LMFDB - Embedded newform 5625.2.a.be.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:	$0$
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			$-1.47435$ 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)

$f(q)$	$=$	$q+2.47435 q^{2} +4.12242 q^{4} +0.973070 q^{7} +5.25163 q^{8} +O(q^{10})$	$q+2.47435 q^{2} +4.12242 q^{4} +0.973070 q^{7} +5.25163 q^{8} +5.38225 q^{11} +1.99670 q^{13} +2.40772 q^{14} +4.74954 q^{16} -2.04301 q^{17} +6.20428 q^{19} +13.3176 q^{22} +1.93813 q^{23} +4.94054 q^{26} +4.01141 q^{28} -4.81812 q^{29} -6.64014 q^{31} +1.24877 q^{32} -5.05512 q^{34} +0.978913 q^{37} +15.3516 q^{38} -2.73319 q^{41} -3.99413 q^{43} +22.1879 q^{44} +4.79562 q^{46} +7.21339 q^{47} -6.05313 q^{49} +8.23125 q^{52} +13.2746 q^{53} +5.11020 q^{56} -11.9217 q^{58} -6.54024 q^{59} +2.72672 q^{61} -16.4301 q^{62} -6.40916 q^{64} -9.56957 q^{67} -8.42215 q^{68} +5.68520 q^{71} +9.35281 q^{73} +2.42218 q^{74} +25.5767 q^{76} +5.23731 q^{77} -3.18171 q^{79} -6.76289 q^{82} +6.11387 q^{83} -9.88290 q^{86} +28.2656 q^{88} +3.00743 q^{89} +1.94293 q^{91} +7.98980 q^{92} +17.8485 q^{94} -5.95526 q^{97} -14.9776 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.47435	1.74963	0.874816	−	0.484455i	$-0.160982\pi$
$2$	2.47435	1.74963	0.874816	+	0.484455i	$0.160982\pi$
$3$	0	0
$3$	0	0
$4$	4.12242	2.06121
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.973070	0.367786	0.183893	−	0.982946i	$-0.441130\pi$
$7$	0.973070	0.367786	0.183893	+	0.982946i	$0.441130\pi$
$8$	5.25163	1.85673
$9$	0	0
$10$	0	0
$11$	5.38225	1.62281	0.811405	−	0.584484i	$-0.198703\pi$
$11$	5.38225	1.62281	0.811405	+	0.584484i	$0.198703\pi$
$12$	0	0
$13$	1.99670	0.553785	0.276892	−	0.960901i	$-0.410695\pi$
$13$	1.99670	0.553785	0.276892	+	0.960901i	$0.410695\pi$
$14$	2.40772	0.643490
$15$	0	0
$16$	4.74954	1.18738
$17$	−2.04301	−0.495502	−0.247751	−	0.968824i	$-0.579692\pi$
$17$	−2.04301	−0.495502	−0.247751	+	0.968824i	$0.579692\pi$
$18$	0	0
$19$	6.20428	1.42336	0.711680	−	0.702504i	$-0.247935\pi$
$19$	6.20428	1.42336	0.711680	+	0.702504i	$0.247935\pi$
$20$	0	0
$21$	0	0
$22$	13.3176	2.83932
$23$	1.93813	0.404128	0.202064	−	0.979372i	$-0.435235\pi$
$23$	1.93813	0.404128	0.202064	+	0.979372i	$0.435235\pi$
$24$	0	0
$25$	0	0
$26$	4.94054	0.968920
$27$	0	0
$28$	4.01141	0.758085
$29$	−4.81812	−0.894702	−0.447351	−	0.894359i	$-0.647632\pi$
$29$	−4.81812	−0.894702	−0.447351	+	0.894359i	$0.647632\pi$
$30$	0	0
$31$	−6.64014	−1.19260	−0.596302	−	0.802760i	$-0.703364\pi$
$31$	−6.64014	−1.19260	−0.596302	+	0.802760i	$0.703364\pi$
$32$	1.24877	0.220754
$33$	0	0
$34$	−5.05512	−0.866947
$35$	0	0
$36$	0	0
$37$	0.978913	0.160932	0.0804662	−	0.996757i	$-0.474359\pi$
$37$	0.978913	0.160932	0.0804662	+	0.996757i	$0.474359\pi$
$38$	15.3516	2.49035
$39$	0	0
$40$	0	0
$41$	−2.73319	−0.426853	−0.213427	−	0.976959i	$-0.568462\pi$
$41$	−2.73319	−0.426853	−0.213427	+	0.976959i	$0.568462\pi$
$42$	0	0
$43$	−3.99413	−0.609100	−0.304550	−	0.952496i	$-0.598506\pi$
$43$	−3.99413	−0.609100	−0.304550	+	0.952496i	$0.598506\pi$
$44$	22.1879	3.34496
$45$	0	0
$46$	4.79562	0.707076
$47$	7.21339	1.05218	0.526091	−	0.850428i	$-0.323657\pi$
$47$	7.21339	1.05218	0.526091	+	0.850428i	$0.323657\pi$
$48$	0	0
$49$	−6.05313	−0.864734
$50$	0	0
$51$	0	0
$52$	8.23125	1.14147
$53$	13.2746	1.82340	0.911700	−	0.410857i	$-0.134770\pi$
$53$	13.2746	1.82340	0.911700	+	0.410857i	$0.134770\pi$
$54$	0	0
$55$	0	0
$56$	5.11020	0.682880
$57$	0	0
$58$	−11.9217	−1.56540
$59$	−6.54024	−0.851467	−0.425733	−	0.904849i	$-0.639984\pi$
$59$	−6.54024	−0.851467	−0.425733	+	0.904849i	$0.639984\pi$
$60$	0	0
$61$	2.72672	0.349121	0.174560	−	0.984646i	$-0.444150\pi$
$61$	2.72672	0.349121	0.174560	+	0.984646i	$0.444150\pi$
$62$	−16.4301	−2.08662
$63$	0	0
$64$	−6.40916	−0.801146
$65$	0	0
$66$	0	0
$67$	−9.56957	−1.16911	−0.584555	−	0.811354i	$-0.698731\pi$
$67$	−9.56957	−1.16911	−0.584555	+	0.811354i	$0.698731\pi$
$68$	−8.42215	−1.02134
$69$	0	0
$70$	0	0
$71$	5.68520	0.674710	0.337355	−	0.941378i	$-0.390468\pi$
$71$	5.68520	0.674710	0.337355	+	0.941378i	$0.390468\pi$
$72$	0	0
$73$	9.35281	1.09466	0.547332	−	0.836916i	$-0.315644\pi$
$73$	9.35281	1.09466	0.547332	+	0.836916i	$0.315644\pi$
$74$	2.42218	0.281572
$75$	0	0
$76$	25.5767	2.93385
$77$	5.23731	0.596847
$78$	0	0
$79$	−3.18171	−0.357970	−0.178985	−	0.983852i	$-0.557281\pi$
$79$	−3.18171	−0.357970	−0.178985	+	0.983852i	$0.557281\pi$
$80$	0	0
$81$	0	0
$82$	−6.76289	−0.746836
$83$	6.11387	0.671085	0.335542	−	0.942025i	$-0.391080\pi$
$83$	6.11387	0.671085	0.335542	+	0.942025i	$0.391080\pi$
$84$	0	0
$85$	0	0
$86$	−9.88290	−1.06570
$87$	0	0
$88$	28.2656	3.01312
$89$	3.00743	0.318787	0.159393	−	0.987215i	$-0.449046\pi$
$89$	3.00743	0.318787	0.159393	+	0.987215i	$0.449046\pi$
$90$	0	0
$91$	1.94293	0.203674
$92$	7.98980	0.832994
$93$	0	0
$94$	17.8485	1.84093
$95$	0	0
$96$	0	0
$97$	−5.95526	−0.604666	−0.302333	−	0.953202i	$-0.597765\pi$
$97$	−5.95526	−0.604666	−0.302333	+	0.953202i	$0.597765\pi$
$98$	−14.9776	−1.51297
$99$	0	0
$100$	0	0
$101$	7.77373	0.773515	0.386758	−	0.922181i	$-0.373595\pi$
$101$	7.77373	0.773515	0.386758	+	0.922181i	$0.373595\pi$
$102$	0	0
$103$	−4.95056	−0.487793	−0.243897	−	0.969801i	$-0.578426\pi$
$103$	−4.95056	−0.487793	−0.243897	+	0.969801i	$0.578426\pi$
$104$	10.4859	1.02823
$105$	0	0
$106$	32.8459	3.19028
$107$	1.16798	0.112913	0.0564567	−	0.998405i	$-0.482020\pi$
$107$	1.16798	0.112913	0.0564567	+	0.998405i	$0.482020\pi$
$108$	0	0
$109$	−17.6879	−1.69420	−0.847099	−	0.531435i	$-0.821653\pi$
$109$	−17.6879	−1.69420	−0.847099	+	0.531435i	$0.821653\pi$
$110$	0	0
$111$	0	0
$112$	4.62163	0.436703
$113$	5.03643	0.473787	0.236894	−	0.971536i	$-0.423871\pi$
$113$	5.03643	0.473787	0.236894	+	0.971536i	$0.423871\pi$
$114$	0	0
$115$	0	0
$116$	−19.8623	−1.84417
$117$	0	0
$118$	−16.1829	−1.48975
$119$	−1.98799	−0.182239
$120$	0	0
$121$	17.9687	1.63351
$122$	6.74687	0.610833
$123$	0	0
$124$	−27.3735	−2.45821
$125$	0	0
$126$	0	0
$127$	−7.79457	−0.691656	−0.345828	−	0.938298i	$-0.612402\pi$
$127$	−7.79457	−0.691656	−0.345828	+	0.938298i	$0.612402\pi$
$128$	−18.3561	−1.62246
$129$	0	0
$130$	0	0
$131$	9.79015	0.855369	0.427685	−	0.903928i	$-0.359329\pi$
$131$	9.79015	0.855369	0.427685	+	0.903928i	$0.359329\pi$
$132$	0	0
$133$	6.03720	0.523491
$134$	−23.6785	−2.04551
$135$	0	0
$136$	−10.7291	−0.920015
$137$	−3.63326	−0.310411	−0.155205	−	0.987882i	$-0.549604\pi$
$137$	−3.63326	−0.310411	−0.155205	+	0.987882i	$0.549604\pi$
$138$	0	0
$139$	−1.86079	−0.157830	−0.0789149	−	0.996881i	$-0.525146\pi$
$139$	−1.86079	−0.157830	−0.0789149	+	0.996881i	$0.525146\pi$
$140$	0	0
$141$	0	0
$142$	14.0672	1.18049
$143$	10.7467	0.898688
$144$	0	0
$145$	0	0
$146$	23.1421	1.91526
$147$	0	0
$148$	4.03550	0.331716
$149$	−16.8530	−1.38066	−0.690328	−	0.723497i	$-0.742534\pi$
$149$	−16.8530	−1.38066	−0.690328	+	0.723497i	$0.742534\pi$
$150$	0	0
$151$	−14.3201	−1.16536	−0.582678	−	0.812703i	$-0.697995\pi$
$151$	−14.3201	−1.16536	−0.582678	+	0.812703i	$0.697995\pi$
$152$	32.5826	2.64280
$153$	0	0
$154$	12.9590	1.04426
$155$	0	0
$156$	0	0
$157$	21.6869	1.73080	0.865400	−	0.501081i	$-0.167064\pi$
$157$	21.6869	1.73080	0.865400	+	0.501081i	$0.167064\pi$
$158$	−7.87266	−0.626315
$159$	0	0
$160$	0	0
$161$	1.88594	0.148633
$162$	0	0
$163$	19.7210	1.54467	0.772334	−	0.635217i	$-0.219089\pi$
$163$	19.7210	1.54467	0.772334	+	0.635217i	$0.219089\pi$
$164$	−11.2674	−0.879835
$165$	0	0
$166$	15.1279	1.17415
$167$	6.07947	0.470444	0.235222	−	0.971942i	$-0.424418\pi$
$167$	6.07947	0.470444	0.235222	+	0.971942i	$0.424418\pi$
$168$	0	0
$169$	−9.01319	−0.693322
$170$	0	0
$171$	0	0
$172$	−16.4655	−1.25548
$173$	15.0309	1.14278	0.571388	−	0.820680i	$-0.306405\pi$
$173$	15.0309	1.14278	0.571388	+	0.820680i	$0.306405\pi$
$174$	0	0
$175$	0	0
$176$	25.5632	1.92690
$177$	0	0
$178$	7.44144	0.557759
$179$	20.0167	1.49612	0.748060	−	0.663631i	$-0.230986\pi$
$179$	20.0167	1.49612	0.748060	+	0.663631i	$0.230986\pi$
$180$	0	0
$181$	20.2993	1.50883	0.754417	−	0.656395i	$-0.227920\pi$
$181$	20.2993	1.50883	0.754417	+	0.656395i	$0.227920\pi$
$182$	4.80749	0.356355
$183$	0	0
$184$	10.1783	0.750357
$185$	0	0
$186$	0	0
$187$	−10.9960	−0.804106
$188$	29.7367	2.16877
$189$	0	0
$190$	0	0
$191$	9.47681	0.685718	0.342859	−	0.939387i	$-0.388605\pi$
$191$	9.47681	0.685718	0.342859	+	0.939387i	$0.388605\pi$
$192$	0	0
$193$	−2.72478	−0.196134	−0.0980671	−	0.995180i	$-0.531266\pi$
$193$	−2.72478	−0.196134	−0.0980671	+	0.995180i	$0.531266\pi$
$194$	−14.7354	−1.05794
$195$	0	0
$196$	−24.9536	−1.78240
$197$	0.780036	0.0555753	0.0277876	−	0.999614i	$-0.491154\pi$
$197$	0.780036	0.0555753	0.0277876	+	0.999614i	$0.491154\pi$
$198$	0	0
$199$	−7.42526	−0.526363	−0.263181	−	0.964746i	$-0.584772\pi$
$199$	−7.42526	−0.526363	−0.263181	+	0.964746i	$0.584772\pi$
$200$	0	0
$201$	0	0
$202$	19.2350	1.35337
$203$	−4.68836	−0.329059
$204$	0	0
$205$	0	0
$206$	−12.2494	−0.853459
$207$	0	0
$208$	9.48340	0.657555
$209$	33.3930	2.30984
$210$	0	0
$211$	−2.98128	−0.205240	−0.102620	−	0.994721i	$-0.532723\pi$
$211$	−2.98128	−0.205240	−0.102620	+	0.994721i	$0.532723\pi$
$212$	54.7233	3.75841
$213$	0	0
$214$	2.89001	0.197557
$215$	0	0
$216$	0	0
$217$	−6.46132	−0.438623
$218$	−43.7662	−2.96422
$219$	0	0
$220$	0	0
$221$	−4.07927	−0.274402
$222$	0	0
$223$	−24.8720	−1.66556	−0.832778	−	0.553607i	$-0.813251\pi$
$223$	−24.8720	−1.66556	−0.832778	+	0.553607i	$0.813251\pi$
$224$	1.21514	0.0811903
$225$	0	0
$226$	12.4619	0.828953
$227$	−17.5594	−1.16546	−0.582729	−	0.812667i	$-0.698015\pi$
$227$	−17.5594	−1.16546	−0.582729	+	0.812667i	$0.698015\pi$
$228$	0	0
$229$	7.01849	0.463795	0.231897	−	0.972740i	$-0.425507\pi$
$229$	7.01849	0.463795	0.231897	+	0.972740i	$0.425507\pi$
$230$	0	0
$231$	0	0
$232$	−25.3030	−1.66122
$233$	−8.69126	−0.569383	−0.284692	−	0.958619i	$-0.591891\pi$
$233$	−8.69126	−0.569383	−0.284692	+	0.958619i	$0.591891\pi$
$234$	0	0
$235$	0	0
$236$	−26.9617	−1.75505
$237$	0	0
$238$	−4.91899	−0.318851
$239$	−21.3430	−1.38057	−0.690283	−	0.723540i	$-0.742514\pi$
$239$	−21.3430	−1.38057	−0.690283	+	0.723540i	$0.742514\pi$
$240$	0	0
$241$	−21.3897	−1.37783	−0.688916	−	0.724841i	$-0.741913\pi$
$241$	−21.3897	−1.37783	−0.688916	+	0.724841i	$0.741913\pi$
$242$	44.4608	2.85805
$243$	0	0
$244$	11.2407	0.719612
$245$	0	0
$246$	0	0
$247$	12.3881	0.788235
$248$	−34.8716	−2.21435
$249$	0	0
$250$	0	0
$251$	−14.9016	−0.940580	−0.470290	−	0.882512i	$-0.655851\pi$
$251$	−14.9016	−0.940580	−0.470290	+	0.882512i	$0.655851\pi$
$252$	0	0
$253$	10.4315	0.655823
$254$	−19.2865	−1.21014
$255$	0	0
$256$	−32.6011	−2.03757
$257$	19.0689	1.18948	0.594741	−	0.803917i	$-0.297254\pi$
$257$	19.0689	1.18948	0.594741	+	0.803917i	$0.297254\pi$
$258$	0	0
$259$	0.952551	0.0591887
$260$	0	0
$261$	0	0
$262$	24.2243	1.49658
$263$	−6.78341	−0.418283	−0.209142	−	0.977885i	$-0.567067\pi$
$263$	−6.78341	−0.418283	−0.209142	+	0.977885i	$0.567067\pi$
$264$	0	0
$265$	0	0
$266$	14.9382	0.915917
$267$	0	0
$268$	−39.4498	−2.40978
$269$	−0.494663	−0.0301601	−0.0150801	−	0.999886i	$-0.504800\pi$
$269$	−0.494663	−0.0301601	−0.0150801	+	0.999886i	$0.504800\pi$
$270$	0	0
$271$	9.82610	0.596893	0.298446	−	0.954426i	$-0.403532\pi$
$271$	9.82610	0.596893	0.298446	+	0.954426i	$0.403532\pi$
$272$	−9.70334	−0.588352
$273$	0	0
$274$	−8.98998	−0.543104
$275$	0	0
$276$	0	0
$277$	−25.1399	−1.51051	−0.755255	−	0.655431i	$-0.772487\pi$
$277$	−25.1399	−1.51051	−0.755255	+	0.655431i	$0.772487\pi$
$278$	−4.60424	−0.276144
$279$	0	0
$280$	0	0
$281$	7.77050	0.463549	0.231775	−	0.972770i	$-0.425547\pi$
$281$	7.77050	0.463549	0.231775	+	0.972770i	$0.425547\pi$
$282$	0	0
$283$	−28.9617	−1.72160	−0.860798	−	0.508946i	$-0.830035\pi$
$283$	−28.9617	−1.72160	−0.860798	+	0.508946i	$0.830035\pi$
$284$	23.4368	1.39072
$285$	0	0
$286$	26.5912	1.57237
$287$	−2.65959	−0.156991
$288$	0	0
$289$	−12.8261	−0.754477
$290$	0	0
$291$	0	0
$292$	38.5562	2.25633
$293$	−0.154329	−0.00901602	−0.00450801	−	0.999990i	$-0.501435\pi$
$293$	−0.154329	−0.00901602	−0.00450801	+	0.999990i	$0.501435\pi$
$294$	0	0
$295$	0	0
$296$	5.14089	0.298808
$297$	0	0
$298$	−41.7004	−2.41564
$299$	3.86987	0.223800
$300$	0	0
$301$	−3.88657	−0.224018
$302$	−35.4330	−2.03894
$303$	0	0
$304$	29.4675	1.69007
$305$	0	0
$306$	0	0
$307$	14.4875	0.826845	0.413423	−	0.910539i	$-0.364333\pi$
$307$	14.4875	0.826845	0.413423	+	0.910539i	$0.364333\pi$
$308$	21.5904	1.23023
$309$	0	0
$310$	0	0
$311$	−18.5385	−1.05122	−0.525610	−	0.850726i	$-0.676163\pi$
$311$	−18.5385	−1.05122	−0.525610	+	0.850726i	$0.676163\pi$
$312$	0	0
$313$	−6.85703	−0.387582	−0.193791	−	0.981043i	$-0.562078\pi$
$313$	−6.85703	−0.387582	−0.193791	+	0.981043i	$0.562078\pi$
$314$	53.6610	3.02826
$315$	0	0
$316$	−13.1163	−0.737852
$317$	−27.2810	−1.53225	−0.766127	−	0.642689i	$-0.777819\pi$
$317$	−27.2810	−1.53225	−0.766127	+	0.642689i	$0.777819\pi$
$318$	0	0
$319$	−25.9323	−1.45193
$320$	0	0
$321$	0	0
$322$	4.66647	0.260052
$323$	−12.6754	−0.705278
$324$	0	0
$325$	0	0
$326$	48.7967	2.70260
$327$	0	0
$328$	−14.3537	−0.792551
$329$	7.01914	0.386978
$330$	0	0
$331$	−13.0429	−0.716901	−0.358450	−	0.933549i	$-0.616695\pi$
$331$	−13.0429	−0.716901	−0.358450	+	0.933549i	$0.616695\pi$
$332$	25.2040	1.38325
$333$	0	0
$334$	15.0428	0.823103
$335$	0	0
$336$	0	0
$337$	19.6507	1.07044	0.535222	−	0.844712i	$-0.320228\pi$
$337$	19.6507	1.07044	0.535222	+	0.844712i	$0.320228\pi$
$338$	−22.3018	−1.21306
$339$	0	0
$340$	0	0
$341$	−35.7389	−1.93537
$342$	0	0
$343$	−12.7016	−0.685823
$344$	−20.9757	−1.13093
$345$	0	0
$346$	37.1917	1.99944
$347$	4.38943	0.235637	0.117818	−	0.993035i	$-0.462410\pi$
$347$	4.38943	0.235637	0.117818	+	0.993035i	$0.462410\pi$
$348$	0	0
$349$	27.1955	1.45574	0.727872	−	0.685713i	$-0.240510\pi$
$349$	27.1955	1.45574	0.727872	+	0.685713i	$0.240510\pi$
$350$	0	0
$351$	0	0
$352$	6.72122	0.358242
$353$	−15.9317	−0.847960	−0.423980	−	0.905672i	$-0.639367\pi$
$353$	−15.9317	−0.847960	−0.423980	+	0.905672i	$0.639367\pi$
$354$	0	0
$355$	0	0
$356$	12.3979	0.657087
$357$	0	0
$358$	49.5284	2.61766
$359$	5.01508	0.264686	0.132343	−	0.991204i	$-0.457750\pi$
$359$	5.01508	0.264686	0.132343	+	0.991204i	$0.457750\pi$
$360$	0	0
$361$	19.4931	1.02595
$362$	50.2276	2.63991
$363$	0	0
$364$	8.00958	0.419816
$365$	0	0
$366$	0	0
$367$	−28.3669	−1.48074	−0.740370	−	0.672200i	$-0.765349\pi$
$367$	−28.3669	−1.48074	−0.740370	+	0.672200i	$0.765349\pi$
$368$	9.20522	0.479855
$369$	0	0
$370$	0	0
$371$	12.9171	0.670621
$372$	0	0
$373$	32.7218	1.69427	0.847135	−	0.531377i	$-0.178325\pi$
$373$	32.7218	1.69427	0.847135	+	0.531377i	$0.178325\pi$
$374$	−27.2080	−1.40689
$375$	0	0
$376$	37.8821	1.95362
$377$	−9.62033	−0.495472
$378$	0	0
$379$	4.48483	0.230370	0.115185	−	0.993344i	$-0.463254\pi$
$379$	4.48483	0.230370	0.115185	+	0.993344i	$0.463254\pi$
$380$	0	0
$381$	0	0
$382$	23.4490	1.19975
$383$	16.2839	0.832069	0.416034	−	0.909349i	$-0.363420\pi$
$383$	16.2839	0.832069	0.416034	+	0.909349i	$0.363420\pi$
$384$	0	0
$385$	0	0
$386$	−6.74208	−0.343163
$387$	0	0
$388$	−24.5501	−1.24634
$389$	−21.4899	−1.08958	−0.544791	−	0.838572i	$-0.683391\pi$
$389$	−21.4899	−1.08958	−0.544791	+	0.838572i	$0.683391\pi$
$390$	0	0
$391$	−3.95962	−0.200246
$392$	−31.7888	−1.60558
$393$	0	0
$394$	1.93008	0.0972363
$395$	0	0
$396$	0	0
$397$	20.8870	1.04829	0.524145	−	0.851629i	$-0.324385\pi$
$397$	20.8870	1.04829	0.524145	+	0.851629i	$0.324385\pi$
$398$	−18.3727	−0.920941
$399$	0	0
$400$	0	0
$401$	0.633423	0.0316316	0.0158158	−	0.999875i	$-0.494965\pi$
$401$	0.633423	0.0316316	0.0158158	+	0.999875i	$0.494965\pi$
$402$	0	0
$403$	−13.2584	−0.660447
$404$	32.0466	1.59438
$405$	0	0
$406$	−11.6007	−0.575732
$407$	5.26876	0.261163
$408$	0	0
$409$	−4.29822	−0.212534	−0.106267	−	0.994338i	$-0.533890\pi$
$409$	−4.29822	−0.212534	−0.106267	+	0.994338i	$0.533890\pi$
$410$	0	0
$411$	0	0
$412$	−20.4083	−1.00545
$413$	−6.36411	−0.313158
$414$	0	0
$415$	0	0
$416$	2.49343	0.122250
$417$	0	0
$418$	82.6261	4.04137
$419$	−33.6215	−1.64252	−0.821258	−	0.570556i	$-0.806728\pi$
$419$	−33.6215	−1.64252	−0.821258	+	0.570556i	$0.806728\pi$
$420$	0	0
$421$	−21.3768	−1.04184	−0.520920	−	0.853605i	$-0.674411\pi$
$421$	−21.3768	−1.04184	−0.520920	+	0.853605i	$0.674411\pi$
$422$	−7.37674	−0.359094
$423$	0	0
$424$	69.7130	3.38556
$425$	0	0
$426$	0	0
$427$	2.65329	0.128402
$428$	4.81493	0.232738
$429$	0	0
$430$	0	0
$431$	−26.0905	−1.25673	−0.628367	−	0.777917i	$-0.716277\pi$
$431$	−26.0905	−1.25673	−0.628367	+	0.777917i	$0.716277\pi$
$432$	0	0
$433$	−11.7676	−0.565513	−0.282757	−	0.959192i	$-0.591249\pi$
$433$	−11.7676	−0.565513	−0.282757	+	0.959192i	$0.591249\pi$
$434$	−15.9876	−0.767429
$435$	0	0
$436$	−72.9172	−3.49210
$437$	12.0247	0.575220
$438$	0	0
$439$	4.92572	0.235092	0.117546	−	0.993067i	$-0.462497\pi$
$439$	4.92572	0.235092	0.117546	+	0.993067i	$0.462497\pi$
$440$	0	0
$441$	0	0
$442$	−10.0936	−0.480102
$443$	9.35961	0.444689	0.222344	−	0.974968i	$-0.428629\pi$
$443$	9.35961	0.444689	0.222344	+	0.974968i	$0.428629\pi$
$444$	0	0
$445$	0	0
$446$	−61.5422	−2.91411
$447$	0	0
$448$	−6.23657	−0.294650
$449$	−7.41602	−0.349983	−0.174992	−	0.984570i	$-0.555990\pi$
$449$	−7.41602	−0.349983	−0.174992	+	0.984570i	$0.555990\pi$
$450$	0	0
$451$	−14.7107	−0.692702
$452$	20.7623	0.976576
$453$	0	0
$454$	−43.4481	−2.03912
$455$	0	0
$456$	0	0
$457$	−8.79714	−0.411513	−0.205756	−	0.978603i	$-0.565965\pi$
$457$	−8.79714	−0.411513	−0.205756	+	0.978603i	$0.565965\pi$
$458$	17.3662	0.811471
$459$	0	0
$460$	0	0
$461$	42.1526	1.96324	0.981621	−	0.190838i	$-0.0611207\pi$
$461$	42.1526	1.96324	0.981621	+	0.190838i	$0.0611207\pi$
$462$	0	0
$463$	−23.0565	−1.07153	−0.535763	−	0.844368i	$-0.679976\pi$
$463$	−23.0565	−1.07153	−0.535763	+	0.844368i	$0.679976\pi$
$464$	−22.8838	−1.06235
$465$	0	0
$466$	−21.5053	−0.996212
$467$	4.37964	0.202666	0.101333	−	0.994853i	$-0.467689\pi$
$467$	4.37964	0.202666	0.101333	+	0.994853i	$0.467689\pi$
$468$	0	0
$469$	−9.31186	−0.429982
$470$	0	0
$471$	0	0
$472$	−34.3469	−1.58095
$473$	−21.4974	−0.988453
$474$	0	0
$475$	0	0
$476$	−8.19534	−0.375633
$477$	0	0
$478$	−52.8102	−2.41548
$479$	−14.1913	−0.648417	−0.324209	−	0.945986i	$-0.605098\pi$
$479$	−14.1913	−0.648417	−0.324209	+	0.945986i	$0.605098\pi$
$480$	0	0
$481$	1.95460	0.0891219
$482$	−52.9257	−2.41070
$483$	0	0
$484$	74.0744	3.36702
$485$	0	0
$486$	0	0
$487$	−22.0694	−1.00006	−0.500029	−	0.866009i	$-0.666677\pi$
$487$	−22.0694	−1.00006	−0.500029	+	0.866009i	$0.666677\pi$
$488$	14.3197	0.648223
$489$	0	0
$490$	0	0
$491$	7.95704	0.359096	0.179548	−	0.983749i	$-0.442536\pi$
$491$	7.95704	0.359096	0.179548	+	0.983749i	$0.442536\pi$
$492$	0	0
$493$	9.84345	0.443327
$494$	30.6525	1.37912
$495$	0	0
$496$	−31.5376	−1.41608
$497$	5.53210	0.248149
$498$	0	0
$499$	−4.09384	−0.183266	−0.0916328	−	0.995793i	$-0.529209\pi$
$499$	−4.09384	−0.183266	−0.0916328	+	0.995793i	$0.529209\pi$
$500$	0	0
$501$	0	0
$502$	−36.8718	−1.64567
$503$	26.9554	1.20188	0.600941	−	0.799293i	$-0.294792\pi$
$503$	26.9554	1.20188	0.600941	+	0.799293i	$0.294792\pi$
$504$	0	0
$505$	0	0
$506$	25.8112	1.14745
$507$	0	0
$508$	−32.1325	−1.42565
$509$	−33.8837	−1.50187	−0.750935	−	0.660376i	$-0.770397\pi$
$509$	−33.8837	−1.50187	−0.750935	+	0.660376i	$0.770397\pi$
$510$	0	0
$511$	9.10093	0.402602
$512$	−43.9545	−1.94253
$513$	0	0
$514$	47.1831	2.08116
$515$	0	0
$516$	0	0
$517$	38.8243	1.70749
$518$	2.35695	0.103558
$519$	0	0
$520$	0	0
$521$	26.6469	1.16742	0.583711	−	0.811962i	$-0.301600\pi$
$521$	26.6469	1.16742	0.583711	+	0.811962i	$0.301600\pi$
$522$	0	0
$523$	1.80935	0.0791172	0.0395586	−	0.999217i	$-0.487405\pi$
$523$	1.80935	0.0791172	0.0395586	+	0.999217i	$0.487405\pi$
$524$	40.3592	1.76310
$525$	0	0
$526$	−16.7846	−0.731842
$527$	13.5659	0.590938
$528$	0	0
$529$	−19.2437	−0.836680
$530$	0	0
$531$	0	0
$532$	24.8879	1.07903
$533$	−5.45737	−0.236385
$534$	0	0
$535$	0	0
$536$	−50.2558	−2.17072
$537$	0	0
$538$	−1.22397	−0.0527691
$539$	−32.5795	−1.40330
$540$	0	0
$541$	−33.5233	−1.44128	−0.720640	−	0.693309i	$-0.756152\pi$
$541$	−33.5233	−1.44128	−0.720640	+	0.693309i	$0.756152\pi$
$542$	24.3132	1.04434
$543$	0	0
$544$	−2.55126	−0.109384
$545$	0	0
$546$	0	0
$547$	27.5146	1.17644	0.588219	−	0.808702i	$-0.299829\pi$
$547$	27.5146	1.17644	0.588219	+	0.808702i	$0.299829\pi$
$548$	−14.9779	−0.639822
$549$	0	0
$550$	0	0
$551$	−29.8929	−1.27348
$552$	0	0
$553$	−3.09602	−0.131656
$554$	−62.2050	−2.64284
$555$	0	0
$556$	−7.67095	−0.325321
$557$	2.16761	0.0918448	0.0459224	−	0.998945i	$-0.485377\pi$
$557$	2.16761	0.0918448	0.0459224	+	0.998945i	$0.485377\pi$
$558$	0	0
$559$	−7.97509	−0.337310
$560$	0	0
$561$	0	0
$562$	19.2270	0.811040
$563$	33.4836	1.41116	0.705582	−	0.708628i	$-0.250686\pi$
$563$	33.4836	1.41116	0.705582	+	0.708628i	$0.250686\pi$
$564$	0	0
$565$	0	0
$566$	−71.6616	−3.01216
$567$	0	0
$568$	29.8566	1.25275
$569$	−17.2323	−0.722415	−0.361208	−	0.932485i	$-0.617635\pi$
$569$	−17.2323	−0.722415	−0.361208	+	0.932485i	$0.617635\pi$
$570$	0	0
$571$	42.7928	1.79082	0.895411	−	0.445240i	$-0.146882\pi$
$571$	42.7928	1.79082	0.895411	+	0.445240i	$0.146882\pi$
$572$	44.3027	1.85239
$573$	0	0
$574$	−6.58076	−0.274676
$575$	0	0
$576$	0	0
$577$	31.8385	1.32546	0.662728	−	0.748860i	$-0.269399\pi$
$577$	31.8385	1.32546	0.662728	+	0.748860i	$0.269399\pi$
$578$	−31.7363	−1.32006
$579$	0	0
$580$	0	0
$581$	5.94923	0.246816
$582$	0	0
$583$	71.4470	2.95903
$584$	49.1175	2.03250
$585$	0	0
$586$	−0.381866	−0.0157747
$587$	−24.8027	−1.02372	−0.511858	−	0.859070i	$-0.671043\pi$
$587$	−24.8027	−1.02372	−0.511858	+	0.859070i	$0.671043\pi$
$588$	0	0
$589$	−41.1973	−1.69750
$590$	0	0
$591$	0	0
$592$	4.64938	0.191089
$593$	17.8431	0.732728	0.366364	−	0.930472i	$-0.380603\pi$
$593$	17.8431	0.732728	0.366364	+	0.930472i	$0.380603\pi$
$594$	0	0
$595$	0	0
$596$	−69.4754	−2.84582
$597$	0	0
$598$	9.57541	0.391568
$599$	28.1028	1.14825	0.574126	−	0.818767i	$-0.305342\pi$
$599$	28.1028	1.14825	0.574126	+	0.818767i	$0.305342\pi$
$600$	0	0
$601$	−26.6296	−1.08624	−0.543121	−	0.839654i	$-0.682758\pi$
$601$	−26.6296	−1.08624	−0.543121	+	0.839654i	$0.682758\pi$
$602$	−9.61675	−0.391949
$603$	0	0
$604$	−59.0336	−2.40204
$605$	0	0
$606$	0	0
$607$	19.2757	0.782375	0.391187	−	0.920311i	$-0.372064\pi$
$607$	19.2757	0.782375	0.391187	+	0.920311i	$0.372064\pi$
$608$	7.74774	0.314213
$609$	0	0
$610$	0	0
$611$	14.4030	0.582682
$612$	0	0
$613$	0.465680	0.0188086	0.00940432	−	0.999956i	$-0.497006\pi$
$613$	0.465680	0.0188086	0.00940432	+	0.999956i	$0.497006\pi$
$614$	35.8472	1.44668
$615$	0	0
$616$	27.5044	1.10818
$617$	−44.2765	−1.78251	−0.891253	−	0.453507i	$-0.850173\pi$
$617$	−44.2765	−1.78251	−0.891253	+	0.453507i	$0.850173\pi$
$618$	0	0
$619$	−36.1620	−1.45347	−0.726736	−	0.686917i	$-0.758964\pi$
$619$	−36.1620	−1.45347	−0.726736	+	0.686917i	$0.758964\pi$
$620$	0	0
$621$	0	0
$622$	−45.8707	−1.83925
$623$	2.92644	0.117245
$624$	0	0
$625$	0	0
$626$	−16.9667	−0.678126
$627$	0	0
$628$	89.4025	3.56755
$629$	−1.99993	−0.0797424
$630$	0	0
$631$	−0.199431	−0.00793921	−0.00396961	−	0.999992i	$-0.501264\pi$
$631$	−0.199431	−0.00793921	−0.00396961	+	0.999992i	$0.501264\pi$
$632$	−16.7091	−0.664654
$633$	0	0
$634$	−67.5028	−2.68088
$635$	0	0
$636$	0	0
$637$	−12.0863	−0.478876
$638$	−64.1657	−2.54035
$639$	0	0
$640$	0	0
$641$	30.9078	1.22079	0.610393	−	0.792099i	$-0.291012\pi$
$641$	30.9078	1.22079	0.610393	+	0.792099i	$0.291012\pi$
$642$	0	0
$643$	22.2489	0.877412	0.438706	−	0.898631i	$-0.355437\pi$
$643$	22.2489	0.877412	0.438706	+	0.898631i	$0.355437\pi$
$644$	7.77463	0.306363
$645$	0	0
$646$	−31.3634	−1.23398
$647$	−35.0696	−1.37873	−0.689364	−	0.724415i	$-0.742110\pi$
$647$	−35.0696	−1.37873	−0.689364	+	0.724415i	$0.742110\pi$
$648$	0	0
$649$	−35.2012	−1.38177
$650$	0	0
$651$	0	0
$652$	81.2983	3.18389
$653$	−4.00012	−0.156537	−0.0782683	−	0.996932i	$-0.524939\pi$
$653$	−4.00012	−0.156537	−0.0782683	+	0.996932i	$0.524939\pi$
$654$	0	0
$655$	0	0
$656$	−12.9814	−0.506839
$657$	0	0
$658$	17.3678	0.677068
$659$	−10.0349	−0.390904	−0.195452	−	0.980713i	$-0.562617\pi$
$659$	−10.0349	−0.390904	−0.195452	+	0.980713i	$0.562617\pi$
$660$	0	0
$661$	−48.5672	−1.88905	−0.944523	−	0.328445i	$-0.893475\pi$
$661$	−48.5672	−1.88905	−0.944523	+	0.328445i	$0.893475\pi$
$662$	−32.2727	−1.25431
$663$	0	0
$664$	32.1078	1.24602
$665$	0	0
$666$	0	0
$667$	−9.33814	−0.361574
$668$	25.0622	0.969684
$669$	0	0
$670$	0	0
$671$	14.6759	0.566557
$672$	0	0
$673$	−5.06306	−0.195167	−0.0975834	−	0.995227i	$-0.531111\pi$
$673$	−5.06306	−0.195167	−0.0975834	+	0.995227i	$0.531111\pi$
$674$	48.6229	1.87288
$675$	0	0
$676$	−37.1562	−1.42908
$677$	33.6669	1.29393	0.646963	−	0.762521i	$-0.276039\pi$
$677$	33.6669	1.29393	0.646963	+	0.762521i	$0.276039\pi$
$678$	0	0
$679$	−5.79489	−0.222387
$680$	0	0
$681$	0	0
$682$	−88.4307	−3.38619
$683$	32.4523	1.24175	0.620877	−	0.783908i	$-0.286777\pi$
$683$	32.4523	1.24175	0.620877	+	0.783908i	$0.286777\pi$
$684$	0	0
$685$	0	0
$686$	−31.4283	−1.19994
$687$	0	0
$688$	−18.9703	−0.723235
$689$	26.5053	1.00977
$690$	0	0
$691$	−30.7811	−1.17097	−0.585484	−	0.810684i	$-0.699095\pi$
$691$	−30.7811	−1.17097	−0.585484	+	0.810684i	$0.699095\pi$
$692$	61.9637	2.35550
$693$	0	0
$694$	10.8610	0.412278
$695$	0	0
$696$	0	0
$697$	5.58394	0.211507
$698$	67.2914	2.54702
$699$	0	0
$700$	0	0
$701$	13.1755	0.497633	0.248817	−	0.968551i	$-0.419958\pi$
$701$	13.1755	0.497633	0.248817	+	0.968551i	$0.419958\pi$
$702$	0	0
$703$	6.07345	0.229065
$704$	−34.4958	−1.30011
$705$	0	0
$706$	−39.4207	−1.48362
$707$	7.56439	0.284488
$708$	0	0
$709$	−10.6307	−0.399246	−0.199623	−	0.979873i	$-0.563972\pi$
$709$	−10.6307	−0.399246	−0.199623	+	0.979873i	$0.563972\pi$
$710$	0	0
$711$	0	0
$712$	15.7939	0.591901
$713$	−12.8695	−0.481965
$714$	0	0
$715$	0	0
$716$	82.5174	3.08382
$717$	0	0
$718$	12.4091	0.463102
$719$	15.3464	0.572324	0.286162	−	0.958181i	$-0.407620\pi$
$719$	15.3464	0.572324	0.286162	+	0.958181i	$0.407620\pi$
$720$	0	0
$721$	−4.81724	−0.179404
$722$	48.2328	1.79504
$723$	0	0
$724$	83.6823	3.11003
$725$	0	0
$726$	0	0
$727$	−23.8156	−0.883272	−0.441636	−	0.897194i	$-0.645602\pi$
$727$	−23.8156	−0.883272	−0.441636	+	0.897194i	$0.645602\pi$
$728$	10.2035	0.378168
$729$	0	0
$730$	0	0
$731$	8.16005	0.301810
$732$	0	0
$733$	4.09180	0.151134	0.0755670	−	0.997141i	$-0.475923\pi$
$733$	4.09180	0.151134	0.0755670	+	0.997141i	$0.475923\pi$
$734$	−70.1897	−2.59075
$735$	0	0
$736$	2.42029	0.0892130
$737$	−51.5059	−1.89724
$738$	0	0
$739$	2.45909	0.0904590	0.0452295	−	0.998977i	$-0.485598\pi$
$739$	2.45909	0.0904590	0.0452295	+	0.998977i	$0.485598\pi$
$740$	0	0
$741$	0	0
$742$	31.9614	1.17334
$743$	−11.1921	−0.410598	−0.205299	−	0.978699i	$-0.565817\pi$
$743$	−11.1921	−0.410598	−0.205299	+	0.978699i	$0.565817\pi$
$744$	0	0
$745$	0	0
$746$	80.9653	2.96435
$747$	0	0
$748$	−45.3301	−1.65743
$749$	1.13653	0.0415279
$750$	0	0
$751$	−10.3616	−0.378101	−0.189051	−	0.981967i	$-0.560541\pi$
$751$	−10.3616	−0.378101	−0.189051	+	0.981967i	$0.560541\pi$
$752$	34.2603	1.24934
$753$	0	0
$754$	−23.8041	−0.866894
$755$	0	0
$756$	0	0
$757$	5.84722	0.212521	0.106260	−	0.994338i	$-0.466112\pi$
$757$	5.84722	0.212521	0.106260	+	0.994338i	$0.466112\pi$
$758$	11.0971	0.403063
$759$	0	0
$760$	0	0
$761$	−41.6931	−1.51137	−0.755687	−	0.654932i	$-0.772697\pi$
$761$	−41.6931	−1.51137	−0.755687	+	0.654932i	$0.772697\pi$
$762$	0	0
$763$	−17.2116	−0.623102
$764$	39.0674	1.41341
$765$	0	0
$766$	40.2921	1.45581
$767$	−13.0589	−0.471530
$768$	0	0
$769$	−17.3076	−0.624129	−0.312065	−	0.950061i	$-0.601021\pi$
$769$	−17.3076	−0.624129	−0.312065	+	0.950061i	$0.601021\pi$
$770$	0	0
$771$	0	0
$772$	−11.2327	−0.404274
$773$	17.8695	0.642723	0.321361	−	0.946957i	$-0.395860\pi$
$773$	17.8695	0.642723	0.321361	+	0.946957i	$0.395860\pi$
$774$	0	0
$775$	0	0
$776$	−31.2748	−1.12270
$777$	0	0
$778$	−53.1736	−1.90637
$779$	−16.9575	−0.607565
$780$	0	0
$781$	30.5992	1.09493
$782$	−9.79749	−0.350358
$783$	0	0
$784$	−28.7496	−1.02677
$785$	0	0
$786$	0	0
$787$	−9.91104	−0.353290	−0.176645	−	0.984275i	$-0.556525\pi$
$787$	−9.91104	−0.353290	−0.176645	+	0.984275i	$0.556525\pi$
$788$	3.21564	0.114552
$789$	0	0
$790$	0	0
$791$	4.90080	0.174252
$792$	0	0
$793$	5.44444	0.193338
$794$	51.6819	1.83412
$795$	0	0
$796$	−30.6101	−1.08495
$797$	39.7590	1.40834	0.704168	−	0.710034i	$-0.251320\pi$
$797$	39.7590	1.40834	0.704168	+	0.710034i	$0.251320\pi$
$798$	0	0
$799$	−14.7370	−0.521359
$800$	0	0
$801$	0	0
$802$	1.56731	0.0553437
$803$	50.3392	1.77643
$804$	0	0
$805$	0	0
$806$	−32.8059	−1.15554
$807$	0	0
$808$	40.8248	1.43621
$809$	5.05110	0.177587	0.0887935	−	0.996050i	$-0.471699\pi$
$809$	5.05110	0.177587	0.0887935	+	0.996050i	$0.471699\pi$
$810$	0	0
$811$	11.7743	0.413452	0.206726	−	0.978399i	$-0.433719\pi$
$811$	11.7743	0.413452	0.206726	+	0.978399i	$0.433719\pi$
$812$	−19.3274	−0.678260
$813$	0	0
$814$	13.0368	0.456939
$815$	0	0
$816$	0	0
$817$	−24.7807	−0.866968
$818$	−10.6353	−0.371855
$819$	0	0
$820$	0	0
$821$	−5.25401	−0.183366	−0.0916832	−	0.995788i	$-0.529225\pi$
$821$	−5.25401	−0.183366	−0.0916832	+	0.995788i	$0.529225\pi$
$822$	0	0
$823$	37.0042	1.28989	0.644943	−	0.764230i	$-0.276881\pi$
$823$	37.0042	1.28989	0.644943	+	0.764230i	$0.276881\pi$
$824$	−25.9985	−0.905701
$825$	0	0
$826$	−15.7471	−0.547910
$827$	28.1147	0.977645	0.488822	−	0.872383i	$-0.337427\pi$
$827$	28.1147	0.977645	0.488822	+	0.872383i	$0.337427\pi$
$828$	0	0
$829$	−14.4532	−0.501979	−0.250990	−	0.967990i	$-0.580756\pi$
$829$	−14.4532	−0.501979	−0.250990	+	0.967990i	$0.580756\pi$
$830$	0	0
$831$	0	0
$832$	−12.7972	−0.443662
$833$	12.3666	0.428477
$834$	0	0
$835$	0	0
$836$	137.660	4.76108
$837$	0	0
$838$	−83.1914	−2.87380
$839$	−16.3694	−0.565134	−0.282567	−	0.959248i	$-0.591186\pi$
$839$	−16.3694	−0.565134	−0.282567	+	0.959248i	$0.591186\pi$
$840$	0	0
$841$	−5.78575	−0.199509
$842$	−52.8937	−1.82284
$843$	0	0
$844$	−12.2901	−0.423043
$845$	0	0
$846$	0	0
$847$	17.4848	0.600784
$848$	63.0480	2.16508
$849$	0	0
$850$	0	0
$851$	1.89726	0.0650373
$852$	0	0
$853$	46.8216	1.60314	0.801571	−	0.597900i	$-0.203998\pi$
$853$	46.8216	1.60314	0.801571	+	0.597900i	$0.203998\pi$
$854$	6.56517	0.224656
$855$	0	0
$856$	6.13382	0.209650
$857$	28.9569	0.989148	0.494574	−	0.869136i	$-0.335324\pi$
$857$	28.9569	0.989148	0.494574	+	0.869136i	$0.335324\pi$
$858$	0	0
$859$	24.4996	0.835917	0.417959	−	0.908466i	$-0.362746\pi$
$859$	24.4996	0.835917	0.417959	+	0.908466i	$0.362746\pi$
$860$	0	0
$861$	0	0
$862$	−64.5571	−2.19882
$863$	10.4696	0.356391	0.178195	−	0.983995i	$-0.442974\pi$
$863$	10.4696	0.356391	0.178195	+	0.983995i	$0.442974\pi$
$864$	0	0
$865$	0	0
$866$	−29.1171	−0.989440
$867$	0	0
$868$	−26.6363	−0.904096
$869$	−17.1247	−0.580917
$870$	0	0
$871$	−19.1076	−0.647435
$872$	−92.8905	−3.14567
$873$	0	0
$874$	29.7534	1.00642
$875$	0	0
$876$	0	0
$877$	−39.9121	−1.34774	−0.673868	−	0.738852i	$-0.735368\pi$
$877$	−39.9121	−1.34774	−0.673868	+	0.738852i	$0.735368\pi$
$878$	12.1880	0.411324
$879$	0	0
$880$	0	0
$881$	7.88965	0.265809	0.132904	−	0.991129i	$-0.457570\pi$
$881$	7.88965	0.265809	0.132904	+	0.991129i	$0.457570\pi$
$882$	0	0
$883$	15.1304	0.509178	0.254589	−	0.967049i	$-0.418060\pi$
$883$	15.1304	0.509178	0.254589	+	0.967049i	$0.418060\pi$
$884$	−16.8165	−0.565600
$885$	0	0
$886$	23.1590	0.778042
$887$	47.6886	1.60123	0.800614	−	0.599180i	$-0.204507\pi$
$887$	47.6886	1.60123	0.800614	+	0.599180i	$0.204507\pi$
$888$	0	0
$889$	−7.58466	−0.254381
$890$	0	0
$891$	0	0
$892$	−102.533	−3.43306
$893$	44.7539	1.49763
$894$	0	0
$895$	0	0
$896$	−17.8618	−0.596719
$897$	0	0
$898$	−18.3498	−0.612342
$899$	31.9930	1.06703
$900$	0	0
$901$	−27.1200	−0.903499
$902$	−36.3996	−1.21197
$903$	0	0
$904$	26.4494	0.879696
$905$	0	0
$906$	0	0
$907$	38.9321	1.29272	0.646359	−	0.763033i	$-0.276291\pi$
$907$	38.9321	1.29272	0.646359	+	0.763033i	$0.276291\pi$
$908$	−72.3873	−2.40226
$909$	0	0
$910$	0	0
$911$	−37.6740	−1.24819	−0.624097	−	0.781347i	$-0.714533\pi$
$911$	−37.6740	−1.24819	−0.624097	+	0.781347i	$0.714533\pi$
$912$	0	0
$913$	32.9064	1.08904
$914$	−21.7672	−0.719996
$915$	0	0
$916$	28.9332	0.955980
$917$	9.52650	0.314593
$918$	0	0
$919$	14.4646	0.477145	0.238572	−	0.971125i	$-0.423321\pi$
$919$	14.4646	0.477145	0.238572	+	0.971125i	$0.423321\pi$
$920$	0	0
$921$	0	0
$922$	104.300	3.43495
$923$	11.3516	0.373644
$924$	0	0
$925$	0	0
$926$	−57.0499	−1.87478
$927$	0	0
$928$	−6.01674	−0.197509
$929$	40.8990	1.34185	0.670927	−	0.741524i	$-0.265897\pi$
$929$	40.8990	1.34185	0.670927	+	0.741524i	$0.265897\pi$
$930$	0	0
$931$	−37.5553	−1.23083
$932$	−35.8291	−1.17362
$933$	0	0
$934$	10.8368	0.354590
$935$	0	0
$936$	0	0
$937$	−33.2563	−1.08644	−0.543218	−	0.839592i	$-0.682794\pi$
$937$	−33.2563	−1.08644	−0.543218	+	0.839592i	$0.682794\pi$
$938$	−23.0408	−0.752310
$939$	0	0
$940$	0	0
$941$	−39.1577	−1.27650	−0.638252	−	0.769827i	$-0.720342\pi$
$941$	−39.1577	−1.27650	−0.638252	+	0.769827i	$0.720342\pi$
$942$	0	0
$943$	−5.29728	−0.172503
$944$	−31.0631	−1.01102
$945$	0	0
$946$	−53.1923	−1.72943
$947$	−28.5840	−0.928854	−0.464427	−	0.885611i	$-0.653740\pi$
$947$	−28.5840	−0.928854	−0.464427	+	0.885611i	$0.653740\pi$
$948$	0	0
$949$	18.6747	0.606208
$950$	0	0
$951$	0	0
$952$	−10.4402	−0.338368
$953$	−43.3568	−1.40446	−0.702232	−	0.711948i	$-0.747813\pi$
$953$	−43.3568	−1.40446	−0.702232	+	0.711948i	$0.747813\pi$
$954$	0	0
$955$	0	0
$956$	−87.9850	−2.84564
$957$	0	0
$958$	−35.1143	−1.13449
$959$	−3.53542	−0.114165
$960$	0	0
$961$	13.0915	0.422306
$962$	4.83636	0.155931
$963$	0	0
$964$	−88.1775	−2.84001
$965$	0	0
$966$	0	0
$967$	−32.2551	−1.03726	−0.518628	−	0.855000i	$-0.673557\pi$
$967$	−32.2551	−1.03726	−0.518628	+	0.855000i	$0.673557\pi$
$968$	94.3647	3.03300
$969$	0	0
$970$	0	0
$971$	−10.9579	−0.351654	−0.175827	−	0.984421i	$-0.556260\pi$
$971$	−10.9579	−0.351654	−0.175827	+	0.984421i	$0.556260\pi$
$972$	0	0
$973$	−1.81068	−0.0580476
$974$	−54.6074	−1.74973
$975$	0	0
$976$	12.9507	0.414540
$977$	−20.1536	−0.644770	−0.322385	−	0.946609i	$-0.604484\pi$
$977$	−20.1536	−0.644770	−0.322385	+	0.946609i	$0.604484\pi$
$978$	0	0
$979$	16.1867	0.517330
$980$	0	0
$981$	0	0
$982$	19.6885	0.628286
$983$	−16.0511	−0.511951	−0.255975	−	0.966683i	$-0.582397\pi$
$983$	−16.0511	−0.511951	−0.255975	+	0.966683i	$0.582397\pi$
$984$	0	0
$985$	0	0
$986$	24.3562	0.775659
$987$	0	0
$988$	51.0689	1.62472
$989$	−7.74115	−0.246154
$990$	0	0
$991$	20.5569	0.653010	0.326505	−	0.945195i	$-0.394129\pi$
$991$	20.5569	0.653010	0.326505	+	0.945195i	$0.394129\pi$
$992$	−8.29204	−0.263272
$993$	0	0
$994$	13.6884	0.434169
$995$	0	0
$996$	0	0
$997$	−21.7296	−0.688183	−0.344092	−	0.938936i	$-0.611813\pi$
$997$	−21.7296	−0.688183	−0.344092	+	0.938936i	$0.611813\pi$
$998$	−10.1296	−0.320647
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.be.1.7		8
3.2	odd	2		625.2.a.e.1.2	✓	8
5.4	even	2		5625.2.a.s.1.2		8
12.11	even	2		10000.2.a.bn.1.6		8
15.2	even	4		625.2.b.d.624.2		16
15.8	even	4		625.2.b.d.624.15		16
15.14	odd	2		625.2.a.g.1.7	yes	8
60.59	even	2		10000.2.a.be.1.3		8
75.2	even	20		625.2.e.k.124.1		32
75.8	even	20		625.2.e.j.374.1		32
75.11	odd	10		625.2.d.p.501.1		16
75.14	odd	10		625.2.d.n.501.4		16
75.17	even	20		625.2.e.j.374.8		32
75.23	even	20		625.2.e.k.124.8		32
75.29	odd	10		625.2.d.m.376.1		16
75.38	even	20		625.2.e.k.499.1		32
75.41	odd	10		625.2.d.p.126.1		16
75.44	odd	10		625.2.d.m.251.1		16
75.47	even	20		625.2.e.j.249.1		32
75.53	even	20		625.2.e.j.249.8		32
75.56	odd	10		625.2.d.q.251.4		16
75.59	odd	10		625.2.d.n.126.4		16
75.62	even	20		625.2.e.k.499.8		32
75.71	odd	10		625.2.d.q.376.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
625.2.a.e.1.2	✓	8	3.2	odd	2
625.2.a.g.1.7	yes	8	15.14	odd	2
625.2.b.d.624.2		16	15.2	even	4
625.2.b.d.624.15		16	15.8	even	4
625.2.d.m.251.1		16	75.44	odd	10
625.2.d.m.376.1		16	75.29	odd	10
625.2.d.n.126.4		16	75.59	odd	10
625.2.d.n.501.4		16	75.14	odd	10
625.2.d.p.126.1		16	75.41	odd	10
625.2.d.p.501.1		16	75.11	odd	10
625.2.d.q.251.4		16	75.56	odd	10
625.2.d.q.376.4		16	75.71	odd	10
625.2.e.j.249.1		32	75.47	even	20
625.2.e.j.249.8		32	75.53	even	20
625.2.e.j.374.1		32	75.8	even	20
625.2.e.j.374.8		32	75.17	even	20
625.2.e.k.124.1		32	75.2	even	20
625.2.e.k.124.8		32	75.23	even	20
625.2.e.k.499.1		32	75.38	even	20
625.2.e.k.499.8		32	75.62	even	20
5625.2.a.s.1.2		8	5.4	even	2
5625.2.a.be.1.7		8	1.1	even	1	trivial
10000.2.a.be.1.3		8	60.59	even	2
10000.2.a.bn.1.6		8	12.11	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)

\(f(q)\)	\(=\)	\(q+2.47435 q^{2} +4.12242 q^{4} +0.973070 q^{7} +5.25163 q^{8} +O(q^{10})\)	\(q+2.47435 q^{2} +4.12242 q^{4} +0.973070 q^{7} +5.25163 q^{8} +5.38225 q^{11} +1.99670 q^{13} +2.40772 q^{14} +4.74954 q^{16} -2.04301 q^{17} +6.20428 q^{19} +13.3176 q^{22} +1.93813 q^{23} +4.94054 q^{26} +4.01141 q^{28} -4.81812 q^{29} -6.64014 q^{31} +1.24877 q^{32} -5.05512 q^{34} +0.978913 q^{37} +15.3516 q^{38} -2.73319 q^{41} -3.99413 q^{43} +22.1879 q^{44} +4.79562 q^{46} +7.21339 q^{47} -6.05313 q^{49} +8.23125 q^{52} +13.2746 q^{53} +5.11020 q^{56} -11.9217 q^{58} -6.54024 q^{59} +2.72672 q^{61} -16.4301 q^{62} -6.40916 q^{64} -9.56957 q^{67} -8.42215 q^{68} +5.68520 q^{71} +9.35281 q^{73} +2.42218 q^{74} +25.5767 q^{76} +5.23731 q^{77} -3.18171 q^{79} -6.76289 q^{82} +6.11387 q^{83} -9.88290 q^{86} +28.2656 q^{88} +3.00743 q^{89} +1.94293 q^{91} +7.98980 q^{92} +17.8485 q^{94} -5.95526 q^{97} -14.9776 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 5 q^{2} + 11 q^{4} - 10 q^{7} + 15 q^{8}+O(q^{10}) \) 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

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