$95$	0	0
$96$	0	0
$97$	−0.952006	−0.0966615	−0.0483308	−	0.998831i	$-0.515390\pi$
$97$	−0.952006	−0.0966615	−0.0483308	+	0.998831i	$0.515390\pi$
$98$	−0.518410	−0.0523673
$99$	0	0
$100$	0	0
$101$	−8.31781	−0.827653	−0.413826	−	0.910356i	$-0.635808\pi$
$101$	−8.31781	−0.827653	−0.413826	+	0.910356i	$0.635808\pi$
$102$	0	0
$103$	12.4091	1.22271	0.611355	−	0.791357i	$-0.290625\pi$
$103$	12.4091	1.22271	0.611355	+	0.791357i	$0.290625\pi$
$104$	1.94050	0.190282
$105$	0	0
$106$	−8.39698	−0.815587
$107$	17.5579	1.69739	0.848695	−	0.528883i	$-0.177389\pi$
$107$	17.5579	1.69739	0.848695	+	0.528883i	$0.177389\pi$
$108$	0	0
$109$	1.00973	0.0967146	0.0483573	−	0.998830i	$-0.484601\pi$
$109$	1.00973	0.0967146	0.0483573	+	0.998830i	$0.484601\pi$
$110$	0	0
$111$	0	0
$112$	−3.36866	−0.318309
$113$	15.5787	1.46552	0.732761	−	0.680487i	$-0.238232\pi$
$113$	15.5787	1.46552	0.732761	+	0.680487i	$0.238232\pi$
$114$	0	0
$115$	0	0
$116$	1.49235	0.138561
$117$	0	0
$118$	−3.17792	−0.292552
$119$	−5.29157	−0.485078
$120$	0	0
$121$	7.21701	0.656091
$122$	3.73992	0.338596
$123$	0	0
$124$	13.1634	1.18211
$125$	0	0
$126$	0	0
$127$	−16.9536	−1.50438	−0.752192	−	0.658943i	$-0.771004\pi$
$127$	−16.9536	−1.50438	−0.752192	+	0.658943i	$0.771004\pi$
$128$	10.4421	0.922961
$129$	0	0
$130$	0	0
$131$	−11.9324	−1.04254	−0.521268	−	0.853393i	$-0.674541\pi$
$131$	−11.9324	−1.04254	−0.521268	+	0.853393i	$0.674541\pi$
$132$	0	0
$133$	−18.6816	−1.61990
$134$	6.05175	0.522792
$135$	0	0
$136$	4.73655	0.406156
$137$	0.239785	0.0204862	0.0102431	−	0.999948i	$-0.496739\pi$
$137$	0.239785	0.0204862	0.0102431	+	0.999948i	$0.496739\pi$
$138$	0	0
$139$	−17.0255	−1.44408	−0.722040	−	0.691851i	$-0.756795\pi$
$139$	−17.0255	−1.44408	−0.722040	+	0.691851i	$0.756795\pi$
$140$	0	0
$141$	0	0
$142$	0.472457	0.0396477
$143$	−3.32853	−0.278346
$144$	0	0
$145$	0	0
$146$	−11.7808	−0.974982
$147$	0	0
$148$	−2.21468	−0.182045
$149$	4.13046	0.338380	0.169190	−	0.985583i	$-0.445885\pi$
$149$	4.13046	0.338380	0.169190	+	0.985583i	$0.445885\pi$
$150$	0	0
$151$	−3.21580	−0.261698	−0.130849	−	0.991402i	$-0.541770\pi$
$151$	−3.21580	−0.261698	−0.130849	+	0.991402i	$0.541770\pi$
$152$	16.7221	1.35635
$153$	0	0
$154$	−8.45362	−0.681212
$155$	0	0
$156$	0	0
$157$	6.84237	0.546081	0.273040	−	0.962003i	$-0.411971\pi$
$157$	6.84237	0.546081	0.273040	+	0.962003i	$0.411971\pi$
$158$	−6.80115	−0.541070
$159$	0	0
$160$	0	0
$161$	6.03696	0.475779
$162$	0	0
$163$	−14.0502	−1.10050	−0.550248	−	0.835001i	$-0.685467\pi$
$163$	−14.0502	−1.10050	−0.550248	+	0.835001i	$0.685467\pi$
$164$	−11.5118	−0.898923
$165$	0	0
$166$	−0.00883480	−0.000685713 0
$167$	−17.8725	−1.38301	−0.691507	−	0.722370i	$-0.743053\pi$
$167$	−17.8725	−1.38301	−0.691507	+	0.722370i	$0.743053\pi$
$168$	0	0
$169$	−12.3918	−0.953217
$170$	0	0
$171$	0	0
$172$	12.2317	0.932656
$173$	2.82518	0.214795	0.107397	−	0.994216i	$-0.465748\pi$
$173$	2.82518	0.214795	0.107397	+	0.994216i	$0.465748\pi$
$174$	0	0
$175$	0	0
$176$	−5.17218	−0.389868
$177$	0	0
$178$	−3.89580	−0.292002
$179$	−20.2829	−1.51601	−0.758006	−	0.652247i	$-0.773826\pi$
$179$	−20.2829	−1.51601	−0.758006	+	0.652247i	$0.773826\pi$
$180$	0	0
$181$	−8.63783	−0.642045	−0.321023	−	0.947072i	$-0.604027\pi$
$181$	−8.63783	−0.642045	−0.321023	+	0.947072i	$0.604027\pi$
$182$	1.54461	0.114494
$183$	0	0
$184$	−5.40376	−0.398370
$185$	0	0
$186$	0	0
$187$	−8.12458	−0.594128
$188$	−7.75099	−0.565299
$189$	0	0
$190$	0	0
$191$	−0.896850	−0.0648938	−0.0324469	−	0.999473i	$-0.510330\pi$
$191$	−0.896850	−0.0648938	−0.0324469	+	0.999473i	$0.510330\pi$
$192$	0	0
$193$	−10.4506	−0.752251	−0.376126	−	0.926569i	$-0.622744\pi$
$193$	−10.4506	−0.752251	−0.376126	+	0.926569i	$0.622744\pi$
$194$	0.678299	0.0486991
$195$	0	0
$196$	−1.08583	−0.0775594
$197$	−9.57740	−0.682361	−0.341181	−	0.939998i	$-0.610827\pi$
$197$	−9.57740	−0.682361	−0.341181	+	0.939998i	$0.610827\pi$
$198$	0	0
$199$	5.61768	0.398227	0.199113	−	0.979976i	$-0.436194\pi$
$199$	5.61768	0.398227	0.199113	+	0.979976i	$0.436194\pi$
$200$	0	0
$201$	0	0
$202$	5.92640	0.416980
$203$	2.77986	0.195108
$204$	0	0
$205$	0	0
$206$	−8.84145	−0.616013
$207$	0	0
$208$	0.945039	0.0655267
$209$	−28.6834	−1.98407
$210$	0	0
$211$	−0.605216	−0.0416648	−0.0208324	−	0.999783i	$-0.506632\pi$
$211$	−0.605216	−0.0416648	−0.0208324	+	0.999783i	$0.506632\pi$
$212$	−17.5878	−1.20794
$213$	0	0
$214$	−12.5099	−0.855162
$215$	0	0
$216$	0	0
$217$	24.5200	1.66453
$218$	−0.719428	−0.0487258
$219$	0	0
$220$	0	0
$221$	1.48449	0.0998575
$222$	0	0
$223$	16.5537	1.10852	0.554260	−	0.832344i	$-0.313001\pi$
$223$	16.5537	1.10852	0.554260	+	0.832344i	$0.313001\pi$
$224$	16.2343	1.08470
$225$	0	0
$226$	−11.0997	−0.738344
$227$	22.0061	1.46060	0.730299	−	0.683128i	$-0.239381\pi$
$227$	22.0061	1.46060	0.730299	+	0.683128i	$0.239381\pi$
$228$	0	0
$229$	−24.5647	−1.62328	−0.811641	−	0.584157i	$-0.801425\pi$
$229$	−24.5647	−1.62328	−0.811641	+	0.584157i	$0.801425\pi$
$230$	0	0
$231$	0	0
$232$	−2.48828	−0.163364
$233$	5.56824	0.364787	0.182394	−	0.983226i	$-0.441615\pi$
$233$	5.56824	0.364787	0.182394	+	0.983226i	$0.441615\pi$
$234$	0	0
$235$	0	0
$236$	−6.65630	−0.433288
$237$	0	0
$238$	3.77022	0.244387
$239$	−6.90640	−0.446738	−0.223369	−	0.974734i	$-0.571705\pi$
$239$	−6.90640	−0.446738	−0.223369	+	0.974734i	$0.571705\pi$
$240$	0	0
$241$	−25.1989	−1.62320	−0.811602	−	0.584210i	$-0.801404\pi$
$241$	−25.1989	−1.62320	−0.811602	+	0.584210i	$0.801404\pi$
$242$	−5.14208	−0.330545
$243$	0	0
$244$	7.83342	0.501483
$245$	0	0
$246$	0	0
$247$	5.24091	0.333471
$248$	−21.9482	−1.39371
$249$	0	0
$250$	0	0
$251$	−28.3687	−1.79062	−0.895309	−	0.445446i	$-0.853045\pi$
$251$	−28.3687	−1.79062	−0.895309	+	0.445446i	$0.853045\pi$
$252$	0	0
$253$	9.26903	0.582739
$254$	12.0793	0.757924
$255$	0	0
$256$	−10.9146	−0.682163
$257$	−8.71100	−0.543377	−0.271688	−	0.962385i	$-0.587582\pi$
$257$	−8.71100	−0.543377	−0.271688	+	0.962385i	$0.587582\pi$
$258$	0	0
$259$	−4.12536	−0.256337
$260$	0	0
$261$	0	0
$262$	8.50175	0.525240
$263$	13.2128	0.814736	0.407368	−	0.913264i	$-0.366447\pi$
$263$	13.2128	0.814736	0.407368	+	0.913264i	$0.366447\pi$
$264$	0	0
$265$	0	0
$266$	13.3106	0.816123
$267$	0	0
$268$	12.6757	0.774289
$269$	10.3447	0.630725	0.315363	−	0.948971i	$-0.397874\pi$
$269$	10.3447	0.630725	0.315363	+	0.948971i	$0.397874\pi$
$270$	0	0
$271$	9.76022	0.592891	0.296445	−	0.955050i	$-0.404199\pi$
$271$	9.76022	0.592891	0.296445	+	0.955050i	$0.404199\pi$
$272$	2.30674	0.139866
$273$	0	0
$274$	−0.170845	−0.0103212
$275$	0	0
$276$	0	0
$277$	1.87503	0.112659	0.0563297	−	0.998412i	$-0.482060\pi$
$277$	1.87503	0.112659	0.0563297	+	0.998412i	$0.482060\pi$
$278$	12.1306	0.727542
$279$	0	0
$280$	0	0
$281$	21.2930	1.27024	0.635118	−	0.772415i	$-0.280951\pi$
$281$	21.2930	1.27024	0.635118	+	0.772415i	$0.280951\pi$
$282$	0	0
$283$	−15.8729	−0.943544	−0.471772	−	0.881721i	$-0.656385\pi$
$283$	−15.8729	−0.943544	−0.471772	+	0.881721i	$0.656385\pi$
$284$	0.989581	0.0587208
$285$	0	0
$286$	2.37156	0.140233
$287$	−21.4435	−1.26577
$288$	0	0
$289$	−13.3765	−0.786855
$290$	0	0
$291$	0	0
$292$	−24.6753	−1.44401
$293$	33.1369	1.93588	0.967938	−	0.251189i	$-0.0808216\pi$
$293$	33.1369	1.93588	0.967938	+	0.251189i	$0.0808216\pi$
$294$	0	0
$295$	0	0
$296$	3.69266	0.214631
$297$	0	0
$298$	−2.94293	−0.170479
$299$	−1.69360	−0.0979433
$300$	0	0
$301$	22.7844	1.31327
$302$	2.29124	0.131846
$303$	0	0
$304$	8.14381	0.467080
$305$	0	0
$306$	0	0
$307$	−17.2605	−0.985109	−0.492555	−	0.870282i	$-0.663937\pi$
$307$	−17.2605	−0.985109	−0.492555	+	0.870282i	$0.663937\pi$
$308$	−17.7065	−1.00892
$309$	0	0
$310$	0	0
$311$	32.5116	1.84356	0.921781	−	0.387711i	$-0.126734\pi$
$311$	32.5116	1.84356	0.921781	+	0.387711i	$0.126734\pi$
$312$	0	0
$313$	−6.26304	−0.354008	−0.177004	−	0.984210i	$-0.556641\pi$
$313$	−6.26304	−0.354008	−0.177004	+	0.984210i	$0.556641\pi$
$314$	−4.87516	−0.275121
$315$	0	0
$316$	−14.2453	−0.801360
$317$	−18.8523	−1.05885	−0.529426	−	0.848356i	$-0.677593\pi$
$317$	−18.8523	−1.05885	−0.529426	+	0.848356i	$0.677593\pi$
$318$	0	0
$319$	4.26814	0.238970
$320$	0	0
$321$	0	0
$322$	−4.30130	−0.239702
$323$	12.7925	0.711793
$324$	0	0
$325$	0	0
$326$	10.0107	0.554441
$327$	0	0
$328$	19.1943	1.05983
$329$	−14.4380	−0.795995
$330$	0	0
$331$	−0.596245	−0.0327726	−0.0163863	−	0.999866i	$-0.505216\pi$
$331$	−0.596245	−0.0327726	−0.0163863	+	0.999866i	$0.505216\pi$
$332$	−0.0185049	−0.00101559
$333$	0	0
$334$	12.7341	0.696776
$335$	0	0
$336$	0	0
$337$	−32.6414	−1.77809	−0.889046	−	0.457818i	$-0.848631\pi$
$337$	−32.6414	−1.77809	−0.889046	+	0.457818i	$0.848631\pi$
$338$	8.82911	0.480240
$339$	0	0
$340$	0	0
$341$	37.6476	2.03873
$342$	0	0
$343$	17.4364	0.941476
$344$	−20.3946	−1.09960
$345$	0	0
$346$	−2.01293	−0.108216
$347$	−13.1985	−0.708531	−0.354265	−	0.935145i	$-0.615269\pi$
$347$	−13.1985	−0.708531	−0.354265	+	0.935145i	$0.615269\pi$
$348$	0	0
$349$	15.6412	0.837255	0.418628	−	0.908158i	$-0.362511\pi$
$349$	15.6412	0.837255	0.418628	+	0.908158i	$0.362511\pi$
$350$	0	0
$351$	0	0
$352$	24.9258	1.32855
$353$	−5.65486	−0.300978	−0.150489	−	0.988612i	$-0.548085\pi$
$353$	−5.65486	−0.300978	−0.150489	+	0.988612i	$0.548085\pi$
$354$	0	0
$355$	0	0
$356$	−8.15991	−0.432475
$357$	0	0
$358$	14.4514	0.763782
$359$	−25.2693	−1.33366	−0.666832	−	0.745208i	$-0.732350\pi$
$359$	−25.2693	−1.33366	−0.666832	+	0.745208i	$0.732350\pi$
$360$	0	0
$361$	26.1632	1.37701
$362$	6.15441	0.323469
$363$	0	0
$364$	3.23525	0.169573
$365$	0	0
$366$	0	0
$367$	26.2788	1.37174	0.685872	−	0.727722i	$-0.259421\pi$
$367$	26.2788	1.37174	0.685872	+	0.727722i	$0.259421\pi$
$368$	−2.63167	−0.137185
$369$	0	0
$370$	0	0
$371$	−32.7615	−1.70089
$372$	0	0
$373$	1.43033	0.0740596	0.0370298	−	0.999314i	$-0.488210\pi$
$373$	1.43033	0.0740596	0.0370298	+	0.999314i	$0.488210\pi$
$374$	5.78872	0.299328
$375$	0	0
$376$	12.9237	0.666487
$377$	−0.779856	−0.0401646
$378$	0	0
$379$	−35.2336	−1.80983	−0.904915	−	0.425591i	$-0.860066\pi$
$379$	−35.2336	−1.80983	−0.904915	+	0.425591i	$0.860066\pi$
$380$	0	0
$381$	0	0
$382$	0.639001	0.0326941
$383$	−32.0701	−1.63870	−0.819352	−	0.573291i	$-0.805666\pi$
$383$	−32.0701	−1.63870	−0.819352	+	0.573291i	$0.805666\pi$
$384$	0	0
$385$	0	0
$386$	7.44601	0.378992
$387$	0	0
$388$	1.42073	0.0721265
$389$	−1.63922	−0.0831117	−0.0415558	−	0.999136i	$-0.513231\pi$
$389$	−1.63922	−0.0831117	−0.0415558	+	0.999136i	$0.513231\pi$
$390$	0	0
$391$	−4.13389	−0.209060
$392$	1.81047	0.0914425
$393$	0	0
$394$	6.82385	0.343781
$395$	0	0
$396$	0	0
$397$	−2.04098	−0.102434	−0.0512170	−	0.998688i	$-0.516310\pi$
$397$	−2.04098	−0.102434	−0.0512170	+	0.998688i	$0.516310\pi$
$398$	−4.00257	−0.200631
$399$	0	0
$400$	0	0
$401$	−17.4108	−0.869455	−0.434727	−	0.900562i	$-0.643155\pi$
$401$	−17.4108	−0.869455	−0.434727	+	0.900562i	$0.643155\pi$
$402$	0	0
$403$	−6.87880	−0.342658
$404$	12.4131	0.617574
$405$	0	0
$406$	−1.98063	−0.0982972
$407$	−6.33400	−0.313965
$408$	0	0
$409$	−37.3010	−1.84441	−0.922207	−	0.386696i	$-0.873616\pi$
$409$	−37.3010	−1.84441	−0.922207	+	0.386696i	$0.873616\pi$
$410$	0	0
$411$	0	0
$412$	−18.5188	−0.912356
$413$	−12.3989	−0.610111
$414$	0	0
$415$	0	0
$416$	−4.55434	−0.223295
$417$	0	0
$418$	20.4368	0.999596
$419$	−4.32941	−0.211506	−0.105753	−	0.994392i	$-0.533725\pi$
$419$	−4.32941	−0.211506	−0.105753	+	0.994392i	$0.533725\pi$
$420$	0	0
$421$	−34.2505	−1.66927	−0.834634	−	0.550806i	$-0.814321\pi$
$421$	−34.2505	−1.66927	−0.834634	+	0.550806i	$0.814321\pi$
$422$	0.431214	0.0209911
$423$	0	0
$424$	29.3252	1.42416
$425$	0	0
$426$	0	0
$427$	14.5916	0.706137
$428$	−26.2026	−1.26655
$429$	0	0
$430$	0	0
$431$	−34.4885	−1.66125	−0.830626	−	0.556831i	$-0.812017\pi$
$431$	−34.4885	−1.66125	−0.830626	+	0.556831i	$0.812017\pi$
$432$	0	0
$433$	−3.02258	−0.145256	−0.0726279	−	0.997359i	$-0.523139\pi$
$433$	−3.02258	−0.145256	−0.0726279	+	0.997359i	$0.523139\pi$
$434$	−17.4704	−0.838606
$435$	0	0
$436$	−1.50687	−0.0721661
$437$	−14.5945	−0.698148
$438$	0	0
$439$	8.37392	0.399665	0.199833	−	0.979830i	$-0.435960\pi$
$439$	8.37392	0.399665	0.199833	+	0.979830i	$0.435960\pi$
$440$	0	0
$441$	0	0
$442$	−1.05769	−0.0503092
$443$	−29.8770	−1.41950	−0.709749	−	0.704454i	$-0.751192\pi$
$443$	−29.8770	−1.41950	−0.709749	+	0.704454i	$0.751192\pi$
$444$	0	0
$445$	0	0
$446$	−11.7945	−0.558484
$447$	0	0
$448$	−4.82952	−0.228174
$449$	−12.0780	−0.569994	−0.284997	−	0.958528i	$-0.591993\pi$
$449$	−12.0780	−0.569994	−0.284997	+	0.958528i	$0.591993\pi$
$450$	0	0
$451$	−32.9240	−1.55033
$452$	−23.2489	−1.09354
$453$	0	0
$454$	−15.6793	−0.735864
$455$	0	0
$456$	0	0
$457$	−3.03357	−0.141905	−0.0709523	−	0.997480i	$-0.522604\pi$
$457$	−3.03357	−0.141905	−0.0709523	+	0.997480i	$0.522604\pi$
$458$	17.5022	0.817826
$459$	0	0
$460$	0	0
$461$	4.44508	0.207028	0.103514	−	0.994628i	$-0.466991\pi$
$461$	4.44508	0.207028	0.103514	+	0.994628i	$0.466991\pi$
$462$	0	0
$463$	9.04875	0.420531	0.210266	−	0.977644i	$-0.432567\pi$
$463$	9.04875	0.420531	0.210266	+	0.977644i	$0.432567\pi$
$464$	−1.21181	−0.0562570
$465$	0	0
$466$	−3.96734	−0.183784
$467$	8.11327	0.375437	0.187719	−	0.982223i	$-0.439891\pi$
$467$	8.11327	0.375437	0.187719	+	0.982223i	$0.439891\pi$
$468$	0	0
$469$	23.6114	1.09027
$470$	0	0
$471$	0	0
$472$	11.0984	0.510847
$473$	34.9827	1.60851
$474$	0	0
$475$	0	0
$476$	7.89688	0.361953
$477$	0	0
$478$	4.92078	0.225071
$479$	−6.62947	−0.302908	−0.151454	−	0.988464i	$-0.548396\pi$
$479$	−6.62947	−0.302908	−0.151454	+	0.988464i	$0.548396\pi$
$480$	0	0
$481$	1.15732	0.0527693
$482$	17.9541	0.817787
$483$	0	0
$484$	−10.7703	−0.489559
$485$	0	0
$486$	0	0
$487$	29.2047	1.32339	0.661695	−	0.749773i	$-0.269837\pi$
$487$	29.2047	1.32339	0.661695	+	0.749773i	$0.269837\pi$
$488$	−13.0611	−0.591249
$489$	0	0
$490$	0	0
$491$	4.32464	0.195168	0.0975841	−	0.995227i	$-0.468889\pi$
$491$	4.32464	0.195168	0.0975841	+	0.995227i	$0.468889\pi$
$492$	0	0
$493$	−1.90354	−0.0857312
$494$	−3.73412	−0.168006
$495$	0	0
$496$	−10.6889	−0.479947
$497$	1.84333	0.0826846
$498$	0	0
$499$	4.29688	0.192355	0.0961774	−	0.995364i	$-0.469338\pi$
$499$	4.29688	0.192355	0.0961774	+	0.995364i	$0.469338\pi$
$500$	0	0
$501$	0	0
$502$	20.2126	0.902131
$503$	−14.2512	−0.635429	−0.317715	−	0.948186i	$-0.602915\pi$
$503$	−14.2512	−0.635429	−0.317715	+	0.948186i	$0.602915\pi$
$504$	0	0
$505$	0	0
$506$	−6.60414	−0.293590
$507$	0	0
$508$	25.3007	1.12254
$509$	19.4974	0.864206	0.432103	−	0.901824i	$-0.357772\pi$
$509$	19.4974	0.864206	0.432103	+	0.901824i	$0.357772\pi$
$510$	0	0
$511$	−45.9635	−2.03331
$512$	−13.1076	−0.579280
$513$	0	0
$514$	6.20654	0.273759
$515$	0	0
$516$	0	0
$517$	−22.1679	−0.974943
$518$	2.93930	0.129145
$519$	0	0
$520$	0	0
$521$	−24.5229	−1.07437	−0.537185	−	0.843465i	$-0.680512\pi$
$521$	−24.5229	−1.07437	−0.537185	+	0.843465i	$0.680512\pi$
$522$	0	0
$523$	10.8566	0.474725	0.237362	−	0.971421i	$-0.423717\pi$
$523$	10.8566	0.474725	0.237362	+	0.971421i	$0.423717\pi$
$524$	17.8073	0.777914
$525$	0	0
$526$	−9.41406	−0.410472
$527$	−16.7904	−0.731401
$528$	0	0
$529$	−18.2838	−0.794948
$530$	0	0
$531$	0	0
$532$	27.8795	1.20873
$533$	6.01572	0.260570
$534$	0	0
$535$	0	0
$536$	−21.1348	−0.912886
$537$	0	0
$538$	−7.37052	−0.317766
$539$	−3.10549	−0.133763
$540$	0	0
$541$	−22.7032	−0.976085	−0.488043	−	0.872820i	$-0.662289\pi$
$541$	−22.7032	−0.976085	−0.488043	+	0.872820i	$0.662289\pi$
$542$	−6.95410	−0.298704
$543$	0	0
$544$	−11.1166	−0.476622
$545$	0	0
$546$	0	0
$547$	32.8229	1.40340	0.701702	−	0.712471i	$-0.252424\pi$
$547$	32.8229	1.40340	0.701702	+	0.712471i	$0.252424\pi$
$548$	−0.357843	−0.0152863
$549$	0	0
$550$	0	0
$551$	−6.72036	−0.286297
$552$	0	0
$553$	−26.5352	−1.12839
$554$	−1.33595	−0.0567590
$555$	0	0
$556$	25.4080	1.07754
$557$	26.3234	1.11536	0.557679	−	0.830057i	$-0.311692\pi$
$557$	26.3234	1.11536	0.557679	+	0.830057i	$0.311692\pi$
$558$	0	0
$559$	−6.39189	−0.270348
$560$	0	0
$561$	0	0
$562$	−15.1712	−0.639958
$563$	28.4750	1.20008	0.600039	−	0.799971i	$-0.295152\pi$
$563$	28.4750	1.20008	0.600039	+	0.799971i	$0.295152\pi$
$564$	0	0
$565$	0	0
$566$	11.3093	0.475367
$567$	0	0
$568$	−1.64999	−0.0692318
$569$	14.7555	0.618582	0.309291	−	0.950967i	$-0.399908\pi$
$569$	14.7555	0.618582	0.309291	+	0.950967i	$0.399908\pi$
$570$	0	0
$571$	−0.204397	−0.00855376	−0.00427688	−	0.999991i	$-0.501361\pi$
$571$	−0.204397	−0.00855376	−0.00427688	+	0.999991i	$0.501361\pi$
$572$	4.96734	0.207695
$573$	0	0
$574$	15.2784	0.637708
$575$	0	0
$576$	0	0
$577$	37.3300	1.55407	0.777035	−	0.629457i	$-0.216723\pi$
$577$	37.3300	1.55407	0.777035	+	0.629457i	$0.216723\pi$
$578$	9.53071	0.396425
$579$	0	0
$580$	0	0
$581$	−0.0344697	−0.00143004
$582$	0	0
$583$	−50.3014	−2.08327
$584$	41.1425	1.70249
$585$	0	0
$586$	−23.6098	−0.975314
$587$	−10.8497	−0.447813	−0.223907	−	0.974611i	$-0.571881\pi$
$587$	−10.8497	−0.447813	−0.223907	+	0.974611i	$0.571881\pi$
$588$	0	0
$589$	−59.2776	−2.44249
$590$	0	0
$591$	0	0
$592$	1.79835	0.0739119
$593$	14.0575	0.577272	0.288636	−	0.957439i	$-0.406798\pi$
$593$	14.0575	0.577272	0.288636	+	0.957439i	$0.406798\pi$
$594$	0	0
$595$	0	0
$596$	−6.16409	−0.252491
$597$	0	0
$598$	1.20668	0.0493448
$599$	5.52446	0.225724	0.112862	−	0.993611i	$-0.463998\pi$
$599$	5.52446	0.225724	0.112862	+	0.993611i	$0.463998\pi$
$600$	0	0
$601$	−6.85394	−0.279578	−0.139789	−	0.990181i	$-0.544642\pi$
$601$	−6.85394	−0.279578	−0.139789	+	0.990181i	$0.544642\pi$
$602$	−16.2337	−0.661638
$603$	0	0
$604$	4.79910	0.195273
$605$	0	0
$606$	0	0
$607$	−30.4979	−1.23787	−0.618935	−	0.785442i	$-0.712436\pi$
$607$	−30.4979	−1.23787	−0.618935	+	0.785442i	$0.712436\pi$
$608$	−39.2467	−1.59166
$609$	0	0
$610$	0	0
$611$	4.05042	0.163863
$612$	0	0
$613$	−29.9211	−1.20850	−0.604250	−	0.796794i	$-0.706527\pi$
$613$	−29.9211	−1.20850	−0.604250	+	0.796794i	$0.706527\pi$
$614$	12.2980	0.496308
$615$	0	0
$616$	29.5230	1.18952
$617$	−1.04305	−0.0419917	−0.0209959	−	0.999780i	$-0.506684\pi$
$617$	−1.04305	−0.0419917	−0.0209959	+	0.999780i	$0.506684\pi$
$618$	0	0
$619$	22.1661	0.890930	0.445465	−	0.895299i	$-0.353038\pi$
$619$	22.1661	0.890930	0.445465	+	0.895299i	$0.353038\pi$
$620$	0	0
$621$	0	0
$622$	−23.1643	−0.928805
$623$	−15.1998	−0.608966
$624$	0	0
$625$	0	0
$626$	4.46238	0.178353
$627$	0	0
$628$	−10.2112	−0.407472
$629$	2.82489	0.112636
$630$	0	0
$631$	20.1089	0.800524	0.400262	−	0.916401i	$-0.368919\pi$
$631$	20.1089	0.800524	0.400262	+	0.916401i	$0.368919\pi$
$632$	23.7520	0.944804
$633$	0	0
$634$	13.4322	0.533460
$635$	0	0
$636$	0	0
$637$	0.567422	0.0224821
$638$	−3.04103	−0.120395
$639$	0	0
$640$	0	0
$641$	−5.93422	−0.234388	−0.117194	−	0.993109i	$-0.537390\pi$
$641$	−5.93422	−0.234388	−0.117194	+	0.993109i	$0.537390\pi$
$642$	0	0
$643$	−0.951317	−0.0375163	−0.0187581	−	0.999824i	$-0.505971\pi$
$643$	−0.951317	−0.0375163	−0.0187581	+	0.999824i	$0.505971\pi$
$644$	−9.00926	−0.355015
$645$	0	0
$646$	−9.11458	−0.358608
$647$	−35.1384	−1.38143	−0.690716	−	0.723126i	$-0.742705\pi$
$647$	−35.1384	−1.38143	−0.690716	+	0.723126i	$0.742705\pi$
$648$	0	0
$649$	−19.0371	−0.747271
$650$	0	0
$651$	0	0
$652$	20.9678	0.821163
$653$	−45.9930	−1.79984	−0.899922	−	0.436050i	$-0.856377\pi$
$653$	−45.9930	−1.79984	−0.899922	+	0.436050i	$0.856377\pi$
$654$	0	0
$655$	0	0
$656$	9.34779	0.364970
$657$	0	0
$658$	10.2870	0.401030
$659$	19.3162	0.752451	0.376226	−	0.926528i	$-0.377222\pi$
$659$	19.3162	0.752451	0.376226	+	0.926528i	$0.377222\pi$
$660$	0	0
$661$	44.3224	1.72394	0.861970	−	0.506959i	$-0.169230\pi$
$661$	44.3224	1.72394	0.861970	+	0.506959i	$0.169230\pi$
$662$	0.424821	0.0165111
$663$	0	0
$664$	0.0308542	0.00119738
$665$	0	0
$666$	0	0
$667$	2.17168	0.0840878
$668$	26.6720	1.03197
$669$	0	0
$670$	0	0
$671$	22.4037	0.864883
$672$	0	0
$673$	−12.7421	−0.491170	−0.245585	−	0.969375i	$-0.578980\pi$
$673$	−12.7421	−0.491170	−0.245585	+	0.969375i	$0.578980\pi$
$674$	23.2569	0.895821
$675$	0	0
$676$	18.4930	0.711267
$677$	−9.41654	−0.361907	−0.180954	−	0.983492i	$-0.557918\pi$
$677$	−9.41654	−0.361907	−0.180954	+	0.983492i	$0.557918\pi$
$678$	0	0
$679$	2.64644	0.101561
$680$	0	0
$681$	0	0
$682$	−26.8237	−1.02713
$683$	6.92316	0.264907	0.132454	−	0.991189i	$-0.457714\pi$
$683$	6.92316	0.264907	0.132454	+	0.991189i	$0.457714\pi$
$684$	0	0
$685$	0	0
$686$	−12.4233	−0.474325
$687$	0	0
$688$	−9.93231	−0.378666
$689$	9.19085	0.350144
$690$	0	0
$691$	−48.7272	−1.85367	−0.926835	−	0.375469i	$-0.877482\pi$
$691$	−48.7272	−1.85367	−0.926835	+	0.375469i	$0.877482\pi$
$692$	−4.21616	−0.160274
$693$	0	0
$694$	9.40384	0.356965
$695$	0	0
$696$	0	0
$697$	14.6837	0.556186
$698$	−11.1443	−0.421818
$699$	0	0
$700$	0	0
$701$	−2.88580	−0.108995	−0.0544975	−	0.998514i	$-0.517356\pi$
$701$	−2.88580	−0.108995	−0.0544975	+	0.998514i	$0.517356\pi$
$702$	0	0
$703$	9.97314	0.376144
$704$	−7.41516	−0.279469
$705$	0	0
$706$	4.02906	0.151636
$707$	23.1223	0.869604
$708$	0	0
$709$	−1.73056	−0.0649924	−0.0324962	−	0.999472i	$-0.510346\pi$
$709$	−1.73056	−0.0649924	−0.0324962	+	0.999472i	$0.510346\pi$
$710$	0	0
$711$	0	0
$712$	13.6055	0.509887
$713$	19.1555	0.717381
$714$	0	0
$715$	0	0
$716$	30.2692	1.13121
$717$	0	0
$718$	18.0043	0.671913
$719$	12.9089	0.481420	0.240710	−	0.970597i	$-0.422620\pi$
$719$	12.9089	0.481420	0.240710	+	0.970597i	$0.422620\pi$
$720$	0	0
$721$	−34.4956	−1.28468
$722$	−18.6412	−0.693752
$723$	0	0
$724$	12.8907	0.479078
$725$	0	0
$726$	0	0
$727$	38.3405	1.42197	0.710985	−	0.703207i	$-0.248249\pi$
$727$	38.3405	1.42197	0.710985	+	0.703207i	$0.248249\pi$
$728$	−5.39431	−0.199927
$729$	0	0
$730$	0	0
$731$	−15.6019	−0.577057
$732$	0	0
$733$	34.8966	1.28894	0.644468	−	0.764631i	$-0.277079\pi$
$733$	34.8966	1.28894	0.644468	+	0.764631i	$0.277079\pi$
$734$	−18.7235	−0.691098
$735$	0	0
$736$	12.6826	0.467485
$737$	36.2525	1.33538
$738$	0	0
$739$	14.2969	0.525921	0.262960	−	0.964807i	$-0.415301\pi$
$739$	14.2969	0.525921	0.262960	+	0.964807i	$0.415301\pi$
$740$	0	0
$741$	0	0
$742$	23.3424	0.856927
$743$	−44.1357	−1.61918	−0.809591	−	0.586995i	$-0.800311\pi$
$743$	−44.1357	−1.61918	−0.809591	+	0.586995i	$0.800311\pi$
$744$	0	0
$745$	0	0
$746$	−1.01910	−0.0373120
$747$	0	0
$748$	12.1247	0.443324
$749$	−48.8085	−1.78343
$750$	0	0
$751$	9.56375	0.348986	0.174493	−	0.984658i	$-0.444171\pi$
$751$	9.56375	0.348986	0.174493	+	0.984658i	$0.444171\pi$
$752$	6.29393	0.229516
$753$	0	0
$754$	0.555643	0.0202353
$755$	0	0
$756$	0	0
$757$	−30.1910	−1.09731	−0.548656	−	0.836048i	$-0.684860\pi$
$757$	−30.1910	−1.09731	−0.548656	+	0.836048i	$0.684860\pi$
$758$	25.1038	0.911811
$759$	0	0
$760$	0	0
$761$	−8.27038	−0.299801	−0.149900	−	0.988701i	$-0.547895\pi$
$761$	−8.27038	−0.299801	−0.149900	+	0.988701i	$0.547895\pi$
$762$	0	0
$763$	−2.80690	−0.101617
$764$	1.33841	0.0484221
$765$	0	0
$766$	22.8498	0.825595
$767$	3.47837	0.125597
$768$	0	0
$769$	−22.4592	−0.809901	−0.404950	−	0.914339i	$-0.632711\pi$
$769$	−22.4592	−0.809901	−0.404950	+	0.914339i	$0.632711\pi$
$770$	0	0
$771$	0	0
$772$	15.5960	0.561311
$773$	−26.9520	−0.969398	−0.484699	−	0.874681i	$-0.661071\pi$
$773$	−26.9520	−0.969398	−0.484699	+	0.874681i	$0.661071\pi$
$774$	0	0
$775$	0	0
$776$	−2.36886	−0.0850371
$777$	0	0
$778$	1.16794	0.0418725
$779$	51.8401	1.85737
$780$	0	0
$781$	2.83021	0.101273
$782$	2.94537	0.105326
$783$	0	0
$784$	0.881713	0.0314897
$785$	0	0
$786$	0	0
$787$	−1.16423	−0.0415004	−0.0207502	−	0.999785i	$-0.506605\pi$
$787$	−1.16423	−0.0415004	−0.0207502	+	0.999785i	$0.506605\pi$
$788$	14.2928	0.509161
$789$	0	0
$790$	0	0
$791$	−43.3065	−1.53980
$792$	0	0
$793$	−4.09350	−0.145364
$794$	1.45419	0.0516073
$795$	0	0
$796$	−8.38354	−0.297147
$797$	15.9624	0.565418	0.282709	−	0.959206i	$-0.408767\pi$
$797$	15.9624	0.565418	0.282709	+	0.959206i	$0.408767\pi$
$798$	0	0
$799$	9.88664	0.349764
$800$	0	0
$801$	0	0
$802$	12.4051	0.438040
$803$	−70.5715	−2.49042
$804$	0	0
$805$	0	0
$806$	4.90111	0.172634
$807$	0	0
$808$	−20.6971	−0.728120
$809$	7.83605	0.275501	0.137750	−	0.990467i	$-0.456013\pi$
$809$	7.83605	0.275501	0.137750	+	0.990467i	$0.456013\pi$
$810$	0	0
$811$	−23.9935	−0.842526	−0.421263	−	0.906939i	$-0.638413\pi$
$811$	−23.9935	−0.842526	−0.421263	+	0.906939i	$0.638413\pi$
$812$	−4.14852	−0.145585
$813$	0	0
$814$	4.51294	0.158179
$815$	0	0
$816$	0	0
$817$	−55.0817	−1.92706
$818$	26.5768	0.929235
$819$	0	0
$820$	0	0
$821$	12.8804	0.449528	0.224764	−	0.974413i	$-0.427839\pi$
$821$	12.8804	0.449528	0.224764	+	0.974413i	$0.427839\pi$
$822$	0	0
$823$	−47.8165	−1.66678	−0.833389	−	0.552687i	$-0.813603\pi$
$823$	−47.8165	−1.66678	−0.833389	+	0.552687i	$0.813603\pi$
$824$	30.8775	1.07567
$825$	0	0
$826$	8.83417	0.307380
$827$	−14.6873	−0.510726	−0.255363	−	0.966845i	$-0.582195\pi$
$827$	−14.6873	−0.510726	−0.255363	+	0.966845i	$0.582195\pi$
$828$	0	0
$829$	41.9876	1.45829	0.729145	−	0.684359i	$-0.239918\pi$
$829$	41.9876	1.45829	0.729145	+	0.684359i	$0.239918\pi$
$830$	0	0
$831$	0	0
$832$	1.35487	0.0469715
$833$	1.38501	0.0479879
$834$	0	0
$835$	0	0
$836$	42.8057	1.48047
$837$	0	0
$838$	3.08468	0.106559
$839$	4.04408	0.139617	0.0698085	−	0.997560i	$-0.477761\pi$
$839$	4.04408	0.139617	0.0698085	+	0.997560i	$0.477761\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	24.4033	0.840994
$843$	0	0
$844$	0.903195	0.0310892
$845$	0	0
$846$	0	0
$847$	−20.0622	−0.689347
$848$	14.2816	0.490432
$849$	0	0
$850$	0	0
$851$	−3.22282	−0.110477
$852$	0	0
$853$	43.8299	1.50071	0.750353	−	0.661037i	$-0.229883\pi$
$853$	43.8299	1.50071	0.750353	+	0.661037i	$0.229883\pi$
$854$	−10.3964	−0.355759
$855$	0	0
$856$	43.6891	1.49326
$857$	2.09275	0.0714868	0.0357434	−	0.999361i	$-0.488620\pi$
$857$	2.09275	0.0714868	0.0357434	+	0.999361i	$0.488620\pi$
$858$	0	0
$859$	46.5885	1.58958	0.794790	−	0.606885i	$-0.207581\pi$
$859$	46.5885	1.58958	0.794790	+	0.606885i	$0.207581\pi$
$860$	0	0
$861$	0	0
$862$	24.5729	0.836955
$863$	32.1466	1.09428	0.547141	−	0.837040i	$-0.315716\pi$
$863$	32.1466	1.09428	0.547141	+	0.837040i	$0.315716\pi$
$864$	0	0
$865$	0	0
$866$	2.15357	0.0731814
$867$	0	0
$868$	−36.5925	−1.24203
$869$	−40.7417	−1.38207
$870$	0	0
$871$	−6.62390	−0.224442
$872$	2.51249	0.0850838
$873$	0	0
$874$	10.3985	0.351734
$875$	0	0
$876$	0	0
$877$	−48.0544	−1.62268	−0.811342	−	0.584572i	$-0.801262\pi$
$877$	−48.0544	−1.62268	−0.811342	+	0.584572i	$0.801262\pi$
$878$	−5.96637	−0.201355
$879$	0	0
$880$	0	0
$881$	30.9846	1.04390	0.521950	−	0.852976i	$-0.325205\pi$
$881$	30.9846	1.04390	0.521950	+	0.852976i	$0.325205\pi$
$882$	0	0
$883$	−9.60378	−0.323193	−0.161597	−	0.986857i	$-0.551664\pi$
$883$	−9.60378	−0.323193	−0.161597	+	0.986857i	$0.551664\pi$
$884$	−2.21538	−0.0745112
$885$	0	0
$886$	21.2872	0.715158
$887$	−29.7279	−0.998165	−0.499083	−	0.866554i	$-0.666330\pi$
$887$	−29.7279	−0.998165	−0.499083	+	0.866554i	$0.666330\pi$
$888$	0	0
$889$	47.1284	1.58064
$890$	0	0
$891$	0	0
$892$	−24.7040	−0.827150
$893$	34.9043	1.16803
$894$	0	0
$895$	0	0
$896$	−29.0276	−0.969743
$897$	0	0
$898$	8.60548	0.287169
$899$	8.82061	0.294184
$900$	0	0
$901$	22.4339	0.747380
$902$	23.4582	0.781071
$903$	0	0
$904$	38.7642	1.28928
$905$	0	0
$906$	0	0
$907$	−4.35876	−0.144730	−0.0723651	−	0.997378i	$-0.523055\pi$
$907$	−4.35876	−0.144730	−0.0723651	+	0.997378i	$0.523055\pi$
$908$	−32.8409	−1.08986
$909$	0	0
$910$	0	0
$911$	0.389317	0.0128986	0.00644932	−	0.999979i	$-0.497947\pi$
$911$	0.389317	0.0128986	0.00644932	+	0.999979i	$0.497947\pi$
$912$	0	0
$913$	−0.0529241	−0.00175153
$914$	2.16141	0.0714930
$915$	0	0
$916$	36.6592	1.21125
$917$	33.1703	1.09538
$918$	0	0
$919$	−54.9915	−1.81400	−0.907001	−	0.421128i	$-0.861634\pi$
$919$	−54.9915	−1.81400	−0.907001	+	0.421128i	$0.861634\pi$
$920$	0	0
$921$	0	0
$922$	−3.16710	−0.104303
$923$	−0.517124	−0.0170213
$924$	0	0
$925$	0	0
$926$	−6.44719	−0.211868
$927$	0	0
$928$	5.83998	0.191707
$929$	−17.4617	−0.572901	−0.286450	−	0.958095i	$-0.592475\pi$
$929$	−17.4617	−0.572901	−0.286450	+	0.958095i	$0.592475\pi$
$930$	0	0
$931$	4.88972	0.160254
$932$	−8.30977	−0.272195
$933$	0	0
$934$	−5.78066	−0.189149
$935$	0	0
$936$	0	0
$937$	−27.4261	−0.895970	−0.447985	−	0.894041i	$-0.647858\pi$
$937$	−27.4261	−0.895970	−0.447985	+	0.894041i	$0.647858\pi$
$938$	−16.8230	−0.549291
$939$	0	0
$940$	0	0
$941$	35.1587	1.14614	0.573070	−	0.819506i	$-0.305752\pi$
$941$	35.1587	1.14614	0.573070	+	0.819506i	$0.305752\pi$
$942$	0	0
$943$	−16.7521	−0.545524
$944$	5.40502	0.175918
$945$	0	0
$946$	−24.9250	−0.810381
$947$	−44.1438	−1.43448	−0.717240	−	0.696827i	$-0.754595\pi$
$947$	−44.1438	−1.43448	−0.717240	+	0.696827i	$0.754595\pi$
$948$	0	0
$949$	12.8945	0.418574
$950$	0	0
$951$	0	0
$952$	−13.1669	−0.426743
$953$	−27.3169	−0.884881	−0.442441	−	0.896798i	$-0.645887\pi$
$953$	−27.3169	−0.884881	−0.442441	+	0.896798i	$0.645887\pi$
$954$	0	0
$955$	0	0
$956$	10.3068	0.333345
$957$	0	0
$958$	4.72346	0.152608
$959$	−0.666567	−0.0215246
$960$	0	0
$961$	46.8031	1.50978
$962$	−0.824585	−0.0265857
$963$	0	0
$964$	37.6056	1.21120
$965$	0	0
$966$	0	0
$967$	−54.3910	−1.74910	−0.874548	−	0.484939i	$-0.838842\pi$
$967$	−54.3910	−1.74910	−0.874548	+	0.484939i	$0.838842\pi$
$968$	17.9579	0.577190
$969$	0	0
$970$	0	0
$971$	14.2906	0.458606	0.229303	−	0.973355i	$-0.426355\pi$
$971$	14.2906	0.458606	0.229303	+	0.973355i	$0.426355\pi$
$972$	0	0
$973$	47.3283	1.51728
$974$	−20.8082	−0.666737
$975$	0	0
$976$	−6.36086	−0.203606
$977$	−8.94905	−0.286306	−0.143153	−	0.989701i	$-0.545724\pi$
$977$	−8.94905	−0.286306	−0.143153	+	0.989701i	$0.545724\pi$
$978$	0	0
$979$	−23.3374	−0.745868
$980$	0	0
$981$	0	0
$982$	−3.08128	−0.0983277
$983$	−22.1981	−0.708010	−0.354005	−	0.935244i	$-0.615180\pi$
$983$	−22.1981	−0.708010	−0.354005	+	0.935244i	$0.615180\pi$
$984$	0	0
$985$	0	0
$986$	1.35626	0.0431923
$987$	0	0
$988$	−7.82128	−0.248828
$989$	17.7996	0.565995
$990$	0	0
$991$	−13.2764	−0.421740	−0.210870	−	0.977514i	$-0.567630\pi$
$991$	−13.2764	−0.421740	−0.210870	+	0.977514i	$0.567630\pi$
$992$	51.5121	1.63551
$993$	0	0
$994$	−1.31336	−0.0416573
$995$	0	0
$996$	0	0
$997$	19.6929	0.623682	0.311841	−	0.950134i	$-0.399054\pi$
$997$	19.6929	0.623682	0.311841	+	0.950134i	$0.399054\pi$
$998$	−3.06150	−0.0969102
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6525.2.a.bs.1.2		5
3.2	odd	2		2175.2.a.w.1.4		5
5.2	odd	4		1305.2.c.j.784.5		10
5.3	odd	4		1305.2.c.j.784.6		10
5.4	even	2		6525.2.a.bl.1.4		5
15.2	even	4		435.2.c.e.349.6	yes	10
15.8	even	4		435.2.c.e.349.5	✓	10
15.14	odd	2		2175.2.a.z.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.c.e.349.5	✓	10	15.8	even	4
435.2.c.e.349.6	yes	10	15.2	even	4
1305.2.c.j.784.5		10	5.2	odd	4
1305.2.c.j.784.6		10	5.3	odd	4
2175.2.a.w.1.4		5	3.2	odd	2
2175.2.a.z.1.2		5	15.14	odd	2
6525.2.a.bl.1.4		5	5.4	even	2
6525.2.a.bs.1.2		5	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:	\( N \)	\(=\)	\( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6525.a (trivial)

\(f(q)\)	\(=\)	\(q-0.712495 q^{2} -1.49235 q^{4} -2.77986 q^{7} +2.48828 q^{8} +O(q^{10})\)	\(q-0.712495 q^{2} -1.49235 q^{4} -2.77986 q^{7} +2.48828 q^{8} -4.26814 q^{11} +0.779856 q^{13} +1.98063 q^{14} +1.21181 q^{16} +1.90354 q^{17} +6.72036 q^{19} +3.04103 q^{22} -2.17168 q^{23} -0.555643 q^{26} +4.14852 q^{28} -1.00000 q^{29} -8.82061 q^{31} -5.83998 q^{32} -1.35626 q^{34} +1.48402 q^{37} -4.78822 q^{38} +7.71389 q^{41} -8.19624 q^{43} +6.36956 q^{44} +1.54731 q^{46} +5.19381 q^{47} +0.727598 q^{49} -1.16382 q^{52} +11.7853 q^{53} -6.91707 q^{56} +0.712495 q^{58} +4.46028 q^{59} -5.24905 q^{61} +6.28464 q^{62} +1.73733 q^{64} -8.49375 q^{67} -2.84075 q^{68} -0.663102 q^{71} +16.5345 q^{73} -1.05736 q^{74} -10.0291 q^{76} +11.8648 q^{77} +9.54554 q^{79} -5.49611 q^{82} +0.0123998 q^{83} +5.83978 q^{86} -10.6203 q^{88} +5.46783 q^{89} -2.16789 q^{91} +3.24091 q^{92} -3.70056 q^{94} -0.952006 q^{97} -0.518410 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{2} + 5 q^{4} - 8 q^{7} + 9 q^{8}+O(q^{10}) \) 5 * q + 3 * q^2 + 5 * q^4 - 8 * q^7 + 9 * q^8	\( 5 q + 3 q^{2} + 5 q^{4} - 8 q^{7} + 9 q^{8} - 12 q^{11} - 2 q^{13} - 6 q^{14} + q^{16} - 2 q^{19} - 14 q^{22} + 8 q^{23} + 6 q^{28} - 5 q^{29} + 2 q^{31} + q^{32} + 4 q^{34} - 16 q^{37} - 14 q^{38} + 14 q^{41} - 20 q^{44} - 6 q^{46} + 2 q^{47} - 7 q^{49} - 16 q^{52} + 26 q^{53} + 2 q^{56} - 3 q^{58} - 4 q^{59} - 12 q^{61} - 9 q^{64} - 12 q^{67} - 20 q^{68} - 30 q^{71} + 12 q^{73} - 2 q^{74} - 44 q^{76} + 18 q^{77} - 18 q^{79} - 10 q^{82} + 2 q^{83} - 30 q^{86} - 42 q^{88} + 22 q^{89} - 12 q^{91} + 20 q^{92} - 50 q^{94} - 20 q^{97} - 9 q^{98}+O(q^{100}) \) 5 * q + 3 * q^2 + 5 * q^4 - 8 * q^7 + 9 * q^8 - 12 * q^11 - 2 * q^13 - 6 * q^14 + q^16 - 2 * q^19 - 14 * q^22 + 8 * q^23 + 6 * q^28 - 5 * q^29 + 2 * q^31 + q^32 + 4 * q^34 - 16 * q^37 - 14 * q^38 + 14 * q^41 - 20 * q^44 - 6 * q^46 + 2 * q^47 - 7 * q^49 - 16 * q^52 + 26 * q^53 + 2 * q^56 - 3 * q^58 - 4 * q^59 - 12 * q^61 - 9 * q^64 - 12 * q^67 - 20 * q^68 - 30 * q^71 + 12 * q^73 - 2 * q^74 - 44 * q^76 + 18 * q^77 - 18 * q^79 - 10 * q^82 + 2 * q^83 - 30 * q^86 - 42 * q^88 + 22 * q^89 - 12 * q^91 + 20 * q^92 - 50 * q^94 - 20 * q^97 - 9 * q^98

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