LMFDB - Embedded newform 2880.3.e.a.2431.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2880,3,Mod(2431,2880)] 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("2880.2431"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2880, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2880.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,-64,0,0,0,40,0,0,0,0,0,0,0,20,0,0,0, -72,0,0,0,0,0,0,0,-112] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(37)] == traces)

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

Self dual:	no
Analytic conductor:	$78.4743161358$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2431.2
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	2880.2431
Dual form			2880.3.e.a.2431.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607 q^{5} +12.1803i q^{7} +5.52786i q^{11} -20.4721 q^{13} +1.05573 q^{17} +12.0000i q^{19} -31.5967i q^{23} +5.00000 q^{25} -44.8328 q^{29} +27.4164i q^{31} -27.2361i q^{35} -23.5279 q^{37} -8.69505 q^{41} +26.6525i q^{43} +44.5410i q^{47} -99.3607 q^{49} +0.695048 q^{53} -12.3607i q^{55} -94.6099i q^{59} +33.1935 q^{61} +45.7771 q^{65} -82.2067i q^{67} +122.249i q^{71} +132.164 q^{73} -67.3313 q^{77} -134.833i q^{79} +19.4590i q^{83} -2.36068 q^{85} +30.0000 q^{89} -249.358i q^{91} -26.8328i q^{95} +12.3344 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 64 q^{13} + 40 q^{17} + 20 q^{25} - 72 q^{29} - 112 q^{37} - 160 q^{41} - 308 q^{49} + 128 q^{53} - 64 q^{61} + 40 q^{65} - 8 q^{73} + 160 q^{77} + 80 q^{85} + 120 q^{89} + 264 q^{97}+O(q^{100}) $ 4 * q - 64 * q^13 + 40 * q^17 + 20 * q^25 - 72 * q^29 - 112 * q^37 - 160 * q^41 - 308 * q^49 + 128 * q^53 - 64 * q^61 + 40 * q^65 - 8 * q^73 + 160 * q^77 + 80 * q^85 + 120 * q^89 + 264 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$5$	−2.23607	−0.447214
$5$	−2.23607	−0.447214
$6$	0	0
$7$	12.1803i	1.74005i	0.493009	+	0.870024i	$0.335897\pi$
$7$	12.1803i	1.74005i	−0.493009	+	0.870024i	$0.664103\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.52786i	0.502533i	0.967918	+	0.251267i	$0.0808471\pi$
$11$	5.52786i	0.502533i	−0.967918	+	0.251267i	$0.919153\pi$
$12$	0	0
$13$	−20.4721	−1.57478	−0.787390	−	0.616455i	$-0.788568\pi$
$13$	−20.4721	−1.57478	−0.787390	+	0.616455i	$0.788568\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.05573	0.0621017	0.0310508	−	0.999518i	$-0.490115\pi$
$17$	1.05573	0.0621017	0.0310508	+	0.999518i	$0.490115\pi$
$18$	0	0
$19$	12.0000i	0.631579i	0.948829	+	0.315789i	$0.102269\pi$
$19$	12.0000i	0.631579i	−0.948829	+	0.315789i	$0.897731\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 31.5967i	− 1.37377i	−0.726765	−	0.686886i	$-0.758977\pi$
$23$	− 31.5967i	− 1.37377i	0.726765	−	0.686886i	$-0.241023\pi$
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−44.8328	−1.54596	−0.772980	−	0.634431i	$-0.781235\pi$
$29$	−44.8328	−1.54596	−0.772980	+	0.634431i	$0.781235\pi$
$30$	0	0
$31$	27.4164i	0.884400i	0.896916	+	0.442200i	$0.145802\pi$
$31$	27.4164i	0.884400i	−0.896916	+	0.442200i	$0.854198\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 27.2361i	− 0.778173i
$36$	0	0
$37$	−23.5279	−0.635888	−0.317944	−	0.948109i	$-0.602992\pi$
$37$	−23.5279	−0.635888	−0.317944	+	0.948109i	$0.602992\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.69505	−0.212074	−0.106037	−	0.994362i	$-0.533816\pi$
$41$	−8.69505	−0.212074	−0.106037	+	0.994362i	$0.533816\pi$
$42$	0	0
$43$	26.6525i	0.619825i	0.950765	+	0.309913i	$0.100300\pi$
$43$	26.6525i	0.619825i	−0.950765	+	0.309913i	$0.899700\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	44.5410i	0.947681i	0.880611	+	0.473841i	$0.157133\pi$
$47$	44.5410i	0.947681i	−0.880611	+	0.473841i	$0.842867\pi$
$48$	0	0
$49$	−99.3607	−2.02777
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.695048	0.0131141	0.00655706	−	0.999979i	$-0.497913\pi$
$53$	0.695048	0.0131141	0.00655706	+	0.999979i	$0.497913\pi$
$54$	0	0
$55$	− 12.3607i	− 0.224740i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 94.6099i	− 1.60356i	−0.597621	−	0.801779i	$-0.703887\pi$
$59$	− 94.6099i	− 1.60356i	0.597621	−	0.801779i	$-0.296113\pi$
$60$	0	0
$61$	33.1935	0.544156	0.272078	−	0.962275i	$-0.412289\pi$
$61$	33.1935	0.544156	0.272078	+	0.962275i	$0.412289\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	45.7771	0.704263
$66$	0	0
$67$	− 82.2067i	− 1.22696i	−0.789708	−	0.613482i	$-0.789768\pi$
$67$	− 82.2067i	− 1.22696i	0.789708	−	0.613482i	$-0.210232\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	122.249i	1.72182i	0.508757	+	0.860910i	$0.330105\pi$
$71$	122.249i	1.72182i	−0.508757	+	0.860910i	$0.669895\pi$
$72$	0	0
$73$	132.164	1.81047	0.905233	−	0.424915i	$-0.139696\pi$
$73$	132.164	1.81047	0.905233	+	0.424915i	$0.139696\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−67.3313	−0.874432
$78$	0	0
$79$	− 134.833i	− 1.70674i	−0.521302	−	0.853372i	$-0.674553\pi$
$79$	− 134.833i	− 1.70674i	0.521302	−	0.853372i	$-0.325447\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	19.4590i	0.234446i	0.993106	+	0.117223i	$0.0373992\pi$
$83$	19.4590i	0.234446i	−0.993106	+	0.117223i	$0.962601\pi$
$84$	0	0
$85$	−2.36068	−0.0277727
$86$	0	0
$87$	0	0
$88$	0	0
$89$	30.0000	0.337079	0.168539	−	0.985695i	$-0.446095\pi$
$89$	30.0000	0.337079	0.168539	+	0.985695i	$0.446095\pi$
$90$	0	0
$91$	− 249.358i	− 2.74019i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 26.8328i	− 0.282451i
$96$	0	0
$97$	12.3344	0.127158	0.0635792	−	0.997977i	$-0.479748\pi$
$97$	12.3344	0.127158	0.0635792	+	0.997977i	$0.479748\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	105.554	1.04509	0.522545	−	0.852611i	$-0.324983\pi$
$101$	105.554	1.04509	0.522545	+	0.852611i	$0.324983\pi$
$102$	0	0
$103$	100.987i	0.980455i	0.871595	+	0.490227i	$0.163086\pi$
$103$	100.987i	0.980455i	−0.871595	+	0.490227i	$0.836914\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	22.8754i	0.213789i	0.994270	+	0.106894i	$0.0340907\pi$
$107$	22.8754i	0.213789i	−0.994270	+	0.106894i	$0.965909\pi$
$108$	0	0
$109$	35.5279	0.325944	0.162972	−	0.986631i	$-0.447892\pi$
$109$	35.5279	0.325944	0.162972	+	0.986631i	$0.447892\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	70.7214	0.625853	0.312926	−	0.949777i	$-0.398691\pi$
$113$	70.7214	0.625853	0.312926	+	0.949777i	$0.398691\pi$
$114$	0	0
$115$	70.6525i	0.614369i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	12.8591i	0.108060i
$120$	0	0
$121$	90.4427	0.747460
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.1803	−0.0894427
$126$	0	0
$127$	− 148.541i	− 1.16961i	−0.811172	−	0.584807i	$-0.801170\pi$
$127$	− 148.541i	− 1.16961i	0.811172	−	0.584807i	$-0.198830\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	34.9180i	0.266549i	0.991079	+	0.133275i	$0.0425492\pi$
$131$	34.9180i	0.266549i	−0.991079	+	0.133275i	$0.957451\pi$
$132$	0	0
$133$	−146.164	−1.09898
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−100.610	−0.734379	−0.367189	−	0.930146i	$-0.619680\pi$
$137$	−100.610	−0.734379	−0.367189	+	0.930146i	$0.619680\pi$
$138$	0	0
$139$	− 198.610i	− 1.42885i	−0.699713	−	0.714424i	$-0.746689\pi$
$139$	− 198.610i	− 1.42885i	0.699713	−	0.714424i	$-0.253311\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 113.167i	− 0.791379i
$144$	0	0
$145$	100.249	0.691374
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−203.082	−1.36297	−0.681483	−	0.731834i	$-0.738665\pi$
$149$	−203.082	−1.36297	−0.681483	+	0.731834i	$0.738665\pi$
$150$	0	0
$151$	− 113.803i	− 0.753665i	−0.926281	−	0.376832i	$-0.877013\pi$
$151$	− 113.803i	− 0.753665i	0.926281	−	0.376832i	$-0.122987\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 61.3050i	− 0.395516i
$156$	0	0
$157$	144.918	0.923044	0.461522	−	0.887129i	$-0.347303\pi$
$157$	144.918	0.923044	0.461522	+	0.887129i	$0.347303\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	384.859	2.39043
$162$	0	0
$163$	203.374i	1.24769i	0.781547	+	0.623846i	$0.214431\pi$
$163$	203.374i	1.24769i	−0.781547	+	0.623846i	$0.785569\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	88.7639i	0.531521i	0.964039	+	0.265760i	$0.0856230\pi$
$167$	88.7639i	0.531521i	−0.964039	+	0.265760i	$0.914377\pi$
$168$	0	0
$169$	250.108	1.47993
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−135.580	−0.783702	−0.391851	−	0.920029i	$-0.628165\pi$
$173$	−135.580	−0.783702	−0.391851	+	0.920029i	$0.628165\pi$
$174$	0	0
$175$	60.9017i	0.348010i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 203.830i	− 1.13871i	−0.822091	−	0.569357i	$-0.807192\pi$
$179$	− 203.830i	− 1.13871i	0.822091	−	0.569357i	$-0.192808\pi$
$180$	0	0
$181$	−22.6687	−0.125242	−0.0626208	−	0.998037i	$-0.519946\pi$
$181$	−22.6687	−0.125242	−0.0626208	+	0.998037i	$0.519946\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	52.6099	0.284378
$186$	0	0
$187$	5.83592i	0.0312081i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	150.748i	0.789255i	0.918841	+	0.394627i	$0.129126\pi$
$191$	150.748i	0.789255i	−0.918841	+	0.394627i	$0.870874\pi$
$192$	0	0
$193$	−210.498	−1.09067	−0.545333	−	0.838220i	$-0.683597\pi$
$193$	−210.498	−1.09067	−0.545333	+	0.838220i	$0.683597\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−97.0232	−0.492504	−0.246252	−	0.969206i	$-0.579199\pi$
$197$	−97.0232	−0.492504	−0.246252	+	0.969206i	$0.579199\pi$
$198$	0	0
$199$	104.892i	0.527094i	0.964647	+	0.263547i	$0.0848924\pi$
$199$	104.892i	0.527094i	−0.964647	+	0.263547i	$0.915108\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 546.079i	− 2.69004i
$204$	0	0
$205$	19.4427	0.0948425
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−66.3344	−0.317389
$210$	0	0
$211$	− 240.584i	− 1.14021i	−0.821573	−	0.570103i	$-0.806903\pi$
$211$	− 240.584i	− 1.14021i	0.821573	−	0.570103i	$-0.193097\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 59.5967i	− 0.277194i
$216$	0	0
$217$	−333.941	−1.53890
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−21.6130	−0.0977964
$222$	0	0
$223$	− 18.2918i	− 0.0820260i	−0.999159	−	0.0410130i	$-0.986941\pi$
$223$	− 18.2918i	− 0.0820260i	0.999159	−	0.0410130i	$-0.0130585\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 92.4559i	− 0.407295i	−0.979044	−	0.203647i	$-0.934720\pi$
$227$	− 92.4559i	− 0.407295i	0.979044	−	0.203647i	$-0.0652796\pi$
$228$	0	0
$229$	−165.161	−0.721227	−0.360613	−	0.932715i	$-0.617433\pi$
$229$	−165.161	−0.721227	−0.360613	+	0.932715i	$0.617433\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−37.3313	−0.160220	−0.0801100	−	0.996786i	$-0.525527\pi$
$233$	−37.3313	−0.160220	−0.0801100	+	0.996786i	$0.525527\pi$
$234$	0	0
$235$	− 99.5967i	− 0.423816i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 397.495i	− 1.66316i	−0.555405	−	0.831580i	$-0.687437\pi$
$239$	− 397.495i	− 1.66316i	0.555405	−	0.831580i	$-0.312563\pi$
$240$	0	0
$241$	237.915	0.987199	0.493599	−	0.869689i	$-0.335681\pi$
$241$	237.915	0.987199	0.493599	+	0.869689i	$0.335681\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	222.177	0.906846
$246$	0	0
$247$	− 245.666i	− 0.994598i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 277.803i	− 1.10679i	−0.832920	−	0.553393i	$-0.813333\pi$
$251$	− 277.803i	− 1.10679i	0.832920	−	0.553393i	$-0.186667\pi$
$252$	0	0
$253$	174.663	0.690366
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−446.498	−1.73735	−0.868674	−	0.495384i	$-0.835027\pi$
$257$	−446.498	−1.73735	−0.868674	+	0.495384i	$0.835027\pi$
$258$	0	0
$259$	− 286.577i	− 1.10648i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 105.039i	− 0.399390i	−0.979858	−	0.199695i	$-0.936005\pi$
$263$	− 105.039i	− 0.399390i	0.979858	−	0.199695i	$-0.0639951\pi$
$264$	0	0
$265$	−1.55418	−0.00586481
$266$	0	0
$267$	0	0
$268$	0	0
$269$	504.354	1.87492	0.937462	−	0.348088i	$-0.113169\pi$
$269$	504.354	1.87492	0.937462	+	0.348088i	$0.113169\pi$
$270$	0	0
$271$	− 164.413i	− 0.606691i	−0.952881	−	0.303346i	$-0.901896\pi$
$271$	− 164.413i	− 0.606691i	0.952881	−	0.303346i	$-0.0981037\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	27.6393i	0.100507i
$276$	0	0
$277$	−315.580	−1.13928	−0.569640	−	0.821894i	$-0.692917\pi$
$277$	−315.580	−1.13928	−0.569640	+	0.821894i	$0.692917\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	207.410	0.738115	0.369057	−	0.929407i	$-0.379681\pi$
$281$	207.410	0.738115	0.369057	+	0.929407i	$0.379681\pi$
$282$	0	0
$283$	− 438.895i	− 1.55087i	−0.631429	−	0.775434i	$-0.717531\pi$
$283$	− 438.895i	− 1.55087i	0.631429	−	0.775434i	$-0.282469\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 105.909i	− 0.369020i
$288$	0	0
$289$	−287.885	−0.996143
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−272.918	−0.931461	−0.465730	−	0.884927i	$-0.654208\pi$
$293$	−272.918	−0.931461	−0.465730	+	0.884927i	$0.654208\pi$
$294$	0	0
$295$	211.554i	0.717133i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	646.853i	2.16339i
$300$	0	0
$301$	−324.636	−1.07853
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−74.2229	−0.243354
$306$	0	0
$307$	251.459i	0.819085i	0.912291	+	0.409542i	$0.134312\pi$
$307$	251.459i	0.819085i	−0.912291	+	0.409542i	$0.865688\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 352.912i	− 1.13476i	−0.823455	−	0.567382i	$-0.807956\pi$
$311$	− 352.912i	− 1.13476i	0.823455	−	0.567382i	$-0.192044\pi$
$312$	0	0
$313$	−435.220	−1.39048	−0.695239	−	0.718778i	$-0.744702\pi$
$313$	−435.220	−1.39048	−0.695239	+	0.718778i	$0.744702\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	41.4164	0.130651	0.0653256	−	0.997864i	$-0.479191\pi$
$317$	41.4164	0.130651	0.0653256	+	0.997864i	$0.479191\pi$
$318$	0	0
$319$	− 247.830i	− 0.776896i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	12.6687i	0.0392221i
$324$	0	0
$325$	−102.361	−0.314956
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−542.525	−1.64901
$330$	0	0
$331$	103.416i	0.312436i	0.987723	+	0.156218i	$0.0499303\pi$
$331$	103.416i	0.312436i	−0.987723	+	0.156218i	$0.950070\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	183.820i	0.548715i
$336$	0	0
$337$	106.997	0.317498	0.158749	−	0.987319i	$-0.449254\pi$
$337$	106.997	0.317498	0.158749	+	0.987319i	$0.449254\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−151.554	−0.444440
$342$	0	0
$343$	− 613.410i	− 1.78837i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	511.426i	1.47385i	0.675974	+	0.736926i	$0.263723\pi$
$347$	511.426i	1.47385i	−0.675974	+	0.736926i	$0.736277\pi$
$348$	0	0
$349$	63.1672	0.180995	0.0904974	−	0.995897i	$-0.471154\pi$
$349$	63.1672	0.180995	0.0904974	+	0.995897i	$0.471154\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−62.7740	−0.177830	−0.0889150	−	0.996039i	$-0.528340\pi$
$353$	−62.7740	−0.177830	−0.0889150	+	0.996039i	$0.528340\pi$
$354$	0	0
$355$	− 273.358i	− 0.770021i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	176.105i	0.490544i	0.969454	+	0.245272i	$0.0788772\pi$
$359$	176.105i	0.490544i	−0.969454	+	0.245272i	$0.921123\pi$
$360$	0	0
$361$	217.000	0.601108
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−295.528	−0.809665
$366$	0	0
$367$	− 190.259i	− 0.518418i	−0.965821	−	0.259209i	$-0.916538\pi$
$367$	− 190.259i	− 0.518418i	0.965821	−	0.259209i	$-0.0834618\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8.46592i	0.0228192i
$372$	0	0
$373$	525.076	1.40771	0.703855	−	0.710344i	$-0.251460\pi$
$373$	525.076	1.40771	0.703855	+	0.710344i	$0.251460\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	917.823	2.43455
$378$	0	0
$379$	530.217i	1.39899i	0.714638	+	0.699494i	$0.246591\pi$
$379$	530.217i	1.39899i	−0.714638	+	0.699494i	$0.753409\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	165.177i	0.431272i	0.976474	+	0.215636i	$0.0691825\pi$
$383$	165.177i	0.431272i	−0.976474	+	0.215636i	$0.930818\pi$
$384$	0	0
$385$	150.557	0.391058
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−532.079	−1.36781	−0.683906	−	0.729570i	$-0.739720\pi$
$389$	−532.079	−1.36781	−0.683906	+	0.729570i	$0.739720\pi$
$390$	0	0
$391$	− 33.3576i	− 0.0853135i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	301.495i	0.763279i
$396$	0	0
$397$	512.407	1.29070	0.645349	−	0.763888i	$-0.276712\pi$
$397$	512.407	1.29070	0.645349	+	0.763888i	$0.276712\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−22.3282	−0.0556812	−0.0278406	−	0.999612i	$-0.508863\pi$
$401$	−22.3282	−0.0556812	−0.0278406	+	0.999612i	$0.508863\pi$
$402$	0	0
$403$	− 561.272i	− 1.39274i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 130.059i	− 0.319555i
$408$	0	0
$409$	−42.4659	−0.103829	−0.0519143	−	0.998652i	$-0.516532\pi$
$409$	−42.4659	−0.103829	−0.0519143	+	0.998652i	$0.516532\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1152.38	2.79027
$414$	0	0
$415$	− 43.5116i	− 0.104847i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 284.381i	− 0.678713i	−0.940658	−	0.339357i	$-0.889791\pi$
$419$	− 284.381i	− 0.678713i	0.940658	−	0.339357i	$-0.110209\pi$
$420$	0	0
$421$	−201.246	−0.478019	−0.239010	−	0.971017i	$-0.576823\pi$
$421$	−201.246	−0.478019	−0.239010	+	0.971017i	$0.576823\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.27864	0.0124203
$426$	0	0
$427$	404.308i	0.946857i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 81.9086i	− 0.190043i	−0.995475	−	0.0950216i	$-0.969708\pi$
$431$	− 81.9086i	− 0.190043i	0.995475	−	0.0950216i	$-0.0302920\pi$
$432$	0	0
$433$	−195.325	−0.451097	−0.225549	−	0.974232i	$-0.572417\pi$
$433$	−195.325	−0.451097	−0.225549	+	0.974232i	$0.572417\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	379.161	0.867645
$438$	0	0
$439$	− 466.780i	− 1.06328i	−0.846970	−	0.531640i	$-0.821576\pi$
$439$	− 466.780i	− 1.06328i	0.846970	−	0.531640i	$-0.178424\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	376.508i	0.849906i	0.905215	+	0.424953i	$0.139709\pi$
$443$	376.508i	0.849906i	−0.905215	+	0.424953i	$0.860291\pi$
$444$	0	0
$445$	−67.0820	−0.150746
$446$	0	0
$447$	0	0
$448$	0	0
$449$	247.029	0.550177	0.275088	−	0.961419i	$-0.411293\pi$
$449$	247.029	0.550177	0.275088	+	0.961419i	$0.411293\pi$
$450$	0	0
$451$	− 48.0650i	− 0.106574i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	557.580i	1.22545i
$456$	0	0
$457$	−229.003	−0.501101	−0.250550	−	0.968104i	$-0.580612\pi$
$457$	−229.003	−0.501101	−0.250550	+	0.968104i	$0.580612\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	62.7740	0.136169	0.0680846	−	0.997680i	$-0.478311\pi$
$461$	62.7740	0.136169	0.0680846	+	0.997680i	$0.478311\pi$
$462$	0	0
$463$	− 204.075i	− 0.440767i	−0.975413	−	0.220383i	$-0.929269\pi$
$463$	− 204.075i	− 0.440767i	0.975413	−	0.220383i	$-0.0707309\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	412.371i	0.883021i	0.897256	+	0.441510i	$0.145557\pi$
$467$	412.371i	0.883021i	−0.897256	+	0.441510i	$0.854443\pi$
$468$	0	0
$469$	1001.30	2.13498
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−147.331	−0.311483
$474$	0	0
$475$	60.0000i	0.126316i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 638.597i	− 1.33319i	−0.745421	−	0.666594i	$-0.767751\pi$
$479$	− 638.597i	− 1.33319i	0.745421	−	0.666594i	$-0.232249\pi$
$480$	0	0
$481$	481.666	1.00138
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−27.5805	−0.0568670
$486$	0	0
$487$	− 24.8692i	− 0.0510661i	−0.999674	−	0.0255330i	$-0.991872\pi$
$487$	− 24.8692i	− 0.0510661i	0.999674	−	0.0255330i	$-0.00812830\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 320.584i	− 0.652920i	−0.945211	−	0.326460i	$-0.894144\pi$
$491$	− 320.584i	− 0.652920i	0.945211	−	0.326460i	$-0.105856\pi$
$492$	0	0
$493$	−47.3313	−0.0960066
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1489.04	−2.99605
$498$	0	0
$499$	168.774i	0.338224i	0.985597	+	0.169112i	$0.0540900\pi$
$499$	168.774i	0.338224i	−0.985597	+	0.169112i	$0.945910\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 662.089i	− 1.31628i	−0.752895	−	0.658140i	$-0.771343\pi$
$503$	− 662.089i	− 1.31628i	0.752895	−	0.658140i	$-0.228657\pi$
$504$	0	0
$505$	−236.026	−0.467379
$506$	0	0
$507$	0	0
$508$	0	0
$509$	651.823	1.28060	0.640298	−	0.768126i	$-0.278811\pi$
$509$	651.823	1.28060	0.640298	+	0.768126i	$0.278811\pi$
$510$	0	0
$511$	1609.80i	3.15030i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 225.813i	− 0.438473i
$516$	0	0
$517$	−246.217	−0.476241
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−750.984	−1.44143	−0.720714	−	0.693232i	$-0.756186\pi$
$521$	−750.984	−1.44143	−0.720714	+	0.693232i	$0.756186\pi$
$522$	0	0
$523$	− 328.128i	− 0.627395i	−0.949523	−	0.313698i	$-0.898432\pi$
$523$	− 328.128i	− 0.627395i	0.949523	−	0.313698i	$-0.101568\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	28.9443i	0.0549227i
$528$	0	0
$529$	−469.354	−0.887249
$530$	0	0
$531$	0	0
$532$	0	0
$533$	178.006	0.333970
$534$	0	0
$535$	− 51.1509i	− 0.0956092i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 549.252i	− 1.01902i
$540$	0	0
$541$	−238.563	−0.440968	−0.220484	−	0.975391i	$-0.570764\pi$
$541$	−238.563	−0.440968	−0.220484	+	0.975391i	$0.570764\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−79.4427	−0.145766
$546$	0	0
$547$	341.453i	0.624228i	0.950045	+	0.312114i	$0.101037\pi$
$547$	341.453i	0.624228i	−0.950045	+	0.312114i	$0.898963\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 537.994i	− 0.976395i
$552$	0	0
$553$	1642.31	2.96982
$554$	0	0
$555$	0	0
$556$	0	0
$557$	349.574	0.627602	0.313801	−	0.949489i	$-0.398398\pi$
$557$	349.574	0.627602	0.313801	+	0.949489i	$0.398398\pi$
$558$	0	0
$559$	− 545.633i	− 0.976088i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	746.778i	1.32643i	0.748431	+	0.663213i	$0.230808\pi$
$563$	746.778i	1.32643i	−0.748431	+	0.663213i	$0.769192\pi$
$564$	0	0
$565$	−158.138	−0.279890
$566$	0	0
$567$	0	0
$568$	0	0
$569$	660.906	1.16152	0.580761	−	0.814074i	$-0.302755\pi$
$569$	660.906	1.16152	0.580761	+	0.814074i	$0.302755\pi$
$570$	0	0
$571$	− 529.240i	− 0.926865i	−0.886132	−	0.463432i	$-0.846618\pi$
$571$	− 529.240i	− 0.926865i	0.886132	−	0.463432i	$-0.153382\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 157.984i	− 0.274754i
$576$	0	0
$577$	−677.830	−1.17475	−0.587374	−	0.809316i	$-0.699838\pi$
$577$	−677.830	−1.17475	−0.587374	+	0.809316i	$0.699838\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−237.017	−0.407947
$582$	0	0
$583$	3.84213i	0.00659028i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 80.6137i	− 0.137332i	−0.997640	−	0.0686659i	$-0.978126\pi$
$587$	− 80.6137i	− 0.137332i	0.997640	−	0.0686659i	$-0.0218742\pi$
$588$	0	0
$589$	−328.997	−0.558569
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−347.548	−0.586084	−0.293042	−	0.956100i	$-0.594668\pi$
$593$	−347.548	−0.586084	−0.293042	+	0.956100i	$0.594668\pi$
$594$	0	0
$595$	− 28.7539i	− 0.0483259i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 78.2817i	− 0.130687i	−0.997863	−	0.0653437i	$-0.979186\pi$
$599$	− 78.2817i	− 0.130687i	0.997863	−	0.0653437i	$-0.0208144\pi$
$600$	0	0
$601$	−39.3050	−0.0653993	−0.0326996	−	0.999465i	$-0.510410\pi$
$601$	−39.3050	−0.0653993	−0.0326996	+	0.999465i	$0.510410\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−202.236	−0.334274
$606$	0	0
$607$	432.443i	0.712427i	0.934405	+	0.356214i	$0.115932\pi$
$607$	432.443i	0.712427i	−0.934405	+	0.356214i	$0.884068\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 911.850i	− 1.49239i
$612$	0	0
$613$	−498.413	−0.813072	−0.406536	−	0.913635i	$-0.633263\pi$
$613$	−498.413	−0.813072	−0.406536	+	0.913635i	$0.633263\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	611.325	0.990802	0.495401	−	0.868664i	$-0.335021\pi$
$617$	611.325	0.990802	0.495401	+	0.868664i	$0.335021\pi$
$618$	0	0
$619$	490.663i	0.792670i	0.918106	+	0.396335i	$0.129718\pi$
$619$	490.663i	0.792670i	−0.918106	+	0.396335i	$0.870282\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	365.410i	0.586533i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−24.8390	−0.0394897
$630$	0	0
$631$	104.636i	0.165826i	0.996557	+	0.0829130i	$0.0264224\pi$
$631$	104.636i	0.165826i	−0.996557	+	0.0829130i	$0.973578\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	332.148i	0.523067i
$636$	0	0
$637$	2034.13	3.19329
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−150.020	−0.234041	−0.117020	−	0.993130i	$-0.537334\pi$
$641$	−150.020	−0.234041	−0.117020	+	0.993130i	$0.537334\pi$
$642$	0	0
$643$	646.810i	1.00593i	0.864308	+	0.502963i	$0.167757\pi$
$643$	646.810i	1.00593i	−0.864308	+	0.502963i	$0.832243\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 694.450i	− 1.07334i	−0.843793	−	0.536669i	$-0.819682\pi$
$647$	− 694.450i	− 1.07334i	0.843793	−	0.536669i	$-0.180318\pi$
$648$	0	0
$649$	522.991	0.805841
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−698.361	−1.06947	−0.534733	−	0.845021i	$-0.679588\pi$
$653$	−698.361	−1.06947	−0.534733	+	0.845021i	$0.679588\pi$
$654$	0	0
$655$	− 78.0789i	− 0.119204i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 834.282i	− 1.26598i	−0.774159	−	0.632991i	$-0.781827\pi$
$659$	− 834.282i	− 1.26598i	0.774159	−	0.632991i	$-0.218173\pi$
$660$	0	0
$661$	−978.584	−1.48046	−0.740230	−	0.672354i	$-0.765283\pi$
$661$	−978.584	−1.48046	−0.740230	+	0.672354i	$0.765283\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	326.833	0.491478
$666$	0	0
$667$	1416.57i	2.12379i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	183.489i	0.273456i
$672$	0	0
$673$	−682.669	−1.01437	−0.507183	−	0.861838i	$-0.669313\pi$
$673$	−682.669	−1.01437	−0.507183	+	0.861838i	$0.669313\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	948.067	1.40039	0.700197	−	0.713950i	$-0.253096\pi$
$677$	948.067	1.40039	0.700197	+	0.713950i	$0.253096\pi$
$678$	0	0
$679$	150.237i	0.221262i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	341.688i	0.500275i	0.968210	+	0.250138i	$0.0804759\pi$
$683$	341.688i	0.500275i	−0.968210	+	0.250138i	$0.919524\pi$
$684$	0	0
$685$	224.971	0.328424
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−14.2291	−0.0206518
$690$	0	0
$691$	− 215.416i	− 0.311746i	−0.987777	−	0.155873i	$-0.950181\pi$
$691$	− 215.416i	− 0.311746i	0.987777	−	0.155873i	$-0.0498190\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	444.105i	0.639000i
$696$	0	0
$697$	−9.17961	−0.0131702
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1137.40	−1.62254	−0.811272	−	0.584668i	$-0.801225\pi$
$701$	−1137.40	−1.62254	−0.811272	+	0.584668i	$0.801225\pi$
$702$	0	0
$703$	− 282.334i	− 0.401614i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1285.69i	1.81851i
$708$	0	0
$709$	853.149	1.20331	0.601656	−	0.798755i	$-0.294508\pi$
$709$	853.149	1.20331	0.601656	+	0.798755i	$0.294508\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	866.269	1.21496
$714$	0	0
$715$	253.050i	0.353915i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 696.616i	− 0.968868i	−0.874828	−	0.484434i	$-0.839026\pi$
$719$	− 696.616i	− 0.968868i	0.874828	−	0.484434i	$-0.160974\pi$
$720$	0	0
$721$	−1230.05	−1.70604
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−224.164	−0.309192
$726$	0	0
$727$	− 588.265i	− 0.809168i	−0.914501	−	0.404584i	$-0.867416\pi$
$727$	− 588.265i	− 0.809168i	0.914501	−	0.404584i	$-0.132584\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	28.1378i	0.0384922i
$732$	0	0
$733$	−1106.86	−1.51004	−0.755020	−	0.655702i	$-0.772373\pi$
$733$	−1106.86	−1.51004	−0.755020	+	0.655702i	$0.772373\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	454.427	0.616590
$738$	0	0
$739$	− 720.053i	− 0.974361i	−0.873301	−	0.487180i	$-0.838025\pi$
$739$	− 720.053i	− 0.974361i	0.873301	−	0.487180i	$-0.161975\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	317.793i	0.427716i	0.976865	+	0.213858i	$0.0686031\pi$
$743$	317.793i	0.427716i	−0.976865	+	0.213858i	$0.931397\pi$
$744$	0	0
$745$	454.105	0.609537
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−278.630	−0.372003
$750$	0	0
$751$	606.512i	0.807606i	0.914846	+	0.403803i	$0.132312\pi$
$751$	606.512i	0.807606i	−0.914846	+	0.403803i	$0.867688\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	254.472i	0.337049i
$756$	0	0
$757$	−359.463	−0.474852	−0.237426	−	0.971406i	$-0.576304\pi$
$757$	−359.463	−0.474852	−0.237426	+	0.971406i	$0.576304\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−239.876	−0.315212	−0.157606	−	0.987502i	$-0.550378\pi$
$761$	−239.876	−0.315212	−0.157606	+	0.987502i	$0.550378\pi$
$762$	0	0
$763$	432.741i	0.567158i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1936.87i	2.52525i
$768$	0	0
$769$	−768.885	−0.999851	−0.499926	−	0.866068i	$-0.666639\pi$
$769$	−768.885	−0.999851	−0.499926	+	0.866068i	$0.666639\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−33.5743	−0.0434337	−0.0217169	−	0.999764i	$-0.506913\pi$
$773$	−33.5743	−0.0434337	−0.0217169	+	0.999764i	$0.506913\pi$
$774$	0	0
$775$	137.082i	0.176880i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 104.341i	− 0.133942i
$780$	0	0
$781$	−675.777	−0.865272
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−324.046	−0.412798
$786$	0	0
$787$	426.738i	0.542233i	0.962546	+	0.271117i	$0.0873929\pi$
$787$	426.738i	0.542233i	−0.962546	+	0.271117i	$0.912607\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	861.410i	1.08901i
$792$	0	0
$793$	−679.542	−0.856925
$794$	0	0
$795$	0	0
$796$	0	0
$797$	194.203	0.243667	0.121834	−	0.992551i	$-0.461123\pi$
$797$	194.203	0.243667	0.121834	+	0.992551i	$0.461123\pi$
$798$	0	0
$799$	47.0232i	0.0588526i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	730.585i	0.909819i
$804$	0	0
$805$	−860.571	−1.06903
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1446.33	−1.78780	−0.893899	−	0.448269i	$-0.852041\pi$
$809$	−1446.33	−1.78780	−0.893899	+	0.448269i	$0.852041\pi$
$810$	0	0
$811$	− 23.3112i	− 0.0287437i	−0.999897	−	0.0143719i	$-0.995425\pi$
$811$	− 23.3112i	− 0.0287437i	0.999897	−	0.0143719i	$-0.00457486\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 454.758i	− 0.557985i
$816$	0	0
$817$	−319.830	−0.391468
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−701.416	−0.854344	−0.427172	−	0.904170i	$-0.640490\pi$
$821$	−701.416	−0.854344	−0.427172	+	0.904170i	$0.640490\pi$
$822$	0	0
$823$	− 270.961i	− 0.329235i	−0.986357	−	0.164618i	$-0.947361\pi$
$823$	− 270.961i	− 0.329235i	0.986357	−	0.164618i	$-0.0526390\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1035.10i	1.25163i	0.779971	+	0.625815i	$0.215234\pi$
$827$	1035.10i	1.25163i	−0.779971	+	0.625815i	$0.784766\pi$
$828$	0	0
$829$	332.735	0.401369	0.200685	−	0.979656i	$-0.435683\pi$
$829$	332.735	0.401369	0.200685	+	0.979656i	$0.435683\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−104.898	−0.125928
$834$	0	0
$835$	− 198.482i	− 0.237703i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 102.177i	− 0.121784i	−0.998144	−	0.0608918i	$-0.980606\pi$
$839$	− 102.177i	− 0.121784i	0.998144	−	0.0608918i	$-0.0193945\pi$
$840$	0	0
$841$	1168.98	1.38999
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−559.259	−0.661845
$846$	0	0
$847$	1101.62i	1.30062i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	743.404i	0.873565i
$852$	0	0
$853$	−358.401	−0.420165	−0.210083	−	0.977684i	$-0.567373\pi$
$853$	−358.401	−0.420165	−0.210083	+	0.977684i	$0.567373\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−33.3839	−0.0389544	−0.0194772	−	0.999810i	$-0.506200\pi$
$857$	−33.3839	−0.0389544	−0.0194772	+	0.999810i	$0.506200\pi$
$858$	0	0
$859$	− 989.483i	− 1.15190i	−0.817485	−	0.575950i	$-0.804632\pi$
$859$	− 989.483i	− 1.15190i	0.817485	−	0.575950i	$-0.195368\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	767.341i	0.889156i	0.895740	+	0.444578i	$0.146646\pi$
$863$	767.341i	0.889156i	−0.895740	+	0.444578i	$0.853354\pi$
$864$	0	0
$865$	303.167	0.350482
$866$	0	0
$867$	0	0
$868$	0	0
$869$	745.337	0.857696
$870$	0	0
$871$	1682.95i	1.93220i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 136.180i	− 0.155635i
$876$	0	0
$877$	1013.80	1.15598	0.577992	−	0.816043i	$-0.303837\pi$
$877$	1013.80	1.15598	0.577992	+	0.816043i	$0.303837\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	280.748	0.318669	0.159335	−	0.987225i	$-0.449065\pi$
$881$	280.748	0.318669	0.159335	+	0.987225i	$0.449065\pi$
$882$	0	0
$883$	1096.87i	1.24221i	0.783728	+	0.621104i	$0.213315\pi$
$883$	1096.87i	1.24221i	−0.783728	+	0.621104i	$0.786685\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1164.82i	1.31321i	0.754235	+	0.656605i	$0.228008\pi$
$887$	1164.82i	1.31321i	−0.754235	+	0.656605i	$0.771992\pi$
$888$	0	0
$889$	1809.28	2.03519
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−534.492	−0.598536
$894$	0	0
$895$	455.777i	0.509248i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 1229.15i	− 1.36725i
$900$	0	0
$901$	0.733782	0.000814408 0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	50.6888	0.0560098
$906$	0	0
$907$	− 1132.82i	− 1.24897i	−0.781036	−	0.624485i	$-0.785309\pi$
$907$	− 1132.82i	− 1.24897i	0.781036	−	0.624485i	$-0.214691\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1793.23i	1.96842i	0.177014	+	0.984208i	$0.443356\pi$
$911$	1793.23i	1.96842i	−0.177014	+	0.984208i	$0.556644\pi$
$912$	0	0
$913$	−107.567	−0.117817
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−425.313	−0.463809
$918$	0	0
$919$	− 205.390i	− 0.223493i	−0.993737	−	0.111747i	$-0.964356\pi$
$919$	− 205.390i	− 0.223493i	0.993737	−	0.111747i	$-0.0356445\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 2502.70i	− 2.71149i
$924$	0	0
$925$	−117.639	−0.127178
$926$	0	0
$927$	0	0
$928$	0	0
$929$	736.237	0.792505	0.396252	−	0.918142i	$-0.370311\pi$
$929$	736.237	0.792505	0.396252	+	0.918142i	$0.370311\pi$
$930$	0	0
$931$	− 1192.33i	− 1.28070i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 13.0495i	− 0.0139567i
$936$	0	0
$937$	−40.3344	−0.0430463	−0.0215231	−	0.999768i	$-0.506852\pi$
$937$	−40.3344	−0.0430463	−0.0215231	+	0.999768i	$0.506852\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	447.115	0.475148	0.237574	−	0.971369i	$-0.423648\pi$
$941$	447.115	0.475148	0.237574	+	0.971369i	$0.423648\pi$
$942$	0	0
$943$	274.735i	0.291342i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	144.553i	0.152644i	0.997083	+	0.0763218i	$0.0243176\pi$
$947$	144.553i	0.152644i	−0.997083	+	0.0763218i	$0.975682\pi$
$948$	0	0
$949$	−2705.68	−2.85109
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−432.006	−0.453312	−0.226656	−	0.973975i	$-0.572779\pi$
$953$	−432.006	−0.453312	−0.226656	+	0.973975i	$0.572779\pi$
$954$	0	0
$955$	− 337.082i	− 0.352965i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1225.46i	− 1.27785i
$960$	0	0
$961$	209.341	0.217836
$962$	0	0
$963$	0	0
$964$	0	0
$965$	470.689	0.487760
$966$	0	0
$967$	1472.95i	1.52322i	0.648035	+	0.761610i	$0.275591\pi$
$967$	1472.95i	1.52322i	−0.648035	+	0.761610i	$0.724409\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 308.190i	− 0.317395i	−0.987327	−	0.158697i	$-0.949271\pi$
$971$	− 308.190i	− 0.317395i	0.987327	−	0.158697i	$-0.0507294\pi$
$972$	0	0
$973$	2419.14	2.48627
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1376.78	1.40919	0.704596	−	0.709609i	$-0.251128\pi$
$977$	1376.78	1.40919	0.704596	+	0.709609i	$0.251128\pi$
$978$	0	0
$979$	165.836i	0.169393i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1250.38i	− 1.27200i	−0.771689	−	0.636000i	$-0.780588\pi$
$983$	− 1250.38i	− 1.27200i	0.771689	−	0.636000i	$-0.219412\pi$
$984$	0	0
$985$	216.950	0.220254
$986$	0	0
$987$	0	0
$988$	0	0
$989$	842.132	0.851498
$990$	0	0
$991$	− 1727.44i	− 1.74313i	−0.490278	−	0.871566i	$-0.663105\pi$
$991$	− 1727.44i	− 1.74313i	0.490278	−	0.871566i	$-0.336895\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 234.545i	− 0.235723i
$996$	0	0
$997$	−1131.90	−1.13530	−0.567651	−	0.823269i	$-0.692148\pi$
$997$	−1131.90	−1.13530	−0.567651	+	0.823269i	$0.692148\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.3.e.a.2431.2		4
3.2	odd	2		320.3.b.b.191.3		4
4.3	odd	2	inner	2880.3.e.a.2431.1		4
8.3	odd	2		1440.3.e.b.991.3		4
8.5	even	2		1440.3.e.b.991.4		4
12.11	even	2		320.3.b.b.191.2		4
15.2	even	4		1600.3.h.m.1599.1		4
15.8	even	4		1600.3.h.d.1599.3		4
15.14	odd	2		1600.3.b.n.1151.2		4
24.5	odd	2		160.3.b.a.31.2	✓	4
24.11	even	2		160.3.b.a.31.3	yes	4
48.5	odd	4		1280.3.g.d.1151.2		4
48.11	even	4		1280.3.g.a.1151.4		4
48.29	odd	4		1280.3.g.a.1151.3		4
48.35	even	4		1280.3.g.d.1151.1		4
60.23	odd	4		1600.3.h.m.1599.2		4
60.47	odd	4		1600.3.h.d.1599.4		4
60.59	even	2		1600.3.b.n.1151.3		4
120.29	odd	2		800.3.b.d.351.3		4
120.53	even	4		800.3.h.j.799.2		4
120.59	even	2		800.3.b.d.351.2		4
120.77	even	4		800.3.h.c.799.4		4
120.83	odd	4		800.3.h.c.799.3		4
120.107	odd	4		800.3.h.j.799.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.3.b.a.31.2	✓	4	24.5	odd	2
160.3.b.a.31.3	yes	4	24.11	even	2
320.3.b.b.191.2		4	12.11	even	2
320.3.b.b.191.3		4	3.2	odd	2
800.3.b.d.351.2		4	120.59	even	2
800.3.b.d.351.3		4	120.29	odd	2
800.3.h.c.799.3		4	120.83	odd	4
800.3.h.c.799.4		4	120.77	even	4
800.3.h.j.799.1		4	120.107	odd	4
800.3.h.j.799.2		4	120.53	even	4
1280.3.g.a.1151.3		4	48.29	odd	4
1280.3.g.a.1151.4		4	48.11	even	4
1280.3.g.d.1151.1		4	48.35	even	4
1280.3.g.d.1151.2		4	48.5	odd	4
1440.3.e.b.991.3		4	8.3	odd	2
1440.3.e.b.991.4		4	8.5	even	2
1600.3.b.n.1151.2		4	15.14	odd	2
1600.3.b.n.1151.3		4	60.59	even	2
1600.3.h.d.1599.3		4	15.8	even	4
1600.3.h.d.1599.4		4	60.47	odd	4
1600.3.h.m.1599.1		4	15.2	even	4
1600.3.h.m.1599.2		4	60.23	odd	4
2880.3.e.a.2431.1		4	4.3	odd	2	inner
2880.3.e.a.2431.2		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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2880.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.23607 q^{5} +12.1803i q^{7} +5.52786i q^{11} -20.4721 q^{13} +1.05573 q^{17} +12.0000i q^{19} -31.5967i q^{23} +5.00000 q^{25} -44.8328 q^{29} +27.4164i q^{31} -27.2361i q^{35} -23.5279 q^{37} -8.69505 q^{41} +26.6525i q^{43} +44.5410i q^{47} -99.3607 q^{49} +0.695048 q^{53} -12.3607i q^{55} -94.6099i q^{59} +33.1935 q^{61} +45.7771 q^{65} -82.2067i q^{67} +122.249i q^{71} +132.164 q^{73} -67.3313 q^{77} -134.833i q^{79} +19.4590i q^{83} -2.36068 q^{85} +30.0000 q^{89} -249.358i q^{91} -26.8328i q^{95} +12.3344 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 64 q^{13} + 40 q^{17} + 20 q^{25} - 72 q^{29} - 112 q^{37} - 160 q^{41} - 308 q^{49} + 128 q^{53} - 64 q^{61} + 40 q^{65} - 8 q^{73} + 160 q^{77} + 80 q^{85} + 120 q^{89} + 264 q^{97}+O(q^{100}) \) 4 * q - 64 * q^13 + 40 * q^17 + 20 * q^25 - 72 * q^29 - 112 * q^37 - 160 * q^41 - 308 * q^49 + 128 * q^53 - 64 * q^61 + 40 * q^65 - 8 * q^73 + 160 * q^77 + 80 * q^85 + 120 * q^89 + 264 * q^97

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