LMFDB - Embedded newform 1560.2.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1560,2,Mod(1,1560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1560.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1560.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:	$12.4566627153$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.940.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 7x - 4 $ x^3 - 7*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.89511$ of defining polynomial
Character	$\chi$	$=$	1560.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +2.40857 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +2.40857 q^{7} +1.00000 q^{9} +5.38164 q^{11} -1.00000 q^{13} +1.00000 q^{15} -7.17185 q^{17} +7.79021 q^{19} +2.40857 q^{21} -2.40857 q^{23} +1.00000 q^{25} +1.00000 q^{27} +4.97307 q^{29} -6.76328 q^{31} +5.38164 q^{33} +2.40857 q^{35} +7.38164 q^{37} -1.00000 q^{39} +3.38164 q^{41} -11.5804 q^{43} +1.00000 q^{45} -10.7633 q^{47} -1.19878 q^{49} -7.17185 q^{51} +1.43550 q^{53} +5.38164 q^{55} +7.79021 q^{57} +5.94614 q^{59} -1.43550 q^{61} +2.40857 q^{63} -1.00000 q^{65} +7.58043 q^{67} -2.40857 q^{69} -2.61836 q^{71} +7.02693 q^{73} +1.00000 q^{75} +12.9621 q^{77} +2.61836 q^{79} +1.00000 q^{81} +1.94614 q^{83} -7.17185 q^{85} +4.97307 q^{87} +0.618358 q^{89} -2.40857 q^{91} -6.76328 q^{93} +7.79021 q^{95} +9.22571 q^{97} +5.38164 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{5} + q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 3 * q^5 + q^7 + 3 * q^9	$ 3 q + 3 q^{3} + 3 q^{5} + q^{7} + 3 q^{9} + 5 q^{11} - 3 q^{13} + 3 q^{15} + 7 q^{17} + 6 q^{19} + q^{21} - q^{23} + 3 q^{25} + 3 q^{27} + 10 q^{29} + 2 q^{31} + 5 q^{33} + q^{35} + 11 q^{37} - 3 q^{39} - q^{41} + 3 q^{45} - 10 q^{47} + 20 q^{49} + 7 q^{51} + 3 q^{53} + 5 q^{55} + 6 q^{57} + 8 q^{59} - 3 q^{61} + q^{63} - 3 q^{65} - 12 q^{67} - q^{69} - 19 q^{71} + 26 q^{73} + 3 q^{75} - 7 q^{77} + 19 q^{79} + 3 q^{81} - 4 q^{83} + 7 q^{85} + 10 q^{87} + 13 q^{89} - q^{91} + 2 q^{93} + 6 q^{95} + 9 q^{97} + 5 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^5 + q^7 + 3 * q^9 + 5 * q^11 - 3 * q^13 + 3 * q^15 + 7 * q^17 + 6 * q^19 + q^21 - q^23 + 3 * q^25 + 3 * q^27 + 10 * q^29 + 2 * q^31 + 5 * q^33 + q^35 + 11 * q^37 - 3 * q^39 - q^41 + 3 * q^45 - 10 * q^47 + 20 * q^49 + 7 * q^51 + 3 * q^53 + 5 * q^55 + 6 * q^57 + 8 * q^59 - 3 * q^61 + q^63 - 3 * q^65 - 12 * q^67 - q^69 - 19 * q^71 + 26 * q^73 + 3 * q^75 - 7 * q^77 + 19 * q^79 + 3 * q^81 - 4 * q^83 + 7 * q^85 + 10 * q^87 + 13 * q^89 - q^91 + 2 * q^93 + 6 * q^95 + 9 * q^97 + 5 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.40857	0.910354	0.455177	−	0.890401i	$-0.349576\pi$
$7$	2.40857	0.910354	0.455177	+	0.890401i	$0.349576\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.38164	1.62263	0.811313	−	0.584612i	$-0.198753\pi$
$11$	5.38164	1.62263	0.811313	+	0.584612i	$0.198753\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−7.17185	−1.73943	−0.869715	−	0.493554i	$-0.835698\pi$
$17$	−7.17185	−1.73943	−0.869715	+	0.493554i	$0.835698\pi$
$18$	0	0
$19$	7.79021	1.78720	0.893599	−	0.448867i	$-0.148172\pi$
$19$	7.79021	1.78720	0.893599	+	0.448867i	$0.148172\pi$
$20$	0	0
$21$	2.40857	0.525593
$22$	0	0
$23$	−2.40857	−0.502222	−0.251111	−	0.967958i	$-0.580796\pi$
$23$	−2.40857	−0.502222	−0.251111	+	0.967958i	$0.580796\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	4.97307	0.923476	0.461738	−	0.887016i	$-0.347226\pi$
$29$	4.97307	0.923476	0.461738	+	0.887016i	$0.347226\pi$
$30$	0	0
$31$	−6.76328	−1.21472	−0.607361	−	0.794426i	$-0.707772\pi$
$31$	−6.76328	−1.21472	−0.607361	+	0.794426i	$0.707772\pi$
$32$	0	0
$33$	5.38164	0.936824
$34$	0	0
$35$	2.40857	0.407123
$36$	0	0
$37$	7.38164	1.21353	0.606767	−	0.794880i	$-0.292466\pi$
$37$	7.38164	1.21353	0.606767	+	0.794880i	$0.292466\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	3.38164	0.528124	0.264062	−	0.964506i	$-0.414938\pi$
$41$	3.38164	0.528124	0.264062	+	0.964506i	$0.414938\pi$
$42$	0	0
$43$	−11.5804	−1.76600	−0.882999	−	0.469374i	$-0.844479\pi$
$43$	−11.5804	−1.76600	−0.882999	+	0.469374i	$0.844479\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−10.7633	−1.56999	−0.784993	−	0.619504i	$-0.787334\pi$
$47$	−10.7633	−1.56999	−0.784993	+	0.619504i	$0.787334\pi$
$48$	0	0
$49$	−1.19878	−0.171255
$50$	0	0
$51$	−7.17185	−1.00426
$52$	0	0
$53$	1.43550	0.197181	0.0985906	−	0.995128i	$-0.468567\pi$
$53$	1.43550	0.197181	0.0985906	+	0.995128i	$0.468567\pi$
$54$	0	0
$55$	5.38164	0.725660
$56$	0	0
$57$	7.79021	1.03184
$58$	0	0
$59$	5.94614	0.774122	0.387061	−	0.922054i	$-0.373490\pi$
$59$	5.94614	0.774122	0.387061	+	0.922054i	$0.373490\pi$
$60$	0	0
$61$	−1.43550	−0.183797	−0.0918985	−	0.995768i	$-0.529294\pi$
$61$	−1.43550	−0.183797	−0.0918985	+	0.995768i	$0.529294\pi$
$62$	0	0
$63$	2.40857	0.303451
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	7.58043	0.926096	0.463048	−	0.886333i	$-0.346756\pi$
$67$	7.58043	0.926096	0.463048	+	0.886333i	$0.346756\pi$
$68$	0	0
$69$	−2.40857	−0.289958
$70$	0	0
$71$	−2.61836	−0.310742	−0.155371	−	0.987856i	$-0.549657\pi$
$71$	−2.61836	−0.310742	−0.155371	+	0.987856i	$0.549657\pi$
$72$	0	0
$73$	7.02693	0.822440	0.411220	−	0.911536i	$-0.365103\pi$
$73$	7.02693	0.822440	0.411220	+	0.911536i	$0.365103\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	12.9621	1.47716
$78$	0	0
$79$	2.61836	0.294588	0.147294	−	0.989093i	$-0.452944\pi$
$79$	2.61836	0.294588	0.147294	+	0.989093i	$0.452944\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.94614	0.213617	0.106808	−	0.994280i	$-0.465937\pi$
$83$	1.94614	0.213617	0.106808	+	0.994280i	$0.465937\pi$
$84$	0	0
$85$	−7.17185	−0.777897
$86$	0	0
$87$	4.97307	0.533169
$88$	0	0
$89$	0.618358	0.0655458	0.0327729	−	0.999463i	$-0.489566\pi$
$89$	0.618358	0.0655458	0.0327729	+	0.999463i	$0.489566\pi$
$90$	0	0
$91$	−2.40857	−0.252487
$92$	0	0
$93$	−6.76328	−0.701320
$94$	0	0
$95$	7.79021	0.799259
$96$	0	0
$97$	9.22571	0.936729	0.468365	−	0.883535i	$-0.344843\pi$
$97$	9.22571	0.936729	0.468365	+	0.883535i	$0.344843\pi$
$98$	0	0
$99$	5.38164	0.540875
$100$	0	0
$101$	−19.7364	−1.96384	−0.981920	−	0.189295i	$-0.939380\pi$
$101$	−19.7364	−1.96384	−0.981920	+	0.189295i	$0.939380\pi$
$102$	0	0
$103$	3.58043	0.352790	0.176395	−	0.984319i	$-0.443556\pi$
$103$	3.58043	0.352790	0.176395	+	0.984319i	$0.443556\pi$
$104$	0	0
$105$	2.40857	0.235052
$106$	0	0
$107$	−16.9621	−1.63978	−0.819892	−	0.572518i	$-0.805967\pi$
$107$	−16.9621	−1.63978	−0.819892	+	0.572518i	$0.805967\pi$
$108$	0	0
$109$	13.7902	1.32086	0.660431	−	0.750886i	$-0.270373\pi$
$109$	13.7902	1.32086	0.660431	+	0.750886i	$0.270373\pi$
$110$	0	0
$111$	7.38164	0.700634
$112$	0	0
$113$	0.973070	0.0915388	0.0457694	−	0.998952i	$-0.485426\pi$
$113$	0.973070	0.0915388	0.0457694	+	0.998952i	$0.485426\pi$
$114$	0	0
$115$	−2.40857	−0.224600
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−17.2739	−1.58350
$120$	0	0
$121$	17.9621	1.63292
$122$	0	0
$123$	3.38164	0.304912
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	14.3437	1.27280	0.636399	−	0.771360i	$-0.280423\pi$
$127$	14.3437	1.27280	0.636399	+	0.771360i	$0.280423\pi$
$128$	0	0
$129$	−11.5804	−1.01960
$130$	0	0
$131$	1.84407	0.161117	0.0805587	−	0.996750i	$-0.474330\pi$
$131$	1.84407	0.161117	0.0805587	+	0.996750i	$0.474330\pi$
$132$	0	0
$133$	18.7633	1.62698
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	−4.14493	−0.351568	−0.175784	−	0.984429i	$-0.556246\pi$
$139$	−4.14493	−0.351568	−0.175784	+	0.984429i	$0.556246\pi$
$140$	0	0
$141$	−10.7633	−0.906432
$142$	0	0
$143$	−5.38164	−0.450035
$144$	0	0
$145$	4.97307	0.412991
$146$	0	0
$147$	−1.19878	−0.0988741
$148$	0	0
$149$	0.618358	0.0506579	0.0253289	−	0.999679i	$-0.491937\pi$
$149$	0.618358	0.0506579	0.0253289	+	0.999679i	$0.491937\pi$
$150$	0	0
$151$	−13.6343	−1.10954	−0.554771	−	0.832003i	$-0.687194\pi$
$151$	−13.6343	−1.10954	−0.554771	+	0.832003i	$0.687194\pi$
$152$	0	0
$153$	−7.17185	−0.579810
$154$	0	0
$155$	−6.76328	−0.543240
$156$	0	0
$157$	−11.5266	−0.919920	−0.459960	−	0.887940i	$-0.652136\pi$
$157$	−11.5266	−0.919920	−0.459960	+	0.887940i	$0.652136\pi$
$158$	0	0
$159$	1.43550	0.113843
$160$	0	0
$161$	−5.80122	−0.457200
$162$	0	0
$163$	−10.1988	−0.798830	−0.399415	−	0.916770i	$-0.630787\pi$
$163$	−10.1988	−0.798830	−0.399415	+	0.916770i	$0.630787\pi$
$164$	0	0
$165$	5.38164	0.418960
$166$	0	0
$167$	−0.419574	−0.0324676	−0.0162338	−	0.999868i	$-0.505168\pi$
$167$	−0.419574	−0.0324676	−0.0162338	+	0.999868i	$0.505168\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	7.79021	0.595732
$172$	0	0
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	2.40857	0.182071
$176$	0	0
$177$	5.94614	0.446939
$178$	0	0
$179$	−9.84407	−0.735780	−0.367890	−	0.929869i	$-0.619920\pi$
$179$	−9.84407	−0.735780	−0.367890	+	0.929869i	$0.619920\pi$
$180$	0	0
$181$	−9.01593	−0.670149	−0.335074	−	0.942192i	$-0.608761\pi$
$181$	−9.01593	−0.670149	−0.335074	+	0.942192i	$0.608761\pi$
$182$	0	0
$183$	−1.43550	−0.106115
$184$	0	0
$185$	7.38164	0.542709
$186$	0	0
$187$	−38.5964	−2.82244
$188$	0	0
$189$	2.40857	0.175198
$190$	0	0
$191$	−9.63429	−0.697112	−0.348556	−	0.937288i	$-0.613328\pi$
$191$	−9.63429	−0.697112	−0.348556	+	0.937288i	$0.613328\pi$
$192$	0	0
$193$	25.9351	1.86685	0.933426	−	0.358770i	$-0.116804\pi$
$193$	25.9351	1.86685	0.933426	+	0.358770i	$0.116804\pi$
$194$	0	0
$195$	−1.00000	−0.0716115
$196$	0	0
$197$	−0.871002	−0.0620563	−0.0310282	−	0.999519i	$-0.509878\pi$
$197$	−0.871002	−0.0620563	−0.0310282	+	0.999519i	$0.509878\pi$
$198$	0	0
$199$	23.5804	1.67157	0.835786	−	0.549055i	$-0.185012\pi$
$199$	23.5804	1.67157	0.835786	+	0.549055i	$0.185012\pi$
$200$	0	0
$201$	7.58043	0.534682
$202$	0	0
$203$	11.9780	0.840690
$204$	0	0
$205$	3.38164	0.236184
$206$	0	0
$207$	−2.40857	−0.167407
$208$	0	0
$209$	41.9241	2.89995
$210$	0	0
$211$	19.1609	1.31909	0.659544	−	0.751666i	$-0.270750\pi$
$211$	19.1609	1.31909	0.659544	+	0.751666i	$0.270750\pi$
$212$	0	0
$213$	−2.61836	−0.179407
$214$	0	0
$215$	−11.5804	−0.789779
$216$	0	0
$217$	−16.2899	−1.10583
$218$	0	0
$219$	7.02693	0.474836
$220$	0	0
$221$	7.17185	0.482431
$222$	0	0
$223$	−18.9731	−1.27053	−0.635265	−	0.772294i	$-0.719109\pi$
$223$	−18.9731	−1.27053	−0.635265	+	0.772294i	$0.719109\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−6.76328	−0.448895	−0.224447	−	0.974486i	$-0.572058\pi$
$227$	−6.76328	−0.448895	−0.224447	+	0.974486i	$0.572058\pi$
$228$	0	0
$229$	3.02693	0.200025	0.100013	−	0.994986i	$-0.468112\pi$
$229$	3.02693	0.200025	0.100013	+	0.994986i	$0.468112\pi$
$230$	0	0
$231$	12.9621	0.852841
$232$	0	0
$233$	−17.9351	−1.17497	−0.587485	−	0.809235i	$-0.699882\pi$
$233$	−17.9351	−1.17497	−0.587485	+	0.809235i	$0.699882\pi$
$234$	0	0
$235$	−10.7633	−0.702119
$236$	0	0
$237$	2.61836	0.170081
$238$	0	0
$239$	3.32778	0.215256	0.107628	−	0.994191i	$-0.465674\pi$
$239$	3.32778	0.215256	0.107628	+	0.994191i	$0.465674\pi$
$240$	0	0
$241$	−21.1609	−1.36309	−0.681545	−	0.731776i	$-0.738692\pi$
$241$	−21.1609	−1.36309	−0.681545	+	0.731776i	$0.738692\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.19878	−0.0765875
$246$	0	0
$247$	−7.79021	−0.495679
$248$	0	0
$249$	1.94614	0.123332
$250$	0	0
$251$	28.1878	1.77920	0.889599	−	0.456743i	$-0.150984\pi$
$251$	28.1878	1.77920	0.889599	+	0.456743i	$0.150984\pi$
$252$	0	0
$253$	−12.9621	−0.814918
$254$	0	0
$255$	−7.17185	−0.449119
$256$	0	0
$257$	−30.1878	−1.88306	−0.941531	−	0.336926i	$-0.890613\pi$
$257$	−30.1878	−1.88306	−0.941531	+	0.336926i	$0.890613\pi$
$258$	0	0
$259$	17.7792	1.10475
$260$	0	0
$261$	4.97307	0.307825
$262$	0	0
$263$	−13.0269	−0.803275	−0.401637	−	0.915799i	$-0.631559\pi$
$263$	−13.0269	−0.803275	−0.401637	+	0.915799i	$0.631559\pi$
$264$	0	0
$265$	1.43550	0.0881821
$266$	0	0
$267$	0.618358	0.0378429
$268$	0	0
$269$	−6.20979	−0.378617	−0.189309	−	0.981918i	$-0.560625\pi$
$269$	−6.20979	−0.378617	−0.189309	+	0.981918i	$0.560625\pi$
$270$	0	0
$271$	−29.9241	−1.81776	−0.908881	−	0.417056i	$-0.863062\pi$
$271$	−29.9241	−1.81776	−0.908881	+	0.417056i	$0.863062\pi$
$272$	0	0
$273$	−2.40857	−0.145773
$274$	0	0
$275$	5.38164	0.324525
$276$	0	0
$277$	5.89228	0.354033	0.177016	−	0.984208i	$-0.443355\pi$
$277$	5.89228	0.354033	0.177016	+	0.984208i	$0.443355\pi$
$278$	0	0
$279$	−6.76328	−0.404907
$280$	0	0
$281$	−15.9461	−0.951267	−0.475634	−	0.879644i	$-0.657781\pi$
$281$	−15.9461	−0.951267	−0.475634	+	0.879644i	$0.657781\pi$
$282$	0	0
$283$	−17.9461	−1.06679	−0.533394	−	0.845867i	$-0.679084\pi$
$283$	−17.9461	−1.06679	−0.533394	+	0.845867i	$0.679084\pi$
$284$	0	0
$285$	7.79021	0.461452
$286$	0	0
$287$	8.14493	0.480780
$288$	0	0
$289$	34.4355	2.02562
$290$	0	0
$291$	9.22571	0.540821
$292$	0	0
$293$	0.763283	0.0445915	0.0222957	−	0.999751i	$-0.492902\pi$
$293$	0.763283	0.0445915	0.0222957	+	0.999751i	$0.492902\pi$
$294$	0	0
$295$	5.94614	0.346198
$296$	0	0
$297$	5.38164	0.312275
$298$	0	0
$299$	2.40857	0.139291
$300$	0	0
$301$	−27.8923	−1.60768
$302$	0	0
$303$	−19.7364	−1.13382
$304$	0	0
$305$	−1.43550	−0.0821965
$306$	0	0
$307$	−16.1449	−0.921440	−0.460720	−	0.887546i	$-0.652409\pi$
$307$	−16.1449	−0.921440	−0.460720	+	0.887546i	$0.652409\pi$
$308$	0	0
$309$	3.58043	0.203683
$310$	0	0
$311$	−29.5266	−1.67430	−0.837149	−	0.546975i	$-0.815779\pi$
$311$	−29.5266	−1.67430	−0.837149	+	0.546975i	$0.815779\pi$
$312$	0	0
$313$	33.5804	1.89808	0.949039	−	0.315159i	$-0.102058\pi$
$313$	33.5804	1.89808	0.949039	+	0.315159i	$0.102058\pi$
$314$	0	0
$315$	2.40857	0.135708
$316$	0	0
$317$	−7.23672	−0.406454	−0.203227	−	0.979132i	$-0.565143\pi$
$317$	−7.23672	−0.406454	−0.203227	+	0.979132i	$0.565143\pi$
$318$	0	0
$319$	26.7633	1.49846
$320$	0	0
$321$	−16.9621	−0.946730
$322$	0	0
$323$	−55.8703	−3.10871
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	13.7902	0.762601
$328$	0	0
$329$	−25.9241	−1.42924
$330$	0	0
$331$	2.55350	0.140353	0.0701764	−	0.997535i	$-0.477644\pi$
$331$	2.55350	0.140353	0.0701764	+	0.997535i	$0.477644\pi$
$332$	0	0
$333$	7.38164	0.404511
$334$	0	0
$335$	7.58043	0.414163
$336$	0	0
$337$	4.34371	0.236617	0.118308	−	0.992977i	$-0.462253\pi$
$337$	4.34371	0.236617	0.118308	+	0.992977i	$0.462253\pi$
$338$	0	0
$339$	0.973070	0.0528499
$340$	0	0
$341$	−36.3976	−1.97104
$342$	0	0
$343$	−19.7474	−1.06626
$344$	0	0
$345$	−2.40857	−0.129673
$346$	0	0
$347$	29.7792	1.59863	0.799316	−	0.600911i	$-0.205195\pi$
$347$	29.7792	1.59863	0.799316	+	0.600911i	$0.205195\pi$
$348$	0	0
$349$	15.4245	0.825654	0.412827	−	0.910809i	$-0.364541\pi$
$349$	15.4245	0.825654	0.412827	+	0.910809i	$0.364541\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	−2.61836	−0.138968
$356$	0	0
$357$	−17.2739	−0.914233
$358$	0	0
$359$	−10.0539	−0.530622	−0.265311	−	0.964163i	$-0.585475\pi$
$359$	−10.0539	−0.530622	−0.265311	+	0.964163i	$0.585475\pi$
$360$	0	0
$361$	41.6874	2.19407
$362$	0	0
$363$	17.9621	0.942764
$364$	0	0
$365$	7.02693	0.367806
$366$	0	0
$367$	−11.5804	−0.604493	−0.302247	−	0.953230i	$-0.597737\pi$
$367$	−11.5804	−0.604493	−0.302247	+	0.953230i	$0.597737\pi$
$368$	0	0
$369$	3.38164	0.176041
$370$	0	0
$371$	3.45751	0.179505
$372$	0	0
$373$	6.39757	0.331254	0.165627	−	0.986189i	$-0.447035\pi$
$373$	6.39757	0.331254	0.165627	+	0.986189i	$0.447035\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−4.97307	−0.256126
$378$	0	0
$379$	−17.4245	−0.895036	−0.447518	−	0.894275i	$-0.647692\pi$
$379$	−17.4245	−0.895036	−0.447518	+	0.894275i	$0.647692\pi$
$380$	0	0
$381$	14.3437	0.734851
$382$	0	0
$383$	−20.8171	−1.06371	−0.531853	−	0.846837i	$-0.678504\pi$
$383$	−20.8171	−1.06371	−0.531853	+	0.846837i	$0.678504\pi$
$384$	0	0
$385$	12.9621	0.660608
$386$	0	0
$387$	−11.5804	−0.588666
$388$	0	0
$389$	−19.7364	−1.00067	−0.500336	−	0.865831i	$-0.666790\pi$
$389$	−19.7364	−1.00067	−0.500336	+	0.865831i	$0.666790\pi$
$390$	0	0
$391$	17.2739	0.873580
$392$	0	0
$393$	1.84407	0.0930211
$394$	0	0
$395$	2.61836	0.131744
$396$	0	0
$397$	14.5425	0.729867	0.364933	−	0.931034i	$-0.381092\pi$
$397$	14.5425	0.729867	0.364933	+	0.931034i	$0.381092\pi$
$398$	0	0
$399$	18.7633	0.939339
$400$	0	0
$401$	−27.2147	−1.35904	−0.679519	−	0.733658i	$-0.737811\pi$
$401$	−27.2147	−1.35904	−0.679519	+	0.733658i	$0.737811\pi$
$402$	0	0
$403$	6.76328	0.336903
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	39.7254	1.96911
$408$	0	0
$409$	2.41957	0.119640	0.0598201	−	0.998209i	$-0.480947\pi$
$409$	2.41957	0.119640	0.0598201	+	0.998209i	$0.480947\pi$
$410$	0	0
$411$	2.00000	0.0986527
$412$	0	0
$413$	14.3217	0.704725
$414$	0	0
$415$	1.94614	0.0955323
$416$	0	0
$417$	−4.14493	−0.202978
$418$	0	0
$419$	−13.3168	−0.650567	−0.325284	−	0.945616i	$-0.605460\pi$
$419$	−13.3168	−0.650567	−0.325284	+	0.945616i	$0.605460\pi$
$420$	0	0
$421$	−29.6825	−1.44664	−0.723318	−	0.690515i	$-0.757384\pi$
$421$	−29.6825	−1.44664	−0.723318	+	0.690515i	$0.757384\pi$
$422$	0	0
$423$	−10.7633	−0.523329
$424$	0	0
$425$	−7.17185	−0.347886
$426$	0	0
$427$	−3.45751	−0.167320
$428$	0	0
$429$	−5.38164	−0.259828
$430$	0	0
$431$	11.8923	0.572831	0.286416	−	0.958105i	$-0.407536\pi$
$431$	11.8923	0.572831	0.286416	+	0.958105i	$0.407536\pi$
$432$	0	0
$433$	5.18286	0.249072	0.124536	−	0.992215i	$-0.460256\pi$
$433$	5.18286	0.249072	0.124536	+	0.992215i	$0.460256\pi$
$434$	0	0
$435$	4.97307	0.238440
$436$	0	0
$437$	−18.7633	−0.897570
$438$	0	0
$439$	−25.3596	−1.21035	−0.605175	−	0.796093i	$-0.706897\pi$
$439$	−25.3596	−1.21035	−0.605175	+	0.796093i	$0.706897\pi$
$440$	0	0
$441$	−1.19878	−0.0570850
$442$	0	0
$443$	−3.01593	−0.143291	−0.0716455	−	0.997430i	$-0.522825\pi$
$443$	−3.01593	−0.143291	−0.0716455	+	0.997430i	$0.522825\pi$
$444$	0	0
$445$	0.618358	0.0293130
$446$	0	0
$447$	0.618358	0.0292473
$448$	0	0
$449$	38.6502	1.82402	0.912008	−	0.410172i	$-0.134531\pi$
$449$	38.6502	1.82402	0.912008	+	0.410172i	$0.134531\pi$
$450$	0	0
$451$	18.1988	0.856947
$452$	0	0
$453$	−13.6343	−0.640595
$454$	0	0
$455$	−2.40857	−0.112916
$456$	0	0
$457$	22.7523	1.06431	0.532153	−	0.846648i	$-0.321383\pi$
$457$	22.7523	1.06431	0.532153	+	0.846648i	$0.321383\pi$
$458$	0	0
$459$	−7.17185	−0.334754
$460$	0	0
$461$	−29.8331	−1.38946	−0.694732	−	0.719268i	$-0.744477\pi$
$461$	−29.8331	−1.38946	−0.694732	+	0.719268i	$0.744477\pi$
$462$	0	0
$463$	−22.8061	−1.05989	−0.529946	−	0.848032i	$-0.677788\pi$
$463$	−22.8061	−1.05989	−0.529946	+	0.848032i	$0.677788\pi$
$464$	0	0
$465$	−6.76328	−0.313640
$466$	0	0
$467$	41.2519	1.90891	0.954456	−	0.298353i	$-0.0964373\pi$
$467$	41.2519	1.90891	0.954456	+	0.298353i	$0.0964373\pi$
$468$	0	0
$469$	18.2580	0.843076
$470$	0	0
$471$	−11.5266	−0.531116
$472$	0	0
$473$	−62.3217	−2.86556
$474$	0	0
$475$	7.79021	0.357439
$476$	0	0
$477$	1.43550	0.0657271
$478$	0	0
$479$	30.0911	1.37490	0.687448	−	0.726234i	$-0.258731\pi$
$479$	30.0911	1.37490	0.687448	+	0.726234i	$0.258731\pi$
$480$	0	0
$481$	−7.38164	−0.336574
$482$	0	0
$483$	−5.80122	−0.263964
$484$	0	0
$485$	9.22571	0.418918
$486$	0	0
$487$	−37.8813	−1.71656	−0.858282	−	0.513178i	$-0.828468\pi$
$487$	−37.8813	−1.71656	−0.858282	+	0.513178i	$0.828468\pi$
$488$	0	0
$489$	−10.1988	−0.461205
$490$	0	0
$491$	−24.0801	−1.08672	−0.543359	−	0.839500i	$-0.682848\pi$
$491$	−24.0801	−1.08672	−0.543359	+	0.839500i	$0.682848\pi$
$492$	0	0
$493$	−35.6661	−1.60632
$494$	0	0
$495$	5.38164	0.241887
$496$	0	0
$497$	−6.30650	−0.282885
$498$	0	0
$499$	15.3706	0.688084	0.344042	−	0.938954i	$-0.388204\pi$
$499$	15.3706	0.688084	0.344042	+	0.938954i	$0.388204\pi$
$500$	0	0
$501$	−0.419574	−0.0187452
$502$	0	0
$503$	−26.5535	−1.18396	−0.591981	−	0.805952i	$-0.701654\pi$
$503$	−26.5535	−1.18396	−0.591981	+	0.805952i	$0.701654\pi$
$504$	0	0
$505$	−19.7364	−0.878256
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−7.80122	−0.345783	−0.172891	−	0.984941i	$-0.555311\pi$
$509$	−7.80122	−0.345783	−0.172891	+	0.984941i	$0.555311\pi$
$510$	0	0
$511$	16.9249	0.748712
$512$	0	0
$513$	7.79021	0.343946
$514$	0	0
$515$	3.58043	0.157772
$516$	0	0
$517$	−57.9241	−2.54750
$518$	0	0
$519$	−6.00000	−0.263371
$520$	0	0
$521$	35.2147	1.54278	0.771392	−	0.636360i	$-0.219561\pi$
$521$	35.2147	1.54278	0.771392	+	0.636360i	$0.219561\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	0	0
$525$	2.40857	0.105119
$526$	0	0
$527$	48.5053	2.11292
$528$	0	0
$529$	−17.1988	−0.747773
$530$	0	0
$531$	5.94614	0.258041
$532$	0	0
$533$	−3.38164	−0.146475
$534$	0	0
$535$	−16.9621	−0.733334
$536$	0	0
$537$	−9.84407	−0.424803
$538$	0	0
$539$	−6.45143	−0.277883
$540$	0	0
$541$	14.4996	0.623388	0.311694	−	0.950183i	$-0.399104\pi$
$541$	14.4996	0.623388	0.311694	+	0.950183i	$0.399104\pi$
$542$	0	0
$543$	−9.01593	−0.386910
$544$	0	0
$545$	13.7902	0.590708
$546$	0	0
$547$	−13.6343	−0.582960	−0.291480	−	0.956577i	$-0.594148\pi$
$547$	−13.6343	−0.582960	−0.291480	+	0.956577i	$0.594148\pi$
$548$	0	0
$549$	−1.43550	−0.0612657
$550$	0	0
$551$	38.7413	1.65043
$552$	0	0
$553$	6.30650	0.268180
$554$	0	0
$555$	7.38164	0.313333
$556$	0	0
$557$	−5.89228	−0.249664	−0.124832	−	0.992178i	$-0.539839\pi$
$557$	−5.89228	−0.249664	−0.124832	+	0.992178i	$0.539839\pi$
$558$	0	0
$559$	11.5804	0.489800
$560$	0	0
$561$	−38.5964	−1.62954
$562$	0	0
$563$	14.6184	0.616090	0.308045	−	0.951372i	$-0.400325\pi$
$563$	14.6184	0.616090	0.308045	+	0.951372i	$0.400325\pi$
$564$	0	0
$565$	0.973070	0.0409374
$566$	0	0
$567$	2.40857	0.101150
$568$	0	0
$569$	45.9780	1.92750	0.963749	−	0.266811i	$-0.0859699\pi$
$569$	45.9780	1.92750	0.963749	+	0.266811i	$0.0859699\pi$
$570$	0	0
$571$	8.25264	0.345362	0.172681	−	0.984978i	$-0.444757\pi$
$571$	8.25264	0.345362	0.172681	+	0.984978i	$0.444757\pi$
$572$	0	0
$573$	−9.63429	−0.402478
$574$	0	0
$575$	−2.40857	−0.100444
$576$	0	0
$577$	−14.3547	−0.597594	−0.298797	−	0.954317i	$-0.596585\pi$
$577$	−14.3547	−0.597594	−0.298797	+	0.954317i	$0.596585\pi$
$578$	0	0
$579$	25.9351	1.07783
$580$	0	0
$581$	4.68742	0.194467
$582$	0	0
$583$	7.72535	0.319951
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	0	0
$587$	17.2367	0.711435	0.355718	−	0.934593i	$-0.384236\pi$
$587$	17.2367	0.711435	0.355718	+	0.934593i	$0.384236\pi$
$588$	0	0
$589$	−52.6874	−2.17095
$590$	0	0
$591$	−0.871002	−0.0358282
$592$	0	0
$593$	33.8703	1.39089	0.695443	−	0.718581i	$-0.255208\pi$
$593$	33.8703	1.39089	0.695443	+	0.718581i	$0.255208\pi$
$594$	0	0
$595$	−17.2739	−0.708162
$596$	0	0
$597$	23.5804	0.965083
$598$	0	0
$599$	−35.0531	−1.43223	−0.716116	−	0.697981i	$-0.754082\pi$
$599$	−35.0531	−1.43223	−0.716116	+	0.697981i	$0.754082\pi$
$600$	0	0
$601$	3.90893	0.159449	0.0797244	−	0.996817i	$-0.474596\pi$
$601$	3.90893	0.159449	0.0797244	+	0.996817i	$0.474596\pi$
$602$	0	0
$603$	7.58043	0.308699
$604$	0	0
$605$	17.9621	0.730262
$606$	0	0
$607$	6.05386	0.245719	0.122859	−	0.992424i	$-0.460794\pi$
$607$	6.05386	0.245719	0.122859	+	0.992424i	$0.460794\pi$
$608$	0	0
$609$	11.9780	0.485373
$610$	0	0
$611$	10.7633	0.435436
$612$	0	0
$613$	−28.1768	−1.13805	−0.569025	−	0.822320i	$-0.692679\pi$
$613$	−28.1768	−1.13805	−0.569025	+	0.822320i	$0.692679\pi$
$614$	0	0
$615$	3.38164	0.136361
$616$	0	0
$617$	46.6874	1.87956	0.939782	−	0.341773i	$-0.111028\pi$
$617$	46.6874	1.87956	0.939782	+	0.341773i	$0.111028\pi$
$618$	0	0
$619$	35.2629	1.41734	0.708668	−	0.705542i	$-0.249296\pi$
$619$	35.2629	1.41734	0.708668	+	0.705542i	$0.249296\pi$
$620$	0	0
$621$	−2.40857	−0.0966526
$622$	0	0
$623$	1.48936	0.0596699
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	41.9241	1.67429
$628$	0	0
$629$	−52.9401	−2.11086
$630$	0	0
$631$	35.5804	1.41643	0.708217	−	0.705995i	$-0.249500\pi$
$631$	35.5804	1.41643	0.708217	+	0.705995i	$0.249500\pi$
$632$	0	0
$633$	19.1609	0.761575
$634$	0	0
$635$	14.3437	0.569213
$636$	0	0
$637$	1.19878	0.0474976
$638$	0	0
$639$	−2.61836	−0.103581
$640$	0	0
$641$	6.39757	0.252689	0.126344	−	0.991986i	$-0.459676\pi$
$641$	6.39757	0.252689	0.126344	+	0.991986i	$0.459676\pi$
$642$	0	0
$643$	−26.6184	−1.04973	−0.524863	−	0.851187i	$-0.675883\pi$
$643$	−26.6184	−1.04973	−0.524863	+	0.851187i	$0.675883\pi$
$644$	0	0
$645$	−11.5804	−0.455979
$646$	0	0
$647$	−3.11800	−0.122581	−0.0612905	−	0.998120i	$-0.519522\pi$
$647$	−3.11800	−0.122581	−0.0612905	+	0.998120i	$0.519522\pi$
$648$	0	0
$649$	32.0000	1.25611
$650$	0	0
$651$	−16.2899	−0.638450
$652$	0	0
$653$	−17.4727	−0.683760	−0.341880	−	0.939744i	$-0.611064\pi$
$653$	−17.4727	−0.683760	−0.341880	+	0.939744i	$0.611064\pi$
$654$	0	0
$655$	1.84407	0.0720539
$656$	0	0
$657$	7.02693	0.274147
$658$	0	0
$659$	−7.28493	−0.283780	−0.141890	−	0.989882i	$-0.545318\pi$
$659$	−7.28493	−0.283780	−0.141890	+	0.989882i	$0.545318\pi$
$660$	0	0
$661$	18.8972	0.735016	0.367508	−	0.930020i	$-0.380211\pi$
$661$	18.8972	0.735016	0.367508	+	0.930020i	$0.380211\pi$
$662$	0	0
$663$	7.17185	0.278532
$664$	0	0
$665$	18.7633	0.727609
$666$	0	0
$667$	−11.9780	−0.463790
$668$	0	0
$669$	−18.9731	−0.733541
$670$	0	0
$671$	−7.72535	−0.298234
$672$	0	0
$673$	−30.7951	−1.18707	−0.593533	−	0.804810i	$-0.702267\pi$
$673$	−30.7951	−1.18707	−0.593533	+	0.804810i	$0.702267\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−34.8323	−1.33872	−0.669358	−	0.742940i	$-0.733431\pi$
$677$	−34.8323	−1.33872	−0.669358	+	0.742940i	$0.733431\pi$
$678$	0	0
$679$	22.2208	0.852756
$680$	0	0
$681$	−6.76328	−0.259170
$682$	0	0
$683$	49.1070	1.87903	0.939513	−	0.342512i	$-0.111278\pi$
$683$	49.1070	1.87903	0.939513	+	0.342512i	$0.111278\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	3.02693	0.115485
$688$	0	0
$689$	−1.43550	−0.0546882
$690$	0	0
$691$	−43.2629	−1.64580	−0.822900	−	0.568187i	$-0.807645\pi$
$691$	−43.2629	−1.64580	−0.822900	+	0.568187i	$0.807645\pi$
$692$	0	0
$693$	12.9621	0.492388
$694$	0	0
$695$	−4.14493	−0.157226
$696$	0	0
$697$	−24.2526	−0.918634
$698$	0	0
$699$	−17.9351	−0.678369
$700$	0	0
$701$	1.79021	0.0676154	0.0338077	−	0.999428i	$-0.489237\pi$
$701$	1.79021	0.0676154	0.0338077	+	0.999428i	$0.489237\pi$
$702$	0	0
$703$	57.5046	2.16883
$704$	0	0
$705$	−10.7633	−0.405369
$706$	0	0
$707$	−47.5364	−1.78779
$708$	0	0
$709$	19.3168	0.725457	0.362728	−	0.931895i	$-0.381845\pi$
$709$	19.3168	0.725457	0.362728	+	0.931895i	$0.381845\pi$
$710$	0	0
$711$	2.61836	0.0981961
$712$	0	0
$713$	16.2899	0.610060
$714$	0	0
$715$	−5.38164	−0.201262
$716$	0	0
$717$	3.32778	0.124278
$718$	0	0
$719$	13.9461	0.520103	0.260052	−	0.965595i	$-0.416260\pi$
$719$	13.9461	0.520103	0.260052	+	0.965595i	$0.416260\pi$
$720$	0	0
$721$	8.62371	0.321164
$722$	0	0
$723$	−21.1609	−0.786981
$724$	0	0
$725$	4.97307	0.184695
$726$	0	0
$727$	−37.9241	−1.40653	−0.703264	−	0.710929i	$-0.748275\pi$
$727$	−37.9241	−1.40653	−0.703264	+	0.710929i	$0.748275\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	83.0531	3.07183
$732$	0	0
$733$	28.0691	1.03675	0.518377	−	0.855152i	$-0.326536\pi$
$733$	28.0691	1.03675	0.518377	+	0.855152i	$0.326536\pi$
$734$	0	0
$735$	−1.19878	−0.0442178
$736$	0	0
$737$	40.7951	1.50271
$738$	0	0
$739$	−22.9511	−0.844269	−0.422134	−	0.906533i	$-0.638719\pi$
$739$	−22.9511	−0.844269	−0.422134	+	0.906533i	$0.638719\pi$
$740$	0	0
$741$	−7.79021	−0.286181
$742$	0	0
$743$	35.9780	1.31990	0.659952	−	0.751307i	$-0.270576\pi$
$743$	35.9780	1.31990	0.659952	+	0.751307i	$0.270576\pi$
$744$	0	0
$745$	0.618358	0.0226549
$746$	0	0
$747$	1.94614	0.0712056
$748$	0	0
$749$	−40.8543	−1.49279
$750$	0	0
$751$	30.5964	1.11648	0.558238	−	0.829681i	$-0.311477\pi$
$751$	30.5964	1.11648	0.558238	+	0.829681i	$0.311477\pi$
$752$	0	0
$753$	28.1878	1.02722
$754$	0	0
$755$	−13.6343	−0.496202
$756$	0	0
$757$	53.5584	1.94661	0.973307	−	0.229507i	$-0.0737115\pi$
$757$	53.5584	1.94661	0.973307	+	0.229507i	$0.0737115\pi$
$758$	0	0
$759$	−12.9621	−0.470493
$760$	0	0
$761$	10.3119	0.373804	0.186902	−	0.982379i	$-0.440155\pi$
$761$	10.3119	0.373804	0.186902	+	0.982379i	$0.440155\pi$
$762$	0	0
$763$	33.2147	1.20245
$764$	0	0
$765$	−7.17185	−0.259299
$766$	0	0
$767$	−5.94614	−0.214703
$768$	0	0
$769$	0.871002	0.0314091	0.0157046	−	0.999877i	$-0.495001\pi$
$769$	0.871002	0.0314091	0.0157046	+	0.999877i	$0.495001\pi$
$770$	0	0
$771$	−30.1878	−1.08719
$772$	0	0
$773$	26.3976	0.949455	0.474727	−	0.880133i	$-0.342547\pi$
$773$	26.3976	0.949455	0.474727	+	0.880133i	$0.342547\pi$
$774$	0	0
$775$	−6.76328	−0.242944
$776$	0	0
$777$	17.7792	0.637826
$778$	0	0
$779$	26.3437	0.943861
$780$	0	0
$781$	−14.0911	−0.504218
$782$	0	0
$783$	4.97307	0.177723
$784$	0	0
$785$	−11.5266	−0.411401
$786$	0	0
$787$	21.5266	0.767339	0.383670	−	0.923470i	$-0.374660\pi$
$787$	21.5266	0.767339	0.383670	+	0.923470i	$0.374660\pi$
$788$	0	0
$789$	−13.0269	−0.463771
$790$	0	0
$791$	2.34371	0.0833327
$792$	0	0
$793$	1.43550	0.0509761
$794$	0	0
$795$	1.43550	0.0509120
$796$	0	0
$797$	14.6722	0.519717	0.259858	−	0.965647i	$-0.416324\pi$
$797$	14.6722	0.519717	0.259858	+	0.965647i	$0.416324\pi$
$798$	0	0
$799$	77.1927	2.73088
$800$	0	0
$801$	0.618358	0.0218486
$802$	0	0
$803$	37.8164	1.33451
$804$	0	0
$805$	−5.80122	−0.204466
$806$	0	0
$807$	−6.20979	−0.218595
$808$	0	0
$809$	24.3657	0.856653	0.428326	−	0.903624i	$-0.359103\pi$
$809$	24.3657	0.856653	0.428326	+	0.903624i	$0.359103\pi$
$810$	0	0
$811$	40.0801	1.40740	0.703701	−	0.710497i	$-0.251530\pi$
$811$	40.0801	1.40740	0.703701	+	0.710497i	$0.251530\pi$
$812$	0	0
$813$	−29.9241	−1.04949
$814$	0	0
$815$	−10.1988	−0.357248
$816$	0	0
$817$	−90.2140	−3.15619
$818$	0	0
$819$	−2.40857	−0.0841623
$820$	0	0
$821$	9.74736	0.340185	0.170093	−	0.985428i	$-0.445593\pi$
$821$	9.74736	0.340185	0.170093	+	0.985428i	$0.445593\pi$
$822$	0	0
$823$	1.94614	0.0678382	0.0339191	−	0.999425i	$-0.489201\pi$
$823$	1.94614	0.0678382	0.0339191	+	0.999425i	$0.489201\pi$
$824$	0	0
$825$	5.38164	0.187365
$826$	0	0
$827$	18.4514	0.641619	0.320810	−	0.947144i	$-0.396045\pi$
$827$	18.4514	0.641619	0.320810	+	0.947144i	$0.396045\pi$
$828$	0	0
$829$	5.16085	0.179244	0.0896219	−	0.995976i	$-0.471434\pi$
$829$	5.16085	0.179244	0.0896219	+	0.995976i	$0.471434\pi$
$830$	0	0
$831$	5.89228	0.204401
$832$	0	0
$833$	8.59751	0.297886
$834$	0	0
$835$	−0.419574	−0.0145200
$836$	0	0
$837$	−6.76328	−0.233773
$838$	0	0
$839$	8.27465	0.285673	0.142836	−	0.989746i	$-0.454378\pi$
$839$	8.27465	0.285673	0.142836	+	0.989746i	$0.454378\pi$
$840$	0	0
$841$	−4.26857	−0.147192
$842$	0	0
$843$	−15.9461	−0.549214
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	43.2629	1.48653
$848$	0	0
$849$	−17.9461	−0.615910
$850$	0	0
$851$	−17.7792	−0.609463
$852$	0	0
$853$	7.09179	0.242818	0.121409	−	0.992603i	$-0.461259\pi$
$853$	7.09179	0.242818	0.121409	+	0.992603i	$0.461259\pi$
$854$	0	0
$855$	7.79021	0.266420
$856$	0	0
$857$	3.17185	0.108348	0.0541742	−	0.998531i	$-0.482747\pi$
$857$	3.17185	0.108348	0.0541742	+	0.998531i	$0.482747\pi$
$858$	0	0
$859$	−27.7254	−0.945977	−0.472988	−	0.881069i	$-0.656825\pi$
$859$	−27.7254	−0.945977	−0.472988	+	0.881069i	$0.656825\pi$
$860$	0	0
$861$	8.14493	0.277578
$862$	0	0
$863$	3.18286	0.108346	0.0541729	−	0.998532i	$-0.482748\pi$
$863$	3.18286	0.108346	0.0541729	+	0.998532i	$0.482748\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	0	0
$867$	34.4355	1.16949
$868$	0	0
$869$	14.0911	0.478007
$870$	0	0
$871$	−7.58043	−0.256853
$872$	0	0
$873$	9.22571	0.312243
$874$	0	0
$875$	2.40857	0.0814246
$876$	0	0
$877$	17.7846	0.600542	0.300271	−	0.953854i	$-0.402923\pi$
$877$	17.7846	0.600542	0.300271	+	0.953854i	$0.402923\pi$
$878$	0	0
$879$	0.763283	0.0257449
$880$	0	0
$881$	−37.3650	−1.25886	−0.629429	−	0.777058i	$-0.716711\pi$
$881$	−37.3650	−1.25886	−0.629429	+	0.777058i	$0.716711\pi$
$882$	0	0
$883$	−32.6874	−1.10002	−0.550010	−	0.835158i	$-0.685376\pi$
$883$	−32.6874	−1.10002	−0.550010	+	0.835158i	$0.685376\pi$
$884$	0	0
$885$	5.94614	0.199877
$886$	0	0
$887$	23.5156	0.789575	0.394788	−	0.918772i	$-0.370818\pi$
$887$	23.5156	0.789575	0.394788	+	0.918772i	$0.370818\pi$
$888$	0	0
$889$	34.5478	1.15870
$890$	0	0
$891$	5.38164	0.180292
$892$	0	0
$893$	−83.8483	−2.80588
$894$	0	0
$895$	−9.84407	−0.329051
$896$	0	0
$897$	2.40857	0.0804199
$898$	0	0
$899$	−33.6343	−1.12177
$900$	0	0
$901$	−10.2952	−0.342983
$902$	0	0
$903$	−27.8923	−0.928197
$904$	0	0
$905$	−9.01593	−0.299700
$906$	0	0
$907$	−9.65629	−0.320632	−0.160316	−	0.987066i	$-0.551251\pi$
$907$	−9.65629	−0.320632	−0.160316	+	0.987066i	$0.551251\pi$
$908$	0	0
$909$	−19.7364	−0.654614
$910$	0	0
$911$	13.9461	0.462056	0.231028	−	0.972947i	$-0.425791\pi$
$911$	13.9461	0.462056	0.231028	+	0.972947i	$0.425791\pi$
$912$	0	0
$913$	10.4734	0.346620
$914$	0	0
$915$	−1.43550	−0.0474562
$916$	0	0
$917$	4.44158	0.146674
$918$	0	0
$919$	55.7254	1.83821	0.919105	−	0.394013i	$-0.128914\pi$
$919$	55.7254	1.83821	0.919105	+	0.394013i	$0.128914\pi$
$920$	0	0
$921$	−16.1449	−0.531993
$922$	0	0
$923$	2.61836	0.0861843
$924$	0	0
$925$	7.38164	0.242707
$926$	0	0
$927$	3.58043	0.117597
$928$	0	0
$929$	47.7792	1.56759	0.783793	−	0.621023i	$-0.213283\pi$
$929$	47.7792	1.56759	0.783793	+	0.621023i	$0.213283\pi$
$930$	0	0
$931$	−9.33879	−0.306066
$932$	0	0
$933$	−29.5266	−0.966656
$934$	0	0
$935$	−38.5964	−1.26224
$936$	0	0
$937$	17.7846	0.580996	0.290498	−	0.956876i	$-0.406179\pi$
$937$	17.7846	0.580996	0.290498	+	0.956876i	$0.406179\pi$
$938$	0	0
$939$	33.5804	1.09586
$940$	0	0
$941$	7.77921	0.253595	0.126798	−	0.991929i	$-0.459530\pi$
$941$	7.77921	0.253595	0.126798	+	0.991929i	$0.459530\pi$
$942$	0	0
$943$	−8.14493	−0.265235
$944$	0	0
$945$	2.40857	0.0783508
$946$	0	0
$947$	−38.6336	−1.25542	−0.627711	−	0.778446i	$-0.716008\pi$
$947$	−38.6336	−1.25542	−0.627711	+	0.778446i	$0.716008\pi$
$948$	0	0
$949$	−7.02693	−0.228104
$950$	0	0
$951$	−7.23672	−0.234667
$952$	0	0
$953$	31.9890	1.03623	0.518113	−	0.855312i	$-0.326635\pi$
$953$	31.9890	1.03623	0.518113	+	0.855312i	$0.326635\pi$
$954$	0	0
$955$	−9.63429	−0.311758
$956$	0	0
$957$	26.7633	0.865134
$958$	0	0
$959$	4.81714	0.155554
$960$	0	0
$961$	14.7420	0.475548
$962$	0	0
$963$	−16.9621	−0.546595
$964$	0	0
$965$	25.9351	0.834882
$966$	0	0
$967$	6.86535	0.220775	0.110387	−	0.993889i	$-0.464791\pi$
$967$	6.86535	0.220775	0.110387	+	0.993889i	$0.464791\pi$
$968$	0	0
$969$	−55.8703	−1.79481
$970$	0	0
$971$	−49.2948	−1.58194	−0.790972	−	0.611852i	$-0.790425\pi$
$971$	−49.2948	−1.58194	−0.790972	+	0.611852i	$0.790425\pi$
$972$	0	0
$973$	−9.98335	−0.320051
$974$	0	0
$975$	−1.00000	−0.0320256
$976$	0	0
$977$	−12.2360	−0.391464	−0.195732	−	0.980657i	$-0.562708\pi$
$977$	−12.2360	−0.391464	−0.195732	+	0.980657i	$0.562708\pi$
$978$	0	0
$979$	3.32778	0.106356
$980$	0	0
$981$	13.7902	0.440288
$982$	0	0
$983$	−50.8490	−1.62183	−0.810916	−	0.585163i	$-0.801030\pi$
$983$	−50.8490	−1.62183	−0.810916	+	0.585163i	$0.801030\pi$
$984$	0	0
$985$	−0.871002	−0.0277524
$986$	0	0
$987$	−25.9241	−0.825174
$988$	0	0
$989$	27.8923	0.886923
$990$	0	0
$991$	19.7474	0.627295	0.313648	−	0.949539i	$-0.398449\pi$
$991$	19.7474	0.627295	0.313648	+	0.949539i	$0.398449\pi$
$992$	0	0
$993$	2.55350	0.0810328
$994$	0	0
$995$	23.5804	0.747550
$996$	0	0
$997$	27.9241	0.884366	0.442183	−	0.896925i	$-0.354204\pi$
$997$	27.9241	0.884366	0.442183	+	0.896925i	$0.354204\pi$
$998$	0	0
$999$	7.38164	0.233545

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1560.2.a.q.1.2	✓	3
3.2	odd	2		4680.2.a.bh.1.2		3
4.3	odd	2		3120.2.a.bi.1.2		3
5.4	even	2		7800.2.a.bi.1.2		3
12.11	even	2		9360.2.a.cy.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.a.q.1.2	✓	3	1.1	even	1	trivial
3120.2.a.bi.1.2		3	4.3	odd	2
4680.2.a.bh.1.2		3	3.2	odd	2
7800.2.a.bi.1.2		3	5.4	even	2
9360.2.a.cy.1.2		3	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:	\( N \)	\(=\)	\( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1560.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +2.40857 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +2.40857 q^{7} +1.00000 q^{9} +5.38164 q^{11} -1.00000 q^{13} +1.00000 q^{15} -7.17185 q^{17} +7.79021 q^{19} +2.40857 q^{21} -2.40857 q^{23} +1.00000 q^{25} +1.00000 q^{27} +4.97307 q^{29} -6.76328 q^{31} +5.38164 q^{33} +2.40857 q^{35} +7.38164 q^{37} -1.00000 q^{39} +3.38164 q^{41} -11.5804 q^{43} +1.00000 q^{45} -10.7633 q^{47} -1.19878 q^{49} -7.17185 q^{51} +1.43550 q^{53} +5.38164 q^{55} +7.79021 q^{57} +5.94614 q^{59} -1.43550 q^{61} +2.40857 q^{63} -1.00000 q^{65} +7.58043 q^{67} -2.40857 q^{69} -2.61836 q^{71} +7.02693 q^{73} +1.00000 q^{75} +12.9621 q^{77} +2.61836 q^{79} +1.00000 q^{81} +1.94614 q^{83} -7.17185 q^{85} +4.97307 q^{87} +0.618358 q^{89} -2.40857 q^{91} -6.76328 q^{93} +7.79021 q^{95} +9.22571 q^{97} +5.38164 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{5} + q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^5 + q^7 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{5} + q^{7} + 3 q^{9} + 5 q^{11} - 3 q^{13} + 3 q^{15} + 7 q^{17} + 6 q^{19} + q^{21} - q^{23} + 3 q^{25} + 3 q^{27} + 10 q^{29} + 2 q^{31} + 5 q^{33} + q^{35} + 11 q^{37} - 3 q^{39} - q^{41} + 3 q^{45} - 10 q^{47} + 20 q^{49} + 7 q^{51} + 3 q^{53} + 5 q^{55} + 6 q^{57} + 8 q^{59} - 3 q^{61} + q^{63} - 3 q^{65} - 12 q^{67} - q^{69} - 19 q^{71} + 26 q^{73} + 3 q^{75} - 7 q^{77} + 19 q^{79} + 3 q^{81} - 4 q^{83} + 7 q^{85} + 10 q^{87} + 13 q^{89} - q^{91} + 2 q^{93} + 6 q^{95} + 9 q^{97} + 5 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^5 + q^7 + 3 * q^9 + 5 * q^11 - 3 * q^13 + 3 * q^15 + 7 * q^17 + 6 * q^19 + q^21 - q^23 + 3 * q^25 + 3 * q^27 + 10 * q^29 + 2 * q^31 + 5 * q^33 + q^35 + 11 * q^37 - 3 * q^39 - q^41 + 3 * q^45 - 10 * q^47 + 20 * q^49 + 7 * q^51 + 3 * q^53 + 5 * q^55 + 6 * q^57 + 8 * q^59 - 3 * q^61 + q^63 - 3 * q^65 - 12 * q^67 - q^69 - 19 * q^71 + 26 * q^73 + 3 * q^75 - 7 * q^77 + 19 * q^79 + 3 * q^81 - 4 * q^83 + 7 * q^85 + 10 * q^87 + 13 * q^89 - q^91 + 2 * q^93 + 6 * q^95 + 9 * q^97 + 5 * q^99

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