LMFDB - Embedded newform 1560.2.g.h.961.2

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			961.2
Root			$1.56155i$ of defining polynomial
Character	$\chi$	$=$	1560.961
Dual form			1560.2.g.h.961.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000i q^{5} +0.561553i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000i q^{5} +0.561553i q^{7} +1.00000 q^{9} +2.56155i q^{11} +(3.56155 - 0.561553i) q^{13} -1.00000i q^{15} +4.56155 q^{17} +3.12311i q^{19} +0.561553i q^{21} -2.56155 q^{23} -1.00000 q^{25} +1.00000 q^{27} -2.00000 q^{29} -2.00000i q^{31} +2.56155i q^{33} +0.561553 q^{35} -1.43845i q^{37} +(3.56155 - 0.561553i) q^{39} +5.68466i q^{41} +9.12311 q^{43} -1.00000i q^{45} +6.68466 q^{49} +4.56155 q^{51} +5.68466 q^{53} +2.56155 q^{55} +3.12311i q^{57} -10.2462i q^{59} +5.68466 q^{61} +0.561553i q^{63} +(-0.561553 - 3.56155i) q^{65} +9.36932i q^{67} -2.56155 q^{69} +5.43845i q^{71} -9.12311i q^{73} -1.00000 q^{75} -1.43845 q^{77} +3.68466 q^{79} +1.00000 q^{81} +5.12311i q^{83} -4.56155i q^{85} -2.00000 q^{87} -4.56155i q^{89} +(0.315342 + 2.00000i) q^{91} -2.00000i q^{93} +3.12311 q^{95} +2.56155i q^{97} +2.56155i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$			0.561553i			0.212247i	0.994353	+	0.106124i	$0.0338439\pi$
$7$			0.561553i			0.212247i	−0.994353	+	0.106124i	$0.966156\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			2.56155i			0.772337i	0.922428	+	0.386169i	$0.126202\pi$
$11$			2.56155i			0.772337i	−0.922428	+	0.386169i	$0.873798\pi$
$12$		0			0
$13$	3.56155	−	0.561553i	0.987797	−	0.155747i
$13$	3.56155	−	0.561553i	0.987797	−	0.155747i
$14$		0			0
$15$		−	1.00000i		−	0.258199i
$16$		0			0
$17$	4.56155			1.10634			0.553170	−	0.833069i	$-0.313418\pi$
$17$	4.56155			1.10634			0.553170	+	0.833069i	$0.313418\pi$
$18$		0			0
$19$			3.12311i			0.716490i	0.933628	+	0.358245i	$0.116625\pi$
$19$			3.12311i			0.716490i	−0.933628	+	0.358245i	$0.883375\pi$
$20$		0			0
$21$			0.561553i			0.122541i
$22$		0			0
$23$	−2.56155			−0.534121			−0.267060	−	0.963680i	$-0.586052\pi$
$23$	−2.56155			−0.534121			−0.267060	+	0.963680i	$0.586052\pi$
$24$		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
$33$			2.56155i			0.445909i
$34$		0			0
$35$	0.561553			0.0949197
$36$		0			0
$37$		−	1.43845i		−	0.236479i	−0.992985	−	0.118240i	$-0.962275\pi$
$37$		−	1.43845i		−	0.236479i	0.992985	−	0.118240i	$-0.0377251\pi$
$38$		0			0
$39$	3.56155	−	0.561553i	0.570305	−	0.0899204i
$40$		0			0
$41$			5.68466i			0.887794i	0.896078	+	0.443897i	$0.146404\pi$
$41$			5.68466i			0.887794i	−0.896078	+	0.443897i	$0.853596\pi$
$42$		0			0
$43$	9.12311			1.39126			0.695630	−	0.718400i	$-0.255125\pi$
$43$	9.12311			1.39126			0.695630	+	0.718400i	$0.255125\pi$
$44$		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
$49$	6.68466			0.954951
$50$		0			0
$51$	4.56155			0.638745
$52$		0			0
$53$	5.68466			0.780848			0.390424	−	0.920635i	$-0.372328\pi$
$53$	5.68466			0.780848			0.390424	+	0.920635i	$0.372328\pi$
$54$		0			0
$55$	2.56155			0.345400
$56$		0			0
$57$			3.12311i			0.413665i
$58$		0			0
$59$		−	10.2462i		−	1.33394i	−0.745083	−	0.666972i	$-0.767590\pi$
$59$		−	10.2462i		−	1.33394i	0.745083	−	0.666972i	$-0.232410\pi$
$60$		0			0
$61$	5.68466			0.727846			0.363923	−	0.931429i	$-0.381437\pi$
$61$	5.68466			0.727846			0.363923	+	0.931429i	$0.381437\pi$
$62$		0			0
$63$			0.561553i			0.0707490i
$64$		0			0
$65$	−0.561553	−	3.56155i	−0.0696521	−	0.441756i
$66$		0			0
$67$			9.36932i			1.14464i	0.820029	+	0.572322i	$0.193957\pi$
$67$			9.36932i			1.14464i	−0.820029	+	0.572322i	$0.806043\pi$
$68$		0			0
$69$	−2.56155			−0.308375
$70$		0			0
$71$			5.43845i			0.645425i	0.946497	+	0.322712i	$0.104595\pi$
$71$			5.43845i			0.645425i	−0.946497	+	0.322712i	$0.895405\pi$
$72$		0			0
$73$		−	9.12311i		−	1.06778i	−0.845554	−	0.533889i	$-0.820730\pi$
$73$		−	9.12311i		−	1.06778i	0.845554	−	0.533889i	$-0.179270\pi$
$74$		0			0
$75$	−1.00000			−0.115470
$76$		0			0
$77$	−1.43845			−0.163926
$78$		0			0
$79$	3.68466			0.414556			0.207278	−	0.978282i	$-0.433539\pi$
$79$	3.68466			0.414556			0.207278	+	0.978282i	$0.433539\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			5.12311i			0.562334i	0.959659	+	0.281167i	$0.0907215\pi$
$83$			5.12311i			0.562334i	−0.959659	+	0.281167i	$0.909279\pi$
$84$		0			0
$85$		−	4.56155i		−	0.494770i
$86$		0			0
$87$	−2.00000			−0.214423
$88$		0			0
$89$		−	4.56155i		−	0.483524i	−0.970336	−	0.241762i	$-0.922275\pi$
$89$		−	4.56155i		−	0.483524i	0.970336	−	0.241762i	$-0.0777253\pi$
$90$		0			0
$91$	0.315342	+	2.00000i	0.0330568	+	0.209657i
$92$		0			0
$93$		−	2.00000i		−	0.207390i
$94$		0			0
$95$	3.12311			0.320424
$96$		0			0
$97$			2.56155i			0.260086i	0.991508	+	0.130043i	$0.0415116\pi$
$97$			2.56155i			0.260086i	−0.991508	+	0.130043i	$0.958488\pi$
$98$		0			0
$99$			2.56155i			0.257446i
$100$		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
$103$	−2.24621			−0.221326			−0.110663	−	0.993858i	$-0.535297\pi$
$103$	−2.24621			−0.221326			−0.110663	+	0.993858i	$0.535297\pi$
$104$		0			0
$105$	0.561553			0.0548019
$106$		0			0
$107$	−3.68466			−0.356209			−0.178105	−	0.984012i	$-0.556997\pi$
$107$	−3.68466			−0.356209			−0.178105	+	0.984012i	$0.556997\pi$
$108$		0			0
$109$		−	17.1231i		−	1.64010i	−0.572295	−	0.820048i	$-0.693947\pi$
$109$		−	17.1231i		−	1.64010i	0.572295	−	0.820048i	$-0.306053\pi$
$110$		0			0
$111$		−	1.43845i		−	0.136531i
$112$		0			0
$113$	−12.2462			−1.15203			−0.576013	−	0.817440i	$-0.695392\pi$
$113$	−12.2462			−1.15203			−0.576013	+	0.817440i	$0.695392\pi$
$114$		0			0
$115$			2.56155i			0.238866i
$116$		0			0
$117$	3.56155	−	0.561553i	0.329266	−	0.0519156i
$118$		0			0
$119$			2.56155i			0.234817i
$120$		0			0
$121$	4.43845			0.403495
$122$		0			0
$123$			5.68466i			0.512568i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$	10.8769			0.965168			0.482584	−	0.875850i	$-0.339698\pi$
$127$	10.8769			0.965168			0.482584	+	0.875850i	$0.339698\pi$
$128$		0			0
$129$	9.12311			0.803245
$130$		0			0
$131$	−12.4924			−1.09147			−0.545734	−	0.837958i	$-0.683749\pi$
$131$	−12.4924			−1.09147			−0.545734	+	0.837958i	$0.683749\pi$
$132$		0			0
$133$	−1.75379			−0.152073
$134$		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
$139$	−5.43845			−0.461283			−0.230642	−	0.973039i	$-0.574082\pi$
$139$	−5.43845			−0.461283			−0.230642	+	0.973039i	$0.574082\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.43845	+	9.12311i	0.120289	+	0.762912i
$144$		0			0
$145$			2.00000i			0.166091i
$146$		0			0
$147$	6.68466			0.551341
$148$		0			0
$149$		−	7.93087i		−	0.649722i	−0.945762	−	0.324861i	$-0.894682\pi$
$149$		−	7.93087i		−	0.649722i	0.945762	−	0.324861i	$-0.105318\pi$
$150$		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
$153$	4.56155			0.368780
$154$		0			0
$155$	−2.00000			−0.160644
$156$		0			0
$157$	−15.1231			−1.20696			−0.603478	−	0.797380i	$-0.706219\pi$
$157$	−15.1231			−1.20696			−0.603478	+	0.797380i	$0.706219\pi$
$158$		0			0
$159$	5.68466			0.450823
$160$		0			0
$161$		−	1.43845i		−	0.113366i
$162$		0			0
$163$			17.0540i			1.33577i	0.744264	+	0.667885i	$0.232800\pi$
$163$			17.0540i			1.33577i	−0.744264	+	0.667885i	$0.767200\pi$
$164$		0			0
$165$	2.56155			0.199417
$166$		0			0
$167$		−	10.2462i		−	0.792876i	−0.918062	−	0.396438i	$-0.870246\pi$
$167$		−	10.2462i		−	0.792876i	0.918062	−	0.396438i	$-0.129754\pi$
$168$		0			0
$169$	12.3693	−	4.00000i	0.951486	−	0.307692i
$170$		0			0
$171$			3.12311i			0.238830i
$172$		0			0
$173$	−18.4924			−1.40595			−0.702976	−	0.711213i	$-0.748146\pi$
$173$	−18.4924			−1.40595			−0.702976	+	0.711213i	$0.748146\pi$
$174$		0			0
$175$		−	0.561553i		−	0.0424494i
$176$		0			0
$177$		−	10.2462i		−	0.770152i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	6.80776			0.506017			0.253009	−	0.967464i	$-0.418580\pi$
$181$	6.80776			0.506017			0.253009	+	0.967464i	$0.418580\pi$
$182$		0			0
$183$	5.68466			0.420222
$184$		0			0
$185$	−1.43845			−0.105757
$186$		0			0
$187$			11.6847i			0.854467i
$188$		0			0
$189$			0.561553i			0.0408470i
$190$		0			0
$191$	−22.2462			−1.60968			−0.804840	−	0.593492i	$-0.797749\pi$
$191$	−22.2462			−1.60968			−0.804840	+	0.593492i	$0.797749\pi$
$192$		0			0
$193$			14.5616i			1.04816i	0.851668	+	0.524082i	$0.175591\pi$
$193$			14.5616i			1.04816i	−0.851668	+	0.524082i	$0.824409\pi$
$194$		0			0
$195$	−0.561553	−	3.56155i	−0.0402136	−	0.255048i
$196$		0			0
$197$		−	18.4924i		−	1.31753i	−0.752349	−	0.658765i	$-0.771079\pi$
$197$		−	18.4924i		−	1.31753i	0.752349	−	0.658765i	$-0.228921\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$			9.36932i			0.660861i
$202$		0			0
$203$		−	1.12311i		−	0.0788266i
$204$		0			0
$205$	5.68466			0.397034
$206$		0			0
$207$	−2.56155			−0.178040
$208$		0			0
$209$	−8.00000			−0.553372
$210$		0			0
$211$	−8.49242			−0.584642			−0.292321	−	0.956320i	$-0.594428\pi$
$211$	−8.49242			−0.584642			−0.292321	+	0.956320i	$0.594428\pi$
$212$		0			0
$213$			5.43845i			0.372636i
$214$		0			0
$215$		−	9.12311i		−	0.622191i
$216$		0			0
$217$	1.12311			0.0762414
$218$		0			0
$219$		−	9.12311i		−	0.616482i
$220$		0			0
$221$	16.2462	−	2.56155i	1.09284	−	0.172309i
$222$		0			0
$223$		−	4.87689i		−	0.326581i	−0.986578	−	0.163291i	$-0.947789\pi$
$223$		−	4.87689i		−	0.326581i	0.986578	−	0.163291i	$-0.0522108\pi$
$224$		0			0
$225$	−1.00000			−0.0666667
$226$		0			0
$227$			9.75379i			0.647382i	0.946163	+	0.323691i	$0.104924\pi$
$227$			9.75379i			0.647382i	−0.946163	+	0.323691i	$0.895076\pi$
$228$		0			0
$229$		−	27.3693i		−	1.80862i	−0.426881	−	0.904308i	$-0.640388\pi$
$229$		−	27.3693i		−	1.80862i	0.426881	−	0.904308i	$-0.359612\pi$
$230$		0			0
$231$	−1.43845			−0.0946429
$232$		0			0
$233$	−14.8078			−0.970089			−0.485044	−	0.874490i	$-0.661197\pi$
$233$	−14.8078			−0.970089			−0.485044	+	0.874490i	$0.661197\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	3.68466			0.239344
$238$		0			0
$239$		−	0.315342i		−	0.0203977i	−0.999948	−	0.0101989i	$-0.996754\pi$
$239$		−	0.315342i		−	0.0203977i	0.999948	−	0.0101989i	$-0.00324646\pi$
$240$		0			0
$241$			1.12311i			0.0723456i	0.999346	+	0.0361728i	$0.0115167\pi$
$241$			1.12311i			0.0723456i	−0.999346	+	0.0361728i	$0.988483\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$		−	6.68466i		−	0.427067i
$246$		0			0
$247$	1.75379	+	11.1231i	0.111591	+	0.707746i
$248$		0			0
$249$			5.12311i			0.324664i
$250$		0			0
$251$	−11.3693			−0.717625			−0.358812	−	0.933410i	$-0.616818\pi$
$251$	−11.3693			−0.717625			−0.358812	+	0.933410i	$0.616818\pi$
$252$		0			0
$253$		−	6.56155i		−	0.412521i
$254$		0			0
$255$		−	4.56155i		−	0.285656i
$256$		0			0
$257$	−10.4924			−0.654499			−0.327250	−	0.944938i	$-0.606122\pi$
$257$	−10.4924			−0.654499			−0.327250	+	0.944938i	$0.606122\pi$
$258$		0			0
$259$	0.807764			0.0501920
$260$		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
$265$		−	5.68466i		−	0.349206i
$266$		0			0
$267$		−	4.56155i		−	0.279162i
$268$		0			0
$269$	−20.8769			−1.27289			−0.636443	−	0.771323i	$-0.719595\pi$
$269$	−20.8769			−1.27289			−0.636443	+	0.771323i	$0.719595\pi$
$270$		0			0
$271$			20.2462i			1.22987i	0.788578	+	0.614935i	$0.210818\pi$
$271$			20.2462i			1.22987i	−0.788578	+	0.614935i	$0.789182\pi$
$272$		0			0
$273$	0.315342	+	2.00000i	0.0190853	+	0.121046i
$274$		0			0
$275$		−	2.56155i		−	0.154467i
$276$		0			0
$277$	21.3693			1.28396			0.641979	−	0.766722i	$-0.278114\pi$
$277$	21.3693			1.28396			0.641979	+	0.766722i	$0.278114\pi$
$278$		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
$283$	6.87689			0.408789			0.204394	−	0.978889i	$-0.434477\pi$
$283$	6.87689			0.408789			0.204394	+	0.978889i	$0.434477\pi$
$284$		0			0
$285$	3.12311			0.184997
$286$		0			0
$287$	−3.19224			−0.188432
$288$		0			0
$289$	3.80776			0.223986
$290$		0			0
$291$			2.56155i			0.150161i
$292$		0			0
$293$		−	10.4924i		−	0.612974i	−0.951875	−	0.306487i	$-0.900846\pi$
$293$		−	10.4924i		−	0.612974i	0.951875	−	0.306487i	$-0.0991536\pi$
$294$		0			0
$295$	−10.2462			−0.596557
$296$		0			0
$297$			2.56155i			0.148636i
$298$		0			0
$299$	−9.12311	+	1.43845i	−0.527603	+	0.0831875i
$300$		0			0
$301$			5.12311i			0.295291i
$302$		0			0
$303$	2.00000			0.114897
$304$		0			0
$305$		−	5.68466i		−	0.325503i
$306$		0			0
$307$			8.56155i			0.488634i	0.969695	+	0.244317i	$0.0785637\pi$
$307$			8.56155i			0.488634i	−0.969695	+	0.244317i	$0.921436\pi$
$308$		0			0
$309$	−2.24621			−0.127782
$310$		0			0
$311$	−17.1231			−0.970962			−0.485481	−	0.874247i	$-0.661356\pi$
$311$	−17.1231			−0.970962			−0.485481	+	0.874247i	$0.661356\pi$
$312$		0			0
$313$	−17.3693			−0.981772			−0.490886	−	0.871224i	$-0.663327\pi$
$313$	−17.3693			−0.981772			−0.490886	+	0.871224i	$0.663327\pi$
$314$		0			0
$315$	0.561553			0.0316399
$316$		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
$319$		−	5.12311i		−	0.286839i
$320$		0			0
$321$	−3.68466			−0.205658
$322$		0			0
$323$			14.2462i			0.792680i
$324$		0			0
$325$	−3.56155	+	0.561553i	−0.197559	+	0.0311493i
$326$		0			0
$327$		−	17.1231i		−	0.946910i
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	5.36932i		−	0.295124i	−0.989053	−	0.147562i	$-0.952857\pi$
$331$		−	5.36932i		−	0.295124i	0.989053	−	0.147562i	$-0.0471427\pi$
$332$		0			0
$333$		−	1.43845i		−	0.0788264i
$334$		0			0
$335$	9.36932			0.511900
$336$		0			0
$337$	−6.49242			−0.353665			−0.176832	−	0.984241i	$-0.556585\pi$
$337$	−6.49242			−0.353665			−0.176832	+	0.984241i	$0.556585\pi$
$338$		0			0
$339$	−12.2462			−0.665123
$340$		0			0
$341$	5.12311			0.277432
$342$		0			0
$343$			7.68466i			0.414933i
$344$		0			0
$345$			2.56155i			0.137909i
$346$		0			0
$347$	−21.4384			−1.15088			−0.575438	−	0.817845i	$-0.695168\pi$
$347$	−21.4384			−1.15088			−0.575438	+	0.817845i	$0.695168\pi$
$348$		0			0
$349$		−	25.6155i		−	1.37117i	−0.727994	−	0.685584i	$-0.759547\pi$
$349$		−	25.6155i		−	1.37117i	0.727994	−	0.685584i	$-0.240453\pi$
$350$		0			0
$351$	3.56155	−	0.561553i	0.190102	−	0.0299735i
$352$		0			0
$353$			31.1231i			1.65652i	0.560347	+	0.828258i	$0.310668\pi$
$353$			31.1231i			1.65652i	−0.560347	+	0.828258i	$0.689332\pi$
$354$		0			0
$355$	5.43845			0.288643
$356$		0			0
$357$			2.56155i			0.135572i
$358$		0			0
$359$		−	20.4924i		−	1.08155i	−0.841168	−	0.540774i	$-0.818131\pi$
$359$		−	20.4924i		−	1.08155i	0.841168	−	0.540774i	$-0.181869\pi$
$360$		0			0
$361$	9.24621			0.486643
$362$		0			0
$363$	4.43845			0.232958
$364$		0			0
$365$	−9.12311			−0.477525
$366$		0			0
$367$	−27.8617			−1.45437			−0.727185	−	0.686441i	$-0.759172\pi$
$367$	−27.8617			−1.45437			−0.727185	+	0.686441i	$0.759172\pi$
$368$		0			0
$369$			5.68466i			0.295931i
$370$		0			0
$371$			3.19224i			0.165733i
$372$		0			0
$373$	−19.1231			−0.990157			−0.495078	−	0.868848i	$-0.664861\pi$
$373$	−19.1231			−0.990157			−0.495078	+	0.868848i	$0.664861\pi$
$374$		0			0
$375$			1.00000i			0.0516398i
$376$		0			0
$377$	−7.12311	+	1.12311i	−0.366859	+	0.0578429i
$378$		0			0
$379$		−	22.4924i		−	1.15536i	−0.816264	−	0.577679i	$-0.803959\pi$
$379$		−	22.4924i		−	1.15536i	0.816264	−	0.577679i	$-0.196041\pi$
$380$		0			0
$381$	10.8769			0.557240
$382$		0			0
$383$		−	5.12311i		−	0.261778i	−0.991397	−	0.130889i	$-0.958217\pi$
$383$		−	5.12311i		−	0.261778i	0.991397	−	0.130889i	$-0.0417832\pi$
$384$		0			0
$385$			1.43845i			0.0733101i
$386$		0			0
$387$	9.12311			0.463754
$388$		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
$391$	−11.6847			−0.590919
$392$		0			0
$393$	−12.4924			−0.630159
$394$		0			0
$395$		−	3.68466i		−	0.185395i
$396$		0			0
$397$		−	7.19224i		−	0.360968i	−0.983578	−	0.180484i	$-0.942234\pi$
$397$		−	7.19224i		−	0.360968i	0.983578	−	0.180484i	$-0.0577664\pi$
$398$		0			0
$399$	−1.75379			−0.0877993
$400$		0			0
$401$			30.4924i			1.52272i	0.648330	+	0.761359i	$0.275468\pi$
$401$			30.4924i			1.52272i	−0.648330	+	0.761359i	$0.724532\pi$
$402$		0			0
$403$	−1.12311	−	7.12311i	−0.0559459	−	0.354827i
$404$		0			0
$405$		−	1.00000i		−	0.0496904i
$406$		0			0
$407$	3.68466			0.182642
$408$		0			0
$409$		−	32.4924i		−	1.60665i	−0.595543	−	0.803323i	$-0.703063\pi$
$409$		−	32.4924i		−	1.60665i	0.595543	−	0.803323i	$-0.296937\pi$
$410$		0			0
$411$			6.00000i			0.295958i
$412$		0			0
$413$	5.75379			0.283125
$414$		0			0
$415$	5.12311			0.251483
$416$		0			0
$417$	−5.43845			−0.266322
$418$		0			0
$419$	21.1231			1.03193			0.515966	−	0.856609i	$-0.327433\pi$
$419$	21.1231			1.03193			0.515966	+	0.856609i	$0.327433\pi$
$420$		0			0
$421$		−	17.1231i		−	0.834529i	−0.908785	−	0.417265i	$-0.862989\pi$
$421$		−	17.1231i		−	0.834529i	0.908785	−	0.417265i	$-0.137011\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−4.56155			−0.221268
$426$		0			0
$427$			3.19224i			0.154483i
$428$		0			0
$429$	1.43845	+	9.12311i	0.0694489	+	0.440468i
$430$		0			0
$431$		−	32.9848i		−	1.58882i	−0.607379	−	0.794412i	$-0.707779\pi$
$431$		−	32.9848i		−	1.58882i	0.607379	−	0.794412i	$-0.292221\pi$
$432$		0			0
$433$	1.50758			0.0724496			0.0362248	−	0.999344i	$-0.488467\pi$
$433$	1.50758			0.0724496			0.0362248	+	0.999344i	$0.488467\pi$
$434$		0			0
$435$			2.00000i			0.0958927i
$436$		0			0
$437$		−	8.00000i		−	0.382692i
$438$		0			0
$439$	−14.5616			−0.694985			−0.347492	−	0.937683i	$-0.612967\pi$
$439$	−14.5616			−0.694985			−0.347492	+	0.937683i	$0.612967\pi$
$440$		0			0
$441$	6.68466			0.318317
$442$		0			0
$443$	−21.9309			−1.04197			−0.520984	−	0.853567i	$-0.674435\pi$
$443$	−21.9309			−1.04197			−0.520984	+	0.853567i	$0.674435\pi$
$444$		0			0
$445$	−4.56155			−0.216238
$446$		0			0
$447$		−	7.93087i		−	0.375117i
$448$		0			0
$449$			14.3153i			0.675583i	0.941221	+	0.337791i	$0.109680\pi$
$449$			14.3153i			0.675583i	−0.941221	+	0.337791i	$0.890320\pi$
$450$		0			0
$451$	−14.5616			−0.685677
$452$		0			0
$453$			14.0000i			0.657777i
$454$		0			0
$455$	2.00000	−	0.315342i	0.0937614	−	0.0147834i
$456$		0			0
$457$			26.5616i			1.24250i	0.783614	+	0.621249i	$0.213374\pi$
$457$			26.5616i			1.24250i	−0.783614	+	0.621249i	$0.786626\pi$
$458$		0			0
$459$	4.56155			0.212915
$460$		0			0
$461$		−	2.31534i		−	0.107836i	−0.998545	−	0.0539181i	$-0.982829\pi$
$461$		−	2.31534i		−	0.107836i	0.998545	−	0.0539181i	$-0.0171710\pi$
$462$		0			0
$463$			39.3002i			1.82643i	0.407473	+	0.913217i	$0.366410\pi$
$463$			39.3002i			1.82643i	−0.407473	+	0.913217i	$0.633590\pi$
$464$		0			0
$465$	−2.00000			−0.0927478
$466$		0			0
$467$	−18.5616			−0.858926			−0.429463	−	0.903084i	$-0.641297\pi$
$467$	−18.5616			−0.858926			−0.429463	+	0.903084i	$0.641297\pi$
$468$		0			0
$469$	−5.26137			−0.242947
$470$		0			0
$471$	−15.1231			−0.696836
$472$		0			0
$473$			23.3693i			1.07452i
$474$		0			0
$475$		−	3.12311i		−	0.143298i
$476$		0			0
$477$	5.68466			0.260283
$478$		0			0
$479$		−	24.3153i		−	1.11100i	−0.831518	−	0.555498i	$-0.812528\pi$
$479$		−	24.3153i		−	1.11100i	0.831518	−	0.555498i	$-0.187472\pi$
$480$		0			0
$481$	−0.807764	−	5.12311i	−0.0368309	−	0.233594i
$482$		0			0
$483$		−	1.43845i		−	0.0654516i
$484$		0			0
$485$	2.56155			0.116314
$486$		0			0
$487$			25.0540i			1.13530i	0.823269	+	0.567652i	$0.192148\pi$
$487$			25.0540i			1.13530i	−0.823269	+	0.567652i	$0.807852\pi$
$488$		0			0
$489$			17.0540i			0.771207i
$490$		0			0
$491$	6.24621			0.281888			0.140944	−	0.990018i	$-0.454986\pi$
$491$	6.24621			0.281888			0.140944	+	0.990018i	$0.454986\pi$
$492$		0			0
$493$	−9.12311			−0.410884
$494$		0			0
$495$	2.56155			0.115133
$496$		0			0
$497$	−3.05398			−0.136990
$498$		0			0
$499$		−	44.1080i		−	1.97454i	−0.159044	−	0.987272i	$-0.550841\pi$
$499$		−	44.1080i		−	1.97454i	0.159044	−	0.987272i	$-0.449159\pi$
$500$		0			0
$501$		−	10.2462i		−	0.457767i
$502$		0			0
$503$	−26.2462			−1.17026			−0.585130	−	0.810939i	$-0.698957\pi$
$503$	−26.2462			−1.17026			−0.585130	+	0.810939i	$0.698957\pi$
$504$		0			0
$505$		−	2.00000i		−	0.0889988i
$506$		0			0
$507$	12.3693	−	4.00000i	0.549341	−	0.177646i
$508$		0			0
$509$			4.06913i			0.180361i	0.995925	+	0.0901805i	$0.0287444\pi$
$509$			4.06913i			0.180361i	−0.995925	+	0.0901805i	$0.971256\pi$
$510$		0			0
$511$	5.12311			0.226633
$512$		0			0
$513$			3.12311i			0.137888i
$514$		0			0
$515$			2.24621i			0.0989799i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−18.4924			−0.811727
$520$		0			0
$521$	−11.6155			−0.508886			−0.254443	−	0.967088i	$-0.581892\pi$
$521$	−11.6155			−0.508886			−0.254443	+	0.967088i	$0.581892\pi$
$522$		0			0
$523$	39.8617			1.74303			0.871516	−	0.490367i	$-0.163137\pi$
$523$	39.8617			1.74303			0.871516	+	0.490367i	$0.163137\pi$
$524$		0			0
$525$		−	0.561553i		−	0.0245082i
$526$		0			0
$527$		−	9.12311i		−	0.397409i
$528$		0			0
$529$	−16.4384			−0.714715
$530$		0			0
$531$		−	10.2462i		−	0.444648i
$532$		0			0
$533$	3.19224	+	20.2462i	0.138271	+	0.876961i
$534$		0			0
$535$			3.68466i			0.159302i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			17.1231i			0.737544i
$540$		0			0
$541$			8.49242i			0.365118i	0.983195	+	0.182559i	$0.0584380\pi$
$541$			8.49242i			0.365118i	−0.983195	+	0.182559i	$0.941562\pi$
$542$		0			0
$543$	6.80776			0.292149
$544$		0			0
$545$	−17.1231			−0.733473
$546$		0			0
$547$	1.12311			0.0480205			0.0240103	−	0.999712i	$-0.492357\pi$
$547$	1.12311			0.0480205			0.0240103	+	0.999712i	$0.492357\pi$
$548$		0			0
$549$	5.68466			0.242615
$550$		0			0
$551$		−	6.24621i		−	0.266098i
$552$		0			0
$553$			2.06913i			0.0879884i
$554$		0			0
$555$	−1.43845			−0.0610587
$556$		0			0
$557$			8.73863i			0.370268i	0.982713	+	0.185134i	$0.0592719\pi$
$557$			8.73863i			0.370268i	−0.982713	+	0.185134i	$0.940728\pi$
$558$		0			0
$559$	32.4924	−	5.12311i	1.37428	−	0.216684i
$560$		0			0
$561$			11.6847i			0.493327i
$562$		0			0
$563$	27.6847			1.16677			0.583385	−	0.812196i	$-0.301728\pi$
$563$	27.6847			1.16677			0.583385	+	0.812196i	$0.301728\pi$
$564$		0			0
$565$			12.2462i			0.515202i
$566$		0			0
$567$			0.561553i			0.0235830i
$568$		0			0
$569$	−4.73863			−0.198654			−0.0993269	−	0.995055i	$-0.531669\pi$
$569$	−4.73863			−0.198654			−0.0993269	+	0.995055i	$0.531669\pi$
$570$		0			0
$571$	−11.1922			−0.468380			−0.234190	−	0.972191i	$-0.575244\pi$
$571$	−11.1922			−0.468380			−0.234190	+	0.972191i	$0.575244\pi$
$572$		0			0
$573$	−22.2462			−0.929349
$574$		0			0
$575$	2.56155			0.106824
$576$		0			0
$577$			22.5616i			0.939250i	0.882866	+	0.469625i	$0.155611\pi$
$577$			22.5616i			0.939250i	−0.882866	+	0.469625i	$0.844389\pi$
$578$		0			0
$579$			14.5616i			0.605157i
$580$		0			0
$581$	−2.87689			−0.119354
$582$		0			0
$583$			14.5616i			0.603078i
$584$		0			0
$585$	−0.561553	−	3.56155i	−0.0232174	−	0.147252i
$586$		0			0
$587$		−	47.3693i		−	1.95514i	−0.210609	−	0.977570i	$-0.567545\pi$
$587$		−	47.3693i		−	1.95514i	0.210609	−	0.977570i	$-0.432455\pi$
$588$		0			0
$589$	6.24621			0.257371
$590$		0			0
$591$		−	18.4924i		−	0.760677i
$592$		0			0
$593$			16.7386i			0.687373i	0.939084	+	0.343687i	$0.111676\pi$
$593$			16.7386i			0.687373i	−0.939084	+	0.343687i	$0.888324\pi$
$594$		0			0
$595$	2.56155			0.105013
$596$		0			0
$597$		0			0
$598$		0			0
$599$	20.4924			0.837298			0.418649	−	0.908148i	$-0.362504\pi$
$599$	20.4924			0.837298			0.418649	+	0.908148i	$0.362504\pi$
$600$		0			0
$601$	17.0540			0.695646			0.347823	−	0.937560i	$-0.386921\pi$
$601$	17.0540			0.695646			0.347823	+	0.937560i	$0.386921\pi$
$602$		0			0
$603$			9.36932i			0.381548i
$604$		0			0
$605$		−	4.43845i		−	0.180449i
$606$		0			0
$607$	−12.4924			−0.507052			−0.253526	−	0.967329i	$-0.581590\pi$
$607$	−12.4924			−0.507052			−0.253526	+	0.967329i	$0.581590\pi$
$608$		0			0
$609$		−	1.12311i		−	0.0455105i
$610$		0			0
$611$		0			0
$612$		0			0
$613$			12.3153i			0.497412i	0.968579	+	0.248706i	$0.0800053\pi$
$613$			12.3153i			0.497412i	−0.968579	+	0.248706i	$0.919995\pi$
$614$		0			0
$615$	5.68466			0.229228
$616$		0			0
$617$		−	24.2462i		−	0.976116i	−0.872811	−	0.488058i	$-0.837706\pi$
$617$		−	24.2462i		−	0.976116i	0.872811	−	0.488058i	$-0.162294\pi$
$618$		0			0
$619$		−	18.4924i		−	0.743273i	−0.928378	−	0.371637i	$-0.878797\pi$
$619$		−	18.4924i		−	0.743273i	0.928378	−	0.371637i	$-0.121203\pi$
$620$		0			0
$621$	−2.56155			−0.102792
$622$		0			0
$623$	2.56155			0.102626
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−8.00000			−0.319489
$628$		0			0
$629$		−	6.56155i		−	0.261626i
$630$		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
$633$	−8.49242			−0.337543
$634$		0			0
$635$		−	10.8769i		−	0.431636i
$636$		0			0
$637$	23.8078	−	3.75379i	0.943298	−	0.148731i
$638$		0			0
$639$			5.43845i			0.215142i
$640$		0			0
$641$	40.8769			1.61454			0.807270	−	0.590182i	$-0.200944\pi$
$641$	40.8769			1.61454			0.807270	+	0.590182i	$0.200944\pi$
$642$		0			0
$643$		−	8.06913i		−	0.318215i	−0.987261	−	0.159108i	$-0.949138\pi$
$643$		−	8.06913i		−	0.318215i	0.987261	−	0.159108i	$-0.0508617\pi$
$644$		0			0
$645$		−	9.12311i		−	0.359222i
$646$		0			0
$647$	35.0540			1.37811			0.689057	−	0.724707i	$-0.258025\pi$
$647$	35.0540			1.37811			0.689057	+	0.724707i	$0.258025\pi$
$648$		0			0
$649$	26.2462			1.03025
$650$		0			0
$651$	1.12311			0.0440180
$652$		0			0
$653$	1.50758			0.0589961			0.0294980	−	0.999565i	$-0.490609\pi$
$653$	1.50758			0.0589961			0.0294980	+	0.999565i	$0.490609\pi$
$654$		0			0
$655$			12.4924i			0.488119i
$656$		0			0
$657$		−	9.12311i		−	0.355926i
$658$		0			0
$659$	13.6155			0.530386			0.265193	−	0.964195i	$-0.414564\pi$
$659$	13.6155			0.530386			0.265193	+	0.964195i	$0.414564\pi$
$660$		0			0
$661$		−	10.2462i		−	0.398531i	−0.979945	−	0.199266i	$-0.936144\pi$
$661$		−	10.2462i		−	0.398531i	0.979945	−	0.199266i	$-0.0638557\pi$
$662$		0			0
$663$	16.2462	−	2.56155i	0.630951	−	0.0994825i
$664$		0			0
$665$			1.75379i			0.0680090i
$666$		0			0
$667$	5.12311			0.198367
$668$		0			0
$669$		−	4.87689i		−	0.188552i
$670$		0			0
$671$			14.5616i			0.562143i
$672$		0			0
$673$	−44.1080			−1.70024			−0.850118	−	0.526592i	$-0.823470\pi$
$673$	−44.1080			−1.70024			−0.850118	+	0.526592i	$0.823470\pi$
$674$		0			0
$675$	−1.00000			−0.0384900
$676$		0			0
$677$	28.4233			1.09240			0.546198	−	0.837656i	$-0.316075\pi$
$677$	28.4233			1.09240			0.546198	+	0.837656i	$0.316075\pi$
$678$		0			0
$679$	−1.43845			−0.0552025
$680$		0			0
$681$			9.75379i			0.373766i
$682$		0			0
$683$			5.61553i			0.214872i	0.994212	+	0.107436i	$0.0342641\pi$
$683$			5.61553i			0.214872i	−0.994212	+	0.107436i	$0.965736\pi$
$684$		0			0
$685$	6.00000			0.229248
$686$		0			0
$687$		−	27.3693i		−	1.04420i
$688$		0			0
$689$	20.2462	−	3.19224i	0.771319	−	0.121615i
$690$		0			0
$691$		−	19.7538i		−	0.751470i	−0.926727	−	0.375735i	$-0.877390\pi$
$691$		−	19.7538i		−	0.751470i	0.926727	−	0.375735i	$-0.122610\pi$
$692$		0			0
$693$	−1.43845			−0.0546421
$694$		0			0
$695$			5.43845i			0.206292i
$696$		0			0
$697$			25.9309i			0.982202i
$698$		0			0
$699$	−14.8078			−0.560081
$700$		0			0
$701$	−10.6307			−0.401515			−0.200758	−	0.979641i	$-0.564340\pi$
$701$	−10.6307			−0.401515			−0.200758	+	0.979641i	$0.564340\pi$
$702$		0			0
$703$	4.49242			0.169435
$704$		0			0
$705$		0			0
$706$		0			0
$707$			1.12311i			0.0422387i
$708$		0			0
$709$			9.75379i			0.366311i	0.983084	+	0.183156i	$0.0586312\pi$
$709$			9.75379i			0.366311i	−0.983084	+	0.183156i	$0.941369\pi$
$710$		0			0
$711$	3.68466			0.138185
$712$		0			0
$713$			5.12311i			0.191862i
$714$		0			0
$715$	9.12311	−	1.43845i	0.341185	−	0.0537949i
$716$		0			0
$717$		−	0.315342i		−	0.0117766i
$718$		0			0
$719$	35.3693			1.31905			0.659526	−	0.751681i	$-0.270757\pi$
$719$	35.3693			1.31905			0.659526	+	0.751681i	$0.270757\pi$
$720$		0			0
$721$		−	1.26137i		−	0.0469757i
$722$		0			0
$723$			1.12311i			0.0417687i
$724$		0			0
$725$	2.00000			0.0742781
$726$		0			0
$727$	−19.8617			−0.736631			−0.368316	−	0.929701i	$-0.620065\pi$
$727$	−19.8617			−0.736631			−0.368316	+	0.929701i	$0.620065\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	41.6155			1.53921
$732$		0			0
$733$		−	17.3002i		−	0.638997i	−0.947587	−	0.319498i	$-0.896486\pi$
$733$		−	17.3002i		−	0.638997i	0.947587	−	0.319498i	$-0.103514\pi$
$734$		0			0
$735$		−	6.68466i		−	0.246567i
$736$		0			0
$737$	−24.0000			−0.884051
$738$		0			0
$739$		−	3.75379i		−	0.138085i	−0.997614	−	0.0690427i	$-0.978006\pi$
$739$		−	3.75379i		−	0.138085i	0.997614	−	0.0690427i	$-0.0219945\pi$
$740$		0			0
$741$	1.75379	+	11.1231i	0.0644270	+	0.408617i
$742$		0			0
$743$			16.4924i			0.605048i	0.953142	+	0.302524i	$0.0978293\pi$
$743$			16.4924i			0.605048i	−0.953142	+	0.302524i	$0.902171\pi$
$744$		0			0
$745$	−7.93087			−0.290565
$746$		0			0
$747$			5.12311i			0.187445i
$748$		0			0
$749$		−	2.06913i		−	0.0756044i
$750$		0			0
$751$	−14.5616			−0.531359			−0.265679	−	0.964061i	$-0.585596\pi$
$751$	−14.5616			−0.531359			−0.265679	+	0.964061i	$0.585596\pi$
$752$		0			0
$753$	−11.3693			−0.414321
$754$		0			0
$755$	14.0000			0.509512
$756$		0			0
$757$	28.8769			1.04955			0.524774	−	0.851241i	$-0.324150\pi$
$757$	28.8769			1.04955			0.524774	+	0.851241i	$0.324150\pi$
$758$		0			0
$759$		−	6.56155i		−	0.238169i
$760$		0			0
$761$			41.2311i			1.49462i	0.664473	+	0.747312i	$0.268656\pi$
$761$			41.2311i			1.49462i	−0.664473	+	0.747312i	$0.731344\pi$
$762$		0			0
$763$	9.61553			0.348105
$764$		0			0
$765$		−	4.56155i		−	0.164923i
$766$		0			0
$767$	−5.75379	−	36.4924i	−0.207757	−	1.31767i
$768$		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
$771$	−10.4924			−0.377875
$772$		0			0
$773$		−	13.5076i		−	0.485834i	−0.970047	−	0.242917i	$-0.921896\pi$
$773$		−	13.5076i		−	0.485834i	0.970047	−	0.242917i	$-0.0781042\pi$
$774$		0			0
$775$			2.00000i			0.0718421i
$776$		0			0
$777$	0.807764			0.0289784
$778$		0			0
$779$	−17.7538			−0.636095
$780$		0			0
$781$	−13.9309			−0.498486
$782$		0			0
$783$	−2.00000			−0.0714742
$784$		0			0
$785$			15.1231i			0.539767i
$786$		0			0
$787$		−	13.3693i		−	0.476565i	−0.971196	−	0.238282i	$-0.923416\pi$
$787$		−	13.3693i		−	0.476565i	0.971196	−	0.238282i	$-0.0765844\pi$
$788$		0			0
$789$	16.0000			0.569615
$790$		0			0
$791$		−	6.87689i		−	0.244514i
$792$		0			0
$793$	20.2462	−	3.19224i	0.718964	−	0.113360i
$794$		0			0
$795$		−	5.68466i		−	0.201614i
$796$		0			0
$797$	48.4233			1.71524			0.857621	−	0.514283i	$-0.171942\pi$
$797$	48.4233			1.71524			0.857621	+	0.514283i	$0.171942\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		−	4.56155i		−	0.161175i
$802$		0			0
$803$	23.3693			0.824685
$804$		0			0
$805$	−1.43845			−0.0506986
$806$		0			0
$807$	−20.8769			−0.734901
$808$		0			0
$809$	35.1231			1.23486			0.617431	−	0.786625i	$-0.288173\pi$
$809$	35.1231			1.23486			0.617431	+	0.786625i	$0.288173\pi$
$810$		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
$813$			20.2462i			0.710066i
$814$		0			0
$815$	17.0540			0.597375
$816$		0			0
$817$			28.4924i			0.996824i
$818$		0			0
$819$	0.315342	+	2.00000i	0.0110189	+	0.0698857i
$820$		0			0
$821$			19.9309i			0.695592i	0.937570	+	0.347796i	$0.113070\pi$
$821$			19.9309i			0.695592i	−0.937570	+	0.347796i	$0.886930\pi$
$822$		0			0
$823$	8.63068			0.300847			0.150423	−	0.988622i	$-0.451936\pi$
$823$	8.63068			0.300847			0.150423	+	0.988622i	$0.451936\pi$
$824$		0			0
$825$		−	2.56155i		−	0.0891818i
$826$		0			0
$827$			22.2462i			0.773577i	0.922169	+	0.386788i	$0.126416\pi$
$827$			22.2462i			0.773577i	−0.922169	+	0.386788i	$0.873584\pi$
$828$		0			0
$829$	37.2311			1.29309			0.646544	−	0.762877i	$-0.276214\pi$
$829$	37.2311			1.29309			0.646544	+	0.762877i	$0.276214\pi$
$830$		0			0
$831$	21.3693			0.741293
$832$		0			0
$833$	30.4924			1.05650
$834$		0			0
$835$	−10.2462			−0.354585
$836$		0			0
$837$		−	2.00000i		−	0.0691301i
$838$		0			0
$839$			27.5464i			0.951007i	0.879714	+	0.475504i	$0.157734\pi$
$839$			27.5464i			0.951007i	−0.879714	+	0.475504i	$0.842266\pi$
$840$		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
$847$			2.49242i			0.0856407i
$848$		0			0
$849$	6.87689			0.236014
$850$		0			0
$851$			3.68466i			0.126308i
$852$		0			0
$853$		−	55.5464i		−	1.90187i	−0.309387	−	0.950936i	$-0.600124\pi$
$853$		−	55.5464i		−	1.90187i	0.309387	−	0.950936i	$-0.399876\pi$
$854$		0			0
$855$	3.12311			0.106808
$856$		0			0
$857$	15.9309			0.544188			0.272094	−	0.962271i	$-0.412284\pi$
$857$	15.9309			0.544188			0.272094	+	0.962271i	$0.412284\pi$
$858$		0			0
$859$	−45.4384			−1.55034			−0.775170	−	0.631753i	$-0.782336\pi$
$859$	−45.4384			−1.55034			−0.775170	+	0.631753i	$0.782336\pi$
$860$		0			0
$861$	−3.19224			−0.108791
$862$		0			0
$863$		−	23.3693i		−	0.795501i	−0.917494	−	0.397750i	$-0.869791\pi$
$863$		−	23.3693i		−	0.795501i	0.917494	−	0.397750i	$-0.130209\pi$
$864$		0			0
$865$			18.4924i			0.628761i
$866$		0			0
$867$	3.80776			0.129318
$868$		0			0
$869$			9.43845i			0.320177i
$870$		0			0
$871$	5.26137	+	33.3693i	0.178275	+	1.13068i
$872$		0			0
$873$			2.56155i			0.0866954i
$874$		0			0
$875$	−0.561553			−0.0189839
$876$		0			0
$877$			21.1231i			0.713277i	0.934243	+	0.356638i	$0.116077\pi$
$877$			21.1231i			0.713277i	−0.934243	+	0.356638i	$0.883923\pi$
$878$		0			0
$879$		−	10.4924i		−	0.353901i
$880$		0			0
$881$	48.1080			1.62080			0.810399	−	0.585878i	$-0.199250\pi$
$881$	48.1080			1.62080			0.810399	+	0.585878i	$0.199250\pi$
$882$		0			0
$883$	47.2311			1.58945			0.794726	−	0.606969i	$-0.207615\pi$
$883$	47.2311			1.58945			0.794726	+	0.606969i	$0.207615\pi$
$884$		0			0
$885$	−10.2462			−0.344423
$886$		0			0
$887$	44.1771			1.48332			0.741661	−	0.670775i	$-0.234039\pi$
$887$	44.1771			1.48332			0.741661	+	0.670775i	$0.234039\pi$
$888$		0			0
$889$			6.10795i			0.204854i
$890$		0			0
$891$			2.56155i			0.0858152i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−9.12311	+	1.43845i	−0.304612	+	0.0480284i
$898$		0			0
$899$			4.00000i			0.133407i
$900$		0			0
$901$	25.9309			0.863883
$902$		0			0
$903$			5.12311i			0.170486i
$904$		0			0
$905$		−	6.80776i		−	0.226298i
$906$		0			0
$907$	−18.7386			−0.622206			−0.311103	−	0.950376i	$-0.600698\pi$
$907$	−18.7386			−0.622206			−0.311103	+	0.950376i	$0.600698\pi$
$908$		0			0
$909$	2.00000			0.0663358
$910$		0			0
$911$	−20.6307			−0.683525			−0.341763	−	0.939786i	$-0.611024\pi$
$911$	−20.6307			−0.683525			−0.341763	+	0.939786i	$0.611024\pi$
$912$		0			0
$913$	−13.1231			−0.434311
$914$		0			0
$915$		−	5.68466i		−	0.187929i
$916$		0			0
$917$		−	7.01515i		−	0.231661i
$918$		0			0
$919$	−11.0540			−0.364637			−0.182318	−	0.983240i	$-0.558360\pi$
$919$	−11.0540			−0.364637			−0.182318	+	0.983240i	$0.558360\pi$
$920$		0			0
$921$			8.56155i			0.282113i
$922$		0			0
$923$	3.05398	+	19.3693i	0.100523	+	0.637549i
$924$		0			0
$925$			1.43845i			0.0472959i
$926$		0			0
$927$	−2.24621			−0.0737753
$928$		0			0
$929$			17.5464i			0.575679i	0.957679	+	0.287839i	$0.0929369\pi$
$929$			17.5464i			0.575679i	−0.957679	+	0.287839i	$0.907063\pi$
$930$		0			0
$931$			20.8769i			0.684213i
$932$		0			0
$933$	−17.1231			−0.560585
$934$		0			0
$935$	11.6847			0.382129
$936$		0			0
$937$	−38.4924			−1.25749			−0.628746	−	0.777610i	$-0.716432\pi$
$937$	−38.4924			−1.25749			−0.628746	+	0.777610i	$0.716432\pi$
$938$		0			0
$939$	−17.3693			−0.566826
$940$		0			0
$941$			14.8078i			0.482719i	0.970436	+	0.241360i	$0.0775933\pi$
$941$			14.8078i			0.482719i	−0.970436	+	0.241360i	$0.922407\pi$
$942$		0			0
$943$		−	14.5616i		−	0.474189i
$944$		0			0
$945$	0.561553			0.0182673
$946$		0			0
$947$		−	12.4924i		−	0.405949i	−0.979184	−	0.202975i	$-0.934939\pi$
$947$		−	12.4924i		−	0.405949i	0.979184	−	0.202975i	$-0.0650609\pi$
$948$		0			0
$949$	−5.12311	−	32.4924i	−0.166303	−	1.05475i
$950$		0			0
$951$			10.0000i			0.324272i
$952$		0			0
$953$	43.7926			1.41858			0.709291	−	0.704916i	$-0.249015\pi$
$953$	43.7926			1.41858			0.709291	+	0.704916i	$0.249015\pi$
$954$		0			0
$955$			22.2462i			0.719870i
$956$		0			0
$957$		−	5.12311i		−	0.165606i
$958$		0			0
$959$	−3.36932			−0.108801
$960$		0			0
$961$	27.0000			0.870968
$962$		0			0
$963$	−3.68466			−0.118736
$964$		0			0
$965$	14.5616			0.468753
$966$		0			0
$967$			51.1231i			1.64401i	0.569482	+	0.822004i	$0.307144\pi$
$967$			51.1231i			1.64401i	−0.569482	+	0.822004i	$0.692856\pi$
$968$		0			0
$969$			14.2462i			0.457654i
$970$		0			0
$971$	37.6155			1.20714			0.603570	−	0.797310i	$-0.293744\pi$
$971$	37.6155			1.20714			0.603570	+	0.797310i	$0.293744\pi$
$972$		0			0
$973$		−	3.05398i		−	0.0979060i
$974$		0			0
$975$	−3.56155	+	0.561553i	−0.114061	+	0.0179841i
$976$		0			0
$977$			5.36932i			0.171780i	0.996305	+	0.0858898i	$0.0273733\pi$
$977$			5.36932i			0.171780i	−0.996305	+	0.0858898i	$0.972627\pi$
$978$		0			0
$979$	11.6847			0.373443
$980$		0			0
$981$		−	17.1231i		−	0.546699i
$982$		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
$985$	−18.4924			−0.589218
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−23.3693			−0.743101
$990$		0			0
$991$	37.3002			1.18488			0.592440	−	0.805615i	$-0.298165\pi$
$991$	37.3002			1.18488			0.592440	+	0.805615i	$0.298165\pi$
$992$		0			0
$993$		−	5.36932i		−	0.170390i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	3.12311			0.0989097			0.0494549	−	0.998776i	$-0.484252\pi$
$997$	3.12311			0.0989097			0.0494549	+	0.998776i	$0.484252\pi$
$998$		0			0
$999$		−	1.43845i		−	0.0455105i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1560.2.g.h.961.2	✓	4
3.2	odd	2		4680.2.g.g.2521.4		4
4.3	odd	2		3120.2.g.p.961.1		4
13.12	even	2	inner	1560.2.g.h.961.3	yes	4
39.38	odd	2		4680.2.g.g.2521.1		4
52.51	odd	2		3120.2.g.p.961.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.g.h.961.2	✓	4	1.1	even	1	trivial
1560.2.g.h.961.3	yes	4	13.12	even	2	inner
3120.2.g.p.961.1		4	4.3	odd	2
3120.2.g.p.961.4		4	52.51	odd	2
4680.2.g.g.2521.1		4	39.38	odd	2
4680.2.g.g.2521.4		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000i q^{5} +0.561553i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000i q^{5} +0.561553i q^{7} +1.00000 q^{9} +2.56155i q^{11} +(3.56155 - 0.561553i) q^{13} -1.00000i q^{15} +4.56155 q^{17} +3.12311i q^{19} +0.561553i q^{21} -2.56155 q^{23} -1.00000 q^{25} +1.00000 q^{27} -2.00000 q^{29} -2.00000i q^{31} +2.56155i q^{33} +0.561553 q^{35} -1.43845i q^{37} +(3.56155 - 0.561553i) q^{39} +5.68466i q^{41} +9.12311 q^{43} -1.00000i q^{45} +6.68466 q^{49} +4.56155 q^{51} +5.68466 q^{53} +2.56155 q^{55} +3.12311i q^{57} -10.2462i q^{59} +5.68466 q^{61} +0.561553i q^{63} +(-0.561553 - 3.56155i) q^{65} +9.36932i q^{67} -2.56155 q^{69} +5.43845i q^{71} -9.12311i q^{73} -1.00000 q^{75} -1.43845 q^{77} +3.68466 q^{79} +1.00000 q^{81} +5.12311i q^{83} -4.56155i q^{85} -2.00000 q^{87} -4.56155i q^{89} +(0.315342 + 2.00000i) q^{91} -2.00000i q^{93} +3.12311 q^{95} +2.56155i q^{97} +2.56155i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more