LMFDB - Embedded newform 5000.2.a.l.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5000,2,Mod(1,5000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5000, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5000 = 2^{3} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5000.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:	$39.9252010106$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 16x^{6} + 22x^{5} + 86x^{4} - 60x^{3} - 155x^{2} + 40x + 80 $ x^8 - 2x^7 - 16x^6 + 22x^5 + 86x^4 - 60x^3 - 155x^2 + 40*x + 80
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 200)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.49708$ of defining polynomial
Character	$\chi$	$=$	5000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.49708 q^{3} -4.94031 q^{7} -0.758738 q^{9} +O(q^{10})$	$q-1.49708 q^{3} -4.94031 q^{7} -0.758738 q^{9} +1.29454 q^{11} +0.820577 q^{13} -2.18987 q^{17} +6.43739 q^{19} +7.39606 q^{21} +2.21475 q^{23} +5.62715 q^{27} -5.33309 q^{29} +0.708310 q^{31} -1.93804 q^{33} -7.15587 q^{37} -1.22847 q^{39} +10.1238 q^{41} +5.26985 q^{43} +8.04526 q^{47} +17.4067 q^{49} +3.27842 q^{51} -13.2228 q^{53} -9.63732 q^{57} +9.59708 q^{59} -6.54215 q^{61} +3.74840 q^{63} -7.32076 q^{67} -3.31567 q^{69} -0.640303 q^{71} +11.9775 q^{73} -6.39544 q^{77} -2.24014 q^{79} -6.14810 q^{81} +11.4552 q^{83} +7.98408 q^{87} -3.17552 q^{89} -4.05390 q^{91} -1.06040 q^{93} -16.4982 q^{97} -0.982218 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{3} - 3 q^{7} + 12 q^{9}+O(q^{10}) $ 8 * q - 2 * q^3 - 3 * q^7 + 12 * q^9	$ 8 q - 2 q^{3} - 3 q^{7} + 12 q^{9} + 5 q^{11} - 7 q^{13} - 7 q^{17} + 5 q^{19} - 4 q^{21} - 7 q^{23} - 26 q^{27} - 25 q^{29} + 3 q^{31} - 27 q^{33} - 15 q^{37} - 8 q^{39} - 26 q^{41} - 21 q^{43} + 2 q^{47} + 9 q^{49} + 50 q^{51} - 12 q^{53} - 32 q^{57} + 36 q^{59} - 33 q^{61} - 9 q^{63} - 7 q^{67} - 11 q^{69} + 12 q^{71} - 2 q^{73} + 14 q^{77} - 16 q^{79} + 12 q^{81} - q^{83} - 9 q^{87} - 30 q^{89} - 11 q^{91} + 20 q^{93} - 32 q^{97} + 61 q^{99}+O(q^{100}) $ 8 * q - 2 * q^3 - 3 * q^7 + 12 * q^9 + 5 * q^11 - 7 * q^13 - 7 * q^17 + 5 * q^19 - 4 * q^21 - 7 * q^23 - 26 * q^27 - 25 * q^29 + 3 * q^31 - 27 * q^33 - 15 * q^37 - 8 * q^39 - 26 * q^41 - 21 * q^43 + 2 * q^47 + 9 * q^49 + 50 * q^51 - 12 * q^53 - 32 * q^57 + 36 * q^59 - 33 * q^61 - 9 * q^63 - 7 * q^67 - 11 * q^69 + 12 * q^71 - 2 * q^73 + 14 * q^77 - 16 * q^79 + 12 * q^81 - q^83 - 9 * q^87 - 30 * q^89 - 11 * q^91 + 20 * q^93 - 32 * q^97 + 61 * q^99

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	0
$2$	0	0
$3$	−1.49708	−0.864342	−0.432171	−	0.901792i	$-0.642252\pi$
$3$	−1.49708	−0.864342	−0.432171	+	0.901792i	$0.642252\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.94031	−1.86726	−0.933631	−	0.358237i	$-0.883378\pi$
$7$	−4.94031	−1.86726	−0.933631	+	0.358237i	$0.883378\pi$
$8$	0	0
$9$	−0.758738	−0.252913
$10$	0	0
$11$	1.29454	0.390319	0.195159	−	0.980772i	$-0.437478\pi$
$11$	1.29454	0.390319	0.195159	+	0.980772i	$0.437478\pi$
$12$	0	0
$13$	0.820577	0.227587	0.113794	−	0.993504i	$-0.463700\pi$
$13$	0.820577	0.227587	0.113794	+	0.993504i	$0.463700\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.18987	−0.531121	−0.265561	−	0.964094i	$-0.585557\pi$
$17$	−2.18987	−0.531121	−0.265561	+	0.964094i	$0.585557\pi$
$18$	0	0
$19$	6.43739	1.47684	0.738420	−	0.674341i	$-0.235572\pi$
$19$	6.43739	1.47684	0.738420	+	0.674341i	$0.235572\pi$
$20$	0	0
$21$	7.39606	1.61395
$22$	0	0
$23$	2.21475	0.461808	0.230904	−	0.972977i	$-0.425832\pi$
$23$	2.21475	0.461808	0.230904	+	0.972977i	$0.425832\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.62715	1.08295
$28$	0	0
$29$	−5.33309	−0.990330	−0.495165	−	0.868799i	$-0.664892\pi$
$29$	−5.33309	−0.990330	−0.495165	+	0.868799i	$0.664892\pi$
$30$	0	0
$31$	0.708310	0.127216	0.0636081	−	0.997975i	$-0.479739\pi$
$31$	0.708310	0.127216	0.0636081	+	0.997975i	$0.479739\pi$
$32$	0	0
$33$	−1.93804	−0.337369
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.15587	−1.17642	−0.588209	−	0.808709i	$-0.700167\pi$
$37$	−7.15587	−1.17642	−0.588209	+	0.808709i	$0.700167\pi$
$38$	0	0
$39$	−1.22847	−0.196713
$40$	0	0
$41$	10.1238	1.58107	0.790533	−	0.612419i	$-0.209803\pi$
$41$	10.1238	1.58107	0.790533	+	0.612419i	$0.209803\pi$
$42$	0	0
$43$	5.26985	0.803645	0.401823	−	0.915718i	$-0.368377\pi$
$43$	5.26985	0.803645	0.401823	+	0.915718i	$0.368377\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.04526	1.17352	0.586761	−	0.809760i	$-0.300403\pi$
$47$	8.04526	1.17352	0.586761	+	0.809760i	$0.300403\pi$
$48$	0	0
$49$	17.4067	2.48667
$50$	0	0
$51$	3.27842	0.459071
$52$	0	0
$53$	−13.2228	−1.81630	−0.908148	−	0.418649i	$-0.862504\pi$
$53$	−13.2228	−1.81630	−0.908148	+	0.418649i	$0.862504\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−9.63732	−1.27649
$58$	0	0
$59$	9.59708	1.24943	0.624716	−	0.780852i	$-0.285215\pi$
$59$	9.59708	1.24943	0.624716	+	0.780852i	$0.285215\pi$
$60$	0	0
$61$	−6.54215	−0.837636	−0.418818	−	0.908070i	$-0.637555\pi$
$61$	−6.54215	−0.837636	−0.418818	+	0.908070i	$0.637555\pi$
$62$	0	0
$63$	3.74840	0.472254
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.32076	−0.894374	−0.447187	−	0.894441i	$-0.647574\pi$
$67$	−7.32076	−0.894374	−0.447187	+	0.894441i	$0.647574\pi$
$68$	0	0
$69$	−3.31567	−0.399160
$70$	0	0
$71$	−0.640303	−0.0759900	−0.0379950	−	0.999278i	$-0.512097\pi$
$71$	−0.640303	−0.0759900	−0.0379950	+	0.999278i	$0.512097\pi$
$72$	0	0
$73$	11.9775	1.40186	0.700929	−	0.713231i	$-0.252769\pi$
$73$	11.9775	1.40186	0.700929	+	0.713231i	$0.252769\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.39544	−0.728828
$78$	0	0
$79$	−2.24014	−0.252035	−0.126018	−	0.992028i	$-0.540220\pi$
$79$	−2.24014	−0.252035	−0.126018	+	0.992028i	$0.540220\pi$
$80$	0	0
$81$	−6.14810	−0.683123
$82$	0	0
$83$	11.4552	1.25737	0.628687	−	0.777659i	$-0.283593\pi$
$83$	11.4552	1.25737	0.628687	+	0.777659i	$0.283593\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	7.98408	0.855984
$88$	0	0
$89$	−3.17552	−0.336605	−0.168302	−	0.985735i	$-0.553828\pi$
$89$	−3.17552	−0.336605	−0.168302	+	0.985735i	$0.553828\pi$
$90$	0	0
$91$	−4.05390	−0.424965
$92$	0	0
$93$	−1.06040	−0.109958
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−16.4982	−1.67513	−0.837567	−	0.546335i	$-0.816023\pi$
$97$	−16.4982	−1.67513	−0.837567	+	0.546335i	$0.816023\pi$
$98$	0	0
$99$	−0.982218	−0.0987166
$100$	0	0
$101$	−10.4077	−1.03561	−0.517804	−	0.855499i	$-0.673250\pi$
$101$	−10.4077	−1.03561	−0.517804	+	0.855499i	$0.673250\pi$
$102$	0	0
$103$	−1.70894	−0.168386	−0.0841932	−	0.996449i	$-0.526831\pi$
$103$	−1.70894	−0.168386	−0.0841932	+	0.996449i	$0.526831\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.83747	0.274308	0.137154	−	0.990550i	$-0.456204\pi$
$107$	2.83747	0.274308	0.137154	+	0.990550i	$0.456204\pi$
$108$	0	0
$109$	8.24315	0.789550	0.394775	−	0.918778i	$-0.370822\pi$
$109$	8.24315	0.789550	0.394775	+	0.918778i	$0.370822\pi$
$110$	0	0
$111$	10.7129	1.01683
$112$	0	0
$113$	3.90738	0.367576	0.183788	−	0.982966i	$-0.441164\pi$
$113$	3.90738	0.367576	0.183788	+	0.982966i	$0.441164\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−0.622603	−0.0575597
$118$	0	0
$119$	10.8186	0.991742
$120$	0	0
$121$	−9.32416	−0.847651
$122$	0	0
$123$	−15.1561	−1.36658
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−14.3298	−1.27156	−0.635780	−	0.771870i	$-0.719321\pi$
$127$	−14.3298	−1.27156	−0.635780	+	0.771870i	$0.719321\pi$
$128$	0	0
$129$	−7.88942	−0.694624
$130$	0	0
$131$	−10.2184	−0.892785	−0.446393	−	0.894837i	$-0.647292\pi$
$131$	−10.2184	−0.892785	−0.446393	+	0.894837i	$0.647292\pi$
$132$	0	0
$133$	−31.8027	−2.75765
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−7.33174	−0.626393	−0.313196	−	0.949688i	$-0.601400\pi$
$137$	−7.33174	−0.626393	−0.313196	+	0.949688i	$0.601400\pi$
$138$	0	0
$139$	6.69504	0.567866	0.283933	−	0.958844i	$-0.408361\pi$
$139$	6.69504	0.567866	0.283933	+	0.958844i	$0.408361\pi$
$140$	0	0
$141$	−12.0444	−1.01432
$142$	0	0
$143$	1.06227	0.0888316
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−26.0592	−2.14933
$148$	0	0
$149$	−1.30424	−0.106847	−0.0534236	−	0.998572i	$-0.517013\pi$
$149$	−1.30424	−0.106847	−0.0534236	+	0.998572i	$0.517013\pi$
$150$	0	0
$151$	0.227988	0.0185534	0.00927670	−	0.999957i	$-0.497047\pi$
$151$	0.227988	0.0185534	0.00927670	+	0.999957i	$0.497047\pi$
$152$	0	0
$153$	1.66154	0.134327
$154$	0	0
$155$	0	0
$156$	0	0
$157$	7.75041	0.618550	0.309275	−	0.950973i	$-0.399914\pi$
$157$	7.75041	0.618550	0.309275	+	0.950973i	$0.399914\pi$
$158$	0	0
$159$	19.7957	1.56990
$160$	0	0
$161$	−10.9416	−0.862316
$162$	0	0
$163$	−8.38560	−0.656811	−0.328406	−	0.944537i	$-0.606511\pi$
$163$	−8.38560	−0.656811	−0.328406	+	0.944537i	$0.606511\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.17950	−0.555567	−0.277783	−	0.960644i	$-0.589600\pi$
$167$	−7.17950	−0.555567	−0.277783	+	0.960644i	$0.589600\pi$
$168$	0	0
$169$	−12.3267	−0.948204
$170$	0	0
$171$	−4.88429	−0.373511
$172$	0	0
$173$	−14.9876	−1.13948	−0.569741	−	0.821824i	$-0.692957\pi$
$173$	−14.9876	−1.13948	−0.569741	+	0.821824i	$0.692957\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−14.3676	−1.07994
$178$	0	0
$179$	−18.4643	−1.38009	−0.690044	−	0.723767i	$-0.742409\pi$
$179$	−18.4643	−1.38009	−0.690044	+	0.723767i	$0.742409\pi$
$180$	0	0
$181$	20.1038	1.49431	0.747153	−	0.664653i	$-0.231420\pi$
$181$	20.1038	1.49431	0.747153	+	0.664653i	$0.231420\pi$
$182$	0	0
$183$	9.79415	0.724004
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2.83488	−0.207307
$188$	0	0
$189$	−27.7999	−2.02214
$190$	0	0
$191$	−22.0308	−1.59409	−0.797045	−	0.603920i	$-0.793605\pi$
$191$	−22.0308	−1.59409	−0.797045	+	0.603920i	$0.793605\pi$
$192$	0	0
$193$	16.9231	1.21815	0.609077	−	0.793111i	$-0.291540\pi$
$193$	16.9231	1.21815	0.609077	+	0.793111i	$0.291540\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3.67725	−0.261994	−0.130997	−	0.991383i	$-0.541818\pi$
$197$	−3.67725	−0.261994	−0.130997	+	0.991383i	$0.541818\pi$
$198$	0	0
$199$	26.4847	1.87745	0.938724	−	0.344669i	$-0.112009\pi$
$199$	26.4847	1.87745	0.938724	+	0.344669i	$0.112009\pi$
$200$	0	0
$201$	10.9598	0.773045
$202$	0	0
$203$	26.3471	1.84920
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.68042	−0.116797
$208$	0	0
$209$	8.33347	0.576438
$210$	0	0
$211$	−3.10303	−0.213622	−0.106811	−	0.994279i	$-0.534064\pi$
$211$	−3.10303	−0.213622	−0.106811	+	0.994279i	$0.534064\pi$
$212$	0	0
$213$	0.958587	0.0656813
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−3.49927	−0.237546
$218$	0	0
$219$	−17.9313	−1.21169
$220$	0	0
$221$	−1.79696	−0.120876
$222$	0	0
$223$	8.17756	0.547610	0.273805	−	0.961785i	$-0.411718\pi$
$223$	8.17756	0.547610	0.273805	+	0.961785i	$0.411718\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−15.0825	−1.00106	−0.500531	−	0.865718i	$-0.666862\pi$
$227$	−15.0825	−1.00106	−0.500531	+	0.865718i	$0.666862\pi$
$228$	0	0
$229$	−8.75982	−0.578865	−0.289433	−	0.957198i	$-0.593467\pi$
$229$	−8.75982	−0.578865	−0.289433	+	0.957198i	$0.593467\pi$
$230$	0	0
$231$	9.57451	0.629956
$232$	0	0
$233$	7.93200	0.519643	0.259821	−	0.965657i	$-0.416336\pi$
$233$	7.93200	0.519643	0.259821	+	0.965657i	$0.416336\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	3.35367	0.217845
$238$	0	0
$239$	0.938488	0.0607058	0.0303529	−	0.999539i	$-0.490337\pi$
$239$	0.938488	0.0607058	0.0303529	+	0.999539i	$0.490337\pi$
$240$	0	0
$241$	10.9564	0.705766	0.352883	−	0.935667i	$-0.385201\pi$
$241$	10.9564	0.705766	0.352883	+	0.935667i	$0.385201\pi$
$242$	0	0
$243$	−7.67721	−0.492494
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.28238	0.336110
$248$	0	0
$249$	−17.1494	−1.08680
$250$	0	0
$251$	−0.246980	−0.0155893	−0.00779463	−	0.999970i	$-0.502481\pi$
$251$	−0.246980	−0.0155893	−0.00779463	+	0.999970i	$0.502481\pi$
$252$	0	0
$253$	2.86709	0.180252
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.97244	0.497307	0.248654	−	0.968592i	$-0.420012\pi$
$257$	7.97244	0.497307	0.248654	+	0.968592i	$0.420012\pi$
$258$	0	0
$259$	35.3522	2.19668
$260$	0	0
$261$	4.04642	0.250467
$262$	0	0
$263$	7.80127	0.481047	0.240523	−	0.970643i	$-0.422681\pi$
$263$	7.80127	0.481047	0.240523	+	0.970643i	$0.422681\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4.75402	0.290942
$268$	0	0
$269$	1.42524	0.0868985	0.0434493	−	0.999056i	$-0.486165\pi$
$269$	1.42524	0.0868985	0.0434493	+	0.999056i	$0.486165\pi$
$270$	0	0
$271$	−25.7128	−1.56194	−0.780970	−	0.624568i	$-0.785275\pi$
$271$	−25.7128	−1.56194	−0.780970	+	0.624568i	$0.785275\pi$
$272$	0	0
$273$	6.06904	0.367315
$274$	0	0
$275$	0	0
$276$	0	0
$277$	12.5625	0.754808	0.377404	−	0.926049i	$-0.376817\pi$
$277$	12.5625	0.754808	0.377404	+	0.926049i	$0.376817\pi$
$278$	0	0
$279$	−0.537422	−0.0321746
$280$	0	0
$281$	−5.80391	−0.346232	−0.173116	−	0.984901i	$-0.555384\pi$
$281$	−5.80391	−0.346232	−0.173116	+	0.984901i	$0.555384\pi$
$282$	0	0
$283$	−3.40917	−0.202654	−0.101327	−	0.994853i	$-0.532309\pi$
$283$	−3.40917	−0.202654	−0.101327	+	0.994853i	$0.532309\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−50.0146	−2.95227
$288$	0	0
$289$	−12.2045	−0.717910
$290$	0	0
$291$	24.6991	1.44789
$292$	0	0
$293$	−0.336923	−0.0196832	−0.00984162	−	0.999952i	$-0.503133\pi$
$293$	−0.336923	−0.0196832	−0.00984162	+	0.999952i	$0.503133\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	7.28458	0.422694
$298$	0	0
$299$	1.81737	0.105101
$300$	0	0
$301$	−26.0347	−1.50062
$302$	0	0
$303$	15.5813	0.895120
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−23.5065	−1.34159	−0.670793	−	0.741645i	$-0.734046\pi$
$307$	−23.5065	−1.34159	−0.670793	+	0.741645i	$0.734046\pi$
$308$	0	0
$309$	2.55842	0.145543
$310$	0	0
$311$	7.26773	0.412115	0.206058	−	0.978540i	$-0.433937\pi$
$311$	7.26773	0.412115	0.206058	+	0.978540i	$0.433937\pi$
$312$	0	0
$313$	−11.8476	−0.669667	−0.334834	−	0.942277i	$-0.608680\pi$
$313$	−11.8476	−0.669667	−0.334834	+	0.942277i	$0.608680\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.07506	−0.285044	−0.142522	−	0.989792i	$-0.545521\pi$
$317$	−5.07506	−0.285044	−0.142522	+	0.989792i	$0.545521\pi$
$318$	0	0
$319$	−6.90390	−0.386544
$320$	0	0
$321$	−4.24793	−0.237096
$322$	0	0
$323$	−14.0971	−0.784381
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−12.3407	−0.682442
$328$	0	0
$329$	−39.7461	−2.19127
$330$	0	0
$331$	1.16971	0.0642932	0.0321466	−	0.999483i	$-0.489766\pi$
$331$	1.16971	0.0642932	0.0321466	+	0.999483i	$0.489766\pi$
$332$	0	0
$333$	5.42943	0.297531
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−15.7121	−0.855892	−0.427946	−	0.903804i	$-0.640763\pi$
$337$	−15.7121	−0.855892	−0.427946	+	0.903804i	$0.640763\pi$
$338$	0	0
$339$	−5.84968	−0.317711
$340$	0	0
$341$	0.916937	0.0496549
$342$	0	0
$343$	−51.4121	−2.77599
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6.33855	−0.340271	−0.170135	−	0.985421i	$-0.554421\pi$
$347$	−6.33855	−0.340271	−0.170135	+	0.985421i	$0.554421\pi$
$348$	0	0
$349$	−32.3947	−1.73405	−0.867023	−	0.498268i	$-0.833970\pi$
$349$	−32.3947	−1.73405	−0.867023	+	0.498268i	$0.833970\pi$
$350$	0	0
$351$	4.61751	0.246464
$352$	0	0
$353$	−10.2379	−0.544910	−0.272455	−	0.962169i	$-0.587836\pi$
$353$	−10.2379	−0.544910	−0.272455	+	0.962169i	$0.587836\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−16.1964	−0.857205
$358$	0	0
$359$	−28.7555	−1.51766	−0.758828	−	0.651291i	$-0.774228\pi$
$359$	−28.7555	−1.51766	−0.758828	+	0.651291i	$0.774228\pi$
$360$	0	0
$361$	22.4400	1.18106
$362$	0	0
$363$	13.9591	0.732661
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−13.9388	−0.727599	−0.363800	−	0.931477i	$-0.618521\pi$
$367$	−13.9388	−0.727599	−0.363800	+	0.931477i	$0.618521\pi$
$368$	0	0
$369$	−7.68129	−0.399872
$370$	0	0
$371$	65.3249	3.39150
$372$	0	0
$373$	13.3450	0.690979	0.345490	−	0.938423i	$-0.387713\pi$
$373$	13.3450	0.690979	0.345490	+	0.938423i	$0.387713\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.37621	−0.225386
$378$	0	0
$379$	24.1004	1.23796	0.618978	−	0.785408i	$-0.287547\pi$
$379$	24.1004	1.23796	0.618978	+	0.785408i	$0.287547\pi$
$380$	0	0
$381$	21.4529	1.09906
$382$	0	0
$383$	34.4007	1.75779	0.878896	−	0.477013i	$-0.158280\pi$
$383$	34.4007	1.75779	0.878896	+	0.477013i	$0.158280\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3.99844	−0.203252
$388$	0	0
$389$	−3.43049	−0.173933	−0.0869665	−	0.996211i	$-0.527717\pi$
$389$	−3.43049	−0.173933	−0.0869665	+	0.996211i	$0.527717\pi$
$390$	0	0
$391$	−4.85002	−0.245276
$392$	0	0
$393$	15.2978	0.771672
$394$	0	0
$395$	0	0
$396$	0	0
$397$	30.0865	1.51000	0.755000	−	0.655725i	$-0.227637\pi$
$397$	30.0865	1.51000	0.755000	+	0.655725i	$0.227637\pi$
$398$	0	0
$399$	47.6114	2.38355
$400$	0	0
$401$	36.3896	1.81721	0.908604	−	0.417658i	$-0.137149\pi$
$401$	36.3896	1.81721	0.908604	+	0.417658i	$0.137149\pi$
$402$	0	0
$403$	0.581223	0.0289528
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9.26357	−0.459178
$408$	0	0
$409$	3.67965	0.181947	0.0909735	−	0.995853i	$-0.471002\pi$
$409$	3.67965	0.181947	0.0909735	+	0.995853i	$0.471002\pi$
$410$	0	0
$411$	10.9762	0.541418
$412$	0	0
$413$	−47.4125	−2.33302
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−10.0230	−0.490830
$418$	0	0
$419$	−6.79223	−0.331822	−0.165911	−	0.986141i	$-0.553056\pi$
$419$	−6.79223	−0.331822	−0.165911	+	0.986141i	$0.553056\pi$
$420$	0	0
$421$	21.7077	1.05797	0.528984	−	0.848632i	$-0.322573\pi$
$421$	21.7077	1.05797	0.528984	+	0.848632i	$0.322573\pi$
$422$	0	0
$423$	−6.10424	−0.296798
$424$	0	0
$425$	0	0
$426$	0	0
$427$	32.3202	1.56409
$428$	0	0
$429$	−1.59031	−0.0767809
$430$	0	0
$431$	−16.7430	−0.806480	−0.403240	−	0.915094i	$-0.632116\pi$
$431$	−16.7430	−0.806480	−0.403240	+	0.915094i	$0.632116\pi$
$432$	0	0
$433$	−24.4481	−1.17490	−0.587451	−	0.809260i	$-0.699868\pi$
$433$	−24.4481	−1.17490	−0.587451	+	0.809260i	$0.699868\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	14.2572	0.682016
$438$	0	0
$439$	−22.8425	−1.09021	−0.545107	−	0.838366i	$-0.683511\pi$
$439$	−22.8425	−1.09021	−0.545107	+	0.838366i	$0.683511\pi$
$440$	0	0
$441$	−13.2071	−0.628909
$442$	0	0
$443$	−8.95402	−0.425418	−0.212709	−	0.977116i	$-0.568229\pi$
$443$	−8.95402	−0.425418	−0.212709	+	0.977116i	$0.568229\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.95255	0.0923525
$448$	0	0
$449$	5.23228	0.246927	0.123463	−	0.992349i	$-0.460600\pi$
$449$	5.23228	0.246927	0.123463	+	0.992349i	$0.460600\pi$
$450$	0	0
$451$	13.1056	0.617120
$452$	0	0
$453$	−0.341317	−0.0160365
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−27.3114	−1.27758	−0.638788	−	0.769383i	$-0.720564\pi$
$457$	−27.3114	−1.27758	−0.638788	+	0.769383i	$0.720564\pi$
$458$	0	0
$459$	−12.3227	−0.575175
$460$	0	0
$461$	−6.18230	−0.287939	−0.143969	−	0.989582i	$-0.545987\pi$
$461$	−6.18230	−0.287939	−0.143969	+	0.989582i	$0.545987\pi$
$462$	0	0
$463$	−6.45833	−0.300144	−0.150072	−	0.988675i	$-0.547951\pi$
$463$	−6.45833	−0.300144	−0.150072	+	0.988675i	$0.547951\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−0.390795	−0.0180838	−0.00904192	−	0.999959i	$-0.502878\pi$
$467$	−0.390795	−0.0180838	−0.00904192	+	0.999959i	$0.502878\pi$
$468$	0	0
$469$	36.1668	1.67003
$470$	0	0
$471$	−11.6030	−0.534639
$472$	0	0
$473$	6.82204	0.313678
$474$	0	0
$475$	0	0
$476$	0	0
$477$	10.0327	0.459364
$478$	0	0
$479$	−29.1556	−1.33215	−0.666077	−	0.745883i	$-0.732028\pi$
$479$	−29.1556	−1.33215	−0.666077	+	0.745883i	$0.732028\pi$
$480$	0	0
$481$	−5.87194	−0.267738
$482$	0	0
$483$	16.3804	0.745336
$484$	0	0
$485$	0	0
$486$	0	0
$487$	20.5764	0.932407	0.466203	−	0.884678i	$-0.345621\pi$
$487$	20.5764	0.932407	0.466203	+	0.884678i	$0.345621\pi$
$488$	0	0
$489$	12.5540	0.567710
$490$	0	0
$491$	32.0224	1.44515	0.722575	−	0.691292i	$-0.242958\pi$
$491$	32.0224	1.44515	0.722575	+	0.691292i	$0.242958\pi$
$492$	0	0
$493$	11.6788	0.525985
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.16329	0.141893
$498$	0	0
$499$	9.59439	0.429504	0.214752	−	0.976669i	$-0.431106\pi$
$499$	9.59439	0.429504	0.214752	+	0.976669i	$0.431106\pi$
$500$	0	0
$501$	10.7483	0.480200
$502$	0	0
$503$	−10.6589	−0.475259	−0.237630	−	0.971356i	$-0.576370\pi$
$503$	−10.6589	−0.475259	−0.237630	+	0.971356i	$0.576370\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	18.4540	0.819573
$508$	0	0
$509$	−17.3268	−0.767997	−0.383998	−	0.923334i	$-0.625453\pi$
$509$	−17.3268	−0.767997	−0.383998	+	0.923334i	$0.625453\pi$
$510$	0	0
$511$	−59.1725	−2.61764
$512$	0	0
$513$	36.2242	1.59934
$514$	0	0
$515$	0	0
$516$	0	0
$517$	10.4149	0.458048
$518$	0	0
$519$	22.4376	0.984903
$520$	0	0
$521$	−32.7610	−1.43528	−0.717642	−	0.696412i	$-0.754779\pi$
$521$	−32.7610	−1.43528	−0.717642	+	0.696412i	$0.754779\pi$
$522$	0	0
$523$	−2.51192	−0.109838	−0.0549192	−	0.998491i	$-0.517490\pi$
$523$	−2.51192	−0.109838	−0.0549192	+	0.998491i	$0.517490\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.55111	−0.0675673
$528$	0	0
$529$	−18.0949	−0.786734
$530$	0	0
$531$	−7.28166	−0.315997
$532$	0	0
$533$	8.30733	0.359830
$534$	0	0
$535$	0	0
$536$	0	0
$537$	27.6427	1.19287
$538$	0	0
$539$	22.5336	0.970593
$540$	0	0
$541$	4.52084	0.194366	0.0971831	−	0.995267i	$-0.469017\pi$
$541$	4.52084	0.194366	0.0971831	+	0.995267i	$0.469017\pi$
$542$	0	0
$543$	−30.0971	−1.29159
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−34.3057	−1.46681	−0.733404	−	0.679793i	$-0.762070\pi$
$547$	−34.3057	−1.46681	−0.733404	+	0.679793i	$0.762070\pi$
$548$	0	0
$549$	4.96377	0.211849
$550$	0	0
$551$	−34.3312	−1.46256
$552$	0	0
$553$	11.0670	0.470615
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.34120	−0.0991997	−0.0495998	−	0.998769i	$-0.515795\pi$
$557$	−2.34120	−0.0991997	−0.0495998	+	0.998769i	$0.515795\pi$
$558$	0	0
$559$	4.32432	0.182899
$560$	0	0
$561$	4.24405	0.179184
$562$	0	0
$563$	1.74065	0.0733598	0.0366799	−	0.999327i	$-0.488322\pi$
$563$	1.74065	0.0733598	0.0366799	+	0.999327i	$0.488322\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	30.3735	1.27557
$568$	0	0
$569$	−41.6976	−1.74805	−0.874027	−	0.485877i	$-0.838500\pi$
$569$	−41.6976	−1.74805	−0.874027	+	0.485877i	$0.838500\pi$
$570$	0	0
$571$	−1.94712	−0.0814845	−0.0407422	−	0.999170i	$-0.512972\pi$
$571$	−1.94712	−0.0814845	−0.0407422	+	0.999170i	$0.512972\pi$
$572$	0	0
$573$	32.9819	1.37784
$574$	0	0
$575$	0	0
$576$	0	0
$577$	13.9171	0.579376	0.289688	−	0.957121i	$-0.406448\pi$
$577$	13.9171	0.579376	0.289688	+	0.957121i	$0.406448\pi$
$578$	0	0
$579$	−25.3354	−1.05290
$580$	0	0
$581$	−56.5923	−2.34785
$582$	0	0
$583$	−17.1175	−0.708935
$584$	0	0
$585$	0	0
$586$	0	0
$587$	17.2840	0.713387	0.356693	−	0.934221i	$-0.383904\pi$
$587$	17.2840	0.713387	0.356693	+	0.934221i	$0.383904\pi$
$588$	0	0
$589$	4.55967	0.187878
$590$	0	0
$591$	5.50516	0.226452
$592$	0	0
$593$	26.1715	1.07474	0.537368	−	0.843348i	$-0.319419\pi$
$593$	26.1715	1.07474	0.537368	+	0.843348i	$0.319419\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−39.6498	−1.62276
$598$	0	0
$599$	−34.0712	−1.39211	−0.696055	−	0.717988i	$-0.745063\pi$
$599$	−34.0712	−1.39211	−0.696055	+	0.717988i	$0.745063\pi$
$600$	0	0
$601$	−37.2599	−1.51986	−0.759931	−	0.650004i	$-0.774767\pi$
$601$	−37.2599	−1.51986	−0.759931	+	0.650004i	$0.774767\pi$
$602$	0	0
$603$	5.55454	0.226198
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−0.110015	−0.00446537	−0.00223268	−	0.999998i	$-0.500711\pi$
$607$	−0.110015	−0.00446537	−0.00223268	+	0.999998i	$0.500711\pi$
$608$	0	0
$609$	−39.4438	−1.59835
$610$	0	0
$611$	6.60176	0.267078
$612$	0	0
$613$	2.97732	0.120253	0.0601264	−	0.998191i	$-0.480850\pi$
$613$	2.97732	0.120253	0.0601264	+	0.998191i	$0.480850\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−36.5291	−1.47060	−0.735302	−	0.677739i	$-0.762960\pi$
$617$	−36.5291	−1.47060	−0.735302	+	0.677739i	$0.762960\pi$
$618$	0	0
$619$	33.5335	1.34783	0.673913	−	0.738811i	$-0.264612\pi$
$619$	33.5335	1.34783	0.673913	+	0.738811i	$0.264612\pi$
$620$	0	0
$621$	12.4627	0.500112
$622$	0	0
$623$	15.6881	0.628529
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−12.4759	−0.498240
$628$	0	0
$629$	15.6704	0.624821
$630$	0	0
$631$	13.0085	0.517861	0.258930	−	0.965896i	$-0.416630\pi$
$631$	13.0085	0.517861	0.258930	+	0.965896i	$0.416630\pi$
$632$	0	0
$633$	4.64550	0.184642
$634$	0	0
$635$	0	0
$636$	0	0
$637$	14.2835	0.565933
$638$	0	0
$639$	0.485822	0.0192188
$640$	0	0
$641$	−20.4313	−0.806988	−0.403494	−	0.914982i	$-0.632204\pi$
$641$	−20.4313	−0.806988	−0.403494	+	0.914982i	$0.632204\pi$
$642$	0	0
$643$	−9.64777	−0.380471	−0.190235	−	0.981738i	$-0.560925\pi$
$643$	−9.64777	−0.380471	−0.190235	+	0.981738i	$0.560925\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	41.6901	1.63901	0.819503	−	0.573074i	$-0.194249\pi$
$647$	41.6901	1.63901	0.819503	+	0.573074i	$0.194249\pi$
$648$	0	0
$649$	12.4238	0.487677
$650$	0	0
$651$	5.23871	0.205321
$652$	0	0
$653$	−35.6904	−1.39667	−0.698336	−	0.715770i	$-0.746076\pi$
$653$	−35.6904	−1.39667	−0.698336	+	0.715770i	$0.746076\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−9.08777	−0.354548
$658$	0	0
$659$	25.6446	0.998974	0.499487	−	0.866321i	$-0.333522\pi$
$659$	25.6446	0.998974	0.499487	+	0.866321i	$0.333522\pi$
$660$	0	0
$661$	−2.60647	−0.101380	−0.0506900	−	0.998714i	$-0.516142\pi$
$661$	−2.60647	−0.101380	−0.0506900	+	0.998714i	$0.516142\pi$
$662$	0	0
$663$	2.69020	0.104479
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−11.8115	−0.457342
$668$	0	0
$669$	−12.2425	−0.473322
$670$	0	0
$671$	−8.46908	−0.326945
$672$	0	0
$673$	−13.3526	−0.514704	−0.257352	−	0.966318i	$-0.582850\pi$
$673$	−13.3526	−0.514704	−0.257352	+	0.966318i	$0.582850\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−15.2346	−0.585513	−0.292757	−	0.956187i	$-0.594573\pi$
$677$	−15.2346	−0.585513	−0.292757	+	0.956187i	$0.594573\pi$
$678$	0	0
$679$	81.5060	3.12791
$680$	0	0
$681$	22.5798	0.865261
$682$	0	0
$683$	22.2180	0.850148	0.425074	−	0.905159i	$-0.360248\pi$
$683$	22.2180	0.850148	0.425074	+	0.905159i	$0.360248\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	13.1142	0.500338
$688$	0	0
$689$	−10.8504	−0.413366
$690$	0	0
$691$	−11.7544	−0.447160	−0.223580	−	0.974686i	$-0.571774\pi$
$691$	−11.7544	−0.447160	−0.223580	+	0.974686i	$0.571774\pi$
$692$	0	0
$693$	4.85246	0.184330
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−22.1697	−0.839738
$698$	0	0
$699$	−11.8749	−0.449149
$700$	0	0
$701$	11.7537	0.443932	0.221966	−	0.975054i	$-0.428753\pi$
$701$	11.7537	0.443932	0.221966	+	0.975054i	$0.428753\pi$
$702$	0	0
$703$	−46.0652	−1.73738
$704$	0	0
$705$	0	0
$706$	0	0
$707$	51.4174	1.93375
$708$	0	0
$709$	−9.01288	−0.338486	−0.169243	−	0.985574i	$-0.554132\pi$
$709$	−9.01288	−0.338486	−0.169243	+	0.985574i	$0.554132\pi$
$710$	0	0
$711$	1.69968	0.0637428
$712$	0	0
$713$	1.56873	0.0587494
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−1.40500	−0.0524705
$718$	0	0
$719$	44.3166	1.65273	0.826365	−	0.563135i	$-0.190405\pi$
$719$	44.3166	1.65273	0.826365	+	0.563135i	$0.190405\pi$
$720$	0	0
$721$	8.44267	0.314421
$722$	0	0
$723$	−16.4027	−0.610024
$724$	0	0
$725$	0	0
$726$	0	0
$727$	5.03563	0.186761	0.0933805	−	0.995630i	$-0.470233\pi$
$727$	5.03563	0.186761	0.0933805	+	0.995630i	$0.470233\pi$
$728$	0	0
$729$	29.9377	1.10881
$730$	0	0
$731$	−11.5403	−0.426833
$732$	0	0
$733$	−6.49296	−0.239823	−0.119911	−	0.992785i	$-0.538261\pi$
$733$	−6.49296	−0.239823	−0.119911	+	0.992785i	$0.538261\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9.47703	−0.349091
$738$	0	0
$739$	6.71004	0.246833	0.123416	−	0.992355i	$-0.460615\pi$
$739$	6.71004	0.246833	0.123416	+	0.992355i	$0.460615\pi$
$740$	0	0
$741$	−7.90817	−0.290514
$742$	0	0
$743$	27.9505	1.02541	0.512703	−	0.858566i	$-0.328644\pi$
$743$	27.9505	1.02541	0.512703	+	0.858566i	$0.328644\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−8.69151	−0.318006
$748$	0	0
$749$	−14.0180	−0.512205
$750$	0	0
$751$	−16.0513	−0.585719	−0.292860	−	0.956155i	$-0.594607\pi$
$751$	−16.0513	−0.585719	−0.292860	+	0.956155i	$0.594607\pi$
$752$	0	0
$753$	0.369751	0.0134745
$754$	0	0
$755$	0	0
$756$	0	0
$757$	37.5640	1.36529	0.682643	−	0.730752i	$-0.260831\pi$
$757$	37.5640	1.36529	0.682643	+	0.730752i	$0.260831\pi$
$758$	0	0
$759$	−4.29227	−0.155800
$760$	0	0
$761$	−9.35886	−0.339258	−0.169629	−	0.985508i	$-0.554257\pi$
$761$	−9.35886	−0.339258	−0.169629	+	0.985508i	$0.554257\pi$
$762$	0	0
$763$	−40.7237	−1.47430
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.87514	0.284355
$768$	0	0
$769$	−28.5093	−1.02807	−0.514036	−	0.857769i	$-0.671850\pi$
$769$	−28.5093	−1.02807	−0.514036	+	0.857769i	$0.671850\pi$
$770$	0	0
$771$	−11.9354	−0.429844
$772$	0	0
$773$	10.7469	0.386537	0.193269	−	0.981146i	$-0.438091\pi$
$773$	10.7469	0.386537	0.193269	+	0.981146i	$0.438091\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−52.9253	−1.89868
$778$	0	0
$779$	65.1707	2.33498
$780$	0	0
$781$	−0.828899	−0.0296603
$782$	0	0
$783$	−30.0101	−1.07247
$784$	0	0
$785$	0	0
$786$	0	0
$787$	14.7372	0.525323	0.262662	−	0.964888i	$-0.415400\pi$
$787$	14.7372	0.525323	0.262662	+	0.964888i	$0.415400\pi$
$788$	0	0
$789$	−11.6792	−0.415789
$790$	0	0
$791$	−19.3037	−0.686360
$792$	0	0
$793$	−5.36833	−0.190635
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.60628	0.198585	0.0992923	−	0.995058i	$-0.468342\pi$
$797$	5.60628	0.198585	0.0992923	+	0.995058i	$0.468342\pi$
$798$	0	0
$799$	−17.6181	−0.623283
$800$	0	0
$801$	2.40939	0.0851315
$802$	0	0
$803$	15.5053	0.547172
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2.13371	−0.0751100
$808$	0	0
$809$	23.2492	0.817399	0.408699	−	0.912669i	$-0.365982\pi$
$809$	23.2492	0.817399	0.408699	+	0.912669i	$0.365982\pi$
$810$	0	0
$811$	−34.6018	−1.21503	−0.607517	−	0.794306i	$-0.707835\pi$
$811$	−34.6018	−1.21503	−0.607517	+	0.794306i	$0.707835\pi$
$812$	0	0
$813$	38.4942	1.35005
$814$	0	0
$815$	0	0
$816$	0	0
$817$	33.9241	1.18685
$818$	0	0
$819$	3.07585	0.107479
$820$	0	0
$821$	−27.3695	−0.955203	−0.477601	−	0.878577i	$-0.658494\pi$
$821$	−27.3695	−0.955203	−0.477601	+	0.878577i	$0.658494\pi$
$822$	0	0
$823$	−43.1615	−1.50451	−0.752257	−	0.658870i	$-0.771035\pi$
$823$	−43.1615	−1.50451	−0.752257	+	0.658870i	$0.771035\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	31.7145	1.10282	0.551411	−	0.834234i	$-0.314090\pi$
$827$	31.7145	1.10282	0.551411	+	0.834234i	$0.314090\pi$
$828$	0	0
$829$	−24.6810	−0.857205	−0.428603	−	0.903493i	$-0.640994\pi$
$829$	−24.6810	−0.857205	−0.428603	+	0.903493i	$0.640994\pi$
$830$	0	0
$831$	−18.8071	−0.652412
$832$	0	0
$833$	−38.1183	−1.32072
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.98577	0.137768
$838$	0	0
$839$	38.7060	1.33628	0.668139	−	0.744036i	$-0.267091\pi$
$839$	38.7060	1.33628	0.668139	+	0.744036i	$0.267091\pi$
$840$	0	0
$841$	−0.558176	−0.0192475
$842$	0	0
$843$	8.68895	0.299263
$844$	0	0
$845$	0	0
$846$	0	0
$847$	46.0642	1.58279
$848$	0	0
$849$	5.10382	0.175163
$850$	0	0
$851$	−15.8485	−0.543279
$852$	0	0
$853$	15.3871	0.526845	0.263422	−	0.964681i	$-0.415149\pi$
$853$	15.3871	0.526845	0.263422	+	0.964681i	$0.415149\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	39.2764	1.34166	0.670829	−	0.741612i	$-0.265939\pi$
$857$	39.2764	1.34166	0.670829	+	0.741612i	$0.265939\pi$
$858$	0	0
$859$	−43.4683	−1.48312	−0.741559	−	0.670887i	$-0.765913\pi$
$859$	−43.4683	−1.48312	−0.741559	+	0.670887i	$0.765913\pi$
$860$	0	0
$861$	74.8760	2.55177
$862$	0	0
$863$	18.3423	0.624380	0.312190	−	0.950020i	$-0.398937\pi$
$863$	18.3423	0.624380	0.312190	+	0.950020i	$0.398937\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	18.2711	0.620520
$868$	0	0
$869$	−2.89995	−0.0983741
$870$	0	0
$871$	−6.00725	−0.203548
$872$	0	0
$873$	12.5178	0.423662
$874$	0	0
$875$	0	0
$876$	0	0
$877$	21.3721	0.721685	0.360842	−	0.932627i	$-0.382489\pi$
$877$	21.3721	0.721685	0.360842	+	0.932627i	$0.382489\pi$
$878$	0	0
$879$	0.504402	0.0170131
$880$	0	0
$881$	−9.96736	−0.335809	−0.167904	−	0.985803i	$-0.553700\pi$
$881$	−9.96736	−0.335809	−0.167904	+	0.985803i	$0.553700\pi$
$882$	0	0
$883$	−5.75705	−0.193740	−0.0968701	−	0.995297i	$-0.530883\pi$
$883$	−5.75705	−0.193740	−0.0968701	+	0.995297i	$0.530883\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−10.6446	−0.357410	−0.178705	−	0.983903i	$-0.557191\pi$
$887$	−10.6446	−0.357410	−0.178705	+	0.983903i	$0.557191\pi$
$888$	0	0
$889$	70.7934	2.37434
$890$	0	0
$891$	−7.95898	−0.266636
$892$	0	0
$893$	51.7905	1.73310
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2.72076	−0.0908436
$898$	0	0
$899$	−3.77748	−0.125986
$900$	0	0
$901$	28.9563	0.964674
$902$	0	0
$903$	38.9762	1.29705
$904$	0	0
$905$	0	0
$906$	0	0
$907$	23.4462	0.778518	0.389259	−	0.921128i	$-0.372731\pi$
$907$	23.4462	0.778518	0.389259	+	0.921128i	$0.372731\pi$
$908$	0	0
$909$	7.89674	0.261918
$910$	0	0
$911$	34.1224	1.13053	0.565263	−	0.824910i	$-0.308774\pi$
$911$	34.1224	1.13053	0.565263	+	0.824910i	$0.308774\pi$
$912$	0	0
$913$	14.8293	0.490777
$914$	0	0
$915$	0	0
$916$	0	0
$917$	50.4820	1.66706
$918$	0	0
$919$	−48.2385	−1.59124	−0.795620	−	0.605796i	$-0.792855\pi$
$919$	−48.2385	−1.59124	−0.795620	+	0.605796i	$0.792855\pi$
$920$	0	0
$921$	35.1912	1.15959
$922$	0	0
$923$	−0.525418	−0.0172943
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.29663	0.0425870
$928$	0	0
$929$	12.4406	0.408163	0.204081	−	0.978954i	$-0.434579\pi$
$929$	12.4406	0.408163	0.204081	+	0.978954i	$0.434579\pi$
$930$	0	0
$931$	112.054	3.67241
$932$	0	0
$933$	−10.8804	−0.356209
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−59.6304	−1.94804	−0.974021	−	0.226459i	$-0.927285\pi$
$937$	−59.6304	−1.94804	−0.974021	+	0.226459i	$0.927285\pi$
$938$	0	0
$939$	17.7369	0.578822
$940$	0	0
$941$	6.38832	0.208253	0.104127	−	0.994564i	$-0.466795\pi$
$941$	6.38832	0.208253	0.104127	+	0.994564i	$0.466795\pi$
$942$	0	0
$943$	22.4216	0.730149
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.27950	−0.0740739	−0.0370370	−	0.999314i	$-0.511792\pi$
$947$	−2.27950	−0.0740739	−0.0370370	+	0.999314i	$0.511792\pi$
$948$	0	0
$949$	9.82845	0.319045
$950$	0	0
$951$	7.59779	0.246375
$952$	0	0
$953$	−4.07015	−0.131845	−0.0659225	−	0.997825i	$-0.520999\pi$
$953$	−4.07015	−0.131845	−0.0659225	+	0.997825i	$0.520999\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	10.3357	0.334107
$958$	0	0
$959$	36.2211	1.16964
$960$	0	0
$961$	−30.4983	−0.983816
$962$	0	0
$963$	−2.15289	−0.0693760
$964$	0	0
$965$	0	0
$966$	0	0
$967$	35.2861	1.13472	0.567362	−	0.823469i	$-0.307964\pi$
$967$	35.2861	1.13472	0.567362	+	0.823469i	$0.307964\pi$
$968$	0	0
$969$	21.1045	0.677974
$970$	0	0
$971$	−17.1938	−0.551774	−0.275887	−	0.961190i	$-0.588972\pi$
$971$	−17.1938	−0.551774	−0.275887	+	0.961190i	$0.588972\pi$
$972$	0	0
$973$	−33.0756	−1.06035
$974$	0	0
$975$	0	0
$976$	0	0
$977$	45.3216	1.44997	0.724983	−	0.688767i	$-0.241848\pi$
$977$	45.3216	1.44997	0.724983	+	0.688767i	$0.241848\pi$
$978$	0	0
$979$	−4.11084	−0.131383
$980$	0	0
$981$	−6.25439	−0.199687
$982$	0	0
$983$	3.95653	0.126194	0.0630968	−	0.998007i	$-0.479902\pi$
$983$	3.95653	0.126194	0.0630968	+	0.998007i	$0.479902\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	59.5032	1.89401
$988$	0	0
$989$	11.6714	0.371129
$990$	0	0
$991$	25.0599	0.796053	0.398027	−	0.917374i	$-0.369695\pi$
$991$	25.0599	0.796053	0.398027	+	0.917374i	$0.369695\pi$
$992$	0	0
$993$	−1.75116	−0.0555713
$994$	0	0
$995$	0	0
$996$	0	0
$997$	51.4073	1.62809	0.814043	−	0.580805i	$-0.197262\pi$
$997$	51.4073	1.62809	0.814043	+	0.580805i	$0.197262\pi$
$998$	0	0
$999$	−40.2671	−1.27400

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5000.2.a.l.1.3		8
4.3	odd	2		10000.2.a.bk.1.6		8
5.4	even	2		5000.2.a.m.1.6		8
20.19	odd	2		10000.2.a.bh.1.3		8
25.3	odd	20		1000.2.q.d.49.6		32
25.4	even	10		1000.2.m.c.201.3		16
25.6	even	5		200.2.m.c.161.2	yes	16
25.8	odd	20		1000.2.q.d.449.3		32
25.17	odd	20		1000.2.q.d.449.6		32
25.19	even	10		1000.2.m.c.801.3		16
25.21	even	5		200.2.m.c.41.2	✓	16
25.22	odd	20		1000.2.q.d.49.3		32
100.31	odd	10		400.2.u.g.161.3		16
100.71	odd	10		400.2.u.g.241.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.2.m.c.41.2	✓	16	25.21	even	5
200.2.m.c.161.2	yes	16	25.6	even	5
400.2.u.g.161.3		16	100.31	odd	10
400.2.u.g.241.3		16	100.71	odd	10
1000.2.m.c.201.3		16	25.4	even	10
1000.2.m.c.801.3		16	25.19	even	10
1000.2.q.d.49.3		32	25.22	odd	20
1000.2.q.d.49.6		32	25.3	odd	20
1000.2.q.d.449.3		32	25.8	odd	20
1000.2.q.d.449.6		32	25.17	odd	20
5000.2.a.l.1.3		8	1.1	even	1	trivial
5000.2.a.m.1.6		8	5.4	even	2
10000.2.a.bh.1.3		8	20.19	odd	2
10000.2.a.bk.1.6		8	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5000 = 2^{3} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5000.a (trivial)

\(f(q)\)	\(=\)	\(q-1.49708 q^{3} -4.94031 q^{7} -0.758738 q^{9} +O(q^{10})\)	\(q-1.49708 q^{3} -4.94031 q^{7} -0.758738 q^{9} +1.29454 q^{11} +0.820577 q^{13} -2.18987 q^{17} +6.43739 q^{19} +7.39606 q^{21} +2.21475 q^{23} +5.62715 q^{27} -5.33309 q^{29} +0.708310 q^{31} -1.93804 q^{33} -7.15587 q^{37} -1.22847 q^{39} +10.1238 q^{41} +5.26985 q^{43} +8.04526 q^{47} +17.4067 q^{49} +3.27842 q^{51} -13.2228 q^{53} -9.63732 q^{57} +9.59708 q^{59} -6.54215 q^{61} +3.74840 q^{63} -7.32076 q^{67} -3.31567 q^{69} -0.640303 q^{71} +11.9775 q^{73} -6.39544 q^{77} -2.24014 q^{79} -6.14810 q^{81} +11.4552 q^{83} +7.98408 q^{87} -3.17552 q^{89} -4.05390 q^{91} -1.06040 q^{93} -16.4982 q^{97} -0.982218 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{3} - 3 q^{7} + 12 q^{9}+O(q^{10}) \) 8 * q - 2 * q^3 - 3 * q^7 + 12 * q^9	\( 8 q - 2 q^{3} - 3 q^{7} + 12 q^{9} + 5 q^{11} - 7 q^{13} - 7 q^{17} + 5 q^{19} - 4 q^{21} - 7 q^{23} - 26 q^{27} - 25 q^{29} + 3 q^{31} - 27 q^{33} - 15 q^{37} - 8 q^{39} - 26 q^{41} - 21 q^{43} + 2 q^{47} + 9 q^{49} + 50 q^{51} - 12 q^{53} - 32 q^{57} + 36 q^{59} - 33 q^{61} - 9 q^{63} - 7 q^{67} - 11 q^{69} + 12 q^{71} - 2 q^{73} + 14 q^{77} - 16 q^{79} + 12 q^{81} - q^{83} - 9 q^{87} - 30 q^{89} - 11 q^{91} + 20 q^{93} - 32 q^{97} + 61 q^{99}+O(q^{100}) \) 8 * q - 2 * q^3 - 3 * q^7 + 12 * q^9 + 5 * q^11 - 7 * q^13 - 7 * q^17 + 5 * q^19 - 4 * q^21 - 7 * q^23 - 26 * q^27 - 25 * q^29 + 3 * q^31 - 27 * q^33 - 15 * q^37 - 8 * q^39 - 26 * q^41 - 21 * q^43 + 2 * q^47 + 9 * q^49 + 50 * q^51 - 12 * q^53 - 32 * q^57 + 36 * q^59 - 33 * q^61 - 9 * q^63 - 7 * q^67 - 11 * q^69 + 12 * q^71 - 2 * q^73 + 14 * q^77 - 16 * q^79 + 12 * q^81 - q^83 - 9 * q^87 - 30 * q^89 - 11 * q^91 + 20 * q^93 - 32 * q^97 + 61 * q^99

Introduction

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