LMFDB - Embedded newform 1960.2.g.e.1569.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1960,2,Mod(1569,1960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1960, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1960.1569");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1960 = 2^{3} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1960.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:	$15.6506787962$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 25x^{10} + 230x^{8} + 950x^{6} + 1657x^{4} + 785x^{2} + 64 $ x^12 + 25x^10 + 230x^8 + 950x^6 + 1657x^4 + 785*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1569.9
Root			$2.03837i$ of defining polynomial
Character	$\chi$	$=$	1960.1569
Dual form			1960.2.g.e.1569.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.03837i q^{3} +(0.240243 + 2.22312i) q^{5} -1.15495 q^{9} +O(q^{10})$	$q+2.03837i q^{3} +(0.240243 + 2.22312i) q^{5} -1.15495 q^{9} +1.04807 q^{11} -1.15495i q^{13} +(-4.53155 + 0.489703i) q^{15} +6.81986i q^{17} -4.75120 q^{19} -3.19331i q^{23} +(-4.88457 + 1.06818i) q^{25} +3.76090i q^{27} -5.14130 q^{29} +5.57372 q^{31} +2.13636i q^{33} +4.91769i q^{37} +2.35420 q^{39} -9.68948 q^{41} -7.44559i q^{43} +(-0.277467 - 2.56759i) q^{45} +9.29130i q^{47} -13.9014 q^{51} +5.60119i q^{53} +(0.251792 + 2.33000i) q^{55} -9.68469i q^{57} +10.8624 q^{59} +3.33443 q^{61} +(2.56759 - 0.277467i) q^{65} +5.45101i q^{67} +6.50915 q^{69} -4.10916 q^{71} -6.46879i q^{73} +(-2.17734 - 9.95655i) q^{75} -0.612744 q^{79} -11.1309 q^{81} -0.275623i q^{83} +(-15.1614 + 1.63842i) q^{85} -10.4799i q^{87} -17.2646 q^{89} +11.3613i q^{93} +(-1.14144 - 10.5625i) q^{95} -7.59854i q^{97} -1.21047 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 14 q^{9}+O(q^{10}) $ 12 * q - 14 * q^9	$ 12 q - 14 q^{9} + 2 q^{11} + 6 q^{15} - 10 q^{19} + 2 q^{25} + 6 q^{29} + 4 q^{31} - 20 q^{39} - 12 q^{41} - 8 q^{45} + 6 q^{55} - 48 q^{59} - 18 q^{61} + 26 q^{65} + 30 q^{69} + 8 q^{71} - 14 q^{75} + 44 q^{79} - 12 q^{81} - 22 q^{85} + 30 q^{89} + 26 q^{95} - 22 q^{99}+O(q^{100}) $ 12 * q - 14 * q^9 + 2 * q^11 + 6 * q^15 - 10 * q^19 + 2 * q^25 + 6 * q^29 + 4 * q^31 - 20 * q^39 - 12 * q^41 - 8 * q^45 + 6 * q^55 - 48 * q^59 - 18 * q^61 + 26 * q^65 + 30 * q^69 + 8 * q^71 - 14 * q^75 + 44 * q^79 - 12 * q^81 - 22 * q^85 + 30 * q^89 + 26 * q^95 - 22 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1960\mathbb{Z}\right)^\times$.

$n$	$981$	$1081$	$1177$	$1471$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$			2.03837i			1.17685i	0.808551	+	0.588426i	$0.200252\pi$
$3$			2.03837i			1.17685i	−0.808551	+	0.588426i	$0.799748\pi$
$4$		0			0
$5$	0.240243	+	2.22312i	0.107440	+	0.994212i
$5$	0.240243	+	2.22312i	0.107440	+	0.994212i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−1.15495			−0.384982
$10$		0			0
$11$	1.04807			0.316006			0.158003	−	0.987439i	$-0.449494\pi$
$11$	1.04807			0.316006			0.158003	+	0.987439i	$0.449494\pi$
$12$		0			0
$13$		−	1.15495i		−	0.320324i	−0.987091	−	0.160162i	$-0.948798\pi$
$13$		−	1.15495i		−	0.320324i	0.987091	−	0.160162i	$-0.0512017\pi$
$14$		0			0
$15$	−4.53155	+	0.489703i	−1.17004	+	0.126441i
$16$		0			0
$17$			6.81986i			1.65406i	0.562158	+	0.827030i	$0.309971\pi$
$17$			6.81986i			1.65406i	−0.562158	+	0.827030i	$0.690029\pi$
$18$		0			0
$19$	−4.75120			−1.09000			−0.545000	−	0.838436i	$-0.683470\pi$
$19$	−4.75120			−1.09000			−0.545000	+	0.838436i	$0.683470\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	3.19331i		−	0.665852i	−0.942953	−	0.332926i	$-0.891964\pi$
$23$		−	3.19331i		−	0.665852i	0.942953	−	0.332926i	$-0.108036\pi$
$24$		0			0
$25$	−4.88457	+	1.06818i	−0.976913	+	0.213636i
$26$		0			0
$27$			3.76090i			0.723786i
$28$		0			0
$29$	−5.14130			−0.954716			−0.477358	−	0.878709i	$-0.658405\pi$
$29$	−5.14130			−0.954716			−0.477358	+	0.878709i	$0.658405\pi$
$30$		0			0
$31$	5.57372			1.00107			0.500534	−	0.865717i	$-0.333137\pi$
$31$	5.57372			1.00107			0.500534	+	0.865717i	$0.333137\pi$
$32$		0			0
$33$			2.13636i			0.371892i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			4.91769i			0.808463i	0.914657	+	0.404232i	$0.132461\pi$
$37$			4.91769i			0.808463i	−0.914657	+	0.404232i	$0.867539\pi$
$38$		0			0
$39$	2.35420			0.376974
$40$		0			0
$41$	−9.68948			−1.51324			−0.756621	−	0.653853i	$-0.773151\pi$
$41$	−9.68948			−1.51324			−0.756621	+	0.653853i	$0.773151\pi$
$42$		0			0
$43$		−	7.44559i		−	1.13544i	−0.823221	−	0.567721i	$-0.807825\pi$
$43$		−	7.44559i		−	1.13544i	0.823221	−	0.567721i	$-0.192175\pi$
$44$		0			0
$45$	−0.277467	−	2.56759i	−0.0413624	−	0.382753i
$46$		0			0
$47$			9.29130i			1.35528i	0.735396	+	0.677638i	$0.236996\pi$
$47$			9.29130i			1.35528i	−0.735396	+	0.677638i	$0.763004\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−13.9014			−1.94658
$52$		0			0
$53$			5.60119i			0.769383i	0.923045	+	0.384692i	$0.125692\pi$
$53$			5.60119i			0.769383i	−0.923045	+	0.384692i	$0.874308\pi$
$54$		0			0
$55$	0.251792	+	2.33000i	0.0339516	+	0.314177i
$56$		0			0
$57$		−	9.68469i		−	1.28277i
$58$		0			0
$59$	10.8624			1.41416			0.707080	−	0.707134i	$-0.250012\pi$
$59$	10.8624			1.41416			0.707080	+	0.707134i	$0.250012\pi$
$60$		0			0
$61$	3.33443			0.426930			0.213465	−	0.976951i	$-0.431525\pi$
$61$	3.33443			0.426930			0.213465	+	0.976951i	$0.431525\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.56759	−	0.277467i	0.318470	−	0.0344156i
$66$		0			0
$67$			5.45101i			0.665947i	0.942936	+	0.332973i	$0.108052\pi$
$67$			5.45101i			0.665947i	−0.942936	+	0.332973i	$0.891948\pi$
$68$		0			0
$69$	6.50915			0.783609
$70$		0			0
$71$	−4.10916			−0.487668			−0.243834	−	0.969817i	$-0.578405\pi$
$71$	−4.10916			−0.487668			−0.243834	+	0.969817i	$0.578405\pi$
$72$		0			0
$73$		−	6.46879i		−	0.757114i	−0.925578	−	0.378557i	$-0.876420\pi$
$73$		−	6.46879i		−	0.757114i	0.925578	−	0.378557i	$-0.123580\pi$
$74$		0			0
$75$	−2.17734	−	9.95655i	−0.251418	−	1.14968i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−0.612744			−0.0689391			−0.0344695	−	0.999406i	$-0.510974\pi$
$79$	−0.612744			−0.0689391			−0.0344695	+	0.999406i	$0.510974\pi$
$80$		0			0
$81$	−11.1309			−1.23677
$82$		0			0
$83$		−	0.275623i		−	0.0302535i	−0.999886	−	0.0151268i	$-0.995185\pi$
$83$		−	0.275623i		−	0.0302535i	0.999886	−	0.0151268i	$-0.00481518\pi$
$84$		0			0
$85$	−15.1614	+	1.63842i	−1.64449	+	0.177712i
$86$		0			0
$87$		−	10.4799i		−	1.12356i
$88$		0			0
$89$	−17.2646			−1.83005			−0.915024	−	0.403399i	$-0.867829\pi$
$89$	−17.2646			−1.83005			−0.915024	+	0.403399i	$0.867829\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$			11.3613i			1.17811i
$94$		0			0
$95$	−1.14144	−	10.5625i	−0.117109	−	1.08369i
$96$		0			0
$97$		−	7.59854i		−	0.771515i	−0.922600	−	0.385757i	$-0.873940\pi$
$97$		−	7.59854i		−	0.771515i	0.922600	−	0.385757i	$-0.126060\pi$
$98$		0			0
$99$	−1.21047			−0.121657
$100$		0			0
$101$	0.564671			0.0561869			0.0280934	−	0.999605i	$-0.491056\pi$
$101$	0.564671			0.0561869			0.0280934	+	0.999605i	$0.491056\pi$
$102$		0			0
$103$		−	15.5819i		−	1.53533i	−0.640848	−	0.767667i	$-0.721417\pi$
$103$		−	15.5819i		−	1.53533i	0.640848	−	0.767667i	$-0.278583\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	14.0241i		−	1.35576i	−0.735172	−	0.677880i	$-0.762899\pi$
$107$		−	14.0241i		−	1.35576i	0.735172	−	0.677880i	$-0.237101\pi$
$108$		0			0
$109$	15.7978			1.51315			0.756577	−	0.653904i	$-0.226870\pi$
$109$	15.7978			1.51315			0.756577	+	0.653904i	$0.226870\pi$
$110$		0			0
$111$	−10.0241			−0.951442
$112$		0			0
$113$			20.3625i			1.91554i	0.287536	+	0.957770i	$0.407164\pi$
$113$			20.3625i			1.91554i	−0.287536	+	0.957770i	$0.592836\pi$
$114$		0			0
$115$	7.09913	−	0.767171i	0.661998	−	0.0715390i
$116$		0			0
$117$			1.33390i			0.123319i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−9.90154			−0.900140
$122$		0			0
$123$		−	19.7507i		−	1.78086i
$124$		0			0
$125$	−3.54818	−	10.6024i	−0.317359	−	0.948306i
$126$		0			0
$127$		−	0.358593i		−	0.0318199i	−0.999873	−	0.0159100i	$-0.994935\pi$
$127$		−	0.358593i		−	0.0318199i	0.999873	−	0.0159100i	$-0.00506451\pi$
$128$		0			0
$129$	15.1769			1.33625
$130$		0			0
$131$	15.5114			1.35524			0.677620	−	0.735412i	$-0.263011\pi$
$131$	15.5114			1.35524			0.677620	+	0.735412i	$0.263011\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−8.36095	+	0.903529i	−0.719596	+	0.0777634i
$136$		0			0
$137$		−	0.842399i		−	0.0719710i	−0.999352	−	0.0359855i	$-0.988543\pi$
$137$		−	0.842399i		−	0.0719710i	0.999352	−	0.0359855i	$-0.0114570\pi$
$138$		0			0
$139$	2.97876			0.252655			0.126327	−	0.991989i	$-0.459681\pi$
$139$	2.97876			0.252655			0.126327	+	0.991989i	$0.459681\pi$
$140$		0			0
$141$	−18.9391			−1.59496
$142$		0			0
$143$		−	1.21047i		−	0.101224i
$144$		0			0
$145$	−1.23516	−	11.4298i	−0.102575	−	0.949190i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.10290			−0.0903533			−0.0451767	−	0.998979i	$-0.514385\pi$
$149$	−1.10290			−0.0903533			−0.0451767	+	0.998979i	$0.514385\pi$
$150$		0			0
$151$	−12.5525			−1.02151			−0.510753	−	0.859728i	$-0.670633\pi$
$151$	−12.5525			−1.02151			−0.510753	+	0.859728i	$0.670633\pi$
$152$		0			0
$153$		−	7.87657i		−	0.636783i
$154$		0			0
$155$	1.33904	+	12.3911i	0.107555	+	0.995274i
$156$		0			0
$157$			21.0712i			1.68166i	0.541298	+	0.840831i	$0.317933\pi$
$157$			21.0712i			1.68166i	−0.541298	+	0.840831i	$0.682067\pi$
$158$		0			0
$159$	−11.4173			−0.905451
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	21.8658i		−	1.71266i	−0.516425	−	0.856332i	$-0.672738\pi$
$163$		−	21.8658i		−	1.71266i	0.516425	−	0.856332i	$-0.327262\pi$
$164$		0			0
$165$	−4.74939	+	0.513245i	−0.369740	+	0.0399561i
$166$		0			0
$167$			10.1680i			0.786821i	0.919363	+	0.393411i	$0.128705\pi$
$167$			10.1680i			0.786821i	−0.919363	+	0.393411i	$0.871295\pi$
$168$		0			0
$169$	11.6661			0.897392
$170$		0			0
$171$	5.48737			0.419630
$172$		0			0
$173$			0.798446i			0.0607048i	0.999539	+	0.0303524i	$0.00966295\pi$
$173$			0.798446i			0.0607048i	−0.999539	+	0.0303524i	$0.990337\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			22.1415i			1.66426i
$178$		0			0
$179$	12.1755			0.910042			0.455021	−	0.890481i	$-0.349632\pi$
$179$	12.1755			0.910042			0.455021	+	0.890481i	$0.349632\pi$
$180$		0			0
$181$	4.00613			0.297773			0.148887	−	0.988854i	$-0.452431\pi$
$181$	4.00613			0.297773			0.148887	+	0.988854i	$0.452431\pi$
$182$		0			0
$183$			6.79680i			0.502434i
$184$		0			0
$185$	−10.9326	+	1.18144i	−0.803784	+	0.0868612i
$186$		0			0
$187$			7.14771i			0.522693i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	16.6616			1.20559			0.602797	−	0.797895i	$-0.294053\pi$
$191$	16.6616			1.20559			0.602797	+	0.797895i	$0.294053\pi$
$192$		0			0
$193$			7.42501i			0.534464i	0.963632	+	0.267232i	$0.0861090\pi$
$193$			7.42501i			0.534464i	−0.963632	+	0.267232i	$0.913891\pi$
$194$		0			0
$195$	0.565581	+	5.23369i	0.0405021	+	0.374792i
$196$		0			0
$197$		−	1.88883i		−	0.134574i	−0.997734	−	0.0672869i	$-0.978566\pi$
$197$		−	1.88883i		−	0.134574i	0.997734	−	0.0672869i	$-0.0214343\pi$
$198$		0			0
$199$	−17.2373			−1.22192			−0.610960	−	0.791662i	$-0.709216\pi$
$199$	−17.2373			−1.22192			−0.610960	+	0.791662i	$0.709216\pi$
$200$		0			0
$201$	−11.1112			−0.783721
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−2.32783	−	21.5409i	−0.162583	−	1.50448i
$206$		0			0
$207$			3.68810i			0.256341i
$208$		0			0
$209$	−4.97960			−0.344446
$210$		0			0
$211$	−5.25478			−0.361754			−0.180877	−	0.983506i	$-0.557894\pi$
$211$	−5.25478			−0.361754			−0.180877	+	0.983506i	$0.557894\pi$
$212$		0			0
$213$		−	8.37598i		−	0.573913i
$214$		0			0
$215$	16.5525	−	1.78875i	1.12887	−	0.121992i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	13.1858			0.891012
$220$		0			0
$221$	7.87657			0.529835
$222$		0			0
$223$			13.2760i			0.889027i	0.895772	+	0.444513i	$0.146623\pi$
$223$			13.2760i			0.889027i	−0.895772	+	0.444513i	$0.853377\pi$
$224$		0			0
$225$	5.64141	−	1.23369i	0.376094	−	0.0822459i
$226$		0			0
$227$			11.8039i			0.783455i	0.920081	+	0.391727i	$0.128122\pi$
$227$			11.8039i			0.783455i	−0.920081	+	0.391727i	$0.871878\pi$
$228$		0			0
$229$	13.4922			0.891592			0.445796	−	0.895135i	$-0.352921\pi$
$229$	13.4922			0.891592			0.445796	+	0.895135i	$0.352921\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	1.14060i		−	0.0747230i	−0.999302	−	0.0373615i	$-0.988105\pi$
$233$		−	1.14060i		−	0.0747230i	0.999302	−	0.0373615i	$-0.0118953\pi$
$234$		0			0
$235$	−20.6557	+	2.23217i	−1.34743	+	0.145611i
$236$		0			0
$237$		−	1.24900i		−	0.0811311i
$238$		0			0
$239$	−2.94796			−0.190688			−0.0953438	−	0.995444i	$-0.530395\pi$
$239$	−2.94796			−0.190688			−0.0953438	+	0.995444i	$0.530395\pi$
$240$		0			0
$241$	−10.6131			−0.683650			−0.341825	−	0.939764i	$-0.611045\pi$
$241$	−10.6131			−0.683650			−0.341825	+	0.939764i	$0.611045\pi$
$242$		0			0
$243$		−	11.4062i		−	0.731711i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			5.48737i			0.349153i
$248$		0			0
$249$	0.561821			0.0356039
$250$		0			0
$251$	12.3119			0.777120			0.388560	−	0.921423i	$-0.372973\pi$
$251$	12.3119			0.777120			0.388560	+	0.921423i	$0.372973\pi$
$252$		0			0
$253$		−	3.34683i		−	0.210413i
$254$		0			0
$255$	−3.33971	−	30.9045i	−0.209141	−	1.93532i
$256$		0			0
$257$			17.7484i			1.10712i	0.832811	+	0.553558i	$0.186730\pi$
$257$			17.7484i			1.10712i	−0.832811	+	0.553558i	$0.813270\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	5.93792			0.367548
$262$		0			0
$263$			4.17570i			0.257485i	0.991678	+	0.128742i	$0.0410940\pi$
$263$			4.17570i			0.257485i	−0.991678	+	0.128742i	$0.958906\pi$
$264$		0			0
$265$	−12.4522	+	1.34565i	−0.764930	+	0.0826624i
$266$		0			0
$267$		−	35.1917i		−	2.15370i
$268$		0			0
$269$	26.2681			1.60160			0.800798	−	0.598935i	$-0.204409\pi$
$269$	26.2681			1.60160			0.800798	+	0.598935i	$0.204409\pi$
$270$		0			0
$271$	−26.3074			−1.59806			−0.799031	−	0.601290i	$-0.794654\pi$
$271$	−26.3074			−1.59806			−0.799031	+	0.601290i	$0.794654\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−5.11938	+	1.11953i	−0.308710	+	0.0675102i
$276$		0			0
$277$			20.9321i			1.25769i	0.777530	+	0.628845i	$0.216472\pi$
$277$			20.9321i			1.25769i	−0.777530	+	0.628845i	$0.783528\pi$
$278$		0			0
$279$	−6.43734			−0.385393
$280$		0			0
$281$	15.7097			0.937162			0.468581	−	0.883420i	$-0.344765\pi$
$281$	15.7097			0.937162			0.468581	+	0.883420i	$0.344765\pi$
$282$		0			0
$283$			10.8087i			0.642512i	0.946992	+	0.321256i	$0.104105\pi$
$283$			10.8087i			0.642512i	−0.946992	+	0.321256i	$0.895895\pi$
$284$		0			0
$285$	21.5303	−	2.32668i	1.27534	−	0.137820i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−29.5105			−1.73591
$290$		0			0
$291$	15.4886			0.907959
$292$		0			0
$293$			9.91887i			0.579467i	0.957107	+	0.289733i	$0.0935667\pi$
$293$			9.91887i			0.579467i	−0.957107	+	0.289733i	$0.906433\pi$
$294$		0			0
$295$	2.60960	+	24.1484i	0.151937	+	1.40597i
$296$		0			0
$297$			3.94170i			0.228721i
$298$		0			0
$299$	−3.68810			−0.213288
$300$		0			0
$301$		0			0
$302$		0			0
$303$			1.15101i			0.0661236i
$304$		0			0
$305$	0.801074	+	7.41286i	0.0458693	+	0.424459i
$306$		0			0
$307$			5.64632i			0.322252i	0.986934	+	0.161126i	$0.0515126\pi$
$307$			5.64632i			0.322252i	−0.986934	+	0.161126i	$0.948487\pi$
$308$		0			0
$309$	31.7617			1.80686
$310$		0			0
$311$	−12.0190			−0.681534			−0.340767	−	0.940148i	$-0.610687\pi$
$311$	−12.0190			−0.681534			−0.340767	+	0.940148i	$0.610687\pi$
$312$		0			0
$313$			18.9785i			1.07273i	0.843987	+	0.536363i	$0.180202\pi$
$313$			18.9785i			1.07273i	−0.843987	+	0.536363i	$0.819798\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			12.7418i			0.715651i	0.933788	+	0.357826i	$0.116482\pi$
$317$			12.7418i			0.715651i	−0.933788	+	0.357826i	$0.883518\pi$
$318$		0			0
$319$	−5.38846			−0.301696
$320$		0			0
$321$	28.5863			1.59553
$322$		0			0
$323$		−	32.4025i		−	1.80292i
$324$		0			0
$325$	1.23369	+	5.64141i	0.0684327	+	0.312929i
$326$		0			0
$327$			32.2017i			1.78076i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	23.8384			1.31027			0.655137	−	0.755510i	$-0.272611\pi$
$331$	23.8384			1.31027			0.655137	+	0.755510i	$0.272611\pi$
$332$		0			0
$333$		−	5.67966i		−	0.311244i
$334$		0			0
$335$	−12.1183	+	1.30957i	−0.662092	+	0.0715492i
$336$		0			0
$337$		−	23.7427i		−	1.29335i	−0.762767	−	0.646673i	$-0.776160\pi$
$337$		−	23.7427i		−	1.29335i	0.762767	−	0.646673i	$-0.223840\pi$
$338$		0			0
$339$	−41.5062			−2.25431
$340$		0			0
$341$	5.84166			0.316344
$342$		0			0
$343$		0			0
$344$		0			0
$345$	1.56378	+	14.4707i	0.0841909	+	0.779074i
$346$		0			0
$347$		−	2.53548i		−	0.136112i	−0.997682	−	0.0680558i	$-0.978320\pi$
$347$		−	2.53548i		−	0.136112i	0.997682	−	0.0680558i	$-0.0216796\pi$
$348$		0			0
$349$	24.9321			1.33459			0.667293	−	0.744795i	$-0.267453\pi$
$349$	24.9321			1.33459			0.667293	+	0.744795i	$0.267453\pi$
$350$		0			0
$351$	4.34363			0.231846
$352$		0			0
$353$			2.35067i			0.125113i	0.998041	+	0.0625567i	$0.0199254\pi$
$353$			2.35067i			0.125113i	−0.998041	+	0.0625567i	$0.980075\pi$
$354$		0			0
$355$	−0.987196	−	9.13517i	−0.0523949	−	0.484845i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	32.3016			1.70481			0.852407	−	0.522880i	$-0.175142\pi$
$359$	32.3016			1.70481			0.852407	+	0.522880i	$0.175142\pi$
$360$		0			0
$361$	3.57387			0.188098
$362$		0			0
$363$		−	20.1830i		−	1.05933i
$364$		0			0
$365$	14.3809	−	1.55408i	0.752732	−	0.0813442i
$366$		0			0
$367$			7.55523i			0.394380i	0.980365	+	0.197190i	$0.0631816\pi$
$367$			7.55523i			0.394380i	−0.980365	+	0.197190i	$0.936818\pi$
$368$		0			0
$369$	11.1908			0.582571
$370$		0			0
$371$		0			0
$372$		0			0
$373$			29.0647i			1.50491i	0.658641	+	0.752457i	$0.271132\pi$
$373$			29.0647i			1.50491i	−0.658641	+	0.752457i	$0.728868\pi$
$374$		0			0
$375$	21.6116	−	7.23249i	1.11602	−	0.373484i
$376$		0			0
$377$			5.93792i			0.305819i
$378$		0			0
$379$	9.36318			0.480954			0.240477	−	0.970655i	$-0.422696\pi$
$379$	9.36318			0.480954			0.240477	+	0.970655i	$0.422696\pi$
$380$		0			0
$381$	0.730944			0.0374474
$382$		0			0
$383$			1.93471i			0.0988589i	0.998778	+	0.0494295i	$0.0157403\pi$
$383$			1.93471i			0.0988589i	−0.998778	+	0.0494295i	$0.984260\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			8.59925i			0.437124i
$388$		0			0
$389$	−0.757390			−0.0384012			−0.0192006	−	0.999816i	$-0.506112\pi$
$389$	−0.757390			−0.0384012			−0.0192006	+	0.999816i	$0.506112\pi$
$390$		0			0
$391$	21.7780			1.10136
$392$		0			0
$393$			31.6180i			1.59492i
$394$		0			0
$395$	−0.147207	−	1.36221i	−0.00740680	−	0.0685400i
$396$		0			0
$397$			33.4580i			1.67921i	0.543199	+	0.839604i	$0.317213\pi$
$397$			33.4580i			1.67921i	−0.543199	+	0.839604i	$0.682787\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	30.5011			1.52315			0.761577	−	0.648074i	$-0.224425\pi$
$401$	30.5011			1.52315			0.761577	+	0.648074i	$0.224425\pi$
$402$		0			0
$403$		−	6.43734i		−	0.320667i
$404$		0			0
$405$	−2.67413	−	24.7455i	−0.132878	−	1.22961i
$406$		0			0
$407$			5.15410i			0.255479i
$408$		0			0
$409$	−18.6737			−0.923353			−0.461676	−	0.887048i	$-0.652752\pi$
$409$	−18.6737			−0.923353			−0.461676	+	0.887048i	$0.652752\pi$
$410$		0			0
$411$	1.71712			0.0846992
$412$		0			0
$413$		0			0
$414$		0			0
$415$	0.612744	−	0.0662164i	0.0300784	−	0.00325044i
$416$		0			0
$417$			6.07180i			0.297338i
$418$		0			0
$419$	−34.8449			−1.70228			−0.851142	−	0.524936i	$-0.824089\pi$
$419$	−34.8449			−1.70228			−0.851142	+	0.524936i	$0.824089\pi$
$420$		0			0
$421$	5.53106			0.269567			0.134784	−	0.990875i	$-0.456966\pi$
$421$	5.53106			0.269567			0.134784	+	0.990875i	$0.456966\pi$
$422$		0			0
$423$		−	10.7309i		−	0.521757i
$424$		0			0
$425$	−7.28484	−	33.3121i	−0.353366	−	1.61587i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	2.46738			0.119126
$430$		0			0
$431$	10.5655			0.508922			0.254461	−	0.967083i	$-0.418102\pi$
$431$	10.5655			0.508922			0.254461	+	0.967083i	$0.418102\pi$
$432$		0			0
$433$		−	14.3654i		−	0.690358i	−0.938537	−	0.345179i	$-0.887818\pi$
$433$		−	14.3654i		−	0.690358i	0.938537	−	0.345179i	$-0.112182\pi$
$434$		0			0
$435$	23.2981	−	2.51771i	1.11706	−	0.120715i
$436$		0			0
$437$			15.1721i			0.725778i
$438$		0			0
$439$	31.6168			1.50899			0.754493	−	0.656308i	$-0.227883\pi$
$439$	31.6168			1.50899			0.754493	+	0.656308i	$0.227883\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	11.9930i		−	0.569804i	−0.958557	−	0.284902i	$-0.908039\pi$
$443$		−	11.9930i		−	0.569804i	0.958557	−	0.284902i	$-0.0919611\pi$
$444$		0			0
$445$	−4.14771	−	38.3814i	−0.196620	−	1.81946i
$446$		0			0
$447$		−	2.24812i		−	0.106333i
$448$		0			0
$449$	4.98107			0.235071			0.117536	−	0.993069i	$-0.462501\pi$
$449$	4.98107			0.235071			0.117536	+	0.993069i	$0.462501\pi$
$450$		0			0
$451$	−10.1553			−0.478194
$452$		0			0
$453$		−	25.5866i		−	1.20216i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	26.9785i		−	1.26200i	−0.775783	−	0.631000i	$-0.782645\pi$
$457$		−	26.9785i		−	1.26200i	0.775783	−	0.631000i	$-0.217355\pi$
$458$		0			0
$459$	−25.6488			−1.19718
$460$		0			0
$461$	−22.6886			−1.05672			−0.528358	−	0.849022i	$-0.677192\pi$
$461$	−22.6886			−1.05672			−0.528358	+	0.849022i	$0.677192\pi$
$462$		0			0
$463$			27.0921i			1.25908i	0.776969	+	0.629539i	$0.216756\pi$
$463$			27.0921i			1.25908i	−0.776969	+	0.629539i	$0.783244\pi$
$464$		0			0
$465$	−25.2576	+	2.72947i	−1.17129	+	0.126576i
$466$		0			0
$467$			4.55482i			0.210772i	0.994431	+	0.105386i	$0.0336078\pi$
$467$			4.55482i			0.210772i	−0.994431	+	0.105386i	$0.966392\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−42.9508			−1.97907
$472$		0			0
$473$		−	7.80352i		−	0.358806i
$474$		0			0
$475$	23.2075	−	5.07513i	1.06483	−	0.232863i
$476$		0			0
$477$		−	6.46907i		−	0.296199i
$478$		0			0
$479$	8.22937			0.376010			0.188005	−	0.982168i	$-0.439798\pi$
$479$	8.22937			0.376010			0.188005	+	0.982168i	$0.439798\pi$
$480$		0			0
$481$	5.67966			0.258970
$482$		0			0
$483$		0			0
$484$		0			0
$485$	16.8925	−	1.82549i	0.767049	−	0.0828914i
$486$		0			0
$487$			19.0063i			0.861259i	0.902529	+	0.430629i	$0.141708\pi$
$487$			19.0063i			0.861259i	−0.902529	+	0.430629i	$0.858292\pi$
$488$		0			0
$489$	44.5706			2.01555
$490$		0			0
$491$	19.2359			0.868102			0.434051	−	0.900888i	$-0.357084\pi$
$491$	19.2359			0.868102			0.434051	+	0.900888i	$0.357084\pi$
$492$		0			0
$493$		−	35.0630i		−	1.57916i
$494$		0			0
$495$	−0.290806	−	2.69102i	−0.0130708	−	0.120952i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−12.2105			−0.546618			−0.273309	−	0.961926i	$-0.588118\pi$
$499$	−12.2105			−0.546618			−0.273309	+	0.961926i	$0.588118\pi$
$500$		0			0
$501$	−20.7261			−0.925972
$502$		0			0
$503$			2.17917i			0.0971646i	0.998819	+	0.0485823i	$0.0154703\pi$
$503$			2.17917i			0.0971646i	−0.998819	+	0.0485823i	$0.984530\pi$
$504$		0			0
$505$	0.135658	+	1.25533i	0.00603671	+	0.0558616i
$506$		0			0
$507$			23.7798i			1.05610i
$508$		0			0
$509$	12.4401			0.551398			0.275699	−	0.961244i	$-0.411091\pi$
$509$	12.4401			0.551398			0.275699	+	0.961244i	$0.411091\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		−	17.8688i		−	0.788926i
$514$		0			0
$515$	34.6406	−	3.74345i	1.52645	−	0.164956i
$516$		0			0
$517$			9.73797i			0.428275i
$518$		0			0
$519$	−1.62753			−0.0714405
$520$		0			0
$521$	−18.8545			−0.826030			−0.413015	−	0.910724i	$-0.635524\pi$
$521$	−18.8545			−0.826030			−0.413015	+	0.910724i	$0.635524\pi$
$522$		0			0
$523$		−	19.0063i		−	0.831088i	−0.909573	−	0.415544i	$-0.863591\pi$
$523$		−	19.0063i		−	0.831088i	0.909573	−	0.415544i	$-0.136409\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			38.0120i			1.65583i
$528$		0			0
$529$	12.8027			0.556641
$530$		0			0
$531$	−12.5454			−0.544426
$532$		0			0
$533$			11.1908i			0.484728i
$534$		0			0
$535$	31.1773	−	3.36919i	1.34791	−	0.145663i
$536$		0			0
$537$			24.8182i			1.07099i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	24.3621			1.04741			0.523703	−	0.851901i	$-0.324550\pi$
$541$	24.3621			1.04741			0.523703	+	0.851901i	$0.324550\pi$
$542$		0			0
$543$			8.16596i			0.350435i
$544$		0			0
$545$	3.79531	+	35.1205i	0.162573	+	1.50440i
$546$		0			0
$547$			3.44667i			0.147369i	0.997282	+	0.0736845i	$0.0234758\pi$
$547$			3.44667i			0.147369i	−0.997282	+	0.0736845i	$0.976524\pi$
$548$		0			0
$549$	−3.85109			−0.164360
$550$		0			0
$551$	24.4273			1.04064
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−2.40821	−	22.2847i	−0.102223	−	0.945935i
$556$		0			0
$557$			38.5215i			1.63221i	0.577905	+	0.816104i	$0.303870\pi$
$557$			38.5215i			1.63221i	−0.577905	+	0.816104i	$0.696130\pi$
$558$		0			0
$559$	−8.59925			−0.363709
$560$		0			0
$561$	−14.5697			−0.615132
$562$		0			0
$563$			10.7673i			0.453788i	0.973919	+	0.226894i	$0.0728571\pi$
$563$			10.7673i			0.453788i	−0.973919	+	0.226894i	$0.927143\pi$
$564$		0			0
$565$	−45.2683	+	4.89194i	−1.90445	+	0.205805i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	23.3706			0.979747			0.489873	−	0.871794i	$-0.337043\pi$
$569$	23.3706			0.979747			0.489873	+	0.871794i	$0.337043\pi$
$570$		0			0
$571$	−11.5645			−0.483960			−0.241980	−	0.970281i	$-0.577797\pi$
$571$	−11.5645			−0.483960			−0.241980	+	0.970281i	$0.577797\pi$
$572$		0			0
$573$			33.9625i			1.41881i
$574$		0			0
$575$	3.41103	+	15.5980i	0.142250	+	0.650480i
$576$		0			0
$577$		−	2.16239i		−	0.0900213i	−0.998987	−	0.0450107i	$-0.985668\pi$
$577$		−	2.16239i		−	0.0900213i	0.998987	−	0.0450107i	$-0.0143322\pi$
$578$		0			0
$579$	−15.1349			−0.628985
$580$		0			0
$581$		0			0
$582$		0			0
$583$			5.87046i			0.243130i
$584$		0			0
$585$	−2.96542	+	0.320460i	−0.122605	+	0.0132494i
$586$		0			0
$587$			1.65367i			0.0682544i	0.999417	+	0.0341272i	$0.0108651\pi$
$587$			1.65367i			0.0682544i	−0.999417	+	0.0341272i	$0.989135\pi$
$588$		0			0
$589$	−26.4818			−1.09116
$590$		0			0
$591$	3.85014			0.158374
$592$		0			0
$593$		−	25.5873i		−	1.05074i	−0.850873	−	0.525372i	$-0.823926\pi$
$593$		−	25.5873i		−	1.05074i	0.850873	−	0.525372i	$-0.176074\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	35.1360i		−	1.43802i
$598$		0			0
$599$	−11.7773			−0.481209			−0.240604	−	0.970623i	$-0.577346\pi$
$599$	−11.7773			−0.481209			−0.240604	+	0.970623i	$0.577346\pi$
$600$		0			0
$601$	−18.5454			−0.756484			−0.378242	−	0.925707i	$-0.623471\pi$
$601$	−18.5454			−0.756484			−0.378242	+	0.925707i	$0.623471\pi$
$602$		0			0
$603$		−	6.29562i		−	0.256377i
$604$		0			0
$605$	−2.37877	−	22.0124i	−0.0967109	−	0.894930i
$606$		0			0
$607$		−	7.14662i		−	0.290072i	−0.989426	−	0.145036i	$-0.953670\pi$
$607$		−	7.14662i		−	0.290072i	0.989426	−	0.145036i	$-0.0463298\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	10.7309			0.434128
$612$		0			0
$613$		−	20.5746i		−	0.831002i	−0.909593	−	0.415501i	$-0.863606\pi$
$613$		−	20.5746i		−	0.831002i	0.909593	−	0.415501i	$-0.136394\pi$
$614$		0			0
$615$	43.9083	−	4.74497i	1.77055	−	0.191336i
$616$		0			0
$617$		−	44.4857i		−	1.79093i	−0.445134	−	0.895464i	$-0.646844\pi$
$617$		−	44.4857i		−	1.79093i	0.445134	−	0.895464i	$-0.353156\pi$
$618$		0			0
$619$	−28.8136			−1.15812			−0.579058	−	0.815286i	$-0.696580\pi$
$619$	−28.8136			−1.15812			−0.579058	+	0.815286i	$0.696580\pi$
$620$		0			0
$621$	12.0097			0.481934
$622$		0			0
$623$		0			0
$624$		0			0
$625$	22.7180	−	10.4352i	0.908719	−	0.417407i
$626$		0			0
$627$		−	10.1503i		−	0.405362i
$628$		0			0
$629$	−33.5380			−1.33725
$630$		0			0
$631$	48.6854			1.93813			0.969067	−	0.246796i	$-0.0793777\pi$
$631$	48.6854			1.93813			0.969067	+	0.246796i	$0.0793777\pi$
$632$		0			0
$633$		−	10.7112i		−	0.425731i
$634$		0			0
$635$	0.797196	−	0.0861493i	0.0316358	−	0.00341873i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	4.74585			0.187743
$640$		0			0
$641$	36.2497			1.43178			0.715888	−	0.698216i	$-0.246022\pi$
$641$	36.2497			1.43178			0.715888	+	0.698216i	$0.246022\pi$
$642$		0			0
$643$		−	17.4717i		−	0.689015i	−0.938784	−	0.344507i	$-0.888046\pi$
$643$		−	17.4717i		−	0.689015i	0.938784	−	0.344507i	$-0.111954\pi$
$644$		0			0
$645$	3.64613	+	33.7400i	0.143566	+	1.32851i
$646$		0			0
$647$		−	6.67004i		−	0.262226i	−0.991367	−	0.131113i	$-0.958145\pi$
$647$		−	6.67004i		−	0.262226i	0.991367	−	0.131113i	$-0.0418551\pi$
$648$		0			0
$649$	11.3845			0.446883
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	16.0250i		−	0.627107i	−0.949570	−	0.313554i	$-0.898480\pi$
$653$		−	16.0250i		−	0.627107i	0.949570	−	0.313554i	$-0.101520\pi$
$654$		0			0
$655$	3.72651	+	34.4839i	0.145607	+	1.34740i
$656$		0			0
$657$			7.47109i			0.291475i
$658$		0			0
$659$	−12.3934			−0.482780			−0.241390	−	0.970428i	$-0.577603\pi$
$659$	−12.3934			−0.482780			−0.241390	+	0.970428i	$0.577603\pi$
$660$		0			0
$661$	−12.3132			−0.478928			−0.239464	−	0.970905i	$-0.576972\pi$
$661$	−12.3132			−0.478928			−0.239464	+	0.970905i	$0.576972\pi$
$662$		0			0
$663$			16.0553i			0.623538i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			16.4178i			0.635699i
$668$		0			0
$669$	−27.0614			−1.04625
$670$		0			0
$671$	3.49473			0.134913
$672$		0			0
$673$			10.3139i			0.397572i	0.980043	+	0.198786i	$0.0636998\pi$
$673$			10.3139i			0.397572i	−0.980043	+	0.198786i	$0.936300\pi$
$674$		0			0
$675$	−4.01732	−	18.3704i	−0.154627	−	0.707076i
$676$		0			0
$677$		−	30.3992i		−	1.16834i	−0.811633	−	0.584168i	$-0.801421\pi$
$677$		−	30.3992i		−	1.16834i	0.811633	−	0.584168i	$-0.198579\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−24.0608			−0.922010
$682$		0			0
$683$			25.7960i			0.987057i	0.869730	+	0.493529i	$0.164293\pi$
$683$			25.7960i			0.987057i	−0.869730	+	0.493529i	$0.835707\pi$
$684$		0			0
$685$	1.87276	−	0.202380i	0.0715544	−	0.00773255i
$686$		0			0
$687$			27.5021i			1.04927i
$688$		0			0
$689$	6.46907			0.246452
$690$		0			0
$691$	15.6453			0.595176			0.297588	−	0.954694i	$-0.403818\pi$
$691$	15.6453			0.595176			0.297588	+	0.954694i	$0.403818\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	0.715625	+	6.62215i	0.0271452	+	0.251192i
$696$		0			0
$697$		−	66.0809i		−	2.50299i
$698$		0			0
$699$	2.32496			0.0879379
$700$		0			0
$701$	30.3083			1.14473			0.572365	−	0.819999i	$-0.306026\pi$
$701$	30.3083			1.14473			0.572365	+	0.819999i	$0.306026\pi$
$702$		0			0
$703$		−	23.3649i		−	0.881224i
$704$		0			0
$705$	−4.54998	−	42.1040i	−0.171362	−	1.58573i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−40.7086			−1.52885			−0.764423	−	0.644715i	$-0.776976\pi$
$709$	−40.7086			−1.52885			−0.764423	+	0.644715i	$0.776976\pi$
$710$		0			0
$711$	0.707686			0.0265403
$712$		0			0
$713$		−	17.7986i		−	0.666564i
$714$		0			0
$715$	2.69102	−	0.290806i	0.100638	−	0.0108755i
$716$		0			0
$717$		−	6.00902i		−	0.224411i
$718$		0			0
$719$	11.2692			0.420269			0.210134	−	0.977673i	$-0.432610\pi$
$719$	11.2692			0.420269			0.210134	+	0.977673i	$0.432610\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		−	21.6334i		−	0.804555i
$724$		0			0
$725$	25.1130	−	5.49183i	0.932675	−	0.203962i
$726$		0			0
$727$			25.5742i			0.948495i	0.880392	+	0.474248i	$0.157280\pi$
$727$			25.5742i			0.948495i	−0.880392	+	0.474248i	$0.842720\pi$
$728$		0			0
$729$	−10.1427			−0.375655
$730$		0			0
$731$	50.7779			1.87809
$732$		0			0
$733$		−	7.86641i		−	0.290552i	−0.989391	−	0.145276i	$-0.953593\pi$
$733$		−	7.86641i		−	0.290552i	0.989391	−	0.145276i	$-0.0464071\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			5.71306i			0.210443i
$738$		0			0
$739$	−14.0515			−0.516892			−0.258446	−	0.966026i	$-0.583210\pi$
$739$	−14.0515			−0.516892			−0.258446	+	0.966026i	$0.583210\pi$
$740$		0			0
$741$	−11.1853			−0.410902
$742$		0			0
$743$			11.3464i			0.416258i	0.978101	+	0.208129i	$0.0667374\pi$
$743$			11.3464i			0.416258i	−0.978101	+	0.208129i	$0.933263\pi$
$744$		0			0
$745$	−0.264964	−	2.45189i	−0.00970754	−	0.0898303i
$746$		0			0
$747$			0.318329i			0.0116471i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−48.1936			−1.75861			−0.879306	−	0.476258i	$-0.841993\pi$
$751$	−48.1936			−1.75861			−0.879306	+	0.476258i	$0.841993\pi$
$752$		0			0
$753$			25.0962i			0.914556i
$754$		0			0
$755$	−3.01564	−	27.9057i	−0.109750	−	1.01559i
$756$		0			0
$757$			20.3236i			0.738676i	0.929295	+	0.369338i	$0.120415\pi$
$757$			20.3236i			0.738676i	−0.929295	+	0.369338i	$0.879585\pi$
$758$		0			0
$759$	6.82206			0.247625
$760$		0			0
$761$	−6.92883			−0.251170			−0.125585	−	0.992083i	$-0.540081\pi$
$761$	−6.92883			−0.251170			−0.125585	+	0.992083i	$0.540081\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	17.5106	−	1.89229i	0.633097	−	0.0684158i
$766$		0			0
$767$		−	12.5454i		−	0.452989i
$768$		0			0
$769$	−32.5653			−1.17434			−0.587168	−	0.809465i	$-0.699757\pi$
$769$	−32.5653			−1.17434			−0.587168	+	0.809465i	$0.699757\pi$
$770$		0			0
$771$	−36.1778			−1.30291
$772$		0			0
$773$			31.4728i			1.13200i	0.824407	+	0.565998i	$0.191509\pi$
$773$			31.4728i			1.13200i	−0.824407	+	0.565998i	$0.808491\pi$
$774$		0			0
$775$	−27.2252	+	5.95373i	−0.977957	+	0.213864i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	46.0366			1.64943
$780$		0			0
$781$	−4.30670			−0.154106
$782$		0			0
$783$		−	19.3359i		−	0.691010i
$784$		0			0
$785$	−46.8438	+	5.06220i	−1.67193	+	0.180678i
$786$		0			0
$787$			45.9267i			1.63711i	0.574428	+	0.818555i	$0.305224\pi$
$787$			45.9267i			1.63711i	−0.574428	+	0.818555i	$0.694776\pi$
$788$		0			0
$789$	−8.51162			−0.303022
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	3.85109i		−	0.136756i
$794$		0			0
$795$	−2.74292	−	25.3821i	−0.0972815	−	0.900209i
$796$		0			0
$797$		−	13.5988i		−	0.481696i	−0.970563	−	0.240848i	$-0.922575\pi$
$797$		−	13.5988i		−	0.481696i	0.970563	−	0.240848i	$-0.0774255\pi$
$798$		0			0
$799$	−63.3654			−2.24171
$800$		0			0
$801$	19.9397			0.704535
$802$		0			0
$803$		−	6.77976i		−	0.239253i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			53.5441i			1.88484i
$808$		0			0
$809$	−17.0751			−0.600329			−0.300165	−	0.953887i	$-0.597042\pi$
$809$	−17.0751			−0.600329			−0.300165	+	0.953887i	$0.597042\pi$
$810$		0			0
$811$	35.7020			1.25367			0.626833	−	0.779153i	$-0.284351\pi$
$811$	35.7020			1.25367			0.626833	+	0.779153i	$0.284351\pi$
$812$		0			0
$813$		−	53.6242i		−	1.88068i
$814$		0			0
$815$	48.6105	−	5.25311i	1.70275	−	0.184008i
$816$		0			0
$817$			35.3755i			1.23763i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−16.4670			−0.574701			−0.287351	−	0.957825i	$-0.592774\pi$
$821$	−16.4670			−0.574701			−0.287351	+	0.957825i	$0.592774\pi$
$822$		0			0
$823$		−	29.0400i		−	1.01227i	−0.862454	−	0.506135i	$-0.831074\pi$
$823$		−	29.0400i		−	1.01227i	0.862454	−	0.506135i	$-0.168926\pi$
$824$		0			0
$825$	−2.28201	−	10.4352i	−0.0794495	−	0.363307i
$826$		0			0
$827$		−	47.8903i		−	1.66531i	−0.553794	−	0.832654i	$-0.686821\pi$
$827$		−	47.8903i		−	1.66531i	0.553794	−	0.832654i	$-0.313179\pi$
$828$		0			0
$829$	24.1186			0.837672			0.418836	−	0.908062i	$-0.362438\pi$
$829$	24.1186			0.837672			0.418836	+	0.908062i	$0.362438\pi$
$830$		0			0
$831$	−42.6674			−1.48012
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−22.6047	+	2.44278i	−0.782267	+	0.0845359i
$836$		0			0
$837$			20.9622i			0.724559i
$838$		0			0
$839$	25.1230			0.867343			0.433672	−	0.901071i	$-0.357218\pi$
$839$	25.1230			0.867343			0.433672	+	0.901071i	$0.357218\pi$
$840$		0			0
$841$	−2.56701			−0.0885175
$842$		0			0
$843$			32.0222i			1.10290i
$844$		0			0
$845$	2.80270	+	25.9352i	0.0964157	+	0.892198i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−22.0321			−0.756141
$850$		0			0
$851$	15.7037			0.538317
$852$		0			0
$853$			2.78647i			0.0954069i	0.998862	+	0.0477035i	$0.0151902\pi$
$853$			2.78647i			0.0954069i	−0.998862	+	0.0477035i	$0.984810\pi$
$854$		0			0
$855$	1.31830	+	12.1991i	0.0450850	+	0.417201i
$856$		0			0
$857$		−	33.7254i		−	1.15204i	−0.817436	−	0.576019i	$-0.804605\pi$
$857$		−	33.7254i		−	1.15204i	0.817436	−	0.576019i	$-0.195395\pi$
$858$		0			0
$859$	19.1171			0.652266			0.326133	−	0.945324i	$-0.394254\pi$
$859$	19.1171			0.652266			0.326133	+	0.945324i	$0.394254\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			33.0437i			1.12482i	0.826859	+	0.562410i	$0.190126\pi$
$863$			33.0437i			1.12482i	−0.826859	+	0.562410i	$0.809874\pi$
$864$		0			0
$865$	−1.77505	+	0.191821i	−0.0603534	+	0.00652211i
$866$		0			0
$867$		−	60.1533i		−	2.04291i
$868$		0			0
$869$	−0.642200			−0.0217852
$870$		0			0
$871$	6.29562			0.213319
$872$		0			0
$873$			8.77590i			0.297019i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	0.185101i		−	0.00625043i	−0.999995	−	0.00312521i	$-0.999005\pi$
$877$		−	0.185101i		−	0.00625043i	0.999995	−	0.00312521i	$-0.000994788\pi$
$878$		0			0
$879$	−20.2183			−0.681947
$880$		0			0
$881$	−43.7063			−1.47250			−0.736252	−	0.676708i	$-0.763406\pi$
$881$	−43.7063			−1.47250			−0.736252	+	0.676708i	$0.763406\pi$
$882$		0			0
$883$			4.70129i			0.158211i	0.996866	+	0.0791055i	$0.0252064\pi$
$883$			4.70129i			0.158211i	−0.996866	+	0.0791055i	$0.974794\pi$
$884$		0			0
$885$	−49.2233	+	5.31934i	−1.65462	+	0.178807i
$886$		0			0
$887$			28.9752i			0.972893i	0.873710	+	0.486447i	$0.161707\pi$
$887$			28.9752i			0.972893i	−0.873710	+	0.486447i	$0.838293\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−11.6660			−0.390827
$892$		0			0
$893$		−	44.1448i		−	1.47725i
$894$		0			0
$895$	2.92508	+	27.0677i	0.0977748	+	0.904774i
$896$		0			0
$897$		−	7.51771i		−	0.251009i
$898$		0			0
$899$	−28.6562			−0.955736
$900$		0			0
$901$	−38.1994			−1.27261
$902$		0			0
$903$		0			0
$904$		0			0
$905$	0.962443	+	8.90612i	0.0319927	+	0.296049i
$906$		0			0
$907$			35.9661i			1.19423i	0.802154	+	0.597117i	$0.203687\pi$
$907$			35.9661i			1.19423i	−0.802154	+	0.597117i	$0.796313\pi$
$908$		0			0
$909$	−0.652164			−0.0216309
$910$		0			0
$911$	29.1414			0.965499			0.482749	−	0.875759i	$-0.339638\pi$
$911$	29.1414			0.965499			0.482749	+	0.875759i	$0.339638\pi$
$912$		0			0
$913$		−	0.288873i		−	0.00956030i
$914$		0			0
$915$	−15.1101	+	1.63288i	−0.499526	+	0.0539814i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−27.6014			−0.910485			−0.455243	−	0.890367i	$-0.650447\pi$
$919$	−27.6014			−0.910485			−0.455243	+	0.890367i	$0.650447\pi$
$920$		0			0
$921$	−11.5093			−0.379244
$922$		0			0
$923$			4.74585i			0.156212i
$924$		0			0
$925$	−5.25298	−	24.0208i	−0.172717	−	0.789799i
$926$		0			0
$927$			17.9963i			0.591076i
$928$		0			0
$929$	41.4287			1.35923			0.679616	−	0.733568i	$-0.262147\pi$
$929$	41.4287			1.35923			0.679616	+	0.733568i	$0.262147\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		−	24.4991i		−	0.802065i
$934$		0			0
$935$	−15.8903	+	1.71719i	−0.519667	+	0.0561580i
$936$		0			0
$937$			17.6450i			0.576438i	0.957564	+	0.288219i	$0.0930632\pi$
$937$			17.6450i			0.576438i	−0.957564	+	0.288219i	$0.906937\pi$
$938$		0			0
$939$	−38.6851			−1.26244
$940$		0			0
$941$	−24.1005			−0.785653			−0.392827	−	0.919613i	$-0.628503\pi$
$941$	−24.1005			−0.785653			−0.392827	+	0.919613i	$0.628503\pi$
$942$		0			0
$943$			30.9415i			1.00760i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	9.64168i		−	0.313312i	−0.987653	−	0.156656i	$-0.949929\pi$
$947$		−	9.64168i		−	0.313312i	0.987653	−	0.156656i	$-0.0500715\pi$
$948$		0			0
$949$	−7.47109			−0.242522
$950$		0			0
$951$	−25.9725			−0.842216
$952$		0			0
$953$			12.3033i			0.398544i	0.979944	+	0.199272i	$0.0638577\pi$
$953$			12.3033i			0.398544i	−0.979944	+	0.199272i	$0.936142\pi$
$954$		0			0
$955$	4.00284	+	37.0409i	0.129529	+	1.19861i
$956$		0			0
$957$		−	10.9837i		−	0.355052i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.0662997			0.00213870
$962$		0			0
$963$			16.1971i			0.521943i
$964$		0			0
$965$	−16.5067	+	1.78380i	−0.531370	+	0.0574227i
$966$		0			0
$967$		−	15.7098i		−	0.505193i	−0.967572	−	0.252597i	$-0.918715\pi$
$967$		−	15.7098i		−	0.505193i	0.967572	−	0.252597i	$-0.0812846\pi$
$968$		0			0
$969$	66.0482			2.12178
$970$		0			0
$971$	4.71582			0.151338			0.0756689	−	0.997133i	$-0.475891\pi$
$971$	4.71582			0.151338			0.0756689	+	0.997133i	$0.475891\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−11.4993	+	2.51471i	−0.368271	+	0.0805352i
$976$		0			0
$977$			29.0115i			0.928159i	0.885794	+	0.464079i	$0.153615\pi$
$977$			29.0115i			0.928159i	−0.885794	+	0.464079i	$0.846385\pi$
$978$		0			0
$979$	−18.0946			−0.578306
$980$		0			0
$981$	−18.2456			−0.582537
$982$		0			0
$983$		−	29.6408i		−	0.945394i	−0.881225	−	0.472697i	$-0.843280\pi$
$983$		−	29.6408i		−	0.945394i	0.881225	−	0.472697i	$-0.156720\pi$
$984$		0			0
$985$	4.19911	−	0.453779i	0.133795	−	0.0144586i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−23.7761			−0.756036
$990$		0			0
$991$	39.0278			1.23976			0.619879	−	0.784697i	$-0.287182\pi$
$991$	39.0278			1.23976			0.619879	+	0.784697i	$0.287182\pi$
$992$		0			0
$993$			48.5913i			1.54200i
$994$		0			0
$995$	−4.14114	−	38.3207i	−0.131283	−	1.21485i
$996$		0			0
$997$		−	12.3710i		−	0.391792i	−0.980625	−	0.195896i	$-0.937238\pi$
$997$		−	12.3710i		−	0.391792i	0.980625	−	0.195896i	$-0.0627615\pi$
$998$		0			0
$999$	−18.4949			−0.585154

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1960.2.g.e.1569.9		12
5.2	odd	4		9800.2.a.cw.1.5		6
5.3	odd	4		9800.2.a.cy.1.2		6
5.4	even	2	inner	1960.2.g.e.1569.4		12
7.3	odd	6		280.2.bg.a.9.4	✓	24
7.5	odd	6		280.2.bg.a.249.9	yes	24
7.6	odd	2		1960.2.g.f.1569.4		12
28.3	even	6		560.2.bw.f.289.9		24
28.19	even	6		560.2.bw.f.529.4		24
35.3	even	12		1400.2.q.o.401.2		12
35.12	even	12		1400.2.q.n.1201.5		12
35.13	even	4		9800.2.a.cv.1.5		6
35.17	even	12		1400.2.q.n.401.5		12
35.19	odd	6		280.2.bg.a.249.4	yes	24
35.24	odd	6		280.2.bg.a.9.9	yes	24
35.27	even	4		9800.2.a.cx.1.2		6
35.33	even	12		1400.2.q.o.1201.2		12
35.34	odd	2		1960.2.g.f.1569.9		12
140.19	even	6		560.2.bw.f.529.9		24
140.59	even	6		560.2.bw.f.289.4		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.bg.a.9.4	✓	24	7.3	odd	6
280.2.bg.a.9.9	yes	24	35.24	odd	6
280.2.bg.a.249.4	yes	24	35.19	odd	6
280.2.bg.a.249.9	yes	24	7.5	odd	6
560.2.bw.f.289.4		24	140.59	even	6
560.2.bw.f.289.9		24	28.3	even	6
560.2.bw.f.529.4		24	28.19	even	6
560.2.bw.f.529.9		24	140.19	even	6
1400.2.q.n.401.5		12	35.17	even	12
1400.2.q.n.1201.5		12	35.12	even	12
1400.2.q.o.401.2		12	35.3	even	12
1400.2.q.o.1201.2		12	35.33	even	12
1960.2.g.e.1569.4		12	5.4	even	2	inner
1960.2.g.e.1569.9		12	1.1	even	1	trivial
1960.2.g.f.1569.4		12	7.6	odd	2
1960.2.g.f.1569.9		12	35.34	odd	2
9800.2.a.cv.1.5		6	35.13	even	4
9800.2.a.cw.1.5		6	5.2	odd	4
9800.2.a.cx.1.2		6	35.27	even	4
9800.2.a.cy.1.2		6	5.3	odd	4

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:	\( N \)	\(=\)	\( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1960.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.03837i q^{3} +(0.240243 + 2.22312i) q^{5} -1.15495 q^{9} +O(q^{10})\)	\(q+2.03837i q^{3} +(0.240243 + 2.22312i) q^{5} -1.15495 q^{9} +1.04807 q^{11} -1.15495i q^{13} +(-4.53155 + 0.489703i) q^{15} +6.81986i q^{17} -4.75120 q^{19} -3.19331i q^{23} +(-4.88457 + 1.06818i) q^{25} +3.76090i q^{27} -5.14130 q^{29} +5.57372 q^{31} +2.13636i q^{33} +4.91769i q^{37} +2.35420 q^{39} -9.68948 q^{41} -7.44559i q^{43} +(-0.277467 - 2.56759i) q^{45} +9.29130i q^{47} -13.9014 q^{51} +5.60119i q^{53} +(0.251792 + 2.33000i) q^{55} -9.68469i q^{57} +10.8624 q^{59} +3.33443 q^{61} +(2.56759 - 0.277467i) q^{65} +5.45101i q^{67} +6.50915 q^{69} -4.10916 q^{71} -6.46879i q^{73} +(-2.17734 - 9.95655i) q^{75} -0.612744 q^{79} -11.1309 q^{81} -0.275623i q^{83} +(-15.1614 + 1.63842i) q^{85} -10.4799i q^{87} -17.2646 q^{89} +11.3613i q^{93} +(-1.14144 - 10.5625i) q^{95} -7.59854i q^{97} -1.21047 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 14 q^{9}+O(q^{10}) \) 12 * q - 14 * q^9	\( 12 q - 14 q^{9} + 2 q^{11} + 6 q^{15} - 10 q^{19} + 2 q^{25} + 6 q^{29} + 4 q^{31} - 20 q^{39} - 12 q^{41} - 8 q^{45} + 6 q^{55} - 48 q^{59} - 18 q^{61} + 26 q^{65} + 30 q^{69} + 8 q^{71} - 14 q^{75} + 44 q^{79} - 12 q^{81} - 22 q^{85} + 30 q^{89} + 26 q^{95} - 22 q^{99}+O(q^{100}) \) 12 * q - 14 * q^9 + 2 * q^11 + 6 * q^15 - 10 * q^19 + 2 * q^25 + 6 * q^29 + 4 * q^31 - 20 * q^39 - 12 * q^41 - 8 * q^45 + 6 * q^55 - 48 * q^59 - 18 * q^61 + 26 * q^65 + 30 * q^69 + 8 * q^71 - 14 * q^75 + 44 * q^79 - 12 * q^81 - 22 * q^85 + 30 * q^89 + 26 * q^95 - 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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