LMFDB - Embedded newform 1560.2.g.h.961.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1560,2,Mod(961,1560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1560.961");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1560.g (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$12.4566627153$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			961.1
Root			$-2.56155i$ of defining polynomial
Character	$\chi$	$=$	1560.961
Dual form			1560.2.g.h.961.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000i q^{5} -3.56155i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000i q^{5} -3.56155i q^{7} +1.00000 q^{9} -1.56155i q^{11} +(-0.561553 + 3.56155i) q^{13} -1.00000i q^{15} +0.438447 q^{17} -5.12311i q^{19} -3.56155i q^{21} +1.56155 q^{23} -1.00000 q^{25} +1.00000 q^{27} -2.00000 q^{29} -2.00000i q^{31} -1.56155i q^{33} -3.56155 q^{35} -5.56155i q^{37} +(-0.561553 + 3.56155i) q^{39} -6.68466i q^{41} +0.876894 q^{43} -1.00000i q^{45} -5.68466 q^{49} +0.438447 q^{51} -6.68466 q^{53} -1.56155 q^{55} -5.12311i q^{57} +6.24621i q^{59} -6.68466 q^{61} -3.56155i q^{63} +(3.56155 + 0.561553i) q^{65} -15.3693i q^{67} +1.56155 q^{69} +9.56155i q^{71} -0.876894i q^{73} -1.00000 q^{75} -5.56155 q^{77} -8.68466 q^{79} +1.00000 q^{81} -3.12311i q^{83} -0.438447i q^{85} -2.00000 q^{87} -0.438447i q^{89} +(12.6847 + 2.00000i) q^{91} -2.00000i q^{93} -5.12311 q^{95} -1.56155i q^{97} -1.56155i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^9	$ 4 q + 4 q^{3} + 4 q^{9} + 6 q^{13} + 10 q^{17} - 2 q^{23} - 4 q^{25} + 4 q^{27} - 8 q^{29} - 6 q^{35} + 6 q^{39} + 20 q^{43} + 2 q^{49} + 10 q^{51} - 2 q^{53} + 2 q^{55} - 2 q^{61} + 6 q^{65} - 2 q^{69} - 4 q^{75} - 14 q^{77} - 10 q^{79} + 4 q^{81} - 8 q^{87} + 26 q^{91} - 4 q^{95}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^9 + 6 * q^13 + 10 * q^17 - 2 * q^23 - 4 * q^25 + 4 * q^27 - 8 * q^29 - 6 * q^35 + 6 * q^39 + 20 * q^43 + 2 * q^49 + 10 * q^51 - 2 * q^53 + 2 * q^55 - 2 * q^61 + 6 * q^65 - 2 * q^69 - 4 * q^75 - 14 * q^77 - 10 * q^79 + 4 * q^81 - 8 * q^87 + 26 * q^91 - 4 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times$.

$n$	$391$	$521$	$781$	$937$	$1081$
$\chi(n)$	$1$	$1$	$1$	$1$	$-1$

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
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$		−	3.56155i		−	1.34614i	−0.739579	−	0.673070i	$-0.764975\pi$
$7$		−	3.56155i		−	1.34614i	0.739579	−	0.673070i	$-0.235025\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	1.56155i		−	0.470826i	−0.971895	−	0.235413i	$-0.924356\pi$
$11$		−	1.56155i		−	0.470826i	0.971895	−	0.235413i	$-0.0756443\pi$
$12$		0			0
$13$	−0.561553	+	3.56155i	−0.155747	+	0.987797i
$13$	−0.561553	+	3.56155i	−0.155747	+	0.987797i
$14$		0			0
$15$		−	1.00000i		−	0.258199i
$16$		0			0
$17$	0.438447			0.106339			0.0531695	−	0.998586i	$-0.483068\pi$
$17$	0.438447			0.106339			0.0531695	+	0.998586i	$0.483068\pi$
$18$		0			0
$19$		−	5.12311i		−	1.17532i	−0.809108	−	0.587661i	$-0.800049\pi$
$19$		−	5.12311i		−	1.17532i	0.809108	−	0.587661i	$-0.199951\pi$
$20$		0			0
$21$		−	3.56155i		−	0.777195i
$22$		0			0
$23$	1.56155			0.325606			0.162803	−	0.986659i	$-0.447946\pi$
$23$	1.56155			0.325606			0.162803	+	0.986659i	$0.447946\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$		−	2.00000i		−	0.359211i	−0.983739	−	0.179605i	$-0.942518\pi$
$31$		−	2.00000i		−	0.359211i	0.983739	−	0.179605i	$-0.0574821\pi$
$32$		0			0
$33$		−	1.56155i		−	0.271831i
$34$		0			0
$35$	−3.56155			−0.602012
$36$		0			0
$37$		−	5.56155i		−	0.914314i	−0.889386	−	0.457157i	$-0.848868\pi$
$37$		−	5.56155i		−	0.914314i	0.889386	−	0.457157i	$-0.151132\pi$
$38$		0			0
$39$	−0.561553	+	3.56155i	−0.0899204	+	0.570305i
$40$		0			0
$41$		−	6.68466i		−	1.04397i	−0.852955	−	0.521984i	$-0.825192\pi$
$41$		−	6.68466i		−	1.04397i	0.852955	−	0.521984i	$-0.174808\pi$
$42$		0			0
$43$	0.876894			0.133725			0.0668626	−	0.997762i	$-0.478701\pi$
$43$	0.876894			0.133725			0.0668626	+	0.997762i	$0.478701\pi$
$44$		0			0
$45$		−	1.00000i		−	0.149071i
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	−5.68466			−0.812094
$50$		0			0
$51$	0.438447			0.0613949
$52$		0			0
$53$	−6.68466			−0.918208			−0.459104	−	0.888382i	$-0.651830\pi$
$53$	−6.68466			−0.918208			−0.459104	+	0.888382i	$0.651830\pi$
$54$		0			0
$55$	−1.56155			−0.210560
$56$		0			0
$57$		−	5.12311i		−	0.678572i
$58$		0			0
$59$			6.24621i			0.813187i	0.913609	+	0.406594i	$0.133284\pi$
$59$			6.24621i			0.813187i	−0.913609	+	0.406594i	$0.866716\pi$
$60$		0			0
$61$	−6.68466			−0.855883			−0.427941	−	0.903806i	$-0.640761\pi$
$61$	−6.68466			−0.855883			−0.427941	+	0.903806i	$0.640761\pi$
$62$		0			0
$63$		−	3.56155i		−	0.448713i
$64$		0			0
$65$	3.56155	+	0.561553i	0.441756	+	0.0696521i
$66$		0			0
$67$		−	15.3693i		−	1.87766i	−0.344380	−	0.938830i	$-0.611911\pi$
$67$		−	15.3693i		−	1.87766i	0.344380	−	0.938830i	$-0.388089\pi$
$68$		0			0
$69$	1.56155			0.187989
$70$		0			0
$71$			9.56155i			1.13475i	0.823460	+	0.567374i	$0.192041\pi$
$71$			9.56155i			1.13475i	−0.823460	+	0.567374i	$0.807959\pi$
$72$		0			0
$73$		−	0.876894i		−	0.102633i	−0.998682	−	0.0513164i	$-0.983658\pi$
$73$		−	0.876894i		−	0.102633i	0.998682	−	0.0513164i	$-0.0163417\pi$
$74$		0			0
$75$	−1.00000			−0.115470
$76$		0			0
$77$	−5.56155			−0.633798
$78$		0			0
$79$	−8.68466			−0.977100			−0.488550	−	0.872536i	$-0.662474\pi$
$79$	−8.68466			−0.977100			−0.488550	+	0.872536i	$0.662474\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	3.12311i		−	0.342805i	−0.985201	−	0.171403i	$-0.945170\pi$
$83$		−	3.12311i		−	0.342805i	0.985201	−	0.171403i	$-0.0548299\pi$
$84$		0			0
$85$		−	0.438447i		−	0.0475563i
$86$		0			0
$87$	−2.00000			−0.214423
$88$		0			0
$89$		−	0.438447i		−	0.0464753i	−0.999730	−	0.0232377i	$-0.992603\pi$
$89$		−	0.438447i		−	0.0464753i	0.999730	−	0.0232377i	$-0.00739744\pi$
$90$		0			0
$91$	12.6847	+	2.00000i	1.32971	+	0.209657i
$92$		0			0
$93$		−	2.00000i		−	0.207390i
$94$		0			0
$95$	−5.12311			−0.525620
$96$		0			0
$97$		−	1.56155i		−	0.158552i	−0.996853	−	0.0792758i	$-0.974739\pi$
$97$		−	1.56155i		−	0.158552i	0.996853	−	0.0792758i	$-0.0252608\pi$
$98$		0			0
$99$		−	1.56155i		−	0.156942i
$100$		0			0
$101$	2.00000			0.199007			0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000			0.199007			0.0995037	+	0.995037i	$0.468274\pi$
$102$		0			0
$103$	14.2462			1.40372			0.701860	−	0.712314i	$-0.252353\pi$
$103$	14.2462			1.40372			0.701860	+	0.712314i	$0.252353\pi$
$104$		0			0
$105$	−3.56155			−0.347572
$106$		0			0
$107$	8.68466			0.839578			0.419789	−	0.907622i	$-0.362104\pi$
$107$	8.68466			0.839578			0.419789	+	0.907622i	$0.362104\pi$
$108$		0			0
$109$		−	8.87689i		−	0.850252i	−0.905134	−	0.425126i	$-0.860230\pi$
$109$		−	8.87689i		−	0.850252i	0.905134	−	0.425126i	$-0.139770\pi$
$110$		0			0
$111$		−	5.56155i		−	0.527879i
$112$		0			0
$113$	4.24621			0.399450			0.199725	−	0.979852i	$-0.435995\pi$
$113$	4.24621			0.399450			0.199725	+	0.979852i	$0.435995\pi$
$114$		0			0
$115$		−	1.56155i		−	0.145616i
$116$		0			0
$117$	−0.561553	+	3.56155i	−0.0519156	+	0.329266i
$118$		0			0
$119$		−	1.56155i		−	0.143147i
$120$		0			0
$121$	8.56155			0.778323
$122$		0			0
$123$		−	6.68466i		−	0.602735i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$	19.1231			1.69690			0.848451	−	0.529275i	$-0.177536\pi$
$127$	19.1231			1.69690			0.848451	+	0.529275i	$0.177536\pi$
$128$		0			0
$129$	0.876894			0.0772062
$130$		0			0
$131$	20.4924			1.79043			0.895216	−	0.445633i	$-0.147021\pi$
$131$	20.4924			1.79043			0.895216	+	0.445633i	$0.147021\pi$
$132$		0			0
$133$	−18.2462			−1.58215
$134$		0			0
$135$		−	1.00000i		−	0.0860663i
$136$		0			0
$137$			6.00000i			0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$			6.00000i			0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$		0			0
$139$	−9.56155			−0.811000			−0.405500	−	0.914095i	$-0.632903\pi$
$139$	−9.56155			−0.811000			−0.405500	+	0.914095i	$0.632903\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	5.56155	+	0.876894i	0.465080	+	0.0733296i
$144$		0			0
$145$			2.00000i			0.166091i
$146$		0			0
$147$	−5.68466			−0.468863
$148$		0			0
$149$			20.9309i			1.71472i	0.514714	+	0.857362i	$0.327898\pi$
$149$			20.9309i			1.71472i	−0.514714	+	0.857362i	$0.672102\pi$
$150$		0			0
$151$			14.0000i			1.13930i	0.821886	+	0.569652i	$0.192922\pi$
$151$			14.0000i			1.13930i	−0.821886	+	0.569652i	$0.807078\pi$
$152$		0			0
$153$	0.438447			0.0354464
$154$		0			0
$155$	−2.00000			−0.160644
$156$		0			0
$157$	−6.87689			−0.548836			−0.274418	−	0.961611i	$-0.588485\pi$
$157$	−6.87689			−0.548836			−0.274418	+	0.961611i	$0.588485\pi$
$158$		0			0
$159$	−6.68466			−0.530128
$160$		0			0
$161$		−	5.56155i		−	0.438312i
$162$		0			0
$163$		−	20.0540i		−	1.57075i	−0.619021	−	0.785374i	$-0.712471\pi$
$163$		−	20.0540i		−	1.57075i	0.619021	−	0.785374i	$-0.287529\pi$
$164$		0			0
$165$	−1.56155			−0.121567
$166$		0			0
$167$			6.24621i			0.483346i	0.970358	+	0.241673i	$0.0776962\pi$
$167$			6.24621i			0.483346i	−0.970358	+	0.241673i	$0.922304\pi$
$168$		0			0
$169$	−12.3693	−	4.00000i	−0.951486	−	0.307692i
$170$		0			0
$171$		−	5.12311i		−	0.391774i
$172$		0			0
$173$	14.4924			1.10184			0.550919	−	0.834559i	$-0.314277\pi$
$173$	14.4924			1.10184			0.550919	+	0.834559i	$0.314277\pi$
$174$		0			0
$175$			3.56155i			0.269228i
$176$		0			0
$177$			6.24621i			0.469494i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	−13.8078			−1.02632			−0.513162	−	0.858292i	$-0.671526\pi$
$181$	−13.8078			−1.02632			−0.513162	+	0.858292i	$0.671526\pi$
$182$		0			0
$183$	−6.68466			−0.494144
$184$		0			0
$185$	−5.56155			−0.408893
$186$		0			0
$187$		−	0.684658i		−	0.0500672i
$188$		0			0
$189$		−	3.56155i		−	0.259065i
$190$		0			0
$191$	−5.75379			−0.416330			−0.208165	−	0.978094i	$-0.566749\pi$
$191$	−5.75379			−0.416330			−0.208165	+	0.978094i	$0.566749\pi$
$192$		0			0
$193$			10.4384i			0.751376i	0.926746	+	0.375688i	$0.122594\pi$
$193$			10.4384i			0.751376i	−0.926746	+	0.375688i	$0.877406\pi$
$194$		0			0
$195$	3.56155	+	0.561553i	0.255048	+	0.0402136i
$196$		0			0
$197$			14.4924i			1.03254i	0.856425	+	0.516271i	$0.172680\pi$
$197$			14.4924i			1.03254i	−0.856425	+	0.516271i	$0.827320\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		−	15.3693i		−	1.08407i
$202$		0			0
$203$			7.12311i			0.499944i
$204$		0			0
$205$	−6.68466			−0.466877
$206$		0			0
$207$	1.56155			0.108535
$208$		0			0
$209$	−8.00000			−0.553372
$210$		0			0
$211$	24.4924			1.68613			0.843064	−	0.537813i	$-0.180749\pi$
$211$	24.4924			1.68613			0.843064	+	0.537813i	$0.180749\pi$
$212$		0			0
$213$			9.56155i			0.655147i
$214$		0			0
$215$		−	0.876894i		−	0.0598037i
$216$		0			0
$217$	−7.12311			−0.483548
$218$		0			0
$219$		−	0.876894i		−	0.0592550i
$220$		0			0
$221$	−0.246211	+	1.56155i	−0.0165620	+	0.105041i
$222$		0			0
$223$		−	13.1231i		−	0.878788i	−0.898294	−	0.439394i	$-0.855193\pi$
$223$		−	13.1231i		−	0.878788i	0.898294	−	0.439394i	$-0.144807\pi$
$224$		0			0
$225$	−1.00000			−0.0666667
$226$		0			0
$227$			26.2462i			1.74202i	0.491263	+	0.871011i	$0.336535\pi$
$227$			26.2462i			1.74202i	−0.491263	+	0.871011i	$0.663465\pi$
$228$		0			0
$229$		−	2.63068i		−	0.173840i	−0.996215	−	0.0869202i	$-0.972297\pi$
$229$		−	2.63068i		−	0.173840i	0.996215	−	0.0869202i	$-0.0277025\pi$
$230$		0			0
$231$	−5.56155			−0.365923
$232$		0			0
$233$	5.80776			0.380479			0.190240	−	0.981738i	$-0.439074\pi$
$233$	5.80776			0.380479			0.190240	+	0.981738i	$0.439074\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−8.68466			−0.564129
$238$		0			0
$239$		−	12.6847i		−	0.820502i	−0.911973	−	0.410251i	$-0.865441\pi$
$239$		−	12.6847i		−	0.820502i	0.911973	−	0.410251i	$-0.134559\pi$
$240$		0			0
$241$		−	7.12311i		−	0.458840i	−0.973328	−	0.229420i	$-0.926317\pi$
$241$		−	7.12311i		−	0.458840i	0.973328	−	0.229420i	$-0.0736829\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$			5.68466i			0.363180i
$246$		0			0
$247$	18.2462	+	2.87689i	1.16098	+	0.183052i
$248$		0			0
$249$		−	3.12311i		−	0.197919i
$250$		0			0
$251$	13.3693			0.843864			0.421932	−	0.906628i	$-0.361352\pi$
$251$	13.3693			0.843864			0.421932	+	0.906628i	$0.361352\pi$
$252$		0			0
$253$		−	2.43845i		−	0.153304i
$254$		0			0
$255$		−	0.438447i		−	0.0274566i
$256$		0			0
$257$	22.4924			1.40304			0.701519	−	0.712650i	$-0.252505\pi$
$257$	22.4924			1.40304			0.701519	+	0.712650i	$0.252505\pi$
$258$		0			0
$259$	−19.8078			−1.23079
$260$		0			0
$261$	−2.00000			−0.123797
$262$		0			0
$263$	16.0000			0.986602			0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000			0.986602			0.493301	+	0.869859i	$0.335790\pi$
$264$		0			0
$265$			6.68466i			0.410635i
$266$		0			0
$267$		−	0.438447i		−	0.0268325i
$268$		0			0
$269$	−29.1231			−1.77567			−0.887834	−	0.460165i	$-0.847790\pi$
$269$	−29.1231			−1.77567			−0.887834	+	0.460165i	$0.847790\pi$
$270$		0			0
$271$			3.75379i			0.228026i	0.993479	+	0.114013i	$0.0363706\pi$
$271$			3.75379i			0.228026i	−0.993479	+	0.114013i	$0.963629\pi$
$272$		0			0
$273$	12.6847	+	2.00000i	0.767710	+	0.121046i
$274$		0			0
$275$			1.56155i			0.0941652i
$276$		0			0
$277$	−3.36932			−0.202443			−0.101221	−	0.994864i	$-0.532275\pi$
$277$	−3.36932			−0.202443			−0.101221	+	0.994864i	$0.532275\pi$
$278$		0			0
$279$		−	2.00000i		−	0.119737i
$280$		0			0
$281$		−	2.00000i		−	0.119310i	−0.998219	−	0.0596550i	$-0.981000\pi$
$281$		−	2.00000i		−	0.119310i	0.998219	−	0.0596550i	$-0.0190001\pi$
$282$		0			0
$283$	15.1231			0.898975			0.449488	−	0.893287i	$-0.351607\pi$
$283$	15.1231			0.898975			0.449488	+	0.893287i	$0.351607\pi$
$284$		0			0
$285$	−5.12311			−0.303467
$286$		0			0
$287$	−23.8078			−1.40533
$288$		0			0
$289$	−16.8078			−0.988692
$290$		0			0
$291$		−	1.56155i		−	0.0915398i
$292$		0			0
$293$			22.4924i			1.31402i	0.753881	+	0.657011i	$0.228179\pi$
$293$			22.4924i			1.31402i	−0.753881	+	0.657011i	$0.771821\pi$
$294$		0			0
$295$	6.24621			0.363668
$296$		0			0
$297$		−	1.56155i		−	0.0906105i
$298$		0			0
$299$	−0.876894	+	5.56155i	−0.0507121	+	0.321633i
$300$		0			0
$301$		−	3.12311i		−	0.180013i
$302$		0			0
$303$	2.00000			0.114897
$304$		0			0
$305$			6.68466i			0.382762i
$306$		0			0
$307$			4.43845i			0.253316i	0.991946	+	0.126658i	$0.0404250\pi$
$307$			4.43845i			0.253316i	−0.991946	+	0.126658i	$0.959575\pi$
$308$		0			0
$309$	14.2462			0.810439
$310$		0			0
$311$	−8.87689			−0.503362			−0.251681	−	0.967810i	$-0.580983\pi$
$311$	−8.87689			−0.503362			−0.251681	+	0.967810i	$0.580983\pi$
$312$		0			0
$313$	7.36932			0.416538			0.208269	−	0.978072i	$-0.433217\pi$
$313$	7.36932			0.416538			0.208269	+	0.978072i	$0.433217\pi$
$314$		0			0
$315$	−3.56155			−0.200671
$316$		0			0
$317$			10.0000i			0.561656i	0.959758	+	0.280828i	$0.0906090\pi$
$317$			10.0000i			0.561656i	−0.959758	+	0.280828i	$0.909391\pi$
$318$		0			0
$319$			3.12311i			0.174860i
$320$		0			0
$321$	8.68466			0.484730
$322$		0			0
$323$		−	2.24621i		−	0.124983i
$324$		0			0
$325$	0.561553	−	3.56155i	0.0311493	−	0.197559i
$326$		0			0
$327$		−	8.87689i		−	0.490893i
$328$		0			0
$329$		0			0
$330$		0			0
$331$			19.3693i			1.06463i	0.846545	+	0.532317i	$0.178679\pi$
$331$			19.3693i			1.06463i	−0.846545	+	0.532317i	$0.821321\pi$
$332$		0			0
$333$		−	5.56155i		−	0.304771i
$334$		0			0
$335$	−15.3693			−0.839715
$336$		0			0
$337$	26.4924			1.44313			0.721567	−	0.692345i	$-0.243422\pi$
$337$	26.4924			1.44313			0.721567	+	0.692345i	$0.243422\pi$
$338$		0			0
$339$	4.24621			0.230623
$340$		0			0
$341$	−3.12311			−0.169126
$342$		0			0
$343$		−	4.68466i		−	0.252948i
$344$		0			0
$345$		−	1.56155i		−	0.0840712i
$346$		0			0
$347$	−25.5616			−1.37222			−0.686108	−	0.727500i	$-0.740682\pi$
$347$	−25.5616			−1.37222			−0.686108	+	0.727500i	$0.740682\pi$
$348$		0			0
$349$			15.6155i			0.835880i	0.908475	+	0.417940i	$0.137248\pi$
$349$			15.6155i			0.835880i	−0.908475	+	0.417940i	$0.862752\pi$
$350$		0			0
$351$	−0.561553	+	3.56155i	−0.0299735	+	0.190102i
$352$		0			0
$353$			22.8769i			1.21761i	0.793318	+	0.608807i	$0.208352\pi$
$353$			22.8769i			1.21761i	−0.793318	+	0.608807i	$0.791648\pi$
$354$		0			0
$355$	9.56155			0.507475
$356$		0			0
$357$		−	1.56155i		−	0.0826461i
$358$		0			0
$359$			12.4924i			0.659325i	0.944099	+	0.329662i	$0.106935\pi$
$359$			12.4924i			0.659325i	−0.944099	+	0.329662i	$0.893065\pi$
$360$		0			0
$361$	−7.24621			−0.381380
$362$		0			0
$363$	8.56155			0.449365
$364$		0			0
$365$	−0.876894			−0.0458987
$366$		0			0
$367$	29.8617			1.55877			0.779385	−	0.626545i	$-0.215532\pi$
$367$	29.8617			1.55877			0.779385	+	0.626545i	$0.215532\pi$
$368$		0			0
$369$		−	6.68466i		−	0.347989i
$370$		0			0
$371$			23.8078i			1.23604i
$372$		0			0
$373$	−10.8769			−0.563184			−0.281592	−	0.959534i	$-0.590863\pi$
$373$	−10.8769			−0.563184			−0.281592	+	0.959534i	$0.590863\pi$
$374$		0			0
$375$			1.00000i			0.0516398i
$376$		0			0
$377$	1.12311	−	7.12311i	0.0578429	−	0.366859i
$378$		0			0
$379$			10.4924i			0.538960i	0.963006	+	0.269480i	$0.0868517\pi$
$379$			10.4924i			0.538960i	−0.963006	+	0.269480i	$0.913148\pi$
$380$		0			0
$381$	19.1231			0.979706
$382$		0			0
$383$			3.12311i			0.159583i	0.996812	+	0.0797916i	$0.0254255\pi$
$383$			3.12311i			0.159583i	−0.996812	+	0.0797916i	$0.974575\pi$
$384$		0			0
$385$			5.56155i			0.283443i
$386$		0			0
$387$	0.876894			0.0445750
$388$		0			0
$389$	−10.0000			−0.507020			−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000			−0.507020			−0.253510	+	0.967333i	$0.581585\pi$
$390$		0			0
$391$	0.684658			0.0346247
$392$		0			0
$393$	20.4924			1.03371
$394$		0			0
$395$			8.68466i			0.436973i
$396$		0			0
$397$		−	27.8078i		−	1.39563i	−0.716277	−	0.697816i	$-0.754155\pi$
$397$		−	27.8078i		−	1.39563i	0.716277	−	0.697816i	$-0.245845\pi$
$398$		0			0
$399$	−18.2462			−0.913453
$400$		0			0
$401$		−	2.49242i		−	0.124466i	−0.998062	−	0.0622328i	$-0.980178\pi$
$401$		−	2.49242i		−	0.124466i	0.998062	−	0.0622328i	$-0.0198221\pi$
$402$		0			0
$403$	7.12311	+	1.12311i	0.354827	+	0.0559459i
$404$		0			0
$405$		−	1.00000i		−	0.0496904i
$406$		0			0
$407$	−8.68466			−0.430483
$408$		0			0
$409$			0.492423i			0.0243487i	0.999926	+	0.0121744i	$0.00387532\pi$
$409$			0.492423i			0.0243487i	−0.999926	+	0.0121744i	$0.996125\pi$
$410$		0			0
$411$			6.00000i			0.295958i
$412$		0			0
$413$	22.2462			1.09466
$414$		0			0
$415$	−3.12311			−0.153307
$416$		0			0
$417$	−9.56155			−0.468231
$418$		0			0
$419$	12.8769			0.629077			0.314539	−	0.949245i	$-0.398150\pi$
$419$	12.8769			0.629077			0.314539	+	0.949245i	$0.398150\pi$
$420$		0			0
$421$		−	8.87689i		−	0.432633i	−0.976323	−	0.216317i	$-0.930596\pi$
$421$		−	8.87689i		−	0.432633i	0.976323	−	0.216317i	$-0.0694044\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.438447			−0.0212678
$426$		0			0
$427$			23.8078i			1.15214i
$428$		0			0
$429$	5.56155	+	0.876894i	0.268514	+	0.0423369i
$430$		0			0
$431$			32.9848i			1.58882i	0.607379	+	0.794412i	$0.292221\pi$
$431$			32.9848i			1.58882i	−0.607379	+	0.794412i	$0.707779\pi$
$432$		0			0
$433$	34.4924			1.65760			0.828800	−	0.559545i	$-0.189024\pi$
$433$	34.4924			1.65760			0.828800	+	0.559545i	$0.189024\pi$
$434$		0			0
$435$			2.00000i			0.0958927i
$436$		0			0
$437$		−	8.00000i		−	0.382692i
$438$		0			0
$439$	−10.4384			−0.498200			−0.249100	−	0.968478i	$-0.580135\pi$
$439$	−10.4384			−0.498200			−0.249100	+	0.968478i	$0.580135\pi$
$440$		0			0
$441$	−5.68466			−0.270698
$442$		0			0
$443$	6.93087			0.329296			0.164648	−	0.986352i	$-0.447351\pi$
$443$	6.93087			0.329296			0.164648	+	0.986352i	$0.447351\pi$
$444$		0			0
$445$	−0.438447			−0.0207844
$446$		0			0
$447$			20.9309i			0.989996i
$448$		0			0
$449$			26.6847i			1.25933i	0.776868	+	0.629663i	$0.216807\pi$
$449$			26.6847i			1.25933i	−0.776868	+	0.629663i	$0.783193\pi$
$450$		0			0
$451$	−10.4384			−0.491527
$452$		0			0
$453$			14.0000i			0.657777i
$454$		0			0
$455$	2.00000	−	12.6847i	0.0937614	−	0.594666i
$456$		0			0
$457$			22.4384i			1.04963i	0.851217	+	0.524813i	$0.175865\pi$
$457$			22.4384i			1.04963i	−0.851217	+	0.524813i	$0.824135\pi$
$458$		0			0
$459$	0.438447			0.0204650
$460$		0			0
$461$		−	14.6847i		−	0.683933i	−0.939712	−	0.341966i	$-0.888907\pi$
$461$		−	14.6847i		−	0.683933i	0.939712	−	0.341966i	$-0.111093\pi$
$462$		0			0
$463$		−	14.3002i		−	0.664586i	−0.943176	−	0.332293i	$-0.892178\pi$
$463$		−	14.3002i		−	0.664586i	0.943176	−	0.332293i	$-0.107822\pi$
$464$		0			0
$465$	−2.00000			−0.0927478
$466$		0			0
$467$	−14.4384			−0.668132			−0.334066	−	0.942550i	$-0.608421\pi$
$467$	−14.4384			−0.668132			−0.334066	+	0.942550i	$0.608421\pi$
$468$		0			0
$469$	−54.7386			−2.52760
$470$		0			0
$471$	−6.87689			−0.316871
$472$		0			0
$473$		−	1.36932i		−	0.0629613i
$474$		0			0
$475$			5.12311i			0.235064i
$476$		0			0
$477$	−6.68466			−0.306069
$478$		0			0
$479$		−	36.6847i		−	1.67616i	−0.545544	−	0.838082i	$-0.683677\pi$
$479$		−	36.6847i		−	1.67616i	0.545544	−	0.838082i	$-0.316323\pi$
$480$		0			0
$481$	19.8078	+	3.12311i	0.903156	+	0.142401i
$482$		0			0
$483$		−	5.56155i		−	0.253059i
$484$		0			0
$485$	−1.56155			−0.0709065
$486$		0			0
$487$		−	12.0540i		−	0.546218i	−0.961983	−	0.273109i	$-0.911948\pi$
$487$		−	12.0540i		−	0.546218i	0.961983	−	0.273109i	$-0.0880519\pi$
$488$		0			0
$489$		−	20.0540i		−	0.906872i
$490$		0			0
$491$	−10.2462			−0.462405			−0.231203	−	0.972906i	$-0.574266\pi$
$491$	−10.2462			−0.462405			−0.231203	+	0.972906i	$0.574266\pi$
$492$		0			0
$493$	−0.876894			−0.0394933
$494$		0			0
$495$	−1.56155			−0.0701866
$496$		0			0
$497$	34.0540			1.52753
$498$		0			0
$499$			30.1080i			1.34782i	0.738815	+	0.673908i	$0.235386\pi$
$499$			30.1080i			1.34782i	−0.738815	+	0.673908i	$0.764614\pi$
$500$		0			0
$501$			6.24621i			0.279060i
$502$		0			0
$503$	−9.75379			−0.434900			−0.217450	−	0.976071i	$-0.569774\pi$
$503$	−9.75379			−0.434900			−0.217450	+	0.976071i	$0.569774\pi$
$504$		0			0
$505$		−	2.00000i		−	0.0889988i
$506$		0			0
$507$	−12.3693	−	4.00000i	−0.549341	−	0.177646i
$508$		0			0
$509$			32.9309i			1.45964i	0.683642	+	0.729818i	$0.260395\pi$
$509$			32.9309i			1.45964i	−0.683642	+	0.729818i	$0.739605\pi$
$510$		0			0
$511$	−3.12311			−0.138158
$512$		0			0
$513$		−	5.12311i		−	0.226191i
$514$		0			0
$515$		−	14.2462i		−	0.627763i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	14.4924			0.636147
$520$		0			0
$521$	29.6155			1.29748			0.648740	−	0.761010i	$-0.275296\pi$
$521$	29.6155			1.29748			0.648740	+	0.761010i	$0.275296\pi$
$522$		0			0
$523$	−17.8617			−0.781039			−0.390520	−	0.920595i	$-0.627705\pi$
$523$	−17.8617			−0.781039			−0.390520	+	0.920595i	$0.627705\pi$
$524$		0			0
$525$			3.56155i			0.155439i
$526$		0			0
$527$		−	0.876894i		−	0.0381981i
$528$		0			0
$529$	−20.5616			−0.893981
$530$		0			0
$531$			6.24621i			0.271062i
$532$		0			0
$533$	23.8078	+	3.75379i	1.03123	+	0.162595i
$534$		0			0
$535$		−	8.68466i		−	0.375471i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			8.87689i			0.382355i
$540$		0			0
$541$		−	24.4924i		−	1.05301i	−0.850172	−	0.526506i	$-0.823502\pi$
$541$		−	24.4924i		−	1.05301i	0.850172	−	0.526506i	$-0.176498\pi$
$542$		0			0
$543$	−13.8078			−0.592548
$544$		0			0
$545$	−8.87689			−0.380244
$546$		0			0
$547$	−7.12311			−0.304562			−0.152281	−	0.988337i	$-0.548662\pi$
$547$	−7.12311			−0.304562			−0.152281	+	0.988337i	$0.548662\pi$
$548$		0			0
$549$	−6.68466			−0.285294
$550$		0			0
$551$			10.2462i			0.436503i
$552$		0			0
$553$			30.9309i			1.31531i
$554$		0			0
$555$	−5.56155			−0.236075
$556$		0			0
$557$		−	40.7386i		−	1.72615i	−0.505075	−	0.863076i	$-0.668535\pi$
$557$		−	40.7386i		−	1.72615i	0.505075	−	0.863076i	$-0.331465\pi$
$558$		0			0
$559$	−0.492423	+	3.12311i	−0.0208273	+	0.132093i
$560$		0			0
$561$		−	0.684658i		−	0.0289063i
$562$		0			0
$563$	15.3153			0.645465			0.322732	−	0.946490i	$-0.395399\pi$
$563$	15.3153			0.645465			0.322732	+	0.946490i	$0.395399\pi$
$564$		0			0
$565$		−	4.24621i		−	0.178639i
$566$		0			0
$567$		−	3.56155i		−	0.149571i
$568$		0			0
$569$	44.7386			1.87554			0.937771	−	0.347256i	$-0.112886\pi$
$569$	44.7386			1.87554			0.937771	+	0.347256i	$0.112886\pi$
$570$		0			0
$571$	−31.8078			−1.33111			−0.665557	−	0.746347i	$-0.731806\pi$
$571$	−31.8078			−1.33111			−0.665557	+	0.746347i	$0.731806\pi$
$572$		0			0
$573$	−5.75379			−0.240368
$574$		0			0
$575$	−1.56155			−0.0651213
$576$		0			0
$577$			18.4384i			0.767603i	0.923416	+	0.383801i	$0.125385\pi$
$577$			18.4384i			0.767603i	−0.923416	+	0.383801i	$0.874615\pi$
$578$		0			0
$579$			10.4384i			0.433807i
$580$		0			0
$581$	−11.1231			−0.461464
$582$		0			0
$583$			10.4384i			0.432316i
$584$		0			0
$585$	3.56155	+	0.561553i	0.147252	+	0.0232174i
$586$		0			0
$587$		−	22.6307i		−	0.934068i	−0.884239	−	0.467034i	$-0.845323\pi$
$587$		−	22.6307i		−	0.934068i	0.884239	−	0.467034i	$-0.154677\pi$
$588$		0			0
$589$	−10.2462			−0.422188
$590$		0			0
$591$			14.4924i			0.596139i
$592$		0			0
$593$		−	32.7386i		−	1.34441i	−0.740363	−	0.672207i	$-0.765346\pi$
$593$		−	32.7386i		−	1.34441i	0.740363	−	0.672207i	$-0.234654\pi$
$594$		0			0
$595$	−1.56155			−0.0640174
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−12.4924			−0.510427			−0.255213	−	0.966885i	$-0.582146\pi$
$599$	−12.4924			−0.510427			−0.255213	+	0.966885i	$0.582146\pi$
$600$		0			0
$601$	−20.0540			−0.818019			−0.409009	−	0.912530i	$-0.634126\pi$
$601$	−20.0540			−0.818019			−0.409009	+	0.912530i	$0.634126\pi$
$602$		0			0
$603$		−	15.3693i		−	0.625887i
$604$		0			0
$605$		−	8.56155i		−	0.348077i
$606$		0			0
$607$	20.4924			0.831762			0.415881	−	0.909419i	$-0.363473\pi$
$607$	20.4924			0.831762			0.415881	+	0.909419i	$0.363473\pi$
$608$		0			0
$609$			7.12311i			0.288643i
$610$		0			0
$611$		0			0
$612$		0			0
$613$			24.6847i			0.997004i	0.866889	+	0.498502i	$0.166116\pi$
$613$			24.6847i			0.997004i	−0.866889	+	0.498502i	$0.833884\pi$
$614$		0			0
$615$	−6.68466			−0.269551
$616$		0			0
$617$		−	7.75379i		−	0.312156i	−0.987745	−	0.156078i	$-0.950115\pi$
$617$		−	7.75379i		−	0.312156i	0.987745	−	0.156078i	$-0.0498851\pi$
$618$		0			0
$619$			14.4924i			0.582500i	0.956647	+	0.291250i	$0.0940711\pi$
$619$			14.4924i			0.582500i	−0.956647	+	0.291250i	$0.905929\pi$
$620$		0			0
$621$	1.56155			0.0626630
$622$		0			0
$623$	−1.56155			−0.0625623
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−8.00000			−0.319489
$628$		0			0
$629$		−	2.43845i		−	0.0972273i
$630$		0			0
$631$		−	18.0000i		−	0.716569i	−0.933613	−	0.358284i	$-0.883362\pi$
$631$		−	18.0000i		−	0.716569i	0.933613	−	0.358284i	$-0.116638\pi$
$632$		0			0
$633$	24.4924			0.973486
$634$		0			0
$635$		−	19.1231i		−	0.758877i
$636$		0			0
$637$	3.19224	−	20.2462i	0.126481	−	0.802184i
$638$		0			0
$639$			9.56155i			0.378249i
$640$		0			0
$641$	49.1231			1.94025			0.970123	−	0.242614i	$-0.0780047\pi$
$641$	49.1231			1.94025			0.970123	+	0.242614i	$0.0780047\pi$
$642$		0			0
$643$		−	36.9309i		−	1.45641i	−0.685359	−	0.728206i	$-0.740355\pi$
$643$		−	36.9309i		−	1.45641i	0.685359	−	0.728206i	$-0.259645\pi$
$644$		0			0
$645$		−	0.876894i		−	0.0345277i
$646$		0			0
$647$	−2.05398			−0.0807501			−0.0403751	−	0.999185i	$-0.512855\pi$
$647$	−2.05398			−0.0807501			−0.0403751	+	0.999185i	$0.512855\pi$
$648$		0			0
$649$	9.75379			0.382870
$650$		0			0
$651$	−7.12311			−0.279177
$652$		0			0
$653$	34.4924			1.34979			0.674896	−	0.737912i	$-0.264188\pi$
$653$	34.4924			1.34979			0.674896	+	0.737912i	$0.264188\pi$
$654$		0			0
$655$		−	20.4924i		−	0.800705i
$656$		0			0
$657$		−	0.876894i		−	0.0342109i
$658$		0			0
$659$	−27.6155			−1.07575			−0.537874	−	0.843025i	$-0.680772\pi$
$659$	−27.6155			−1.07575			−0.537874	+	0.843025i	$0.680772\pi$
$660$		0			0
$661$			6.24621i			0.242949i	0.992595	+	0.121475i	$0.0387623\pi$
$661$			6.24621i			0.242949i	−0.992595	+	0.121475i	$0.961238\pi$
$662$		0			0
$663$	−0.246211	+	1.56155i	−0.00956205	+	0.0606457i
$664$		0			0
$665$			18.2462i			0.707558i
$666$		0			0
$667$	−3.12311			−0.120927
$668$		0			0
$669$		−	13.1231i		−	0.507369i
$670$		0			0
$671$			10.4384i			0.402972i
$672$		0			0
$673$	30.1080			1.16058			0.580288	−	0.814411i	$-0.302940\pi$
$673$	30.1080			1.16058			0.580288	+	0.814411i	$0.302940\pi$
$674$		0			0
$675$	−1.00000			−0.0384900
$676$		0			0
$677$	−33.4233			−1.28456			−0.642281	−	0.766469i	$-0.722012\pi$
$677$	−33.4233			−1.28456			−0.642281	+	0.766469i	$0.722012\pi$
$678$		0			0
$679$	−5.56155			−0.213433
$680$		0			0
$681$			26.2462i			1.00576i
$682$		0			0
$683$		−	35.6155i		−	1.36279i	−0.731916	−	0.681395i	$-0.761374\pi$
$683$		−	35.6155i		−	1.36279i	0.731916	−	0.681395i	$-0.238626\pi$
$684$		0			0
$685$	6.00000			0.229248
$686$		0			0
$687$		−	2.63068i		−	0.100367i
$688$		0			0
$689$	3.75379	−	23.8078i	0.143008	−	0.907004i
$690$		0			0
$691$		−	36.2462i		−	1.37887i	−0.724347	−	0.689435i	$-0.757859\pi$
$691$		−	36.2462i		−	1.37887i	0.724347	−	0.689435i	$-0.242141\pi$
$692$		0			0
$693$	−5.56155			−0.211266
$694$		0			0
$695$			9.56155i			0.362690i
$696$		0			0
$697$		−	2.93087i		−	0.111015i
$698$		0			0
$699$	5.80776			0.219670
$700$		0			0
$701$	−35.3693			−1.33588			−0.667940	−	0.744215i	$-0.732824\pi$
$701$	−35.3693			−1.33588			−0.667940	+	0.744215i	$0.732824\pi$
$702$		0			0
$703$	−28.4924			−1.07461
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	7.12311i		−	0.267892i
$708$		0			0
$709$			26.2462i			0.985697i	0.870115	+	0.492849i	$0.164044\pi$
$709$			26.2462i			0.985697i	−0.870115	+	0.492849i	$0.835956\pi$
$710$		0			0
$711$	−8.68466			−0.325700
$712$		0			0
$713$		−	3.12311i		−	0.116961i
$714$		0			0
$715$	0.876894	−	5.56155i	0.0327940	−	0.207990i
$716$		0			0
$717$		−	12.6847i		−	0.473717i
$718$		0			0
$719$	10.6307			0.396458			0.198229	−	0.980156i	$-0.436481\pi$
$719$	10.6307			0.396458			0.198229	+	0.980156i	$0.436481\pi$
$720$		0			0
$721$		−	50.7386i		−	1.88961i
$722$		0			0
$723$		−	7.12311i		−	0.264911i
$724$		0			0
$725$	2.00000			0.0742781
$726$		0			0
$727$	37.8617			1.40421			0.702107	−	0.712071i	$-0.252243\pi$
$727$	37.8617			1.40421			0.702107	+	0.712071i	$0.252243\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	0.384472			0.0142202
$732$		0			0
$733$			36.3002i			1.34078i	0.742010	+	0.670389i	$0.233873\pi$
$733$			36.3002i			1.34078i	−0.742010	+	0.670389i	$0.766127\pi$
$734$		0			0
$735$			5.68466i			0.209682i
$736$		0			0
$737$	−24.0000			−0.884051
$738$		0			0
$739$		−	20.2462i		−	0.744769i	−0.928079	−	0.372384i	$-0.878540\pi$
$739$		−	20.2462i		−	0.744769i	0.928079	−	0.372384i	$-0.121460\pi$
$740$		0			0
$741$	18.2462	+	2.87689i	0.670291	+	0.105685i
$742$		0			0
$743$		−	16.4924i		−	0.605048i	−0.953142	−	0.302524i	$-0.902171\pi$
$743$		−	16.4924i		−	0.605048i	0.953142	−	0.302524i	$-0.0978293\pi$
$744$		0			0
$745$	20.9309			0.766848
$746$		0			0
$747$		−	3.12311i		−	0.114268i
$748$		0			0
$749$		−	30.9309i		−	1.13019i
$750$		0			0
$751$	−10.4384			−0.380904			−0.190452	−	0.981696i	$-0.560995\pi$
$751$	−10.4384			−0.380904			−0.190452	+	0.981696i	$0.560995\pi$
$752$		0			0
$753$	13.3693			0.487205
$754$		0			0
$755$	14.0000			0.509512
$756$		0			0
$757$	37.1231			1.34926			0.674631	−	0.738155i	$-0.264303\pi$
$757$	37.1231			1.34926			0.674631	+	0.738155i	$0.264303\pi$
$758$		0			0
$759$		−	2.43845i		−	0.0885100i
$760$		0			0
$761$		−	41.2311i		−	1.49462i	−0.664473	−	0.747312i	$-0.731344\pi$
$761$		−	41.2311i		−	1.49462i	0.664473	−	0.747312i	$-0.268656\pi$
$762$		0			0
$763$	−31.6155			−1.14456
$764$		0			0
$765$		−	0.438447i		−	0.0158521i
$766$		0			0
$767$	−22.2462	−	3.50758i	−0.803264	−	0.126651i
$768$		0			0
$769$			24.0000i			0.865462i	0.901523	+	0.432731i	$0.142450\pi$
$769$			24.0000i			0.865462i	−0.901523	+	0.432731i	$0.857550\pi$
$770$		0			0
$771$	22.4924			0.810045
$772$		0			0
$773$		−	46.4924i		−	1.67222i	−0.548565	−	0.836108i	$-0.684826\pi$
$773$		−	46.4924i		−	1.67222i	0.548565	−	0.836108i	$-0.315174\pi$
$774$		0			0
$775$			2.00000i			0.0718421i
$776$		0			0
$777$	−19.8078			−0.710600
$778$		0			0
$779$	−34.2462			−1.22700
$780$		0			0
$781$	14.9309			0.534269
$782$		0			0
$783$	−2.00000			−0.0714742
$784$		0			0
$785$			6.87689i			0.245447i
$786$		0			0
$787$			11.3693i			0.405272i	0.979254	+	0.202636i	$0.0649509\pi$
$787$			11.3693i			0.405272i	−0.979254	+	0.202636i	$0.935049\pi$
$788$		0			0
$789$	16.0000			0.569615
$790$		0			0
$791$		−	15.1231i		−	0.537716i
$792$		0			0
$793$	3.75379	−	23.8078i	0.133301	−	0.845438i
$794$		0			0
$795$			6.68466i			0.237080i
$796$		0			0
$797$	−13.4233			−0.475477			−0.237739	−	0.971329i	$-0.576406\pi$
$797$	−13.4233			−0.475477			−0.237739	+	0.971329i	$0.576406\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		−	0.438447i		−	0.0154918i
$802$		0			0
$803$	−1.36932			−0.0483221
$804$		0			0
$805$	−5.56155			−0.196019
$806$		0			0
$807$	−29.1231			−1.02518
$808$		0			0
$809$	26.8769			0.944941			0.472471	−	0.881346i	$-0.343362\pi$
$809$	26.8769			0.944941			0.472471	+	0.881346i	$0.343362\pi$
$810$		0			0
$811$		−	10.0000i		−	0.351147i	−0.984466	−	0.175574i	$-0.943822\pi$
$811$		−	10.0000i		−	0.351147i	0.984466	−	0.175574i	$-0.0561780\pi$
$812$		0			0
$813$			3.75379i			0.131651i
$814$		0			0
$815$	−20.0540			−0.702460
$816$		0			0
$817$		−	4.49242i		−	0.157170i
$818$		0			0
$819$	12.6847	+	2.00000i	0.443238	+	0.0698857i
$820$		0			0
$821$		−	8.93087i		−	0.311690i	−0.987782	−	0.155845i	$-0.950190\pi$
$821$		−	8.93087i		−	0.311690i	0.987782	−	0.155845i	$-0.0498100\pi$
$822$		0			0
$823$	33.3693			1.16318			0.581591	−	0.813482i	$-0.302431\pi$
$823$	33.3693			1.16318			0.581591	+	0.813482i	$0.302431\pi$
$824$		0			0
$825$			1.56155i			0.0543663i
$826$		0			0
$827$			5.75379i			0.200079i	0.994983	+	0.100039i	$0.0318969\pi$
$827$			5.75379i			0.200079i	−0.994983	+	0.100039i	$0.968103\pi$
$828$		0			0
$829$	−45.2311			−1.57094			−0.785470	−	0.618900i	$-0.787579\pi$
$829$	−45.2311			−1.57094			−0.785470	+	0.618900i	$0.787579\pi$
$830$		0			0
$831$	−3.36932			−0.116880
$832$		0			0
$833$	−2.49242			−0.0863573
$834$		0			0
$835$	6.24621			0.216159
$836$		0			0
$837$		−	2.00000i		−	0.0691301i
$838$		0			0
$839$		−	42.5464i		−	1.46886i	−0.678682	−	0.734432i	$-0.737448\pi$
$839$		−	42.5464i		−	1.46886i	0.678682	−	0.734432i	$-0.262552\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$		−	2.00000i		−	0.0688837i
$844$		0			0
$845$	−4.00000	+	12.3693i	−0.137604	+	0.425517i
$846$		0			0
$847$		−	30.4924i		−	1.04773i
$848$		0			0
$849$	15.1231			0.519024
$850$		0			0
$851$		−	8.68466i		−	0.297706i
$852$		0			0
$853$			14.5464i			0.498059i	0.968496	+	0.249030i	$0.0801116\pi$
$853$			14.5464i			0.498059i	−0.968496	+	0.249030i	$0.919888\pi$
$854$		0			0
$855$	−5.12311			−0.175207
$856$		0			0
$857$	−12.9309			−0.441710			−0.220855	−	0.975307i	$-0.570885\pi$
$857$	−12.9309			−0.441710			−0.220855	+	0.975307i	$0.570885\pi$
$858$		0			0
$859$	−49.5616			−1.69102			−0.845509	−	0.533961i	$-0.820703\pi$
$859$	−49.5616			−1.69102			−0.845509	+	0.533961i	$0.820703\pi$
$860$		0			0
$861$	−23.8078			−0.811366
$862$		0			0
$863$			1.36932i			0.0466121i	0.999728	+	0.0233060i	$0.00741922\pi$
$863$			1.36932i			0.0466121i	−0.999728	+	0.0233060i	$0.992581\pi$
$864$		0			0
$865$		−	14.4924i		−	0.492757i
$866$		0			0
$867$	−16.8078			−0.570822
$868$		0			0
$869$			13.5616i			0.460044i
$870$		0			0
$871$	54.7386	+	8.63068i	1.85475	+	0.292440i
$872$		0			0
$873$		−	1.56155i		−	0.0528506i
$874$		0			0
$875$	3.56155			0.120402
$876$		0			0
$877$			12.8769i			0.434822i	0.976080	+	0.217411i	$0.0697612\pi$
$877$			12.8769i			0.434822i	−0.976080	+	0.217411i	$0.930239\pi$
$878$		0			0
$879$			22.4924i			0.758651i
$880$		0			0
$881$	−26.1080			−0.879599			−0.439800	−	0.898096i	$-0.644951\pi$
$881$	−26.1080			−0.879599			−0.439800	+	0.898096i	$0.644951\pi$
$882$		0			0
$883$	−35.2311			−1.18562			−0.592810	−	0.805343i	$-0.701981\pi$
$883$	−35.2311			−1.18562			−0.592810	+	0.805343i	$0.701981\pi$
$884$		0			0
$885$	6.24621			0.209964
$886$		0			0
$887$	−1.17708			−0.0395225			−0.0197613	−	0.999805i	$-0.506291\pi$
$887$	−1.17708			−0.0395225			−0.0197613	+	0.999805i	$0.506291\pi$
$888$		0			0
$889$		−	68.1080i		−	2.28427i
$890$		0			0
$891$		−	1.56155i		−	0.0523140i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−0.876894	+	5.56155i	−0.0292787	+	0.185695i
$898$		0			0
$899$			4.00000i			0.133407i
$900$		0			0
$901$	−2.93087			−0.0976414
$902$		0			0
$903$		−	3.12311i		−	0.103930i
$904$		0			0
$905$			13.8078i			0.458986i
$906$		0			0
$907$	30.7386			1.02066			0.510330	−	0.859979i	$-0.329523\pi$
$907$	30.7386			1.02066			0.510330	+	0.859979i	$0.329523\pi$
$908$		0			0
$909$	2.00000			0.0663358
$910$		0			0
$911$	−45.3693			−1.50315			−0.751576	−	0.659646i	$-0.770706\pi$
$911$	−45.3693			−1.50315			−0.751576	+	0.659646i	$0.770706\pi$
$912$		0			0
$913$	−4.87689			−0.161402
$914$		0			0
$915$			6.68466i			0.220988i
$916$		0			0
$917$		−	72.9848i		−	2.41017i
$918$		0			0
$919$	26.0540			0.859441			0.429721	−	0.902962i	$-0.358612\pi$
$919$	26.0540			0.859441			0.429721	+	0.902962i	$0.358612\pi$
$920$		0			0
$921$			4.43845i			0.146252i
$922$		0			0
$923$	−34.0540	−	5.36932i	−1.12090	−	0.176733i
$924$		0			0
$925$			5.56155i			0.182863i
$926$		0			0
$927$	14.2462			0.467907
$928$		0			0
$929$		−	52.5464i		−	1.72399i	−0.506916	−	0.861996i	$-0.669214\pi$
$929$		−	52.5464i		−	1.72399i	0.506916	−	0.861996i	$-0.330786\pi$
$930$		0			0
$931$			29.1231i			0.954471i
$932$		0			0
$933$	−8.87689			−0.290616
$934$		0			0
$935$	−0.684658			−0.0223907
$936$		0			0
$937$	−5.50758			−0.179925			−0.0899624	−	0.995945i	$-0.528675\pi$
$937$	−5.50758			−0.179925			−0.0899624	+	0.995945i	$0.528675\pi$
$938$		0			0
$939$	7.36932			0.240489
$940$		0			0
$941$		−	5.80776i		−	0.189328i	−0.995509	−	0.0946638i	$-0.969822\pi$
$941$		−	5.80776i		−	0.189328i	0.995509	−	0.0946638i	$-0.0301776\pi$
$942$		0			0
$943$		−	10.4384i		−	0.339923i
$944$		0			0
$945$	−3.56155			−0.115857
$946$		0			0
$947$			20.4924i			0.665914i	0.942942	+	0.332957i	$0.108046\pi$
$947$			20.4924i			0.665914i	−0.942942	+	0.332957i	$0.891954\pi$
$948$		0			0
$949$	3.12311	+	0.492423i	0.101380	+	0.0159847i
$950$		0			0
$951$			10.0000i			0.324272i
$952$		0			0
$953$	−42.7926			−1.38619			−0.693094	−	0.720847i	$-0.743753\pi$
$953$	−42.7926			−1.38619			−0.693094	+	0.720847i	$0.743753\pi$
$954$		0			0
$955$			5.75379i			0.186188i
$956$		0			0
$957$			3.12311i			0.100956i
$958$		0			0
$959$	21.3693			0.690051
$960$		0			0
$961$	27.0000			0.870968
$962$		0			0
$963$	8.68466			0.279859
$964$		0			0
$965$	10.4384			0.336026
$966$		0			0
$967$			42.8769i			1.37883i	0.724368	+	0.689414i	$0.242132\pi$
$967$			42.8769i			1.37883i	−0.724368	+	0.689414i	$0.757868\pi$
$968$		0			0
$969$		−	2.24621i		−	0.0721587i
$970$		0			0
$971$	−3.61553			−0.116028			−0.0580139	−	0.998316i	$-0.518477\pi$
$971$	−3.61553			−0.116028			−0.0580139	+	0.998316i	$0.518477\pi$
$972$		0			0
$973$			34.0540i			1.09172i
$974$		0			0
$975$	0.561553	−	3.56155i	0.0179841	−	0.114061i
$976$		0			0
$977$		−	19.3693i		−	0.619679i	−0.950789	−	0.309840i	$-0.899725\pi$
$977$		−	19.3693i		−	0.619679i	0.950789	−	0.309840i	$-0.100275\pi$
$978$		0			0
$979$	−0.684658			−0.0218818
$980$		0			0
$981$		−	8.87689i		−	0.283417i
$982$		0			0
$983$			20.0000i			0.637901i	0.947771	+	0.318950i	$0.103330\pi$
$983$			20.0000i			0.637901i	−0.947771	+	0.318950i	$0.896670\pi$
$984$		0			0
$985$	14.4924			0.461767
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.36932			0.0435417
$990$		0			0
$991$	−16.3002			−0.517792			−0.258896	−	0.965905i	$-0.583359\pi$
$991$	−16.3002			−0.517792			−0.258896	+	0.965905i	$0.583359\pi$
$992$		0			0
$993$			19.3693i			0.614667i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−5.12311			−0.162250			−0.0811252	−	0.996704i	$-0.525851\pi$
$997$	−5.12311			−0.162250			−0.0811252	+	0.996704i	$0.525851\pi$
$998$		0			0
$999$		−	5.56155i		−	0.175960i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1560.2.g.h.961.1	✓	4
3.2	odd	2		4680.2.g.g.2521.3		4
4.3	odd	2		3120.2.g.p.961.2		4
13.12	even	2	inner	1560.2.g.h.961.4	yes	4
39.38	odd	2		4680.2.g.g.2521.2		4
52.51	odd	2		3120.2.g.p.961.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.g.h.961.1	✓	4	1.1	even	1	trivial
1560.2.g.h.961.4	yes	4	13.12	even	2	inner
3120.2.g.p.961.2		4	4.3	odd	2
3120.2.g.p.961.3		4	52.51	odd	2
4680.2.g.g.2521.2		4	39.38	odd	2
4680.2.g.g.2521.3		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1560.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000i q^{5} -3.56155i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000i q^{5} -3.56155i q^{7} +1.00000 q^{9} -1.56155i q^{11} +(-0.561553 + 3.56155i) q^{13} -1.00000i q^{15} +0.438447 q^{17} -5.12311i q^{19} -3.56155i q^{21} +1.56155 q^{23} -1.00000 q^{25} +1.00000 q^{27} -2.00000 q^{29} -2.00000i q^{31} -1.56155i q^{33} -3.56155 q^{35} -5.56155i q^{37} +(-0.561553 + 3.56155i) q^{39} -6.68466i q^{41} +0.876894 q^{43} -1.00000i q^{45} -5.68466 q^{49} +0.438447 q^{51} -6.68466 q^{53} -1.56155 q^{55} -5.12311i q^{57} +6.24621i q^{59} -6.68466 q^{61} -3.56155i q^{63} +(3.56155 + 0.561553i) q^{65} -15.3693i q^{67} +1.56155 q^{69} +9.56155i q^{71} -0.876894i q^{73} -1.00000 q^{75} -5.56155 q^{77} -8.68466 q^{79} +1.00000 q^{81} -3.12311i q^{83} -0.438447i q^{85} -2.00000 q^{87} -0.438447i q^{89} +(12.6847 + 2.00000i) q^{91} -2.00000i q^{93} -5.12311 q^{95} -1.56155i q^{97} -1.56155i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^9	\( 4 q + 4 q^{3} + 4 q^{9} + 6 q^{13} + 10 q^{17} - 2 q^{23} - 4 q^{25} + 4 q^{27} - 8 q^{29} - 6 q^{35} + 6 q^{39} + 20 q^{43} + 2 q^{49} + 10 q^{51} - 2 q^{53} + 2 q^{55} - 2 q^{61} + 6 q^{65} - 2 q^{69} - 4 q^{75} - 14 q^{77} - 10 q^{79} + 4 q^{81} - 8 q^{87} + 26 q^{91} - 4 q^{95}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^9 + 6 * q^13 + 10 * q^17 - 2 * q^23 - 4 * q^25 + 4 * q^27 - 8 * q^29 - 6 * q^35 + 6 * q^39 + 20 * q^43 + 2 * q^49 + 10 * q^51 - 2 * q^53 + 2 * q^55 - 2 * q^61 + 6 * q^65 - 2 * q^69 - 4 * q^75 - 14 * q^77 - 10 * q^79 + 4 * q^81 - 8 * q^87 + 26 * q^91 - 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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