LMFDB - Embedded newform 5625.2.a.s.1.7

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:	$1$
Dimension:	$8$
Coefficient field:	8.8.6152203125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 20x^{5} + 26x^{4} - 35x^{3} - 27x^{2} + 16x + 1 $ x^8 - 3x^7 - 8x^6 + 20x^5 + 26x^4 - 35x^3 - 27x^2 + 16*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 625)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$2.68341$ 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)

$f(q)$	$=$	$q+1.68341 q^{2} +0.833870 q^{4} +4.59110 q^{7} -1.96307 q^{8} +O(q^{10})$	$q+1.68341 q^{2} +0.833870 q^{4} +4.59110 q^{7} -1.96307 q^{8} -3.91479 q^{11} +0.572939 q^{13} +7.72871 q^{14} -4.97240 q^{16} -0.232611 q^{17} -5.55010 q^{19} -6.59020 q^{22} -4.93267 q^{23} +0.964492 q^{26} +3.82839 q^{28} -4.13062 q^{29} -3.49531 q^{31} -4.44444 q^{32} -0.391580 q^{34} +5.41648 q^{37} -9.34309 q^{38} -10.4227 q^{41} -1.38833 q^{43} -3.26443 q^{44} -8.30370 q^{46} +0.920418 q^{47} +14.0782 q^{49} +0.477757 q^{52} +1.23118 q^{53} -9.01268 q^{56} -6.95353 q^{58} -4.50780 q^{59} -11.6588 q^{61} -5.88405 q^{62} +2.46298 q^{64} +2.95447 q^{67} -0.193968 q^{68} -3.20551 q^{71} +10.2922 q^{73} +9.11816 q^{74} -4.62806 q^{76} -17.9732 q^{77} -9.61509 q^{79} -17.5457 q^{82} -10.4834 q^{83} -2.33712 q^{86} +7.68502 q^{88} -7.25828 q^{89} +2.63042 q^{91} -4.11320 q^{92} +1.54944 q^{94} +8.31971 q^{97} +23.6994 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{2} + 11 q^{4} + 10 q^{7} - 15 q^{8}+O(q^{10}) $ 8 * q - 5 * q^2 + 11 * q^4 + 10 * q^7 - 15 * q^8	$ 8 q - 5 q^{2} + 11 q^{4} + 10 q^{7} - 15 q^{8} - q^{11} + 10 q^{13} + 8 q^{14} + 13 q^{16} - 15 q^{17} - 10 q^{19} - 5 q^{22} - 30 q^{23} - 11 q^{26} - 5 q^{28} - 10 q^{29} - 9 q^{31} - 30 q^{32} + 7 q^{34} - 10 q^{37} - 20 q^{38} + 4 q^{41} + 18 q^{44} - 9 q^{46} - 30 q^{47} - 4 q^{49} + 5 q^{52} - 10 q^{53} - 30 q^{58} + 5 q^{59} + 6 q^{61} - 10 q^{62} - 9 q^{64} + 10 q^{67} - 40 q^{68} + 9 q^{71} + 18 q^{74} - 10 q^{76} - 5 q^{77} - 20 q^{79} - 45 q^{82} - 40 q^{83} + 24 q^{86} - 40 q^{88} + 5 q^{89} + 6 q^{91} - 15 q^{92} + 47 q^{94} + 30 q^{98}+O(q^{100}) $ 8 * q - 5 * q^2 + 11 * q^4 + 10 * q^7 - 15 * q^8 - q^11 + 10 * q^13 + 8 * q^14 + 13 * q^16 - 15 * q^17 - 10 * q^19 - 5 * q^22 - 30 * q^23 - 11 * q^26 - 5 * q^28 - 10 * q^29 - 9 * q^31 - 30 * q^32 + 7 * q^34 - 10 * q^37 - 20 * q^38 + 4 * q^41 + 18 * q^44 - 9 * q^46 - 30 * q^47 - 4 * q^49 + 5 * q^52 - 10 * q^53 - 30 * q^58 + 5 * q^59 + 6 * q^61 - 10 * q^62 - 9 * q^64 + 10 * q^67 - 40 * q^68 + 9 * q^71 + 18 * q^74 - 10 * q^76 - 5 * q^77 - 20 * q^79 - 45 * q^82 - 40 * q^83 + 24 * q^86 - 40 * q^88 + 5 * q^89 + 6 * q^91 - 15 * q^92 + 47 * q^94 + 30 * 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 $

$2$	1.68341	1.19035	0.595175	−	0.803596i	$-0.297083\pi$
$2$	1.68341	1.19035	0.595175	+	0.803596i	$0.297083\pi$
$3$	0	0
$3$	0	0
$4$	0.833870	0.416935
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.59110	1.73527	0.867637	−	0.497198i	$-0.165638\pi$
$7$	4.59110	1.73527	0.867637	+	0.497198i	$0.165638\pi$
$8$	−1.96307	−0.694052
$9$	0	0
$10$	0	0
$11$	−3.91479	−1.18035	−0.590177	−	0.807274i	$-0.700942\pi$
$11$	−3.91479	−1.18035	−0.590177	+	0.807274i	$0.700942\pi$
$12$	0	0
$13$	0.572939	0.158905	0.0794524	−	0.996839i	$-0.474683\pi$
$13$	0.572939	0.158905	0.0794524	+	0.996839i	$0.474683\pi$
$14$	7.72871	2.06559
$15$	0	0
$16$	−4.97240	−1.24310
$17$	−0.232611	−0.0564165	−0.0282082	−	0.999602i	$-0.508980\pi$
$17$	−0.232611	−0.0564165	−0.0282082	+	0.999602i	$0.508980\pi$
$18$	0	0
$19$	−5.55010	−1.27328	−0.636640	−	0.771161i	$-0.719676\pi$
$19$	−5.55010	−1.27328	−0.636640	+	0.771161i	$0.719676\pi$
$20$	0	0
$21$	0	0
$22$	−6.59020	−1.40503
$23$	−4.93267	−1.02853	−0.514266	−	0.857631i	$-0.671936\pi$
$23$	−4.93267	−1.02853	−0.514266	+	0.857631i	$0.671936\pi$
$24$	0	0
$25$	0	0
$26$	0.964492	0.189152
$27$	0	0
$28$	3.82839	0.723497
$29$	−4.13062	−0.767037	−0.383519	−	0.923533i	$-0.625288\pi$
$29$	−4.13062	−0.767037	−0.383519	+	0.923533i	$0.625288\pi$
$30$	0	0
$31$	−3.49531	−0.627777	−0.313888	−	0.949460i	$-0.601632\pi$
$31$	−3.49531	−0.627777	−0.313888	+	0.949460i	$0.601632\pi$
$32$	−4.44444	−0.785674
$33$	0	0
$34$	−0.391580	−0.0671554
$35$	0	0
$36$	0	0
$37$	5.41648	0.890464	0.445232	−	0.895415i	$-0.353121\pi$
$37$	5.41648	0.890464	0.445232	+	0.895415i	$0.353121\pi$
$38$	−9.34309	−1.51565
$39$	0	0
$40$	0	0
$41$	−10.4227	−1.62775	−0.813876	−	0.581039i	$-0.802646\pi$
$41$	−10.4227	−1.62775	−0.813876	+	0.581039i	$0.802646\pi$
$42$	0	0
$43$	−1.38833	−0.211718	−0.105859	−	0.994381i	$-0.533759\pi$
$43$	−1.38833	−0.211718	−0.105859	+	0.994381i	$0.533759\pi$
$44$	−3.26443	−0.492131
$45$	0	0
$46$	−8.30370	−1.22431
$47$	0.920418	0.134257	0.0671284	−	0.997744i	$-0.478616\pi$
$47$	0.920418	0.134257	0.0671284	+	0.997744i	$0.478616\pi$
$48$	0	0
$49$	14.0782	2.01118
$50$	0	0
$51$	0	0
$52$	0.477757	0.0662530
$53$	1.23118	0.169115	0.0845576	−	0.996419i	$-0.473052\pi$
$53$	1.23118	0.169115	0.0845576	+	0.996419i	$0.473052\pi$
$54$	0	0
$55$	0	0
$56$	−9.01268	−1.20437
$57$	0	0
$58$	−6.95353	−0.913043
$59$	−4.50780	−0.586866	−0.293433	−	0.955980i	$-0.594798\pi$
$59$	−4.50780	−0.586866	−0.293433	+	0.955980i	$0.594798\pi$
$60$	0	0
$61$	−11.6588	−1.49275	−0.746376	−	0.665525i	$-0.768208\pi$
$61$	−11.6588	−1.49275	−0.746376	+	0.665525i	$0.768208\pi$
$62$	−5.88405	−0.747275
$63$	0	0
$64$	2.46298	0.307873
$65$	0	0
$66$	0	0
$67$	2.95447	0.360946	0.180473	−	0.983580i	$-0.442237\pi$
$67$	2.95447	0.360946	0.180473	+	0.983580i	$0.442237\pi$
$68$	−0.193968	−0.0235220
$69$	0	0
$70$	0	0
$71$	−3.20551	−0.380424	−0.190212	−	0.981743i	$-0.560918\pi$
$71$	−3.20551	−0.380424	−0.190212	+	0.981743i	$0.560918\pi$
$72$	0	0
$73$	10.2922	1.20461	0.602306	−	0.798266i	$-0.294249\pi$
$73$	10.2922	1.20461	0.602306	+	0.798266i	$0.294249\pi$
$74$	9.11816	1.05996
$75$	0	0
$76$	−4.62806	−0.530875
$77$	−17.9732	−2.04824
$78$	0	0
$79$	−9.61509	−1.08178	−0.540891	−	0.841093i	$-0.681913\pi$
$79$	−9.61509	−1.08178	−0.540891	+	0.841093i	$0.681913\pi$
$80$	0	0
$81$	0	0
$82$	−17.5457	−1.93759
$83$	−10.4834	−1.15070	−0.575351	−	0.817906i	$-0.695135\pi$
$83$	−10.4834	−1.15070	−0.575351	+	0.817906i	$0.695135\pi$
$84$	0	0
$85$	0	0
$86$	−2.33712	−0.252018
$87$	0	0
$88$	7.68502	0.819226
$89$	−7.25828	−0.769376	−0.384688	−	0.923047i	$-0.625691\pi$
$89$	−7.25828	−0.769376	−0.384688	+	0.923047i	$0.625691\pi$
$90$	0	0
$91$	2.63042	0.275743
$92$	−4.11320	−0.428831
$93$	0	0
$94$	1.54944	0.159813
$95$	0	0
$96$	0	0
$97$	8.31971	0.844739	0.422369	−	0.906424i	$-0.361199\pi$
$97$	8.31971	0.844739	0.422369	+	0.906424i	$0.361199\pi$
$98$	23.6994	2.39401
$99$	0	0
$100$	0	0
$101$	3.56513	0.354744	0.177372	−	0.984144i	$-0.443240\pi$
$101$	3.56513	0.354744	0.177372	+	0.984144i	$0.443240\pi$
$102$	0	0
$103$	0.399323	0.0393465	0.0196732	−	0.999806i	$-0.493737\pi$
$103$	0.399323	0.0393465	0.0196732	+	0.999806i	$0.493737\pi$
$104$	−1.12472	−0.110288
$105$	0	0
$106$	2.07258	0.201306
$107$	−1.64372	−0.158904	−0.0794522	−	0.996839i	$-0.525317\pi$
$107$	−1.64372	−0.158904	−0.0794522	+	0.996839i	$0.525317\pi$
$108$	0	0
$109$	0.0749154	0.00717560	0.00358780	−	0.999994i	$-0.498858\pi$
$109$	0.0749154	0.00717560	0.00358780	+	0.999994i	$0.498858\pi$
$110$	0	0
$111$	0	0
$112$	−22.8288	−2.15712
$113$	14.1328	1.32951	0.664753	−	0.747063i	$-0.268537\pi$
$113$	14.1328	1.32951	0.664753	+	0.747063i	$0.268537\pi$
$114$	0	0
$115$	0	0
$116$	−3.44440	−0.319805
$117$	0	0
$118$	−7.58848	−0.698576
$119$	−1.06794	−0.0978981
$120$	0	0
$121$	4.32557	0.393234
$122$	−19.6265	−1.77690
$123$	0	0
$124$	−2.91464	−0.261742
$125$	0	0
$126$	0	0
$127$	−11.8124	−1.04819	−0.524093	−	0.851661i	$-0.675595\pi$
$127$	−11.8124	−1.04819	−0.524093	+	0.851661i	$0.675595\pi$
$128$	13.0351	1.15215
$129$	0	0
$130$	0	0
$131$	16.7373	1.46234	0.731170	−	0.682195i	$-0.238974\pi$
$131$	16.7373	1.46234	0.731170	+	0.682195i	$0.238974\pi$
$132$	0	0
$133$	−25.4811	−2.20949
$134$	4.97359	0.429653
$135$	0	0
$136$	0.456633	0.0391560
$137$	10.4221	0.890423	0.445211	−	0.895425i	$-0.353128\pi$
$137$	10.4221	0.890423	0.445211	+	0.895425i	$0.353128\pi$
$138$	0	0
$139$	6.65993	0.564888	0.282444	−	0.959284i	$-0.408855\pi$
$139$	6.65993	0.564888	0.282444	+	0.959284i	$0.408855\pi$
$140$	0	0
$141$	0	0
$142$	−5.39619	−0.452838
$143$	−2.24294	−0.187564
$144$	0	0
$145$	0	0
$146$	17.3260	1.43391
$147$	0	0
$148$	4.51664	0.371266
$149$	12.0316	0.985667	0.492834	−	0.870124i	$-0.335961\pi$
$149$	12.0316	0.985667	0.492834	+	0.870124i	$0.335961\pi$
$150$	0	0
$151$	−1.54218	−0.125501	−0.0627505	−	0.998029i	$-0.519987\pi$
$151$	−1.54218	−0.125501	−0.0627505	+	0.998029i	$0.519987\pi$
$152$	10.8953	0.883722
$153$	0	0
$154$	−30.2563	−2.43812
$155$	0	0
$156$	0	0
$157$	9.82482	0.784106	0.392053	−	0.919943i	$-0.371765\pi$
$157$	9.82482	0.784106	0.392053	+	0.919943i	$0.371765\pi$
$158$	−16.1861	−1.28770
$159$	0	0
$160$	0	0
$161$	−22.6464	−1.78478
$162$	0	0
$163$	5.58107	0.437143	0.218572	−	0.975821i	$-0.429860\pi$
$163$	5.58107	0.437143	0.218572	+	0.975821i	$0.429860\pi$
$164$	−8.69117	−0.678667
$165$	0	0
$166$	−17.6479	−1.36974
$167$	−21.8607	−1.69163	−0.845817	−	0.533473i	$-0.820887\pi$
$167$	−21.8607	−1.69163	−0.845817	+	0.533473i	$0.820887\pi$
$168$	0	0
$169$	−12.6717	−0.974749
$170$	0	0
$171$	0	0
$172$	−1.15768	−0.0882726
$173$	23.4385	1.78200	0.891000	−	0.454004i	$-0.150005\pi$
$173$	23.4385	1.78200	0.891000	+	0.454004i	$0.150005\pi$
$174$	0	0
$175$	0	0
$176$	19.4659	1.46730
$177$	0	0
$178$	−12.2187	−0.915827
$179$	6.31873	0.472284	0.236142	−	0.971719i	$-0.424117\pi$
$179$	6.31873	0.472284	0.236142	+	0.971719i	$0.424117\pi$
$180$	0	0
$181$	−13.3377	−0.991385	−0.495693	−	0.868498i	$-0.665086\pi$
$181$	−13.3377	−0.991385	−0.495693	+	0.868498i	$0.665086\pi$
$182$	4.42808	0.328231
$183$	0	0
$184$	9.68319	0.713854
$185$	0	0
$186$	0	0
$187$	0.910624	0.0665914
$188$	0.767510	0.0559764
$189$	0	0
$190$	0	0
$191$	2.78083	0.201214	0.100607	−	0.994926i	$-0.467922\pi$
$191$	2.78083	0.201214	0.100607	+	0.994926i	$0.467922\pi$
$192$	0	0
$193$	−22.5667	−1.62438	−0.812192	−	0.583391i	$-0.801726\pi$
$193$	−22.5667	−1.62438	−0.812192	+	0.583391i	$0.801726\pi$
$194$	14.0055	1.00554
$195$	0	0
$196$	11.7394	0.838530
$197$	1.27182	0.0906135	0.0453068	−	0.998973i	$-0.485573\pi$
$197$	1.27182	0.0906135	0.0453068	+	0.998973i	$0.485573\pi$
$198$	0	0
$199$	−8.62648	−0.611515	−0.305757	−	0.952109i	$-0.598910\pi$
$199$	−8.62648	−0.611515	−0.305757	+	0.952109i	$0.598910\pi$
$200$	0	0
$201$	0	0
$202$	6.00158	0.422270
$203$	−18.9641	−1.33102
$204$	0	0
$205$	0	0
$206$	0.672225	0.0468361
$207$	0	0
$208$	−2.84888	−0.197535
$209$	21.7275	1.50292
$210$	0	0
$211$	−21.0316	−1.44787	−0.723937	−	0.689866i	$-0.757670\pi$
$211$	−21.0316	−1.44787	−0.723937	+	0.689866i	$0.757670\pi$
$212$	1.02664	0.0705101
$213$	0	0
$214$	−2.76705	−0.189152
$215$	0	0
$216$	0	0
$217$	−16.0473	−1.08936
$218$	0.126113	0.00854148
$219$	0	0
$220$	0	0
$221$	−0.133272	−0.00896485
$222$	0	0
$223$	6.33537	0.424248	0.212124	−	0.977243i	$-0.431962\pi$
$223$	6.33537	0.424248	0.212124	+	0.977243i	$0.431962\pi$
$224$	−20.4049	−1.36336
$225$	0	0
$226$	23.7914	1.58258
$227$	−11.2624	−0.747512	−0.373756	−	0.927527i	$-0.621930\pi$
$227$	−11.2624	−0.747512	−0.373756	+	0.927527i	$0.621930\pi$
$228$	0	0
$229$	15.0408	0.993927	0.496964	−	0.867771i	$-0.334448\pi$
$229$	15.0408	0.993927	0.496964	+	0.867771i	$0.334448\pi$
$230$	0	0
$231$	0	0
$232$	8.10872	0.532363
$233$	−7.82749	−0.512796	−0.256398	−	0.966571i	$-0.582536\pi$
$233$	−7.82749	−0.512796	−0.256398	+	0.966571i	$0.582536\pi$
$234$	0	0
$235$	0	0
$236$	−3.75892	−0.244685
$237$	0	0
$238$	−1.79778	−0.116533
$239$	−8.73886	−0.565270	−0.282635	−	0.959228i	$-0.591208\pi$
$239$	−8.73886	−0.565270	−0.282635	+	0.959228i	$0.591208\pi$
$240$	0	0
$241$	−0.600892	−0.0387068	−0.0193534	−	0.999813i	$-0.506161\pi$
$241$	−0.600892	−0.0387068	−0.0193534	+	0.999813i	$0.506161\pi$
$242$	7.28171	0.468086
$243$	0	0
$244$	−9.72190	−0.622381
$245$	0	0
$246$	0	0
$247$	−3.17987	−0.202330
$248$	6.86156	0.435710
$249$	0	0
$250$	0	0
$251$	14.1908	0.895712	0.447856	−	0.894106i	$-0.352188\pi$
$251$	14.1908	0.895712	0.447856	+	0.894106i	$0.352188\pi$
$252$	0	0
$253$	19.3103	1.21403
$254$	−19.8852	−1.24771
$255$	0	0
$256$	17.0174	1.06359
$257$	−17.6859	−1.10322	−0.551609	−	0.834103i	$-0.685986\pi$
$257$	−17.6859	−1.10322	−0.551609	+	0.834103i	$0.685986\pi$
$258$	0	0
$259$	24.8676	1.54520
$260$	0	0
$261$	0	0
$262$	28.1757	1.74070
$263$	−24.3348	−1.50055	−0.750274	−	0.661127i	$-0.770078\pi$
$263$	−24.3348	−1.50055	−0.750274	+	0.661127i	$0.770078\pi$
$264$	0	0
$265$	0	0
$266$	−42.8951	−2.63007
$267$	0	0
$268$	2.46365	0.150491
$269$	−29.9819	−1.82803	−0.914013	−	0.405685i	$-0.867033\pi$
$269$	−29.9819	−1.82803	−0.914013	+	0.405685i	$0.867033\pi$
$270$	0	0
$271$	−27.6981	−1.68254	−0.841271	−	0.540613i	$-0.818192\pi$
$271$	−27.6981	−1.68254	−0.841271	+	0.540613i	$0.818192\pi$
$272$	1.15664	0.0701314
$273$	0	0
$274$	17.5447	1.05992
$275$	0	0
$276$	0	0
$277$	−2.29407	−0.137838	−0.0689188	−	0.997622i	$-0.521955\pi$
$277$	−2.29407	−0.137838	−0.0689188	+	0.997622i	$0.521955\pi$
$278$	11.2114	0.672415
$279$	0	0
$280$	0	0
$281$	1.61829	0.0965388	0.0482694	−	0.998834i	$-0.484629\pi$
$281$	1.61829	0.0965388	0.0482694	+	0.998834i	$0.484629\pi$
$282$	0	0
$283$	12.9922	0.772306	0.386153	−	0.922435i	$-0.373804\pi$
$283$	12.9922	0.772306	0.386153	+	0.922435i	$0.373804\pi$
$284$	−2.67298	−0.158612
$285$	0	0
$286$	−3.77578	−0.223267
$287$	−47.8517	−2.82459
$288$	0	0
$289$	−16.9459	−0.996817
$290$	0	0
$291$	0	0
$292$	8.58236	0.502245
$293$	−11.7009	−0.683572	−0.341786	−	0.939778i	$-0.611032\pi$
$293$	−11.7009	−0.683572	−0.341786	+	0.939778i	$0.611032\pi$
$294$	0	0
$295$	0	0
$296$	−10.6330	−0.618028
$297$	0	0
$298$	20.2541	1.17329
$299$	−2.82612	−0.163439
$300$	0	0
$301$	−6.37395	−0.367388
$302$	−2.59613	−0.149390
$303$	0	0
$304$	27.5973	1.58281
$305$	0	0
$306$	0	0
$307$	−26.5673	−1.51628	−0.758138	−	0.652094i	$-0.773891\pi$
$307$	−26.5673	−1.51628	−0.758138	+	0.652094i	$0.773891\pi$
$308$	−14.9873	−0.853982
$309$	0	0
$310$	0	0
$311$	13.4910	0.765005	0.382502	−	0.923955i	$-0.375062\pi$
$311$	13.4910	0.765005	0.382502	+	0.923955i	$0.375062\pi$
$312$	0	0
$313$	−16.9944	−0.960578	−0.480289	−	0.877110i	$-0.659468\pi$
$313$	−16.9944	−0.960578	−0.480289	+	0.877110i	$0.659468\pi$
$314$	16.5392	0.933361
$315$	0	0
$316$	−8.01774	−0.451033
$317$	16.7959	0.943351	0.471675	−	0.881772i	$-0.343649\pi$
$317$	16.7959	0.943351	0.471675	+	0.881772i	$0.343649\pi$
$318$	0	0
$319$	16.1705	0.905375
$320$	0	0
$321$	0	0
$322$	−38.1231	−2.12452
$323$	1.29101	0.0718340
$324$	0	0
$325$	0	0
$326$	9.39524	0.520354
$327$	0	0
$328$	20.4605	1.12974
$329$	4.22574	0.232972
$330$	0	0
$331$	12.8344	0.705442	0.352721	−	0.935729i	$-0.385257\pi$
$331$	12.8344	0.705442	0.352721	+	0.935729i	$0.385257\pi$
$332$	−8.74180	−0.479768
$333$	0	0
$334$	−36.8006	−2.01364
$335$	0	0
$336$	0	0
$337$	−21.2375	−1.15688	−0.578440	−	0.815725i	$-0.696338\pi$
$337$	−21.2375	−1.15688	−0.578440	+	0.815725i	$0.696338\pi$
$338$	−21.3317	−1.16029
$339$	0	0
$340$	0	0
$341$	13.6834	0.740998
$342$	0	0
$343$	32.4969	1.75467
$344$	2.72539	0.146943
$345$	0	0
$346$	39.4567	2.12120
$347$	28.6281	1.53684	0.768419	−	0.639947i	$-0.221044\pi$
$347$	28.6281	1.53684	0.768419	+	0.639947i	$0.221044\pi$
$348$	0	0
$349$	19.9124	1.06588	0.532942	−	0.846152i	$-0.321086\pi$
$349$	19.9124	1.06588	0.532942	+	0.846152i	$0.321086\pi$
$350$	0	0
$351$	0	0
$352$	17.3990	0.927372
$353$	24.6916	1.31420	0.657099	−	0.753804i	$-0.271783\pi$
$353$	24.6916	1.31420	0.657099	+	0.753804i	$0.271783\pi$
$354$	0	0
$355$	0	0
$356$	−6.05246	−0.320780
$357$	0	0
$358$	10.6370	0.562184
$359$	−21.5011	−1.13478	−0.567391	−	0.823448i	$-0.692047\pi$
$359$	−21.5011	−1.13478	−0.567391	+	0.823448i	$0.692047\pi$
$360$	0	0
$361$	11.8036	0.621241
$362$	−22.4529	−1.18010
$363$	0	0
$364$	2.19343	0.114967
$365$	0	0
$366$	0	0
$367$	1.14383	0.0597075	0.0298538	−	0.999554i	$-0.490496\pi$
$367$	1.14383	0.0597075	0.0298538	+	0.999554i	$0.490496\pi$
$368$	24.5272	1.27857
$369$	0	0
$370$	0	0
$371$	5.65246	0.293461
$372$	0	0
$373$	−11.5395	−0.597493	−0.298747	−	0.954332i	$-0.596569\pi$
$373$	−11.5395	−0.597493	−0.298747	+	0.954332i	$0.596569\pi$
$374$	1.53295	0.0792671
$375$	0	0
$376$	−1.80685	−0.0931812
$377$	−2.36660	−0.121886
$378$	0	0
$379$	21.1053	1.08410	0.542052	−	0.840345i	$-0.317648\pi$
$379$	21.1053	1.08410	0.542052	+	0.840345i	$0.317648\pi$
$380$	0	0
$381$	0	0
$382$	4.68127	0.239515
$383$	−0.858129	−0.0438483	−0.0219242	−	0.999760i	$-0.506979\pi$
$383$	−0.858129	−0.0438483	−0.0219242	+	0.999760i	$0.506979\pi$
$384$	0	0
$385$	0	0
$386$	−37.9889	−1.93359
$387$	0	0
$388$	6.93756	0.352201
$389$	33.9346	1.72055	0.860277	−	0.509827i	$-0.170290\pi$
$389$	33.9346	1.72055	0.860277	+	0.509827i	$0.170290\pi$
$390$	0	0
$391$	1.14739	0.0580262
$392$	−27.6366	−1.39586
$393$	0	0
$394$	2.14100	0.107862
$395$	0	0
$396$	0	0
$397$	26.8286	1.34649	0.673245	−	0.739420i	$-0.264900\pi$
$397$	26.8286	1.34649	0.673245	+	0.739420i	$0.264900\pi$
$398$	−14.5219	−0.727917
$399$	0	0
$400$	0	0
$401$	−3.79757	−0.189642	−0.0948208	−	0.995494i	$-0.530228\pi$
$401$	−3.79757	−0.189642	−0.0948208	+	0.995494i	$0.530228\pi$
$402$	0	0
$403$	−2.00260	−0.0997567
$404$	2.97286	0.147905
$405$	0	0
$406$	−31.9244	−1.58438
$407$	−21.2044	−1.05106
$408$	0	0
$409$	−14.1754	−0.700927	−0.350463	−	0.936576i	$-0.613976\pi$
$409$	−14.1754	−0.700927	−0.350463	+	0.936576i	$0.613976\pi$
$410$	0	0
$411$	0	0
$412$	0.332984	0.0164049
$413$	−20.6958	−1.01837
$414$	0	0
$415$	0	0
$416$	−2.54640	−0.124847
$417$	0	0
$418$	36.5762	1.78900
$419$	19.5969	0.957369	0.478685	−	0.877987i	$-0.341114\pi$
$419$	19.5969	0.957369	0.478685	+	0.877987i	$0.341114\pi$
$420$	0	0
$421$	−25.7840	−1.25664	−0.628318	−	0.777957i	$-0.716256\pi$
$421$	−25.7840	−1.25664	−0.628318	+	0.777957i	$0.716256\pi$
$422$	−35.4048	−1.72348
$423$	0	0
$424$	−2.41689	−0.117375
$425$	0	0
$426$	0	0
$427$	−53.5266	−2.59033
$428$	−1.37065	−0.0662528
$429$	0	0
$430$	0	0
$431$	−9.42533	−0.454002	−0.227001	−	0.973894i	$-0.572892\pi$
$431$	−9.42533	−0.454002	−0.227001	+	0.973894i	$0.572892\pi$
$432$	0	0
$433$	1.75161	0.0841770	0.0420885	−	0.999114i	$-0.486599\pi$
$433$	1.75161	0.0841770	0.0420885	+	0.999114i	$0.486599\pi$
$434$	−27.0143	−1.29673
$435$	0	0
$436$	0.0624697	0.00299176
$437$	27.3768	1.30961
$438$	0	0
$439$	28.0830	1.34033	0.670165	−	0.742212i	$-0.266223\pi$
$439$	28.0830	1.34033	0.670165	+	0.742212i	$0.266223\pi$
$440$	0	0
$441$	0	0
$442$	−0.224352	−0.0106713
$443$	29.9110	1.42111	0.710557	−	0.703640i	$-0.248443\pi$
$443$	29.9110	1.42111	0.710557	+	0.703640i	$0.248443\pi$
$444$	0	0
$445$	0	0
$446$	10.6650	0.505004
$447$	0	0
$448$	11.3078	0.534244
$449$	−6.29974	−0.297303	−0.148652	−	0.988890i	$-0.547493\pi$
$449$	−6.29974	−0.297303	−0.148652	+	0.988890i	$0.547493\pi$
$450$	0	0
$451$	40.8026	1.92132
$452$	11.7850	0.554318
$453$	0	0
$454$	−18.9593	−0.889802
$455$	0	0
$456$	0	0
$457$	19.1809	0.897243	0.448622	−	0.893722i	$-0.351915\pi$
$457$	19.1809	0.897243	0.448622	+	0.893722i	$0.351915\pi$
$458$	25.3199	1.18312
$459$	0	0
$460$	0	0
$461$	−7.07110	−0.329334	−0.164667	−	0.986349i	$-0.552655\pi$
$461$	−7.07110	−0.329334	−0.164667	+	0.986349i	$0.552655\pi$
$462$	0	0
$463$	−9.61842	−0.447006	−0.223503	−	0.974703i	$-0.571749\pi$
$463$	−9.61842	−0.447006	−0.223503	+	0.974703i	$0.571749\pi$
$464$	20.5391	0.953504
$465$	0	0
$466$	−13.1769	−0.610407
$467$	19.2220	0.889488	0.444744	−	0.895658i	$-0.353295\pi$
$467$	19.2220	0.889488	0.444744	+	0.895658i	$0.353295\pi$
$468$	0	0
$469$	13.5643	0.626341
$470$	0	0
$471$	0	0
$472$	8.84915	0.407315
$473$	5.43500	0.249902
$474$	0	0
$475$	0	0
$476$	−0.890525	−0.0408172
$477$	0	0
$478$	−14.7111	−0.672869
$479$	37.9996	1.73625	0.868123	−	0.496350i	$-0.165327\pi$
$479$	37.9996	1.73625	0.868123	+	0.496350i	$0.165327\pi$
$480$	0	0
$481$	3.10332	0.141499
$482$	−1.01155	−0.0460747
$483$	0	0
$484$	3.60696	0.163953
$485$	0	0
$486$	0	0
$487$	−16.7490	−0.758969	−0.379485	−	0.925198i	$-0.623899\pi$
$487$	−16.7490	−0.758969	−0.379485	+	0.925198i	$0.623899\pi$
$488$	22.8870	1.03605
$489$	0	0
$490$	0	0
$491$	8.95055	0.403933	0.201966	−	0.979392i	$-0.435267\pi$
$491$	8.95055	0.403933	0.201966	+	0.979392i	$0.435267\pi$
$492$	0	0
$493$	0.960829	0.0432736
$494$	−5.35302	−0.240844
$495$	0	0
$496$	17.3801	0.780389
$497$	−14.7168	−0.660140
$498$	0	0
$499$	−36.3310	−1.62640	−0.813200	−	0.581985i	$-0.802276\pi$
$499$	−36.3310	−1.62640	−0.813200	+	0.581985i	$0.802276\pi$
$500$	0	0
$501$	0	0
$502$	23.8889	1.06621
$503$	−12.2945	−0.548184	−0.274092	−	0.961703i	$-0.588377\pi$
$503$	−12.2945	−0.548184	−0.274092	+	0.961703i	$0.588377\pi$
$504$	0	0
$505$	0	0
$506$	32.5072	1.44512
$507$	0	0
$508$	−9.85005	−0.437025
$509$	31.1292	1.37978	0.689889	−	0.723915i	$-0.257659\pi$
$509$	31.1292	1.37978	0.689889	+	0.723915i	$0.257659\pi$
$510$	0	0
$511$	47.2526	2.09033
$512$	2.57716	0.113895
$513$	0	0
$514$	−29.7727	−1.31322
$515$	0	0
$516$	0	0
$517$	−3.60324	−0.158470
$518$	41.8624	1.83933
$519$	0	0
$520$	0	0
$521$	−16.2169	−0.710476	−0.355238	−	0.934776i	$-0.615600\pi$
$521$	−16.2169	−0.710476	−0.355238	+	0.934776i	$0.615600\pi$
$522$	0	0
$523$	−30.5932	−1.33775	−0.668873	−	0.743377i	$-0.733223\pi$
$523$	−30.5932	−1.33775	−0.668873	+	0.743377i	$0.733223\pi$
$524$	13.9567	0.609701
$525$	0	0
$526$	−40.9655	−1.78618
$527$	0.813049	0.0354170
$528$	0	0
$529$	1.33119	0.0578776
$530$	0	0
$531$	0	0
$532$	−21.2479	−0.921214
$533$	−5.97157	−0.258657
$534$	0	0
$535$	0	0
$536$	−5.79985	−0.250515
$537$	0	0
$538$	−50.4718	−2.17599
$539$	−55.1133	−2.37390
$540$	0	0
$541$	18.3014	0.786840	0.393420	−	0.919359i	$-0.371292\pi$
$541$	18.3014	0.786840	0.393420	+	0.919359i	$0.371292\pi$
$542$	−46.6273	−2.00282
$543$	0	0
$544$	1.03383	0.0443250
$545$	0	0
$546$	0	0
$547$	−6.55603	−0.280316	−0.140158	−	0.990129i	$-0.544761\pi$
$547$	−6.55603	−0.280316	−0.140158	+	0.990129i	$0.544761\pi$
$548$	8.69071	0.371249
$549$	0	0
$550$	0	0
$551$	22.9254	0.976653
$552$	0	0
$553$	−44.1439	−1.87719
$554$	−3.86187	−0.164075
$555$	0	0
$556$	5.55352	0.235522
$557$	35.5383	1.50581	0.752904	−	0.658131i	$-0.228653\pi$
$557$	35.5383	1.50581	0.752904	+	0.658131i	$0.228653\pi$
$558$	0	0
$559$	−0.795427	−0.0336430
$560$	0	0
$561$	0	0
$562$	2.72424	0.114915
$563$	−38.2479	−1.61196	−0.805979	−	0.591944i	$-0.798361\pi$
$563$	−38.2479	−1.61196	−0.805979	+	0.591944i	$0.798361\pi$
$564$	0	0
$565$	0	0
$566$	21.8712	0.919315
$567$	0	0
$568$	6.29266	0.264034
$569$	−7.55897	−0.316889	−0.158444	−	0.987368i	$-0.550648\pi$
$569$	−7.55897	−0.316889	−0.158444	+	0.987368i	$0.550648\pi$
$570$	0	0
$571$	23.0262	0.963617	0.481809	−	0.876277i	$-0.339980\pi$
$571$	23.0262	0.963617	0.481809	+	0.876277i	$0.339980\pi$
$572$	−1.87032	−0.0782019
$573$	0	0
$574$	−80.5540	−3.36226
$575$	0	0
$576$	0	0
$577$	22.3168	0.929059	0.464529	−	0.885558i	$-0.346223\pi$
$577$	22.3168	0.929059	0.464529	+	0.885558i	$0.346223\pi$
$578$	−28.5269	−1.18656
$579$	0	0
$580$	0	0
$581$	−48.1304	−1.99678
$582$	0	0
$583$	−4.81980	−0.199616
$584$	−20.2044	−0.836062
$585$	0	0
$586$	−19.6974	−0.813691
$587$	21.0786	0.870007	0.435004	−	0.900429i	$-0.356747\pi$
$587$	21.0786	0.870007	0.435004	+	0.900429i	$0.356747\pi$
$588$	0	0
$589$	19.3993	0.799335
$590$	0	0
$591$	0	0
$592$	−26.9329	−1.10694
$593$	−34.3547	−1.41078	−0.705390	−	0.708819i	$-0.749228\pi$
$593$	−34.3547	−1.41078	−0.705390	+	0.708819i	$0.749228\pi$
$594$	0	0
$595$	0	0
$596$	10.0328	0.410959
$597$	0	0
$598$	−4.75752	−0.194549
$599$	−0.498231	−0.0203572	−0.0101786	−	0.999948i	$-0.503240\pi$
$599$	−0.498231	−0.0203572	−0.0101786	+	0.999948i	$0.503240\pi$
$600$	0	0
$601$	27.8635	1.13657	0.568287	−	0.822830i	$-0.307606\pi$
$601$	27.8635	1.13657	0.568287	+	0.822830i	$0.307606\pi$
$602$	−10.7300	−0.437321
$603$	0	0
$604$	−1.28598	−0.0523258
$605$	0	0
$606$	0	0
$607$	−14.1000	−0.572303	−0.286152	−	0.958184i	$-0.592376\pi$
$607$	−14.1000	−0.572303	−0.286152	+	0.958184i	$0.592376\pi$
$608$	24.6671	1.00038
$609$	0	0
$610$	0	0
$611$	0.527344	0.0213340
$612$	0	0
$613$	−31.8257	−1.28543	−0.642714	−	0.766106i	$-0.722192\pi$
$613$	−31.8257	−1.28543	−0.642714	+	0.766106i	$0.722192\pi$
$614$	−44.7237	−1.80490
$615$	0	0
$616$	35.2827	1.42158
$617$	33.9662	1.36743	0.683715	−	0.729749i	$-0.260363\pi$
$617$	33.9662	1.36743	0.683715	+	0.729749i	$0.260363\pi$
$618$	0	0
$619$	1.12842	0.0453550	0.0226775	−	0.999743i	$-0.492781\pi$
$619$	1.12842	0.0453550	0.0226775	+	0.999743i	$0.492781\pi$
$620$	0	0
$621$	0	0
$622$	22.7109	0.910624
$623$	−33.3235	−1.33508
$624$	0	0
$625$	0	0
$626$	−28.6085	−1.14343
$627$	0	0
$628$	8.19262	0.326921
$629$	−1.25993	−0.0502369
$630$	0	0
$631$	32.7801	1.30496	0.652478	−	0.757807i	$-0.273729\pi$
$631$	32.7801	1.30496	0.652478	+	0.757807i	$0.273729\pi$
$632$	18.8751	0.750813
$633$	0	0
$634$	28.2744	1.12292
$635$	0	0
$636$	0	0
$637$	8.06597	0.319586
$638$	27.2216	1.07771
$639$	0	0
$640$	0	0
$641$	−30.4126	−1.20123	−0.600613	−	0.799540i	$-0.705077\pi$
$641$	−30.4126	−1.20123	−0.600613	+	0.799540i	$0.705077\pi$
$642$	0	0
$643$	1.06932	0.0421700	0.0210850	−	0.999778i	$-0.493288\pi$
$643$	1.06932	0.0421700	0.0210850	+	0.999778i	$0.493288\pi$
$644$	−18.8841	−0.744140
$645$	0	0
$646$	2.17331	0.0855076
$647$	−30.5680	−1.20175	−0.600876	−	0.799343i	$-0.705181\pi$
$647$	−30.5680	−1.20175	−0.600876	+	0.799343i	$0.705181\pi$
$648$	0	0
$649$	17.6471	0.692709
$650$	0	0
$651$	0	0
$652$	4.65389	0.182260
$653$	33.3502	1.30509	0.652546	−	0.757749i	$-0.273701\pi$
$653$	33.3502	1.30509	0.652546	+	0.757749i	$0.273701\pi$
$654$	0	0
$655$	0	0
$656$	51.8258	2.02346
$657$	0	0
$658$	7.11365	0.277319
$659$	10.1150	0.394026	0.197013	−	0.980401i	$-0.436876\pi$
$659$	10.1150	0.394026	0.197013	+	0.980401i	$0.436876\pi$
$660$	0	0
$661$	−30.8383	−1.19947	−0.599735	−	0.800199i	$-0.704727\pi$
$661$	−30.8383	−1.19947	−0.599735	+	0.800199i	$0.704727\pi$
$662$	21.6055	0.839723
$663$	0	0
$664$	20.5797	0.798647
$665$	0	0
$666$	0	0
$667$	20.3750	0.788922
$668$	−18.2290	−0.705302
$669$	0	0
$670$	0	0
$671$	45.6416	1.76197
$672$	0	0
$673$	−30.8250	−1.18821	−0.594107	−	0.804386i	$-0.702495\pi$
$673$	−30.8250	−1.18821	−0.594107	+	0.804386i	$0.702495\pi$
$674$	−35.7514	−1.37709
$675$	0	0
$676$	−10.5666	−0.406407
$677$	−41.2521	−1.58545	−0.792723	−	0.609582i	$-0.791337\pi$
$677$	−41.2521	−1.58545	−0.792723	+	0.609582i	$0.791337\pi$
$678$	0	0
$679$	38.1967	1.46585
$680$	0	0
$681$	0	0
$682$	23.0348	0.882048
$683$	−22.3514	−0.855254	−0.427627	−	0.903955i	$-0.640650\pi$
$683$	−22.3514	−0.855254	−0.427627	+	0.903955i	$0.640650\pi$
$684$	0	0
$685$	0	0
$686$	54.7056	2.08867
$687$	0	0
$688$	6.90332	0.263186
$689$	0.705390	0.0268732
$690$	0	0
$691$	9.18901	0.349566	0.174783	−	0.984607i	$-0.444078\pi$
$691$	9.18901	0.349566	0.174783	+	0.984607i	$0.444078\pi$
$692$	19.5447	0.742978
$693$	0	0
$694$	48.1929	1.82938
$695$	0	0
$696$	0	0
$697$	2.42443	0.0918320
$698$	33.5207	1.26878
$699$	0	0
$700$	0	0
$701$	35.9929	1.35943	0.679717	−	0.733475i	$-0.262103\pi$
$701$	35.9929	1.35943	0.679717	+	0.733475i	$0.262103\pi$
$702$	0	0
$703$	−30.0620	−1.13381
$704$	−9.64206	−0.363399
$705$	0	0
$706$	41.5660	1.56436
$707$	16.3679	0.615578
$708$	0	0
$709$	20.2461	0.760359	0.380180	−	0.924913i	$-0.375862\pi$
$709$	20.2461	0.760359	0.380180	+	0.924913i	$0.375862\pi$
$710$	0	0
$711$	0	0
$712$	14.2485	0.533987
$713$	17.2412	0.645688
$714$	0	0
$715$	0	0
$716$	5.26900	0.196912
$717$	0	0
$718$	−36.1951	−1.35079
$719$	37.0265	1.38086	0.690428	−	0.723401i	$-0.257422\pi$
$719$	37.0265	1.38086	0.690428	+	0.723401i	$0.257422\pi$
$720$	0	0
$721$	1.83333	0.0682769
$722$	19.8703	0.739495
$723$	0	0
$724$	−11.1219	−0.413343
$725$	0	0
$726$	0	0
$727$	−12.0790	−0.447985	−0.223993	−	0.974591i	$-0.571909\pi$
$727$	−12.0790	−0.447985	−0.223993	+	0.974591i	$0.571909\pi$
$728$	−5.16372	−0.191380
$729$	0	0
$730$	0	0
$731$	0.322940	0.0119444
$732$	0	0
$733$	44.6116	1.64777	0.823884	−	0.566759i	$-0.191803\pi$
$733$	44.6116	1.64777	0.823884	+	0.566759i	$0.191803\pi$
$734$	1.92554	0.0710729
$735$	0	0
$736$	21.9229	0.808090
$737$	−11.5661	−0.426044
$738$	0	0
$739$	−14.6463	−0.538774	−0.269387	−	0.963032i	$-0.586821\pi$
$739$	−14.6463	−0.538774	−0.269387	+	0.963032i	$0.586821\pi$
$740$	0	0
$741$	0	0
$742$	9.51542	0.349322
$743$	−24.4397	−0.896604	−0.448302	−	0.893882i	$-0.647971\pi$
$743$	−24.4397	−0.896604	−0.448302	+	0.893882i	$0.647971\pi$
$744$	0	0
$745$	0	0
$746$	−19.4257	−0.711227
$747$	0	0
$748$	0.759342	0.0277643
$749$	−7.54648	−0.275743
$750$	0	0
$751$	31.2863	1.14165	0.570826	−	0.821071i	$-0.306623\pi$
$751$	31.2863	1.14165	0.570826	+	0.821071i	$0.306623\pi$
$752$	−4.57669	−0.166895
$753$	0	0
$754$	−3.98395	−0.145087
$755$	0	0
$756$	0	0
$757$	−11.3251	−0.411617	−0.205808	−	0.978592i	$-0.565982\pi$
$757$	−11.3251	−0.411617	−0.205808	+	0.978592i	$0.565982\pi$
$758$	35.5288	1.29046
$759$	0	0
$760$	0	0
$761$	41.3338	1.49835	0.749174	−	0.662373i	$-0.230451\pi$
$761$	41.3338	1.49835	0.749174	+	0.662373i	$0.230451\pi$
$762$	0	0
$763$	0.343944	0.0124516
$764$	2.31885	0.0838930
$765$	0	0
$766$	−1.44458	−0.0521949
$767$	−2.58270	−0.0932558
$768$	0	0
$769$	22.6491	0.816747	0.408374	−	0.912815i	$-0.366096\pi$
$769$	22.6491	0.816747	0.408374	+	0.912815i	$0.366096\pi$
$770$	0	0
$771$	0	0
$772$	−18.8177	−0.677263
$773$	−16.8541	−0.606200	−0.303100	−	0.952959i	$-0.598022\pi$
$773$	−16.8541	−0.606200	−0.303100	+	0.952959i	$0.598022\pi$
$774$	0	0
$775$	0	0
$776$	−16.3322	−0.586292
$777$	0	0
$778$	57.1259	2.04806
$779$	57.8470	2.07258
$780$	0	0
$781$	12.5489	0.449035
$782$	1.93153	0.0690715
$783$	0	0
$784$	−70.0026	−2.50009
$785$	0	0
$786$	0	0
$787$	20.1087	0.716798	0.358399	−	0.933568i	$-0.383323\pi$
$787$	20.1087	0.716798	0.358399	+	0.933568i	$0.383323\pi$
$788$	1.06053	0.0377800
$789$	0	0
$790$	0	0
$791$	64.8854	2.30706
$792$	0	0
$793$	−6.67976	−0.237205
$794$	45.1635	1.60279
$795$	0	0
$796$	−7.19336	−0.254962
$797$	54.1727	1.91890	0.959448	−	0.281887i	$-0.0909602\pi$
$797$	54.1727	1.91890	0.959448	+	0.281887i	$0.0909602\pi$
$798$	0	0
$799$	−0.214100	−0.00757430
$800$	0	0
$801$	0	0
$802$	−6.39287	−0.225740
$803$	−40.2918	−1.42187
$804$	0	0
$805$	0	0
$806$	−3.37120	−0.118745
$807$	0	0
$808$	−6.99862	−0.246211
$809$	−15.3366	−0.539206	−0.269603	−	0.962972i	$-0.586892\pi$
$809$	−15.3366	−0.539206	−0.269603	+	0.962972i	$0.586892\pi$
$810$	0	0
$811$	11.5666	0.406158	0.203079	−	0.979162i	$-0.434905\pi$
$811$	11.5666	0.406158	0.203079	+	0.979162i	$0.434905\pi$
$812$	−15.8136	−0.554949
$813$	0	0
$814$	−35.6957	−1.25113
$815$	0	0
$816$	0	0
$817$	7.70535	0.269576
$818$	−23.8630	−0.834349
$819$	0	0
$820$	0	0
$821$	−33.3675	−1.16453	−0.582267	−	0.812997i	$-0.697834\pi$
$821$	−33.3675	−1.16453	−0.582267	+	0.812997i	$0.697834\pi$
$822$	0	0
$823$	29.9821	1.04511	0.522554	−	0.852606i	$-0.324979\pi$
$823$	29.9821	1.04511	0.522554	+	0.852606i	$0.324979\pi$
$824$	−0.783901	−0.0273085
$825$	0	0
$826$	−34.8395	−1.21222
$827$	24.7230	0.859702	0.429851	−	0.902900i	$-0.358566\pi$
$827$	24.7230	0.859702	0.429851	+	0.902900i	$0.358566\pi$
$828$	0	0
$829$	−0.211406	−0.00734242	−0.00367121	−	0.999993i	$-0.501169\pi$
$829$	−0.211406	−0.00734242	−0.00367121	+	0.999993i	$0.501169\pi$
$830$	0	0
$831$	0	0
$832$	1.41114	0.0489225
$833$	−3.27475	−0.113464
$834$	0	0
$835$	0	0
$836$	18.1179	0.626620
$837$	0	0
$838$	32.9895	1.13961
$839$	5.51714	0.190473	0.0952365	−	0.995455i	$-0.469639\pi$
$839$	5.51714	0.190473	0.0952365	+	0.995455i	$0.469639\pi$
$840$	0	0
$841$	−11.9380	−0.411654
$842$	−43.4051	−1.49584
$843$	0	0
$844$	−17.5376	−0.603670
$845$	0	0
$846$	0	0
$847$	19.8591	0.682368
$848$	−6.12191	−0.210227
$849$	0	0
$850$	0	0
$851$	−26.7177	−0.915871
$852$	0	0
$853$	1.53946	0.0527100	0.0263550	−	0.999653i	$-0.491610\pi$
$853$	1.53946	0.0527100	0.0263550	+	0.999653i	$0.491610\pi$
$854$	−90.1072	−3.08341
$855$	0	0
$856$	3.22674	0.110288
$857$	−45.3407	−1.54881	−0.774404	−	0.632691i	$-0.781950\pi$
$857$	−45.3407	−1.54881	−0.774404	+	0.632691i	$0.781950\pi$
$858$	0	0
$859$	−21.7964	−0.743685	−0.371842	−	0.928296i	$-0.621274\pi$
$859$	−21.7964	−0.743685	−0.371842	+	0.928296i	$0.621274\pi$
$860$	0	0
$861$	0	0
$862$	−15.8667	−0.540422
$863$	−12.7882	−0.435314	−0.217657	−	0.976025i	$-0.569841\pi$
$863$	−12.7882	−0.435314	−0.217657	+	0.976025i	$0.569841\pi$
$864$	0	0
$865$	0	0
$866$	2.94868	0.100200
$867$	0	0
$868$	−13.3814	−0.454194
$869$	37.6410	1.27688
$870$	0	0
$871$	1.69273	0.0573561
$872$	−0.147065	−0.00498023
$873$	0	0
$874$	46.0863	1.55889
$875$	0	0
$876$	0	0
$877$	−9.55340	−0.322595	−0.161298	−	0.986906i	$-0.551568\pi$
$877$	−9.55340	−0.322595	−0.161298	+	0.986906i	$0.551568\pi$
$878$	47.2753	1.59546
$879$	0	0
$880$	0	0
$881$	39.1333	1.31843	0.659217	−	0.751953i	$-0.270888\pi$
$881$	39.1333	1.31843	0.659217	+	0.751953i	$0.270888\pi$
$882$	0	0
$883$	35.0254	1.17870	0.589349	−	0.807879i	$-0.299384\pi$
$883$	35.0254	1.17870	0.589349	+	0.807879i	$0.299384\pi$
$884$	−0.111132	−0.00373776
$885$	0	0
$886$	50.3525	1.69162
$887$	36.2438	1.21695	0.608474	−	0.793574i	$-0.291782\pi$
$887$	36.2438	1.21695	0.608474	+	0.793574i	$0.291782\pi$
$888$	0	0
$889$	−54.2322	−1.81889
$890$	0	0
$891$	0	0
$892$	5.28288	0.176884
$893$	−5.10841	−0.170946
$894$	0	0
$895$	0	0
$896$	59.8455	1.99930
$897$	0	0
$898$	−10.6051	−0.353895
$899$	14.4378	0.481528
$900$	0	0
$901$	−0.286386	−0.00954089
$902$	68.6876	2.28705
$903$	0	0
$904$	−27.7438	−0.922746
$905$	0	0
$906$	0	0
$907$	40.8532	1.35651	0.678255	−	0.734827i	$-0.262737\pi$
$907$	40.8532	1.35651	0.678255	+	0.734827i	$0.262737\pi$
$908$	−9.39139	−0.311664
$909$	0	0
$910$	0	0
$911$	−12.7468	−0.422320	−0.211160	−	0.977452i	$-0.567724\pi$
$911$	−12.7468	−0.422320	−0.211160	+	0.977452i	$0.567724\pi$
$912$	0	0
$913$	41.0403	1.35824
$914$	32.2893	1.06803
$915$	0	0
$916$	12.5421	0.414403
$917$	76.8425	2.53756
$918$	0	0
$919$	9.67206	0.319052	0.159526	−	0.987194i	$-0.449003\pi$
$919$	9.67206	0.319052	0.159526	+	0.987194i	$0.449003\pi$
$920$	0	0
$921$	0	0
$922$	−11.9036	−0.392023
$923$	−1.83656	−0.0604512
$924$	0	0
$925$	0	0
$926$	−16.1917	−0.532094
$927$	0	0
$928$	18.3583	0.602641
$929$	−10.9195	−0.358257	−0.179128	−	0.983826i	$-0.557328\pi$
$929$	−10.9195	−0.358257	−0.179128	+	0.983826i	$0.557328\pi$
$930$	0	0
$931$	−78.1356	−2.56079
$932$	−6.52711	−0.213803
$933$	0	0
$934$	32.3585	1.05880
$935$	0	0
$936$	0	0
$937$	−15.9466	−0.520954	−0.260477	−	0.965480i	$-0.583880\pi$
$937$	−15.9466	−0.520954	−0.260477	+	0.965480i	$0.583880\pi$
$938$	22.8343	0.745565
$939$	0	0
$940$	0	0
$941$	−5.92701	−0.193215	−0.0966074	−	0.995323i	$-0.530799\pi$
$941$	−5.92701	−0.193215	−0.0966074	+	0.995323i	$0.530799\pi$
$942$	0	0
$943$	51.4117	1.67419
$944$	22.4146	0.729533
$945$	0	0
$946$	9.14934	0.297471
$947$	−0.990382	−0.0321831	−0.0160916	−	0.999871i	$-0.505122\pi$
$947$	−0.990382	−0.0321831	−0.0160916	+	0.999871i	$0.505122\pi$
$948$	0	0
$949$	5.89681	0.191418
$950$	0	0
$951$	0	0
$952$	2.09645	0.0679463
$953$	−53.5993	−1.73625	−0.868125	−	0.496345i	$-0.834675\pi$
$953$	−53.5993	−1.73625	−0.868125	+	0.496345i	$0.834675\pi$
$954$	0	0
$955$	0	0
$956$	−7.28708	−0.235681
$957$	0	0
$958$	63.9689	2.06674
$959$	47.8491	1.54513
$960$	0	0
$961$	−18.7828	−0.605896
$962$	5.22415	0.168433
$963$	0	0
$964$	−0.501066	−0.0161382
$965$	0	0
$966$	0	0
$967$	29.7350	0.956212	0.478106	−	0.878302i	$-0.341323\pi$
$967$	29.7350	0.956212	0.478106	+	0.878302i	$0.341323\pi$
$968$	−8.49141	−0.272924
$969$	0	0
$970$	0	0
$971$	−60.6166	−1.94528	−0.972640	−	0.232317i	$-0.925369\pi$
$971$	−60.6166	−1.94528	−0.972640	+	0.232317i	$0.925369\pi$
$972$	0	0
$973$	30.5764	0.980236
$974$	−28.1954	−0.903440
$975$	0	0
$976$	57.9720	1.85564
$977$	44.5130	1.42410	0.712048	−	0.702131i	$-0.247768\pi$
$977$	44.5130	1.42410	0.712048	+	0.702131i	$0.247768\pi$
$978$	0	0
$979$	28.4146	0.908135
$980$	0	0
$981$	0	0
$982$	15.0675	0.480822
$983$	38.2401	1.21967	0.609834	−	0.792529i	$-0.291236\pi$
$983$	38.2401	1.21967	0.609834	+	0.792529i	$0.291236\pi$
$984$	0	0
$985$	0	0
$986$	1.61747	0.0515107
$987$	0	0
$988$	−2.65160	−0.0843586
$989$	6.84815	0.217759
$990$	0	0
$991$	22.7082	0.721349	0.360675	−	0.932692i	$-0.382547\pi$
$991$	22.7082	0.721349	0.360675	+	0.932692i	$0.382547\pi$
$992$	15.5347	0.493228
$993$	0	0
$994$	−24.7745	−0.785799
$995$	0	0
$996$	0	0
$997$	−20.6284	−0.653309	−0.326654	−	0.945144i	$-0.605921\pi$
$997$	−20.6284	−0.653309	−0.326654	+	0.945144i	$0.605921\pi$
$998$	−61.1600	−1.93599
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.s.1.7		8
3.2	odd	2		625.2.a.g.1.2	yes	8
5.4	even	2		5625.2.a.be.1.2		8
12.11	even	2		10000.2.a.be.1.5		8
15.2	even	4		625.2.b.d.624.5		16
15.8	even	4		625.2.b.d.624.12		16
15.14	odd	2		625.2.a.e.1.7	✓	8
60.59	even	2		10000.2.a.bn.1.4		8
75.2	even	20		625.2.e.k.124.3		32
75.8	even	20		625.2.e.j.374.3		32
75.11	odd	10		625.2.d.n.501.1		16
75.14	odd	10		625.2.d.p.501.4		16
75.17	even	20		625.2.e.j.374.6		32
75.23	even	20		625.2.e.k.124.6		32
75.29	odd	10		625.2.d.q.376.1		16
75.38	even	20		625.2.e.k.499.3		32
75.41	odd	10		625.2.d.n.126.1		16
75.44	odd	10		625.2.d.q.251.1		16
75.47	even	20		625.2.e.j.249.3		32
75.53	even	20		625.2.e.j.249.6		32
75.56	odd	10		625.2.d.m.251.4		16
75.59	odd	10		625.2.d.p.126.4		16
75.62	even	20		625.2.e.k.499.6		32
75.71	odd	10		625.2.d.m.376.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
625.2.a.e.1.7	✓	8	15.14	odd	2
625.2.a.g.1.2	yes	8	3.2	odd	2
625.2.b.d.624.5		16	15.2	even	4
625.2.b.d.624.12		16	15.8	even	4
625.2.d.m.251.4		16	75.56	odd	10
625.2.d.m.376.4		16	75.71	odd	10
625.2.d.n.126.1		16	75.41	odd	10
625.2.d.n.501.1		16	75.11	odd	10
625.2.d.p.126.4		16	75.59	odd	10
625.2.d.p.501.4		16	75.14	odd	10
625.2.d.q.251.1		16	75.44	odd	10
625.2.d.q.376.1		16	75.29	odd	10
625.2.e.j.249.3		32	75.47	even	20
625.2.e.j.249.6		32	75.53	even	20
625.2.e.j.374.3		32	75.8	even	20
625.2.e.j.374.6		32	75.17	even	20
625.2.e.k.124.3		32	75.2	even	20
625.2.e.k.124.6		32	75.23	even	20
625.2.e.k.499.3		32	75.38	even	20
625.2.e.k.499.6		32	75.62	even	20
5625.2.a.s.1.7		8	1.1	even	1	trivial
5625.2.a.be.1.2		8	5.4	even	2
10000.2.a.be.1.5		8	12.11	even	2
10000.2.a.bn.1.4		8	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)

\(f(q)\)	\(=\)	\(q+1.68341 q^{2} +0.833870 q^{4} +4.59110 q^{7} -1.96307 q^{8} +O(q^{10})\)	\(q+1.68341 q^{2} +0.833870 q^{4} +4.59110 q^{7} -1.96307 q^{8} -3.91479 q^{11} +0.572939 q^{13} +7.72871 q^{14} -4.97240 q^{16} -0.232611 q^{17} -5.55010 q^{19} -6.59020 q^{22} -4.93267 q^{23} +0.964492 q^{26} +3.82839 q^{28} -4.13062 q^{29} -3.49531 q^{31} -4.44444 q^{32} -0.391580 q^{34} +5.41648 q^{37} -9.34309 q^{38} -10.4227 q^{41} -1.38833 q^{43} -3.26443 q^{44} -8.30370 q^{46} +0.920418 q^{47} +14.0782 q^{49} +0.477757 q^{52} +1.23118 q^{53} -9.01268 q^{56} -6.95353 q^{58} -4.50780 q^{59} -11.6588 q^{61} -5.88405 q^{62} +2.46298 q^{64} +2.95447 q^{67} -0.193968 q^{68} -3.20551 q^{71} +10.2922 q^{73} +9.11816 q^{74} -4.62806 q^{76} -17.9732 q^{77} -9.61509 q^{79} -17.5457 q^{82} -10.4834 q^{83} -2.33712 q^{86} +7.68502 q^{88} -7.25828 q^{89} +2.63042 q^{91} -4.11320 q^{92} +1.54944 q^{94} +8.31971 q^{97} +23.6994 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{2} + 11 q^{4} + 10 q^{7} - 15 q^{8}+O(q^{10}) \) 8 * q - 5 * q^2 + 11 * q^4 + 10 * q^7 - 15 * q^8	\( 8 q - 5 q^{2} + 11 q^{4} + 10 q^{7} - 15 q^{8} - q^{11} + 10 q^{13} + 8 q^{14} + 13 q^{16} - 15 q^{17} - 10 q^{19} - 5 q^{22} - 30 q^{23} - 11 q^{26} - 5 q^{28} - 10 q^{29} - 9 q^{31} - 30 q^{32} + 7 q^{34} - 10 q^{37} - 20 q^{38} + 4 q^{41} + 18 q^{44} - 9 q^{46} - 30 q^{47} - 4 q^{49} + 5 q^{52} - 10 q^{53} - 30 q^{58} + 5 q^{59} + 6 q^{61} - 10 q^{62} - 9 q^{64} + 10 q^{67} - 40 q^{68} + 9 q^{71} + 18 q^{74} - 10 q^{76} - 5 q^{77} - 20 q^{79} - 45 q^{82} - 40 q^{83} + 24 q^{86} - 40 q^{88} + 5 q^{89} + 6 q^{91} - 15 q^{92} + 47 q^{94} + 30 q^{98}+O(q^{100}) \) 8 * q - 5 * q^2 + 11 * q^4 + 10 * q^7 - 15 * q^8 - q^11 + 10 * q^13 + 8 * q^14 + 13 * q^16 - 15 * q^17 - 10 * q^19 - 5 * q^22 - 30 * q^23 - 11 * q^26 - 5 * q^28 - 10 * q^29 - 9 * q^31 - 30 * q^32 + 7 * q^34 - 10 * q^37 - 20 * q^38 + 4 * q^41 + 18 * q^44 - 9 * q^46 - 30 * q^47 - 4 * q^49 + 5 * q^52 - 10 * q^53 - 30 * q^58 + 5 * q^59 + 6 * q^61 - 10 * q^62 - 9 * q^64 + 10 * q^67 - 40 * q^68 + 9 * q^71 + 18 * q^74 - 10 * q^76 - 5 * q^77 - 20 * q^79 - 45 * q^82 - 40 * q^83 + 24 * q^86 - 40 * q^88 + 5 * q^89 + 6 * q^91 - 15 * q^92 + 47 * q^94 + 30 * q^98

Introduction

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