LMFDB - Embedded newform 625.2.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [625,2,Mod(1,625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(625, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 625 = 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	625.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:	$4.99065012633$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 25)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} +0.618034 q^{6} -1.61803 q^{7} -2.23607 q^{8} -2.00000 q^{9} -0.763932 q^{11} -1.61803 q^{12} -4.85410 q^{13} -1.00000 q^{14} +1.85410 q^{16} -0.763932 q^{17} -1.23607 q^{18} -5.85410 q^{19} -1.61803 q^{21} -0.472136 q^{22} +8.23607 q^{23} -2.23607 q^{24} -3.00000 q^{26} -5.00000 q^{27} +2.61803 q^{28} -1.38197 q^{29} -3.00000 q^{31} +5.61803 q^{32} -0.763932 q^{33} -0.472136 q^{34} +3.23607 q^{36} +4.23607 q^{37} -3.61803 q^{38} -4.85410 q^{39} -5.23607 q^{41} -1.00000 q^{42} +1.85410 q^{43} +1.23607 q^{44} +5.09017 q^{46} -1.61803 q^{47} +1.85410 q^{48} -4.38197 q^{49} -0.763932 q^{51} +7.85410 q^{52} +5.47214 q^{53} -3.09017 q^{54} +3.61803 q^{56} -5.85410 q^{57} -0.854102 q^{58} -4.14590 q^{59} -4.70820 q^{61} -1.85410 q^{62} +3.23607 q^{63} -0.236068 q^{64} -0.472136 q^{66} +9.23607 q^{67} +1.23607 q^{68} +8.23607 q^{69} -4.38197 q^{71} +4.47214 q^{72} -9.00000 q^{73} +2.61803 q^{74} +9.47214 q^{76} +1.23607 q^{77} -3.00000 q^{78} +3.09017 q^{79} +1.00000 q^{81} -3.23607 q^{82} -1.76393 q^{83} +2.61803 q^{84} +1.14590 q^{86} -1.38197 q^{87} +1.70820 q^{88} +8.94427 q^{89} +7.85410 q^{91} -13.3262 q^{92} -3.00000 q^{93} -1.00000 q^{94} +5.61803 q^{96} +2.85410 q^{97} -2.70820 q^{98} +1.52786 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 2 q^{3} - q^{4} - q^{6} - q^{7} - 4 q^{9} - 6 q^{11} - q^{12} - 3 q^{13} - 2 q^{14} - 3 q^{16} - 6 q^{17} + 2 q^{18} - 5 q^{19} - q^{21} + 8 q^{22} + 12 q^{23} - 6 q^{26} - 10 q^{27}+ \cdots + 12 q^{99}+O(q^{100}) $ 2 * q - q^2 + 2 * q^3 - q^4 - q^6 - q^7 - 4 * q^9 - 6 * q^11 - q^12 - 3 * q^13 - 2 * q^14 - 3 * q^16 - 6 * q^17 + 2 * q^18 - 5 * q^19 - q^21 + 8 * q^22 + 12 * q^23 - 6 * q^26 - 10 * q^27 + 3 * q^28 - 5 * q^29 - 6 * q^31 + 9 * q^32 - 6 * q^33 + 8 * q^34 + 2 * q^36 + 4 * q^37 - 5 * q^38 - 3 * q^39 - 6 * q^41 - 2 * q^42 - 3 * q^43 - 2 * q^44 - q^46 - q^47 - 3 * q^48 - 11 * q^49 - 6 * q^51 + 9 * q^52 + 2 * q^53 + 5 * q^54 + 5 * q^56 - 5 * q^57 + 5 * q^58 - 15 * q^59 + 4 * q^61 + 3 * q^62 + 2 * q^63 + 4 * q^64 + 8 * q^66 + 14 * q^67 - 2 * q^68 + 12 * q^69 - 11 * q^71 - 18 * q^73 + 3 * q^74 + 10 * q^76 - 2 * q^77 - 6 * q^78 - 5 * q^79 + 2 * q^81 - 2 * q^82 - 8 * q^83 + 3 * q^84 + 9 * q^86 - 5 * q^87 - 10 * q^88 + 9 * q^91 - 11 * q^92 - 6 * q^93 - 2 * q^94 + 9 * q^96 - q^97 + 8 * q^98 + 12 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.618034	0.437016	0.218508	−	0.975835i	$-0.429881\pi$
$2$	0.618034	0.437016	0.218508	+	0.975835i	$0.429881\pi$
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	−1.61803	−0.809017
$5$	0	0
$5$	0	0
$6$	0.618034	0.252311
$7$	−1.61803	−0.611559	−0.305780	−	0.952102i	$-0.598917\pi$
$7$	−1.61803	−0.611559	−0.305780	+	0.952102i	$0.598917\pi$
$8$	−2.23607	−0.790569
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−0.763932	−0.230334	−0.115167	−	0.993346i	$-0.536740\pi$
$11$	−0.763932	−0.230334	−0.115167	+	0.993346i	$0.536740\pi$
$12$	−1.61803	−0.467086
$13$	−4.85410	−1.34629	−0.673143	−	0.739512i	$-0.735056\pi$
$13$	−4.85410	−1.34629	−0.673143	+	0.739512i	$0.735056\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.85410	0.463525
$17$	−0.763932	−0.185281	−0.0926404	−	0.995700i	$-0.529531\pi$
$17$	−0.763932	−0.185281	−0.0926404	+	0.995700i	$0.529531\pi$
$18$	−1.23607	−0.291344
$19$	−5.85410	−1.34302	−0.671512	−	0.740994i	$-0.734355\pi$
$19$	−5.85410	−1.34302	−0.671512	+	0.740994i	$0.734355\pi$
$20$	0	0
$21$	−1.61803	−0.353084
$22$	−0.472136	−0.100660
$23$	8.23607	1.71734	0.858669	−	0.512530i	$-0.171292\pi$
$23$	8.23607	1.71734	0.858669	+	0.512530i	$0.171292\pi$
$24$	−2.23607	−0.456435
$25$	0	0
$26$	−3.00000	−0.588348
$27$	−5.00000	−0.962250
$28$	2.61803	0.494762
$29$	−1.38197	−0.256625	−0.128312	−	0.991734i	$-0.540956\pi$
$29$	−1.38197	−0.256625	−0.128312	+	0.991734i	$0.540956\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	5.61803	0.993137
$33$	−0.763932	−0.132983
$34$	−0.472136	−0.0809706
$35$	0	0
$36$	3.23607	0.539345
$37$	4.23607	0.696405	0.348203	−	0.937419i	$-0.386792\pi$
$37$	4.23607	0.696405	0.348203	+	0.937419i	$0.386792\pi$
$38$	−3.61803	−0.586923
$39$	−4.85410	−0.777278
$40$	0	0
$41$	−5.23607	−0.817736	−0.408868	−	0.912593i	$-0.634076\pi$
$41$	−5.23607	−0.817736	−0.408868	+	0.912593i	$0.634076\pi$
$42$	−1.00000	−0.154303
$43$	1.85410	0.282748	0.141374	−	0.989956i	$-0.454848\pi$
$43$	1.85410	0.282748	0.141374	+	0.989956i	$0.454848\pi$
$44$	1.23607	0.186344
$45$	0	0
$46$	5.09017	0.750505
$47$	−1.61803	−0.236015	−0.118007	−	0.993013i	$-0.537651\pi$
$47$	−1.61803	−0.236015	−0.118007	+	0.993013i	$0.537651\pi$
$48$	1.85410	0.267617
$49$	−4.38197	−0.625995
$50$	0	0
$51$	−0.763932	−0.106972
$52$	7.85410	1.08917
$53$	5.47214	0.751656	0.375828	−	0.926690i	$-0.377358\pi$
$53$	5.47214	0.751656	0.375828	+	0.926690i	$0.377358\pi$
$54$	−3.09017	−0.420519
$55$	0	0
$56$	3.61803	0.483480
$57$	−5.85410	−0.775395
$58$	−0.854102	−0.112149
$59$	−4.14590	−0.539750	−0.269875	−	0.962895i	$-0.586982\pi$
$59$	−4.14590	−0.539750	−0.269875	+	0.962895i	$0.586982\pi$
$60$	0	0
$61$	−4.70820	−0.602824	−0.301412	−	0.953494i	$-0.597458\pi$
$61$	−4.70820	−0.602824	−0.301412	+	0.953494i	$0.597458\pi$
$62$	−1.85410	−0.235471
$63$	3.23607	0.407706
$64$	−0.236068	−0.0295085
$65$	0	0
$66$	−0.472136	−0.0581159
$67$	9.23607	1.12837	0.564183	−	0.825650i	$-0.309191\pi$
$67$	9.23607	1.12837	0.564183	+	0.825650i	$0.309191\pi$
$68$	1.23607	0.149895
$69$	8.23607	0.991506
$70$	0	0
$71$	−4.38197	−0.520044	−0.260022	−	0.965603i	$-0.583730\pi$
$71$	−4.38197	−0.520044	−0.260022	+	0.965603i	$0.583730\pi$
$72$	4.47214	0.527046
$73$	−9.00000	−1.05337	−0.526685	−	0.850060i	$-0.676565\pi$
$73$	−9.00000	−1.05337	−0.526685	+	0.850060i	$0.676565\pi$
$74$	2.61803	0.304340
$75$	0	0
$76$	9.47214	1.08653
$77$	1.23607	0.140863
$78$	−3.00000	−0.339683
$79$	3.09017	0.347671	0.173836	−	0.984775i	$-0.444384\pi$
$79$	3.09017	0.347671	0.173836	+	0.984775i	$0.444384\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−3.23607	−0.357364
$83$	−1.76393	−0.193617	−0.0968083	−	0.995303i	$-0.530863\pi$
$83$	−1.76393	−0.193617	−0.0968083	+	0.995303i	$0.530863\pi$
$84$	2.61803	0.285651
$85$	0	0
$86$	1.14590	0.123565
$87$	−1.38197	−0.148162
$88$	1.70820	0.182095
$89$	8.94427	0.948091	0.474045	−	0.880500i	$-0.342793\pi$
$89$	8.94427	0.948091	0.474045	+	0.880500i	$0.342793\pi$
$90$	0	0
$91$	7.85410	0.823334
$92$	−13.3262	−1.38936
$93$	−3.00000	−0.311086
$94$	−1.00000	−0.103142
$95$	0	0
$96$	5.61803	0.573388
$97$	2.85410	0.289790	0.144895	−	0.989447i	$-0.453716\pi$
$97$	2.85410	0.289790	0.144895	+	0.989447i	$0.453716\pi$
$98$	−2.70820	−0.273570
$99$	1.52786	0.153556
$100$	0	0
$101$	−7.47214	−0.743505	−0.371753	−	0.928332i	$-0.621243\pi$
$101$	−7.47214	−0.743505	−0.371753	+	0.928332i	$0.621243\pi$
$102$	−0.472136	−0.0467484
$103$	−11.5623	−1.13927	−0.569634	−	0.821899i	$-0.692915\pi$
$103$	−11.5623	−1.13927	−0.569634	+	0.821899i	$0.692915\pi$
$104$	10.8541	1.06433
$105$	0	0
$106$	3.38197	0.328486
$107$	10.4164	1.00699	0.503496	−	0.863998i	$-0.332047\pi$
$107$	10.4164	1.00699	0.503496	+	0.863998i	$0.332047\pi$
$108$	8.09017	0.778477
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	4.23607	0.402070
$112$	−3.00000	−0.283473
$113$	10.1459	0.954446	0.477223	−	0.878782i	$-0.341643\pi$
$113$	10.1459	0.954446	0.477223	+	0.878782i	$0.341643\pi$
$114$	−3.61803	−0.338860
$115$	0	0
$116$	2.23607	0.207614
$117$	9.70820	0.897524
$118$	−2.56231	−0.235879
$119$	1.23607	0.113310
$120$	0	0
$121$	−10.4164	−0.946946
$122$	−2.90983	−0.263444
$123$	−5.23607	−0.472120
$124$	4.85410	0.435911
$125$	0	0
$126$	2.00000	0.178174
$127$	−15.8885	−1.40988	−0.704940	−	0.709267i	$-0.749026\pi$
$127$	−15.8885	−1.40988	−0.704940	+	0.709267i	$0.749026\pi$
$128$	−11.3820	−1.00603
$129$	1.85410	0.163245
$130$	0	0
$131$	−17.7984	−1.55505	−0.777526	−	0.628851i	$-0.783525\pi$
$131$	−17.7984	−1.55505	−0.777526	+	0.628851i	$0.783525\pi$
$132$	1.23607	0.107586
$133$	9.47214	0.821338
$134$	5.70820	0.493114
$135$	0	0
$136$	1.70820	0.146477
$137$	5.94427	0.507853	0.253927	−	0.967223i	$-0.418278\pi$
$137$	5.94427	0.507853	0.253927	+	0.967223i	$0.418278\pi$
$138$	5.09017	0.433304
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	0	0
$141$	−1.61803	−0.136263
$142$	−2.70820	−0.227267
$143$	3.70820	0.310096
$144$	−3.70820	−0.309017
$145$	0	0
$146$	−5.56231	−0.460340
$147$	−4.38197	−0.361418
$148$	−6.85410	−0.563404
$149$	13.9443	1.14236	0.571180	−	0.820825i	$-0.306486\pi$
$149$	13.9443	1.14236	0.571180	+	0.820825i	$0.306486\pi$
$150$	0	0
$151$	−5.56231	−0.452654	−0.226327	−	0.974051i	$-0.572672\pi$
$151$	−5.56231	−0.452654	−0.226327	+	0.974051i	$0.572672\pi$
$152$	13.0902	1.06175
$153$	1.52786	0.123520
$154$	0.763932	0.0615594
$155$	0	0
$156$	7.85410	0.628831
$157$	−9.18034	−0.732671	−0.366335	−	0.930483i	$-0.619388\pi$
$157$	−9.18034	−0.732671	−0.366335	+	0.930483i	$0.619388\pi$
$158$	1.90983	0.151938
$159$	5.47214	0.433969
$160$	0	0
$161$	−13.3262	−1.05025
$162$	0.618034	0.0485573
$163$	11.0000	0.861586	0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000	0.861586	0.430793	+	0.902451i	$0.358234\pi$
$164$	8.47214	0.661563
$165$	0	0
$166$	−1.09017	−0.0846136
$167$	−5.56231	−0.430424	−0.215212	−	0.976567i	$-0.569044\pi$
$167$	−5.56231	−0.430424	−0.215212	+	0.976567i	$0.569044\pi$
$168$	3.61803	0.279137
$169$	10.5623	0.812485
$170$	0	0
$171$	11.7082	0.895349
$172$	−3.00000	−0.228748
$173$	−16.8885	−1.28401	−0.642006	−	0.766700i	$-0.721898\pi$
$173$	−16.8885	−1.28401	−0.642006	+	0.766700i	$0.721898\pi$
$174$	−0.854102	−0.0647493
$175$	0	0
$176$	−1.41641	−0.106766
$177$	−4.14590	−0.311625
$178$	5.52786	0.414331
$179$	−9.47214	−0.707981	−0.353990	−	0.935249i	$-0.615175\pi$
$179$	−9.47214	−0.707981	−0.353990	+	0.935249i	$0.615175\pi$
$180$	0	0
$181$	13.7082	1.01892	0.509461	−	0.860494i	$-0.329845\pi$
$181$	13.7082	1.01892	0.509461	+	0.860494i	$0.329845\pi$
$182$	4.85410	0.359810
$183$	−4.70820	−0.348040
$184$	−18.4164	−1.35768
$185$	0	0
$186$	−1.85410	−0.135949
$187$	0.583592	0.0426765
$188$	2.61803	0.190940
$189$	8.09017	0.588473
$190$	0	0
$191$	−24.1803	−1.74963	−0.874814	−	0.484459i	$-0.839016\pi$
$191$	−24.1803	−1.74963	−0.874814	+	0.484459i	$0.839016\pi$
$192$	−0.236068	−0.0170367
$193$	−5.70820	−0.410886	−0.205443	−	0.978669i	$-0.565863\pi$
$193$	−5.70820	−0.410886	−0.205443	+	0.978669i	$0.565863\pi$
$194$	1.76393	0.126643
$195$	0	0
$196$	7.09017	0.506441
$197$	−9.70820	−0.691681	−0.345840	−	0.938293i	$-0.612406\pi$
$197$	−9.70820	−0.691681	−0.345840	+	0.938293i	$0.612406\pi$
$198$	0.944272	0.0671065
$199$	2.56231	0.181637	0.0908185	−	0.995867i	$-0.471052\pi$
$199$	2.56231	0.181637	0.0908185	+	0.995867i	$0.471052\pi$
$200$	0	0
$201$	9.23607	0.651462
$202$	−4.61803	−0.324924
$203$	2.23607	0.156941
$204$	1.23607	0.0865421
$205$	0	0
$206$	−7.14590	−0.497878
$207$	−16.4721	−1.14489
$208$	−9.00000	−0.624038
$209$	4.47214	0.309344
$210$	0	0
$211$	13.1803	0.907372	0.453686	−	0.891162i	$-0.350109\pi$
$211$	13.1803	0.907372	0.453686	+	0.891162i	$0.350109\pi$
$212$	−8.85410	−0.608102
$213$	−4.38197	−0.300247
$214$	6.43769	0.440072
$215$	0	0
$216$	11.1803	0.760726
$217$	4.85410	0.329518
$218$	6.18034	0.418585
$219$	−9.00000	−0.608164
$220$	0	0
$221$	3.70820	0.249441
$222$	2.61803	0.175711
$223$	22.1803	1.48531	0.742653	−	0.669677i	$-0.233567\pi$
$223$	22.1803	1.48531	0.742653	+	0.669677i	$0.233567\pi$
$224$	−9.09017	−0.607363
$225$	0	0
$226$	6.27051	0.417108
$227$	19.2361	1.27674	0.638371	−	0.769729i	$-0.279608\pi$
$227$	19.2361	1.27674	0.638371	+	0.769729i	$0.279608\pi$
$228$	9.47214	0.627308
$229$	8.29180	0.547937	0.273969	−	0.961739i	$-0.411664\pi$
$229$	8.29180	0.547937	0.273969	+	0.961739i	$0.411664\pi$
$230$	0	0
$231$	1.23607	0.0813273
$232$	3.09017	0.202880
$233$	14.9443	0.979032	0.489516	−	0.871994i	$-0.337174\pi$
$233$	14.9443	0.979032	0.489516	+	0.871994i	$0.337174\pi$
$234$	6.00000	0.392232
$235$	0	0
$236$	6.70820	0.436667
$237$	3.09017	0.200728
$238$	0.763932	0.0495184
$239$	−29.4721	−1.90639	−0.953197	−	0.302350i	$-0.902229\pi$
$239$	−29.4721	−1.90639	−0.953197	+	0.302350i	$0.902229\pi$
$240$	0	0
$241$	11.4721	0.738985	0.369493	−	0.929234i	$-0.379532\pi$
$241$	11.4721	0.738985	0.369493	+	0.929234i	$0.379532\pi$
$242$	−6.43769	−0.413831
$243$	16.0000	1.02640
$244$	7.61803	0.487695
$245$	0	0
$246$	−3.23607	−0.206324
$247$	28.4164	1.80809
$248$	6.70820	0.425971
$249$	−1.76393	−0.111785
$250$	0	0
$251$	−6.81966	−0.430453	−0.215227	−	0.976564i	$-0.569049\pi$
$251$	−6.81966	−0.430453	−0.215227	+	0.976564i	$0.569049\pi$
$252$	−5.23607	−0.329841
$253$	−6.29180	−0.395562
$254$	−9.81966	−0.616140
$255$	0	0
$256$	−6.56231	−0.410144
$257$	16.1459	1.00715	0.503577	−	0.863951i	$-0.332017\pi$
$257$	16.1459	1.00715	0.503577	+	0.863951i	$0.332017\pi$
$258$	1.14590	0.0713405
$259$	−6.85410	−0.425893
$260$	0	0
$261$	2.76393	0.171083
$262$	−11.0000	−0.679582
$263$	−22.0902	−1.36214	−0.681069	−	0.732219i	$-0.738485\pi$
$263$	−22.0902	−1.36214	−0.681069	+	0.732219i	$0.738485\pi$
$264$	1.70820	0.105133
$265$	0	0
$266$	5.85410	0.358938
$267$	8.94427	0.547381
$268$	−14.9443	−0.912867
$269$	−17.2361	−1.05090	−0.525451	−	0.850824i	$-0.676103\pi$
$269$	−17.2361	−1.05090	−0.525451	+	0.850824i	$0.676103\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	−1.41641	−0.0858823
$273$	7.85410	0.475352
$274$	3.67376	0.221940
$275$	0	0
$276$	−13.3262	−0.802145
$277$	−11.2918	−0.678458	−0.339229	−	0.940704i	$-0.610166\pi$
$277$	−11.2918	−0.678458	−0.339229	+	0.940704i	$0.610166\pi$
$278$	3.09017	0.185336
$279$	6.00000	0.359211
$280$	0	0
$281$	−1.09017	−0.0650341	−0.0325170	−	0.999471i	$-0.510352\pi$
$281$	−1.09017	−0.0650341	−0.0325170	+	0.999471i	$0.510352\pi$
$282$	−1.00000	−0.0595491
$283$	−23.1459	−1.37588	−0.687940	−	0.725767i	$-0.741485\pi$
$283$	−23.1459	−1.37588	−0.687940	+	0.725767i	$0.741485\pi$
$284$	7.09017	0.420724
$285$	0	0
$286$	2.29180	0.135517
$287$	8.47214	0.500094
$288$	−11.2361	−0.662092
$289$	−16.4164	−0.965671
$290$	0	0
$291$	2.85410	0.167310
$292$	14.5623	0.852194
$293$	−28.4721	−1.66336	−0.831680	−	0.555255i	$-0.812621\pi$
$293$	−28.4721	−1.66336	−0.831680	+	0.555255i	$0.812621\pi$
$294$	−2.70820	−0.157946
$295$	0	0
$296$	−9.47214	−0.550557
$297$	3.81966	0.221639
$298$	8.61803	0.499229
$299$	−39.9787	−2.31203
$300$	0	0
$301$	−3.00000	−0.172917
$302$	−3.43769	−0.197817
$303$	−7.47214	−0.429263
$304$	−10.8541	−0.622525
$305$	0	0
$306$	0.944272	0.0539804
$307$	4.76393	0.271892	0.135946	−	0.990716i	$-0.456593\pi$
$307$	4.76393	0.271892	0.135946	+	0.990716i	$0.456593\pi$
$308$	−2.00000	−0.113961
$309$	−11.5623	−0.657757
$310$	0	0
$311$	−29.5066	−1.67316	−0.836582	−	0.547841i	$-0.815450\pi$
$311$	−29.5066	−1.67316	−0.836582	+	0.547841i	$0.815450\pi$
$312$	10.8541	0.614493
$313$	−21.2361	−1.20033	−0.600167	−	0.799875i	$-0.704899\pi$
$313$	−21.2361	−1.20033	−0.600167	+	0.799875i	$0.704899\pi$
$314$	−5.67376	−0.320189
$315$	0	0
$316$	−5.00000	−0.281272
$317$	−23.6525	−1.32846	−0.664228	−	0.747530i	$-0.731239\pi$
$317$	−23.6525	−1.32846	−0.664228	+	0.747530i	$0.731239\pi$
$318$	3.38197	0.189651
$319$	1.05573	0.0591094
$320$	0	0
$321$	10.4164	0.581387
$322$	−8.23607	−0.458978
$323$	4.47214	0.248836
$324$	−1.61803	−0.0898908
$325$	0	0
$326$	6.79837	0.376527
$327$	10.0000	0.553001
$328$	11.7082	0.646477
$329$	2.61803	0.144337
$330$	0	0
$331$	17.1246	0.941254	0.470627	−	0.882332i	$-0.344028\pi$
$331$	17.1246	0.941254	0.470627	+	0.882332i	$0.344028\pi$
$332$	2.85410	0.156639
$333$	−8.47214	−0.464270
$334$	−3.43769	−0.188102
$335$	0	0
$336$	−3.00000	−0.163663
$337$	1.14590	0.0624210	0.0312105	−	0.999513i	$-0.490064\pi$
$337$	1.14590	0.0624210	0.0312105	+	0.999513i	$0.490064\pi$
$338$	6.52786	0.355069
$339$	10.1459	0.551050
$340$	0	0
$341$	2.29180	0.124108
$342$	7.23607	0.391282
$343$	18.4164	0.994393
$344$	−4.14590	−0.223532
$345$	0	0
$346$	−10.4377	−0.561134
$347$	−31.0902	−1.66901	−0.834504	−	0.551002i	$-0.814246\pi$
$347$	−31.0902	−1.66901	−0.834504	+	0.551002i	$0.814246\pi$
$348$	2.23607	0.119866
$349$	8.29180	0.443850	0.221925	−	0.975064i	$-0.428766\pi$
$349$	8.29180	0.443850	0.221925	+	0.975064i	$0.428766\pi$
$350$	0	0
$351$	24.2705	1.29546
$352$	−4.29180	−0.228753
$353$	24.0902	1.28219	0.641095	−	0.767461i	$-0.278480\pi$
$353$	24.0902	1.28219	0.641095	+	0.767461i	$0.278480\pi$
$354$	−2.56231	−0.136185
$355$	0	0
$356$	−14.4721	−0.767022
$357$	1.23607	0.0654197
$358$	−5.85410	−0.309399
$359$	−28.7426	−1.51698	−0.758489	−	0.651685i	$-0.774062\pi$
$359$	−28.7426	−1.51698	−0.758489	+	0.651685i	$0.774062\pi$
$360$	0	0
$361$	15.2705	0.803711
$362$	8.47214	0.445286
$363$	−10.4164	−0.546720
$364$	−12.7082	−0.666091
$365$	0	0
$366$	−2.90983	−0.152099
$367$	−5.43769	−0.283845	−0.141923	−	0.989878i	$-0.545328\pi$
$367$	−5.43769	−0.283845	−0.141923	+	0.989878i	$0.545328\pi$
$368$	15.2705	0.796030
$369$	10.4721	0.545158
$370$	0	0
$371$	−8.85410	−0.459682
$372$	4.85410	0.251673
$373$	5.27051	0.272897	0.136448	−	0.990647i	$-0.456431\pi$
$373$	5.27051	0.272897	0.136448	+	0.990647i	$0.456431\pi$
$374$	0.360680	0.0186503
$375$	0	0
$376$	3.61803	0.186586
$377$	6.70820	0.345490
$378$	5.00000	0.257172
$379$	−34.5967	−1.77712	−0.888558	−	0.458765i	$-0.848292\pi$
$379$	−34.5967	−1.77712	−0.888558	+	0.458765i	$0.848292\pi$
$380$	0	0
$381$	−15.8885	−0.813995
$382$	−14.9443	−0.764615
$383$	−11.3607	−0.580504	−0.290252	−	0.956950i	$-0.593739\pi$
$383$	−11.3607	−0.580504	−0.290252	+	0.956950i	$0.593739\pi$
$384$	−11.3820	−0.580834
$385$	0	0
$386$	−3.52786	−0.179564
$387$	−3.70820	−0.188499
$388$	−4.61803	−0.234445
$389$	15.0000	0.760530	0.380265	−	0.924878i	$-0.375833\pi$
$389$	15.0000	0.760530	0.380265	+	0.924878i	$0.375833\pi$
$390$	0	0
$391$	−6.29180	−0.318190
$392$	9.79837	0.494893
$393$	−17.7984	−0.897809
$394$	−6.00000	−0.302276
$395$	0	0
$396$	−2.47214	−0.124230
$397$	−0.0344419	−0.00172859	−0.000864294	−	1.00000i	$-0.500275\pi$
$397$	−0.0344419	−0.00172859	−0.000864294		1.00000i	$0.500275\pi$
$398$	1.58359	0.0793783
$399$	9.47214	0.474200
$400$	0	0
$401$	−22.5967	−1.12843	−0.564214	−	0.825629i	$-0.690821\pi$
$401$	−22.5967	−1.12843	−0.564214	+	0.825629i	$0.690821\pi$
$402$	5.70820	0.284699
$403$	14.5623	0.725400
$404$	12.0902	0.601508
$405$	0	0
$406$	1.38197	0.0685858
$407$	−3.23607	−0.160406
$408$	1.70820	0.0845687
$409$	28.4164	1.40510	0.702550	−	0.711634i	$-0.252045\pi$
$409$	28.4164	1.40510	0.702550	+	0.711634i	$0.252045\pi$
$410$	0	0
$411$	5.94427	0.293209
$412$	18.7082	0.921687
$413$	6.70820	0.330089
$414$	−10.1803	−0.500336
$415$	0	0
$416$	−27.2705	−1.33705
$417$	5.00000	0.244851
$418$	2.76393	0.135188
$419$	−0.527864	−0.0257878	−0.0128939	−	0.999917i	$-0.504104\pi$
$419$	−0.527864	−0.0257878	−0.0128939	+	0.999917i	$0.504104\pi$
$420$	0	0
$421$	32.0000	1.55958	0.779792	−	0.626038i	$-0.215325\pi$
$421$	32.0000	1.55958	0.779792	+	0.626038i	$0.215325\pi$
$422$	8.14590	0.396536
$423$	3.23607	0.157343
$424$	−12.2361	−0.594236
$425$	0	0
$426$	−2.70820	−0.131213
$427$	7.61803	0.368663
$428$	−16.8541	−0.814674
$429$	3.70820	0.179034
$430$	0	0
$431$	23.8328	1.14799	0.573993	−	0.818860i	$-0.305394\pi$
$431$	23.8328	1.14799	0.573993	+	0.818860i	$0.305394\pi$
$432$	−9.27051	−0.446028
$433$	20.1459	0.968150	0.484075	−	0.875026i	$-0.339156\pi$
$433$	20.1459	0.968150	0.484075	+	0.875026i	$0.339156\pi$
$434$	3.00000	0.144005
$435$	0	0
$436$	−16.1803	−0.774898
$437$	−48.2148	−2.30643
$438$	−5.56231	−0.265777
$439$	5.97871	0.285348	0.142674	−	0.989770i	$-0.454430\pi$
$439$	5.97871	0.285348	0.142674	+	0.989770i	$0.454430\pi$
$440$	0	0
$441$	8.76393	0.417330
$442$	2.29180	0.109010
$443$	12.0557	0.572785	0.286392	−	0.958112i	$-0.407544\pi$
$443$	12.0557	0.572785	0.286392	+	0.958112i	$0.407544\pi$
$444$	−6.85410	−0.325281
$445$	0	0
$446$	13.7082	0.649102
$447$	13.9443	0.659541
$448$	0.381966	0.0180462
$449$	20.3262	0.959254	0.479627	−	0.877472i	$-0.340772\pi$
$449$	20.3262	0.959254	0.479627	+	0.877472i	$0.340772\pi$
$450$	0	0
$451$	4.00000	0.188353
$452$	−16.4164	−0.772163
$453$	−5.56231	−0.261340
$454$	11.8885	0.557957
$455$	0	0
$456$	13.0902	0.613003
$457$	5.41641	0.253369	0.126684	−	0.991943i	$-0.459566\pi$
$457$	5.41641	0.253369	0.126684	+	0.991943i	$0.459566\pi$
$458$	5.12461	0.239457
$459$	3.81966	0.178286
$460$	0	0
$461$	23.1803	1.07962	0.539808	−	0.841788i	$-0.318497\pi$
$461$	23.1803	1.07962	0.539808	+	0.841788i	$0.318497\pi$
$462$	0.763932	0.0355413
$463$	16.1246	0.749374	0.374687	−	0.927151i	$-0.377750\pi$
$463$	16.1246	0.749374	0.374687	+	0.927151i	$0.377750\pi$
$464$	−2.56231	−0.118952
$465$	0	0
$466$	9.23607	0.427853
$467$	−28.4508	−1.31655	−0.658274	−	0.752778i	$-0.728713\pi$
$467$	−28.4508	−1.31655	−0.658274	+	0.752778i	$0.728713\pi$
$468$	−15.7082	−0.726112
$469$	−14.9443	−0.690062
$470$	0	0
$471$	−9.18034	−0.423008
$472$	9.27051	0.426710
$473$	−1.41641	−0.0651265
$474$	1.90983	0.0877214
$475$	0	0
$476$	−2.00000	−0.0916698
$477$	−10.9443	−0.501104
$478$	−18.2148	−0.833125
$479$	−4.14590	−0.189431	−0.0947155	−	0.995504i	$-0.530194\pi$
$479$	−4.14590	−0.189431	−0.0947155	+	0.995504i	$0.530194\pi$
$480$	0	0
$481$	−20.5623	−0.937560
$482$	7.09017	0.322948
$483$	−13.3262	−0.606365
$484$	16.8541	0.766096
$485$	0	0
$486$	9.88854	0.448553
$487$	−9.58359	−0.434274	−0.217137	−	0.976141i	$-0.569672\pi$
$487$	−9.58359	−0.434274	−0.217137	+	0.976141i	$0.569672\pi$
$488$	10.5279	0.476574
$489$	11.0000	0.497437
$490$	0	0
$491$	37.2492	1.68103	0.840517	−	0.541785i	$-0.182251\pi$
$491$	37.2492	1.68103	0.840517	+	0.541785i	$0.182251\pi$
$492$	8.47214	0.381953
$493$	1.05573	0.0475476
$494$	17.5623	0.790165
$495$	0	0
$496$	−5.56231	−0.249755
$497$	7.09017	0.318038
$498$	−1.09017	−0.0488517
$499$	−12.5623	−0.562366	−0.281183	−	0.959654i	$-0.590727\pi$
$499$	−12.5623	−0.562366	−0.281183	+	0.959654i	$0.590727\pi$
$500$	0	0
$501$	−5.56231	−0.248506
$502$	−4.21478	−0.188115
$503$	−10.5836	−0.471899	−0.235950	−	0.971765i	$-0.575820\pi$
$503$	−10.5836	−0.471899	−0.235950	+	0.971765i	$0.575820\pi$
$504$	−7.23607	−0.322320
$505$	0	0
$506$	−3.88854	−0.172867
$507$	10.5623	0.469088
$508$	25.7082	1.14062
$509$	−4.67376	−0.207161	−0.103580	−	0.994621i	$-0.533030\pi$
$509$	−4.67376	−0.207161	−0.103580	+	0.994621i	$0.533030\pi$
$510$	0	0
$511$	14.5623	0.644198
$512$	18.7082	0.826794
$513$	29.2705	1.29232
$514$	9.97871	0.440142
$515$	0	0
$516$	−3.00000	−0.132068
$517$	1.23607	0.0543622
$518$	−4.23607	−0.186122
$519$	−16.8885	−0.741325
$520$	0	0
$521$	−15.3607	−0.672964	−0.336482	−	0.941690i	$-0.609237\pi$
$521$	−15.3607	−0.672964	−0.336482	+	0.941690i	$0.609237\pi$
$522$	1.70820	0.0747661
$523$	−19.8541	−0.868159	−0.434080	−	0.900875i	$-0.642926\pi$
$523$	−19.8541	−0.868159	−0.434080	+	0.900875i	$0.642926\pi$
$524$	28.7984	1.25806
$525$	0	0
$526$	−13.6525	−0.595276
$527$	2.29180	0.0998322
$528$	−1.41641	−0.0616412
$529$	44.8328	1.94925
$530$	0	0
$531$	8.29180	0.359833
$532$	−15.3262	−0.664477
$533$	25.4164	1.10091
$534$	5.52786	0.239214
$535$	0	0
$536$	−20.6525	−0.892051
$537$	−9.47214	−0.408753
$538$	−10.6525	−0.459261
$539$	3.34752	0.144188
$540$	0	0
$541$	−13.1246	−0.564271	−0.282136	−	0.959375i	$-0.591043\pi$
$541$	−13.1246	−0.564271	−0.282136	+	0.959375i	$0.591043\pi$
$542$	−4.94427	−0.212375
$543$	13.7082	0.588275
$544$	−4.29180	−0.184009
$545$	0	0
$546$	4.85410	0.207736
$547$	−34.7082	−1.48402	−0.742008	−	0.670391i	$-0.766126\pi$
$547$	−34.7082	−1.48402	−0.742008	+	0.670391i	$0.766126\pi$
$548$	−9.61803	−0.410862
$549$	9.41641	0.401882
$550$	0	0
$551$	8.09017	0.344653
$552$	−18.4164	−0.783854
$553$	−5.00000	−0.212622
$554$	−6.97871	−0.296497
$555$	0	0
$556$	−8.09017	−0.343100
$557$	9.23607	0.391345	0.195672	−	0.980669i	$-0.437311\pi$
$557$	9.23607	0.391345	0.195672	+	0.980669i	$0.437311\pi$
$558$	3.70820	0.156981
$559$	−9.00000	−0.380659
$560$	0	0
$561$	0.583592	0.0246393
$562$	−0.673762	−0.0284209
$563$	9.61803	0.405352	0.202676	−	0.979246i	$-0.435036\pi$
$563$	9.61803	0.405352	0.202676	+	0.979246i	$0.435036\pi$
$564$	2.61803	0.110239
$565$	0	0
$566$	−14.3050	−0.601282
$567$	−1.61803	−0.0679510
$568$	9.79837	0.411131
$569$	29.4721	1.23554	0.617768	−	0.786360i	$-0.288037\pi$
$569$	29.4721	1.23554	0.617768	+	0.786360i	$0.288037\pi$
$570$	0	0
$571$	32.1246	1.34437	0.672187	−	0.740382i	$-0.265355\pi$
$571$	32.1246	1.34437	0.672187	+	0.740382i	$0.265355\pi$
$572$	−6.00000	−0.250873
$573$	−24.1803	−1.01015
$574$	5.23607	0.218549
$575$	0	0
$576$	0.472136	0.0196723
$577$	37.7771	1.57268	0.786340	−	0.617794i	$-0.211973\pi$
$577$	37.7771	1.57268	0.786340	+	0.617794i	$0.211973\pi$
$578$	−10.1459	−0.422014
$579$	−5.70820	−0.237225
$580$	0	0
$581$	2.85410	0.118408
$582$	1.76393	0.0731173
$583$	−4.18034	−0.173132
$584$	20.1246	0.832762
$585$	0	0
$586$	−17.5967	−0.726915
$587$	18.7082	0.772170	0.386085	−	0.922463i	$-0.373827\pi$
$587$	18.7082	0.772170	0.386085	+	0.922463i	$0.373827\pi$
$588$	7.09017	0.292394
$589$	17.5623	0.723642
$590$	0	0
$591$	−9.70820	−0.399342
$592$	7.85410	0.322802
$593$	−22.0902	−0.907135	−0.453567	−	0.891222i	$-0.649849\pi$
$593$	−22.0902	−0.907135	−0.453567	+	0.891222i	$0.649849\pi$
$594$	2.36068	0.0968599
$595$	0	0
$596$	−22.5623	−0.924188
$597$	2.56231	0.104868
$598$	−24.7082	−1.01039
$599$	−0.527864	−0.0215679	−0.0107840	−	0.999942i	$-0.503433\pi$
$599$	−0.527864	−0.0215679	−0.0107840	+	0.999942i	$0.503433\pi$
$600$	0	0
$601$	36.2705	1.47950	0.739752	−	0.672879i	$-0.234943\pi$
$601$	36.2705	1.47950	0.739752	+	0.672879i	$0.234943\pi$
$602$	−1.85410	−0.0755676
$603$	−18.4721	−0.752244
$604$	9.00000	0.366205
$605$	0	0
$606$	−4.61803	−0.187595
$607$	−15.4377	−0.626597	−0.313298	−	0.949655i	$-0.601434\pi$
$607$	−15.4377	−0.626597	−0.313298	+	0.949655i	$0.601434\pi$
$608$	−32.8885	−1.33381
$609$	2.23607	0.0906100
$610$	0	0
$611$	7.85410	0.317743
$612$	−2.47214	−0.0999302
$613$	31.9787	1.29161	0.645804	−	0.763503i	$-0.276522\pi$
$613$	31.9787	1.29161	0.645804	+	0.763503i	$0.276522\pi$
$614$	2.94427	0.118821
$615$	0	0
$616$	−2.76393	−0.111362
$617$	9.76393	0.393081	0.196541	−	0.980496i	$-0.437029\pi$
$617$	9.76393	0.393081	0.196541	+	0.980496i	$0.437029\pi$
$618$	−7.14590	−0.287450
$619$	39.4721	1.58652	0.793260	−	0.608884i	$-0.208382\pi$
$619$	39.4721	1.58652	0.793260	+	0.608884i	$0.208382\pi$
$620$	0	0
$621$	−41.1803	−1.65251
$622$	−18.2361	−0.731200
$623$	−14.4721	−0.579814
$624$	−9.00000	−0.360288
$625$	0	0
$626$	−13.1246	−0.524565
$627$	4.47214	0.178600
$628$	14.8541	0.592743
$629$	−3.23607	−0.129030
$630$	0	0
$631$	−5.76393	−0.229459	−0.114729	−	0.993397i	$-0.536600\pi$
$631$	−5.76393	−0.229459	−0.114729	+	0.993397i	$0.536600\pi$
$632$	−6.90983	−0.274858
$633$	13.1803	0.523871
$634$	−14.6180	−0.580556
$635$	0	0
$636$	−8.85410	−0.351088
$637$	21.2705	0.842768
$638$	0.652476	0.0258318
$639$	8.76393	0.346696
$640$	0	0
$641$	10.0902	0.398538	0.199269	−	0.979945i	$-0.436143\pi$
$641$	10.0902	0.398538	0.199269	+	0.979945i	$0.436143\pi$
$642$	6.43769	0.254076
$643$	22.8328	0.900438	0.450219	−	0.892918i	$-0.351346\pi$
$643$	22.8328	0.900438	0.450219	+	0.892918i	$0.351346\pi$
$644$	21.5623	0.849674
$645$	0	0
$646$	2.76393	0.108745
$647$	30.5410	1.20069	0.600346	−	0.799741i	$-0.295030\pi$
$647$	30.5410	1.20069	0.600346	+	0.799741i	$0.295030\pi$
$648$	−2.23607	−0.0878410
$649$	3.16718	0.124323
$650$	0	0
$651$	4.85410	0.190247
$652$	−17.7984	−0.697038
$653$	7.90983	0.309536	0.154768	−	0.987951i	$-0.450537\pi$
$653$	7.90983	0.309536	0.154768	+	0.987951i	$0.450537\pi$
$654$	6.18034	0.241670
$655$	0	0
$656$	−9.70820	−0.379042
$657$	18.0000	0.702247
$658$	1.61803	0.0630775
$659$	24.4721	0.953299	0.476650	−	0.879093i	$-0.341851\pi$
$659$	24.4721	0.953299	0.476650	+	0.879093i	$0.341851\pi$
$660$	0	0
$661$	−40.6869	−1.58254	−0.791269	−	0.611468i	$-0.790579\pi$
$661$	−40.6869	−1.58254	−0.791269	+	0.611468i	$0.790579\pi$
$662$	10.5836	0.411343
$663$	3.70820	0.144015
$664$	3.94427	0.153067
$665$	0	0
$666$	−5.23607	−0.202894
$667$	−11.3820	−0.440711
$668$	9.00000	0.348220
$669$	22.1803	0.857541
$670$	0	0
$671$	3.59675	0.138851
$672$	−9.09017	−0.350661
$673$	−10.1803	−0.392423	−0.196212	−	0.980562i	$-0.562864\pi$
$673$	−10.1803	−0.392423	−0.196212	+	0.980562i	$0.562864\pi$
$674$	0.708204	0.0272790
$675$	0	0
$676$	−17.0902	−0.657314
$677$	8.38197	0.322145	0.161073	−	0.986943i	$-0.448505\pi$
$677$	8.38197	0.322145	0.161073	+	0.986943i	$0.448505\pi$
$678$	6.27051	0.240817
$679$	−4.61803	−0.177224
$680$	0	0
$681$	19.2361	0.737128
$682$	1.41641	0.0542371
$683$	−4.52786	−0.173254	−0.0866270	−	0.996241i	$-0.527609\pi$
$683$	−4.52786	−0.173254	−0.0866270	+	0.996241i	$0.527609\pi$
$684$	−18.9443	−0.724352
$685$	0	0
$686$	11.3820	0.434565
$687$	8.29180	0.316352
$688$	3.43769	0.131061
$689$	−26.5623	−1.01194
$690$	0	0
$691$	2.72949	0.103835	0.0519173	−	0.998651i	$-0.483467\pi$
$691$	2.72949	0.103835	0.0519173	+	0.998651i	$0.483467\pi$
$692$	27.3262	1.03879
$693$	−2.47214	−0.0939087
$694$	−19.2148	−0.729383
$695$	0	0
$696$	3.09017	0.117133
$697$	4.00000	0.151511
$698$	5.12461	0.193969
$699$	14.9443	0.565244
$700$	0	0
$701$	35.0132	1.32243	0.661214	−	0.750197i	$-0.270041\pi$
$701$	35.0132	1.32243	0.661214	+	0.750197i	$0.270041\pi$
$702$	15.0000	0.566139
$703$	−24.7984	−0.935288
$704$	0.180340	0.00679682
$705$	0	0
$706$	14.8885	0.560338
$707$	12.0902	0.454698
$708$	6.70820	0.252110
$709$	33.5410	1.25966	0.629830	−	0.776733i	$-0.283125\pi$
$709$	33.5410	1.25966	0.629830	+	0.776733i	$0.283125\pi$
$710$	0	0
$711$	−6.18034	−0.231781
$712$	−20.0000	−0.749532
$713$	−24.7082	−0.925330
$714$	0.763932	0.0285894
$715$	0	0
$716$	15.3262	0.572768
$717$	−29.4721	−1.10066
$718$	−17.7639	−0.662944
$719$	−36.7082	−1.36899	−0.684493	−	0.729020i	$-0.739976\pi$
$719$	−36.7082	−1.36899	−0.684493	+	0.729020i	$0.739976\pi$
$720$	0	0
$721$	18.7082	0.696730
$722$	9.43769	0.351235
$723$	11.4721	0.426653
$724$	−22.1803	−0.824326
$725$	0	0
$726$	−6.43769	−0.238925
$727$	4.43769	0.164585	0.0822925	−	0.996608i	$-0.473776\pi$
$727$	4.43769	0.164585	0.0822925	+	0.996608i	$0.473776\pi$
$728$	−17.5623	−0.650902
$729$	13.0000	0.481481
$730$	0	0
$731$	−1.41641	−0.0523877
$732$	7.61803	0.281571
$733$	26.9787	0.996482	0.498241	−	0.867039i	$-0.333980\pi$
$733$	26.9787	0.996482	0.498241	+	0.867039i	$0.333980\pi$
$734$	−3.36068	−0.124045
$735$	0	0
$736$	46.2705	1.70555
$737$	−7.05573	−0.259901
$738$	6.47214	0.238243
$739$	−30.9787	−1.13957	−0.569785	−	0.821794i	$-0.692974\pi$
$739$	−30.9787	−1.13957	−0.569785	+	0.821794i	$0.692974\pi$
$740$	0	0
$741$	28.4164	1.04390
$742$	−5.47214	−0.200888
$743$	−16.3607	−0.600215	−0.300108	−	0.953905i	$-0.597023\pi$
$743$	−16.3607	−0.600215	−0.300108	+	0.953905i	$0.597023\pi$
$744$	6.70820	0.245935
$745$	0	0
$746$	3.25735	0.119260
$747$	3.52786	0.129078
$748$	−0.944272	−0.0345260
$749$	−16.8541	−0.615835
$750$	0	0
$751$	−40.8885	−1.49204	−0.746022	−	0.665921i	$-0.768039\pi$
$751$	−40.8885	−1.49204	−0.746022	+	0.665921i	$0.768039\pi$
$752$	−3.00000	−0.109399
$753$	−6.81966	−0.248522
$754$	4.14590	0.150985
$755$	0	0
$756$	−13.0902	−0.476085
$757$	3.58359	0.130248	0.0651239	−	0.997877i	$-0.479256\pi$
$757$	3.58359	0.130248	0.0651239	+	0.997877i	$0.479256\pi$
$758$	−21.3820	−0.776628
$759$	−6.29180	−0.228378
$760$	0	0
$761$	37.4508	1.35759	0.678796	−	0.734327i	$-0.262502\pi$
$761$	37.4508	1.35759	0.678796	+	0.734327i	$0.262502\pi$
$762$	−9.81966	−0.355729
$763$	−16.1803	−0.585768
$764$	39.1246	1.41548
$765$	0	0
$766$	−7.02129	−0.253689
$767$	20.1246	0.726658
$768$	−6.56231	−0.236797
$769$	−13.4164	−0.483808	−0.241904	−	0.970300i	$-0.577772\pi$
$769$	−13.4164	−0.483808	−0.241904	+	0.970300i	$0.577772\pi$
$770$	0	0
$771$	16.1459	0.581480
$772$	9.23607	0.332413
$773$	33.1591	1.19265	0.596324	−	0.802744i	$-0.296627\pi$
$773$	33.1591	1.19265	0.596324	+	0.802744i	$0.296627\pi$
$774$	−2.29180	−0.0823769
$775$	0	0
$776$	−6.38197	−0.229099
$777$	−6.85410	−0.245890
$778$	9.27051	0.332364
$779$	30.6525	1.09824
$780$	0	0
$781$	3.34752	0.119784
$782$	−3.88854	−0.139054
$783$	6.90983	0.246937
$784$	−8.12461	−0.290165
$785$	0	0
$786$	−11.0000	−0.392357
$787$	−34.1803	−1.21840	−0.609199	−	0.793018i	$-0.708509\pi$
$787$	−34.1803	−1.21840	−0.609199	+	0.793018i	$0.708509\pi$
$788$	15.7082	0.559582
$789$	−22.0902	−0.786431
$790$	0	0
$791$	−16.4164	−0.583700
$792$	−3.41641	−0.121397
$793$	22.8541	0.811573
$794$	−0.0212862	−0.000755420 0
$795$	0	0
$796$	−4.14590	−0.146947
$797$	14.2361	0.504267	0.252134	−	0.967692i	$-0.418868\pi$
$797$	14.2361	0.504267	0.252134	+	0.967692i	$0.418868\pi$
$798$	5.85410	0.207233
$799$	1.23607	0.0437289
$800$	0	0
$801$	−17.8885	−0.632061
$802$	−13.9656	−0.493141
$803$	6.87539	0.242627
$804$	−14.9443	−0.527044
$805$	0	0
$806$	9.00000	0.317011
$807$	−17.2361	−0.606738
$808$	16.7082	0.587793
$809$	−15.9787	−0.561782	−0.280891	−	0.959740i	$-0.590630\pi$
$809$	−15.9787	−0.561782	−0.280891	+	0.959740i	$0.590630\pi$
$810$	0	0
$811$	−1.29180	−0.0453611	−0.0226805	−	0.999743i	$-0.507220\pi$
$811$	−1.29180	−0.0453611	−0.0226805	+	0.999743i	$0.507220\pi$
$812$	−3.61803	−0.126968
$813$	−8.00000	−0.280572
$814$	−2.00000	−0.0701000
$815$	0	0
$816$	−1.41641	−0.0495842
$817$	−10.8541	−0.379737
$818$	17.5623	0.614052
$819$	−15.7082	−0.548889
$820$	0	0
$821$	19.6869	0.687078	0.343539	−	0.939138i	$-0.388374\pi$
$821$	19.6869	0.687078	0.343539	+	0.939138i	$0.388374\pi$
$822$	3.67376	0.128137
$823$	34.2918	1.19534	0.597668	−	0.801743i	$-0.296094\pi$
$823$	34.2918	1.19534	0.597668	+	0.801743i	$0.296094\pi$
$824$	25.8541	0.900670
$825$	0	0
$826$	4.14590	0.144254
$827$	−30.0344	−1.04440	−0.522200	−	0.852823i	$-0.674889\pi$
$827$	−30.0344	−1.04440	−0.522200	+	0.852823i	$0.674889\pi$
$828$	26.6525	0.926238
$829$	−29.1459	−1.01228	−0.506139	−	0.862452i	$-0.668928\pi$
$829$	−29.1459	−1.01228	−0.506139	+	0.862452i	$0.668928\pi$
$830$	0	0
$831$	−11.2918	−0.391708
$832$	1.14590	0.0397269
$833$	3.34752	0.115985
$834$	3.09017	0.107004
$835$	0	0
$836$	−7.23607	−0.250265
$837$	15.0000	0.518476
$838$	−0.326238	−0.0112697
$839$	4.14590	0.143132	0.0715661	−	0.997436i	$-0.477200\pi$
$839$	4.14590	0.143132	0.0715661	+	0.997436i	$0.477200\pi$
$840$	0	0
$841$	−27.0902	−0.934144
$842$	19.7771	0.681563
$843$	−1.09017	−0.0375474
$844$	−21.3262	−0.734079
$845$	0	0
$846$	2.00000	0.0687614
$847$	16.8541	0.579114
$848$	10.1459	0.348412
$849$	−23.1459	−0.794365
$850$	0	0
$851$	34.8885	1.19596
$852$	7.09017	0.242905
$853$	47.3050	1.61969	0.809845	−	0.586643i	$-0.199551\pi$
$853$	47.3050	1.61969	0.809845	+	0.586643i	$0.199551\pi$
$854$	4.70820	0.161111
$855$	0	0
$856$	−23.2918	−0.796097
$857$	−40.6869	−1.38984	−0.694919	−	0.719088i	$-0.744560\pi$
$857$	−40.6869	−1.38984	−0.694919	+	0.719088i	$0.744560\pi$
$858$	2.29180	0.0782406
$859$	−28.4164	−0.969555	−0.484778	−	0.874637i	$-0.661099\pi$
$859$	−28.4164	−0.969555	−0.484778	+	0.874637i	$0.661099\pi$
$860$	0	0
$861$	8.47214	0.288730
$862$	14.7295	0.501688
$863$	−41.5623	−1.41480	−0.707399	−	0.706815i	$-0.750131\pi$
$863$	−41.5623	−1.41480	−0.707399	+	0.706815i	$0.750131\pi$
$864$	−28.0902	−0.955647
$865$	0	0
$866$	12.4508	0.423097
$867$	−16.4164	−0.557530
$868$	−7.85410	−0.266586
$869$	−2.36068	−0.0800806
$870$	0	0
$871$	−44.8328	−1.51910
$872$	−22.3607	−0.757228
$873$	−5.70820	−0.193193
$874$	−29.7984	−1.00795
$875$	0	0
$876$	14.5623	0.492015
$877$	30.5410	1.03130	0.515648	−	0.856800i	$-0.327551\pi$
$877$	30.5410	1.03130	0.515648	+	0.856800i	$0.327551\pi$
$878$	3.69505	0.124702
$879$	−28.4721	−0.960341
$880$	0	0
$881$	4.36068	0.146915	0.0734575	−	0.997298i	$-0.476597\pi$
$881$	4.36068	0.146915	0.0734575	+	0.997298i	$0.476597\pi$
$882$	5.41641	0.182380
$883$	−47.4164	−1.59569	−0.797845	−	0.602863i	$-0.794026\pi$
$883$	−47.4164	−1.59569	−0.797845	+	0.602863i	$0.794026\pi$
$884$	−6.00000	−0.201802
$885$	0	0
$886$	7.45085	0.250316
$887$	−5.88854	−0.197718	−0.0988590	−	0.995101i	$-0.531519\pi$
$887$	−5.88854	−0.197718	−0.0988590	+	0.995101i	$0.531519\pi$
$888$	−9.47214	−0.317864
$889$	25.7082	0.862225
$890$	0	0
$891$	−0.763932	−0.0255927
$892$	−35.8885	−1.20164
$893$	9.47214	0.316973
$894$	8.61803	0.288230
$895$	0	0
$896$	18.4164	0.615249
$897$	−39.9787	−1.33485
$898$	12.5623	0.419210
$899$	4.14590	0.138273
$900$	0	0
$901$	−4.18034	−0.139267
$902$	2.47214	0.0823131
$903$	−3.00000	−0.0998337
$904$	−22.6869	−0.754556
$905$	0	0
$906$	−3.43769	−0.114210
$907$	47.2492	1.56888	0.784442	−	0.620202i	$-0.212949\pi$
$907$	47.2492	1.56888	0.784442	+	0.620202i	$0.212949\pi$
$908$	−31.1246	−1.03291
$909$	14.9443	0.495670
$910$	0	0
$911$	−35.7639	−1.18491	−0.592456	−	0.805603i	$-0.701842\pi$
$911$	−35.7639	−1.18491	−0.592456	+	0.805603i	$0.701842\pi$
$912$	−10.8541	−0.359415
$913$	1.34752	0.0445965
$914$	3.34752	0.110726
$915$	0	0
$916$	−13.4164	−0.443291
$917$	28.7984	0.951006
$918$	2.36068	0.0779140
$919$	−1.78522	−0.0588889	−0.0294445	−	0.999566i	$-0.509374\pi$
$919$	−1.78522	−0.0588889	−0.0294445	+	0.999566i	$0.509374\pi$
$920$	0	0
$921$	4.76393	0.156977
$922$	14.3262	0.471810
$923$	21.2705	0.700127
$924$	−2.00000	−0.0657952
$925$	0	0
$926$	9.96556	0.327489
$927$	23.1246	0.759512
$928$	−7.76393	−0.254864
$929$	−36.6312	−1.20183	−0.600915	−	0.799313i	$-0.705197\pi$
$929$	−36.6312	−1.20183	−0.600915	+	0.799313i	$0.705197\pi$
$930$	0	0
$931$	25.6525	0.840726
$932$	−24.1803	−0.792053
$933$	−29.5066	−0.966002
$934$	−17.5836	−0.575353
$935$	0	0
$936$	−21.7082	−0.709555
$937$	51.2705	1.67493	0.837467	−	0.546487i	$-0.184035\pi$
$937$	51.2705	1.67493	0.837467	+	0.546487i	$0.184035\pi$
$938$	−9.23607	−0.301568
$939$	−21.2361	−0.693013
$940$	0	0
$941$	−19.5836	−0.638407	−0.319203	−	0.947686i	$-0.603415\pi$
$941$	−19.5836	−0.638407	−0.319203	+	0.947686i	$0.603415\pi$
$942$	−5.67376	−0.184861
$943$	−43.1246	−1.40433
$944$	−7.68692	−0.250188
$945$	0	0
$946$	−0.875388	−0.0284613
$947$	−28.6525	−0.931080	−0.465540	−	0.885027i	$-0.654140\pi$
$947$	−28.6525	−0.931080	−0.465540	+	0.885027i	$0.654140\pi$
$948$	−5.00000	−0.162392
$949$	43.6869	1.41814
$950$	0	0
$951$	−23.6525	−0.766984
$952$	−2.76393	−0.0895796
$953$	34.7426	1.12542	0.562712	−	0.826653i	$-0.309758\pi$
$953$	34.7426	1.12542	0.562712	+	0.826653i	$0.309758\pi$
$954$	−6.76393	−0.218990
$955$	0	0
$956$	47.6869	1.54231
$957$	1.05573	0.0341268
$958$	−2.56231	−0.0827843
$959$	−9.61803	−0.310583
$960$	0	0
$961$	−22.0000	−0.709677
$962$	−12.7082	−0.409729
$963$	−20.8328	−0.671328
$964$	−18.5623	−0.597852
$965$	0	0
$966$	−8.23607	−0.264991
$967$	39.8885	1.28273	0.641365	−	0.767236i	$-0.278369\pi$
$967$	39.8885	1.28273	0.641365	+	0.767236i	$0.278369\pi$
$968$	23.2918	0.748627
$969$	4.47214	0.143666
$970$	0	0
$971$	3.38197	0.108532	0.0542662	−	0.998527i	$-0.482718\pi$
$971$	3.38197	0.108532	0.0542662	+	0.998527i	$0.482718\pi$
$972$	−25.8885	−0.830375
$973$	−8.09017	−0.259359
$974$	−5.92299	−0.189785
$975$	0	0
$976$	−8.72949	−0.279424
$977$	−33.6525	−1.07664	−0.538319	−	0.842741i	$-0.680940\pi$
$977$	−33.6525	−1.07664	−0.538319	+	0.842741i	$0.680940\pi$
$978$	6.79837	0.217388
$979$	−6.83282	−0.218378
$980$	0	0
$981$	−20.0000	−0.638551
$982$	23.0213	0.734639
$983$	7.38197	0.235448	0.117724	−	0.993046i	$-0.462440\pi$
$983$	7.38197	0.235448	0.117724	+	0.993046i	$0.462440\pi$
$984$	11.7082	0.373244
$985$	0	0
$986$	0.652476	0.0207791
$987$	2.61803	0.0833329
$988$	−45.9787	−1.46278
$989$	15.2705	0.485574
$990$	0	0
$991$	29.3607	0.932673	0.466336	−	0.884607i	$-0.345574\pi$
$991$	29.3607	0.932673	0.466336	+	0.884607i	$0.345574\pi$
$992$	−16.8541	−0.535118
$993$	17.1246	0.543433
$994$	4.38197	0.138988
$995$	0	0
$996$	2.85410	0.0904357
$997$	−10.8885	−0.344844	−0.172422	−	0.985023i	$-0.555159\pi$
$997$	−10.8885	−0.344844	−0.172422	+	0.985023i	$0.555159\pi$
$998$	−7.76393	−0.245763
$999$	−21.1803	−0.670116

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	625.2.a.b.1.2		2
3.2	odd	2		5625.2.a.f.1.1		2
4.3	odd	2		10000.2.a.c.1.2		2
5.2	odd	4		625.2.b.a.624.3		4
5.3	odd	4		625.2.b.a.624.2		4
5.4	even	2		625.2.a.c.1.1		2
15.14	odd	2		5625.2.a.d.1.2		2
20.19	odd	2		10000.2.a.l.1.1		2
25.2	odd	20		625.2.e.c.124.2		8
25.3	odd	20		125.2.e.a.49.1		8
25.4	even	10		125.2.d.a.76.1		4
25.6	even	5		25.2.d.a.11.1	✓	4
25.8	odd	20		125.2.e.a.74.2		8
25.9	even	10		625.2.d.b.126.1		4
25.11	even	5		625.2.d.h.501.1		4
25.12	odd	20		625.2.e.c.499.1		8
25.13	odd	20		625.2.e.c.499.2		8
25.14	even	10		625.2.d.b.501.1		4
25.16	even	5		625.2.d.h.126.1		4
25.17	odd	20		125.2.e.a.74.1		8
25.19	even	10		125.2.d.a.51.1		4
25.21	even	5		25.2.d.a.16.1	yes	4
25.22	odd	20		125.2.e.a.49.2		8
25.23	odd	20		625.2.e.c.124.1		8
75.56	odd	10		225.2.h.b.136.1		4
75.71	odd	10		225.2.h.b.91.1		4
100.31	odd	10		400.2.u.b.161.1		4
100.71	odd	10		400.2.u.b.241.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.2.d.a.11.1	✓	4	25.6	even	5
25.2.d.a.16.1	yes	4	25.21	even	5
125.2.d.a.51.1		4	25.19	even	10
125.2.d.a.76.1		4	25.4	even	10
125.2.e.a.49.1		8	25.3	odd	20
125.2.e.a.49.2		8	25.22	odd	20
125.2.e.a.74.1		8	25.17	odd	20
125.2.e.a.74.2		8	25.8	odd	20
225.2.h.b.91.1		4	75.71	odd	10
225.2.h.b.136.1		4	75.56	odd	10
400.2.u.b.161.1		4	100.31	odd	10
400.2.u.b.241.1		4	100.71	odd	10
625.2.a.b.1.2		2	1.1	even	1	trivial
625.2.a.c.1.1		2	5.4	even	2
625.2.b.a.624.2		4	5.3	odd	4
625.2.b.a.624.3		4	5.2	odd	4
625.2.d.b.126.1		4	25.9	even	10
625.2.d.b.501.1		4	25.14	even	10
625.2.d.h.126.1		4	25.16	even	5
625.2.d.h.501.1		4	25.11	even	5
625.2.e.c.124.1		8	25.23	odd	20
625.2.e.c.124.2		8	25.2	odd	20
625.2.e.c.499.1		8	25.12	odd	20
625.2.e.c.499.2		8	25.13	odd	20
5625.2.a.d.1.2		2	15.14	odd	2
5625.2.a.f.1.1		2	3.2	odd	2
10000.2.a.c.1.2		2	4.3	odd	2
10000.2.a.l.1.1		2	20.19	odd	2

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:	\( N \)	\(=\)	\( 625 = 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	625.a (trivial)

\(f(q)\)	\(=\)	\(q+0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} +0.618034 q^{6} -1.61803 q^{7} -2.23607 q^{8} -2.00000 q^{9} -0.763932 q^{11} -1.61803 q^{12} -4.85410 q^{13} -1.00000 q^{14} +1.85410 q^{16} -0.763932 q^{17} -1.23607 q^{18} -5.85410 q^{19} -1.61803 q^{21} -0.472136 q^{22} +8.23607 q^{23} -2.23607 q^{24} -3.00000 q^{26} -5.00000 q^{27} +2.61803 q^{28} -1.38197 q^{29} -3.00000 q^{31} +5.61803 q^{32} -0.763932 q^{33} -0.472136 q^{34} +3.23607 q^{36} +4.23607 q^{37} -3.61803 q^{38} -4.85410 q^{39} -5.23607 q^{41} -1.00000 q^{42} +1.85410 q^{43} +1.23607 q^{44} +5.09017 q^{46} -1.61803 q^{47} +1.85410 q^{48} -4.38197 q^{49} -0.763932 q^{51} +7.85410 q^{52} +5.47214 q^{53} -3.09017 q^{54} +3.61803 q^{56} -5.85410 q^{57} -0.854102 q^{58} -4.14590 q^{59} -4.70820 q^{61} -1.85410 q^{62} +3.23607 q^{63} -0.236068 q^{64} -0.472136 q^{66} +9.23607 q^{67} +1.23607 q^{68} +8.23607 q^{69} -4.38197 q^{71} +4.47214 q^{72} -9.00000 q^{73} +2.61803 q^{74} +9.47214 q^{76} +1.23607 q^{77} -3.00000 q^{78} +3.09017 q^{79} +1.00000 q^{81} -3.23607 q^{82} -1.76393 q^{83} +2.61803 q^{84} +1.14590 q^{86} -1.38197 q^{87} +1.70820 q^{88} +8.94427 q^{89} +7.85410 q^{91} -13.3262 q^{92} -3.00000 q^{93} -1.00000 q^{94} +5.61803 q^{96} +2.85410 q^{97} -2.70820 q^{98} +1.52786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 2 q^{3} - q^{4} - q^{6} - q^{7} - 4 q^{9} - 6 q^{11} - q^{12} - 3 q^{13} - 2 q^{14} - 3 q^{16} - 6 q^{17} + 2 q^{18} - 5 q^{19} - q^{21} + 8 q^{22} + 12 q^{23} - 6 q^{26} - 10 q^{27}+ \cdots + 12 q^{99}+O(q^{100}) \) 2 * q - q^2 + 2 * q^3 - q^4 - q^6 - q^7 - 4 * q^9 - 6 * q^11 - q^12 - 3 * q^13 - 2 * q^14 - 3 * q^16 - 6 * q^17 + 2 * q^18 - 5 * q^19 - q^21 + 8 * q^22 + 12 * q^23 - 6 * q^26 - 10 * q^27 + 3 * q^28 - 5 * q^29 - 6 * q^31 + 9 * q^32 - 6 * q^33 + 8 * q^34 + 2 * q^36 + 4 * q^37 - 5 * q^38 - 3 * q^39 - 6 * q^41 - 2 * q^42 - 3 * q^43 - 2 * q^44 - q^46 - q^47 - 3 * q^48 - 11 * q^49 - 6 * q^51 + 9 * q^52 + 2 * q^53 + 5 * q^54 + 5 * q^56 - 5 * q^57 + 5 * q^58 - 15 * q^59 + 4 * q^61 + 3 * q^62 + 2 * q^63 + 4 * q^64 + 8 * q^66 + 14 * q^67 - 2 * q^68 + 12 * q^69 - 11 * q^71 - 18 * q^73 + 3 * q^74 + 10 * q^76 - 2 * q^77 - 6 * q^78 - 5 * q^79 + 2 * q^81 - 2 * q^82 - 8 * q^83 + 3 * q^84 + 9 * q^86 - 5 * q^87 - 10 * q^88 + 9 * q^91 - 11 * q^92 - 6 * q^93 - 2 * q^94 + 9 * q^96 - q^97 + 8 * q^98 + 12 * q^99

Introduction

L-functions

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