LMFDB - Embedded newform 5625.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5625,2,Mod(1,5625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, 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("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.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:	$44.9158511370$
Analytic rank:	$0$
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			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	5625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.618034	−0.437016	−0.218508	−	0.975835i	$-0.570119\pi$
$2$	−0.618034	−0.437016	−0.218508	+	0.975835i	$0.570119\pi$
$3$	0	0
$3$	0	0
$4$	−1.61803	−0.809017
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−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
$9$	0	0
$10$	0	0
$11$	0.763932	0.230334	0.115167	−	0.993346i	$-0.463260\pi$
$11$	0.763932	0.230334	0.115167	+	0.993346i	$0.463260\pi$
$12$	0	0
$13$	−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	$-0.470469\pi$
$17$	0.763932	0.185281	0.0926404	+	0.995700i	$0.470469\pi$
$18$	0	0
$19$	−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
$21$	0	0
$22$	−0.472136	−0.100660
$23$	−8.23607	−1.71734	−0.858669	−	0.512530i	$-0.828708\pi$
$23$	−8.23607	−1.71734	−0.858669	+	0.512530i	$0.828708\pi$
$24$	0	0
$25$	0	0
$26$	3.00000	0.588348
$27$	0	0
$28$	2.61803	0.494762
$29$	1.38197	0.256625	0.128312	−	0.991734i	$-0.459044\pi$
$29$	1.38197	0.256625	0.128312	+	0.991734i	$0.459044\pi$
$30$	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	0
$34$	−0.472136	−0.0809706
$35$	0	0
$36$	0	0
$37$	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
$39$	0	0
$40$	0	0
$41$	5.23607	0.817736	0.408868	−	0.912593i	$-0.365924\pi$
$41$	5.23607	0.817736	0.408868	+	0.912593i	$0.365924\pi$
$42$	0	0
$43$	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	$-0.462349\pi$
$47$	1.61803	0.236015	0.118007	+	0.993013i	$0.462349\pi$
$48$	0	0
$49$	−4.38197	−0.625995
$50$	0	0
$51$	0	0
$52$	7.85410	1.08917
$53$	−5.47214	−0.751656	−0.375828	−	0.926690i	$-0.622642\pi$
$53$	−5.47214	−0.751656	−0.375828	+	0.926690i	$0.622642\pi$
$54$	0	0
$55$	0	0
$56$	−3.61803	−0.483480
$57$	0	0
$58$	−0.854102	−0.112149
$59$	4.14590	0.539750	0.269875	−	0.962895i	$-0.413018\pi$
$59$	4.14590	0.539750	0.269875	+	0.962895i	$0.413018\pi$
$60$	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
$63$	0	0
$64$	−0.236068	−0.0295085
$65$	0	0
$66$	0	0
$67$	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
$69$	0	0
$70$	0	0
$71$	4.38197	0.520044	0.260022	−	0.965603i	$-0.416270\pi$
$71$	4.38197	0.520044	0.260022	+	0.965603i	$0.416270\pi$
$72$	0	0
$73$	−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
$78$	0	0
$79$	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
$81$	0	0
$82$	−3.23607	−0.357364
$83$	1.76393	0.193617	0.0968083	−	0.995303i	$-0.469137\pi$
$83$	1.76393	0.193617	0.0968083	+	0.995303i	$0.469137\pi$
$84$	0	0
$85$	0	0
$86$	−1.14590	−0.123565
$87$	0	0
$88$	1.70820	0.182095
$89$	−8.94427	−0.948091	−0.474045	−	0.880500i	$-0.657207\pi$
$89$	−8.94427	−0.948091	−0.474045	+	0.880500i	$0.657207\pi$
$90$	0	0
$91$	7.85410	0.823334
$92$	13.3262	1.38936
$93$	0	0
$94$	−1.00000	−0.103142
$95$	0	0
$96$	0	0
$97$	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
$99$	0	0
$100$	0	0
$101$	7.47214	0.743505	0.371753	−	0.928332i	$-0.378757\pi$
$101$	7.47214	0.743505	0.371753	+	0.928332i	$0.378757\pi$
$102$	0	0
$103$	−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	$-0.667953\pi$
$107$	−10.4164	−1.00699	−0.503496	+	0.863998i	$0.667953\pi$
$108$	0	0
$109$	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
$111$	0	0
$112$	−3.00000	−0.283473
$113$	−10.1459	−0.954446	−0.477223	−	0.878782i	$-0.658357\pi$
$113$	−10.1459	−0.954446	−0.477223	+	0.878782i	$0.658357\pi$
$114$	0	0
$115$	0	0
$116$	−2.23607	−0.207614
$117$	0	0
$118$	−2.56231	−0.235879
$119$	−1.23607	−0.113310
$120$	0	0
$121$	−10.4164	−0.946946
$122$	2.90983	0.263444
$123$	0	0
$124$	4.85410	0.435911
$125$	0	0
$126$	0	0
$127$	−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
$129$	0	0
$130$	0	0
$131$	17.7984	1.55505	0.777526	−	0.628851i	$-0.216475\pi$
$131$	17.7984	1.55505	0.777526	+	0.628851i	$0.216475\pi$
$132$	0	0
$133$	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	$-0.581722\pi$
$137$	−5.94427	−0.507853	−0.253927	+	0.967223i	$0.581722\pi$
$138$	0	0
$139$	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
$141$	0	0
$142$	−2.70820	−0.227267
$143$	−3.70820	−0.310096
$144$	0	0
$145$	0	0
$146$	5.56231	0.460340
$147$	0	0
$148$	−6.85410	−0.563404
$149$	−13.9443	−1.14236	−0.571180	−	0.820825i	$-0.693514\pi$
$149$	−13.9443	−1.14236	−0.571180	+	0.820825i	$0.693514\pi$
$150$	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
$153$	0	0
$154$	0.763932	0.0615594
$155$	0	0
$156$	0	0
$157$	−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
$159$	0	0
$160$	0	0
$161$	13.3262	1.05025
$162$	0	0
$163$	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	$-0.430956\pi$
$167$	5.56231	0.430424	0.215212	+	0.976567i	$0.430956\pi$
$168$	0	0
$169$	10.5623	0.812485
$170$	0	0
$171$	0	0
$172$	−3.00000	−0.228748
$173$	16.8885	1.28401	0.642006	−	0.766700i	$-0.278102\pi$
$173$	16.8885	1.28401	0.642006	+	0.766700i	$0.278102\pi$
$174$	0	0
$175$	0	0
$176$	1.41641	0.106766
$177$	0	0
$178$	5.52786	0.414331
$179$	9.47214	0.707981	0.353990	−	0.935249i	$-0.384825\pi$
$179$	9.47214	0.707981	0.353990	+	0.935249i	$0.384825\pi$
$180$	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
$183$	0	0
$184$	−18.4164	−1.35768
$185$	0	0
$186$	0	0
$187$	0.583592	0.0426765
$188$	−2.61803	−0.190940
$189$	0	0
$190$	0	0
$191$	24.1803	1.74963	0.874814	−	0.484459i	$-0.160984\pi$
$191$	24.1803	1.74963	0.874814	+	0.484459i	$0.160984\pi$
$192$	0	0
$193$	−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	$-0.387594\pi$
$197$	9.70820	0.691681	0.345840	+	0.938293i	$0.387594\pi$
$198$	0	0
$199$	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
$201$	0	0
$202$	−4.61803	−0.324924
$203$	−2.23607	−0.156941
$204$	0	0
$205$	0	0
$206$	7.14590	0.497878
$207$	0	0
$208$	−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
$213$	0	0
$214$	6.43769	0.440072
$215$	0	0
$216$	0	0
$217$	4.85410	0.329518
$218$	−6.18034	−0.418585
$219$	0	0
$220$	0	0
$221$	−3.70820	−0.249441
$222$	0	0
$223$	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	$-0.720392\pi$
$227$	−19.2361	−1.27674	−0.638371	+	0.769729i	$0.720392\pi$
$228$	0	0
$229$	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
$231$	0	0
$232$	3.09017	0.202880
$233$	−14.9443	−0.979032	−0.489516	−	0.871994i	$-0.662826\pi$
$233$	−14.9443	−0.979032	−0.489516	+	0.871994i	$0.662826\pi$
$234$	0	0
$235$	0	0
$236$	−6.70820	−0.436667
$237$	0	0
$238$	0.763932	0.0495184
$239$	29.4721	1.90639	0.953197	−	0.302350i	$-0.0977711\pi$
$239$	29.4721	1.90639	0.953197	+	0.302350i	$0.0977711\pi$
$240$	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
$243$	0	0
$244$	7.61803	0.487695
$245$	0	0
$246$	0	0
$247$	28.4164	1.80809
$248$	−6.70820	−0.425971
$249$	0	0
$250$	0	0
$251$	6.81966	0.430453	0.215227	−	0.976564i	$-0.430951\pi$
$251$	6.81966	0.430453	0.215227	+	0.976564i	$0.430951\pi$
$252$	0	0
$253$	−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	$-0.667983\pi$
$257$	−16.1459	−1.00715	−0.503577	+	0.863951i	$0.667983\pi$
$258$	0	0
$259$	−6.85410	−0.425893
$260$	0	0
$261$	0	0
$262$	−11.0000	−0.679582
$263$	22.0902	1.36214	0.681069	−	0.732219i	$-0.261515\pi$
$263$	22.0902	1.36214	0.681069	+	0.732219i	$0.261515\pi$
$264$	0	0
$265$	0	0
$266$	−5.85410	−0.358938
$267$	0	0
$268$	−14.9443	−0.912867
$269$	17.2361	1.05090	0.525451	−	0.850824i	$-0.323897\pi$
$269$	17.2361	1.05090	0.525451	+	0.850824i	$0.323897\pi$
$270$	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
$273$	0	0
$274$	3.67376	0.221940
$275$	0	0
$276$	0	0
$277$	−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
$279$	0	0
$280$	0	0
$281$	1.09017	0.0650341	0.0325170	−	0.999471i	$-0.489648\pi$
$281$	1.09017	0.0650341	0.0325170	+	0.999471i	$0.489648\pi$
$282$	0	0
$283$	−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
$288$	0	0
$289$	−16.4164	−0.965671
$290$	0	0
$291$	0	0
$292$	14.5623	0.852194
$293$	28.4721	1.66336	0.831680	−	0.555255i	$-0.187379\pi$
$293$	28.4721	1.66336	0.831680	+	0.555255i	$0.187379\pi$
$294$	0	0
$295$	0	0
$296$	9.47214	0.550557
$297$	0	0
$298$	8.61803	0.499229
$299$	39.9787	2.31203
$300$	0	0
$301$	−3.00000	−0.172917
$302$	3.43769	0.197817
$303$	0	0
$304$	−10.8541	−0.622525
$305$	0	0
$306$	0	0
$307$	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
$309$	0	0
$310$	0	0
$311$	29.5066	1.67316	0.836582	−	0.547841i	$-0.184550\pi$
$311$	29.5066	1.67316	0.836582	+	0.547841i	$0.184550\pi$
$312$	0	0
$313$	−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	$-0.268761\pi$
$317$	23.6525	1.32846	0.664228	+	0.747530i	$0.268761\pi$
$318$	0	0
$319$	1.05573	0.0591094
$320$	0	0
$321$	0	0
$322$	−8.23607	−0.458978
$323$	−4.47214	−0.248836
$324$	0	0
$325$	0	0
$326$	−6.79837	−0.376527
$327$	0	0
$328$	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
$333$	0	0
$334$	−3.43769	−0.188102
$335$	0	0
$336$	0	0
$337$	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
$339$	0	0
$340$	0	0
$341$	−2.29180	−0.124108
$342$	0	0
$343$	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	$-0.185754\pi$
$347$	31.0902	1.66901	0.834504	+	0.551002i	$0.185754\pi$
$348$	0	0
$349$	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
$351$	0	0
$352$	−4.29180	−0.228753
$353$	−24.0902	−1.28219	−0.641095	−	0.767461i	$-0.721520\pi$
$353$	−24.0902	−1.28219	−0.641095	+	0.767461i	$0.721520\pi$
$354$	0	0
$355$	0	0
$356$	14.4721	0.767022
$357$	0	0
$358$	−5.85410	−0.309399
$359$	28.7426	1.51698	0.758489	−	0.651685i	$-0.225938\pi$
$359$	28.7426	1.51698	0.758489	+	0.651685i	$0.225938\pi$
$360$	0	0
$361$	15.2705	0.803711
$362$	−8.47214	−0.445286
$363$	0	0
$364$	−12.7082	−0.666091
$365$	0	0
$366$	0	0
$367$	−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
$369$	0	0
$370$	0	0
$371$	8.85410	0.459682
$372$	0	0
$373$	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
$378$	0	0
$379$	−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
$381$	0	0
$382$	−14.9443	−0.764615
$383$	11.3607	0.580504	0.290252	−	0.956950i	$-0.406261\pi$
$383$	11.3607	0.580504	0.290252	+	0.956950i	$0.406261\pi$
$384$	0	0
$385$	0	0
$386$	3.52786	0.179564
$387$	0	0
$388$	−4.61803	−0.234445
$389$	−15.0000	−0.760530	−0.380265	−	0.924878i	$-0.624167\pi$
$389$	−15.0000	−0.760530	−0.380265	+	0.924878i	$0.624167\pi$
$390$	0	0
$391$	−6.29180	−0.318190
$392$	−9.79837	−0.494893
$393$	0	0
$394$	−6.00000	−0.302276
$395$	0	0
$396$	0	0
$397$	−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
$399$	0	0
$400$	0	0
$401$	22.5967	1.12843	0.564214	−	0.825629i	$-0.309179\pi$
$401$	22.5967	1.12843	0.564214	+	0.825629i	$0.309179\pi$
$402$	0	0
$403$	14.5623	0.725400
$404$	−12.0902	−0.601508
$405$	0	0
$406$	1.38197	0.0685858
$407$	3.23607	0.160406
$408$	0	0
$409$	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
$411$	0	0
$412$	18.7082	0.921687
$413$	−6.70820	−0.330089
$414$	0	0
$415$	0	0
$416$	27.2705	1.33705
$417$	0	0
$418$	2.76393	0.135188
$419$	0.527864	0.0257878	0.0128939	−	0.999917i	$-0.495896\pi$
$419$	0.527864	0.0257878	0.0128939	+	0.999917i	$0.495896\pi$
$420$	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
$423$	0	0
$424$	−12.2361	−0.594236
$425$	0	0
$426$	0	0
$427$	7.61803	0.368663
$428$	16.8541	0.814674
$429$	0	0
$430$	0	0
$431$	−23.8328	−1.14799	−0.573993	−	0.818860i	$-0.694606\pi$
$431$	−23.8328	−1.14799	−0.573993	+	0.818860i	$0.694606\pi$
$432$	0	0
$433$	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
$438$	0	0
$439$	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
$441$	0	0
$442$	2.29180	0.109010
$443$	−12.0557	−0.572785	−0.286392	−	0.958112i	$-0.592456\pi$
$443$	−12.0557	−0.572785	−0.286392	+	0.958112i	$0.592456\pi$
$444$	0	0
$445$	0	0
$446$	−13.7082	−0.649102
$447$	0	0
$448$	0.381966	0.0180462
$449$	−20.3262	−0.959254	−0.479627	−	0.877472i	$-0.659228\pi$
$449$	−20.3262	−0.959254	−0.479627	+	0.877472i	$0.659228\pi$
$450$	0	0
$451$	4.00000	0.188353
$452$	16.4164	0.772163
$453$	0	0
$454$	11.8885	0.557957
$455$	0	0
$456$	0	0
$457$	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
$459$	0	0
$460$	0	0
$461$	−23.1803	−1.07962	−0.539808	−	0.841788i	$-0.681503\pi$
$461$	−23.1803	−1.07962	−0.539808	+	0.841788i	$0.681503\pi$
$462$	0	0
$463$	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	$-0.271287\pi$
$467$	28.4508	1.31655	0.658274	+	0.752778i	$0.271287\pi$
$468$	0	0
$469$	−14.9443	−0.690062
$470$	0	0
$471$	0	0
$472$	9.27051	0.426710
$473$	1.41641	0.0651265
$474$	0	0
$475$	0	0
$476$	2.00000	0.0916698
$477$	0	0
$478$	−18.2148	−0.833125
$479$	4.14590	0.189431	0.0947155	−	0.995504i	$-0.469806\pi$
$479$	4.14590	0.189431	0.0947155	+	0.995504i	$0.469806\pi$
$480$	0	0
$481$	−20.5623	−0.937560
$482$	−7.09017	−0.322948
$483$	0	0
$484$	16.8541	0.766096
$485$	0	0
$486$	0	0
$487$	−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
$489$	0	0
$490$	0	0
$491$	−37.2492	−1.68103	−0.840517	−	0.541785i	$-0.817749\pi$
$491$	−37.2492	−1.68103	−0.840517	+	0.541785i	$0.817749\pi$
$492$	0	0
$493$	1.05573	0.0475476
$494$	−17.5623	−0.790165
$495$	0	0
$496$	−5.56231	−0.249755
$497$	−7.09017	−0.318038
$498$	0	0
$499$	−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
$501$	0	0
$502$	−4.21478	−0.188115
$503$	10.5836	0.471899	0.235950	−	0.971765i	$-0.424180\pi$
$503$	10.5836	0.471899	0.235950	+	0.971765i	$0.424180\pi$
$504$	0	0
$505$	0	0
$506$	3.88854	0.172867
$507$	0	0
$508$	25.7082	1.14062
$509$	4.67376	0.207161	0.103580	−	0.994621i	$-0.466970\pi$
$509$	4.67376	0.207161	0.103580	+	0.994621i	$0.466970\pi$
$510$	0	0
$511$	14.5623	0.644198
$512$	−18.7082	−0.826794
$513$	0	0
$514$	9.97871	0.440142
$515$	0	0
$516$	0	0
$517$	1.23607	0.0543622
$518$	4.23607	0.186122
$519$	0	0
$520$	0	0
$521$	15.3607	0.672964	0.336482	−	0.941690i	$-0.390763\pi$
$521$	15.3607	0.672964	0.336482	+	0.941690i	$0.390763\pi$
$522$	0	0
$523$	−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
$528$	0	0
$529$	44.8328	1.94925
$530$	0	0
$531$	0	0
$532$	−15.3262	−0.664477
$533$	−25.4164	−1.10091
$534$	0	0
$535$	0	0
$536$	20.6525	0.892051
$537$	0	0
$538$	−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
$543$	0	0
$544$	−4.29180	−0.184009
$545$	0	0
$546$	0	0
$547$	−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
$549$	0	0
$550$	0	0
$551$	−8.09017	−0.344653
$552$	0	0
$553$	−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	$-0.562689\pi$
$557$	−9.23607	−0.391345	−0.195672	+	0.980669i	$0.562689\pi$
$558$	0	0
$559$	−9.00000	−0.380659
$560$	0	0
$561$	0	0
$562$	−0.673762	−0.0284209
$563$	−9.61803	−0.405352	−0.202676	−	0.979246i	$-0.564964\pi$
$563$	−9.61803	−0.405352	−0.202676	+	0.979246i	$0.564964\pi$
$564$	0	0
$565$	0	0
$566$	14.3050	0.601282
$567$	0	0
$568$	9.79837	0.411131
$569$	−29.4721	−1.23554	−0.617768	−	0.786360i	$-0.711963\pi$
$569$	−29.4721	−1.23554	−0.617768	+	0.786360i	$0.711963\pi$
$570$	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
$573$	0	0
$574$	5.23607	0.218549
$575$	0	0
$576$	0	0
$577$	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
$579$	0	0
$580$	0	0
$581$	−2.85410	−0.118408
$582$	0	0
$583$	−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	$-0.626173\pi$
$587$	−18.7082	−0.772170	−0.386085	+	0.922463i	$0.626173\pi$
$588$	0	0
$589$	17.5623	0.723642
$590$	0	0
$591$	0	0
$592$	7.85410	0.322802
$593$	22.0902	0.907135	0.453567	−	0.891222i	$-0.350151\pi$
$593$	22.0902	0.907135	0.453567	+	0.891222i	$0.350151\pi$
$594$	0	0
$595$	0	0
$596$	22.5623	0.924188
$597$	0	0
$598$	−24.7082	−1.01039
$599$	0.527864	0.0215679	0.0107840	−	0.999942i	$-0.496567\pi$
$599$	0.527864	0.0215679	0.0107840	+	0.999942i	$0.496567\pi$
$600$	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
$603$	0	0
$604$	9.00000	0.366205
$605$	0	0
$606$	0	0
$607$	−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
$609$	0	0
$610$	0	0
$611$	−7.85410	−0.317743
$612$	0	0
$613$	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	$-0.562971\pi$
$617$	−9.76393	−0.393081	−0.196541	+	0.980496i	$0.562971\pi$
$618$	0	0
$619$	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
$621$	0	0
$622$	−18.2361	−0.731200
$623$	14.4721	0.579814
$624$	0	0
$625$	0	0
$626$	13.1246	0.524565
$627$	0	0
$628$	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
$633$	0	0
$634$	−14.6180	−0.580556
$635$	0	0
$636$	0	0
$637$	21.2705	0.842768
$638$	−0.652476	−0.0258318
$639$	0	0
$640$	0	0
$641$	−10.0902	−0.398538	−0.199269	−	0.979945i	$-0.563857\pi$
$641$	−10.0902	−0.398538	−0.199269	+	0.979945i	$0.563857\pi$
$642$	0	0
$643$	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	$-0.704970\pi$
$647$	−30.5410	−1.20069	−0.600346	+	0.799741i	$0.704970\pi$
$648$	0	0
$649$	3.16718	0.124323
$650$	0	0
$651$	0	0
$652$	−17.7984	−0.697038
$653$	−7.90983	−0.309536	−0.154768	−	0.987951i	$-0.549463\pi$
$653$	−7.90983	−0.309536	−0.154768	+	0.987951i	$0.549463\pi$
$654$	0	0
$655$	0	0
$656$	9.70820	0.379042
$657$	0	0
$658$	1.61803	0.0630775
$659$	−24.4721	−0.953299	−0.476650	−	0.879093i	$-0.658149\pi$
$659$	−24.4721	−0.953299	−0.476650	+	0.879093i	$0.658149\pi$
$660$	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
$663$	0	0
$664$	3.94427	0.153067
$665$	0	0
$666$	0	0
$667$	−11.3820	−0.440711
$668$	−9.00000	−0.348220
$669$	0	0
$670$	0	0
$671$	−3.59675	−0.138851
$672$	0	0
$673$	−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	$-0.551495\pi$
$677$	−8.38197	−0.322145	−0.161073	+	0.986943i	$0.551495\pi$
$678$	0	0
$679$	−4.61803	−0.177224
$680$	0	0
$681$	0	0
$682$	1.41641	0.0542371
$683$	4.52786	0.173254	0.0866270	−	0.996241i	$-0.472391\pi$
$683$	4.52786	0.173254	0.0866270	+	0.996241i	$0.472391\pi$
$684$	0	0
$685$	0	0
$686$	−11.3820	−0.434565
$687$	0	0
$688$	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
$693$	0	0
$694$	−19.2148	−0.729383
$695$	0	0
$696$	0	0
$697$	4.00000	0.151511
$698$	−5.12461	−0.193969
$699$	0	0
$700$	0	0
$701$	−35.0132	−1.32243	−0.661214	−	0.750197i	$-0.729959\pi$
$701$	−35.0132	−1.32243	−0.661214	+	0.750197i	$0.729959\pi$
$702$	0	0
$703$	−24.7984	−0.935288
$704$	−0.180340	−0.00679682
$705$	0	0
$706$	14.8885	0.560338
$707$	−12.0902	−0.454698
$708$	0	0
$709$	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
$711$	0	0
$712$	−20.0000	−0.749532
$713$	24.7082	0.925330
$714$	0	0
$715$	0	0
$716$	−15.3262	−0.572768
$717$	0	0
$718$	−17.7639	−0.662944
$719$	36.7082	1.36899	0.684493	−	0.729020i	$-0.260024\pi$
$719$	36.7082	1.36899	0.684493	+	0.729020i	$0.260024\pi$
$720$	0	0
$721$	18.7082	0.696730
$722$	−9.43769	−0.351235
$723$	0	0
$724$	−22.1803	−0.824326
$725$	0	0
$726$	0	0
$727$	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
$729$	0	0
$730$	0	0
$731$	1.41641	0.0523877
$732$	0	0
$733$	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
$738$	0	0
$739$	−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
$741$	0	0
$742$	−5.47214	−0.200888
$743$	16.3607	0.600215	0.300108	−	0.953905i	$-0.402977\pi$
$743$	16.3607	0.600215	0.300108	+	0.953905i	$0.402977\pi$
$744$	0	0
$745$	0	0
$746$	−3.25735	−0.119260
$747$	0	0
$748$	−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
$753$	0	0
$754$	4.14590	0.150985
$755$	0	0
$756$	0	0
$757$	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
$759$	0	0
$760$	0	0
$761$	−37.4508	−1.35759	−0.678796	−	0.734327i	$-0.737498\pi$
$761$	−37.4508	−1.35759	−0.678796	+	0.734327i	$0.737498\pi$
$762$	0	0
$763$	−16.1803	−0.585768
$764$	−39.1246	−1.41548
$765$	0	0
$766$	−7.02129	−0.253689
$767$	−20.1246	−0.726658
$768$	0	0
$769$	−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
$771$	0	0
$772$	9.23607	0.332413
$773$	−33.1591	−1.19265	−0.596324	−	0.802744i	$-0.703373\pi$
$773$	−33.1591	−1.19265	−0.596324	+	0.802744i	$0.703373\pi$
$774$	0	0
$775$	0	0
$776$	6.38197	0.229099
$777$	0	0
$778$	9.27051	0.332364
$779$	−30.6525	−1.09824
$780$	0	0
$781$	3.34752	0.119784
$782$	3.88854	0.139054
$783$	0	0
$784$	−8.12461	−0.290165
$785$	0	0
$786$	0	0
$787$	−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
$789$	0	0
$790$	0	0
$791$	16.4164	0.583700
$792$	0	0
$793$	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	$-0.581132\pi$
$797$	−14.2361	−0.504267	−0.252134	+	0.967692i	$0.581132\pi$
$798$	0	0
$799$	1.23607	0.0437289
$800$	0	0
$801$	0	0
$802$	−13.9656	−0.493141
$803$	−6.87539	−0.242627
$804$	0	0
$805$	0	0
$806$	−9.00000	−0.317011
$807$	0	0
$808$	16.7082	0.587793
$809$	15.9787	0.561782	0.280891	−	0.959740i	$-0.409370\pi$
$809$	15.9787	0.561782	0.280891	+	0.959740i	$0.409370\pi$
$810$	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
$813$	0	0
$814$	−2.00000	−0.0701000
$815$	0	0
$816$	0	0
$817$	−10.8541	−0.379737
$818$	−17.5623	−0.614052
$819$	0	0
$820$	0	0
$821$	−19.6869	−0.687078	−0.343539	−	0.939138i	$-0.611626\pi$
$821$	−19.6869	−0.687078	−0.343539	+	0.939138i	$0.611626\pi$
$822$	0	0
$823$	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	$-0.325111\pi$
$827$	30.0344	1.04440	0.522200	+	0.852823i	$0.325111\pi$
$828$	0	0
$829$	−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
$831$	0	0
$832$	1.14590	0.0397269
$833$	−3.34752	−0.115985
$834$	0	0
$835$	0	0
$836$	7.23607	0.250265
$837$	0	0
$838$	−0.326238	−0.0112697
$839$	−4.14590	−0.143132	−0.0715661	−	0.997436i	$-0.522800\pi$
$839$	−4.14590	−0.143132	−0.0715661	+	0.997436i	$0.522800\pi$
$840$	0	0
$841$	−27.0902	−0.934144
$842$	−19.7771	−0.681563
$843$	0	0
$844$	−21.3262	−0.734079
$845$	0	0
$846$	0	0
$847$	16.8541	0.579114
$848$	−10.1459	−0.348412
$849$	0	0
$850$	0	0
$851$	−34.8885	−1.19596
$852$	0	0
$853$	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	$-0.255440\pi$
$857$	40.6869	1.38984	0.694919	+	0.719088i	$0.255440\pi$
$858$	0	0
$859$	−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
$861$	0	0
$862$	14.7295	0.501688
$863$	41.5623	1.41480	0.707399	−	0.706815i	$-0.249869\pi$
$863$	41.5623	1.41480	0.707399	+	0.706815i	$0.249869\pi$
$864$	0	0
$865$	0	0
$866$	−12.4508	−0.423097
$867$	0	0
$868$	−7.85410	−0.266586
$869$	2.36068	0.0800806
$870$	0	0
$871$	−44.8328	−1.51910
$872$	22.3607	0.757228
$873$	0	0
$874$	−29.7984	−1.00795
$875$	0	0
$876$	0	0
$877$	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
$879$	0	0
$880$	0	0
$881$	−4.36068	−0.146915	−0.0734575	−	0.997298i	$-0.523403\pi$
$881$	−4.36068	−0.146915	−0.0734575	+	0.997298i	$0.523403\pi$
$882$	0	0
$883$	−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	$-0.468481\pi$
$887$	5.88854	0.197718	0.0988590	+	0.995101i	$0.468481\pi$
$888$	0	0
$889$	25.7082	0.862225
$890$	0	0
$891$	0	0
$892$	−35.8885	−1.20164
$893$	−9.47214	−0.316973
$894$	0	0
$895$	0	0
$896$	−18.4164	−0.615249
$897$	0	0
$898$	12.5623	0.419210
$899$	−4.14590	−0.138273
$900$	0	0
$901$	−4.18034	−0.139267
$902$	−2.47214	−0.0823131
$903$	0	0
$904$	−22.6869	−0.754556
$905$	0	0
$906$	0	0
$907$	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
$909$	0	0
$910$	0	0
$911$	35.7639	1.18491	0.592456	−	0.805603i	$-0.298158\pi$
$911$	35.7639	1.18491	0.592456	+	0.805603i	$0.298158\pi$
$912$	0	0
$913$	1.34752	0.0445965
$914$	−3.34752	−0.110726
$915$	0	0
$916$	−13.4164	−0.443291
$917$	−28.7984	−0.951006
$918$	0	0
$919$	−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
$921$	0	0
$922$	14.3262	0.471810
$923$	−21.2705	−0.700127
$924$	0	0
$925$	0	0
$926$	−9.96556	−0.327489
$927$	0	0
$928$	−7.76393	−0.254864
$929$	36.6312	1.20183	0.600915	−	0.799313i	$-0.294803\pi$
$929$	36.6312	1.20183	0.600915	+	0.799313i	$0.294803\pi$
$930$	0	0
$931$	25.6525	0.840726
$932$	24.1803	0.792053
$933$	0	0
$934$	−17.5836	−0.575353
$935$	0	0
$936$	0	0
$937$	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
$939$	0	0
$940$	0	0
$941$	19.5836	0.638407	0.319203	−	0.947686i	$-0.396585\pi$
$941$	19.5836	0.638407	0.319203	+	0.947686i	$0.396585\pi$
$942$	0	0
$943$	−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	$-0.345860\pi$
$947$	28.6525	0.931080	0.465540	+	0.885027i	$0.345860\pi$
$948$	0	0
$949$	43.6869	1.41814
$950$	0	0
$951$	0	0
$952$	−2.76393	−0.0895796
$953$	−34.7426	−1.12542	−0.562712	−	0.826653i	$-0.690242\pi$
$953$	−34.7426	−1.12542	−0.562712	+	0.826653i	$0.690242\pi$
$954$	0	0
$955$	0	0
$956$	−47.6869	−1.54231
$957$	0	0
$958$	−2.56231	−0.0827843
$959$	9.61803	0.310583
$960$	0	0
$961$	−22.0000	−0.709677
$962$	12.7082	0.409729
$963$	0	0
$964$	−18.5623	−0.597852
$965$	0	0
$966$	0	0
$967$	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
$969$	0	0
$970$	0	0
$971$	−3.38197	−0.108532	−0.0542662	−	0.998527i	$-0.517282\pi$
$971$	−3.38197	−0.108532	−0.0542662	+	0.998527i	$0.517282\pi$
$972$	0	0
$973$	−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	$-0.319060\pi$
$977$	33.6525	1.07664	0.538319	+	0.842741i	$0.319060\pi$
$978$	0	0
$979$	−6.83282	−0.218378
$980$	0	0
$981$	0	0
$982$	23.0213	0.734639
$983$	−7.38197	−0.235448	−0.117724	−	0.993046i	$-0.537560\pi$
$983$	−7.38197	−0.235448	−0.117724	+	0.993046i	$0.537560\pi$
$984$	0	0
$985$	0	0
$986$	−0.652476	−0.0207791
$987$	0	0
$988$	−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
$993$	0	0
$994$	4.38197	0.138988
$995$	0	0
$996$	0	0
$997$	−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
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.f.1.1		2
3.2	odd	2		625.2.a.b.1.2		2
5.4	even	2		5625.2.a.d.1.2		2
12.11	even	2		10000.2.a.c.1.2		2
15.2	even	4		625.2.b.a.624.3		4
15.8	even	4		625.2.b.a.624.2		4
15.14	odd	2		625.2.a.c.1.1		2
25.6	even	5		225.2.h.b.136.1		4
25.21	even	5		225.2.h.b.91.1		4
60.59	even	2		10000.2.a.l.1.1		2
75.2	even	20		625.2.e.c.124.2		8
75.8	even	20		125.2.e.a.74.2		8
75.11	odd	10		625.2.d.h.501.1		4
75.14	odd	10		625.2.d.b.501.1		4
75.17	even	20		125.2.e.a.74.1		8
75.23	even	20		625.2.e.c.124.1		8
75.29	odd	10		125.2.d.a.76.1		4
75.38	even	20		625.2.e.c.499.2		8
75.41	odd	10		625.2.d.h.126.1		4
75.44	odd	10		125.2.d.a.51.1		4
75.47	even	20		125.2.e.a.49.2		8
75.53	even	20		125.2.e.a.49.1		8
75.56	odd	10		25.2.d.a.11.1	✓	4
75.59	odd	10		625.2.d.b.126.1		4
75.62	even	20		625.2.e.c.499.1		8
75.71	odd	10		25.2.d.a.16.1	yes	4
300.71	even	10		400.2.u.b.241.1		4
300.131	even	10		400.2.u.b.161.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.2.d.a.11.1	✓	4	75.56	odd	10
25.2.d.a.16.1	yes	4	75.71	odd	10
125.2.d.a.51.1		4	75.44	odd	10
125.2.d.a.76.1		4	75.29	odd	10
125.2.e.a.49.1		8	75.53	even	20
125.2.e.a.49.2		8	75.47	even	20
125.2.e.a.74.1		8	75.17	even	20
125.2.e.a.74.2		8	75.8	even	20
225.2.h.b.91.1		4	25.21	even	5
225.2.h.b.136.1		4	25.6	even	5
400.2.u.b.161.1		4	300.131	even	10
400.2.u.b.241.1		4	300.71	even	10
625.2.a.b.1.2		2	3.2	odd	2
625.2.a.c.1.1		2	15.14	odd	2
625.2.b.a.624.2		4	15.8	even	4
625.2.b.a.624.3		4	15.2	even	4
625.2.d.b.126.1		4	75.59	odd	10
625.2.d.b.501.1		4	75.14	odd	10
625.2.d.h.126.1		4	75.41	odd	10
625.2.d.h.501.1		4	75.11	odd	10
625.2.e.c.124.1		8	75.23	even	20
625.2.e.c.124.2		8	75.2	even	20
625.2.e.c.499.1		8	75.62	even	20
625.2.e.c.499.2		8	75.38	even	20
5625.2.a.d.1.2		2	5.4	even	2
5625.2.a.f.1.1		2	1.1	even	1	trivial
10000.2.a.c.1.2		2	12.11	even	2
10000.2.a.l.1.1		2	60.59	even	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:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

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

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