LMFDB - Embedded newform 1080.6.a.c.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1080,6,Mod(1,1080)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1080.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1080, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 1080 = 2^{3} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1080.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-50,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$173.214525398$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{11}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 11 $ x^2 - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.31662$ of defining polynomial
Character	$\chi$	$=$	1080.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-25.0000 q^{5} +101.499 q^{7} +254.095 q^{11} +1120.10 q^{13} +2168.28 q^{17} +486.083 q^{19} -231.581 q^{23} +625.000 q^{25} +8568.22 q^{29} -763.151 q^{31} -2537.47 q^{35} +3232.80 q^{37} +4426.31 q^{41} -11685.9 q^{43} +14009.0 q^{47} -6505.01 q^{49} -25320.1 q^{53} -6352.38 q^{55} +14810.2 q^{59} -18101.1 q^{61} -28002.4 q^{65} -8828.46 q^{67} -24891.7 q^{71} +38234.7 q^{73} +25790.3 q^{77} -71930.4 q^{79} +57290.5 q^{83} -54207.1 q^{85} -5294.34 q^{89} +113688. q^{91} -12152.1 q^{95} +32086.1 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 50 q^{5} + 4 q^{7} - 248 q^{11} + 1484 q^{13} + 1670 q^{17} - 1774 q^{19} + 2482 q^{23} + 1250 q^{25} + 3724 q^{29} - 9526 q^{31} - 100 q^{35} + 5988 q^{37} + 9768 q^{41} - 20944 q^{43} - 3344 q^{47}+ \cdots - 54112 q^{97}+O(q^{100}) $ 2 * q - 50 * q^5 + 4 * q^7 - 248 * q^11 + 1484 * q^13 + 1670 * q^17 - 1774 * q^19 + 2482 * q^23 + 1250 * q^25 + 3724 * q^29 - 9526 * q^31 - 100 * q^35 + 5988 * q^37 + 9768 * q^41 - 20944 * q^43 - 3344 * q^47 - 13806 * q^49 - 13746 * q^53 + 6200 * q^55 - 11572 * q^59 - 22750 * q^61 - 37100 * q^65 + 5188 * q^67 - 52848 * q^71 + 112488 * q^73 + 74744 * q^77 - 128538 * q^79 + 60374 * q^83 - 41750 * q^85 - 79800 * q^89 + 78208 * q^91 + 44350 * q^95 - 54112 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	101.499	0.782917	0.391458	−	0.920196i	$-0.371971\pi$
$7$	101.499	0.782917	0.391458	+	0.920196i	$0.371971\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	254.095	0.633162	0.316581	−	0.948565i	$-0.397465\pi$
$11$	254.095	0.633162	0.316581	+	0.948565i	$0.397465\pi$
$12$	0	0
$13$	1120.10	1.83822	0.919108	−	0.394006i	$-0.128911\pi$
$13$	1120.10	1.83822	0.919108	+	0.394006i	$0.128911\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2168.28	1.81967	0.909837	−	0.414965i	$-0.136206\pi$
$17$	2168.28	1.81967	0.909837	+	0.414965i	$0.136206\pi$
$18$	0	0
$19$	486.083	0.308906	0.154453	−	0.988000i	$-0.450639\pi$
$19$	486.083	0.308906	0.154453	+	0.988000i	$0.450639\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−231.581	−0.0912818	−0.0456409	−	0.998958i	$-0.514533\pi$
$23$	−231.581	−0.0912818	−0.0456409	+	0.998958i	$0.514533\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8568.22	1.89189	0.945944	−	0.324330i	$-0.105139\pi$
$29$	8568.22	1.89189	0.945944	+	0.324330i	$0.105139\pi$
$30$	0	0
$31$	−763.151	−0.142628	−0.0713142	−	0.997454i	$-0.522719\pi$
$31$	−763.151	−0.142628	−0.0713142	+	0.997454i	$0.522719\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2537.47	−0.350131
$36$	0	0
$37$	3232.80	0.388217	0.194108	−	0.980980i	$-0.437819\pi$
$37$	3232.80	0.388217	0.194108	+	0.980980i	$0.437819\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4426.31	0.411227	0.205614	−	0.978633i	$-0.434081\pi$
$41$	4426.31	0.411227	0.205614	+	0.978633i	$0.434081\pi$
$42$	0	0
$43$	−11685.9	−0.963808	−0.481904	−	0.876224i	$-0.660055\pi$
$43$	−11685.9	−0.963808	−0.481904	+	0.876224i	$0.660055\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	14009.0	0.925044	0.462522	−	0.886608i	$-0.346945\pi$
$47$	14009.0	0.925044	0.462522	+	0.886608i	$0.346945\pi$
$48$	0	0
$49$	−6505.01	−0.387041
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−25320.1	−1.23816	−0.619078	−	0.785330i	$-0.712493\pi$
$53$	−25320.1	−1.23816	−0.619078	+	0.785330i	$0.712493\pi$
$54$	0	0
$55$	−6352.38	−0.283159
$56$	0	0
$57$	0	0
$58$	0	0
$59$	14810.2	0.553901	0.276950	−	0.960884i	$-0.410676\pi$
$59$	14810.2	0.553901	0.276950	+	0.960884i	$0.410676\pi$
$60$	0	0
$61$	−18101.1	−0.622846	−0.311423	−	0.950271i	$-0.600806\pi$
$61$	−18101.1	−0.622846	−0.311423	+	0.950271i	$0.600806\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−28002.4	−0.822075
$66$	0	0
$67$	−8828.46	−0.240269	−0.120134	−	0.992758i	$-0.538333\pi$
$67$	−8828.46	−0.240269	−0.120134	+	0.992758i	$0.538333\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−24891.7	−0.586015	−0.293008	−	0.956110i	$-0.594656\pi$
$71$	−24891.7	−0.586015	−0.293008	+	0.956110i	$0.594656\pi$
$72$	0	0
$73$	38234.7	0.839752	0.419876	−	0.907582i	$-0.362074\pi$
$73$	38234.7	0.839752	0.419876	+	0.907582i	$0.362074\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	25790.3	0.495713
$78$	0	0
$79$	−71930.4	−1.29672	−0.648358	−	0.761336i	$-0.724544\pi$
$79$	−71930.4	−1.29672	−0.648358	+	0.761336i	$0.724544\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	57290.5	0.912824	0.456412	−	0.889769i	$-0.349134\pi$
$83$	57290.5	0.912824	0.456412	+	0.889769i	$0.349134\pi$
$84$	0	0
$85$	−54207.1	−0.813783
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−5294.34	−0.0708495	−0.0354247	−	0.999372i	$-0.511278\pi$
$89$	−5294.34	−0.0708495	−0.0354247	+	0.999372i	$0.511278\pi$
$90$	0	0
$91$	113688.	1.43917
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−12152.1	−0.138147
$96$	0	0
$97$	32086.1	0.346248	0.173124	−	0.984900i	$-0.444614\pi$
$97$	32086.1	0.346248	0.173124	+	0.984900i	$0.444614\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−32724.3	−0.319203	−0.159601	−	0.987182i	$-0.551021\pi$
$101$	−32724.3	−0.319203	−0.159601	+	0.987182i	$0.551021\pi$
$102$	0	0
$103$	127296.	1.18228	0.591142	−	0.806567i	$-0.298677\pi$
$103$	127296.	1.18228	0.591142	+	0.806567i	$0.298677\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	107983.	0.911794	0.455897	−	0.890033i	$-0.349319\pi$
$107$	107983.	0.911794	0.455897	+	0.890033i	$0.349319\pi$
$108$	0	0
$109$	−150394.	−1.21245	−0.606225	−	0.795293i	$-0.707317\pi$
$109$	−150394.	−1.21245	−0.606225	+	0.795293i	$0.707317\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	4065.65	0.0299526	0.0149763	−	0.999888i	$-0.495233\pi$
$113$	4065.65	0.0299526	0.0149763	+	0.999888i	$0.495233\pi$
$114$	0	0
$115$	5789.54	0.0408225
$116$	0	0
$117$	0	0
$118$	0	0
$119$	220078.	1.42465
$120$	0	0
$121$	−96486.6	−0.599106
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	168442.	0.926703	0.463352	−	0.886175i	$-0.346647\pi$
$127$	168442.	0.926703	0.463352	+	0.886175i	$0.346647\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−351179.	−1.78793	−0.893964	−	0.448139i	$-0.852087\pi$
$131$	−351179.	−1.78793	−0.893964	+	0.448139i	$0.852087\pi$
$132$	0	0
$133$	49336.8	0.241848
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−47249.1	−0.215076	−0.107538	−	0.994201i	$-0.534297\pi$
$137$	−47249.1	−0.215076	−0.107538	+	0.994201i	$0.534297\pi$
$138$	0	0
$139$	−314219.	−1.37941	−0.689707	−	0.724088i	$-0.742261\pi$
$139$	−314219.	−1.37941	−0.689707	+	0.724088i	$0.742261\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	284611.	1.16389
$144$	0	0
$145$	−214205.	−0.846078
$146$	0	0
$147$	0	0
$148$	0	0
$149$	248250.	0.916060	0.458030	−	0.888937i	$-0.348555\pi$
$149$	248250.	0.916060	0.458030	+	0.888937i	$0.348555\pi$
$150$	0	0
$151$	129307.	0.461507	0.230754	−	0.973012i	$-0.425881\pi$
$151$	129307.	0.461507	0.230754	+	0.973012i	$0.425881\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	19078.8	0.0637854
$156$	0	0
$157$	−86956.8	−0.281549	−0.140775	−	0.990042i	$-0.544959\pi$
$157$	−86956.8	−0.281549	−0.140775	+	0.990042i	$0.544959\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−23505.2	−0.0714660
$162$	0	0
$163$	343607.	1.01296	0.506480	−	0.862252i	$-0.330946\pi$
$163$	343607.	1.01296	0.506480	+	0.862252i	$0.330946\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−472717.	−1.31163	−0.655813	−	0.754923i	$-0.727674\pi$
$167$	−472717.	−1.31163	−0.655813	+	0.754923i	$0.727674\pi$
$168$	0	0
$169$	883320.	2.37904
$170$	0	0
$171$	0	0
$172$	0	0
$173$	257127.	0.653179	0.326589	−	0.945166i	$-0.394101\pi$
$173$	257127.	0.653179	0.326589	+	0.945166i	$0.394101\pi$
$174$	0	0
$175$	63436.7	0.156583
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−170170.	−0.396963	−0.198481	−	0.980105i	$-0.563601\pi$
$179$	−170170.	−0.396963	−0.198481	+	0.980105i	$0.563601\pi$
$180$	0	0
$181$	−159984.	−0.362979	−0.181489	−	0.983393i	$-0.558092\pi$
$181$	−159984.	−0.362979	−0.181489	+	0.983393i	$0.558092\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−80819.9	−0.173616
$186$	0	0
$187$	550950.	1.15215
$188$	0	0
$189$	0	0
$190$	0	0
$191$	627362.	1.24433	0.622164	−	0.782887i	$-0.286254\pi$
$191$	627362.	1.24433	0.622164	+	0.782887i	$0.286254\pi$
$192$	0	0
$193$	682959.	1.31978	0.659890	−	0.751362i	$-0.270603\pi$
$193$	682959.	1.31978	0.659890	+	0.751362i	$0.270603\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	572449.	1.05092	0.525462	−	0.850817i	$-0.323893\pi$
$197$	572449.	1.05092	0.525462	+	0.850817i	$0.323893\pi$
$198$	0	0
$199$	−905955.	−1.62171	−0.810856	−	0.585245i	$-0.800998\pi$
$199$	−905955.	−1.62171	−0.810856	+	0.585245i	$0.800998\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	869663.	1.48119
$204$	0	0
$205$	−110658.	−0.183906
$206$	0	0
$207$	0	0
$208$	0	0
$209$	123511.	0.195587
$210$	0	0
$211$	−1.23786e6	−1.91411	−0.957054	−	0.289910i	$-0.906375\pi$
$211$	−1.23786e6	−1.91411	−0.957054	+	0.289910i	$0.906375\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	292147.	0.431028
$216$	0	0
$217$	−77458.8	−0.111666
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.42868e6	3.34495
$222$	0	0
$223$	−1.32540e6	−1.78479	−0.892393	−	0.451258i	$-0.850975\pi$
$223$	−1.32540e6	−1.78479	−0.892393	+	0.451258i	$0.850975\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	37893.2	0.0488086	0.0244043	−	0.999702i	$-0.492231\pi$
$227$	37893.2	0.0488086	0.0244043	+	0.999702i	$0.492231\pi$
$228$	0	0
$229$	96404.7	0.121481	0.0607406	−	0.998154i	$-0.480654\pi$
$229$	96404.7	0.121481	0.0607406	+	0.998154i	$0.480654\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.36944e6	−1.65255	−0.826274	−	0.563268i	$-0.809544\pi$
$233$	−1.36944e6	−1.65255	−0.826274	+	0.563268i	$0.809544\pi$
$234$	0	0
$235$	−350225.	−0.413692
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.42820e6	1.61732	0.808658	−	0.588280i	$-0.200195\pi$
$239$	1.42820e6	1.61732	0.808658	+	0.588280i	$0.200195\pi$
$240$	0	0
$241$	758335.	0.841043	0.420522	−	0.907283i	$-0.361847\pi$
$241$	758335.	0.841043	0.420522	+	0.907283i	$0.361847\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	162625.	0.173090
$246$	0	0
$247$	544459.	0.567836
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.13225e6	1.13438	0.567188	−	0.823588i	$-0.308031\pi$
$251$	1.13225e6	1.13438	0.567188	+	0.823588i	$0.308031\pi$
$252$	0	0
$253$	−58843.7	−0.0577961
$254$	0	0
$255$	0	0
$256$	0	0
$257$	55508.7	0.0524238	0.0262119	−	0.999656i	$-0.491656\pi$
$257$	55508.7	0.0524238	0.0262119	+	0.999656i	$0.491656\pi$
$258$	0	0
$259$	328125.	0.303941
$260$	0	0
$261$	0	0
$262$	0	0
$263$	946554.	0.843832	0.421916	−	0.906635i	$-0.361358\pi$
$263$	946554.	0.843832	0.421916	+	0.906635i	$0.361358\pi$
$264$	0	0
$265$	633002.	0.553720
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−537772.	−0.453124	−0.226562	−	0.973997i	$-0.572749\pi$
$269$	−537772.	−0.453124	−0.226562	+	0.973997i	$0.572749\pi$
$270$	0	0
$271$	−1.42760e6	−1.18082	−0.590408	−	0.807105i	$-0.701033\pi$
$271$	−1.42760e6	−1.18082	−0.590408	+	0.807105i	$0.701033\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	158810.	0.126632
$276$	0	0
$277$	−451652.	−0.353675	−0.176837	−	0.984240i	$-0.556587\pi$
$277$	−451652.	−0.353675	−0.176837	+	0.984240i	$0.556587\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.11232e6	0.840355	0.420178	−	0.907442i	$-0.361968\pi$
$281$	1.11232e6	0.840355	0.420178	+	0.907442i	$0.361968\pi$
$282$	0	0
$283$	1.03211e6	0.766052	0.383026	−	0.923737i	$-0.374882\pi$
$283$	1.03211e6	0.766052	0.383026	+	0.923737i	$0.374882\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	449264.	0.321957
$288$	0	0
$289$	3.28159e6	2.31122
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.17904e6	−0.802341	−0.401171	−	0.916003i	$-0.631397\pi$
$293$	−1.17904e6	−0.802341	−0.401171	+	0.916003i	$0.631397\pi$
$294$	0	0
$295$	−370256.	−0.247712
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−259393.	−0.167796
$300$	0	0
$301$	−1.18610e6	−0.754581
$302$	0	0
$303$	0	0
$304$	0	0
$305$	452528.	0.278545
$306$	0	0
$307$	535779.	0.324444	0.162222	−	0.986754i	$-0.448134\pi$
$307$	535779.	0.324444	0.162222	+	0.986754i	$0.448134\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2.03141e6	−1.19096	−0.595479	−	0.803371i	$-0.703038\pi$
$311$	−2.03141e6	−1.19096	−0.595479	+	0.803371i	$0.703038\pi$
$312$	0	0
$313$	−3.04001e6	−1.75394	−0.876969	−	0.480546i	$-0.840438\pi$
$313$	−3.04001e6	−1.75394	−0.876969	+	0.480546i	$0.840438\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.44246e6	0.806224	0.403112	−	0.915151i	$-0.367928\pi$
$317$	1.44246e6	0.806224	0.403112	+	0.915151i	$0.367928\pi$
$318$	0	0
$319$	2.17714e6	1.19787
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.05396e6	0.562108
$324$	0	0
$325$	700060.	0.367643
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.42190e6	0.724233
$330$	0	0
$331$	82656.7	0.0414675	0.0207337	−	0.999785i	$-0.493400\pi$
$331$	82656.7	0.0414675	0.0207337	+	0.999785i	$0.493400\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	220711.	0.107452
$336$	0	0
$337$	−485901.	−0.233063	−0.116531	−	0.993187i	$-0.537178\pi$
$337$	−485901.	−0.233063	−0.116531	+	0.993187i	$0.537178\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−193913.	−0.0903069
$342$	0	0
$343$	−2.36614e6	−1.08594
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.31240e6	−0.585115	−0.292557	−	0.956248i	$-0.594506\pi$
$347$	−1.31240e6	−0.585115	−0.292557	+	0.956248i	$0.594506\pi$
$348$	0	0
$349$	−1.50340e6	−0.660710	−0.330355	−	0.943857i	$-0.607168\pi$
$349$	−1.50340e6	−0.660710	−0.330355	+	0.943857i	$0.607168\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.45396e6	1.47530	0.737650	−	0.675183i	$-0.235935\pi$
$353$	3.45396e6	1.47530	0.737650	+	0.675183i	$0.235935\pi$
$354$	0	0
$355$	622293.	0.262074
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−548375.	−0.224565	−0.112282	−	0.993676i	$-0.535816\pi$
$359$	−548375.	−0.224565	−0.112282	+	0.993676i	$0.535816\pi$
$360$	0	0
$361$	−2.23982e6	−0.904577
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−955868.	−0.375548
$366$	0	0
$367$	−2.41160e6	−0.934633	−0.467316	−	0.884090i	$-0.654779\pi$
$367$	−2.41160e6	−0.934633	−0.467316	+	0.884090i	$0.654779\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.56995e6	−0.969373
$372$	0	0
$373$	2.46980e6	0.919155	0.459578	−	0.888138i	$-0.348001\pi$
$373$	2.46980e6	0.919155	0.459578	+	0.888138i	$0.348001\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.59722e6	3.47770
$378$	0	0
$379$	929815.	0.332505	0.166253	−	0.986083i	$-0.446833\pi$
$379$	929815.	0.332505	0.166253	+	0.986083i	$0.446833\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5.57604e6	1.94236	0.971179	−	0.238352i	$-0.0766071\pi$
$383$	5.57604e6	1.94236	0.971179	+	0.238352i	$0.0766071\pi$
$384$	0	0
$385$	−644759.	−0.221690
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.57190e6	0.861747	0.430874	−	0.902412i	$-0.358205\pi$
$389$	2.57190e6	0.861747	0.430874	+	0.902412i	$0.358205\pi$
$390$	0	0
$391$	−502134.	−0.166103
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.79826e6	0.579909
$396$	0	0
$397$	2.70282e6	0.860677	0.430338	−	0.902668i	$-0.358394\pi$
$397$	2.70282e6	0.860677	0.430338	+	0.902668i	$0.358394\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.36521e6	0.423974	0.211987	−	0.977272i	$-0.432007\pi$
$401$	1.36521e6	0.423974	0.211987	+	0.977272i	$0.432007\pi$
$402$	0	0
$403$	−854801.	−0.262182
$404$	0	0
$405$	0	0
$406$	0	0
$407$	821438.	0.245804
$408$	0	0
$409$	3.89695e6	1.15190	0.575952	−	0.817484i	$-0.304632\pi$
$409$	3.89695e6	1.15190	0.575952	+	0.817484i	$0.304632\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.50322e6	0.433658
$414$	0	0
$415$	−1.43226e6	−0.408227
$416$	0	0
$417$	0	0
$418$	0	0
$419$	5.03961e6	1.40237	0.701184	−	0.712980i	$-0.252655\pi$
$419$	5.03961e6	1.40237	0.701184	+	0.712980i	$0.252655\pi$
$420$	0	0
$421$	5.49218e6	1.51022	0.755109	−	0.655600i	$-0.227584\pi$
$421$	5.49218e6	1.51022	0.755109	+	0.655600i	$0.227584\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.35518e6	0.363935
$426$	0	0
$427$	−1.83724e6	−0.487637
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4.75587e6	1.23321	0.616604	−	0.787273i	$-0.288508\pi$
$431$	4.75587e6	1.23321	0.616604	+	0.787273i	$0.288508\pi$
$432$	0	0
$433$	−5.67962e6	−1.45579	−0.727897	−	0.685687i	$-0.759502\pi$
$433$	−5.67962e6	−1.45579	−0.727897	+	0.685687i	$0.759502\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−112568.	−0.0281975
$438$	0	0
$439$	−2.85947e6	−0.708147	−0.354074	−	0.935218i	$-0.615204\pi$
$439$	−2.85947e6	−0.708147	−0.354074	+	0.935218i	$0.615204\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−202730.	−0.0490804	−0.0245402	−	0.999699i	$-0.507812\pi$
$443$	−202730.	−0.0490804	−0.0245402	+	0.999699i	$0.507812\pi$
$444$	0	0
$445$	132358.	0.0316848
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4.55132e6	−1.06542	−0.532711	−	0.846298i	$-0.678827\pi$
$449$	−4.55132e6	−1.06542	−0.532711	+	0.846298i	$0.678827\pi$
$450$	0	0
$451$	1.12470e6	0.260373
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.84221e6	−0.643616
$456$	0	0
$457$	6.43082e6	1.44037	0.720187	−	0.693780i	$-0.244056\pi$
$457$	6.43082e6	1.44037	0.720187	+	0.693780i	$0.244056\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−3.31375e6	−0.726219	−0.363110	−	0.931746i	$-0.618285\pi$
$461$	−3.31375e6	−0.726219	−0.363110	+	0.931746i	$0.618285\pi$
$462$	0	0
$463$	6.05290e6	1.31223	0.656116	−	0.754660i	$-0.272198\pi$
$463$	6.05290e6	1.31223	0.656116	+	0.754660i	$0.272198\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.11777e6	1.51026	0.755130	−	0.655575i	$-0.227574\pi$
$467$	7.11777e6	1.51026	0.755130	+	0.655575i	$0.227574\pi$
$468$	0	0
$469$	−896077.	−0.188111
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2.96933e6	−0.610246
$474$	0	0
$475$	303802.	0.0617812
$476$	0	0
$477$	0	0
$478$	0	0
$479$	4.76924e6	0.949752	0.474876	−	0.880053i	$-0.342493\pi$
$479$	4.76924e6	0.949752	0.474876	+	0.880053i	$0.342493\pi$
$480$	0	0
$481$	3.62104e6	0.713626
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−802151.	−0.154847
$486$	0	0
$487$	−1.02959e7	−1.96718	−0.983588	−	0.180428i	$-0.942252\pi$
$487$	−1.02959e7	−1.96718	−0.983588	+	0.180428i	$0.942252\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	2.73418e6	0.511827	0.255913	−	0.966700i	$-0.417624\pi$
$491$	2.73418e6	0.511827	0.255913	+	0.966700i	$0.417624\pi$
$492$	0	0
$493$	1.85783e7	3.44262
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.52648e6	−0.458801
$498$	0	0
$499$	−4.43017e6	−0.796470	−0.398235	−	0.917284i	$-0.630377\pi$
$499$	−4.43017e6	−0.796470	−0.398235	+	0.917284i	$0.630377\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1.06852e7	−1.88306	−0.941528	−	0.336935i	$-0.890610\pi$
$503$	−1.06852e7	−1.88306	−0.941528	+	0.336935i	$0.890610\pi$
$504$	0	0
$505$	818107.	0.142752
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.21388e6	0.378756	0.189378	−	0.981904i	$-0.439353\pi$
$509$	2.21388e6	0.378756	0.189378	+	0.981904i	$0.439353\pi$
$510$	0	0
$511$	3.88078e6	0.657456
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3.18240e6	−0.528734
$516$	0	0
$517$	3.55962e6	0.585703
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−6.57432e6	−1.06110	−0.530550	−	0.847654i	$-0.678015\pi$
$521$	−6.57432e6	−1.06110	−0.530550	+	0.847654i	$0.678015\pi$
$522$	0	0
$523$	630333.	0.100766	0.0503832	−	0.998730i	$-0.483956\pi$
$523$	630333.	0.100766	0.0503832	+	0.998730i	$0.483956\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.65473e6	−0.259537
$528$	0	0
$529$	−6.38271e6	−0.991668
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.95788e6	0.755924
$534$	0	0
$535$	−2.69958e6	−0.407767
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.65289e6	−0.245060
$540$	0	0
$541$	−4.57583e6	−0.672167	−0.336083	−	0.941832i	$-0.609102\pi$
$541$	−4.57583e6	−0.672167	−0.336083	+	0.941832i	$0.609102\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3.75984e6	0.542224
$546$	0	0
$547$	−1.35268e7	−1.93297	−0.966486	−	0.256721i	$-0.917358\pi$
$547$	−1.35268e7	−1.93297	−0.966486	+	0.256721i	$0.917358\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.16486e6	0.584415
$552$	0	0
$553$	−7.30085e6	−1.01522
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.36802e6	0.186833	0.0934166	−	0.995627i	$-0.470221\pi$
$557$	1.36802e6	0.186833	0.0934166	+	0.995627i	$0.470221\pi$
$558$	0	0
$559$	−1.30893e7	−1.77169
$560$	0	0
$561$	0	0
$562$	0	0
$563$	287983.	0.0382909	0.0191454	−	0.999817i	$-0.493905\pi$
$563$	287983.	0.0382909	0.0191454	+	0.999817i	$0.493905\pi$
$564$	0	0
$565$	−101641.	−0.0133952
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.02239e7	−1.32384	−0.661920	−	0.749575i	$-0.730258\pi$
$569$	−1.02239e7	−1.32384	−0.661920	+	0.749575i	$0.730258\pi$
$570$	0	0
$571$	6.38462e6	0.819493	0.409746	−	0.912199i	$-0.365617\pi$
$571$	6.38462e6	0.819493	0.409746	+	0.912199i	$0.365617\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−144738.	−0.0182564
$576$	0	0
$577$	−4.03899e6	−0.505048	−0.252524	−	0.967591i	$-0.581261\pi$
$577$	−4.03899e6	−0.505048	−0.252524	+	0.967591i	$0.581261\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	5.81491e6	0.714665
$582$	0	0
$583$	−6.43371e6	−0.783953
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8.79929e6	−1.05403	−0.527014	−	0.849856i	$-0.676689\pi$
$587$	−8.79929e6	−1.05403	−0.527014	+	0.849856i	$0.676689\pi$
$588$	0	0
$589$	−370954.	−0.0440587
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−3.67271e6	−0.428894	−0.214447	−	0.976736i	$-0.568795\pi$
$593$	−3.67271e6	−0.428894	−0.214447	+	0.976736i	$0.568795\pi$
$594$	0	0
$595$	−5.50195e6	−0.637124
$596$	0	0
$597$	0	0
$598$	0	0
$599$	3.22693e6	0.367470	0.183735	−	0.982976i	$-0.441181\pi$
$599$	3.22693e6	0.367470	0.183735	+	0.982976i	$0.441181\pi$
$600$	0	0
$601$	9.26294e6	1.04607	0.523037	−	0.852310i	$-0.324799\pi$
$601$	9.26294e6	1.04607	0.523037	+	0.852310i	$0.324799\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.41217e6	0.267928
$606$	0	0
$607$	−2.45733e6	−0.270702	−0.135351	−	0.990798i	$-0.543216\pi$
$607$	−2.45733e6	−0.270702	−0.135351	+	0.990798i	$0.543216\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.56914e7	1.70043
$612$	0	0
$613$	1.23201e7	1.32423	0.662114	−	0.749403i	$-0.269659\pi$
$613$	1.23201e7	1.32423	0.662114	+	0.749403i	$0.269659\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.30851e6	−0.667135	−0.333568	−	0.942726i	$-0.608253\pi$
$617$	−6.30851e6	−0.667135	−0.333568	+	0.942726i	$0.608253\pi$
$618$	0	0
$619$	1.43375e7	1.50399	0.751996	−	0.659167i	$-0.229091\pi$
$619$	1.43375e7	1.50399	0.751996	+	0.659167i	$0.229091\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−537369.	−0.0554692
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	7.00962e6	0.706428
$630$	0	0
$631$	−7.09131e6	−0.709011	−0.354506	−	0.935054i	$-0.615351\pi$
$631$	−7.09131e6	−0.709011	−0.354506	+	0.935054i	$0.615351\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−4.21105e6	−0.414434
$636$	0	0
$637$	−7.28623e6	−0.711466
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−3.41831e6	−0.328599	−0.164299	−	0.986411i	$-0.552536\pi$
$641$	−3.41831e6	−0.328599	−0.164299	+	0.986411i	$0.552536\pi$
$642$	0	0
$643$	−5.61084e6	−0.535181	−0.267590	−	0.963533i	$-0.586227\pi$
$643$	−5.61084e6	−0.535181	−0.267590	+	0.963533i	$0.586227\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2.05386e7	−1.92890	−0.964452	−	0.264257i	$-0.914873\pi$
$647$	−2.05386e7	−1.92890	−0.964452	+	0.264257i	$0.914873\pi$
$648$	0	0
$649$	3.76321e6	0.350709
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.28565e7	1.17988	0.589941	−	0.807446i	$-0.299151\pi$
$653$	1.28565e7	1.17988	0.589941	+	0.807446i	$0.299151\pi$
$654$	0	0
$655$	8.77947e6	0.799586
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.21433e6	−0.108924	−0.0544620	−	0.998516i	$-0.517344\pi$
$659$	−1.21433e6	−0.108924	−0.0544620	+	0.998516i	$0.517344\pi$
$660$	0	0
$661$	1.68972e7	1.50422	0.752108	−	0.659040i	$-0.229037\pi$
$661$	1.68972e7	1.50422	0.752108	+	0.659040i	$0.229037\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.23342e6	−0.108158
$666$	0	0
$667$	−1.98424e6	−0.172695
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.59941e6	−0.394362
$672$	0	0
$673$	5.15783e6	0.438964	0.219482	−	0.975617i	$-0.429563\pi$
$673$	5.15783e6	0.438964	0.219482	+	0.975617i	$0.429563\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−2.04545e7	−1.71521	−0.857605	−	0.514309i	$-0.828048\pi$
$677$	−2.04545e7	−1.71521	−0.857605	+	0.514309i	$0.828048\pi$
$678$	0	0
$679$	3.25669e6	0.271083
$680$	0	0
$681$	0	0
$682$	0	0
$683$	36424.8	0.00298776	0.00149388	−	0.999999i	$-0.499524\pi$
$683$	36424.8	0.00298776	0.00149388	+	0.999999i	$0.499524\pi$
$684$	0	0
$685$	1.18123e6	0.0961849
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2.83609e7	−2.27600
$690$	0	0
$691$	−1.92415e7	−1.53301	−0.766503	−	0.642241i	$-0.778005\pi$
$691$	−1.92415e7	−1.53301	−0.766503	+	0.642241i	$0.778005\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.85547e6	0.616893
$696$	0	0
$697$	9.59748e6	0.748300
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−3.36694e6	−0.258786	−0.129393	−	0.991593i	$-0.541303\pi$
$701$	−3.36694e6	−0.258786	−0.129393	+	0.991593i	$0.541303\pi$
$702$	0	0
$703$	1.57141e6	0.119922
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.32147e6	−0.249909
$708$	0	0
$709$	4.06268e6	0.303527	0.151764	−	0.988417i	$-0.451505\pi$
$709$	4.06268e6	0.303527	0.151764	+	0.988417i	$0.451505\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	176731.	0.0130194
$714$	0	0
$715$	−7.11527e6	−0.520507
$716$	0	0
$717$	0	0
$718$	0	0
$719$	152035.	0.0109678	0.00548392	−	0.999985i	$-0.498254\pi$
$719$	152035.	0.0109678	0.00548392	+	0.999985i	$0.498254\pi$
$720$	0	0
$721$	1.29204e7	0.925630
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5.35513e6	0.378378
$726$	0	0
$727$	−1.10583e7	−0.775984	−0.387992	−	0.921663i	$-0.626831\pi$
$727$	−1.10583e7	−0.775984	−0.387992	+	0.921663i	$0.626831\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−2.53383e7	−1.75382
$732$	0	0
$733$	1.46140e7	1.00464	0.502319	−	0.864683i	$-0.332480\pi$
$733$	1.46140e7	1.00464	0.502319	+	0.864683i	$0.332480\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.24327e6	−0.152129
$738$	0	0
$739$	−1.95198e7	−1.31481	−0.657405	−	0.753537i	$-0.728346\pi$
$739$	−1.95198e7	−1.31481	−0.657405	+	0.753537i	$0.728346\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−2.56749e7	−1.70622	−0.853112	−	0.521728i	$-0.825288\pi$
$743$	−2.56749e7	−1.70622	−0.853112	+	0.521728i	$0.825288\pi$
$744$	0	0
$745$	−6.20626e6	−0.409675
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.09602e7	0.713859
$750$	0	0
$751$	98418.0	0.00636758	0.00318379	−	0.999995i	$-0.498987\pi$
$751$	98418.0	0.00636758	0.00318379	+	0.999995i	$0.498987\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−3.23267e6	−0.206392
$756$	0	0
$757$	−2.27659e7	−1.44393	−0.721963	−	0.691932i	$-0.756760\pi$
$757$	−2.27659e7	−1.44393	−0.721963	+	0.691932i	$0.756760\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.03666e7	−0.648895	−0.324448	−	0.945904i	$-0.605178\pi$
$761$	−1.03666e7	−0.648895	−0.324448	+	0.945904i	$0.605178\pi$
$762$	0	0
$763$	−1.52648e7	−0.949247
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.65889e7	1.01819
$768$	0	0
$769$	3.19043e7	1.94551	0.972754	−	0.231840i	$-0.0744745\pi$
$769$	3.19043e7	1.94551	0.972754	+	0.231840i	$0.0744745\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−4.44636e6	−0.267643	−0.133821	−	0.991005i	$-0.542725\pi$
$773$	−4.44636e6	−0.267643	−0.133821	+	0.991005i	$0.542725\pi$
$774$	0	0
$775$	−476969.	−0.0285257
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.15155e6	0.127030
$780$	0	0
$781$	−6.32487e6	−0.371043
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.17392e6	0.125913
$786$	0	0
$787$	6.67710e6	0.384283	0.192141	−	0.981367i	$-0.438457\pi$
$787$	6.67710e6	0.384283	0.192141	+	0.981367i	$0.438457\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	412658.	0.0234504
$792$	0	0
$793$	−2.02750e7	−1.14493
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−9.01010e6	−0.502440	−0.251220	−	0.967930i	$-0.580832\pi$
$797$	−9.01010e6	−0.502440	−0.251220	+	0.967930i	$0.580832\pi$
$798$	0	0
$799$	3.03755e7	1.68328
$800$	0	0
$801$	0	0
$802$	0	0
$803$	9.71526e6	0.531699
$804$	0	0
$805$	587631.	0.0319606
$806$	0	0
$807$	0	0
$808$	0	0
$809$	9.62357e6	0.516969	0.258485	−	0.966015i	$-0.416777\pi$
$809$	9.62357e6	0.516969	0.258485	+	0.966015i	$0.416777\pi$
$810$	0	0
$811$	−2.65601e7	−1.41800	−0.709002	−	0.705207i	$-0.750854\pi$
$811$	−2.65601e7	−1.41800	−0.709002	+	0.705207i	$0.750854\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−8.59016e6	−0.453010
$816$	0	0
$817$	−5.68031e6	−0.297726
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−8.39304e6	−0.434572	−0.217286	−	0.976108i	$-0.569720\pi$
$821$	−8.39304e6	−0.434572	−0.217286	+	0.976108i	$0.569720\pi$
$822$	0	0
$823$	−1.80623e7	−0.929550	−0.464775	−	0.885429i	$-0.653865\pi$
$823$	−1.80623e7	−0.929550	−0.464775	+	0.885429i	$0.653865\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.07639e7	−0.547275	−0.273638	−	0.961833i	$-0.588227\pi$
$827$	−1.07639e7	−0.547275	−0.273638	+	0.961833i	$0.588227\pi$
$828$	0	0
$829$	2.80469e7	1.41742	0.708710	−	0.705500i	$-0.249278\pi$
$829$	2.80469e7	1.41742	0.708710	+	0.705500i	$0.249278\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.41047e7	−0.704289
$834$	0	0
$835$	1.18179e7	0.586577
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−2.51483e7	−1.23340	−0.616700	−	0.787198i	$-0.711531\pi$
$839$	−2.51483e7	−1.23340	−0.616700	+	0.787198i	$0.711531\pi$
$840$	0	0
$841$	5.29032e7	2.57924
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2.20830e7	−1.06394
$846$	0	0
$847$	−9.79327e6	−0.469050
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−748656.	−0.0354371
$852$	0	0
$853$	2.50392e7	1.17828	0.589139	−	0.808032i	$-0.299467\pi$
$853$	2.50392e7	1.17828	0.589139	+	0.808032i	$0.299467\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	4.20247e7	1.95458	0.977288	−	0.211917i	$-0.0679706\pi$
$857$	4.20247e7	1.95458	0.977288	+	0.211917i	$0.0679706\pi$
$858$	0	0
$859$	2.54461e7	1.17662	0.588312	−	0.808634i	$-0.299793\pi$
$859$	2.54461e7	1.17662	0.588312	+	0.808634i	$0.299793\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.27150e7	−0.581151	−0.290576	−	0.956852i	$-0.593847\pi$
$863$	−1.27150e7	−0.581151	−0.290576	+	0.956852i	$0.593847\pi$
$864$	0	0
$865$	−6.42817e6	−0.292110
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.82772e7	−0.821031
$870$	0	0
$871$	−9.88871e6	−0.441666
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.58592e6	−0.0700262
$876$	0	0
$877$	−1.74338e7	−0.765409	−0.382704	−	0.923871i	$-0.625007\pi$
$877$	−1.74338e7	−0.765409	−0.382704	+	0.923871i	$0.625007\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.31070e7	−0.568937	−0.284469	−	0.958685i	$-0.591817\pi$
$881$	−1.31070e7	−0.568937	−0.284469	+	0.958685i	$0.591817\pi$
$882$	0	0
$883$	−1.80224e7	−0.777876	−0.388938	−	0.921264i	$-0.627158\pi$
$883$	−1.80224e7	−0.777876	−0.388938	+	0.921264i	$0.627158\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.96494e7	1.69211	0.846054	−	0.533098i	$-0.178972\pi$
$887$	3.96494e7	1.69211	0.846054	+	0.533098i	$0.178972\pi$
$888$	0	0
$889$	1.70966e7	0.725532
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.80953e6	0.285752
$894$	0	0
$895$	4.25424e6	0.177527
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−6.53884e6	−0.269837
$900$	0	0
$901$	−5.49011e7	−2.25304
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.99961e6	0.162329
$906$	0	0
$907$	1.99541e7	0.805405	0.402703	−	0.915331i	$-0.368071\pi$
$907$	1.99541e7	0.805405	0.402703	+	0.915331i	$0.368071\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−2.89836e7	−1.15706	−0.578530	−	0.815661i	$-0.696374\pi$
$911$	−2.89836e7	−1.15706	−0.578530	+	0.815661i	$0.696374\pi$
$912$	0	0
$913$	1.45572e7	0.577965
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.56442e7	−1.39980
$918$	0	0
$919$	−1.96413e7	−0.767151	−0.383575	−	0.923510i	$-0.625307\pi$
$919$	−1.96413e7	−0.767151	−0.383575	+	0.923510i	$0.625307\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−2.78811e7	−1.07722
$924$	0	0
$925$	2.02050e6	0.0776433
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−2.80349e6	−0.106576	−0.0532881	−	0.998579i	$-0.516970\pi$
$929$	−2.80349e6	−0.106576	−0.0532881	+	0.998579i	$0.516970\pi$
$930$	0	0
$931$	−3.16197e6	−0.119559
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1.37738e7	−0.515257
$936$	0	0
$937$	−3.56858e7	−1.32784	−0.663921	−	0.747802i	$-0.731109\pi$
$937$	−3.56858e7	−1.32784	−0.663921	+	0.747802i	$0.731109\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−8.92762e6	−0.328671	−0.164336	−	0.986404i	$-0.552548\pi$
$941$	−8.92762e6	−0.328671	−0.164336	+	0.986404i	$0.552548\pi$
$942$	0	0
$943$	−1.02505e6	−0.0375375
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.02382e7	−1.09567	−0.547836	−	0.836586i	$-0.684548\pi$
$947$	−3.02382e7	−1.09567	−0.547836	+	0.836586i	$0.684548\pi$
$948$	0	0
$949$	4.28265e7	1.54364
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−3.06486e7	−1.09314	−0.546572	−	0.837412i	$-0.684068\pi$
$953$	−3.06486e7	−1.09314	−0.546572	+	0.837412i	$0.684068\pi$
$954$	0	0
$955$	−1.56840e7	−0.556480
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−4.79572e6	−0.168387
$960$	0	0
$961$	−2.80468e7	−0.979657
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.70740e7	−0.590224
$966$	0	0
$967$	1.75738e7	0.604365	0.302183	−	0.953250i	$-0.402285\pi$
$967$	1.75738e7	0.604365	0.302183	+	0.953250i	$0.402285\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−2.40755e7	−0.819458	−0.409729	−	0.912207i	$-0.634377\pi$
$971$	−2.40755e7	−0.819458	−0.409729	+	0.912207i	$0.634377\pi$
$972$	0	0
$973$	−3.18928e7	−1.07997
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.73572e7	1.25210	0.626049	−	0.779784i	$-0.284671\pi$
$977$	3.73572e7	1.25210	0.626049	+	0.779784i	$0.284671\pi$
$978$	0	0
$979$	−1.34527e6	−0.0448592
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−1.90868e7	−0.630013	−0.315007	−	0.949089i	$-0.602007\pi$
$983$	−1.90868e7	−0.630013	−0.315007	+	0.949089i	$0.602007\pi$
$984$	0	0
$985$	−1.43112e7	−0.469987
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.70623e6	0.0879781
$990$	0	0
$991$	3.82344e7	1.23672	0.618359	−	0.785896i	$-0.287798\pi$
$991$	3.82344e7	1.23672	0.618359	+	0.785896i	$0.287798\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	2.26489e7	0.725252
$996$	0	0
$997$	1.82342e7	0.580964	0.290482	−	0.956880i	$-0.406184\pi$
$997$	1.82342e7	0.580964	0.290482	+	0.956880i	$0.406184\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.6.a.c.1.2	✓	2
3.2	odd	2		1080.6.a.d.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.6.a.c.1.2	✓	2	1.1	even	1	trivial
1080.6.a.d.1.2	yes	2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1080.a (trivial)

\(f(q)\)	\(=\)	\(q-25.0000 q^{5} +101.499 q^{7} +254.095 q^{11} +1120.10 q^{13} +2168.28 q^{17} +486.083 q^{19} -231.581 q^{23} +625.000 q^{25} +8568.22 q^{29} -763.151 q^{31} -2537.47 q^{35} +3232.80 q^{37} +4426.31 q^{41} -11685.9 q^{43} +14009.0 q^{47} -6505.01 q^{49} -25320.1 q^{53} -6352.38 q^{55} +14810.2 q^{59} -18101.1 q^{61} -28002.4 q^{65} -8828.46 q^{67} -24891.7 q^{71} +38234.7 q^{73} +25790.3 q^{77} -71930.4 q^{79} +57290.5 q^{83} -54207.1 q^{85} -5294.34 q^{89} +113688. q^{91} -12152.1 q^{95} +32086.1 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 50 q^{5} + 4 q^{7} - 248 q^{11} + 1484 q^{13} + 1670 q^{17} - 1774 q^{19} + 2482 q^{23} + 1250 q^{25} + 3724 q^{29} - 9526 q^{31} - 100 q^{35} + 5988 q^{37} + 9768 q^{41} - 20944 q^{43} - 3344 q^{47}+ \cdots - 54112 q^{97}+O(q^{100}) \) 2 * q - 50 * q^5 + 4 * q^7 - 248 * q^11 + 1484 * q^13 + 1670 * q^17 - 1774 * q^19 + 2482 * q^23 + 1250 * q^25 + 3724 * q^29 - 9526 * q^31 - 100 * q^35 + 5988 * q^37 + 9768 * q^41 - 20944 * q^43 - 3344 * q^47 - 13806 * q^49 - 13746 * q^53 + 6200 * q^55 - 11572 * q^59 - 22750 * q^61 - 37100 * q^65 + 5188 * q^67 - 52848 * q^71 + 112488 * q^73 + 74744 * q^77 - 128538 * q^79 + 60374 * q^83 - 41750 * q^85 - 79800 * q^89 + 78208 * q^91 + 44350 * q^95 - 54112 * q^97

Introduction

L-functions

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