LMFDB - Embedded newform 960.2.k.e.481.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,2,Mod(481,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("960.481");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	960.k (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:	$7.66563859404$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			481.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	960.481
Dual form			960.2.k.e.481.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +1.00000i q^{5} -2.00000 q^{7} -1.00000 q^{9} -1.46410i q^{11} +1.46410i q^{13} +1.00000 q^{15} -3.46410 q^{17} -6.92820i q^{19} +2.00000i q^{21} -4.00000 q^{23} -1.00000 q^{25} +1.00000i q^{27} -4.92820i q^{29} -7.46410 q^{31} -1.46410 q^{33} -2.00000i q^{35} +2.53590i q^{37} +1.46410 q^{39} -8.92820 q^{41} -6.92820i q^{43} -1.00000i q^{45} +4.00000 q^{47} -3.00000 q^{49} +3.46410i q^{51} +12.9282i q^{53} +1.46410 q^{55} -6.92820 q^{57} -2.53590i q^{59} -4.00000i q^{61} +2.00000 q^{63} -1.46410 q^{65} -6.92820i q^{67} +4.00000i q^{69} -6.92820 q^{71} +10.0000 q^{73} +1.00000i q^{75} +2.92820i q^{77} +6.39230 q^{79} +1.00000 q^{81} +8.00000i q^{83} -3.46410i q^{85} -4.92820 q^{87} -12.9282 q^{89} -2.92820i q^{91} +7.46410i q^{93} +6.92820 q^{95} +0.928203 q^{97} +1.46410i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{7} - 4 q^{9} + 4 q^{15} - 16 q^{23} - 4 q^{25} - 16 q^{31} + 8 q^{33} - 8 q^{39} - 8 q^{41} + 16 q^{47} - 12 q^{49} - 8 q^{55} + 8 q^{63} + 8 q^{65} + 40 q^{73} - 16 q^{79} + 4 q^{81} + 8 q^{87}+ \cdots - 24 q^{97}+O(q^{100}) $ 4 * q - 8 * q^7 - 4 * q^9 + 4 * q^15 - 16 * q^23 - 4 * q^25 - 16 * q^31 + 8 * q^33 - 8 * q^39 - 8 * q^41 + 16 * q^47 - 12 * q^49 - 8 * q^55 + 8 * q^63 + 8 * q^65 + 40 * q^73 - 16 * q^79 + 4 * q^81 + 8 * q^87 - 24 * q^89 - 24 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 1.46410i	− 0.441443i	−0.975337	−	0.220722i	$-0.929159\pi$
$11$	− 1.46410i	− 0.441443i	0.975337	−	0.220722i	$-0.0708412\pi$
$12$	0	0
$13$	1.46410i	0.406069i	0.979172	+	0.203034i	$0.0650803\pi$
$13$	1.46410i	0.406069i	−0.979172	+	0.203034i	$0.934920\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	− 6.92820i	− 1.58944i	−0.606977	−	0.794719i	$-0.707618\pi$
$19$	− 6.92820i	− 1.58944i	0.606977	−	0.794719i	$-0.292382\pi$
$20$	0	0
$21$	2.00000i	0.436436i
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	− 4.92820i	− 0.915144i	−0.889172	−	0.457572i	$-0.848719\pi$
$29$	− 4.92820i	− 0.915144i	0.889172	−	0.457572i	$-0.151281\pi$
$30$	0	0
$31$	−7.46410	−1.34059	−0.670296	−	0.742094i	$-0.733833\pi$
$31$	−7.46410	−1.34059	−0.670296	+	0.742094i	$0.733833\pi$
$32$	0	0
$33$	−1.46410	−0.254867
$34$	0	0
$35$	− 2.00000i	− 0.338062i
$36$	0	0
$37$	2.53590i	0.416899i	0.978033	+	0.208450i	$0.0668417\pi$
$37$	2.53590i	0.416899i	−0.978033	+	0.208450i	$0.933158\pi$
$38$	0	0
$39$	1.46410	0.234444
$40$	0	0
$41$	−8.92820	−1.39435	−0.697176	−	0.716900i	$-0.745560\pi$
$41$	−8.92820	−1.39435	−0.697176	+	0.716900i	$0.745560\pi$
$42$	0	0
$43$	− 6.92820i	− 1.05654i	−0.849076	−	0.528271i	$-0.822841\pi$
$43$	− 6.92820i	− 1.05654i	0.849076	−	0.528271i	$-0.177159\pi$
$44$	0	0
$45$	− 1.00000i	− 0.149071i
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	3.46410i	0.485071i
$52$	0	0
$53$	12.9282i	1.77583i	0.460012	+	0.887913i	$0.347845\pi$
$53$	12.9282i	1.77583i	−0.460012	+	0.887913i	$0.652155\pi$
$54$	0	0
$55$	1.46410	0.197419
$56$	0	0
$57$	−6.92820	−0.917663
$58$	0	0
$59$	− 2.53590i	− 0.330146i	−0.986281	−	0.165073i	$-0.947214\pi$
$59$	− 2.53590i	− 0.330146i	0.986281	−	0.165073i	$-0.0527859\pi$
$60$	0	0
$61$	− 4.00000i	− 0.512148i	−0.966657	−	0.256074i	$-0.917571\pi$
$61$	− 4.00000i	− 0.512148i	0.966657	−	0.256074i	$-0.0824290\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	−1.46410	−0.181599
$66$	0	0
$67$	− 6.92820i	− 0.846415i	−0.906033	−	0.423207i	$-0.860904\pi$
$67$	− 6.92820i	− 0.846415i	0.906033	−	0.423207i	$-0.139096\pi$
$68$	0	0
$69$	4.00000i	0.481543i
$70$	0	0
$71$	−6.92820	−0.822226	−0.411113	−	0.911584i	$-0.634860\pi$
$71$	−6.92820	−0.822226	−0.411113	+	0.911584i	$0.634860\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	1.00000i	0.115470i
$76$	0	0
$77$	2.92820i	0.333700i
$78$	0	0
$79$	6.39230	0.719190	0.359595	−	0.933108i	$-0.382915\pi$
$79$	6.39230	0.719190	0.359595	+	0.933108i	$0.382915\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.00000i	0.878114i	0.898459	+	0.439057i	$0.144687\pi$
$83$	8.00000i	0.878114i	−0.898459	+	0.439057i	$0.855313\pi$
$84$	0	0
$85$	− 3.46410i	− 0.375735i
$86$	0	0
$87$	−4.92820	−0.528359
$88$	0	0
$89$	−12.9282	−1.37039	−0.685193	−	0.728361i	$-0.740282\pi$
$89$	−12.9282	−1.37039	−0.685193	+	0.728361i	$0.740282\pi$
$90$	0	0
$91$	− 2.92820i	− 0.306959i
$92$	0	0
$93$	7.46410i	0.773991i
$94$	0	0
$95$	6.92820	0.710819
$96$	0	0
$97$	0.928203	0.0942448	0.0471224	−	0.998889i	$-0.484995\pi$
$97$	0.928203	0.0942448	0.0471224	+	0.998889i	$0.484995\pi$
$98$	0	0
$99$	1.46410i	0.147148i
$100$	0	0
$101$	10.0000i	0.995037i	0.867453	+	0.497519i	$0.165755\pi$
$101$	10.0000i	0.995037i	−0.867453	+	0.497519i	$0.834245\pi$
$102$	0	0
$103$	−8.92820	−0.879722	−0.439861	−	0.898066i	$-0.644972\pi$
$103$	−8.92820	−0.879722	−0.439861	+	0.898066i	$0.644972\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	0	0
$107$	6.92820i	0.669775i	0.942258	+	0.334887i	$0.108698\pi$
$107$	6.92820i	0.669775i	−0.942258	+	0.334887i	$0.891302\pi$
$108$	0	0
$109$	− 18.9282i	− 1.81299i	−0.422213	−	0.906497i	$-0.638747\pi$
$109$	− 18.9282i	− 1.81299i	0.422213	−	0.906497i	$-0.361253\pi$
$110$	0	0
$111$	2.53590	0.240697
$112$	0	0
$113$	0.535898	0.0504131	0.0252065	−	0.999682i	$-0.491976\pi$
$113$	0.535898	0.0504131	0.0252065	+	0.999682i	$0.491976\pi$
$114$	0	0
$115$	− 4.00000i	− 0.373002i
$116$	0	0
$117$	− 1.46410i	− 0.135356i
$118$	0	0
$119$	6.92820	0.635107
$120$	0	0
$121$	8.85641	0.805128
$122$	0	0
$123$	8.92820i	0.805029i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	−20.9282	−1.85708	−0.928539	−	0.371235i	$-0.878934\pi$
$127$	−20.9282	−1.85708	−0.928539	+	0.371235i	$0.878934\pi$
$128$	0	0
$129$	−6.92820	−0.609994
$130$	0	0
$131$	5.46410i	0.477401i	0.971093	+	0.238700i	$0.0767214\pi$
$131$	5.46410i	0.477401i	−0.971093	+	0.238700i	$0.923279\pi$
$132$	0	0
$133$	13.8564i	1.20150i
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	22.3923	1.91310	0.956552	−	0.291562i	$-0.0941750\pi$
$137$	22.3923	1.91310	0.956552	+	0.291562i	$0.0941750\pi$
$138$	0	0
$139$	− 12.0000i	− 1.01783i	−0.860818	−	0.508913i	$-0.830047\pi$
$139$	− 12.0000i	− 1.01783i	0.860818	−	0.508913i	$-0.169953\pi$
$140$	0	0
$141$	− 4.00000i	− 0.336861i
$142$	0	0
$143$	2.14359	0.179256
$144$	0	0
$145$	4.92820	0.409265
$146$	0	0
$147$	3.00000i	0.247436i
$148$	0	0
$149$	− 18.7846i	− 1.53890i	−0.638710	−	0.769448i	$-0.720532\pi$
$149$	− 18.7846i	− 1.53890i	0.638710	−	0.769448i	$-0.279468\pi$
$150$	0	0
$151$	4.53590	0.369126	0.184563	−	0.982821i	$-0.440913\pi$
$151$	4.53590	0.369126	0.184563	+	0.982821i	$0.440913\pi$
$152$	0	0
$153$	3.46410	0.280056
$154$	0	0
$155$	− 7.46410i	− 0.599531i
$156$	0	0
$157$	− 21.4641i	− 1.71302i	−0.516129	−	0.856511i	$-0.672627\pi$
$157$	− 21.4641i	− 1.71302i	0.516129	−	0.856511i	$-0.327373\pi$
$158$	0	0
$159$	12.9282	1.02527
$160$	0	0
$161$	8.00000	0.630488
$162$	0	0
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	0	0
$165$	− 1.46410i	− 0.113980i
$166$	0	0
$167$	13.8564	1.07224	0.536120	−	0.844141i	$-0.319889\pi$
$167$	13.8564	1.07224	0.536120	+	0.844141i	$0.319889\pi$
$168$	0	0
$169$	10.8564	0.835108
$170$	0	0
$171$	6.92820i	0.529813i
$172$	0	0
$173$	− 11.0718i	− 0.841773i	−0.907113	−	0.420887i	$-0.861719\pi$
$173$	− 11.0718i	− 0.841773i	0.907113	−	0.420887i	$-0.138281\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	0	0
$177$	−2.53590	−0.190610
$178$	0	0
$179$	16.3923i	1.22522i	0.790386	+	0.612609i	$0.209880\pi$
$179$	16.3923i	1.22522i	−0.790386	+	0.612609i	$0.790120\pi$
$180$	0	0
$181$	2.92820i	0.217652i	0.994061	+	0.108826i	$0.0347091\pi$
$181$	2.92820i	0.217652i	−0.994061	+	0.108826i	$0.965291\pi$
$182$	0	0
$183$	−4.00000	−0.295689
$184$	0	0
$185$	−2.53590	−0.186443
$186$	0	0
$187$	5.07180i	0.370887i
$188$	0	0
$189$	− 2.00000i	− 0.145479i
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	18.7846	1.35215	0.676073	−	0.736835i	$-0.263680\pi$
$193$	18.7846	1.35215	0.676073	+	0.736835i	$0.263680\pi$
$194$	0	0
$195$	1.46410i	0.104846i
$196$	0	0
$197$	23.8564i	1.69970i	0.527026	+	0.849849i	$0.323307\pi$
$197$	23.8564i	1.69970i	−0.527026	+	0.849849i	$0.676693\pi$
$198$	0	0
$199$	25.3205	1.79492	0.897462	−	0.441093i	$-0.145409\pi$
$199$	25.3205	1.79492	0.897462	+	0.441093i	$0.145409\pi$
$200$	0	0
$201$	−6.92820	−0.488678
$202$	0	0
$203$	9.85641i	0.691784i
$204$	0	0
$205$	− 8.92820i	− 0.623573i
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	−10.1436	−0.701647
$210$	0	0
$211$	4.00000i	0.275371i	0.990476	+	0.137686i	$0.0439664\pi$
$211$	4.00000i	0.275371i	−0.990476	+	0.137686i	$0.956034\pi$
$212$	0	0
$213$	6.92820i	0.474713i
$214$	0	0
$215$	6.92820	0.472500
$216$	0	0
$217$	14.9282	1.01339
$218$	0	0
$219$	− 10.0000i	− 0.675737i
$220$	0	0
$221$	− 5.07180i	− 0.341166i
$222$	0	0
$223$	−20.9282	−1.40146	−0.700728	−	0.713428i	$-0.747141\pi$
$223$	−20.9282	−1.40146	−0.700728	+	0.713428i	$0.747141\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	− 13.0718i	− 0.867606i	−0.901008	−	0.433803i	$-0.857171\pi$
$227$	− 13.0718i	− 0.867606i	0.901008	−	0.433803i	$-0.142829\pi$
$228$	0	0
$229$	2.14359i	0.141653i	0.997489	+	0.0708263i	$0.0225636\pi$
$229$	2.14359i	0.141653i	−0.997489	+	0.0708263i	$0.977436\pi$
$230$	0	0
$231$	2.92820	0.192662
$232$	0	0
$233$	4.53590	0.297157	0.148578	−	0.988901i	$-0.452530\pi$
$233$	4.53590	0.297157	0.148578	+	0.988901i	$0.452530\pi$
$234$	0	0
$235$	4.00000i	0.260931i
$236$	0	0
$237$	− 6.39230i	− 0.415225i
$238$	0	0
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	− 3.00000i	− 0.191663i
$246$	0	0
$247$	10.1436	0.645421
$248$	0	0
$249$	8.00000	0.506979
$250$	0	0
$251$	23.3205i	1.47198i	0.676994	+	0.735989i	$0.263282\pi$
$251$	23.3205i	1.47198i	−0.676994	+	0.735989i	$0.736718\pi$
$252$	0	0
$253$	5.85641i	0.368189i
$254$	0	0
$255$	−3.46410	−0.216930
$256$	0	0
$257$	−7.46410	−0.465598	−0.232799	−	0.972525i	$-0.574788\pi$
$257$	−7.46410	−0.465598	−0.232799	+	0.972525i	$0.574788\pi$
$258$	0	0
$259$	− 5.07180i	− 0.315146i
$260$	0	0
$261$	4.92820i	0.305048i
$262$	0	0
$263$	−27.7128	−1.70885	−0.854423	−	0.519579i	$-0.826089\pi$
$263$	−27.7128	−1.70885	−0.854423	+	0.519579i	$0.826089\pi$
$264$	0	0
$265$	−12.9282	−0.794173
$266$	0	0
$267$	12.9282i	0.791193i
$268$	0	0
$269$	− 26.7846i	− 1.63309i	−0.577284	−	0.816543i	$-0.695888\pi$
$269$	− 26.7846i	− 1.63309i	0.577284	−	0.816543i	$-0.304112\pi$
$270$	0	0
$271$	−11.4641	−0.696395	−0.348197	−	0.937421i	$-0.613206\pi$
$271$	−11.4641	−0.696395	−0.348197	+	0.937421i	$0.613206\pi$
$272$	0	0
$273$	−2.92820	−0.177223
$274$	0	0
$275$	1.46410i	0.0882886i
$276$	0	0
$277$	− 6.53590i	− 0.392704i	−0.980533	−	0.196352i	$-0.937090\pi$
$277$	− 6.53590i	− 0.392704i	0.980533	−	0.196352i	$-0.0629095\pi$
$278$	0	0
$279$	7.46410	0.446864
$280$	0	0
$281$	0.928203	0.0553720	0.0276860	−	0.999617i	$-0.491186\pi$
$281$	0.928203	0.0553720	0.0276860	+	0.999617i	$0.491186\pi$
$282$	0	0
$283$	22.9282i	1.36294i	0.731846	+	0.681470i	$0.238659\pi$
$283$	22.9282i	1.36294i	−0.731846	+	0.681470i	$0.761341\pi$
$284$	0	0
$285$	− 6.92820i	− 0.410391i
$286$	0	0
$287$	17.8564	1.05403
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	− 0.928203i	− 0.0544122i
$292$	0	0
$293$	4.14359i	0.242071i	0.992648	+	0.121036i	$0.0386215\pi$
$293$	4.14359i	0.242071i	−0.992648	+	0.121036i	$0.961378\pi$
$294$	0	0
$295$	2.53590	0.147646
$296$	0	0
$297$	1.46410	0.0849558
$298$	0	0
$299$	− 5.85641i	− 0.338685i
$300$	0	0
$301$	13.8564i	0.798670i
$302$	0	0
$303$	10.0000	0.574485
$304$	0	0
$305$	4.00000	0.229039
$306$	0	0
$307$	9.85641i	0.562535i	0.959629	+	0.281267i	$0.0907548\pi$
$307$	9.85641i	0.562535i	−0.959629	+	0.281267i	$0.909245\pi$
$308$	0	0
$309$	8.92820i	0.507908i
$310$	0	0
$311$	2.14359	0.121552	0.0607760	−	0.998151i	$-0.480642\pi$
$311$	2.14359	0.121552	0.0607760	+	0.998151i	$0.480642\pi$
$312$	0	0
$313$	−4.92820	−0.278559	−0.139279	−	0.990253i	$-0.544479\pi$
$313$	−4.92820	−0.278559	−0.139279	+	0.990253i	$0.544479\pi$
$314$	0	0
$315$	2.00000i	0.112687i
$316$	0	0
$317$	11.8564i	0.665922i	0.942941	+	0.332961i	$0.108048\pi$
$317$	11.8564i	0.665922i	−0.942941	+	0.332961i	$0.891952\pi$
$318$	0	0
$319$	−7.21539	−0.403984
$320$	0	0
$321$	6.92820	0.386695
$322$	0	0
$323$	24.0000i	1.33540i
$324$	0	0
$325$	− 1.46410i	− 0.0812137i
$326$	0	0
$327$	−18.9282	−1.04673
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	− 9.07180i	− 0.498631i	−0.968422	−	0.249316i	$-0.919794\pi$
$331$	− 9.07180i	− 0.498631i	0.968422	−	0.249316i	$-0.0802056\pi$
$332$	0	0
$333$	− 2.53590i	− 0.138966i
$334$	0	0
$335$	6.92820	0.378528
$336$	0	0
$337$	−23.8564	−1.29954	−0.649771	−	0.760130i	$-0.725135\pi$
$337$	−23.8564	−1.29954	−0.649771	+	0.760130i	$0.725135\pi$
$338$	0	0
$339$	− 0.535898i	− 0.0291060i
$340$	0	0
$341$	10.9282i	0.591795i
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	−4.00000	−0.215353
$346$	0	0
$347$	− 2.92820i	− 0.157194i	−0.996906	−	0.0785971i	$-0.974956\pi$
$347$	− 2.92820i	− 0.157194i	0.996906	−	0.0785971i	$-0.0250441\pi$
$348$	0	0
$349$	− 4.00000i	− 0.214115i	−0.994253	−	0.107058i	$-0.965857\pi$
$349$	− 4.00000i	− 0.214115i	0.994253	−	0.107058i	$-0.0341429\pi$
$350$	0	0
$351$	−1.46410	−0.0781480
$352$	0	0
$353$	3.46410	0.184376	0.0921878	−	0.995742i	$-0.470614\pi$
$353$	3.46410	0.184376	0.0921878	+	0.995742i	$0.470614\pi$
$354$	0	0
$355$	− 6.92820i	− 0.367711i
$356$	0	0
$357$	− 6.92820i	− 0.366679i
$358$	0	0
$359$	−22.9282	−1.21010	−0.605052	−	0.796186i	$-0.706848\pi$
$359$	−22.9282	−1.21010	−0.605052	+	0.796186i	$0.706848\pi$
$360$	0	0
$361$	−29.0000	−1.52632
$362$	0	0
$363$	− 8.85641i	− 0.464841i
$364$	0	0
$365$	10.0000i	0.523424i
$366$	0	0
$367$	2.00000	0.104399	0.0521996	−	0.998637i	$-0.483377\pi$
$367$	2.00000	0.104399	0.0521996	+	0.998637i	$0.483377\pi$
$368$	0	0
$369$	8.92820	0.464784
$370$	0	0
$371$	− 25.8564i	− 1.34240i
$372$	0	0
$373$	− 8.39230i	− 0.434537i	−0.976112	−	0.217269i	$-0.930285\pi$
$373$	− 8.39230i	− 0.434537i	0.976112	−	0.217269i	$-0.0697147\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	7.21539	0.371612
$378$	0	0
$379$	− 20.0000i	− 1.02733i	−0.857991	−	0.513665i	$-0.828287\pi$
$379$	− 20.0000i	− 1.02733i	0.857991	−	0.513665i	$-0.171713\pi$
$380$	0	0
$381$	20.9282i	1.07218i
$382$	0	0
$383$	−37.8564	−1.93437	−0.967186	−	0.254069i	$-0.918231\pi$
$383$	−37.8564	−1.93437	−0.967186	+	0.254069i	$0.918231\pi$
$384$	0	0
$385$	−2.92820	−0.149235
$386$	0	0
$387$	6.92820i	0.352180i
$388$	0	0
$389$	− 31.8564i	− 1.61518i	−0.589742	−	0.807592i	$-0.700770\pi$
$389$	− 31.8564i	− 1.61518i	0.589742	−	0.807592i	$-0.299230\pi$
$390$	0	0
$391$	13.8564	0.700749
$392$	0	0
$393$	5.46410	0.275627
$394$	0	0
$395$	6.39230i	0.321632i
$396$	0	0
$397$	15.6077i	0.783328i	0.920108	+	0.391664i	$0.128100\pi$
$397$	15.6077i	0.783328i	−0.920108	+	0.391664i	$0.871900\pi$
$398$	0	0
$399$	13.8564	0.693688
$400$	0	0
$401$	22.0000	1.09863	0.549314	−	0.835616i	$-0.314889\pi$
$401$	22.0000	1.09863	0.549314	+	0.835616i	$0.314889\pi$
$402$	0	0
$403$	− 10.9282i	− 0.544373i
$404$	0	0
$405$	1.00000i	0.0496904i
$406$	0	0
$407$	3.71281	0.184037
$408$	0	0
$409$	−3.85641	−0.190687	−0.0953435	−	0.995444i	$-0.530395\pi$
$409$	−3.85641	−0.190687	−0.0953435	+	0.995444i	$0.530395\pi$
$410$	0	0
$411$	− 22.3923i	− 1.10453i
$412$	0	0
$413$	5.07180i	0.249567i
$414$	0	0
$415$	−8.00000	−0.392705
$416$	0	0
$417$	−12.0000	−0.587643
$418$	0	0
$419$	− 15.3205i	− 0.748456i	−0.927337	−	0.374228i	$-0.877908\pi$
$419$	− 15.3205i	− 0.748456i	0.927337	−	0.374228i	$-0.122092\pi$
$420$	0	0
$421$	16.0000i	0.779792i	0.920859	+	0.389896i	$0.127489\pi$
$421$	16.0000i	0.779792i	−0.920859	+	0.389896i	$0.872511\pi$
$422$	0	0
$423$	−4.00000	−0.194487
$424$	0	0
$425$	3.46410	0.168034
$426$	0	0
$427$	8.00000i	0.387147i
$428$	0	0
$429$	− 2.14359i	− 0.103494i
$430$	0	0
$431$	30.9282	1.48976	0.744880	−	0.667199i	$-0.232507\pi$
$431$	30.9282	1.48976	0.744880	+	0.667199i	$0.232507\pi$
$432$	0	0
$433$	−7.85641	−0.377555	−0.188777	−	0.982020i	$-0.560452\pi$
$433$	−7.85641	−0.377555	−0.188777	+	0.982020i	$0.560452\pi$
$434$	0	0
$435$	− 4.92820i	− 0.236289i
$436$	0	0
$437$	27.7128i	1.32568i
$438$	0	0
$439$	33.3205	1.59030	0.795151	−	0.606412i	$-0.207392\pi$
$439$	33.3205	1.59030	0.795151	+	0.606412i	$0.207392\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	0	0
$443$	− 30.6410i	− 1.45580i	−0.685684	−	0.727899i	$-0.740497\pi$
$443$	− 30.6410i	− 1.45580i	0.685684	−	0.727899i	$-0.259503\pi$
$444$	0	0
$445$	− 12.9282i	− 0.612856i
$446$	0	0
$447$	−18.7846	−0.888482
$448$	0	0
$449$	−37.7128	−1.77978	−0.889889	−	0.456177i	$-0.849218\pi$
$449$	−37.7128	−1.77978	−0.889889	+	0.456177i	$0.849218\pi$
$450$	0	0
$451$	13.0718i	0.615527i
$452$	0	0
$453$	− 4.53590i	− 0.213115i
$454$	0	0
$455$	2.92820	0.137276
$456$	0	0
$457$	−8.92820	−0.417644	−0.208822	−	0.977954i	$-0.566963\pi$
$457$	−8.92820	−0.417644	−0.208822	+	0.977954i	$0.566963\pi$
$458$	0	0
$459$	− 3.46410i	− 0.161690i
$460$	0	0
$461$	− 16.9282i	− 0.788425i	−0.919019	−	0.394213i	$-0.871017\pi$
$461$	− 16.9282i	− 0.788425i	0.919019	−	0.394213i	$-0.128983\pi$
$462$	0	0
$463$	−15.8564	−0.736910	−0.368455	−	0.929646i	$-0.620113\pi$
$463$	−15.8564	−0.736910	−0.368455	+	0.929646i	$0.620113\pi$
$464$	0	0
$465$	−7.46410	−0.346139
$466$	0	0
$467$	− 37.8564i	− 1.75179i	−0.482506	−	0.875893i	$-0.660273\pi$
$467$	− 37.8564i	− 1.75179i	0.482506	−	0.875893i	$-0.339727\pi$
$468$	0	0
$469$	13.8564i	0.639829i
$470$	0	0
$471$	−21.4641	−0.989014
$472$	0	0
$473$	−10.1436	−0.466403
$474$	0	0
$475$	6.92820i	0.317888i
$476$	0	0
$477$	− 12.9282i	− 0.591942i
$478$	0	0
$479$	−36.7846	−1.68073	−0.840366	−	0.542020i	$-0.817660\pi$
$479$	−36.7846	−1.68073	−0.840366	+	0.542020i	$0.817660\pi$
$480$	0	0
$481$	−3.71281	−0.169290
$482$	0	0
$483$	− 8.00000i	− 0.364013i
$484$	0	0
$485$	0.928203i	0.0421475i
$486$	0	0
$487$	−18.7846	−0.851212	−0.425606	−	0.904909i	$-0.639939\pi$
$487$	−18.7846	−0.851212	−0.425606	+	0.904909i	$0.639939\pi$
$488$	0	0
$489$	4.00000	0.180886
$490$	0	0
$491$	− 3.32051i	− 0.149852i	−0.997189	−	0.0749262i	$-0.976128\pi$
$491$	− 3.32051i	− 0.149852i	0.997189	−	0.0749262i	$-0.0238721\pi$
$492$	0	0
$493$	17.0718i	0.768875i
$494$	0	0
$495$	−1.46410	−0.0658065
$496$	0	0
$497$	13.8564	0.621545
$498$	0	0
$499$	9.07180i	0.406109i	0.979167	+	0.203055i	$0.0650869\pi$
$499$	9.07180i	0.406109i	−0.979167	+	0.203055i	$0.934913\pi$
$500$	0	0
$501$	− 13.8564i	− 0.619059i
$502$	0	0
$503$	−4.00000	−0.178351	−0.0891756	−	0.996016i	$-0.528423\pi$
$503$	−4.00000	−0.178351	−0.0891756	+	0.996016i	$0.528423\pi$
$504$	0	0
$505$	−10.0000	−0.444994
$506$	0	0
$507$	− 10.8564i	− 0.482150i
$508$	0	0
$509$	16.1436i	0.715552i	0.933807	+	0.357776i	$0.116465\pi$
$509$	16.1436i	0.715552i	−0.933807	+	0.357776i	$0.883535\pi$
$510$	0	0
$511$	−20.0000	−0.884748
$512$	0	0
$513$	6.92820	0.305888
$514$	0	0
$515$	− 8.92820i	− 0.393424i
$516$	0	0
$517$	− 5.85641i	− 0.257564i
$518$	0	0
$519$	−11.0718	−0.485998
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	− 20.0000i	− 0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$	− 20.0000i	− 0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$	0	0
$525$	− 2.00000i	− 0.0872872i
$526$	0	0
$527$	25.8564	1.12632
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	2.53590i	0.110049i
$532$	0	0
$533$	− 13.0718i	− 0.566202i
$534$	0	0
$535$	−6.92820	−0.299532
$536$	0	0
$537$	16.3923	0.707380
$538$	0	0
$539$	4.39230i	0.189190i
$540$	0	0
$541$	− 2.92820i	− 0.125893i	−0.998017	−	0.0629466i	$-0.979950\pi$
$541$	− 2.92820i	− 0.125893i	0.998017	−	0.0629466i	$-0.0200498\pi$
$542$	0	0
$543$	2.92820	0.125661
$544$	0	0
$545$	18.9282	0.810795
$546$	0	0
$547$	9.07180i	0.387882i	0.981013	+	0.193941i	$0.0621270\pi$
$547$	9.07180i	0.387882i	−0.981013	+	0.193941i	$0.937873\pi$
$548$	0	0
$549$	4.00000i	0.170716i
$550$	0	0
$551$	−34.1436	−1.45457
$552$	0	0
$553$	−12.7846	−0.543657
$554$	0	0
$555$	2.53590i	0.107643i
$556$	0	0
$557$	26.7846i	1.13490i	0.823408	+	0.567450i	$0.192070\pi$
$557$	26.7846i	1.13490i	−0.823408	+	0.567450i	$0.807930\pi$
$558$	0	0
$559$	10.1436	0.429028
$560$	0	0
$561$	5.07180	0.214131
$562$	0	0
$563$	18.9282i	0.797729i	0.917010	+	0.398864i	$0.130596\pi$
$563$	18.9282i	0.797729i	−0.917010	+	0.398864i	$0.869404\pi$
$564$	0	0
$565$	0.535898i	0.0225454i
$566$	0	0
$567$	−2.00000	−0.0839921
$568$	0	0
$569$	31.8564	1.33549	0.667745	−	0.744390i	$-0.267260\pi$
$569$	31.8564	1.33549	0.667745	+	0.744390i	$0.267260\pi$
$570$	0	0
$571$	− 1.07180i	− 0.0448533i	−0.999748	−	0.0224266i	$-0.992861\pi$
$571$	− 1.07180i	− 0.0448533i	0.999748	−	0.0224266i	$-0.00713922\pi$
$572$	0	0
$573$	8.00000i	0.334205i
$574$	0	0
$575$	4.00000	0.166812
$576$	0	0
$577$	−28.9282	−1.20430	−0.602148	−	0.798384i	$-0.705688\pi$
$577$	−28.9282	−1.20430	−0.602148	+	0.798384i	$0.705688\pi$
$578$	0	0
$579$	− 18.7846i	− 0.780662i
$580$	0	0
$581$	− 16.0000i	− 0.663792i
$582$	0	0
$583$	18.9282	0.783926
$584$	0	0
$585$	1.46410	0.0605332
$586$	0	0
$587$	− 34.6410i	− 1.42979i	−0.699233	−	0.714894i	$-0.746475\pi$
$587$	− 34.6410i	− 1.42979i	0.699233	−	0.714894i	$-0.253525\pi$
$588$	0	0
$589$	51.7128i	2.13079i
$590$	0	0
$591$	23.8564	0.981321
$592$	0	0
$593$	−37.3205	−1.53257	−0.766285	−	0.642501i	$-0.777897\pi$
$593$	−37.3205	−1.53257	−0.766285	+	0.642501i	$0.777897\pi$
$594$	0	0
$595$	6.92820i	0.284029i
$596$	0	0
$597$	− 25.3205i	− 1.03630i
$598$	0	0
$599$	−34.6410	−1.41539	−0.707697	−	0.706516i	$-0.750266\pi$
$599$	−34.6410	−1.41539	−0.707697	+	0.706516i	$0.750266\pi$
$600$	0	0
$601$	−47.5692	−1.94039	−0.970194	−	0.242328i	$-0.922089\pi$
$601$	−47.5692	−1.94039	−0.970194	+	0.242328i	$0.922089\pi$
$602$	0	0
$603$	6.92820i	0.282138i
$604$	0	0
$605$	8.85641i	0.360064i
$606$	0	0
$607$	−18.0000	−0.730597	−0.365299	−	0.930890i	$-0.619033\pi$
$607$	−18.0000	−0.730597	−0.365299	+	0.930890i	$0.619033\pi$
$608$	0	0
$609$	9.85641	0.399402
$610$	0	0
$611$	5.85641i	0.236925i
$612$	0	0
$613$	22.2487i	0.898617i	0.893377	+	0.449308i	$0.148330\pi$
$613$	22.2487i	0.898617i	−0.893377	+	0.449308i	$0.851670\pi$
$614$	0	0
$615$	−8.92820	−0.360020
$616$	0	0
$617$	14.6795	0.590974	0.295487	−	0.955347i	$-0.404518\pi$
$617$	14.6795	0.590974	0.295487	+	0.955347i	$0.404518\pi$
$618$	0	0
$619$	33.8564i	1.36080i	0.732839	+	0.680402i	$0.238195\pi$
$619$	33.8564i	1.36080i	−0.732839	+	0.680402i	$0.761805\pi$
$620$	0	0
$621$	− 4.00000i	− 0.160514i
$622$	0	0
$623$	25.8564	1.03592
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	10.1436i	0.405096i
$628$	0	0
$629$	− 8.78461i	− 0.350265i
$630$	0	0
$631$	9.60770	0.382476	0.191238	−	0.981544i	$-0.438750\pi$
$631$	9.60770	0.382476	0.191238	+	0.981544i	$0.438750\pi$
$632$	0	0
$633$	4.00000	0.158986
$634$	0	0
$635$	− 20.9282i	− 0.830510i
$636$	0	0
$637$	− 4.39230i	− 0.174029i
$638$	0	0
$639$	6.92820	0.274075
$640$	0	0
$641$	44.6410	1.76321	0.881607	−	0.471984i	$-0.156462\pi$
$641$	44.6410	1.76321	0.881607	+	0.471984i	$0.156462\pi$
$642$	0	0
$643$	15.7128i	0.619653i	0.950793	+	0.309826i	$0.100271\pi$
$643$	15.7128i	0.619653i	−0.950793	+	0.309826i	$0.899729\pi$
$644$	0	0
$645$	− 6.92820i	− 0.272798i
$646$	0	0
$647$	9.85641	0.387495	0.193748	−	0.981051i	$-0.437936\pi$
$647$	9.85641	0.387495	0.193748	+	0.981051i	$0.437936\pi$
$648$	0	0
$649$	−3.71281	−0.145741
$650$	0	0
$651$	− 14.9282i	− 0.585082i
$652$	0	0
$653$	− 19.8564i	− 0.777041i	−0.921440	−	0.388521i	$-0.872986\pi$
$653$	− 19.8564i	− 0.777041i	0.921440	−	0.388521i	$-0.127014\pi$
$654$	0	0
$655$	−5.46410	−0.213500
$656$	0	0
$657$	−10.0000	−0.390137
$658$	0	0
$659$	− 0.392305i	− 0.0152820i	−0.999971	−	0.00764101i	$-0.997568\pi$
$659$	− 0.392305i	− 0.0152820i	0.999971	−	0.00764101i	$-0.00243223\pi$
$660$	0	0
$661$	− 4.00000i	− 0.155582i	−0.996970	−	0.0777910i	$-0.975213\pi$
$661$	− 4.00000i	− 0.155582i	0.996970	−	0.0777910i	$-0.0247867\pi$
$662$	0	0
$663$	−5.07180	−0.196972
$664$	0	0
$665$	−13.8564	−0.537328
$666$	0	0
$667$	19.7128i	0.763283i
$668$	0	0
$669$	20.9282i	0.809131i
$670$	0	0
$671$	−5.85641	−0.226084
$672$	0	0
$673$	−29.7128	−1.14534	−0.572672	−	0.819784i	$-0.694093\pi$
$673$	−29.7128	−1.14534	−0.572672	+	0.819784i	$0.694093\pi$
$674$	0	0
$675$	− 1.00000i	− 0.0384900i
$676$	0	0
$677$	− 4.14359i	− 0.159251i	−0.996825	−	0.0796256i	$-0.974628\pi$
$677$	− 4.14359i	− 0.159251i	0.996825	−	0.0796256i	$-0.0253725\pi$
$678$	0	0
$679$	−1.85641	−0.0712423
$680$	0	0
$681$	−13.0718	−0.500912
$682$	0	0
$683$	− 18.1436i	− 0.694245i	−0.937820	−	0.347123i	$-0.887159\pi$
$683$	− 18.1436i	− 0.694245i	0.937820	−	0.347123i	$-0.112841\pi$
$684$	0	0
$685$	22.3923i	0.855566i
$686$	0	0
$687$	2.14359	0.0817832
$688$	0	0
$689$	−18.9282	−0.721107
$690$	0	0
$691$	− 9.07180i	− 0.345107i	−0.985000	−	0.172554i	$-0.944798\pi$
$691$	− 9.07180i	− 0.345107i	0.985000	−	0.172554i	$-0.0552018\pi$
$692$	0	0
$693$	− 2.92820i	− 0.111233i
$694$	0	0
$695$	12.0000	0.455186
$696$	0	0
$697$	30.9282	1.17149
$698$	0	0
$699$	− 4.53590i	− 0.171563i
$700$	0	0
$701$	24.1436i	0.911891i	0.890008	+	0.455945i	$0.150699\pi$
$701$	24.1436i	0.911891i	−0.890008	+	0.455945i	$0.849301\pi$
$702$	0	0
$703$	17.5692	0.662636
$704$	0	0
$705$	4.00000	0.150649
$706$	0	0
$707$	− 20.0000i	− 0.752177i
$708$	0	0
$709$	3.71281i	0.139438i	0.997567	+	0.0697188i	$0.0222102\pi$
$709$	3.71281i	0.139438i	−0.997567	+	0.0697188i	$0.977790\pi$
$710$	0	0
$711$	−6.39230	−0.239730
$712$	0	0
$713$	29.8564	1.11813
$714$	0	0
$715$	2.14359i	0.0801659i
$716$	0	0
$717$	− 16.0000i	− 0.597531i
$718$	0	0
$719$	3.21539	0.119914	0.0599569	−	0.998201i	$-0.480904\pi$
$719$	3.21539	0.119914	0.0599569	+	0.998201i	$0.480904\pi$
$720$	0	0
$721$	17.8564	0.665007
$722$	0	0
$723$	− 26.0000i	− 0.966950i
$724$	0	0
$725$	4.92820i	0.183029i
$726$	0	0
$727$	8.92820	0.331129	0.165564	−	0.986199i	$-0.447055\pi$
$727$	8.92820	0.331129	0.165564	+	0.986199i	$0.447055\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	24.0000i	0.887672i
$732$	0	0
$733$	− 1.46410i	− 0.0540778i	−0.999634	−	0.0270389i	$-0.991392\pi$
$733$	− 1.46410i	− 0.0540778i	0.999634	−	0.0270389i	$-0.00860780\pi$
$734$	0	0
$735$	−3.00000	−0.110657
$736$	0	0
$737$	−10.1436	−0.373644
$738$	0	0
$739$	26.6410i	0.980006i	0.871721	+	0.490003i	$0.163004\pi$
$739$	26.6410i	0.980006i	−0.871721	+	0.490003i	$0.836996\pi$
$740$	0	0
$741$	− 10.1436i	− 0.372634i
$742$	0	0
$743$	19.7128	0.723193	0.361596	−	0.932335i	$-0.382232\pi$
$743$	19.7128	0.723193	0.361596	+	0.932335i	$0.382232\pi$
$744$	0	0
$745$	18.7846	0.688215
$746$	0	0
$747$	− 8.00000i	− 0.292705i
$748$	0	0
$749$	− 13.8564i	− 0.506302i
$750$	0	0
$751$	46.3923	1.69288	0.846440	−	0.532485i	$-0.178742\pi$
$751$	46.3923	1.69288	0.846440	+	0.532485i	$0.178742\pi$
$752$	0	0
$753$	23.3205	0.849847
$754$	0	0
$755$	4.53590i	0.165078i
$756$	0	0
$757$	− 48.1051i	− 1.74841i	−0.485557	−	0.874205i	$-0.661383\pi$
$757$	− 48.1051i	− 1.74841i	0.485557	−	0.874205i	$-0.338617\pi$
$758$	0	0
$759$	5.85641	0.212574
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	37.8564i	1.37049i
$764$	0	0
$765$	3.46410i	0.125245i
$766$	0	0
$767$	3.71281	0.134062
$768$	0	0
$769$	−11.8564	−0.427553	−0.213776	−	0.976883i	$-0.568576\pi$
$769$	−11.8564	−0.427553	−0.213776	+	0.976883i	$0.568576\pi$
$770$	0	0
$771$	7.46410i	0.268813i
$772$	0	0
$773$	− 41.7128i	− 1.50031i	−0.661265	−	0.750153i	$-0.729980\pi$
$773$	− 41.7128i	− 1.50031i	0.661265	−	0.750153i	$-0.270020\pi$
$774$	0	0
$775$	7.46410	0.268118
$776$	0	0
$777$	−5.07180	−0.181950
$778$	0	0
$779$	61.8564i	2.21624i
$780$	0	0
$781$	10.1436i	0.362966i
$782$	0	0
$783$	4.92820	0.176120
$784$	0	0
$785$	21.4641	0.766087
$786$	0	0
$787$	23.7128i	0.845270i	0.906300	+	0.422635i	$0.138895\pi$
$787$	23.7128i	0.845270i	−0.906300	+	0.422635i	$0.861105\pi$
$788$	0	0
$789$	27.7128i	0.986602i
$790$	0	0
$791$	−1.07180	−0.0381087
$792$	0	0
$793$	5.85641	0.207967
$794$	0	0
$795$	12.9282i	0.458516i
$796$	0	0
$797$	18.7846i	0.665385i	0.943035	+	0.332693i	$0.107957\pi$
$797$	18.7846i	0.665385i	−0.943035	+	0.332693i	$0.892043\pi$
$798$	0	0
$799$	−13.8564	−0.490204
$800$	0	0
$801$	12.9282	0.456796
$802$	0	0
$803$	− 14.6410i	− 0.516670i
$804$	0	0
$805$	8.00000i	0.281963i
$806$	0	0
$807$	−26.7846	−0.942863
$808$	0	0
$809$	18.7846	0.660432	0.330216	−	0.943905i	$-0.392878\pi$
$809$	18.7846	0.660432	0.330216	+	0.943905i	$0.392878\pi$
$810$	0	0
$811$	17.8564i	0.627023i	0.949584	+	0.313512i	$0.101505\pi$
$811$	17.8564i	0.627023i	−0.949584	+	0.313512i	$0.898495\pi$
$812$	0	0
$813$	11.4641i	0.402064i
$814$	0	0
$815$	−4.00000	−0.140114
$816$	0	0
$817$	−48.0000	−1.67931
$818$	0	0
$819$	2.92820i	0.102320i
$820$	0	0
$821$	− 38.7846i	− 1.35359i	−0.736171	−	0.676796i	$-0.763368\pi$
$821$	− 38.7846i	− 1.35359i	0.736171	−	0.676796i	$-0.236632\pi$
$822$	0	0
$823$	54.7846	1.90967	0.954836	−	0.297134i	$-0.0960309\pi$
$823$	54.7846	1.90967	0.954836	+	0.297134i	$0.0960309\pi$
$824$	0	0
$825$	1.46410	0.0509735
$826$	0	0
$827$	37.5692i	1.30641i	0.757181	+	0.653205i	$0.226576\pi$
$827$	37.5692i	1.30641i	−0.757181	+	0.653205i	$0.773424\pi$
$828$	0	0
$829$	2.92820i	0.101701i	0.998706	+	0.0508504i	$0.0161932\pi$
$829$	2.92820i	0.101701i	−0.998706	+	0.0508504i	$0.983807\pi$
$830$	0	0
$831$	−6.53590	−0.226728
$832$	0	0
$833$	10.3923	0.360072
$834$	0	0
$835$	13.8564i	0.479521i
$836$	0	0
$837$	− 7.46410i	− 0.257997i
$838$	0	0
$839$	−42.6410	−1.47213	−0.736066	−	0.676910i	$-0.763319\pi$
$839$	−42.6410	−1.47213	−0.736066	+	0.676910i	$0.763319\pi$
$840$	0	0
$841$	4.71281	0.162511
$842$	0	0
$843$	− 0.928203i	− 0.0319690i
$844$	0	0
$845$	10.8564i	0.373472i
$846$	0	0
$847$	−17.7128	−0.608619
$848$	0	0
$849$	22.9282	0.786894
$850$	0	0
$851$	− 10.1436i	− 0.347718i
$852$	0	0
$853$	− 35.3205i	− 1.20935i	−0.796472	−	0.604676i	$-0.793303\pi$
$853$	− 35.3205i	− 1.20935i	0.796472	−	0.604676i	$-0.206697\pi$
$854$	0	0
$855$	−6.92820	−0.236940
$856$	0	0
$857$	53.0333	1.81158	0.905792	−	0.423723i	$-0.139277\pi$
$857$	53.0333	1.81158	0.905792	+	0.423723i	$0.139277\pi$
$858$	0	0
$859$	− 39.7128i	− 1.35498i	−0.735530	−	0.677492i	$-0.763067\pi$
$859$	− 39.7128i	− 1.35498i	0.735530	−	0.677492i	$-0.236933\pi$
$860$	0	0
$861$	− 17.8564i	− 0.608545i
$862$	0	0
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	11.0718	0.376452
$866$	0	0
$867$	5.00000i	0.169809i
$868$	0	0
$869$	− 9.35898i	− 0.317482i
$870$	0	0
$871$	10.1436	0.343703
$872$	0	0
$873$	−0.928203	−0.0314149
$874$	0	0
$875$	2.00000i	0.0676123i
$876$	0	0
$877$	18.5359i	0.625913i	0.949767	+	0.312956i	$0.101319\pi$
$877$	18.5359i	0.625913i	−0.949767	+	0.312956i	$0.898681\pi$
$878$	0	0
$879$	4.14359	0.139760
$880$	0	0
$881$	−12.6410	−0.425887	−0.212943	−	0.977065i	$-0.568305\pi$
$881$	−12.6410	−0.425887	−0.212943	+	0.977065i	$0.568305\pi$
$882$	0	0
$883$	− 49.8564i	− 1.67780i	−0.544284	−	0.838901i	$-0.683199\pi$
$883$	− 49.8564i	− 1.67780i	0.544284	−	0.838901i	$-0.316801\pi$
$884$	0	0
$885$	− 2.53590i	− 0.0852433i
$886$	0	0
$887$	10.1436	0.340589	0.170294	−	0.985393i	$-0.445528\pi$
$887$	10.1436	0.340589	0.170294	+	0.985393i	$0.445528\pi$
$888$	0	0
$889$	41.8564	1.40382
$890$	0	0
$891$	− 1.46410i	− 0.0490492i
$892$	0	0
$893$	− 27.7128i	− 0.927374i
$894$	0	0
$895$	−16.3923	−0.547934
$896$	0	0
$897$	−5.85641	−0.195540
$898$	0	0
$899$	36.7846i	1.22684i
$900$	0	0
$901$	− 44.7846i	− 1.49199i
$902$	0	0
$903$	13.8564	0.461112
$904$	0	0
$905$	−2.92820	−0.0973368
$906$	0	0
$907$	− 1.07180i	− 0.0355884i	−0.999842	−	0.0177942i	$-0.994336\pi$
$907$	− 1.07180i	− 0.0355884i	0.999842	−	0.0177942i	$-0.00566437\pi$
$908$	0	0
$909$	− 10.0000i	− 0.331679i
$910$	0	0
$911$	−27.7128	−0.918166	−0.459083	−	0.888393i	$-0.651822\pi$
$911$	−27.7128	−0.918166	−0.459083	+	0.888393i	$0.651822\pi$
$912$	0	0
$913$	11.7128	0.387638
$914$	0	0
$915$	− 4.00000i	− 0.132236i
$916$	0	0
$917$	− 10.9282i	− 0.360881i
$918$	0	0
$919$	−14.3923	−0.474758	−0.237379	−	0.971417i	$-0.576288\pi$
$919$	−14.3923	−0.474758	−0.237379	+	0.971417i	$0.576288\pi$
$920$	0	0
$921$	9.85641	0.324780
$922$	0	0
$923$	− 10.1436i	− 0.333880i
$924$	0	0
$925$	− 2.53590i	− 0.0833798i
$926$	0	0
$927$	8.92820	0.293241
$928$	0	0
$929$	39.5692	1.29822	0.649112	−	0.760693i	$-0.275141\pi$
$929$	39.5692	1.29822	0.649112	+	0.760693i	$0.275141\pi$
$930$	0	0
$931$	20.7846i	0.681188i
$932$	0	0
$933$	− 2.14359i	− 0.0701781i
$934$	0	0
$935$	−5.07180	−0.165865
$936$	0	0
$937$	0.143594	0.00469100	0.00234550	−	0.999997i	$-0.499253\pi$
$937$	0.143594	0.00469100	0.00234550	+	0.999997i	$0.499253\pi$
$938$	0	0
$939$	4.92820i	0.160826i
$940$	0	0
$941$	− 6.00000i	− 0.195594i	−0.995206	−	0.0977972i	$-0.968820\pi$
$941$	− 6.00000i	− 0.195594i	0.995206	−	0.0977972i	$-0.0311797\pi$
$942$	0	0
$943$	35.7128	1.16297
$944$	0	0
$945$	2.00000	0.0650600
$946$	0	0
$947$	− 1.85641i	− 0.0603251i	−0.999545	−	0.0301626i	$-0.990398\pi$
$947$	− 1.85641i	− 0.0603251i	0.999545	−	0.0301626i	$-0.00960249\pi$
$948$	0	0
$949$	14.6410i	0.475267i
$950$	0	0
$951$	11.8564	0.384470
$952$	0	0
$953$	−29.3205	−0.949784	−0.474892	−	0.880044i	$-0.657513\pi$
$953$	−29.3205	−0.949784	−0.474892	+	0.880044i	$0.657513\pi$
$954$	0	0
$955$	− 8.00000i	− 0.258874i
$956$	0	0
$957$	7.21539i	0.233240i
$958$	0	0
$959$	−44.7846	−1.44617
$960$	0	0
$961$	24.7128	0.797188
$962$	0	0
$963$	− 6.92820i	− 0.223258i
$964$	0	0
$965$	18.7846i	0.604698i
$966$	0	0
$967$	18.7846	0.604072	0.302036	−	0.953296i	$-0.402334\pi$
$967$	18.7846	0.604072	0.302036	+	0.953296i	$0.402334\pi$
$968$	0	0
$969$	24.0000	0.770991
$970$	0	0
$971$	− 4.39230i	− 0.140956i	−0.997513	−	0.0704779i	$-0.977548\pi$
$971$	− 4.39230i	− 0.140956i	0.997513	−	0.0704779i	$-0.0224524\pi$
$972$	0	0
$973$	24.0000i	0.769405i
$974$	0	0
$975$	−1.46410	−0.0468888
$976$	0	0
$977$	34.3923	1.10031	0.550154	−	0.835063i	$-0.314569\pi$
$977$	34.3923	1.10031	0.550154	+	0.835063i	$0.314569\pi$
$978$	0	0
$979$	18.9282i	0.604948i
$980$	0	0
$981$	18.9282i	0.604331i
$982$	0	0
$983$	−13.8564	−0.441951	−0.220975	−	0.975279i	$-0.570924\pi$
$983$	−13.8564	−0.441951	−0.220975	+	0.975279i	$0.570924\pi$
$984$	0	0
$985$	−23.8564	−0.760128
$986$	0	0
$987$	8.00000i	0.254643i
$988$	0	0
$989$	27.7128i	0.881216i
$990$	0	0
$991$	−14.6795	−0.466309	−0.233155	−	0.972440i	$-0.574905\pi$
$991$	−14.6795	−0.466309	−0.233155	+	0.972440i	$0.574905\pi$
$992$	0	0
$993$	−9.07180	−0.287885
$994$	0	0
$995$	25.3205i	0.802714i
$996$	0	0
$997$	− 2.53590i	− 0.0803127i	−0.999193	−	0.0401564i	$-0.987214\pi$
$997$	− 2.53590i	− 0.0803127i	0.999193	−	0.0401564i	$-0.0127856\pi$
$998$	0	0
$999$	−2.53590	−0.0802323

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.2.k.e.481.1	✓	4
3.2	odd	2		2880.2.k.f.1441.2		4
4.3	odd	2		960.2.k.f.481.4	yes	4
5.2	odd	4		4800.2.d.i.1249.1		4
5.3	odd	4		4800.2.d.r.1249.3		4
5.4	even	2		4800.2.k.o.2401.3		4
8.3	odd	2		960.2.k.f.481.1	yes	4
8.5	even	2	inner	960.2.k.e.481.4	yes	4
12.11	even	2		2880.2.k.k.1441.1		4
16.3	odd	4		3840.2.a.bf.1.2		2
16.5	even	4		3840.2.a.be.1.2		2
16.11	odd	4		3840.2.a.bi.1.1		2
16.13	even	4		3840.2.a.bn.1.1		2
20.3	even	4		4800.2.d.m.1249.2		4
20.7	even	4		4800.2.d.n.1249.4		4
20.19	odd	2		4800.2.k.i.2401.2		4
24.5	odd	2		2880.2.k.f.1441.3		4
24.11	even	2		2880.2.k.k.1441.4		4
40.3	even	4		4800.2.d.n.1249.1		4
40.13	odd	4		4800.2.d.i.1249.4		4
40.19	odd	2		4800.2.k.i.2401.3		4
40.27	even	4		4800.2.d.m.1249.3		4
40.29	even	2		4800.2.k.o.2401.2		4
40.37	odd	4		4800.2.d.r.1249.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
960.2.k.e.481.1	✓	4	1.1	even	1	trivial
960.2.k.e.481.4	yes	4	8.5	even	2	inner
960.2.k.f.481.1	yes	4	8.3	odd	2
960.2.k.f.481.4	yes	4	4.3	odd	2
2880.2.k.f.1441.2		4	3.2	odd	2
2880.2.k.f.1441.3		4	24.5	odd	2
2880.2.k.k.1441.1		4	12.11	even	2
2880.2.k.k.1441.4		4	24.11	even	2
3840.2.a.be.1.2		2	16.5	even	4
3840.2.a.bf.1.2		2	16.3	odd	4
3840.2.a.bi.1.1		2	16.11	odd	4
3840.2.a.bn.1.1		2	16.13	even	4
4800.2.d.i.1249.1		4	5.2	odd	4
4800.2.d.i.1249.4		4	40.13	odd	4
4800.2.d.m.1249.2		4	20.3	even	4
4800.2.d.m.1249.3		4	40.27	even	4
4800.2.d.n.1249.1		4	40.3	even	4
4800.2.d.n.1249.4		4	20.7	even	4
4800.2.d.r.1249.2		4	40.37	odd	4
4800.2.d.r.1249.3		4	5.3	odd	4
4800.2.k.i.2401.2		4	20.19	odd	2
4800.2.k.i.2401.3		4	40.19	odd	2
4800.2.k.o.2401.2		4	40.29	even	2
4800.2.k.o.2401.3		4	5.4	even	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:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	960.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +1.00000i q^{5} -2.00000 q^{7} -1.00000 q^{9} -1.46410i q^{11} +1.46410i q^{13} +1.00000 q^{15} -3.46410 q^{17} -6.92820i q^{19} +2.00000i q^{21} -4.00000 q^{23} -1.00000 q^{25} +1.00000i q^{27} -4.92820i q^{29} -7.46410 q^{31} -1.46410 q^{33} -2.00000i q^{35} +2.53590i q^{37} +1.46410 q^{39} -8.92820 q^{41} -6.92820i q^{43} -1.00000i q^{45} +4.00000 q^{47} -3.00000 q^{49} +3.46410i q^{51} +12.9282i q^{53} +1.46410 q^{55} -6.92820 q^{57} -2.53590i q^{59} -4.00000i q^{61} +2.00000 q^{63} -1.46410 q^{65} -6.92820i q^{67} +4.00000i q^{69} -6.92820 q^{71} +10.0000 q^{73} +1.00000i q^{75} +2.92820i q^{77} +6.39230 q^{79} +1.00000 q^{81} +8.00000i q^{83} -3.46410i q^{85} -4.92820 q^{87} -12.9282 q^{89} -2.92820i q^{91} +7.46410i q^{93} +6.92820 q^{95} +0.928203 q^{97} +1.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{7} - 4 q^{9} + 4 q^{15} - 16 q^{23} - 4 q^{25} - 16 q^{31} + 8 q^{33} - 8 q^{39} - 8 q^{41} + 16 q^{47} - 12 q^{49} - 8 q^{55} + 8 q^{63} + 8 q^{65} + 40 q^{73} - 16 q^{79} + 4 q^{81} + 8 q^{87}+ \cdots - 24 q^{97}+O(q^{100}) \) 4 * q - 8 * q^7 - 4 * q^9 + 4 * q^15 - 16 * q^23 - 4 * q^25 - 16 * q^31 + 8 * q^33 - 8 * q^39 - 8 * q^41 + 16 * q^47 - 12 * q^49 - 8 * q^55 + 8 * q^63 + 8 * q^65 + 40 * q^73 - 16 * q^79 + 4 * q^81 + 8 * q^87 - 24 * q^89 - 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more