LMFDB - Embedded newform 7800.2.a.bh.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7800,2,Mod(1,7800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, 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("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7800.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:	$62.2833135766$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.3732.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 13x + 19 $ x^3 - x^2 - 13*x + 19
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.56943$ of defining polynomial
Character	$\chi$	$=$	7800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +4.96747 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +4.96747 q^{7} +1.00000 q^{9} +0.430572 q^{11} +1.00000 q^{13} -3.00000 q^{17} +3.39804 q^{19} -4.96747 q^{21} +5.39804 q^{23} -1.00000 q^{27} -4.13886 q^{29} +0.430572 q^{31} -0.430572 q^{33} +8.53690 q^{37} -1.00000 q^{39} -6.53690 q^{41} +6.53690 q^{43} -12.1063 q^{47} +17.6758 q^{49} +3.00000 q^{51} +4.39804 q^{53} -3.39804 q^{57} +12.1063 q^{59} +6.13886 q^{61} +4.96747 q^{63} -9.56943 q^{67} -5.39804 q^{69} +9.67575 q^{71} -9.67575 q^{73} +2.13886 q^{77} +10.5369 q^{79} +1.00000 q^{81} +7.82861 q^{83} +4.13886 q^{87} +2.86114 q^{89} +4.96747 q^{91} -0.430572 q^{93} +5.93494 q^{97} +0.430572 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - q^7 + 3 * q^9	$ 3 q - 3 q^{3} - q^{7} + 3 q^{9} + 5 q^{11} + 3 q^{13} - 9 q^{17} - 2 q^{19} + q^{21} + 4 q^{23} - 3 q^{27} - 5 q^{29} + 5 q^{31} - 5 q^{33} + 6 q^{37} - 3 q^{39} - 13 q^{47} + 26 q^{49} + 9 q^{51} + q^{53} + 2 q^{57} + 13 q^{59} + 11 q^{61} - q^{63} - 25 q^{67} - 4 q^{69} + 2 q^{71} - 2 q^{73} - q^{77} + 12 q^{79} + 3 q^{81} + 15 q^{83} + 5 q^{87} + 16 q^{89} - q^{91} - 5 q^{93} - 14 q^{97} + 5 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - q^7 + 3 * q^9 + 5 * q^11 + 3 * q^13 - 9 * q^17 - 2 * q^19 + q^21 + 4 * q^23 - 3 * q^27 - 5 * q^29 + 5 * q^31 - 5 * q^33 + 6 * q^37 - 3 * q^39 - 13 * q^47 + 26 * q^49 + 9 * q^51 + q^53 + 2 * q^57 + 13 * q^59 + 11 * q^61 - q^63 - 25 * q^67 - 4 * q^69 + 2 * q^71 - 2 * q^73 - q^77 + 12 * q^79 + 3 * q^81 + 15 * q^83 + 5 * q^87 + 16 * q^89 - q^91 - 5 * q^93 - 14 * 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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.96747	1.87753	0.938763	−	0.344562i	$-0.111973\pi$
$7$	4.96747	1.87753	0.938763	+	0.344562i	$0.111973\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.430572	0.129822	0.0649112	−	0.997891i	$-0.479324\pi$
$11$	0.430572	0.129822	0.0649112	+	0.997891i	$0.479324\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	3.39804	0.779564	0.389782	−	0.920907i	$-0.372550\pi$
$19$	3.39804	0.779564	0.389782	+	0.920907i	$0.372550\pi$
$20$	0	0
$21$	−4.96747	−1.08399
$22$	0	0
$23$	5.39804	1.12557	0.562785	−	0.826603i	$-0.309730\pi$
$23$	5.39804	1.12557	0.562785	+	0.826603i	$0.309730\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.13886	−0.768566	−0.384283	−	0.923215i	$-0.625551\pi$
$29$	−4.13886	−0.768566	−0.384283	+	0.923215i	$0.625551\pi$
$30$	0	0
$31$	0.430572	0.0773331	0.0386665	−	0.999252i	$-0.487689\pi$
$31$	0.430572	0.0773331	0.0386665	+	0.999252i	$0.487689\pi$
$32$	0	0
$33$	−0.430572	−0.0749530
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.53690	1.40346	0.701729	−	0.712444i	$-0.252412\pi$
$37$	8.53690	1.40346	0.701729	+	0.712444i	$0.252412\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−6.53690	−1.02089	−0.510446	−	0.859910i	$-0.670520\pi$
$41$	−6.53690	−1.02089	−0.510446	+	0.859910i	$0.670520\pi$
$42$	0	0
$43$	6.53690	0.996867	0.498434	−	0.866928i	$-0.333909\pi$
$43$	6.53690	0.996867	0.498434	+	0.866928i	$0.333909\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.1063	−1.76589	−0.882944	−	0.469477i	$-0.844442\pi$
$47$	−12.1063	−1.76589	−0.882944	+	0.469477i	$0.844442\pi$
$48$	0	0
$49$	17.6758	2.52511
$50$	0	0
$51$	3.00000	0.420084
$52$	0	0
$53$	4.39804	0.604118	0.302059	−	0.953289i	$-0.402326\pi$
$53$	4.39804	0.604118	0.302059	+	0.953289i	$0.402326\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.39804	−0.450082
$58$	0	0
$59$	12.1063	1.57611	0.788055	−	0.615605i	$-0.211088\pi$
$59$	12.1063	1.57611	0.788055	+	0.615605i	$0.211088\pi$
$60$	0	0
$61$	6.13886	0.786000	0.393000	−	0.919538i	$-0.371437\pi$
$61$	6.13886	0.786000	0.393000	+	0.919538i	$0.371437\pi$
$62$	0	0
$63$	4.96747	0.625842
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−9.56943	−1.16909	−0.584546	−	0.811361i	$-0.698727\pi$
$67$	−9.56943	−1.16909	−0.584546	+	0.811361i	$0.698727\pi$
$68$	0	0
$69$	−5.39804	−0.649848
$70$	0	0
$71$	9.67575	1.14830	0.574150	−	0.818750i	$-0.305333\pi$
$71$	9.67575	1.14830	0.574150	+	0.818750i	$0.305333\pi$
$72$	0	0
$73$	−9.67575	−1.13246	−0.566231	−	0.824247i	$-0.691599\pi$
$73$	−9.67575	−1.13246	−0.566231	+	0.824247i	$0.691599\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.13886	0.243745
$78$	0	0
$79$	10.5369	1.18549	0.592747	−	0.805389i	$-0.298043\pi$
$79$	10.5369	1.18549	0.592747	+	0.805389i	$0.298043\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.82861	0.859302	0.429651	−	0.902995i	$-0.358637\pi$
$83$	7.82861	0.859302	0.429651	+	0.902995i	$0.358637\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.13886	0.443732
$88$	0	0
$89$	2.86114	0.303281	0.151640	−	0.988436i	$-0.451544\pi$
$89$	2.86114	0.303281	0.151640	+	0.988436i	$0.451544\pi$
$90$	0	0
$91$	4.96747	0.520732
$92$	0	0
$93$	−0.430572	−0.0446483
$94$	0	0
$95$	0	0
$96$	0	0
$97$	5.93494	0.602602	0.301301	−	0.953529i	$-0.402579\pi$
$97$	5.93494	0.602602	0.301301	+	0.953529i	$0.402579\pi$
$98$	0	0
$99$	0.430572	0.0432742
$100$	0	0
$101$	−16.3330	−1.62519	−0.812596	−	0.582827i	$-0.801946\pi$
$101$	−16.3330	−1.62519	−0.812596	+	0.582827i	$0.801946\pi$
$102$	0	0
$103$	8.53690	0.841165	0.420583	−	0.907254i	$-0.361826\pi$
$103$	8.53690	0.841165	0.420583	+	0.907254i	$0.361826\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.6572	−1.12695	−0.563473	−	0.826134i	$-0.690535\pi$
$107$	−11.6572	−1.12695	−0.563473	+	0.826134i	$0.690535\pi$
$108$	0	0
$109$	−3.67575	−0.352073	−0.176037	−	0.984384i	$-0.556328\pi$
$109$	−3.67575	−0.352073	−0.176037	+	0.984384i	$0.556328\pi$
$110$	0	0
$111$	−8.53690	−0.810286
$112$	0	0
$113$	3.13886	0.295279	0.147639	−	0.989041i	$-0.452833\pi$
$113$	3.13886	0.295279	0.147639	+	0.989041i	$0.452833\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−14.9024	−1.36610
$120$	0	0
$121$	−10.8146	−0.983146
$122$	0	0
$123$	6.53690	0.589412
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−19.6758	−1.74594	−0.872970	−	0.487773i	$-0.837809\pi$
$127$	−19.6758	−1.74594	−0.872970	+	0.487773i	$0.837809\pi$
$128$	0	0
$129$	−6.53690	−0.575542
$130$	0	0
$131$	3.39804	0.296888	0.148444	−	0.988921i	$-0.452573\pi$
$131$	3.39804	0.296888	0.148444	+	0.988921i	$0.452573\pi$
$132$	0	0
$133$	16.8797	1.46365
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.65723	0.483330	0.241665	−	0.970360i	$-0.422307\pi$
$137$	5.65723	0.483330	0.241665	+	0.970360i	$0.422307\pi$
$138$	0	0
$139$	17.8699	1.51570	0.757852	−	0.652427i	$-0.226249\pi$
$139$	17.8699	1.51570	0.757852	+	0.652427i	$0.226249\pi$
$140$	0	0
$141$	12.1063	1.01954
$142$	0	0
$143$	0.430572	0.0360063
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−17.6758	−1.45787
$148$	0	0
$149$	16.2126	1.32819	0.664096	−	0.747647i	$-0.268817\pi$
$149$	16.2126	1.32819	0.664096	+	0.747647i	$0.268817\pi$
$150$	0	0
$151$	−5.89368	−0.479621	−0.239810	−	0.970820i	$-0.577085\pi$
$151$	−5.89368	−0.479621	−0.239810	+	0.970820i	$0.577085\pi$
$152$	0	0
$153$	−3.00000	−0.242536
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.39804	0.351002	0.175501	−	0.984479i	$-0.443845\pi$
$157$	4.39804	0.351002	0.175501	+	0.984479i	$0.443845\pi$
$158$	0	0
$159$	−4.39804	−0.348787
$160$	0	0
$161$	26.8146	2.11329
$162$	0	0
$163$	−16.4718	−1.29017	−0.645087	−	0.764109i	$-0.723179\pi$
$163$	−16.4718	−1.29017	−0.645087	+	0.764109i	$0.723179\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−12.5369	−0.970134	−0.485067	−	0.874477i	$-0.661205\pi$
$167$	−12.5369	−0.970134	−0.485067	+	0.874477i	$0.661205\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	3.39804	0.259855
$172$	0	0
$173$	7.53690	0.573020	0.286510	−	0.958077i	$-0.407505\pi$
$173$	7.53690	0.573020	0.286510	+	0.958077i	$0.407505\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−12.1063	−0.909967
$178$	0	0
$179$	−25.6107	−1.91423	−0.957116	−	0.289703i	$-0.906443\pi$
$179$	−25.6107	−1.91423	−0.957116	+	0.289703i	$0.906443\pi$
$180$	0	0
$181$	9.25919	0.688230	0.344115	−	0.938928i	$-0.388179\pi$
$181$	9.25919	0.688230	0.344115	+	0.938928i	$0.388179\pi$
$182$	0	0
$183$	−6.13886	−0.453797
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.29172	−0.0944597
$188$	0	0
$189$	−4.96747	−0.361330
$190$	0	0
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	0	0
$193$	−6.47184	−0.465853	−0.232926	−	0.972494i	$-0.574830\pi$
$193$	−6.47184	−0.465853	−0.232926	+	0.972494i	$0.574830\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.259187	0.0184663	0.00923314	−	0.999957i	$-0.497061\pi$
$197$	0.259187	0.0184663	0.00923314	+	0.999957i	$0.497061\pi$
$198$	0	0
$199$	−8.79608	−0.623538	−0.311769	−	0.950158i	$-0.600921\pi$
$199$	−8.79608	−0.623538	−0.311769	+	0.950158i	$0.600921\pi$
$200$	0	0
$201$	9.56943	0.674975
$202$	0	0
$203$	−20.5596	−1.44300
$204$	0	0
$205$	0	0
$206$	0	0
$207$	5.39804	0.375190
$208$	0	0
$209$	1.46310	0.101205
$210$	0	0
$211$	19.6758	1.35453	0.677267	−	0.735737i	$-0.263164\pi$
$211$	19.6758	1.35453	0.677267	+	0.735737i	$0.263164\pi$
$212$	0	0
$213$	−9.67575	−0.662972
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.13886	0.145195
$218$	0	0
$219$	9.67575	0.653827
$220$	0	0
$221$	−3.00000	−0.201802
$222$	0	0
$223$	−14.1941	−0.950509	−0.475254	−	0.879848i	$-0.657644\pi$
$223$	−14.1941	−0.950509	−0.475254	+	0.879848i	$0.657644\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−0.495634	−0.0328964	−0.0164482	−	0.999865i	$-0.505236\pi$
$227$	−0.495634	−0.0328964	−0.0164482	+	0.999865i	$0.505236\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	−2.13886	−0.140726
$232$	0	0
$233$	−11.1389	−0.729731	−0.364865	−	0.931060i	$-0.618885\pi$
$233$	−11.1389	−0.729731	−0.364865	+	0.931060i	$0.618885\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−10.5369	−0.684445
$238$	0	0
$239$	−17.9762	−1.16278	−0.581392	−	0.813624i	$-0.697492\pi$
$239$	−17.9762	−1.16278	−0.581392	+	0.813624i	$0.697492\pi$
$240$	0	0
$241$	30.4718	1.96286	0.981432	−	0.191812i	$-0.0614363\pi$
$241$	30.4718	1.96286	0.981432	+	0.191812i	$0.0614363\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.39804	0.216212
$248$	0	0
$249$	−7.82861	−0.496118
$250$	0	0
$251$	23.3330	1.47276	0.736382	−	0.676566i	$-0.236532\pi$
$251$	23.3330	1.47276	0.736382	+	0.676566i	$0.236532\pi$
$252$	0	0
$253$	2.32425	0.146124
$254$	0	0
$255$	0	0
$256$	0	0
$257$	22.6758	1.41447	0.707237	−	0.706976i	$-0.249941\pi$
$257$	22.6758	1.41447	0.707237	+	0.706976i	$0.249941\pi$
$258$	0	0
$259$	42.4068	2.63503
$260$	0	0
$261$	−4.13886	−0.256189
$262$	0	0
$263$	−4.53690	−0.279757	−0.139879	−	0.990169i	$-0.544671\pi$
$263$	−4.53690	−0.279757	−0.139879	+	0.990169i	$0.544671\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−2.86114	−0.175099
$268$	0	0
$269$	12.8699	0.784690	0.392345	−	0.919818i	$-0.371664\pi$
$269$	12.8699	0.784690	0.392345	+	0.919818i	$0.371664\pi$
$270$	0	0
$271$	−8.68976	−0.527865	−0.263933	−	0.964541i	$-0.585020\pi$
$271$	−8.68976	−0.527865	−0.263933	+	0.964541i	$0.585020\pi$
$272$	0	0
$273$	−4.96747	−0.300645
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−5.93494	−0.356596	−0.178298	−	0.983977i	$-0.557059\pi$
$277$	−5.93494	−0.356596	−0.178298	+	0.983977i	$0.557059\pi$
$278$	0	0
$279$	0.430572	0.0257777
$280$	0	0
$281$	11.2039	0.668370	0.334185	−	0.942508i	$-0.391539\pi$
$281$	11.2039	0.668370	0.334185	+	0.942508i	$0.391539\pi$
$282$	0	0
$283$	7.20392	0.428228	0.214114	−	0.976809i	$-0.431314\pi$
$283$	7.20392	0.428228	0.214114	+	0.976809i	$0.431314\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−32.4718	−1.91675
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−5.93494	−0.347912
$292$	0	0
$293$	−4.27771	−0.249907	−0.124953	−	0.992163i	$-0.539878\pi$
$293$	−4.27771	−0.249907	−0.124953	+	0.992163i	$0.539878\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−0.430572	−0.0249843
$298$	0	0
$299$	5.39804	0.312177
$300$	0	0
$301$	32.4718	1.87165
$302$	0	0
$303$	16.3330	0.938305
$304$	0	0
$305$	0	0
$306$	0	0
$307$	14.8146	0.845514	0.422757	−	0.906243i	$-0.361062\pi$
$307$	14.8146	0.845514	0.422757	+	0.906243i	$0.361062\pi$
$308$	0	0
$309$	−8.53690	−0.485647
$310$	0	0
$311$	−17.3515	−0.983914	−0.491957	−	0.870620i	$-0.663718\pi$
$311$	−17.3515	−0.983914	−0.491957	+	0.870620i	$0.663718\pi$
$312$	0	0
$313$	22.3980	1.26601	0.633006	−	0.774147i	$-0.281821\pi$
$313$	22.3980	1.26601	0.633006	+	0.774147i	$0.281821\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−27.2679	−1.53152	−0.765759	−	0.643127i	$-0.777637\pi$
$317$	−27.2679	−1.53152	−0.765759	+	0.643127i	$0.777637\pi$
$318$	0	0
$319$	−1.78208	−0.0997771
$320$	0	0
$321$	11.6572	0.650643
$322$	0	0
$323$	−10.1941	−0.567216
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3.67575	0.203270
$328$	0	0
$329$	−60.1378	−3.31550
$330$	0	0
$331$	−10.8797	−0.598001	−0.299000	−	0.954253i	$-0.596653\pi$
$331$	−10.8797	−0.598001	−0.299000	+	0.954253i	$0.596653\pi$
$332$	0	0
$333$	8.53690	0.467819
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−34.2029	−1.86315	−0.931574	−	0.363551i	$-0.881564\pi$
$337$	−34.2029	−1.86315	−0.931574	+	0.363551i	$0.881564\pi$
$338$	0	0
$339$	−3.13886	−0.170479
$340$	0	0
$341$	0.185393	0.0100396
$342$	0	0
$343$	53.0315	2.86343
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−28.2777	−1.51803	−0.759014	−	0.651075i	$-0.774318\pi$
$347$	−28.2777	−1.51803	−0.759014	+	0.651075i	$0.774318\pi$
$348$	0	0
$349$	27.3330	1.46310	0.731550	−	0.681787i	$-0.238797\pi$
$349$	27.3330	1.46310	0.731550	+	0.681787i	$0.238797\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−3.18539	−0.169541	−0.0847707	−	0.996400i	$-0.527016\pi$
$353$	−3.18539	−0.169541	−0.0847707	+	0.996400i	$0.527016\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	14.9024	0.788719
$358$	0	0
$359$	14.1063	0.744503	0.372252	−	0.928132i	$-0.378586\pi$
$359$	14.1063	0.744503	0.372252	+	0.928132i	$0.378586\pi$
$360$	0	0
$361$	−7.45331	−0.392280
$362$	0	0
$363$	10.8146	0.567620
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−25.5922	−1.33590	−0.667950	−	0.744206i	$-0.732828\pi$
$367$	−25.5922	−1.33590	−0.667950	+	0.744206i	$0.732828\pi$
$368$	0	0
$369$	−6.53690	−0.340297
$370$	0	0
$371$	21.8471	1.13425
$372$	0	0
$373$	−15.4068	−0.797733	−0.398866	−	0.917009i	$-0.630596\pi$
$373$	−15.4068	−0.797733	−0.398866	+	0.917009i	$0.630596\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.13886	−0.213162
$378$	0	0
$379$	21.7821	1.11887	0.559435	−	0.828874i	$-0.311018\pi$
$379$	21.7821	1.11887	0.559435	+	0.828874i	$0.311018\pi$
$380$	0	0
$381$	19.6758	1.00802
$382$	0	0
$383$	−4.47184	−0.228500	−0.114250	−	0.993452i	$-0.536447\pi$
$383$	−4.47184	−0.228500	−0.114250	+	0.993452i	$0.536447\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.53690	0.332289
$388$	0	0
$389$	13.4166	0.680247	0.340123	−	0.940381i	$-0.389531\pi$
$389$	13.4166	0.680247	0.340123	+	0.940381i	$0.389531\pi$
$390$	0	0
$391$	−16.1941	−0.818972
$392$	0	0
$393$	−3.39804	−0.171409
$394$	0	0
$395$	0	0
$396$	0	0
$397$	36.7310	1.84348	0.921739	−	0.387812i	$-0.126769\pi$
$397$	36.7310	1.84348	0.921739	+	0.387812i	$0.126769\pi$
$398$	0	0
$399$	−16.8797	−0.845040
$400$	0	0
$401$	27.8514	1.39083	0.695415	−	0.718608i	$-0.255221\pi$
$401$	27.8514	1.39083	0.695415	+	0.718608i	$0.255221\pi$
$402$	0	0
$403$	0.430572	0.0214483
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.67575	0.182200
$408$	0	0
$409$	25.8514	1.27827	0.639134	−	0.769096i	$-0.279293\pi$
$409$	25.8514	1.27827	0.639134	+	0.769096i	$0.279293\pi$
$410$	0	0
$411$	−5.65723	−0.279050
$412$	0	0
$413$	60.1378	2.95919
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−17.8699	−0.875092
$418$	0	0
$419$	28.5369	1.39412	0.697059	−	0.717013i	$-0.254491\pi$
$419$	28.5369	1.39412	0.697059	+	0.717013i	$0.254491\pi$
$420$	0	0
$421$	4.51837	0.220212	0.110106	−	0.993920i	$-0.464881\pi$
$421$	4.51837	0.220212	0.110106	+	0.993920i	$0.464881\pi$
$422$	0	0
$423$	−12.1063	−0.588630
$424$	0	0
$425$	0	0
$426$	0	0
$427$	30.4946	1.47574
$428$	0	0
$429$	−0.430572	−0.0207882
$430$	0	0
$431$	34.1941	1.64707	0.823537	−	0.567263i	$-0.191998\pi$
$431$	34.1941	1.64707	0.823537	+	0.567263i	$0.191998\pi$
$432$	0	0
$433$	11.6572	0.560211	0.280105	−	0.959969i	$-0.409631\pi$
$433$	11.6572	0.560211	0.280105	+	0.959969i	$0.409631\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	18.3428	0.877454
$438$	0	0
$439$	−36.6194	−1.74775	−0.873875	−	0.486151i	$-0.838401\pi$
$439$	−36.6194	−1.74775	−0.873875	+	0.486151i	$0.838401\pi$
$440$	0	0
$441$	17.6758	0.841702
$442$	0	0
$443$	4.25919	0.202360	0.101180	−	0.994868i	$-0.467738\pi$
$443$	4.25919	0.202360	0.101180	+	0.994868i	$0.467738\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−16.2126	−0.766832
$448$	0	0
$449$	24.9437	1.17716	0.588582	−	0.808437i	$-0.299686\pi$
$449$	24.9437	1.17716	0.588582	+	0.808437i	$0.299686\pi$
$450$	0	0
$451$	−2.81461	−0.132535
$452$	0	0
$453$	5.89368	0.276909
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−7.05527	−0.330032	−0.165016	−	0.986291i	$-0.552768\pi$
$457$	−7.05527	−0.330032	−0.165016	+	0.986291i	$0.552768\pi$
$458$	0	0
$459$	3.00000	0.140028
$460$	0	0
$461$	−5.35150	−0.249244	−0.124622	−	0.992204i	$-0.539772\pi$
$461$	−5.35150	−0.249244	−0.124622	+	0.992204i	$0.539772\pi$
$462$	0	0
$463$	20.0878	0.933559	0.466780	−	0.884374i	$-0.345414\pi$
$463$	20.0878	0.933559	0.466780	+	0.884374i	$0.345414\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	9.00873	0.416874	0.208437	−	0.978036i	$-0.433162\pi$
$467$	9.00873	0.416874	0.208437	+	0.978036i	$0.433162\pi$
$468$	0	0
$469$	−47.5358	−2.19500
$470$	0	0
$471$	−4.39804	−0.202651
$472$	0	0
$473$	2.81461	0.129416
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.39804	0.201373
$478$	0	0
$479$	19.5044	0.891177	0.445589	−	0.895238i	$-0.352994\pi$
$479$	19.5044	0.891177	0.445589	+	0.895238i	$0.352994\pi$
$480$	0	0
$481$	8.53690	0.389249
$482$	0	0
$483$	−26.8146	−1.22011
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−14.9860	−0.679080	−0.339540	−	0.940592i	$-0.610271\pi$
$487$	−14.9860	−0.679080	−0.339540	+	0.940592i	$0.610271\pi$
$488$	0	0
$489$	16.4718	0.744882
$490$	0	0
$491$	22.0000	0.992846	0.496423	−	0.868081i	$-0.334646\pi$
$491$	22.0000	0.992846	0.496423	+	0.868081i	$0.334646\pi$
$492$	0	0
$493$	12.4166	0.559214
$494$	0	0
$495$	0	0
$496$	0	0
$497$	48.0640	2.15597
$498$	0	0
$499$	−12.5596	−0.562247	−0.281123	−	0.959672i	$-0.590707\pi$
$499$	−12.5596	−0.562247	−0.281123	+	0.959672i	$0.590707\pi$
$500$	0	0
$501$	12.5369	0.560107
$502$	0	0
$503$	−29.6758	−1.32318	−0.661588	−	0.749867i	$-0.730117\pi$
$503$	−29.6758	−1.32318	−0.661588	+	0.749867i	$0.730117\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	30.3417	1.34487	0.672436	−	0.740155i	$-0.265248\pi$
$509$	30.3417	1.34487	0.672436	+	0.740155i	$0.265248\pi$
$510$	0	0
$511$	−48.0640	−2.12623
$512$	0	0
$513$	−3.39804	−0.150027
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−5.21265	−0.229252
$518$	0	0
$519$	−7.53690	−0.330833
$520$	0	0
$521$	−7.72229	−0.338320	−0.169160	−	0.985589i	$-0.554105\pi$
$521$	−7.72229	−0.338320	−0.169160	+	0.985589i	$0.554105\pi$
$522$	0	0
$523$	27.6572	1.20937	0.604683	−	0.796466i	$-0.293300\pi$
$523$	27.6572	1.20937	0.604683	+	0.796466i	$0.293300\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.29172	−0.0562681
$528$	0	0
$529$	6.13886	0.266907
$530$	0	0
$531$	12.1063	0.525370
$532$	0	0
$533$	−6.53690	−0.283144
$534$	0	0
$535$	0	0
$536$	0	0
$537$	25.6107	1.10518
$538$	0	0
$539$	7.61069	0.327816
$540$	0	0
$541$	−13.6758	−0.587967	−0.293983	−	0.955811i	$-0.594981\pi$
$541$	−13.6758	−0.587967	−0.293983	+	0.955811i	$0.594981\pi$
$542$	0	0
$543$	−9.25919	−0.397350
$544$	0	0
$545$	0	0
$546$	0	0
$547$	14.1291	0.604115	0.302058	−	0.953290i	$-0.402327\pi$
$547$	14.1291	0.604115	0.302058	+	0.953290i	$0.402327\pi$
$548$	0	0
$549$	6.13886	0.262000
$550$	0	0
$551$	−14.0640	−0.599147
$552$	0	0
$553$	52.3417	2.22580
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−10.4718	−0.443706	−0.221853	−	0.975080i	$-0.571210\pi$
$557$	−10.4718	−0.443706	−0.221853	+	0.975080i	$0.571210\pi$
$558$	0	0
$559$	6.53690	0.276481
$560$	0	0
$561$	1.29172	0.0545363
$562$	0	0
$563$	5.95346	0.250909	0.125454	−	0.992099i	$-0.459961\pi$
$563$	5.95346	0.250909	0.125454	+	0.992099i	$0.459961\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.96747	0.208614
$568$	0	0
$569$	−22.4631	−0.941702	−0.470851	−	0.882213i	$-0.656053\pi$
$569$	−22.4631	−0.941702	−0.470851	+	0.882213i	$0.656053\pi$
$570$	0	0
$571$	−33.2679	−1.39222	−0.696110	−	0.717936i	$-0.745087\pi$
$571$	−33.2679	−1.39222	−0.696110	+	0.717936i	$0.745087\pi$
$572$	0	0
$573$	−6.00000	−0.250654
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−14.4904	−0.603242	−0.301621	−	0.953428i	$-0.597528\pi$
$577$	−14.4904	−0.603242	−0.301621	+	0.953428i	$0.597528\pi$
$578$	0	0
$579$	6.47184	0.268960
$580$	0	0
$581$	38.8884	1.61336
$582$	0	0
$583$	1.89368	0.0784280
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19.3742	0.799661	0.399830	−	0.916589i	$-0.369069\pi$
$587$	19.3742	0.799661	0.399830	+	0.916589i	$0.369069\pi$
$588$	0	0
$589$	1.46310	0.0602861
$590$	0	0
$591$	−0.259187	−0.0106615
$592$	0	0
$593$	26.9437	1.10644	0.553222	−	0.833034i	$-0.313398\pi$
$593$	26.9437	1.10644	0.553222	+	0.833034i	$0.313398\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8.79608	0.360000
$598$	0	0
$599$	3.91641	0.160020	0.0800102	−	0.996794i	$-0.474505\pi$
$599$	3.91641	0.160020	0.0800102	+	0.996794i	$0.474505\pi$
$600$	0	0
$601$	9.87967	0.403000	0.201500	−	0.979489i	$-0.435418\pi$
$601$	9.87967	0.403000	0.201500	+	0.979489i	$0.435418\pi$
$602$	0	0
$603$	−9.56943	−0.389697
$604$	0	0
$605$	0	0
$606$	0	0
$607$	27.0087	1.09625	0.548125	−	0.836396i	$-0.315342\pi$
$607$	27.0087	1.09625	0.548125	+	0.836396i	$0.315342\pi$
$608$	0	0
$609$	20.5596	0.833119
$610$	0	0
$611$	−12.1063	−0.489769
$612$	0	0
$613$	16.2777	0.657451	0.328725	−	0.944426i	$-0.393381\pi$
$613$	16.2777	0.657451	0.328725	+	0.944426i	$0.393381\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−4.99127	−0.200941	−0.100470	−	0.994940i	$-0.532035\pi$
$617$	−4.99127	−0.200941	−0.100470	+	0.994940i	$0.532035\pi$
$618$	0	0
$619$	−27.8884	−1.12093	−0.560465	−	0.828178i	$-0.689377\pi$
$619$	−27.8884	−1.12093	−0.560465	+	0.828178i	$0.689377\pi$
$620$	0	0
$621$	−5.39804	−0.216616
$622$	0	0
$623$	14.2126	0.569418
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−1.46310	−0.0584307
$628$	0	0
$629$	−25.6107	−1.02117
$630$	0	0
$631$	−4.66702	−0.185791	−0.0928956	−	0.995676i	$-0.529612\pi$
$631$	−4.66702	−0.185791	−0.0928956	+	0.995676i	$0.529612\pi$
$632$	0	0
$633$	−19.6758	−0.782041
$634$	0	0
$635$	0	0
$636$	0	0
$637$	17.6758	0.700339
$638$	0	0
$639$	9.67575	0.382767
$640$	0	0
$641$	−21.1476	−0.835280	−0.417640	−	0.908613i	$-0.637143\pi$
$641$	−21.1476	−0.835280	−0.417640	+	0.908613i	$0.637143\pi$
$642$	0	0
$643$	34.7495	1.37039	0.685194	−	0.728360i	$-0.259717\pi$
$643$	34.7495	1.37039	0.685194	+	0.728360i	$0.259717\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−4.45331	−0.175078	−0.0875389	−	0.996161i	$-0.527900\pi$
$647$	−4.45331	−0.175078	−0.0875389	+	0.996161i	$0.527900\pi$
$648$	0	0
$649$	5.21265	0.204614
$650$	0	0
$651$	−2.13886	−0.0838283
$652$	0	0
$653$	9.00000	0.352197	0.176099	−	0.984373i	$-0.443652\pi$
$653$	9.00000	0.352197	0.176099	+	0.984373i	$0.443652\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−9.67575	−0.377487
$658$	0	0
$659$	35.4806	1.38213	0.691063	−	0.722794i	$-0.257143\pi$
$659$	35.4806	1.38213	0.691063	+	0.722794i	$0.257143\pi$
$660$	0	0
$661$	−1.33298	−0.0518469	−0.0259235	−	0.999664i	$-0.508253\pi$
$661$	−1.33298	−0.0518469	−0.0259235	+	0.999664i	$0.508253\pi$
$662$	0	0
$663$	3.00000	0.116510
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−22.3417	−0.865075
$668$	0	0
$669$	14.1941	0.548777
$670$	0	0
$671$	2.64322	0.102040
$672$	0	0
$673$	31.6845	1.22135	0.610674	−	0.791882i	$-0.290899\pi$
$673$	31.6845	1.22135	0.610674	+	0.791882i	$0.290899\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	23.2039	0.891799	0.445899	−	0.895083i	$-0.352884\pi$
$677$	23.2039	0.891799	0.445899	+	0.895083i	$0.352884\pi$
$678$	0	0
$679$	29.4816	1.13140
$680$	0	0
$681$	0.495634	0.0189927
$682$	0	0
$683$	−13.0965	−0.501125	−0.250562	−	0.968100i	$-0.580616\pi$
$683$	−13.0965	−0.501125	−0.250562	+	0.968100i	$0.580616\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−2.00000	−0.0763048
$688$	0	0
$689$	4.39804	0.167552
$690$	0	0
$691$	−38.2539	−1.45525	−0.727624	−	0.685976i	$-0.759375\pi$
$691$	−38.2539	−1.45525	−0.727624	+	0.685976i	$0.759375\pi$
$692$	0	0
$693$	2.13886	0.0812484
$694$	0	0
$695$	0	0
$696$	0	0
$697$	19.6107	0.742808
$698$	0	0
$699$	11.1389	0.421310
$700$	0	0
$701$	34.1563	1.29007	0.645033	−	0.764155i	$-0.276844\pi$
$701$	34.1563	1.29007	0.645033	+	0.764155i	$0.276844\pi$
$702$	0	0
$703$	29.0087	1.09409
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−81.1336	−3.05134
$708$	0	0
$709$	21.3330	0.801177	0.400588	−	0.916258i	$-0.368806\pi$
$709$	21.3330	0.801177	0.400588	+	0.916258i	$0.368806\pi$
$710$	0	0
$711$	10.5369	0.395165
$712$	0	0
$713$	2.32425	0.0870438
$714$	0	0
$715$	0	0
$716$	0	0
$717$	17.9762	0.671334
$718$	0	0
$719$	0.0185238	0.000690822 0	0.000345411	−	1.00000i	$-0.499890\pi$
$719$	0.0185238	0.000690822 0	0.000345411		1.00000i	$0.499890\pi$
$720$	0	0
$721$	42.4068	1.57931
$722$	0	0
$723$	−30.4718	−1.13326
$724$	0	0
$725$	0	0
$726$	0	0
$727$	12.6670	0.469794	0.234897	−	0.972020i	$-0.424525\pi$
$727$	12.6670	0.469794	0.234897	+	0.972020i	$0.424525\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−19.6107	−0.725328
$732$	0	0
$733$	8.51837	0.314633	0.157317	−	0.987548i	$-0.449716\pi$
$733$	8.51837	0.314633	0.157317	+	0.987548i	$0.449716\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.12033	−0.151774
$738$	0	0
$739$	11.9122	0.438197	0.219099	−	0.975703i	$-0.429688\pi$
$739$	11.9122	0.438197	0.219099	+	0.975703i	$0.429688\pi$
$740$	0	0
$741$	−3.39804	−0.124830
$742$	0	0
$743$	−21.3568	−0.783504	−0.391752	−	0.920071i	$-0.628131\pi$
$743$	−21.3568	−0.783504	−0.391752	+	0.920071i	$0.628131\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	7.82861	0.286434
$748$	0	0
$749$	−57.9069	−2.11587
$750$	0	0
$751$	18.9902	0.692963	0.346481	−	0.938057i	$-0.387376\pi$
$751$	18.9902	0.692963	0.346481	+	0.938057i	$0.387376\pi$
$752$	0	0
$753$	−23.3330	−0.850301
$754$	0	0
$755$	0	0
$756$	0	0
$757$	2.72229	0.0989433	0.0494716	−	0.998776i	$-0.484246\pi$
$757$	2.72229	0.0989433	0.0494716	+	0.998776i	$0.484246\pi$
$758$	0	0
$759$	−2.32425	−0.0843649
$760$	0	0
$761$	3.52816	0.127896	0.0639479	−	0.997953i	$-0.479631\pi$
$761$	3.52816	0.127896	0.0639479	+	0.997953i	$0.479631\pi$
$762$	0	0
$763$	−18.2592	−0.661027
$764$	0	0
$765$	0	0
$766$	0	0
$767$	12.1063	0.437134
$768$	0	0
$769$	−20.0651	−0.723565	−0.361782	−	0.932263i	$-0.617832\pi$
$769$	−20.0651	−0.723565	−0.361782	+	0.932263i	$0.617832\pi$
$770$	0	0
$771$	−22.6758	−0.816647
$772$	0	0
$773$	−29.8048	−1.07200	−0.536002	−	0.844216i	$-0.680066\pi$
$773$	−29.8048	−1.07200	−0.536002	+	0.844216i	$0.680066\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−42.4068	−1.52133
$778$	0	0
$779$	−22.2126	−0.795851
$780$	0	0
$781$	4.16611	0.149075
$782$	0	0
$783$	4.13886	0.147911
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−13.8471	−0.493597	−0.246799	−	0.969067i	$-0.579379\pi$
$787$	−13.8471	−0.493597	−0.246799	+	0.969067i	$0.579379\pi$
$788$	0	0
$789$	4.53690	0.161518
$790$	0	0
$791$	15.5922	0.554394
$792$	0	0
$793$	6.13886	0.217997
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−14.5271	−0.514576	−0.257288	−	0.966335i	$-0.582829\pi$
$797$	−14.5271	−0.514576	−0.257288	+	0.966335i	$0.582829\pi$
$798$	0	0
$799$	36.3190	1.28487
$800$	0	0
$801$	2.86114	0.101094
$802$	0	0
$803$	−4.16611	−0.147019
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−12.8699	−0.453041
$808$	0	0
$809$	−0.342772	−0.0120512	−0.00602560	−	0.999982i	$-0.501918\pi$
$809$	−0.342772	−0.0120512	−0.00602560	+	0.999982i	$0.501918\pi$
$810$	0	0
$811$	17.5880	0.617597	0.308798	−	0.951128i	$-0.400073\pi$
$811$	17.5880	0.617597	0.308798	+	0.951128i	$0.400073\pi$
$812$	0	0
$813$	8.68976	0.304763
$814$	0	0
$815$	0	0
$816$	0	0
$817$	22.2126	0.777122
$818$	0	0
$819$	4.96747	0.173577
$820$	0	0
$821$	5.22244	0.182264	0.0911322	−	0.995839i	$-0.470951\pi$
$821$	5.22244	0.182264	0.0911322	+	0.995839i	$0.470951\pi$
$822$	0	0
$823$	1.69428	0.0590587	0.0295294	−	0.999564i	$-0.490599\pi$
$823$	1.69428	0.0590587	0.0295294	+	0.999564i	$0.490599\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−21.2637	−0.739411	−0.369706	−	0.929149i	$-0.620541\pi$
$827$	−21.2637	−0.739411	−0.369706	+	0.929149i	$0.620541\pi$
$828$	0	0
$829$	−0.804816	−0.0279524	−0.0139762	−	0.999902i	$-0.504449\pi$
$829$	−0.804816	−0.0279524	−0.0139762	+	0.999902i	$0.504449\pi$
$830$	0	0
$831$	5.93494	0.205881
$832$	0	0
$833$	−53.0273	−1.83729
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−0.430572	−0.0148828
$838$	0	0
$839$	−43.9535	−1.51744	−0.758721	−	0.651416i	$-0.774175\pi$
$839$	−43.9535	−1.51744	−0.758721	+	0.651416i	$0.774175\pi$
$840$	0	0
$841$	−11.8699	−0.409306
$842$	0	0
$843$	−11.2039	−0.385883
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−53.7212	−1.84588
$848$	0	0
$849$	−7.20392	−0.247238
$850$	0	0
$851$	46.0825	1.57969
$852$	0	0
$853$	49.4620	1.69355	0.846774	−	0.531953i	$-0.178542\pi$
$853$	49.4620	1.69355	0.846774	+	0.531953i	$0.178542\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	49.7398	1.69908	0.849539	−	0.527526i	$-0.176880\pi$
$857$	49.7398	1.69908	0.849539	+	0.527526i	$0.176880\pi$
$858$	0	0
$859$	−30.9622	−1.05642	−0.528208	−	0.849115i	$-0.677136\pi$
$859$	−30.9622	−1.05642	−0.528208	+	0.849115i	$0.677136\pi$
$860$	0	0
$861$	32.4718	1.10664
$862$	0	0
$863$	38.4480	1.30879	0.654393	−	0.756154i	$-0.272924\pi$
$863$	38.4480	1.30879	0.654393	+	0.756154i	$0.272924\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8.00000	0.271694
$868$	0	0
$869$	4.53690	0.153904
$870$	0	0
$871$	−9.56943	−0.324248
$872$	0	0
$873$	5.93494	0.200867
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−13.6572	−0.461172	−0.230586	−	0.973052i	$-0.574064\pi$
$877$	−13.6572	−0.461172	−0.230586	+	0.973052i	$0.574064\pi$
$878$	0	0
$879$	4.27771	0.144284
$880$	0	0
$881$	−12.8059	−0.431441	−0.215720	−	0.976455i	$-0.569210\pi$
$881$	−12.8059	−0.431441	−0.215720	+	0.976455i	$0.569210\pi$
$882$	0	0
$883$	3.50964	0.118109	0.0590544	−	0.998255i	$-0.481191\pi$
$883$	3.50964	0.118109	0.0590544	+	0.998255i	$0.481191\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−41.3700	−1.38907	−0.694535	−	0.719459i	$-0.744390\pi$
$887$	−41.3700	−1.38907	−0.694535	+	0.719459i	$0.744390\pi$
$888$	0	0
$889$	−97.7387	−3.27805
$890$	0	0
$891$	0.430572	0.0144247
$892$	0	0
$893$	−41.1378	−1.37662
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−5.39804	−0.180235
$898$	0	0
$899$	−1.78208	−0.0594356
$900$	0	0
$901$	−13.1941	−0.439560
$902$	0	0
$903$	−32.4718	−1.08060
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−25.4620	−0.845453	−0.422727	−	0.906257i	$-0.638927\pi$
$907$	−25.4620	−0.845453	−0.422727	+	0.906257i	$0.638927\pi$
$908$	0	0
$909$	−16.3330	−0.541731
$910$	0	0
$911$	−53.1378	−1.76053	−0.880267	−	0.474479i	$-0.842637\pi$
$911$	−53.1378	−1.76053	−0.880267	+	0.474479i	$0.842637\pi$
$912$	0	0
$913$	3.37079	0.111557
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.8797	0.557416
$918$	0	0
$919$	−8.86114	−0.292302	−0.146151	−	0.989262i	$-0.546689\pi$
$919$	−8.86114	−0.292302	−0.146151	+	0.989262i	$0.546689\pi$
$920$	0	0
$921$	−14.8146	−0.488158
$922$	0	0
$923$	9.67575	0.318481
$924$	0	0
$925$	0	0
$926$	0	0
$927$	8.53690	0.280388
$928$	0	0
$929$	−6.51837	−0.213861	−0.106930	−	0.994267i	$-0.534102\pi$
$929$	−6.51837	−0.213861	−0.106930	+	0.994267i	$0.534102\pi$
$930$	0	0
$931$	60.0629	1.96848
$932$	0	0
$933$	17.3515	0.568063
$934$	0	0
$935$	0	0
$936$	0	0
$937$	3.58343	0.117066	0.0585328	−	0.998285i	$-0.481358\pi$
$937$	3.58343	0.117066	0.0585328	+	0.998285i	$0.481358\pi$
$938$	0	0
$939$	−22.3980	−0.730932
$940$	0	0
$941$	−53.5456	−1.74554	−0.872769	−	0.488134i	$-0.837678\pi$
$941$	−53.5456	−1.74554	−0.872769	+	0.488134i	$0.837678\pi$
$942$	0	0
$943$	−35.2864	−1.14908
$944$	0	0
$945$	0	0
$946$	0	0
$947$	33.9947	1.10468	0.552340	−	0.833619i	$-0.313735\pi$
$947$	33.9947	1.10468	0.552340	+	0.833619i	$0.313735\pi$
$948$	0	0
$949$	−9.67575	−0.314088
$950$	0	0
$951$	27.2679	0.884223
$952$	0	0
$953$	−18.4631	−0.598079	−0.299039	−	0.954241i	$-0.596666\pi$
$953$	−18.4631	−0.598079	−0.299039	+	0.954241i	$0.596666\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.78208	0.0576064
$958$	0	0
$959$	28.1021	0.907464
$960$	0	0
$961$	−30.8146	−0.994020
$962$	0	0
$963$	−11.6572	−0.375649
$964$	0	0
$965$	0	0
$966$	0	0
$967$	20.9209	0.672772	0.336386	−	0.941724i	$-0.390795\pi$
$967$	20.9209	0.672772	0.336386	+	0.941724i	$0.390795\pi$
$968$	0	0
$969$	10.1941	0.327482
$970$	0	0
$971$	23.0087	0.738385	0.369193	−	0.929353i	$-0.379634\pi$
$971$	23.0087	0.738385	0.369193	+	0.929353i	$0.379634\pi$
$972$	0	0
$973$	88.7681	2.84577
$974$	0	0
$975$	0	0
$976$	0	0
$977$	12.5184	0.400498	0.200249	−	0.979745i	$-0.435825\pi$
$977$	12.5184	0.400498	0.200249	+	0.979745i	$0.435825\pi$
$978$	0	0
$979$	1.23193	0.0393727
$980$	0	0
$981$	−3.67575	−0.117358
$982$	0	0
$983$	−60.1703	−1.91914	−0.959568	−	0.281478i	$-0.909175\pi$
$983$	−60.1703	−1.91914	−0.959568	+	0.281478i	$0.909175\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	60.1378	1.91421
$988$	0	0
$989$	35.2864	1.12204
$990$	0	0
$991$	−59.5827	−1.89271	−0.946353	−	0.323134i	$-0.895263\pi$
$991$	−59.5827	−1.89271	−0.946353	+	0.323134i	$0.895263\pi$
$992$	0	0
$993$	10.8797	0.345256
$994$	0	0
$995$	0	0
$996$	0	0
$997$	60.1193	1.90400	0.951998	−	0.306103i	$-0.0990254\pi$
$997$	60.1193	1.90400	0.951998	+	0.306103i	$0.0990254\pi$
$998$	0	0
$999$	−8.53690	−0.270095

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.bh.1.3	✓	3
5.4	even	2		7800.2.a.bs.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7800.2.a.bh.1.3	✓	3	1.1	even	1	trivial
7800.2.a.bs.1.1	yes	3	5.4	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:	\( N \)	\(=\)	\( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7800.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +4.96747 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +4.96747 q^{7} +1.00000 q^{9} +0.430572 q^{11} +1.00000 q^{13} -3.00000 q^{17} +3.39804 q^{19} -4.96747 q^{21} +5.39804 q^{23} -1.00000 q^{27} -4.13886 q^{29} +0.430572 q^{31} -0.430572 q^{33} +8.53690 q^{37} -1.00000 q^{39} -6.53690 q^{41} +6.53690 q^{43} -12.1063 q^{47} +17.6758 q^{49} +3.00000 q^{51} +4.39804 q^{53} -3.39804 q^{57} +12.1063 q^{59} +6.13886 q^{61} +4.96747 q^{63} -9.56943 q^{67} -5.39804 q^{69} +9.67575 q^{71} -9.67575 q^{73} +2.13886 q^{77} +10.5369 q^{79} +1.00000 q^{81} +7.82861 q^{83} +4.13886 q^{87} +2.86114 q^{89} +4.96747 q^{91} -0.430572 q^{93} +5.93494 q^{97} +0.430572 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - q^7 + 3 * q^9	\( 3 q - 3 q^{3} - q^{7} + 3 q^{9} + 5 q^{11} + 3 q^{13} - 9 q^{17} - 2 q^{19} + q^{21} + 4 q^{23} - 3 q^{27} - 5 q^{29} + 5 q^{31} - 5 q^{33} + 6 q^{37} - 3 q^{39} - 13 q^{47} + 26 q^{49} + 9 q^{51} + q^{53} + 2 q^{57} + 13 q^{59} + 11 q^{61} - q^{63} - 25 q^{67} - 4 q^{69} + 2 q^{71} - 2 q^{73} - q^{77} + 12 q^{79} + 3 q^{81} + 15 q^{83} + 5 q^{87} + 16 q^{89} - q^{91} - 5 q^{93} - 14 q^{97} + 5 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - q^7 + 3 * q^9 + 5 * q^11 + 3 * q^13 - 9 * q^17 - 2 * q^19 + q^21 + 4 * q^23 - 3 * q^27 - 5 * q^29 + 5 * q^31 - 5 * q^33 + 6 * q^37 - 3 * q^39 - 13 * q^47 + 26 * q^49 + 9 * q^51 + q^53 + 2 * q^57 + 13 * q^59 + 11 * q^61 - q^63 - 25 * q^67 - 4 * q^69 + 2 * q^71 - 2 * q^73 - q^77 + 12 * q^79 + 3 * q^81 + 15 * q^83 + 5 * q^87 + 16 * q^89 - q^91 - 5 * q^93 - 14 * q^97 + 5 * q^99

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