LMFDB - Embedded newform 2368.2.g.h.961.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2368,2,Mod(961,2368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2368.961");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2368 = 2^{6} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2368.g (of order $2$, degree $1$, not 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:	$18.9085751986$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{21})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 11x^{2} + 25 $ x^4 + 11*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 74)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			961.3
Root			$-1.79129i$ of defining polynomial
Character	$\chi$	$=$	2368.961
Dual form			2368.2.g.h.961.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.79129 q^{3} -0.791288i q^{5} +2.00000 q^{7} +0.208712 q^{9} +O(q^{10})$	$q+1.79129 q^{3} -0.791288i q^{5} +2.00000 q^{7} +0.208712 q^{9} -0.791288 q^{11} +3.79129i q^{13} -1.41742i q^{15} +7.58258i q^{17} +1.58258i q^{19} +3.58258 q^{21} +3.79129i q^{23} +4.37386 q^{25} -5.00000 q^{27} -3.79129i q^{29} +8.37386i q^{31} -1.41742 q^{33} -1.58258i q^{35} +(-4.00000 + 4.58258i) q^{37} +6.79129i q^{39} +9.79129 q^{41} -6.00000i q^{43} -0.165151i q^{45} +7.58258 q^{47} -3.00000 q^{49} +13.5826i q^{51} +1.58258 q^{53} +0.626136i q^{55} +2.83485i q^{57} +1.58258i q^{59} -12.7913i q^{61} +0.417424 q^{63} +3.00000 q^{65} +6.37386 q^{67} +6.79129i q^{69} +9.16515 q^{71} -4.37386 q^{73} +7.83485 q^{75} -1.58258 q^{77} -8.20871i q^{79} -9.58258 q^{81} +15.1652 q^{83} +6.00000 q^{85} -6.79129i q^{87} +6.00000i q^{89} +7.58258i q^{91} +15.0000i q^{93} +1.25227 q^{95} -13.5826i q^{97} -0.165151 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 8 q^{7} + 10 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 + 8 * q^7 + 10 * q^9	$ 4 q - 2 q^{3} + 8 q^{7} + 10 q^{9} + 6 q^{11} - 4 q^{21} - 10 q^{25} - 20 q^{27} - 24 q^{33} - 16 q^{37} + 30 q^{41} + 12 q^{47} - 12 q^{49} - 12 q^{53} + 20 q^{63} + 12 q^{65} - 2 q^{67} + 10 q^{73} + 68 q^{75} + 12 q^{77} - 20 q^{81} + 24 q^{83} + 24 q^{85} + 60 q^{95} + 36 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 + 8 * q^7 + 10 * q^9 + 6 * q^11 - 4 * q^21 - 10 * q^25 - 20 * q^27 - 24 * q^33 - 16 * q^37 + 30 * q^41 + 12 * q^47 - 12 * q^49 - 12 * q^53 + 20 * q^63 + 12 * q^65 - 2 * q^67 + 10 * q^73 + 68 * q^75 + 12 * q^77 - 20 * q^81 + 24 * q^83 + 24 * q^85 + 60 * q^95 + 36 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2368\mathbb{Z}\right)^\times$.

$n$	$705$	$1407$	$1925$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	1.79129			1.03420			0.517100	−	0.855925i	$-0.327011\pi$
$3$	1.79129			1.03420			0.517100	+	0.855925i	$0.327011\pi$
$4$		0			0
$5$		−	0.791288i		−	0.353875i	−0.984222	−	0.176937i	$-0.943381\pi$
$5$		−	0.791288i		−	0.353875i	0.984222	−	0.176937i	$-0.0566190\pi$
$6$		0			0
$7$	2.00000			0.755929			0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000			0.755929			0.377964	+	0.925820i	$0.376624\pi$
$8$		0			0
$9$	0.208712			0.0695707
$10$		0			0
$11$	−0.791288			−0.238582			−0.119291	−	0.992859i	$-0.538062\pi$
$11$	−0.791288			−0.238582			−0.119291	+	0.992859i	$0.538062\pi$
$12$		0			0
$13$			3.79129i			1.05151i	0.850635	+	0.525757i	$0.176218\pi$
$13$			3.79129i			1.05151i	−0.850635	+	0.525757i	$0.823782\pi$
$14$		0			0
$15$		−	1.41742i		−	0.365977i
$16$		0			0
$17$			7.58258i			1.83904i	0.393038	+	0.919522i	$0.371424\pi$
$17$			7.58258i			1.83904i	−0.393038	+	0.919522i	$0.628576\pi$
$18$		0			0
$19$			1.58258i			0.363068i	0.983385	+	0.181534i	$0.0581062\pi$
$19$			1.58258i			0.363068i	−0.983385	+	0.181534i	$0.941894\pi$
$20$		0			0
$21$	3.58258			0.781782
$22$		0			0
$23$			3.79129i			0.790538i	0.918565	+	0.395269i	$0.129349\pi$
$23$			3.79129i			0.790538i	−0.918565	+	0.395269i	$0.870651\pi$
$24$		0			0
$25$	4.37386			0.874773
$26$		0			0
$27$	−5.00000			−0.962250
$28$		0			0
$29$		−	3.79129i		−	0.704024i	−0.935995	−	0.352012i	$-0.885498\pi$
$29$		−	3.79129i		−	0.704024i	0.935995	−	0.352012i	$-0.114502\pi$
$30$		0			0
$31$			8.37386i			1.50399i	0.659169	+	0.751995i	$0.270908\pi$
$31$			8.37386i			1.50399i	−0.659169	+	0.751995i	$0.729092\pi$
$32$		0			0
$33$	−1.41742			−0.246742
$34$		0			0
$35$		−	1.58258i		−	0.267504i
$36$		0			0
$37$	−4.00000	+	4.58258i	−0.657596	+	0.753371i
$37$	−4.00000	+	4.58258i	−0.657596	+	0.753371i
$38$		0			0
$39$			6.79129i			1.08748i
$40$		0			0
$41$	9.79129			1.52914			0.764571	−	0.644539i	$-0.222951\pi$
$41$	9.79129			1.52914			0.764571	+	0.644539i	$0.222951\pi$
$42$		0			0
$43$		−	6.00000i		−	0.914991i	−0.889212	−	0.457496i	$-0.848747\pi$
$43$		−	6.00000i		−	0.914991i	0.889212	−	0.457496i	$-0.151253\pi$
$44$		0			0
$45$		−	0.165151i		−	0.0246193i
$46$		0			0
$47$	7.58258			1.10603			0.553016	−	0.833171i	$-0.313477\pi$
$47$	7.58258			1.10603			0.553016	+	0.833171i	$0.313477\pi$
$48$		0			0
$49$	−3.00000			−0.428571
$50$		0			0
$51$			13.5826i			1.90194i
$52$		0			0
$53$	1.58258			0.217383			0.108692	−	0.994076i	$-0.465334\pi$
$53$	1.58258			0.217383			0.108692	+	0.994076i	$0.465334\pi$
$54$		0			0
$55$			0.626136i			0.0844282i
$56$		0			0
$57$			2.83485i			0.375485i
$58$		0			0
$59$			1.58258i			0.206034i	0.994680	+	0.103017i	$0.0328496\pi$
$59$			1.58258i			0.206034i	−0.994680	+	0.103017i	$0.967150\pi$
$60$		0			0
$61$		−	12.7913i		−	1.63776i	−0.573967	−	0.818878i	$-0.694596\pi$
$61$		−	12.7913i		−	1.63776i	0.573967	−	0.818878i	$-0.305404\pi$
$62$		0			0
$63$	0.417424			0.0525905
$64$		0			0
$65$	3.00000			0.372104
$66$		0			0
$67$	6.37386			0.778691			0.389346	−	0.921092i	$-0.372701\pi$
$67$	6.37386			0.778691			0.389346	+	0.921092i	$0.372701\pi$
$68$		0			0
$69$			6.79129i			0.817575i
$70$		0			0
$71$	9.16515			1.08770			0.543852	−	0.839181i	$-0.316965\pi$
$71$	9.16515			1.08770			0.543852	+	0.839181i	$0.316965\pi$
$72$		0			0
$73$	−4.37386			−0.511922			−0.255961	−	0.966687i	$-0.582392\pi$
$73$	−4.37386			−0.511922			−0.255961	+	0.966687i	$0.582392\pi$
$74$		0			0
$75$	7.83485			0.904690
$76$		0			0
$77$	−1.58258			−0.180351
$78$		0			0
$79$		−	8.20871i		−	0.923552i	−0.886997	−	0.461776i	$-0.847212\pi$
$79$		−	8.20871i		−	0.923552i	0.886997	−	0.461776i	$-0.152788\pi$
$80$		0			0
$81$	−9.58258			−1.06473
$82$		0			0
$83$	15.1652			1.66459			0.832296	−	0.554332i	$-0.187026\pi$
$83$	15.1652			1.66459			0.832296	+	0.554332i	$0.187026\pi$
$84$		0			0
$85$	6.00000			0.650791
$86$		0			0
$87$		−	6.79129i		−	0.728102i
$88$		0			0
$89$			6.00000i			0.635999i	0.948091	+	0.317999i	$0.103011\pi$
$89$			6.00000i			0.635999i	−0.948091	+	0.317999i	$0.896989\pi$
$90$		0			0
$91$			7.58258i			0.794870i
$92$		0			0
$93$			15.0000i			1.55543i
$94$		0			0
$95$	1.25227			0.128480
$96$		0			0
$97$		−	13.5826i		−	1.37910i	−0.724237	−	0.689551i	$-0.757808\pi$
$97$		−	13.5826i		−	1.37910i	0.724237	−	0.689551i	$-0.242192\pi$
$98$		0			0
$99$	−0.165151			−0.0165983
$100$		0			0
$101$	−7.58258			−0.754494			−0.377247	−	0.926113i	$-0.623129\pi$
$101$	−7.58258			−0.754494			−0.377247	+	0.926113i	$0.623129\pi$
$102$		0			0
$103$		−	6.79129i		−	0.669165i	−0.942366	−	0.334583i	$-0.891405\pi$
$103$		−	6.79129i		−	0.669165i	0.942366	−	0.334583i	$-0.108595\pi$
$104$		0			0
$105$		−	2.83485i		−	0.276653i
$106$		0			0
$107$	−5.37386			−0.519511			−0.259755	−	0.965674i	$-0.583642\pi$
$107$	−5.37386			−0.519511			−0.259755	+	0.965674i	$0.583642\pi$
$108$		0			0
$109$		−	6.00000i		−	0.574696i	−0.957826	−	0.287348i	$-0.907226\pi$
$109$		−	6.00000i		−	0.574696i	0.957826	−	0.287348i	$-0.0927736\pi$
$110$		0			0
$111$	−7.16515	+	8.20871i	−0.680086	+	0.779136i
$112$		0			0
$113$		−	10.4174i		−	0.979989i	−0.871726	−	0.489994i	$-0.836999\pi$
$113$		−	10.4174i		−	0.979989i	0.871726	−	0.489994i	$-0.163001\pi$
$114$		0			0
$115$	3.00000			0.279751
$116$		0			0
$117$			0.791288i			0.0731546i
$118$		0			0
$119$			15.1652i			1.39019i
$120$		0			0
$121$	−10.3739			−0.943079
$122$		0			0
$123$	17.5390			1.58144
$124$		0			0
$125$		−	7.41742i		−	0.663435i
$126$		0			0
$127$	−8.00000			−0.709885			−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000			−0.709885			−0.354943	+	0.934888i	$0.615500\pi$
$128$		0			0
$129$		−	10.7477i		−	0.946285i
$130$		0			0
$131$			16.7477i			1.46326i	0.681704	+	0.731628i	$0.261239\pi$
$131$			16.7477i			1.46326i	−0.681704	+	0.731628i	$0.738761\pi$
$132$		0			0
$133$			3.16515i			0.274453i
$134$		0			0
$135$			3.95644i			0.340516i
$136$		0			0
$137$	−3.62614			−0.309802			−0.154901	−	0.987930i	$-0.549506\pi$
$137$	−3.62614			−0.309802			−0.154901	+	0.987930i	$0.549506\pi$
$138$		0			0
$139$	−13.3739			−1.13436			−0.567178	−	0.823595i	$-0.691965\pi$
$139$	−13.3739			−1.13436			−0.567178	+	0.823595i	$0.691965\pi$
$140$		0			0
$141$	13.5826			1.14386
$142$		0			0
$143$		−	3.00000i		−	0.250873i
$144$		0			0
$145$	−3.00000			−0.249136
$146$		0			0
$147$	−5.37386			−0.443229
$148$		0			0
$149$	4.41742			0.361889			0.180945	−	0.983493i	$-0.442084\pi$
$149$	4.41742			0.361889			0.180945	+	0.983493i	$0.442084\pi$
$150$		0			0
$151$	14.7477			1.20015			0.600077	−	0.799943i	$-0.295137\pi$
$151$	14.7477			1.20015			0.600077	+	0.799943i	$0.295137\pi$
$152$		0			0
$153$			1.58258i			0.127944i
$154$		0			0
$155$	6.62614			0.532224
$156$		0			0
$157$	2.00000			0.159617			0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000			0.159617			0.0798087	+	0.996810i	$0.474569\pi$
$158$		0			0
$159$	2.83485			0.224818
$160$		0			0
$161$			7.58258i			0.597591i
$162$		0			0
$163$			19.5826i			1.53383i	0.641751	+	0.766913i	$0.278208\pi$
$163$			19.5826i			1.53383i	−0.641751	+	0.766913i	$0.721792\pi$
$164$		0			0
$165$			1.12159i			0.0873157i
$166$		0			0
$167$			21.9564i			1.69904i	0.527556	+	0.849520i	$0.323108\pi$
$167$			21.9564i			1.69904i	−0.527556	+	0.849520i	$0.676892\pi$
$168$		0			0
$169$	−1.37386			−0.105682
$170$		0			0
$171$			0.330303i			0.0252589i
$172$		0			0
$173$	−15.1652			−1.15299			−0.576493	−	0.817102i	$-0.695579\pi$
$173$	−15.1652			−1.15299			−0.576493	+	0.817102i	$0.695579\pi$
$174$		0			0
$175$	8.74773			0.661266
$176$		0			0
$177$			2.83485i			0.213080i
$178$		0			0
$179$			1.58258i			0.118287i	0.998249	+	0.0591436i	$0.0188370\pi$
$179$			1.58258i			0.118287i	−0.998249	+	0.0591436i	$0.981163\pi$
$180$		0			0
$181$	−8.74773			−0.650213			−0.325107	−	0.945677i	$-0.605400\pi$
$181$	−8.74773			−0.650213			−0.325107	+	0.945677i	$0.605400\pi$
$182$		0			0
$183$		−	22.9129i		−	1.69377i
$184$		0			0
$185$	3.62614	+	3.16515i	0.266599	+	0.232707i
$186$		0			0
$187$		−	6.00000i		−	0.438763i
$188$		0			0
$189$	−10.0000			−0.727393
$190$		0			0
$191$		−	8.37386i		−	0.605912i	−0.953005	−	0.302956i	$-0.902027\pi$
$191$		−	8.37386i		−	0.605912i	0.953005	−	0.302956i	$-0.0979734\pi$
$192$		0			0
$193$		−	18.3303i		−	1.31944i	−0.751510	−	0.659722i	$-0.770674\pi$
$193$		−	18.3303i		−	1.31944i	0.751510	−	0.659722i	$-0.229326\pi$
$194$		0			0
$195$	5.37386			0.384830
$196$		0			0
$197$	19.9129			1.41873			0.709367	−	0.704839i	$-0.248981\pi$
$197$	19.9129			1.41873			0.709367	+	0.704839i	$0.248981\pi$
$198$		0			0
$199$			15.1652i			1.07503i	0.843254	+	0.537515i	$0.180637\pi$
$199$			15.1652i			1.07503i	−0.843254	+	0.537515i	$0.819363\pi$
$200$		0			0
$201$	11.4174			0.805323
$202$		0			0
$203$		−	7.58258i		−	0.532192i
$204$		0			0
$205$		−	7.74773i		−	0.541125i
$206$		0			0
$207$			0.791288i			0.0549983i
$208$		0			0
$209$		−	1.25227i		−	0.0866215i
$210$		0			0
$211$	10.3739			0.714166			0.357083	−	0.934073i	$-0.383771\pi$
$211$	10.3739			0.714166			0.357083	+	0.934073i	$0.383771\pi$
$212$		0			0
$213$	16.4174			1.12490
$214$		0			0
$215$	−4.74773			−0.323792
$216$		0			0
$217$			16.7477i			1.13691i
$218$		0			0
$219$	−7.83485			−0.529430
$220$		0			0
$221$	−28.7477			−1.93378
$222$		0			0
$223$	−14.0000			−0.937509			−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000			−0.937509			−0.468755	+	0.883328i	$0.655297\pi$
$224$		0			0
$225$	0.912878			0.0608586
$226$		0			0
$227$		−	25.9129i		−	1.71990i	−0.510380	−	0.859949i	$-0.670495\pi$
$227$		−	25.9129i		−	1.71990i	0.510380	−	0.859949i	$-0.329505\pi$
$228$		0			0
$229$	−6.74773			−0.445902			−0.222951	−	0.974830i	$-0.571569\pi$
$229$	−6.74773			−0.445902			−0.222951	+	0.974830i	$0.571569\pi$
$230$		0			0
$231$	−2.83485			−0.186519
$232$		0			0
$233$	−16.1216			−1.05616			−0.528080	−	0.849194i	$-0.677088\pi$
$233$	−16.1216			−1.05616			−0.528080	+	0.849194i	$0.677088\pi$
$234$		0			0
$235$		−	6.00000i		−	0.391397i
$236$		0			0
$237$		−	14.7042i		−	0.955138i
$238$		0			0
$239$		−	2.04356i		−	0.132187i	−0.997813	−	0.0660935i	$-0.978946\pi$
$239$		−	2.04356i		−	0.132187i	0.997813	−	0.0660935i	$-0.0210536\pi$
$240$		0			0
$241$			4.41742i			0.284551i	0.989827	+	0.142276i	$0.0454419\pi$
$241$			4.41742i			0.284551i	−0.989827	+	0.142276i	$0.954558\pi$
$242$		0			0
$243$	−2.16515			−0.138895
$244$		0			0
$245$			2.37386i			0.151661i
$246$		0			0
$247$	−6.00000			−0.381771
$248$		0			0
$249$	27.1652			1.72152
$250$		0			0
$251$			4.41742i			0.278825i	0.990234	+	0.139413i	$0.0445215\pi$
$251$			4.41742i			0.278825i	−0.990234	+	0.139413i	$0.955479\pi$
$252$		0			0
$253$		−	3.00000i		−	0.188608i
$254$		0			0
$255$	10.7477			0.673049
$256$		0			0
$257$		−	4.74773i		−	0.296155i	−0.988976	−	0.148078i	$-0.952691\pi$
$257$		−	4.74773i		−	0.296155i	0.988976	−	0.148078i	$-0.0473085\pi$
$258$		0			0
$259$	−8.00000	+	9.16515i	−0.497096	+	0.569495i
$260$		0			0
$261$		−	0.791288i		−	0.0489795i
$262$		0			0
$263$	27.1652			1.67507			0.837537	−	0.546380i	$-0.183994\pi$
$263$	27.1652			1.67507			0.837537	+	0.546380i	$0.183994\pi$
$264$		0			0
$265$		−	1.25227i		−	0.0769265i
$266$		0			0
$267$			10.7477i			0.657750i
$268$		0			0
$269$	−16.7477			−1.02113			−0.510563	−	0.859840i	$-0.670563\pi$
$269$	−16.7477			−1.02113			−0.510563	+	0.859840i	$0.670563\pi$
$270$		0			0
$271$	−22.0000			−1.33640			−0.668202	−	0.743980i	$-0.732936\pi$
$271$	−22.0000			−1.33640			−0.668202	+	0.743980i	$0.732936\pi$
$272$		0			0
$273$			13.5826i			0.822055i
$274$		0			0
$275$	−3.46099			−0.208705
$276$		0			0
$277$			9.62614i			0.578378i	0.957272	+	0.289189i	$0.0933857\pi$
$277$			9.62614i			0.578378i	−0.957272	+	0.289189i	$0.906614\pi$
$278$		0			0
$279$			1.74773i			0.104634i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$			24.0000i			1.42665i	0.700832	+	0.713326i	$0.252812\pi$
$283$			24.0000i			1.42665i	−0.700832	+	0.713326i	$0.747188\pi$
$284$		0			0
$285$	2.24318			0.132875
$286$		0			0
$287$	19.5826			1.15592
$288$		0			0
$289$	−40.4955			−2.38209
$290$		0			0
$291$		−	24.3303i		−	1.42627i
$292$		0			0
$293$	6.00000			0.350524			0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000			0.350524			0.175262	+	0.984522i	$0.443923\pi$
$294$		0			0
$295$	1.25227			0.0729101
$296$		0			0
$297$	3.95644			0.229576
$298$		0			0
$299$	−14.3739			−0.831262
$300$		0			0
$301$		−	12.0000i		−	0.691669i
$302$		0			0
$303$	−13.5826			−0.780299
$304$		0			0
$305$	−10.1216			−0.579561
$306$		0			0
$307$	1.37386			0.0784105			0.0392053	−	0.999231i	$-0.487517\pi$
$307$	1.37386			0.0784105			0.0392053	+	0.999231i	$0.487517\pi$
$308$		0			0
$309$		−	12.1652i		−	0.692051i
$310$		0			0
$311$			23.3739i			1.32541i	0.748880	+	0.662705i	$0.230592\pi$
$311$			23.3739i			1.32541i	−0.748880	+	0.662705i	$0.769408\pi$
$312$		0			0
$313$		−	14.8348i		−	0.838515i	−0.907867	−	0.419258i	$-0.862290\pi$
$313$		−	14.8348i		−	0.838515i	0.907867	−	0.419258i	$-0.137710\pi$
$314$		0			0
$315$		−	0.330303i		−	0.0186105i
$316$		0			0
$317$	28.7477			1.61463			0.807317	−	0.590119i	$-0.200919\pi$
$317$	28.7477			1.61463			0.807317	+	0.590119i	$0.200919\pi$
$318$		0			0
$319$			3.00000i			0.167968i
$320$		0			0
$321$	−9.62614			−0.537279
$322$		0			0
$323$	−12.0000			−0.667698
$324$		0			0
$325$			16.5826i			0.919836i
$326$		0			0
$327$		−	10.7477i		−	0.594351i
$328$		0			0
$329$	15.1652			0.836082
$330$		0			0
$331$		−	25.5826i		−	1.40615i	−0.711118	−	0.703073i	$-0.751811\pi$
$331$		−	25.5826i		−	1.40615i	0.711118	−	0.703073i	$-0.248189\pi$
$332$		0			0
$333$	−0.834849	+	0.956439i	−0.0457494	+	0.0524125i
$334$		0			0
$335$		−	5.04356i		−	0.275559i
$336$		0			0
$337$	33.1216			1.80425			0.902124	−	0.431477i	$-0.142007\pi$
$337$	33.1216			1.80425			0.902124	+	0.431477i	$0.142007\pi$
$338$		0			0
$339$		−	18.6606i		−	1.01350i
$340$		0			0
$341$		−	6.62614i		−	0.358825i
$342$		0			0
$343$	−20.0000			−1.07990
$344$		0			0
$345$	5.37386			0.289319
$346$		0			0
$347$			7.58258i			0.407054i	0.979069	+	0.203527i	$0.0652405\pi$
$347$			7.58258i			0.407054i	−0.979069	+	0.203527i	$0.934760\pi$
$348$		0			0
$349$	−10.0000			−0.535288			−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000			−0.535288			−0.267644	+	0.963518i	$0.586245\pi$
$350$		0			0
$351$		−	18.9564i		−	1.01182i
$352$		0			0
$353$		−	1.58258i		−	0.0842320i	−0.999113	−	0.0421160i	$-0.986590\pi$
$353$		−	1.58258i		−	0.0842320i	0.999113	−	0.0421160i	$-0.0134099\pi$
$354$		0			0
$355$		−	7.25227i		−	0.384911i
$356$		0			0
$357$			27.1652i			1.43773i
$358$		0			0
$359$	8.83485			0.466285			0.233143	−	0.972443i	$-0.425099\pi$
$359$	8.83485			0.466285			0.233143	+	0.972443i	$0.425099\pi$
$360$		0			0
$361$	16.4955			0.868182
$362$		0			0
$363$	−18.5826			−0.975332
$364$		0			0
$365$			3.46099i			0.181156i
$366$		0			0
$367$	5.25227			0.274166			0.137083	−	0.990560i	$-0.456227\pi$
$367$	5.25227			0.274166			0.137083	+	0.990560i	$0.456227\pi$
$368$		0			0
$369$	2.04356			0.106384
$370$		0			0
$371$	3.16515			0.164326
$372$		0			0
$373$	12.7477			0.660052			0.330026	−	0.943972i	$-0.392942\pi$
$373$	12.7477			0.660052			0.330026	+	0.943972i	$0.392942\pi$
$374$		0			0
$375$		−	13.2867i		−	0.686124i
$376$		0			0
$377$	14.3739			0.740292
$378$		0			0
$379$	20.1216			1.03358			0.516788	−	0.856113i	$-0.327127\pi$
$379$	20.1216			1.03358			0.516788	+	0.856113i	$0.327127\pi$
$380$		0			0
$381$	−14.3303			−0.734164
$382$		0			0
$383$			18.3303i			0.936635i	0.883560	+	0.468317i	$0.155140\pi$
$383$			18.3303i			0.936635i	−0.883560	+	0.468317i	$0.844860\pi$
$384$		0			0
$385$			1.25227i			0.0638217i
$386$		0			0
$387$		−	1.25227i		−	0.0636566i
$388$		0			0
$389$		−	32.8693i		−	1.66654i	−0.552866	−	0.833270i	$-0.686466\pi$
$389$		−	32.8693i		−	1.66654i	0.552866	−	0.833270i	$-0.313534\pi$
$390$		0			0
$391$	−28.7477			−1.45384
$392$		0			0
$393$			30.0000i			1.51330i
$394$		0			0
$395$	−6.49545			−0.326822
$396$		0			0
$397$	−14.7477			−0.740167			−0.370084	−	0.928998i	$-0.620671\pi$
$397$	−14.7477			−0.740167			−0.370084	+	0.928998i	$0.620671\pi$
$398$		0			0
$399$			5.66970i			0.283840i
$400$		0			0
$401$		−	16.7477i		−	0.836342i	−0.908368	−	0.418171i	$-0.862671\pi$
$401$		−	16.7477i		−	0.836342i	0.908368	−	0.418171i	$-0.137329\pi$
$402$		0			0
$403$	−31.7477			−1.58147
$404$		0			0
$405$			7.58258i			0.376781i
$406$		0			0
$407$	3.16515	−	3.62614i	0.156891	−	0.179741i
$408$		0			0
$409$			27.1652i			1.34323i	0.740900	+	0.671615i	$0.234399\pi$
$409$			27.1652i			1.34323i	−0.740900	+	0.671615i	$0.765601\pi$
$410$		0			0
$411$	−6.49545			−0.320397
$412$		0			0
$413$			3.16515i			0.155747i
$414$		0			0
$415$		−	12.0000i		−	0.589057i
$416$		0			0
$417$	−23.9564			−1.17315
$418$		0			0
$419$	2.20871			0.107903			0.0539513	−	0.998544i	$-0.482818\pi$
$419$	2.20871			0.107903			0.0539513	+	0.998544i	$0.482818\pi$
$420$		0			0
$421$			18.9564i			0.923880i	0.886911	+	0.461940i	$0.152847\pi$
$421$			18.9564i			0.923880i	−0.886911	+	0.461940i	$0.847153\pi$
$422$		0			0
$423$	1.58258			0.0769475
$424$		0			0
$425$			33.1652i			1.60875i
$426$		0			0
$427$		−	25.5826i		−	1.23803i
$428$		0			0
$429$		−	5.37386i		−	0.259453i
$430$		0			0
$431$			8.83485i			0.425560i	0.977100	+	0.212780i	$0.0682517\pi$
$431$			8.83485i			0.425560i	−0.977100	+	0.212780i	$0.931748\pi$
$432$		0			0
$433$	10.6261			0.510660			0.255330	−	0.966854i	$-0.417816\pi$
$433$	10.6261			0.510660			0.255330	+	0.966854i	$0.417816\pi$
$434$		0			0
$435$	−5.37386			−0.257657
$436$		0			0
$437$	−6.00000			−0.287019
$438$		0			0
$439$		−	12.6261i		−	0.602613i	−0.953527	−	0.301306i	$-0.902577\pi$
$439$		−	12.6261i		−	0.602613i	0.953527	−	0.301306i	$-0.0974227\pi$
$440$		0			0
$441$	−0.626136			−0.0298160
$442$		0			0
$443$	27.9564			1.32825			0.664125	−	0.747621i	$-0.268804\pi$
$443$	27.9564			1.32825			0.664125	+	0.747621i	$0.268804\pi$
$444$		0			0
$445$	4.74773			0.225064
$446$		0			0
$447$	7.91288			0.374266
$448$		0			0
$449$		−	2.83485i		−	0.133785i	−0.997760	−	0.0668924i	$-0.978692\pi$
$449$		−	2.83485i		−	0.133785i	0.997760	−	0.0668924i	$-0.0213084\pi$
$450$		0			0
$451$	−7.74773			−0.364826
$452$		0			0
$453$	26.4174			1.24120
$454$		0			0
$455$	6.00000			0.281284
$456$		0			0
$457$		−	21.4955i		−	1.00551i	−0.864428	−	0.502757i	$-0.832319\pi$
$457$		−	21.4955i		−	1.00551i	0.864428	−	0.502757i	$-0.167681\pi$
$458$		0			0
$459$		−	37.9129i		−	1.76962i
$460$		0			0
$461$		−	12.3303i		−	0.574279i	−0.957889	−	0.287140i	$-0.907296\pi$
$461$		−	12.3303i		−	0.574279i	0.957889	−	0.287140i	$-0.0927044\pi$
$462$		0			0
$463$		−	29.7042i		−	1.38047i	−0.723585	−	0.690235i	$-0.757507\pi$
$463$		−	29.7042i		−	1.38047i	0.723585	−	0.690235i	$-0.242493\pi$
$464$		0			0
$465$	11.8693			0.550426
$466$		0			0
$467$			16.4174i			0.759708i	0.925046	+	0.379854i	$0.124026\pi$
$467$			16.4174i			0.759708i	−0.925046	+	0.379854i	$0.875974\pi$
$468$		0			0
$469$	12.7477			0.588635
$470$		0			0
$471$	3.58258			0.165076
$472$		0			0
$473$			4.74773i			0.218301i
$474$		0			0
$475$			6.92197i			0.317602i
$476$		0			0
$477$	0.330303			0.0151235
$478$		0			0
$479$		−	3.79129i		−	0.173228i	−0.996242	−	0.0866142i	$-0.972395\pi$
$479$		−	3.79129i		−	0.173228i	0.996242	−	0.0866142i	$-0.0276047\pi$
$480$		0			0
$481$	−17.3739	−	15.1652i	−0.792180	−	0.691471i
$482$		0			0
$483$			13.5826i			0.618029i
$484$		0			0
$485$	−10.7477			−0.488029
$486$		0			0
$487$		−	36.6606i		−	1.66125i	−0.556832	−	0.830625i	$-0.687983\pi$
$487$		−	36.6606i		−	1.66125i	0.556832	−	0.830625i	$-0.312017\pi$
$488$		0			0
$489$			35.0780i			1.58628i
$490$		0			0
$491$	5.37386			0.242519			0.121260	−	0.992621i	$-0.461307\pi$
$491$	5.37386			0.242519			0.121260	+	0.992621i	$0.461307\pi$
$492$		0			0
$493$	28.7477			1.29473
$494$		0			0
$495$			0.130682i			0.00587373i
$496$		0			0
$497$	18.3303			0.822226
$498$		0			0
$499$		−	10.7477i		−	0.481134i	−0.970632	−	0.240567i	$-0.922667\pi$
$499$		−	10.7477i		−	0.481134i	0.970632	−	0.240567i	$-0.0773334\pi$
$500$		0			0
$501$			39.3303i			1.75715i
$502$		0			0
$503$		−	15.6261i		−	0.696735i	−0.937358	−	0.348367i	$-0.886736\pi$
$503$		−	15.6261i		−	0.696735i	0.937358	−	0.348367i	$-0.113264\pi$
$504$		0			0
$505$			6.00000i			0.266996i
$506$		0			0
$507$	−2.46099			−0.109296
$508$		0			0
$509$	−4.41742			−0.195799			−0.0978994	−	0.995196i	$-0.531212\pi$
$509$	−4.41742			−0.195799			−0.0978994	+	0.995196i	$0.531212\pi$
$510$		0			0
$511$	−8.74773			−0.386977
$512$		0			0
$513$		−	7.91288i		−	0.349362i
$514$		0			0
$515$	−5.37386			−0.236801
$516$		0			0
$517$	−6.00000			−0.263880
$518$		0			0
$519$	−27.1652			−1.19242
$520$		0			0
$521$	−9.16515			−0.401533			−0.200766	−	0.979639i	$-0.564343\pi$
$521$	−9.16515			−0.401533			−0.200766	+	0.979639i	$0.564343\pi$
$522$		0			0
$523$			15.1652i			0.663126i	0.943433	+	0.331563i	$0.107576\pi$
$523$			15.1652i			0.663126i	−0.943433	+	0.331563i	$0.892424\pi$
$524$		0			0
$525$	15.6697			0.683882
$526$		0			0
$527$	−63.4955			−2.76591
$528$		0			0
$529$	8.62614			0.375049
$530$		0			0
$531$			0.330303i			0.0143339i
$532$		0			0
$533$			37.1216i			1.60791i
$534$		0			0
$535$			4.25227i			0.183842i
$536$		0			0
$537$			2.83485i			0.122333i
$538$		0			0
$539$	2.37386			0.102250
$540$		0			0
$541$			12.7913i			0.549940i	0.961453	+	0.274970i	$0.0886680\pi$
$541$			12.7913i			0.549940i	−0.961453	+	0.274970i	$0.911332\pi$
$542$		0			0
$543$	−15.6697			−0.672451
$544$		0			0
$545$	−4.74773			−0.203370
$546$		0			0
$547$		−	25.9129i		−	1.10795i	−0.832532	−	0.553977i	$-0.813109\pi$
$547$		−	25.9129i		−	1.10795i	0.832532	−	0.553977i	$-0.186891\pi$
$548$		0			0
$549$		−	2.66970i		−	0.113940i
$550$		0			0
$551$	6.00000			0.255609
$552$		0			0
$553$		−	16.4174i		−	0.698140i
$554$		0			0
$555$	6.49545	+	5.66970i	0.275717	+	0.240665i
$556$		0			0
$557$			8.70417i			0.368807i	0.982851	+	0.184404i	$0.0590354\pi$
$557$			8.70417i			0.368807i	−0.982851	+	0.184404i	$0.940965\pi$
$558$		0			0
$559$	22.7477			0.962126
$560$		0			0
$561$		−	10.7477i		−	0.453769i
$562$		0			0
$563$			10.7477i			0.452963i	0.974016	+	0.226481i	$0.0727222\pi$
$563$			10.7477i			0.452963i	−0.974016	+	0.226481i	$0.927278\pi$
$564$		0			0
$565$	−8.24318			−0.346793
$566$		0			0
$567$	−19.1652			−0.804861
$568$		0			0
$569$		−	45.1652i		−	1.89342i	−0.322085	−	0.946711i	$-0.604384\pi$
$569$		−	45.1652i		−	1.89342i	0.322085	−	0.946711i	$-0.395616\pi$
$570$		0			0
$571$	−14.6261			−0.612085			−0.306042	−	0.952018i	$-0.599005\pi$
$571$	−14.6261			−0.612085			−0.306042	+	0.952018i	$0.599005\pi$
$572$		0			0
$573$		−	15.0000i		−	0.626634i
$574$		0			0
$575$			16.5826i			0.691541i
$576$		0			0
$577$		−	13.5826i		−	0.565450i	−0.959201	−	0.282725i	$-0.908762\pi$
$577$		−	13.5826i		−	0.565450i	0.959201	−	0.282725i	$-0.0912384\pi$
$578$		0			0
$579$		−	32.8348i		−	1.36457i
$580$		0			0
$581$	30.3303			1.25831
$582$		0			0
$583$	−1.25227			−0.0518638
$584$		0			0
$585$	0.626136			0.0258876
$586$		0			0
$587$			19.9129i			0.821892i	0.911660	+	0.410946i	$0.134802\pi$
$587$			19.9129i			0.821892i	−0.911660	+	0.410946i	$0.865198\pi$
$588$		0			0
$589$	−13.2523			−0.546050
$590$		0			0
$591$	35.6697			1.46726
$592$		0			0
$593$	−18.7913			−0.771666			−0.385833	−	0.922569i	$-0.626086\pi$
$593$	−18.7913			−0.771666			−0.385833	+	0.922569i	$0.626086\pi$
$594$		0			0
$595$	12.0000			0.491952
$596$		0			0
$597$			27.1652i			1.11180i
$598$		0			0
$599$	−8.83485			−0.360982			−0.180491	−	0.983577i	$-0.557769\pi$
$599$	−8.83485			−0.360982			−0.180491	+	0.983577i	$0.557769\pi$
$600$		0			0
$601$	−33.1216			−1.35106			−0.675529	−	0.737333i	$-0.736085\pi$
$601$	−33.1216			−1.35106			−0.675529	+	0.737333i	$0.736085\pi$
$602$		0			0
$603$	1.33030			0.0541741
$604$		0			0
$605$			8.20871i			0.333732i
$606$		0			0
$607$		−	26.5390i		−	1.07719i	−0.842566	−	0.538593i	$-0.818956\pi$
$607$		−	26.5390i		−	1.07719i	0.842566	−	0.538593i	$-0.181044\pi$
$608$		0			0
$609$		−	13.5826i		−	0.550394i
$610$		0			0
$611$			28.7477i			1.16301i
$612$		0			0
$613$	49.4955			1.99910			0.999551	−	0.0299539i	$-0.00953603\pi$
$613$	49.4955			1.99910			0.999551	+	0.0299539i	$0.00953603\pi$
$614$		0			0
$615$		−	13.8784i		−	0.559631i
$616$		0			0
$617$	−23.0436			−0.927699			−0.463849	−	0.885914i	$-0.653532\pi$
$617$	−23.0436			−0.927699			−0.463849	+	0.885914i	$0.653532\pi$
$618$		0			0
$619$	−35.1216			−1.41166			−0.705828	−	0.708383i	$-0.749425\pi$
$619$	−35.1216			−1.41166			−0.705828	+	0.708383i	$0.749425\pi$
$620$		0			0
$621$		−	18.9564i		−	0.760696i
$622$		0			0
$623$			12.0000i			0.480770i
$624$		0			0
$625$	16.0000			0.640000
$626$		0			0
$627$		−	2.24318i		−	0.0895840i
$628$		0			0
$629$	−34.7477	−	30.3303i	−1.38548	−	1.20935i
$630$		0			0
$631$		−	18.9564i		−	0.754644i	−0.926082	−	0.377322i	$-0.876845\pi$
$631$		−	18.9564i		−	0.754644i	0.926082	−	0.377322i	$-0.123155\pi$
$632$		0			0
$633$	18.5826			0.738591
$634$		0			0
$635$			6.33030i			0.251210i
$636$		0			0
$637$		−	11.3739i		−	0.450649i
$638$		0			0
$639$	1.91288			0.0756723
$640$		0			0
$641$	−3.46099			−0.136701			−0.0683503	−	0.997661i	$-0.521774\pi$
$641$	−3.46099			−0.136701			−0.0683503	+	0.997661i	$0.521774\pi$
$642$		0			0
$643$		−	39.4955i		−	1.55755i	−0.627304	−	0.778774i	$-0.715842\pi$
$643$		−	39.4955i		−	1.55755i	0.627304	−	0.778774i	$-0.284158\pi$
$644$		0			0
$645$	−8.50455			−0.334866
$646$		0			0
$647$		−	17.7042i		−	0.696023i	−0.937490	−	0.348011i	$-0.886857\pi$
$647$		−	17.7042i		−	0.696023i	0.937490	−	0.348011i	$-0.113143\pi$
$648$		0			0
$649$		−	1.25227i		−	0.0491560i
$650$		0			0
$651$			30.0000i			1.17579i
$652$		0			0
$653$		−	0.626136i		−	0.0245026i	−0.999925	−	0.0122513i	$-0.996100\pi$
$653$		−	0.626136i		−	0.0245026i	0.999925	−	0.0122513i	$-0.00389981\pi$
$654$		0			0
$655$	13.2523			0.517809
$656$		0			0
$657$	−0.912878			−0.0356148
$658$		0			0
$659$	11.0436			0.430196			0.215098	−	0.976592i	$-0.430993\pi$
$659$	11.0436			0.430196			0.215098	+	0.976592i	$0.430993\pi$
$660$		0			0
$661$		−	32.2087i		−	1.25277i	−0.779512	−	0.626387i	$-0.784533\pi$
$661$		−	32.2087i		−	1.25277i	0.779512	−	0.626387i	$-0.215467\pi$
$662$		0			0
$663$	−51.4955			−1.99992
$664$		0			0
$665$	2.50455			0.0971221
$666$		0			0
$667$	14.3739			0.556558
$668$		0			0
$669$	−25.0780			−0.969573
$670$		0			0
$671$			10.1216i			0.390740i
$672$		0			0
$673$	−36.1216			−1.39238			−0.696192	−	0.717855i	$-0.745124\pi$
$673$	−36.1216			−1.39238			−0.696192	+	0.717855i	$0.745124\pi$
$674$		0			0
$675$	−21.8693			−0.841750
$676$		0			0
$677$	−9.16515			−0.352245			−0.176123	−	0.984368i	$-0.556356\pi$
$677$	−9.16515			−0.352245			−0.176123	+	0.984368i	$0.556356\pi$
$678$		0			0
$679$		−	27.1652i		−	1.04250i
$680$		0			0
$681$		−	46.4174i		−	1.77872i
$682$		0			0
$683$		−	31.5826i		−	1.20847i	−0.796805	−	0.604237i	$-0.793478\pi$
$683$		−	31.5826i		−	1.20847i	0.796805	−	0.604237i	$-0.206522\pi$
$684$		0			0
$685$			2.86932i			0.109631i
$686$		0			0
$687$	−12.0871			−0.461152
$688$		0			0
$689$			6.00000i			0.228582i
$690$		0			0
$691$	25.4955			0.969893			0.484946	−	0.874544i	$-0.338839\pi$
$691$	25.4955			0.969893			0.484946	+	0.874544i	$0.338839\pi$
$692$		0			0
$693$	−0.330303			−0.0125472
$694$		0			0
$695$			10.5826i			0.401420i
$696$		0			0
$697$			74.2432i			2.81216i
$698$		0			0
$699$	−28.8784			−1.09228
$700$		0			0
$701$		−	3.95644i		−	0.149433i	−0.997205	−	0.0747163i	$-0.976195\pi$
$701$		−	3.95644i		−	0.149433i	0.997205	−	0.0747163i	$-0.0238051\pi$
$702$		0			0
$703$	−7.25227	−	6.33030i	−0.273525	−	0.238752i
$704$		0			0
$705$		−	10.7477i		−	0.404783i
$706$		0			0
$707$	−15.1652			−0.570344
$708$		0			0
$709$		−	23.2087i		−	0.871621i	−0.900038	−	0.435811i	$-0.856462\pi$
$709$		−	23.2087i		−	0.871621i	0.900038	−	0.435811i	$-0.143538\pi$
$710$		0			0
$711$		−	1.71326i		−	0.0642522i
$712$		0			0
$713$	−31.7477			−1.18896
$714$		0			0
$715$	−2.37386			−0.0887775
$716$		0			0
$717$		−	3.66061i		−	0.136708i
$718$		0			0
$719$	21.1652			0.789327			0.394663	−	0.918826i	$-0.370861\pi$
$719$	21.1652			0.789327			0.394663	+	0.918826i	$0.370861\pi$
$720$		0			0
$721$		−	13.5826i		−	0.505842i
$722$		0			0
$723$			7.91288i			0.294283i
$724$		0			0
$725$		−	16.5826i		−	0.615861i
$726$		0			0
$727$			13.1216i			0.486653i	0.969944	+	0.243326i	$0.0782386\pi$
$727$			13.1216i			0.486653i	−0.969944	+	0.243326i	$0.921761\pi$
$728$		0			0
$729$	24.8693			0.921086
$730$		0			0
$731$	45.4955			1.68271
$732$		0			0
$733$	−47.4955			−1.75428			−0.877142	−	0.480231i	$-0.840553\pi$
$733$	−47.4955			−1.75428			−0.877142	+	0.480231i	$0.840553\pi$
$734$		0			0
$735$			4.25227i			0.156847i
$736$		0			0
$737$	−5.04356			−0.185782
$738$		0			0
$739$	−35.1216			−1.29197			−0.645984	−	0.763351i	$-0.723553\pi$
$739$	−35.1216			−1.29197			−0.645984	+	0.763351i	$0.723553\pi$
$740$		0			0
$741$	−10.7477			−0.394828
$742$		0			0
$743$	−15.1652			−0.556355			−0.278178	−	0.960530i	$-0.589730\pi$
$743$	−15.1652			−0.556355			−0.278178	+	0.960530i	$0.589730\pi$
$744$		0			0
$745$		−	3.49545i		−	0.128064i
$746$		0			0
$747$	3.16515			0.115807
$748$		0			0
$749$	−10.7477			−0.392713
$750$		0			0
$751$	41.4955			1.51419			0.757095	−	0.653304i	$-0.226618\pi$
$751$	41.4955			1.51419			0.757095	+	0.653304i	$0.226618\pi$
$752$		0			0
$753$			7.91288i			0.288361i
$754$		0			0
$755$		−	11.6697i		−	0.424704i
$756$		0			0
$757$			35.2087i			1.27968i	0.768507	+	0.639841i	$0.221000\pi$
$757$			35.2087i			1.27968i	−0.768507	+	0.639841i	$0.779000\pi$
$758$		0			0
$759$		−	5.37386i		−	0.195059i
$760$		0			0
$761$	14.2087			0.515065			0.257533	−	0.966270i	$-0.417090\pi$
$761$	14.2087			0.515065			0.257533	+	0.966270i	$0.417090\pi$
$762$		0			0
$763$		−	12.0000i		−	0.434429i
$764$		0			0
$765$	1.25227			0.0452760
$766$		0			0
$767$	−6.00000			−0.216647
$768$		0			0
$769$		−	11.6697i		−	0.420820i	−0.977613	−	0.210410i	$-0.932520\pi$
$769$		−	11.6697i		−	0.420820i	0.977613	−	0.210410i	$-0.0674799\pi$
$770$		0			0
$771$		−	8.50455i		−	0.306284i
$772$		0			0
$773$	−31.9129			−1.14783			−0.573913	−	0.818916i	$-0.694575\pi$
$773$	−31.9129			−1.14783			−0.573913	+	0.818916i	$0.694575\pi$
$774$		0			0
$775$			36.6261i			1.31565i
$776$		0			0
$777$	−14.3303	+	16.4174i	−0.514097	+	0.588972i
$778$		0			0
$779$			15.4955i			0.555182i
$780$		0			0
$781$	−7.25227			−0.259507
$782$		0			0
$783$			18.9564i			0.677448i
$784$		0			0
$785$		−	1.58258i		−	0.0564845i
$786$		0			0
$787$	8.00000			0.285169			0.142585	−	0.989783i	$-0.454459\pi$
$787$	8.00000			0.285169			0.142585	+	0.989783i	$0.454459\pi$
$788$		0			0
$789$	48.6606			1.73236
$790$		0			0
$791$		−	20.8348i		−	0.740802i
$792$		0			0
$793$	48.4955			1.72212
$794$		0			0
$795$		−	2.24318i		−	0.0795574i
$796$		0			0
$797$		−	35.3739i		−	1.25301i	−0.779419	−	0.626503i	$-0.784486\pi$
$797$		−	35.3739i		−	1.25301i	0.779419	−	0.626503i	$-0.215514\pi$
$798$		0			0
$799$			57.4955i			2.03404i
$800$		0			0
$801$			1.25227i			0.0442469i
$802$		0			0
$803$	3.46099			0.122136
$804$		0			0
$805$	6.00000			0.211472
$806$		0			0
$807$	−30.0000			−1.05605
$808$		0			0
$809$		−	53.0780i		−	1.86612i	−0.359715	−	0.933062i	$-0.617126\pi$
$809$		−	53.0780i		−	1.86612i	0.359715	−	0.933062i	$-0.382874\pi$
$810$		0			0
$811$	−19.8693			−0.697706			−0.348853	−	0.937177i	$-0.613429\pi$
$811$	−19.8693			−0.697706			−0.348853	+	0.937177i	$0.613429\pi$
$812$		0			0
$813$	−39.4083			−1.38211
$814$		0			0
$815$	15.4955			0.542782
$816$		0			0
$817$	9.49545			0.332204
$818$		0			0
$819$			1.58258i			0.0552997i
$820$		0			0
$821$	−3.16515			−0.110465			−0.0552323	−	0.998474i	$-0.517590\pi$
$821$	−3.16515			−0.110465			−0.0552323	+	0.998474i	$0.517590\pi$
$822$		0			0
$823$	16.0000			0.557725			0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000			0.557725			0.278862	+	0.960331i	$0.410043\pi$
$824$		0			0
$825$	−6.19962			−0.215843
$826$		0			0
$827$			33.1652i			1.15327i	0.817004	+	0.576633i	$0.195634\pi$
$827$			33.1652i			1.15327i	−0.817004	+	0.576633i	$0.804366\pi$
$828$		0			0
$829$		−	17.0436i		−	0.591947i	−0.955196	−	0.295974i	$-0.904356\pi$
$829$		−	17.0436i		−	0.591947i	0.955196	−	0.295974i	$-0.0956441\pi$
$830$		0			0
$831$			17.2432i			0.598159i
$832$		0			0
$833$		−	22.7477i		−	0.788162i
$834$		0			0
$835$	17.3739			0.601247
$836$		0			0
$837$		−	41.8693i		−	1.44722i
$838$		0			0
$839$	−3.49545			−0.120676			−0.0603382	−	0.998178i	$-0.519218\pi$
$839$	−3.49545			−0.120676			−0.0603382	+	0.998178i	$0.519218\pi$
$840$		0			0
$841$	14.6261			0.504350
$842$		0			0
$843$		0			0
$844$		0			0
$845$			1.08712i			0.0373981i
$846$		0			0
$847$	−20.7477			−0.712900
$848$		0			0
$849$			42.9909i			1.47544i
$850$		0			0
$851$	−17.3739	−	15.1652i	−0.595568	−	0.519855i
$852$		0			0
$853$		−	44.7042i		−	1.53064i	−0.643649	−	0.765321i	$-0.722580\pi$
$853$		−	44.7042i		−	1.53064i	0.643649	−	0.765321i	$-0.277420\pi$
$854$		0			0
$855$	0.261365			0.00893848
$856$		0			0
$857$			33.1652i			1.13290i	0.824096	+	0.566450i	$0.191684\pi$
$857$			33.1652i			1.13290i	−0.824096	+	0.566450i	$0.808316\pi$
$858$		0			0
$859$		−	18.6606i		−	0.636692i	−0.947975	−	0.318346i	$-0.896873\pi$
$859$		−	18.6606i		−	0.636692i	0.947975	−	0.318346i	$-0.103127\pi$
$860$		0			0
$861$	35.0780			1.19546
$862$		0			0
$863$	19.2523			0.655355			0.327677	−	0.944790i	$-0.393734\pi$
$863$	19.2523			0.655355			0.327677	+	0.944790i	$0.393734\pi$
$864$		0			0
$865$			12.0000i			0.408012i
$866$		0			0
$867$	−72.5390			−2.46355
$868$		0			0
$869$			6.49545i			0.220343i
$870$		0			0
$871$			24.1652i			0.818805i
$872$		0			0
$873$		−	2.83485i		−	0.0959451i
$874$		0			0
$875$		−	14.8348i		−	0.501509i
$876$		0			0
$877$	−11.2523			−0.379962			−0.189981	−	0.981788i	$-0.560843\pi$
$877$	−11.2523			−0.379962			−0.189981	+	0.981788i	$0.560843\pi$
$878$		0			0
$879$	10.7477			0.362512
$880$		0			0
$881$	−13.1216			−0.442078			−0.221039	−	0.975265i	$-0.570945\pi$
$881$	−13.1216			−0.442078			−0.221039	+	0.975265i	$0.570945\pi$
$882$		0			0
$883$			31.9129i			1.07395i	0.843597	+	0.536977i	$0.180434\pi$
$883$			31.9129i			1.07395i	−0.843597	+	0.536977i	$0.819566\pi$
$884$		0			0
$885$	2.24318			0.0754037
$886$		0			0
$887$	57.8258			1.94160			0.970799	−	0.239893i	$-0.0771122\pi$
$887$	57.8258			1.94160			0.970799	+	0.239893i	$0.0771122\pi$
$888$		0			0
$889$	−16.0000			−0.536623
$890$		0			0
$891$	7.58258			0.254026
$892$		0			0
$893$			12.0000i			0.401565i
$894$		0			0
$895$	1.25227			0.0418589
$896$		0			0
$897$	−25.7477			−0.859692
$898$		0			0
$899$	31.7477			1.05885
$900$		0			0
$901$			12.0000i			0.399778i
$902$		0			0
$903$		−	21.4955i		−	0.715324i
$904$		0			0
$905$			6.92197i			0.230094i
$906$		0			0
$907$		−	30.3303i		−	1.00710i	−0.863966	−	0.503551i	$-0.832027\pi$
$907$		−	30.3303i		−	1.00710i	0.863966	−	0.503551i	$-0.167973\pi$
$908$		0			0
$909$	−1.58258			−0.0524907
$910$		0			0
$911$		−	17.6697i		−	0.585423i	−0.956201	−	0.292712i	$-0.905442\pi$
$911$		−	17.6697i		−	0.585423i	0.956201	−	0.292712i	$-0.0945576\pi$
$912$		0			0
$913$	−12.0000			−0.397142
$914$		0			0
$915$	−18.1307			−0.599382
$916$		0			0
$917$			33.4955i			1.10612i
$918$		0			0
$919$		−	36.0000i		−	1.18753i	−0.804638	−	0.593765i	$-0.797641\pi$
$919$		−	36.0000i		−	1.18753i	0.804638	−	0.593765i	$-0.202359\pi$
$920$		0			0
$921$	2.46099			0.0810922
$922$		0			0
$923$			34.7477i			1.14374i
$924$		0			0
$925$	−17.4955	+	20.0436i	−0.575247	+	0.659028i
$926$		0			0
$927$		−	1.41742i		−	0.0465543i
$928$		0			0
$929$	8.37386			0.274738			0.137369	−	0.990520i	$-0.456135\pi$
$929$	8.37386			0.274738			0.137369	+	0.990520i	$0.456135\pi$
$930$		0			0
$931$		−	4.74773i		−	0.155600i
$932$		0			0
$933$			41.8693i			1.37074i
$934$		0			0
$935$	−4.74773			−0.155267
$936$		0			0
$937$	34.8693			1.13913			0.569565	−	0.821946i	$-0.307111\pi$
$937$	34.8693			1.13913			0.569565	+	0.821946i	$0.307111\pi$
$938$		0			0
$939$		−	26.5735i		−	0.867193i
$940$		0			0
$941$	21.4955			0.700732			0.350366	−	0.936613i	$-0.386057\pi$
$941$	21.4955			0.700732			0.350366	+	0.936613i	$0.386057\pi$
$942$		0			0
$943$			37.1216i			1.20885i
$944$		0			0
$945$			7.91288i			0.257406i
$946$		0			0
$947$		−	1.25227i		−	0.0406934i	−0.999793	−	0.0203467i	$-0.993523\pi$
$947$		−	1.25227i		−	0.0406934i	0.999793	−	0.0203467i	$-0.00647700\pi$
$948$		0			0
$949$		−	16.5826i		−	0.538293i
$950$		0			0
$951$	51.4955			1.66985
$952$		0			0
$953$	31.4519			1.01883			0.509413	−	0.860522i	$-0.329862\pi$
$953$	31.4519			1.01883			0.509413	+	0.860522i	$0.329862\pi$
$954$		0			0
$955$	−6.62614			−0.214417
$956$		0			0
$957$			5.37386i			0.173712i
$958$		0			0
$959$	−7.25227			−0.234188
$960$		0			0
$961$	−39.1216			−1.26199
$962$		0			0
$963$	−1.12159			−0.0361428
$964$		0			0
$965$	−14.5045			−0.466918
$966$		0			0
$967$			21.9564i			0.706071i	0.935610	+	0.353036i	$0.114851\pi$
$967$			21.9564i			0.706071i	−0.935610	+	0.353036i	$0.885149\pi$
$968$		0			0
$969$	−21.4955			−0.690533
$970$		0			0
$971$	50.3739			1.61657			0.808287	−	0.588789i	$-0.200395\pi$
$971$	50.3739			1.61657			0.808287	+	0.588789i	$0.200395\pi$
$972$		0			0
$973$	−26.7477			−0.857493
$974$		0			0
$975$			29.7042i			0.951295i
$976$		0			0
$977$			41.0780i			1.31420i	0.753802	+	0.657101i	$0.228218\pi$
$977$			41.0780i			1.31420i	−0.753802	+	0.657101i	$0.771782\pi$
$978$		0			0
$979$		−	4.74773i		−	0.151738i
$980$		0			0
$981$		−	1.25227i		−	0.0399820i
$982$		0			0
$983$	18.3303			0.584646			0.292323	−	0.956320i	$-0.405572\pi$
$983$	18.3303			0.584646			0.292323	+	0.956320i	$0.405572\pi$
$984$		0			0
$985$		−	15.7568i		−	0.502054i
$986$		0			0
$987$	27.1652			0.864676
$988$		0			0
$989$	22.7477			0.723336
$990$		0			0
$991$		−	18.9564i		−	0.602171i	−0.953597	−	0.301086i	$-0.902651\pi$
$991$		−	18.9564i		−	0.602171i	0.953597	−	0.301086i	$-0.0973490\pi$
$992$		0			0
$993$		−	45.8258i		−	1.45424i
$994$		0			0
$995$	12.0000			0.380426
$996$		0			0
$997$			18.0000i			0.570066i	0.958518	+	0.285033i	$0.0920045\pi$
$997$			18.0000i			0.570066i	−0.958518	+	0.285033i	$0.907995\pi$
$998$		0			0
$999$	20.0000	−	22.9129i	0.632772	−	0.724931i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2368.2.g.h.961.3		4
4.3	odd	2		2368.2.g.j.961.1		4
8.3	odd	2		74.2.b.a.73.2	✓	4
8.5	even	2		592.2.g.c.369.2		4
24.5	odd	2		5328.2.h.m.2737.2		4
24.11	even	2		666.2.c.b.73.3		4
37.36	even	2	inner	2368.2.g.h.961.4		4
40.3	even	4		1850.2.c.g.1849.3		4
40.19	odd	2		1850.2.d.e.1701.3		4
40.27	even	4		1850.2.c.h.1849.2		4
148.147	odd	2		2368.2.g.j.961.2		4
296.43	even	4		2738.2.a.h.1.1		2
296.147	odd	2		74.2.b.a.73.4	yes	4
296.179	even	4		2738.2.a.k.1.1		2
296.221	even	2		592.2.g.c.369.1		4
888.221	odd	2		5328.2.h.m.2737.3		4
888.443	even	2		666.2.c.b.73.2		4
1480.147	even	4		1850.2.c.g.1849.2		4
1480.443	even	4		1850.2.c.h.1849.3		4
1480.739	odd	2		1850.2.d.e.1701.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.2.b.a.73.2	✓	4	8.3	odd	2
74.2.b.a.73.4	yes	4	296.147	odd	2
592.2.g.c.369.1		4	296.221	even	2
592.2.g.c.369.2		4	8.5	even	2
666.2.c.b.73.2		4	888.443	even	2
666.2.c.b.73.3		4	24.11	even	2
1850.2.c.g.1849.2		4	1480.147	even	4
1850.2.c.g.1849.3		4	40.3	even	4
1850.2.c.h.1849.2		4	40.27	even	4
1850.2.c.h.1849.3		4	1480.443	even	4
1850.2.d.e.1701.1		4	1480.739	odd	2
1850.2.d.e.1701.3		4	40.19	odd	2
2368.2.g.h.961.3		4	1.1	even	1	trivial
2368.2.g.h.961.4		4	37.36	even	2	inner
2368.2.g.j.961.1		4	4.3	odd	2
2368.2.g.j.961.2		4	148.147	odd	2
2738.2.a.h.1.1		2	296.43	even	4
2738.2.a.k.1.1		2	296.179	even	4
5328.2.h.m.2737.2		4	24.5	odd	2
5328.2.h.m.2737.3		4	888.221	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:	\( N \)	\(=\)	\( 2368 = 2^{6} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2368.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.79129 q^{3} -0.791288i q^{5} +2.00000 q^{7} +0.208712 q^{9} +O(q^{10})\)	\(q+1.79129 q^{3} -0.791288i q^{5} +2.00000 q^{7} +0.208712 q^{9} -0.791288 q^{11} +3.79129i q^{13} -1.41742i q^{15} +7.58258i q^{17} +1.58258i q^{19} +3.58258 q^{21} +3.79129i q^{23} +4.37386 q^{25} -5.00000 q^{27} -3.79129i q^{29} +8.37386i q^{31} -1.41742 q^{33} -1.58258i q^{35} +(-4.00000 + 4.58258i) q^{37} +6.79129i q^{39} +9.79129 q^{41} -6.00000i q^{43} -0.165151i q^{45} +7.58258 q^{47} -3.00000 q^{49} +13.5826i q^{51} +1.58258 q^{53} +0.626136i q^{55} +2.83485i q^{57} +1.58258i q^{59} -12.7913i q^{61} +0.417424 q^{63} +3.00000 q^{65} +6.37386 q^{67} +6.79129i q^{69} +9.16515 q^{71} -4.37386 q^{73} +7.83485 q^{75} -1.58258 q^{77} -8.20871i q^{79} -9.58258 q^{81} +15.1652 q^{83} +6.00000 q^{85} -6.79129i q^{87} +6.00000i q^{89} +7.58258i q^{91} +15.0000i q^{93} +1.25227 q^{95} -13.5826i q^{97} -0.165151 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 8 q^{7} + 10 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 + 8 * q^7 + 10 * q^9	\( 4 q - 2 q^{3} + 8 q^{7} + 10 q^{9} + 6 q^{11} - 4 q^{21} - 10 q^{25} - 20 q^{27} - 24 q^{33} - 16 q^{37} + 30 q^{41} + 12 q^{47} - 12 q^{49} - 12 q^{53} + 20 q^{63} + 12 q^{65} - 2 q^{67} + 10 q^{73} + 68 q^{75} + 12 q^{77} - 20 q^{81} + 24 q^{83} + 24 q^{85} + 60 q^{95} + 36 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 + 8 * q^7 + 10 * q^9 + 6 * q^11 - 4 * q^21 - 10 * q^25 - 20 * q^27 - 24 * q^33 - 16 * q^37 + 30 * q^41 + 12 * q^47 - 12 * q^49 - 12 * q^53 + 20 * q^63 + 12 * q^65 - 2 * q^67 + 10 * q^73 + 68 * q^75 + 12 * q^77 - 20 * q^81 + 24 * q^83 + 24 * q^85 + 60 * q^95 + 36 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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