LMFDB - Embedded newform 5625.2.a.w.1.4

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:	8.8.33620000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 10x^{6} + 30x^{4} - 25x^{2} + 5 $ x^8 - 10x^6 + 30x^4 - 25*x^2 + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 225)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.541884$ 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)

$f(q)$	$=$	$q-0.541884 q^{2} -1.70636 q^{4} +1.08833 q^{7} +2.00842 q^{8} +O(q^{10})$	$q-0.541884 q^{2} -1.70636 q^{4} +1.08833 q^{7} +2.00842 q^{8} +2.91478 q^{11} +0.966262 q^{13} -0.589747 q^{14} +2.32440 q^{16} +2.65994 q^{17} -1.96626 q^{19} -1.57947 q^{22} -6.83428 q^{23} -0.523601 q^{26} -1.85708 q^{28} -6.81599 q^{29} +2.48817 q^{31} -5.27639 q^{32} -1.44138 q^{34} -8.22827 q^{37} +1.06548 q^{38} -10.5657 q^{41} -7.72721 q^{43} -4.97368 q^{44} +3.70338 q^{46} +9.02009 q^{47} -5.81554 q^{49} -1.64879 q^{52} -5.53384 q^{53} +2.18582 q^{56} +3.69348 q^{58} +13.2266 q^{59} +3.64879 q^{61} -1.34830 q^{62} -1.78961 q^{64} +8.84814 q^{67} -4.53882 q^{68} +7.81002 q^{71} +15.6298 q^{73} +4.45876 q^{74} +3.35515 q^{76} +3.17224 q^{77} -6.60515 q^{79} +5.72537 q^{82} +1.89878 q^{83} +4.18725 q^{86} +5.85410 q^{88} +1.14554 q^{89} +1.05161 q^{91} +11.6618 q^{92} -4.88784 q^{94} -6.19437 q^{97} +3.15135 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{4}+O(q^{10}) $ 8 * q + 4 * q^4	$ 8 q + 4 q^{4} + 10 q^{13} - 8 q^{16} - 18 q^{19} - 30 q^{28} - 26 q^{31} - 20 q^{34} - 40 q^{43} - 30 q^{46} - 16 q^{49} + 40 q^{52} + 10 q^{58} - 24 q^{61} - 34 q^{64} - 40 q^{67} + 40 q^{73} - 44 q^{76} - 42 q^{79} + 60 q^{82} + 20 q^{88} - 40 q^{91} - 10 q^{94} - 40 q^{97}+O(q^{100}) $ 8 * q + 4 * q^4 + 10 * q^13 - 8 * q^16 - 18 * q^19 - 30 * q^28 - 26 * q^31 - 20 * q^34 - 40 * q^43 - 30 * q^46 - 16 * q^49 + 40 * q^52 + 10 * q^58 - 24 * q^61 - 34 * q^64 - 40 * q^67 + 40 * q^73 - 44 * q^76 - 42 * q^79 + 60 * q^82 + 20 * q^88 - 40 * q^91 - 10 * q^94 - 40 * q^97

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 $

$2$	−0.541884	−0.383170	−0.191585	−	0.981476i	$-0.561363\pi$
$2$	−0.541884	−0.383170	−0.191585	+	0.981476i	$0.561363\pi$
$3$	0	0
$3$	0	0
$4$	−1.70636	−0.853181
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.08833	0.411349	0.205675	−	0.978620i	$-0.434061\pi$
$7$	1.08833	0.411349	0.205675	+	0.978620i	$0.434061\pi$
$8$	2.00842	0.710083
$9$	0	0
$10$	0	0
$11$	2.91478	0.878841	0.439420	−	0.898282i	$-0.355184\pi$
$11$	2.91478	0.878841	0.439420	+	0.898282i	$0.355184\pi$
$12$	0	0
$13$	0.966262	0.267993	0.133996	−	0.990982i	$-0.457219\pi$
$13$	0.966262	0.267993	0.133996	+	0.990982i	$0.457219\pi$
$14$	−0.589747	−0.157617
$15$	0	0
$16$	2.32440	0.581099
$17$	2.65994	0.645130	0.322565	−	0.946547i	$-0.395455\pi$
$17$	2.65994	0.645130	0.322565	+	0.946547i	$0.395455\pi$
$18$	0	0
$19$	−1.96626	−0.451091	−0.225546	−	0.974233i	$-0.572416\pi$
$19$	−1.96626	−0.451091	−0.225546	+	0.974233i	$0.572416\pi$
$20$	0	0
$21$	0	0
$22$	−1.57947	−0.336745
$23$	−6.83428	−1.42505	−0.712523	−	0.701649i	$-0.752447\pi$
$23$	−6.83428	−1.42505	−0.712523	+	0.701649i	$0.752447\pi$
$24$	0	0
$25$	0	0
$26$	−0.523601	−0.102687
$27$	0	0
$28$	−1.85708	−0.350956
$29$	−6.81599	−1.26570	−0.632849	−	0.774275i	$-0.718115\pi$
$29$	−6.81599	−1.26570	−0.632849	+	0.774275i	$0.718115\pi$
$30$	0	0
$31$	2.48817	0.446888	0.223444	−	0.974717i	$-0.428270\pi$
$31$	2.48817	0.446888	0.223444	+	0.974717i	$0.428270\pi$
$32$	−5.27639	−0.932742
$33$	0	0
$34$	−1.44138	−0.247194
$35$	0	0
$36$	0	0
$37$	−8.22827	−1.35272	−0.676359	−	0.736572i	$-0.736443\pi$
$37$	−8.22827	−1.35272	−0.676359	+	0.736572i	$0.736443\pi$
$38$	1.06548	0.172844
$39$	0	0
$40$	0	0
$41$	−10.5657	−1.65008	−0.825041	−	0.565072i	$-0.808848\pi$
$41$	−10.5657	−1.65008	−0.825041	+	0.565072i	$0.808848\pi$
$42$	0	0
$43$	−7.72721	−1.17839	−0.589195	−	0.807991i	$-0.700555\pi$
$43$	−7.72721	−1.17839	−0.589195	+	0.807991i	$0.700555\pi$
$44$	−4.97368	−0.749810
$45$	0	0
$46$	3.70338	0.546034
$47$	9.02009	1.31572	0.657858	−	0.753142i	$-0.271463\pi$
$47$	9.02009	1.31572	0.657858	+	0.753142i	$0.271463\pi$
$48$	0	0
$49$	−5.81554	−0.830792
$50$	0	0
$51$	0	0
$52$	−1.64879	−0.228646
$53$	−5.53384	−0.760132	−0.380066	−	0.924959i	$-0.624099\pi$
$53$	−5.53384	−0.760132	−0.380066	+	0.924959i	$0.624099\pi$
$54$	0	0
$55$	0	0
$56$	2.18582	0.292092
$57$	0	0
$58$	3.69348	0.484977
$59$	13.2266	1.72196	0.860980	−	0.508639i	$-0.169851\pi$
$59$	13.2266	1.72196	0.860980	+	0.508639i	$0.169851\pi$
$60$	0	0
$61$	3.64879	0.467180	0.233590	−	0.972335i	$-0.424953\pi$
$61$	3.64879	0.467180	0.233590	+	0.972335i	$0.424953\pi$
$62$	−1.34830	−0.171234
$63$	0	0
$64$	−1.78961	−0.223701
$65$	0	0
$66$	0	0
$67$	8.84814	1.08097	0.540486	−	0.841353i	$-0.318240\pi$
$67$	8.84814	1.08097	0.540486	+	0.841353i	$0.318240\pi$
$68$	−4.53882	−0.550413
$69$	0	0
$70$	0	0
$71$	7.81002	0.926879	0.463439	−	0.886129i	$-0.346615\pi$
$71$	7.81002	0.926879	0.463439	+	0.886129i	$0.346615\pi$
$72$	0	0
$73$	15.6298	1.82933	0.914664	−	0.404215i	$-0.132455\pi$
$73$	15.6298	1.82933	0.914664	+	0.404215i	$0.132455\pi$
$74$	4.45876	0.518321
$75$	0	0
$76$	3.35515	0.384863
$77$	3.17224	0.361511
$78$	0	0
$79$	−6.60515	−0.743137	−0.371569	−	0.928406i	$-0.621180\pi$
$79$	−6.60515	−0.743137	−0.371569	+	0.928406i	$0.621180\pi$
$80$	0	0
$81$	0	0
$82$	5.72537	0.632262
$83$	1.89878	0.208418	0.104209	−	0.994555i	$-0.466769\pi$
$83$	1.89878	0.208418	0.104209	+	0.994555i	$0.466769\pi$
$84$	0	0
$85$	0	0
$86$	4.18725	0.451523
$87$	0	0
$88$	5.85410	0.624049
$89$	1.14554	0.121427	0.0607136	−	0.998155i	$-0.480662\pi$
$89$	1.14554	0.121427	0.0607136	+	0.998155i	$0.480662\pi$
$90$	0	0
$91$	1.05161	0.110239
$92$	11.6618	1.21582
$93$	0	0
$94$	−4.88784	−0.504142
$95$	0	0
$96$	0	0
$97$	−6.19437	−0.628942	−0.314471	−	0.949267i	$-0.601827\pi$
$97$	−6.19437	−0.628942	−0.314471	+	0.949267i	$0.601827\pi$
$98$	3.15135	0.318334
$99$	0	0
$100$	0	0
$101$	−3.88891	−0.386961	−0.193481	−	0.981104i	$-0.561978\pi$
$101$	−3.88891	−0.386961	−0.193481	+	0.981104i	$0.561978\pi$
$102$	0	0
$103$	−17.7858	−1.75248	−0.876241	−	0.481873i	$-0.839957\pi$
$103$	−17.7858	−1.75248	−0.876241	+	0.481873i	$0.839957\pi$
$104$	1.94066	0.190297
$105$	0	0
$106$	2.99870	0.291259
$107$	7.02996	0.679612	0.339806	−	0.940496i	$-0.389639\pi$
$107$	7.02996	0.679612	0.339806	+	0.940496i	$0.389639\pi$
$108$	0	0
$109$	−9.02594	−0.864528	−0.432264	−	0.901747i	$-0.642285\pi$
$109$	−9.02594	−0.864528	−0.432264	+	0.901747i	$0.642285\pi$
$110$	0	0
$111$	0	0
$112$	2.52971	0.239035
$113$	12.9865	1.22166	0.610831	−	0.791761i	$-0.290835\pi$
$113$	12.9865	1.22166	0.610831	+	0.791761i	$0.290835\pi$
$114$	0	0
$115$	0	0
$116$	11.6306	1.07987
$117$	0	0
$118$	−7.16729	−0.659803
$119$	2.89489	0.265374
$120$	0	0
$121$	−2.50403	−0.227639
$122$	−1.97722	−0.179009
$123$	0	0
$124$	−4.24571	−0.381276
$125$	0	0
$126$	0	0
$127$	−1.69830	−0.150699	−0.0753497	−	0.997157i	$-0.524007\pi$
$127$	−1.69830	−0.150699	−0.0753497	+	0.997157i	$0.524007\pi$
$128$	11.5225	1.01846
$129$	0	0
$130$	0	0
$131$	−13.4197	−1.17248	−0.586242	−	0.810136i	$-0.699393\pi$
$131$	−13.4197	−1.17248	−0.586242	+	0.810136i	$0.699393\pi$
$132$	0	0
$133$	−2.13994	−0.185556
$134$	−4.79466	−0.414196
$135$	0	0
$136$	5.34227	0.458096
$137$	5.70265	0.487210	0.243605	−	0.969875i	$-0.421670\pi$
$137$	5.70265	0.487210	0.243605	+	0.969875i	$0.421670\pi$
$138$	0	0
$139$	−19.2976	−1.63680	−0.818400	−	0.574649i	$-0.805139\pi$
$139$	−19.2976	−1.63680	−0.818400	+	0.574649i	$0.805139\pi$
$140$	0	0
$141$	0	0
$142$	−4.23212	−0.355152
$143$	2.81644	0.235523
$144$	0	0
$145$	0	0
$146$	−8.46952	−0.700943
$147$	0	0
$148$	14.0404	1.15411
$149$	−17.0295	−1.39511	−0.697555	−	0.716531i	$-0.745729\pi$
$149$	−17.0295	−1.39511	−0.697555	+	0.716531i	$0.745729\pi$
$150$	0	0
$151$	−0.691471	−0.0562711	−0.0281355	−	0.999604i	$-0.508957\pi$
$151$	−0.691471	−0.0562711	−0.0281355	+	0.999604i	$0.508957\pi$
$152$	−3.94907	−0.320312
$153$	0	0
$154$	−1.71899	−0.138520
$155$	0	0
$156$	0	0
$157$	−0.448304	−0.0357786	−0.0178893	−	0.999840i	$-0.505695\pi$
$157$	−0.448304	−0.0357786	−0.0178893	+	0.999840i	$0.505695\pi$
$158$	3.57922	0.284748
$159$	0	0
$160$	0	0
$161$	−7.43794	−0.586191
$162$	0	0
$163$	9.91378	0.776507	0.388253	−	0.921553i	$-0.373079\pi$
$163$	9.91378	0.776507	0.388253	+	0.921553i	$0.373079\pi$
$164$	18.0289	1.40782
$165$	0	0
$166$	−1.02892	−0.0798594
$167$	−18.9197	−1.46405	−0.732024	−	0.681279i	$-0.761424\pi$
$167$	−18.9197	−1.46405	−0.732024	+	0.681279i	$0.761424\pi$
$168$	0	0
$169$	−12.0663	−0.928180
$170$	0	0
$171$	0	0
$172$	13.1854	1.00538
$173$	12.0026	0.912544	0.456272	−	0.889840i	$-0.349184\pi$
$173$	12.0026	0.912544	0.456272	+	0.889840i	$0.349184\pi$
$174$	0	0
$175$	0	0
$176$	6.77511	0.510693
$177$	0	0
$178$	−0.620751	−0.0465272
$179$	9.10884	0.680826	0.340413	−	0.940276i	$-0.389433\pi$
$179$	9.10884	0.680826	0.340413	+	0.940276i	$0.389433\pi$
$180$	0	0
$181$	−6.09315	−0.452900	−0.226450	−	0.974023i	$-0.572712\pi$
$181$	−6.09315	−0.452900	−0.226450	+	0.974023i	$0.572712\pi$
$182$	−0.569850	−0.0422401
$183$	0	0
$184$	−13.7261	−1.01190
$185$	0	0
$186$	0	0
$187$	7.75315	0.566966
$188$	−15.3915	−1.12254
$189$	0	0
$190$	0	0
$191$	9.25234	0.669476	0.334738	−	0.942311i	$-0.391352\pi$
$191$	9.25234	0.669476	0.334738	+	0.942311i	$0.391352\pi$
$192$	0	0
$193$	−24.2638	−1.74655	−0.873275	−	0.487228i	$-0.838008\pi$
$193$	−24.2638	−1.74655	−0.873275	+	0.487228i	$0.838008\pi$
$194$	3.35662	0.240992
$195$	0	0
$196$	9.92342	0.708816
$197$	−12.6628	−0.902190	−0.451095	−	0.892476i	$-0.648967\pi$
$197$	−12.6628	−0.902190	−0.451095	+	0.892476i	$0.648967\pi$
$198$	0	0
$199$	−13.8284	−0.980271	−0.490136	−	0.871646i	$-0.663053\pi$
$199$	−13.8284	−0.980271	−0.490136	+	0.871646i	$0.663053\pi$
$200$	0	0
$201$	0	0
$202$	2.10734	0.148272
$203$	−7.41804	−0.520644
$204$	0	0
$205$	0	0
$206$	9.63781	0.671498
$207$	0	0
$208$	2.24597	0.155730
$209$	−5.73123	−0.396437
$210$	0	0
$211$	−14.1143	−0.971666	−0.485833	−	0.874052i	$-0.661484\pi$
$211$	−14.1143	−0.971666	−0.485833	+	0.874052i	$0.661484\pi$
$212$	9.44274	0.648530
$213$	0	0
$214$	−3.80942	−0.260407
$215$	0	0
$216$	0	0
$217$	2.70794	0.183827
$218$	4.89101	0.331261
$219$	0	0
$220$	0	0
$221$	2.57020	0.172890
$222$	0	0
$223$	18.5973	1.24537	0.622685	−	0.782472i	$-0.286042\pi$
$223$	18.5973	1.24537	0.622685	+	0.782472i	$0.286042\pi$
$224$	−5.74244	−0.383683
$225$	0	0
$226$	−7.03714	−0.468104
$227$	15.4829	1.02764	0.513818	−	0.857899i	$-0.328231\pi$
$227$	15.4829	1.02764	0.513818	+	0.857899i	$0.328231\pi$
$228$	0	0
$229$	−15.6041	−1.03115	−0.515574	−	0.856845i	$-0.672421\pi$
$229$	−15.6041	−1.03115	−0.515574	+	0.856845i	$0.672421\pi$
$230$	0	0
$231$	0	0
$232$	−13.6894	−0.898750
$233$	−21.6264	−1.41679	−0.708396	−	0.705815i	$-0.750581\pi$
$233$	−21.6264	−1.41679	−0.708396	+	0.705815i	$0.750581\pi$
$234$	0	0
$235$	0	0
$236$	−22.5694	−1.46914
$237$	0	0
$238$	−1.56869	−0.101683
$239$	−15.6301	−1.01103	−0.505514	−	0.862819i	$-0.668697\pi$
$239$	−15.6301	−1.01103	−0.505514	+	0.862819i	$0.668697\pi$
$240$	0	0
$241$	−10.4496	−0.673118	−0.336559	−	0.941662i	$-0.609263\pi$
$241$	−10.4496	−0.673118	−0.336559	+	0.941662i	$0.609263\pi$
$242$	1.35689	0.0872245
$243$	0	0
$244$	−6.22616	−0.398589
$245$	0	0
$246$	0	0
$247$	−1.89992	−0.120889
$248$	4.99727	0.317327
$249$	0	0
$250$	0	0
$251$	4.30657	0.271829	0.135914	−	0.990721i	$-0.456603\pi$
$251$	4.30657	0.271829	0.135914	+	0.990721i	$0.456603\pi$
$252$	0	0
$253$	−19.9204	−1.25239
$254$	0.920279	0.0577435
$255$	0	0
$256$	−2.66466	−0.166541
$257$	28.0650	1.75065	0.875323	−	0.483539i	$-0.160649\pi$
$257$	28.0650	1.75065	0.875323	+	0.483539i	$0.160649\pi$
$258$	0	0
$259$	−8.95505	−0.556440
$260$	0	0
$261$	0	0
$262$	7.27191	0.449260
$263$	−25.0433	−1.54423	−0.772117	−	0.635481i	$-0.780802\pi$
$263$	−25.0433	−1.54423	−0.772117	+	0.635481i	$0.780802\pi$
$264$	0	0
$265$	0	0
$266$	1.15960	0.0710995
$267$	0	0
$268$	−15.0981	−0.922265
$269$	−4.33184	−0.264117	−0.132058	−	0.991242i	$-0.542159\pi$
$269$	−4.33184	−0.264117	−0.132058	+	0.991242i	$0.542159\pi$
$270$	0	0
$271$	−23.5398	−1.42994	−0.714970	−	0.699155i	$-0.753560\pi$
$271$	−23.5398	−1.42994	−0.714970	+	0.699155i	$0.753560\pi$
$272$	6.18275	0.374884
$273$	0	0
$274$	−3.09017	−0.186684
$275$	0	0
$276$	0	0
$277$	15.9663	0.959320	0.479660	−	0.877455i	$-0.340760\pi$
$277$	15.9663	0.959320	0.479660	+	0.877455i	$0.340760\pi$
$278$	10.4570	0.627172
$279$	0	0
$280$	0	0
$281$	16.3146	0.973246	0.486623	−	0.873612i	$-0.338229\pi$
$281$	16.3146	0.973246	0.486623	+	0.873612i	$0.338229\pi$
$282$	0	0
$283$	17.2719	1.02671	0.513354	−	0.858177i	$-0.328403\pi$
$283$	17.2719	1.02671	0.513354	+	0.858177i	$0.328403\pi$
$284$	−13.3267	−0.790795
$285$	0	0
$286$	−1.52618	−0.0902452
$287$	−11.4989	−0.678761
$288$	0	0
$289$	−9.92472	−0.583807
$290$	0	0
$291$	0	0
$292$	−26.6701	−1.56075
$293$	20.4862	1.19681	0.598407	−	0.801192i	$-0.295801\pi$
$293$	20.4862	1.19681	0.598407	+	0.801192i	$0.295801\pi$
$294$	0	0
$295$	0	0
$296$	−16.5258	−0.960542
$297$	0	0
$298$	9.22800	0.534564
$299$	−6.60370	−0.381902
$300$	0	0
$301$	−8.40974	−0.484730
$302$	0.374697	0.0215614
$303$	0	0
$304$	−4.57037	−0.262129
$305$	0	0
$306$	0	0
$307$	1.57747	0.0900309	0.0450155	−	0.998986i	$-0.485666\pi$
$307$	1.57747	0.0900309	0.0450155	+	0.998986i	$0.485666\pi$
$308$	−5.41299	−0.308434
$309$	0	0
$310$	0	0
$311$	13.1745	0.747059	0.373530	−	0.927618i	$-0.378147\pi$
$311$	13.1745	0.747059	0.373530	+	0.927618i	$0.378147\pi$
$312$	0	0
$313$	18.4391	1.04224	0.521120	−	0.853484i	$-0.325514\pi$
$313$	18.4391	1.04224	0.521120	+	0.853484i	$0.325514\pi$
$314$	0.242929	0.0137093
$315$	0	0
$316$	11.2708	0.634031
$317$	−0.707984	−0.0397644	−0.0198822	−	0.999802i	$-0.506329\pi$
$317$	−0.707984	−0.0397644	−0.0198822	+	0.999802i	$0.506329\pi$
$318$	0	0
$319$	−19.8672	−1.11235
$320$	0	0
$321$	0	0
$322$	4.03050	0.224611
$323$	−5.23014	−0.291013
$324$	0	0
$325$	0	0
$326$	−5.37211	−0.297534
$327$	0	0
$328$	−21.2203	−1.17170
$329$	9.81682	0.541219
$330$	0	0
$331$	−8.25026	−0.453475	−0.226738	−	0.973956i	$-0.572806\pi$
$331$	−8.25026	−0.453475	−0.226738	+	0.973956i	$0.572806\pi$
$332$	−3.24000	−0.177818
$333$	0	0
$334$	10.2523	0.560979
$335$	0	0
$336$	0	0
$337$	−2.58711	−0.140929	−0.0704645	−	0.997514i	$-0.522448\pi$
$337$	−2.58711	−0.140929	−0.0704645	+	0.997514i	$0.522448\pi$
$338$	6.53855	0.355650
$339$	0	0
$340$	0	0
$341$	7.25247	0.392743
$342$	0	0
$343$	−13.9475	−0.753095
$344$	−15.5195	−0.836754
$345$	0	0
$346$	−6.50403	−0.349659
$347$	−15.3862	−0.825976	−0.412988	−	0.910736i	$-0.635515\pi$
$347$	−15.3862	−0.825976	−0.412988	+	0.910736i	$0.635515\pi$
$348$	0	0
$349$	1.86531	0.0998477	0.0499239	−	0.998753i	$-0.484102\pi$
$349$	1.86531	0.0998477	0.0499239	+	0.998753i	$0.484102\pi$
$350$	0	0
$351$	0	0
$352$	−15.3795	−0.819732
$353$	−3.27395	−0.174255	−0.0871275	−	0.996197i	$-0.527769\pi$
$353$	−3.27395	−0.174255	−0.0871275	+	0.996197i	$0.527769\pi$
$354$	0	0
$355$	0	0
$356$	−1.95471	−0.103599
$357$	0	0
$358$	−4.93593	−0.260872
$359$	7.47189	0.394351	0.197176	−	0.980368i	$-0.436823\pi$
$359$	7.47189	0.394351	0.197176	+	0.980368i	$0.436823\pi$
$360$	0	0
$361$	−15.1338	−0.796517
$362$	3.30178	0.173538
$363$	0	0
$364$	−1.79443	−0.0940535
$365$	0	0
$366$	0	0
$367$	−4.76902	−0.248941	−0.124470	−	0.992223i	$-0.539723\pi$
$367$	−4.76902	−0.248941	−0.124470	+	0.992223i	$0.539723\pi$
$368$	−15.8856	−0.828092
$369$	0	0
$370$	0	0
$371$	−6.02264	−0.312680
$372$	0	0
$373$	27.7024	1.43438	0.717188	−	0.696880i	$-0.245429\pi$
$373$	27.7024	1.43438	0.717188	+	0.696880i	$0.245429\pi$
$374$	−4.20130	−0.217244
$375$	0	0
$376$	18.1161	0.934267
$377$	−6.58603	−0.339198
$378$	0	0
$379$	−16.8433	−0.865183	−0.432592	−	0.901590i	$-0.642401\pi$
$379$	−16.8433	−0.865183	−0.432592	+	0.901590i	$0.642401\pi$
$380$	0	0
$381$	0	0
$382$	−5.01369	−0.256523
$383$	13.1971	0.674342	0.337171	−	0.941443i	$-0.390530\pi$
$383$	13.1971	0.674342	0.337171	+	0.941443i	$0.390530\pi$
$384$	0	0
$385$	0	0
$386$	13.1482	0.669225
$387$	0	0
$388$	10.5698	0.536602
$389$	14.8723	0.754053	0.377027	−	0.926202i	$-0.376946\pi$
$389$	14.8723	0.754053	0.377027	+	0.926202i	$0.376946\pi$
$390$	0	0
$391$	−18.1788	−0.919339
$392$	−11.6800	−0.589931
$393$	0	0
$394$	6.86179	0.345692
$395$	0	0
$396$	0	0
$397$	−9.85112	−0.494414	−0.247207	−	0.968963i	$-0.579513\pi$
$397$	−9.85112	−0.494414	−0.247207	+	0.968963i	$0.579513\pi$
$398$	7.49340	0.375610
$399$	0	0
$400$	0	0
$401$	1.90738	0.0952498	0.0476249	−	0.998865i	$-0.484835\pi$
$401$	1.90738	0.0952498	0.0476249	+	0.998865i	$0.484835\pi$
$402$	0	0
$403$	2.40422	0.119763
$404$	6.63589	0.330148
$405$	0	0
$406$	4.01971	0.199495
$407$	−23.9836	−1.18882
$408$	0	0
$409$	−20.2898	−1.00327	−0.501633	−	0.865081i	$-0.667267\pi$
$409$	−20.2898	−1.00327	−0.501633	+	0.865081i	$0.667267\pi$
$410$	0	0
$411$	0	0
$412$	30.3489	1.49518
$413$	14.3949	0.708327
$414$	0	0
$415$	0	0
$416$	−5.09837	−0.249968
$417$	0	0
$418$	3.10566	0.151903
$419$	35.5353	1.73601	0.868005	−	0.496555i	$-0.165402\pi$
$419$	35.5353	1.73601	0.868005	+	0.496555i	$0.165402\pi$
$420$	0	0
$421$	7.80450	0.380368	0.190184	−	0.981748i	$-0.439092\pi$
$421$	7.80450	0.380368	0.190184	+	0.981748i	$0.439092\pi$
$422$	7.64829	0.372313
$423$	0	0
$424$	−11.1143	−0.539756
$425$	0	0
$426$	0	0
$427$	3.97108	0.192174
$428$	−11.9957	−0.579832
$429$	0	0
$430$	0	0
$431$	−16.6440	−0.801715	−0.400857	−	0.916141i	$-0.631288\pi$
$431$	−16.6440	−0.801715	−0.400857	+	0.916141i	$0.631288\pi$
$432$	0	0
$433$	−0.0305956	−0.00147033	−0.000735166	−	1.00000i	$-0.500234\pi$
$433$	−0.0305956	−0.00147033	−0.000735166		1.00000i	$0.500234\pi$
$434$	−1.46739	−0.0704369
$435$	0	0
$436$	15.4015	0.737599
$437$	13.4380	0.642826
$438$	0	0
$439$	12.3604	0.589930	0.294965	−	0.955508i	$-0.404692\pi$
$439$	12.3604	0.589930	0.294965	+	0.955508i	$0.404692\pi$
$440$	0	0
$441$	0	0
$442$	−1.39275	−0.0662463
$443$	−29.8515	−1.41829	−0.709143	−	0.705065i	$-0.750918\pi$
$443$	−29.8515	−1.41829	−0.709143	+	0.705065i	$0.750918\pi$
$444$	0	0
$445$	0	0
$446$	−10.0776	−0.477188
$447$	0	0
$448$	−1.94768	−0.0920191
$449$	31.5217	1.48760	0.743799	−	0.668403i	$-0.233022\pi$
$449$	31.5217	1.48760	0.743799	+	0.668403i	$0.233022\pi$
$450$	0	0
$451$	−30.7967	−1.45016
$452$	−22.1596	−1.04230
$453$	0	0
$454$	−8.38993	−0.393759
$455$	0	0
$456$	0	0
$457$	−14.8832	−0.696206	−0.348103	−	0.937456i	$-0.613174\pi$
$457$	−14.8832	−0.696206	−0.348103	+	0.937456i	$0.613174\pi$
$458$	8.45561	0.395105
$459$	0	0
$460$	0	0
$461$	−10.4909	−0.488608	−0.244304	−	0.969699i	$-0.578559\pi$
$461$	−10.4909	−0.488608	−0.244304	+	0.969699i	$0.578559\pi$
$462$	0	0
$463$	−18.9169	−0.879144	−0.439572	−	0.898207i	$-0.644870\pi$
$463$	−18.9169	−0.879144	−0.439572	+	0.898207i	$0.644870\pi$
$464$	−15.8431	−0.735496
$465$	0	0
$466$	11.7190	0.542872
$467$	12.6041	0.583247	0.291623	−	0.956533i	$-0.405805\pi$
$467$	12.6041	0.583247	0.291623	+	0.956533i	$0.405805\pi$
$468$	0	0
$469$	9.62968	0.444657
$470$	0	0
$471$	0	0
$472$	26.5646	1.22273
$473$	−22.5232	−1.03562
$474$	0	0
$475$	0	0
$476$	−4.93973	−0.226412
$477$	0	0
$478$	8.46970	0.387395
$479$	−23.0358	−1.05253	−0.526267	−	0.850319i	$-0.676409\pi$
$479$	−23.0358	−1.05253	−0.526267	+	0.850319i	$0.676409\pi$
$480$	0	0
$481$	−7.95066	−0.362519
$482$	5.66247	0.257918
$483$	0	0
$484$	4.27279	0.194218
$485$	0	0
$486$	0	0
$487$	−29.2131	−1.32377	−0.661886	−	0.749604i	$-0.730244\pi$
$487$	−29.2131	−1.32377	−0.661886	+	0.749604i	$0.730244\pi$
$488$	7.32830	0.331736
$489$	0	0
$490$	0	0
$491$	−37.7392	−1.70315	−0.851573	−	0.524235i	$-0.824351\pi$
$491$	−37.7392	−1.70315	−0.851573	+	0.524235i	$0.824351\pi$
$492$	0	0
$493$	−18.1301	−0.816540
$494$	1.02954	0.0463211
$495$	0	0
$496$	5.78348	0.259686
$497$	8.49986	0.381271
$498$	0	0
$499$	−34.5121	−1.54497	−0.772487	−	0.635031i	$-0.780987\pi$
$499$	−34.5121	−1.54497	−0.772487	+	0.635031i	$0.780987\pi$
$500$	0	0
$501$	0	0
$502$	−2.33366	−0.104156
$503$	−1.67674	−0.0747623	−0.0373812	−	0.999301i	$-0.511902\pi$
$503$	−1.67674	−0.0747623	−0.0373812	+	0.999301i	$0.511902\pi$
$504$	0	0
$505$	0	0
$506$	10.7946	0.479877
$507$	0	0
$508$	2.89791	0.128574
$509$	−9.18843	−0.407270	−0.203635	−	0.979047i	$-0.565276\pi$
$509$	−9.18843	−0.407270	−0.203635	+	0.979047i	$0.565276\pi$
$510$	0	0
$511$	17.0103	0.752493
$512$	−21.6011	−0.954644
$513$	0	0
$514$	−15.2080	−0.670794
$515$	0	0
$516$	0	0
$517$	26.2916	1.15630
$518$	4.85260	0.213211
$519$	0	0
$520$	0	0
$521$	39.0181	1.70942	0.854708	−	0.519109i	$-0.173736\pi$
$521$	39.0181	1.70942	0.854708	+	0.519109i	$0.173736\pi$
$522$	0	0
$523$	−5.53399	−0.241984	−0.120992	−	0.992653i	$-0.538608\pi$
$523$	−5.53399	−0.241984	−0.120992	+	0.992653i	$0.538608\pi$
$524$	22.8989	1.00034
$525$	0	0
$526$	13.5705	0.591703
$527$	6.61837	0.288301
$528$	0	0
$529$	23.7073	1.03075
$530$	0	0
$531$	0	0
$532$	3.65151	0.158313
$533$	−10.2092	−0.442210
$534$	0	0
$535$	0	0
$536$	17.7708	0.767580
$537$	0	0
$538$	2.34735	0.101202
$539$	−16.9510	−0.730133
$540$	0	0
$541$	−13.1293	−0.564471	−0.282236	−	0.959345i	$-0.591076\pi$
$541$	−13.1293	−0.564471	−0.282236	+	0.959345i	$0.591076\pi$
$542$	12.7558	0.547909
$543$	0	0
$544$	−14.0349	−0.601740
$545$	0	0
$546$	0	0
$547$	−25.6298	−1.09585	−0.547925	−	0.836527i	$-0.684582\pi$
$547$	−25.6298	−1.09585	−0.547925	+	0.836527i	$0.684582\pi$
$548$	−9.73078	−0.415678
$549$	0	0
$550$	0	0
$551$	13.4020	0.570946
$552$	0	0
$553$	−7.18857	−0.305689
$554$	−8.65186	−0.367582
$555$	0	0
$556$	32.9287	1.39649
$557$	−38.4002	−1.62707	−0.813534	−	0.581517i	$-0.802459\pi$
$557$	−38.4002	−1.62707	−0.813534	+	0.581517i	$0.802459\pi$
$558$	0	0
$559$	−7.46651	−0.315800
$560$	0	0
$561$	0	0
$562$	−8.84060	−0.372918
$563$	22.5165	0.948956	0.474478	−	0.880267i	$-0.342637\pi$
$563$	22.5165	0.948956	0.474478	+	0.880267i	$0.342637\pi$
$564$	0	0
$565$	0	0
$566$	−9.35937	−0.393403
$567$	0	0
$568$	15.6858	0.658160
$569$	−24.2768	−1.01774	−0.508868	−	0.860845i	$-0.669936\pi$
$569$	−24.2768	−1.01774	−0.508868	+	0.860845i	$0.669936\pi$
$570$	0	0
$571$	15.3457	0.642196	0.321098	−	0.947046i	$-0.395948\pi$
$571$	15.3457	0.642196	0.321098	+	0.947046i	$0.395948\pi$
$572$	−4.80587	−0.200944
$573$	0	0
$574$	6.23108	0.260080
$575$	0	0
$576$	0	0
$577$	−10.2461	−0.426552	−0.213276	−	0.976992i	$-0.568413\pi$
$577$	−10.2461	−0.426552	−0.213276	+	0.976992i	$0.568413\pi$
$578$	5.37804	0.223697
$579$	0	0
$580$	0	0
$581$	2.06649	0.0857326
$582$	0	0
$583$	−16.1300	−0.668035
$584$	31.3911	1.29897
$585$	0	0
$586$	−11.1011	−0.458583
$587$	−36.2362	−1.49563	−0.747814	−	0.663908i	$-0.768897\pi$
$587$	−36.2362	−1.49563	−0.747814	+	0.663908i	$0.768897\pi$
$588$	0	0
$589$	−4.89238	−0.201587
$590$	0	0
$591$	0	0
$592$	−19.1258	−0.786063
$593$	−5.53608	−0.227339	−0.113670	−	0.993519i	$-0.536261\pi$
$593$	−5.53608	−0.227339	−0.113670	+	0.993519i	$0.536261\pi$
$594$	0	0
$595$	0	0
$596$	29.0585	1.19028
$597$	0	0
$598$	3.57844	0.146333
$599$	22.4935	0.919059	0.459530	−	0.888162i	$-0.348018\pi$
$599$	22.4935	0.919059	0.459530	+	0.888162i	$0.348018\pi$
$600$	0	0
$601$	8.51096	0.347169	0.173585	−	0.984819i	$-0.444465\pi$
$601$	8.51096	0.347169	0.173585	+	0.984819i	$0.444465\pi$
$602$	4.55710	0.185734
$603$	0	0
$604$	1.17990	0.0480094
$605$	0	0
$606$	0	0
$607$	28.6926	1.16460	0.582298	−	0.812975i	$-0.302154\pi$
$607$	28.6926	1.16460	0.582298	+	0.812975i	$0.302154\pi$
$608$	10.3748	0.420752
$609$	0	0
$610$	0	0
$611$	8.71577	0.352602
$612$	0	0
$613$	16.4772	0.665509	0.332754	−	0.943014i	$-0.392022\pi$
$613$	16.4772	0.665509	0.332754	+	0.943014i	$0.392022\pi$
$614$	−0.854805	−0.0344971
$615$	0	0
$616$	6.37118	0.256702
$617$	9.99431	0.402356	0.201178	−	0.979555i	$-0.435523\pi$
$617$	9.99431	0.402356	0.201178	+	0.979555i	$0.435523\pi$
$618$	0	0
$619$	−0.110481	−0.00444059	−0.00222029	−	0.999998i	$-0.500707\pi$
$619$	−0.110481	−0.00444059	−0.00222029	+	0.999998i	$0.500707\pi$
$620$	0	0
$621$	0	0
$622$	−7.13906	−0.286250
$623$	1.24673	0.0499490
$624$	0	0
$625$	0	0
$626$	−9.99185	−0.399355
$627$	0	0
$628$	0.764970	0.0305256
$629$	−21.8867	−0.872680
$630$	0	0
$631$	−27.7410	−1.10435	−0.552175	−	0.833728i	$-0.686202\pi$
$631$	−27.7410	−1.10435	−0.552175	+	0.833728i	$0.686202\pi$
$632$	−13.2659	−0.527689
$633$	0	0
$634$	0.383645	0.0152365
$635$	0	0
$636$	0	0
$637$	−5.61933	−0.222646
$638$	10.7657	0.426217
$639$	0	0
$640$	0	0
$641$	26.5314	1.04793	0.523964	−	0.851741i	$-0.324453\pi$
$641$	26.5314	1.04793	0.523964	+	0.851741i	$0.324453\pi$
$642$	0	0
$643$	26.7379	1.05444	0.527220	−	0.849729i	$-0.323234\pi$
$643$	26.7379	1.05444	0.527220	+	0.849729i	$0.323234\pi$
$644$	12.6918	0.500127
$645$	0	0
$646$	2.83413	0.111507
$647$	29.6395	1.16525	0.582625	−	0.812741i	$-0.302026\pi$
$647$	29.6395	1.16525	0.582625	+	0.812741i	$0.302026\pi$
$648$	0	0
$649$	38.5528	1.51333
$650$	0	0
$651$	0	0
$652$	−16.9165	−0.662501
$653$	−13.5621	−0.530728	−0.265364	−	0.964148i	$-0.585492\pi$
$653$	−13.5621	−0.530728	−0.265364	+	0.964148i	$0.585492\pi$
$654$	0	0
$655$	0	0
$656$	−24.5588	−0.958862
$657$	0	0
$658$	−5.31957	−0.207379
$659$	0.634910	0.0247326	0.0123663	−	0.999924i	$-0.496064\pi$
$659$	0.634910	0.0247326	0.0123663	+	0.999924i	$0.496064\pi$
$660$	0	0
$661$	−10.9856	−0.427292	−0.213646	−	0.976911i	$-0.568534\pi$
$661$	−10.9856	−0.427292	−0.213646	+	0.976911i	$0.568534\pi$
$662$	4.47068	0.173758
$663$	0	0
$664$	3.81354	0.147994
$665$	0	0
$666$	0	0
$667$	46.5824	1.80368
$668$	32.2838	1.24910
$669$	0	0
$670$	0	0
$671$	10.6354	0.410577
$672$	0	0
$673$	−14.5368	−0.560352	−0.280176	−	0.959949i	$-0.590393\pi$
$673$	−14.5368	−0.560352	−0.280176	+	0.959949i	$0.590393\pi$
$674$	1.40191	0.0539997
$675$	0	0
$676$	20.5895	0.791906
$677$	−26.2373	−1.00838	−0.504190	−	0.863593i	$-0.668209\pi$
$677$	−26.2373	−1.00838	−0.504190	+	0.863593i	$0.668209\pi$
$678$	0	0
$679$	−6.74150	−0.258715
$680$	0	0
$681$	0	0
$682$	−3.92999	−0.150487
$683$	40.0139	1.53109	0.765544	−	0.643384i	$-0.222470\pi$
$683$	40.0139	1.53109	0.765544	+	0.643384i	$0.222470\pi$
$684$	0	0
$685$	0	0
$686$	7.55793	0.288563
$687$	0	0
$688$	−17.9611	−0.684761
$689$	−5.34714	−0.203710
$690$	0	0
$691$	27.0675	1.02970	0.514848	−	0.857282i	$-0.327849\pi$
$691$	27.0675	1.02970	0.514848	+	0.857282i	$0.327849\pi$
$692$	−20.4808	−0.778565
$693$	0	0
$694$	8.33755	0.316489
$695$	0	0
$696$	0	0
$697$	−28.1041	−1.06452
$698$	−1.01078	−0.0382586
$699$	0	0
$700$	0	0
$701$	−34.7236	−1.31149	−0.655745	−	0.754982i	$-0.727646\pi$
$701$	−34.7236	−1.31149	−0.655745	+	0.754982i	$0.727646\pi$
$702$	0	0
$703$	16.1789	0.610200
$704$	−5.21631	−0.196597
$705$	0	0
$706$	1.77410	0.0667692
$707$	−4.23241	−0.159176
$708$	0	0
$709$	37.8680	1.42216	0.711080	−	0.703111i	$-0.248206\pi$
$709$	37.8680	1.42216	0.711080	+	0.703111i	$0.248206\pi$
$710$	0	0
$711$	0	0
$712$	2.30073	0.0862234
$713$	−17.0048	−0.636835
$714$	0	0
$715$	0	0
$716$	−15.5430	−0.580868
$717$	0	0
$718$	−4.04889	−0.151103
$719$	−12.9493	−0.482926	−0.241463	−	0.970410i	$-0.577627\pi$
$719$	−12.9493	−0.482926	−0.241463	+	0.970410i	$0.577627\pi$
$720$	0	0
$721$	−19.3567	−0.720882
$722$	8.20077	0.305201
$723$	0	0
$724$	10.3971	0.386406
$725$	0	0
$726$	0	0
$727$	40.0931	1.48697	0.743486	−	0.668751i	$-0.233171\pi$
$727$	40.0931	1.48697	0.743486	+	0.668751i	$0.233171\pi$
$728$	2.11207	0.0782786
$729$	0	0
$730$	0	0
$731$	−20.5539	−0.760214
$732$	0	0
$733$	15.3021	0.565197	0.282599	−	0.959238i	$-0.408804\pi$
$733$	15.3021	0.565197	0.282599	+	0.959238i	$0.408804\pi$
$734$	2.58425	0.0953865
$735$	0	0
$736$	36.0603	1.32920
$737$	25.7904	0.950003
$738$	0	0
$739$	15.4571	0.568598	0.284299	−	0.958736i	$-0.408239\pi$
$739$	15.4571	0.568598	0.284299	+	0.958736i	$0.408239\pi$
$740$	0	0
$741$	0	0
$742$	3.26357	0.119809
$743$	2.02309	0.0742199	0.0371100	−	0.999311i	$-0.488185\pi$
$743$	2.02309	0.0742199	0.0371100	+	0.999311i	$0.488185\pi$
$744$	0	0
$745$	0	0
$746$	−15.0115	−0.549609
$747$	0	0
$748$	−13.2297	−0.483725
$749$	7.65090	0.279558
$750$	0	0
$751$	5.78809	0.211210	0.105605	−	0.994408i	$-0.466322\pi$
$751$	5.78809	0.211210	0.105605	+	0.994408i	$0.466322\pi$
$752$	20.9663	0.764561
$753$	0	0
$754$	3.56886	0.129970
$755$	0	0
$756$	0	0
$757$	44.7826	1.62765	0.813826	−	0.581109i	$-0.197381\pi$
$757$	44.7826	1.62765	0.813826	+	0.581109i	$0.197381\pi$
$758$	9.12712	0.331512
$759$	0	0
$760$	0	0
$761$	−13.6970	−0.496518	−0.248259	−	0.968694i	$-0.579858\pi$
$761$	−13.6970	−0.496518	−0.248259	+	0.968694i	$0.579858\pi$
$762$	0	0
$763$	−9.82318	−0.355623
$764$	−15.7878	−0.571184
$765$	0	0
$766$	−7.15131	−0.258387
$767$	12.7804	0.461473
$768$	0	0
$769$	20.4077	0.735922	0.367961	−	0.929841i	$-0.380056\pi$
$769$	20.4077	0.735922	0.367961	+	0.929841i	$0.380056\pi$
$770$	0	0
$771$	0	0
$772$	41.4029	1.49012
$773$	−4.07699	−0.146639	−0.0733196	−	0.997308i	$-0.523359\pi$
$773$	−4.07699	−0.146639	−0.0733196	+	0.997308i	$0.523359\pi$
$774$	0	0
$775$	0	0
$776$	−12.4409	−0.446601
$777$	0	0
$778$	−8.05903	−0.288930
$779$	20.7749	0.744338
$780$	0	0
$781$	22.7645	0.814579
$782$	9.85077	0.352263
$783$	0	0
$784$	−13.5176	−0.482772
$785$	0	0
$786$	0	0
$787$	−31.5066	−1.12309	−0.561544	−	0.827447i	$-0.689793\pi$
$787$	−31.5066	−1.12309	−0.561544	+	0.827447i	$0.689793\pi$
$788$	21.6074	0.769732
$789$	0	0
$790$	0	0
$791$	14.1335	0.502530
$792$	0	0
$793$	3.52569	0.125201
$794$	5.33816	0.189444
$795$	0	0
$796$	23.5963	0.836349
$797$	−35.9093	−1.27197	−0.635986	−	0.771701i	$-0.719406\pi$
$797$	−35.9093	−1.27197	−0.635986	+	0.771701i	$0.719406\pi$
$798$	0	0
$799$	23.9929	0.848808
$800$	0	0
$801$	0	0
$802$	−1.03358	−0.0364968
$803$	45.5574	1.60769
$804$	0	0
$805$	0	0
$806$	−1.30281	−0.0458894
$807$	0	0
$808$	−7.81056	−0.274774
$809$	−52.5637	−1.84804	−0.924020	−	0.382344i	$-0.875117\pi$
$809$	−52.5637	−1.84804	−0.924020	+	0.382344i	$0.875117\pi$
$810$	0	0
$811$	−14.2701	−0.501092	−0.250546	−	0.968105i	$-0.580610\pi$
$811$	−14.2701	−0.501092	−0.250546	+	0.968105i	$0.580610\pi$
$812$	12.6579	0.444204
$813$	0	0
$814$	12.9963	0.455521
$815$	0	0
$816$	0	0
$817$	15.1937	0.531561
$818$	10.9947	0.384421
$819$	0	0
$820$	0	0
$821$	−23.4763	−0.819329	−0.409664	−	0.912236i	$-0.634354\pi$
$821$	−23.4763	−0.819329	−0.409664	+	0.912236i	$0.634354\pi$
$822$	0	0
$823$	−36.8357	−1.28401	−0.642006	−	0.766700i	$-0.721898\pi$
$823$	−36.8357	−1.28401	−0.642006	+	0.766700i	$0.721898\pi$
$824$	−35.7212	−1.24441
$825$	0	0
$826$	−7.80036	−0.271409
$827$	−15.2948	−0.531853	−0.265926	−	0.963993i	$-0.585678\pi$
$827$	−15.2948	−0.531853	−0.265926	+	0.963993i	$0.585678\pi$
$828$	0	0
$829$	31.5748	1.09664	0.548318	−	0.836270i	$-0.315268\pi$
$829$	31.5748	1.09664	0.548318	+	0.836270i	$0.315268\pi$
$830$	0	0
$831$	0	0
$832$	−1.72923	−0.0599502
$833$	−15.4690	−0.535969
$834$	0	0
$835$	0	0
$836$	9.77955	0.338233
$837$	0	0
$838$	−19.2560	−0.665187
$839$	−12.8719	−0.444388	−0.222194	−	0.975003i	$-0.571322\pi$
$839$	−12.8719	−0.444388	−0.222194	+	0.975003i	$0.571322\pi$
$840$	0	0
$841$	17.4578	0.601992
$842$	−4.22913	−0.145745
$843$	0	0
$844$	24.0840	0.829007
$845$	0	0
$846$	0	0
$847$	−2.72521	−0.0936393
$848$	−12.8628	−0.441712
$849$	0	0
$850$	0	0
$851$	56.2342	1.92768
$852$	0	0
$853$	29.0757	0.995533	0.497767	−	0.867311i	$-0.334154\pi$
$853$	29.0757	0.995533	0.497767	+	0.867311i	$0.334154\pi$
$854$	−2.15187	−0.0736353
$855$	0	0
$856$	14.1191	0.482580
$857$	5.81897	0.198772	0.0993862	−	0.995049i	$-0.468312\pi$
$857$	5.81897	0.198772	0.0993862	+	0.995049i	$0.468312\pi$
$858$	0	0
$859$	7.73702	0.263984	0.131992	−	0.991251i	$-0.457863\pi$
$859$	7.73702	0.263984	0.131992	+	0.991251i	$0.457863\pi$
$860$	0	0
$861$	0	0
$862$	9.01912	0.307193
$863$	−45.5128	−1.54927	−0.774636	−	0.632407i	$-0.782067\pi$
$863$	−45.5128	−1.54927	−0.774636	+	0.632407i	$0.782067\pi$
$864$	0	0
$865$	0	0
$866$	0.0165793	0.000563387 0
$867$	0	0
$868$	−4.62073	−0.156838
$869$	−19.2526	−0.653099
$870$	0	0
$871$	8.54962	0.289693
$872$	−18.1278	−0.613886
$873$	0	0
$874$	−7.28182	−0.246311
$875$	0	0
$876$	0	0
$877$	−14.8342	−0.500916	−0.250458	−	0.968127i	$-0.580581\pi$
$877$	−14.8342	−0.500916	−0.250458	+	0.968127i	$0.580581\pi$
$878$	−6.69790	−0.226043
$879$	0	0
$880$	0	0
$881$	17.5392	0.590912	0.295456	−	0.955356i	$-0.404528\pi$
$881$	17.5392	0.590912	0.295456	+	0.955356i	$0.404528\pi$
$882$	0	0
$883$	12.6917	0.427111	0.213555	−	0.976931i	$-0.431496\pi$
$883$	12.6917	0.427111	0.213555	+	0.976931i	$0.431496\pi$
$884$	−4.38569	−0.147507
$885$	0	0
$886$	16.1760	0.543444
$887$	−22.5608	−0.757517	−0.378758	−	0.925496i	$-0.623649\pi$
$887$	−22.5608	−0.757517	−0.378758	+	0.925496i	$0.623649\pi$
$888$	0	0
$889$	−1.84830	−0.0619901
$890$	0	0
$891$	0	0
$892$	−31.7338	−1.06253
$893$	−17.7359	−0.593508
$894$	0	0
$895$	0	0
$896$	12.5403	0.418942
$897$	0	0
$898$	−17.0811	−0.570003
$899$	−16.9593	−0.565625
$900$	0	0
$901$	−14.7197	−0.490384
$902$	16.6882	0.555657
$903$	0	0
$904$	26.0822	0.867481
$905$	0	0
$906$	0	0
$907$	−40.3527	−1.33989	−0.669945	−	0.742411i	$-0.733682\pi$
$907$	−40.3527	−1.33989	−0.669945	+	0.742411i	$0.733682\pi$
$908$	−26.4194	−0.876760
$909$	0	0
$910$	0	0
$911$	−30.9207	−1.02445	−0.512224	−	0.858852i	$-0.671178\pi$
$911$	−30.9207	−1.02445	−0.512224	+	0.858852i	$0.671178\pi$
$912$	0	0
$913$	5.53453	0.183166
$914$	8.06495	0.266765
$915$	0	0
$916$	26.6263	0.879756
$917$	−14.6050	−0.482301
$918$	0	0
$919$	−24.7171	−0.815341	−0.407671	−	0.913129i	$-0.633659\pi$
$919$	−24.7171	−0.815341	−0.407671	+	0.913129i	$0.633659\pi$
$920$	0	0
$921$	0	0
$922$	5.68482	0.187220
$923$	7.54652	0.248397
$924$	0	0
$925$	0	0
$926$	10.2508	0.336861
$927$	0	0
$928$	35.9638	1.18057
$929$	44.1871	1.44973	0.724866	−	0.688890i	$-0.241902\pi$
$929$	44.1871	1.44973	0.724866	+	0.688890i	$0.241902\pi$
$930$	0	0
$931$	11.4349	0.374763
$932$	36.9025	1.20878
$933$	0	0
$934$	−6.82994	−0.223482
$935$	0	0
$936$	0	0
$937$	−39.8074	−1.30045	−0.650226	−	0.759741i	$-0.725326\pi$
$937$	−39.8074	−1.30045	−0.650226	+	0.759741i	$0.725326\pi$
$938$	−5.21817	−0.170379
$939$	0	0
$940$	0	0
$941$	16.5667	0.540058	0.270029	−	0.962852i	$-0.412967\pi$
$941$	16.5667	0.540058	0.270029	+	0.962852i	$0.412967\pi$
$942$	0	0
$943$	72.2088	2.35144
$944$	30.7439	1.00063
$945$	0	0
$946$	12.2049	0.396817
$947$	2.20243	0.0715693	0.0357847	−	0.999360i	$-0.488607\pi$
$947$	2.20243	0.0715693	0.0357847	+	0.999360i	$0.488607\pi$
$948$	0	0
$949$	15.1025	0.490247
$950$	0	0
$951$	0	0
$952$	5.81414	0.188437
$953$	−28.8960	−0.936033	−0.468017	−	0.883720i	$-0.655031\pi$
$953$	−28.8960	−0.936033	−0.468017	+	0.883720i	$0.655031\pi$
$954$	0	0
$955$	0	0
$956$	26.6706	0.862589
$957$	0	0
$958$	12.4827	0.403299
$959$	6.20635	0.200413
$960$	0	0
$961$	−24.8090	−0.800291
$962$	4.30833	0.138906
$963$	0	0
$964$	17.8308	0.574292
$965$	0	0
$966$	0	0
$967$	14.3641	0.461918	0.230959	−	0.972963i	$-0.425814\pi$
$967$	14.3641	0.461918	0.230959	+	0.972963i	$0.425814\pi$
$968$	−5.02914	−0.161643
$969$	0	0
$970$	0	0
$971$	−48.9228	−1.57001	−0.785003	−	0.619492i	$-0.787339\pi$
$971$	−48.9228	−1.57001	−0.785003	+	0.619492i	$0.787339\pi$
$972$	0	0
$973$	−21.0021	−0.673297
$974$	15.8301	0.507229
$975$	0	0
$976$	8.48124	0.271478
$977$	−42.1234	−1.34765	−0.673823	−	0.738893i	$-0.735349\pi$
$977$	−42.1234	−1.34765	−0.673823	+	0.738893i	$0.735349\pi$
$978$	0	0
$979$	3.33901	0.106715
$980$	0	0
$981$	0	0
$982$	20.4503	0.652594
$983$	−23.8715	−0.761383	−0.380691	−	0.924702i	$-0.624314\pi$
$983$	−23.8715	−0.761383	−0.380691	+	0.924702i	$0.624314\pi$
$984$	0	0
$985$	0	0
$986$	9.82442	0.312873
$987$	0	0
$988$	3.24196	0.103140
$989$	52.8099	1.67926
$990$	0	0
$991$	−60.0723	−1.90826	−0.954130	−	0.299394i	$-0.903216\pi$
$991$	−60.0723	−1.90826	−0.954130	+	0.299394i	$0.903216\pi$
$992$	−13.1285	−0.416831
$993$	0	0
$994$	−4.60594	−0.146091
$995$	0	0
$996$	0	0
$997$	−26.9668	−0.854047	−0.427023	−	0.904241i	$-0.640438\pi$
$997$	−26.9668	−0.854047	−0.427023	+	0.904241i	$0.640438\pi$
$998$	18.7015	0.591987
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.w.1.4		8
3.2	odd	2	inner	5625.2.a.w.1.5		8
5.4	even	2		5625.2.a.v.1.5		8
15.14	odd	2		5625.2.a.v.1.4		8
25.11	even	5		225.2.h.e.46.2	✓	16
25.16	even	5		225.2.h.e.181.2	yes	16
75.11	odd	10		225.2.h.e.46.3	yes	16
75.41	odd	10		225.2.h.e.181.3	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.2.h.e.46.2	✓	16	25.11	even	5
225.2.h.e.46.3	yes	16	75.11	odd	10
225.2.h.e.181.2	yes	16	25.16	even	5
225.2.h.e.181.3	yes	16	75.41	odd	10
5625.2.a.v.1.4		8	15.14	odd	2
5625.2.a.v.1.5		8	5.4	even	2
5625.2.a.w.1.4		8	1.1	even	1	trivial
5625.2.a.w.1.5		8	3.2	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-0.541884 q^{2} -1.70636 q^{4} +1.08833 q^{7} +2.00842 q^{8} +O(q^{10})\)	\(q-0.541884 q^{2} -1.70636 q^{4} +1.08833 q^{7} +2.00842 q^{8} +2.91478 q^{11} +0.966262 q^{13} -0.589747 q^{14} +2.32440 q^{16} +2.65994 q^{17} -1.96626 q^{19} -1.57947 q^{22} -6.83428 q^{23} -0.523601 q^{26} -1.85708 q^{28} -6.81599 q^{29} +2.48817 q^{31} -5.27639 q^{32} -1.44138 q^{34} -8.22827 q^{37} +1.06548 q^{38} -10.5657 q^{41} -7.72721 q^{43} -4.97368 q^{44} +3.70338 q^{46} +9.02009 q^{47} -5.81554 q^{49} -1.64879 q^{52} -5.53384 q^{53} +2.18582 q^{56} +3.69348 q^{58} +13.2266 q^{59} +3.64879 q^{61} -1.34830 q^{62} -1.78961 q^{64} +8.84814 q^{67} -4.53882 q^{68} +7.81002 q^{71} +15.6298 q^{73} +4.45876 q^{74} +3.35515 q^{76} +3.17224 q^{77} -6.60515 q^{79} +5.72537 q^{82} +1.89878 q^{83} +4.18725 q^{86} +5.85410 q^{88} +1.14554 q^{89} +1.05161 q^{91} +11.6618 q^{92} -4.88784 q^{94} -6.19437 q^{97} +3.15135 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{4}+O(q^{10}) \) 8 * q + 4 * q^4	\( 8 q + 4 q^{4} + 10 q^{13} - 8 q^{16} - 18 q^{19} - 30 q^{28} - 26 q^{31} - 20 q^{34} - 40 q^{43} - 30 q^{46} - 16 q^{49} + 40 q^{52} + 10 q^{58} - 24 q^{61} - 34 q^{64} - 40 q^{67} + 40 q^{73} - 44 q^{76} - 42 q^{79} + 60 q^{82} + 20 q^{88} - 40 q^{91} - 10 q^{94} - 40 q^{97}+O(q^{100}) \) 8 * q + 4 * q^4 + 10 * q^13 - 8 * q^16 - 18 * q^19 - 30 * q^28 - 26 * q^31 - 20 * q^34 - 40 * q^43 - 30 * q^46 - 16 * q^49 + 40 * q^52 + 10 * q^58 - 24 * q^61 - 34 * q^64 - 40 * q^67 + 40 * q^73 - 44 * q^76 - 42 * q^79 + 60 * q^82 + 20 * q^88 - 40 * q^91 - 10 * q^94 - 40 * q^97

Introduction

L-functions

Modular forms

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Database

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