LMFDB - Embedded newform 6480.2.a.cb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6480,2,Mod(1,6480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6480, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6480.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6480 = 2^{4} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6480.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:	$51.7430605098$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.29268.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 9x^{2} + 5x + 16 $ x^4 - x^3 - 9x^2 + 5x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 360)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.43292$ of defining polynomial
Character	$\chi$	$=$	6480.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -3.03717 q^{7} -5.27114 q^{11} -0.513812 q^{13} +2.80320 q^{17} -8.29877 q^{19} -5.03717 q^{23} +1.00000 q^{25} -4.78496 q^{29} +8.58816 q^{31} -3.03717 q^{35} -6.58816 q^{37} +7.98175 q^{41} +1.19680 q^{43} +9.62533 q^{47} +2.22442 q^{49} -0.467941 q^{53} -5.27114 q^{55} +0.757332 q^{59} -0.271144 q^{61} -0.513812 q^{65} +7.26159 q^{67} -1.48619 q^{71} +5.31701 q^{73} +16.0094 q^{77} +15.9635 q^{79} -12.0648 q^{83} +2.80320 q^{85} +15.2529 q^{89} +1.56053 q^{91} -8.29877 q^{95} -6.36374 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + q^{7} - q^{11} + 4 q^{13} + 5 q^{17} - q^{19} - 7 q^{23} + 4 q^{25} + 7 q^{29} + 2 q^{31} + q^{35} + 6 q^{37} + 12 q^{41} + 11 q^{43} - 7 q^{47} + 3 q^{49} + 12 q^{53} - q^{55} - 11 q^{59}+ \cdots + q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 + q^7 - q^11 + 4 * q^13 + 5 * q^17 - q^19 - 7 * q^23 + 4 * q^25 + 7 * q^29 + 2 * q^31 + q^35 + 6 * q^37 + 12 * q^41 + 11 * q^43 - 7 * q^47 + 3 * q^49 + 12 * q^53 - q^55 - 11 * q^59 + 19 * q^61 + 4 * q^65 + 10 * q^67 - 12 * q^71 + 9 * q^73 + 32 * q^77 + 24 * q^79 - 23 * q^83 + 5 * q^85 + 21 * q^89 - 14 * q^91 - q^95 + 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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.03717	−1.14794	−0.573972	−	0.818875i	$-0.694598\pi$
$7$	−3.03717	−1.14794	−0.573972	+	0.818875i	$0.694598\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.27114	−1.58931	−0.794655	−	0.607062i	$-0.792348\pi$
$11$	−5.27114	−1.58931	−0.794655	+	0.607062i	$0.792348\pi$
$12$	0	0
$13$	−0.513812	−0.142506	−0.0712528	−	0.997458i	$-0.522700\pi$
$13$	−0.513812	−0.142506	−0.0712528	+	0.997458i	$0.522700\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.80320	0.679876	0.339938	−	0.940448i	$-0.389594\pi$
$17$	2.80320	0.679876	0.339938	+	0.940448i	$0.389594\pi$
$18$	0	0
$19$	−8.29877	−1.90387	−0.951934	−	0.306304i	$-0.900908\pi$
$19$	−8.29877	−1.90387	−0.951934	+	0.306304i	$0.900908\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.03717	−1.05032	−0.525162	−	0.851003i	$-0.675995\pi$
$23$	−5.03717	−1.05032	−0.525162	+	0.851003i	$0.675995\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.78496	−0.888544	−0.444272	−	0.895892i	$-0.646538\pi$
$29$	−4.78496	−0.888544	−0.444272	+	0.895892i	$0.646538\pi$
$30$	0	0
$31$	8.58816	1.54248	0.771239	−	0.636545i	$-0.219637\pi$
$31$	8.58816	1.54248	0.771239	+	0.636545i	$0.219637\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.03717	−0.513376
$36$	0	0
$37$	−6.58816	−1.08309	−0.541543	−	0.840673i	$-0.682160\pi$
$37$	−6.58816	−1.08309	−0.541543	+	0.840673i	$0.682160\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.98175	1.24654	0.623270	−	0.782007i	$-0.285804\pi$
$41$	7.98175	1.24654	0.623270	+	0.782007i	$0.285804\pi$
$42$	0	0
$43$	1.19680	0.182510	0.0912549	−	0.995828i	$-0.470912\pi$
$43$	1.19680	0.182510	0.0912549	+	0.995828i	$0.470912\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.62533	1.40400	0.701999	−	0.712178i	$-0.252291\pi$
$47$	9.62533	1.40400	0.701999	+	0.712178i	$0.252291\pi$
$48$	0	0
$49$	2.22442	0.317774
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−0.467941	−0.0642766	−0.0321383	−	0.999483i	$-0.510232\pi$
$53$	−0.467941	−0.0642766	−0.0321383	+	0.999483i	$0.510232\pi$
$54$	0	0
$55$	−5.27114	−0.710761
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.757332	0.0985962	0.0492981	−	0.998784i	$-0.484302\pi$
$59$	0.757332	0.0985962	0.0492981	+	0.998784i	$0.484302\pi$
$60$	0	0
$61$	−0.271144	−0.0347164	−0.0173582	−	0.999849i	$-0.505526\pi$
$61$	−0.271144	−0.0347164	−0.0173582	+	0.999849i	$0.505526\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.513812	−0.0637305
$66$	0	0
$67$	7.26159	0.887145	0.443572	−	0.896239i	$-0.353711\pi$
$67$	7.26159	0.887145	0.443572	+	0.896239i	$0.353711\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.48619	−0.176378	−0.0881891	−	0.996104i	$-0.528108\pi$
$71$	−1.48619	−0.176378	−0.0881891	+	0.996104i	$0.528108\pi$
$72$	0	0
$73$	5.31701	0.622309	0.311155	−	0.950359i	$-0.399284\pi$
$73$	5.31701	0.622309	0.311155	+	0.950359i	$0.399284\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	16.0094	1.82444
$78$	0	0
$79$	15.9635	1.79603	0.898017	−	0.439960i	$-0.145007\pi$
$79$	15.9635	1.79603	0.898017	+	0.439960i	$0.145007\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−12.0648	−1.32428	−0.662142	−	0.749379i	$-0.730352\pi$
$83$	−12.0648	−1.32428	−0.662142	+	0.749379i	$0.730352\pi$
$84$	0	0
$85$	2.80320	0.304050
$86$	0	0
$87$	0	0
$88$	0	0
$89$	15.2529	1.61680	0.808402	−	0.588631i	$-0.200333\pi$
$89$	15.2529	1.61680	0.808402	+	0.588631i	$0.200333\pi$
$90$	0	0
$91$	1.56053	0.163588
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−8.29877	−0.851436
$96$	0	0
$97$	−6.36374	−0.646140	−0.323070	−	0.946375i	$-0.604715\pi$
$97$	−6.36374	−0.646140	−0.323070	+	0.946375i	$0.604715\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−2.07435	−0.206405	−0.103203	−	0.994660i	$-0.532909\pi$
$101$	−2.07435	−0.206405	−0.103203	+	0.994660i	$0.532909\pi$
$102$	0	0
$103$	−5.05610	−0.498192	−0.249096	−	0.968479i	$-0.580134\pi$
$103$	−5.05610	−0.498192	−0.249096	+	0.968479i	$0.580134\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.2901	1.28480	0.642400	−	0.766370i	$-0.277939\pi$
$107$	13.2901	1.28480	0.642400	+	0.766370i	$0.277939\pi$
$108$	0	0
$109$	8.34549	0.799353	0.399676	−	0.916656i	$-0.369122\pi$
$109$	8.34549	0.799353	0.399676	+	0.916656i	$0.369122\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−5.10197	−0.479953	−0.239976	−	0.970779i	$-0.577140\pi$
$113$	−5.10197	−0.479953	−0.239976	+	0.970779i	$0.577140\pi$
$114$	0	0
$115$	−5.03717	−0.469719
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−8.51381	−0.780460
$120$	0	0
$121$	16.7850	1.52591
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	11.0830	0.983461	0.491731	−	0.870747i	$-0.336365\pi$
$127$	11.0830	0.983461	0.491731	+	0.870747i	$0.336365\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.55116	−0.135525	−0.0677627	−	0.997701i	$-0.521586\pi$
$131$	−1.55116	−0.135525	−0.0677627	+	0.997701i	$0.521586\pi$
$132$	0	0
$133$	25.2048	2.18553
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.81258	−0.411166	−0.205583	−	0.978640i	$-0.565909\pi$
$137$	−4.81258	−0.411166	−0.205583	+	0.978640i	$0.565909\pi$
$138$	0	0
$139$	21.4657	1.82070	0.910349	−	0.413842i	$-0.135813\pi$
$139$	21.4657	1.82070	0.910349	+	0.413842i	$0.135813\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.70837	0.226486
$144$	0	0
$145$	−4.78496	−0.397369
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−14.7943	−1.21200	−0.605999	−	0.795465i	$-0.707227\pi$
$149$	−14.7943	−1.21200	−0.605999	+	0.795465i	$0.707227\pi$
$150$	0	0
$151$	−0.981753	−0.0798939	−0.0399469	−	0.999202i	$-0.512719\pi$
$151$	−0.981753	−0.0798939	−0.0399469	+	0.999202i	$0.512719\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.58816	0.689817
$156$	0	0
$157$	−6.86154	−0.547610	−0.273805	−	0.961785i	$-0.588282\pi$
$157$	−6.86154	−0.547610	−0.273805	+	0.961785i	$0.588282\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	15.2988	1.20571
$162$	0	0
$163$	4.90741	0.384378	0.192189	−	0.981358i	$-0.438441\pi$
$163$	4.90741	0.384378	0.192189	+	0.981358i	$0.438441\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.47664	−0.269030	−0.134515	−	0.990912i	$-0.542948\pi$
$167$	−3.47664	−0.269030	−0.134515	+	0.990912i	$0.542948\pi$
$168$	0	0
$169$	−12.7360	−0.979692
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.5423	0.801515	0.400758	−	0.916184i	$-0.368747\pi$
$173$	10.5423	0.801515	0.400758	+	0.916184i	$0.368747\pi$
$174$	0	0
$175$	−3.03717	−0.229589
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4.90741	−0.366797	−0.183398	−	0.983039i	$-0.558710\pi$
$179$	−4.90741	−0.366797	−0.183398	+	0.983039i	$0.558710\pi$
$180$	0	0
$181$	5.21504	0.387631	0.193816	−	0.981038i	$-0.437914\pi$
$181$	5.21504	0.387631	0.193816	+	0.981038i	$0.437914\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.58816	−0.484371
$186$	0	0
$187$	−14.7761	−1.08053
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−25.6901	−1.85887	−0.929436	−	0.368983i	$-0.879706\pi$
$191$	−25.6901	−1.85887	−0.929436	+	0.368983i	$0.879706\pi$
$192$	0	0
$193$	3.19680	0.230111	0.115055	−	0.993359i	$-0.463295\pi$
$193$	3.19680	0.230111	0.115055	+	0.993359i	$0.463295\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.6808	0.974713	0.487357	−	0.873203i	$-0.337961\pi$
$197$	13.6808	0.974713	0.487357	+	0.873203i	$0.337961\pi$
$198$	0	0
$199$	5.95413	0.422077	0.211039	−	0.977478i	$-0.432315\pi$
$199$	5.95413	0.422077	0.211039	+	0.977478i	$0.432315\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	14.5327	1.02000
$204$	0	0
$205$	7.98175	0.557470
$206$	0	0
$207$	0	0
$208$	0	0
$209$	43.7440	3.02584
$210$	0	0
$211$	−14.5882	−1.00429	−0.502145	−	0.864783i	$-0.667456\pi$
$211$	−14.5882	−1.00429	−0.502145	+	0.864783i	$0.667456\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.19680	0.0816209
$216$	0	0
$217$	−26.0837	−1.77068
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.44032	−0.0968863
$222$	0	0
$223$	27.7740	1.85989	0.929943	−	0.367703i	$-0.119856\pi$
$223$	27.7740	1.85989	0.929943	+	0.367703i	$0.119856\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−12.8411	−0.852291	−0.426145	−	0.904655i	$-0.640129\pi$
$227$	−12.8411	−0.852291	−0.426145	+	0.904655i	$0.640129\pi$
$228$	0	0
$229$	25.9060	1.71192	0.855959	−	0.517043i	$-0.172967\pi$
$229$	25.9060	1.71192	0.855959	+	0.517043i	$0.172967\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5.71998	0.374729	0.187364	−	0.982290i	$-0.440006\pi$
$233$	5.71998	0.374729	0.187364	+	0.982290i	$0.440006\pi$
$234$	0	0
$235$	9.62533	0.627887
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−11.0467	−0.714553	−0.357277	−	0.933999i	$-0.616295\pi$
$239$	−11.0467	−0.714553	−0.357277	+	0.933999i	$0.616295\pi$
$240$	0	0
$241$	12.4862	0.804306	0.402153	−	0.915572i	$-0.368262\pi$
$241$	12.4862	0.804306	0.402153	+	0.915572i	$0.368262\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.22442	0.142113
$246$	0	0
$247$	4.26400	0.271312
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−8.22442	−0.519121	−0.259560	−	0.965727i	$-0.583578\pi$
$251$	−8.22442	−0.519121	−0.259560	+	0.965727i	$0.583578\pi$
$252$	0	0
$253$	26.5517	1.66929
$254$	0	0
$255$	0	0
$256$	0	0
$257$	16.8317	1.04993	0.524966	−	0.851123i	$-0.324078\pi$
$257$	16.8317	1.04993	0.524966	+	0.851123i	$0.324078\pi$
$258$	0	0
$259$	20.0094	1.24332
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−12.1202	−0.747365	−0.373682	−	0.927557i	$-0.621905\pi$
$263$	−12.1202	−0.747365	−0.373682	+	0.927557i	$0.621905\pi$
$264$	0	0
$265$	−0.467941	−0.0287454
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−22.5859	−1.37709	−0.688544	−	0.725195i	$-0.741750\pi$
$269$	−22.5859	−1.37709	−0.688544	+	0.725195i	$0.741750\pi$
$270$	0	0
$271$	10.9818	0.667094	0.333547	−	0.942733i	$-0.391754\pi$
$271$	10.9818	0.667094	0.333547	+	0.942733i	$0.391754\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.27114	−0.317862
$276$	0	0
$277$	28.7178	1.72548	0.862741	−	0.505646i	$-0.168746\pi$
$277$	28.7178	1.72548	0.862741	+	0.505646i	$0.168746\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−24.3090	−1.45015	−0.725077	−	0.688668i	$-0.758196\pi$
$281$	−24.3090	−1.45015	−0.725077	+	0.688668i	$0.758196\pi$
$282$	0	0
$283$	−14.6245	−0.869335	−0.434668	−	0.900591i	$-0.643134\pi$
$283$	−14.6245	−0.869335	−0.434668	+	0.900591i	$0.643134\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−24.2420	−1.43096
$288$	0	0
$289$	−9.14206	−0.537768
$290$	0	0
$291$	0	0
$292$	0	0
$293$	15.6901	0.916627	0.458314	−	0.888791i	$-0.348454\pi$
$293$	15.6901	0.916627	0.458314	+	0.888791i	$0.348454\pi$
$294$	0	0
$295$	0.757332	0.0440936
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.58816	0.149677
$300$	0	0
$301$	−3.63488	−0.209511
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−0.271144	−0.0155256
$306$	0	0
$307$	26.3920	1.50627	0.753137	−	0.657864i	$-0.228540\pi$
$307$	26.3920	1.50627	0.753137	+	0.657864i	$0.228540\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	14.0285	0.795482	0.397741	−	0.917498i	$-0.369794\pi$
$311$	14.0285	0.795482	0.397741	+	0.917498i	$0.369794\pi$
$312$	0	0
$313$	−34.4110	−1.94502	−0.972511	−	0.232855i	$-0.925193\pi$
$313$	−34.4110	−1.94502	−0.972511	+	0.232855i	$0.925193\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.19456	−0.123259	−0.0616295	−	0.998099i	$-0.519630\pi$
$317$	−2.19456	−0.123259	−0.0616295	+	0.998099i	$0.519630\pi$
$318$	0	0
$319$	25.2222	1.41217
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−23.2631	−1.29439
$324$	0	0
$325$	−0.513812	−0.0285011
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−29.2338	−1.61171
$330$	0	0
$331$	−12.8980	−0.708940	−0.354470	−	0.935067i	$-0.615339\pi$
$331$	−12.8980	−0.708940	−0.354470	+	0.935067i	$0.615339\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	7.26159	0.396743
$336$	0	0
$337$	12.8126	0.697946	0.348973	−	0.937133i	$-0.386531\pi$
$337$	12.8126	0.697946	0.348973	+	0.937133i	$0.386531\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−45.2694	−2.45148
$342$	0	0
$343$	14.5043	0.783157
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−8.18793	−0.439551	−0.219775	−	0.975550i	$-0.570532\pi$
$347$	−8.18793	−0.439551	−0.219775	+	0.975550i	$0.570532\pi$
$348$	0	0
$349$	5.34464	0.286092	0.143046	−	0.989716i	$-0.454310\pi$
$349$	5.34464	0.286092	0.143046	+	0.989716i	$0.454310\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−14.8317	−0.789411	−0.394705	−	0.918808i	$-0.629153\pi$
$353$	−14.8317	−0.789411	−0.394705	+	0.918808i	$0.629153\pi$
$354$	0	0
$355$	−1.48619	−0.0788787
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6.55031	−0.345712	−0.172856	−	0.984947i	$-0.555300\pi$
$359$	−6.55031	−0.345712	−0.172856	+	0.984947i	$0.555300\pi$
$360$	0	0
$361$	49.8695	2.62471
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5.31701	0.278305
$366$	0	0
$367$	28.1202	1.46786	0.733932	−	0.679223i	$-0.237683\pi$
$367$	28.1202	1.46786	0.733932	+	0.679223i	$0.237683\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.42122	0.0737860
$372$	0	0
$373$	−17.7360	−0.918335	−0.459168	−	0.888350i	$-0.651852\pi$
$373$	−17.7360	−0.918335	−0.459168	+	0.888350i	$0.651852\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.45857	0.126623
$378$	0	0
$379$	−15.8228	−0.812763	−0.406381	−	0.913703i	$-0.633210\pi$
$379$	−15.8228	−0.812763	−0.406381	+	0.913703i	$0.633210\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7.11135	0.363373	0.181686	−	0.983357i	$-0.441844\pi$
$383$	7.11135	0.363373	0.181686	+	0.983357i	$0.441844\pi$
$384$	0	0
$385$	16.0094	0.815913
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5.74761	0.291415	0.145708	−	0.989328i	$-0.453454\pi$
$389$	5.74761	0.291415	0.145708	+	0.989328i	$0.453454\pi$
$390$	0	0
$391$	−14.1202	−0.714090
$392$	0	0
$393$	0	0
$394$	0	0
$395$	15.9635	0.803211
$396$	0	0
$397$	6.35710	0.319054	0.159527	−	0.987194i	$-0.449003\pi$
$397$	6.35710	0.319054	0.159527	+	0.987194i	$0.449003\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−35.1100	−1.75331	−0.876654	−	0.481121i	$-0.840230\pi$
$401$	−35.1100	−1.75331	−0.876654	+	0.481121i	$0.840230\pi$
$402$	0	0
$403$	−4.41269	−0.219812
$404$	0	0
$405$	0	0
$406$	0	0
$407$	34.7271	1.72136
$408$	0	0
$409$	24.8504	1.22877	0.614387	−	0.789005i	$-0.289403\pi$
$409$	24.8504	1.22877	0.614387	+	0.789005i	$0.289403\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2.30015	−0.113183
$414$	0	0
$415$	−12.0648	−0.592238
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−3.37535	−0.164897	−0.0824483	−	0.996595i	$-0.526274\pi$
$419$	−3.37535	−0.164897	−0.0824483	+	0.996595i	$0.526274\pi$
$420$	0	0
$421$	−3.96351	−0.193169	−0.0965847	−	0.995325i	$-0.530792\pi$
$421$	−3.96351	−0.193169	−0.0965847	+	0.995325i	$0.530792\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.80320	0.135975
$426$	0	0
$427$	0.823510	0.0398524
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−7.62516	−0.367291	−0.183645	−	0.982993i	$-0.558790\pi$
$431$	−7.62516	−0.367291	−0.183645	+	0.982993i	$0.558790\pi$
$432$	0	0
$433$	−7.29024	−0.350347	−0.175173	−	0.984538i	$-0.556049\pi$
$433$	−7.29024	−0.350347	−0.175173	+	0.984538i	$0.556049\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	41.8023	1.99968
$438$	0	0
$439$	−36.3905	−1.73682	−0.868411	−	0.495844i	$-0.834859\pi$
$439$	−36.3905	−1.73682	−0.868411	+	0.495844i	$0.834859\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	35.3738	1.68066	0.840330	−	0.542076i	$-0.182361\pi$
$443$	35.3738	1.68066	0.840330	+	0.542076i	$0.182361\pi$
$444$	0	0
$445$	15.2529	0.723057
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−14.9425	−0.705181	−0.352591	−	0.935778i	$-0.614699\pi$
$449$	−14.9425	−0.705181	−0.352591	+	0.935778i	$0.614699\pi$
$450$	0	0
$451$	−42.0730	−1.98114
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.56053	0.0731590
$456$	0	0
$457$	−12.7841	−0.598015	−0.299008	−	0.954251i	$-0.596656\pi$
$457$	−12.7841	−0.598015	−0.299008	+	0.954251i	$0.596656\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.11443	0.377927	0.188963	−	0.981984i	$-0.439487\pi$
$461$	8.11443	0.377927	0.188963	+	0.981984i	$0.439487\pi$
$462$	0	0
$463$	21.5605	1.00200	0.501002	−	0.865446i	$-0.332965\pi$
$463$	21.5605	1.00200	0.501002	+	0.865446i	$0.332965\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	30.8602	1.42804	0.714019	−	0.700127i	$-0.246873\pi$
$467$	30.8602	1.42804	0.714019	+	0.700127i	$0.246873\pi$
$468$	0	0
$469$	−22.0547	−1.01839
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−6.30849	−0.290065
$474$	0	0
$475$	−8.29877	−0.380774
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1.06412	0.0486208	0.0243104	−	0.999704i	$-0.492261\pi$
$479$	1.06412	0.0486208	0.0243104	+	0.999704i	$0.492261\pi$
$480$	0	0
$481$	3.38507	0.154346
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.36374	−0.288962
$486$	0	0
$487$	27.0182	1.22431	0.612157	−	0.790736i	$-0.290302\pi$
$487$	27.0182	1.22431	0.612157	+	0.790736i	$0.290302\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−16.2338	−0.732621	−0.366310	−	0.930493i	$-0.619379\pi$
$491$	−16.2338	−0.732621	−0.366310	+	0.930493i	$0.619379\pi$
$492$	0	0
$493$	−13.4132	−0.604100
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.51381	0.202472
$498$	0	0
$499$	−20.7208	−0.927592	−0.463796	−	0.885942i	$-0.653513\pi$
$499$	−20.7208	−0.927592	−0.463796	+	0.885942i	$0.653513\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	5.47834	0.244267	0.122134	−	0.992514i	$-0.461026\pi$
$503$	5.47834	0.244267	0.122134	+	0.992514i	$0.461026\pi$
$504$	0	0
$505$	−2.07435	−0.0923072
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−8.71061	−0.386091	−0.193045	−	0.981190i	$-0.561836\pi$
$509$	−8.71061	−0.386091	−0.193045	+	0.981190i	$0.561836\pi$
$510$	0	0
$511$	−16.1487	−0.714376
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−5.05610	−0.222798
$516$	0	0
$517$	−50.7365	−2.23139
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−2.87041	−0.125755	−0.0628774	−	0.998021i	$-0.520028\pi$
$521$	−2.87041	−0.125755	−0.0628774	+	0.998021i	$0.520028\pi$
$522$	0	0
$523$	6.65295	0.290913	0.145457	−	0.989365i	$-0.453535\pi$
$523$	6.65295	0.290913	0.145457	+	0.989365i	$0.453535\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	24.0743	1.04869
$528$	0	0
$529$	2.37311	0.103179
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−4.10112	−0.177639
$534$	0	0
$535$	13.2901	0.574580
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−11.7252	−0.505042
$540$	0	0
$541$	7.79348	0.335068	0.167534	−	0.985866i	$-0.446420\pi$
$541$	7.79348	0.335068	0.167534	+	0.985866i	$0.446420\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	8.34549	0.357482
$546$	0	0
$547$	11.7580	0.502736	0.251368	−	0.967892i	$-0.419120\pi$
$547$	11.7580	0.502736	0.251368	+	0.967892i	$0.419120\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	39.7092	1.69167
$552$	0	0
$553$	−48.4839	−2.06175
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16.7827	0.711107	0.355553	−	0.934656i	$-0.384292\pi$
$557$	16.7827	0.711107	0.355553	+	0.934656i	$0.384292\pi$
$558$	0	0
$559$	−0.614928	−0.0260087
$560$	0	0
$561$	0	0
$562$	0	0
$563$	25.7486	1.08518	0.542588	−	0.839999i	$-0.317445\pi$
$563$	25.7486	1.08518	0.542588	+	0.839999i	$0.317445\pi$
$564$	0	0
$565$	−5.10197	−0.214641
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5.11308	−0.214351	−0.107176	−	0.994240i	$-0.534181\pi$
$569$	−5.11308	−0.214351	−0.107176	+	0.994240i	$0.534181\pi$
$570$	0	0
$571$	−25.4657	−1.06571	−0.532853	−	0.846208i	$-0.678880\pi$
$571$	−25.4657	−1.06571	−0.532853	+	0.846208i	$0.678880\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−5.03717	−0.210065
$576$	0	0
$577$	44.7164	1.86157	0.930783	−	0.365571i	$-0.119126\pi$
$577$	44.7164	1.86157	0.930783	+	0.365571i	$0.119126\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	36.6429	1.52020
$582$	0	0
$583$	2.46658	0.102155
$584$	0	0
$585$	0	0
$586$	0	0
$587$	40.9156	1.68877	0.844383	−	0.535740i	$-0.179967\pi$
$587$	40.9156	1.68877	0.844383	+	0.535740i	$0.179967\pi$
$588$	0	0
$589$	−71.2711	−2.93668
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−11.9729	−0.491667	−0.245834	−	0.969312i	$-0.579062\pi$
$593$	−11.9729	−0.491667	−0.245834	+	0.969312i	$0.579062\pi$
$594$	0	0
$595$	−8.51381	−0.349032
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−14.3571	−0.586615	−0.293308	−	0.956018i	$-0.594756\pi$
$599$	−14.3571	−0.586615	−0.293308	+	0.956018i	$0.594756\pi$
$600$	0	0
$601$	44.6420	1.82099	0.910493	−	0.413524i	$-0.135702\pi$
$601$	44.6420	1.82099	0.910493	+	0.413524i	$0.135702\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	16.7850	0.682405
$606$	0	0
$607$	29.7740	1.20849	0.604245	−	0.796798i	$-0.293475\pi$
$607$	29.7740	1.20849	0.604245	+	0.796798i	$0.293475\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.94561	−0.200078
$612$	0	0
$613$	33.4132	1.34955	0.674773	−	0.738025i	$-0.264241\pi$
$613$	33.4132	1.34955	0.674773	+	0.738025i	$0.264241\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−44.5881	−1.79505	−0.897525	−	0.440963i	$-0.854637\pi$
$617$	−44.5881	−1.79505	−0.897525	+	0.440963i	$0.854637\pi$
$618$	0	0
$619$	36.9533	1.48528	0.742638	−	0.669693i	$-0.233574\pi$
$619$	36.9533	1.48528	0.742638	+	0.669693i	$0.233574\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−46.3257	−1.85600
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−18.4679	−0.736365
$630$	0	0
$631$	1.10197	0.0438687	0.0219344	−	0.999759i	$-0.493018\pi$
$631$	1.10197	0.0438687	0.0219344	+	0.999759i	$0.493018\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	11.0830	0.439817
$636$	0	0
$637$	−1.14293	−0.0452847
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−13.5691	−0.535946	−0.267973	−	0.963426i	$-0.586354\pi$
$641$	−13.5691	−0.535946	−0.267973	+	0.963426i	$0.586354\pi$
$642$	0	0
$643$	−9.72953	−0.383695	−0.191848	−	0.981425i	$-0.561448\pi$
$643$	−9.72953	−0.383695	−0.191848	+	0.981425i	$0.561448\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	38.5786	1.51668	0.758341	−	0.651858i	$-0.226010\pi$
$647$	38.5786	1.51668	0.758341	+	0.651858i	$0.226010\pi$
$648$	0	0
$649$	−3.99201	−0.156700
$650$	0	0
$651$	0	0
$652$	0	0
$653$	17.1304	0.670366	0.335183	−	0.942153i	$-0.391202\pi$
$653$	17.1304	0.670366	0.335183	+	0.942153i	$0.391202\pi$
$654$	0	0
$655$	−1.55116	−0.0606088
$656$	0	0
$657$	0	0
$658$	0	0
$659$	23.6630	0.921780	0.460890	−	0.887457i	$-0.347530\pi$
$659$	23.6630	0.921780	0.460890	+	0.887457i	$0.347530\pi$
$660$	0	0
$661$	18.4301	0.716847	0.358424	−	0.933559i	$-0.383314\pi$
$661$	18.4301	0.716847	0.358424	+	0.933559i	$0.383314\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	25.2048	0.977400
$666$	0	0
$667$	24.1026	0.933258
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.42924	0.0551751
$672$	0	0
$673$	−16.4964	−0.635890	−0.317945	−	0.948109i	$-0.602993\pi$
$673$	−16.4964	−0.635890	−0.317945	+	0.948109i	$0.602993\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10.1028	0.388283	0.194141	−	0.980974i	$-0.437808\pi$
$677$	10.1028	0.388283	0.194141	+	0.980974i	$0.437808\pi$
$678$	0	0
$679$	19.3278	0.741732
$680$	0	0
$681$	0	0
$682$	0	0
$683$	8.16917	0.312585	0.156292	−	0.987711i	$-0.450046\pi$
$683$	8.16917	0.312585	0.156292	+	0.987711i	$0.450046\pi$
$684$	0	0
$685$	−4.81258	−0.183879
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0.240434	0.00915979
$690$	0	0
$691$	3.51466	0.133704	0.0668521	−	0.997763i	$-0.478704\pi$
$691$	3.51466	0.133704	0.0668521	+	0.997763i	$0.478704\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	21.4657	0.814241
$696$	0	0
$697$	22.3745	0.847493
$698$	0	0
$699$	0	0
$700$	0	0
$701$	33.9897	1.28378	0.641888	−	0.766799i	$-0.278152\pi$
$701$	33.9897	1.28378	0.641888	+	0.766799i	$0.278152\pi$
$702$	0	0
$703$	54.6736	2.06205
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.30015	0.236941
$708$	0	0
$709$	−3.99777	−0.150139	−0.0750696	−	0.997178i	$-0.523918\pi$
$709$	−3.99777	−0.150139	−0.0750696	+	0.997178i	$0.523918\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−43.2600	−1.62010
$714$	0	0
$715$	2.70837	0.101287
$716$	0	0
$717$	0	0
$718$	0	0
$719$	33.2142	1.23868	0.619340	−	0.785123i	$-0.287400\pi$
$719$	33.2142	1.23868	0.619340	+	0.785123i	$0.287400\pi$
$720$	0	0
$721$	15.3562	0.571897
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.78496	−0.177709
$726$	0	0
$727$	27.5888	1.02321	0.511607	−	0.859220i	$-0.329051\pi$
$727$	27.5888	1.02321	0.511607	+	0.859220i	$0.329051\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3.35487	0.124084
$732$	0	0
$733$	9.71860	0.358965	0.179482	−	0.983761i	$-0.442558\pi$
$733$	9.71860	0.358965	0.179482	+	0.983761i	$0.442558\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−38.2769	−1.40995
$738$	0	0
$739$	−36.2355	−1.33294	−0.666472	−	0.745530i	$-0.732197\pi$
$739$	−36.2355	−1.33294	−0.666472	+	0.745530i	$0.732197\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−20.6814	−0.758728	−0.379364	−	0.925247i	$-0.623857\pi$
$743$	−20.6814	−0.758728	−0.379364	+	0.925247i	$0.623857\pi$
$744$	0	0
$745$	−14.7943	−0.542022
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−40.3642	−1.47488
$750$	0	0
$751$	−29.0749	−1.06096	−0.530478	−	0.847699i	$-0.677988\pi$
$751$	−29.0749	−1.06096	−0.530478	+	0.847699i	$0.677988\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−0.981753	−0.0357296
$756$	0	0
$757$	−22.1202	−0.803973	−0.401986	−	0.915646i	$-0.631680\pi$
$757$	−22.1202	−0.803973	−0.401986	+	0.915646i	$0.631680\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	15.2907	0.554289	0.277145	−	0.960828i	$-0.410612\pi$
$761$	15.2907	0.554289	0.277145	+	0.960828i	$0.410612\pi$
$762$	0	0
$763$	−25.3467	−0.917612
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−0.389126	−0.0140505
$768$	0	0
$769$	27.9978	1.00963	0.504813	−	0.863229i	$-0.331562\pi$
$769$	27.9978	1.00963	0.504813	+	0.863229i	$0.331562\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−18.4853	−0.664871	−0.332436	−	0.943126i	$-0.607870\pi$
$773$	−18.4853	−0.664871	−0.332436	+	0.943126i	$0.607870\pi$
$774$	0	0
$775$	8.58816	0.308496
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−66.2387	−2.37325
$780$	0	0
$781$	7.83391	0.280319
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−6.86154	−0.244899
$786$	0	0
$787$	−8.92616	−0.318183	−0.159092	−	0.987264i	$-0.550857\pi$
$787$	−8.92616	−0.318183	−0.159092	+	0.987264i	$0.550857\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	15.4956	0.550959
$792$	0	0
$793$	0.139317	0.00494728
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−32.4555	−1.14963	−0.574816	−	0.818283i	$-0.694926\pi$
$797$	−32.4555	−1.14963	−0.574816	+	0.818283i	$0.694926\pi$
$798$	0	0
$799$	26.9818	0.954546
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−28.0267	−0.989042
$804$	0	0
$805$	15.2988	0.539211
$806$	0	0
$807$	0	0
$808$	0	0
$809$	12.4759	0.438631	0.219315	−	0.975654i	$-0.429618\pi$
$809$	12.4759	0.438631	0.219315	+	0.975654i	$0.429618\pi$
$810$	0	0
$811$	16.0632	0.564057	0.282028	−	0.959406i	$-0.408993\pi$
$811$	16.0632	0.564057	0.282028	+	0.959406i	$0.408993\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	4.90741	0.171899
$816$	0	0
$817$	−9.93194	−0.347475
$818$	0	0
$819$	0	0
$820$	0	0
$821$	47.6149	1.66177	0.830886	−	0.556443i	$-0.187834\pi$
$821$	47.6149	1.66177	0.830886	+	0.556443i	$0.187834\pi$
$822$	0	0
$823$	−17.7553	−0.618910	−0.309455	−	0.950914i	$-0.600147\pi$
$823$	−17.7553	−0.618910	−0.309455	+	0.950914i	$0.600147\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−24.2420	−0.842976	−0.421488	−	0.906834i	$-0.638492\pi$
$827$	−24.2420	−0.842976	−0.421488	+	0.906834i	$0.638492\pi$
$828$	0	0
$829$	−30.3816	−1.05520	−0.527599	−	0.849494i	$-0.676908\pi$
$829$	−30.3816	−1.05520	−0.527599	+	0.849494i	$0.676908\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.23550	0.216047
$834$	0	0
$835$	−3.47664	−0.120314
$836$	0	0
$837$	0	0
$838$	0	0
$839$	10.6434	0.367451	0.183726	−	0.982978i	$-0.441184\pi$
$839$	10.6434	0.367451	0.183726	+	0.982978i	$0.441184\pi$
$840$	0	0
$841$	−6.10420	−0.210490
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−12.7360	−0.438132
$846$	0	0
$847$	−50.9788	−1.75165
$848$	0	0
$849$	0	0
$850$	0	0
$851$	33.1857	1.13759
$852$	0	0
$853$	27.1492	0.929571	0.464785	−	0.885423i	$-0.346131\pi$
$853$	27.1492	0.929571	0.464785	+	0.885423i	$0.346131\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	43.1666	1.47454	0.737271	−	0.675597i	$-0.236114\pi$
$857$	43.1666	1.47454	0.737271	+	0.675597i	$0.236114\pi$
$858$	0	0
$859$	6.61972	0.225862	0.112931	−	0.993603i	$-0.463976\pi$
$859$	6.61972	0.225862	0.112931	+	0.993603i	$0.463976\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−5.31055	−0.180773	−0.0903866	−	0.995907i	$-0.528810\pi$
$863$	−5.31055	−0.180773	−0.0903866	+	0.995907i	$0.528810\pi$
$864$	0	0
$865$	10.5423	0.358449
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−84.1459	−2.85446
$870$	0	0
$871$	−3.73109	−0.126423
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.03717	−0.102675
$876$	0	0
$877$	−2.12022	−0.0715946	−0.0357973	−	0.999359i	$-0.511397\pi$
$877$	−2.12022	−0.0715946	−0.0357973	+	0.999359i	$0.511397\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	22.5668	0.760296	0.380148	−	0.924926i	$-0.375873\pi$
$881$	22.5668	0.760296	0.380148	+	0.924926i	$0.375873\pi$
$882$	0	0
$883$	−41.5399	−1.39793	−0.698964	−	0.715157i	$-0.746355\pi$
$883$	−41.5399	−1.39793	−0.698964	+	0.715157i	$0.746355\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	43.4497	1.45890	0.729449	−	0.684035i	$-0.239777\pi$
$887$	43.4497	1.45890	0.729449	+	0.684035i	$0.239777\pi$
$888$	0	0
$889$	−33.6611	−1.12896
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−79.8784	−2.67303
$894$	0	0
$895$	−4.90741	−0.164037
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−41.0940	−1.37056
$900$	0	0
$901$	−1.31173	−0.0437002
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5.21504	0.173354
$906$	0	0
$907$	−4.38266	−0.145524	−0.0727620	−	0.997349i	$-0.523181\pi$
$907$	−4.38266	−0.145524	−0.0727620	+	0.997349i	$0.523181\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−14.6719	−0.486101	−0.243051	−	0.970014i	$-0.578148\pi$
$911$	−14.6719	−0.486101	−0.243051	+	0.970014i	$0.578148\pi$
$912$	0	0
$913$	63.5953	2.10470
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.71114	0.155575
$918$	0	0
$919$	37.3233	1.23118	0.615591	−	0.788066i	$-0.288917\pi$
$919$	37.3233	1.23118	0.615591	+	0.788066i	$0.288917\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0.763621	0.0251349
$924$	0	0
$925$	−6.58816	−0.216617
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−19.3250	−0.634033	−0.317016	−	0.948420i	$-0.602681\pi$
$929$	−19.3250	−0.634033	−0.317016	+	0.948420i	$0.602681\pi$
$930$	0	0
$931$	−18.4599	−0.605000
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−14.7761	−0.483230
$936$	0	0
$937$	−15.8339	−0.517271	−0.258636	−	0.965975i	$-0.583273\pi$
$937$	−15.8339	−0.517271	−0.258636	+	0.965975i	$0.583273\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−4.35933	−0.142110	−0.0710551	−	0.997472i	$-0.522637\pi$
$941$	−4.35933	−0.142110	−0.0710551	+	0.997472i	$0.522637\pi$
$942$	0	0
$943$	−40.2055	−1.30927
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−39.6316	−1.28785	−0.643927	−	0.765087i	$-0.722696\pi$
$947$	−39.6316	−1.28785	−0.643927	+	0.765087i	$0.722696\pi$
$948$	0	0
$949$	−2.73194	−0.0886826
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−38.3922	−1.24365	−0.621823	−	0.783158i	$-0.713608\pi$
$953$	−38.3922	−1.24365	−0.621823	+	0.783158i	$0.713608\pi$
$954$	0	0
$955$	−25.6901	−0.831313
$956$	0	0
$957$	0	0
$958$	0	0
$959$	14.6166	0.471996
$960$	0	0
$961$	42.7565	1.37924
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.19680	0.102909
$966$	0	0
$967$	52.3257	1.68268	0.841340	−	0.540506i	$-0.181767\pi$
$967$	52.3257	1.68268	0.841340	+	0.540506i	$0.181767\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	45.4831	1.45962	0.729811	−	0.683649i	$-0.239608\pi$
$971$	45.4831	1.45962	0.729811	+	0.683649i	$0.239608\pi$
$972$	0	0
$973$	−65.1951	−2.09006
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.03511	0.0651090	0.0325545	−	0.999470i	$-0.489636\pi$
$977$	2.03511	0.0651090	0.0325545	+	0.999470i	$0.489636\pi$
$978$	0	0
$979$	−80.4002	−2.56960
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−25.4130	−0.810550	−0.405275	−	0.914195i	$-0.632824\pi$
$983$	−25.4130	−0.810550	−0.405275	+	0.914195i	$0.632824\pi$
$984$	0	0
$985$	13.6808	0.435905
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.02848	−0.191694
$990$	0	0
$991$	0.259533	0.00824435	0.00412218	−	0.999992i	$-0.498688\pi$
$991$	0.259533	0.00824435	0.00412218	+	0.999992i	$0.498688\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	5.95413	0.188759
$996$	0	0
$997$	−48.9140	−1.54912	−0.774561	−	0.632499i	$-0.782029\pi$
$997$	−48.9140	−1.54912	−0.774561	+	0.632499i	$0.782029\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6480.2.a.cb.1.1		4
3.2	odd	2		6480.2.a.bz.1.1		4
4.3	odd	2		3240.2.a.u.1.4		4
9.2	odd	6		2160.2.q.l.1441.4		8
9.4	even	3		720.2.q.l.241.2		8
9.5	odd	6		2160.2.q.l.721.4		8
9.7	even	3		720.2.q.l.481.2		8
12.11	even	2		3240.2.a.s.1.4		4
36.7	odd	6		360.2.q.e.121.3	✓	8
36.11	even	6		1080.2.q.e.361.1		8
36.23	even	6		1080.2.q.e.721.1		8
36.31	odd	6		360.2.q.e.241.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.q.e.121.3	✓	8	36.7	odd	6
360.2.q.e.241.3	yes	8	36.31	odd	6
720.2.q.l.241.2		8	9.4	even	3
720.2.q.l.481.2		8	9.7	even	3
1080.2.q.e.361.1		8	36.11	even	6
1080.2.q.e.721.1		8	36.23	even	6
2160.2.q.l.721.4		8	9.5	odd	6
2160.2.q.l.1441.4		8	9.2	odd	6
3240.2.a.s.1.4		4	12.11	even	2
3240.2.a.u.1.4		4	4.3	odd	2
6480.2.a.bz.1.1		4	3.2	odd	2
6480.2.a.cb.1.1		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:	\( N \)	\(=\)	\( 6480 = 2^{4} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6480.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -3.03717 q^{7} -5.27114 q^{11} -0.513812 q^{13} +2.80320 q^{17} -8.29877 q^{19} -5.03717 q^{23} +1.00000 q^{25} -4.78496 q^{29} +8.58816 q^{31} -3.03717 q^{35} -6.58816 q^{37} +7.98175 q^{41} +1.19680 q^{43} +9.62533 q^{47} +2.22442 q^{49} -0.467941 q^{53} -5.27114 q^{55} +0.757332 q^{59} -0.271144 q^{61} -0.513812 q^{65} +7.26159 q^{67} -1.48619 q^{71} +5.31701 q^{73} +16.0094 q^{77} +15.9635 q^{79} -12.0648 q^{83} +2.80320 q^{85} +15.2529 q^{89} +1.56053 q^{91} -8.29877 q^{95} -6.36374 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + q^{7} - q^{11} + 4 q^{13} + 5 q^{17} - q^{19} - 7 q^{23} + 4 q^{25} + 7 q^{29} + 2 q^{31} + q^{35} + 6 q^{37} + 12 q^{41} + 11 q^{43} - 7 q^{47} + 3 q^{49} + 12 q^{53} - q^{55} - 11 q^{59}+ \cdots + q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 + q^7 - q^11 + 4 * q^13 + 5 * q^17 - q^19 - 7 * q^23 + 4 * q^25 + 7 * q^29 + 2 * q^31 + q^35 + 6 * q^37 + 12 * q^41 + 11 * q^43 - 7 * q^47 + 3 * q^49 + 12 * q^53 - q^55 - 11 * q^59 + 19 * q^61 + 4 * q^65 + 10 * q^67 - 12 * q^71 + 9 * q^73 + 32 * q^77 + 24 * q^79 - 23 * q^83 + 5 * q^85 + 21 * q^89 - 14 * q^91 - q^95 + q^97

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