LMFDB - Embedded newform 6025.2.a.k.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6025,2,Mod(1,6025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6025, 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("6025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6025 = 5^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6025.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:	$48.1098672178$
Analytic rank:	$0$
Dimension:	$25$
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	6025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.32584 q^{2} +2.02466 q^{3} -0.242137 q^{4} -2.68439 q^{6} -3.04433 q^{7} +2.97272 q^{8} +1.09926 q^{9} +O(q^{10})$	\(q-1.32584 q^{2} +2.02466 q^{3} -0.242137 q^{4} -2.68439 q^{6} -3.04433 q^{7} +2.97272 q^{8} +1.09926 q^{9} +1.79607 q^{11} -0.490247 q^{12} +0.100045 q^{13} +4.03630 q^{14} -3.45709 q^{16} +0.353668 q^{17} -1.45745 q^{18} -2.02746 q^{19} -6.16373 q^{21} -2.38130 q^{22} +4.43970 q^{23} +6.01877 q^{24} -0.132644 q^{26} -3.84836 q^{27} +0.737145 q^{28} +8.69145 q^{29} +6.94070 q^{31} -1.36188 q^{32} +3.63643 q^{33} -0.468909 q^{34} -0.266172 q^{36} -3.30485 q^{37} +2.68809 q^{38} +0.202557 q^{39} +2.34644 q^{41} +8.17215 q^{42} -2.63050 q^{43} -0.434895 q^{44} -5.88635 q^{46} -13.2780 q^{47} -6.99945 q^{48} +2.26792 q^{49} +0.716059 q^{51} -0.0242246 q^{52} -8.92645 q^{53} +5.10232 q^{54} -9.04994 q^{56} -4.10492 q^{57} -11.5235 q^{58} -0.729581 q^{59} +14.0485 q^{61} -9.20229 q^{62} -3.34651 q^{63} +8.71983 q^{64} -4.82134 q^{66} +8.35024 q^{67} -0.0856363 q^{68} +8.98890 q^{69} -7.35552 q^{71} +3.26780 q^{72} -6.30530 q^{73} +4.38171 q^{74} +0.490923 q^{76} -5.46781 q^{77} -0.268559 q^{78} +1.38656 q^{79} -11.0894 q^{81} -3.11101 q^{82} +10.8019 q^{83} +1.49247 q^{84} +3.48763 q^{86} +17.5973 q^{87} +5.33921 q^{88} -3.55647 q^{89} -0.304569 q^{91} -1.07502 q^{92} +14.0526 q^{93} +17.6045 q^{94} -2.75735 q^{96} +13.9809 q^{97} -3.00690 q^{98} +1.97435 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 4 q^{2} - 9 q^{3} + 36 q^{4} + 7 q^{6} - 7 q^{7} + 15 q^{8} + 36 q^{9}+O(q^{10}) $ 25 * q + 4 * q^2 - 9 * q^3 + 36 * q^4 + 7 * q^6 - 7 * q^7 + 15 * q^8 + 36 * q^9	$ 25 q + 4 q^{2} - 9 q^{3} + 36 q^{4} + 7 q^{6} - 7 q^{7} + 15 q^{8} + 36 q^{9} + 10 q^{11} - 22 q^{12} - 10 q^{13} + 13 q^{14} + 54 q^{16} - q^{17} + 13 q^{18} + 50 q^{19} + 9 q^{21} - 11 q^{22} + 31 q^{23} + 22 q^{24} + 8 q^{26} - 42 q^{27} - 14 q^{28} + 4 q^{29} + 34 q^{31} + 44 q^{32} - 28 q^{33} + 33 q^{34} + 83 q^{36} - 14 q^{37} + 10 q^{38} + 23 q^{39} + 11 q^{41} - 23 q^{42} - 49 q^{43} + 20 q^{44} + 27 q^{46} + 28 q^{47} - 30 q^{48} + 66 q^{49} + 49 q^{51} - 39 q^{52} + 16 q^{53} + 5 q^{54} + 51 q^{56} - 10 q^{57} + 8 q^{58} + 30 q^{59} + 35 q^{61} + 18 q^{62} + 73 q^{64} - 13 q^{66} - 37 q^{67} - 11 q^{68} - 4 q^{69} + 12 q^{71} + 90 q^{72} - 36 q^{73} - 12 q^{74} + 57 q^{76} + 31 q^{77} + 9 q^{78} + 16 q^{79} + 65 q^{81} + 11 q^{82} - 43 q^{83} - 62 q^{84} - 9 q^{86} + 22 q^{87} - 20 q^{88} + 38 q^{89} + 86 q^{91} + 119 q^{92} - 10 q^{93} - 18 q^{94} - 34 q^{96} - 17 q^{97} + 32 q^{98} + 26 q^{99}+O(q^{100}) $ 25 * q + 4 * q^2 - 9 * q^3 + 36 * q^4 + 7 * q^6 - 7 * q^7 + 15 * q^8 + 36 * q^9 + 10 * q^11 - 22 * q^12 - 10 * q^13 + 13 * q^14 + 54 * q^16 - q^17 + 13 * q^18 + 50 * q^19 + 9 * q^21 - 11 * q^22 + 31 * q^23 + 22 * q^24 + 8 * q^26 - 42 * q^27 - 14 * q^28 + 4 * q^29 + 34 * q^31 + 44 * q^32 - 28 * q^33 + 33 * q^34 + 83 * q^36 - 14 * q^37 + 10 * q^38 + 23 * q^39 + 11 * q^41 - 23 * q^42 - 49 * q^43 + 20 * q^44 + 27 * q^46 + 28 * q^47 - 30 * q^48 + 66 * q^49 + 49 * q^51 - 39 * q^52 + 16 * q^53 + 5 * q^54 + 51 * q^56 - 10 * q^57 + 8 * q^58 + 30 * q^59 + 35 * q^61 + 18 * q^62 + 73 * q^64 - 13 * q^66 - 37 * q^67 - 11 * q^68 - 4 * q^69 + 12 * q^71 + 90 * q^72 - 36 * q^73 - 12 * q^74 + 57 * q^76 + 31 * q^77 + 9 * q^78 + 16 * q^79 + 65 * q^81 + 11 * q^82 - 43 * q^83 - 62 * q^84 - 9 * q^86 + 22 * q^87 - 20 * q^88 + 38 * q^89 + 86 * q^91 + 119 * q^92 - 10 * q^93 - 18 * q^94 - 34 * q^96 - 17 * q^97 + 32 * q^98 + 26 * q^99

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 $

$2$	−1.32584	−0.937513	−0.468757	−	0.883327i	$-0.655298\pi$
$2$	−1.32584	−0.937513	−0.468757	+	0.883327i	$0.655298\pi$
$3$	2.02466	1.16894	0.584470	−	0.811415i	$-0.301302\pi$
$3$	2.02466	1.16894	0.584470	+	0.811415i	$0.301302\pi$
$4$	−0.242137	−0.121069
$5$	0	0
$5$	0	0
$6$	−2.68439	−1.09590
$7$	−3.04433	−1.15065	−0.575323	−	0.817926i	$-0.695124\pi$
$7$	−3.04433	−1.15065	−0.575323	+	0.817926i	$0.695124\pi$
$8$	2.97272	1.05102
$9$	1.09926	0.366420
$10$	0	0
$11$	1.79607	0.541534	0.270767	−	0.962645i	$-0.412723\pi$
$11$	1.79607	0.541534	0.270767	+	0.962645i	$0.412723\pi$
$12$	−0.490247	−0.141522
$13$	0.100045	0.0277474	0.0138737	−	0.999904i	$-0.495584\pi$
$13$	0.100045	0.0277474	0.0138737	+	0.999904i	$0.495584\pi$
$14$	4.03630	1.07875
$15$	0	0
$16$	−3.45709	−0.864274
$17$	0.353668	0.0857771	0.0428886	−	0.999080i	$-0.486344\pi$
$17$	0.353668	0.0857771	0.0428886	+	0.999080i	$0.486344\pi$
$18$	−1.45745	−0.343524
$19$	−2.02746	−0.465130	−0.232565	−	0.972581i	$-0.574712\pi$
$19$	−2.02746	−0.465130	−0.232565	+	0.972581i	$0.574712\pi$
$20$	0	0
$21$	−6.16373	−1.34504
$22$	−2.38130	−0.507696
$23$	4.43970	0.925741	0.462871	−	0.886426i	$-0.346819\pi$
$23$	4.43970	0.925741	0.462871	+	0.886426i	$0.346819\pi$
$24$	6.01877	1.22858
$25$	0	0
$26$	−0.132644	−0.0260136
$27$	−3.84836	−0.740617
$28$	0.737145	0.139307
$29$	8.69145	1.61396	0.806981	−	0.590577i	$-0.201100\pi$
$29$	8.69145	1.61396	0.806981	+	0.590577i	$0.201100\pi$
$30$	0	0
$31$	6.94070	1.24659	0.623293	−	0.781988i	$-0.285794\pi$
$31$	6.94070	1.24659	0.623293	+	0.781988i	$0.285794\pi$
$32$	−1.36188	−0.240749
$33$	3.63643	0.633021
$34$	−0.468909	−0.0804172
$35$	0	0
$36$	−0.266172	−0.0443620
$37$	−3.30485	−0.543313	−0.271657	−	0.962394i	$-0.587572\pi$
$37$	−3.30485	−0.543313	−0.271657	+	0.962394i	$0.587572\pi$
$38$	2.68809	0.436066
$39$	0.202557	0.0324351
$40$	0	0
$41$	2.34644	0.366452	0.183226	−	0.983071i	$-0.441346\pi$
$41$	2.34644	0.366452	0.183226	+	0.983071i	$0.441346\pi$
$42$	8.17215	1.26099
$43$	−2.63050	−0.401147	−0.200574	−	0.979679i	$-0.564281\pi$
$43$	−2.63050	−0.401147	−0.200574	+	0.979679i	$0.564281\pi$
$44$	−0.434895	−0.0655628
$45$	0	0
$46$	−5.88635	−0.867895
$47$	−13.2780	−1.93679	−0.968395	−	0.249421i	$-0.919760\pi$
$47$	−13.2780	−1.93679	−0.968395	+	0.249421i	$0.919760\pi$
$48$	−6.99945	−1.01028
$49$	2.26792	0.323988
$50$	0	0
$51$	0.716059	0.100268
$52$	−0.0242246	−0.00335934
$53$	−8.92645	−1.22614	−0.613071	−	0.790028i	$-0.710066\pi$
$53$	−8.92645	−1.22614	−0.613071	+	0.790028i	$0.710066\pi$
$54$	5.10232	0.694338
$55$	0	0
$56$	−9.04994	−1.20935
$57$	−4.10492	−0.543710
$58$	−11.5235	−1.51311
$59$	−0.729581	−0.0949834	−0.0474917	−	0.998872i	$-0.515123\pi$
$59$	−0.729581	−0.0949834	−0.0474917	+	0.998872i	$0.515123\pi$
$60$	0	0
$61$	14.0485	1.79873	0.899363	−	0.437202i	$-0.144031\pi$
$61$	14.0485	1.79873	0.899363	+	0.437202i	$0.144031\pi$
$62$	−9.20229	−1.16869
$63$	−3.34651	−0.421620
$64$	8.71983	1.08998
$65$	0	0
$66$	−4.82134	−0.593466
$67$	8.35024	1.02014	0.510072	−	0.860132i	$-0.329619\pi$
$67$	8.35024	1.02014	0.510072	+	0.860132i	$0.329619\pi$
$68$	−0.0856363	−0.0103849
$69$	8.98890	1.08214
$70$	0	0
$71$	−7.35552	−0.872940	−0.436470	−	0.899719i	$-0.643772\pi$
$71$	−7.35552	−0.872940	−0.436470	+	0.899719i	$0.643772\pi$
$72$	3.26780	0.385114
$73$	−6.30530	−0.737979	−0.368990	−	0.929433i	$-0.620296\pi$
$73$	−6.30530	−0.737979	−0.368990	+	0.929433i	$0.620296\pi$
$74$	4.38171	0.509363
$75$	0	0
$76$	0.490923	0.0563127
$77$	−5.46781	−0.623115
$78$	−0.268559	−0.0304083
$79$	1.38656	0.156000	0.0780000	−	0.996953i	$-0.475147\pi$
$79$	1.38656	0.156000	0.0780000	+	0.996953i	$0.475147\pi$
$80$	0	0
$81$	−11.0894	−1.23216
$82$	−3.11101	−0.343554
$83$	10.8019	1.18566	0.592830	−	0.805328i	$-0.298011\pi$
$83$	10.8019	1.18566	0.592830	+	0.805328i	$0.298011\pi$
$84$	1.49247	0.162842
$85$	0	0
$86$	3.48763	0.376081
$87$	17.5973	1.88663
$88$	5.33921	0.569162
$89$	−3.55647	−0.376985	−0.188492	−	0.982075i	$-0.560360\pi$
$89$	−3.55647	−0.376985	−0.188492	+	0.982075i	$0.560360\pi$
$90$	0	0
$91$	−0.304569	−0.0319275
$92$	−1.07502	−0.112078
$93$	14.0526	1.45718
$94$	17.6045	1.81577
$95$	0	0
$96$	−2.75735	−0.281421
$97$	13.9809	1.41955	0.709774	−	0.704429i	$-0.248797\pi$
$97$	13.9809	1.41955	0.709774	+	0.704429i	$0.248797\pi$
$98$	−3.00690	−0.303743
$99$	1.97435	0.198429
$100$	0	0
$101$	−12.1770	−1.21166	−0.605830	−	0.795594i	$-0.707159\pi$
$101$	−12.1770	−1.21166	−0.605830	+	0.795594i	$0.707159\pi$
$102$	−0.949382	−0.0940029
$103$	4.12347	0.406297	0.203149	−	0.979148i	$-0.434883\pi$
$103$	4.12347	0.406297	0.203149	+	0.979148i	$0.434883\pi$
$104$	0.297405	0.0291630
$105$	0	0
$106$	11.8351	1.14952
$107$	0.755504	0.0730374	0.0365187	−	0.999333i	$-0.488373\pi$
$107$	0.755504	0.0730374	0.0365187	+	0.999333i	$0.488373\pi$
$108$	0.931831	0.0896655
$109$	14.2543	1.36532	0.682659	−	0.730737i	$-0.260823\pi$
$109$	14.2543	1.36532	0.682659	+	0.730737i	$0.260823\pi$
$110$	0	0
$111$	−6.69120	−0.635101
$112$	10.5245	0.994474
$113$	−0.609697	−0.0573555	−0.0286777	−	0.999589i	$-0.509130\pi$
$113$	−0.609697	−0.0573555	−0.0286777	+	0.999589i	$0.509130\pi$
$114$	5.44248	0.509735
$115$	0	0
$116$	−2.10453	−0.195400
$117$	0.109975	0.0101672
$118$	0.967311	0.0890482
$119$	−1.07668	−0.0986992
$120$	0	0
$121$	−7.77415	−0.706740
$122$	−18.6261	−1.68633
$123$	4.75075	0.428361
$124$	−1.68060	−0.150923
$125$	0	0
$126$	4.43695	0.395275
$127$	−17.9310	−1.59112	−0.795558	−	0.605877i	$-0.792822\pi$
$127$	−17.9310	−1.59112	−0.795558	+	0.605877i	$0.792822\pi$
$128$	−8.83738	−0.781121
$129$	−5.32587	−0.468917
$130$	0	0
$131$	17.5465	1.53304	0.766521	−	0.642219i	$-0.221986\pi$
$131$	17.5465	1.53304	0.766521	+	0.642219i	$0.221986\pi$
$132$	−0.880515	−0.0766390
$133$	6.17224	0.535201
$134$	−11.0711	−0.956399
$135$	0	0
$136$	1.05136	0.0901532
$137$	20.6648	1.76551	0.882757	−	0.469829i	$-0.155685\pi$
$137$	20.6648	1.76551	0.882757	+	0.469829i	$0.155685\pi$
$138$	−11.9179	−1.01452
$139$	19.1212	1.62184	0.810921	−	0.585155i	$-0.198966\pi$
$139$	19.1212	1.62184	0.810921	+	0.585155i	$0.198966\pi$
$140$	0	0
$141$	−26.8834	−2.26399
$142$	9.75228	0.818393
$143$	0.179687	0.0150262
$144$	−3.80025	−0.316687
$145$	0	0
$146$	8.35984	0.691866
$147$	4.59176	0.378722
$148$	0.800227	0.0657782
$149$	−9.67814	−0.792864	−0.396432	−	0.918064i	$-0.629752\pi$
$149$	−9.67814	−0.792864	−0.396432	+	0.918064i	$0.629752\pi$
$150$	0	0
$151$	10.7919	0.878235	0.439118	−	0.898430i	$-0.355291\pi$
$151$	10.7919	0.878235	0.439118	+	0.898430i	$0.355291\pi$
$152$	−6.02707	−0.488860
$153$	0.388774	0.0314305
$154$	7.24946	0.584178
$155$	0	0
$156$	−0.0490466	−0.00392687
$157$	−5.75595	−0.459375	−0.229687	−	0.973264i	$-0.573770\pi$
$157$	−5.75595	−0.459375	−0.229687	+	0.973264i	$0.573770\pi$
$158$	−1.83836	−0.146252
$159$	−18.0731	−1.43329
$160$	0	0
$161$	−13.5159	−1.06520
$162$	14.7028	1.15516
$163$	3.54323	0.277527	0.138764	−	0.990326i	$-0.455687\pi$
$163$	3.54323	0.277527	0.138764	+	0.990326i	$0.455687\pi$
$164$	−0.568161	−0.0443659
$165$	0	0
$166$	−14.3216	−1.11157
$167$	−5.31456	−0.411253	−0.205626	−	0.978631i	$-0.565923\pi$
$167$	−5.31456	−0.411253	−0.205626	+	0.978631i	$0.565923\pi$
$168$	−18.3231	−1.41366
$169$	−12.9900	−0.999230
$170$	0	0
$171$	−2.22870	−0.170433
$172$	0.636942	0.0485664
$173$	4.17054	0.317080	0.158540	−	0.987353i	$-0.449321\pi$
$173$	4.17054	0.317080	0.158540	+	0.987353i	$0.449321\pi$
$174$	−23.3312	−1.76874
$175$	0	0
$176$	−6.20917	−0.468034
$177$	−1.47716	−0.111030
$178$	4.71532	0.353428
$179$	3.52748	0.263656	0.131828	−	0.991273i	$-0.457915\pi$
$179$	3.52748	0.263656	0.131828	+	0.991273i	$0.457915\pi$
$180$	0	0
$181$	16.4705	1.22425	0.612123	−	0.790763i	$-0.290316\pi$
$181$	16.4705	1.22425	0.612123	+	0.790763i	$0.290316\pi$
$182$	0.403811	0.0299324
$183$	28.4435	2.10260
$184$	13.1980	0.972970
$185$	0	0
$186$	−18.6315	−1.36613
$187$	0.635211	0.0464513
$188$	3.21509	0.234485
$189$	11.7156	0.852188
$190$	0	0
$191$	8.82609	0.638633	0.319317	−	0.947648i	$-0.396547\pi$
$191$	8.82609	0.638633	0.319317	+	0.947648i	$0.396547\pi$
$192$	17.6547	1.27412
$193$	−2.05717	−0.148078	−0.0740391	−	0.997255i	$-0.523589\pi$
$193$	−2.05717	−0.148078	−0.0740391	+	0.997255i	$0.523589\pi$
$194$	−18.5365	−1.33085
$195$	0	0
$196$	−0.549147	−0.0392248
$197$	15.8660	1.13040	0.565202	−	0.824953i	$-0.308798\pi$
$197$	15.8660	1.13040	0.565202	+	0.824953i	$0.308798\pi$
$198$	−2.61767	−0.186030
$199$	5.91176	0.419074	0.209537	−	0.977801i	$-0.432804\pi$
$199$	5.91176	0.419074	0.209537	+	0.977801i	$0.432804\pi$
$200$	0	0
$201$	16.9064	1.19249
$202$	16.1448	1.13595
$203$	−26.4596	−1.85710
$204$	−0.173385	−0.0121393
$205$	0	0
$206$	−5.46708	−0.380909
$207$	4.88039	0.339210
$208$	−0.345864	−0.0239814
$209$	−3.64145	−0.251884
$210$	0	0
$211$	13.3940	0.922082	0.461041	−	0.887379i	$-0.347476\pi$
$211$	13.3940	0.922082	0.461041	+	0.887379i	$0.347476\pi$
$212$	2.16143	0.148447
$213$	−14.8925	−1.02041
$214$	−1.00168	−0.0684735
$215$	0	0
$216$	−11.4401	−0.778400
$217$	−21.1297	−1.43438
$218$	−18.8990	−1.28000
$219$	−12.7661	−0.862654
$220$	0	0
$221$	0.0353826	0.00238009
$222$	8.87149	0.595415
$223$	12.4802	0.835733	0.417866	−	0.908508i	$-0.362778\pi$
$223$	12.4802	0.835733	0.417866	+	0.908508i	$0.362778\pi$
$224$	4.14601	0.277017
$225$	0	0
$226$	0.808364	0.0537715
$227$	14.2706	0.947170	0.473585	−	0.880748i	$-0.342960\pi$
$227$	14.2706	0.947170	0.473585	+	0.880748i	$0.342960\pi$
$228$	0.993954	0.0658262
$229$	13.8257	0.913629	0.456815	−	0.889562i	$-0.348990\pi$
$229$	13.8257	0.913629	0.456815	+	0.889562i	$0.348990\pi$
$230$	0	0
$231$	−11.0705	−0.728384
$232$	25.8373	1.69630
$233$	0.638926	0.0418575	0.0209287	−	0.999781i	$-0.493338\pi$
$233$	0.638926	0.0418575	0.0209287	+	0.999781i	$0.493338\pi$
$234$	−0.145810	−0.00953190
$235$	0	0
$236$	0.176659	0.0114995
$237$	2.80731	0.182355
$238$	1.42751	0.0925318
$239$	5.56073	0.359694	0.179847	−	0.983695i	$-0.442440\pi$
$239$	5.56073	0.359694	0.179847	+	0.983695i	$0.442440\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	10.3073	0.662579
$243$	−10.9072	−0.699700
$244$	−3.40167	−0.217769
$245$	0	0
$246$	−6.29876	−0.401594
$247$	−0.202836	−0.0129062
$248$	20.6328	1.31018
$249$	21.8702	1.38597
$250$	0	0
$251$	16.9313	1.06869	0.534347	−	0.845265i	$-0.320558\pi$
$251$	16.9313	1.06869	0.534347	+	0.845265i	$0.320558\pi$
$252$	0.810314	0.0510450
$253$	7.97400	0.501321
$254$	23.7737	1.49169
$255$	0	0
$256$	−5.72268	−0.357667
$257$	11.9189	0.743479	0.371740	−	0.928337i	$-0.378761\pi$
$257$	11.9189	0.743479	0.371740	+	0.928337i	$0.378761\pi$
$258$	7.06128	0.439616
$259$	10.0610	0.625162
$260$	0	0
$261$	9.55418	0.591389
$262$	−23.2639	−1.43725
$263$	13.4726	0.830753	0.415377	−	0.909650i	$-0.363650\pi$
$263$	13.4726	0.830753	0.415377	+	0.909650i	$0.363650\pi$
$264$	10.8101	0.665316
$265$	0	0
$266$	−8.18342	−0.501758
$267$	−7.20065	−0.440673
$268$	−2.02190	−0.123507
$269$	13.9885	0.852897	0.426448	−	0.904512i	$-0.359765\pi$
$269$	13.9885	0.852897	0.426448	+	0.904512i	$0.359765\pi$
$270$	0	0
$271$	19.9386	1.21118	0.605591	−	0.795776i	$-0.292937\pi$
$271$	19.9386	1.21118	0.605591	+	0.795776i	$0.292937\pi$
$272$	−1.22266	−0.0741349
$273$	−0.616649	−0.0373213
$274$	−27.3983	−1.65519
$275$	0	0
$276$	−2.17655	−0.131013
$277$	31.1614	1.87231	0.936154	−	0.351590i	$-0.114359\pi$
$277$	31.1614	1.87231	0.936154	+	0.351590i	$0.114359\pi$
$278$	−25.3518	−1.52050
$279$	7.62964	0.456775
$280$	0	0
$281$	−8.57349	−0.511452	−0.255726	−	0.966749i	$-0.582314\pi$
$281$	−8.57349	−0.511452	−0.255726	+	0.966749i	$0.582314\pi$
$282$	35.6432	2.12252
$283$	−30.9341	−1.83884	−0.919420	−	0.393278i	$-0.871341\pi$
$283$	−30.9341	−1.83884	−0.919420	+	0.393278i	$0.871341\pi$
$284$	1.78105	0.105686
$285$	0	0
$286$	−0.238237	−0.0140872
$287$	−7.14333	−0.421657
$288$	−1.49706	−0.0882152
$289$	−16.8749	−0.992642
$290$	0	0
$291$	28.3067	1.65937
$292$	1.52675	0.0893462
$293$	11.0505	0.645579	0.322789	−	0.946471i	$-0.395379\pi$
$293$	11.0505	0.645579	0.322789	+	0.946471i	$0.395379\pi$
$294$	−6.08796	−0.355057
$295$	0	0
$296$	−9.82440	−0.571031
$297$	−6.91190	−0.401069
$298$	12.8317	0.743321
$299$	0.444169	0.0256869
$300$	0	0
$301$	8.00809	0.461579
$302$	−14.3084	−0.823357
$303$	−24.6544	−1.41636
$304$	7.00911	0.402000
$305$	0	0
$306$	−0.515453	−0.0294665
$307$	11.9129	0.679907	0.339954	−	0.940442i	$-0.389589\pi$
$307$	11.9129	0.679907	0.339954	+	0.940442i	$0.389589\pi$
$308$	1.32396	0.0754397
$309$	8.34864	0.474937
$310$	0	0
$311$	21.5262	1.22064	0.610320	−	0.792155i	$-0.291041\pi$
$311$	21.5262	1.22064	0.610320	+	0.792155i	$0.291041\pi$
$312$	0.602146	0.0340898
$313$	4.47097	0.252714	0.126357	−	0.991985i	$-0.459672\pi$
$313$	4.47097	0.252714	0.126357	+	0.991985i	$0.459672\pi$
$314$	7.63150	0.430670
$315$	0	0
$316$	−0.335737	−0.0188867
$317$	3.55620	0.199736	0.0998679	−	0.995001i	$-0.468158\pi$
$317$	3.55620	0.199736	0.0998679	+	0.995001i	$0.468158\pi$
$318$	23.9621	1.34373
$319$	15.6104	0.874016
$320$	0	0
$321$	1.52964	0.0853763
$322$	17.9200	0.998640
$323$	−0.717047	−0.0398975
$324$	2.68516	0.149176
$325$	0	0
$326$	−4.69777	−0.260186
$327$	28.8602	1.59597
$328$	6.97532	0.385148
$329$	40.4224	2.22856
$330$	0	0
$331$	−22.0961	−1.21451	−0.607255	−	0.794507i	$-0.707729\pi$
$331$	−22.0961	−1.21451	−0.607255	+	0.794507i	$0.707729\pi$
$332$	−2.61554	−0.143546
$333$	−3.63289	−0.199081
$334$	7.04627	0.385555
$335$	0	0
$336$	21.3086	1.16248
$337$	−13.2631	−0.722486	−0.361243	−	0.932472i	$-0.617648\pi$
$337$	−13.2631	−0.722486	−0.361243	+	0.932472i	$0.617648\pi$
$338$	17.2227	0.936792
$339$	−1.23443	−0.0670451
$340$	0	0
$341$	12.4660	0.675070
$342$	2.95491	0.159783
$343$	14.4060	0.777851
$344$	−7.81975	−0.421612
$345$	0	0
$346$	−5.52948	−0.297267
$347$	−31.6800	−1.70067	−0.850336	−	0.526240i	$-0.823601\pi$
$347$	−31.6800	−1.70067	−0.850336	+	0.526240i	$0.823601\pi$
$348$	−4.26096	−0.228411
$349$	17.1323	0.917070	0.458535	−	0.888676i	$-0.348374\pi$
$349$	17.1323	0.917070	0.458535	+	0.888676i	$0.348374\pi$
$350$	0	0
$351$	−0.385008	−0.0205502
$352$	−2.44603	−0.130374
$353$	−12.1058	−0.644327	−0.322163	−	0.946684i	$-0.604410\pi$
$353$	−12.1058	−0.644327	−0.322163	+	0.946684i	$0.604410\pi$
$354$	1.95848	0.104092
$355$	0	0
$356$	0.861154	0.0456411
$357$	−2.17992	−0.115373
$358$	−4.67689	−0.247181
$359$	−32.1356	−1.69605	−0.848025	−	0.529956i	$-0.822209\pi$
$359$	−32.1356	−1.69605	−0.848025	+	0.529956i	$0.822209\pi$
$360$	0	0
$361$	−14.8894	−0.783654
$362$	−21.8374	−1.14775
$363$	−15.7400	−0.826137
$364$	0.0737475	0.00386542
$365$	0	0
$366$	−37.7116	−1.97122
$367$	−7.48127	−0.390519	−0.195259	−	0.980752i	$-0.562555\pi$
$367$	−7.48127	−0.390519	−0.195259	+	0.980752i	$0.562555\pi$
$368$	−15.3485	−0.800094
$369$	2.57935	0.134276
$370$	0	0
$371$	27.1750	1.41086
$372$	−3.40265	−0.176419
$373$	−23.2022	−1.20137	−0.600683	−	0.799487i	$-0.705105\pi$
$373$	−23.2022	−1.20137	−0.600683	+	0.799487i	$0.705105\pi$
$374$	−0.842191	−0.0435487
$375$	0	0
$376$	−39.4717	−2.03560
$377$	0.869534	0.0447833
$378$	−15.5331	−0.798938
$379$	−13.0230	−0.668948	−0.334474	−	0.942405i	$-0.608559\pi$
$379$	−13.0230	−0.668948	−0.334474	+	0.942405i	$0.608559\pi$
$380$	0	0
$381$	−36.3042	−1.85992
$382$	−11.7020	−0.598727
$383$	8.52674	0.435696	0.217848	−	0.975983i	$-0.430096\pi$
$383$	8.52674	0.435696	0.217848	+	0.975983i	$0.430096\pi$
$384$	−17.8927	−0.913083
$385$	0	0
$386$	2.72749	0.138825
$387$	−2.89160	−0.146988
$388$	−3.38531	−0.171863
$389$	−9.47661	−0.480483	−0.240242	−	0.970713i	$-0.577227\pi$
$389$	−9.47661	−0.480483	−0.240242	+	0.970713i	$0.577227\pi$
$390$	0	0
$391$	1.57018	0.0794074
$392$	6.74189	0.340517
$393$	35.5257	1.79203
$394$	−21.0358	−1.05977
$395$	0	0
$396$	−0.478063	−0.0240236
$397$	21.7239	1.09029	0.545145	−	0.838341i	$-0.316474\pi$
$397$	21.7239	1.09029	0.545145	+	0.838341i	$0.316474\pi$
$398$	−7.83807	−0.392887
$399$	12.4967	0.625618
$400$	0	0
$401$	−18.0902	−0.903381	−0.451691	−	0.892175i	$-0.649179\pi$
$401$	−18.0902	−0.903381	−0.451691	+	0.892175i	$0.649179\pi$
$402$	−22.4153	−1.11797
$403$	0.694381	0.0345896
$404$	2.94851	0.146694
$405$	0	0
$406$	35.0813	1.74106
$407$	−5.93572	−0.294223
$408$	2.12865	0.105384
$409$	−2.85438	−0.141140	−0.0705701	−	0.997507i	$-0.522482\pi$
$409$	−2.85438	−0.141140	−0.0705701	+	0.997507i	$0.522482\pi$
$410$	0	0
$411$	41.8393	2.06378
$412$	−0.998446	−0.0491899
$413$	2.22108	0.109292
$414$	−6.47063	−0.318014
$415$	0	0
$416$	−0.136249	−0.00668016
$417$	38.7141	1.89584
$418$	4.82799	0.236145
$419$	30.9992	1.51441	0.757205	−	0.653178i	$-0.226565\pi$
$419$	30.9992	1.51441	0.757205	+	0.653178i	$0.226565\pi$
$420$	0	0
$421$	18.9371	0.922939	0.461470	−	0.887156i	$-0.347322\pi$
$421$	18.9371	0.922939	0.461470	+	0.887156i	$0.347322\pi$
$422$	−17.7584	−0.864465
$423$	−14.5959	−0.709679
$424$	−26.5359	−1.28870
$425$	0	0
$426$	19.7451	0.956652
$427$	−42.7682	−2.06970
$428$	−0.182936	−0.00884254
$429$	0.363806	0.0175647
$430$	0	0
$431$	14.8655	0.716047	0.358023	−	0.933713i	$-0.383451\pi$
$431$	14.8655	0.716047	0.358023	+	0.933713i	$0.383451\pi$
$432$	13.3041	0.640095
$433$	−14.5262	−0.698083	−0.349041	−	0.937107i	$-0.613493\pi$
$433$	−14.5262	−0.698083	−0.349041	+	0.937107i	$0.613493\pi$
$434$	28.0148	1.34475
$435$	0	0
$436$	−3.45151	−0.165297
$437$	−9.00130	−0.430590
$438$	16.9259	0.808749
$439$	−12.0153	−0.573461	−0.286731	−	0.958011i	$-0.592568\pi$
$439$	−12.0153	−0.573461	−0.286731	+	0.958011i	$0.592568\pi$
$440$	0	0
$441$	2.49303	0.118716
$442$	−0.0469119	−0.00223137
$443$	12.2129	0.580252	0.290126	−	0.956988i	$-0.406303\pi$
$443$	12.2129	0.580252	0.290126	+	0.956988i	$0.406303\pi$
$444$	1.62019	0.0768908
$445$	0	0
$446$	−16.5467	−0.783511
$447$	−19.5950	−0.926811
$448$	−26.5460	−1.25418
$449$	23.0221	1.08648	0.543241	−	0.839577i	$-0.317197\pi$
$449$	23.0221	1.08648	0.543241	+	0.839577i	$0.317197\pi$
$450$	0	0
$451$	4.21436	0.198447
$452$	0.147630	0.00694395
$453$	21.8500	1.02660
$454$	−18.9205	−0.887984
$455$	0	0
$456$	−12.2028	−0.571448
$457$	4.97076	0.232523	0.116261	−	0.993219i	$-0.462909\pi$
$457$	4.97076	0.232523	0.116261	+	0.993219i	$0.462909\pi$
$458$	−18.3307	−0.856539
$459$	−1.36104	−0.0635279
$460$	0	0
$461$	22.2971	1.03848	0.519241	−	0.854628i	$-0.326215\pi$
$461$	22.2971	1.03848	0.519241	+	0.854628i	$0.326215\pi$
$462$	14.6777	0.682869
$463$	−7.31113	−0.339777	−0.169888	−	0.985463i	$-0.554341\pi$
$463$	−7.31113	−0.339777	−0.169888	+	0.985463i	$0.554341\pi$
$464$	−30.0472	−1.39491
$465$	0	0
$466$	−0.847117	−0.0392419
$467$	37.5955	1.73971	0.869855	−	0.493307i	$-0.164212\pi$
$467$	37.5955	1.73971	0.869855	+	0.493307i	$0.164212\pi$
$468$	−0.0266291	−0.00123093
$469$	−25.4208	−1.17383
$470$	0	0
$471$	−11.6539	−0.536982
$472$	−2.16884	−0.0998291
$473$	−4.72455	−0.217235
$474$	−3.72206	−0.170960
$475$	0	0
$476$	0.260705	0.0119494
$477$	−9.81250	−0.449283
$478$	−7.37266	−0.337218
$479$	19.1132	0.873306	0.436653	−	0.899630i	$-0.356164\pi$
$479$	19.1132	0.873306	0.436653	+	0.899630i	$0.356164\pi$
$480$	0	0
$481$	−0.330632	−0.0150755
$482$	−1.32584	−0.0603905
$483$	−27.3651	−1.24516
$484$	1.88241	0.0855641
$485$	0	0
$486$	14.4613	0.655978
$487$	21.8759	0.991292	0.495646	−	0.868525i	$-0.334931\pi$
$487$	21.8759	0.991292	0.495646	+	0.868525i	$0.334931\pi$
$488$	41.7623	1.89049
$489$	7.17385	0.324413
$490$	0	0
$491$	−29.2959	−1.32211	−0.661054	−	0.750339i	$-0.729890\pi$
$491$	−29.2959	−1.32211	−0.661054	+	0.750339i	$0.729890\pi$
$492$	−1.15033	−0.0518611
$493$	3.07389	0.138441
$494$	0.268929	0.0120997
$495$	0	0
$496$	−23.9947	−1.07739
$497$	22.3926	1.00445
$498$	−28.9964	−1.29936
$499$	4.44924	0.199175	0.0995876	−	0.995029i	$-0.468248\pi$
$499$	4.44924	0.199175	0.0995876	+	0.995029i	$0.468248\pi$
$500$	0	0
$501$	−10.7602	−0.480730
$502$	−22.4482	−1.00191
$503$	−28.9684	−1.29164	−0.645820	−	0.763490i	$-0.723484\pi$
$503$	−28.9684	−1.29164	−0.645820	+	0.763490i	$0.723484\pi$
$504$	−9.94825	−0.443130
$505$	0	0
$506$	−10.5723	−0.469995
$507$	−26.3004	−1.16804
$508$	4.34176	0.192634
$509$	−19.0593	−0.844791	−0.422395	−	0.906412i	$-0.638811\pi$
$509$	−19.0593	−0.844791	−0.422395	+	0.906412i	$0.638811\pi$
$510$	0	0
$511$	19.1954	0.849154
$512$	25.2621	1.11644
$513$	7.80238	0.344483
$514$	−15.8026	−0.697022
$515$	0	0
$516$	1.28959	0.0567712
$517$	−23.8481	−1.04884
$518$	−13.3394	−0.586097
$519$	8.44393	0.370647
$520$	0	0
$521$	−10.2343	−0.448373	−0.224187	−	0.974546i	$-0.571973\pi$
$521$	−10.2343	−0.448373	−0.224187	+	0.974546i	$0.571973\pi$
$522$	−12.6673	−0.554435
$523$	−11.2295	−0.491033	−0.245517	−	0.969392i	$-0.578958\pi$
$523$	−11.2295	−0.491033	−0.245517	+	0.969392i	$0.578958\pi$
$524$	−4.24866	−0.185603
$525$	0	0
$526$	−17.8625	−0.778842
$527$	2.45470	0.106929
$528$	−12.5715	−0.547104
$529$	−3.28907	−0.143003
$530$	0	0
$531$	−0.802000	−0.0348038
$532$	−1.49453	−0.0647961
$533$	0.234749	0.0101681
$534$	9.54694	0.413137
$535$	0	0
$536$	24.8230	1.07219
$537$	7.14195	0.308198
$538$	−18.5466	−0.799602
$539$	4.07333	0.175451
$540$	0	0
$541$	22.3237	0.959769	0.479885	−	0.877332i	$-0.340679\pi$
$541$	22.3237	0.959769	0.479885	+	0.877332i	$0.340679\pi$
$542$	−26.4354	−1.13550
$543$	33.3473	1.43107
$544$	−0.481654	−0.0206507
$545$	0	0
$546$	0.817581	0.0349892
$547$	−19.2895	−0.824758	−0.412379	−	0.911012i	$-0.635302\pi$
$547$	−19.2895	−0.824758	−0.412379	+	0.911012i	$0.635302\pi$
$548$	−5.00372	−0.213749
$549$	15.4430	0.659090
$550$	0	0
$551$	−17.6215	−0.750703
$552$	26.7215	1.13734
$553$	−4.22113	−0.179501
$554$	−41.3152	−1.75531
$555$	0	0
$556$	−4.62997	−0.196354
$557$	−18.3152	−0.776038	−0.388019	−	0.921651i	$-0.626841\pi$
$557$	−18.3152	−0.776038	−0.388019	+	0.921651i	$0.626841\pi$
$558$	−10.1157	−0.428232
$559$	−0.263168	−0.0111308
$560$	0	0
$561$	1.28609	0.0542987
$562$	11.3671	0.479493
$563$	7.82106	0.329618	0.164809	−	0.986325i	$-0.447299\pi$
$563$	7.82106	0.329618	0.164809	+	0.986325i	$0.447299\pi$
$564$	6.50948	0.274098
$565$	0	0
$566$	41.0138	1.72394
$567$	33.7598	1.41778
$568$	−21.8659	−0.917475
$569$	−28.1832	−1.18150	−0.590751	−	0.806854i	$-0.701169\pi$
$569$	−28.1832	−1.18150	−0.590751	+	0.806854i	$0.701169\pi$
$570$	0	0
$571$	25.2412	1.05631	0.528155	−	0.849148i	$-0.322884\pi$
$571$	25.2412	1.05631	0.528155	+	0.849148i	$0.322884\pi$
$572$	−0.0435089	−0.00181920
$573$	17.8699	0.746524
$574$	9.47094	0.395309
$575$	0	0
$576$	9.58537	0.399390
$577$	19.3497	0.805537	0.402768	−	0.915302i	$-0.368048\pi$
$577$	19.3497	0.805537	0.402768	+	0.915302i	$0.368048\pi$
$578$	22.3735	0.930615
$579$	−4.16507	−0.173095
$580$	0	0
$581$	−32.8844	−1.36428
$582$	−37.5303	−1.55568
$583$	−16.0325	−0.663998
$584$	−18.7439	−0.775629
$585$	0	0
$586$	−14.6513	−0.605239
$587$	10.0485	0.414747	0.207373	−	0.978262i	$-0.433508\pi$
$587$	10.0485	0.414747	0.207373	+	0.978262i	$0.433508\pi$
$588$	−1.11184	−0.0458514
$589$	−14.0720	−0.579825
$590$	0	0
$591$	32.1233	1.32137
$592$	11.4252	0.469571
$593$	38.4349	1.57833	0.789166	−	0.614179i	$-0.210513\pi$
$593$	38.4349	1.57833	0.789166	+	0.614179i	$0.210513\pi$
$594$	9.16411	0.376008
$595$	0	0
$596$	2.34344	0.0959910
$597$	11.9693	0.489872
$598$	−0.588898	−0.0240818
$599$	−0.0858525	−0.00350784	−0.00175392	−	0.999998i	$-0.500558\pi$
$599$	−0.0858525	−0.00350784	−0.00175392	+	0.999998i	$0.500558\pi$
$600$	0	0
$601$	10.5160	0.428955	0.214477	−	0.976729i	$-0.431195\pi$
$601$	10.5160	0.428955	0.214477	+	0.976729i	$0.431195\pi$
$602$	−10.6175	−0.432736
$603$	9.17909	0.373801
$604$	−2.61313	−0.106327
$605$	0	0
$606$	32.6879	1.32785
$607$	11.5035	0.466914	0.233457	−	0.972367i	$-0.424996\pi$
$607$	11.5035	0.466914	0.233457	+	0.972367i	$0.424996\pi$
$608$	2.76115	0.111980
$609$	−53.5718	−2.17084
$610$	0	0
$611$	−1.32839	−0.0537409
$612$	−0.0941366	−0.00380525
$613$	13.1760	0.532175	0.266087	−	0.963949i	$-0.414269\pi$
$613$	13.1760	0.532175	0.266087	+	0.963949i	$0.414269\pi$
$614$	−15.7947	−0.637422
$615$	0	0
$616$	−16.2543	−0.654904
$617$	−11.8470	−0.476941	−0.238471	−	0.971150i	$-0.576646\pi$
$617$	−11.8470	−0.476941	−0.238471	+	0.971150i	$0.576646\pi$
$618$	−11.0690	−0.445260
$619$	−12.7910	−0.514114	−0.257057	−	0.966396i	$-0.582753\pi$
$619$	−12.7910	−0.514114	−0.257057	+	0.966396i	$0.582753\pi$
$620$	0	0
$621$	−17.0855	−0.685619
$622$	−28.5404	−1.14437
$623$	10.8270	0.433776
$624$	−0.700258	−0.0280328
$625$	0	0
$626$	−5.92781	−0.236923
$627$	−7.37270	−0.294437
$628$	1.39373	0.0556159
$629$	−1.16882	−0.0466038
$630$	0	0
$631$	14.1203	0.562121	0.281060	−	0.959690i	$-0.409314\pi$
$631$	14.1203	0.562121	0.281060	+	0.959690i	$0.409314\pi$
$632$	4.12185	0.163959
$633$	27.1184	1.07786
$634$	−4.71496	−0.187255
$635$	0	0
$636$	4.37616	0.173526
$637$	0.226893	0.00898983
$638$	−20.6970	−0.819402
$639$	−8.08564	−0.319863
$640$	0	0
$641$	24.7254	0.976593	0.488297	−	0.872678i	$-0.337618\pi$
$641$	24.7254	0.976593	0.488297	+	0.872678i	$0.337618\pi$
$642$	−2.02807	−0.0800414
$643$	16.9975	0.670318	0.335159	−	0.942162i	$-0.391210\pi$
$643$	16.9975	0.670318	0.335159	+	0.942162i	$0.391210\pi$
$644$	3.27270	0.128962
$645$	0	0
$646$	0.950692	0.0374045
$647$	1.38493	0.0544470	0.0272235	−	0.999629i	$-0.491333\pi$
$647$	1.38493	0.0544470	0.0272235	+	0.999629i	$0.491333\pi$
$648$	−32.9658	−1.29502
$649$	−1.31038	−0.0514368
$650$	0	0
$651$	−42.7806	−1.67670
$652$	−0.857949	−0.0335999
$653$	24.3154	0.951534	0.475767	−	0.879571i	$-0.342171\pi$
$653$	24.3154	0.951534	0.475767	+	0.879571i	$0.342171\pi$
$654$	−38.2642	−1.49625
$655$	0	0
$656$	−8.11187	−0.316715
$657$	−6.93117	−0.270411
$658$	−53.5939	−2.08931
$659$	35.7071	1.39095	0.695475	−	0.718550i	$-0.255194\pi$
$659$	35.7071	1.39095	0.695475	+	0.718550i	$0.255194\pi$
$660$	0	0
$661$	30.4094	1.18279	0.591395	−	0.806382i	$-0.298578\pi$
$661$	30.4094	1.18279	0.591395	+	0.806382i	$0.298578\pi$
$662$	29.2960	1.13862
$663$	0.0716379	0.00278219
$664$	32.1110	1.24615
$665$	0	0
$666$	4.81664	0.186641
$667$	38.5874	1.49411
$668$	1.28685	0.0497898
$669$	25.2681	0.976921
$670$	0	0
$671$	25.2320	0.974072
$672$	8.39427	0.323816
$673$	10.4201	0.401666	0.200833	−	0.979625i	$-0.435635\pi$
$673$	10.4201	0.401666	0.200833	+	0.979625i	$0.435635\pi$
$674$	17.5848	0.677340
$675$	0	0
$676$	3.14536	0.120975
$677$	−24.0844	−0.925640	−0.462820	−	0.886452i	$-0.653162\pi$
$677$	−24.0844	−0.925640	−0.462820	+	0.886452i	$0.653162\pi$
$678$	1.63666	0.0628557
$679$	−42.5625	−1.63340
$680$	0	0
$681$	28.8931	1.10718
$682$	−16.5279	−0.632887
$683$	22.2056	0.849674	0.424837	−	0.905270i	$-0.360331\pi$
$683$	22.2056	0.849674	0.424837	+	0.905270i	$0.360331\pi$
$684$	0.539652	0.0206341
$685$	0	0
$686$	−19.1001	−0.729246
$687$	27.9924	1.06798
$688$	9.09388	0.346701
$689$	−0.893045	−0.0340223
$690$	0	0
$691$	−22.8898	−0.870768	−0.435384	−	0.900245i	$-0.643387\pi$
$691$	−22.8898	−0.870768	−0.435384	+	0.900245i	$0.643387\pi$
$692$	−1.00984	−0.0383885
$693$	−6.01055	−0.228322
$694$	42.0028	1.59440
$695$	0	0
$696$	52.3118	1.98287
$697$	0.829861	0.0314332
$698$	−22.7147	−0.859765
$699$	1.29361	0.0489289
$700$	0	0
$701$	22.2327	0.839718	0.419859	−	0.907589i	$-0.362080\pi$
$701$	22.2327	0.839718	0.419859	+	0.907589i	$0.362080\pi$
$702$	0.510460	0.0192661
$703$	6.70043	0.252712
$704$	15.6614	0.590261
$705$	0	0
$706$	16.0504	0.604065
$707$	37.0708	1.39419
$708$	0.357675	0.0134422
$709$	−36.0658	−1.35448	−0.677241	−	0.735762i	$-0.736824\pi$
$709$	−36.0658	−1.35448	−0.677241	+	0.735762i	$0.736824\pi$
$710$	0	0
$711$	1.52419	0.0571615
$712$	−10.5724	−0.396217
$713$	30.8146	1.15402
$714$	2.89023	0.108164
$715$	0	0
$716$	−0.854134	−0.0319205
$717$	11.2586	0.420460
$718$	42.6068	1.59007
$719$	−41.4592	−1.54617	−0.773083	−	0.634305i	$-0.781286\pi$
$719$	−41.4592	−1.54617	−0.773083	+	0.634305i	$0.781286\pi$
$720$	0	0
$721$	−12.5532	−0.467505
$722$	19.7410	0.734686
$723$	2.02466	0.0752980
$724$	−3.98813	−0.148218
$725$	0	0
$726$	20.8688	0.774515
$727$	−36.9815	−1.37157	−0.685784	−	0.727805i	$-0.740541\pi$
$727$	−36.9815	−1.37157	−0.685784	+	0.727805i	$0.740541\pi$
$728$	−0.905399	−0.0335563
$729$	11.1847	0.414249
$730$	0	0
$731$	−0.930323	−0.0344093
$732$	−6.88723	−0.254559
$733$	−13.8389	−0.511152	−0.255576	−	0.966789i	$-0.582265\pi$
$733$	−13.8389	−0.511152	−0.255576	+	0.966789i	$0.582265\pi$
$734$	9.91899	0.366117
$735$	0	0
$736$	−6.04634	−0.222871
$737$	14.9976	0.552443
$738$	−3.41982	−0.125885
$739$	34.7385	1.27788	0.638939	−	0.769257i	$-0.279374\pi$
$739$	34.7385	1.27788	0.638939	+	0.769257i	$0.279374\pi$
$740$	0	0
$741$	−0.410675	−0.0150865
$742$	−36.0298	−1.32270
$743$	−8.59167	−0.315198	−0.157599	−	0.987503i	$-0.550375\pi$
$743$	−8.59167	−0.315198	−0.157599	+	0.987503i	$0.550375\pi$
$744$	41.7745	1.53153
$745$	0	0
$746$	30.7625	1.12630
$747$	11.8741	0.434450
$748$	−0.153808	−0.00562379
$749$	−2.30000	−0.0840402
$750$	0	0
$751$	21.3458	0.778919	0.389459	−	0.921044i	$-0.372662\pi$
$751$	21.3458	0.778919	0.389459	+	0.921044i	$0.372662\pi$
$752$	45.9032	1.67392
$753$	34.2802	1.24924
$754$	−1.15287	−0.0419849
$755$	0	0
$756$	−2.83680	−0.103173
$757$	−49.0973	−1.78447	−0.892237	−	0.451568i	$-0.850865\pi$
$757$	−49.0973	−1.78447	−0.892237	+	0.451568i	$0.850865\pi$
$758$	17.2665	0.627148
$759$	16.1447	0.586014
$760$	0	0
$761$	−49.9688	−1.81137	−0.905683	−	0.423955i	$-0.860642\pi$
$761$	−49.9688	−1.81137	−0.905683	+	0.423955i	$0.860642\pi$
$762$	48.1337	1.74370
$763$	−43.3948	−1.57100
$764$	−2.13713	−0.0773185
$765$	0	0
$766$	−11.3051	−0.408471
$767$	−0.0729907	−0.00263554
$768$	−11.5865	−0.418092
$769$	−19.4741	−0.702255	−0.351128	−	0.936328i	$-0.614202\pi$
$769$	−19.4741	−0.702255	−0.351128	+	0.936328i	$0.614202\pi$
$770$	0	0
$771$	24.1317	0.869083
$772$	0.498117	0.0179276
$773$	7.87928	0.283398	0.141699	−	0.989910i	$-0.454743\pi$
$773$	7.87928	0.283398	0.141699	+	0.989910i	$0.454743\pi$
$774$	3.83382	0.137804
$775$	0	0
$776$	41.5615	1.49197
$777$	20.3702	0.730776
$778$	12.5645	0.450459
$779$	−4.75731	−0.170448
$780$	0	0
$781$	−13.2110	−0.472727
$782$	−2.08181	−0.0744455
$783$	−33.4478	−1.19533
$784$	−7.84040	−0.280014
$785$	0	0
$786$	−47.1015	−1.68006
$787$	−48.7404	−1.73741	−0.868703	−	0.495333i	$-0.835046\pi$
$787$	−48.7404	−1.73741	−0.868703	+	0.495333i	$0.835046\pi$
$788$	−3.84174	−0.136856
$789$	27.2774	0.971101
$790$	0	0
$791$	1.85612	0.0659959
$792$	5.86919	0.208552
$793$	1.40548	0.0499100
$794$	−28.8025	−1.02216
$795$	0	0
$796$	−1.43146	−0.0507367
$797$	2.61718	0.0927051	0.0463526	−	0.998925i	$-0.485240\pi$
$797$	2.61718	0.0927051	0.0463526	+	0.998925i	$0.485240\pi$
$798$	−16.5687	−0.586525
$799$	−4.69599	−0.166132
$800$	0	0
$801$	−3.90949	−0.138135
$802$	23.9848	0.846932
$803$	−11.3247	−0.399641
$804$	−4.09368	−0.144373
$805$	0	0
$806$	−0.920641	−0.0324282
$807$	28.3221	0.996985
$808$	−36.1989	−1.27347
$809$	−41.0761	−1.44416	−0.722080	−	0.691809i	$-0.756814\pi$
$809$	−41.0761	−1.44416	−0.722080	+	0.691809i	$0.756814\pi$
$810$	0	0
$811$	18.6245	0.653993	0.326996	−	0.945026i	$-0.393963\pi$
$811$	18.6245	0.653993	0.326996	+	0.945026i	$0.393963\pi$
$812$	6.40686	0.224837
$813$	40.3689	1.41580
$814$	7.86984	0.275838
$815$	0	0
$816$	−2.47548	−0.0866592
$817$	5.33322	0.186586
$818$	3.78447	0.132321
$819$	−0.334801	−0.0116989
$820$	0	0
$821$	−38.4848	−1.34313	−0.671564	−	0.740946i	$-0.734377\pi$
$821$	−38.4848	−1.34313	−0.671564	+	0.740946i	$0.734377\pi$
$822$	−55.4724	−1.93482
$823$	−29.9573	−1.04425	−0.522123	−	0.852870i	$-0.674860\pi$
$823$	−29.9573	−1.04425	−0.522123	+	0.852870i	$0.674860\pi$
$824$	12.2579	0.427026
$825$	0	0
$826$	−2.94481	−0.102463
$827$	17.3939	0.604844	0.302422	−	0.953174i	$-0.402205\pi$
$827$	17.3939	0.604844	0.302422	+	0.953174i	$0.402205\pi$
$828$	−1.18172	−0.0410678
$829$	41.4644	1.44012	0.720060	−	0.693912i	$-0.244114\pi$
$829$	41.4644	1.44012	0.720060	+	0.693912i	$0.244114\pi$
$830$	0	0
$831$	63.0914	2.18862
$832$	0.872373	0.0302441
$833$	0.802089	0.0277907
$834$	−51.3288	−1.77737
$835$	0	0
$836$	0.881730	0.0304953
$837$	−26.7103	−0.923243
$838$	−41.1001	−1.41978
$839$	12.5173	0.432145	0.216072	−	0.976377i	$-0.430675\pi$
$839$	12.5173	0.432145	0.216072	+	0.976377i	$0.430675\pi$
$840$	0	0
$841$	46.5414	1.60487
$842$	−25.1077	−0.865268
$843$	−17.3584	−0.597856
$844$	−3.24319	−0.111635
$845$	0	0
$846$	19.3520	0.665334
$847$	23.6670	0.813209
$848$	30.8596	1.05972
$849$	−62.6311	−2.14949
$850$	0	0
$851$	−14.6725	−0.502968
$852$	3.60602	0.123540
$853$	−40.5026	−1.38678	−0.693392	−	0.720561i	$-0.743885\pi$
$853$	−40.5026	−1.38678	−0.693392	+	0.720561i	$0.743885\pi$
$854$	56.7040	1.94037
$855$	0	0
$856$	2.24591	0.0767635
$857$	−4.42595	−0.151188	−0.0755939	−	0.997139i	$-0.524085\pi$
$857$	−4.42595	−0.151188	−0.0755939	+	0.997139i	$0.524085\pi$
$858$	−0.482350	−0.0164671
$859$	34.9029	1.19087	0.595437	−	0.803402i	$-0.296979\pi$
$859$	34.9029	1.19087	0.595437	+	0.803402i	$0.296979\pi$
$860$	0	0
$861$	−14.4628	−0.492892
$862$	−19.7094	−0.671303
$863$	10.9055	0.371227	0.185614	−	0.982623i	$-0.440573\pi$
$863$	10.9055	0.371227	0.185614	+	0.982623i	$0.440573\pi$
$864$	5.24100	0.178302
$865$	0	0
$866$	19.2594	0.654462
$867$	−34.1660	−1.16034
$868$	5.11630	0.173659
$869$	2.49035	0.0844793
$870$	0	0
$871$	0.835397	0.0283064
$872$	42.3742	1.43497
$873$	15.3687	0.520152
$874$	11.9343	0.403684
$875$	0	0
$876$	3.09115	0.104440
$877$	−8.18855	−0.276508	−0.138254	−	0.990397i	$-0.544149\pi$
$877$	−8.18855	−0.276508	−0.138254	+	0.990397i	$0.544149\pi$
$878$	15.9305	0.537628
$879$	22.3736	0.754643
$880$	0	0
$881$	8.93204	0.300928	0.150464	−	0.988615i	$-0.451923\pi$
$881$	8.93204	0.300928	0.150464	+	0.988615i	$0.451923\pi$
$882$	−3.30537	−0.111298
$883$	−40.4510	−1.36129	−0.680643	−	0.732615i	$-0.738299\pi$
$883$	−40.4510	−1.36129	−0.680643	+	0.732615i	$0.738299\pi$
$884$	−0.00856746	−0.000288155 0
$885$	0	0
$886$	−16.1924	−0.543994
$887$	−58.8126	−1.97473	−0.987366	−	0.158455i	$-0.949349\pi$
$887$	−58.8126	−1.97473	−0.987366	+	0.158455i	$0.949349\pi$
$888$	−19.8911	−0.667501
$889$	54.5877	1.83081
$890$	0	0
$891$	−19.9173	−0.667255
$892$	−3.02191	−0.101181
$893$	26.9205	0.900860
$894$	25.9799	0.868898
$895$	0	0
$896$	26.9038	0.898794
$897$	0.899292	0.0300265
$898$	−30.5238	−1.01859
$899$	60.3248	2.01194
$900$	0	0
$901$	−3.15700	−0.105175
$902$	−5.58759	−0.186046
$903$	16.2137	0.539558
$904$	−1.81246	−0.0602816
$905$	0	0
$906$	−28.9697	−0.962455
$907$	19.5914	0.650522	0.325261	−	0.945624i	$-0.394548\pi$
$907$	19.5914	0.650522	0.325261	+	0.945624i	$0.394548\pi$
$908$	−3.45543	−0.114673
$909$	−13.3857	−0.443977
$910$	0	0
$911$	−36.8647	−1.22138	−0.610692	−	0.791868i	$-0.709109\pi$
$911$	−36.8647	−1.22138	−0.610692	+	0.791868i	$0.709109\pi$
$912$	14.1911	0.469914
$913$	19.4009	0.642076
$914$	−6.59046	−0.217993
$915$	0	0
$916$	−3.34772	−0.110612
$917$	−53.4172	−1.76399
$918$	1.80453	0.0595583
$919$	21.4022	0.705995	0.352997	−	0.935624i	$-0.385162\pi$
$919$	21.4022	0.705995	0.352997	+	0.935624i	$0.385162\pi$
$920$	0	0
$921$	24.1197	0.794771
$922$	−29.5625	−0.973590
$923$	−0.735881	−0.0242218
$924$	2.68057	0.0881844
$925$	0	0
$926$	9.69342	0.318545
$927$	4.53277	0.148876
$928$	−11.8367	−0.388559
$929$	12.6133	0.413828	0.206914	−	0.978359i	$-0.433658\pi$
$929$	12.6133	0.413828	0.206914	+	0.978359i	$0.433658\pi$
$930$	0	0
$931$	−4.59810	−0.150697
$932$	−0.154708	−0.00506763
$933$	43.5834	1.42686
$934$	−49.8457	−1.63100
$935$	0	0
$936$	0.326926	0.0106859
$937$	56.2291	1.83693	0.918463	−	0.395507i	$-0.129431\pi$
$937$	56.2291	1.83693	0.918463	+	0.395507i	$0.129431\pi$
$938$	33.7041	1.10048
$939$	9.05220	0.295408
$940$	0	0
$941$	−9.84947	−0.321083	−0.160542	−	0.987029i	$-0.551324\pi$
$941$	−9.84947	−0.321083	−0.160542	+	0.987029i	$0.551324\pi$
$942$	15.4512	0.503428
$943$	10.4175	0.339240
$944$	2.52223	0.0820916
$945$	0	0
$946$	6.26402	0.203661
$947$	3.66056	0.118952	0.0594761	−	0.998230i	$-0.481057\pi$
$947$	3.66056	0.118952	0.0594761	+	0.998230i	$0.481057\pi$
$948$	−0.679755	−0.0220774
$949$	−0.630812	−0.0204770
$950$	0	0
$951$	7.20010	0.233479
$952$	−3.20068	−0.103734
$953$	48.1748	1.56054	0.780268	−	0.625446i	$-0.215083\pi$
$953$	48.1748	1.56054	0.780268	+	0.625446i	$0.215083\pi$
$954$	13.0098	0.421209
$955$	0	0
$956$	−1.34646	−0.0435476
$957$	31.6059	1.02167
$958$	−25.3412	−0.818736
$959$	−62.9104	−2.03148
$960$	0	0
$961$	17.1733	0.553978
$962$	0.438367	0.0141335
$963$	0.830496	0.0267624
$964$	−0.242137	−0.00779872
$965$	0	0
$966$	36.2819	1.16735
$967$	9.98807	0.321195	0.160597	−	0.987020i	$-0.448658\pi$
$967$	9.98807	0.321195	0.160597	+	0.987020i	$0.448658\pi$
$968$	−23.1104	−0.742796
$969$	−1.45178	−0.0466378
$970$	0	0
$971$	27.4950	0.882356	0.441178	−	0.897420i	$-0.354561\pi$
$971$	27.4950	0.882356	0.441178	+	0.897420i	$0.354561\pi$
$972$	2.64105	0.0847118
$973$	−58.2113	−1.86617
$974$	−29.0041	−0.929350
$975$	0	0
$976$	−48.5670	−1.55459
$977$	−31.8959	−1.02044	−0.510220	−	0.860044i	$-0.670436\pi$
$977$	−31.8959	−1.02044	−0.510220	+	0.860044i	$0.670436\pi$
$978$	−9.51141	−0.304141
$979$	−6.38765	−0.204150
$980$	0	0
$981$	15.6692	0.500280
$982$	38.8418	1.23949
$983$	22.8367	0.728377	0.364189	−	0.931325i	$-0.381346\pi$
$983$	22.8367	0.728377	0.364189	+	0.931325i	$0.381346\pi$
$984$	14.1227	0.450214
$985$	0	0
$986$	−4.07550	−0.129790
$987$	81.8418	2.60505
$988$	0.0491143	0.00156253
$989$	−11.6786	−0.371359
$990$	0	0
$991$	33.8840	1.07636	0.538180	−	0.842830i	$-0.319112\pi$
$991$	33.8840	1.07636	0.538180	+	0.842830i	$0.319112\pi$
$992$	−9.45240	−0.300114
$993$	−44.7371	−1.41969
$994$	−29.6891	−0.941681
$995$	0	0
$996$	−5.29558	−0.167797
$997$	12.3737	0.391878	0.195939	−	0.980616i	$-0.437224\pi$
$997$	12.3737	0.391878	0.195939	+	0.980616i	$0.437224\pi$
$998$	−5.89900	−0.186730
$999$	12.7182	0.402387

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.k.1.8		25
5.4	even	2		1205.2.a.d.1.18	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.a.d.1.18	✓	25	5.4	even	2
6025.2.a.k.1.8		25	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:	\( N \)	\(=\)	\( 6025 = 5^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6025.a (trivial)

\(f(q)\)	\(=\)	\(q-1.32584 q^{2} +2.02466 q^{3} -0.242137 q^{4} -2.68439 q^{6} -3.04433 q^{7} +2.97272 q^{8} +1.09926 q^{9} +O(q^{10})\)	\(q-1.32584 q^{2} +2.02466 q^{3} -0.242137 q^{4} -2.68439 q^{6} -3.04433 q^{7} +2.97272 q^{8} +1.09926 q^{9} +1.79607 q^{11} -0.490247 q^{12} +0.100045 q^{13} +4.03630 q^{14} -3.45709 q^{16} +0.353668 q^{17} -1.45745 q^{18} -2.02746 q^{19} -6.16373 q^{21} -2.38130 q^{22} +4.43970 q^{23} +6.01877 q^{24} -0.132644 q^{26} -3.84836 q^{27} +0.737145 q^{28} +8.69145 q^{29} +6.94070 q^{31} -1.36188 q^{32} +3.63643 q^{33} -0.468909 q^{34} -0.266172 q^{36} -3.30485 q^{37} +2.68809 q^{38} +0.202557 q^{39} +2.34644 q^{41} +8.17215 q^{42} -2.63050 q^{43} -0.434895 q^{44} -5.88635 q^{46} -13.2780 q^{47} -6.99945 q^{48} +2.26792 q^{49} +0.716059 q^{51} -0.0242246 q^{52} -8.92645 q^{53} +5.10232 q^{54} -9.04994 q^{56} -4.10492 q^{57} -11.5235 q^{58} -0.729581 q^{59} +14.0485 q^{61} -9.20229 q^{62} -3.34651 q^{63} +8.71983 q^{64} -4.82134 q^{66} +8.35024 q^{67} -0.0856363 q^{68} +8.98890 q^{69} -7.35552 q^{71} +3.26780 q^{72} -6.30530 q^{73} +4.38171 q^{74} +0.490923 q^{76} -5.46781 q^{77} -0.268559 q^{78} +1.38656 q^{79} -11.0894 q^{81} -3.11101 q^{82} +10.8019 q^{83} +1.49247 q^{84} +3.48763 q^{86} +17.5973 q^{87} +5.33921 q^{88} -3.55647 q^{89} -0.304569 q^{91} -1.07502 q^{92} +14.0526 q^{93} +17.6045 q^{94} -2.75735 q^{96} +13.9809 q^{97} -3.00690 q^{98} +1.97435 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 4 q^{2} - 9 q^{3} + 36 q^{4} + 7 q^{6} - 7 q^{7} + 15 q^{8} + 36 q^{9}+O(q^{10}) \) 25 * q + 4 * q^2 - 9 * q^3 + 36 * q^4 + 7 * q^6 - 7 * q^7 + 15 * q^8 + 36 * q^9	\( 25 q + 4 q^{2} - 9 q^{3} + 36 q^{4} + 7 q^{6} - 7 q^{7} + 15 q^{8} + 36 q^{9} + 10 q^{11} - 22 q^{12} - 10 q^{13} + 13 q^{14} + 54 q^{16} - q^{17} + 13 q^{18} + 50 q^{19} + 9 q^{21} - 11 q^{22} + 31 q^{23} + 22 q^{24} + 8 q^{26} - 42 q^{27} - 14 q^{28} + 4 q^{29} + 34 q^{31} + 44 q^{32} - 28 q^{33} + 33 q^{34} + 83 q^{36} - 14 q^{37} + 10 q^{38} + 23 q^{39} + 11 q^{41} - 23 q^{42} - 49 q^{43} + 20 q^{44} + 27 q^{46} + 28 q^{47} - 30 q^{48} + 66 q^{49} + 49 q^{51} - 39 q^{52} + 16 q^{53} + 5 q^{54} + 51 q^{56} - 10 q^{57} + 8 q^{58} + 30 q^{59} + 35 q^{61} + 18 q^{62} + 73 q^{64} - 13 q^{66} - 37 q^{67} - 11 q^{68} - 4 q^{69} + 12 q^{71} + 90 q^{72} - 36 q^{73} - 12 q^{74} + 57 q^{76} + 31 q^{77} + 9 q^{78} + 16 q^{79} + 65 q^{81} + 11 q^{82} - 43 q^{83} - 62 q^{84} - 9 q^{86} + 22 q^{87} - 20 q^{88} + 38 q^{89} + 86 q^{91} + 119 q^{92} - 10 q^{93} - 18 q^{94} - 34 q^{96} - 17 q^{97} + 32 q^{98} + 26 q^{99}+O(q^{100}) \) 25 * q + 4 * q^2 - 9 * q^3 + 36 * q^4 + 7 * q^6 - 7 * q^7 + 15 * q^8 + 36 * q^9 + 10 * q^11 - 22 * q^12 - 10 * q^13 + 13 * q^14 + 54 * q^16 - q^17 + 13 * q^18 + 50 * q^19 + 9 * q^21 - 11 * q^22 + 31 * q^23 + 22 * q^24 + 8 * q^26 - 42 * q^27 - 14 * q^28 + 4 * q^29 + 34 * q^31 + 44 * q^32 - 28 * q^33 + 33 * q^34 + 83 * q^36 - 14 * q^37 + 10 * q^38 + 23 * q^39 + 11 * q^41 - 23 * q^42 - 49 * q^43 + 20 * q^44 + 27 * q^46 + 28 * q^47 - 30 * q^48 + 66 * q^49 + 49 * q^51 - 39 * q^52 + 16 * q^53 + 5 * q^54 + 51 * q^56 - 10 * q^57 + 8 * q^58 + 30 * q^59 + 35 * q^61 + 18 * q^62 + 73 * q^64 - 13 * q^66 - 37 * q^67 - 11 * q^68 - 4 * q^69 + 12 * q^71 + 90 * q^72 - 36 * q^73 - 12 * q^74 + 57 * q^76 + 31 * q^77 + 9 * q^78 + 16 * q^79 + 65 * q^81 + 11 * q^82 - 43 * q^83 - 62 * q^84 - 9 * q^86 + 22 * q^87 - 20 * q^88 + 38 * q^89 + 86 * q^91 + 119 * q^92 - 10 * q^93 - 18 * q^94 - 34 * q^96 - 17 * q^97 + 32 * q^98 + 26 * q^99

Introduction

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