LMFDB - Embedded newform 6025.2.a.o.1.7

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:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
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)

$f(q)$	$=$	$q-1.77260 q^{2} -0.796316 q^{3} +1.14211 q^{4} +1.41155 q^{6} -3.30701 q^{7} +1.52069 q^{8} -2.36588 q^{9} +O(q^{10})$	\(q-1.77260 q^{2} -0.796316 q^{3} +1.14211 q^{4} +1.41155 q^{6} -3.30701 q^{7} +1.52069 q^{8} -2.36588 q^{9} +2.56471 q^{11} -0.909481 q^{12} +4.28352 q^{13} +5.86201 q^{14} -4.97980 q^{16} +3.96351 q^{17} +4.19376 q^{18} -3.96039 q^{19} +2.63342 q^{21} -4.54620 q^{22} +2.74854 q^{23} -1.21095 q^{24} -7.59297 q^{26} +4.27294 q^{27} -3.77697 q^{28} -3.80858 q^{29} +4.48219 q^{31} +5.78582 q^{32} -2.04232 q^{33} -7.02572 q^{34} -2.70210 q^{36} -1.87643 q^{37} +7.02018 q^{38} -3.41103 q^{39} +1.91969 q^{41} -4.66801 q^{42} +12.3260 q^{43} +2.92918 q^{44} -4.87207 q^{46} -1.93004 q^{47} +3.96550 q^{48} +3.93631 q^{49} -3.15621 q^{51} +4.89226 q^{52} +1.47102 q^{53} -7.57421 q^{54} -5.02895 q^{56} +3.15372 q^{57} +6.75109 q^{58} -10.0661 q^{59} -2.78683 q^{61} -7.94513 q^{62} +7.82399 q^{63} -0.296329 q^{64} +3.62021 q^{66} +4.70122 q^{67} +4.52677 q^{68} -2.18871 q^{69} +0.780100 q^{71} -3.59778 q^{72} +1.79037 q^{73} +3.32615 q^{74} -4.52320 q^{76} -8.48152 q^{77} +6.04640 q^{78} +13.9652 q^{79} +3.69504 q^{81} -3.40285 q^{82} -17.8217 q^{83} +3.00766 q^{84} -21.8491 q^{86} +3.03283 q^{87} +3.90013 q^{88} -15.7162 q^{89} -14.1656 q^{91} +3.13914 q^{92} -3.56924 q^{93} +3.42118 q^{94} -4.60734 q^{96} +9.66645 q^{97} -6.97751 q^{98} -6.06780 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9}+O(q^{10}) $ 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9	$ 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9} + q^{11} + 14 q^{12} + 9 q^{13} - q^{14} + 43 q^{16} + 12 q^{17} + 42 q^{18} + 2 q^{21} + 5 q^{22} + 77 q^{23} - 2 q^{24} + 2 q^{26} + 38 q^{27} + 42 q^{28} + 2 q^{29} + q^{31} + 72 q^{32} + 20 q^{33} + 5 q^{34} + 32 q^{36} + 28 q^{37} + 23 q^{38} - 2 q^{39} - 2 q^{41} + 37 q^{42} + 31 q^{43} + 3 q^{44} + 14 q^{46} + 96 q^{47} + 13 q^{48} + 40 q^{49} - 10 q^{51} + 42 q^{52} + 54 q^{53} + 4 q^{54} - 15 q^{56} + 37 q^{57} + 27 q^{58} + q^{59} + 5 q^{61} + 39 q^{62} + 70 q^{63} + 65 q^{64} - 52 q^{66} + 34 q^{67} + 52 q^{68} + 21 q^{69} - 9 q^{71} + 70 q^{72} + 25 q^{73} + 22 q^{74} - 47 q^{76} + 54 q^{77} + 58 q^{78} + 13 q^{79} + 12 q^{81} - 5 q^{82} + 63 q^{83} + 95 q^{84} - 18 q^{86} + 47 q^{87} + 13 q^{88} + 19 q^{89} - 31 q^{91} + 137 q^{92} + 52 q^{93} + 120 q^{94} - 49 q^{96} + 36 q^{97} + 64 q^{98} + 16 q^{99}+O(q^{100}) $ 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9 + q^11 + 14 * q^12 + 9 * q^13 - q^14 + 43 * q^16 + 12 * q^17 + 42 * q^18 + 2 * q^21 + 5 * q^22 + 77 * q^23 - 2 * q^24 + 2 * q^26 + 38 * q^27 + 42 * q^28 + 2 * q^29 + q^31 + 72 * q^32 + 20 * q^33 + 5 * q^34 + 32 * q^36 + 28 * q^37 + 23 * q^38 - 2 * q^39 - 2 * q^41 + 37 * q^42 + 31 * q^43 + 3 * q^44 + 14 * q^46 + 96 * q^47 + 13 * q^48 + 40 * q^49 - 10 * q^51 + 42 * q^52 + 54 * q^53 + 4 * q^54 - 15 * q^56 + 37 * q^57 + 27 * q^58 + q^59 + 5 * q^61 + 39 * q^62 + 70 * q^63 + 65 * q^64 - 52 * q^66 + 34 * q^67 + 52 * q^68 + 21 * q^69 - 9 * q^71 + 70 * q^72 + 25 * q^73 + 22 * q^74 - 47 * q^76 + 54 * q^77 + 58 * q^78 + 13 * q^79 + 12 * q^81 - 5 * q^82 + 63 * q^83 + 95 * q^84 - 18 * q^86 + 47 * q^87 + 13 * q^88 + 19 * q^89 - 31 * q^91 + 137 * q^92 + 52 * q^93 + 120 * q^94 - 49 * q^96 + 36 * q^97 + 64 * q^98 + 16 * 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 $

$2$	−1.77260	−1.25342	−0.626709	−	0.779254i	$-0.715598\pi$
$2$	−1.77260	−1.25342	−0.626709	+	0.779254i	$0.715598\pi$
$3$	−0.796316	−0.459753	−0.229877	−	0.973220i	$-0.573832\pi$
$3$	−0.796316	−0.459753	−0.229877	+	0.973220i	$0.573832\pi$
$4$	1.14211	0.571056
$5$	0	0
$5$	0	0
$6$	1.41155	0.576263
$7$	−3.30701	−1.24993	−0.624966	−	0.780652i	$-0.714887\pi$
$7$	−3.30701	−1.24993	−0.624966	+	0.780652i	$0.714887\pi$
$8$	1.52069	0.537646
$9$	−2.36588	−0.788627
$10$	0	0
$11$	2.56471	0.773289	0.386644	−	0.922229i	$-0.373634\pi$
$11$	2.56471	0.773289	0.386644	+	0.922229i	$0.373634\pi$
$12$	−0.909481	−0.262545
$13$	4.28352	1.18803	0.594017	−	0.804452i	$-0.297541\pi$
$13$	4.28352	1.18803	0.594017	+	0.804452i	$0.297541\pi$
$14$	5.86201	1.56669
$15$	0	0
$16$	−4.97980	−1.24495
$17$	3.96351	0.961293	0.480646	−	0.876915i	$-0.340402\pi$
$17$	3.96351	0.961293	0.480646	+	0.876915i	$0.340402\pi$
$18$	4.19376	0.988479
$19$	−3.96039	−0.908575	−0.454288	−	0.890855i	$-0.650106\pi$
$19$	−3.96039	−0.908575	−0.454288	+	0.890855i	$0.650106\pi$
$20$	0	0
$21$	2.63342	0.574660
$22$	−4.54620	−0.969254
$23$	2.74854	0.573111	0.286555	−	0.958064i	$-0.407490\pi$
$23$	2.74854	0.573111	0.286555	+	0.958064i	$0.407490\pi$
$24$	−1.21095	−0.247185
$25$	0	0
$26$	−7.59297	−1.48910
$27$	4.27294	0.822327
$28$	−3.77697	−0.713781
$29$	−3.80858	−0.707235	−0.353618	−	0.935390i	$-0.615049\pi$
$29$	−3.80858	−0.707235	−0.353618	+	0.935390i	$0.615049\pi$
$30$	0	0
$31$	4.48219	0.805025	0.402513	−	0.915414i	$-0.368137\pi$
$31$	4.48219	0.805025	0.402513	+	0.915414i	$0.368137\pi$
$32$	5.78582	1.02280
$33$	−2.04232	−0.355522
$34$	−7.02572	−1.20490
$35$	0	0
$36$	−2.70210	−0.450350
$37$	−1.87643	−0.308482	−0.154241	−	0.988033i	$-0.549293\pi$
$37$	−1.87643	−0.308482	−0.154241	+	0.988033i	$0.549293\pi$
$38$	7.02018	1.13882
$39$	−3.41103	−0.546203
$40$	0	0
$41$	1.91969	0.299806	0.149903	−	0.988701i	$-0.452104\pi$
$41$	1.91969	0.299806	0.149903	+	0.988701i	$0.452104\pi$
$42$	−4.66801	−0.720289
$43$	12.3260	1.87970	0.939848	−	0.341592i	$-0.110966\pi$
$43$	12.3260	1.87970	0.939848	+	0.341592i	$0.110966\pi$
$44$	2.92918	0.441591
$45$	0	0
$46$	−4.87207	−0.718347
$47$	−1.93004	−0.281525	−0.140762	−	0.990043i	$-0.544955\pi$
$47$	−1.93004	−0.281525	−0.140762	+	0.990043i	$0.544955\pi$
$48$	3.96550	0.572370
$49$	3.93631	0.562331
$50$	0	0
$51$	−3.15621	−0.441957
$52$	4.89226	0.678434
$53$	1.47102	0.202060	0.101030	−	0.994883i	$-0.467786\pi$
$53$	1.47102	0.202060	0.101030	+	0.994883i	$0.467786\pi$
$54$	−7.57421	−1.03072
$55$	0	0
$56$	−5.02895	−0.672021
$57$	3.15372	0.417720
$58$	6.75109	0.886461
$59$	−10.0661	−1.31049	−0.655246	−	0.755415i	$-0.727435\pi$
$59$	−10.0661	−1.31049	−0.655246	+	0.755415i	$0.727435\pi$
$60$	0	0
$61$	−2.78683	−0.356816	−0.178408	−	0.983957i	$-0.557095\pi$
$61$	−2.78683	−0.356816	−0.178408	+	0.983957i	$0.557095\pi$
$62$	−7.94513	−1.00903
$63$	7.82399	0.985730
$64$	−0.296329	−0.0370412
$65$	0	0
$66$	3.62021	0.445617
$67$	4.70122	0.574345	0.287173	−	0.957879i	$-0.407285\pi$
$67$	4.70122	0.574345	0.287173	+	0.957879i	$0.407285\pi$
$68$	4.52677	0.548952
$69$	−2.18871	−0.263489
$70$	0	0
$71$	0.780100	0.0925809	0.0462904	−	0.998928i	$-0.485260\pi$
$71$	0.780100	0.0925809	0.0462904	+	0.998928i	$0.485260\pi$
$72$	−3.59778	−0.424002
$73$	1.79037	0.209547	0.104773	−	0.994496i	$-0.466588\pi$
$73$	1.79037	0.209547	0.104773	+	0.994496i	$0.466588\pi$
$74$	3.32615	0.386657
$75$	0	0
$76$	−4.52320	−0.518847
$77$	−8.48152	−0.966558
$78$	6.04640	0.684620
$79$	13.9652	1.57121	0.785603	−	0.618730i	$-0.212353\pi$
$79$	13.9652	1.57121	0.785603	+	0.618730i	$0.212353\pi$
$80$	0	0
$81$	3.69504	0.410560
$82$	−3.40285	−0.375782
$83$	−17.8217	−1.95618	−0.978091	−	0.208178i	$-0.933247\pi$
$83$	−17.8217	−1.95618	−0.978091	+	0.208178i	$0.933247\pi$
$84$	3.00766	0.328163
$85$	0	0
$86$	−21.8491	−2.35605
$87$	3.03283	0.325154
$88$	3.90013	0.415756
$89$	−15.7162	−1.66591	−0.832955	−	0.553341i	$-0.813353\pi$
$89$	−15.7162	−1.66591	−0.832955	+	0.553341i	$0.813353\pi$
$90$	0	0
$91$	−14.1656	−1.48496
$92$	3.13914	0.327278
$93$	−3.56924	−0.370113
$94$	3.42118	0.352868
$95$	0	0
$96$	−4.60734	−0.470234
$97$	9.66645	0.981479	0.490740	−	0.871306i	$-0.336727\pi$
$97$	9.66645	0.981479	0.490740	+	0.871306i	$0.336727\pi$
$98$	−6.97751	−0.704835
$99$	−6.06780	−0.609836
$100$	0	0
$101$	2.38561	0.237377	0.118688	−	0.992932i	$-0.462131\pi$
$101$	2.38561	0.237377	0.118688	+	0.992932i	$0.462131\pi$
$102$	5.59469	0.553957
$103$	−11.8914	−1.17169	−0.585846	−	0.810422i	$-0.699238\pi$
$103$	−11.8914	−1.17169	−0.585846	+	0.810422i	$0.699238\pi$
$104$	6.51392	0.638742
$105$	0	0
$106$	−2.60753	−0.253266
$107$	3.36985	0.325775	0.162888	−	0.986645i	$-0.447919\pi$
$107$	3.36985	0.325775	0.162888	+	0.986645i	$0.447919\pi$
$108$	4.88017	0.469595
$109$	−3.39143	−0.324840	−0.162420	−	0.986722i	$-0.551930\pi$
$109$	−3.39143	−0.324840	−0.162420	+	0.986722i	$0.551930\pi$
$110$	0	0
$111$	1.49423	0.141826
$112$	16.4683	1.55610
$113$	5.25894	0.494719	0.247360	−	0.968924i	$-0.420437\pi$
$113$	5.25894	0.494719	0.247360	+	0.968924i	$0.420437\pi$
$114$	−5.59028	−0.523578
$115$	0	0
$116$	−4.34982	−0.403871
$117$	−10.1343	−0.936916
$118$	17.8431	1.64259
$119$	−13.1074	−1.20155
$120$	0	0
$121$	−4.42227	−0.402025
$122$	4.93993	0.447240
$123$	−1.52868	−0.137837
$124$	5.11916	0.459714
$125$	0	0
$126$	−13.8688	−1.23553
$127$	−6.80731	−0.604051	−0.302026	−	0.953300i	$-0.597663\pi$
$127$	−6.80731	−0.604051	−0.302026	+	0.953300i	$0.597663\pi$
$128$	−11.0464	−0.976369
$129$	−9.81539	−0.864197
$130$	0	0
$131$	−10.5564	−0.922319	−0.461159	−	0.887317i	$-0.652566\pi$
$131$	−10.5564	−0.922319	−0.461159	+	0.887317i	$0.652566\pi$
$132$	−2.33255	−0.203023
$133$	13.0970	1.13566
$134$	−8.33338	−0.719895
$135$	0	0
$136$	6.02728	0.516835
$137$	2.52007	0.215304	0.107652	−	0.994189i	$-0.465667\pi$
$137$	2.52007	0.215304	0.107652	+	0.994189i	$0.465667\pi$
$138$	3.87970	0.330262
$139$	2.95174	0.250364	0.125182	−	0.992134i	$-0.460049\pi$
$139$	2.95174	0.250364	0.125182	+	0.992134i	$0.460049\pi$
$140$	0	0
$141$	1.53692	0.129432
$142$	−1.38281	−0.116042
$143$	10.9860	0.918694
$144$	11.7816	0.981802
$145$	0	0
$146$	−3.17360	−0.262649
$147$	−3.13455	−0.258533
$148$	−2.14309	−0.176161
$149$	4.23010	0.346543	0.173272	−	0.984874i	$-0.444566\pi$
$149$	4.23010	0.346543	0.173272	+	0.984874i	$0.444566\pi$
$150$	0	0
$151$	20.7552	1.68903	0.844515	−	0.535532i	$-0.179889\pi$
$151$	20.7552	1.68903	0.844515	+	0.535532i	$0.179889\pi$
$152$	−6.02253	−0.488492
$153$	−9.37720	−0.758101
$154$	15.0343	1.21150
$155$	0	0
$156$	−3.89578	−0.311912
$157$	−22.8964	−1.82733	−0.913665	−	0.406468i	$-0.866760\pi$
$157$	−22.8964	−1.82733	−0.913665	+	0.406468i	$0.866760\pi$
$158$	−24.7547	−1.96938
$159$	−1.17140	−0.0928979
$160$	0	0
$161$	−9.08946	−0.716350
$162$	−6.54982	−0.514603
$163$	1.21699	0.0953220	0.0476610	−	0.998864i	$-0.484823\pi$
$163$	1.21699	0.0953220	0.0476610	+	0.998864i	$0.484823\pi$
$164$	2.19250	0.171206
$165$	0	0
$166$	31.5907	2.45191
$167$	9.22616	0.713942	0.356971	−	0.934115i	$-0.383809\pi$
$167$	9.22616	0.713942	0.356971	+	0.934115i	$0.383809\pi$
$168$	4.00463	0.308964
$169$	5.34855	0.411427
$170$	0	0
$171$	9.36980	0.716527
$172$	14.0777	1.07341
$173$	−8.39463	−0.638232	−0.319116	−	0.947716i	$-0.603386\pi$
$173$	−8.39463	−0.638232	−0.319116	+	0.947716i	$0.603386\pi$
$174$	−5.37600	−0.407553
$175$	0	0
$176$	−12.7717	−0.962707
$177$	8.01578	0.602503
$178$	27.8585	2.08808
$179$	16.4890	1.23244	0.616222	−	0.787572i	$-0.288662\pi$
$179$	16.4890	1.23244	0.616222	+	0.787572i	$0.288662\pi$
$180$	0	0
$181$	−13.0357	−0.968938	−0.484469	−	0.874808i	$-0.660987\pi$
$181$	−13.0357	−0.968938	−0.484469	+	0.874808i	$0.660987\pi$
$182$	25.1100	1.86128
$183$	2.21919	0.164047
$184$	4.17969	0.308131
$185$	0	0
$186$	6.32683	0.463906
$187$	10.1652	0.743357
$188$	−2.20432	−0.160766
$189$	−14.1306	−1.02785
$190$	0	0
$191$	8.38373	0.606625	0.303313	−	0.952891i	$-0.401907\pi$
$191$	8.38373	0.606625	0.303313	+	0.952891i	$0.401907\pi$
$192$	0.235972	0.0170298
$193$	13.3567	0.961433	0.480717	−	0.876876i	$-0.340377\pi$
$193$	13.3567	0.961433	0.480717	+	0.876876i	$0.340377\pi$
$194$	−17.1347	−1.23020
$195$	0	0
$196$	4.49571	0.321122
$197$	−2.57467	−0.183438	−0.0917189	−	0.995785i	$-0.529236\pi$
$197$	−2.57467	−0.183438	−0.0917189	+	0.995785i	$0.529236\pi$
$198$	10.7558	0.764380
$199$	−16.7748	−1.18914	−0.594568	−	0.804046i	$-0.702677\pi$
$199$	−16.7748	−1.18914	−0.594568	+	0.804046i	$0.702677\pi$
$200$	0	0
$201$	−3.74366	−0.264057
$202$	−4.22873	−0.297532
$203$	12.5950	0.883996
$204$	−3.60474	−0.252382
$205$	0	0
$206$	21.0787	1.46862
$207$	−6.50273	−0.451971
$208$	−21.3311	−1.47905
$209$	−10.1572	−0.702591
$210$	0	0
$211$	−24.2834	−1.67174	−0.835868	−	0.548930i	$-0.815035\pi$
$211$	−24.2834	−1.67174	−0.835868	+	0.548930i	$0.815035\pi$
$212$	1.68007	0.115388
$213$	−0.621206	−0.0425643
$214$	−5.97339	−0.408332
$215$	0	0
$216$	6.49782	0.442121
$217$	−14.8226	−1.00623
$218$	6.01164	0.407160
$219$	−1.42570	−0.0963397
$220$	0	0
$221$	16.9778	1.14205
$222$	−2.64867	−0.177767
$223$	−4.98461	−0.333794	−0.166897	−	0.985974i	$-0.553375\pi$
$223$	−4.98461	−0.333794	−0.166897	+	0.985974i	$0.553375\pi$
$224$	−19.1337	−1.27843
$225$	0	0
$226$	−9.32199	−0.620090
$227$	7.73727	0.513541	0.256770	−	0.966472i	$-0.417342\pi$
$227$	7.73727	0.513541	0.256770	+	0.966472i	$0.417342\pi$
$228$	3.60190	0.238542
$229$	20.0085	1.32220	0.661099	−	0.750299i	$-0.270090\pi$
$229$	20.0085	1.32220	0.661099	+	0.750299i	$0.270090\pi$
$230$	0	0
$231$	6.75396	0.444378
$232$	−5.79168	−0.380242
$233$	17.0679	1.11815	0.559077	−	0.829116i	$-0.311156\pi$
$233$	17.0679	1.11815	0.559077	+	0.829116i	$0.311156\pi$
$234$	17.9641	1.17435
$235$	0	0
$236$	−11.4966	−0.748364
$237$	−11.1207	−0.722367
$238$	23.2341	1.50604
$239$	27.0637	1.75061	0.875303	−	0.483574i	$-0.160662\pi$
$239$	27.0637	1.75061	0.875303	+	0.483574i	$0.160662\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	7.83892	0.503905
$243$	−15.7612	−1.01108
$244$	−3.18287	−0.203762
$245$	0	0
$246$	2.70974	0.172767
$247$	−16.9644	−1.07942
$248$	6.81604	0.432819
$249$	14.1917	0.899361
$250$	0	0
$251$	20.6116	1.30099	0.650496	−	0.759510i	$-0.274561\pi$
$251$	20.6116	1.30099	0.650496	+	0.759510i	$0.274561\pi$
$252$	8.93587	0.562907
$253$	7.04921	0.443180
$254$	12.0666	0.757128
$255$	0	0
$256$	20.1734	1.26084
$257$	12.3516	0.770474	0.385237	−	0.922818i	$-0.374120\pi$
$257$	12.3516	0.770474	0.385237	+	0.922818i	$0.374120\pi$
$258$	17.3988	1.08320
$259$	6.20536	0.385582
$260$	0	0
$261$	9.01064	0.557745
$262$	18.7123	1.15605
$263$	24.1294	1.48788	0.743940	−	0.668247i	$-0.232955\pi$
$263$	24.1294	1.48788	0.743940	+	0.668247i	$0.232955\pi$
$264$	−3.10574	−0.191145
$265$	0	0
$266$	−23.2158	−1.42345
$267$	12.5150	0.765907
$268$	5.36932	0.327983
$269$	1.62567	0.0991190	0.0495595	−	0.998771i	$-0.484218\pi$
$269$	1.62567	0.0991190	0.0495595	+	0.998771i	$0.484218\pi$
$270$	0	0
$271$	−23.3981	−1.42133	−0.710667	−	0.703529i	$-0.751607\pi$
$271$	−23.3981	−1.42133	−0.710667	+	0.703529i	$0.751607\pi$
$272$	−19.7375	−1.19676
$273$	11.2803	0.682716
$274$	−4.46708	−0.269866
$275$	0	0
$276$	−2.49975	−0.150467
$277$	17.8453	1.07222	0.536111	−	0.844147i	$-0.319893\pi$
$277$	17.8453	1.07222	0.536111	+	0.844147i	$0.319893\pi$
$278$	−5.23226	−0.313810
$279$	−10.6043	−0.634865
$280$	0	0
$281$	−23.3034	−1.39016	−0.695082	−	0.718930i	$-0.744632\pi$
$281$	−23.3034	−1.39016	−0.695082	+	0.718930i	$0.744632\pi$
$282$	−2.72434	−0.162232
$283$	−18.9223	−1.12481	−0.562407	−	0.826860i	$-0.690125\pi$
$283$	−18.9223	−1.12481	−0.562407	+	0.826860i	$0.690125\pi$
$284$	0.890961	0.0528688
$285$	0	0
$286$	−19.4738	−1.15151
$287$	−6.34845	−0.374737
$288$	−13.6886	−0.806606
$289$	−1.29058	−0.0759165
$290$	0	0
$291$	−7.69755	−0.451238
$292$	2.04480	0.119663
$293$	2.94041	0.171781	0.0858904	−	0.996305i	$-0.472626\pi$
$293$	2.94041	0.171781	0.0858904	+	0.996305i	$0.472626\pi$
$294$	5.55630	0.324050
$295$	0	0
$296$	−2.85347	−0.165854
$297$	10.9588	0.635896
$298$	−7.49828	−0.434364
$299$	11.7734	0.680876
$300$	0	0
$301$	−40.7622	−2.34949
$302$	−36.7906	−2.11706
$303$	−1.89970	−0.109135
$304$	19.7220	1.13113
$305$	0	0
$306$	16.6220	0.950218
$307$	2.51734	0.143672	0.0718361	−	0.997416i	$-0.477114\pi$
$307$	2.51734	0.143672	0.0718361	+	0.997416i	$0.477114\pi$
$308$	−9.68684	−0.551959
$309$	9.46929	0.538689
$310$	0	0
$311$	−19.4246	−1.10147	−0.550734	−	0.834681i	$-0.685652\pi$
$311$	−19.4246	−1.10147	−0.550734	+	0.834681i	$0.685652\pi$
$312$	−5.18714	−0.293664
$313$	2.12047	0.119856	0.0599279	−	0.998203i	$-0.480913\pi$
$313$	2.12047	0.119856	0.0599279	+	0.998203i	$0.480913\pi$
$314$	40.5861	2.29041
$315$	0	0
$316$	15.9498	0.897247
$317$	−10.1140	−0.568060	−0.284030	−	0.958815i	$-0.591672\pi$
$317$	−10.1140	−0.568060	−0.284030	+	0.958815i	$0.591672\pi$
$318$	2.07642	0.116440
$319$	−9.76789	−0.546897
$320$	0	0
$321$	−2.68346	−0.149776
$322$	16.1120	0.897885
$323$	−15.6970	−0.873406
$324$	4.22015	0.234453
$325$	0	0
$326$	−2.15724	−0.119478
$327$	2.70065	0.149346
$328$	2.91927	0.161189
$329$	6.38265	0.351887
$330$	0	0
$331$	3.08883	0.169778	0.0848889	−	0.996390i	$-0.472946\pi$
$331$	3.08883	0.169778	0.0848889	+	0.996390i	$0.472946\pi$
$332$	−20.3543	−1.11709
$333$	4.43940	0.243278
$334$	−16.3543	−0.894868
$335$	0	0
$336$	−13.1139	−0.715424
$337$	6.76875	0.368718	0.184359	−	0.982859i	$-0.440979\pi$
$337$	6.76875	0.368718	0.184359	+	0.982859i	$0.440979\pi$
$338$	−9.48083	−0.515689
$339$	−4.18777	−0.227449
$340$	0	0
$341$	11.4955	0.622517
$342$	−16.6089	−0.898107
$343$	10.1316	0.547057
$344$	18.7441	1.01061
$345$	0	0
$346$	14.8803	0.799971
$347$	−8.73326	−0.468826	−0.234413	−	0.972137i	$-0.575317\pi$
$347$	−8.73326	−0.468826	−0.234413	+	0.972137i	$0.575317\pi$
$348$	3.46383	0.185681
$349$	−23.4652	−1.25606	−0.628031	−	0.778188i	$-0.716139\pi$
$349$	−23.4652	−1.25606	−0.628031	+	0.778188i	$0.716139\pi$
$350$	0	0
$351$	18.3032	0.976953
$352$	14.8389	0.790918
$353$	19.2180	1.02287	0.511436	−	0.859322i	$-0.329114\pi$
$353$	19.2180	1.02287	0.511436	+	0.859322i	$0.329114\pi$
$354$	−14.2088	−0.755188
$355$	0	0
$356$	−17.9496	−0.951328
$357$	10.4376	0.552417
$358$	−29.2284	−1.54477
$359$	8.87741	0.468532	0.234266	−	0.972173i	$-0.424731\pi$
$359$	8.87741	0.468532	0.234266	+	0.972173i	$0.424731\pi$
$360$	0	0
$361$	−3.31534	−0.174491
$362$	23.1071	1.21448
$363$	3.52152	0.184832
$364$	−16.1787	−0.847997
$365$	0	0
$366$	−3.93374	−0.205620
$367$	−11.8767	−0.619960	−0.309980	−	0.950743i	$-0.600322\pi$
$367$	−11.8767	−0.619960	−0.309980	+	0.950743i	$0.600322\pi$
$368$	−13.6872	−0.713495
$369$	−4.54177	−0.236435
$370$	0	0
$371$	−4.86468	−0.252562
$372$	−4.07647	−0.211355
$373$	−21.2968	−1.10270	−0.551352	−	0.834273i	$-0.685888\pi$
$373$	−21.2968	−1.10270	−0.551352	+	0.834273i	$0.685888\pi$
$374$	−18.0189	−0.931736
$375$	0	0
$376$	−2.93499	−0.151361
$377$	−16.3141	−0.840220
$378$	25.0480	1.28833
$379$	29.0320	1.49127	0.745636	−	0.666354i	$-0.232146\pi$
$379$	29.0320	1.49127	0.745636	+	0.666354i	$0.232146\pi$
$380$	0	0
$381$	5.42077	0.277714
$382$	−14.8610	−0.760355
$383$	21.6635	1.10695	0.553477	−	0.832864i	$-0.313301\pi$
$383$	21.6635	1.10695	0.553477	+	0.832864i	$0.313301\pi$
$384$	8.79639	0.448889
$385$	0	0
$386$	−23.6760	−1.20508
$387$	−29.1618	−1.48238
$388$	11.0402	0.560479
$389$	−22.8807	−1.16010	−0.580049	−	0.814582i	$-0.696967\pi$
$389$	−22.8807	−1.16010	−0.580049	+	0.814582i	$0.696967\pi$
$390$	0	0
$391$	10.8939	0.550927
$392$	5.98593	0.302335
$393$	8.40625	0.424039
$394$	4.56386	0.229924
$395$	0	0
$396$	−6.93010	−0.348251
$397$	2.87432	0.144258	0.0721289	−	0.997395i	$-0.477021\pi$
$397$	2.87432	0.144258	0.0721289	+	0.997395i	$0.477021\pi$
$398$	29.7350	1.49048
$399$	−10.4294	−0.522122
$400$	0	0
$401$	−26.7749	−1.33707	−0.668537	−	0.743678i	$-0.733079\pi$
$401$	−26.7749	−1.33707	−0.668537	+	0.743678i	$0.733079\pi$
$402$	6.63600	0.330974
$403$	19.1995	0.956398
$404$	2.72463	0.135555
$405$	0	0
$406$	−22.3259	−1.10802
$407$	−4.81248	−0.238546
$408$	−4.79962	−0.237617
$409$	0.0486720	0.00240668	0.00120334	−	0.999999i	$-0.499617\pi$
$409$	0.0486720	0.00240668	0.00120334	+	0.999999i	$0.499617\pi$
$410$	0	0
$411$	−2.00677	−0.0989867
$412$	−13.5813	−0.669102
$413$	33.2886	1.63803
$414$	11.5267	0.566508
$415$	0	0
$416$	24.7837	1.21512
$417$	−2.35052	−0.115105
$418$	18.0047	0.880640
$419$	13.0598	0.638013	0.319007	−	0.947752i	$-0.396651\pi$
$419$	13.0598	0.638013	0.319007	+	0.947752i	$0.396651\pi$
$420$	0	0
$421$	14.5716	0.710178	0.355089	−	0.934833i	$-0.384451\pi$
$421$	14.5716	0.710178	0.355089	+	0.934833i	$0.384451\pi$
$422$	43.0447	2.09538
$423$	4.56624	0.222018
$424$	2.23697	0.108637
$425$	0	0
$426$	1.10115	0.0533509
$427$	9.21606	0.445996
$428$	3.84874	0.186036
$429$	−8.74831	−0.422372
$430$	0	0
$431$	19.6270	0.945401	0.472701	−	0.881223i	$-0.343279\pi$
$431$	19.6270	0.945401	0.472701	+	0.881223i	$0.343279\pi$
$432$	−21.2784	−1.02376
$433$	19.0439	0.915191	0.457595	−	0.889161i	$-0.348711\pi$
$433$	19.0439	0.915191	0.457595	+	0.889161i	$0.348711\pi$
$434$	26.2746	1.26122
$435$	0	0
$436$	−3.87339	−0.185502
$437$	−10.8853	−0.520714
$438$	2.52719	0.120754
$439$	15.7621	0.752284	0.376142	−	0.926562i	$-0.377251\pi$
$439$	15.7621	0.752284	0.376142	+	0.926562i	$0.377251\pi$
$440$	0	0
$441$	−9.31285	−0.443469
$442$	−30.0948	−1.43146
$443$	36.0763	1.71404	0.857018	−	0.515286i	$-0.172314\pi$
$443$	36.0763	1.71404	0.857018	+	0.515286i	$0.172314\pi$
$444$	1.70657	0.0809904
$445$	0	0
$446$	8.83571	0.418383
$447$	−3.36850	−0.159324
$448$	0.979964	0.0462990
$449$	−14.5290	−0.685668	−0.342834	−	0.939396i	$-0.611387\pi$
$449$	−14.5290	−0.685668	−0.342834	+	0.939396i	$0.611387\pi$
$450$	0	0
$451$	4.92346	0.231836
$452$	6.00629	0.282512
$453$	−16.5277	−0.776537
$454$	−13.7151	−0.643681
$455$	0	0
$456$	4.79584	0.224586
$457$	21.0002	0.982349	0.491175	−	0.871061i	$-0.336568\pi$
$457$	21.0002	0.982349	0.491175	+	0.871061i	$0.336568\pi$
$458$	−35.4670	−1.65727
$459$	16.9358	0.790497
$460$	0	0
$461$	5.66457	0.263825	0.131913	−	0.991261i	$-0.457888\pi$
$461$	5.66457	0.263825	0.131913	+	0.991261i	$0.457888\pi$
$462$	−11.9721	−0.556992
$463$	4.18732	0.194601	0.0973007	−	0.995255i	$-0.468979\pi$
$463$	4.18732	0.194601	0.0973007	+	0.995255i	$0.468979\pi$
$464$	18.9660	0.880473
$465$	0	0
$466$	−30.2545	−1.40151
$467$	−17.1363	−0.792973	−0.396486	−	0.918041i	$-0.629771\pi$
$467$	−17.1363	−0.792973	−0.396486	+	0.918041i	$0.629771\pi$
$468$	−11.5745	−0.535031
$469$	−15.5470	−0.717893
$470$	0	0
$471$	18.2327	0.840121
$472$	−15.3074	−0.704582
$473$	31.6126	1.45355
$474$	19.7126	0.905428
$475$	0	0
$476$	−14.9701	−0.686152
$477$	−3.48026	−0.159350
$478$	−47.9732	−2.19424
$479$	41.9817	1.91819	0.959095	−	0.283083i	$-0.0913571\pi$
$479$	41.9817	1.91819	0.959095	+	0.283083i	$0.0913571\pi$
$480$	0	0
$481$	−8.03770	−0.366488
$482$	1.77260	0.0807397
$483$	7.23808	0.329344
$484$	−5.05073	−0.229578
$485$	0	0
$486$	27.9383	1.26731
$487$	10.7782	0.488408	0.244204	−	0.969724i	$-0.421473\pi$
$487$	10.7782	0.488408	0.244204	+	0.969724i	$0.421473\pi$
$488$	−4.23791	−0.191841
$489$	−0.969109	−0.0438246
$490$	0	0
$491$	38.4407	1.73481	0.867403	−	0.497607i	$-0.165788\pi$
$491$	38.4407	1.73481	0.867403	+	0.497607i	$0.165788\pi$
$492$	−1.74593	−0.0787124
$493$	−15.0953	−0.679860
$494$	30.0711	1.35296
$495$	0	0
$496$	−22.3204	−1.00222
$497$	−2.57980	−0.115720
$498$	−25.1562	−1.12727
$499$	−3.75040	−0.167891	−0.0839455	−	0.996470i	$-0.526752\pi$
$499$	−3.75040	−0.167891	−0.0839455	+	0.996470i	$0.526752\pi$
$500$	0	0
$501$	−7.34694	−0.328237
$502$	−36.5361	−1.63069
$503$	−3.19744	−0.142567	−0.0712833	−	0.997456i	$-0.522709\pi$
$503$	−3.19744	−0.142567	−0.0712833	+	0.997456i	$0.522709\pi$
$504$	11.8979	0.529974
$505$	0	0
$506$	−12.4954	−0.555490
$507$	−4.25913	−0.189155
$508$	−7.77471	−0.344947
$509$	8.05962	0.357236	0.178618	−	0.983918i	$-0.442837\pi$
$509$	8.05962	0.357236	0.178618	+	0.983918i	$0.442837\pi$
$510$	0	0
$511$	−5.92076	−0.261919
$512$	−13.6667	−0.603989
$513$	−16.9225	−0.747146
$514$	−21.8945	−0.965725
$515$	0	0
$516$	−11.2103	−0.493504
$517$	−4.94998	−0.217700
$518$	−10.9996	−0.483295
$519$	6.68478	0.293429
$520$	0	0
$521$	10.2415	0.448687	0.224344	−	0.974510i	$-0.427976\pi$
$521$	10.2415	0.448687	0.224344	+	0.974510i	$0.427976\pi$
$522$	−15.9723	−0.699087
$523$	13.5351	0.591849	0.295924	−	0.955211i	$-0.404372\pi$
$523$	13.5351	0.591849	0.295924	+	0.955211i	$0.404372\pi$
$524$	−12.0566	−0.526696
$525$	0	0
$526$	−42.7717	−1.86493
$527$	17.7652	0.773865
$528$	10.1703	0.442607
$529$	−15.4455	−0.671544
$530$	0	0
$531$	23.8152	1.03349
$532$	14.9583	0.648524
$533$	8.22305	0.356180
$534$	−22.1841	−0.960002
$535$	0	0
$536$	7.14911	0.308795
$537$	−13.1304	−0.566620
$538$	−2.88167	−0.124238
$539$	10.0955	0.434844
$540$	0	0
$541$	−14.9257	−0.641707	−0.320854	−	0.947129i	$-0.603970\pi$
$541$	−14.9257	−0.641707	−0.320854	+	0.947129i	$0.603970\pi$
$542$	41.4755	1.78152
$543$	10.3806	0.445472
$544$	22.9321	0.983208
$545$	0	0
$546$	−19.9955	−0.855729
$547$	36.3057	1.55232	0.776159	−	0.630537i	$-0.217165\pi$
$547$	36.3057	1.55232	0.776159	+	0.630537i	$0.217165\pi$
$548$	2.87820	0.122951
$549$	6.59330	0.281395
$550$	0	0
$551$	15.0834	0.642576
$552$	−3.32835	−0.141664
$553$	−46.1830	−1.96390
$554$	−31.6327	−1.34394
$555$	0	0
$556$	3.37122	0.142972
$557$	31.2471	1.32398	0.661991	−	0.749512i	$-0.269712\pi$
$557$	31.2471	1.32398	0.661991	+	0.749512i	$0.269712\pi$
$558$	18.7972	0.795750
$559$	52.7987	2.23315
$560$	0	0
$561$	−8.09475	−0.341761
$562$	41.3076	1.74246
$563$	−9.19266	−0.387425	−0.193712	−	0.981058i	$-0.562053\pi$
$563$	−9.19266	−0.387425	−0.193712	+	0.981058i	$0.562053\pi$
$564$	1.75533	0.0739128
$565$	0	0
$566$	33.5417	1.40986
$567$	−12.2195	−0.513172
$568$	1.18629	0.0497758
$569$	13.0870	0.548637	0.274318	−	0.961639i	$-0.411548\pi$
$569$	13.0870	0.548637	0.274318	+	0.961639i	$0.411548\pi$
$570$	0	0
$571$	20.6382	0.863680	0.431840	−	0.901950i	$-0.357864\pi$
$571$	20.6382	0.863680	0.431840	+	0.901950i	$0.357864\pi$
$572$	12.5472	0.524625
$573$	−6.67610	−0.278898
$574$	11.2533	0.469702
$575$	0	0
$576$	0.701080	0.0292117
$577$	28.3730	1.18118	0.590591	−	0.806971i	$-0.298895\pi$
$577$	28.3730	1.18118	0.590591	+	0.806971i	$0.298895\pi$
$578$	2.28768	0.0951551
$579$	−10.6361	−0.442022
$580$	0	0
$581$	58.9364	2.44509
$582$	13.6447	0.565590
$583$	3.77274	0.156251
$584$	2.72260	0.112662
$585$	0	0
$586$	−5.21218	−0.215313
$587$	−39.4096	−1.62661	−0.813304	−	0.581840i	$-0.802333\pi$
$587$	−39.4096	−1.62661	−0.813304	+	0.581840i	$0.802333\pi$
$588$	−3.58000	−0.147637
$589$	−17.7512	−0.731426
$590$	0	0
$591$	2.05025	0.0843361
$592$	9.34423	0.384045
$593$	−24.4717	−1.00493	−0.502467	−	0.864597i	$-0.667574\pi$
$593$	−24.4717	−1.00493	−0.502467	+	0.864597i	$0.667574\pi$
$594$	−19.4256	−0.797043
$595$	0	0
$596$	4.83125	0.197896
$597$	13.3581	0.546709
$598$	−20.8696	−0.853421
$599$	3.63096	0.148357	0.0741785	−	0.997245i	$-0.476367\pi$
$599$	3.63096	0.148357	0.0741785	+	0.997245i	$0.476367\pi$
$600$	0	0
$601$	27.1033	1.10557	0.552783	−	0.833326i	$-0.313566\pi$
$601$	27.1033	1.10557	0.552783	+	0.833326i	$0.313566\pi$
$602$	72.2551	2.94490
$603$	−11.1225	−0.452944
$604$	23.7047	0.964531
$605$	0	0
$606$	3.36740	0.136791
$607$	−18.7909	−0.762699	−0.381350	−	0.924431i	$-0.624541\pi$
$607$	−18.7909	−0.762699	−0.381350	+	0.924431i	$0.624541\pi$
$608$	−22.9141	−0.929288
$609$	−10.0296	−0.406420
$610$	0	0
$611$	−8.26735	−0.334461
$612$	−10.7098	−0.432918
$613$	13.8395	0.558971	0.279486	−	0.960150i	$-0.409836\pi$
$613$	13.8395	0.558971	0.279486	+	0.960150i	$0.409836\pi$
$614$	−4.46224	−0.180081
$615$	0	0
$616$	−12.8978	−0.519667
$617$	29.9600	1.20614	0.603072	−	0.797687i	$-0.293943\pi$
$617$	29.9600	1.20614	0.603072	+	0.797687i	$0.293943\pi$
$618$	−16.7853	−0.675203
$619$	33.8918	1.36223	0.681114	−	0.732177i	$-0.261496\pi$
$619$	33.8918	1.36223	0.681114	+	0.732177i	$0.261496\pi$
$620$	0	0
$621$	11.7443	0.471284
$622$	34.4320	1.38060
$623$	51.9735	2.08227
$624$	16.9863	0.679996
$625$	0	0
$626$	−3.75874	−0.150229
$627$	8.08837	0.323018
$628$	−26.1502	−1.04351
$629$	−7.43723	−0.296542
$630$	0	0
$631$	27.8266	1.10776	0.553879	−	0.832597i	$-0.313147\pi$
$631$	27.8266	1.10776	0.553879	+	0.832597i	$0.313147\pi$
$632$	21.2368	0.844753
$633$	19.3372	0.768586
$634$	17.9281	0.712017
$635$	0	0
$636$	−1.33787	−0.0530499
$637$	16.8613	0.668068
$638$	17.3146	0.685490
$639$	−1.84562	−0.0730118
$640$	0	0
$641$	−19.2173	−0.759036	−0.379518	−	0.925184i	$-0.623910\pi$
$641$	−19.2173	−0.759036	−0.379518	+	0.925184i	$0.623910\pi$
$642$	4.75670	0.187732
$643$	−13.2242	−0.521510	−0.260755	−	0.965405i	$-0.583971\pi$
$643$	−13.2242	−0.521510	−0.260755	+	0.965405i	$0.583971\pi$
$644$	−10.3812	−0.409076
$645$	0	0
$646$	27.8246	1.09474
$647$	37.5450	1.47605	0.738023	−	0.674775i	$-0.235759\pi$
$647$	37.5450	1.47605	0.738023	+	0.674775i	$0.235759\pi$
$648$	5.61902	0.220736
$649$	−25.8166	−1.01339
$650$	0	0
$651$	11.8035	0.462616
$652$	1.38994	0.0544342
$653$	−9.94796	−0.389294	−0.194647	−	0.980873i	$-0.562356\pi$
$653$	−9.94796	−0.389294	−0.194647	+	0.980873i	$0.562356\pi$
$654$	−4.78716	−0.187193
$655$	0	0
$656$	−9.55970	−0.373244
$657$	−4.23580	−0.165254
$658$	−11.3139	−0.441061
$659$	10.0094	0.389912	0.194956	−	0.980812i	$-0.437544\pi$
$659$	10.0094	0.389912	0.194956	+	0.980812i	$0.437544\pi$
$660$	0	0
$661$	24.5012	0.952987	0.476493	−	0.879178i	$-0.341908\pi$
$661$	24.5012	0.952987	0.476493	+	0.879178i	$0.341908\pi$
$662$	−5.47527	−0.212802
$663$	−13.5197	−0.525061
$664$	−27.1013	−1.05173
$665$	0	0
$666$	−7.86928	−0.304928
$667$	−10.4680	−0.405324
$668$	10.5373	0.407701
$669$	3.96932	0.153463
$670$	0	0
$671$	−7.14739	−0.275922
$672$	15.2365	0.587761
$673$	−19.3677	−0.746571	−0.373286	−	0.927716i	$-0.621769\pi$
$673$	−19.3677	−0.746571	−0.373286	+	0.927716i	$0.621769\pi$
$674$	−11.9983	−0.462157
$675$	0	0
$676$	6.10863	0.234947
$677$	14.7273	0.566015	0.283008	−	0.959118i	$-0.408668\pi$
$677$	14.7273	0.566015	0.283008	+	0.959118i	$0.408668\pi$
$678$	7.42325	0.285088
$679$	−31.9670	−1.22678
$680$	0	0
$681$	−6.16131	−0.236102
$682$	−20.3769	−0.780273
$683$	−8.71906	−0.333626	−0.166813	−	0.985989i	$-0.553348\pi$
$683$	−8.71906	−0.333626	−0.166813	+	0.985989i	$0.553348\pi$
$684$	10.7014	0.409177
$685$	0	0
$686$	−17.9593	−0.685691
$687$	−15.9331	−0.607884
$688$	−61.3811	−2.34013
$689$	6.30115	0.240055
$690$	0	0
$691$	43.2710	1.64611	0.823053	−	0.567964i	$-0.192269\pi$
$691$	43.2710	1.64611	0.823053	+	0.567964i	$0.192269\pi$
$692$	−9.58760	−0.364466
$693$	20.0663	0.762254
$694$	15.4806	0.587635
$695$	0	0
$696$	4.61201	0.174818
$697$	7.60873	0.288201
$698$	41.5944	1.57437
$699$	−13.5914	−0.514075
$700$	0	0
$701$	27.7127	1.04670	0.523348	−	0.852119i	$-0.324683\pi$
$701$	27.7127	1.04670	0.523348	+	0.852119i	$0.324683\pi$
$702$	−32.4443	−1.22453
$703$	7.43137	0.280279
$704$	−0.759999	−0.0286435
$705$	0	0
$706$	−34.0658	−1.28208
$707$	−7.88923	−0.296705
$708$	9.15492	0.344063
$709$	−20.9050	−0.785103	−0.392552	−	0.919730i	$-0.628408\pi$
$709$	−20.9050	−0.785103	−0.392552	+	0.919730i	$0.628408\pi$
$710$	0	0
$711$	−33.0400	−1.23910
$712$	−23.8995	−0.895670
$713$	12.3195	0.461369
$714$	−18.5017	−0.692409
$715$	0	0
$716$	18.8323	0.703794
$717$	−21.5513	−0.804847
$718$	−15.7361	−0.587266
$719$	−14.5646	−0.543168	−0.271584	−	0.962415i	$-0.587547\pi$
$719$	−14.5646	−0.543168	−0.271584	+	0.962415i	$0.587547\pi$
$720$	0	0
$721$	39.3249	1.46454
$722$	5.87677	0.218711
$723$	0.796316	0.0296153
$724$	−14.8883	−0.553318
$725$	0	0
$726$	−6.24225	−0.231672
$727$	36.4381	1.35141	0.675707	−	0.737170i	$-0.263838\pi$
$727$	36.4381	1.35141	0.675707	+	0.737170i	$0.263838\pi$
$728$	−21.5416	−0.798385
$729$	1.46580	0.0542888
$730$	0	0
$731$	48.8542	1.80694
$732$	2.53457	0.0936802
$733$	−1.57042	−0.0580046	−0.0290023	−	0.999579i	$-0.509233\pi$
$733$	−1.57042	−0.0580046	−0.0290023	+	0.999579i	$0.509233\pi$
$734$	21.0527	0.777069
$735$	0	0
$736$	15.9026	0.586176
$737$	12.0573	0.444135
$738$	8.05074	0.296352
$739$	40.9054	1.50473	0.752366	−	0.658746i	$-0.228913\pi$
$739$	40.9054	1.50473	0.752366	+	0.658746i	$0.228913\pi$
$740$	0	0
$741$	13.5090	0.496266
$742$	8.62314	0.316565
$743$	35.2420	1.29290	0.646452	−	0.762955i	$-0.276252\pi$
$743$	35.2420	1.29290	0.646452	+	0.762955i	$0.276252\pi$
$744$	−5.42772	−0.198990
$745$	0	0
$746$	37.7506	1.38215
$747$	42.1639	1.54270
$748$	11.6098	0.424498
$749$	−11.1441	−0.407197
$750$	0	0
$751$	44.3013	1.61658	0.808288	−	0.588787i	$-0.200394\pi$
$751$	44.3013	1.61658	0.808288	+	0.588787i	$0.200394\pi$
$752$	9.61120	0.350485
$753$	−16.4133	−0.598135
$754$	28.9184	1.05315
$755$	0	0
$756$	−16.1388	−0.586961
$757$	26.2395	0.953691	0.476846	−	0.878987i	$-0.341780\pi$
$757$	26.2395	0.953691	0.476846	+	0.878987i	$0.341780\pi$
$758$	−51.4621	−1.86919
$759$	−5.61340	−0.203753
$760$	0	0
$761$	−30.5909	−1.10892	−0.554459	−	0.832211i	$-0.687075\pi$
$761$	−30.5909	−1.10892	−0.554459	+	0.832211i	$0.687075\pi$
$762$	−9.60886	−0.348092
$763$	11.2155	0.406027
$764$	9.57515	0.346417
$765$	0	0
$766$	−38.4008	−1.38748
$767$	−43.1183	−1.55691
$768$	−16.0644	−0.579675
$769$	19.3812	0.698904	0.349452	−	0.936954i	$-0.386368\pi$
$769$	19.3812	0.698904	0.349452	+	0.936954i	$0.386368\pi$
$770$	0	0
$771$	−9.83580	−0.354228
$772$	15.2548	0.549032
$773$	38.3721	1.38015	0.690074	−	0.723739i	$-0.257578\pi$
$773$	38.3721	1.38015	0.690074	+	0.723739i	$0.257578\pi$
$774$	51.6923	1.85804
$775$	0	0
$776$	14.6997	0.527689
$777$	−4.94142	−0.177273
$778$	40.5583	1.45409
$779$	−7.60273	−0.272396
$780$	0	0
$781$	2.00073	0.0715917
$782$	−19.3105	−0.690542
$783$	−16.2738	−0.581579
$784$	−19.6021	−0.700074
$785$	0	0
$786$	−14.9009	−0.531498
$787$	7.53832	0.268712	0.134356	−	0.990933i	$-0.457103\pi$
$787$	7.53832	0.268712	0.134356	+	0.990933i	$0.457103\pi$
$788$	−2.94056	−0.104753
$789$	−19.2146	−0.684057
$790$	0	0
$791$	−17.3914	−0.618365
$792$	−9.22726	−0.327876
$793$	−11.9374	−0.423910
$794$	−5.09501	−0.180815
$795$	0	0
$796$	−19.1587	−0.679063
$797$	−34.1050	−1.20806	−0.604031	−	0.796961i	$-0.706440\pi$
$797$	−34.1050	−1.20806	−0.604031	+	0.796961i	$0.706440\pi$
$798$	18.4871	0.654437
$799$	−7.64972	−0.270628
$800$	0	0
$801$	37.1826	1.31378
$802$	47.4612	1.67591
$803$	4.59177	0.162040
$804$	−4.27567	−0.150791
$805$	0	0
$806$	−34.0331	−1.19877
$807$	−1.29455	−0.0455703
$808$	3.62778	0.127625
$809$	−26.3506	−0.926437	−0.463219	−	0.886244i	$-0.653306\pi$
$809$	−26.3506	−0.926437	−0.463219	+	0.886244i	$0.653306\pi$
$810$	0	0
$811$	−39.0591	−1.37155	−0.685776	−	0.727813i	$-0.740537\pi$
$811$	−39.0591	−1.37155	−0.685776	+	0.727813i	$0.740537\pi$
$812$	14.3849	0.504811
$813$	18.6323	0.653463
$814$	8.53061	0.298998
$815$	0	0
$816$	15.7173	0.550215
$817$	−48.8157	−1.70785
$818$	−0.0862760	−0.00301657
$819$	33.5142	1.17108
$820$	0	0
$821$	37.9337	1.32390	0.661948	−	0.749550i	$-0.269730\pi$
$821$	37.9337	1.32390	0.661948	+	0.749550i	$0.269730\pi$
$822$	3.55720	0.124072
$823$	−51.0178	−1.77837	−0.889184	−	0.457549i	$-0.848727\pi$
$823$	−51.0178	−1.77837	−0.889184	+	0.457549i	$0.848727\pi$
$824$	−18.0831	−0.629956
$825$	0	0
$826$	−59.0075	−2.05313
$827$	38.3128	1.33227	0.666133	−	0.745833i	$-0.267948\pi$
$827$	38.3128	1.33227	0.666133	+	0.745833i	$0.267948\pi$
$828$	−7.42684	−0.258100
$829$	9.38935	0.326106	0.163053	−	0.986617i	$-0.447866\pi$
$829$	9.38935	0.326106	0.163053	+	0.986617i	$0.447866\pi$
$830$	0	0
$831$	−14.2105	−0.492958
$832$	−1.26933	−0.0440062
$833$	15.6016	0.540564
$834$	4.16653	0.144275
$835$	0	0
$836$	−11.6007	−0.401219
$837$	19.1521	0.661994
$838$	−23.1498	−0.799697
$839$	−8.99167	−0.310427	−0.155213	−	0.987881i	$-0.549607\pi$
$839$	−8.99167	−0.310427	−0.155213	+	0.987881i	$0.549607\pi$
$840$	0	0
$841$	−14.4947	−0.499818
$842$	−25.8297	−0.890150
$843$	18.5569	0.639132
$844$	−27.7343	−0.954655
$845$	0	0
$846$	−8.09411	−0.278281
$847$	14.6245	0.502504
$848$	−7.32540	−0.251555
$849$	15.0681	0.517137
$850$	0	0
$851$	−5.15743	−0.176795
$852$	−0.709487	−0.0243066
$853$	18.1351	0.620935	0.310468	−	0.950584i	$-0.399514\pi$
$853$	18.1351	0.620935	0.310468	+	0.950584i	$0.399514\pi$
$854$	−16.3364	−0.559020
$855$	0	0
$856$	5.12450	0.175152
$857$	38.0394	1.29940	0.649701	−	0.760190i	$-0.274894\pi$
$857$	38.0394	1.29940	0.649701	+	0.760190i	$0.274894\pi$
$858$	15.5073	0.529409
$859$	2.06557	0.0704763	0.0352381	−	0.999379i	$-0.488781\pi$
$859$	2.06557	0.0704763	0.0352381	+	0.999379i	$0.488781\pi$
$860$	0	0
$861$	5.05537	0.172286
$862$	−34.7909	−1.18498
$863$	−54.7974	−1.86533	−0.932663	−	0.360749i	$-0.882521\pi$
$863$	−54.7974	−1.86533	−0.932663	+	0.360749i	$0.882521\pi$
$864$	24.7224	0.841074
$865$	0	0
$866$	−33.7572	−1.14712
$867$	1.02771	0.0349029
$868$	−16.9291	−0.574612
$869$	35.8166	1.21500
$870$	0	0
$871$	20.1378	0.682342
$872$	−5.15732	−0.174649
$873$	−22.8697	−0.774021
$874$	19.2953	0.652672
$875$	0	0
$876$	−1.62831	−0.0550153
$877$	−22.7227	−0.767293	−0.383646	−	0.923480i	$-0.625332\pi$
$877$	−22.7227	−0.767293	−0.383646	+	0.923480i	$0.625332\pi$
$878$	−27.9399	−0.942926
$879$	−2.34150	−0.0789768
$880$	0	0
$881$	−5.14358	−0.173291	−0.0866457	−	0.996239i	$-0.527615\pi$
$881$	−5.14358	−0.173291	−0.0866457	+	0.996239i	$0.527615\pi$
$882$	16.5080	0.555852
$883$	−38.7936	−1.30551	−0.652755	−	0.757569i	$-0.726387\pi$
$883$	−38.7936	−1.30551	−0.652755	+	0.757569i	$0.726387\pi$
$884$	19.3905	0.652174
$885$	0	0
$886$	−63.9488	−2.14840
$887$	5.38059	0.180663	0.0903313	−	0.995912i	$-0.471207\pi$
$887$	5.38059	0.180663	0.0903313	+	0.995912i	$0.471207\pi$
$888$	2.27226	0.0762521
$889$	22.5118	0.755023
$890$	0	0
$891$	9.47669	0.317481
$892$	−5.69298	−0.190615
$893$	7.64369	0.255786
$894$	5.97100	0.199700
$895$	0	0
$896$	36.5304	1.22040
$897$	−9.37538	−0.313035
$898$	25.7542	0.859428
$899$	−17.0708	−0.569342
$900$	0	0
$901$	5.83041	0.194239
$902$	−8.72732	−0.290588
$903$	32.4596	1.08019
$904$	7.99723	0.265984
$905$	0	0
$906$	29.2969	0.973325
$907$	−30.8726	−1.02511	−0.512554	−	0.858655i	$-0.671300\pi$
$907$	−30.8726	−1.02511	−0.512554	+	0.858655i	$0.671300\pi$
$908$	8.83683	0.293260
$909$	−5.64406	−0.187202
$910$	0	0
$911$	25.3272	0.839127	0.419563	−	0.907726i	$-0.362183\pi$
$911$	25.3272	0.839127	0.419563	+	0.907726i	$0.362183\pi$
$912$	−15.7049	−0.520041
$913$	−45.7074	−1.51269
$914$	−37.2250	−1.23129
$915$	0	0
$916$	22.8519	0.755048
$917$	34.9102	1.15284
$918$	−30.0205	−0.990823
$919$	28.1700	0.929244	0.464622	−	0.885509i	$-0.346190\pi$
$919$	28.1700	0.929244	0.464622	+	0.885509i	$0.346190\pi$
$920$	0	0
$921$	−2.00460	−0.0660537
$922$	−10.0410	−0.330683
$923$	3.34157	0.109989
$924$	7.71378	0.253765
$925$	0	0
$926$	−7.42245	−0.243917
$927$	28.1336	0.924028
$928$	−22.0357	−0.723358
$929$	11.6096	0.380900	0.190450	−	0.981697i	$-0.439005\pi$
$929$	11.6096	0.380900	0.190450	+	0.981697i	$0.439005\pi$
$930$	0	0
$931$	−15.5893	−0.510919
$932$	19.4934	0.638529
$933$	15.4681	0.506403
$934$	30.3758	0.993926
$935$	0	0
$936$	−15.4112	−0.503730
$937$	16.0752	0.525154	0.262577	−	0.964911i	$-0.415428\pi$
$937$	16.0752	0.525154	0.262577	+	0.964911i	$0.415428\pi$
$938$	27.5586	0.899820
$939$	−1.68856	−0.0551041
$940$	0	0
$941$	−1.48903	−0.0485411	−0.0242705	−	0.999705i	$-0.507726\pi$
$941$	−1.48903	−0.0485411	−0.0242705	+	0.999705i	$0.507726\pi$
$942$	−32.3194	−1.05302
$943$	5.27636	0.171822
$944$	50.1271	1.63150
$945$	0	0
$946$	−56.0365	−1.82190
$947$	11.7424	0.381578	0.190789	−	0.981631i	$-0.438895\pi$
$947$	11.7424	0.381578	0.190789	+	0.981631i	$0.438895\pi$
$948$	−12.7011	−0.412512
$949$	7.66907	0.248949
$950$	0	0
$951$	8.05396	0.261167
$952$	−19.9323	−0.646009
$953$	−14.1905	−0.459677	−0.229838	−	0.973229i	$-0.573820\pi$
$953$	−14.1905	−0.459677	−0.229838	+	0.973229i	$0.573820\pi$
$954$	6.16911	0.199732
$955$	0	0
$956$	30.9098	0.999694
$957$	7.77833	0.251438
$958$	−74.4167	−2.40429
$959$	−8.33390	−0.269116
$960$	0	0
$961$	−10.9100	−0.351935
$962$	14.2476	0.459362
$963$	−7.97265	−0.256915
$964$	−1.14211	−0.0367849
$965$	0	0
$966$	−12.8302	−0.412806
$967$	−4.65270	−0.149621	−0.0748104	−	0.997198i	$-0.523835\pi$
$967$	−4.65270	−0.149621	−0.0748104	+	0.997198i	$0.523835\pi$
$968$	−6.72492	−0.216147
$969$	12.4998	0.401551
$970$	0	0
$971$	28.1958	0.904847	0.452424	−	0.891803i	$-0.350560\pi$
$971$	28.1958	0.904847	0.452424	+	0.891803i	$0.350560\pi$
$972$	−18.0011	−0.577385
$973$	−9.76145	−0.312938
$974$	−19.1055	−0.612179
$975$	0	0
$976$	13.8778	0.444219
$977$	−10.1118	−0.323506	−0.161753	−	0.986831i	$-0.551715\pi$
$977$	−10.1118	−0.323506	−0.161753	+	0.986831i	$0.551715\pi$
$978$	1.71784	0.0549305
$979$	−40.3074	−1.28823
$980$	0	0
$981$	8.02371	0.256177
$982$	−68.1400	−2.17444
$983$	46.3152	1.47723	0.738613	−	0.674130i	$-0.235481\pi$
$983$	46.3152	1.47723	0.738613	+	0.674130i	$0.235481\pi$
$984$	−2.32466	−0.0741074
$985$	0	0
$986$	26.7580	0.852148
$987$	−5.08260	−0.161781
$988$	−19.3752	−0.616408
$989$	33.8785	1.07727
$990$	0	0
$991$	−36.8551	−1.17074	−0.585371	−	0.810766i	$-0.699051\pi$
$991$	−36.8551	−1.17074	−0.585371	+	0.810766i	$0.699051\pi$
$992$	25.9331	0.823377
$993$	−2.45969	−0.0780558
$994$	4.57295	0.145045
$995$	0	0
$996$	16.2085	0.513585
$997$	−18.1282	−0.574126	−0.287063	−	0.957912i	$-0.592679\pi$
$997$	−18.1282	−0.574126	−0.287063	+	0.957912i	$0.592679\pi$
$998$	6.64797	0.210438
$999$	−8.01784	−0.253673

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.o.1.7	yes	40
5.4	even	2		6025.2.a.l.1.34	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6025.2.a.l.1.34	✓	40	5.4	even	2
6025.2.a.o.1.7	yes	40	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)

\(f(q)\)	\(=\)	\(q-1.77260 q^{2} -0.796316 q^{3} +1.14211 q^{4} +1.41155 q^{6} -3.30701 q^{7} +1.52069 q^{8} -2.36588 q^{9} +O(q^{10})\)	\(q-1.77260 q^{2} -0.796316 q^{3} +1.14211 q^{4} +1.41155 q^{6} -3.30701 q^{7} +1.52069 q^{8} -2.36588 q^{9} +2.56471 q^{11} -0.909481 q^{12} +4.28352 q^{13} +5.86201 q^{14} -4.97980 q^{16} +3.96351 q^{17} +4.19376 q^{18} -3.96039 q^{19} +2.63342 q^{21} -4.54620 q^{22} +2.74854 q^{23} -1.21095 q^{24} -7.59297 q^{26} +4.27294 q^{27} -3.77697 q^{28} -3.80858 q^{29} +4.48219 q^{31} +5.78582 q^{32} -2.04232 q^{33} -7.02572 q^{34} -2.70210 q^{36} -1.87643 q^{37} +7.02018 q^{38} -3.41103 q^{39} +1.91969 q^{41} -4.66801 q^{42} +12.3260 q^{43} +2.92918 q^{44} -4.87207 q^{46} -1.93004 q^{47} +3.96550 q^{48} +3.93631 q^{49} -3.15621 q^{51} +4.89226 q^{52} +1.47102 q^{53} -7.57421 q^{54} -5.02895 q^{56} +3.15372 q^{57} +6.75109 q^{58} -10.0661 q^{59} -2.78683 q^{61} -7.94513 q^{62} +7.82399 q^{63} -0.296329 q^{64} +3.62021 q^{66} +4.70122 q^{67} +4.52677 q^{68} -2.18871 q^{69} +0.780100 q^{71} -3.59778 q^{72} +1.79037 q^{73} +3.32615 q^{74} -4.52320 q^{76} -8.48152 q^{77} +6.04640 q^{78} +13.9652 q^{79} +3.69504 q^{81} -3.40285 q^{82} -17.8217 q^{83} +3.00766 q^{84} -21.8491 q^{86} +3.03283 q^{87} +3.90013 q^{88} -15.7162 q^{89} -14.1656 q^{91} +3.13914 q^{92} -3.56924 q^{93} +3.42118 q^{94} -4.60734 q^{96} +9.66645 q^{97} -6.97751 q^{98} -6.06780 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9}+O(q^{10}) \) 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9	\( 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9} + q^{11} + 14 q^{12} + 9 q^{13} - q^{14} + 43 q^{16} + 12 q^{17} + 42 q^{18} + 2 q^{21} + 5 q^{22} + 77 q^{23} - 2 q^{24} + 2 q^{26} + 38 q^{27} + 42 q^{28} + 2 q^{29} + q^{31} + 72 q^{32} + 20 q^{33} + 5 q^{34} + 32 q^{36} + 28 q^{37} + 23 q^{38} - 2 q^{39} - 2 q^{41} + 37 q^{42} + 31 q^{43} + 3 q^{44} + 14 q^{46} + 96 q^{47} + 13 q^{48} + 40 q^{49} - 10 q^{51} + 42 q^{52} + 54 q^{53} + 4 q^{54} - 15 q^{56} + 37 q^{57} + 27 q^{58} + q^{59} + 5 q^{61} + 39 q^{62} + 70 q^{63} + 65 q^{64} - 52 q^{66} + 34 q^{67} + 52 q^{68} + 21 q^{69} - 9 q^{71} + 70 q^{72} + 25 q^{73} + 22 q^{74} - 47 q^{76} + 54 q^{77} + 58 q^{78} + 13 q^{79} + 12 q^{81} - 5 q^{82} + 63 q^{83} + 95 q^{84} - 18 q^{86} + 47 q^{87} + 13 q^{88} + 19 q^{89} - 31 q^{91} + 137 q^{92} + 52 q^{93} + 120 q^{94} - 49 q^{96} + 36 q^{97} + 64 q^{98} + 16 q^{99}+O(q^{100}) \) 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9 + q^11 + 14 * q^12 + 9 * q^13 - q^14 + 43 * q^16 + 12 * q^17 + 42 * q^18 + 2 * q^21 + 5 * q^22 + 77 * q^23 - 2 * q^24 + 2 * q^26 + 38 * q^27 + 42 * q^28 + 2 * q^29 + q^31 + 72 * q^32 + 20 * q^33 + 5 * q^34 + 32 * q^36 + 28 * q^37 + 23 * q^38 - 2 * q^39 - 2 * q^41 + 37 * q^42 + 31 * q^43 + 3 * q^44 + 14 * q^46 + 96 * q^47 + 13 * q^48 + 40 * q^49 - 10 * q^51 + 42 * q^52 + 54 * q^53 + 4 * q^54 - 15 * q^56 + 37 * q^57 + 27 * q^58 + q^59 + 5 * q^61 + 39 * q^62 + 70 * q^63 + 65 * q^64 - 52 * q^66 + 34 * q^67 + 52 * q^68 + 21 * q^69 - 9 * q^71 + 70 * q^72 + 25 * q^73 + 22 * q^74 - 47 * q^76 + 54 * q^77 + 58 * q^78 + 13 * q^79 + 12 * q^81 - 5 * q^82 + 63 * q^83 + 95 * q^84 - 18 * q^86 + 47 * q^87 + 13 * q^88 + 19 * q^89 - 31 * q^91 + 137 * q^92 + 52 * q^93 + 120 * q^94 - 49 * q^96 + 36 * q^97 + 64 * q^98 + 16 * q^99

Introduction

L-functions

Modular forms

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Database

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