LMFDB - Embedded newform 1560.2.g.c.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:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			961.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1560.961
Dual form			1560.2.g.c.961.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000i q^{5} -4.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000i q^{5} -4.00000i q^{7} +1.00000 q^{9} -4.00000i q^{11} +(-3.00000 + 2.00000i) q^{13} -1.00000i q^{15} +8.00000 q^{17} +4.00000i q^{21} -6.00000 q^{23} -1.00000 q^{25} -1.00000 q^{27} -8.00000 q^{29} +2.00000i q^{31} +4.00000i q^{33} +4.00000 q^{35} +4.00000i q^{37} +(3.00000 - 2.00000i) q^{39} -10.0000i q^{41} -4.00000 q^{43} +1.00000i q^{45} +8.00000i q^{47} -9.00000 q^{49} -8.00000 q^{51} -10.0000 q^{53} +4.00000 q^{55} -12.0000i q^{59} -2.00000 q^{61} -4.00000i q^{63} +(-2.00000 - 3.00000i) q^{65} -2.00000i q^{67} +6.00000 q^{69} -12.0000i q^{71} +2.00000i q^{73} +1.00000 q^{75} -16.0000 q^{77} +8.00000 q^{79} +1.00000 q^{81} -8.00000i q^{83} +8.00000i q^{85} +8.00000 q^{87} +6.00000i q^{89} +(8.00000 + 12.0000i) q^{91} -2.00000i q^{93} -10.0000i q^{97} -4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{9} - 6 q^{13} + 16 q^{17} - 12 q^{23} - 2 q^{25} - 2 q^{27} - 16 q^{29} + 8 q^{35} + 6 q^{39} - 8 q^{43} - 18 q^{49} - 16 q^{51} - 20 q^{53} + 8 q^{55} - 4 q^{61} - 4 q^{65} + 12 q^{69} + 2 q^{75} - 32 q^{77} + 16 q^{79} + 2 q^{81} + 16 q^{87} + 16 q^{91}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^9 - 6 * q^13 + 16 * q^17 - 12 * q^23 - 2 * q^25 - 2 * q^27 - 16 * q^29 + 8 * q^35 + 6 * q^39 - 8 * q^43 - 18 * q^49 - 16 * q^51 - 20 * q^53 + 8 * q^55 - 4 * q^61 - 4 * q^65 + 12 * q^69 + 2 * q^75 - 32 * q^77 + 16 * q^79 + 2 * q^81 + 16 * q^87 + 16 * q^91

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
$7$		−	4.00000i		−	1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$		−	4.00000i		−	1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	4.00000i		−	1.20605i	−0.797724	−	0.603023i	$-0.793963\pi$
$11$		−	4.00000i		−	1.20605i	0.797724	−	0.603023i	$-0.206037\pi$
$12$		0			0
$13$	−3.00000	+	2.00000i	−0.832050	+	0.554700i
$13$	−3.00000	+	2.00000i	−0.832050	+	0.554700i
$14$		0			0
$15$		−	1.00000i		−	0.258199i
$16$		0			0
$17$	8.00000			1.94029			0.970143	−	0.242536i	$-0.0779791\pi$
$17$	8.00000			1.94029			0.970143	+	0.242536i	$0.0779791\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$			4.00000i			0.872872i
$22$		0			0
$23$	−6.00000			−1.25109			−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000			−1.25109			−0.625543	+	0.780189i	$0.715123\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	−8.00000			−1.48556			−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000			−1.48556			−0.742781	+	0.669534i	$0.766494\pi$
$30$		0			0
$31$			2.00000i			0.359211i	0.983739	+	0.179605i	$0.0574821\pi$
$31$			2.00000i			0.359211i	−0.983739	+	0.179605i	$0.942518\pi$
$32$		0			0
$33$			4.00000i			0.696311i
$34$		0			0
$35$	4.00000			0.676123
$36$		0			0
$37$			4.00000i			0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$			4.00000i			0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$		0			0
$39$	3.00000	−	2.00000i	0.480384	−	0.320256i
$40$		0			0
$41$		−	10.0000i		−	1.56174i	−0.624695	−	0.780869i	$-0.714777\pi$
$41$		−	10.0000i		−	1.56174i	0.624695	−	0.780869i	$-0.285223\pi$
$42$		0			0
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$		0			0
$45$			1.00000i			0.149071i
$46$		0			0
$47$			8.00000i			1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$			8.00000i			1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$		0			0
$49$	−9.00000			−1.28571
$50$		0			0
$51$	−8.00000			−1.12022
$52$		0			0
$53$	−10.0000			−1.37361			−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000			−1.37361			−0.686803	+	0.726844i	$0.740986\pi$
$54$		0			0
$55$	4.00000			0.539360
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	12.0000i		−	1.56227i	−0.624364	−	0.781133i	$-0.714642\pi$
$59$		−	12.0000i		−	1.56227i	0.624364	−	0.781133i	$-0.285358\pi$
$60$		0			0
$61$	−2.00000			−0.256074			−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000			−0.256074			−0.128037	+	0.991769i	$0.540868\pi$
$62$		0			0
$63$		−	4.00000i		−	0.503953i
$64$		0			0
$65$	−2.00000	−	3.00000i	−0.248069	−	0.372104i
$66$		0			0
$67$		−	2.00000i		−	0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$		−	2.00000i		−	0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$		0			0
$69$	6.00000			0.722315
$70$		0			0
$71$		−	12.0000i		−	1.42414i	−0.702109	−	0.712069i	$-0.747758\pi$
$71$		−	12.0000i		−	1.42414i	0.702109	−	0.712069i	$-0.252242\pi$
$72$		0			0
$73$			2.00000i			0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$			2.00000i			0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$		0			0
$75$	1.00000			0.115470
$76$		0			0
$77$	−16.0000			−1.82337
$78$		0			0
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	8.00000i		−	0.878114i	−0.898459	−	0.439057i	$-0.855313\pi$
$83$		−	8.00000i		−	0.878114i	0.898459	−	0.439057i	$-0.144687\pi$
$84$		0			0
$85$			8.00000i			0.867722i
$86$		0			0
$87$	8.00000			0.857690
$88$		0			0
$89$			6.00000i			0.635999i	0.948091	+	0.317999i	$0.103011\pi$
$89$			6.00000i			0.635999i	−0.948091	+	0.317999i	$0.896989\pi$
$90$		0			0
$91$	8.00000	+	12.0000i	0.838628	+	1.25794i
$92$		0			0
$93$		−	2.00000i		−	0.207390i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	10.0000i		−	1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$		−	10.0000i		−	1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$		0			0
$99$		−	4.00000i		−	0.402015i
$100$		0			0
$101$	−4.00000			−0.398015			−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000			−0.398015			−0.199007	+	0.979998i	$0.563772\pi$
$102$		0			0
$103$	4.00000			0.394132			0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000			0.394132			0.197066	+	0.980390i	$0.436859\pi$
$104$		0			0
$105$	−4.00000			−0.390360
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$		−	14.0000i		−	1.34096i	−0.741929	−	0.670478i	$-0.766089\pi$
$109$		−	14.0000i		−	1.34096i	0.741929	−	0.670478i	$-0.233911\pi$
$110$		0			0
$111$		−	4.00000i		−	0.379663i
$112$		0			0
$113$	−16.0000			−1.50515			−0.752577	−	0.658505i	$-0.771189\pi$
$113$	−16.0000			−1.50515			−0.752577	+	0.658505i	$0.771189\pi$
$114$		0			0
$115$		−	6.00000i		−	0.559503i
$116$		0			0
$117$	−3.00000	+	2.00000i	−0.277350	+	0.184900i
$118$		0			0
$119$		−	32.0000i		−	2.93344i
$120$		0			0
$121$	−5.00000			−0.454545
$122$		0			0
$123$			10.0000i			0.901670i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$	−16.0000			−1.41977			−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000			−1.41977			−0.709885	+	0.704317i	$0.751253\pi$
$128$		0			0
$129$	4.00000			0.352180
$130$		0			0
$131$	−22.0000			−1.92215			−0.961074	−	0.276289i	$-0.910895\pi$
$131$	−22.0000			−1.92215			−0.961074	+	0.276289i	$0.910895\pi$
$132$		0			0
$133$		0			0
$134$		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
$139$	−16.0000			−1.35710			−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000			−1.35710			−0.678551	+	0.734553i	$0.737392\pi$
$140$		0			0
$141$		−	8.00000i		−	0.673722i
$142$		0			0
$143$	8.00000	+	12.0000i	0.668994	+	1.00349i
$144$		0			0
$145$		−	8.00000i		−	0.664364i
$146$		0			0
$147$	9.00000			0.742307
$148$		0			0
$149$			6.00000i			0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$			6.00000i			0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$		0			0
$151$			2.00000i			0.162758i	0.996683	+	0.0813788i	$0.0259324\pi$
$151$			2.00000i			0.162758i	−0.996683	+	0.0813788i	$0.974068\pi$
$152$		0			0
$153$	8.00000			0.646762
$154$		0			0
$155$	−2.00000			−0.160644
$156$		0			0
$157$	2.00000			0.159617			0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000			0.159617			0.0798087	+	0.996810i	$0.474569\pi$
$158$		0			0
$159$	10.0000			0.793052
$160$		0			0
$161$			24.0000i			1.89146i
$162$		0			0
$163$			14.0000i			1.09656i	0.836293	+	0.548282i	$0.184718\pi$
$163$			14.0000i			1.09656i	−0.836293	+	0.548282i	$0.815282\pi$
$164$		0			0
$165$	−4.00000			−0.311400
$166$		0			0
$167$			8.00000i			0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$			8.00000i			0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$		0			0
$169$	5.00000	−	12.0000i	0.384615	−	0.923077i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2.00000			−0.152057			−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000			−0.152057			−0.0760286	+	0.997106i	$0.524224\pi$
$174$		0			0
$175$			4.00000i			0.302372i
$176$		0			0
$177$			12.0000i			0.901975i
$178$		0			0
$179$	2.00000			0.149487			0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000			0.149487			0.0747435	+	0.997203i	$0.476186\pi$
$180$		0			0
$181$	22.0000			1.63525			0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000			1.63525			0.817624	+	0.575753i	$0.195291\pi$
$182$		0			0
$183$	2.00000			0.147844
$184$		0			0
$185$	−4.00000			−0.294086
$186$		0			0
$187$		−	32.0000i		−	2.34007i
$188$		0			0
$189$			4.00000i			0.290957i
$190$		0			0
$191$	4.00000			0.289430			0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000			0.289430			0.144715	+	0.989473i	$0.453773\pi$
$192$		0			0
$193$			2.00000i			0.143963i	0.997406	+	0.0719816i	$0.0229323\pi$
$193$			2.00000i			0.143963i	−0.997406	+	0.0719816i	$0.977068\pi$
$194$		0			0
$195$	2.00000	+	3.00000i	0.143223	+	0.214834i
$196$		0			0
$197$		−	2.00000i		−	0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$		−	2.00000i		−	0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$			2.00000i			0.141069i
$202$		0			0
$203$			32.0000i			2.24596i
$204$		0			0
$205$	10.0000			0.698430
$206$		0			0
$207$	−6.00000			−0.417029
$208$		0			0
$209$		0			0
$210$		0			0
$211$	24.0000			1.65223			0.826114	−	0.563503i	$-0.190547\pi$
$211$	24.0000			1.65223			0.826114	+	0.563503i	$0.190547\pi$
$212$		0			0
$213$			12.0000i			0.822226i
$214$		0			0
$215$		−	4.00000i		−	0.272798i
$216$		0			0
$217$	8.00000			0.543075
$218$		0			0
$219$		−	2.00000i		−	0.135147i
$220$		0			0
$221$	−24.0000	+	16.0000i	−1.61441	+	1.07628i
$222$		0			0
$223$			16.0000i			1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$			16.0000i			1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$		0			0
$225$	−1.00000			−0.0666667
$226$		0			0
$227$		−	24.0000i		−	1.59294i	−0.604681	−	0.796468i	$-0.706699\pi$
$227$		−	24.0000i		−	1.59294i	0.604681	−	0.796468i	$-0.293301\pi$
$228$		0			0
$229$			22.0000i			1.45380i	0.686743	+	0.726900i	$0.259040\pi$
$229$			22.0000i			1.45380i	−0.686743	+	0.726900i	$0.740960\pi$
$230$		0			0
$231$	16.0000			1.05272
$232$		0			0
$233$	−4.00000			−0.262049			−0.131024	−	0.991379i	$-0.541827\pi$
$233$	−4.00000			−0.262049			−0.131024	+	0.991379i	$0.541827\pi$
$234$		0			0
$235$	−8.00000			−0.521862
$236$		0			0
$237$	−8.00000			−0.519656
$238$		0			0
$239$		−	8.00000i		−	0.517477i	−0.965947	−	0.258738i	$-0.916693\pi$
$239$		−	8.00000i		−	0.517477i	0.965947	−	0.258738i	$-0.0833068\pi$
$240$		0			0
$241$			4.00000i			0.257663i	0.991667	+	0.128831i	$0.0411226\pi$
$241$			4.00000i			0.257663i	−0.991667	+	0.128831i	$0.958877\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$		−	9.00000i		−	0.574989i
$246$		0			0
$247$		0			0
$248$		0			0
$249$			8.00000i			0.506979i
$250$		0			0
$251$	−18.0000			−1.13615			−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000			−1.13615			−0.568075	+	0.822977i	$0.692312\pi$
$252$		0			0
$253$			24.0000i			1.50887i
$254$		0			0
$255$		−	8.00000i		−	0.500979i
$256$		0			0
$257$	16.0000			0.998053			0.499026	−	0.866587i	$-0.333691\pi$
$257$	16.0000			0.998053			0.499026	+	0.866587i	$0.333691\pi$
$258$		0			0
$259$	16.0000			0.994192
$260$		0			0
$261$	−8.00000			−0.495188
$262$		0			0
$263$	6.00000			0.369976			0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000			0.369976			0.184988	+	0.982741i	$0.440775\pi$
$264$		0			0
$265$		−	10.0000i		−	0.614295i
$266$		0			0
$267$		−	6.00000i		−	0.367194i
$268$		0			0
$269$	12.0000			0.731653			0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000			0.731653			0.365826	+	0.930683i	$0.380786\pi$
$270$		0			0
$271$		−	10.0000i		−	0.607457i	−0.952759	−	0.303728i	$-0.901768\pi$
$271$		−	10.0000i		−	0.607457i	0.952759	−	0.303728i	$-0.0982315\pi$
$272$		0			0
$273$	−8.00000	−	12.0000i	−0.484182	−	0.726273i
$274$		0			0
$275$			4.00000i			0.241209i
$276$		0			0
$277$	26.0000			1.56219			0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000			1.56219			0.781094	+	0.624413i	$0.214662\pi$
$278$		0			0
$279$			2.00000i			0.119737i
$280$		0			0
$281$		−	30.0000i		−	1.78965i	−0.446417	−	0.894825i	$-0.647300\pi$
$281$		−	30.0000i		−	1.78965i	0.446417	−	0.894825i	$-0.352700\pi$
$282$		0			0
$283$	−4.00000			−0.237775			−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000			−0.237775			−0.118888	+	0.992908i	$0.537933\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−40.0000			−2.36113
$288$		0			0
$289$	47.0000			2.76471
$290$		0			0
$291$			10.0000i			0.586210i
$292$		0			0
$293$		−	26.0000i		−	1.51894i	−0.650545	−	0.759468i	$-0.725459\pi$
$293$		−	26.0000i		−	1.51894i	0.650545	−	0.759468i	$-0.274541\pi$
$294$		0			0
$295$	12.0000			0.698667
$296$		0			0
$297$			4.00000i			0.232104i
$298$		0			0
$299$	18.0000	−	12.0000i	1.04097	−	0.693978i
$300$		0			0
$301$			16.0000i			0.922225i
$302$		0			0
$303$	4.00000			0.229794
$304$		0			0
$305$		−	2.00000i		−	0.114520i
$306$		0			0
$307$			6.00000i			0.342438i	0.985233	+	0.171219i	$0.0547706\pi$
$307$			6.00000i			0.342438i	−0.985233	+	0.171219i	$0.945229\pi$
$308$		0			0
$309$	−4.00000			−0.227552
$310$		0			0
$311$	28.0000			1.58773			0.793867	−	0.608091i	$-0.208065\pi$
$311$	28.0000			1.58773			0.793867	+	0.608091i	$0.208065\pi$
$312$		0			0
$313$	−26.0000			−1.46961			−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000			−1.46961			−0.734803	+	0.678280i	$0.762726\pi$
$314$		0			0
$315$	4.00000			0.225374
$316$		0			0
$317$			6.00000i			0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$			6.00000i			0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$		0			0
$319$			32.0000i			1.79166i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	3.00000	−	2.00000i	0.166410	−	0.110940i
$326$		0			0
$327$			14.0000i			0.774202i
$328$		0			0
$329$	32.0000			1.76422
$330$		0			0
$331$			12.0000i			0.659580i	0.944054	+	0.329790i	$0.106978\pi$
$331$			12.0000i			0.659580i	−0.944054	+	0.329790i	$0.893022\pi$
$332$		0			0
$333$			4.00000i			0.219199i
$334$		0			0
$335$	2.00000			0.109272
$336$		0			0
$337$	−22.0000			−1.19842			−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000			−1.19842			−0.599208	+	0.800593i	$0.704518\pi$
$338$		0			0
$339$	16.0000			0.869001
$340$		0			0
$341$	8.00000			0.433224
$342$		0			0
$343$			8.00000i			0.431959i
$344$		0			0
$345$			6.00000i			0.323029i
$346$		0			0
$347$	8.00000			0.429463			0.214731	−	0.976673i	$-0.431112\pi$
$347$	8.00000			0.429463			0.214731	+	0.976673i	$0.431112\pi$
$348$		0			0
$349$		−	14.0000i		−	0.749403i	−0.927146	−	0.374701i	$-0.877745\pi$
$349$		−	14.0000i		−	0.749403i	0.927146	−	0.374701i	$-0.122255\pi$
$350$		0			0
$351$	3.00000	−	2.00000i	0.160128	−	0.106752i
$352$		0			0
$353$			18.0000i			0.958043i	0.877803	+	0.479022i	$0.159008\pi$
$353$			18.0000i			0.958043i	−0.877803	+	0.479022i	$0.840992\pi$
$354$		0			0
$355$	12.0000			0.636894
$356$		0			0
$357$			32.0000i			1.69362i
$358$		0			0
$359$		−	24.0000i		−	1.26667i	−0.773877	−	0.633336i	$-0.781685\pi$
$359$		−	24.0000i		−	1.26667i	0.773877	−	0.633336i	$-0.218315\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$		0			0
$363$	5.00000			0.262432
$364$		0			0
$365$	−2.00000			−0.104685
$366$		0			0
$367$	8.00000			0.417597			0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000			0.417597			0.208798	+	0.977959i	$0.433045\pi$
$368$		0			0
$369$		−	10.0000i		−	0.520579i
$370$		0			0
$371$			40.0000i			2.07670i
$372$		0			0
$373$	6.00000			0.310668			0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000			0.310668			0.155334	+	0.987862i	$0.450355\pi$
$374$		0			0
$375$			1.00000i			0.0516398i
$376$		0			0
$377$	24.0000	−	16.0000i	1.23606	−	0.824042i
$378$		0			0
$379$			20.0000i			1.02733i	0.857991	+	0.513665i	$0.171713\pi$
$379$			20.0000i			1.02733i	−0.857991	+	0.513665i	$0.828287\pi$
$380$		0			0
$381$	16.0000			0.819705
$382$		0			0
$383$		−	24.0000i		−	1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$		−	24.0000i		−	1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$		0			0
$385$		−	16.0000i		−	0.815436i
$386$		0			0
$387$	−4.00000			−0.203331
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$	−48.0000			−2.42746
$392$		0			0
$393$	22.0000			1.10975
$394$		0			0
$395$			8.00000i			0.402524i
$396$		0			0
$397$			4.00000i			0.200754i	0.994949	+	0.100377i	$0.0320049\pi$
$397$			4.00000i			0.200754i	−0.994949	+	0.100377i	$0.967995\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			14.0000i			0.699127i	0.936913	+	0.349563i	$0.113670\pi$
$401$			14.0000i			0.699127i	−0.936913	+	0.349563i	$0.886330\pi$
$402$		0			0
$403$	−4.00000	−	6.00000i	−0.199254	−	0.298881i
$404$		0			0
$405$			1.00000i			0.0496904i
$406$		0			0
$407$	16.0000			0.793091
$408$		0			0
$409$		−	32.0000i		−	1.58230i	−0.611623	−	0.791149i	$-0.709483\pi$
$409$		−	32.0000i		−	1.58230i	0.611623	−	0.791149i	$-0.290517\pi$
$410$		0			0
$411$		−	6.00000i		−	0.295958i
$412$		0			0
$413$	−48.0000			−2.36193
$414$		0			0
$415$	8.00000			0.392705
$416$		0			0
$417$	16.0000			0.783523
$418$		0			0
$419$	26.0000			1.27018			0.635092	−	0.772437i	$-0.280962\pi$
$419$	26.0000			1.27018			0.635092	+	0.772437i	$0.280962\pi$
$420$		0			0
$421$			22.0000i			1.07221i	0.844150	+	0.536107i	$0.180106\pi$
$421$			22.0000i			1.07221i	−0.844150	+	0.536107i	$0.819894\pi$
$422$		0			0
$423$			8.00000i			0.388973i
$424$		0			0
$425$	−8.00000			−0.388057
$426$		0			0
$427$			8.00000i			0.387147i
$428$		0			0
$429$	−8.00000	−	12.0000i	−0.386244	−	0.579365i
$430$		0			0
$431$		−	36.0000i		−	1.73406i	−0.498257	−	0.867029i	$-0.666026\pi$
$431$		−	36.0000i		−	1.73406i	0.498257	−	0.867029i	$-0.333974\pi$
$432$		0			0
$433$	26.0000			1.24948			0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000			1.24948			0.624740	+	0.780833i	$0.285205\pi$
$434$		0			0
$435$			8.00000i			0.383571i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−24.0000			−1.14546			−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000			−1.14546			−0.572729	+	0.819745i	$0.694115\pi$
$440$		0			0
$441$	−9.00000			−0.428571
$442$		0			0
$443$	20.0000			0.950229			0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000			0.950229			0.475114	+	0.879924i	$0.342407\pi$
$444$		0			0
$445$	−6.00000			−0.284427
$446$		0			0
$447$		−	6.00000i		−	0.283790i
$448$		0			0
$449$		−	34.0000i		−	1.60456i	−0.596948	−	0.802280i	$-0.703620\pi$
$449$		−	34.0000i		−	1.60456i	0.596948	−	0.802280i	$-0.296380\pi$
$450$		0			0
$451$	−40.0000			−1.88353
$452$		0			0
$453$		−	2.00000i		−	0.0939682i
$454$		0			0
$455$	−12.0000	+	8.00000i	−0.562569	+	0.375046i
$456$		0			0
$457$			22.0000i			1.02912i	0.857455	+	0.514558i	$0.172044\pi$
$457$			22.0000i			1.02912i	−0.857455	+	0.514558i	$0.827956\pi$
$458$		0			0
$459$	−8.00000			−0.373408
$460$		0			0
$461$			18.0000i			0.838344i	0.907907	+	0.419172i	$0.137680\pi$
$461$			18.0000i			0.838344i	−0.907907	+	0.419172i	$0.862320\pi$
$462$		0			0
$463$			24.0000i			1.11537i	0.830051	+	0.557687i	$0.188311\pi$
$463$			24.0000i			1.11537i	−0.830051	+	0.557687i	$0.811689\pi$
$464$		0			0
$465$	2.00000			0.0927478
$466$		0			0
$467$	−36.0000			−1.66588			−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000			−1.66588			−0.832941	+	0.553362i	$0.813345\pi$
$468$		0			0
$469$	−8.00000			−0.369406
$470$		0			0
$471$	−2.00000			−0.0921551
$472$		0			0
$473$			16.0000i			0.735681i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−10.0000			−0.457869
$478$		0			0
$479$		−	8.00000i		−	0.365529i	−0.983157	−	0.182765i	$-0.941495\pi$
$479$		−	8.00000i		−	0.365529i	0.983157	−	0.182765i	$-0.0585046\pi$
$480$		0			0
$481$	−8.00000	−	12.0000i	−0.364769	−	0.547153i
$482$		0			0
$483$		−	24.0000i		−	1.09204i
$484$		0			0
$485$	10.0000			0.454077
$486$		0			0
$487$			12.0000i			0.543772i	0.962329	+	0.271886i	$0.0876473\pi$
$487$			12.0000i			0.543772i	−0.962329	+	0.271886i	$0.912353\pi$
$488$		0			0
$489$		−	14.0000i		−	0.633102i
$490$		0			0
$491$	26.0000			1.17336			0.586682	−	0.809818i	$-0.300434\pi$
$491$	26.0000			1.17336			0.586682	+	0.809818i	$0.300434\pi$
$492$		0			0
$493$	−64.0000			−2.88242
$494$		0			0
$495$	4.00000			0.179787
$496$		0			0
$497$	−48.0000			−2.15309
$498$		0			0
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$		0			0
$501$		−	8.00000i		−	0.357414i
$502$		0			0
$503$	14.0000			0.624229			0.312115	−	0.950044i	$-0.398963\pi$
$503$	14.0000			0.624229			0.312115	+	0.950044i	$0.398963\pi$
$504$		0			0
$505$		−	4.00000i		−	0.177998i
$506$		0			0
$507$	−5.00000	+	12.0000i	−0.222058	+	0.532939i
$508$		0			0
$509$			2.00000i			0.0886484i	0.999017	+	0.0443242i	$0.0141135\pi$
$509$			2.00000i			0.0886484i	−0.999017	+	0.0443242i	$0.985887\pi$
$510$		0			0
$511$	8.00000			0.353899
$512$		0			0
$513$		0			0
$514$		0			0
$515$			4.00000i			0.176261i
$516$		0			0
$517$	32.0000			1.40736
$518$		0			0
$519$	2.00000			0.0877903
$520$		0			0
$521$	−6.00000			−0.262865			−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000			−0.262865			−0.131432	+	0.991325i	$0.541958\pi$
$522$		0			0
$523$	4.00000			0.174908			0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000			0.174908			0.0874539	+	0.996169i	$0.472127\pi$
$524$		0			0
$525$		−	4.00000i		−	0.174574i
$526$		0			0
$527$			16.0000i			0.696971i
$528$		0			0
$529$	13.0000			0.565217
$530$		0			0
$531$		−	12.0000i		−	0.520756i
$532$		0			0
$533$	20.0000	+	30.0000i	0.866296	+	1.29944i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−2.00000			−0.0863064
$538$		0			0
$539$			36.0000i			1.55063i
$540$		0			0
$541$			10.0000i			0.429934i	0.976621	+	0.214967i	$0.0689643\pi$
$541$			10.0000i			0.429934i	−0.976621	+	0.214967i	$0.931036\pi$
$542$		0			0
$543$	−22.0000			−0.944110
$544$		0			0
$545$	14.0000			0.599694
$546$		0			0
$547$	44.0000			1.88130			0.940652	−	0.339372i	$-0.110215\pi$
$547$	44.0000			1.88130			0.940652	+	0.339372i	$0.110215\pi$
$548$		0			0
$549$	−2.00000			−0.0853579
$550$		0			0
$551$		0			0
$552$		0			0
$553$		−	32.0000i		−	1.36078i
$554$		0			0
$555$	4.00000			0.169791
$556$		0			0
$557$			46.0000i			1.94908i	0.224208	+	0.974541i	$0.428020\pi$
$557$			46.0000i			1.94908i	−0.224208	+	0.974541i	$0.571980\pi$
$558$		0			0
$559$	12.0000	−	8.00000i	0.507546	−	0.338364i
$560$		0			0
$561$			32.0000i			1.35104i
$562$		0			0
$563$	12.0000			0.505740			0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000			0.505740			0.252870	+	0.967500i	$0.418626\pi$
$564$		0			0
$565$		−	16.0000i		−	0.673125i
$566$		0			0
$567$		−	4.00000i		−	0.167984i
$568$		0			0
$569$	−26.0000			−1.08998			−0.544988	−	0.838444i	$-0.683466\pi$
$569$	−26.0000			−1.08998			−0.544988	+	0.838444i	$0.683466\pi$
$570$		0			0
$571$	−40.0000			−1.67395			−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000			−1.67395			−0.836974	+	0.547243i	$0.815677\pi$
$572$		0			0
$573$	−4.00000			−0.167102
$574$		0			0
$575$	6.00000			0.250217
$576$		0			0
$577$		−	10.0000i		−	0.416305i	−0.978096	−	0.208153i	$-0.933255\pi$
$577$		−	10.0000i		−	0.416305i	0.978096	−	0.208153i	$-0.0667451\pi$
$578$		0			0
$579$		−	2.00000i		−	0.0831172i
$580$		0			0
$581$	−32.0000			−1.32758
$582$		0			0
$583$			40.0000i			1.65663i
$584$		0			0
$585$	−2.00000	−	3.00000i	−0.0826898	−	0.124035i
$586$		0			0
$587$			20.0000i			0.825488i	0.910847	+	0.412744i	$0.135430\pi$
$587$			20.0000i			0.825488i	−0.910847	+	0.412744i	$0.864570\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$			2.00000i			0.0822690i
$592$		0			0
$593$		−	18.0000i		−	0.739171i	−0.929197	−	0.369586i	$-0.879500\pi$
$593$		−	18.0000i		−	0.739171i	0.929197	−	0.369586i	$-0.120500\pi$
$594$		0			0
$595$	32.0000			1.31187
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−48.0000			−1.96123			−0.980613	−	0.195952i	$-0.937220\pi$
$599$	−48.0000			−1.96123			−0.980613	+	0.195952i	$0.937220\pi$
$600$		0			0
$601$	26.0000			1.06056			0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000			1.06056			0.530281	+	0.847822i	$0.322086\pi$
$602$		0			0
$603$		−	2.00000i		−	0.0814463i
$604$		0			0
$605$		−	5.00000i		−	0.203279i
$606$		0			0
$607$	−8.00000			−0.324710			−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000			−0.324710			−0.162355	+	0.986732i	$0.551909\pi$
$608$		0			0
$609$		−	32.0000i		−	1.29671i
$610$		0			0
$611$	−16.0000	−	24.0000i	−0.647291	−	0.970936i
$612$		0			0
$613$		−	20.0000i		−	0.807792i	−0.914805	−	0.403896i	$-0.867656\pi$
$613$		−	20.0000i		−	0.807792i	0.914805	−	0.403896i	$-0.132344\pi$
$614$		0			0
$615$	−10.0000			−0.403239
$616$		0			0
$617$		−	26.0000i		−	1.04672i	−0.852111	−	0.523360i	$-0.824678\pi$
$617$		−	26.0000i		−	1.04672i	0.852111	−	0.523360i	$-0.175322\pi$
$618$		0			0
$619$		−	28.0000i		−	1.12542i	−0.826656	−	0.562708i	$-0.809760\pi$
$619$		−	28.0000i		−	1.12542i	0.826656	−	0.562708i	$-0.190240\pi$
$620$		0			0
$621$	6.00000			0.240772
$622$		0			0
$623$	24.0000			0.961540
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$			32.0000i			1.27592i
$630$		0			0
$631$		−	10.0000i		−	0.398094i	−0.979990	−	0.199047i	$-0.936215\pi$
$631$		−	10.0000i		−	0.398094i	0.979990	−	0.199047i	$-0.0637846\pi$
$632$		0			0
$633$	−24.0000			−0.953914
$634$		0			0
$635$		−	16.0000i		−	0.634941i
$636$		0			0
$637$	27.0000	−	18.0000i	1.06978	−	0.713186i
$638$		0			0
$639$		−	12.0000i		−	0.474713i
$640$		0			0
$641$	38.0000			1.50091			0.750455	−	0.660922i	$-0.229834\pi$
$641$	38.0000			1.50091			0.750455	+	0.660922i	$0.229834\pi$
$642$		0			0
$643$			6.00000i			0.236617i	0.992977	+	0.118308i	$0.0377472\pi$
$643$			6.00000i			0.236617i	−0.992977	+	0.118308i	$0.962253\pi$
$644$		0			0
$645$			4.00000i			0.157500i
$646$		0			0
$647$	−46.0000			−1.80845			−0.904223	−	0.427060i	$-0.859549\pi$
$647$	−46.0000			−1.80845			−0.904223	+	0.427060i	$0.859549\pi$
$648$		0			0
$649$	−48.0000			−1.88416
$650$		0			0
$651$	−8.00000			−0.313545
$652$		0			0
$653$	6.00000			0.234798			0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000			0.234798			0.117399	+	0.993085i	$0.462544\pi$
$654$		0			0
$655$		−	22.0000i		−	0.859611i
$656$		0			0
$657$			2.00000i			0.0780274i
$658$		0			0
$659$	−26.0000			−1.01282			−0.506408	−	0.862294i	$-0.669027\pi$
$659$	−26.0000			−1.01282			−0.506408	+	0.862294i	$0.669027\pi$
$660$		0			0
$661$		−	10.0000i		−	0.388955i	−0.980907	−	0.194477i	$-0.937699\pi$
$661$		−	10.0000i		−	0.388955i	0.980907	−	0.194477i	$-0.0623011\pi$
$662$		0			0
$663$	24.0000	−	16.0000i	0.932083	−	0.621389i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	48.0000			1.85857
$668$		0			0
$669$		−	16.0000i		−	0.618596i
$670$		0			0
$671$			8.00000i			0.308837i
$672$		0			0
$673$	30.0000			1.15642			0.578208	−	0.815890i	$-0.303752\pi$
$673$	30.0000			1.15642			0.578208	+	0.815890i	$0.303752\pi$
$674$		0			0
$675$	1.00000			0.0384900
$676$		0			0
$677$	18.0000			0.691796			0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000			0.691796			0.345898	+	0.938272i	$0.387574\pi$
$678$		0			0
$679$	−40.0000			−1.53506
$680$		0			0
$681$			24.0000i			0.919682i
$682$		0			0
$683$		−	12.0000i		−	0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$		−	12.0000i		−	0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$		0			0
$685$	−6.00000			−0.229248
$686$		0			0
$687$		−	22.0000i		−	0.839352i
$688$		0			0
$689$	30.0000	−	20.0000i	1.14291	−	0.761939i
$690$		0			0
$691$			20.0000i			0.760836i	0.924815	+	0.380418i	$0.124220\pi$
$691$			20.0000i			0.760836i	−0.924815	+	0.380418i	$0.875780\pi$
$692$		0			0
$693$	−16.0000			−0.607790
$694$		0			0
$695$		−	16.0000i		−	0.606915i
$696$		0			0
$697$		−	80.0000i		−	3.03022i
$698$		0			0
$699$	4.00000			0.151294
$700$		0			0
$701$	48.0000			1.81293			0.906467	−	0.422276i	$-0.138769\pi$
$701$	48.0000			1.81293			0.906467	+	0.422276i	$0.138769\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	8.00000			0.301297
$706$		0			0
$707$			16.0000i			0.601742i
$708$		0			0
$709$		−	42.0000i		−	1.57734i	−0.614815	−	0.788672i	$-0.710769\pi$
$709$		−	42.0000i		−	1.57734i	0.614815	−	0.788672i	$-0.289231\pi$
$710$		0			0
$711$	8.00000			0.300023
$712$		0			0
$713$		−	12.0000i		−	0.449404i
$714$		0			0
$715$	−12.0000	+	8.00000i	−0.448775	+	0.299183i
$716$		0			0
$717$			8.00000i			0.298765i
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		−	16.0000i		−	0.595871i
$722$		0			0
$723$		−	4.00000i		−	0.148762i
$724$		0			0
$725$	8.00000			0.297113
$726$		0			0
$727$	−12.0000			−0.445055			−0.222528	−	0.974926i	$-0.571431\pi$
$727$	−12.0000			−0.445055			−0.222528	+	0.974926i	$0.571431\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−32.0000			−1.18356
$732$		0			0
$733$			16.0000i			0.590973i	0.955347	+	0.295487i	$0.0954818\pi$
$733$			16.0000i			0.590973i	−0.955347	+	0.295487i	$0.904518\pi$
$734$		0			0
$735$			9.00000i			0.331970i
$736$		0			0
$737$	−8.00000			−0.294684
$738$		0			0
$739$			4.00000i			0.147142i	0.997290	+	0.0735712i	$0.0234396\pi$
$739$			4.00000i			0.147142i	−0.997290	+	0.0735712i	$0.976560\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$	−6.00000			−0.219823
$746$		0			0
$747$		−	8.00000i		−	0.292705i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−40.0000			−1.45962			−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000			−1.45962			−0.729810	+	0.683650i	$0.760392\pi$
$752$		0			0
$753$	18.0000			0.655956
$754$		0			0
$755$	−2.00000			−0.0727875
$756$		0			0
$757$	26.0000			0.944986			0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000			0.944986			0.472493	+	0.881334i	$0.343354\pi$
$758$		0			0
$759$		−	24.0000i		−	0.871145i
$760$		0			0
$761$		−	6.00000i		−	0.217500i	−0.994069	−	0.108750i	$-0.965315\pi$
$761$		−	6.00000i		−	0.217500i	0.994069	−	0.108750i	$-0.0346848\pi$
$762$		0			0
$763$	−56.0000			−2.02734
$764$		0			0
$765$			8.00000i			0.289241i
$766$		0			0
$767$	24.0000	+	36.0000i	0.866590	+	1.29988i
$768$		0			0
$769$			40.0000i			1.44244i	0.692708	+	0.721218i	$0.256418\pi$
$769$			40.0000i			1.44244i	−0.692708	+	0.721218i	$0.743582\pi$
$770$		0			0
$771$	−16.0000			−0.576226
$772$		0			0
$773$		−	46.0000i		−	1.65451i	−0.561830	−	0.827253i	$-0.689903\pi$
$773$		−	46.0000i		−	1.65451i	0.561830	−	0.827253i	$-0.310097\pi$
$774$		0			0
$775$		−	2.00000i		−	0.0718421i
$776$		0			0
$777$	−16.0000			−0.573997
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−48.0000			−1.71758
$782$		0			0
$783$	8.00000			0.285897
$784$		0			0
$785$			2.00000i			0.0713831i
$786$		0			0
$787$		−	38.0000i		−	1.35455i	−0.735728	−	0.677277i	$-0.763160\pi$
$787$		−	38.0000i		−	1.35455i	0.735728	−	0.677277i	$-0.236840\pi$
$788$		0			0
$789$	−6.00000			−0.213606
$790$		0			0
$791$			64.0000i			2.27558i
$792$		0			0
$793$	6.00000	−	4.00000i	0.213066	−	0.142044i
$794$		0			0
$795$			10.0000i			0.354663i
$796$		0			0
$797$	18.0000			0.637593			0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000			0.637593			0.318796	+	0.947823i	$0.396721\pi$
$798$		0			0
$799$			64.0000i			2.26416i
$800$		0			0
$801$			6.00000i			0.212000i
$802$		0			0
$803$	8.00000			0.282314
$804$		0			0
$805$	−24.0000			−0.845889
$806$		0			0
$807$	−12.0000			−0.422420
$808$		0			0
$809$	46.0000			1.61727			0.808637	−	0.588308i	$-0.200206\pi$
$809$	46.0000			1.61727			0.808637	+	0.588308i	$0.200206\pi$
$810$		0			0
$811$		−	32.0000i		−	1.12367i	−0.827249	−	0.561836i	$-0.810095\pi$
$811$		−	32.0000i		−	1.12367i	0.827249	−	0.561836i	$-0.189905\pi$
$812$		0			0
$813$			10.0000i			0.350715i
$814$		0			0
$815$	−14.0000			−0.490399
$816$		0			0
$817$		0			0
$818$		0			0
$819$	8.00000	+	12.0000i	0.279543	+	0.419314i
$820$		0			0
$821$			38.0000i			1.32621i	0.748527	+	0.663105i	$0.230762\pi$
$821$			38.0000i			1.32621i	−0.748527	+	0.663105i	$0.769238\pi$
$822$		0			0
$823$	36.0000			1.25488			0.627441	−	0.778664i	$-0.284103\pi$
$823$	36.0000			1.25488			0.627441	+	0.778664i	$0.284103\pi$
$824$		0			0
$825$		−	4.00000i		−	0.139262i
$826$		0			0
$827$			4.00000i			0.139094i	0.997579	+	0.0695468i	$0.0221553\pi$
$827$			4.00000i			0.139094i	−0.997579	+	0.0695468i	$0.977845\pi$
$828$		0			0
$829$	−6.00000			−0.208389			−0.104194	−	0.994557i	$-0.533226\pi$
$829$	−6.00000			−0.208389			−0.104194	+	0.994557i	$0.533226\pi$
$830$		0			0
$831$	−26.0000			−0.901930
$832$		0			0
$833$	−72.0000			−2.49465
$834$		0			0
$835$	−8.00000			−0.276851
$836$		0			0
$837$		−	2.00000i		−	0.0691301i
$838$		0			0
$839$		−	40.0000i		−	1.38095i	−0.723355	−	0.690477i	$-0.757401\pi$
$839$		−	40.0000i		−	1.38095i	0.723355	−	0.690477i	$-0.242599\pi$
$840$		0			0
$841$	35.0000			1.20690
$842$		0			0
$843$			30.0000i			1.03325i
$844$		0			0
$845$	12.0000	+	5.00000i	0.412813	+	0.172005i
$846$		0			0
$847$			20.0000i			0.687208i
$848$		0			0
$849$	4.00000			0.137280
$850$		0			0
$851$		−	24.0000i		−	0.822709i
$852$		0			0
$853$		−	44.0000i		−	1.50653i	−0.657716	−	0.753266i	$-0.728477\pi$
$853$		−	44.0000i		−	1.50653i	0.657716	−	0.753266i	$-0.271523\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−24.0000			−0.819824			−0.409912	−	0.912125i	$-0.634441\pi$
$857$	−24.0000			−0.819824			−0.409912	+	0.912125i	$0.634441\pi$
$858$		0			0
$859$	−16.0000			−0.545913			−0.272956	−	0.962026i	$-0.588002\pi$
$859$	−16.0000			−0.545913			−0.272956	+	0.962026i	$0.588002\pi$
$860$		0			0
$861$	40.0000			1.36320
$862$		0			0
$863$			32.0000i			1.08929i	0.838666	+	0.544646i	$0.183336\pi$
$863$			32.0000i			1.08929i	−0.838666	+	0.544646i	$0.816664\pi$
$864$		0			0
$865$		−	2.00000i		−	0.0680020i
$866$		0			0
$867$	−47.0000			−1.59620
$868$		0			0
$869$		−	32.0000i		−	1.08553i
$870$		0			0
$871$	4.00000	+	6.00000i	0.135535	+	0.203302i
$872$		0			0
$873$		−	10.0000i		−	0.338449i
$874$		0			0
$875$	−4.00000			−0.135225
$876$		0			0
$877$		−	32.0000i		−	1.08056i	−0.841484	−	0.540282i	$-0.818318\pi$
$877$		−	32.0000i		−	1.08056i	0.841484	−	0.540282i	$-0.181682\pi$
$878$		0			0
$879$			26.0000i			0.876958i
$880$		0			0
$881$	14.0000			0.471672			0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000			0.471672			0.235836	+	0.971793i	$0.424217\pi$
$882$		0			0
$883$	−12.0000			−0.403832			−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000			−0.403832			−0.201916	+	0.979403i	$0.564717\pi$
$884$		0			0
$885$	−12.0000			−0.403376
$886$		0			0
$887$	14.0000			0.470074			0.235037	−	0.971986i	$-0.424479\pi$
$887$	14.0000			0.470074			0.235037	+	0.971986i	$0.424479\pi$
$888$		0			0
$889$			64.0000i			2.14649i
$890$		0			0
$891$		−	4.00000i		−	0.134005i
$892$		0			0
$893$		0			0
$894$		0			0
$895$			2.00000i			0.0668526i
$896$		0			0
$897$	−18.0000	+	12.0000i	−0.601003	+	0.400668i
$898$		0			0
$899$		−	16.0000i		−	0.533630i
$900$		0			0
$901$	−80.0000			−2.66519
$902$		0			0
$903$		−	16.0000i		−	0.532447i
$904$		0			0
$905$			22.0000i			0.731305i
$906$		0			0
$907$	−36.0000			−1.19536			−0.597680	−	0.801735i	$-0.703911\pi$
$907$	−36.0000			−1.19536			−0.597680	+	0.801735i	$0.703911\pi$
$908$		0			0
$909$	−4.00000			−0.132672
$910$		0			0
$911$	−24.0000			−0.795155			−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000			−0.795155			−0.397578	+	0.917568i	$0.630149\pi$
$912$		0			0
$913$	−32.0000			−1.05905
$914$		0			0
$915$			2.00000i			0.0661180i
$916$		0			0
$917$			88.0000i			2.90602i
$918$		0			0
$919$	−16.0000			−0.527791			−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000			−0.527791			−0.263896	+	0.964551i	$0.585007\pi$
$920$		0			0
$921$		−	6.00000i		−	0.197707i
$922$		0			0
$923$	24.0000	+	36.0000i	0.789970	+	1.18495i
$924$		0			0
$925$		−	4.00000i		−	0.131519i
$926$		0			0
$927$	4.00000			0.131377
$928$		0			0
$929$			6.00000i			0.196854i	0.995144	+	0.0984268i	$0.0313810\pi$
$929$			6.00000i			0.196854i	−0.995144	+	0.0984268i	$0.968619\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−28.0000			−0.916679
$934$		0			0
$935$	32.0000			1.04651
$936$		0			0
$937$	2.00000			0.0653372			0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000			0.0653372			0.0326686	+	0.999466i	$0.489599\pi$
$938$		0			0
$939$	26.0000			0.848478
$940$		0			0
$941$		−	30.0000i		−	0.977972i	−0.872292	−	0.488986i	$-0.837367\pi$
$941$		−	30.0000i		−	0.977972i	0.872292	−	0.488986i	$-0.162633\pi$
$942$		0			0
$943$			60.0000i			1.95387i
$944$		0			0
$945$	−4.00000			−0.130120
$946$		0			0
$947$		−	40.0000i		−	1.29983i	−0.760009	−	0.649913i	$-0.774805\pi$
$947$		−	40.0000i		−	1.29983i	0.760009	−	0.649913i	$-0.225195\pi$
$948$		0			0
$949$	−4.00000	−	6.00000i	−0.129845	−	0.194768i
$950$		0			0
$951$		−	6.00000i		−	0.194563i
$952$		0			0
$953$	−36.0000			−1.16615			−0.583077	−	0.812417i	$-0.698151\pi$
$953$	−36.0000			−1.16615			−0.583077	+	0.812417i	$0.698151\pi$
$954$		0			0
$955$			4.00000i			0.129437i
$956$		0			0
$957$		−	32.0000i		−	1.03441i
$958$		0			0
$959$	24.0000			0.775000
$960$		0			0
$961$	27.0000			0.870968
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2.00000			−0.0643823
$966$		0			0
$967$		−	28.0000i		−	0.900419i	−0.892923	−	0.450210i	$-0.851349\pi$
$967$		−	28.0000i		−	0.900419i	0.892923	−	0.450210i	$-0.148651\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	2.00000			0.0641831			0.0320915	−	0.999485i	$-0.489783\pi$
$971$	2.00000			0.0641831			0.0320915	+	0.999485i	$0.489783\pi$
$972$		0			0
$973$			64.0000i			2.05175i
$974$		0			0
$975$	−3.00000	+	2.00000i	−0.0960769	+	0.0640513i
$976$		0			0
$977$		−	30.0000i		−	0.959785i	−0.877327	−	0.479893i	$-0.840676\pi$
$977$		−	30.0000i		−	0.959785i	0.877327	−	0.479893i	$-0.159324\pi$
$978$		0			0
$979$	24.0000			0.767043
$980$		0			0
$981$		−	14.0000i		−	0.446986i
$982$		0			0
$983$		−	16.0000i		−	0.510321i	−0.966899	−	0.255160i	$-0.917872\pi$
$983$		−	16.0000i		−	0.510321i	0.966899	−	0.255160i	$-0.0821283\pi$
$984$		0			0
$985$	2.00000			0.0637253
$986$		0			0
$987$	−32.0000			−1.01857
$988$		0			0
$989$	24.0000			0.763156
$990$		0			0
$991$	8.00000			0.254128			0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000			0.254128			0.127064	+	0.991894i	$0.459445\pi$
$992$		0			0
$993$		−	12.0000i		−	0.380808i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−54.0000			−1.71020			−0.855099	−	0.518465i	$-0.826503\pi$
$997$	−54.0000			−1.71020			−0.855099	+	0.518465i	$0.826503\pi$
$998$		0			0
$999$		−	4.00000i		−	0.126554i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1560.2.g.c.961.2	yes	2
3.2	odd	2		4680.2.g.a.2521.1		2
4.3	odd	2		3120.2.g.g.961.2		2
13.12	even	2	inner	1560.2.g.c.961.1	✓	2
39.38	odd	2		4680.2.g.a.2521.2		2
52.51	odd	2		3120.2.g.g.961.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.g.c.961.1	✓	2	13.12	even	2	inner
1560.2.g.c.961.2	yes	2	1.1	even	1	trivial
3120.2.g.g.961.1		2	52.51	odd	2
3120.2.g.g.961.2		2	4.3	odd	2
4680.2.g.a.2521.1		2	3.2	odd	2
4680.2.g.a.2521.2		2	39.38	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000i q^{5} -4.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000i q^{5} -4.00000i q^{7} +1.00000 q^{9} -4.00000i q^{11} +(-3.00000 + 2.00000i) q^{13} -1.00000i q^{15} +8.00000 q^{17} +4.00000i q^{21} -6.00000 q^{23} -1.00000 q^{25} -1.00000 q^{27} -8.00000 q^{29} +2.00000i q^{31} +4.00000i q^{33} +4.00000 q^{35} +4.00000i q^{37} +(3.00000 - 2.00000i) q^{39} -10.0000i q^{41} -4.00000 q^{43} +1.00000i q^{45} +8.00000i q^{47} -9.00000 q^{49} -8.00000 q^{51} -10.0000 q^{53} +4.00000 q^{55} -12.0000i q^{59} -2.00000 q^{61} -4.00000i q^{63} +(-2.00000 - 3.00000i) q^{65} -2.00000i q^{67} +6.00000 q^{69} -12.0000i q^{71} +2.00000i q^{73} +1.00000 q^{75} -16.0000 q^{77} +8.00000 q^{79} +1.00000 q^{81} -8.00000i q^{83} +8.00000i q^{85} +8.00000 q^{87} +6.00000i q^{89} +(8.00000 + 12.0000i) q^{91} -2.00000i q^{93} -10.0000i q^{97} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{9} - 6 q^{13} + 16 q^{17} - 12 q^{23} - 2 q^{25} - 2 q^{27} - 16 q^{29} + 8 q^{35} + 6 q^{39} - 8 q^{43} - 18 q^{49} - 16 q^{51} - 20 q^{53} + 8 q^{55} - 4 q^{61} - 4 q^{65} + 12 q^{69} + 2 q^{75} - 32 q^{77} + 16 q^{79} + 2 q^{81} + 16 q^{87} + 16 q^{91}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^9 - 6 * q^13 + 16 * q^17 - 12 * q^23 - 2 * q^25 - 2 * q^27 - 16 * q^29 + 8 * q^35 + 6 * q^39 - 8 * q^43 - 18 * q^49 - 16 * q^51 - 20 * q^53 + 8 * q^55 - 4 * q^61 - 4 * q^65 + 12 * q^69 + 2 * q^75 - 32 * q^77 + 16 * q^79 + 2 * q^81 + 16 * q^87 + 16 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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