LMFDB - Embedded newform 2352.2.k.j.881.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2352,2,Mod(881,2352)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2352, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2352.881"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2352.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,8,0,0,0,0, 0,0,0,0,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(37)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$18.7808145554$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 1176)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.8
Character	$\chi$	$=$	2352.881
Dual form			2352.2.k.j.881.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.698115 + 1.58513i) q^{3} -0.489061 q^{5} +(-2.02527 - 2.21321i) q^{9} +1.52274i q^{11} +0.652068i q^{13} +(0.341421 - 0.775225i) q^{15} +7.55538 q^{17} +0.420989i q^{19} +5.90997i q^{23} -4.76082 q^{25} +(4.92209 - 1.66524i) q^{27} -7.16356i q^{29} +7.38015i q^{31} +(-2.41375 - 1.06305i) q^{33} -8.09219 q^{37} +(-1.03361 - 0.455218i) q^{39} +2.53185 q^{41} +5.56181 q^{43} +(0.990481 + 1.08239i) q^{45} +9.31441 q^{47} +(-5.27452 + 11.9763i) q^{51} +1.46232i q^{53} -0.744715i q^{55} +(-0.667322 - 0.293899i) q^{57} -7.35687 q^{59} +11.1496i q^{61} -0.318901i q^{65} -14.6240 q^{67} +(-9.36807 - 4.12584i) q^{69} +12.8513i q^{71} +9.24519i q^{73} +(3.32360 - 7.54652i) q^{75} -1.21440 q^{79} +(-0.796558 + 8.96468i) q^{81} -9.03219 q^{83} -3.69504 q^{85} +(11.3552 + 5.00099i) q^{87} -9.55993 q^{89} +(-11.6985 - 5.15220i) q^{93} -0.205889i q^{95} -4.95430i q^{97} +(3.37015 - 3.08397i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 16 q^{15} + 8 q^{25} + 16 q^{37} + 64 q^{39} - 16 q^{43} - 48 q^{51} + 48 q^{57} - 16 q^{67} + 80 q^{81} - 64 q^{85} - 32 q^{93} + 56 q^{99}+O(q^{100}) $ 24 * q + 16 * q^15 + 8 * q^25 + 16 * q^37 + 64 * q^39 - 16 * q^43 - 48 * q^51 + 48 * q^57 - 16 * q^67 + 80 * q^81 - 64 * q^85 - 32 * q^93 + 56 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$3$	−0.698115	+	1.58513i	−0.403057	+	0.915175i
$3$	−0.698115	+	1.58513i	−0.403057	+	0.915175i
$4$		0			0
$5$	−0.489061			−0.218715			−0.109357	−	0.994003i	$-0.534879\pi$
$5$	−0.489061			−0.218715			−0.109357	+	0.994003i	$0.534879\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−2.02527	−	2.21321i	−0.675090	−	0.737735i
$10$		0			0
$11$			1.52274i			0.459125i	0.973294	+	0.229562i	$0.0737295\pi$
$11$			1.52274i			0.459125i	−0.973294	+	0.229562i	$0.926271\pi$
$12$		0			0
$13$			0.652068i			0.180851i	0.995903	+	0.0904255i	$0.0288227\pi$
$13$			0.652068i			0.180851i	−0.995903	+	0.0904255i	$0.971177\pi$
$14$		0			0
$15$	0.341421	−	0.775225i	0.0881545	−	0.200162i
$16$		0			0
$17$	7.55538			1.83245			0.916224	−	0.400666i	$-0.131221\pi$
$17$	7.55538			1.83245			0.916224	+	0.400666i	$0.131221\pi$
$18$		0			0
$19$			0.420989i			0.0965814i	0.998833	+	0.0482907i	$0.0153774\pi$
$19$			0.420989i			0.0965814i	−0.998833	+	0.0482907i	$0.984623\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			5.90997i			1.23231i	0.787623	+	0.616157i	$0.211312\pi$
$23$			5.90997i			1.23231i	−0.787623	+	0.616157i	$0.788688\pi$
$24$		0			0
$25$	−4.76082			−0.952164
$26$		0			0
$27$	4.92209	−	1.66524i	0.947257	−	0.320476i
$28$		0			0
$29$		−	7.16356i		−	1.33024i	−0.746737	−	0.665120i	$-0.768381\pi$
$29$		−	7.16356i		−	1.33024i	0.746737	−	0.665120i	$-0.231619\pi$
$30$		0			0
$31$			7.38015i			1.32551i	0.748834	+	0.662757i	$0.230614\pi$
$31$			7.38015i			1.32551i	−0.748834	+	0.662757i	$0.769386\pi$
$32$		0			0
$33$	−2.41375	−	1.06305i	−0.420179	−	0.185053i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−8.09219			−1.33035			−0.665174	−	0.746689i	$-0.731642\pi$
$37$	−8.09219			−1.33035			−0.665174	+	0.746689i	$0.731642\pi$
$38$		0			0
$39$	−1.03361	−	0.455218i	−0.165510	−	0.0728933i
$40$		0			0
$41$	2.53185			0.395409			0.197704	−	0.980262i	$-0.436651\pi$
$41$	2.53185			0.395409			0.197704	+	0.980262i	$0.436651\pi$
$42$		0			0
$43$	5.56181			0.848168			0.424084	−	0.905623i	$-0.360596\pi$
$43$	5.56181			0.848168			0.424084	+	0.905623i	$0.360596\pi$
$44$		0			0
$45$	0.990481	+	1.08239i	0.147652	+	0.161354i
$46$		0			0
$47$	9.31441			1.35865			0.679323	−	0.733839i	$-0.262273\pi$
$47$	9.31441			1.35865			0.679323	+	0.733839i	$0.262273\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−5.27452	+	11.9763i	−0.738581	+	1.67701i
$52$		0			0
$53$			1.46232i			0.200864i	0.994944	+	0.100432i	$0.0320226\pi$
$53$			1.46232i			0.200864i	−0.994944	+	0.100432i	$0.967977\pi$
$54$		0			0
$55$		−	0.744715i		−	0.100417i
$56$		0			0
$57$	−0.667322	−	0.293899i	−0.0883889	−	0.0389278i
$58$		0			0
$59$	−7.35687			−0.957783			−0.478892	−	0.877874i	$-0.658961\pi$
$59$	−7.35687			−0.957783			−0.478892	+	0.877874i	$0.658961\pi$
$60$		0			0
$61$			11.1496i			1.42757i	0.700367	+	0.713783i	$0.253020\pi$
$61$			11.1496i			1.42757i	−0.700367	+	0.713783i	$0.746980\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	0.318901i		−	0.0395548i
$66$		0			0
$67$	−14.6240			−1.78660			−0.893302	−	0.449457i	$-0.851617\pi$
$67$	−14.6240			−1.78660			−0.893302	+	0.449457i	$0.851617\pi$
$68$		0			0
$69$	−9.36807	−	4.12584i	−1.12778	−	0.496693i
$70$		0			0
$71$			12.8513i			1.52516i	0.646891	+	0.762582i	$0.276069\pi$
$71$			12.8513i			1.52516i	−0.646891	+	0.762582i	$0.723931\pi$
$72$		0			0
$73$			9.24519i			1.08207i	0.841001	+	0.541034i	$0.181967\pi$
$73$			9.24519i			1.08207i	−0.841001	+	0.541034i	$0.818033\pi$
$74$		0			0
$75$	3.32360	−	7.54652i	0.383776	−	0.871397i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−1.21440			−0.136631			−0.0683154	−	0.997664i	$-0.521762\pi$
$79$	−1.21440			−0.136631			−0.0683154	+	0.997664i	$0.521762\pi$
$80$		0			0
$81$	−0.796558	+	8.96468i	−0.0885064	+	0.996076i
$82$		0			0
$83$	−9.03219			−0.991412			−0.495706	−	0.868491i	$-0.665091\pi$
$83$	−9.03219			−0.991412			−0.495706	+	0.868491i	$0.665091\pi$
$84$		0			0
$85$	−3.69504			−0.400783
$86$		0			0
$87$	11.3552	+	5.00099i	1.21740	+	0.536162i
$88$		0			0
$89$	−9.55993			−1.01335			−0.506675	−	0.862137i	$-0.669126\pi$
$89$	−9.55993			−1.01335			−0.506675	+	0.862137i	$0.669126\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−11.6985	−	5.15220i	−1.21308	−	0.534258i
$94$		0			0
$95$		−	0.205889i		−	0.0211238i
$96$		0			0
$97$		−	4.95430i		−	0.503033i	−0.967853	−	0.251517i	$-0.919071\pi$
$97$		−	4.95430i		−	0.503033i	0.967853	−	0.251517i	$-0.0809293\pi$
$98$		0			0
$99$	3.37015	−	3.08397i	0.338712	−	0.309951i
$100$		0			0
$101$	5.51027			0.548292			0.274146	−	0.961688i	$-0.411605\pi$
$101$	5.51027			0.548292			0.274146	+	0.961688i	$0.411605\pi$
$102$		0			0
$103$			9.17989i			0.904522i	0.891886	+	0.452261i	$0.149382\pi$
$103$			9.17989i			0.904522i	−0.891886	+	0.452261i	$0.850618\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	4.78936i		−	0.463005i	−0.972834	−	0.231503i	$-0.925636\pi$
$107$		−	4.78936i		−	0.463005i	0.972834	−	0.231503i	$-0.0743642\pi$
$108$		0			0
$109$	−13.3763			−1.28122			−0.640608	−	0.767868i	$-0.721318\pi$
$109$	−13.3763			−1.28122			−0.640608	+	0.767868i	$0.721318\pi$
$110$		0			0
$111$	5.64928	−	12.8272i	0.536206	−	1.21750i
$112$		0			0
$113$		−	12.5764i		−	1.18309i	−0.806271	−	0.591546i	$-0.798518\pi$
$113$		−	12.5764i		−	1.18309i	0.806271	−	0.591546i	$-0.201482\pi$
$114$		0			0
$115$		−	2.89034i		−	0.269525i
$116$		0			0
$117$	1.44316	−	1.32061i	0.133420	−	0.122091i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	8.68125			0.789204
$122$		0			0
$123$	−1.76752	+	4.01331i	−0.159372	+	0.361868i
$124$		0			0
$125$	4.77363			0.426967
$126$		0			0
$127$	−0.264143			−0.0234389			−0.0117194	−	0.999931i	$-0.503731\pi$
$127$	−0.264143			−0.0234389			−0.0117194	+	0.999931i	$0.503731\pi$
$128$		0			0
$129$	−3.88278	+	8.81619i	−0.341860	+	0.776222i
$130$		0			0
$131$	−7.51917			−0.656953			−0.328476	−	0.944512i	$-0.606535\pi$
$131$	−7.51917			−0.656953			−0.328476	+	0.944512i	$0.606535\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−2.40720	+	0.814406i	−0.207179	+	0.0700929i
$136$		0			0
$137$		−	0.0198048i		−	0.00169204i	−1.00000		0.000846019i	$-0.999731\pi$
$137$		−	0.0198048i		−	0.00169204i	1.00000		0.000846019i	$-0.000269296\pi$
$138$		0			0
$139$		−	10.1198i		−	0.858349i	−0.903222	−	0.429174i	$-0.858805\pi$
$139$		−	10.1198i		−	0.858349i	0.903222	−	0.429174i	$-0.141195\pi$
$140$		0			0
$141$	−6.50253	+	14.7645i	−0.547612	+	1.24340i
$142$		0			0
$143$	−0.992933			−0.0830332
$144$		0			0
$145$			3.50342i			0.290943i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			14.0863i			1.15399i	0.816746	+	0.576997i	$0.195776\pi$
$149$			14.0863i			1.15399i	−0.816746	+	0.576997i	$0.804224\pi$
$150$		0			0
$151$	−12.5804			−1.02377			−0.511887	−	0.859053i	$-0.671053\pi$
$151$	−12.5804			−1.02377			−0.511887	+	0.859053i	$0.671053\pi$
$152$		0			0
$153$	−15.3017	−	16.7216i	−1.23707	−	1.35186i
$154$		0			0
$155$		−	3.60934i		−	0.289910i
$156$		0			0
$157$		−	8.67035i		−	0.691969i	−0.938240	−	0.345984i	$-0.887545\pi$
$157$		−	8.67035i		−	0.691969i	0.938240	−	0.345984i	$-0.112455\pi$
$158$		0			0
$159$	−2.31796	−	1.02086i	−0.183826	−	0.0809598i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	3.35163			0.262520			0.131260	−	0.991348i	$-0.458098\pi$
$163$	3.35163			0.262520			0.131260	+	0.991348i	$0.458098\pi$
$164$		0			0
$165$	1.18047	+	0.519897i	0.0918994	+	0.0404739i
$166$		0			0
$167$	−23.4014			−1.81085			−0.905426	−	0.424505i	$-0.860448\pi$
$167$	−23.4014			−1.81085			−0.905426	+	0.424505i	$0.860448\pi$
$168$		0			0
$169$	12.5748			0.967293
$170$		0			0
$171$	0.931735	−	0.852616i	0.0712515	−	0.0652012i
$172$		0			0
$173$	14.2413			1.08274			0.541371	−	0.840784i	$-0.317905\pi$
$173$	14.2413			1.08274			0.541371	+	0.840784i	$0.317905\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	5.13595	−	11.6616i	0.386041	−	0.876539i
$178$		0			0
$179$			20.3264i			1.51926i	0.650354	+	0.759632i	$0.274621\pi$
$179$			20.3264i			1.51926i	−0.650354	+	0.759632i	$0.725379\pi$
$180$		0			0
$181$			0.798480i			0.0593506i	0.999560	+	0.0296753i	$0.00944732\pi$
$181$			0.798480i			0.0593506i	−0.999560	+	0.0296753i	$0.990553\pi$
$182$		0			0
$183$	−17.6736	−	7.78374i	−1.30647	−	0.575390i
$184$		0			0
$185$	3.95757			0.290966
$186$		0			0
$187$			11.5049i			0.841322i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	21.0641i		−	1.52414i	−0.647493	−	0.762071i	$-0.724182\pi$
$191$		−	21.0641i		−	1.52414i	0.647493	−	0.762071i	$-0.275818\pi$
$192$		0			0
$193$	−17.9015			−1.28858			−0.644289	−	0.764782i	$-0.722847\pi$
$193$	−17.9015			−1.28858			−0.644289	+	0.764782i	$0.722847\pi$
$194$		0			0
$195$	0.505499	+	0.222629i	0.0361995	+	0.0159428i
$196$		0			0
$197$			19.1354i			1.36334i	0.731658	+	0.681672i	$0.238747\pi$
$197$			19.1354i			1.36334i	−0.731658	+	0.681672i	$0.761253\pi$
$198$		0			0
$199$			12.5562i			0.890086i	0.895509	+	0.445043i	$0.146812\pi$
$199$			12.5562i			0.890086i	−0.895509	+	0.445043i	$0.853188\pi$
$200$		0			0
$201$	10.2092	−	23.1809i	0.720103	−	1.63505i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−1.23823			−0.0864817
$206$		0			0
$207$	13.0800	−	11.9693i	0.909122	−	0.831924i
$208$		0			0
$209$	−0.641058			−0.0443429
$210$		0			0
$211$	−8.96713			−0.617322			−0.308661	−	0.951172i	$-0.599881\pi$
$211$	−8.96713			−0.617322			−0.308661	+	0.951172i	$0.599881\pi$
$212$		0			0
$213$	−20.3709	−	8.97166i	−1.39579	−	0.614728i
$214$		0			0
$215$	−2.72006			−0.185507
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−14.6548	−	6.45421i	−0.990281	−	0.436135i
$220$		0			0
$221$			4.92662i			0.331400i
$222$		0			0
$223$			10.9815i			0.735376i	0.929949	+	0.367688i	$0.119851\pi$
$223$			10.9815i			0.735376i	−0.929949	+	0.367688i	$0.880149\pi$
$224$		0			0
$225$	9.64195	+	10.5367i	0.642797	+	0.702445i
$226$		0			0
$227$	−5.45063			−0.361771			−0.180886	−	0.983504i	$-0.557896\pi$
$227$	−5.45063			−0.361771			−0.180886	+	0.983504i	$0.557896\pi$
$228$		0			0
$229$			12.8004i			0.845874i	0.906159	+	0.422937i	$0.139001\pi$
$229$			12.8004i			0.845874i	−0.906159	+	0.422937i	$0.860999\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			16.1543i			1.05830i	0.848528	+	0.529150i	$0.177489\pi$
$233$			16.1543i			1.05830i	−0.848528	+	0.529150i	$0.822511\pi$
$234$		0			0
$235$	−4.55531			−0.297156
$236$		0			0
$237$	0.847792	−	1.92498i	0.0550700	−	0.125041i
$238$		0			0
$239$		−	1.09590i		−	0.0708882i	−0.999372	−	0.0354441i	$-0.988715\pi$
$239$		−	1.09590i		−	0.0708882i	0.999372	−	0.0354441i	$-0.0112846\pi$
$240$		0			0
$241$			8.56015i			0.551408i	0.961243	+	0.275704i	$0.0889109\pi$
$241$			8.56015i			0.551408i	−0.961243	+	0.275704i	$0.911089\pi$
$242$		0			0
$243$	−13.6541	−	7.52103i	−0.875910	−	0.482474i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−0.274513			−0.0174669
$248$		0			0
$249$	6.30551	−	14.3172i	0.399595	−	0.907315i
$250$		0			0
$251$	−3.88353			−0.245127			−0.122563	−	0.992461i	$-0.539111\pi$
$251$	−3.88353			−0.245127			−0.122563	+	0.992461i	$0.539111\pi$
$252$		0			0
$253$	−8.99938			−0.565786
$254$		0			0
$255$	2.57956	−	5.85712i	0.161538	−	0.366787i
$256$		0			0
$257$	17.8118			1.11107			0.555536	−	0.831492i	$-0.312513\pi$
$257$	17.8118			1.11107			0.555536	+	0.831492i	$0.312513\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−15.8544	+	14.5081i	−0.981364	+	0.898031i
$262$		0			0
$263$			5.47585i			0.337655i	0.985646	+	0.168828i	$0.0539982\pi$
$263$			5.47585i			0.337655i	−0.985646	+	0.168828i	$0.946002\pi$
$264$		0			0
$265$		−	0.715161i		−	0.0439320i
$266$		0			0
$267$	6.67393	−	15.1537i	0.408438	−	0.927393i
$268$		0			0
$269$	−12.1544			−0.741068			−0.370534	−	0.928819i	$-0.620825\pi$
$269$	−12.1544			−0.741068			−0.370534	+	0.928819i	$0.620825\pi$
$270$		0			0
$271$			24.1732i			1.46842i	0.678925	+	0.734208i	$0.262446\pi$
$271$			24.1732i			1.46842i	−0.678925	+	0.734208i	$0.737554\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	7.24951i		−	0.437162i
$276$		0			0
$277$	2.93440			0.176311			0.0881556	−	0.996107i	$-0.471903\pi$
$277$	2.93440			0.176311			0.0881556	+	0.996107i	$0.471903\pi$
$278$		0			0
$279$	16.3338	−	14.9468i	0.977879	−	0.894842i
$280$		0			0
$281$			1.97450i			0.117789i	0.998264	+	0.0588944i	$0.0187575\pi$
$281$			1.97450i			0.117789i	−0.998264	+	0.0588944i	$0.981242\pi$
$282$		0			0
$283$		−	24.1545i		−	1.43584i	−0.696127	−	0.717919i	$-0.745095\pi$
$283$		−	24.1545i		−	1.43584i	0.696127	−	0.717919i	$-0.254905\pi$
$284$		0			0
$285$	0.326361	+	0.143734i	0.0193320	+	0.00851409i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	40.0837			2.35787
$290$		0			0
$291$	7.85321	+	3.45867i	0.460363	+	0.202751i
$292$		0			0
$293$	22.8829			1.33683			0.668417	−	0.743787i	$-0.266972\pi$
$293$	22.8829			1.33683			0.668417	+	0.743787i	$0.266972\pi$
$294$		0			0
$295$	3.59796			0.209481
$296$		0			0
$297$	2.53574	+	7.49508i	0.147139	+	0.434909i
$298$		0			0
$299$	−3.85370			−0.222865
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−3.84680	+	8.73449i	−0.220993	+	0.501783i
$304$		0			0
$305$		−	5.45286i		−	0.312230i
$306$		0			0
$307$			9.74121i			0.555961i	0.960587	+	0.277980i	$0.0896650\pi$
$307$			9.74121i			0.555961i	−0.960587	+	0.277980i	$0.910335\pi$
$308$		0			0
$309$	−14.5513	−	6.40862i	−0.827796	−	0.364574i
$310$		0			0
$311$	22.0678			1.25135			0.625676	−	0.780083i	$-0.284823\pi$
$311$	22.0678			1.25135			0.625676	+	0.780083i	$0.284823\pi$
$312$		0			0
$313$		−	15.5962i		−	0.881551i	−0.897617	−	0.440776i	$-0.854703\pi$
$313$		−	15.5962i		−	0.881551i	0.897617	−	0.440776i	$-0.145297\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			17.6841i			0.993238i	0.867969	+	0.496619i	$0.165425\pi$
$317$			17.6841i			0.993238i	−0.867969	+	0.496619i	$0.834575\pi$
$318$		0			0
$319$	10.9083			0.610746
$320$		0			0
$321$	7.59176	+	3.34353i	0.423731	+	0.186617i
$322$		0			0
$323$			3.18073i			0.176980i
$324$		0			0
$325$		−	3.10438i		−	0.172200i
$326$		0			0
$327$	9.33819	−	21.2031i	0.516403	−	1.17254i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	12.6650			0.696133			0.348066	−	0.937470i	$-0.386838\pi$
$331$	12.6650			0.696133			0.348066	+	0.937470i	$0.386838\pi$
$332$		0			0
$333$	16.3889	+	17.9097i	0.898104	+	0.981444i
$334$		0			0
$335$	7.15202			0.390756
$336$		0			0
$337$	25.7617			1.40333			0.701663	−	0.712509i	$-0.252441\pi$
$337$	25.7617			1.40333			0.701663	+	0.712509i	$0.252441\pi$
$338$		0			0
$339$	19.9353	+	8.77981i	1.08274	+	0.476854i
$340$		0			0
$341$	−11.2381			−0.608577
$342$		0			0
$343$		0			0
$344$		0			0
$345$	4.58156	+	2.01779i	0.246663	+	0.108634i
$346$		0			0
$347$		−	33.0043i		−	1.77177i	−0.463909	−	0.885883i	$-0.653554\pi$
$347$		−	33.0043i		−	1.77177i	0.463909	−	0.885883i	$-0.346446\pi$
$348$		0			0
$349$			29.1507i			1.56040i	0.625528	+	0.780202i	$0.284884\pi$
$349$			29.1507i			1.56040i	−0.625528	+	0.780202i	$0.715116\pi$
$350$		0			0
$351$	1.08585	+	3.20954i	0.0579585	+	0.171312i
$352$		0			0
$353$	−4.58114			−0.243829			−0.121915	−	0.992541i	$-0.538903\pi$
$353$	−4.58114			−0.243829			−0.121915	+	0.992541i	$0.538903\pi$
$354$		0			0
$355$		−	6.28505i		−	0.333576i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	18.4776i		−	0.975213i	−0.873064	−	0.487606i	$-0.837870\pi$
$359$		−	18.4776i		−	0.975213i	0.873064	−	0.487606i	$-0.162130\pi$
$360$		0			0
$361$	18.8228			0.990672
$362$		0			0
$363$	−6.06051	+	13.7609i	−0.318094	+	0.722260i
$364$		0			0
$365$		−	4.52146i		−	0.236664i
$366$		0			0
$367$		−	11.2971i		−	0.589706i	−0.955543	−	0.294853i	$-0.904729\pi$
$367$		−	11.2971i		−	0.589706i	0.955543	−	0.294853i	$-0.0952707\pi$
$368$		0			0
$369$	−5.12768	−	5.60351i	−0.266937	−	0.291707i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−18.6299			−0.964622			−0.482311	−	0.876000i	$-0.660202\pi$
$373$	−18.6299			−0.964622			−0.482311	+	0.876000i	$0.660202\pi$
$374$		0			0
$375$	−3.33255	+	7.56683i	−0.172092	+	0.390749i
$376$		0			0
$377$	4.67113			0.240575
$378$		0			0
$379$	5.62097			0.288730			0.144365	−	0.989525i	$-0.453886\pi$
$379$	5.62097			0.288730			0.144365	+	0.989525i	$0.453886\pi$
$380$		0			0
$381$	0.184402	−	0.418701i	0.00944721	−	0.0214507i
$382$		0			0
$383$	−20.1071			−1.02743			−0.513713	−	0.857962i	$-0.671730\pi$
$383$	−20.1071			−1.02743			−0.513713	+	0.857962i	$0.671730\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−11.2642	−	12.3094i	−0.572590	−	0.625723i
$388$		0			0
$389$		−	23.3657i		−	1.18469i	−0.805685	−	0.592345i	$-0.798202\pi$
$389$		−	23.3657i		−	1.18469i	0.805685	−	0.592345i	$-0.201798\pi$
$390$		0			0
$391$			44.6521i			2.25815i
$392$		0			0
$393$	5.24924	−	11.9188i	0.264789	−	0.601226i
$394$		0			0
$395$	0.593916			0.0298832
$396$		0			0
$397$			32.8394i			1.64816i	0.566470	+	0.824082i	$0.308309\pi$
$397$			32.8394i			1.64816i	−0.566470	+	0.824082i	$0.691691\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	32.9812i		−	1.64700i	−0.567315	−	0.823501i	$-0.692018\pi$
$401$		−	32.9812i		−	1.64700i	0.567315	−	0.823501i	$-0.307982\pi$
$402$		0			0
$403$	−4.81236			−0.239721
$404$		0			0
$405$	0.389565	−	4.38427i	0.0193576	−	0.217856i
$406$		0			0
$407$		−	12.3223i		−	0.610795i
$408$		0			0
$409$			16.7483i			0.828149i	0.910243	+	0.414075i	$0.135895\pi$
$409$			16.7483i			0.828149i	−0.910243	+	0.414075i	$0.864105\pi$
$410$		0			0
$411$	0.0313932	+	0.0138260i	0.00154851	+	0.000681987i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	4.41729			0.216836
$416$		0			0
$417$	16.0412	+	7.06477i	0.785539	+	0.345963i
$418$		0			0
$419$	22.5610			1.10218			0.551089	−	0.834446i	$-0.314212\pi$
$419$	22.5610			1.10218			0.551089	+	0.834446i	$0.314212\pi$
$420$		0			0
$421$	−13.7973			−0.672441			−0.336220	−	0.941783i	$-0.609149\pi$
$421$	−13.7973			−0.672441			−0.336220	+	0.941783i	$0.609149\pi$
$422$		0			0
$423$	−18.8642	−	20.6147i	−0.917209	−	1.00232i
$424$		0			0
$425$	−35.9698			−1.74479
$426$		0			0
$427$		0			0
$428$		0			0
$429$	0.693181	−	1.57393i	0.0334671	−	0.0759899i
$430$		0			0
$431$		−	7.86848i		−	0.379011i	−0.981880	−	0.189506i	$-0.939311\pi$
$431$		−	7.86848i		−	0.379011i	0.981880	−	0.189506i	$-0.0606885\pi$
$432$		0			0
$433$			23.7844i			1.14300i	0.820601	+	0.571502i	$0.193639\pi$
$433$			23.7844i			1.14300i	−0.820601	+	0.571502i	$0.806361\pi$
$434$		0			0
$435$	−5.55337	−	2.44579i	−0.266264	−	0.117267i
$436$		0			0
$437$	−2.48803			−0.119019
$438$		0			0
$439$		−	22.9160i		−	1.09372i	−0.837224	−	0.546860i	$-0.815823\pi$
$439$		−	22.9160i		−	1.09372i	0.837224	−	0.546860i	$-0.184177\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			15.1481i			0.719709i	0.933008	+	0.359855i	$0.117174\pi$
$443$			15.1481i			0.719709i	−0.933008	+	0.359855i	$0.882826\pi$
$444$		0			0
$445$	4.67539			0.221635
$446$		0			0
$447$	−22.3286	−	9.83386i	−1.05611	−	0.465125i
$448$		0			0
$449$		−	15.2140i		−	0.717993i	−0.933339	−	0.358997i	$-0.883119\pi$
$449$		−	15.2140i		−	0.717993i	0.933339	−	0.358997i	$-0.116881\pi$
$450$		0			0
$451$			3.85536i			0.181542i
$452$		0			0
$453$	8.78253	−	19.9415i	0.412639	−	0.936933i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	23.1398			1.08243			0.541216	−	0.840883i	$-0.317964\pi$
$457$	23.1398			1.08243			0.541216	+	0.840883i	$0.317964\pi$
$458$		0			0
$459$	37.1882	−	12.5815i	1.73580	−	0.587256i
$460$		0			0
$461$	−2.52672			−0.117681			−0.0588405	−	0.998267i	$-0.518740\pi$
$461$	−2.52672			−0.117681			−0.0588405	+	0.998267i	$0.518740\pi$
$462$		0			0
$463$	37.3811			1.73725			0.868624	−	0.495472i	$-0.165005\pi$
$463$	37.3811			1.73725			0.868624	+	0.495472i	$0.165005\pi$
$464$		0			0
$465$	5.72128	+	2.51974i	0.265318	+	0.116850i
$466$		0			0
$467$	26.2777			1.21599			0.607994	−	0.793942i	$-0.291975\pi$
$467$	26.2777			1.21599			0.607994	+	0.793942i	$0.291975\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	13.7436	+	6.05290i	0.633273	+	0.278903i
$472$		0			0
$473$			8.46922i			0.389415i
$474$		0			0
$475$		−	2.00425i		−	0.0919614i
$476$		0			0
$477$	3.23640	−	2.96158i	0.148185	−	0.135602i
$478$		0			0
$479$	−10.4953			−0.479543			−0.239772	−	0.970829i	$-0.577073\pi$
$479$	−10.4953			−0.479543			−0.239772	+	0.970829i	$0.577073\pi$
$480$		0			0
$481$		−	5.27665i		−	0.240595i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			2.42296i			0.110021i
$486$		0			0
$487$	14.8614			0.673435			0.336718	−	0.941606i	$-0.390683\pi$
$487$	14.8614			0.673435			0.336718	+	0.941606i	$0.390683\pi$
$488$		0			0
$489$	−2.33983	+	5.31277i	−0.105811	+	0.240252i
$490$		0			0
$491$			13.2061i			0.595981i	0.954569	+	0.297990i	$0.0963164\pi$
$491$			13.2061i			0.595981i	−0.954569	+	0.297990i	$0.903684\pi$
$492$		0			0
$493$		−	54.1234i		−	2.43759i
$494$		0			0
$495$	−1.64821	+	1.50825i	−0.0740814	+	0.0677907i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	32.2745			1.44481			0.722404	−	0.691472i	$-0.243037\pi$
$499$	32.2745			1.44481			0.722404	+	0.691472i	$0.243037\pi$
$500$		0			0
$501$	16.3368	−	37.0942i	0.729876	−	1.65725i
$502$		0			0
$503$	35.5206			1.58379			0.791894	−	0.610659i	$-0.209095\pi$
$503$	35.5206			1.58379			0.791894	+	0.610659i	$0.209095\pi$
$504$		0			0
$505$	−2.69486			−0.119919
$506$		0			0
$507$	−8.77866	+	19.9327i	−0.389874	+	0.885242i
$508$		0			0
$509$	31.7393			1.40682			0.703411	−	0.710784i	$-0.251660\pi$
$509$	31.7393			1.40682			0.703411	+	0.710784i	$0.251660\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	0.701049	+	2.07214i	0.0309521	+	0.0914874i
$514$		0			0
$515$		−	4.48953i		−	0.197832i
$516$		0			0
$517$			14.1835i			0.623788i
$518$		0			0
$519$	−9.94204	+	22.5742i	−0.436407	+	0.990899i
$520$		0			0
$521$	12.8629			0.563532			0.281766	−	0.959483i	$-0.409080\pi$
$521$	12.8629			0.563532			0.281766	+	0.959483i	$0.409080\pi$
$522$		0			0
$523$			29.6046i			1.29452i	0.762270	+	0.647260i	$0.224085\pi$
$523$			29.6046i			1.29452i	−0.762270	+	0.647260i	$0.775915\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			55.7598i			2.42894i
$528$		0			0
$529$	−11.9278			−0.518600
$530$		0			0
$531$	14.8997	+	16.2823i	0.646590	+	0.706591i
$532$		0			0
$533$			1.65094i			0.0715101i
$534$		0			0
$535$			2.34229i			0.101266i
$536$		0			0
$537$	−32.2199	−	14.1901i	−1.39039	−	0.612350i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−9.99026			−0.429515			−0.214757	−	0.976667i	$-0.568896\pi$
$541$	−9.99026			−0.429515			−0.214757	+	0.976667i	$0.568896\pi$
$542$		0			0
$543$	−1.26569	−	0.557431i	−0.0543162	−	0.0239217i
$544$		0			0
$545$	6.54182			0.280221
$546$		0			0
$547$	−36.1746			−1.54671			−0.773357	−	0.633970i	$-0.781424\pi$
$547$	−36.1746			−1.54671			−0.773357	+	0.633970i	$0.781424\pi$
$548$		0			0
$549$	24.6765	−	22.5811i	1.05317	−	0.963736i
$550$		0			0
$551$	3.01578			0.128476
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−2.76284	+	6.27326i	−0.117276	+	0.266285i
$556$		0			0
$557$		−	7.39255i		−	0.313233i	−0.987660	−	0.156616i	$-0.949941\pi$
$557$		−	7.39255i		−	0.313233i	0.987660	−	0.156616i	$-0.0500586\pi$
$558$		0			0
$559$			3.62668i			0.153392i
$560$		0			0
$561$	−18.2368	−	8.03175i	−0.769957	−	0.339101i
$562$		0			0
$563$	−13.4904			−0.568551			−0.284275	−	0.958743i	$-0.591753\pi$
$563$	−13.4904			−0.568551			−0.284275	+	0.958743i	$0.591753\pi$
$564$		0			0
$565$			6.15065i			0.258760i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	34.5088i		−	1.44668i	−0.690490	−	0.723342i	$-0.742605\pi$
$569$		−	34.5088i		−	1.44668i	0.690490	−	0.723342i	$-0.257395\pi$
$570$		0			0
$571$	22.1255			0.925923			0.462962	−	0.886378i	$-0.346787\pi$
$571$	22.1255			0.925923			0.462962	+	0.886378i	$0.346787\pi$
$572$		0			0
$573$	33.3893	+	14.7051i	1.39486	+	0.614316i
$574$		0			0
$575$		−	28.1363i		−	1.17337i
$576$		0			0
$577$		−	2.50202i		−	0.104160i	−0.998643	−	0.0520802i	$-0.983415\pi$
$577$		−	2.50202i		−	0.104160i	0.998643	−	0.0520802i	$-0.0165852\pi$
$578$		0			0
$579$	12.4973	−	28.3762i	0.519371	−	1.17927i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−2.22673			−0.0922218
$584$		0			0
$585$	−0.705793	+	0.645861i	−0.0291810	+	0.0267030i
$586$		0			0
$587$	4.36635			0.180219			0.0901093	−	0.995932i	$-0.471278\pi$
$587$	4.36635			0.180219			0.0901093	+	0.995932i	$0.471278\pi$
$588$		0			0
$589$	−3.10696			−0.128020
$590$		0			0
$591$	−30.3321	−	13.3587i	−1.24770	−	0.549505i
$592$		0			0
$593$	−6.55038			−0.268992			−0.134496	−	0.990914i	$-0.542942\pi$
$593$	−6.55038			−0.268992			−0.134496	+	0.990914i	$0.542942\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−19.9032	−	8.76568i	−0.814584	−	0.358755i
$598$		0			0
$599$			8.43233i			0.344536i	0.985050	+	0.172268i	$0.0551095\pi$
$599$			8.43233i			0.344536i	−0.985050	+	0.172268i	$0.944891\pi$
$600$		0			0
$601$		−	3.05803i		−	0.124740i	−0.998053	−	0.0623699i	$-0.980134\pi$
$601$		−	3.05803i		−	0.124740i	0.998053	−	0.0623699i	$-0.0198659\pi$
$602$		0			0
$603$	29.6175	+	32.3659i	1.20612	+	1.31804i
$604$		0			0
$605$	−4.24566			−0.172611
$606$		0			0
$607$		−	2.07026i		−	0.0840291i	−0.999117	−	0.0420146i	$-0.986622\pi$
$607$		−	2.07026i		−	0.0840291i	0.999117	−	0.0420146i	$-0.0133776\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			6.07363i			0.245713i
$612$		0			0
$613$	28.1973			1.13888			0.569440	−	0.822033i	$-0.307160\pi$
$613$	28.1973			1.13888			0.569440	+	0.822033i	$0.307160\pi$
$614$		0			0
$615$	0.864427	−	1.96275i	0.0348570	−	0.0791459i
$616$		0			0
$617$			22.8152i			0.918505i	0.888306	+	0.459253i	$0.151883\pi$
$617$			22.8152i			0.918505i	−0.888306	+	0.459253i	$0.848117\pi$
$618$		0			0
$619$		−	3.59092i		−	0.144331i	−0.997393	−	0.0721657i	$-0.977009\pi$
$619$		−	3.59092i		−	0.144331i	0.997393	−	0.0721657i	$-0.0229911\pi$
$620$		0			0
$621$	9.84155	+	29.0894i	0.394928	+	1.16732i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	21.4695			0.858780
$626$		0			0
$627$	0.447532	−	1.01616i	0.0178727	−	0.0405815i
$628$		0			0
$629$	−61.1395			−2.43779
$630$		0			0
$631$	−14.4106			−0.573678			−0.286839	−	0.957979i	$-0.592605\pi$
$631$	−14.4106			−0.573678			−0.286839	+	0.957979i	$0.592605\pi$
$632$		0			0
$633$	6.26009	−	14.2141i	0.248816	−	0.564958i
$634$		0			0
$635$	0.129182			0.00512643
$636$		0			0
$637$		0			0
$638$		0			0
$639$	28.4425	−	26.0273i	1.12517	−	1.02962i
$640$		0			0
$641$			14.0910i			0.556562i	0.960500	+	0.278281i	$0.0897647\pi$
$641$			14.0910i			0.556562i	−0.960500	+	0.278281i	$0.910235\pi$
$642$		0			0
$643$		−	8.06888i		−	0.318205i	−0.987262	−	0.159103i	$-0.949140\pi$
$643$		−	8.06888i		−	0.318205i	0.987262	−	0.159103i	$-0.0508601\pi$
$644$		0			0
$645$	1.89892	−	4.31165i	0.0747698	−	0.169771i
$646$		0			0
$647$	−43.2431			−1.70006			−0.850031	−	0.526732i	$-0.823417\pi$
$647$	−43.2431			−1.70006			−0.850031	+	0.526732i	$0.823417\pi$
$648$		0			0
$649$		−	11.2026i		−	0.439742i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	43.9405i		−	1.71952i	−0.510695	−	0.859762i	$-0.670612\pi$
$653$		−	43.9405i		−	1.71952i	0.510695	−	0.859762i	$-0.329388\pi$
$654$		0			0
$655$	3.67733			0.143685
$656$		0			0
$657$	20.4615	−	18.7240i	0.798279	−	0.730493i
$658$		0			0
$659$			24.6275i			0.959351i	0.877446	+	0.479676i	$0.159246\pi$
$659$			24.6275i			0.959351i	−0.877446	+	0.479676i	$0.840754\pi$
$660$		0			0
$661$		−	22.2931i		−	0.867103i	−0.901129	−	0.433551i	$-0.857260\pi$
$661$		−	22.2931i		−	0.867103i	0.901129	−	0.433551i	$-0.142740\pi$
$662$		0			0
$663$	−7.80933	−	3.43935i	−0.303289	−	0.133573i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	42.3364			1.63927
$668$		0			0
$669$	−17.4071	−	7.66635i	−0.672997	−	0.296398i
$670$		0			0
$671$	−16.9781			−0.655431
$672$		0			0
$673$	14.5042			0.559096			0.279548	−	0.960132i	$-0.409815\pi$
$673$	14.5042			0.559096			0.279548	+	0.960132i	$0.409815\pi$
$674$		0			0
$675$	−23.4332	+	7.92793i	−0.901943	+	0.305146i
$676$		0			0
$677$	−41.5727			−1.59777			−0.798884	−	0.601486i	$-0.794576\pi$
$677$	−41.5727			−1.59777			−0.798884	+	0.601486i	$0.794576\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	3.80517	−	8.63995i	0.145814	−	0.331084i
$682$		0			0
$683$			7.61022i			0.291197i	0.989344	+	0.145599i	$0.0465108\pi$
$683$			7.61022i			0.291197i	−0.989344	+	0.145599i	$0.953489\pi$
$684$		0			0
$685$			0.00968575i			0.000370073i
$686$		0			0
$687$	−20.2903	−	8.93614i	−0.774122	−	0.340935i
$688$		0			0
$689$	−0.953529			−0.0363266
$690$		0			0
$691$		−	2.10683i		−	0.0801476i	−0.999197	−	0.0400738i	$-0.987241\pi$
$691$		−	2.10683i		−	0.0801476i	0.999197	−	0.0400738i	$-0.0127593\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			4.94919i			0.187733i
$696$		0			0
$697$	19.1291			0.724566
$698$		0			0
$699$	−25.6066	−	11.2775i	−0.968530	−	0.426555i
$700$		0			0
$701$		−	0.0201617i		−	0.000761498i	−1.00000		0.000380749i	$-0.999879\pi$
$701$		−	0.0201617i		−	0.000761498i	1.00000		0.000380749i	$-0.000121196\pi$
$702$		0			0
$703$		−	3.40672i		−	0.128487i
$704$		0			0
$705$	3.18013	−	7.22076i	0.119771	−	0.271950i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−5.18331			−0.194663			−0.0973317	−	0.995252i	$-0.531031\pi$
$709$	−5.18331			−0.194663			−0.0973317	+	0.995252i	$0.531031\pi$
$710$		0			0
$711$	2.45949	+	2.68772i	0.0922382	+	0.100797i
$712$		0			0
$713$	−43.6165			−1.63345
$714$		0			0
$715$	0.485605			0.0181606
$716$		0			0
$717$	1.73715	+	0.765068i	0.0648751	+	0.0285720i
$718$		0			0
$719$	26.1395			0.974838			0.487419	−	0.873168i	$-0.337938\pi$
$719$	26.1395			0.974838			0.487419	+	0.873168i	$0.337938\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−13.5689	−	5.97597i	−0.504634	−	0.222249i
$724$		0			0
$725$			34.1044i			1.26661i
$726$		0			0
$727$			20.9197i			0.775867i	0.921687	+	0.387934i	$0.126811\pi$
$727$			20.9197i			0.775867i	−0.921687	+	0.387934i	$0.873189\pi$
$728$		0			0
$729$	21.4539	−	16.3930i	0.794590	−	0.607147i
$730$		0			0
$731$	42.0216			1.55422
$732$		0			0
$733$			1.64571i			0.0607858i	0.999538	+	0.0303929i	$0.00967585\pi$
$733$			1.64571i			0.0607858i	−0.999538	+	0.0303929i	$0.990324\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	22.2686i		−	0.820274i
$738$		0			0
$739$	3.65420			0.134422			0.0672109	−	0.997739i	$-0.478590\pi$
$739$	3.65420			0.134422			0.0672109	+	0.997739i	$0.478590\pi$
$740$		0			0
$741$	0.191642	−	0.435139i	0.00704014	−	0.0159852i
$742$		0			0
$743$			23.6421i			0.867346i	0.901070	+	0.433673i	$0.142783\pi$
$743$			23.6421i			0.867346i	−0.901070	+	0.433673i	$0.857217\pi$
$744$		0			0
$745$		−	6.88906i		−	0.252395i
$746$		0			0
$747$	18.2926	+	19.9901i	0.669292	+	0.731399i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−18.4859			−0.674561			−0.337280	−	0.941404i	$-0.609507\pi$
$751$	−18.4859			−0.674561			−0.337280	+	0.941404i	$0.609507\pi$
$752$		0			0
$753$	2.71115	−	6.15590i	0.0987999	−	0.224334i
$754$		0			0
$755$	6.15256			0.223914
$756$		0			0
$757$	−39.2439			−1.42634			−0.713172	−	0.700989i	$-0.752742\pi$
$757$	−39.2439			−1.42634			−0.713172	+	0.700989i	$0.752742\pi$
$758$		0			0
$759$	6.28260	−	14.2652i	0.228044	−	0.517793i
$760$		0			0
$761$	24.3151			0.881423			0.440711	−	0.897649i	$-0.354726\pi$
$761$	24.3151			0.881423			0.440711	+	0.897649i	$0.354726\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	7.48345	+	8.17788i	0.270565	+	0.295672i
$766$		0			0
$767$		−	4.79718i		−	0.173216i
$768$		0			0
$769$			2.38223i			0.0859053i	0.999077	+	0.0429526i	$0.0136765\pi$
$769$			2.38223i			0.0859053i	−0.999077	+	0.0429526i	$0.986324\pi$
$770$		0			0
$771$	−12.4347	+	28.2341i	−0.447826	+	1.01683i
$772$		0			0
$773$	−35.9401			−1.29267			−0.646337	−	0.763052i	$-0.723700\pi$
$773$	−35.9401			−1.29267			−0.646337	+	0.763052i	$0.723700\pi$
$774$		0			0
$775$		−	35.1356i		−	1.26211i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			1.06588i			0.0381891i
$780$		0			0
$781$	−19.5692			−0.700241
$782$		0			0
$783$	−11.9291	−	35.2597i	−0.426310	−	1.26008i
$784$		0			0
$785$			4.24033i			0.151344i
$786$		0			0
$787$		−	8.45824i		−	0.301504i	−0.988572	−	0.150752i	$-0.951831\pi$
$787$		−	8.45824i		−	0.301504i	0.988572	−	0.150752i	$-0.0481694\pi$
$788$		0			0
$789$	−8.67993	−	3.82277i	−0.309014	−	0.136094i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−7.27033			−0.258177
$794$		0			0
$795$	1.13362	+	0.499265i	0.0402055	+	0.0177071i
$796$		0			0
$797$	22.7299			0.805134			0.402567	−	0.915390i	$-0.368118\pi$
$797$	22.7299			0.805134			0.402567	+	0.915390i	$0.368118\pi$
$798$		0			0
$799$	70.3739			2.48965
$800$		0			0
$801$	19.3615	+	21.1581i	0.684103	+	0.747585i
$802$		0			0
$803$	−14.0781			−0.496804
$804$		0			0
$805$		0			0
$806$		0			0
$807$	8.48518	−	19.2663i	0.298693	−	0.678207i
$808$		0			0
$809$		−	40.2926i		−	1.41661i	−0.705905	−	0.708307i	$-0.749459\pi$
$809$		−	40.2926i		−	1.41661i	0.705905	−	0.708307i	$-0.250541\pi$
$810$		0			0
$811$		−	47.7723i		−	1.67751i	−0.544507	−	0.838756i	$-0.683283\pi$
$811$		−	47.7723i		−	1.67751i	0.544507	−	0.838756i	$-0.316717\pi$
$812$		0			0
$813$	−38.3176	−	16.8757i	−1.34386	−	0.591855i
$814$		0			0
$815$	−1.63915			−0.0574170
$816$		0			0
$817$			2.34146i			0.0819173i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			2.95782i			0.103229i	0.998667	+	0.0516144i	$0.0164367\pi$
$821$			2.95782i			0.103229i	−0.998667	+	0.0516144i	$0.983563\pi$
$822$		0			0
$823$	−48.7101			−1.69793			−0.848964	−	0.528451i	$-0.822773\pi$
$823$	−48.7101			−1.69793			−0.848964	+	0.528451i	$0.822773\pi$
$824$		0			0
$825$	11.4914	+	5.06099i	0.400080	+	0.176201i
$826$		0			0
$827$		−	53.0570i		−	1.84497i	−0.386028	−	0.922487i	$-0.626153\pi$
$827$		−	53.0570i		−	1.84497i	0.386028	−	0.922487i	$-0.373847\pi$
$828$		0			0
$829$			41.1362i			1.42872i	0.699779	+	0.714360i	$0.253282\pi$
$829$			41.1362i			1.42872i	−0.699779	+	0.714360i	$0.746718\pi$
$830$		0			0
$831$	−2.04855	+	4.65141i	−0.0710635	+	0.161356i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	11.4447			0.396060
$836$		0			0
$837$	12.2898	+	36.3258i	0.424796	+	1.25560i
$838$		0			0
$839$	42.0861			1.45297			0.726486	−	0.687181i	$-0.241152\pi$
$839$	42.0861			1.45297			0.726486	+	0.687181i	$0.241152\pi$
$840$		0			0
$841$	−22.3166			−0.769536
$842$		0			0
$843$	−3.12984	−	1.37843i	−0.107797	−	0.0474756i
$844$		0			0
$845$	−6.14985			−0.211561
$846$		0			0
$847$		0			0
$848$		0			0
$849$	38.2880	+	16.8626i	1.31404	+	0.578724i
$850$		0			0
$851$		−	47.8246i		−	1.63941i
$852$		0			0
$853$		−	21.7333i		−	0.744134i	−0.928206	−	0.372067i	$-0.878649\pi$
$853$		−	21.7333i		−	0.744134i	0.928206	−	0.372067i	$-0.121351\pi$
$854$		0			0
$855$	−0.455675	+	0.416981i	−0.0155838	+	0.0142605i
$856$		0			0
$857$	−5.42154			−0.185196			−0.0925982	−	0.995704i	$-0.529517\pi$
$857$	−5.42154			−0.185196			−0.0925982	+	0.995704i	$0.529517\pi$
$858$		0			0
$859$		−	2.76401i		−	0.0943069i	−0.998888	−	0.0471535i	$-0.984985\pi$
$859$		−	2.76401i		−	0.0943069i	0.998888	−	0.0471535i	$-0.0150150\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	5.08854i		−	0.173216i	−0.996242	−	0.0866080i	$-0.972397\pi$
$863$		−	5.08854i		−	0.173216i	0.996242	−	0.0866080i	$-0.0276028\pi$
$864$		0			0
$865$	−6.96484			−0.236812
$866$		0			0
$867$	−27.9830	+	63.5379i	−0.950354	+	2.15786i
$868$		0			0
$869$		−	1.84922i		−	0.0627306i
$870$		0			0
$871$		−	9.53583i		−	0.323109i
$872$		0			0
$873$	−10.9649	+	10.0338i	−0.371105	+	0.339593i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	8.27933			0.279573			0.139787	−	0.990182i	$-0.455358\pi$
$877$	8.27933			0.279573			0.139787	+	0.990182i	$0.455358\pi$
$878$		0			0
$879$	−15.9749	+	36.2724i	−0.538820	+	1.22344i
$880$		0			0
$881$	−14.3278			−0.482714			−0.241357	−	0.970436i	$-0.577593\pi$
$881$	−14.3278			−0.482714			−0.241357	+	0.970436i	$0.577593\pi$
$882$		0			0
$883$	−55.7910			−1.87752			−0.938758	−	0.344576i	$-0.888023\pi$
$883$	−55.7910			−1.87752			−0.938758	+	0.344576i	$0.888023\pi$
$884$		0			0
$885$	−2.51179	+	5.70323i	−0.0844329	+	0.191712i
$886$		0			0
$887$	−33.5247			−1.12565			−0.562824	−	0.826577i	$-0.690285\pi$
$887$	−33.5247			−1.12565			−0.562824	+	0.826577i	$0.690285\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−13.6509	−	1.21295i	−0.457323	−	0.0406355i
$892$		0			0
$893$			3.92126i			0.131220i
$894$		0			0
$895$		−	9.94083i		−	0.332285i
$896$		0			0
$897$	2.69033	−	6.10862i	0.0898275	−	0.203961i
$898$		0			0
$899$	52.8682			1.76325
$900$		0			0
$901$			11.0483i			0.368074i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	0.390505i		−	0.0129808i
$906$		0			0
$907$	−6.78703			−0.225360			−0.112680	−	0.993631i	$-0.535943\pi$
$907$	−6.78703			−0.225360			−0.112680	+	0.993631i	$0.535943\pi$
$908$		0			0
$909$	−11.1598	−	12.1954i	−0.370147	−	0.404494i
$910$		0			0
$911$		−	5.62793i		−	0.186462i	−0.995645	−	0.0932308i	$-0.970281\pi$
$911$		−	5.62793i		−	0.186462i	0.995645	−	0.0932308i	$-0.0297194\pi$
$912$		0			0
$913$		−	13.7537i		−	0.455182i
$914$		0			0
$915$	8.64348	+	3.80672i	0.285745	+	0.125846i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−28.7234			−0.947496			−0.473748	−	0.880660i	$-0.657099\pi$
$919$	−28.7234			−0.947496			−0.473748	+	0.880660i	$0.657099\pi$
$920$		0			0
$921$	−15.4411	−	6.80049i	−0.508801	−	0.224084i
$922$		0			0
$923$	−8.37990			−0.275828
$924$		0			0
$925$	38.5254			1.26671
$926$		0			0
$927$	20.3170	−	18.5918i	0.667298	−	0.610634i
$928$		0			0
$929$	10.4848			0.343995			0.171998	−	0.985097i	$-0.444978\pi$
$929$	10.4848			0.343995			0.171998	+	0.985097i	$0.444978\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−15.4059	+	34.9804i	−0.504366	+	1.14521i
$934$		0			0
$935$		−	5.62660i		−	0.184010i
$936$		0			0
$937$		−	45.3531i		−	1.48162i	−0.671713	−	0.740811i	$-0.734441\pi$
$937$		−	45.3531i		−	1.48162i	0.671713	−	0.740811i	$-0.265559\pi$
$938$		0			0
$939$	24.7220	+	10.8880i	0.806773	+	0.355315i
$940$		0			0
$941$	−29.4694			−0.960675			−0.480338	−	0.877084i	$-0.659486\pi$
$941$	−29.4694			−0.960675			−0.480338	+	0.877084i	$0.659486\pi$
$942$		0			0
$943$			14.9632i			0.487268i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	20.0605i		−	0.651877i	−0.945391	−	0.325939i	$-0.894320\pi$
$947$		−	20.0605i		−	0.651877i	0.945391	−	0.325939i	$-0.105680\pi$
$948$		0			0
$949$	−6.02849			−0.195693
$950$		0			0
$951$	−28.0316	−	12.3455i	−0.908986	−	0.400331i
$952$		0			0
$953$		−	4.51738i		−	0.146332i	−0.997320	−	0.0731661i	$-0.976690\pi$
$953$		−	4.51738i		−	0.146332i	0.997320	−	0.0731661i	$-0.0233103\pi$
$954$		0			0
$955$			10.3016i			0.333352i
$956$		0			0
$957$	−7.61523	+	17.2910i	−0.246165	+	0.558939i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−23.4667			−0.756989
$962$		0			0
$963$	−10.5998	+	9.69975i	−0.341575	+	0.312570i
$964$		0			0
$965$	8.75493			0.281831
$966$		0			0
$967$	−28.2943			−0.909883			−0.454942	−	0.890521i	$-0.650340\pi$
$967$	−28.2943			−0.909883			−0.454942	+	0.890521i	$0.650340\pi$
$968$		0			0
$969$	−5.04187	−	2.22051i	−0.161968	−	0.0713332i
$970$		0			0
$971$	44.9316			1.44192			0.720962	−	0.692975i	$-0.243700\pi$
$971$	44.9316			1.44192			0.720962	+	0.692975i	$0.243700\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	4.92084	+	2.16721i	0.157593	+	0.0694064i
$976$		0			0
$977$			39.6506i			1.26853i	0.773114	+	0.634267i	$0.218698\pi$
$977$			39.6506i			1.26853i	−0.773114	+	0.634267i	$0.781302\pi$
$978$		0			0
$979$		−	14.5573i		−	0.465254i
$980$		0			0
$981$	27.0906	+	29.6045i	0.864936	+	0.945198i
$982$		0			0
$983$	1.09263			0.0348494			0.0174247	−	0.999848i	$-0.494453\pi$
$983$	1.09263			0.0348494			0.0174247	+	0.999848i	$0.494453\pi$
$984$		0			0
$985$		−	9.35839i		−	0.298183i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			32.8702i			1.04521i
$990$		0			0
$991$	48.6223			1.54454			0.772269	−	0.635296i	$-0.219122\pi$
$991$	48.6223			1.54454			0.772269	+	0.635296i	$0.219122\pi$
$992$		0			0
$993$	−8.84164	+	20.0757i	−0.280581	+	0.637083i
$994$		0			0
$995$		−	6.14075i		−	0.194675i
$996$		0			0
$997$			30.0811i			0.952677i	0.879262	+	0.476338i	$0.158036\pi$
$997$			30.0811i			0.952677i	−0.879262	+	0.476338i	$0.841964\pi$
$998$		0			0
$999$	−39.8305	+	13.4755i	−1.26018	+	0.426345i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.2.k.j.881.8		24
3.2	odd	2	inner	2352.2.k.j.881.18		24
4.3	odd	2		1176.2.k.b.881.17	yes	24
7.6	odd	2	inner	2352.2.k.j.881.17		24
12.11	even	2		1176.2.k.b.881.7	✓	24
21.20	even	2	inner	2352.2.k.j.881.7		24
28.3	even	6		1176.2.u.c.1097.24		48
28.11	odd	6		1176.2.u.c.1097.1		48
28.19	even	6		1176.2.u.c.521.9		48
28.23	odd	6		1176.2.u.c.521.16		48
28.27	even	2		1176.2.k.b.881.8	yes	24
84.11	even	6		1176.2.u.c.1097.9		48
84.23	even	6		1176.2.u.c.521.24		48
84.47	odd	6		1176.2.u.c.521.1		48
84.59	odd	6		1176.2.u.c.1097.16		48
84.83	odd	2		1176.2.k.b.881.18	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.2.k.b.881.7	✓	24	12.11	even	2
1176.2.k.b.881.8	yes	24	28.27	even	2
1176.2.k.b.881.17	yes	24	4.3	odd	2
1176.2.k.b.881.18	yes	24	84.83	odd	2
1176.2.u.c.521.1		48	84.47	odd	6
1176.2.u.c.521.9		48	28.19	even	6
1176.2.u.c.521.16		48	28.23	odd	6
1176.2.u.c.521.24		48	84.23	even	6
1176.2.u.c.1097.1		48	28.11	odd	6
1176.2.u.c.1097.9		48	84.11	even	6
1176.2.u.c.1097.16		48	84.59	odd	6
1176.2.u.c.1097.24		48	28.3	even	6
2352.2.k.j.881.7		24	21.20	even	2	inner
2352.2.k.j.881.8		24	1.1	even	1	trivial
2352.2.k.j.881.17		24	7.6	odd	2	inner
2352.2.k.j.881.18		24	3.2	odd	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2352.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.698115 + 1.58513i) q^{3} -0.489061 q^{5} +(-2.02527 - 2.21321i) q^{9} +1.52274i q^{11} +0.652068i q^{13} +(0.341421 - 0.775225i) q^{15} +7.55538 q^{17} +0.420989i q^{19} +5.90997i q^{23} -4.76082 q^{25} +(4.92209 - 1.66524i) q^{27} -7.16356i q^{29} +7.38015i q^{31} +(-2.41375 - 1.06305i) q^{33} -8.09219 q^{37} +(-1.03361 - 0.455218i) q^{39} +2.53185 q^{41} +5.56181 q^{43} +(0.990481 + 1.08239i) q^{45} +9.31441 q^{47} +(-5.27452 + 11.9763i) q^{51} +1.46232i q^{53} -0.744715i q^{55} +(-0.667322 - 0.293899i) q^{57} -7.35687 q^{59} +11.1496i q^{61} -0.318901i q^{65} -14.6240 q^{67} +(-9.36807 - 4.12584i) q^{69} +12.8513i q^{71} +9.24519i q^{73} +(3.32360 - 7.54652i) q^{75} -1.21440 q^{79} +(-0.796558 + 8.96468i) q^{81} -9.03219 q^{83} -3.69504 q^{85} +(11.3552 + 5.00099i) q^{87} -9.55993 q^{89} +(-11.6985 - 5.15220i) q^{93} -0.205889i q^{95} -4.95430i q^{97} +(3.37015 - 3.08397i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 16 q^{15} + 8 q^{25} + 16 q^{37} + 64 q^{39} - 16 q^{43} - 48 q^{51} + 48 q^{57} - 16 q^{67} + 80 q^{81} - 64 q^{85} - 32 q^{93} + 56 q^{99}+O(q^{100}) \) 24 * q + 16 * q^15 + 8 * q^25 + 16 * q^37 + 64 * q^39 - 16 * q^43 - 48 * q^51 + 48 * q^57 - 16 * q^67 + 80 * q^81 - 64 * q^85 - 32 * q^93 + 56 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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