LMFDB - Embedded newform 7800.2.a.by.1.2

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:	$4$
Coefficient field:	4.4.30056.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 11x^{2} + 26 $ x^4 - 11*x^2 + 26
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1560)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.74983$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +0.633776 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.633776 q^{7} +1.00000 q^{9} -0.177949 q^{11} -1.00000 q^{13} -4.48933 q^{17} +0.633776 q^{21} +6.48933 q^{23} +1.00000 q^{27} -3.49966 q^{29} -0.177949 q^{33} -1.75688 q^{37} -1.00000 q^{39} +7.67760 q^{41} +7.49966 q^{43} +3.18827 q^{47} -6.59833 q^{49} -4.48933 q^{51} +1.75688 q^{53} +3.93483 q^{59} +3.01033 q^{61} +0.633776 q^{63} +1.62345 q^{67} +6.48933 q^{69} -4.41005 q^{71} +3.85555 q^{73} -0.112780 q^{77} +9.63309 q^{79} +1.00000 q^{81} +1.72338 q^{83} -3.49966 q^{87} -0.589258 q^{89} -0.633776 q^{91} +4.45457 q^{97} -0.177949 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 7 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 7 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 7 q^{7} + 4 q^{9} + q^{11} - 4 q^{13} + 3 q^{17} + 7 q^{21} + 5 q^{23} + 4 q^{27} + 8 q^{29} + q^{33} + 5 q^{37} - 4 q^{39} + 7 q^{41} + 8 q^{43} + 10 q^{47} + 9 q^{49} + 3 q^{51} - 5 q^{53} + 2 q^{59} + 11 q^{61} + 7 q^{63} + 12 q^{67} + 5 q^{69} + 15 q^{71} - 10 q^{73} + 15 q^{77} - q^{79} + 4 q^{81} + 22 q^{83} + 8 q^{87} + 9 q^{89} - 7 q^{91} + q^{97} + q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 7 * q^7 + 4 * q^9 + q^11 - 4 * q^13 + 3 * q^17 + 7 * q^21 + 5 * q^23 + 4 * q^27 + 8 * q^29 + q^33 + 5 * q^37 - 4 * q^39 + 7 * q^41 + 8 * q^43 + 10 * q^47 + 9 * q^49 + 3 * q^51 - 5 * q^53 + 2 * q^59 + 11 * q^61 + 7 * q^63 + 12 * q^67 + 5 * q^69 + 15 * q^71 - 10 * q^73 + 15 * q^77 - q^79 + 4 * q^81 + 22 * q^83 + 8 * q^87 + 9 * q^89 - 7 * q^91 + q^97 + 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
$7$	0.633776	0.239545	0.119772	−	0.992801i	$-0.461784\pi$
$7$	0.633776	0.239545	0.119772	+	0.992801i	$0.461784\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.177949	−0.0536537	−0.0268269	−	0.999640i	$-0.508540\pi$
$11$	−0.177949	−0.0536537	−0.0268269	+	0.999640i	$0.508540\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.48933	−1.08882	−0.544411	−	0.838818i	$-0.683247\pi$
$17$	−4.48933	−1.08882	−0.544411	+	0.838818i	$0.683247\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0.633776	0.138301
$22$	0	0
$23$	6.48933	1.35312	0.676559	−	0.736388i	$-0.263470\pi$
$23$	6.48933	1.35312	0.676559	+	0.736388i	$0.263470\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.49966	−0.649870	−0.324935	−	0.945736i	$-0.605342\pi$
$29$	−3.49966	−0.649870	−0.324935	+	0.945736i	$0.605342\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−0.177949	−0.0309770
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.75688	−0.288830	−0.144415	−	0.989517i	$-0.546130\pi$
$37$	−1.75688	−0.288830	−0.144415	+	0.989517i	$0.546130\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	7.67760	1.19904	0.599520	−	0.800360i	$-0.295358\pi$
$41$	7.67760	1.19904	0.599520	+	0.800360i	$0.295358\pi$
$42$	0	0
$43$	7.49966	1.14369	0.571843	−	0.820363i	$-0.306228\pi$
$43$	7.49966	1.14369	0.571843	+	0.820363i	$0.306228\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.18827	0.465058	0.232529	−	0.972589i	$-0.425300\pi$
$47$	3.18827	0.465058	0.232529	+	0.972589i	$0.425300\pi$
$48$	0	0
$49$	−6.59833	−0.942618
$50$	0	0
$51$	−4.48933	−0.628632
$52$	0	0
$53$	1.75688	0.241326	0.120663	−	0.992694i	$-0.461498\pi$
$53$	1.75688	0.241326	0.120663	+	0.992694i	$0.461498\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.93483	0.512271	0.256136	−	0.966641i	$-0.417551\pi$
$59$	3.93483	0.512271	0.256136	+	0.966641i	$0.417551\pi$
$60$	0	0
$61$	3.01033	0.385433	0.192716	−	0.981255i	$-0.438270\pi$
$61$	3.01033	0.385433	0.192716	+	0.981255i	$0.438270\pi$
$62$	0	0
$63$	0.633776	0.0798482
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.62345	0.198336	0.0991680	−	0.995071i	$-0.468382\pi$
$67$	1.62345	0.198336	0.0991680	+	0.995071i	$0.468382\pi$
$68$	0	0
$69$	6.48933	0.781224
$70$	0	0
$71$	−4.41005	−0.523377	−0.261689	−	0.965152i	$-0.584279\pi$
$71$	−4.41005	−0.523377	−0.261689	+	0.965152i	$0.584279\pi$
$72$	0	0
$73$	3.85555	0.451258	0.225629	−	0.974213i	$-0.427556\pi$
$73$	3.85555	0.451258	0.225629	+	0.974213i	$0.427556\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.112780	−0.0128525
$78$	0	0
$79$	9.63309	1.08381	0.541903	−	0.840441i	$-0.317704\pi$
$79$	9.63309	1.08381	0.541903	+	0.840441i	$0.317704\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.72338	0.189165	0.0945827	−	0.995517i	$-0.469848\pi$
$83$	1.72338	0.189165	0.0945827	+	0.995517i	$0.469848\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−3.49966	−0.375202
$88$	0	0
$89$	−0.589258	−0.0624612	−0.0312306	−	0.999512i	$-0.509943\pi$
$89$	−0.589258	−0.0624612	−0.0312306	+	0.999512i	$0.509943\pi$
$90$	0	0
$91$	−0.633776	−0.0664378
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	4.45457	0.452293	0.226147	−	0.974093i	$-0.427387\pi$
$97$	4.45457	0.452293	0.226147	+	0.974093i	$0.427387\pi$
$98$	0	0
$99$	−0.177949	−0.0178846
$100$	0	0
$101$	16.4783	1.63965	0.819827	−	0.572612i	$-0.194070\pi$
$101$	16.4783	1.63965	0.819827	+	0.572612i	$0.194070\pi$
$102$	0	0
$103$	−5.86966	−0.578355	−0.289177	−	0.957276i	$-0.593382\pi$
$103$	−5.86966	−0.578355	−0.289177	+	0.957276i	$0.593382\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.509981	−0.0493017	−0.0246509	−	0.999696i	$-0.507847\pi$
$107$	−0.509981	−0.0493017	−0.0246509	+	0.999696i	$0.507847\pi$
$108$	0	0
$109$	6.35590	0.608785	0.304392	−	0.952547i	$-0.401547\pi$
$109$	6.35590	0.608785	0.304392	+	0.952547i	$0.401547\pi$
$110$	0	0
$111$	−1.75688	−0.166756
$112$	0	0
$113$	9.71111	0.913544	0.456772	−	0.889584i	$-0.349005\pi$
$113$	9.71111	0.913544	0.456772	+	0.889584i	$0.349005\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−2.84523	−0.260822
$120$	0	0
$121$	−10.9683	−0.997121
$122$	0	0
$123$	7.67760	0.692266
$124$	0	0
$125$	0	0
$126$	0	0
$127$	11.1231	0.987016	0.493508	−	0.869741i	$-0.335714\pi$
$127$	11.1231	0.987016	0.493508	+	0.869741i	$0.335714\pi$
$128$	0	0
$129$	7.49966	0.660308
$130$	0	0
$131$	12.9439	1.13091	0.565457	−	0.824778i	$-0.308700\pi$
$131$	12.9439	1.13091	0.565457	+	0.824778i	$0.308700\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.7904	1.00732	0.503660	−	0.863902i	$-0.331986\pi$
$137$	11.7904	1.00732	0.503660	+	0.863902i	$0.331986\pi$
$138$	0	0
$139$	0.112780	0.00956587	0.00478293	−	0.999989i	$-0.498478\pi$
$139$	0.112780	0.00956587	0.00478293	+	0.999989i	$0.498478\pi$
$140$	0	0
$141$	3.18827	0.268501
$142$	0	0
$143$	0.177949	0.0148809
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−6.59833	−0.544221
$148$	0	0
$149$	2.30829	0.189102	0.0945512	−	0.995520i	$-0.469858\pi$
$149$	2.30829	0.189102	0.0945512	+	0.995520i	$0.469858\pi$
$150$	0	0
$151$	−17.7318	−1.44299	−0.721495	−	0.692420i	$-0.756545\pi$
$151$	−17.7318	−1.44299	−0.721495	+	0.692420i	$0.756545\pi$
$152$	0	0
$153$	−4.48933	−0.362941
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−12.8549	−1.02593	−0.512965	−	0.858410i	$-0.671453\pi$
$157$	−12.8549	−1.02593	−0.512965	+	0.858410i	$0.671453\pi$
$158$	0	0
$159$	1.75688	0.139330
$160$	0	0
$161$	4.11278	0.324132
$162$	0	0
$163$	14.3308	1.12247	0.561237	−	0.827655i	$-0.310326\pi$
$163$	14.3308	1.12247	0.561237	+	0.827655i	$0.310326\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.43449	−0.111004	−0.0555019	−	0.998459i	$-0.517676\pi$
$167$	−1.43449	−0.111004	−0.0555019	+	0.998459i	$0.517676\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.24690	0.0948001	0.0474000	−	0.998876i	$-0.484906\pi$
$173$	1.24690	0.0948001	0.0474000	+	0.998876i	$0.484906\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.93483	0.295760
$178$	0	0
$179$	−7.87621	−0.588695	−0.294348	−	0.955698i	$-0.595102\pi$
$179$	−7.87621	−0.588695	−0.294348	+	0.955698i	$0.595102\pi$
$180$	0	0
$181$	10.3796	0.771513	0.385756	−	0.922601i	$-0.373941\pi$
$181$	10.3796	0.771513	0.385756	+	0.922601i	$0.373941\pi$
$182$	0	0
$183$	3.01033	0.222530
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.798873	0.0584194
$188$	0	0
$189$	0.633776	0.0461004
$190$	0	0
$191$	13.7318	0.993595	0.496798	−	0.867866i	$-0.334509\pi$
$191$	13.7318	0.993595	0.496798	+	0.867866i	$0.334509\pi$
$192$	0	0
$193$	6.34488	0.456715	0.228357	−	0.973577i	$-0.426665\pi$
$193$	6.34488	0.456715	0.228357	+	0.973577i	$0.426665\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.2966	1.51732	0.758659	−	0.651487i	$-0.225855\pi$
$197$	21.2966	1.51732	0.758659	+	0.651487i	$0.225855\pi$
$198$	0	0
$199$	11.2108	0.794710	0.397355	−	0.917665i	$-0.369928\pi$
$199$	11.2108	0.794710	0.397355	+	0.917665i	$0.369928\pi$
$200$	0	0
$201$	1.62345	0.114509
$202$	0	0
$203$	−2.21800	−0.155673
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.48933	0.451040
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−20.2810	−1.39620	−0.698100	−	0.716000i	$-0.745971\pi$
$211$	−20.2810	−1.39620	−0.698100	+	0.716000i	$0.745971\pi$
$212$	0	0
$213$	−4.41005	−0.302172
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	3.85555	0.260534
$220$	0	0
$221$	4.48933	0.301985
$222$	0	0
$223$	−11.3900	−0.762729	−0.381364	−	0.924425i	$-0.624546\pi$
$223$	−11.3900	−0.762729	−0.381364	+	0.924425i	$0.624546\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	14.7020	0.975809	0.487904	−	0.872897i	$-0.337762\pi$
$227$	14.7020	0.975809	0.487904	+	0.872897i	$0.337762\pi$
$228$	0	0
$229$	9.84145	0.650341	0.325171	−	0.945655i	$-0.394578\pi$
$229$	9.84145	0.650341	0.325171	+	0.945655i	$0.394578\pi$
$230$	0	0
$231$	−0.112780	−0.00742038
$232$	0	0
$233$	−22.8439	−1.49655	−0.748275	−	0.663388i	$-0.769118\pi$
$233$	−22.8439	−1.49655	−0.748275	+	0.663388i	$0.769118\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	9.63309	0.625736
$238$	0	0
$239$	−24.1907	−1.56476	−0.782382	−	0.622798i	$-0.785996\pi$
$239$	−24.1907	−1.56476	−0.782382	+	0.622798i	$0.785996\pi$
$240$	0	0
$241$	−22.8896	−1.47445	−0.737225	−	0.675647i	$-0.763864\pi$
$241$	−22.8896	−1.47445	−0.737225	+	0.675647i	$0.763864\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	1.72338	0.109215
$250$	0	0
$251$	−11.6763	−0.737005	−0.368502	−	0.929627i	$-0.620129\pi$
$251$	−11.6763	−0.737005	−0.368502	+	0.929627i	$0.620129\pi$
$252$	0	0
$253$	−1.15477	−0.0725999
$254$	0	0
$255$	0	0
$256$	0	0
$257$	24.9993	1.55941	0.779707	−	0.626144i	$-0.215368\pi$
$257$	24.9993	1.55941	0.779707	+	0.626144i	$0.215368\pi$
$258$	0	0
$259$	−1.11347	−0.0691876
$260$	0	0
$261$	−3.49966	−0.216623
$262$	0	0
$263$	7.12965	0.439633	0.219817	−	0.975541i	$-0.429454\pi$
$263$	7.12965	0.439633	0.219817	+	0.975541i	$0.429454\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−0.589258	−0.0360620
$268$	0	0
$269$	−8.47832	−0.516932	−0.258466	−	0.966020i	$-0.583217\pi$
$269$	−8.47832	−0.516932	−0.258466	+	0.966020i	$0.583217\pi$
$270$	0	0
$271$	5.60142	0.340262	0.170131	−	0.985421i	$-0.445581\pi$
$271$	5.60142	0.340262	0.170131	+	0.985421i	$0.445581\pi$
$272$	0	0
$273$	−0.633776	−0.0383579
$274$	0	0
$275$	0	0
$276$	0	0
$277$	24.2991	1.45999	0.729996	−	0.683451i	$-0.239522\pi$
$277$	24.2991	1.45999	0.729996	+	0.683451i	$0.239522\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	11.6318	0.693897	0.346948	−	0.937884i	$-0.387218\pi$
$281$	11.6318	0.693897	0.346948	+	0.937884i	$0.387218\pi$
$282$	0	0
$283$	−0.197345	−0.0117310	−0.00586549	−	0.999983i	$-0.501867\pi$
$283$	−0.197345	−0.0117310	−0.00586549	+	0.999983i	$0.501867\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.86588	0.287224
$288$	0	0
$289$	3.15408	0.185534
$290$	0	0
$291$	4.45457	0.261132
$292$	0	0
$293$	−29.3918	−1.71709	−0.858544	−	0.512740i	$-0.828630\pi$
$293$	−29.3918	−1.71709	−0.858544	+	0.512740i	$0.828630\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−0.177949	−0.0103257
$298$	0	0
$299$	−6.48933	−0.375288
$300$	0	0
$301$	4.75310	0.273964
$302$	0	0
$303$	16.4783	0.946654
$304$	0	0
$305$	0	0
$306$	0	0
$307$	29.1997	1.66652	0.833259	−	0.552883i	$-0.186472\pi$
$307$	29.1997	1.66652	0.833259	+	0.552883i	$0.186472\pi$
$308$	0	0
$309$	−5.86966	−0.333913
$310$	0	0
$311$	−13.3346	−0.756133	−0.378067	−	0.925778i	$-0.623411\pi$
$311$	−13.3346	−0.756133	−0.378067	+	0.925778i	$0.623411\pi$
$312$	0	0
$313$	5.16441	0.291910	0.145955	−	0.989291i	$-0.453375\pi$
$313$	5.16441	0.291910	0.145955	+	0.989291i	$0.453375\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	28.3461	1.59208	0.796039	−	0.605245i	$-0.206925\pi$
$317$	28.3461	1.59208	0.796039	+	0.605245i	$0.206925\pi$
$318$	0	0
$319$	0.622761	0.0348679
$320$	0	0
$321$	−0.509981	−0.0284644
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	6.35590	0.351482
$328$	0	0
$329$	2.02065	0.111402
$330$	0	0
$331$	−11.1296	−0.611741	−0.305870	−	0.952073i	$-0.598947\pi$
$331$	−11.1296	−0.611741	−0.305870	+	0.952073i	$0.598947\pi$
$332$	0	0
$333$	−1.75688	−0.0962765
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−12.0529	−0.656563	−0.328282	−	0.944580i	$-0.606470\pi$
$337$	−12.0529	−0.656563	−0.328282	+	0.944580i	$0.606470\pi$
$338$	0	0
$339$	9.71111	0.527435
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−8.61829	−0.465344
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7.62654	−0.409414	−0.204707	−	0.978823i	$-0.565624\pi$
$347$	−7.62654	−0.409414	−0.204707	+	0.978823i	$0.565624\pi$
$348$	0	0
$349$	−15.2455	−0.816074	−0.408037	−	0.912965i	$-0.633787\pi$
$349$	−15.2455	−0.816074	−0.408037	+	0.912965i	$0.633787\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	4.83238	0.257201	0.128601	−	0.991696i	$-0.458951\pi$
$353$	4.83238	0.257201	0.128601	+	0.991696i	$0.458951\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−2.84523	−0.150585
$358$	0	0
$359$	7.25528	0.382919	0.191460	−	0.981501i	$-0.438678\pi$
$359$	7.25528	0.382919	0.191460	+	0.981501i	$0.438678\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	−10.9683	−0.575688
$364$	0	0
$365$	0	0
$366$	0	0
$367$	33.5808	1.75290	0.876451	−	0.481491i	$-0.159905\pi$
$367$	33.5808	1.75290	0.876451	+	0.481491i	$0.159905\pi$
$368$	0	0
$369$	7.67760	0.399680
$370$	0	0
$371$	1.11347	0.0578084
$372$	0	0
$373$	−2.78131	−0.144011	−0.0720055	−	0.997404i	$-0.522940\pi$
$373$	−2.78131	−0.144011	−0.0720055	+	0.997404i	$0.522940\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.49966	0.180241
$378$	0	0
$379$	−34.0268	−1.74784	−0.873921	−	0.486069i	$-0.838430\pi$
$379$	−34.0268	−1.74784	−0.873921	+	0.486069i	$0.838430\pi$
$380$	0	0
$381$	11.1231	0.569854
$382$	0	0
$383$	8.45332	0.431944	0.215972	−	0.976400i	$-0.430708\pi$
$383$	8.45332	0.431944	0.215972	+	0.976400i	$0.430708\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7.49966	0.381229
$388$	0	0
$389$	−14.2991	−0.724994	−0.362497	−	0.931985i	$-0.618076\pi$
$389$	−14.2991	−0.724994	−0.362497	+	0.931985i	$0.618076\pi$
$390$	0	0
$391$	−29.1327	−1.47331
$392$	0	0
$393$	12.9439	0.652933
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−3.97488	−0.199493	−0.0997467	−	0.995013i	$-0.531803\pi$
$397$	−3.97488	−0.199493	−0.0997467	+	0.995013i	$0.531803\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−10.2560	−0.512159	−0.256079	−	0.966656i	$-0.582431\pi$
$401$	−10.2560	−0.512159	−0.256079	+	0.966656i	$0.582431\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.312636	0.0154968
$408$	0	0
$409$	19.6296	0.970623	0.485311	−	0.874341i	$-0.338706\pi$
$409$	19.6296	0.970623	0.485311	+	0.874341i	$0.338706\pi$
$410$	0	0
$411$	11.7904	0.581577
$412$	0	0
$413$	2.49380	0.122712
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0.112780	0.00552286
$418$	0	0
$419$	23.7238	1.15899	0.579493	−	0.814977i	$-0.303251\pi$
$419$	23.7238	1.15899	0.579493	+	0.814977i	$0.303251\pi$
$420$	0	0
$421$	17.8697	0.870914	0.435457	−	0.900210i	$-0.356587\pi$
$421$	17.8697	0.870914	0.435457	+	0.900210i	$0.356587\pi$
$422$	0	0
$423$	3.18827	0.155019
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.90787	0.0923284
$428$	0	0
$429$	0.177949	0.00859147
$430$	0	0
$431$	35.9476	1.73153	0.865767	−	0.500448i	$-0.166831\pi$
$431$	35.9476	1.73153	0.865767	+	0.500448i	$0.166831\pi$
$432$	0	0
$433$	8.21396	0.394738	0.197369	−	0.980329i	$-0.436760\pi$
$433$	8.21396	0.394738	0.197369	+	0.980329i	$0.436760\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−25.0914	−1.19755	−0.598775	−	0.800918i	$-0.704345\pi$
$439$	−25.0914	−1.19755	−0.598775	+	0.800918i	$0.704345\pi$
$440$	0	0
$441$	−6.59833	−0.314206
$442$	0	0
$443$	−2.75619	−0.130951	−0.0654753	−	0.997854i	$-0.520856\pi$
$443$	−2.75619	−0.130951	−0.0654753	+	0.997854i	$0.520856\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.30829	0.109178
$448$	0	0
$449$	39.7639	1.87657	0.938287	−	0.345858i	$-0.112412\pi$
$449$	39.7639	1.87657	0.938287	+	0.345858i	$0.112412\pi$
$450$	0	0
$451$	−1.36622	−0.0643330
$452$	0	0
$453$	−17.7318	−0.833111
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8.72143	−0.407971	−0.203986	−	0.978974i	$-0.565390\pi$
$457$	−8.72143	−0.407971	−0.203986	+	0.978974i	$0.565390\pi$
$458$	0	0
$459$	−4.48933	−0.209544
$460$	0	0
$461$	17.5332	0.816601	0.408300	−	0.912848i	$-0.366122\pi$
$461$	17.5332	0.816601	0.408300	+	0.912848i	$0.366122\pi$
$462$	0	0
$463$	−18.1049	−0.841404	−0.420702	−	0.907199i	$-0.638216\pi$
$463$	−18.1049	−0.841404	−0.420702	+	0.907199i	$0.638216\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−11.0451	−0.511106	−0.255553	−	0.966795i	$-0.582257\pi$
$467$	−11.0451	−0.511106	−0.255553	+	0.966795i	$0.582257\pi$
$468$	0	0
$469$	1.02890	0.0475103
$470$	0	0
$471$	−12.8549	−0.592321
$472$	0	0
$473$	−1.33456	−0.0613631
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.75688	0.0804421
$478$	0	0
$479$	−33.3887	−1.52557	−0.762785	−	0.646653i	$-0.776168\pi$
$479$	−33.3887	−1.52557	−0.762785	+	0.646653i	$0.776168\pi$
$480$	0	0
$481$	1.75688	0.0801069
$482$	0	0
$483$	4.11278	0.187138
$484$	0	0
$485$	0	0
$486$	0	0
$487$	35.7414	1.61960	0.809799	−	0.586708i	$-0.199576\pi$
$487$	35.7414	1.61960	0.809799	+	0.586708i	$0.199576\pi$
$488$	0	0
$489$	14.3308	0.648060
$490$	0	0
$491$	36.9914	1.66940	0.834699	−	0.550706i	$-0.185642\pi$
$491$	36.9914	1.66940	0.834699	+	0.550706i	$0.185642\pi$
$492$	0	0
$493$	15.7111	0.707593
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.79498	−0.125372
$498$	0	0
$499$	−20.0670	−0.898323	−0.449161	−	0.893451i	$-0.648277\pi$
$499$	−20.0670	−0.898323	−0.449161	+	0.893451i	$0.648277\pi$
$500$	0	0
$501$	−1.43449	−0.0640881
$502$	0	0
$503$	15.3332	0.683673	0.341836	−	0.939759i	$-0.388951\pi$
$503$	15.3332	0.683673	0.341836	+	0.939759i	$0.388951\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	30.0476	1.33184	0.665918	−	0.746025i	$-0.268040\pi$
$509$	30.0476	1.33184	0.665918	+	0.746025i	$0.268040\pi$
$510$	0	0
$511$	2.44356	0.108097
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−0.567351	−0.0249521
$518$	0	0
$519$	1.24690	0.0547328
$520$	0	0
$521$	−30.8263	−1.35052	−0.675262	−	0.737578i	$-0.735970\pi$
$521$	−30.8263	−1.35052	−0.675262	+	0.737578i	$0.735970\pi$
$522$	0	0
$523$	12.0141	0.525340	0.262670	−	0.964886i	$-0.415397\pi$
$523$	12.0141	0.525340	0.262670	+	0.964886i	$0.415397\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	19.1114	0.830931
$530$	0	0
$531$	3.93483	0.170757
$532$	0	0
$533$	−7.67760	−0.332554
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−7.87621	−0.339883
$538$	0	0
$539$	1.17417	0.0505750
$540$	0	0
$541$	16.3132	0.701360	0.350680	−	0.936495i	$-0.385950\pi$
$541$	16.3132	0.701360	0.350680	+	0.936495i	$0.385950\pi$
$542$	0	0
$543$	10.3796	0.445433
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0.239667	0.0102474	0.00512371	−	0.999987i	$-0.498369\pi$
$547$	0.239667	0.0102474	0.00512371	+	0.999987i	$0.498369\pi$
$548$	0	0
$549$	3.01033	0.128478
$550$	0	0
$551$	0	0
$552$	0	0
$553$	6.10522	0.259620
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10.0304	0.425002	0.212501	−	0.977161i	$-0.431839\pi$
$557$	10.0304	0.425002	0.212501	+	0.977161i	$0.431839\pi$
$558$	0	0
$559$	−7.49966	−0.317202
$560$	0	0
$561$	0.798873	0.0337284
$562$	0	0
$563$	−0.310125	−0.0130702	−0.00653511	−	0.999979i	$-0.502080\pi$
$563$	−0.310125	−0.0130702	−0.00653511	+	0.999979i	$0.502080\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0.633776	0.0266161
$568$	0	0
$569$	12.2850	0.515014	0.257507	−	0.966276i	$-0.417099\pi$
$569$	12.2850	0.515014	0.257507	+	0.966276i	$0.417099\pi$
$570$	0	0
$571$	−31.6124	−1.32294	−0.661470	−	0.749972i	$-0.730067\pi$
$571$	−31.6124	−1.32294	−0.661470	+	0.749972i	$0.730067\pi$
$572$	0	0
$573$	13.7318	0.573652
$574$	0	0
$575$	0	0
$576$	0	0
$577$	6.36691	0.265058	0.132529	−	0.991179i	$-0.457690\pi$
$577$	6.36691	0.265058	0.132529	+	0.991179i	$0.457690\pi$
$578$	0	0
$579$	6.34488	0.263684
$580$	0	0
$581$	1.09224	0.0453136
$582$	0	0
$583$	−0.312636	−0.0129481
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.8103	0.611288	0.305644	−	0.952146i	$-0.401128\pi$
$587$	14.8103	0.611288	0.305644	+	0.952146i	$0.401128\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	21.2966	0.876024
$592$	0	0
$593$	33.9864	1.39565	0.697826	−	0.716267i	$-0.254151\pi$
$593$	33.9864	1.39565	0.697826	+	0.716267i	$0.254151\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	11.2108	0.458826
$598$	0	0
$599$	−1.10900	−0.0453124	−0.0226562	−	0.999743i	$-0.507212\pi$
$599$	−1.10900	−0.0453124	−0.0226562	+	0.999743i	$0.507212\pi$
$600$	0	0
$601$	−47.9670	−1.95661	−0.978306	−	0.207163i	$-0.933577\pi$
$601$	−47.9670	−1.95661	−0.978306	+	0.207163i	$0.933577\pi$
$602$	0	0
$603$	1.62345	0.0661120
$604$	0	0
$605$	0	0
$606$	0	0
$607$	35.5319	1.44220	0.721098	−	0.692833i	$-0.243638\pi$
$607$	35.5319	1.44220	0.721098	+	0.692833i	$0.243638\pi$
$608$	0	0
$609$	−2.21800	−0.0898778
$610$	0	0
$611$	−3.18827	−0.128984
$612$	0	0
$613$	20.4955	0.827806	0.413903	−	0.910321i	$-0.364165\pi$
$613$	20.4955	0.827806	0.413903	+	0.910321i	$0.364165\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	10.6119	0.427218	0.213609	−	0.976919i	$-0.431478\pi$
$617$	10.6119	0.427218	0.213609	+	0.976919i	$0.431478\pi$
$618$	0	0
$619$	−28.5312	−1.14677	−0.573383	−	0.819287i	$-0.694369\pi$
$619$	−28.5312	−1.14677	−0.573383	+	0.819287i	$0.694369\pi$
$620$	0	0
$621$	6.48933	0.260408
$622$	0	0
$623$	−0.373457	−0.0149623
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	7.88722	0.314484
$630$	0	0
$631$	−7.71111	−0.306974	−0.153487	−	0.988151i	$-0.549050\pi$
$631$	−7.71111	−0.306974	−0.153487	+	0.988151i	$0.549050\pi$
$632$	0	0
$633$	−20.2810	−0.806096
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.59833	0.261435
$638$	0	0
$639$	−4.41005	−0.174459
$640$	0	0
$641$	−27.5124	−1.08667	−0.543337	−	0.839515i	$-0.682839\pi$
$641$	−27.5124	−1.08667	−0.543337	+	0.839515i	$0.682839\pi$
$642$	0	0
$643$	−24.1798	−0.953558	−0.476779	−	0.879023i	$-0.658196\pi$
$643$	−24.1798	−0.953558	−0.476779	+	0.879023i	$0.658196\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−27.9347	−1.09823	−0.549113	−	0.835748i	$-0.685034\pi$
$647$	−27.9347	−1.09823	−0.549113	+	0.835748i	$0.685034\pi$
$648$	0	0
$649$	−0.700200	−0.0274853
$650$	0	0
$651$	0	0
$652$	0	0
$653$	27.2868	1.06782	0.533908	−	0.845543i	$-0.320723\pi$
$653$	27.2868	1.06782	0.533908	+	0.845543i	$0.320723\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.85555	0.150419
$658$	0	0
$659$	−30.3198	−1.18109	−0.590545	−	0.807005i	$-0.701087\pi$
$659$	−30.3198	−1.18109	−0.590545	+	0.807005i	$0.701087\pi$
$660$	0	0
$661$	16.7104	0.649960	0.324980	−	0.945721i	$-0.394642\pi$
$661$	16.7104	0.649960	0.324980	+	0.945721i	$0.394642\pi$
$662$	0	0
$663$	4.48933	0.174351
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−22.7104	−0.879351
$668$	0	0
$669$	−11.3900	−0.440362
$670$	0	0
$671$	−0.535685	−0.0206799
$672$	0	0
$673$	−0.950445	−0.0366370	−0.0183185	−	0.999832i	$-0.505831\pi$
$673$	−0.950445	−0.0366370	−0.0183185	+	0.999832i	$0.505831\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−34.1991	−1.31438	−0.657188	−	0.753726i	$-0.728254\pi$
$677$	−34.1991	−1.31438	−0.657188	+	0.753726i	$0.728254\pi$
$678$	0	0
$679$	2.82320	0.108344
$680$	0	0
$681$	14.7020	0.563383
$682$	0	0
$683$	−5.68208	−0.217419	−0.108709	−	0.994074i	$-0.534672\pi$
$683$	−5.68208	−0.217419	−0.108709	+	0.994074i	$0.534672\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	9.84145	0.375475
$688$	0	0
$689$	−1.75688	−0.0669319
$690$	0	0
$691$	−13.7795	−0.524197	−0.262098	−	0.965041i	$-0.584414\pi$
$691$	−13.7795	−0.524197	−0.262098	+	0.965041i	$0.584414\pi$
$692$	0	0
$693$	−0.112780	−0.00428416
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−34.4673	−1.30554
$698$	0	0
$699$	−22.8439	−0.864034
$700$	0	0
$701$	20.4113	0.770924	0.385462	−	0.922724i	$-0.374042\pi$
$701$	20.4113	0.770924	0.385462	+	0.922724i	$0.374042\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.4436	0.392770
$708$	0	0
$709$	3.17852	0.119372	0.0596858	−	0.998217i	$-0.480990\pi$
$709$	3.17852	0.119372	0.0596858	+	0.998217i	$0.480990\pi$
$710$	0	0
$711$	9.63309	0.361269
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−24.1907	−0.903417
$718$	0	0
$719$	−40.5312	−1.51156	−0.755780	−	0.654826i	$-0.772742\pi$
$719$	−40.5312	−1.51156	−0.755780	+	0.654826i	$0.772742\pi$
$720$	0	0
$721$	−3.72005	−0.138542
$722$	0	0
$723$	−22.8896	−0.851274
$724$	0	0
$725$	0	0
$726$	0	0
$727$	4.39066	0.162840	0.0814202	−	0.996680i	$-0.474054\pi$
$727$	4.39066	0.162840	0.0814202	+	0.996680i	$0.474054\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−33.6684	−1.24527
$732$	0	0
$733$	30.1823	1.11481	0.557404	−	0.830241i	$-0.311797\pi$
$733$	30.1823	1.11481	0.557404	+	0.830241i	$0.311797\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.288892	−0.0106415
$738$	0	0
$739$	−49.1565	−1.80825	−0.904125	−	0.427267i	$-0.859476\pi$
$739$	−49.1565	−1.80825	−0.904125	+	0.427267i	$0.859476\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−21.8317	−0.800927	−0.400463	−	0.916313i	$-0.631151\pi$
$743$	−21.8317	−0.800927	−0.400463	+	0.916313i	$0.631151\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.72338	0.0630551
$748$	0	0
$749$	−0.323214	−0.0118100
$750$	0	0
$751$	−22.8170	−0.832605	−0.416302	−	0.909226i	$-0.636674\pi$
$751$	−22.8170	−0.832605	−0.416302	+	0.909226i	$0.636674\pi$
$752$	0	0
$753$	−11.6763	−0.425510
$754$	0	0
$755$	0	0
$756$	0	0
$757$	11.4649	0.416699	0.208349	−	0.978054i	$-0.433191\pi$
$757$	11.4649	0.416699	0.208349	+	0.978054i	$0.433191\pi$
$758$	0	0
$759$	−1.15477	−0.0419156
$760$	0	0
$761$	−26.9005	−0.975143	−0.487571	−	0.873083i	$-0.662117\pi$
$761$	−26.9005	−0.975143	−0.487571	+	0.873083i	$0.662117\pi$
$762$	0	0
$763$	4.02821	0.145831
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.93483	−0.142079
$768$	0	0
$769$	12.9298	0.466260	0.233130	−	0.972446i	$-0.425103\pi$
$769$	12.9298	0.466260	0.233130	+	0.972446i	$0.425103\pi$
$770$	0	0
$771$	24.9993	0.900328
$772$	0	0
$773$	7.56231	0.271998	0.135999	−	0.990709i	$-0.456576\pi$
$773$	7.56231	0.271998	0.135999	+	0.990709i	$0.456576\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1.11347	−0.0399455
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0.784766	0.0280811
$782$	0	0
$783$	−3.49966	−0.125067
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−53.3318	−1.90107	−0.950537	−	0.310612i	$-0.899466\pi$
$787$	−53.3318	−1.90107	−0.950537	+	0.310612i	$0.899466\pi$
$788$	0	0
$789$	7.12965	0.253822
$790$	0	0
$791$	6.15466	0.218835
$792$	0	0
$793$	−3.01033	−0.106900
$794$	0	0
$795$	0	0
$796$	0	0
$797$	46.9391	1.66267	0.831334	−	0.555774i	$-0.187578\pi$
$797$	46.9391	1.66267	0.831334	+	0.555774i	$0.187578\pi$
$798$	0	0
$799$	−14.3132	−0.506365
$800$	0	0
$801$	−0.589258	−0.0208204
$802$	0	0
$803$	−0.686093	−0.0242117
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−8.47832	−0.298451
$808$	0	0
$809$	18.1365	0.637646	0.318823	−	0.947814i	$-0.396712\pi$
$809$	18.1365	0.637646	0.318823	+	0.947814i	$0.396712\pi$
$810$	0	0
$811$	50.6664	1.77914	0.889568	−	0.456802i	$-0.151005\pi$
$811$	50.6664	1.77914	0.889568	+	0.456802i	$0.151005\pi$
$812$	0	0
$813$	5.60142	0.196450
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−0.633776	−0.0221459
$820$	0	0
$821$	36.1986	1.26334	0.631670	−	0.775237i	$-0.282370\pi$
$821$	36.1986	1.26334	0.631670	+	0.775237i	$0.282370\pi$
$822$	0	0
$823$	−0.307033	−0.0107025	−0.00535124	−	0.999986i	$-0.501703\pi$
$823$	−0.307033	−0.0107025	−0.00535124	+	0.999986i	$0.501703\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−55.9733	−1.94638	−0.973191	−	0.230000i	$-0.926127\pi$
$827$	−55.9733	−1.94638	−0.973191	+	0.230000i	$0.926127\pi$
$828$	0	0
$829$	−37.1912	−1.29171	−0.645853	−	0.763462i	$-0.723498\pi$
$829$	−37.1912	−1.29171	−0.645853	+	0.763462i	$0.723498\pi$
$830$	0	0
$831$	24.2991	0.842927
$832$	0	0
$833$	29.6221	1.02634
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	31.2570	1.07911	0.539556	−	0.841950i	$-0.318592\pi$
$839$	31.2570	1.07911	0.539556	+	0.841950i	$0.318592\pi$
$840$	0	0
$841$	−16.7524	−0.577669
$842$	0	0
$843$	11.6318	0.400622
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−6.95146	−0.238855
$848$	0	0
$849$	−0.197345	−0.00677288
$850$	0	0
$851$	−11.4010	−0.390821
$852$	0	0
$853$	−37.7066	−1.29105	−0.645525	−	0.763739i	$-0.723362\pi$
$853$	−37.7066	−1.29105	−0.645525	+	0.763739i	$0.723362\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19.5364	0.667350	0.333675	−	0.942688i	$-0.391711\pi$
$857$	19.5364	0.667350	0.333675	+	0.942688i	$0.391711\pi$
$858$	0	0
$859$	34.0340	1.16122	0.580612	−	0.814180i	$-0.302813\pi$
$859$	34.0340	1.16122	0.580612	+	0.814180i	$0.302813\pi$
$860$	0	0
$861$	4.86588	0.165829
$862$	0	0
$863$	16.9482	0.576925	0.288463	−	0.957491i	$-0.406856\pi$
$863$	16.9482	0.576925	0.288463	+	0.957491i	$0.406856\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.15408	0.107118
$868$	0	0
$869$	−1.71420	−0.0581503
$870$	0	0
$871$	−1.62345	−0.0550085
$872$	0	0
$873$	4.45457	0.150764
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−30.3339	−1.02430	−0.512151	−	0.858895i	$-0.671151\pi$
$877$	−30.3339	−1.02430	−0.512151	+	0.858895i	$0.671151\pi$
$878$	0	0
$879$	−29.3918	−0.991361
$880$	0	0
$881$	−8.66544	−0.291946	−0.145973	−	0.989289i	$-0.546631\pi$
$881$	−8.66544	−0.291946	−0.145973	+	0.989289i	$0.546631\pi$
$882$	0	0
$883$	12.6898	0.427045	0.213522	−	0.976938i	$-0.431506\pi$
$883$	12.6898	0.427045	0.213522	+	0.976938i	$0.431506\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−41.8638	−1.40565	−0.702825	−	0.711363i	$-0.748078\pi$
$887$	−41.8638	−1.40565	−0.702825	+	0.711363i	$0.748078\pi$
$888$	0	0
$889$	7.04955	0.236434
$890$	0	0
$891$	−0.177949	−0.00596153
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−6.48933	−0.216672
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−7.88722	−0.262761
$902$	0	0
$903$	4.75310	0.158173
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−19.2531	−0.639288	−0.319644	−	0.947538i	$-0.603563\pi$
$907$	−19.2531	−0.639288	−0.319644	+	0.947538i	$0.603563\pi$
$908$	0	0
$909$	16.4783	0.546551
$910$	0	0
$911$	−9.30885	−0.308416	−0.154208	−	0.988038i	$-0.549283\pi$
$911$	−9.30885	−0.308416	−0.154208	+	0.988038i	$0.549283\pi$
$912$	0	0
$913$	−0.306674	−0.0101494
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.20353	0.270904
$918$	0	0
$919$	−13.9282	−0.459448	−0.229724	−	0.973256i	$-0.573782\pi$
$919$	−13.9282	−0.459448	−0.229724	+	0.973256i	$0.573782\pi$
$920$	0	0
$921$	29.1997	0.962164
$922$	0	0
$923$	4.41005	0.145159
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−5.86966	−0.192785
$928$	0	0
$929$	−10.4734	−0.343621	−0.171810	−	0.985130i	$-0.554962\pi$
$929$	−10.4734	−0.343621	−0.171810	+	0.985130i	$0.554962\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−13.3346	−0.436554
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−6.63825	−0.216862	−0.108431	−	0.994104i	$-0.534583\pi$
$937$	−6.63825	−0.216862	−0.108431	+	0.994104i	$0.534583\pi$
$938$	0	0
$939$	5.16441	0.168534
$940$	0	0
$941$	43.5538	1.41981	0.709907	−	0.704296i	$-0.248737\pi$
$941$	43.5538	1.41981	0.709907	+	0.704296i	$0.248737\pi$
$942$	0	0
$943$	49.8225	1.62244
$944$	0	0
$945$	0	0
$946$	0	0
$947$	29.8379	0.969601	0.484800	−	0.874625i	$-0.338892\pi$
$947$	29.8379	0.969601	0.484800	+	0.874625i	$0.338892\pi$
$948$	0	0
$949$	−3.85555	−0.125157
$950$	0	0
$951$	28.3461	0.919187
$952$	0	0
$953$	−16.4285	−0.532172	−0.266086	−	0.963949i	$-0.585730\pi$
$953$	−16.4285	−0.532172	−0.266086	+	0.963949i	$0.585730\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0.622761	0.0201310
$958$	0	0
$959$	7.47246	0.241298
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−0.509981	−0.0164339
$964$	0	0
$965$	0	0
$966$	0	0
$967$	39.1011	1.25741	0.628703	−	0.777646i	$-0.283586\pi$
$967$	39.1011	1.25741	0.628703	+	0.777646i	$0.283586\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	45.0240	1.44489	0.722444	−	0.691429i	$-0.243019\pi$
$971$	45.0240	1.44489	0.722444	+	0.691429i	$0.243019\pi$
$972$	0	0
$973$	0.0714772	0.00229145
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−26.5616	−0.849782	−0.424891	−	0.905245i	$-0.639688\pi$
$977$	−26.5616	−0.849782	−0.424891	+	0.905245i	$0.639688\pi$
$978$	0	0
$979$	0.104858	0.00335128
$980$	0	0
$981$	6.35590	0.202928
$982$	0	0
$983$	12.7420	0.406405	0.203203	−	0.979137i	$-0.434865\pi$
$983$	12.7420	0.406405	0.203203	+	0.979137i	$0.434865\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2.02065	0.0643180
$988$	0	0
$989$	48.6677	1.54754
$990$	0	0
$991$	−34.0639	−1.08208	−0.541038	−	0.840998i	$-0.681968\pi$
$991$	−34.0639	−1.08208	−0.541038	+	0.840998i	$0.681968\pi$
$992$	0	0
$993$	−11.1296	−0.353189
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−17.5887	−0.557039	−0.278520	−	0.960431i	$-0.589844\pi$
$997$	−17.5887	−0.557039	−0.278520	+	0.960431i	$0.589844\pi$
$998$	0	0
$999$	−1.75688	−0.0555853

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.by.1.2		4
5.2	odd	4		1560.2.l.d.1249.4	✓	8
5.3	odd	4		1560.2.l.d.1249.8	yes	8
5.4	even	2		7800.2.a.bt.1.3		4
15.2	even	4		4680.2.l.g.2809.2		8
15.8	even	4		4680.2.l.g.2809.1		8
20.3	even	4		3120.2.l.n.1249.4		8
20.7	even	4		3120.2.l.n.1249.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.l.d.1249.4	✓	8	5.2	odd	4
1560.2.l.d.1249.8	yes	8	5.3	odd	4
3120.2.l.n.1249.4		8	20.3	even	4
3120.2.l.n.1249.8		8	20.7	even	4
4680.2.l.g.2809.1		8	15.8	even	4
4680.2.l.g.2809.2		8	15.2	even	4
7800.2.a.bt.1.3		4	5.4	even	2
7800.2.a.by.1.2		4	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.633776 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.633776 q^{7} +1.00000 q^{9} -0.177949 q^{11} -1.00000 q^{13} -4.48933 q^{17} +0.633776 q^{21} +6.48933 q^{23} +1.00000 q^{27} -3.49966 q^{29} -0.177949 q^{33} -1.75688 q^{37} -1.00000 q^{39} +7.67760 q^{41} +7.49966 q^{43} +3.18827 q^{47} -6.59833 q^{49} -4.48933 q^{51} +1.75688 q^{53} +3.93483 q^{59} +3.01033 q^{61} +0.633776 q^{63} +1.62345 q^{67} +6.48933 q^{69} -4.41005 q^{71} +3.85555 q^{73} -0.112780 q^{77} +9.63309 q^{79} +1.00000 q^{81} +1.72338 q^{83} -3.49966 q^{87} -0.589258 q^{89} -0.633776 q^{91} +4.45457 q^{97} -0.177949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 7 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 7 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 7 q^{7} + 4 q^{9} + q^{11} - 4 q^{13} + 3 q^{17} + 7 q^{21} + 5 q^{23} + 4 q^{27} + 8 q^{29} + q^{33} + 5 q^{37} - 4 q^{39} + 7 q^{41} + 8 q^{43} + 10 q^{47} + 9 q^{49} + 3 q^{51} - 5 q^{53} + 2 q^{59} + 11 q^{61} + 7 q^{63} + 12 q^{67} + 5 q^{69} + 15 q^{71} - 10 q^{73} + 15 q^{77} - q^{79} + 4 q^{81} + 22 q^{83} + 8 q^{87} + 9 q^{89} - 7 q^{91} + q^{97} + q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 7 * q^7 + 4 * q^9 + q^11 - 4 * q^13 + 3 * q^17 + 7 * q^21 + 5 * q^23 + 4 * q^27 + 8 * q^29 + q^33 + 5 * q^37 - 4 * q^39 + 7 * q^41 + 8 * q^43 + 10 * q^47 + 9 * q^49 + 3 * q^51 - 5 * q^53 + 2 * q^59 + 11 * q^61 + 7 * q^63 + 12 * q^67 + 5 * q^69 + 15 * q^71 - 10 * q^73 + 15 * q^77 - q^79 + 4 * q^81 + 22 * q^83 + 8 * q^87 + 9 * q^89 - 7 * q^91 + q^97 + q^99

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