LMFDB - Embedded newform 4680.2.a.bl.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4680,2,Mod(1,4680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4680, 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("4680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4680.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:	$37.3699881460$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1849.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 14x - 8 $ x^3 - x^2 - 14*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			1.3
Root			$-2.88824$ of defining polynomial
Character	$\chi$	$=$	4680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +4.88824 q^{7} +O(q^{10})$	$q+1.00000 q^{5} +4.88824 q^{7} -6.11838 q^{11} +1.00000 q^{13} +2.88824 q^{17} -7.00662 q^{19} -4.88824 q^{23} +1.00000 q^{25} +3.23014 q^{29} +9.77647 q^{31} +4.88824 q^{35} -9.89485 q^{37} +9.89485 q^{41} +4.00000 q^{43} +5.77647 q^{47} +16.8949 q^{49} +5.65809 q^{53} -6.11838 q^{55} +12.1184 q^{61} +1.00000 q^{65} +6.46029 q^{67} +11.8949 q^{71} +11.2301 q^{73} -29.9081 q^{77} -1.88162 q^{79} +2.46029 q^{83} +2.88824 q^{85} -11.4346 q^{89} +4.88824 q^{91} -7.00662 q^{95} +5.11176 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} + 5 q^{7}+O(q^{10}) $ 3 * q + 3 * q^5 + 5 * q^7	$ 3 q + 3 q^{5} + 5 q^{7} - 3 q^{11} + 3 q^{13} - q^{17} + 4 q^{19} - 5 q^{23} + 3 q^{25} + 4 q^{29} + 10 q^{31} + 5 q^{35} + 5 q^{37} - 5 q^{41} + 12 q^{43} - 2 q^{47} + 16 q^{49} + 13 q^{53} - 3 q^{55} + 21 q^{61} + 3 q^{65} + 8 q^{67} + q^{71} + 28 q^{73} - 5 q^{77} - 21 q^{79} - 4 q^{83} - q^{85} - 11 q^{89} + 5 q^{91} + 4 q^{95} + 25 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 + 5 * q^7 - 3 * q^11 + 3 * q^13 - q^17 + 4 * q^19 - 5 * q^23 + 3 * q^25 + 4 * q^29 + 10 * q^31 + 5 * q^35 + 5 * q^37 - 5 * q^41 + 12 * q^43 - 2 * q^47 + 16 * q^49 + 13 * q^53 - 3 * q^55 + 21 * q^61 + 3 * q^65 + 8 * q^67 + q^71 + 28 * q^73 - 5 * q^77 - 21 * q^79 - 4 * q^83 - q^85 - 11 * q^89 + 5 * q^91 + 4 * q^95 + 25 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.88824	1.84758	0.923790	−	0.382900i	$-0.125075\pi$
$7$	4.88824	1.84758	0.923790	+	0.382900i	$0.125075\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−6.11838	−1.84476	−0.922380	−	0.386283i	$-0.873759\pi$
$11$	−6.11838	−1.84476	−0.922380	+	0.386283i	$0.873759\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.88824	0.700500	0.350250	−	0.936656i	$-0.386097\pi$
$17$	2.88824	0.700500	0.350250	+	0.936656i	$0.386097\pi$
$18$	0	0
$19$	−7.00662	−1.60743	−0.803714	−	0.595016i	$-0.797146\pi$
$19$	−7.00662	−1.60743	−0.803714	+	0.595016i	$0.797146\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.88824	−1.01927	−0.509634	−	0.860391i	$-0.670219\pi$
$23$	−4.88824	−1.01927	−0.509634	+	0.860391i	$0.670219\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.23014	0.599823	0.299911	−	0.953967i	$-0.403043\pi$
$29$	3.23014	0.599823	0.299911	+	0.953967i	$0.403043\pi$
$30$	0	0
$31$	9.77647	1.75591	0.877953	−	0.478747i	$-0.158909\pi$
$31$	9.77647	1.75591	0.877953	+	0.478747i	$0.158909\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.88824	0.826263
$36$	0	0
$37$	−9.89485	−1.62670	−0.813352	−	0.581772i	$-0.802360\pi$
$37$	−9.89485	−1.62670	−0.813352	+	0.581772i	$0.802360\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.89485	1.54532	0.772658	−	0.634822i	$-0.218927\pi$
$41$	9.89485	1.54532	0.772658	+	0.634822i	$0.218927\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.77647	0.842585	0.421293	−	0.906925i	$-0.361577\pi$
$47$	5.77647	0.842585	0.421293	+	0.906925i	$0.361577\pi$
$48$	0	0
$49$	16.8949	2.41355
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.65809	0.777199	0.388599	−	0.921407i	$-0.372959\pi$
$53$	5.65809	0.777199	0.388599	+	0.921407i	$0.372959\pi$
$54$	0	0
$55$	−6.11838	−0.825002
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	12.1184	1.55160	0.775800	−	0.630979i	$-0.217347\pi$
$61$	12.1184	1.55160	0.775800	+	0.630979i	$0.217347\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	6.46029	0.789250	0.394625	−	0.918842i	$-0.370875\pi$
$67$	6.46029	0.789250	0.394625	+	0.918842i	$0.370875\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	11.8949	1.41166	0.705830	−	0.708382i	$-0.250574\pi$
$71$	11.8949	1.41166	0.705830	+	0.708382i	$0.250574\pi$
$72$	0	0
$73$	11.2301	1.31439	0.657194	−	0.753721i	$-0.271743\pi$
$73$	11.2301	1.31439	0.657194	+	0.753721i	$0.271743\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−29.9081	−3.40834
$78$	0	0
$79$	−1.88162	−0.211699	−0.105849	−	0.994382i	$-0.533756\pi$
$79$	−1.88162	−0.211699	−0.105849	+	0.994382i	$0.533756\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.46029	0.270052	0.135026	−	0.990842i	$-0.456888\pi$
$83$	2.46029	0.270052	0.135026	+	0.990842i	$0.456888\pi$
$84$	0	0
$85$	2.88824	0.313273
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−11.4346	−1.21206	−0.606031	−	0.795441i	$-0.707239\pi$
$89$	−11.4346	−1.21206	−0.606031	+	0.795441i	$0.707239\pi$
$90$	0	0
$91$	4.88824	0.512426
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−7.00662	−0.718864
$96$	0	0
$97$	5.11176	0.519021	0.259510	−	0.965740i	$-0.416439\pi$
$97$	5.11176	0.519021	0.259510	+	0.965740i	$0.416439\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	5.00662	0.498177	0.249088	−	0.968481i	$-0.419869\pi$
$101$	5.00662	0.498177	0.249088	+	0.968481i	$0.419869\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.11838	−0.204791	−0.102396	−	0.994744i	$-0.532651\pi$
$107$	−2.11838	−0.204791	−0.102396	+	0.994744i	$0.532651\pi$
$108$	0	0
$109$	−2.54633	−0.243894	−0.121947	−	0.992537i	$-0.538914\pi$
$109$	−2.54633	−0.243894	−0.121947	+	0.992537i	$0.538914\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−0.769857	−0.0724220	−0.0362110	−	0.999344i	$-0.511529\pi$
$113$	−0.769857	−0.0724220	−0.0362110	+	0.999344i	$0.511529\pi$
$114$	0	0
$115$	−4.88824	−0.455830
$116$	0	0
$117$	0	0
$118$	0	0
$119$	14.1184	1.29423
$120$	0	0
$121$	26.4346	2.40314
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−12.2368	−1.08584	−0.542918	−	0.839785i	$-0.682681\pi$
$127$	−12.2368	−1.08584	−0.542918	+	0.839785i	$0.682681\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0.993385	0.0867924	0.0433962	−	0.999058i	$-0.486182\pi$
$131$	0.993385	0.0867924	0.0433962	+	0.999058i	$0.486182\pi$
$132$	0	0
$133$	−34.2500	−2.96985
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.46029	0.381068	0.190534	−	0.981681i	$-0.438978\pi$
$137$	4.46029	0.381068	0.190534	+	0.981681i	$0.438978\pi$
$138$	0	0
$139$	−4.10515	−0.348194	−0.174097	−	0.984728i	$-0.555701\pi$
$139$	−4.10515	−0.348194	−0.174097	+	0.984728i	$0.555701\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−6.11838	−0.511645
$144$	0	0
$145$	3.23014	0.268249
$146$	0	0
$147$	0	0
$148$	0	0
$149$	9.65809	0.791222	0.395611	−	0.918418i	$-0.370533\pi$
$149$	9.65809	0.791222	0.395611	+	0.918418i	$0.370533\pi$
$150$	0	0
$151$	14.0132	1.14038	0.570190	−	0.821513i	$-0.306869\pi$
$151$	14.0132	1.14038	0.570190	+	0.821513i	$0.306869\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	9.77647	0.785265
$156$	0	0
$157$	−24.0132	−1.91646	−0.958232	−	0.285991i	$-0.907677\pi$
$157$	−24.0132	−1.91646	−0.958232	+	0.285991i	$0.907677\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−23.8949	−1.88318
$162$	0	0
$163$	13.6713	1.07082	0.535410	−	0.844592i	$-0.320157\pi$
$163$	13.6713	1.07082	0.535410	+	0.844592i	$0.320157\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.00000	0.309529	0.154765	−	0.987951i	$-0.450538\pi$
$167$	4.00000	0.309529	0.154765	+	0.987951i	$0.450538\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−12.0132	−0.913349	−0.456675	−	0.889634i	$-0.650960\pi$
$173$	−12.0132	−0.913349	−0.456675	+	0.889634i	$0.650960\pi$
$174$	0	0
$175$	4.88824	0.369516
$176$	0	0
$177$	0	0
$178$	0	0
$179$	15.0066	1.12165	0.560824	−	0.827935i	$-0.310485\pi$
$179$	15.0066	1.12165	0.560824	+	0.827935i	$0.310485\pi$
$180$	0	0
$181$	−1.89485	−0.140843	−0.0704216	−	0.997517i	$-0.522434\pi$
$181$	−1.89485	−0.140843	−0.0704216	+	0.997517i	$0.522434\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−9.89485	−0.727484
$186$	0	0
$187$	−17.6713	−1.29226
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3.55294	−0.257082	−0.128541	−	0.991704i	$-0.541029\pi$
$191$	−3.55294	−0.257082	−0.128541	+	0.991704i	$0.541029\pi$
$192$	0	0
$193$	6.88824	0.495826	0.247913	−	0.968782i	$-0.420255\pi$
$193$	6.88824	0.495826	0.247913	+	0.968782i	$0.420255\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3.77647	−0.269062	−0.134531	−	0.990909i	$-0.542953\pi$
$197$	−3.77647	−0.269062	−0.134531	+	0.990909i	$0.542953\pi$
$198$	0	0
$199$	−27.5529	−1.95318	−0.976588	−	0.215117i	$-0.930987\pi$
$199$	−27.5529	−1.95318	−0.976588	+	0.215117i	$0.930987\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	15.7897	1.10822
$204$	0	0
$205$	9.89485	0.691086
$206$	0	0
$207$	0	0
$208$	0	0
$209$	42.8691	2.96532
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	47.7897	3.24418
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.88824	0.194284
$222$	0	0
$223$	−10.3228	−0.691266	−0.345633	−	0.938370i	$-0.612336\pi$
$223$	−10.3228	−0.691266	−0.345633	+	0.938370i	$0.612336\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−21.3294	−1.41568	−0.707842	−	0.706371i	$-0.750331\pi$
$227$	−21.3294	−1.41568	−0.707842	+	0.706371i	$0.750331\pi$
$228$	0	0
$229$	−3.23014	−0.213454	−0.106727	−	0.994288i	$-0.534037\pi$
$229$	−3.23014	−0.213454	−0.106727	+	0.994288i	$0.534037\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	18.6779	1.22363	0.611816	−	0.791000i	$-0.290439\pi$
$233$	18.6779	1.22363	0.611816	+	0.791000i	$0.290439\pi$
$234$	0	0
$235$	5.77647	0.376815
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−24.1316	−1.56094	−0.780472	−	0.625191i	$-0.785021\pi$
$239$	−24.1316	−1.56094	−0.780472	+	0.625191i	$0.785021\pi$
$240$	0	0
$241$	−9.55294	−0.615359	−0.307680	−	0.951490i	$-0.599552\pi$
$241$	−9.55294	−0.615359	−0.307680	+	0.951490i	$0.599552\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	16.8949	1.07937
$246$	0	0
$247$	−7.00662	−0.445820
$248$	0	0
$249$	0	0
$250$	0	0
$251$	27.2434	1.71959	0.859793	−	0.510642i	$-0.170592\pi$
$251$	27.2434	1.71959	0.859793	+	0.510642i	$0.170592\pi$
$252$	0	0
$253$	29.9081	1.88030
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.69043	−0.354959	−0.177480	−	0.984124i	$-0.556794\pi$
$257$	−5.69043	−0.354959	−0.177480	+	0.984124i	$0.556794\pi$
$258$	0	0
$259$	−48.3684	−3.00546
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−12.7831	−0.788239	−0.394119	−	0.919059i	$-0.628950\pi$
$263$	−12.7831	−0.788239	−0.394119	+	0.919059i	$0.628950\pi$
$264$	0	0
$265$	5.65809	0.347574
$266$	0	0
$267$	0	0
$268$	0	0
$269$	13.0066	0.793027	0.396514	−	0.918029i	$-0.370220\pi$
$269$	13.0066	0.793027	0.396514	+	0.918029i	$0.370220\pi$
$270$	0	0
$271$	−26.2500	−1.59457	−0.797287	−	0.603601i	$-0.793732\pi$
$271$	−26.2500	−1.59457	−0.797287	+	0.603601i	$0.793732\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6.11838	−0.368952
$276$	0	0
$277$	8.46029	0.508329	0.254165	−	0.967161i	$-0.418199\pi$
$277$	8.46029	0.508329	0.254165	+	0.967161i	$0.418199\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5.55294	0.331261	0.165630	−	0.986188i	$-0.447034\pi$
$281$	5.55294	0.331261	0.165630	+	0.986188i	$0.447034\pi$
$282$	0	0
$283$	−7.55294	−0.448976	−0.224488	−	0.974477i	$-0.572071\pi$
$283$	−7.55294	−0.448976	−0.224488	+	0.974477i	$0.572071\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	48.3684	2.85509
$288$	0	0
$289$	−8.65809	−0.509300
$290$	0	0
$291$	0	0
$292$	0	0
$293$	23.7765	1.38904	0.694518	−	0.719475i	$-0.255618\pi$
$293$	23.7765	1.38904	0.694518	+	0.719475i	$0.255618\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.88824	−0.282694
$300$	0	0
$301$	19.5529	1.12701
$302$	0	0
$303$	0	0
$304$	0	0
$305$	12.1184	0.693896
$306$	0	0
$307$	19.2110	1.09643	0.548216	−	0.836337i	$-0.315307\pi$
$307$	19.2110	1.09643	0.548216	+	0.836337i	$0.315307\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−8.46029	−0.478204	−0.239102	−	0.970994i	$-0.576853\pi$
$313$	−8.46029	−0.478204	−0.239102	+	0.970994i	$0.576853\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.2368	0.574954	0.287477	−	0.957788i	$-0.407184\pi$
$317$	10.2368	0.574954	0.287477	+	0.957788i	$0.407184\pi$
$318$	0	0
$319$	−19.7632	−1.10653
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−20.2368	−1.12600
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	28.2368	1.55674
$330$	0	0
$331$	−14.3228	−0.787252	−0.393626	−	0.919271i	$-0.628780\pi$
$331$	−14.3228	−0.787252	−0.393626	+	0.919271i	$0.628780\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6.46029	0.352963
$336$	0	0
$337$	31.3294	1.70662	0.853311	−	0.521402i	$-0.174591\pi$
$337$	31.3294	1.70662	0.853311	+	0.521402i	$0.174591\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−59.8162	−3.23923
$342$	0	0
$343$	48.3684	2.61165
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−16.1316	−0.865990	−0.432995	−	0.901396i	$-0.642543\pi$
$347$	−16.1316	−0.865990	−0.432995	+	0.901396i	$0.642543\pi$
$348$	0	0
$349$	−6.99338	−0.374347	−0.187174	−	0.982327i	$-0.559933\pi$
$349$	−6.99338	−0.374347	−0.187174	+	0.982327i	$0.559933\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−7.09266	−0.377504	−0.188752	−	0.982025i	$-0.560444\pi$
$353$	−7.09266	−0.377504	−0.188752	+	0.982025i	$0.560444\pi$
$354$	0	0
$355$	11.8949	0.631313
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−18.0132	−0.950702	−0.475351	−	0.879796i	$-0.657679\pi$
$359$	−18.0132	−0.950702	−0.475351	+	0.879796i	$0.657679\pi$
$360$	0	0
$361$	30.0927	1.58382
$362$	0	0
$363$	0	0
$364$	0	0
$365$	11.2301	0.587813
$366$	0	0
$367$	16.4735	0.859911	0.429955	−	0.902850i	$-0.358529\pi$
$367$	16.4735	0.859911	0.429955	+	0.902850i	$0.358529\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	27.6581	1.43594
$372$	0	0
$373$	−3.77647	−0.195538	−0.0977692	−	0.995209i	$-0.531171\pi$
$373$	−3.77647	−0.195538	−0.0977692	+	0.995209i	$0.531171\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.23014	0.166361
$378$	0	0
$379$	2.08604	0.107153	0.0535764	−	0.998564i	$-0.482938\pi$
$379$	2.08604	0.107153	0.0535764	+	0.998564i	$0.482938\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−24.2368	−1.23844	−0.619220	−	0.785217i	$-0.712551\pi$
$383$	−24.2368	−1.23844	−0.619220	+	0.785217i	$0.712551\pi$
$384$	0	0
$385$	−29.9081	−1.52426
$386$	0	0
$387$	0	0
$388$	0	0
$389$	29.0066	1.47069	0.735347	−	0.677691i	$-0.237019\pi$
$389$	29.0066	1.47069	0.735347	+	0.677691i	$0.237019\pi$
$390$	0	0
$391$	−14.1184	−0.713997
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.88162	−0.0946746
$396$	0	0
$397$	−17.8949	−0.898117	−0.449058	−	0.893502i	$-0.648240\pi$
$397$	−17.8949	−0.898117	−0.449058	+	0.893502i	$0.648240\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.09266	0.154440	0.0772200	−	0.997014i	$-0.475396\pi$
$401$	3.09266	0.154440	0.0772200	+	0.997014i	$0.475396\pi$
$402$	0	0
$403$	9.77647	0.487001
$404$	0	0
$405$	0	0
$406$	0	0
$407$	60.5405	3.00088
$408$	0	0
$409$	−20.0132	−0.989591	−0.494795	−	0.869010i	$-0.664757\pi$
$409$	−20.0132	−0.989591	−0.494795	+	0.869010i	$0.664757\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	2.46029	0.120771
$416$	0	0
$417$	0	0
$418$	0	0
$419$	9.46690	0.462488	0.231244	−	0.972896i	$-0.425720\pi$
$419$	9.46690	0.462488	0.231244	+	0.972896i	$0.425720\pi$
$420$	0	0
$421$	11.4669	0.558863	0.279431	−	0.960166i	$-0.409854\pi$
$421$	11.4669	0.558863	0.279431	+	0.960166i	$0.409854\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.88824	0.140100
$426$	0	0
$427$	59.2375	2.86670
$428$	0	0
$429$	0	0
$430$	0	0
$431$	21.0927	1.01600	0.507999	−	0.861358i	$-0.330385\pi$
$431$	21.0927	1.01600	0.507999	+	0.861358i	$0.330385\pi$
$432$	0	0
$433$	−7.77647	−0.373713	−0.186857	−	0.982387i	$-0.559830\pi$
$433$	−7.77647	−0.373713	−0.186857	+	0.982387i	$0.559830\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	34.2500	1.63840
$438$	0	0
$439$	−33.6713	−1.60704	−0.803522	−	0.595275i	$-0.797043\pi$
$439$	−33.6713	−1.60704	−0.803522	+	0.595275i	$0.797043\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−20.5787	−0.977722	−0.488861	−	0.872362i	$-0.662587\pi$
$443$	−20.5787	−0.977722	−0.488861	+	0.872362i	$0.662587\pi$
$444$	0	0
$445$	−11.4346	−0.542050
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0.328677	0.0155112	0.00775561	−	0.999970i	$-0.497531\pi$
$449$	0.328677	0.0155112	0.00775561	+	0.999970i	$0.497531\pi$
$450$	0	0
$451$	−60.5405	−2.85074
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4.88824	0.229164
$456$	0	0
$457$	37.5853	1.75817	0.879083	−	0.476669i	$-0.158156\pi$
$457$	37.5853	1.75817	0.879083	+	0.476669i	$0.158156\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	20.8022	0.968855	0.484427	−	0.874831i	$-0.339028\pi$
$461$	20.8022	0.968855	0.484427	+	0.874831i	$0.339028\pi$
$462$	0	0
$463$	10.4279	0.484628	0.242314	−	0.970198i	$-0.422094\pi$
$463$	10.4279	0.484628	0.242314	+	0.970198i	$0.422094\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	28.9875	1.34138	0.670691	−	0.741737i	$-0.265998\pi$
$467$	28.9875	1.34138	0.670691	+	0.741737i	$0.265998\pi$
$468$	0	0
$469$	31.5794	1.45820
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−24.4735	−1.12529
$474$	0	0
$475$	−7.00662	−0.321486
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−16.3419	−0.746681	−0.373340	−	0.927694i	$-0.621788\pi$
$479$	−16.3419	−0.746681	−0.373340	+	0.927694i	$0.621788\pi$
$480$	0	0
$481$	−9.89485	−0.451166
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.11176	0.232113
$486$	0	0
$487$	−2.90147	−0.131478	−0.0657390	−	0.997837i	$-0.520940\pi$
$487$	−2.90147	−0.131478	−0.0657390	+	0.997837i	$0.520940\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−10.7699	−0.486037	−0.243018	−	0.970022i	$-0.578138\pi$
$491$	−10.7699	−0.486037	−0.243018	+	0.970022i	$0.578138\pi$
$492$	0	0
$493$	9.32942	0.420176
$494$	0	0
$495$	0	0
$496$	0	0
$497$	58.1448	2.60815
$498$	0	0
$499$	−8.99338	−0.402599	−0.201300	−	0.979530i	$-0.564516\pi$
$499$	−8.99338	−0.402599	−0.201300	+	0.979530i	$0.564516\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	34.7963	1.55149	0.775746	−	0.631046i	$-0.217374\pi$
$503$	34.7963	1.55149	0.775746	+	0.631046i	$0.217374\pi$
$504$	0	0
$505$	5.00662	0.222791
$506$	0	0
$507$	0	0
$508$	0	0
$509$	31.6713	1.40381	0.701903	−	0.712272i	$-0.252334\pi$
$509$	31.6713	1.40381	0.701903	+	0.712272i	$0.252334\pi$
$510$	0	0
$511$	54.8956	2.42844
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−35.3426	−1.55437
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−3.09266	−0.135492	−0.0677459	−	0.997703i	$-0.521581\pi$
$521$	−3.09266	−0.135492	−0.0677459	+	0.997703i	$0.521581\pi$
$522$	0	0
$523$	−15.5529	−0.680083	−0.340041	−	0.940410i	$-0.610441\pi$
$523$	−15.5529	−0.680083	−0.340041	+	0.940410i	$0.610441\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	28.2368	1.23001
$528$	0	0
$529$	0.894851	0.0389066
$530$	0	0
$531$	0	0
$532$	0	0
$533$	9.89485	0.428594
$534$	0	0
$535$	−2.11838	−0.0915855
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−103.369	−4.45242
$540$	0	0
$541$	7.23014	0.310848	0.155424	−	0.987848i	$-0.450326\pi$
$541$	7.23014	0.310848	0.155424	+	0.987848i	$0.450326\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2.54633	−0.109073
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−22.6324	−0.964171
$552$	0	0
$553$	−9.19781	−0.391131
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−17.1059	−0.724800	−0.362400	−	0.932023i	$-0.618043\pi$
$557$	−17.1059	−0.724800	−0.362400	+	0.932023i	$0.618043\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.8154	1.04585	0.522923	−	0.852380i	$-0.324842\pi$
$563$	24.8154	1.04585	0.522923	+	0.852380i	$0.324842\pi$
$564$	0	0
$565$	−0.769857	−0.0323881
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−32.6970	−1.37073	−0.685366	−	0.728199i	$-0.740358\pi$
$569$	−32.6970	−1.37073	−0.685366	+	0.728199i	$0.740358\pi$
$570$	0	0
$571$	0.131610	0.00550770	0.00275385	−	0.999996i	$-0.499123\pi$
$571$	0.131610	0.00550770	0.00275385	+	0.999996i	$0.499123\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4.88824	−0.203854
$576$	0	0
$577$	7.09853	0.295516	0.147758	−	0.989024i	$-0.452794\pi$
$577$	7.09853	0.295516	0.147758	+	0.989024i	$0.452794\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	12.0265	0.498942
$582$	0	0
$583$	−34.6184	−1.43375
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−33.7765	−1.39410	−0.697052	−	0.717020i	$-0.745505\pi$
$587$	−33.7765	−1.39410	−0.697052	+	0.717020i	$0.745505\pi$
$588$	0	0
$589$	−68.5000	−2.82249
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−45.7897	−1.88036	−0.940179	−	0.340681i	$-0.889342\pi$
$593$	−45.7897	−1.88036	−0.940179	+	0.340681i	$0.889342\pi$
$594$	0	0
$595$	14.1184	0.578797
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4.92057	0.201049	0.100525	−	0.994935i	$-0.467948\pi$
$599$	4.92057	0.201049	0.100525	+	0.994935i	$0.467948\pi$
$600$	0	0
$601$	−42.1316	−1.71858	−0.859292	−	0.511485i	$-0.829096\pi$
$601$	−42.1316	−1.71858	−0.859292	+	0.511485i	$0.829096\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	26.4346	1.07472
$606$	0	0
$607$	−16.4735	−0.668639	−0.334320	−	0.942460i	$-0.608507\pi$
$607$	−16.4735	−0.668639	−0.334320	+	0.942460i	$0.608507\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.77647	0.233691
$612$	0	0
$613$	−25.6846	−1.03739	−0.518695	−	0.854960i	$-0.673582\pi$
$613$	−25.6846	−1.03739	−0.518695	+	0.854960i	$0.673582\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−20.0132	−0.805702	−0.402851	−	0.915266i	$-0.631981\pi$
$617$	−20.0132	−0.805702	−0.402851	+	0.915266i	$0.631981\pi$
$618$	0	0
$619$	−20.5463	−0.825827	−0.412913	−	0.910770i	$-0.635489\pi$
$619$	−20.5463	−0.825827	−0.412913	+	0.910770i	$0.635489\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−55.8949	−2.23938
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−28.5787	−1.13951
$630$	0	0
$631$	−27.5529	−1.09687	−0.548433	−	0.836195i	$-0.684775\pi$
$631$	−27.5529	−1.09687	−0.548433	+	0.836195i	$0.684775\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−12.2368	−0.485601
$636$	0	0
$637$	16.8949	0.669398
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−42.2632	−1.66930	−0.834648	−	0.550783i	$-0.814329\pi$
$641$	−42.2632	−1.66930	−0.834648	+	0.550783i	$0.814329\pi$
$642$	0	0
$643$	15.6581	0.617495	0.308747	−	0.951144i	$-0.400090\pi$
$643$	15.6581	0.617495	0.308747	+	0.951144i	$0.400090\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−50.6912	−1.99288	−0.996438	−	0.0843316i	$-0.973125\pi$
$647$	−50.6912	−1.99288	−0.996438	+	0.0843316i	$0.973125\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−22.0000	−0.860927	−0.430463	−	0.902608i	$-0.641650\pi$
$653$	−22.0000	−0.860927	−0.430463	+	0.902608i	$0.641650\pi$
$654$	0	0
$655$	0.993385	0.0388148
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−15.6904	−0.611212	−0.305606	−	0.952158i	$-0.598859\pi$
$659$	−15.6904	−0.611212	−0.305606	+	0.952158i	$0.598859\pi$
$660$	0	0
$661$	3.46690	0.134847	0.0674234	−	0.997724i	$-0.478522\pi$
$661$	3.46690	0.134847	0.0674234	+	0.997724i	$0.478522\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−34.2500	−1.32816
$666$	0	0
$667$	−15.7897	−0.611380
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−74.1448	−2.86233
$672$	0	0
$673$	−2.92057	−0.112580	−0.0562899	−	0.998414i	$-0.517927\pi$
$673$	−2.92057	−0.112580	−0.0562899	+	0.998414i	$0.517927\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−3.90808	−0.150200	−0.0750999	−	0.997176i	$-0.523928\pi$
$677$	−3.90808	−0.150200	−0.0750999	+	0.997176i	$0.523928\pi$
$678$	0	0
$679$	24.9875	0.958933
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−17.5662	−0.672151	−0.336075	−	0.941835i	$-0.609100\pi$
$683$	−17.5662	−0.672151	−0.336075	+	0.941835i	$0.609100\pi$
$684$	0	0
$685$	4.46029	0.170419
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5.65809	0.215556
$690$	0	0
$691$	43.6522	1.66061	0.830304	−	0.557310i	$-0.188167\pi$
$691$	43.6522	1.66061	0.830304	+	0.557310i	$0.188167\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.10515	−0.155717
$696$	0	0
$697$	28.5787	1.08249
$698$	0	0
$699$	0	0
$700$	0	0
$701$	33.0331	1.24764	0.623821	−	0.781567i	$-0.285579\pi$
$701$	33.0331	1.24764	0.623821	+	0.781567i	$0.285579\pi$
$702$	0	0
$703$	69.3294	2.61481
$704$	0	0
$705$	0	0
$706$	0	0
$707$	24.4735	0.920421
$708$	0	0
$709$	28.5596	1.07258	0.536288	−	0.844035i	$-0.319826\pi$
$709$	28.5596	1.07258	0.536288	+	0.844035i	$0.319826\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−47.7897	−1.78974
$714$	0	0
$715$	−6.11838	−0.228814
$716$	0	0
$717$	0	0
$718$	0	0
$719$	30.0132	1.11930	0.559652	−	0.828728i	$-0.310935\pi$
$719$	30.0132	1.11930	0.559652	+	0.828728i	$0.310935\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.23014	0.119965
$726$	0	0
$727$	3.14410	0.116608	0.0583041	−	0.998299i	$-0.481431\pi$
$727$	3.14410	0.116608	0.0583041	+	0.998299i	$0.481431\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	11.5529	0.427301
$732$	0	0
$733$	−22.3419	−0.825217	−0.412609	−	0.910908i	$-0.635382\pi$
$733$	−22.3419	−0.825217	−0.412609	+	0.910908i	$0.635382\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−39.5265	−1.45598
$738$	0	0
$739$	47.4801	1.74658	0.873292	−	0.487196i	$-0.161980\pi$
$739$	47.4801	1.74658	0.873292	+	0.487196i	$0.161980\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−19.3162	−0.708642	−0.354321	−	0.935124i	$-0.615288\pi$
$743$	−19.3162	−0.708642	−0.354321	+	0.935124i	$0.615288\pi$
$744$	0	0
$745$	9.65809	0.353845
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−10.3551	−0.378368
$750$	0	0
$751$	14.8022	0.540140	0.270070	−	0.962841i	$-0.412953\pi$
$751$	14.8022	0.540140	0.270070	+	0.962841i	$0.412953\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	14.0132	0.509994
$756$	0	0
$757$	5.78970	0.210430	0.105215	−	0.994449i	$-0.466447\pi$
$757$	5.78970	0.210430	0.105215	+	0.994449i	$0.466447\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	27.5662	0.999273	0.499637	−	0.866235i	$-0.333467\pi$
$761$	27.5662	0.999273	0.499637	+	0.866235i	$0.333467\pi$
$762$	0	0
$763$	−12.4471	−0.450614
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	1.78970	0.0645383	0.0322692	−	0.999479i	$-0.489727\pi$
$769$	1.78970	0.0645383	0.0322692	+	0.999479i	$0.489727\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	48.2500	1.73543	0.867716	−	0.497061i	$-0.165587\pi$
$773$	48.2500	1.73543	0.867716	+	0.497061i	$0.165587\pi$
$774$	0	0
$775$	9.77647	0.351181
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−69.3294	−2.48398
$780$	0	0
$781$	−72.7772	−2.60417
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−24.0132	−0.857069
$786$	0	0
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3.76324	−0.133805
$792$	0	0
$793$	12.1184	0.430336
$794$	0	0
$795$	0	0
$796$	0	0
$797$	17.4213	0.617095	0.308548	−	0.951209i	$-0.400157\pi$
$797$	17.4213	0.617095	0.308548	+	0.951209i	$0.400157\pi$
$798$	0	0
$799$	16.6838	0.590231
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−68.7103	−2.42473
$804$	0	0
$805$	−23.8949	−0.842183
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	−11.2434	−0.394808	−0.197404	−	0.980322i	$-0.563251\pi$
$811$	−11.2434	−0.394808	−0.197404	+	0.980322i	$0.563251\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	13.6713	0.478886
$816$	0	0
$817$	−28.0265	−0.980522
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−38.1316	−1.33080	−0.665401	−	0.746486i	$-0.731740\pi$
$821$	−38.1316	−1.33080	−0.665401	+	0.746486i	$0.731740\pi$
$822$	0	0
$823$	−39.1059	−1.36315	−0.681573	−	0.731750i	$-0.738704\pi$
$823$	−39.1059	−1.36315	−0.681573	+	0.731750i	$0.738704\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13.8029	0.479975	0.239988	−	0.970776i	$-0.422857\pi$
$827$	13.8029	0.479975	0.239988	+	0.970776i	$0.422857\pi$
$828$	0	0
$829$	30.0000	1.04194	0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000	1.04194	0.520972	+	0.853574i	$0.325570\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	48.7963	1.69069
$834$	0	0
$835$	4.00000	0.138426
$836$	0	0
$837$	0	0
$838$	0	0
$839$	30.9743	1.06935	0.534675	−	0.845058i	$-0.320434\pi$
$839$	30.9743	1.06935	0.534675	+	0.845058i	$0.320434\pi$
$840$	0	0
$841$	−18.5662	−0.640213
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	129.218	4.44000
$848$	0	0
$849$	0	0
$850$	0	0
$851$	48.3684	1.65805
$852$	0	0
$853$	53.4743	1.83092	0.915462	−	0.402405i	$-0.131826\pi$
$853$	53.4743	1.83092	0.915462	+	0.402405i	$0.131826\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−11.8088	−0.403381	−0.201691	−	0.979449i	$-0.564644\pi$
$857$	−11.8088	−0.403381	−0.201691	+	0.979449i	$0.564644\pi$
$858$	0	0
$859$	−29.1978	−0.996216	−0.498108	−	0.867115i	$-0.665972\pi$
$859$	−29.1978	−0.996216	−0.498108	+	0.867115i	$0.665972\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	42.8956	1.46018	0.730091	−	0.683349i	$-0.239477\pi$
$863$	42.8956	1.46018	0.730091	+	0.683349i	$0.239477\pi$
$864$	0	0
$865$	−12.0132	−0.408462
$866$	0	0
$867$	0	0
$868$	0	0
$869$	11.5125	0.390534
$870$	0	0
$871$	6.46029	0.218898
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.88824	0.165253
$876$	0	0
$877$	24.1853	0.816680	0.408340	−	0.912830i	$-0.366108\pi$
$877$	24.1853	0.816680	0.408340	+	0.912830i	$0.366108\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−23.5397	−0.793073	−0.396537	−	0.918019i	$-0.629788\pi$
$881$	−23.5397	−0.793073	−0.396537	+	0.918019i	$0.629788\pi$
$882$	0	0
$883$	−31.5529	−1.06184	−0.530921	−	0.847422i	$-0.678154\pi$
$883$	−31.5529	−1.06184	−0.530921	+	0.847422i	$0.678154\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−7.75737	−0.260467	−0.130233	−	0.991483i	$-0.541573\pi$
$887$	−7.75737	−0.260467	−0.130233	+	0.991483i	$0.541573\pi$
$888$	0	0
$889$	−59.8162	−2.00617
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−40.4735	−1.35439
$894$	0	0
$895$	15.0066	0.501616
$896$	0	0
$897$	0	0
$898$	0	0
$899$	31.5794	1.05323
$900$	0	0
$901$	16.3419	0.544428
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.89485	−0.0629870
$906$	0	0
$907$	27.7897	0.922742	0.461371	−	0.887207i	$-0.347358\pi$
$907$	27.7897	0.922742	0.461371	+	0.887207i	$0.347358\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−58.9338	−1.95256	−0.976282	−	0.216503i	$-0.930535\pi$
$911$	−58.9338	−1.95256	−0.976282	+	0.216503i	$0.930535\pi$
$912$	0	0
$913$	−15.0530	−0.498180
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.85590	0.160356
$918$	0	0
$919$	13.2243	0.436228	0.218114	−	0.975923i	$-0.430009\pi$
$919$	13.2243	0.436228	0.218114	+	0.975923i	$0.430009\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	11.8949	0.391524
$924$	0	0
$925$	−9.89485	−0.325341
$926$	0	0
$927$	0	0
$928$	0	0
$929$	9.48601	0.311226	0.155613	−	0.987818i	$-0.450265\pi$
$929$	9.48601	0.311226	0.155613	+	0.987818i	$0.450265\pi$
$930$	0	0
$931$	−118.376	−3.87961
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−17.6713	−0.577914
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−32.5272	−1.06036	−0.530179	−	0.847886i	$-0.677875\pi$
$941$	−32.5272	−1.06036	−0.530179	+	0.847886i	$0.677875\pi$
$942$	0	0
$943$	−48.3684	−1.57509
$944$	0	0
$945$	0	0
$946$	0	0
$947$	38.4867	1.25065	0.625326	−	0.780364i	$-0.284966\pi$
$947$	38.4867	1.25065	0.625326	+	0.780364i	$0.284966\pi$
$948$	0	0
$949$	11.2301	0.364546
$950$	0	0
$951$	0	0
$952$	0	0
$953$	25.9941	0.842032	0.421016	−	0.907053i	$-0.361674\pi$
$953$	25.9941	0.842032	0.421016	+	0.907053i	$0.361674\pi$
$954$	0	0
$955$	−3.55294	−0.114971
$956$	0	0
$957$	0	0
$958$	0	0
$959$	21.8029	0.704053
$960$	0	0
$961$	64.5794	2.08321
$962$	0	0
$963$	0	0
$964$	0	0
$965$	6.88824	0.221740
$966$	0	0
$967$	0.137486	0.00442124	0.00221062	−	0.999998i	$-0.499296\pi$
$967$	0.137486	0.00442124	0.00221062	+	0.999998i	$0.499296\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	30.3228	0.973105	0.486552	−	0.873651i	$-0.338254\pi$
$971$	30.3228	0.973105	0.486552	+	0.873651i	$0.338254\pi$
$972$	0	0
$973$	−20.0669	−0.643316
$974$	0	0
$975$	0	0
$976$	0	0
$977$	17.7897	0.569143	0.284572	−	0.958655i	$-0.408149\pi$
$977$	17.7897	0.569143	0.284572	+	0.958655i	$0.408149\pi$
$978$	0	0
$979$	69.9610	2.23596
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−43.1059	−1.37486	−0.687432	−	0.726249i	$-0.741262\pi$
$983$	−43.1059	−1.37486	−0.687432	+	0.726249i	$0.741262\pi$
$984$	0	0
$985$	−3.77647	−0.120328
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−19.5529	−0.621747
$990$	0	0
$991$	8.51399	0.270456	0.135228	−	0.990815i	$-0.456823\pi$
$991$	8.51399	0.270456	0.135228	+	0.990815i	$0.456823\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−27.5529	−0.873487
$996$	0	0
$997$	54.8824	1.73814	0.869071	−	0.494688i	$-0.164718\pi$
$997$	54.8824	1.73814	0.869071	+	0.494688i	$0.164718\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4680.2.a.bl.1.3		3
3.2	odd	2		1560.2.a.p.1.3	✓	3
4.3	odd	2		9360.2.a.db.1.1		3
12.11	even	2		3120.2.a.bh.1.1		3
15.14	odd	2		7800.2.a.bf.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.a.p.1.3	✓	3	3.2	odd	2
3120.2.a.bh.1.1		3	12.11	even	2
4680.2.a.bl.1.3		3	1.1	even	1	trivial
7800.2.a.bf.1.1		3	15.14	odd	2
9360.2.a.db.1.1		3	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4680.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +4.88824 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} +4.88824 q^{7} -6.11838 q^{11} +1.00000 q^{13} +2.88824 q^{17} -7.00662 q^{19} -4.88824 q^{23} +1.00000 q^{25} +3.23014 q^{29} +9.77647 q^{31} +4.88824 q^{35} -9.89485 q^{37} +9.89485 q^{41} +4.00000 q^{43} +5.77647 q^{47} +16.8949 q^{49} +5.65809 q^{53} -6.11838 q^{55} +12.1184 q^{61} +1.00000 q^{65} +6.46029 q^{67} +11.8949 q^{71} +11.2301 q^{73} -29.9081 q^{77} -1.88162 q^{79} +2.46029 q^{83} +2.88824 q^{85} -11.4346 q^{89} +4.88824 q^{91} -7.00662 q^{95} +5.11176 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} + 5 q^{7}+O(q^{10}) \) 3 * q + 3 * q^5 + 5 * q^7	\( 3 q + 3 q^{5} + 5 q^{7} - 3 q^{11} + 3 q^{13} - q^{17} + 4 q^{19} - 5 q^{23} + 3 q^{25} + 4 q^{29} + 10 q^{31} + 5 q^{35} + 5 q^{37} - 5 q^{41} + 12 q^{43} - 2 q^{47} + 16 q^{49} + 13 q^{53} - 3 q^{55} + 21 q^{61} + 3 q^{65} + 8 q^{67} + q^{71} + 28 q^{73} - 5 q^{77} - 21 q^{79} - 4 q^{83} - q^{85} - 11 q^{89} + 5 q^{91} + 4 q^{95} + 25 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 + 5 * q^7 - 3 * q^11 + 3 * q^13 - q^17 + 4 * q^19 - 5 * q^23 + 3 * q^25 + 4 * q^29 + 10 * q^31 + 5 * q^35 + 5 * q^37 - 5 * q^41 + 12 * q^43 - 2 * q^47 + 16 * q^49 + 13 * q^53 - 3 * q^55 + 21 * q^61 + 3 * q^65 + 8 * q^67 + q^71 + 28 * q^73 - 5 * q^77 - 21 * q^79 - 4 * q^83 - q^85 - 11 * q^89 + 5 * q^91 + 4 * q^95 + 25 * q^97

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