LMFDB - Embedded newform 1560.2.g.f.961.4

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:	$\Q(i, \sqrt{41})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 21x^{2} + 100 $ x^4 + 21*x^2 + 100
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			961.4
Root			$3.70156i$ of defining polynomial
Character	$\chi$	$=$	1560.961
Dual form			1560.2.g.f.961.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000i q^{5} +4.70156i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000i q^{5} +4.70156i q^{7} +1.00000 q^{9} -2.70156i q^{11} +(2.00000 - 3.00000i) q^{13} -1.00000i q^{15} +6.70156 q^{17} -7.40312i q^{19} -4.70156i q^{21} +2.70156 q^{23} -1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{29} +9.40312i q^{31} +2.70156i q^{33} -4.70156 q^{35} +2.70156i q^{37} +(-2.00000 + 3.00000i) q^{39} +8.70156i q^{41} -1.40312 q^{43} +1.00000i q^{45} +8.00000i q^{47} -15.1047 q^{49} -6.70156 q^{51} +8.70156 q^{53} +2.70156 q^{55} +7.40312i q^{57} +0.596876i q^{59} +6.70156 q^{61} +4.70156i q^{63} +(3.00000 + 2.00000i) q^{65} +8.00000i q^{67} -2.70156 q^{69} -0.701562i q^{71} -12.8062i q^{73} +1.00000 q^{75} +12.7016 q^{77} -3.29844 q^{79} +1.00000 q^{81} +9.40312i q^{83} +6.70156i q^{85} -2.00000 q^{87} +4.70156i q^{89} +(14.1047 + 9.40312i) q^{91} -9.40312i q^{93} +7.40312 q^{95} -16.1047i q^{97} -2.70156i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 8 q^{13} + 14 q^{17} - 2 q^{23} - 4 q^{25} - 4 q^{27} + 8 q^{29} - 6 q^{35} - 8 q^{39} + 20 q^{43} - 22 q^{49} - 14 q^{51} + 22 q^{53} - 2 q^{55} + 14 q^{61} + 12 q^{65} + 2 q^{69} + 4 q^{75} + 38 q^{77} - 26 q^{79} + 4 q^{81} - 8 q^{87} + 18 q^{91} + 4 q^{95}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^9 + 8 * q^13 + 14 * q^17 - 2 * q^23 - 4 * q^25 - 4 * q^27 + 8 * q^29 - 6 * q^35 - 8 * q^39 + 20 * q^43 - 22 * q^49 - 14 * q^51 + 22 * q^53 - 2 * q^55 + 14 * q^61 + 12 * q^65 + 2 * q^69 + 4 * q^75 + 38 * q^77 - 26 * q^79 + 4 * q^81 - 8 * q^87 + 18 * 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
$7$			4.70156i			1.77702i	0.458854	+	0.888512i	$0.348260\pi$
$7$			4.70156i			1.77702i	−0.458854	+	0.888512i	$0.651740\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	2.70156i		−	0.814552i	−0.913305	−	0.407276i	$-0.866479\pi$
$11$		−	2.70156i		−	0.814552i	0.913305	−	0.407276i	$-0.133521\pi$
$12$		0			0
$13$	2.00000	−	3.00000i	0.554700	−	0.832050i
$13$	2.00000	−	3.00000i	0.554700	−	0.832050i
$14$		0			0
$15$		−	1.00000i		−	0.258199i
$16$		0			0
$17$	6.70156			1.62537			0.812684	−	0.582705i	$-0.198006\pi$
$17$	6.70156			1.62537			0.812684	+	0.582705i	$0.198006\pi$
$18$		0			0
$19$		−	7.40312i		−	1.69839i	−0.528077	−	0.849197i	$-0.677087\pi$
$19$		−	7.40312i		−	1.69839i	0.528077	−	0.849197i	$-0.322913\pi$
$20$		0			0
$21$		−	4.70156i		−	1.02596i
$22$		0			0
$23$	2.70156			0.563315			0.281657	−	0.959515i	$-0.409116\pi$
$23$	2.70156			0.563315			0.281657	+	0.959515i	$0.409116\pi$
$24$		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	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$			9.40312i			1.68885i	0.535673	+	0.844425i	$0.320058\pi$
$31$			9.40312i			1.68885i	−0.535673	+	0.844425i	$0.679942\pi$
$32$		0			0
$33$			2.70156i			0.470282i
$34$		0			0
$35$	−4.70156			−0.794709
$36$		0			0
$37$			2.70156i			0.444134i	0.975031	+	0.222067i	$0.0712804\pi$
$37$			2.70156i			0.444134i	−0.975031	+	0.222067i	$0.928720\pi$
$38$		0			0
$39$	−2.00000	+	3.00000i	−0.320256	+	0.480384i
$40$		0			0
$41$			8.70156i			1.35896i	0.733696	+	0.679478i	$0.237794\pi$
$41$			8.70156i			1.35896i	−0.733696	+	0.679478i	$0.762206\pi$
$42$		0			0
$43$	−1.40312			−0.213974			−0.106987	−	0.994260i	$-0.534120\pi$
$43$	−1.40312			−0.213974			−0.106987	+	0.994260i	$0.534120\pi$
$44$		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
$49$	−15.1047			−2.15781
$50$		0			0
$51$	−6.70156			−0.938406
$52$		0			0
$53$	8.70156			1.19525			0.597626	−	0.801775i	$-0.296111\pi$
$53$	8.70156			1.19525			0.597626	+	0.801775i	$0.296111\pi$
$54$		0			0
$55$	2.70156			0.364279
$56$		0			0
$57$			7.40312i			0.980568i
$58$		0			0
$59$			0.596876i			0.0777066i	0.999245	+	0.0388533i	$0.0123705\pi$
$59$			0.596876i			0.0777066i	−0.999245	+	0.0388533i	$0.987629\pi$
$60$		0			0
$61$	6.70156			0.858047			0.429024	−	0.903293i	$-0.358858\pi$
$61$	6.70156			0.858047			0.429024	+	0.903293i	$0.358858\pi$
$62$		0			0
$63$			4.70156i			0.592341i
$64$		0			0
$65$	3.00000	+	2.00000i	0.372104	+	0.248069i
$66$		0			0
$67$			8.00000i			0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$			8.00000i			0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$		0			0
$69$	−2.70156			−0.325230
$70$		0			0
$71$		−	0.701562i		−	0.0832601i	−0.999133	−	0.0416301i	$-0.986745\pi$
$71$		−	0.701562i		−	0.0832601i	0.999133	−	0.0416301i	$-0.0132551\pi$
$72$		0			0
$73$		−	12.8062i		−	1.49886i	−0.662085	−	0.749429i	$-0.730328\pi$
$73$		−	12.8062i		−	1.49886i	0.662085	−	0.749429i	$-0.269672\pi$
$74$		0			0
$75$	1.00000			0.115470
$76$		0			0
$77$	12.7016			1.44748
$78$		0			0
$79$	−3.29844			−0.371103			−0.185552	−	0.982635i	$-0.559407\pi$
$79$	−3.29844			−0.371103			−0.185552	+	0.982635i	$0.559407\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			9.40312i			1.03213i	0.856550	+	0.516063i	$0.172603\pi$
$83$			9.40312i			1.03213i	−0.856550	+	0.516063i	$0.827397\pi$
$84$		0			0
$85$			6.70156i			0.726886i
$86$		0			0
$87$	−2.00000			−0.214423
$88$		0			0
$89$			4.70156i			0.498365i	0.968457	+	0.249182i	$0.0801618\pi$
$89$			4.70156i			0.498365i	−0.968457	+	0.249182i	$0.919838\pi$
$90$		0			0
$91$	14.1047	+	9.40312i	1.47857	+	0.985715i
$92$		0			0
$93$		−	9.40312i		−	0.975059i
$94$		0			0
$95$	7.40312			0.759545
$96$		0			0
$97$		−	16.1047i		−	1.63518i	−0.575799	−	0.817592i	$-0.695309\pi$
$97$		−	16.1047i		−	1.63518i	0.575799	−	0.817592i	$-0.304691\pi$
$98$		0			0
$99$		−	2.70156i		−	0.271517i
$100$		0			0
$101$	−8.80625			−0.876254			−0.438127	−	0.898913i	$-0.644358\pi$
$101$	−8.80625			−0.876254			−0.438127	+	0.898913i	$0.644358\pi$
$102$		0			0
$103$	−3.40312			−0.335320			−0.167660	−	0.985845i	$-0.553621\pi$
$103$	−3.40312			−0.335320			−0.167660	+	0.985845i	$0.553621\pi$
$104$		0			0
$105$	4.70156			0.458825
$106$		0			0
$107$	−6.10469			−0.590162			−0.295081	−	0.955472i	$-0.595347\pi$
$107$	−6.10469			−0.590162			−0.295081	+	0.955472i	$0.595347\pi$
$108$		0			0
$109$			6.00000i			0.574696i	0.957826	+	0.287348i	$0.0927736\pi$
$109$			6.00000i			0.574696i	−0.957826	+	0.287348i	$0.907226\pi$
$110$		0			0
$111$		−	2.70156i		−	0.256421i
$112$		0			0
$113$	14.0000			1.31701			0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000			1.31701			0.658505	+	0.752577i	$0.271189\pi$
$114$		0			0
$115$			2.70156i			0.251922i
$116$		0			0
$117$	2.00000	−	3.00000i	0.184900	−	0.277350i
$118$		0			0
$119$			31.5078i			2.88832i
$120$		0			0
$121$	3.70156			0.336506
$122$		0			0
$123$		−	8.70156i		−	0.784593i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$	8.80625			0.781428			0.390714	−	0.920512i	$-0.372228\pi$
$127$	8.80625			0.781428			0.390714	+	0.920512i	$0.372228\pi$
$128$		0			0
$129$	1.40312			0.123538
$130$		0			0
$131$	−12.0000			−1.04844			−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$		0			0
$133$	34.8062			3.01808
$134$		0			0
$135$		−	1.00000i		−	0.0860663i
$136$		0			0
$137$		−	8.80625i		−	0.752369i	−0.926545	−	0.376184i	$-0.877236\pi$
$137$		−	8.80625i		−	0.752369i	0.926545	−	0.376184i	$-0.122764\pi$
$138$		0			0
$139$	10.1047			0.857068			0.428534	−	0.903526i	$-0.359030\pi$
$139$	10.1047			0.857068			0.428534	+	0.903526i	$0.359030\pi$
$140$		0			0
$141$		−	8.00000i		−	0.673722i
$142$		0			0
$143$	−8.10469	−	5.40312i	−0.677748	−	0.451832i
$144$		0			0
$145$			2.00000i			0.166091i
$146$		0			0
$147$	15.1047			1.24581
$148$		0			0
$149$			14.7016i			1.20440i	0.798346	+	0.602199i	$0.205709\pi$
$149$			14.7016i			1.20440i	−0.798346	+	0.602199i	$0.794291\pi$
$150$		0			0
$151$		−	5.40312i		−	0.439700i	−0.975534	−	0.219850i	$-0.929443\pi$
$151$		−	5.40312i		−	0.439700i	0.975534	−	0.219850i	$-0.0705568\pi$
$152$		0			0
$153$	6.70156			0.541789
$154$		0			0
$155$	−9.40312			−0.755277
$156$		0			0
$157$	−22.8062			−1.82014			−0.910068	−	0.414458i	$-0.863971\pi$
$157$	−22.8062			−1.82014			−0.910068	+	0.414458i	$0.863971\pi$
$158$		0			0
$159$	−8.70156			−0.690079
$160$		0			0
$161$			12.7016i			1.00102i
$162$		0			0
$163$		−	7.29844i		−	0.571658i	−0.958281	−	0.285829i	$-0.907731\pi$
$163$		−	7.29844i		−	0.571658i	0.958281	−	0.285829i	$-0.0922689\pi$
$164$		0			0
$165$	−2.70156			−0.210316
$166$		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
$171$		−	7.40312i		−	0.566131i
$172$		0			0
$173$	10.5969			0.805666			0.402833	−	0.915274i	$-0.368026\pi$
$173$	10.5969			0.805666			0.402833	+	0.915274i	$0.368026\pi$
$174$		0			0
$175$		−	4.70156i		−	0.355405i
$176$		0			0
$177$		−	0.596876i		−	0.0448639i
$178$		0			0
$179$	6.80625			0.508723			0.254361	−	0.967109i	$-0.418135\pi$
$179$	6.80625			0.508723			0.254361	+	0.967109i	$0.418135\pi$
$180$		0			0
$181$	8.10469			0.602417			0.301208	−	0.953558i	$-0.402610\pi$
$181$	8.10469			0.602417			0.301208	+	0.953558i	$0.402610\pi$
$182$		0			0
$183$	−6.70156			−0.495394
$184$		0			0
$185$	−2.70156			−0.198623
$186$		0			0
$187$		−	18.1047i		−	1.32395i
$188$		0			0
$189$		−	4.70156i		−	0.341988i
$190$		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
$193$			25.5078i			1.83609i	0.396473	+	0.918046i	$0.370234\pi$
$193$			25.5078i			1.83609i	−0.396473	+	0.918046i	$0.629766\pi$
$194$		0			0
$195$	−3.00000	−	2.00000i	−0.214834	−	0.143223i
$196$		0			0
$197$			12.8062i			0.912407i	0.889875	+	0.456204i	$0.150791\pi$
$197$			12.8062i			0.912407i	−0.889875	+	0.456204i	$0.849209\pi$
$198$		0			0
$199$	−14.8062			−1.04959			−0.524794	−	0.851230i	$-0.675857\pi$
$199$	−14.8062			−1.04959			−0.524794	+	0.851230i	$0.675857\pi$
$200$		0			0
$201$		−	8.00000i		−	0.564276i
$202$		0			0
$203$			9.40312i			0.659970i
$204$		0			0
$205$	−8.70156			−0.607743
$206$		0			0
$207$	2.70156			0.187772
$208$		0			0
$209$	−20.0000			−1.38343
$210$		0			0
$211$	−25.6125			−1.76324			−0.881619	−	0.471963i	$-0.843546\pi$
$211$	−25.6125			−1.76324			−0.881619	+	0.471963i	$0.843546\pi$
$212$		0			0
$213$			0.701562i			0.0480702i
$214$		0			0
$215$		−	1.40312i		−	0.0956923i
$216$		0			0
$217$	−44.2094			−3.00113
$218$		0			0
$219$			12.8062i			0.865366i
$220$		0			0
$221$	13.4031	−	20.1047i	0.901592	−	1.35239i
$222$		0			0
$223$		−	24.0000i		−	1.60716i	−0.595198	−	0.803579i	$-0.702926\pi$
$223$		−	24.0000i		−	1.60716i	0.595198	−	0.803579i	$-0.297074\pi$
$224$		0			0
$225$	−1.00000			−0.0666667
$226$		0			0
$227$			10.8062i			0.717236i	0.933484	+	0.358618i	$0.116752\pi$
$227$			10.8062i			0.717236i	−0.933484	+	0.358618i	$0.883248\pi$
$228$		0			0
$229$			16.8062i			1.11059i	0.831654	+	0.555294i	$0.187394\pi$
$229$			16.8062i			1.11059i	−0.831654	+	0.555294i	$0.812606\pi$
$230$		0			0
$231$	−12.7016			−0.835701
$232$		0			0
$233$	−2.70156			−0.176985			−0.0884926	−	0.996077i	$-0.528205\pi$
$233$	−2.70156			−0.176985			−0.0884926	+	0.996077i	$0.528205\pi$
$234$		0			0
$235$	−8.00000			−0.521862
$236$		0			0
$237$	3.29844			0.214257
$238$		0			0
$239$			3.29844i			0.213358i	0.994294	+	0.106679i	$0.0340218\pi$
$239$			3.29844i			0.213358i	−0.994294	+	0.106679i	$0.965978\pi$
$240$		0			0
$241$			1.40312i			0.0903832i	0.998978	+	0.0451916i	$0.0143898\pi$
$241$			1.40312i			0.0903832i	−0.998978	+	0.0451916i	$0.985610\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$		−	15.1047i		−	0.965003i
$246$		0			0
$247$	−22.2094	−	14.8062i	−1.41315	−	0.942099i
$248$		0			0
$249$		−	9.40312i		−	0.595899i
$250$		0			0
$251$	29.4031			1.85591			0.927954	−	0.372694i	$-0.121566\pi$
$251$	29.4031			1.85591			0.927954	+	0.372694i	$0.121566\pi$
$252$		0			0
$253$		−	7.29844i		−	0.458849i
$254$		0			0
$255$		−	6.70156i		−	0.419668i
$256$		0			0
$257$	20.8062			1.29786			0.648929	−	0.760849i	$-0.275217\pi$
$257$	20.8062			1.29786			0.648929	+	0.760849i	$0.275217\pi$
$258$		0			0
$259$	−12.7016			−0.789237
$260$		0			0
$261$	2.00000			0.123797
$262$		0			0
$263$	18.2094			1.12284			0.561419	−	0.827532i	$-0.310256\pi$
$263$	18.2094			1.12284			0.561419	+	0.827532i	$0.310256\pi$
$264$		0			0
$265$			8.70156i			0.534533i
$266$		0			0
$267$		−	4.70156i		−	0.287731i
$268$		0			0
$269$	−0.596876			−0.0363922			−0.0181961	−	0.999834i	$-0.505792\pi$
$269$	−0.596876			−0.0363922			−0.0181961	+	0.999834i	$0.505792\pi$
$270$		0			0
$271$		−	2.59688i		−	0.157749i	−0.996885	−	0.0788745i	$-0.974867\pi$
$271$		−	2.59688i		−	0.157749i	0.996885	−	0.0788745i	$-0.0251326\pi$
$272$		0			0
$273$	−14.1047	−	9.40312i	−0.853654	−	0.569103i
$274$		0			0
$275$			2.70156i			0.162910i
$276$		0			0
$277$	10.8062			0.649285			0.324642	−	0.945837i	$-0.394756\pi$
$277$	10.8062			0.649285			0.324642	+	0.945837i	$0.394756\pi$
$278$		0			0
$279$			9.40312i			0.562950i
$280$		0			0
$281$		−	17.4031i		−	1.03818i	−0.854719	−	0.519092i	$-0.826270\pi$
$281$		−	17.4031i		−	1.03818i	0.854719	−	0.519092i	$-0.173730\pi$
$282$		0			0
$283$	13.4031			0.796733			0.398367	−	0.917226i	$-0.369577\pi$
$283$	13.4031			0.796733			0.398367	+	0.917226i	$0.369577\pi$
$284$		0			0
$285$	−7.40312			−0.438523
$286$		0			0
$287$	−40.9109			−2.41490
$288$		0			0
$289$	27.9109			1.64182
$290$		0			0
$291$			16.1047i			0.944073i
$292$		0			0
$293$		−	6.00000i		−	0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$		−	6.00000i		−	0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$		0			0
$295$	−0.596876			−0.0347515
$296$		0			0
$297$			2.70156i			0.156761i
$298$		0			0
$299$	5.40312	−	8.10469i	0.312471	−	0.468706i
$300$		0			0
$301$		−	6.59688i		−	0.380238i
$302$		0			0
$303$	8.80625			0.505906
$304$		0			0
$305$			6.70156i			0.383730i
$306$		0			0
$307$		−	10.1047i		−	0.576705i	−0.957524	−	0.288352i	$-0.906893\pi$
$307$		−	10.1047i		−	0.576705i	0.957524	−	0.288352i	$-0.0931075\pi$
$308$		0			0
$309$	3.40312			0.193597
$310$		0			0
$311$	5.40312			0.306383			0.153192	−	0.988197i	$-0.451045\pi$
$311$	5.40312			0.306383			0.153192	+	0.988197i	$0.451045\pi$
$312$		0			0
$313$	6.20937			0.350974			0.175487	−	0.984482i	$-0.443850\pi$
$313$	6.20937			0.350974			0.175487	+	0.984482i	$0.443850\pi$
$314$		0			0
$315$	−4.70156			−0.264903
$316$		0			0
$317$		−	28.8062i		−	1.61792i	−0.587864	−	0.808960i	$-0.700031\pi$
$317$		−	28.8062i		−	1.61792i	0.587864	−	0.808960i	$-0.299969\pi$
$318$		0			0
$319$		−	5.40312i		−	0.302517i
$320$		0			0
$321$	6.10469			0.340730
$322$		0			0
$323$		−	49.6125i		−	2.76051i
$324$		0			0
$325$	−2.00000	+	3.00000i	−0.110940	+	0.166410i
$326$		0			0
$327$		−	6.00000i		−	0.331801i
$328$		0			0
$329$	−37.6125			−2.07364
$330$		0			0
$331$		−	15.4031i		−	0.846632i	−0.905982	−	0.423316i	$-0.860866\pi$
$331$		−	15.4031i		−	0.846632i	0.905982	−	0.423316i	$-0.139134\pi$
$332$		0			0
$333$			2.70156i			0.148045i
$334$		0			0
$335$	−8.00000			−0.437087
$336$		0			0
$337$	−7.19375			−0.391869			−0.195934	−	0.980617i	$-0.562774\pi$
$337$	−7.19375			−0.391869			−0.195934	+	0.980617i	$0.562774\pi$
$338$		0			0
$339$	−14.0000			−0.760376
$340$		0			0
$341$	25.4031			1.37566
$342$		0			0
$343$		−	38.1047i		−	2.05746i
$344$		0			0
$345$		−	2.70156i		−	0.145447i
$346$		0			0
$347$	31.5078			1.69143			0.845714	−	0.533637i	$-0.179175\pi$
$347$	31.5078			1.69143			0.845714	+	0.533637i	$0.179175\pi$
$348$		0			0
$349$		−	8.80625i		−	0.471388i	−0.971827	−	0.235694i	$-0.924264\pi$
$349$		−	8.80625i		−	0.471388i	0.971827	−	0.235694i	$-0.0757363\pi$
$350$		0			0
$351$	−2.00000	+	3.00000i	−0.106752	+	0.160128i
$352$		0			0
$353$			30.2094i			1.60788i	0.594709	+	0.803941i	$0.297267\pi$
$353$			30.2094i			1.60788i	−0.594709	+	0.803941i	$0.702733\pi$
$354$		0			0
$355$	0.701562			0.0372351
$356$		0			0
$357$		−	31.5078i		−	1.66757i
$358$		0			0
$359$			16.0000i			0.844448i	0.906492	+	0.422224i	$0.138750\pi$
$359$			16.0000i			0.844448i	−0.906492	+	0.422224i	$0.861250\pi$
$360$		0			0
$361$	−35.8062			−1.88454
$362$		0			0
$363$	−3.70156			−0.194282
$364$		0			0
$365$	12.8062			0.670310
$366$		0			0
$367$	18.0000			0.939592			0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000			0.939592			0.469796	+	0.882775i	$0.344327\pi$
$368$		0			0
$369$			8.70156i			0.452985i
$370$		0			0
$371$			40.9109i			2.12399i
$372$		0			0
$373$	−33.6125			−1.74039			−0.870195	−	0.492708i	$-0.836007\pi$
$373$	−33.6125			−1.74039			−0.870195	+	0.492708i	$0.836007\pi$
$374$		0			0
$375$			1.00000i			0.0516398i
$376$		0			0
$377$	4.00000	−	6.00000i	0.206010	−	0.309016i
$378$		0			0
$379$			4.80625i			0.246880i	0.992352	+	0.123440i	$0.0393927\pi$
$379$			4.80625i			0.246880i	−0.992352	+	0.123440i	$0.960607\pi$
$380$		0			0
$381$	−8.80625			−0.451158
$382$		0			0
$383$		−	6.59688i		−	0.337085i	−0.985694	−	0.168542i	$-0.946094\pi$
$383$		−	6.59688i		−	0.337085i	0.985694	−	0.168542i	$-0.0539060\pi$
$384$		0			0
$385$			12.7016i			0.647332i
$386$		0			0
$387$	−1.40312			−0.0713248
$388$		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
$391$	18.1047			0.915593
$392$		0			0
$393$	12.0000			0.605320
$394$		0			0
$395$		−	3.29844i		−	0.165962i
$396$		0			0
$397$		−	17.2984i		−	0.868184i	−0.900869	−	0.434092i	$-0.857069\pi$
$397$		−	17.2984i		−	0.868184i	0.900869	−	0.434092i	$-0.142931\pi$
$398$		0			0
$399$	−34.8062			−1.74249
$400$		0			0
$401$		−	28.2094i		−	1.40871i	−0.709848	−	0.704354i	$-0.751237\pi$
$401$		−	28.2094i		−	1.40871i	0.709848	−	0.704354i	$-0.248763\pi$
$402$		0			0
$403$	28.2094	+	18.8062i	1.40521	+	0.936806i
$404$		0			0
$405$			1.00000i			0.0496904i
$406$		0			0
$407$	7.29844			0.361770
$408$		0			0
$409$		−	6.80625i		−	0.336547i	−0.985740	−	0.168274i	$-0.946181\pi$
$409$		−	6.80625i		−	0.336547i	0.985740	−	0.168274i	$-0.0538192\pi$
$410$		0			0
$411$			8.80625i			0.434380i
$412$		0			0
$413$	−2.80625			−0.138086
$414$		0			0
$415$	−9.40312			−0.461581
$416$		0			0
$417$	−10.1047			−0.494829
$418$		0			0
$419$	−21.4031			−1.04561			−0.522806	−	0.852452i	$-0.675115\pi$
$419$	−21.4031			−1.04561			−0.522806	+	0.852452i	$0.675115\pi$
$420$		0			0
$421$		−	12.8062i		−	0.624138i	−0.950059	−	0.312069i	$-0.898978\pi$
$421$		−	12.8062i		−	0.624138i	0.950059	−	0.312069i	$-0.101022\pi$
$422$		0			0
$423$			8.00000i			0.388973i
$424$		0			0
$425$	−6.70156			−0.325074
$426$		0			0
$427$			31.5078i			1.52477i
$428$		0			0
$429$	8.10469	+	5.40312i	0.391298	+	0.260865i
$430$		0			0
$431$		−	30.8062i		−	1.48388i	−0.670464	−	0.741942i	$-0.733905\pi$
$431$		−	30.8062i		−	1.48388i	0.670464	−	0.741942i	$-0.266095\pi$
$432$		0			0
$433$	−23.6125			−1.13474			−0.567372	−	0.823462i	$-0.692040\pi$
$433$	−23.6125			−1.13474			−0.567372	+	0.823462i	$0.692040\pi$
$434$		0			0
$435$		−	2.00000i		−	0.0958927i
$436$		0			0
$437$		−	20.0000i		−	0.956730i
$438$		0			0
$439$	9.89531			0.472278			0.236139	−	0.971719i	$-0.424118\pi$
$439$	9.89531			0.472278			0.236139	+	0.971719i	$0.424118\pi$
$440$		0			0
$441$	−15.1047			−0.719271
$442$		0			0
$443$	−11.2984			−0.536805			−0.268402	−	0.963307i	$-0.586496\pi$
$443$	−11.2984			−0.536805			−0.268402	+	0.963307i	$0.586496\pi$
$444$		0			0
$445$	−4.70156			−0.222875
$446$		0			0
$447$		−	14.7016i		−	0.695360i
$448$		0			0
$449$			16.9109i			0.798076i	0.916934	+	0.399038i	$0.130656\pi$
$449$			16.9109i			0.798076i	−0.916934	+	0.399038i	$0.869344\pi$
$450$		0			0
$451$	23.5078			1.10694
$452$		0			0
$453$			5.40312i			0.253861i
$454$		0			0
$455$	−9.40312	+	14.1047i	−0.440825	+	0.661238i
$456$		0			0
$457$			15.8953i			0.743551i	0.928323	+	0.371776i	$0.121251\pi$
$457$			15.8953i			0.743551i	−0.928323	+	0.371776i	$0.878749\pi$
$458$		0			0
$459$	−6.70156			−0.312802
$460$		0			0
$461$			33.7172i			1.57037i	0.619264	+	0.785183i	$0.287431\pi$
$461$			33.7172i			1.57037i	−0.619264	+	0.785183i	$0.712569\pi$
$462$		0			0
$463$			7.50781i			0.348918i	0.984664	+	0.174459i	$0.0558176\pi$
$463$			7.50781i			0.348918i	−0.984664	+	0.174459i	$0.944182\pi$
$464$		0			0
$465$	9.40312			0.436059
$466$		0			0
$467$	30.1047			1.39308			0.696539	−	0.717519i	$-0.254722\pi$
$467$	30.1047			1.39308			0.696539	+	0.717519i	$0.254722\pi$
$468$		0			0
$469$	−37.6125			−1.73678
$470$		0			0
$471$	22.8062			1.05086
$472$		0			0
$473$			3.79063i			0.174293i
$474$		0			0
$475$			7.40312i			0.339679i
$476$		0			0
$477$	8.70156			0.398417
$478$		0			0
$479$		−	26.3141i		−	1.20232i	−0.799129	−	0.601160i	$-0.794705\pi$
$479$		−	26.3141i		−	1.20232i	0.799129	−	0.601160i	$-0.205295\pi$
$480$		0			0
$481$	8.10469	+	5.40312i	0.369542	+	0.246361i
$482$		0			0
$483$		−	12.7016i		−	0.577941i
$484$		0			0
$485$	16.1047			0.731276
$486$		0			0
$487$			15.5078i			0.702726i	0.936239	+	0.351363i	$0.114282\pi$
$487$			15.5078i			0.702726i	−0.936239	+	0.351363i	$0.885718\pi$
$488$		0			0
$489$			7.29844i			0.330047i
$490$		0			0
$491$	−33.6125			−1.51691			−0.758455	−	0.651725i	$-0.774046\pi$
$491$	−33.6125			−1.51691			−0.758455	+	0.651725i	$0.774046\pi$
$492$		0			0
$493$	13.4031			0.603646
$494$		0			0
$495$	2.70156			0.121426
$496$		0			0
$497$	3.29844			0.147955
$498$		0			0
$499$		−	27.4031i		−	1.22673i	−0.789799	−	0.613366i	$-0.789815\pi$
$499$		−	27.4031i		−	1.22673i	0.789799	−	0.613366i	$-0.210185\pi$
$500$		0			0
$501$		−	8.00000i		−	0.357414i
$502$		0			0
$503$	−23.4031			−1.04349			−0.521747	−	0.853100i	$-0.674719\pi$
$503$	−23.4031			−1.04349			−0.521747	+	0.853100i	$0.674719\pi$
$504$		0			0
$505$		−	8.80625i		−	0.391873i
$506$		0			0
$507$	5.00000	+	12.0000i	0.222058	+	0.532939i
$508$		0			0
$509$		−	33.7172i		−	1.49449i	−0.664550	−	0.747244i	$-0.731377\pi$
$509$		−	33.7172i		−	1.49449i	0.664550	−	0.747244i	$-0.268623\pi$
$510$		0			0
$511$	60.2094			2.66351
$512$		0			0
$513$			7.40312i			0.326856i
$514$		0			0
$515$		−	3.40312i		−	0.149960i
$516$		0			0
$517$	21.6125			0.950517
$518$		0			0
$519$	−10.5969			−0.465151
$520$		0			0
$521$	6.20937			0.272038			0.136019	−	0.990706i	$-0.456569\pi$
$521$	6.20937			0.272038			0.136019	+	0.990706i	$0.456569\pi$
$522$		0			0
$523$	6.59688			0.288461			0.144231	−	0.989544i	$-0.453929\pi$
$523$	6.59688			0.288461			0.144231	+	0.989544i	$0.453929\pi$
$524$		0			0
$525$			4.70156i			0.205193i
$526$		0			0
$527$			63.0156i			2.74500i
$528$		0			0
$529$	−15.7016			−0.682677
$530$		0			0
$531$			0.596876i			0.0259022i
$532$		0			0
$533$	26.1047	+	17.4031i	1.13072	+	0.753813i
$534$		0			0
$535$		−	6.10469i		−	0.263929i
$536$		0			0
$537$	−6.80625			−0.293711
$538$		0			0
$539$			40.8062i			1.75765i
$540$		0			0
$541$		−	27.4031i		−	1.17815i	−0.808078	−	0.589076i	$-0.799492\pi$
$541$		−	27.4031i		−	1.17815i	0.808078	−	0.589076i	$-0.200508\pi$
$542$		0			0
$543$	−8.10469			−0.347805
$544$		0			0
$545$	−6.00000			−0.257012
$546$		0			0
$547$	−18.5969			−0.795145			−0.397572	−	0.917571i	$-0.630147\pi$
$547$	−18.5969			−0.795145			−0.397572	+	0.917571i	$0.630147\pi$
$548$		0			0
$549$	6.70156			0.286016
$550$		0			0
$551$		−	14.8062i		−	0.630767i
$552$		0			0
$553$		−	15.5078i		−	0.659459i
$554$		0			0
$555$	2.70156			0.114675
$556$		0			0
$557$		−	14.0000i		−	0.593199i	−0.955002	−	0.296600i	$-0.904147\pi$
$557$		−	14.0000i		−	0.593199i	0.955002	−	0.296600i	$-0.0958526\pi$
$558$		0			0
$559$	−2.80625	+	4.20937i	−0.118692	+	0.178037i
$560$		0			0
$561$			18.1047i			0.764380i
$562$		0			0
$563$	38.1047			1.60592			0.802961	−	0.596032i	$-0.203257\pi$
$563$	38.1047			1.60592			0.802961	+	0.596032i	$0.203257\pi$
$564$		0			0
$565$			14.0000i			0.588984i
$566$		0			0
$567$			4.70156i			0.197447i
$568$		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
$571$	−38.3141			−1.60339			−0.801697	−	0.597731i	$-0.796069\pi$
$571$	−38.3141			−1.60339			−0.801697	+	0.597731i	$0.796069\pi$
$572$		0			0
$573$	−4.00000			−0.167102
$574$		0			0
$575$	−2.70156			−0.112663
$576$		0			0
$577$			33.5078i			1.39495i	0.716610	+	0.697474i	$0.245693\pi$
$577$			33.5078i			1.39495i	−0.716610	+	0.697474i	$0.754307\pi$
$578$		0			0
$579$		−	25.5078i		−	1.06007i
$580$		0			0
$581$	−44.2094			−1.83411
$582$		0			0
$583$		−	23.5078i		−	0.973594i
$584$		0			0
$585$	3.00000	+	2.00000i	0.124035	+	0.0826898i
$586$		0			0
$587$		−	32.2094i		−	1.32942i	−0.747100	−	0.664712i	$-0.768554\pi$
$587$		−	32.2094i		−	1.32942i	0.747100	−	0.664712i	$-0.231446\pi$
$588$		0			0
$589$	69.6125			2.86833
$590$		0			0
$591$		−	12.8062i		−	0.526779i
$592$		0			0
$593$			22.0000i			0.903432i	0.892162	+	0.451716i	$0.149188\pi$
$593$			22.0000i			0.903432i	−0.892162	+	0.451716i	$0.850812\pi$
$594$		0			0
$595$	−31.5078			−1.29169
$596$		0			0
$597$	14.8062			0.605979
$598$		0			0
$599$	−42.8062			−1.74902			−0.874508	−	0.485011i	$-0.838816\pi$
$599$	−42.8062			−1.74902			−0.874508	+	0.485011i	$0.838816\pi$
$600$		0			0
$601$	−47.1203			−1.92208			−0.961039	−	0.276414i	$-0.910854\pi$
$601$	−47.1203			−1.92208			−0.961039	+	0.276414i	$0.910854\pi$
$602$		0			0
$603$			8.00000i			0.325785i
$604$		0			0
$605$			3.70156i			0.150490i
$606$		0			0
$607$	34.2094			1.38852			0.694258	−	0.719726i	$-0.255733\pi$
$607$	34.2094			1.38852			0.694258	+	0.719726i	$0.255733\pi$
$608$		0			0
$609$		−	9.40312i		−	0.381034i
$610$		0			0
$611$	24.0000	+	16.0000i	0.970936	+	0.647291i
$612$		0			0
$613$		−	16.1047i		−	0.650462i	−0.945635	−	0.325231i	$-0.894558\pi$
$613$		−	16.1047i		−	0.650462i	0.945635	−	0.325231i	$-0.105442\pi$
$614$		0			0
$615$	8.70156			0.350881
$616$		0			0
$617$			23.6125i			0.950603i	0.879823	+	0.475302i	$0.157661\pi$
$617$			23.6125i			0.950603i	−0.879823	+	0.475302i	$0.842339\pi$
$618$		0			0
$619$			22.0000i			0.884255i	0.896952	+	0.442127i	$0.145776\pi$
$619$			22.0000i			0.884255i	−0.896952	+	0.442127i	$0.854224\pi$
$620$		0			0
$621$	−2.70156			−0.108410
$622$		0			0
$623$	−22.1047			−0.885606
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	20.0000			0.798723
$628$		0			0
$629$			18.1047i			0.721881i
$630$		0			0
$631$		−	32.2094i		−	1.28223i	−0.767443	−	0.641117i	$-0.778471\pi$
$631$		−	32.2094i		−	1.28223i	0.767443	−	0.641117i	$-0.221529\pi$
$632$		0			0
$633$	25.6125			1.01801
$634$		0			0
$635$			8.80625i			0.349465i
$636$		0			0
$637$	−30.2094	+	45.3141i	−1.19694	+	1.79541i
$638$		0			0
$639$		−	0.701562i		−	0.0277534i
$640$		0			0
$641$	45.0156			1.77801			0.889005	−	0.457897i	$-0.151397\pi$
$641$	45.0156			1.77801			0.889005	+	0.457897i	$0.151397\pi$
$642$		0			0
$643$		−	7.50781i		−	0.296079i	−0.988981	−	0.148040i	$-0.952704\pi$
$643$		−	7.50781i		−	0.296079i	0.988981	−	0.148040i	$-0.0472963\pi$
$644$		0			0
$645$			1.40312i			0.0552480i
$646$		0			0
$647$	14.9109			0.586209			0.293105	−	0.956080i	$-0.405312\pi$
$647$	14.9109			0.586209			0.293105	+	0.956080i	$0.405312\pi$
$648$		0			0
$649$	1.61250			0.0632960
$650$		0			0
$651$	44.2094			1.73270
$652$		0			0
$653$	−1.40312			−0.0549085			−0.0274542	−	0.999623i	$-0.508740\pi$
$653$	−1.40312			−0.0549085			−0.0274542	+	0.999623i	$0.508740\pi$
$654$		0			0
$655$		−	12.0000i		−	0.468879i
$656$		0			0
$657$		−	12.8062i		−	0.499619i
$658$		0			0
$659$	−43.0156			−1.67565			−0.837825	−	0.545938i	$-0.816173\pi$
$659$	−43.0156			−1.67565			−0.837825	+	0.545938i	$0.816173\pi$
$660$		0			0
$661$			12.5969i			0.489962i	0.969528	+	0.244981i	$0.0787817\pi$
$661$			12.5969i			0.489962i	−0.969528	+	0.244981i	$0.921218\pi$
$662$		0			0
$663$	−13.4031	+	20.1047i	−0.520534	+	0.780801i
$664$		0			0
$665$			34.8062i			1.34973i
$666$		0			0
$667$	5.40312			0.209210
$668$		0			0
$669$			24.0000i			0.927894i
$670$		0			0
$671$		−	18.1047i		−	0.698924i
$672$		0			0
$673$	27.4031			1.05631			0.528156	−	0.849147i	$-0.322883\pi$
$673$	27.4031			1.05631			0.528156	+	0.849147i	$0.322883\pi$
$674$		0			0
$675$	1.00000			0.0384900
$676$		0			0
$677$	−3.29844			−0.126769			−0.0633846	−	0.997989i	$-0.520189\pi$
$677$	−3.29844			−0.126769			−0.0633846	+	0.997989i	$0.520189\pi$
$678$		0			0
$679$	75.7172			2.90576
$680$		0			0
$681$		−	10.8062i		−	0.414096i
$682$		0			0
$683$		−	49.4031i		−	1.89036i	−0.326553	−	0.945179i	$-0.605887\pi$
$683$		−	49.4031i		−	1.89036i	0.326553	−	0.945179i	$-0.394113\pi$
$684$		0			0
$685$	8.80625			0.336469
$686$		0			0
$687$		−	16.8062i		−	0.641198i
$688$		0			0
$689$	17.4031	−	26.1047i	0.663006	−	0.994509i
$690$		0			0
$691$		−	19.6125i		−	0.746095i	−0.927812	−	0.373047i	$-0.878313\pi$
$691$		−	19.6125i		−	0.746095i	0.927812	−	0.373047i	$-0.121687\pi$
$692$		0			0
$693$	12.7016			0.482492
$694$		0			0
$695$			10.1047i			0.383293i
$696$		0			0
$697$			58.3141i			2.20880i
$698$		0			0
$699$	2.70156			0.102182
$700$		0			0
$701$	−29.0156			−1.09590			−0.547952	−	0.836509i	$-0.684593\pi$
$701$	−29.0156			−1.09590			−0.547952	+	0.836509i	$0.684593\pi$
$702$		0			0
$703$	20.0000			0.754314
$704$		0			0
$705$	8.00000			0.301297
$706$		0			0
$707$		−	41.4031i		−	1.55712i
$708$		0			0
$709$			10.2094i			0.383421i	0.981452	+	0.191711i	$0.0614035\pi$
$709$			10.2094i			0.383421i	−0.981452	+	0.191711i	$0.938597\pi$
$710$		0			0
$711$	−3.29844			−0.123701
$712$		0			0
$713$			25.4031i			0.951354i
$714$		0			0
$715$	5.40312	−	8.10469i	0.202065	−	0.303098i
$716$		0			0
$717$		−	3.29844i		−	0.123182i
$718$		0			0
$719$	2.59688			0.0968471			0.0484236	−	0.998827i	$-0.484580\pi$
$719$	2.59688			0.0968471			0.0484236	+	0.998827i	$0.484580\pi$
$720$		0			0
$721$		−	16.0000i		−	0.595871i
$722$		0			0
$723$		−	1.40312i		−	0.0521828i
$724$		0			0
$725$	−2.00000			−0.0742781
$726$		0			0
$727$	−36.8062			−1.36507			−0.682534	−	0.730854i	$-0.739122\pi$
$727$	−36.8062			−1.36507			−0.682534	+	0.730854i	$0.739122\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−9.40312			−0.347787
$732$		0			0
$733$			37.2984i			1.37765i	0.724928	+	0.688825i	$0.241873\pi$
$733$			37.2984i			1.37765i	−0.724928	+	0.688825i	$0.758127\pi$
$734$		0			0
$735$			15.1047i			0.557145i
$736$		0			0
$737$	21.6125			0.796107
$738$		0			0
$739$		−	11.1938i		−	0.411769i	−0.978576	−	0.205884i	$-0.933993\pi$
$739$		−	11.1938i		−	0.411769i	0.978576	−	0.205884i	$-0.0660071\pi$
$740$		0			0
$741$	22.2094	+	14.8062i	0.815882	+	0.543921i
$742$		0			0
$743$		−	40.0000i		−	1.46746i	−0.679442	−	0.733729i	$-0.737778\pi$
$743$		−	40.0000i		−	1.46746i	0.679442	−	0.733729i	$-0.262222\pi$
$744$		0			0
$745$	−14.7016			−0.538624
$746$		0			0
$747$			9.40312i			0.344042i
$748$		0			0
$749$		−	28.7016i		−	1.04873i
$750$		0			0
$751$	48.7016			1.77715			0.888573	−	0.458736i	$-0.151698\pi$
$751$	48.7016			1.77715			0.888573	+	0.458736i	$0.151698\pi$
$752$		0			0
$753$	−29.4031			−1.07151
$754$		0			0
$755$	5.40312			0.196640
$756$		0			0
$757$	−18.8062			−0.683525			−0.341762	−	0.939786i	$-0.611024\pi$
$757$	−18.8062			−0.683525			−0.341762	+	0.939786i	$0.611024\pi$
$758$		0			0
$759$			7.29844i			0.264917i
$760$		0			0
$761$			1.40312i			0.0508632i	0.999677	+	0.0254316i	$0.00809600\pi$
$761$			1.40312i			0.0508632i	−0.999677	+	0.0254316i	$0.991904\pi$
$762$		0			0
$763$	−28.2094			−1.02125
$764$		0			0
$765$			6.70156i			0.242295i
$766$		0			0
$767$	1.79063	+	1.19375i	0.0646558	+	0.0431039i
$768$		0			0
$769$			5.19375i			0.187291i	0.995606	+	0.0936457i	$0.0298521\pi$
$769$			5.19375i			0.187291i	−0.995606	+	0.0936457i	$0.970148\pi$
$770$		0			0
$771$	−20.8062			−0.749319
$772$		0			0
$773$			19.1938i			0.690351i	0.938538	+	0.345176i	$0.112181\pi$
$773$			19.1938i			0.690351i	−0.938538	+	0.345176i	$0.887819\pi$
$774$		0			0
$775$		−	9.40312i		−	0.337770i
$776$		0			0
$777$	12.7016			0.455666
$778$		0			0
$779$	64.4187			2.30804
$780$		0			0
$781$	−1.89531			−0.0678197
$782$		0			0
$783$	−2.00000			−0.0714742
$784$		0			0
$785$		−	22.8062i		−	0.813990i
$786$		0			0
$787$			52.0000i			1.85360i	0.375555	+	0.926800i	$0.377452\pi$
$787$			52.0000i			1.85360i	−0.375555	+	0.926800i	$0.622548\pi$
$788$		0			0
$789$	−18.2094			−0.648271
$790$		0			0
$791$			65.8219i			2.34036i
$792$		0			0
$793$	13.4031	−	20.1047i	0.475959	−	0.713938i
$794$		0			0
$795$		−	8.70156i		−	0.308613i
$796$		0			0
$797$	−42.5234			−1.50626			−0.753129	−	0.657873i	$-0.771456\pi$
$797$	−42.5234			−1.50626			−0.753129	+	0.657873i	$0.771456\pi$
$798$		0			0
$799$			53.6125i			1.89667i
$800$		0			0
$801$			4.70156i			0.166122i
$802$		0			0
$803$	−34.5969			−1.22090
$804$		0			0
$805$	−12.7016			−0.447671
$806$		0			0
$807$	0.596876			0.0210110
$808$		0			0
$809$	−16.5969			−0.583515			−0.291758	−	0.956492i	$-0.594240\pi$
$809$	−16.5969			−0.583515			−0.291758	+	0.956492i	$0.594240\pi$
$810$		0			0
$811$		−	7.19375i		−	0.252607i	−0.991992	−	0.126303i	$-0.959689\pi$
$811$		−	7.19375i		−	0.252607i	0.991992	−	0.126303i	$-0.0403113\pi$
$812$		0			0
$813$			2.59688i			0.0910764i
$814$		0			0
$815$	7.29844			0.255653
$816$		0			0
$817$			10.3875i			0.363413i
$818$		0			0
$819$	14.1047	+	9.40312i	0.492858	+	0.328572i
$820$		0			0
$821$		−	25.5078i		−	0.890229i	−0.895474	−	0.445114i	$-0.853163\pi$
$821$		−	25.5078i		−	0.890229i	0.895474	−	0.445114i	$-0.146837\pi$
$822$		0			0
$823$	−3.61250			−0.125924			−0.0629619	−	0.998016i	$-0.520055\pi$
$823$	−3.61250			−0.125924			−0.0629619	+	0.998016i	$0.520055\pi$
$824$		0			0
$825$		−	2.70156i		−	0.0940563i
$826$		0			0
$827$		−	30.8062i		−	1.07124i	−0.844460	−	0.535619i	$-0.820078\pi$
$827$		−	30.8062i		−	1.07124i	0.844460	−	0.535619i	$-0.179922\pi$
$828$		0			0
$829$	14.0000			0.486240			0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000			0.486240			0.243120	+	0.969996i	$0.421829\pi$
$830$		0			0
$831$	−10.8062			−0.374865
$832$		0			0
$833$	−101.225			−3.50724
$834$		0			0
$835$	−8.00000			−0.276851
$836$		0			0
$837$		−	9.40312i		−	0.325020i
$838$		0			0
$839$			8.70156i			0.300411i	0.988655	+	0.150206i	$0.0479936\pi$
$839$			8.70156i			0.300411i	−0.988655	+	0.150206i	$0.952006\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$			17.4031i			0.599395i
$844$		0			0
$845$	12.0000	−	5.00000i	0.412813	−	0.172005i
$846$		0			0
$847$			17.4031i			0.597978i
$848$		0			0
$849$	−13.4031			−0.459994
$850$		0			0
$851$			7.29844i			0.250187i
$852$		0			0
$853$			12.1047i			0.414457i	0.978293	+	0.207228i	$0.0664443\pi$
$853$			12.1047i			0.414457i	−0.978293	+	0.207228i	$0.933556\pi$
$854$		0			0
$855$	7.40312			0.253182
$856$		0			0
$857$	−12.3141			−0.420640			−0.210320	−	0.977633i	$-0.567451\pi$
$857$	−12.3141			−0.420640			−0.210320	+	0.977633i	$0.567451\pi$
$858$		0			0
$859$	−19.5078			−0.665598			−0.332799	−	0.942998i	$-0.607993\pi$
$859$	−19.5078			−0.665598			−0.332799	+	0.942998i	$0.607993\pi$
$860$		0			0
$861$	40.9109			1.39424
$862$		0			0
$863$		−	30.5969i		−	1.04153i	−0.853700	−	0.520765i	$-0.825647\pi$
$863$		−	30.5969i		−	1.04153i	0.853700	−	0.520765i	$-0.174353\pi$
$864$		0			0
$865$			10.5969i			0.360305i
$866$		0			0
$867$	−27.9109			−0.947905
$868$		0			0
$869$			8.91093i			0.302283i
$870$		0			0
$871$	24.0000	+	16.0000i	0.813209	+	0.542139i
$872$		0			0
$873$		−	16.1047i		−	0.545061i
$874$		0			0
$875$	4.70156			0.158942
$876$		0			0
$877$		−	51.6125i		−	1.74283i	−0.490546	−	0.871415i	$-0.663203\pi$
$877$		−	51.6125i		−	1.74283i	0.490546	−	0.871415i	$-0.336797\pi$
$878$		0			0
$879$			6.00000i			0.202375i
$880$		0			0
$881$	−38.2094			−1.28731			−0.643653	−	0.765317i	$-0.722582\pi$
$881$	−38.2094			−1.28731			−0.643653	+	0.765317i	$0.722582\pi$
$882$		0			0
$883$	32.4187			1.09098			0.545489	−	0.838118i	$-0.316344\pi$
$883$	32.4187			1.09098			0.545489	+	0.838118i	$0.316344\pi$
$884$		0			0
$885$	0.596876			0.0200638
$886$		0			0
$887$	52.3141			1.75653			0.878267	−	0.478170i	$-0.158700\pi$
$887$	52.3141			1.75653			0.878267	+	0.478170i	$0.158700\pi$
$888$		0			0
$889$			41.4031i			1.38862i
$890$		0			0
$891$		−	2.70156i		−	0.0905057i
$892$		0			0
$893$	59.2250			1.98189
$894$		0			0
$895$			6.80625i			0.227508i
$896$		0			0
$897$	−5.40312	+	8.10469i	−0.180405	+	0.270608i
$898$		0			0
$899$			18.8062i			0.627224i
$900$		0			0
$901$	58.3141			1.94272
$902$		0			0
$903$			6.59688i			0.219530i
$904$		0			0
$905$			8.10469i			0.269409i
$906$		0			0
$907$	24.0000			0.796907			0.398453	−	0.917189i	$-0.369547\pi$
$907$	24.0000			0.796907			0.398453	+	0.917189i	$0.369547\pi$
$908$		0			0
$909$	−8.80625			−0.292085
$910$		0			0
$911$	37.8219			1.25309			0.626547	−	0.779383i	$-0.284468\pi$
$911$	37.8219			1.25309			0.626547	+	0.779383i	$0.284468\pi$
$912$		0			0
$913$	25.4031			0.840721
$914$		0			0
$915$		−	6.70156i		−	0.221547i
$916$		0			0
$917$		−	56.4187i		−	1.86311i
$918$		0			0
$919$	−7.29844			−0.240753			−0.120377	−	0.992728i	$-0.538410\pi$
$919$	−7.29844			−0.240753			−0.120377	+	0.992728i	$0.538410\pi$
$920$		0			0
$921$			10.1047i			0.332961i
$922$		0			0
$923$	−2.10469	−	1.40312i	−0.0692766	−	0.0461844i
$924$		0			0
$925$		−	2.70156i		−	0.0888268i
$926$		0			0
$927$	−3.40312			−0.111773
$928$		0			0
$929$		−	7.50781i		−	0.246323i	−0.992387	−	0.123162i	$-0.960697\pi$
$929$		−	7.50781i		−	0.246323i	0.992387	−	0.123162i	$-0.0393034\pi$
$930$		0			0
$931$			111.822i			3.66481i
$932$		0			0
$933$	−5.40312			−0.176890
$934$		0			0
$935$	18.1047			0.592087
$936$		0			0
$937$	−8.38750			−0.274008			−0.137004	−	0.990571i	$-0.543747\pi$
$937$	−8.38750			−0.274008			−0.137004	+	0.990571i	$0.543747\pi$
$938$		0			0
$939$	−6.20937			−0.202635
$940$		0			0
$941$			33.5078i			1.09232i	0.837680	+	0.546162i	$0.183912\pi$
$941$			33.5078i			1.09232i	−0.837680	+	0.546162i	$0.816088\pi$
$942$		0			0
$943$			23.5078i			0.765520i
$944$		0			0
$945$	4.70156			0.152942
$946$		0			0
$947$		−	20.0000i		−	0.649913i	−0.945729	−	0.324956i	$-0.894650\pi$
$947$		−	20.0000i		−	0.649913i	0.945729	−	0.324956i	$-0.105350\pi$
$948$		0			0
$949$	−38.4187	−	25.6125i	−1.24713	−	0.831417i
$950$		0			0
$951$			28.8062i			0.934107i
$952$		0			0
$953$	27.1203			0.878513			0.439256	−	0.898362i	$-0.355242\pi$
$953$	27.1203			0.878513			0.439256	+	0.898362i	$0.355242\pi$
$954$		0			0
$955$			4.00000i			0.129437i
$956$		0			0
$957$			5.40312i			0.174658i
$958$		0			0
$959$	41.4031			1.33698
$960$		0			0
$961$	−57.4187			−1.85222
$962$		0			0
$963$	−6.10469			−0.196721
$964$		0			0
$965$	−25.5078			−0.821125
$966$		0			0
$967$		−	8.00000i		−	0.257263i	−0.991692	−	0.128631i	$-0.958942\pi$
$967$		−	8.00000i		−	0.257263i	0.991692	−	0.128631i	$-0.0410584\pi$
$968$		0			0
$969$			49.6125i			1.59378i
$970$		0			0
$971$	−0.209373			−0.00671909			−0.00335955	−	0.999994i	$-0.501069\pi$
$971$	−0.209373			−0.00671909			−0.00335955	+	0.999994i	$0.501069\pi$
$972$		0			0
$973$			47.5078i			1.52303i
$974$		0			0
$975$	2.00000	−	3.00000i	0.0640513	−	0.0960769i
$976$		0			0
$977$			12.5969i			0.403010i	0.979488	+	0.201505i	$0.0645832\pi$
$977$			12.5969i			0.403010i	−0.979488	+	0.201505i	$0.935417\pi$
$978$		0			0
$979$	12.7016			0.405944
$980$		0			0
$981$			6.00000i			0.191565i
$982$		0			0
$983$			48.4187i			1.54432i	0.635429	+	0.772159i	$0.280823\pi$
$983$			48.4187i			1.54432i	−0.635429	+	0.772159i	$0.719177\pi$
$984$		0			0
$985$	−12.8062			−0.408041
$986$		0			0
$987$	37.6125			1.19722
$988$		0			0
$989$	−3.79063			−0.120535
$990$		0			0
$991$	34.1047			1.08337			0.541686	−	0.840581i	$-0.317786\pi$
$991$	34.1047			1.08337			0.541686	+	0.840581i	$0.317786\pi$
$992$		0			0
$993$			15.4031i			0.488803i
$994$		0			0
$995$		−	14.8062i		−	0.469390i
$996$		0			0
$997$	−44.0000			−1.39349			−0.696747	−	0.717317i	$-0.745370\pi$
$997$	−44.0000			−1.39349			−0.696747	+	0.717317i	$0.745370\pi$
$998$		0			0
$999$		−	2.70156i		−	0.0854736i

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000i q^{5} +4.70156i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000i q^{5} +4.70156i q^{7} +1.00000 q^{9} -2.70156i q^{11} +(2.00000 - 3.00000i) q^{13} -1.00000i q^{15} +6.70156 q^{17} -7.40312i q^{19} -4.70156i q^{21} +2.70156 q^{23} -1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{29} +9.40312i q^{31} +2.70156i q^{33} -4.70156 q^{35} +2.70156i q^{37} +(-2.00000 + 3.00000i) q^{39} +8.70156i q^{41} -1.40312 q^{43} +1.00000i q^{45} +8.00000i q^{47} -15.1047 q^{49} -6.70156 q^{51} +8.70156 q^{53} +2.70156 q^{55} +7.40312i q^{57} +0.596876i q^{59} +6.70156 q^{61} +4.70156i q^{63} +(3.00000 + 2.00000i) q^{65} +8.00000i q^{67} -2.70156 q^{69} -0.701562i q^{71} -12.8062i q^{73} +1.00000 q^{75} +12.7016 q^{77} -3.29844 q^{79} +1.00000 q^{81} +9.40312i q^{83} +6.70156i q^{85} -2.00000 q^{87} +4.70156i q^{89} +(14.1047 + 9.40312i) q^{91} -9.40312i q^{93} +7.40312 q^{95} -16.1047i q^{97} -2.70156i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 4 * q^9	\( 4 q - 4 q^{3} + 4 q^{9} + 8 q^{13} + 14 q^{17} - 2 q^{23} - 4 q^{25} - 4 q^{27} + 8 q^{29} - 6 q^{35} - 8 q^{39} + 20 q^{43} - 22 q^{49} - 14 q^{51} + 22 q^{53} - 2 q^{55} + 14 q^{61} + 12 q^{65} + 2 q^{69} + 4 q^{75} + 38 q^{77} - 26 q^{79} + 4 q^{81} - 8 q^{87} + 18 q^{91} + 4 q^{95}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^9 + 8 * q^13 + 14 * q^17 - 2 * q^23 - 4 * q^25 - 4 * q^27 + 8 * q^29 - 6 * q^35 - 8 * q^39 + 20 * q^43 - 22 * q^49 - 14 * q^51 + 22 * q^53 - 2 * q^55 + 14 * q^61 + 12 * q^65 + 2 * q^69 + 4 * q^75 + 38 * q^77 - 26 * q^79 + 4 * q^81 - 8 * q^87 + 18 * q^91 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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