LMFDB - Embedded newform 6025.2.a.j.1.20

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:	$1$
Dimension:	$25$
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
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)

$f(q)$	$=$	$q+1.58885 q^{2} -2.11200 q^{3} +0.524458 q^{4} -3.35566 q^{6} -3.95616 q^{7} -2.34442 q^{8} +1.46053 q^{9} +O(q^{10})$	\(q+1.58885 q^{2} -2.11200 q^{3} +0.524458 q^{4} -3.35566 q^{6} -3.95616 q^{7} -2.34442 q^{8} +1.46053 q^{9} +2.44935 q^{11} -1.10765 q^{12} +4.82916 q^{13} -6.28576 q^{14} -4.77386 q^{16} -2.96026 q^{17} +2.32057 q^{18} +6.29318 q^{19} +8.35539 q^{21} +3.89165 q^{22} -2.58210 q^{23} +4.95141 q^{24} +7.67282 q^{26} +3.25135 q^{27} -2.07484 q^{28} +6.54429 q^{29} -7.41024 q^{31} -2.89612 q^{32} -5.17301 q^{33} -4.70342 q^{34} +0.765987 q^{36} +4.28353 q^{37} +9.99895 q^{38} -10.1992 q^{39} -2.65056 q^{41} +13.2755 q^{42} -12.4355 q^{43} +1.28458 q^{44} -4.10258 q^{46} +12.4795 q^{47} +10.0824 q^{48} +8.65117 q^{49} +6.25205 q^{51} +2.53269 q^{52} +14.2288 q^{53} +5.16593 q^{54} +9.27490 q^{56} -13.2912 q^{57} +10.3979 q^{58} -4.42842 q^{59} +1.55551 q^{61} -11.7738 q^{62} -5.77809 q^{63} +4.94620 q^{64} -8.21916 q^{66} -5.91349 q^{67} -1.55253 q^{68} +5.45338 q^{69} -2.90562 q^{71} -3.42410 q^{72} -12.6887 q^{73} +6.80590 q^{74} +3.30051 q^{76} -9.69000 q^{77} -16.2050 q^{78} -1.75221 q^{79} -11.2484 q^{81} -4.21135 q^{82} +2.97974 q^{83} +4.38205 q^{84} -19.7582 q^{86} -13.8215 q^{87} -5.74230 q^{88} +11.4345 q^{89} -19.1049 q^{91} -1.35420 q^{92} +15.6504 q^{93} +19.8281 q^{94} +6.11660 q^{96} -10.9038 q^{97} +13.7455 q^{98} +3.57735 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 6 q^{2} - 15 q^{3} + 32 q^{4} - q^{6} - 19 q^{7} - 15 q^{8} + 32 q^{9}+O(q^{10}) $ 25 * q - 6 * q^2 - 15 * q^3 + 32 * q^4 - q^6 - 19 * q^7 - 15 * q^8 + 32 * q^9	$ 25 q - 6 q^{2} - 15 q^{3} + 32 q^{4} - q^{6} - 19 q^{7} - 15 q^{8} + 32 q^{9} + 2 q^{11} - 20 q^{12} - 14 q^{13} - 5 q^{14} + 38 q^{16} - 7 q^{17} - 9 q^{18} + 30 q^{19} + q^{21} - q^{22} - 43 q^{23} - 6 q^{24} - 22 q^{26} - 42 q^{27} - 32 q^{28} - 4 q^{29} + 14 q^{31} - 26 q^{32} - 4 q^{33} + 7 q^{34} + 15 q^{36} - 16 q^{37} - 14 q^{38} - 21 q^{39} - q^{41} + 25 q^{42} - 35 q^{43} - 52 q^{44} - 27 q^{46} - 50 q^{47} - 26 q^{48} + 46 q^{49} - 7 q^{51} - 3 q^{52} - 4 q^{53} - 31 q^{54} - 51 q^{56} - 2 q^{58} + 6 q^{59} + 19 q^{61} - 28 q^{63} + 49 q^{64} - 27 q^{66} - 65 q^{67} + 25 q^{68} + 2 q^{69} - 34 q^{71} + 10 q^{72} - 8 q^{73} - 42 q^{74} + 71 q^{76} - q^{77} + 59 q^{78} - 12 q^{79} + 29 q^{81} - 11 q^{82} - 41 q^{83} - 10 q^{84} - 13 q^{86} - 40 q^{87} + 52 q^{88} - 24 q^{89} + 46 q^{91} - 85 q^{92} + 30 q^{93} + 14 q^{94} - 30 q^{96} - 9 q^{97} + 64 q^{98} + 2 q^{99}+O(q^{100}) $ 25 * q - 6 * q^2 - 15 * q^3 + 32 * q^4 - q^6 - 19 * q^7 - 15 * q^8 + 32 * q^9 + 2 * q^11 - 20 * q^12 - 14 * q^13 - 5 * q^14 + 38 * q^16 - 7 * q^17 - 9 * q^18 + 30 * q^19 + q^21 - q^22 - 43 * q^23 - 6 * q^24 - 22 * q^26 - 42 * q^27 - 32 * q^28 - 4 * q^29 + 14 * q^31 - 26 * q^32 - 4 * q^33 + 7 * q^34 + 15 * q^36 - 16 * q^37 - 14 * q^38 - 21 * q^39 - q^41 + 25 * q^42 - 35 * q^43 - 52 * q^44 - 27 * q^46 - 50 * q^47 - 26 * q^48 + 46 * q^49 - 7 * q^51 - 3 * q^52 - 4 * q^53 - 31 * q^54 - 51 * q^56 - 2 * q^58 + 6 * q^59 + 19 * q^61 - 28 * q^63 + 49 * q^64 - 27 * q^66 - 65 * q^67 + 25 * q^68 + 2 * q^69 - 34 * q^71 + 10 * q^72 - 8 * q^73 - 42 * q^74 + 71 * q^76 - q^77 + 59 * q^78 - 12 * q^79 + 29 * q^81 - 11 * q^82 - 41 * q^83 - 10 * q^84 - 13 * q^86 - 40 * q^87 + 52 * q^88 - 24 * q^89 + 46 * q^91 - 85 * q^92 + 30 * q^93 + 14 * q^94 - 30 * q^96 - 9 * q^97 + 64 * q^98 + 2 * 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 $

$2$	1.58885	1.12349	0.561745	−	0.827311i	$-0.310130\pi$
$2$	1.58885	1.12349	0.561745	+	0.827311i	$0.310130\pi$
$3$	−2.11200	−1.21936	−0.609681	−	0.792647i	$-0.708702\pi$
$3$	−2.11200	−1.21936	−0.609681	+	0.792647i	$0.708702\pi$
$4$	0.524458	0.262229
$5$	0	0
$5$	0	0
$6$	−3.35566	−1.36994
$7$	−3.95616	−1.49529	−0.747643	−	0.664101i	$-0.768815\pi$
$7$	−3.95616	−1.49529	−0.747643	+	0.664101i	$0.768815\pi$
$8$	−2.34442	−0.828878
$9$	1.46053	0.486844
$10$	0	0
$11$	2.44935	0.738506	0.369253	−	0.929329i	$-0.379614\pi$
$11$	2.44935	0.738506	0.369253	+	0.929329i	$0.379614\pi$
$12$	−1.10765	−0.319752
$13$	4.82916	1.33937	0.669683	−	0.742647i	$-0.266430\pi$
$13$	4.82916	1.33937	0.669683	+	0.742647i	$0.266430\pi$
$14$	−6.28576	−1.67994
$15$	0	0
$16$	−4.77386	−1.19346
$17$	−2.96026	−0.717968	−0.358984	−	0.933344i	$-0.616877\pi$
$17$	−2.96026	−0.717968	−0.358984	+	0.933344i	$0.616877\pi$
$18$	2.32057	0.546964
$19$	6.29318	1.44375	0.721877	−	0.692021i	$-0.243279\pi$
$19$	6.29318	1.44375	0.721877	+	0.692021i	$0.243279\pi$
$20$	0	0
$21$	8.35539	1.82330
$22$	3.89165	0.829704
$23$	−2.58210	−0.538404	−0.269202	−	0.963084i	$-0.586760\pi$
$23$	−2.58210	−0.538404	−0.269202	+	0.963084i	$0.586760\pi$
$24$	4.95141	1.01070
$25$	0	0
$26$	7.67282	1.50476
$27$	3.25135	0.625723
$28$	−2.07484	−0.392107
$29$	6.54429	1.21524	0.607622	−	0.794226i	$-0.292123\pi$
$29$	6.54429	1.21524	0.607622	+	0.794226i	$0.292123\pi$
$30$	0	0
$31$	−7.41024	−1.33092	−0.665460	−	0.746434i	$-0.731764\pi$
$31$	−7.41024	−1.33092	−0.665460	+	0.746434i	$0.731764\pi$
$32$	−2.89612	−0.511967
$33$	−5.17301	−0.900506
$34$	−4.70342	−0.806629
$35$	0	0
$36$	0.765987	0.127664
$37$	4.28353	0.704208	0.352104	−	0.935961i	$-0.385466\pi$
$37$	4.28353	0.704208	0.352104	+	0.935961i	$0.385466\pi$
$38$	9.99895	1.62204
$39$	−10.1992	−1.63317
$40$	0	0
$41$	−2.65056	−0.413948	−0.206974	−	0.978346i	$-0.566362\pi$
$41$	−2.65056	−0.413948	−0.206974	+	0.978346i	$0.566362\pi$
$42$	13.2755	2.04845
$43$	−12.4355	−1.89640	−0.948198	−	0.317681i	$-0.897096\pi$
$43$	−12.4355	−1.89640	−0.948198	+	0.317681i	$0.897096\pi$
$44$	1.28458	0.193657
$45$	0	0
$46$	−4.10258	−0.604892
$47$	12.4795	1.82032	0.910159	−	0.414258i	$-0.135959\pi$
$47$	12.4795	1.82032	0.910159	+	0.414258i	$0.135959\pi$
$48$	10.0824	1.45527
$49$	8.65117	1.23588
$50$	0	0
$51$	6.25205	0.875462
$52$	2.53269	0.351221
$53$	14.2288	1.95447	0.977236	−	0.212155i	$-0.0680481\pi$
$53$	14.2288	1.95447	0.977236	+	0.212155i	$0.0680481\pi$
$54$	5.16593	0.702994
$55$	0	0
$56$	9.27490	1.23941
$57$	−13.2912	−1.76046
$58$	10.3979	1.36531
$59$	−4.42842	−0.576532	−0.288266	−	0.957550i	$-0.593079\pi$
$59$	−4.42842	−0.576532	−0.288266	+	0.957550i	$0.593079\pi$
$60$	0	0
$61$	1.55551	0.199163	0.0995813	−	0.995029i	$-0.468250\pi$
$61$	1.55551	0.199163	0.0995813	+	0.995029i	$0.468250\pi$
$62$	−11.7738	−1.49527
$63$	−5.77809	−0.727971
$64$	4.94620	0.618275
$65$	0	0
$66$	−8.21916	−1.01171
$67$	−5.91349	−0.722447	−0.361224	−	0.932479i	$-0.617641\pi$
$67$	−5.91349	−0.722447	−0.361224	+	0.932479i	$0.617641\pi$
$68$	−1.55253	−0.188272
$69$	5.45338	0.656510
$70$	0	0
$71$	−2.90562	−0.344834	−0.172417	−	0.985024i	$-0.555158\pi$
$71$	−2.90562	−0.344834	−0.172417	+	0.985024i	$0.555158\pi$
$72$	−3.42410	−0.403534
$73$	−12.6887	−1.48510	−0.742551	−	0.669790i	$-0.766384\pi$
$73$	−12.6887	−1.48510	−0.742551	+	0.669790i	$0.766384\pi$
$74$	6.80590	0.791170
$75$	0	0
$76$	3.30051	0.378594
$77$	−9.69000	−1.10428
$78$	−16.2050	−1.83485
$79$	−1.75221	−0.197139	−0.0985693	−	0.995130i	$-0.531427\pi$
$79$	−1.75221	−0.197139	−0.0985693	+	0.995130i	$0.531427\pi$
$80$	0	0
$81$	−11.2484	−1.24983
$82$	−4.21135	−0.465066
$83$	2.97974	0.327069	0.163535	−	0.986538i	$-0.447710\pi$
$83$	2.97974	0.327069	0.163535	+	0.986538i	$0.447710\pi$
$84$	4.38205	0.478121
$85$	0	0
$86$	−19.7582	−2.13058
$87$	−13.8215	−1.48182
$88$	−5.74230	−0.612131
$89$	11.4345	1.21205	0.606025	−	0.795445i	$-0.292763\pi$
$89$	11.4345	1.21205	0.606025	+	0.795445i	$0.292763\pi$
$90$	0	0
$91$	−19.1049	−2.00274
$92$	−1.35420	−0.141185
$93$	15.6504	1.62287
$94$	19.8281	2.04511
$95$	0	0
$96$	6.11660	0.624273
$97$	−10.9038	−1.10711	−0.553556	−	0.832812i	$-0.686729\pi$
$97$	−10.9038	−1.10711	−0.553556	+	0.832812i	$0.686729\pi$
$98$	13.7455	1.38850
$99$	3.57735	0.359537
$100$	0	0
$101$	−7.85635	−0.781736	−0.390868	−	0.920447i	$-0.627825\pi$
$101$	−7.85635	−0.781736	−0.390868	+	0.920447i	$0.627825\pi$
$102$	9.93360	0.983573
$103$	2.64038	0.260164	0.130082	−	0.991503i	$-0.458476\pi$
$103$	2.64038	0.260164	0.130082	+	0.991503i	$0.458476\pi$
$104$	−11.3216	−1.11017
$105$	0	0
$106$	22.6074	2.19583
$107$	−3.36366	−0.325177	−0.162589	−	0.986694i	$-0.551984\pi$
$107$	−3.36366	−0.325177	−0.162589	+	0.986694i	$0.551984\pi$
$108$	1.70520	0.164083
$109$	11.9162	1.14137	0.570683	−	0.821171i	$-0.306679\pi$
$109$	11.9162	1.14137	0.570683	+	0.821171i	$0.306679\pi$
$110$	0	0
$111$	−9.04680	−0.858684
$112$	18.8861	1.78457
$113$	6.35405	0.597739	0.298869	−	0.954294i	$-0.403391\pi$
$113$	6.35405	0.597739	0.298869	+	0.954294i	$0.403391\pi$
$114$	−21.1177	−1.97786
$115$	0	0
$116$	3.43220	0.318672
$117$	7.05313	0.652062
$118$	−7.03612	−0.647727
$119$	11.7112	1.07357
$120$	0	0
$121$	−5.00070	−0.454609
$122$	2.47148	0.223757
$123$	5.59797	0.504752
$124$	−3.88636	−0.349005
$125$	0	0
$126$	−9.18054	−0.817868
$127$	3.04486	0.270188	0.135094	−	0.990833i	$-0.456866\pi$
$127$	3.04486	0.270188	0.135094	+	0.990833i	$0.456866\pi$
$128$	13.6510	1.20659
$129$	26.2637	2.31239
$130$	0	0
$131$	−14.3622	−1.25483	−0.627416	−	0.778684i	$-0.715887\pi$
$131$	−14.3622	−1.25483	−0.627416	+	0.778684i	$0.715887\pi$
$132$	−2.71303	−0.236139
$133$	−24.8968	−2.15883
$134$	−9.39567	−0.811662
$135$	0	0
$136$	6.94009	0.595108
$137$	−15.3768	−1.31373	−0.656866	−	0.754008i	$-0.728118\pi$
$137$	−15.3768	−1.31373	−0.656866	+	0.754008i	$0.728118\pi$
$138$	8.66463	0.737582
$139$	−19.8845	−1.68658	−0.843291	−	0.537458i	$-0.819385\pi$
$139$	−19.8845	−1.68658	−0.843291	+	0.537458i	$0.819385\pi$
$140$	0	0
$141$	−26.3566	−2.21963
$142$	−4.61661	−0.387417
$143$	11.8283	0.989130
$144$	−6.97237	−0.581031
$145$	0	0
$146$	−20.1605	−1.66850
$147$	−18.2713	−1.50699
$148$	2.24653	0.184664
$149$	6.91808	0.566751	0.283376	−	0.959009i	$-0.408546\pi$
$149$	6.91808	0.566751	0.283376	+	0.959009i	$0.408546\pi$
$150$	0	0
$151$	8.82455	0.718131	0.359066	−	0.933312i	$-0.383095\pi$
$151$	8.82455	0.718131	0.359066	+	0.933312i	$0.383095\pi$
$152$	−14.7539	−1.19670
$153$	−4.32355	−0.349538
$154$	−15.3960	−1.24064
$155$	0	0
$156$	−5.34903	−0.428265
$157$	−17.3366	−1.38361	−0.691804	−	0.722085i	$-0.743184\pi$
$157$	−17.3366	−1.38361	−0.691804	+	0.722085i	$0.743184\pi$
$158$	−2.78400	−0.221483
$159$	−30.0511	−2.38321
$160$	0	0
$161$	10.2152	0.805069
$162$	−17.8721	−1.40417
$163$	−19.0616	−1.49302	−0.746510	−	0.665374i	$-0.768272\pi$
$163$	−19.0616	−1.49302	−0.746510	+	0.665374i	$0.768272\pi$
$164$	−1.39011	−0.108549
$165$	0	0
$166$	4.73438	0.367459
$167$	−16.0450	−1.24160	−0.620801	−	0.783968i	$-0.713192\pi$
$167$	−16.0450	−1.24160	−0.620801	+	0.783968i	$0.713192\pi$
$168$	−19.5886	−1.51129
$169$	10.3207	0.793903
$170$	0	0
$171$	9.19139	0.702883
$172$	−6.52189	−0.497289
$173$	−11.8261	−0.899124	−0.449562	−	0.893249i	$-0.648420\pi$
$173$	−11.8261	−0.899124	−0.449562	+	0.893249i	$0.648420\pi$
$174$	−21.9604	−1.66481
$175$	0	0
$176$	−11.6928	−0.881381
$177$	9.35282	0.703001
$178$	18.1677	1.36173
$179$	−9.26442	−0.692455	−0.346228	−	0.938151i	$-0.612538\pi$
$179$	−9.26442	−0.692455	−0.346228	+	0.938151i	$0.612538\pi$
$180$	0	0
$181$	−12.0259	−0.893881	−0.446941	−	0.894564i	$-0.647486\pi$
$181$	−12.0259	−0.893881	−0.446941	+	0.894564i	$0.647486\pi$
$182$	−30.3549	−2.25005
$183$	−3.28523	−0.242851
$184$	6.05352	0.446272
$185$	0	0
$186$	24.8662	1.82328
$187$	−7.25069	−0.530223
$188$	6.54496	0.477340
$189$	−12.8629	−0.935636
$190$	0	0
$191$	−0.291425	−0.0210868	−0.0105434	−	0.999944i	$-0.503356\pi$
$191$	−0.291425	−0.0210868	−0.0105434	+	0.999944i	$0.503356\pi$
$192$	−10.4464	−0.753901
$193$	5.11396	0.368111	0.184055	−	0.982916i	$-0.441077\pi$
$193$	5.11396	0.368111	0.184055	+	0.982916i	$0.441077\pi$
$194$	−17.3245	−1.24383
$195$	0	0
$196$	4.53717	0.324084
$197$	5.25064	0.374093	0.187046	−	0.982351i	$-0.440109\pi$
$197$	5.25064	0.374093	0.187046	+	0.982351i	$0.440109\pi$
$198$	5.68388	0.403936
$199$	26.5144	1.87955	0.939776	−	0.341790i	$-0.111033\pi$
$199$	26.5144	1.87955	0.939776	+	0.341790i	$0.111033\pi$
$200$	0	0
$201$	12.4893	0.880925
$202$	−12.4826	−0.878273
$203$	−25.8902	−1.81714
$204$	3.27894	0.229571
$205$	0	0
$206$	4.19518	0.292292
$207$	−3.77123	−0.262119
$208$	−23.0537	−1.59849
$209$	15.4142	1.06622
$210$	0	0
$211$	−25.6874	−1.76839	−0.884196	−	0.467117i	$-0.845293\pi$
$211$	−25.6874	−1.76839	−0.884196	+	0.467117i	$0.845293\pi$
$212$	7.46239	0.512519
$213$	6.13666	0.420477
$214$	−5.34437	−0.365333
$215$	0	0
$216$	−7.62254	−0.518648
$217$	29.3161	1.99011
$218$	18.9331	1.28231
$219$	26.7985	1.81088
$220$	0	0
$221$	−14.2955	−0.961622
$222$	−14.3740	−0.964723
$223$	14.1378	0.946739	0.473370	−	0.880864i	$-0.343037\pi$
$223$	14.1378	0.946739	0.473370	+	0.880864i	$0.343037\pi$
$224$	11.4575	0.765537
$225$	0	0
$226$	10.0957	0.671553
$227$	3.95817	0.262713	0.131356	−	0.991335i	$-0.458067\pi$
$227$	3.95817	0.262713	0.131356	+	0.991335i	$0.458067\pi$
$228$	−6.97066	−0.461643
$229$	−17.4905	−1.15580	−0.577902	−	0.816106i	$-0.696128\pi$
$229$	−17.4905	−1.15580	−0.577902	+	0.816106i	$0.696128\pi$
$230$	0	0
$231$	20.4652	1.34651
$232$	−15.3426	−1.00729
$233$	6.93325	0.454212	0.227106	−	0.973870i	$-0.427074\pi$
$233$	6.93325	0.454212	0.227106	+	0.973870i	$0.427074\pi$
$234$	11.2064	0.732585
$235$	0	0
$236$	−2.32252	−0.151183
$237$	3.70065	0.240383
$238$	18.6074	1.20614
$239$	−28.9342	−1.87160	−0.935799	−	0.352535i	$-0.885320\pi$
$239$	−28.9342	−1.87160	−0.935799	+	0.352535i	$0.885320\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	−7.94538	−0.510749
$243$	14.0026	0.898268
$244$	0.815799	0.0522262
$245$	0	0
$246$	8.89436	0.567084
$247$	30.3907	1.93372
$248$	17.3727	1.10317
$249$	−6.29321	−0.398816
$250$	0	0
$251$	−6.98146	−0.440666	−0.220333	−	0.975425i	$-0.570714\pi$
$251$	−6.98146	−0.440666	−0.220333	+	0.975425i	$0.570714\pi$
$252$	−3.03036	−0.190895
$253$	−6.32445	−0.397615
$254$	4.83784	0.303553
$255$	0	0
$256$	11.7971	0.737319
$257$	−5.90262	−0.368196	−0.184098	−	0.982908i	$-0.558936\pi$
$257$	−5.90262	−0.368196	−0.184098	+	0.982908i	$0.558936\pi$
$258$	41.7292	2.59795
$259$	−16.9463	−1.05299
$260$	0	0
$261$	9.55814	0.591634
$262$	−22.8195	−1.40979
$263$	10.0001	0.616635	0.308318	−	0.951283i	$-0.400234\pi$
$263$	10.0001	0.616635	0.308318	+	0.951283i	$0.400234\pi$
$264$	12.1277	0.746410
$265$	0	0
$266$	−39.5574	−2.42542
$267$	−24.1495	−1.47793
$268$	−3.10137	−0.189447
$269$	3.58143	0.218364	0.109182	−	0.994022i	$-0.465177\pi$
$269$	3.58143	0.218364	0.109182	+	0.994022i	$0.465177\pi$
$270$	0	0
$271$	3.45106	0.209637	0.104819	−	0.994491i	$-0.466574\pi$
$271$	3.45106	0.209637	0.104819	+	0.994491i	$0.466574\pi$
$272$	14.1318	0.856869
$273$	40.3495	2.44206
$274$	−24.4315	−1.47596
$275$	0	0
$276$	2.86007	0.172156
$277$	12.2520	0.736150	0.368075	−	0.929796i	$-0.380017\pi$
$277$	12.2520	0.736150	0.368075	+	0.929796i	$0.380017\pi$
$278$	−31.5936	−1.89486
$279$	−10.8229	−0.647950
$280$	0	0
$281$	−14.8076	−0.883347	−0.441674	−	0.897176i	$-0.645615\pi$
$281$	−14.8076	−0.883347	−0.441674	+	0.897176i	$0.645615\pi$
$282$	−41.8768	−2.49373
$283$	5.46878	0.325085	0.162543	−	0.986702i	$-0.448031\pi$
$283$	5.46878	0.325085	0.162543	+	0.986702i	$0.448031\pi$
$284$	−1.52387	−0.0904253
$285$	0	0
$286$	18.7934	1.11128
$287$	10.4860	0.618971
$288$	−4.22988	−0.249248
$289$	−8.23688	−0.484522
$290$	0	0
$291$	23.0288	1.34997
$292$	−6.65469	−0.389436
$293$	14.5691	0.851134	0.425567	−	0.904927i	$-0.360075\pi$
$293$	14.5691	0.851134	0.425567	+	0.904927i	$0.360075\pi$
$294$	−29.0304	−1.69308
$295$	0	0
$296$	−10.0424	−0.583702
$297$	7.96369	0.462100
$298$	10.9918	0.636739
$299$	−12.4693	−0.721121
$300$	0	0
$301$	49.1968	2.83565
$302$	14.0209	0.806813
$303$	16.5926	0.953220
$304$	−30.0428	−1.72307
$305$	0	0
$306$	−6.86949	−0.392702
$307$	17.1669	0.979768	0.489884	−	0.871788i	$-0.337039\pi$
$307$	17.1669	0.979768	0.489884	+	0.871788i	$0.337039\pi$
$308$	−5.08199	−0.289573
$309$	−5.57648	−0.317235
$310$	0	0
$311$	−12.3598	−0.700859	−0.350429	−	0.936589i	$-0.613964\pi$
$311$	−12.3598	−0.700859	−0.350429	+	0.936589i	$0.613964\pi$
$312$	23.9111	1.35370
$313$	−24.5732	−1.38896	−0.694478	−	0.719514i	$-0.744365\pi$
$313$	−24.5732	−1.38896	−0.694478	+	0.719514i	$0.744365\pi$
$314$	−27.5453	−1.55447
$315$	0	0
$316$	−0.918957	−0.0516954
$317$	30.1919	1.69575	0.847873	−	0.530199i	$-0.177883\pi$
$317$	30.1919	1.69575	0.847873	+	0.530199i	$0.177883\pi$
$318$	−47.7469	−2.67751
$319$	16.0292	0.897465
$320$	0	0
$321$	7.10404	0.396509
$322$	16.2304	0.904487
$323$	−18.6294	−1.03657
$324$	−5.89933	−0.327741
$325$	0	0
$326$	−30.2861	−1.67739
$327$	−25.1670	−1.39174
$328$	6.21403	0.343112
$329$	−49.3708	−2.72190
$330$	0	0
$331$	11.3078	0.621533	0.310767	−	0.950486i	$-0.399414\pi$
$331$	11.3078	0.621533	0.310767	+	0.950486i	$0.399414\pi$
$332$	1.56275	0.0857669
$333$	6.25623	0.342839
$334$	−25.4932	−1.39493
$335$	0	0
$336$	−39.8875	−2.17604
$337$	6.30144	0.343261	0.171631	−	0.985161i	$-0.445096\pi$
$337$	6.30144	0.343261	0.171631	+	0.985161i	$0.445096\pi$
$338$	16.3982	0.891942
$339$	−13.4197	−0.728860
$340$	0	0
$341$	−18.1503	−0.982892
$342$	14.6038	0.789682
$343$	−6.53230	−0.352711
$344$	29.1540	1.57188
$345$	0	0
$346$	−18.7900	−1.01016
$347$	−25.1324	−1.34918	−0.674590	−	0.738193i	$-0.735680\pi$
$347$	−25.1324	−1.34918	−0.674590	+	0.738193i	$0.735680\pi$
$348$	−7.24880	−0.388577
$349$	8.05833	0.431352	0.215676	−	0.976465i	$-0.430804\pi$
$349$	8.05833	0.431352	0.215676	+	0.976465i	$0.430804\pi$
$350$	0	0
$351$	15.7013	0.838073
$352$	−7.09361	−0.378091
$353$	23.5613	1.25404	0.627020	−	0.779003i	$-0.284274\pi$
$353$	23.5613	1.25404	0.627020	+	0.779003i	$0.284274\pi$
$354$	14.8603	0.789814
$355$	0	0
$356$	5.99689	0.317835
$357$	−24.7341	−1.30907
$358$	−14.7198	−0.777966
$359$	6.37521	0.336471	0.168235	−	0.985747i	$-0.446193\pi$
$359$	6.37521	0.336471	0.168235	+	0.985747i	$0.446193\pi$
$360$	0	0
$361$	20.6041	1.08443
$362$	−19.1075	−1.00427
$363$	10.5615	0.554333
$364$	−10.0197	−0.525175
$365$	0	0
$366$	−5.21975	−0.272841
$367$	−7.21397	−0.376566	−0.188283	−	0.982115i	$-0.560292\pi$
$367$	−7.21397	−0.376566	−0.188283	+	0.982115i	$0.560292\pi$
$368$	12.3266	0.642567
$369$	−3.87122	−0.201528
$370$	0	0
$371$	−56.2912	−2.92250
$372$	8.20798	0.425564
$373$	−13.7583	−0.712376	−0.356188	−	0.934414i	$-0.615924\pi$
$373$	−13.7583	−0.712376	−0.356188	+	0.934414i	$0.615924\pi$
$374$	−11.5203	−0.595700
$375$	0	0
$376$	−29.2572	−1.50882
$377$	31.6034	1.62766
$378$	−20.4372	−1.05118
$379$	10.9764	0.563819	0.281910	−	0.959441i	$-0.409032\pi$
$379$	10.9764	0.563819	0.281910	+	0.959441i	$0.409032\pi$
$380$	0	0
$381$	−6.43074	−0.329457
$382$	−0.463031	−0.0236907
$383$	0.840298	0.0429372	0.0214686	−	0.999770i	$-0.493166\pi$
$383$	0.840298	0.0429372	0.0214686	+	0.999770i	$0.493166\pi$
$384$	−28.8310	−1.47127
$385$	0	0
$386$	8.12533	0.413568
$387$	−18.1624	−0.923248
$388$	−5.71857	−0.290316
$389$	−15.4877	−0.785256	−0.392628	−	0.919697i	$-0.628434\pi$
$389$	−15.4877	−0.785256	−0.392628	+	0.919697i	$0.628434\pi$
$390$	0	0
$391$	7.64367	0.386557
$392$	−20.2820	−1.02440
$393$	30.3329	1.53009
$394$	8.34251	0.420290
$395$	0	0
$396$	1.87617	0.0942809
$397$	−5.77451	−0.289814	−0.144907	−	0.989445i	$-0.546288\pi$
$397$	−5.77451	−0.289814	−0.144907	+	0.989445i	$0.546288\pi$
$398$	42.1274	2.11166
$399$	52.5820	2.63239
$400$	0	0
$401$	−38.8656	−1.94086	−0.970429	−	0.241387i	$-0.922398\pi$
$401$	−38.8656	−1.94086	−0.970429	+	0.241387i	$0.922398\pi$
$402$	19.8436	0.989710
$403$	−35.7852	−1.78259
$404$	−4.12032	−0.204994
$405$	0	0
$406$	−41.1358	−2.04154
$407$	10.4918	0.520062
$408$	−14.6574	−0.725652
$409$	25.4772	1.25977	0.629884	−	0.776689i	$-0.283102\pi$
$409$	25.4772	1.25977	0.629884	+	0.776689i	$0.283102\pi$
$410$	0	0
$411$	32.4758	1.60191
$412$	1.38477	0.0682226
$413$	17.5195	0.862080
$414$	−5.99194	−0.294488
$415$	0	0
$416$	−13.9858	−0.685712
$417$	41.9960	2.05655
$418$	24.4909	1.19789
$419$	25.9379	1.26715	0.633574	−	0.773682i	$-0.281587\pi$
$419$	25.9379	1.26715	0.633574	+	0.773682i	$0.281587\pi$
$420$	0	0
$421$	−9.95932	−0.485388	−0.242694	−	0.970103i	$-0.578031\pi$
$421$	−9.95932	−0.485388	−0.242694	+	0.970103i	$0.578031\pi$
$422$	−40.8135	−1.98677
$423$	18.2267	0.886211
$424$	−33.3582	−1.62002
$425$	0	0
$426$	9.75026	0.472402
$427$	−6.15384	−0.297805
$428$	−1.76410	−0.0852709
$429$	−24.9813	−1.20611
$430$	0	0
$431$	18.5993	0.895898	0.447949	−	0.894059i	$-0.352155\pi$
$431$	18.5993	0.895898	0.447949	+	0.894059i	$0.352155\pi$
$432$	−15.5215	−0.746779
$433$	−23.3358	−1.12145	−0.560724	−	0.828002i	$-0.689477\pi$
$433$	−23.3358	−1.12145	−0.560724	+	0.828002i	$0.689477\pi$
$434$	46.5790	2.23586
$435$	0	0
$436$	6.24954	0.299299
$437$	−16.2496	−0.777324
$438$	42.5790	2.03450
$439$	−31.4126	−1.49924	−0.749621	−	0.661867i	$-0.769764\pi$
$439$	−31.4126	−1.49924	−0.749621	+	0.661867i	$0.769764\pi$
$440$	0	0
$441$	12.6353	0.601681
$442$	−22.7135	−1.08037
$443$	−14.6049	−0.693900	−0.346950	−	0.937884i	$-0.612783\pi$
$443$	−14.6049	−0.693900	−0.346950	+	0.937884i	$0.612783\pi$
$444$	−4.74466	−0.225172
$445$	0	0
$446$	22.4630	1.06365
$447$	−14.6110	−0.691075
$448$	−19.5679	−0.924499
$449$	−21.4681	−1.01314	−0.506571	−	0.862198i	$-0.669087\pi$
$449$	−21.4681	−1.01314	−0.506571	+	0.862198i	$0.669087\pi$
$450$	0	0
$451$	−6.49214	−0.305703
$452$	3.33243	0.156744
$453$	−18.6374	−0.875662
$454$	6.28895	0.295155
$455$	0	0
$456$	31.1601	1.45921
$457$	20.9663	0.980761	0.490381	−	0.871508i	$-0.336858\pi$
$457$	20.9663	0.980761	0.490381	+	0.871508i	$0.336858\pi$
$458$	−27.7898	−1.29853
$459$	−9.62484	−0.449249
$460$	0	0
$461$	28.6751	1.33553	0.667766	−	0.744371i	$-0.267251\pi$
$461$	28.6751	1.33553	0.667766	+	0.744371i	$0.267251\pi$
$462$	32.5163	1.51280
$463$	36.0970	1.67757	0.838785	−	0.544462i	$-0.183266\pi$
$463$	36.0970	1.67757	0.838785	+	0.544462i	$0.183266\pi$
$464$	−31.2415	−1.45035
$465$	0	0
$466$	11.0159	0.510303
$467$	25.5140	1.18064	0.590322	−	0.807168i	$-0.299001\pi$
$467$	25.5140	1.18064	0.590322	+	0.807168i	$0.299001\pi$
$468$	3.69907	0.170990
$469$	23.3947	1.08027
$470$	0	0
$471$	36.6148	1.68712
$472$	10.3821	0.477874
$473$	−30.4588	−1.40050
$474$	5.87980	0.270068
$475$	0	0
$476$	6.14205	0.281520
$477$	20.7816	0.951523
$478$	−45.9722	−2.10272
$479$	3.66623	0.167514	0.0837572	−	0.996486i	$-0.473308\pi$
$479$	3.66623	0.167514	0.0837572	+	0.996486i	$0.473308\pi$
$480$	0	0
$481$	20.6858	0.943192
$482$	−1.58885	−0.0723703
$483$	−21.5744	−0.981671
$484$	−2.62266	−0.119212
$485$	0	0
$486$	22.2481	1.00919
$487$	37.5444	1.70130	0.850649	−	0.525734i	$-0.176209\pi$
$487$	37.5444	1.70130	0.850649	+	0.525734i	$0.176209\pi$
$488$	−3.64677	−0.165082
$489$	40.2581	1.82053
$490$	0	0
$491$	−36.9795	−1.66886	−0.834430	−	0.551114i	$-0.814203\pi$
$491$	−36.9795	−1.66886	−0.834430	+	0.551114i	$0.814203\pi$
$492$	2.93590	0.132361
$493$	−19.3728	−0.872506
$494$	48.2865	2.17251
$495$	0	0
$496$	35.3755	1.58841
$497$	11.4951	0.515625
$498$	−9.99899	−0.448065
$499$	−24.5030	−1.09691	−0.548454	−	0.836181i	$-0.684783\pi$
$499$	−24.5030	−1.09691	−0.548454	+	0.836181i	$0.684783\pi$
$500$	0	0
$501$	33.8870	1.51396
$502$	−11.0925	−0.495084
$503$	−34.3536	−1.53175	−0.765875	−	0.642990i	$-0.777694\pi$
$503$	−34.3536	−1.53175	−0.765875	+	0.642990i	$0.777694\pi$
$504$	13.5463	0.603399
$505$	0	0
$506$	−10.0486	−0.446716
$507$	−21.7974	−0.968055
$508$	1.59690	0.0708510
$509$	30.5178	1.35268	0.676340	−	0.736590i	$-0.263565\pi$
$509$	30.5178	1.35268	0.676340	+	0.736590i	$0.263565\pi$
$510$	0	0
$511$	50.1985	2.22065
$512$	−8.55820	−0.378222
$513$	20.4614	0.903391
$514$	−9.37841	−0.413664
$515$	0	0
$516$	13.7742	0.606376
$517$	30.5666	1.34432
$518$	−26.9252	−1.18303
$519$	24.9767	1.09636
$520$	0	0
$521$	−5.80889	−0.254492	−0.127246	−	0.991871i	$-0.540614\pi$
$521$	−5.80889	−0.254492	−0.127246	+	0.991871i	$0.540614\pi$
$522$	15.1865	0.664695
$523$	1.01320	0.0443042	0.0221521	−	0.999755i	$-0.492948\pi$
$523$	1.01320	0.0443042	0.0221521	+	0.999755i	$0.492948\pi$
$524$	−7.53237	−0.329053
$525$	0	0
$526$	15.8888	0.692783
$527$	21.9362	0.955557
$528$	24.6952	1.07472
$529$	−16.3328	−0.710121
$530$	0	0
$531$	−6.46785	−0.280681
$532$	−13.0573	−0.566107
$533$	−12.8000	−0.554428
$534$	−38.3701	−1.66044
$535$	0	0
$536$	13.8637	0.598821
$537$	19.5664	0.844354
$538$	5.69037	0.245329
$539$	21.1897	0.912706
$540$	0	0
$541$	13.6671	0.587596	0.293798	−	0.955868i	$-0.405081\pi$
$541$	13.6671	0.587596	0.293798	+	0.955868i	$0.405081\pi$
$542$	5.48324	0.235525
$543$	25.3988	1.08996
$544$	8.57327	0.367576
$545$	0	0
$546$	64.1094	2.74363
$547$	−22.2843	−0.952806	−0.476403	−	0.879227i	$-0.658060\pi$
$547$	−22.2843	−0.952806	−0.476403	+	0.879227i	$0.658060\pi$
$548$	−8.06450	−0.344498
$549$	2.27187	0.0969610
$550$	0	0
$551$	41.1844	1.75452
$552$	−12.7850	−0.544167
$553$	6.93200	0.294779
$554$	19.4666	0.827057
$555$	0	0
$556$	−10.4286	−0.442270
$557$	40.9844	1.73656	0.868282	−	0.496070i	$-0.165224\pi$
$557$	40.9844	1.73656	0.868282	+	0.496070i	$0.165224\pi$
$558$	−17.1960	−0.727965
$559$	−60.0529	−2.53997
$560$	0	0
$561$	15.3134	0.646534
$562$	−23.5271	−0.992432
$563$	−16.0544	−0.676613	−0.338306	−	0.941036i	$-0.609854\pi$
$563$	−16.0544	−0.676613	−0.338306	+	0.941036i	$0.609854\pi$
$564$	−13.8229	−0.582050
$565$	0	0
$566$	8.68909	0.365230
$567$	44.5006	1.86885
$568$	6.81200	0.285825
$569$	−36.0698	−1.51212	−0.756062	−	0.654499i	$-0.772879\pi$
$569$	−36.0698	−1.51212	−0.756062	+	0.654499i	$0.772879\pi$
$570$	0	0
$571$	−44.0870	−1.84498	−0.922491	−	0.386019i	$-0.873850\pi$
$571$	−44.0870	−1.84498	−0.922491	+	0.386019i	$0.873850\pi$
$572$	6.20343	0.259378
$573$	0.615488	0.0257124
$574$	16.6608	0.695407
$575$	0	0
$576$	7.22408	0.301003
$577$	−29.4680	−1.22677	−0.613384	−	0.789785i	$-0.710192\pi$
$577$	−29.4680	−1.22677	−0.613384	+	0.789785i	$0.710192\pi$
$578$	−13.0872	−0.544356
$579$	−10.8007	−0.448860
$580$	0	0
$581$	−11.7883	−0.489062
$582$	36.5893	1.51668
$583$	34.8512	1.44339
$584$	29.7477	1.23097
$585$	0	0
$586$	23.1481	0.956240
$587$	−11.7475	−0.484870	−0.242435	−	0.970168i	$-0.577946\pi$
$587$	−11.7475	−0.484870	−0.242435	+	0.970168i	$0.577946\pi$
$588$	−9.58250	−0.395176
$589$	−46.6340	−1.92152
$590$	0	0
$591$	−11.0893	−0.456155
$592$	−20.4490	−0.840447
$593$	17.0838	0.701548	0.350774	−	0.936460i	$-0.385919\pi$
$593$	17.0838	0.701548	0.350774	+	0.936460i	$0.385919\pi$
$594$	12.6531	0.519165
$595$	0	0
$596$	3.62824	0.148619
$597$	−55.9982	−2.29186
$598$	−19.8120	−0.810172
$599$	−11.1735	−0.456535	−0.228267	−	0.973598i	$-0.573306\pi$
$599$	−11.1735	−0.456535	−0.228267	+	0.973598i	$0.573306\pi$
$600$	0	0
$601$	10.7681	0.439241	0.219620	−	0.975585i	$-0.429518\pi$
$601$	10.7681	0.439241	0.219620	+	0.975585i	$0.429518\pi$
$602$	78.1665	3.18583
$603$	−8.63683	−0.351719
$604$	4.62810	0.188315
$605$	0	0
$606$	26.3632	1.07093
$607$	4.98933	0.202511	0.101255	−	0.994860i	$-0.467714\pi$
$607$	4.98933	0.202511	0.101255	+	0.994860i	$0.467714\pi$
$608$	−18.2258	−0.739155
$609$	54.6801	2.21575
$610$	0	0
$611$	60.2653	2.43807
$612$	−2.26752	−0.0916589
$613$	−29.9743	−1.21065	−0.605324	−	0.795979i	$-0.706957\pi$
$613$	−29.9743	−1.21065	−0.605324	+	0.795979i	$0.706957\pi$
$614$	27.2757	1.10076
$615$	0	0
$616$	22.7174	0.915312
$617$	23.7949	0.957947	0.478973	−	0.877829i	$-0.341009\pi$
$617$	23.7949	0.957947	0.478973	+	0.877829i	$0.341009\pi$
$618$	−8.86021	−0.356410
$619$	26.1679	1.05178	0.525888	−	0.850554i	$-0.323733\pi$
$619$	26.1679	1.05178	0.525888	+	0.850554i	$0.323733\pi$
$620$	0	0
$621$	−8.39531	−0.336892
$622$	−19.6379	−0.787407
$623$	−45.2365	−1.81236
$624$	48.6894	1.94913
$625$	0	0
$626$	−39.0432	−1.56048
$627$	−32.5547	−1.30011
$628$	−9.09229	−0.362822
$629$	−12.6803	−0.505598
$630$	0	0
$631$	−22.7740	−0.906618	−0.453309	−	0.891353i	$-0.649757\pi$
$631$	−22.7740	−0.906618	−0.453309	+	0.891353i	$0.649757\pi$
$632$	4.10791	0.163404
$633$	54.2516	2.15631
$634$	47.9705	1.90515
$635$	0	0
$636$	−15.7605	−0.624946
$637$	41.7779	1.65530
$638$	25.4681	1.00829
$639$	−4.24375	−0.167880
$640$	0	0
$641$	16.9241	0.668463	0.334231	−	0.942491i	$-0.391523\pi$
$641$	16.9241	0.668463	0.334231	+	0.942491i	$0.391523\pi$
$642$	11.2873	0.445474
$643$	−17.7965	−0.701825	−0.350913	−	0.936408i	$-0.614129\pi$
$643$	−17.7965	−0.701825	−0.350913	+	0.936408i	$0.614129\pi$
$644$	5.35743	0.211112
$645$	0	0
$646$	−29.5994	−1.16457
$647$	7.34387	0.288717	0.144359	−	0.989525i	$-0.453888\pi$
$647$	7.34387	0.288717	0.144359	+	0.989525i	$0.453888\pi$
$648$	26.3711	1.03595
$649$	−10.8467	−0.425772
$650$	0	0
$651$	−61.9155	−2.42666
$652$	−9.99701	−0.391513
$653$	−18.2904	−0.715758	−0.357879	−	0.933768i	$-0.616500\pi$
$653$	−18.2904	−0.715758	−0.357879	+	0.933768i	$0.616500\pi$
$654$	−39.9867	−1.56360
$655$	0	0
$656$	12.6534	0.494032
$657$	−18.5323	−0.723013
$658$	−78.4429	−3.05802
$659$	−41.2541	−1.60703	−0.803515	−	0.595285i	$-0.797039\pi$
$659$	−41.2541	−1.60703	−0.803515	+	0.595285i	$0.797039\pi$
$660$	0	0
$661$	−42.9591	−1.67091	−0.835457	−	0.549556i	$-0.814797\pi$
$661$	−42.9591	−1.67091	−0.835457	+	0.549556i	$0.814797\pi$
$662$	17.9665	0.698286
$663$	30.1921	1.17257
$664$	−6.98577	−0.271101
$665$	0	0
$666$	9.94023	0.385176
$667$	−16.8980	−0.654293
$668$	−8.41494	−0.325584
$669$	−29.8591	−1.15442
$670$	0	0
$671$	3.80998	0.147083
$672$	−24.1982	−0.933467
$673$	−9.72876	−0.375016	−0.187508	−	0.982263i	$-0.560041\pi$
$673$	−9.72876	−0.375016	−0.187508	+	0.982263i	$0.560041\pi$
$674$	10.0121	0.385650
$675$	0	0
$676$	5.41279	0.208184
$677$	−28.5338	−1.09664	−0.548322	−	0.836267i	$-0.684733\pi$
$677$	−28.5338	−1.09664	−0.548322	+	0.836267i	$0.684733\pi$
$678$	−21.3220	−0.818867
$679$	43.1371	1.65545
$680$	0	0
$681$	−8.35964	−0.320342
$682$	−28.8381	−1.10427
$683$	7.41861	0.283865	0.141933	−	0.989876i	$-0.454668\pi$
$683$	7.41861	0.283865	0.141933	+	0.989876i	$0.454668\pi$
$684$	4.82049	0.184316
$685$	0	0
$686$	−10.3789	−0.396267
$687$	36.9398	1.40934
$688$	59.3653	2.26328
$689$	68.7129	2.61775
$690$	0	0
$691$	20.4070	0.776318	0.388159	−	0.921592i	$-0.373111\pi$
$691$	20.4070	0.776318	0.388159	+	0.921592i	$0.373111\pi$
$692$	−6.20230	−0.235776
$693$	−14.1525	−0.537611
$694$	−39.9318	−1.51579
$695$	0	0
$696$	32.4035	1.22825
$697$	7.84633	0.297201
$698$	12.8035	0.484620
$699$	−14.6430	−0.553849
$700$	0	0
$701$	16.4358	0.620771	0.310386	−	0.950611i	$-0.399542\pi$
$701$	16.4358	0.620771	0.310386	+	0.950611i	$0.399542\pi$
$702$	24.9471	0.941566
$703$	26.9570	1.01670
$704$	12.1150	0.456600
$705$	0	0
$706$	37.4354	1.40890
$707$	31.0810	1.16892
$708$	4.90516	0.184347
$709$	−19.8152	−0.744177	−0.372088	−	0.928197i	$-0.621358\pi$
$709$	−19.8152	−0.744177	−0.372088	+	0.928197i	$0.621358\pi$
$710$	0	0
$711$	−2.55915	−0.0959756
$712$	−26.8072	−1.00464
$713$	19.1340	0.716573
$714$	−39.2989	−1.47072
$715$	0	0
$716$	−4.85880	−0.181582
$717$	61.1089	2.28215
$718$	10.1293	0.378021
$719$	12.2724	0.457685	0.228843	−	0.973463i	$-0.426506\pi$
$719$	12.2724	0.457685	0.228843	+	0.973463i	$0.426506\pi$
$720$	0	0
$721$	−10.4458	−0.389020
$722$	32.7369	1.21834
$723$	2.11200	0.0785460
$724$	−6.30710	−0.234401
$725$	0	0
$726$	16.7806	0.622787
$727$	−42.2544	−1.56713	−0.783565	−	0.621310i	$-0.786601\pi$
$727$	−42.2544	−1.56713	−0.783565	+	0.621310i	$0.786601\pi$
$728$	44.7899	1.66003
$729$	4.17184	0.154513
$730$	0	0
$731$	36.8123	1.36155
$732$	−1.72296	−0.0636826
$733$	−39.9107	−1.47414	−0.737068	−	0.675819i	$-0.763790\pi$
$733$	−39.9107	−1.47414	−0.737068	+	0.675819i	$0.763790\pi$
$734$	−11.4619	−0.423068
$735$	0	0
$736$	7.47807	0.275645
$737$	−14.4842	−0.533532
$738$	−6.15081	−0.226414
$739$	40.4640	1.48849	0.744246	−	0.667906i	$-0.232809\pi$
$739$	40.4640	1.48849	0.744246	+	0.667906i	$0.232809\pi$
$740$	0	0
$741$	−64.1852	−2.35790
$742$	−89.4386	−3.28339
$743$	−39.9803	−1.46674	−0.733368	−	0.679832i	$-0.762053\pi$
$743$	−39.9803	−1.46674	−0.733368	+	0.679832i	$0.762053\pi$
$744$	−36.6912	−1.34516
$745$	0	0
$746$	−21.8599	−0.800348
$747$	4.35201	0.159232
$748$	−3.80268	−0.139040
$749$	13.3072	0.486233
$750$	0	0
$751$	37.5040	1.36854	0.684271	−	0.729228i	$-0.260121\pi$
$751$	37.5040	1.36854	0.684271	+	0.729228i	$0.260121\pi$
$752$	−59.5753	−2.17249
$753$	14.7448	0.537331
$754$	50.2132	1.82866
$755$	0	0
$756$	−6.74603	−0.245351
$757$	−29.5638	−1.07451	−0.537257	−	0.843418i	$-0.680540\pi$
$757$	−29.5638	−1.07451	−0.537257	+	0.843418i	$0.680540\pi$
$758$	17.4399	0.633445
$759$	13.3572	0.484836
$760$	0	0
$761$	23.7939	0.862529	0.431265	−	0.902225i	$-0.358068\pi$
$761$	23.7939	0.862529	0.431265	+	0.902225i	$0.358068\pi$
$762$	−10.2175	−0.370141
$763$	−47.1424	−1.70667
$764$	−0.152840	−0.00552955
$765$	0	0
$766$	1.33511	0.0482395
$767$	−21.3855	−0.772187
$768$	−24.9155	−0.899059
$769$	33.0919	1.19332	0.596662	−	0.802493i	$-0.296493\pi$
$769$	33.0919	1.19332	0.596662	+	0.802493i	$0.296493\pi$
$770$	0	0
$771$	12.4663	0.448964
$772$	2.68205	0.0965292
$773$	45.8905	1.65057	0.825284	−	0.564718i	$-0.191015\pi$
$773$	45.8905	1.65057	0.825284	+	0.564718i	$0.191015\pi$
$774$	−28.8574	−1.03726
$775$	0	0
$776$	25.5631	0.917661
$777$	35.7906	1.28398
$778$	−24.6077	−0.882227
$779$	−16.6804	−0.597639
$780$	0	0
$781$	−7.11687	−0.254662
$782$	12.1447	0.434293
$783$	21.2778	0.760407
$784$	−41.2995	−1.47498
$785$	0	0
$786$	48.1946	1.71905
$787$	−25.7252	−0.917004	−0.458502	−	0.888693i	$-0.651614\pi$
$787$	−25.7252	−0.917004	−0.458502	+	0.888693i	$0.651614\pi$
$788$	2.75374	0.0980979
$789$	−21.1203	−0.751902
$790$	0	0
$791$	−25.1376	−0.893791
$792$	−8.38681	−0.298012
$793$	7.51180	0.266752
$794$	−9.17485	−0.325603
$795$	0	0
$796$	13.9057	0.492873
$797$	−4.27991	−0.151602	−0.0758011	−	0.997123i	$-0.524151\pi$
$797$	−4.27991	−0.151602	−0.0758011	+	0.997123i	$0.524151\pi$
$798$	83.5451	2.95746
$799$	−36.9424	−1.30693
$800$	0	0
$801$	16.7004	0.590079
$802$	−61.7518	−2.18053
$803$	−31.0791	−1.09676
$804$	6.55009	0.231004
$805$	0	0
$806$	−56.8575	−2.00272
$807$	−7.56397	−0.266264
$808$	18.4186	0.647964
$809$	9.00128	0.316468	0.158234	−	0.987402i	$-0.449420\pi$
$809$	9.00128	0.316468	0.158234	+	0.987402i	$0.449420\pi$
$810$	0	0
$811$	−31.5963	−1.10950	−0.554748	−	0.832018i	$-0.687185\pi$
$811$	−31.5963	−1.10950	−0.554748	+	0.832018i	$0.687185\pi$
$812$	−13.5783	−0.476506
$813$	−7.28864	−0.255624
$814$	16.6700	0.584284
$815$	0	0
$816$	−29.8464	−1.04483
$817$	−78.2588	−2.73793
$818$	40.4796	1.41534
$819$	−27.9033	−0.975020
$820$	0	0
$821$	10.7548	0.375344	0.187672	−	0.982232i	$-0.439906\pi$
$821$	10.7548	0.375344	0.187672	+	0.982232i	$0.439906\pi$
$822$	51.5994	1.79973
$823$	−2.79994	−0.0975997	−0.0487999	−	0.998809i	$-0.515540\pi$
$823$	−2.79994	−0.0975997	−0.0487999	+	0.998809i	$0.515540\pi$
$824$	−6.19017	−0.215645
$825$	0	0
$826$	27.8360	0.968538
$827$	−26.8049	−0.932098	−0.466049	−	0.884759i	$-0.654323\pi$
$827$	−26.8049	−0.932098	−0.466049	+	0.884759i	$0.654323\pi$
$828$	−1.97785	−0.0687351
$829$	−36.4823	−1.26708	−0.633542	−	0.773708i	$-0.718400\pi$
$829$	−36.4823	−1.26708	−0.633542	+	0.773708i	$0.718400\pi$
$830$	0	0
$831$	−25.8761	−0.897634
$832$	23.8860	0.828097
$833$	−25.6097	−0.887323
$834$	66.7255	2.31052
$835$	0	0
$836$	8.08408	0.279594
$837$	−24.0933	−0.832787
$838$	41.2115	1.42363
$839$	−50.9296	−1.75828	−0.879142	−	0.476559i	$-0.841884\pi$
$839$	−50.9296	−1.75828	−0.879142	+	0.476559i	$0.841884\pi$
$840$	0	0
$841$	13.8278	0.476820
$842$	−15.8239	−0.545328
$843$	31.2736	1.07712
$844$	−13.4719	−0.463723
$845$	0	0
$846$	28.9595	0.995649
$847$	19.7836	0.679771
$848$	−67.9262	−2.33259
$849$	−11.5500	−0.396396
$850$	0	0
$851$	−11.0605	−0.379149
$852$	3.21842	0.110261
$853$	44.7630	1.53266	0.766328	−	0.642450i	$-0.222082\pi$
$853$	44.7630	1.53266	0.766328	+	0.642450i	$0.222082\pi$
$854$	−9.77755	−0.334581
$855$	0	0
$856$	7.88584	0.269533
$857$	18.6544	0.637221	0.318610	−	0.947886i	$-0.396784\pi$
$857$	18.6544	0.637221	0.318610	+	0.947886i	$0.396784\pi$
$858$	−39.6916	−1.35505
$859$	43.6734	1.49012	0.745058	−	0.667000i	$-0.232422\pi$
$859$	43.6734	1.49012	0.745058	+	0.667000i	$0.232422\pi$
$860$	0	0
$861$	−22.1465	−0.754749
$862$	29.5516	1.00653
$863$	2.32338	0.0790886	0.0395443	−	0.999218i	$-0.487409\pi$
$863$	2.32338	0.0790886	0.0395443	+	0.999218i	$0.487409\pi$
$864$	−9.41632	−0.320350
$865$	0	0
$866$	−37.0772	−1.25994
$867$	17.3963	0.590808
$868$	15.3750	0.521863
$869$	−4.29176	−0.145588
$870$	0	0
$871$	−28.5571	−0.967622
$872$	−27.9366	−0.946053
$873$	−15.9253	−0.538990
$874$	−25.8182	−0.873315
$875$	0	0
$876$	14.0547	0.474864
$877$	8.77712	0.296382	0.148191	−	0.988959i	$-0.452655\pi$
$877$	8.77712	0.296382	0.148191	+	0.988959i	$0.452655\pi$
$878$	−49.9101	−1.68438
$879$	−30.7698	−1.03784
$880$	0	0
$881$	−16.3354	−0.550352	−0.275176	−	0.961394i	$-0.588736\pi$
$881$	−16.3354	−0.550352	−0.275176	+	0.961394i	$0.588736\pi$
$882$	20.0757	0.675983
$883$	−28.3586	−0.954342	−0.477171	−	0.878811i	$-0.658338\pi$
$883$	−28.3586	−0.954342	−0.477171	+	0.878811i	$0.658338\pi$
$884$	−7.49740	−0.252165
$885$	0	0
$886$	−23.2051	−0.779590
$887$	−37.9929	−1.27568	−0.637839	−	0.770170i	$-0.720171\pi$
$887$	−37.9929	−1.27568	−0.637839	+	0.770170i	$0.720171\pi$
$888$	21.2095	0.711745
$889$	−12.0459	−0.404008
$890$	0	0
$891$	−27.5513	−0.923004
$892$	7.41469	0.248262
$893$	78.5356	2.62809
$894$	−23.2147	−0.776416
$895$	0	0
$896$	−54.0056	−1.80420
$897$	26.3352	0.879308
$898$	−34.1096	−1.13825
$899$	−48.4948	−1.61739
$900$	0	0
$901$	−42.1208	−1.40325
$902$	−10.3151	−0.343454
$903$	−103.903	−3.45769
$904$	−14.8966	−0.495453
$905$	0	0
$906$	−29.6121	−0.983797
$907$	32.3136	1.07296	0.536478	−	0.843914i	$-0.319754\pi$
$907$	32.3136	1.07296	0.536478	+	0.843914i	$0.319754\pi$
$908$	2.07589	0.0688909
$909$	−11.4745	−0.380583
$910$	0	0
$911$	−37.9482	−1.25728	−0.628639	−	0.777697i	$-0.716388\pi$
$911$	−37.9482	−1.25728	−0.628639	+	0.777697i	$0.716388\pi$
$912$	63.4502	2.10105
$913$	7.29842	0.241542
$914$	33.3124	1.10188
$915$	0	0
$916$	−9.17302	−0.303085
$917$	56.8191	1.87633
$918$	−15.2925	−0.504727
$919$	−22.0659	−0.727885	−0.363943	−	0.931421i	$-0.618570\pi$
$919$	−22.0659	−0.727885	−0.363943	+	0.931421i	$0.618570\pi$
$920$	0	0
$921$	−36.2565	−1.19469
$922$	45.5606	1.50046
$923$	−14.0317	−0.461859
$924$	10.7332	0.353095
$925$	0	0
$926$	57.3529	1.88473
$927$	3.85636	0.126659
$928$	−18.9531	−0.622165
$929$	31.8097	1.04364	0.521822	−	0.853055i	$-0.325253\pi$
$929$	31.8097	1.04364	0.521822	+	0.853055i	$0.325253\pi$
$930$	0	0
$931$	54.4434	1.78431
$932$	3.63619	0.119107
$933$	26.1038	0.854600
$934$	40.5379	1.32644
$935$	0	0
$936$	−16.5355	−0.540480
$937$	26.4463	0.863963	0.431981	−	0.901883i	$-0.357815\pi$
$937$	26.4463	0.863963	0.431981	+	0.901883i	$0.357815\pi$
$938$	37.1707	1.21367
$939$	51.8984	1.69364
$940$	0	0
$941$	25.1644	0.820336	0.410168	−	0.912010i	$-0.365470\pi$
$941$	25.1644	0.820336	0.410168	+	0.912010i	$0.365470\pi$
$942$	58.1755	1.89546
$943$	6.84400	0.222871
$944$	21.1407	0.688070
$945$	0	0
$946$	−48.3947	−1.57345
$947$	27.6651	0.898995	0.449497	−	0.893282i	$-0.351603\pi$
$947$	27.6651	0.898995	0.449497	+	0.893282i	$0.351603\pi$
$948$	1.94084	0.0630354
$949$	−61.2758	−1.98910
$950$	0	0
$951$	−63.7652	−2.06773
$952$	−27.4561	−0.889857
$953$	−11.4428	−0.370669	−0.185335	−	0.982675i	$-0.559337\pi$
$953$	−11.4428	−0.370669	−0.185335	+	0.982675i	$0.559337\pi$
$954$	33.0189	1.06903
$955$	0	0
$956$	−15.1748	−0.490787
$957$	−33.8537	−1.09434
$958$	5.82511	0.188201
$959$	60.8332	1.96441
$960$	0	0
$961$	23.9117	0.771346
$962$	32.8668	1.05967
$963$	−4.91273	−0.158311
$964$	−0.524458	−0.0168916
$965$	0	0
$966$	−34.2786	−1.10290
$967$	−13.2578	−0.426343	−0.213172	−	0.977015i	$-0.568379\pi$
$967$	−13.2578	−0.426343	−0.213172	+	0.977015i	$0.568379\pi$
$968$	11.7238	0.376816
$969$	39.3453	1.26395
$970$	0	0
$971$	−5.44620	−0.174777	−0.0873884	−	0.996174i	$-0.527852\pi$
$971$	−5.44620	−0.174777	−0.0873884	+	0.996174i	$0.527852\pi$
$972$	7.34378	0.235552
$973$	78.6662	2.52192
$974$	59.6526	1.91139
$975$	0	0
$976$	−7.42578	−0.237694
$977$	−38.2354	−1.22326	−0.611629	−	0.791145i	$-0.709485\pi$
$977$	−38.2354	−1.22326	−0.611629	+	0.791145i	$0.709485\pi$
$978$	63.9642	2.04535
$979$	28.0070	0.895106
$980$	0	0
$981$	17.4040	0.555667
$982$	−58.7550	−1.87495
$983$	9.27554	0.295844	0.147922	−	0.988999i	$-0.452742\pi$
$983$	9.27554	0.295844	0.147922	+	0.988999i	$0.452742\pi$
$984$	−13.1240	−0.418378
$985$	0	0
$986$	−30.7805	−0.980252
$987$	104.271	3.31898
$988$	15.9387	0.507076
$989$	32.1097	1.02103
$990$	0	0
$991$	−51.0104	−1.62040	−0.810199	−	0.586155i	$-0.800641\pi$
$991$	−51.0104	−1.62040	−0.810199	+	0.586155i	$0.800641\pi$
$992$	21.4610	0.681387
$993$	−23.8821	−0.757874
$994$	18.2640	0.579300
$995$	0	0
$996$	−3.30052	−0.104581
$997$	27.1383	0.859479	0.429739	−	0.902953i	$-0.358605\pi$
$997$	27.1383	0.859479	0.429739	+	0.902953i	$0.358605\pi$
$998$	−38.9318	−1.23236
$999$	13.9273	0.440639

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.j.1.20		25
5.4	even	2		1205.2.a.e.1.6	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.a.e.1.6	✓	25	5.4	even	2
6025.2.a.j.1.20		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)

\(f(q)\)	\(=\)	\(q+1.58885 q^{2} -2.11200 q^{3} +0.524458 q^{4} -3.35566 q^{6} -3.95616 q^{7} -2.34442 q^{8} +1.46053 q^{9} +O(q^{10})\)	\(q+1.58885 q^{2} -2.11200 q^{3} +0.524458 q^{4} -3.35566 q^{6} -3.95616 q^{7} -2.34442 q^{8} +1.46053 q^{9} +2.44935 q^{11} -1.10765 q^{12} +4.82916 q^{13} -6.28576 q^{14} -4.77386 q^{16} -2.96026 q^{17} +2.32057 q^{18} +6.29318 q^{19} +8.35539 q^{21} +3.89165 q^{22} -2.58210 q^{23} +4.95141 q^{24} +7.67282 q^{26} +3.25135 q^{27} -2.07484 q^{28} +6.54429 q^{29} -7.41024 q^{31} -2.89612 q^{32} -5.17301 q^{33} -4.70342 q^{34} +0.765987 q^{36} +4.28353 q^{37} +9.99895 q^{38} -10.1992 q^{39} -2.65056 q^{41} +13.2755 q^{42} -12.4355 q^{43} +1.28458 q^{44} -4.10258 q^{46} +12.4795 q^{47} +10.0824 q^{48} +8.65117 q^{49} +6.25205 q^{51} +2.53269 q^{52} +14.2288 q^{53} +5.16593 q^{54} +9.27490 q^{56} -13.2912 q^{57} +10.3979 q^{58} -4.42842 q^{59} +1.55551 q^{61} -11.7738 q^{62} -5.77809 q^{63} +4.94620 q^{64} -8.21916 q^{66} -5.91349 q^{67} -1.55253 q^{68} +5.45338 q^{69} -2.90562 q^{71} -3.42410 q^{72} -12.6887 q^{73} +6.80590 q^{74} +3.30051 q^{76} -9.69000 q^{77} -16.2050 q^{78} -1.75221 q^{79} -11.2484 q^{81} -4.21135 q^{82} +2.97974 q^{83} +4.38205 q^{84} -19.7582 q^{86} -13.8215 q^{87} -5.74230 q^{88} +11.4345 q^{89} -19.1049 q^{91} -1.35420 q^{92} +15.6504 q^{93} +19.8281 q^{94} +6.11660 q^{96} -10.9038 q^{97} +13.7455 q^{98} +3.57735 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 6 q^{2} - 15 q^{3} + 32 q^{4} - q^{6} - 19 q^{7} - 15 q^{8} + 32 q^{9}+O(q^{10}) \) 25 * q - 6 * q^2 - 15 * q^3 + 32 * q^4 - q^6 - 19 * q^7 - 15 * q^8 + 32 * q^9	\( 25 q - 6 q^{2} - 15 q^{3} + 32 q^{4} - q^{6} - 19 q^{7} - 15 q^{8} + 32 q^{9} + 2 q^{11} - 20 q^{12} - 14 q^{13} - 5 q^{14} + 38 q^{16} - 7 q^{17} - 9 q^{18} + 30 q^{19} + q^{21} - q^{22} - 43 q^{23} - 6 q^{24} - 22 q^{26} - 42 q^{27} - 32 q^{28} - 4 q^{29} + 14 q^{31} - 26 q^{32} - 4 q^{33} + 7 q^{34} + 15 q^{36} - 16 q^{37} - 14 q^{38} - 21 q^{39} - q^{41} + 25 q^{42} - 35 q^{43} - 52 q^{44} - 27 q^{46} - 50 q^{47} - 26 q^{48} + 46 q^{49} - 7 q^{51} - 3 q^{52} - 4 q^{53} - 31 q^{54} - 51 q^{56} - 2 q^{58} + 6 q^{59} + 19 q^{61} - 28 q^{63} + 49 q^{64} - 27 q^{66} - 65 q^{67} + 25 q^{68} + 2 q^{69} - 34 q^{71} + 10 q^{72} - 8 q^{73} - 42 q^{74} + 71 q^{76} - q^{77} + 59 q^{78} - 12 q^{79} + 29 q^{81} - 11 q^{82} - 41 q^{83} - 10 q^{84} - 13 q^{86} - 40 q^{87} + 52 q^{88} - 24 q^{89} + 46 q^{91} - 85 q^{92} + 30 q^{93} + 14 q^{94} - 30 q^{96} - 9 q^{97} + 64 q^{98} + 2 q^{99}+O(q^{100}) \) 25 * q - 6 * q^2 - 15 * q^3 + 32 * q^4 - q^6 - 19 * q^7 - 15 * q^8 + 32 * q^9 + 2 * q^11 - 20 * q^12 - 14 * q^13 - 5 * q^14 + 38 * q^16 - 7 * q^17 - 9 * q^18 + 30 * q^19 + q^21 - q^22 - 43 * q^23 - 6 * q^24 - 22 * q^26 - 42 * q^27 - 32 * q^28 - 4 * q^29 + 14 * q^31 - 26 * q^32 - 4 * q^33 + 7 * q^34 + 15 * q^36 - 16 * q^37 - 14 * q^38 - 21 * q^39 - q^41 + 25 * q^42 - 35 * q^43 - 52 * q^44 - 27 * q^46 - 50 * q^47 - 26 * q^48 + 46 * q^49 - 7 * q^51 - 3 * q^52 - 4 * q^53 - 31 * q^54 - 51 * q^56 - 2 * q^58 + 6 * q^59 + 19 * q^61 - 28 * q^63 + 49 * q^64 - 27 * q^66 - 65 * q^67 + 25 * q^68 + 2 * q^69 - 34 * q^71 + 10 * q^72 - 8 * q^73 - 42 * q^74 + 71 * q^76 - q^77 + 59 * q^78 - 12 * q^79 + 29 * q^81 - 11 * q^82 - 41 * q^83 - 10 * q^84 - 13 * q^86 - 40 * q^87 + 52 * q^88 - 24 * q^89 + 46 * q^91 - 85 * q^92 + 30 * q^93 + 14 * q^94 - 30 * q^96 - 9 * q^97 + 64 * q^98 + 2 * q^99

Introduction

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