LMFDB - Embedded newform 7800.2.a.bq.1.1

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:	3.3.2700.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 15x - 20 $ x^3 - 15*x - 20
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.41883$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -4.41883 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -4.41883 q^{7} +1.00000 q^{9} -4.41883 q^{11} +1.00000 q^{13} +1.31158 q^{17} +0.311583 q^{19} -4.41883 q^{21} -1.37683 q^{23} +1.00000 q^{27} +1.00000 q^{29} -4.41883 q^{31} -4.41883 q^{33} +3.68842 q^{37} +1.00000 q^{39} +1.68842 q^{41} -2.00000 q^{43} -2.73042 q^{47} +12.5261 q^{49} +1.31158 q^{51} +2.37683 q^{53} +0.311583 q^{57} -1.04200 q^{59} -4.06525 q^{61} -4.41883 q^{63} -6.10725 q^{67} -1.37683 q^{69} +3.68842 q^{71} -12.8377 q^{73} +19.5261 q^{77} +14.5261 q^{79} +1.00000 q^{81} -5.04200 q^{83} +1.00000 q^{87} +3.37683 q^{89} -4.41883 q^{91} -4.41883 q^{93} +5.37683 q^{97} -4.41883 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 3 * q^9	$ 3 q + 3 q^{3} + 3 q^{9} + 3 q^{13} + 6 q^{17} + 3 q^{19} + 3 q^{27} + 3 q^{29} + 9 q^{37} + 3 q^{39} + 3 q^{41} - 6 q^{43} + 3 q^{47} + 9 q^{49} + 6 q^{51} + 3 q^{53} + 3 q^{57} + 6 q^{59} - 6 q^{61} - 3 q^{67} + 9 q^{71} - 12 q^{73} + 30 q^{77} + 15 q^{79} + 3 q^{81} - 6 q^{83} + 3 q^{87} + 6 q^{89} + 12 q^{97}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^9 + 3 * q^13 + 6 * q^17 + 3 * q^19 + 3 * q^27 + 3 * q^29 + 9 * q^37 + 3 * q^39 + 3 * q^41 - 6 * q^43 + 3 * q^47 + 9 * q^49 + 6 * q^51 + 3 * q^53 + 3 * q^57 + 6 * q^59 - 6 * q^61 - 3 * q^67 + 9 * q^71 - 12 * q^73 + 30 * q^77 + 15 * q^79 + 3 * q^81 - 6 * q^83 + 3 * q^87 + 6 * q^89 + 12 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	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
$7$	−4.41883	−1.67016	−0.835081	−	0.550127i	$-0.814579\pi$
$7$	−4.41883	−1.67016	−0.835081	+	0.550127i	$0.814579\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.41883	−1.33233	−0.666164	−	0.745805i	$-0.732065\pi$
$11$	−4.41883	−1.33233	−0.666164	+	0.745805i	$0.732065\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.31158	0.318106	0.159053	−	0.987270i	$-0.449156\pi$
$17$	1.31158	0.318106	0.159053	+	0.987270i	$0.449156\pi$
$18$	0	0
$19$	0.311583	0.0714821	0.0357410	−	0.999361i	$-0.488621\pi$
$19$	0.311583	0.0714821	0.0357410	+	0.999361i	$0.488621\pi$
$20$	0	0
$21$	−4.41883	−0.964268
$22$	0	0
$23$	−1.37683	−0.287090	−0.143545	−	0.989644i	$-0.545850\pi$
$23$	−1.37683	−0.287090	−0.143545	+	0.989644i	$0.545850\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	−4.41883	−0.793646	−0.396823	−	0.917895i	$-0.629887\pi$
$31$	−4.41883	−0.793646	−0.396823	+	0.917895i	$0.629887\pi$
$32$	0	0
$33$	−4.41883	−0.769220
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.68842	0.606372	0.303186	−	0.952931i	$-0.401950\pi$
$37$	3.68842	0.606372	0.303186	+	0.952931i	$0.401950\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	1.68842	0.263686	0.131843	−	0.991271i	$-0.457910\pi$
$41$	1.68842	0.263686	0.131843	+	0.991271i	$0.457910\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.73042	−0.398272	−0.199136	−	0.979972i	$-0.563814\pi$
$47$	−2.73042	−0.398272	−0.199136	+	0.979972i	$0.563814\pi$
$48$	0	0
$49$	12.5261	1.78944
$50$	0	0
$51$	1.31158	0.183658
$52$	0	0
$53$	2.37683	0.326483	0.163242	−	0.986586i	$-0.447805\pi$
$53$	2.37683	0.326483	0.163242	+	0.986586i	$0.447805\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.311583	0.0412702
$58$	0	0
$59$	−1.04200	−0.135657	−0.0678284	−	0.997697i	$-0.521607\pi$
$59$	−1.04200	−0.135657	−0.0678284	+	0.997697i	$0.521607\pi$
$60$	0	0
$61$	−4.06525	−0.520502	−0.260251	−	0.965541i	$-0.583805\pi$
$61$	−4.06525	−0.520502	−0.260251	+	0.965541i	$0.583805\pi$
$62$	0	0
$63$	−4.41883	−0.556721
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−6.10725	−0.746119	−0.373060	−	0.927807i	$-0.621691\pi$
$67$	−6.10725	−0.746119	−0.373060	+	0.927807i	$0.621691\pi$
$68$	0	0
$69$	−1.37683	−0.165751
$70$	0	0
$71$	3.68842	0.437735	0.218867	−	0.975755i	$-0.429764\pi$
$71$	3.68842	0.437735	0.218867	+	0.975755i	$0.429764\pi$
$72$	0	0
$73$	−12.8377	−1.50254	−0.751268	−	0.659998i	$-0.770557\pi$
$73$	−12.8377	−1.50254	−0.751268	+	0.659998i	$0.770557\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	19.5261	2.22520
$78$	0	0
$79$	14.5261	1.63431	0.817156	−	0.576417i	$-0.195549\pi$
$79$	14.5261	1.63431	0.817156	+	0.576417i	$0.195549\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.04200	−0.553431	−0.276716	−	0.960952i	$-0.589246\pi$
$83$	−5.04200	−0.553431	−0.276716	+	0.960952i	$0.589246\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	3.37683	0.357944	0.178972	−	0.983854i	$-0.442723\pi$
$89$	3.37683	0.357944	0.178972	+	0.983854i	$0.442723\pi$
$90$	0	0
$91$	−4.41883	−0.463220
$92$	0	0
$93$	−4.41883	−0.458212
$94$	0	0
$95$	0	0
$96$	0	0
$97$	5.37683	0.545935	0.272967	−	0.962023i	$-0.411995\pi$
$97$	5.37683	0.545935	0.272967	+	0.962023i	$0.411995\pi$
$98$	0	0
$99$	−4.41883	−0.444109
$100$	0	0
$101$	14.9029	1.48290	0.741448	−	0.671011i	$-0.234139\pi$
$101$	14.9029	1.48290	0.741448	+	0.671011i	$0.234139\pi$
$102$	0	0
$103$	6.21450	0.612333	0.306166	−	0.951978i	$-0.400954\pi$
$103$	6.21450	0.612333	0.306166	+	0.951978i	$0.400954\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.311583	−0.0301219	−0.0150609	−	0.999887i	$-0.504794\pi$
$107$	−0.311583	−0.0301219	−0.0150609	+	0.999887i	$0.504794\pi$
$108$	0	0
$109$	19.9869	1.91440	0.957200	−	0.289429i	$-0.0934653\pi$
$109$	19.9869	1.91440	0.957200	+	0.289429i	$0.0934653\pi$
$110$	0	0
$111$	3.68842	0.350089
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−5.79567	−0.531288
$120$	0	0
$121$	8.52608	0.775098
$122$	0	0
$123$	1.68842	0.152239
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−7.14925	−0.634393	−0.317197	−	0.948360i	$-0.602742\pi$
$127$	−7.14925	−0.634393	−0.317197	+	0.948360i	$0.602742\pi$
$128$	0	0
$129$	−2.00000	−0.176090
$130$	0	0
$131$	6.31158	0.551446	0.275723	−	0.961237i	$-0.411083\pi$
$131$	6.31158	0.551446	0.275723	+	0.961237i	$0.411083\pi$
$132$	0	0
$133$	−1.37683	−0.119387
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.7724	1.51840	0.759200	−	0.650857i	$-0.225590\pi$
$137$	17.7724	1.51840	0.759200	+	0.650857i	$0.225590\pi$
$138$	0	0
$139$	−4.83767	−0.410325	−0.205163	−	0.978728i	$-0.565772\pi$
$139$	−4.83767	−0.410325	−0.205163	+	0.978728i	$0.565772\pi$
$140$	0	0
$141$	−2.73042	−0.229942
$142$	0	0
$143$	−4.41883	−0.369521
$144$	0	0
$145$	0	0
$146$	0	0
$147$	12.5261	1.03313
$148$	0	0
$149$	−19.0522	−1.56081	−0.780407	−	0.625272i	$-0.784988\pi$
$149$	−19.0522	−1.56081	−0.780407	+	0.625272i	$0.784988\pi$
$150$	0	0
$151$	15.1725	1.23472	0.617360	−	0.786681i	$-0.288202\pi$
$151$	15.1725	1.23472	0.617360	+	0.786681i	$0.288202\pi$
$152$	0	0
$153$	1.31158	0.106035
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1.93475	0.154410	0.0772049	−	0.997015i	$-0.475400\pi$
$157$	1.93475	0.154410	0.0772049	+	0.997015i	$0.475400\pi$
$158$	0	0
$159$	2.37683	0.188495
$160$	0	0
$161$	6.08400	0.479486
$162$	0	0
$163$	0.837665	0.0656110	0.0328055	−	0.999462i	$-0.489556\pi$
$163$	0.837665	0.0656110	0.0328055	+	0.999462i	$0.489556\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.934749	0.0723331	0.0361665	−	0.999346i	$-0.488485\pi$
$167$	0.934749	0.0723331	0.0361665	+	0.999346i	$0.488485\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0.311583	0.0238274
$172$	0	0
$173$	15.9217	1.21050	0.605251	−	0.796035i	$-0.293073\pi$
$173$	15.9217	1.21050	0.605251	+	0.796035i	$0.293073\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1.04200	−0.0783214
$178$	0	0
$179$	−10.2145	−0.763468	−0.381734	−	0.924272i	$-0.624673\pi$
$179$	−10.2145	−0.763468	−0.381734	+	0.924272i	$0.624673\pi$
$180$	0	0
$181$	18.9869	1.41129	0.705643	−	0.708567i	$-0.250658\pi$
$181$	18.9869	1.41129	0.705643	+	0.708567i	$0.250658\pi$
$182$	0	0
$183$	−4.06525	−0.300512
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−5.79567	−0.423821
$188$	0	0
$189$	−4.41883	−0.321423
$190$	0	0
$191$	−4.21450	−0.304950	−0.152475	−	0.988307i	$-0.548724\pi$
$191$	−4.21450	−0.304950	−0.152475	+	0.988307i	$0.548724\pi$
$192$	0	0
$193$	−15.4608	−1.11290	−0.556448	−	0.830883i	$-0.687836\pi$
$193$	−15.4608	−1.11290	−0.556448	+	0.830883i	$0.687836\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	22.2145	1.58272	0.791359	−	0.611352i	$-0.209374\pi$
$197$	22.2145	1.58272	0.791359	+	0.611352i	$0.209374\pi$
$198$	0	0
$199$	26.6101	1.88634	0.943169	−	0.332313i	$-0.107829\pi$
$199$	26.6101	1.88634	0.943169	+	0.332313i	$0.107829\pi$
$200$	0	0
$201$	−6.10725	−0.430772
$202$	0	0
$203$	−4.41883	−0.310141
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.37683	−0.0956966
$208$	0	0
$209$	−1.37683	−0.0952376
$210$	0	0
$211$	2.62317	0.180586	0.0902931	−	0.995915i	$-0.471220\pi$
$211$	2.62317	0.180586	0.0902931	+	0.995915i	$0.471220\pi$
$212$	0	0
$213$	3.68842	0.252726
$214$	0	0
$215$	0	0
$216$	0	0
$217$	19.5261	1.32552
$218$	0	0
$219$	−12.8377	−0.867489
$220$	0	0
$221$	1.31158	0.0882266
$222$	0	0
$223$	−6.53917	−0.437895	−0.218948	−	0.975737i	$-0.570262\pi$
$223$	−6.53917	−0.437895	−0.218948	+	0.975737i	$0.570262\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.87966	0.124758	0.0623789	−	0.998053i	$-0.480131\pi$
$227$	1.87966	0.124758	0.0623789	+	0.998053i	$0.480131\pi$
$228$	0	0
$229$	21.2797	1.40621	0.703103	−	0.711088i	$-0.251797\pi$
$229$	21.2797	1.40621	0.703103	+	0.711088i	$0.251797\pi$
$230$	0	0
$231$	19.5261	1.28472
$232$	0	0
$233$	15.0522	0.986100	0.493050	−	0.870001i	$-0.335882\pi$
$233$	15.0522	0.986100	0.493050	+	0.870001i	$0.335882\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	14.5261	0.943570
$238$	0	0
$239$	23.4710	1.51821	0.759106	−	0.650967i	$-0.225636\pi$
$239$	23.4710	1.51821	0.759106	+	0.650967i	$0.225636\pi$
$240$	0	0
$241$	−8.70716	−0.560878	−0.280439	−	0.959872i	$-0.590480\pi$
$241$	−8.70716	−0.560878	−0.280439	+	0.959872i	$0.590480\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.311583	0.0198256
$248$	0	0
$249$	−5.04200	−0.319524
$250$	0	0
$251$	28.6101	1.80585	0.902926	−	0.429796i	$-0.141414\pi$
$251$	28.6101	1.80585	0.902926	+	0.429796i	$0.141414\pi$
$252$	0	0
$253$	6.08400	0.382498
$254$	0	0
$255$	0	0
$256$	0	0
$257$	16.0653	1.00212	0.501061	−	0.865412i	$-0.332943\pi$
$257$	16.0653	1.00212	0.501061	+	0.865412i	$0.332943\pi$
$258$	0	0
$259$	−16.2985	−1.01274
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	17.5913	1.08473	0.542364	−	0.840144i	$-0.317529\pi$
$263$	17.5913	1.08473	0.542364	+	0.840144i	$0.317529\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3.37683	0.206659
$268$	0	0
$269$	−8.67533	−0.528944	−0.264472	−	0.964393i	$-0.585198\pi$
$269$	−8.67533	−0.528944	−0.264472	+	0.964393i	$0.585198\pi$
$270$	0	0
$271$	16.4188	0.997373	0.498687	−	0.866782i	$-0.333816\pi$
$271$	16.4188	0.997373	0.498687	+	0.866782i	$0.333816\pi$
$272$	0	0
$273$	−4.41883	−0.267440
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−7.67533	−0.461166	−0.230583	−	0.973053i	$-0.574063\pi$
$277$	−7.67533	−0.461166	−0.230583	+	0.973053i	$0.574063\pi$
$278$	0	0
$279$	−4.41883	−0.264549
$280$	0	0
$281$	1.47392	0.0879266	0.0439633	−	0.999033i	$-0.486002\pi$
$281$	1.47392	0.0879266	0.0439633	+	0.999033i	$0.486002\pi$
$282$	0	0
$283$	18.5130	1.10048	0.550242	−	0.835005i	$-0.314536\pi$
$283$	18.5130	1.10048	0.550242	+	0.835005i	$0.314536\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−7.46083	−0.440399
$288$	0	0
$289$	−15.2797	−0.898809
$290$	0	0
$291$	5.37683	0.315196
$292$	0	0
$293$	−19.0522	−1.11304	−0.556520	−	0.830834i	$-0.687864\pi$
$293$	−19.0522	−1.11304	−0.556520	+	0.830834i	$0.687864\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−4.41883	−0.256407
$298$	0	0
$299$	−1.37683	−0.0796244
$300$	0	0
$301$	8.83767	0.509395
$302$	0	0
$303$	14.9029	0.856150
$304$	0	0
$305$	0	0
$306$	0	0
$307$	16.1174	0.919869	0.459935	−	0.887953i	$-0.347873\pi$
$307$	16.1174	0.919869	0.459935	+	0.887953i	$0.347873\pi$
$308$	0	0
$309$	6.21450	0.353531
$310$	0	0
$311$	3.78550	0.214656	0.107328	−	0.994224i	$-0.465770\pi$
$311$	3.78550	0.214656	0.107328	+	0.994224i	$0.465770\pi$
$312$	0	0
$313$	−15.0000	−0.847850	−0.423925	−	0.905697i	$-0.639348\pi$
$313$	−15.0000	−0.847850	−0.423925	+	0.905697i	$0.639348\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−22.5130	−1.26446	−0.632228	−	0.774782i	$-0.717859\pi$
$317$	−22.5130	−1.26446	−0.632228	+	0.774782i	$0.717859\pi$
$318$	0	0
$319$	−4.41883	−0.247407
$320$	0	0
$321$	−0.311583	−0.0173909
$322$	0	0
$323$	0.408667	0.0227389
$324$	0	0
$325$	0	0
$326$	0	0
$327$	19.9869	1.10528
$328$	0	0
$329$	12.0653	0.665179
$330$	0	0
$331$	12.8377	0.705622	0.352811	−	0.935695i	$-0.385226\pi$
$331$	12.8377	0.705622	0.352811	+	0.935695i	$0.385226\pi$
$332$	0	0
$333$	3.68842	0.202124
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−0.279750	−0.0152389	−0.00761947	−	0.999971i	$-0.502425\pi$
$337$	−0.279750	−0.0152389	−0.00761947	+	0.999971i	$0.502425\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	19.5261	1.05740
$342$	0	0
$343$	−24.4188	−1.31849
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.7406	0.683950	0.341975	−	0.939709i	$-0.388904\pi$
$347$	12.7406	0.683950	0.341975	+	0.939709i	$0.388904\pi$
$348$	0	0
$349$	−8.21450	−0.439712	−0.219856	−	0.975532i	$-0.570559\pi$
$349$	−8.21450	−0.439712	−0.219856	+	0.975532i	$0.570559\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	3.36375	0.179034	0.0895171	−	0.995985i	$-0.471468\pi$
$353$	3.36375	0.179034	0.0895171	+	0.995985i	$0.471468\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−5.79567	−0.306739
$358$	0	0
$359$	24.4057	1.28809	0.644043	−	0.764989i	$-0.277256\pi$
$359$	24.4057	1.28809	0.644043	+	0.764989i	$0.277256\pi$
$360$	0	0
$361$	−18.9029	−0.994890
$362$	0	0
$363$	8.52608	0.447503
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−13.7724	−0.718914	−0.359457	−	0.933162i	$-0.617038\pi$
$367$	−13.7724	−0.718914	−0.359457	+	0.933162i	$0.617038\pi$
$368$	0	0
$369$	1.68842	0.0878955
$370$	0	0
$371$	−10.5028	−0.545280
$372$	0	0
$373$	−12.5579	−0.650224	−0.325112	−	0.945675i	$-0.605402\pi$
$373$	−12.5579	−0.650224	−0.325112	+	0.945675i	$0.605402\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.00000	0.0515026
$378$	0	0
$379$	−8.71733	−0.447779	−0.223890	−	0.974615i	$-0.571875\pi$
$379$	−8.71733	−0.447779	−0.223890	+	0.974615i	$0.571875\pi$
$380$	0	0
$381$	−7.14925	−0.366267
$382$	0	0
$383$	−20.7406	−1.05979	−0.529897	−	0.848062i	$-0.677769\pi$
$383$	−20.7406	−1.05979	−0.529897	+	0.848062i	$0.677769\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.00000	−0.101666
$388$	0	0
$389$	−33.4477	−1.69587	−0.847934	−	0.530102i	$-0.822154\pi$
$389$	−33.4477	−1.69587	−0.847934	+	0.530102i	$0.822154\pi$
$390$	0	0
$391$	−1.80583	−0.0913248
$392$	0	0
$393$	6.31158	0.318377
$394$	0	0
$395$	0	0
$396$	0	0
$397$	15.1492	0.760319	0.380159	−	0.924921i	$-0.375869\pi$
$397$	15.1492	0.760319	0.380159	+	0.924921i	$0.375869\pi$
$398$	0	0
$399$	−1.37683	−0.0689279
$400$	0	0
$401$	−4.21450	−0.210462	−0.105231	−	0.994448i	$-0.533558\pi$
$401$	−4.21450	−0.210462	−0.105231	+	0.994448i	$0.533558\pi$
$402$	0	0
$403$	−4.41883	−0.220118
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−16.2985	−0.807887
$408$	0	0
$409$	12.5392	0.620022	0.310011	−	0.950733i	$-0.399667\pi$
$409$	12.5392	0.620022	0.310011	+	0.950733i	$0.399667\pi$
$410$	0	0
$411$	17.7724	0.876649
$412$	0	0
$413$	4.60442	0.226569
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−4.83767	−0.236901
$418$	0	0
$419$	11.3637	0.555155	0.277578	−	0.960703i	$-0.410468\pi$
$419$	11.3637	0.555155	0.277578	+	0.960703i	$0.410468\pi$
$420$	0	0
$421$	−12.6232	−0.615215	−0.307608	−	0.951513i	$-0.599528\pi$
$421$	−12.6232	−0.615215	−0.307608	+	0.951513i	$0.599528\pi$
$422$	0	0
$423$	−2.73042	−0.132757
$424$	0	0
$425$	0	0
$426$	0	0
$427$	17.9637	0.869323
$428$	0	0
$429$	−4.41883	−0.213343
$430$	0	0
$431$	13.8189	0.665634	0.332817	−	0.942991i	$-0.392001\pi$
$431$	13.8189	0.665634	0.332817	+	0.942991i	$0.392001\pi$
$432$	0	0
$433$	15.5782	0.748643	0.374321	−	0.927299i	$-0.377876\pi$
$433$	15.5782	0.748643	0.374321	+	0.927299i	$0.377876\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−0.428998	−0.0205218
$438$	0	0
$439$	13.9029	0.663550	0.331775	−	0.943359i	$-0.392353\pi$
$439$	13.9029	0.663550	0.331775	+	0.943359i	$0.392353\pi$
$440$	0	0
$441$	12.5261	0.596480
$442$	0	0
$443$	−33.4477	−1.58915	−0.794575	−	0.607166i	$-0.792306\pi$
$443$	−33.4477	−1.58915	−0.794575	+	0.607166i	$0.792306\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−19.0522	−0.901136
$448$	0	0
$449$	17.7724	0.838732	0.419366	−	0.907817i	$-0.362252\pi$
$449$	17.7724	0.838732	0.419366	+	0.907817i	$0.362252\pi$
$450$	0	0
$451$	−7.46083	−0.351317
$452$	0	0
$453$	15.1725	0.712866
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−26.2145	−1.22626	−0.613131	−	0.789981i	$-0.710090\pi$
$457$	−26.2145	−1.22626	−0.613131	+	0.789981i	$0.710090\pi$
$458$	0	0
$459$	1.31158	0.0612195
$460$	0	0
$461$	−24.7275	−1.15167	−0.575837	−	0.817564i	$-0.695324\pi$
$461$	−24.7275	−1.15167	−0.575837	+	0.817564i	$0.695324\pi$
$462$	0	0
$463$	−31.7957	−1.47767	−0.738835	−	0.673886i	$-0.764624\pi$
$463$	−31.7957	−1.47767	−0.738835	+	0.673886i	$0.764624\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.85075	−0.317015	−0.158507	−	0.987358i	$-0.550668\pi$
$467$	−6.85075	−0.317015	−0.158507	+	0.987358i	$0.550668\pi$
$468$	0	0
$469$	26.9869	1.24614
$470$	0	0
$471$	1.93475	0.0891485
$472$	0	0
$473$	8.83767	0.406356
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.37683	0.108828
$478$	0	0
$479$	−13.8928	−0.634776	−0.317388	−	0.948296i	$-0.602806\pi$
$479$	−13.8928	−0.634776	−0.317388	+	0.948296i	$0.602806\pi$
$480$	0	0
$481$	3.68842	0.168177
$482$	0	0
$483$	6.08400	0.276831
$484$	0	0
$485$	0	0
$486$	0	0
$487$	5.47100	0.247915	0.123957	−	0.992288i	$-0.460441\pi$
$487$	5.47100	0.247915	0.123957	+	0.992288i	$0.460441\pi$
$488$	0	0
$489$	0.837665	0.0378805
$490$	0	0
$491$	1.91600	0.0864680	0.0432340	−	0.999065i	$-0.486234\pi$
$491$	1.91600	0.0864680	0.0432340	+	0.999065i	$0.486234\pi$
$492$	0	0
$493$	1.31158	0.0590707
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−16.2985	−0.731088
$498$	0	0
$499$	38.0811	1.70474	0.852372	−	0.522937i	$-0.175164\pi$
$499$	38.0811	1.70474	0.852372	+	0.522937i	$0.175164\pi$
$500$	0	0
$501$	0.934749	0.0417615
$502$	0	0
$503$	−35.8898	−1.60025	−0.800124	−	0.599834i	$-0.795233\pi$
$503$	−35.8898	−1.60025	−0.800124	+	0.599834i	$0.795233\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	33.1362	1.46873	0.734367	−	0.678752i	$-0.237479\pi$
$509$	33.1362	1.46873	0.734367	+	0.678752i	$0.237479\pi$
$510$	0	0
$511$	56.7275	2.50948
$512$	0	0
$513$	0.311583	0.0137567
$514$	0	0
$515$	0	0
$516$	0	0
$517$	12.0653	0.530629
$518$	0	0
$519$	15.9217	0.698883
$520$	0	0
$521$	37.8058	1.65630	0.828152	−	0.560504i	$-0.189393\pi$
$521$	37.8058	1.65630	0.828152	+	0.560504i	$0.189393\pi$
$522$	0	0
$523$	−12.2145	−0.534103	−0.267051	−	0.963682i	$-0.586049\pi$
$523$	−12.2145	−0.534103	−0.267051	+	0.963682i	$0.586049\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.79567	−0.252463
$528$	0	0
$529$	−21.1043	−0.917580
$530$	0	0
$531$	−1.04200	−0.0452189
$532$	0	0
$533$	1.68842	0.0731335
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−10.2145	−0.440788
$538$	0	0
$539$	−55.3507	−2.38412
$540$	0	0
$541$	−39.5652	−1.70104	−0.850520	−	0.525943i	$-0.823712\pi$
$541$	−39.5652	−1.70104	−0.850520	+	0.525943i	$0.823712\pi$
$542$	0	0
$543$	18.9869	0.814806
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−18.8377	−0.805440	−0.402720	−	0.915323i	$-0.631935\pi$
$547$	−18.8377	−0.805440	−0.402720	+	0.915323i	$0.631935\pi$
$548$	0	0
$549$	−4.06525	−0.173501
$550$	0	0
$551$	0.311583	0.0132739
$552$	0	0
$553$	−64.1883	−2.72957
$554$	0	0
$555$	0	0
$556$	0	0
$557$	21.5913	0.914854	0.457427	−	0.889247i	$-0.348771\pi$
$557$	21.5913	0.914854	0.457427	+	0.889247i	$0.348771\pi$
$558$	0	0
$559$	−2.00000	−0.0845910
$560$	0	0
$561$	−5.79567	−0.244693
$562$	0	0
$563$	5.01875	0.211515	0.105757	−	0.994392i	$-0.466273\pi$
$563$	5.01875	0.211515	0.105757	+	0.994392i	$0.466273\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−4.41883	−0.185574
$568$	0	0
$569$	43.2854	1.81462	0.907309	−	0.420464i	$-0.138133\pi$
$569$	43.2854	1.81462	0.907309	+	0.420464i	$0.138133\pi$
$570$	0	0
$571$	12.1305	0.507646	0.253823	−	0.967251i	$-0.418312\pi$
$571$	12.1305	0.507646	0.253823	+	0.967251i	$0.418312\pi$
$572$	0	0
$573$	−4.21450	−0.176063
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−7.86950	−0.327612	−0.163806	−	0.986493i	$-0.552377\pi$
$577$	−7.86950	−0.327612	−0.163806	+	0.986493i	$0.552377\pi$
$578$	0	0
$579$	−15.4608	−0.642530
$580$	0	0
$581$	22.2797	0.924320
$582$	0	0
$583$	−10.5028	−0.434983
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20.4188	−0.842775	−0.421388	−	0.906881i	$-0.638457\pi$
$587$	−20.4188	−0.842775	−0.421388	+	0.906881i	$0.638457\pi$
$588$	0	0
$589$	−1.37683	−0.0567314
$590$	0	0
$591$	22.2145	0.913782
$592$	0	0
$593$	−24.0334	−0.986934	−0.493467	−	0.869764i	$-0.664271\pi$
$593$	−24.0334	−0.986934	−0.493467	+	0.869764i	$0.664271\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	26.6101	1.08908
$598$	0	0
$599$	11.8898	0.485805	0.242903	−	0.970051i	$-0.421900\pi$
$599$	11.8898	0.485805	0.242903	+	0.970051i	$0.421900\pi$
$600$	0	0
$601$	−16.3768	−0.668025	−0.334012	−	0.942569i	$-0.608403\pi$
$601$	−16.3768	−0.668025	−0.334012	+	0.942569i	$0.608403\pi$
$602$	0	0
$603$	−6.10725	−0.248706
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−15.0653	−0.611480	−0.305740	−	0.952115i	$-0.598904\pi$
$607$	−15.0653	−0.611480	−0.305740	+	0.952115i	$0.598904\pi$
$608$	0	0
$609$	−4.41883	−0.179060
$610$	0	0
$611$	−2.73042	−0.110461
$612$	0	0
$613$	−31.6753	−1.27935	−0.639677	−	0.768644i	$-0.720932\pi$
$613$	−31.6753	−1.27935	−0.639677	+	0.768644i	$0.720932\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	45.4477	1.82966	0.914829	−	0.403842i	$-0.132326\pi$
$617$	45.4477	1.82966	0.914829	+	0.403842i	$0.132326\pi$
$618$	0	0
$619$	−29.2667	−1.17633	−0.588163	−	0.808742i	$-0.700149\pi$
$619$	−29.2667	−1.17633	−0.588163	+	0.808742i	$0.700149\pi$
$620$	0	0
$621$	−1.37683	−0.0552504
$622$	0	0
$623$	−14.9217	−0.597824
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−1.37683	−0.0549854
$628$	0	0
$629$	4.83767	0.192890
$630$	0	0
$631$	−25.0987	−0.999162	−0.499581	−	0.866267i	$-0.666513\pi$
$631$	−25.0987	−0.999162	−0.499581	+	0.866267i	$0.666513\pi$
$632$	0	0
$633$	2.62317	0.104261
$634$	0	0
$635$	0	0
$636$	0	0
$637$	12.5261	0.496301
$638$	0	0
$639$	3.68842	0.145912
$640$	0	0
$641$	−34.7927	−1.37423	−0.687115	−	0.726548i	$-0.741123\pi$
$641$	−34.7927	−1.37423	−0.687115	+	0.726548i	$0.741123\pi$
$642$	0	0
$643$	−41.5317	−1.63785	−0.818926	−	0.573899i	$-0.805430\pi$
$643$	−41.5317	−1.63785	−0.818926	+	0.573899i	$0.805430\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−9.50733	−0.373772	−0.186886	−	0.982382i	$-0.559839\pi$
$647$	−9.50733	−0.373772	−0.186886	+	0.982382i	$0.559839\pi$
$648$	0	0
$649$	4.60442	0.180739
$650$	0	0
$651$	19.5261	0.765287
$652$	0	0
$653$	36.1492	1.41463	0.707315	−	0.706899i	$-0.249906\pi$
$653$	36.1492	1.41463	0.707315	+	0.706899i	$0.249906\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−12.8377	−0.500845
$658$	0	0
$659$	−7.77241	−0.302770	−0.151385	−	0.988475i	$-0.548373\pi$
$659$	−7.77241	−0.302770	−0.151385	+	0.988475i	$0.548373\pi$
$660$	0	0
$661$	−19.3637	−0.753162	−0.376581	−	0.926384i	$-0.622900\pi$
$661$	−19.3637	−0.753162	−0.376581	+	0.926384i	$0.622900\pi$
$662$	0	0
$663$	1.31158	0.0509377
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.37683	−0.0533112
$668$	0	0
$669$	−6.53917	−0.252819
$670$	0	0
$671$	17.9637	0.693479
$672$	0	0
$673$	−22.6753	−0.874070	−0.437035	−	0.899445i	$-0.643971\pi$
$673$	−22.6753	−0.874070	−0.437035	+	0.899445i	$0.643971\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−4.92166	−0.189155	−0.0945774	−	0.995518i	$-0.530150\pi$
$677$	−4.92166	−0.189155	−0.0945774	+	0.995518i	$0.530150\pi$
$678$	0	0
$679$	−23.7593	−0.911799
$680$	0	0
$681$	1.87966	0.0720289
$682$	0	0
$683$	−6.50283	−0.248824	−0.124412	−	0.992231i	$-0.539704\pi$
$683$	−6.50283	−0.248824	−0.124412	+	0.992231i	$0.539704\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	21.2797	0.811873
$688$	0	0
$689$	2.37683	0.0905502
$690$	0	0
$691$	17.0289	0.647810	0.323905	−	0.946090i	$-0.395004\pi$
$691$	17.0289	0.647810	0.323905	+	0.946090i	$0.395004\pi$
$692$	0	0
$693$	19.5261	0.741735
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.21450	0.0838801
$698$	0	0
$699$	15.0522	0.569325
$700$	0	0
$701$	49.8246	1.88185	0.940924	−	0.338617i	$-0.109959\pi$
$701$	49.8246	1.88185	0.940924	+	0.338617i	$0.109959\pi$
$702$	0	0
$703$	1.14925	0.0433447
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−65.8535	−2.47668
$708$	0	0
$709$	39.2667	1.47469	0.737345	−	0.675516i	$-0.236079\pi$
$709$	39.2667	1.47469	0.737345	+	0.675516i	$0.236079\pi$
$710$	0	0
$711$	14.5261	0.544771
$712$	0	0
$713$	6.08400	0.227848
$714$	0	0
$715$	0	0
$716$	0	0
$717$	23.4710	0.876540
$718$	0	0
$719$	37.5652	1.40094	0.700472	−	0.713680i	$-0.252973\pi$
$719$	37.5652	1.40094	0.700472	+	0.713680i	$0.252973\pi$
$720$	0	0
$721$	−27.4608	−1.02269
$722$	0	0
$723$	−8.70716	−0.323823
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−19.6288	−0.727993	−0.363996	−	0.931400i	$-0.618588\pi$
$727$	−19.6288	−0.727993	−0.363996	+	0.931400i	$0.618588\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−2.62317	−0.0970213
$732$	0	0
$733$	46.3694	1.71269	0.856347	−	0.516402i	$-0.172729\pi$
$733$	46.3694	1.71269	0.856347	+	0.516402i	$0.172729\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	26.9869	0.994076
$738$	0	0
$739$	−32.4057	−1.19206	−0.596032	−	0.802960i	$-0.703257\pi$
$739$	−32.4057	−1.19206	−0.596032	+	0.802960i	$0.703257\pi$
$740$	0	0
$741$	0.311583	0.0114463
$742$	0	0
$743$	46.9652	1.72299	0.861494	−	0.507768i	$-0.169529\pi$
$743$	46.9652	1.72299	0.861494	+	0.507768i	$0.169529\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−5.04200	−0.184477
$748$	0	0
$749$	1.37683	0.0503084
$750$	0	0
$751$	50.0709	1.82711	0.913557	−	0.406711i	$-0.133324\pi$
$751$	50.0709	1.82711	0.913557	+	0.406711i	$0.133324\pi$
$752$	0	0
$753$	28.6101	1.04261
$754$	0	0
$755$	0	0
$756$	0	0
$757$	9.47958	0.344541	0.172271	−	0.985050i	$-0.444890\pi$
$757$	9.47958	0.344541	0.172271	+	0.985050i	$0.444890\pi$
$758$	0	0
$759$	6.08400	0.220835
$760$	0	0
$761$	4.31158	0.156295	0.0781474	−	0.996942i	$-0.475100\pi$
$761$	4.31158	0.156295	0.0781474	+	0.996942i	$0.475100\pi$
$762$	0	0
$763$	−88.3188	−3.19736
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.04200	−0.0376244
$768$	0	0
$769$	−37.0260	−1.33519	−0.667596	−	0.744524i	$-0.732677\pi$
$769$	−37.0260	−1.33519	−0.667596	+	0.744524i	$0.732677\pi$
$770$	0	0
$771$	16.0653	0.578576
$772$	0	0
$773$	−39.6492	−1.42608	−0.713041	−	0.701123i	$-0.752682\pi$
$773$	−39.6492	−1.42608	−0.713041	+	0.701123i	$0.752682\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−16.2985	−0.584705
$778$	0	0
$779$	0.526082	0.0188489
$780$	0	0
$781$	−16.2985	−0.583206
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	0	0
$786$	0	0
$787$	44.7377	1.59473	0.797363	−	0.603500i	$-0.206228\pi$
$787$	44.7377	1.59473	0.797363	+	0.603500i	$0.206228\pi$
$788$	0	0
$789$	17.5913	0.626268
$790$	0	0
$791$	26.5130	0.942694
$792$	0	0
$793$	−4.06525	−0.144361
$794$	0	0
$795$	0	0
$796$	0	0
$797$	26.7724	0.948328	0.474164	−	0.880437i	$-0.342750\pi$
$797$	26.7724	0.948328	0.474164	+	0.880437i	$0.342750\pi$
$798$	0	0
$799$	−3.58117	−0.126693
$800$	0	0
$801$	3.37683	0.119315
$802$	0	0
$803$	56.7275	2.00187
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−8.67533	−0.305386
$808$	0	0
$809$	47.0260	1.65335	0.826673	−	0.562683i	$-0.190231\pi$
$809$	47.0260	1.65335	0.826673	+	0.562683i	$0.190231\pi$
$810$	0	0
$811$	−37.9637	−1.33308	−0.666542	−	0.745467i	$-0.732226\pi$
$811$	−37.9637	−1.33308	−0.666542	+	0.745467i	$0.732226\pi$
$812$	0	0
$813$	16.4188	0.575834
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−0.623166	−0.0218018
$818$	0	0
$819$	−4.41883	−0.154407
$820$	0	0
$821$	23.1362	0.807458	0.403729	−	0.914879i	$-0.367714\pi$
$821$	23.1362	0.807458	0.403729	+	0.914879i	$0.367714\pi$
$822$	0	0
$823$	−37.1492	−1.29494	−0.647471	−	0.762090i	$-0.724173\pi$
$823$	−37.1492	−1.29494	−0.647471	+	0.762090i	$0.724173\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.91150	0.240336	0.120168	−	0.992754i	$-0.461657\pi$
$827$	6.91150	0.240336	0.120168	+	0.992754i	$0.461657\pi$
$828$	0	0
$829$	15.6101	0.542160	0.271080	−	0.962557i	$-0.412619\pi$
$829$	15.6101	0.542160	0.271080	+	0.962557i	$0.412619\pi$
$830$	0	0
$831$	−7.67533	−0.266254
$832$	0	0
$833$	16.4290	0.569231
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−4.41883	−0.152737
$838$	0	0
$839$	−52.1883	−1.80174	−0.900871	−	0.434088i	$-0.857071\pi$
$839$	−52.1883	−1.80174	−0.900871	+	0.434088i	$0.857071\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	1.47392	0.0507644
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−37.6753	−1.29454
$848$	0	0
$849$	18.5130	0.635364
$850$	0	0
$851$	−5.07834	−0.174083
$852$	0	0
$853$	−3.01875	−0.103360	−0.0516800	−	0.998664i	$-0.516458\pi$
$853$	−3.01875	−0.103360	−0.0516800	+	0.998664i	$0.516458\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	44.1043	1.50658	0.753288	−	0.657691i	$-0.228467\pi$
$857$	44.1043	1.50658	0.753288	+	0.657691i	$0.228467\pi$
$858$	0	0
$859$	18.1680	0.619884	0.309942	−	0.950755i	$-0.399690\pi$
$859$	18.1680	0.619884	0.309942	+	0.950755i	$0.399690\pi$
$860$	0	0
$861$	−7.46083	−0.254264
$862$	0	0
$863$	−16.6464	−0.566651	−0.283325	−	0.959024i	$-0.591438\pi$
$863$	−16.6464	−0.566651	−0.283325	+	0.959024i	$0.591438\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−15.2797	−0.518928
$868$	0	0
$869$	−64.1883	−2.17744
$870$	0	0
$871$	−6.10725	−0.206936
$872$	0	0
$873$	5.37683	0.181978
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−13.7724	−0.465061	−0.232531	−	0.972589i	$-0.574701\pi$
$877$	−13.7724	−0.465061	−0.232531	+	0.972589i	$0.574701\pi$
$878$	0	0
$879$	−19.0522	−0.642614
$880$	0	0
$881$	−8.06525	−0.271725	−0.135863	−	0.990728i	$-0.543381\pi$
$881$	−8.06525	−0.271725	−0.135863	+	0.990728i	$0.543381\pi$
$882$	0	0
$883$	−33.9363	−1.14205	−0.571024	−	0.820933i	$-0.693454\pi$
$883$	−33.9363	−1.14205	−0.571024	+	0.820933i	$0.693454\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32.2985	1.08448	0.542239	−	0.840224i	$-0.317577\pi$
$887$	32.2985	1.08448	0.542239	+	0.840224i	$0.317577\pi$
$888$	0	0
$889$	31.5913	1.05954
$890$	0	0
$891$	−4.41883	−0.148036
$892$	0	0
$893$	−0.850752	−0.0284693
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.37683	−0.0459711
$898$	0	0
$899$	−4.41883	−0.147376
$900$	0	0
$901$	3.11742	0.103856
$902$	0	0
$903$	8.83767	0.294099
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−23.8058	−0.790460	−0.395230	−	0.918582i	$-0.629335\pi$
$907$	−23.8058	−0.790460	−0.395230	+	0.918582i	$0.629335\pi$
$908$	0	0
$909$	14.9029	0.494299
$910$	0	0
$911$	−18.5970	−0.616146	−0.308073	−	0.951363i	$-0.599684\pi$
$911$	−18.5970	−0.616146	−0.308073	+	0.951363i	$0.599684\pi$
$912$	0	0
$913$	22.2797	0.737352
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−27.8898	−0.921003
$918$	0	0
$919$	14.4421	0.476400	0.238200	−	0.971216i	$-0.423443\pi$
$919$	14.4421	0.476400	0.238200	+	0.971216i	$0.423443\pi$
$920$	0	0
$921$	16.1174	0.531087
$922$	0	0
$923$	3.68842	0.121406
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6.21450	0.204111
$928$	0	0
$929$	−16.8246	−0.551997	−0.275998	−	0.961158i	$-0.589008\pi$
$929$	−16.8246	−0.551997	−0.275998	+	0.961158i	$0.589008\pi$
$930$	0	0
$931$	3.90292	0.127913
$932$	0	0
$933$	3.78550	0.123932
$934$	0	0
$935$	0	0
$936$	0	0
$937$	38.1231	1.24543	0.622713	−	0.782450i	$-0.286030\pi$
$937$	38.1231	1.24543	0.622713	+	0.782450i	$0.286030\pi$
$938$	0	0
$939$	−15.0000	−0.489506
$940$	0	0
$941$	−0.408667	−0.0133222	−0.00666108	−	0.999978i	$-0.502120\pi$
$941$	−0.408667	−0.0133222	−0.00666108	+	0.999978i	$0.502120\pi$
$942$	0	0
$943$	−2.32467	−0.0757016
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−30.5028	−0.991209	−0.495604	−	0.868548i	$-0.665053\pi$
$947$	−30.5028	−0.991209	−0.495604	+	0.868548i	$0.665053\pi$
$948$	0	0
$949$	−12.8377	−0.416728
$950$	0	0
$951$	−22.5130	−0.730034
$952$	0	0
$953$	24.4942	0.793447	0.396723	−	0.917938i	$-0.370147\pi$
$953$	24.4942	0.793447	0.396723	+	0.917938i	$0.370147\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−4.41883	−0.142841
$958$	0	0
$959$	−78.5333	−2.53597
$960$	0	0
$961$	−11.4739	−0.370126
$962$	0	0
$963$	−0.311583	−0.0100406
$964$	0	0
$965$	0	0
$966$	0	0
$967$	16.5868	0.533396	0.266698	−	0.963780i	$-0.414067\pi$
$967$	16.5868	0.533396	0.266698	+	0.963780i	$0.414067\pi$
$968$	0	0
$969$	0.408667	0.0131283
$970$	0	0
$971$	−54.8711	−1.76090	−0.880448	−	0.474142i	$-0.842758\pi$
$971$	−54.8711	−1.76090	−0.880448	+	0.474142i	$0.842758\pi$
$972$	0	0
$973$	21.3768	0.685310
$974$	0	0
$975$	0	0
$976$	0	0
$977$	19.5448	0.625294	0.312647	−	0.949869i	$-0.398784\pi$
$977$	19.5448	0.625294	0.312647	+	0.949869i	$0.398784\pi$
$978$	0	0
$979$	−14.9217	−0.476898
$980$	0	0
$981$	19.9869	0.638133
$982$	0	0
$983$	58.8217	1.87612	0.938060	−	0.346473i	$-0.112621\pi$
$983$	58.8217	1.87612	0.938060	+	0.346473i	$0.112621\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	12.0653	0.384041
$988$	0	0
$989$	2.75367	0.0875615
$990$	0	0
$991$	−10.2651	−0.326081	−0.163041	−	0.986619i	$-0.552130\pi$
$991$	−10.2651	−0.326081	−0.163041	+	0.986619i	$0.552130\pi$
$992$	0	0
$993$	12.8377	0.407391
$994$	0	0
$995$	0	0
$996$	0	0
$997$	23.6101	0.747739	0.373869	−	0.927481i	$-0.378031\pi$
$997$	23.6101	0.747739	0.373869	+	0.927481i	$0.378031\pi$
$998$	0	0
$999$	3.68842	0.116696

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7800.2.a.bk.1.3	✓	3	5.4	even	2
7800.2.a.bq.1.1	yes	3	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -4.41883 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -4.41883 q^{7} +1.00000 q^{9} -4.41883 q^{11} +1.00000 q^{13} +1.31158 q^{17} +0.311583 q^{19} -4.41883 q^{21} -1.37683 q^{23} +1.00000 q^{27} +1.00000 q^{29} -4.41883 q^{31} -4.41883 q^{33} +3.68842 q^{37} +1.00000 q^{39} +1.68842 q^{41} -2.00000 q^{43} -2.73042 q^{47} +12.5261 q^{49} +1.31158 q^{51} +2.37683 q^{53} +0.311583 q^{57} -1.04200 q^{59} -4.06525 q^{61} -4.41883 q^{63} -6.10725 q^{67} -1.37683 q^{69} +3.68842 q^{71} -12.8377 q^{73} +19.5261 q^{77} +14.5261 q^{79} +1.00000 q^{81} -5.04200 q^{83} +1.00000 q^{87} +3.37683 q^{89} -4.41883 q^{91} -4.41883 q^{93} +5.37683 q^{97} -4.41883 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{9} + 3 q^{13} + 6 q^{17} + 3 q^{19} + 3 q^{27} + 3 q^{29} + 9 q^{37} + 3 q^{39} + 3 q^{41} - 6 q^{43} + 3 q^{47} + 9 q^{49} + 6 q^{51} + 3 q^{53} + 3 q^{57} + 6 q^{59} - 6 q^{61} - 3 q^{67} + 9 q^{71} - 12 q^{73} + 30 q^{77} + 15 q^{79} + 3 q^{81} - 6 q^{83} + 3 q^{87} + 6 q^{89} + 12 q^{97}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^9 + 3 * q^13 + 6 * q^17 + 3 * q^19 + 3 * q^27 + 3 * q^29 + 9 * q^37 + 3 * q^39 + 3 * q^41 - 6 * q^43 + 3 * q^47 + 9 * q^49 + 6 * q^51 + 3 * q^53 + 3 * q^57 + 6 * q^59 - 6 * q^61 - 3 * q^67 + 9 * q^71 - 12 * q^73 + 30 * q^77 + 15 * q^79 + 3 * q^81 - 6 * q^83 + 3 * q^87 + 6 * q^89 + 12 * q^97

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