LMFDB - Embedded newform 2352.2.k.j.881.13

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)

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);

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")

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			881.13
Character	$\chi$	$=$	2352.881
Dual form			2352.2.k.j.881.14

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.0935860 - 1.72952i) q^{3} -3.34363 q^{5} +(-2.98248 - 0.323718i) q^{9} -3.53079i q^{11} +5.57150i q^{13} +(-0.312917 + 5.78288i) q^{15} -0.803762 q^{17} -0.615091i q^{19} +8.47981i q^{23} +6.17985 q^{25} +(-0.838995 + 5.12797i) q^{27} -7.09383i q^{29} +3.25217i q^{31} +(-6.10657 - 0.330432i) q^{33} +7.31658 q^{37} +(9.63603 + 0.521414i) q^{39} -10.8204 q^{41} +6.16756 q^{43} +(9.97232 + 1.08239i) q^{45} +7.19094 q^{47} +(-0.0752209 + 1.39012i) q^{51} -2.45137i q^{53} +11.8056i q^{55} +(-1.06381 - 0.0575639i) q^{57} -8.40304 q^{59} -2.68141i q^{61} -18.6290i q^{65} +4.00789 q^{67} +(14.6660 + 0.793591i) q^{69} -6.75843i q^{71} -10.9705i q^{73} +(0.578348 - 10.6882i) q^{75} +9.97718 q^{79} +(8.79041 + 1.93097i) q^{81} +11.6052 q^{83} +2.68748 q^{85} +(-12.2689 - 0.663883i) q^{87} +9.76545 q^{89} +(5.62470 + 0.304358i) q^{93} +2.05664i q^{95} +13.7962i q^{97} +(-1.14298 + 10.5305i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.0935860	−	1.72952i	0.0540319	−	0.998539i
$3$	0.0935860	−	1.72952i	0.0540319	−	0.998539i
$4$		0			0
$5$	−3.34363			−1.49532			−0.747658	−	0.664084i	$-0.768822\pi$
$5$	−3.34363			−1.49532			−0.747658	+	0.664084i	$0.768822\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−2.98248	−	0.323718i	−0.994161	−	0.107906i
$10$		0			0
$11$		−	3.53079i		−	1.06457i	−0.846565	−	0.532286i	$-0.821333\pi$
$11$		−	3.53079i		−	1.06457i	0.846565	−	0.532286i	$-0.178667\pi$
$12$		0			0
$13$			5.57150i			1.54526i	0.634859	+	0.772628i	$0.281058\pi$
$13$			5.57150i			1.54526i	−0.634859	+	0.772628i	$0.718942\pi$
$14$		0			0
$15$	−0.312917	+	5.78288i	−0.0807948	+	1.49313i
$16$		0			0
$17$	−0.803762			−0.194941			−0.0974705	−	0.995238i	$-0.531075\pi$
$17$	−0.803762			−0.194941			−0.0974705	+	0.995238i	$0.531075\pi$
$18$		0			0
$19$		−	0.615091i		−	0.141111i	−0.997508	−	0.0705557i	$-0.977523\pi$
$19$		−	0.615091i		−	0.141111i	0.997508	−	0.0705557i	$-0.0224773\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			8.47981i			1.76816i	0.467334	+	0.884081i	$0.345215\pi$
$23$			8.47981i			1.76816i	−0.467334	+	0.884081i	$0.654785\pi$
$24$		0			0
$25$	6.17985			1.23597
$26$		0			0
$27$	−0.838995	+	5.12797i	−0.161465	+	0.986878i
$28$		0			0
$29$		−	7.09383i		−	1.31729i	−0.752453	−	0.658646i	$-0.771130\pi$
$29$		−	7.09383i		−	1.31729i	0.752453	−	0.658646i	$-0.228870\pi$
$30$		0			0
$31$			3.25217i			0.584107i	0.956402	+	0.292054i	$0.0943386\pi$
$31$			3.25217i			0.584107i	−0.956402	+	0.292054i	$0.905661\pi$
$32$		0			0
$33$	−6.10657	−	0.330432i	−1.06302	−	0.0575208i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	7.31658			1.20284			0.601420	−	0.798933i	$-0.294602\pi$
$37$	7.31658			1.20284			0.601420	+	0.798933i	$0.294602\pi$
$38$		0			0
$39$	9.63603	+	0.521414i	1.54300	+	0.0834931i
$40$		0			0
$41$	−10.8204			−1.68986			−0.844928	−	0.534881i	$-0.820357\pi$
$41$	−10.8204			−1.68986			−0.844928	+	0.534881i	$0.820357\pi$
$42$		0			0
$43$	6.16756			0.940544			0.470272	−	0.882521i	$-0.344156\pi$
$43$	6.16756			0.940544			0.470272	+	0.882521i	$0.344156\pi$
$44$		0			0
$45$	9.97232	+	1.08239i	1.48659	+	0.161354i
$46$		0			0
$47$	7.19094			1.04891			0.524453	−	0.851439i	$-0.324270\pi$
$47$	7.19094			1.04891			0.524453	+	0.851439i	$0.324270\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−0.0752209	+	1.39012i	−0.0105330	+	0.194656i
$52$		0			0
$53$		−	2.45137i		−	0.336721i	−0.985725	−	0.168361i	$-0.946153\pi$
$53$		−	2.45137i		−	0.336721i	0.985725	−	0.168361i	$-0.0538473\pi$
$54$		0			0
$55$			11.8056i			1.59187i
$56$		0			0
$57$	−1.06381	−	0.0575639i	−0.140905	−	0.00762452i
$58$		0			0
$59$	−8.40304			−1.09398			−0.546992	−	0.837138i	$-0.684227\pi$
$59$	−8.40304			−1.09398			−0.546992	+	0.837138i	$0.684227\pi$
$60$		0			0
$61$		−	2.68141i		−	0.343319i	−0.985156	−	0.171660i	$-0.945087\pi$
$61$		−	2.68141i		−	0.343319i	0.985156	−	0.171660i	$-0.0549130\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	18.6290i		−	2.31065i
$66$		0			0
$67$	4.00789			0.489642			0.244821	−	0.969568i	$-0.421271\pi$
$67$	4.00789			0.489642			0.244821	+	0.969568i	$0.421271\pi$
$68$		0			0
$69$	14.6660	+	0.793591i	1.76558	+	0.0955372i
$70$		0			0
$71$		−	6.75843i		−	0.802078i	−0.916061	−	0.401039i	$-0.868649\pi$
$71$		−	6.75843i		−	0.802078i	0.916061	−	0.401039i	$-0.131351\pi$
$72$		0			0
$73$		−	10.9705i		−	1.28400i	−0.766703	−	0.642002i	$-0.778104\pi$
$73$		−	10.9705i		−	1.28400i	0.766703	−	0.642002i	$-0.221896\pi$
$74$		0			0
$75$	0.578348	−	10.6882i	0.0667819	−	1.23417i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	9.97718			1.12252			0.561260	−	0.827639i	$-0.310317\pi$
$79$	9.97718			1.12252			0.561260	+	0.827639i	$0.310317\pi$
$80$		0			0
$81$	8.79041	+	1.93097i	0.976713	+	0.214552i
$82$		0			0
$83$	11.6052			1.27383			0.636916	−	0.770933i	$-0.280210\pi$
$83$	11.6052			1.27383			0.636916	+	0.770933i	$0.280210\pi$
$84$		0			0
$85$	2.68748			0.291498
$86$		0			0
$87$	−12.2689	−	0.663883i	−1.31537	−	0.0711758i
$88$		0			0
$89$	9.76545			1.03514			0.517568	−	0.855642i	$-0.326838\pi$
$89$	9.76545			1.03514			0.517568	+	0.855642i	$0.326838\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	5.62470	+	0.304358i	0.583254	+	0.0315604i
$94$		0			0
$95$			2.05664i			0.211006i
$96$		0			0
$97$			13.7962i			1.40079i	0.713755	+	0.700395i	$0.246993\pi$
$97$			13.7962i			1.40079i	−0.713755	+	0.700395i	$0.753007\pi$
$98$		0			0
$99$	−1.14298	+	10.5305i	−0.114874	+	1.05836i
$100$		0			0
$101$	3.74267			0.372409			0.186205	−	0.982511i	$-0.440381\pi$
$101$	3.74267			0.372409			0.186205	+	0.982511i	$0.440381\pi$
$102$		0			0
$103$			7.47003i			0.736044i	0.929817	+	0.368022i	$0.119965\pi$
$103$			7.47003i			0.736044i	−0.929817	+	0.368022i	$0.880035\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			6.19165i			0.598570i	0.954164	+	0.299285i	$0.0967481\pi$
$107$			6.19165i			0.598570i	−0.954164	+	0.299285i	$0.903252\pi$
$108$		0			0
$109$	3.15393			0.302091			0.151046	−	0.988527i	$-0.451736\pi$
$109$	3.15393			0.302091			0.151046	+	0.988527i	$0.451736\pi$
$110$		0			0
$111$	0.684730	−	12.6542i	0.0649917	−	1.20108i
$112$		0			0
$113$			9.03372i			0.849821i	0.905235	+	0.424911i	$0.139694\pi$
$113$			9.03372i			0.849821i	−0.905235	+	0.424911i	$0.860306\pi$
$114$		0			0
$115$		−	28.3533i		−	2.64396i
$116$		0			0
$117$	1.80359	−	16.6169i	0.166742	−	1.53623i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.46644			−0.133313
$122$		0			0
$123$	−1.01263	+	18.7140i	−0.0913061	+	1.68739i
$124$		0			0
$125$	−3.94500			−0.352851
$126$		0			0
$127$	−17.1353			−1.52051			−0.760257	−	0.649622i	$-0.774927\pi$
$127$	−17.1353			−1.52051			−0.760257	+	0.649622i	$0.774927\pi$
$128$		0			0
$129$	0.577197	−	10.6669i	0.0508194	−	0.939170i
$130$		0			0
$131$	6.31011			0.551317			0.275658	−	0.961256i	$-0.411104\pi$
$131$	6.31011			0.551317			0.275658	+	0.961256i	$0.411104\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	2.80529	−	17.1460i	0.241441	−	1.47570i
$136$		0			0
$137$			6.49594i			0.554986i	0.960728	+	0.277493i	$0.0895035\pi$
$137$			6.49594i			0.554986i	−0.960728	+	0.277493i	$0.910497\pi$
$138$		0			0
$139$			4.41953i			0.374859i	0.982278	+	0.187430i	$0.0600157\pi$
$139$			4.41953i			0.374859i	−0.982278	+	0.187430i	$0.939984\pi$
$140$		0			0
$141$	0.672972	−	12.4369i	0.0566744	−	1.04737i
$142$		0			0
$143$	19.6718			1.64504
$144$		0			0
$145$			23.7191i			1.96977i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			3.55652i			0.291362i	0.989332	+	0.145681i	$0.0465373\pi$
$149$			3.55652i			0.291362i	−0.989332	+	0.145681i	$0.953463\pi$
$150$		0			0
$151$	17.5979			1.43210			0.716050	−	0.698049i	$-0.245948\pi$
$151$	17.5979			1.43210			0.716050	+	0.698049i	$0.245948\pi$
$152$		0			0
$153$	2.39721	+	0.260192i	0.193803	+	0.0210353i
$154$		0			0
$155$		−	10.8741i		−	0.873425i
$156$		0			0
$157$			1.13406i			0.0905077i	0.998976	+	0.0452539i	$0.0144097\pi$
$157$			1.13406i			0.0905077i	−0.998976	+	0.0452539i	$0.985590\pi$
$158$		0			0
$159$	−4.23969	−	0.229414i	−0.336229	−	0.0181937i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	19.8818			1.55727			0.778633	−	0.627479i	$-0.215913\pi$
$163$	19.8818			1.55727			0.778633	+	0.627479i	$0.215913\pi$
$164$		0			0
$165$	20.4181	+	1.10484i	1.58955	+	0.0860118i
$166$		0			0
$167$	2.40572			0.186160			0.0930802	−	0.995659i	$-0.470329\pi$
$167$	2.40572			0.186160			0.0930802	+	0.995659i	$0.470329\pi$
$168$		0			0
$169$	−18.0416			−1.38782
$170$		0			0
$171$	−0.199116	+	1.83450i	−0.0152268	+	0.140288i
$172$		0			0
$173$	−1.15099			−0.0875079			−0.0437540	−	0.999042i	$-0.513932\pi$
$173$	−1.15099			−0.0875079			−0.0437540	+	0.999042i	$0.513932\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−0.786407	+	14.5332i	−0.0591100	+	1.09238i
$178$		0			0
$179$		−	18.6044i		−	1.39056i	−0.718738	−	0.695281i	$-0.755280\pi$
$179$		−	18.6044i		−	1.39056i	0.718738	−	0.695281i	$-0.244720\pi$
$180$		0			0
$181$		−	13.1502i		−	0.977450i	−0.872438	−	0.488725i	$-0.837462\pi$
$181$		−	13.1502i		−	0.977450i	0.872438	−	0.488725i	$-0.162538\pi$
$182$		0			0
$183$	−4.63756	−	0.250942i	−0.342818	−	0.0185502i
$184$		0			0
$185$	−24.4639			−1.79862
$186$		0			0
$187$			2.83791i			0.207529i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	5.05936i		−	0.366083i	−0.983105	−	0.183041i	$-0.941406\pi$
$191$		−	5.05936i		−	0.366083i	0.983105	−	0.183041i	$-0.0585942\pi$
$192$		0			0
$193$	−17.1986			−1.23798			−0.618990	−	0.785398i	$-0.712458\pi$
$193$	−17.1986			−1.23798			−0.618990	+	0.785398i	$0.712458\pi$
$194$		0			0
$195$	−32.2193	−	1.74342i	−2.30727	−	0.124849i
$196$		0			0
$197$			17.0069i			1.21169i	0.795583	+	0.605844i	$0.207165\pi$
$197$			17.0069i			1.21169i	−0.795583	+	0.605844i	$0.792835\pi$
$198$		0			0
$199$			8.06593i			0.571779i	0.958263	+	0.285889i	$0.0922890\pi$
$199$			8.06593i			0.571779i	−0.958263	+	0.285889i	$0.907711\pi$
$200$		0			0
$201$	0.375082	−	6.93173i	0.0264563	−	0.488926i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	36.1792			2.52687
$206$		0			0
$207$	2.74506	−	25.2909i	0.190795	−	1.75784i
$208$		0			0
$209$	−2.17175			−0.150223
$210$		0			0
$211$	9.66474			0.665348			0.332674	−	0.943042i	$-0.392049\pi$
$211$	9.66474			0.665348			0.332674	+	0.943042i	$0.392049\pi$
$212$		0			0
$213$	−11.6888	−	0.632494i	−0.800906	−	0.0433378i
$214$		0			0
$215$	−20.6220			−1.40641
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−18.9738	−	1.02669i	−1.28213	−	0.0693772i
$220$		0			0
$221$		−	4.47816i		−	0.301234i
$222$		0			0
$223$			12.3980i			0.830232i	0.909769	+	0.415116i	$0.136259\pi$
$223$			12.3980i			0.830232i	−0.909769	+	0.415116i	$0.863741\pi$
$224$		0			0
$225$	−18.4313	−	2.00053i	−1.22875	−	0.133369i
$226$		0			0
$227$	22.9204			1.52128			0.760641	−	0.649173i	$-0.224885\pi$
$227$	22.9204			1.52128			0.760641	+	0.649173i	$0.224885\pi$
$228$		0			0
$229$		−	8.91504i		−	0.589123i	−0.955633	−	0.294561i	$-0.904826\pi$
$229$		−	8.91504i		−	0.589123i	0.955633	−	0.294561i	$-0.0951735\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			20.0482i			1.31340i	0.754152	+	0.656699i	$0.228048\pi$
$233$			20.0482i			1.31340i	−0.754152	+	0.656699i	$0.771952\pi$
$234$		0			0
$235$	−24.0438			−1.56845
$236$		0			0
$237$	0.933724	−	17.2557i	0.0606519	−	1.12088i
$238$		0			0
$239$			6.54949i			0.423652i	0.977307	+	0.211826i	$0.0679409\pi$
$239$			6.54949i			0.423652i	−0.977307	+	0.211826i	$0.932059\pi$
$240$		0			0
$241$		−	15.0266i		−	0.967947i	−0.875083	−	0.483973i	$-0.839193\pi$
$241$		−	15.0266i		−	0.967947i	0.875083	−	0.483973i	$-0.160807\pi$
$242$		0			0
$243$	4.16230	−	15.0225i	0.267012	−	0.963693i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	3.42698			0.218053
$248$		0			0
$249$	1.08608	−	20.0714i	0.0688275	−	1.27197i
$250$		0			0
$251$	10.2682			0.648123			0.324062	−	0.946036i	$-0.394951\pi$
$251$	10.2682			0.648123			0.324062	+	0.946036i	$0.394951\pi$
$252$		0			0
$253$	29.9404			1.88234
$254$		0			0
$255$	0.251511	−	4.64806i	0.0157502	−	0.291073i
$256$		0			0
$257$	−9.05973			−0.565130			−0.282565	−	0.959248i	$-0.591185\pi$
$257$	−9.05973			−0.565130			−0.282565	+	0.959248i	$0.591185\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−2.29640	+	21.1572i	−0.142144	+	1.30960i
$262$		0			0
$263$			16.9461i			1.04494i	0.852657	+	0.522470i	$0.174990\pi$
$263$			16.9461i			1.04494i	−0.852657	+	0.522470i	$0.825010\pi$
$264$		0			0
$265$			8.19646i			0.503505i
$266$		0			0
$267$	0.913909	−	16.8895i	0.0559303	−	1.03362i
$268$		0			0
$269$	−18.5118			−1.12868			−0.564341	−	0.825542i	$-0.690870\pi$
$269$	−18.5118			−1.12868			−0.564341	+	0.825542i	$0.690870\pi$
$270$		0			0
$271$			6.78659i			0.412256i	0.978525	+	0.206128i	$0.0660863\pi$
$271$			6.78659i			0.412256i	−0.978525	+	0.206128i	$0.933914\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	21.8197i		−	1.31578i
$276$		0			0
$277$	−8.70920			−0.523285			−0.261642	−	0.965165i	$-0.584264\pi$
$277$	−8.70920			−0.523285			−0.261642	+	0.965165i	$0.584264\pi$
$278$		0			0
$279$	1.05279	−	9.69955i	0.0630286	−	0.580697i
$280$		0			0
$281$			11.7672i			0.701972i	0.936381	+	0.350986i	$0.114154\pi$
$281$			11.7672i			0.701972i	−0.936381	+	0.350986i	$0.885846\pi$
$282$		0			0
$283$			1.46990i			0.0873767i	0.999045	+	0.0436884i	$0.0139109\pi$
$283$			1.46990i			0.0873767i	−0.999045	+	0.0436884i	$0.986089\pi$
$284$		0			0
$285$	3.55699	+	0.192472i	0.210698	+	0.0114011i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−16.3540			−0.961998
$290$		0			0
$291$	23.8608	+	1.29113i	1.39874	+	0.0756874i
$292$		0			0
$293$	−14.4559			−0.844523			−0.422261	−	0.906474i	$-0.638764\pi$
$293$	−14.4559			−0.844523			−0.422261	+	0.906474i	$0.638764\pi$
$294$		0			0
$295$	28.0967			1.63585
$296$		0			0
$297$	18.1058	+	2.96231i	1.05060	+	0.171891i
$298$		0			0
$299$	−47.2453			−2.73226
$300$		0			0
$301$		0			0
$302$		0			0
$303$	0.350261	−	6.47302i	0.0201220	−	0.371865i
$304$		0			0
$305$			8.96564i			0.513371i
$306$		0			0
$307$		−	7.50958i		−	0.428594i	−0.976769	−	0.214297i	$-0.931254\pi$
$307$		−	7.50958i		−	0.428594i	0.976769	−	0.214297i	$-0.0687461\pi$
$308$		0			0
$309$	12.9196	+	0.699090i	0.734969	+	0.0397698i
$310$		0			0
$311$	0.0307314			0.00174262			0.000871308	−	1.00000i	$-0.499723\pi$
$311$	0.0307314			0.00174262			0.000871308		1.00000i	$0.499723\pi$
$312$		0			0
$313$			4.45310i			0.251704i	0.992049	+	0.125852i	$0.0401664\pi$
$313$			4.45310i			0.251704i	−0.992049	+	0.125852i	$0.959834\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	6.54231i		−	0.367453i	−0.982977	−	0.183726i	$-0.941184\pi$
$317$		−	6.54231i		−	0.367453i	0.982977	−	0.183726i	$-0.0588161\pi$
$318$		0			0
$319$	−25.0468			−1.40235
$320$		0			0
$321$	10.7086	+	0.579452i	0.597695	+	0.0323419i
$322$		0			0
$323$			0.494387i			0.0275084i
$324$		0			0
$325$			34.4311i			1.90989i
$326$		0			0
$327$	0.295163	−	5.45478i	0.0163226	−	0.301650i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	2.45365			0.134865			0.0674324	−	0.997724i	$-0.478519\pi$
$331$	2.45365			0.134865			0.0674324	+	0.997724i	$0.478519\pi$
$332$		0			0
$333$	−21.8216	−	2.36851i	−1.19582	−	0.129793i
$334$		0			0
$335$	−13.4009			−0.732169
$336$		0			0
$337$	−4.26290			−0.232215			−0.116108	−	0.993237i	$-0.537042\pi$
$337$	−4.26290			−0.232215			−0.116108	+	0.993237i	$0.537042\pi$
$338$		0			0
$339$	15.6240	+	0.845430i	0.848580	+	0.0459175i
$340$		0			0
$341$	11.4827			0.621824
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−49.0377	−	2.65347i	−2.64010	−	0.142858i
$346$		0			0
$347$			15.8736i			0.852142i	0.904690	+	0.426071i	$0.140103\pi$
$347$			15.8736i			0.852142i	−0.904690	+	0.426071i	$0.859897\pi$
$348$		0			0
$349$			27.2794i			1.46024i	0.683322	+	0.730118i	$0.260535\pi$
$349$			27.2794i			1.46024i	−0.683322	+	0.730118i	$0.739465\pi$
$350$		0			0
$351$	−28.5705	−	4.67446i	−1.52498	−	0.249504i
$352$		0			0
$353$	22.4639			1.19563			0.597816	−	0.801634i	$-0.296035\pi$
$353$	22.4639			1.19563			0.597816	+	0.801634i	$0.296035\pi$
$354$		0			0
$355$			22.5977i			1.19936i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			25.3300i			1.33686i	0.743773	+	0.668432i	$0.233034\pi$
$359$			25.3300i			1.33686i	−0.743773	+	0.668432i	$0.766966\pi$
$360$		0			0
$361$	18.6217			0.980088
$362$		0			0
$363$	−0.137239	+	2.53625i	−0.00720316	+	0.133118i
$364$		0			0
$365$			36.6814i			1.91999i
$366$		0			0
$367$			21.7825i			1.13704i	0.822671	+	0.568518i	$0.192483\pi$
$367$			21.7825i			1.13704i	−0.822671	+	0.568518i	$0.807517\pi$
$368$		0			0
$369$	32.2715	+	3.50274i	1.67999	+	0.182345i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	9.10573			0.471477			0.235738	−	0.971817i	$-0.424249\pi$
$373$	9.10573			0.471477			0.235738	+	0.971817i	$0.424249\pi$
$374$		0			0
$375$	−0.369196	+	6.82295i	−0.0190652	+	0.352336i
$376$		0			0
$377$	39.5233			2.03555
$378$		0			0
$379$	−34.6413			−1.77941			−0.889703	−	0.456540i	$-0.849089\pi$
$379$	−34.6413			−1.77941			−0.889703	+	0.456540i	$0.849089\pi$
$380$		0			0
$381$	−1.60363	+	29.6359i	−0.0821563	+	1.51829i
$382$		0			0
$383$	30.0473			1.53535			0.767673	−	0.640842i	$-0.221414\pi$
$383$	30.0473			1.53535			0.767673	+	0.640842i	$0.221414\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−18.3946	−	1.99655i	−0.935052	−	0.101490i
$388$		0			0
$389$			16.7660i			0.850072i	0.905177	+	0.425036i	$0.139739\pi$
$389$			16.7660i			0.850072i	−0.905177	+	0.425036i	$0.860261\pi$
$390$		0			0
$391$		−	6.81575i		−	0.344687i
$392$		0			0
$393$	0.590538	−	10.9135i	0.0297887	−	0.550511i
$394$		0			0
$395$	−33.3600			−1.67852
$396$		0			0
$397$			16.2443i			0.815279i	0.913143	+	0.407639i	$0.133648\pi$
$397$			16.2443i			0.815279i	−0.913143	+	0.407639i	$0.866352\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			20.7920i			1.03830i	0.854683	+	0.519151i	$0.173752\pi$
$401$			20.7920i			1.03830i	−0.854683	+	0.519151i	$0.826248\pi$
$402$		0			0
$403$	−18.1195			−0.902595
$404$		0			0
$405$	−29.3919	−	6.45643i	−1.46049	−	0.320823i
$406$		0			0
$407$		−	25.8333i		−	1.28051i
$408$		0			0
$409$		−	7.69676i		−	0.380580i	−0.981728	−	0.190290i	$-0.939057\pi$
$409$		−	7.69676i		−	0.380580i	0.981728	−	0.190290i	$-0.0609428\pi$
$410$		0			0
$411$	11.2349	+	0.607929i	0.554175	+	0.0299869i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−38.8033			−1.90478
$416$		0			0
$417$	7.64366	+	0.413606i	0.374312	+	0.0202544i
$418$		0			0
$419$	25.7986			1.26034			0.630171	−	0.776456i	$-0.282985\pi$
$419$	25.7986			1.26034			0.630171	+	0.776456i	$0.282985\pi$
$420$		0			0
$421$	12.1973			0.594459			0.297229	−	0.954806i	$-0.403937\pi$
$421$	12.1973			0.594459			0.297229	+	0.954806i	$0.403937\pi$
$422$		0			0
$423$	−21.4469	−	2.32784i	−1.04278	−	0.113183i
$424$		0			0
$425$	−4.96713			−0.240941
$426$		0			0
$427$		0			0
$428$		0			0
$429$	1.84100	−	34.0227i	0.0888844	−	1.64263i
$430$		0			0
$431$			13.0024i			0.626306i	0.949703	+	0.313153i	$0.101385\pi$
$431$			13.0024i			0.626306i	−0.949703	+	0.313153i	$0.898615\pi$
$432$		0			0
$433$			2.87672i			0.138246i	0.997608	+	0.0691232i	$0.0220202\pi$
$433$			2.87672i			0.138246i	−0.997608	+	0.0691232i	$0.977980\pi$
$434$		0			0
$435$	41.0228	+	2.21978i	1.96689	+	0.106430i
$436$		0			0
$437$	5.21585			0.249508
$438$		0			0
$439$			22.3517i			1.06679i	0.845867	+	0.533394i	$0.179084\pi$
$439$			22.3517i			1.06679i	−0.845867	+	0.533394i	$0.820916\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	18.9382i		−	0.899781i	−0.893084	−	0.449891i	$-0.851463\pi$
$443$		−	18.9382i		−	0.899781i	0.893084	−	0.449891i	$-0.148537\pi$
$444$		0			0
$445$	−32.6520			−1.54785
$446$		0			0
$447$	6.15108	+	0.332841i	0.290936	+	0.0157428i
$448$		0			0
$449$		−	39.4278i		−	1.86071i	−0.366658	−	0.930356i	$-0.619498\pi$
$449$		−	39.4278i		−	1.86071i	0.366658	−	0.930356i	$-0.380502\pi$
$450$		0			0
$451$			38.2043i			1.79897i
$452$		0			0
$453$	1.64692	−	30.4360i	0.0773791	−	1.43001i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−38.0098			−1.77802			−0.889012	−	0.457884i	$-0.848607\pi$
$457$	−38.0098			−1.77802			−0.889012	+	0.457884i	$0.848607\pi$
$458$		0			0
$459$	0.674353	−	4.12167i	0.0314761	−	0.192383i
$460$		0			0
$461$	7.70698			0.358950			0.179475	−	0.983763i	$-0.442560\pi$
$461$	7.70698			0.358950			0.179475	+	0.983763i	$0.442560\pi$
$462$		0			0
$463$	36.4771			1.69523			0.847617	−	0.530608i	$-0.178036\pi$
$463$	36.4771			1.69523			0.847617	+	0.530608i	$0.178036\pi$
$464$		0			0
$465$	−18.8069	−	1.01766i	−0.872149	−	0.0471928i
$466$		0			0
$467$	31.2288			1.44510			0.722549	−	0.691319i	$-0.242970\pi$
$467$	31.2288			1.44510			0.722549	+	0.691319i	$0.242970\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	1.96138	+	0.106132i	0.0903755	+	0.00489030i
$472$		0			0
$473$		−	21.7763i		−	1.00128i
$474$		0			0
$475$		−	3.80117i		−	0.174410i
$476$		0			0
$477$	−0.793551	+	7.31116i	−0.0363342	+	0.334755i
$478$		0			0
$479$	16.6401			0.760307			0.380154	−	0.924923i	$-0.375871\pi$
$479$	16.6401			0.760307			0.380154	+	0.924923i	$0.375871\pi$
$480$		0			0
$481$			40.7644i			1.85869i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	46.1293i		−	2.09463i
$486$		0			0
$487$	−8.49622			−0.385000			−0.192500	−	0.981297i	$-0.561660\pi$
$487$	−8.49622			−0.385000			−0.192500	+	0.981297i	$0.561660\pi$
$488$		0			0
$489$	1.86066	−	34.3861i	0.0841421	−	1.55499i
$490$		0			0
$491$			21.6613i			0.977561i	0.872407	+	0.488781i	$0.162558\pi$
$491$			21.6613i			0.977561i	−0.872407	+	0.488781i	$0.837442\pi$
$492$		0			0
$493$			5.70176i			0.256794i
$494$		0			0
$495$	3.82169	−	35.2101i	0.171772	−	1.58258i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−13.2488			−0.593100			−0.296550	−	0.955017i	$-0.595836\pi$
$499$	−13.2488			−0.593100			−0.296550	+	0.955017i	$0.595836\pi$
$500$		0			0
$501$	0.225142	−	4.16075i	0.0100586	−	0.185889i
$502$		0			0
$503$	0.808467			0.0360478			0.0180239	−	0.999838i	$-0.494263\pi$
$503$	0.808467			0.0360478			0.0180239	+	0.999838i	$0.494263\pi$
$504$		0			0
$505$	−12.5141			−0.556869
$506$		0			0
$507$	−1.68844	+	31.2034i	−0.0749864	+	1.38579i
$508$		0			0
$509$	5.95966			0.264157			0.132079	−	0.991239i	$-0.457835\pi$
$509$	5.95966			0.264157			0.132079	+	0.991239i	$0.457835\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	3.15417	+	0.516058i	0.139260	+	0.0227845i
$514$		0			0
$515$		−	24.9770i		−	1.10062i
$516$		0			0
$517$		−	25.3897i		−	1.11664i
$518$		0			0
$519$	−0.107716	+	1.99066i	−0.00472822	+	0.0873801i
$520$		0			0
$521$	25.0692			1.09830			0.549151	−	0.835723i	$-0.314951\pi$
$521$	25.0692			1.09830			0.549151	+	0.835723i	$0.314951\pi$
$522$		0			0
$523$		−	8.54452i		−	0.373626i	−0.982396	−	0.186813i	$-0.940184\pi$
$523$		−	8.54452i		−	0.373626i	0.982396	−	0.186813i	$-0.0598158\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	2.61397i		−	0.113866i
$528$		0			0
$529$	−48.9071			−2.12640
$530$		0			0
$531$	25.0619	+	2.72021i	1.08760	+	0.118047i
$532$		0			0
$533$		−	60.2856i		−	2.61126i
$534$		0			0
$535$		−	20.7026i		−	0.895051i
$536$		0			0
$537$	−32.1768	−	1.74112i	−1.38853	−	0.0751347i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	18.2099			0.782906			0.391453	−	0.920198i	$-0.371973\pi$
$541$	18.2099			0.782906			0.391453	+	0.920198i	$0.371973\pi$
$542$		0			0
$543$	−22.7436	−	1.23068i	−0.976022	−	0.0528135i
$544$		0			0
$545$	−10.5456			−0.451722
$546$		0			0
$547$	−6.72942			−0.287729			−0.143865	−	0.989597i	$-0.545953\pi$
$547$	−6.72942			−0.287729			−0.143865	+	0.989597i	$0.545953\pi$
$548$		0			0
$549$	−0.868020	+	7.99726i	−0.0370462	+	0.341315i
$550$		0			0
$551$	−4.36335			−0.185885
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−2.28948	+	42.3109i	−0.0971831	+	1.79600i
$556$		0			0
$557$			26.0369i			1.10322i	0.834102	+	0.551610i	$0.185986\pi$
$557$			26.0369i			1.10322i	−0.834102	+	0.551610i	$0.814014\pi$
$558$		0			0
$559$			34.3626i			1.45338i
$560$		0			0
$561$	4.90823	+	0.265589i	0.207226	+	0.0112132i
$562$		0			0
$563$	−11.4784			−0.483757			−0.241879	−	0.970307i	$-0.577764\pi$
$563$	−11.4784			−0.483757			−0.241879	+	0.970307i	$0.577764\pi$
$564$		0			0
$565$		−	30.2054i		−	1.27075i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	8.66559i		−	0.363280i	−0.983365	−	0.181640i	$-0.941859\pi$
$569$		−	8.66559i		−	0.363280i	0.983365	−	0.181640i	$-0.0581406\pi$
$570$		0			0
$571$	−13.7659			−0.576084			−0.288042	−	0.957618i	$-0.593004\pi$
$571$	−13.7659			−0.576084			−0.288042	+	0.957618i	$0.593004\pi$
$572$		0			0
$573$	−8.75028	−	0.473486i	−0.365548	−	0.0197801i
$574$		0			0
$575$			52.4040i			2.18540i
$576$		0			0
$577$			5.80268i			0.241569i	0.992679	+	0.120784i	$0.0385410\pi$
$577$			5.80268i			0.241569i	−0.992679	+	0.120784i	$0.961459\pi$
$578$		0			0
$579$	−1.60955	+	29.7453i	−0.0668905	+	1.23617i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−8.65525			−0.358464
$584$		0			0
$585$	−6.03055	+	55.5608i	−0.249333	+	2.29716i
$586$		0			0
$587$	−43.7617			−1.80624			−0.903120	−	0.429388i	$-0.858729\pi$
$587$	−43.7617			−1.80624			−0.903120	+	0.429388i	$0.858729\pi$
$588$		0			0
$589$	2.00038			0.0824242
$590$		0			0
$591$	29.4137	+	1.59160i	1.20992	+	0.0654698i
$592$		0			0
$593$	−7.81402			−0.320884			−0.160442	−	0.987045i	$-0.551292\pi$
$593$	−7.81402			−0.320884			−0.160442	+	0.987045i	$0.551292\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	13.9502	+	0.754858i	0.570944	+	0.0308943i
$598$		0			0
$599$			4.93318i			0.201564i	0.994909	+	0.100782i	$0.0321345\pi$
$599$			4.93318i			0.201564i	−0.994909	+	0.100782i	$0.967866\pi$
$600$		0			0
$601$		−	16.9723i		−	0.692314i	−0.938177	−	0.346157i	$-0.887486\pi$
$601$		−	16.9723i		−	0.692314i	0.938177	−	0.346157i	$-0.112514\pi$
$602$		0			0
$603$	−11.9535	−	1.29743i	−0.486783	−	0.0528352i
$604$		0			0
$605$	4.90324			0.199345
$606$		0			0
$607$		−	16.9118i		−	0.686427i	−0.939257	−	0.343214i	$-0.888485\pi$
$607$		−	16.9118i		−	0.686427i	0.939257	−	0.343214i	$-0.111515\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			40.0643i			1.62083i
$612$		0			0
$613$	−4.26984			−0.172457			−0.0862285	−	0.996275i	$-0.527482\pi$
$613$	−4.26984			−0.172457			−0.0862285	+	0.996275i	$0.527482\pi$
$614$		0			0
$615$	3.38587	−	62.5728i	0.136531	−	2.52318i
$616$		0			0
$617$			2.71133i			0.109154i	0.998510	+	0.0545770i	$0.0173810\pi$
$617$			2.71133i			0.109154i	−0.998510	+	0.0545770i	$0.982619\pi$
$618$		0			0
$619$		−	13.3992i		−	0.538559i	−0.963062	−	0.269279i	$-0.913215\pi$
$619$		−	13.3992i		−	0.538559i	0.963062	−	0.269279i	$-0.0867855\pi$
$620$		0			0
$621$	−43.4842	−	7.11452i	−1.74496	−	0.285496i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−17.7087			−0.708347
$626$		0			0
$627$	−0.203246	+	3.75609i	−0.00811685	+	0.150004i
$628$		0			0
$629$	−5.88079			−0.234483
$630$		0			0
$631$	23.2980			0.927479			0.463739	−	0.885972i	$-0.346507\pi$
$631$	23.2980			0.927479			0.463739	+	0.885972i	$0.346507\pi$
$632$		0			0
$633$	0.904485	−	16.7154i	0.0359500	−	0.664376i
$634$		0			0
$635$	57.2942			2.27365
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−2.18782	+	20.1569i	−0.0865490	+	0.797395i
$640$		0			0
$641$		−	27.3946i		−	1.08202i	−0.841016	−	0.541010i	$-0.818042\pi$
$641$		−	27.3946i		−	1.08202i	0.841016	−	0.541010i	$-0.181958\pi$
$642$		0			0
$643$		−	44.6514i		−	1.76088i	−0.474156	−	0.880441i	$-0.657247\pi$
$643$		−	44.6514i		−	1.76088i	0.474156	−	0.880441i	$-0.342753\pi$
$644$		0			0
$645$	−1.92993	+	35.6662i	−0.0759910	+	1.40436i
$646$		0			0
$647$	−27.8564			−1.09515			−0.547575	−	0.836757i	$-0.684449\pi$
$647$	−27.8564			−1.09515			−0.547575	+	0.836757i	$0.684449\pi$
$648$		0			0
$649$			29.6693i			1.16462i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			45.6201i			1.78525i	0.450799	+	0.892625i	$0.351139\pi$
$653$			45.6201i			1.78525i	−0.450799	+	0.892625i	$0.648861\pi$
$654$		0			0
$655$	−21.0987			−0.824393
$656$		0			0
$657$	−3.55136	+	32.7195i	−0.138552	+	1.27651i
$658$		0			0
$659$			26.2506i			1.02258i	0.859409	+	0.511289i	$0.170832\pi$
$659$			26.2506i			1.02258i	−0.859409	+	0.511289i	$0.829168\pi$
$660$		0			0
$661$			32.7790i			1.27495i	0.770470	+	0.637477i	$0.220022\pi$
$661$			32.7790i			1.27495i	−0.770470	+	0.637477i	$0.779978\pi$
$662$		0			0
$663$	−7.74507	−	0.419093i	−0.300794	−	0.0162762i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	60.1543			2.32919
$668$		0			0
$669$	21.4426	+	1.16028i	0.829019	+	0.0448590i
$670$		0			0
$671$	−9.46749			−0.365488
$672$		0			0
$673$	16.0001			0.616759			0.308379	−	0.951263i	$-0.400213\pi$
$673$	16.0001			0.616759			0.308379	+	0.951263i	$0.400213\pi$
$674$		0			0
$675$	−5.18487	+	31.6901i	−0.199566	+	1.21975i
$676$		0			0
$677$	−47.0589			−1.80862			−0.904310	−	0.426876i	$-0.859614\pi$
$677$	−47.0589			−1.80862			−0.904310	+	0.426876i	$0.859614\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	2.14503	−	39.6413i	0.0821977	−	1.51906i
$682$		0			0
$683$		−	16.8998i		−	0.646652i	−0.946288	−	0.323326i	$-0.895199\pi$
$683$		−	16.8998i		−	0.646652i	0.946288	−	0.323326i	$-0.104801\pi$
$684$		0			0
$685$		−	21.7200i		−	0.829879i
$686$		0			0
$687$	−15.4187	−	0.834323i	−0.588262	−	0.0318314i
$688$		0			0
$689$	13.6578			0.520321
$690$		0			0
$691$			24.9992i			0.951013i	0.879712	+	0.475507i	$0.157735\pi$
$691$			24.9992i			0.951013i	−0.879712	+	0.475507i	$0.842265\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	14.7773i		−	0.560533i
$696$		0			0
$697$	8.69699			0.329422
$698$		0			0
$699$	34.6737	+	1.87623i	1.31148	+	0.0709654i
$700$		0			0
$701$		−	13.6579i		−	0.515850i	−0.966165	−	0.257925i	$-0.916961\pi$
$701$		−	13.6579i		−	0.515850i	0.966165	−	0.257925i	$-0.0830388\pi$
$702$		0			0
$703$		−	4.50036i		−	0.169734i
$704$		0			0
$705$	−2.25017	+	41.5843i	−0.0847462	+	1.56616i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−23.4268			−0.879814			−0.439907	−	0.898043i	$-0.644989\pi$
$709$	−23.4268			−0.879814			−0.439907	+	0.898043i	$0.644989\pi$
$710$		0			0
$711$	−29.7568	−	3.22979i	−1.11597	−	0.121127i
$712$		0			0
$713$	−27.5778			−1.03280
$714$		0			0
$715$	−65.7751			−2.45985
$716$		0			0
$717$	11.3275	+	0.612941i	0.423033	+	0.0228907i
$718$		0			0
$719$	−26.5505			−0.990167			−0.495084	−	0.868845i	$-0.664863\pi$
$719$	−26.5505			−0.990167			−0.495084	+	0.868845i	$0.664863\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−25.9888	−	1.40628i	−0.966533	−	0.0523000i
$724$		0			0
$725$		−	43.8389i		−	1.62813i
$726$		0			0
$727$		−	18.8035i		−	0.697385i	−0.937237	−	0.348693i	$-0.886626\pi$
$727$		−	18.8035i		−	0.697385i	0.937237	−	0.348693i	$-0.113374\pi$
$728$		0			0
$729$	−25.5922	−	8.60469i	−0.947858	−	0.318692i
$730$		0			0
$731$	−4.95725			−0.183351
$732$		0			0
$733$		−	16.2120i		−	0.598804i	−0.954127	−	0.299402i	$-0.903213\pi$
$733$		−	16.2120i		−	0.598804i	0.954127	−	0.299402i	$-0.0967871\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	14.1510i		−	0.521259i
$738$		0			0
$739$	37.3329			1.37331			0.686656	−	0.726982i	$-0.259078\pi$
$739$	37.3329			1.37331			0.686656	+	0.726982i	$0.259078\pi$
$740$		0			0
$741$	0.320717	−	5.92703i	0.0117818	−	0.217735i
$742$		0			0
$743$		−	9.87946i		−	0.362442i	−0.983442	−	0.181221i	$-0.941995\pi$
$743$		−	9.87946i		−	0.362442i	0.983442	−	0.181221i	$-0.0580050\pi$
$744$		0			0
$745$		−	11.8917i		−	0.435678i
$746$		0			0
$747$	−34.6122	−	3.75680i	−1.26639	−	0.137454i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	21.0332			0.767513			0.383756	−	0.923434i	$-0.374630\pi$
$751$	21.0332			0.767513			0.383756	+	0.923434i	$0.374630\pi$
$752$		0			0
$753$	0.960960	−	17.7591i	0.0350193	−	0.647177i
$754$		0			0
$755$	−58.8410			−2.14144
$756$		0			0
$757$	−26.4152			−0.960076			−0.480038	−	0.877248i	$-0.659377\pi$
$757$	−26.4152			−0.960076			−0.480038	+	0.877248i	$0.659377\pi$
$758$		0			0
$759$	2.80200	−	51.7825i	0.101706	−	1.87959i
$760$		0			0
$761$	−9.24248			−0.335039			−0.167520	−	0.985869i	$-0.553576\pi$
$761$	−9.24248			−0.335039			−0.167520	+	0.985869i	$0.553576\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−8.01537	−	0.869986i	−0.289796	−	0.0314544i
$766$		0			0
$767$		−	46.8176i		−	1.69048i
$768$		0			0
$769$			1.52391i			0.0549536i	0.999622	+	0.0274768i	$0.00874724\pi$
$769$			1.52391i			0.0549536i	−0.999622	+	0.0274768i	$0.991253\pi$
$770$		0			0
$771$	−0.847863	+	15.6690i	−0.0305351	+	0.564305i
$772$		0			0
$773$	23.5055			0.845433			0.422716	−	0.906262i	$-0.361077\pi$
$773$	23.5055			0.845433			0.422716	+	0.906262i	$0.361077\pi$
$774$		0			0
$775$			20.0979i			0.721939i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			6.65550i			0.238458i
$780$		0			0
$781$	−23.8626			−0.853869
$782$		0			0
$783$	36.3770	+	5.95169i	1.30001	+	0.212696i
$784$		0			0
$785$		−	3.79187i		−	0.135338i
$786$		0			0
$787$		−	18.1870i		−	0.648295i	−0.946007	−	0.324147i	$-0.894923\pi$
$787$		−	18.1870i		−	0.648295i	0.946007	−	0.324147i	$-0.105077\pi$
$788$		0			0
$789$	29.3086	+	1.58592i	1.04341	+	0.0564601i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	14.9395			0.530517
$794$		0			0
$795$	14.1760	+	0.767074i	0.502769	+	0.0272053i
$796$		0			0
$797$	27.9561			0.990256			0.495128	−	0.868820i	$-0.335121\pi$
$797$	27.9561			0.990256			0.495128	+	0.868820i	$0.335121\pi$
$798$		0			0
$799$	−5.77981			−0.204475
$800$		0			0
$801$	−29.1253	−	3.16125i	−1.02909	−	0.111697i
$802$		0			0
$803$	−38.7346			−1.36692
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−1.73244	+	32.0165i	−0.0609848	+	1.12703i
$808$		0			0
$809$		−	3.44368i		−	0.121073i	−0.998166	−	0.0605367i	$-0.980719\pi$
$809$		−	3.44368i		−	0.121073i	0.998166	−	0.0605367i	$-0.0192812\pi$
$810$		0			0
$811$		−	14.3286i		−	0.503144i	−0.967839	−	0.251572i	$-0.919052\pi$
$811$		−	14.3286i		−	0.503144i	0.967839	−	0.251572i	$-0.0809475\pi$
$812$		0			0
$813$	11.7376	+	0.635130i	0.411654	+	0.0222750i
$814$		0			0
$815$	−66.4775			−2.32861
$816$		0			0
$817$		−	3.79361i		−	0.132722i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	11.2673i		−	0.393231i	−0.980481	−	0.196615i	$-0.937005\pi$
$821$		−	11.2673i		−	0.393231i	0.980481	−	0.196615i	$-0.0629950\pi$
$822$		0			0
$823$	5.56223			0.193887			0.0969437	−	0.995290i	$-0.469093\pi$
$823$	5.56223			0.193887			0.0969437	+	0.995290i	$0.469093\pi$
$824$		0			0
$825$	−37.7377	−	2.04202i	−1.31386	−	0.0710941i
$826$		0			0
$827$		−	35.5033i		−	1.23457i	−0.786739	−	0.617285i	$-0.788232\pi$
$827$		−	35.5033i		−	1.23457i	0.786739	−	0.617285i	$-0.211768\pi$
$828$		0			0
$829$		−	4.04828i		−	0.140603i	−0.997526	−	0.0703013i	$-0.977604\pi$
$829$		−	4.04828i		−	0.140603i	0.997526	−	0.0703013i	$-0.0223961\pi$
$830$		0			0
$831$	−0.815059	+	15.0627i	−0.0282741	+	0.522520i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−8.04385			−0.278369
$836$		0			0
$837$	−16.6770	−	2.72856i	−0.576443	−	0.0943127i
$838$		0			0
$839$	−16.0096			−0.552712			−0.276356	−	0.961055i	$-0.589127\pi$
$839$	−16.0096			−0.552712			−0.276356	+	0.961055i	$0.589127\pi$
$840$		0			0
$841$	−21.3225			−0.735257
$842$		0			0
$843$	20.3516	+	1.10124i	0.700947	+	0.0379289i
$844$		0			0
$845$	60.3245			2.07523
$846$		0			0
$847$		0			0
$848$		0			0
$849$	2.54223	+	0.137562i	0.0872491	+	0.00472113i
$850$		0			0
$851$			62.0432i			2.12681i
$852$		0			0
$853$		−	30.8458i		−	1.05614i	−0.849201	−	0.528070i	$-0.822916\pi$
$853$		−	30.8458i		−	1.05614i	0.849201	−	0.528070i	$-0.177084\pi$
$854$		0			0
$855$	0.665769	−	6.13388i	0.0227688	−	0.209774i
$856$		0			0
$857$	3.35224			0.114510			0.0572552	−	0.998360i	$-0.481765\pi$
$857$	3.35224			0.114510			0.0572552	+	0.998360i	$0.481765\pi$
$858$		0			0
$859$		−	24.2441i		−	0.827199i	−0.910459	−	0.413599i	$-0.864271\pi$
$859$		−	24.2441i		−	0.827199i	0.910459	−	0.413599i	$-0.135729\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	14.4740i		−	0.492700i	−0.969181	−	0.246350i	$-0.920769\pi$
$863$		−	14.4740i		−	0.492700i	0.969181	−	0.246350i	$-0.0792313\pi$
$864$		0			0
$865$	3.84847			0.130852
$866$		0			0
$867$	−1.53050	+	28.2845i	−0.0519786	+	0.960593i
$868$		0			0
$869$		−	35.2273i		−	1.19500i
$870$		0			0
$871$			22.3300i			0.756622i
$872$		0			0
$873$	4.46607	−	41.1469i	0.151154	−	1.39261i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	44.7027			1.50950			0.754752	−	0.656010i	$-0.227757\pi$
$877$	44.7027			1.50950			0.754752	+	0.656010i	$0.227757\pi$
$878$		0			0
$879$	−1.35287	+	25.0018i	−0.0456312	+	0.843289i
$880$		0			0
$881$	13.4140			0.451929			0.225964	−	0.974136i	$-0.427447\pi$
$881$	13.4140			0.451929			0.225964	+	0.974136i	$0.427447\pi$
$882$		0			0
$883$	38.1278			1.28310			0.641552	−	0.767080i	$-0.278291\pi$
$883$	38.1278			1.28310			0.641552	+	0.767080i	$0.278291\pi$
$884$		0			0
$885$	2.62945	−	48.5938i	0.0883881	−	1.63346i
$886$		0			0
$887$	56.2789			1.88966			0.944831	−	0.327559i	$-0.106226\pi$
$887$	56.2789			1.88966			0.944831	+	0.327559i	$0.106226\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	6.81783	−	31.0371i	0.228406	−	1.03978i
$892$		0			0
$893$		−	4.42308i		−	0.148013i
$894$		0			0
$895$			62.2064i			2.07933i
$896$		0			0
$897$	−4.42149	+	81.7117i	−0.147629	+	2.72827i
$898$		0			0
$899$	23.0704			0.769440
$900$		0			0
$901$			1.97032i			0.0656408i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			43.9695i			1.46160i
$906$		0			0
$907$	−18.5207			−0.614969			−0.307485	−	0.951553i	$-0.599487\pi$
$907$	−18.5207			−0.614969			−0.307485	+	0.951553i	$0.599487\pi$
$908$		0			0
$909$	−11.1624	−	1.21157i	−0.370235	−	0.0401852i
$910$		0			0
$911$		−	38.9950i		−	1.29196i	−0.763354	−	0.645981i	$-0.776449\pi$
$911$		−	38.9950i		−	1.29196i	0.763354	−	0.645981i	$-0.223551\pi$
$912$		0			0
$913$		−	40.9753i		−	1.35608i
$914$		0			0
$915$	15.5063	+	0.839059i	0.512621	+	0.0277384i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−47.0708			−1.55272			−0.776362	−	0.630288i	$-0.782937\pi$
$919$	−47.0708			−1.55272			−0.776362	+	0.630288i	$0.782937\pi$
$920$		0			0
$921$	−12.9880	−	0.702791i	−0.427968	−	0.0231578i
$922$		0			0
$923$	37.6546			1.23942
$924$		0			0
$925$	45.2154			1.48667
$926$		0			0
$927$	2.41818	−	22.2792i	0.0794235	−	0.731746i
$928$		0			0
$929$	33.2785			1.09183			0.545917	−	0.837839i	$-0.316181\pi$
$929$	33.2785			1.09183			0.545917	+	0.837839i	$0.316181\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	0.00287603	−	0.0531505i	9.41568e−5	−	0.00174007i
$934$		0			0
$935$		−	9.48892i		−	0.310321i
$936$		0			0
$937$			20.0613i			0.655376i	0.944786	+	0.327688i	$0.106269\pi$
$937$			20.0613i			0.655376i	−0.944786	+	0.327688i	$0.893731\pi$
$938$		0			0
$939$	7.70172	+	0.416748i	0.251336	+	0.0136000i
$940$		0			0
$941$	12.4078			0.404483			0.202241	−	0.979336i	$-0.435177\pi$
$941$	12.4078			0.404483			0.202241	+	0.979336i	$0.435177\pi$
$942$		0			0
$943$		−	91.7545i		−	2.98794i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	20.7424i		−	0.674039i	−0.941498	−	0.337019i	$-0.890581\pi$
$947$		−	20.7424i		−	0.674039i	0.941498	−	0.337019i	$-0.109419\pi$
$948$		0			0
$949$	61.1224			1.98412
$950$		0			0
$951$	−11.3151	−	0.612269i	−0.366916	−	0.0198542i
$952$		0			0
$953$			19.7250i			0.638957i	0.947594	+	0.319478i	$0.103508\pi$
$953$			19.7250i			0.638957i	−0.947594	+	0.319478i	$0.896492\pi$
$954$		0			0
$955$			16.9166i			0.547410i
$956$		0			0
$957$	−2.34403	+	43.3190i	−0.0757717	+	1.40030i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	20.4234			0.658819
$962$		0			0
$963$	2.00435	−	18.4665i	0.0645892	−	0.595075i
$964$		0			0
$965$	57.5057			1.85117
$966$		0			0
$967$	41.3604			1.33006			0.665031	−	0.746816i	$-0.268418\pi$
$967$	41.3604			1.33006			0.665031	+	0.746816i	$0.268418\pi$
$968$		0			0
$969$	0.855052	+	0.0462677i	0.0274682	+	0.00148633i
$970$		0			0
$971$	−18.9841			−0.609229			−0.304615	−	0.952476i	$-0.598528\pi$
$971$	−18.9841			−0.609229			−0.304615	+	0.952476i	$0.598528\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	59.5492	+	3.22227i	1.90710	+	0.103195i
$976$		0			0
$977$			37.3298i			1.19429i	0.802134	+	0.597144i	$0.203698\pi$
$977$			37.3298i			1.19429i	−0.802134	+	0.597144i	$0.796302\pi$
$978$		0			0
$979$		−	34.4797i		−	1.10198i
$980$		0			0
$981$	−9.40653	−	1.02098i	−0.300328	−	0.0325975i
$982$		0			0
$983$	6.64713			0.212010			0.106005	−	0.994366i	$-0.466194\pi$
$983$	6.64713			0.212010			0.106005	+	0.994366i	$0.466194\pi$
$984$		0			0
$985$		−	56.8646i		−	1.81186i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			52.2997i			1.66303i
$990$		0			0
$991$	38.6422			1.22751			0.613755	−	0.789496i	$-0.289658\pi$
$991$	38.6422			1.22751			0.613755	+	0.789496i	$0.289658\pi$
$992$		0			0
$993$	0.229627	−	4.24364i	0.00728701	−	0.134668i
$994$		0			0
$995$		−	26.9695i		−	0.854990i
$996$		0			0
$997$		−	20.7492i		−	0.657135i	−0.944480	−	0.328568i	$-0.893434\pi$
$997$		−	20.7492i		−	0.657135i	0.944480	−	0.328568i	$-0.106566\pi$
$998$		0			0
$999$	−6.13858	+	37.5192i	−0.194216	+	1.18706i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.2.k.j.881.13		24
3.2	odd	2	inner	2352.2.k.j.881.11		24
4.3	odd	2		1176.2.k.b.881.12	yes	24
7.6	odd	2	inner	2352.2.k.j.881.12		24
12.11	even	2		1176.2.k.b.881.14	yes	24
21.20	even	2	inner	2352.2.k.j.881.14		24
28.3	even	6		1176.2.u.c.1097.3		48
28.11	odd	6		1176.2.u.c.1097.22		48
28.19	even	6		1176.2.u.c.521.20		48
28.23	odd	6		1176.2.u.c.521.5		48
28.27	even	2		1176.2.k.b.881.13	yes	24
84.11	even	6		1176.2.u.c.1097.20		48
84.23	even	6		1176.2.u.c.521.3		48
84.47	odd	6		1176.2.u.c.521.22		48
84.59	odd	6		1176.2.u.c.1097.5		48
84.83	odd	2		1176.2.k.b.881.11	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.2.k.b.881.11	✓	24	84.83	odd	2
1176.2.k.b.881.12	yes	24	4.3	odd	2
1176.2.k.b.881.13	yes	24	28.27	even	2
1176.2.k.b.881.14	yes	24	12.11	even	2
1176.2.u.c.521.3		48	84.23	even	6
1176.2.u.c.521.5		48	28.23	odd	6
1176.2.u.c.521.20		48	28.19	even	6
1176.2.u.c.521.22		48	84.47	odd	6
1176.2.u.c.1097.3		48	28.3	even	6
1176.2.u.c.1097.5		48	84.59	odd	6
1176.2.u.c.1097.20		48	84.11	even	6
1176.2.u.c.1097.22		48	28.11	odd	6
2352.2.k.j.881.11		24	3.2	odd	2	inner
2352.2.k.j.881.12		24	7.6	odd	2	inner
2352.2.k.j.881.13		24	1.1	even	1	trivial
2352.2.k.j.881.14		24	21.20	even	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)

\(f(q)\)	\(=\)	\(q+(0.0935860 - 1.72952i) q^{3} -3.34363 q^{5} +(-2.98248 - 0.323718i) q^{9} -3.53079i q^{11} +5.57150i q^{13} +(-0.312917 + 5.78288i) q^{15} -0.803762 q^{17} -0.615091i q^{19} +8.47981i q^{23} +6.17985 q^{25} +(-0.838995 + 5.12797i) q^{27} -7.09383i q^{29} +3.25217i q^{31} +(-6.10657 - 0.330432i) q^{33} +7.31658 q^{37} +(9.63603 + 0.521414i) q^{39} -10.8204 q^{41} +6.16756 q^{43} +(9.97232 + 1.08239i) q^{45} +7.19094 q^{47} +(-0.0752209 + 1.39012i) q^{51} -2.45137i q^{53} +11.8056i q^{55} +(-1.06381 - 0.0575639i) q^{57} -8.40304 q^{59} -2.68141i q^{61} -18.6290i q^{65} +4.00789 q^{67} +(14.6660 + 0.793591i) q^{69} -6.75843i q^{71} -10.9705i q^{73} +(0.578348 - 10.6882i) q^{75} +9.97718 q^{79} +(8.79041 + 1.93097i) q^{81} +11.6052 q^{83} +2.68748 q^{85} +(-12.2689 - 0.663883i) q^{87} +9.76545 q^{89} +(5.62470 + 0.304358i) q^{93} +2.05664i q^{95} +13.7962i q^{97} +(-1.14298 + 10.5305i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 16 q^{15} + 8 q^{25} + 16 q^{37} + 64 q^{39} - 16 q^{43} - 48 q^{51} + 48 q^{57} - 16 q^{67} + 80 q^{81} - 64 q^{85} - 32 q^{93} + 56 q^{99}+O(q^{100}) \) 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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