LMFDB - Embedded newform 2736.2.f.h.1025.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(1025,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2736.1025");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.f (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:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 18x^{6} + 37x^{4} + 22x^{2} + 4 $ x^8 + 18x^6 + 37x^4 + 22*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1368)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1025.4
Root			$-3.96694i$ of defining polynomial
Character	$\chi$	$=$	2736.1025
Dual form			2736.2.f.h.1025.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0959660i q^{5} +3.22941 q^{7} +O(q^{10})$	$q-0.0959660i q^{5} +3.22941 q^{7} -0.634345i q^{11} +1.00833i q^{13} -3.46277i q^{17} +(-3.89710 + 1.95259i) q^{19} -6.51966i q^{23} +4.99079 q^{25} +6.76138 q^{29} +9.20257i q^{31} -0.309913i q^{35} -10.9542i q^{37} +10.2530 q^{41} +0.906309 q^{43} -6.82958i q^{47} +3.42907 q^{49} -4.18738 q^{53} -0.0608756 q^{55} -11.4916 q^{59} +7.41679 q^{61} +0.0967658 q^{65} +4.28905i q^{67} +5.42600 q^{71} -1.75831 q^{73} -2.04856i q^{77} +10.5704i q^{79} +3.62281i q^{83} -0.332308 q^{85} -3.30425 q^{89} +3.25632i q^{91} +(0.187383 + 0.373989i) q^{95} -6.30572i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 4 q^{19} - 8 q^{25} + 32 q^{29} - 24 q^{41} + 28 q^{43} + 4 q^{49} + 8 q^{53} - 12 q^{55} - 8 q^{59} - 8 q^{61} - 24 q^{65} + 24 q^{71} + 4 q^{73} - 4 q^{85} + 16 q^{89} - 40 q^{95}+O(q^{100}) $ 8 * q - 4 * q^19 - 8 * q^25 + 32 * q^29 - 24 * q^41 + 28 * q^43 + 4 * q^49 + 8 * q^53 - 12 * q^55 - 8 * q^59 - 8 * q^61 - 24 * q^65 + 24 * q^71 + 4 * q^73 - 4 * q^85 + 16 * q^89 - 40 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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			0
$3$		0			0
$4$		0			0
$5$		−	0.0959660i		−	0.0429173i	−0.999770	−	0.0214587i	$-0.993169\pi$
$5$		−	0.0959660i		−	0.0429173i	0.999770	−	0.0214587i	$-0.00683103\pi$
$6$		0			0
$7$	3.22941			1.22060			0.610301	−	0.792170i	$-0.291049\pi$
$7$	3.22941			1.22060			0.610301	+	0.792170i	$0.291049\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	0.634345i		−	0.191262i	−0.995417	−	0.0956311i	$-0.969513\pi$
$11$		−	0.634345i		−	0.191262i	0.995417	−	0.0956311i	$-0.0304869\pi$
$12$		0			0
$13$			1.00833i			0.279662i	0.990175	+	0.139831i	$0.0446559\pi$
$13$			1.00833i			0.279662i	−0.990175	+	0.139831i	$0.955344\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	3.46277i		−	0.839846i	−0.907560	−	0.419923i	$-0.862057\pi$
$17$		−	3.46277i		−	0.839846i	0.907560	−	0.419923i	$-0.137943\pi$
$18$		0			0
$19$	−3.89710	+	1.95259i	−0.894056	+	0.447955i
$19$	−3.89710	+	1.95259i	−0.894056	+	0.447955i
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	6.51966i		−	1.35944i	−0.733470	−	0.679722i	$-0.762100\pi$
$23$		−	6.51966i		−	1.35944i	0.733470	−	0.679722i	$-0.237900\pi$
$24$		0			0
$25$	4.99079			0.998158
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.76138			1.25556			0.627779	−	0.778392i	$-0.283964\pi$
$29$	6.76138			1.25556			0.627779	+	0.778392i	$0.283964\pi$
$30$		0			0
$31$			9.20257i			1.65283i	0.563061	+	0.826415i	$0.309623\pi$
$31$			9.20257i			1.65283i	−0.563061	+	0.826415i	$0.690377\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	0.309913i		−	0.0523849i
$36$		0			0
$37$		−	10.9542i		−	1.80087i	−0.434995	−	0.900433i	$-0.643250\pi$
$37$		−	10.9542i		−	1.80087i	0.434995	−	0.900433i	$-0.356750\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	10.2530			1.60125			0.800626	−	0.599165i	$-0.204500\pi$
$41$	10.2530			1.60125			0.800626	+	0.599165i	$0.204500\pi$
$42$		0			0
$43$	0.906309			0.138211			0.0691054	−	0.997609i	$-0.477986\pi$
$43$	0.906309			0.138211			0.0691054	+	0.997609i	$0.477986\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	6.82958i		−	0.996196i	−0.867121	−	0.498098i	$-0.834032\pi$
$47$		−	6.82958i		−	0.996196i	0.867121	−	0.498098i	$-0.165968\pi$
$48$		0			0
$49$	3.42907			0.489868
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.18738			−0.575181			−0.287591	−	0.957753i	$-0.592854\pi$
$53$	−4.18738			−0.575181			−0.287591	+	0.957753i	$0.592854\pi$
$54$		0			0
$55$	−0.0608756			−0.00820846
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−11.4916			−1.49608			−0.748041	−	0.663652i	$-0.769006\pi$
$59$	−11.4916			−1.49608			−0.748041	+	0.663652i	$0.769006\pi$
$60$		0			0
$61$	7.41679			0.949623			0.474811	−	0.880088i	$-0.342516\pi$
$61$	7.41679			0.949623			0.474811	+	0.880088i	$0.342516\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.0967658			0.0120023
$66$		0			0
$67$			4.28905i			0.523991i	0.965069	+	0.261995i	$0.0843805\pi$
$67$			4.28905i			0.523991i	−0.965069	+	0.261995i	$0.915619\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	5.42600			0.643948			0.321974	−	0.946749i	$-0.395654\pi$
$71$	5.42600			0.643948			0.321974	+	0.946749i	$0.395654\pi$
$72$		0			0
$73$	−1.75831			−0.205794			−0.102897	−	0.994692i	$-0.532811\pi$
$73$	−1.75831			−0.205794			−0.102897	+	0.994692i	$0.532811\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	2.04856i		−	0.233455i
$78$		0			0
$79$			10.5704i			1.18926i	0.804000	+	0.594630i	$0.202701\pi$
$79$			10.5704i			1.18926i	−0.804000	+	0.594630i	$0.797299\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			3.62281i			0.397655i	0.980034	+	0.198828i	$0.0637134\pi$
$83$			3.62281i			0.397655i	−0.980034	+	0.198828i	$0.936287\pi$
$84$		0			0
$85$	−0.332308			−0.0360439
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.30425			−0.350250			−0.175125	−	0.984546i	$-0.556033\pi$
$89$	−3.30425			−0.350250			−0.175125	+	0.984546i	$0.556033\pi$
$90$		0			0
$91$			3.25632i			0.341355i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.187383	+	0.373989i	0.0192250	+	0.0383705i
$96$		0			0
$97$		−	6.30572i		−	0.640249i	−0.947376	−	0.320124i	$-0.896275\pi$
$97$		−	6.30572i		−	0.640249i	0.947376	−	0.320124i	$-0.103725\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			3.49931i			0.348194i	0.984729	+	0.174097i	$0.0557006\pi$
$101$			3.49931i			0.348194i	−0.984729	+	0.174097i	$0.944299\pi$
$102$		0			0
$103$		−	12.9709i		−	1.27806i	−0.769181	−	0.639031i	$-0.779336\pi$
$103$		−	12.9709i		−	1.27806i	0.769181	−	0.639031i	$-0.220664\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.09677			0.396049			0.198025	−	0.980197i	$-0.436547\pi$
$107$	4.09677			0.396049			0.198025	+	0.980197i	$0.436547\pi$
$108$		0			0
$109$			6.66519i			0.638409i	0.947686	+	0.319205i	$0.103416\pi$
$109$			6.66519i			0.638409i	−0.947686	+	0.319205i	$0.896584\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	13.7101			1.28974			0.644871	−	0.764291i	$-0.276911\pi$
$113$	13.7101			1.28974			0.644871	+	0.764291i	$0.276911\pi$
$114$		0			0
$115$	−0.625666			−0.0583437
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	11.1827i		−	1.02512i
$120$		0			0
$121$	10.5976			0.963419
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	0.958776i		−	0.0857556i
$126$		0			0
$127$		−	6.66519i		−	0.591440i	−0.955275	−	0.295720i	$-0.904440\pi$
$127$		−	6.66519i		−	0.591440i	0.955275	−	0.295720i	$-0.0955595\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	13.9517i		−	1.21897i	−0.792799	−	0.609483i	$-0.791377\pi$
$131$		−	13.9517i		−	1.21897i	0.792799	−	0.609483i	$-0.208623\pi$
$132$		0			0
$133$	−12.5853	+	6.30572i	−1.09129	+	0.546775i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	19.2444i		−	1.64416i	−0.569370	−	0.822082i	$-0.692813\pi$
$137$		−	19.2444i		−	1.64416i	0.569370	−	0.822082i	$-0.307187\pi$
$138$		0			0
$139$	9.78499			0.829952			0.414976	−	0.909832i	$-0.363790\pi$
$139$	9.78499			0.829952			0.414976	+	0.909832i	$0.363790\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.639632			0.0534887
$144$		0			0
$145$		−	0.648863i		−	0.0538851i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			3.10330i			0.254232i	0.991888	+	0.127116i	$0.0405721\pi$
$149$			3.10330i			0.254232i	−0.991888	+	0.127116i	$0.959428\pi$
$150$		0			0
$151$			19.0116i			1.54714i	0.633708	+	0.773572i	$0.281532\pi$
$151$			19.0116i			1.54714i	−0.633708	+	0.773572i	$0.718468\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.883134			0.0709350
$156$		0			0
$157$	13.4260			1.07151			0.535756	−	0.844373i	$-0.320027\pi$
$157$	13.4260			1.07151			0.535756	+	0.844373i	$0.320027\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	21.0547i		−	1.65934i
$162$		0			0
$163$	3.25133			0.254664			0.127332	−	0.991860i	$-0.459359\pi$
$163$	3.25133			0.254664			0.127332	+	0.991860i	$0.459359\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−5.96887			−0.461885			−0.230942	−	0.972967i	$-0.574181\pi$
$167$	−5.96887			−0.461885			−0.230942	+	0.972967i	$0.574181\pi$
$168$		0			0
$169$	11.9833			0.921789
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.73025			0.359634			0.179817	−	0.983700i	$-0.442449\pi$
$173$	4.73025			0.359634			0.179817	+	0.983700i	$0.442449\pi$
$174$		0			0
$175$	16.1173			1.21835
$176$		0			0
$177$		0			0
$178$		0			0
$179$	18.0656			1.35029			0.675144	−	0.737686i	$-0.264081\pi$
$179$	18.0656			1.35029			0.675144	+	0.737686i	$0.264081\pi$
$180$		0			0
$181$		−	7.04905i		−	0.523952i	−0.965074	−	0.261976i	$-0.915626\pi$
$181$		−	7.04905i		−	0.523952i	0.965074	−	0.261976i	$-0.0843741\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.05123			−0.0772883
$186$		0			0
$187$	−2.19659			−0.160631
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	9.17503i		−	0.663882i	−0.943300	−	0.331941i	$-0.892297\pi$
$191$		−	9.17503i		−	0.663882i	0.943300	−	0.331941i	$-0.107703\pi$
$192$		0			0
$193$			2.53738i			0.182645i	0.995821	+	0.0913223i	$0.0291093\pi$
$193$			2.53738i			0.182645i	−0.995821	+	0.0913223i	$0.970891\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			14.7690i			1.05225i	0.850408	+	0.526123i	$0.176355\pi$
$197$			14.7690i			1.05225i	−0.850408	+	0.526123i	$0.823645\pi$
$198$		0			0
$199$	−3.09369			−0.219306			−0.109653	−	0.993970i	$-0.534974\pi$
$199$	−3.09369			−0.219306			−0.109653	+	0.993970i	$0.534974\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	21.8353			1.53253
$204$		0			0
$205$		−	0.983941i		−	0.0687214i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.23862	+	2.47211i	0.0856769	+	0.170999i
$210$		0			0
$211$			16.1328i			1.11062i	0.831642	+	0.555312i	$0.187401\pi$
$211$			16.1328i			1.11062i	−0.831642	+	0.555312i	$0.812599\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	0.0869748i		−	0.00593164i
$216$		0			0
$217$			29.7188i			2.01745i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3.49163			0.234873
$222$		0			0
$223$		−	20.2513i		−	1.35613i	−0.735004	−	0.678063i	$-0.762820\pi$
$223$		−	20.2513i		−	1.35613i	0.735004	−	0.678063i	$-0.237180\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−20.3748			−1.35232			−0.676160	−	0.736755i	$-0.736357\pi$
$227$	−20.3748			−1.35232			−0.676160	+	0.736755i	$0.736357\pi$
$228$		0			0
$229$	−8.02192			−0.530104			−0.265052	−	0.964234i	$-0.585389\pi$
$229$	−8.02192			−0.530104			−0.265052	+	0.964234i	$0.585389\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	26.2561i		−	1.72009i	−0.510216	−	0.860046i	$-0.670435\pi$
$233$		−	26.2561i		−	1.72009i	0.510216	−	0.860046i	$-0.329565\pi$
$234$		0			0
$235$	−0.655407			−0.0427541
$236$		0			0
$237$		0			0
$238$		0			0
$239$			8.84625i			0.572216i	0.958197	+	0.286108i	$0.0923616\pi$
$239$			8.84625i			0.572216i	−0.958197	+	0.286108i	$0.907638\pi$
$240$		0			0
$241$			11.5787i			0.745850i	0.927861	+	0.372925i	$0.121645\pi$
$241$			11.5787i			0.745850i	−0.927861	+	0.372925i	$0.878355\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	0.329075i		−	0.0210238i
$246$		0			0
$247$	−1.96887	−	3.92958i	−0.125276	−	0.250033i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	16.0011i		−	1.00998i	−0.863124	−	0.504991i	$-0.831496\pi$
$251$		−	16.0011i		−	1.00998i	0.863124	−	0.504991i	$-0.168504\pi$
$252$		0			0
$253$	−4.13572			−0.260010
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−12.2219			−0.762380			−0.381190	−	0.924497i	$-0.624486\pi$
$257$	−12.2219			−0.762380			−0.381190	+	0.924497i	$0.624486\pi$
$258$		0			0
$259$		−	35.3757i		−	2.19814i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			16.1426i			0.995397i	0.867350	+	0.497699i	$0.165821\pi$
$263$			16.1426i			0.995397i	−0.867350	+	0.497699i	$0.834179\pi$
$264$		0			0
$265$			0.401846i			0.0246852i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−21.2985			−1.29860			−0.649298	−	0.760534i	$-0.724937\pi$
$269$	−21.2985			−1.29860			−0.649298	+	0.760534i	$0.724937\pi$
$270$		0			0
$271$	20.3748			1.23768			0.618839	−	0.785518i	$-0.287603\pi$
$271$	20.3748			1.23768			0.618839	+	0.785518i	$0.287603\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	3.16588i		−	0.190910i
$276$		0			0
$277$	−21.0892			−1.26713			−0.633565	−	0.773690i	$-0.718409\pi$
$277$	−21.0892			−1.26713			−0.633565	+	0.773690i	$0.718409\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	15.4010			0.918747			0.459374	−	0.888243i	$-0.348074\pi$
$281$	15.4010			0.918747			0.459374	+	0.888243i	$0.348074\pi$
$282$		0			0
$283$	14.0770			0.836788			0.418394	−	0.908266i	$-0.362593\pi$
$283$	14.0770			0.836788			0.418394	+	0.908266i	$0.362593\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	33.1112			1.95449
$288$		0			0
$289$	5.00921			0.294659
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−32.6934			−1.90997			−0.954985	−	0.296655i	$-0.904129\pi$
$293$	−32.6934			−1.90997			−0.954985	+	0.296655i	$0.904129\pi$
$294$		0			0
$295$			1.10281i			0.0642079i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.57400			0.380184
$300$		0			0
$301$	2.92684			0.168700
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	0.711760i		−	0.0407553i
$306$		0			0
$307$		−	11.8344i		−	0.675426i	−0.941249	−	0.337713i	$-0.890347\pi$
$307$		−	11.8344i		−	0.675426i	0.941249	−	0.337713i	$-0.109653\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			13.0622i			0.740691i	0.928894	+	0.370346i	$0.120761\pi$
$311$			13.0622i			0.740691i	−0.928894	+	0.370346i	$0.879239\pi$
$312$		0			0
$313$	2.47723			0.140022			0.0700108	−	0.997546i	$-0.477697\pi$
$313$	2.47723			0.140022			0.0700108	+	0.997546i	$0.477697\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.187383			0.0105245			0.00526223	−	0.999986i	$-0.498325\pi$
$317$	0.187383			0.0105245			0.00526223	+	0.999986i	$0.498325\pi$
$318$		0			0
$319$		−	4.28905i		−	0.240141i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	6.76138	+	13.4948i	0.376213	+	0.750869i
$324$		0			0
$325$			5.03238i			0.279146i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	22.0555i		−	1.21596i
$330$		0			0
$331$		−	10.9298i		−	0.600759i	−0.953820	−	0.300379i	$-0.902887\pi$
$331$		−	10.9298i		−	0.600759i	0.953820	−	0.300379i	$-0.0971133\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	0.411603			0.0224883
$336$		0			0
$337$			31.0867i			1.69340i	0.532072	+	0.846699i	$0.321413\pi$
$337$			31.0867i			1.69340i	−0.532072	+	0.846699i	$0.678587\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	5.83760			0.316124
$342$		0			0
$343$	−11.5320			−0.622668
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	7.57291i		−	0.406535i	−0.979123	−	0.203268i	$-0.934844\pi$
$347$		−	7.57291i		−	0.406535i	0.979123	−	0.203268i	$-0.0651562\pi$
$348$		0			0
$349$	−34.4579			−1.84449			−0.922244	−	0.386609i	$-0.873646\pi$
$349$	−34.4579			−1.84449			−0.922244	+	0.386609i	$0.873646\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			31.5262i			1.67797i	0.544154	+	0.838985i	$0.316851\pi$
$353$			31.5262i			1.67797i	−0.544154	+	0.838985i	$0.683149\pi$
$354$		0			0
$355$		−	0.520712i		−	0.0276365i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			26.3375i			1.39004i	0.718990	+	0.695021i	$0.244605\pi$
$359$			26.3375i			1.39004i	−0.718990	+	0.695021i	$0.755395\pi$
$360$		0			0
$361$	11.3748	−	15.2189i	0.598672	−	0.800994i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			0.168738i			0.00883214i
$366$		0			0
$367$	−10.6085			−0.553759			−0.276880	−	0.960905i	$-0.589300\pi$
$367$	−10.6085			−0.553759			−0.276880	+	0.960905i	$0.589300\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−13.5228			−0.702067
$372$		0			0
$373$			20.2513i			1.04857i	0.851542	+	0.524286i	$0.175668\pi$
$373$			20.2513i			1.04857i	−0.851542	+	0.524286i	$0.824332\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			6.81773i			0.351131i
$378$		0			0
$379$			14.9539i			0.768130i	0.923306	+	0.384065i	$0.125476\pi$
$379$			14.9539i			0.768130i	−0.923306	+	0.384065i	$0.874524\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−18.3436			−0.937316			−0.468658	−	0.883380i	$-0.655262\pi$
$383$	−18.3436			−0.937316			−0.468658	+	0.883380i	$0.655262\pi$
$384$		0			0
$385$	−0.196592			−0.0100193
$386$		0			0
$387$		0			0
$388$		0			0
$389$			37.3762i			1.89505i	0.319684	+	0.947524i	$0.396423\pi$
$389$			37.3762i			1.89505i	−0.319684	+	0.947524i	$0.603577\pi$
$390$		0			0
$391$	−22.5761			−1.14172
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.01440			0.0510398
$396$		0			0
$397$	−22.9912			−1.15390			−0.576948	−	0.816781i	$-0.695757\pi$
$397$	−22.9912			−1.15390			−0.576948	+	0.816781i	$0.695757\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	6.25301			0.312261			0.156130	−	0.987736i	$-0.450098\pi$
$401$	6.25301			0.312261			0.156130	+	0.987736i	$0.450098\pi$
$402$		0			0
$403$	−9.27926			−0.462233
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−6.94877			−0.344438
$408$		0			0
$409$		−	18.0033i		−	0.890205i	−0.895480	−	0.445103i	$-0.853167\pi$
$409$		−	18.0033i		−	0.890205i	0.895480	−	0.445103i	$-0.146833\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−37.1112			−1.82612
$414$		0			0
$415$	0.347667			0.0170663
$416$		0			0
$417$		0			0
$418$		0			0
$419$			23.6515i			1.15545i	0.816232	+	0.577725i	$0.196059\pi$
$419$			23.6515i			1.15545i	−0.816232	+	0.577725i	$0.803941\pi$
$420$		0			0
$421$		−	30.4802i		−	1.48551i	−0.669562	−	0.742756i	$-0.733518\pi$
$421$		−	30.4802i		−	1.48551i	0.669562	−	0.742756i	$-0.266482\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	17.2820i		−	0.838299i
$426$		0			0
$427$	23.9518			1.15911
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−21.2924			−1.02562			−0.512809	−	0.858503i	$-0.671395\pi$
$431$	−21.2924			−1.02562			−0.512809	+	0.858503i	$0.671395\pi$
$432$		0			0
$433$		−	14.4332i		−	0.693615i	−0.937936	−	0.346807i	$-0.887266\pi$
$433$		−	14.4332i		−	0.693615i	0.937936	−	0.346807i	$-0.112734\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	12.7302	+	25.4078i	0.608970	+	1.21542i
$438$		0			0
$439$		−	26.0543i		−	1.24350i	−0.783215	−	0.621751i	$-0.786422\pi$
$439$		−	26.0543i		−	1.24350i	0.783215	−	0.621751i	$-0.213578\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	12.8489i		−	0.610469i	−0.952277	−	0.305235i	$-0.901265\pi$
$443$		−	12.8489i		−	0.610469i	0.952277	−	0.305235i	$-0.0987349\pi$
$444$		0			0
$445$			0.317096i			0.0150318i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−27.6790			−1.30625			−0.653127	−	0.757249i	$-0.726543\pi$
$449$	−27.6790			−1.30625			−0.653127	+	0.757249i	$0.726543\pi$
$450$		0			0
$451$		−	6.50395i		−	0.306259i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	0.312496			0.0146501
$456$		0			0
$457$	−2.61646			−0.122393			−0.0611963	−	0.998126i	$-0.519492\pi$
$457$	−2.61646			−0.122393			−0.0611963	+	0.998126i	$0.519492\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	1.69776i		−	0.0790727i	−0.999218	−	0.0395364i	$-0.987412\pi$
$461$		−	1.69776i		−	0.0790727i	0.999218	−	0.0395364i	$-0.0125881\pi$
$462$		0			0
$463$	−21.6558			−1.00643			−0.503216	−	0.864161i	$-0.667850\pi$
$463$	−21.6558			−1.00643			−0.503216	+	0.864161i	$0.667850\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			39.6305i			1.83388i	0.399025	+	0.916940i	$0.369349\pi$
$467$			39.6305i			1.83388i	−0.399025	+	0.916940i	$0.630651\pi$
$468$		0			0
$469$			13.8511i			0.639584i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	0.574912i		−	0.0264345i
$474$		0			0
$475$	−19.4496	+	9.74498i	−0.892409	+	0.447130i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	6.81071i		−	0.311189i	−0.987821	−	0.155595i	$-0.950271\pi$
$479$		−	6.81071i		−	0.311189i	0.987821	−	0.155595i	$-0.0497294\pi$
$480$		0			0
$481$	11.0455			0.503633
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−0.605135			−0.0274777
$486$		0			0
$487$		−	8.28871i		−	0.375597i	−0.982208	−	0.187799i	$-0.939865\pi$
$487$		−	8.28871i		−	0.375597i	0.982208	−	0.187799i	$-0.0601352\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	21.9792i		−	0.991906i	−0.868349	−	0.495953i	$-0.834819\pi$
$491$		−	21.9792i		−	0.991906i	0.868349	−	0.495953i	$-0.165181\pi$
$492$		0			0
$493$		−	23.4131i		−	1.05447i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	17.5228			0.786003
$498$		0			0
$499$	9.59146			0.429373			0.214686	−	0.976683i	$-0.431127\pi$
$499$	9.59146			0.429373			0.214686	+	0.976683i	$0.431127\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			11.1665i			0.497891i	0.968518	+	0.248945i	$0.0800840\pi$
$503$			11.1665i			0.497891i	−0.968518	+	0.248945i	$0.919916\pi$
$504$		0			0
$505$	0.335814			0.0149435
$506$		0			0
$507$		0			0
$508$		0			0
$509$	20.6335			0.914563			0.457282	−	0.889322i	$-0.348823\pi$
$509$	20.6335			0.914563			0.457282	+	0.889322i	$0.348823\pi$
$510$		0			0
$511$	−5.67829			−0.251193
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.24477			−0.0548510
$516$		0			0
$517$	−4.33231			−0.190535
$518$		0			0
$519$		0			0
$520$		0			0
$521$	10.5310			0.461372			0.230686	−	0.973028i	$-0.425903\pi$
$521$	10.5310			0.461372			0.230686	+	0.973028i	$0.425903\pi$
$522$		0			0
$523$			15.0820i			0.659492i	0.944070	+	0.329746i	$0.106963\pi$
$523$			15.0820i			0.659492i	−0.944070	+	0.329746i	$0.893037\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	31.8664			1.38812
$528$		0			0
$529$	−19.5060			−0.848088
$530$		0			0
$531$		0			0
$532$		0			0
$533$			10.3385i			0.447809i
$534$		0			0
$535$		−	0.393150i		−	0.0169974i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	2.17522i		−	0.0936932i
$540$		0			0
$541$	14.6165			0.628411			0.314205	−	0.949355i	$-0.398262\pi$
$541$	14.6165			0.628411			0.314205	+	0.949355i	$0.398262\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	0.639632			0.0273988
$546$		0			0
$547$			30.6848i			1.31199i	0.754766	+	0.655994i	$0.227750\pi$
$547$			30.6848i			1.31199i	−0.754766	+	0.655994i	$0.772250\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−26.3498	+	13.2022i	−1.12254	+	0.562434i
$552$		0			0
$553$			34.1360i			1.45161i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			19.7958i			0.838777i	0.907807	+	0.419388i	$0.137755\pi$
$557$			19.7958i			0.838777i	−0.907807	+	0.419388i	$0.862245\pi$
$558$		0			0
$559$			0.913862i			0.0386522i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	5.46284			0.230231			0.115116	−	0.993352i	$-0.463276\pi$
$563$	5.46284			0.230231			0.115116	+	0.993352i	$0.463276\pi$
$564$		0			0
$565$		−	1.31571i		−	0.0553523i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	6.38662			0.267741			0.133870	−	0.990999i	$-0.457259\pi$
$569$	6.38662			0.267741			0.133870	+	0.990999i	$0.457259\pi$
$570$		0			0
$571$	14.6085			0.611347			0.305673	−	0.952136i	$-0.401118\pi$
$571$	14.6085			0.611347			0.305673	+	0.952136i	$0.401118\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	32.5383i		−	1.35694i
$576$		0			0
$577$	−1.46634			−0.0610447			−0.0305223	−	0.999534i	$-0.509717\pi$
$577$	−1.46634			−0.0610447			−0.0305223	+	0.999534i	$0.509717\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			11.6995i			0.485379i
$582$		0			0
$583$			2.65625i			0.110010i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			9.43971i			0.389619i	0.980841	+	0.194809i	$0.0624088\pi$
$587$			9.43971i			0.389619i	−0.980841	+	0.194809i	$0.937591\pi$
$588$		0			0
$589$	−17.9689	−	35.8633i	−0.740394	−	1.47772i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	16.8541i		−	0.692114i	−0.938214	−	0.346057i	$-0.887520\pi$
$593$		−	16.8541i		−	0.692114i	0.938214	−	0.346057i	$-0.112480\pi$
$594$		0			0
$595$	−1.07316			−0.0439953
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−19.6852			−0.804314			−0.402157	−	0.915571i	$-0.631739\pi$
$599$	−19.6852			−0.804314			−0.402157	+	0.915571i	$0.631739\pi$
$600$		0			0
$601$			21.3878i			0.872425i	0.899844	+	0.436213i	$0.143680\pi$
$601$			21.3878i			0.872425i	−0.899844	+	0.436213i	$0.856320\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	1.01701i		−	0.0413473i
$606$		0			0
$607$			15.1945i			0.616726i	0.951269	+	0.308363i	$0.0997811\pi$
$607$			15.1945i			0.616726i	−0.951269	+	0.308363i	$0.900219\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	6.88650			0.278598
$612$		0			0
$613$	14.2872			0.577055			0.288527	−	0.957472i	$-0.406834\pi$
$613$	14.2872			0.577055			0.288527	+	0.957472i	$0.406834\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	32.9587i		−	1.32687i	−0.748236	−	0.663433i	$-0.769099\pi$
$617$		−	32.9587i		−	1.32687i	0.748236	−	0.663433i	$-0.230901\pi$
$618$		0			0
$619$	−43.3203			−1.74119			−0.870595	−	0.492000i	$-0.836266\pi$
$619$	−43.3203			−1.74119			−0.870595	+	0.492000i	$0.836266\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−10.6708			−0.427515
$624$		0			0
$625$	24.8619			0.994478
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−37.9320			−1.51245
$630$		0			0
$631$	10.2495			0.408026			0.204013	−	0.978968i	$-0.434601\pi$
$631$	10.2495			0.408026			0.204013	+	0.978968i	$0.434601\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−0.639632			−0.0253830
$636$		0			0
$637$			3.45765i			0.136997i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.97501			0.0780084			0.0390042	−	0.999239i	$-0.487581\pi$
$641$	1.97501			0.0780084			0.0390042	+	0.999239i	$0.487581\pi$
$642$		0			0
$643$	−2.13922			−0.0843627			−0.0421813	−	0.999110i	$-0.513431\pi$
$643$	−2.13922			−0.0843627			−0.0421813	+	0.999110i	$0.513431\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			17.6003i			0.691938i	0.938246	+	0.345969i	$0.112450\pi$
$647$			17.6003i			0.691938i	−0.938246	+	0.345969i	$0.887550\pi$
$648$		0			0
$649$			7.28966i			0.286144i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			44.4429i			1.73919i	0.493770	+	0.869593i	$0.335619\pi$
$653$			44.4429i			1.73919i	−0.493770	+	0.869593i	$0.664381\pi$
$654$		0			0
$655$	−1.33889			−0.0523147
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−32.0944			−1.25022			−0.625111	−	0.780536i	$-0.714946\pi$
$659$	−32.0944			−1.25022			−0.625111	+	0.780536i	$0.714946\pi$
$660$		0			0
$661$		−	50.8747i		−	1.97880i	−0.145228	−	0.989398i	$-0.546392\pi$
$661$		−	50.8747i		−	1.97880i	0.145228	−	0.989398i	$-0.453608\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.605135	+	1.20776i	0.0234661	+	0.0468351i
$666$		0			0
$667$		−	44.0819i		−	1.70686i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	4.70480i		−	0.181627i
$672$		0			0
$673$		−	29.3838i		−	1.13266i	−0.824178	−	0.566330i	$-0.808363\pi$
$673$		−	29.3838i		−	1.13266i	0.824178	−	0.566330i	$-0.191637\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−32.1874			−1.23706			−0.618531	−	0.785761i	$-0.712272\pi$
$677$	−32.1874			−1.23706			−0.618531	+	0.785761i	$0.712272\pi$
$678$		0			0
$679$		−	20.3637i		−	0.781488i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−11.8065			−0.451762			−0.225881	−	0.974155i	$-0.572526\pi$
$683$	−11.8065			−0.451762			−0.225881	+	0.974155i	$0.572526\pi$
$684$		0			0
$685$	−1.84681			−0.0705631
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	4.22228i		−	0.160856i
$690$		0			0
$691$	−39.7933			−1.51381			−0.756903	−	0.653527i	$-0.773289\pi$
$691$	−39.7933			−1.51381			−0.756903	+	0.653527i	$0.773289\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	0.939027i		−	0.0356193i
$696$		0			0
$697$		−	35.5039i		−	1.34480i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			39.5499i			1.49378i	0.664948	+	0.746890i	$0.268454\pi$
$701$			39.5499i			1.49378i	−0.664948	+	0.746890i	$0.731546\pi$
$702$		0			0
$703$	21.3892	+	42.6898i	0.806708	+	1.61007i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			11.3007i			0.425006i
$708$		0			0
$709$	−42.6373			−1.60128			−0.800639	−	0.599148i	$-0.795506\pi$
$709$	−42.6373			−1.60128			−0.800639	+	0.599148i	$0.795506\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	59.9977			2.24693
$714$		0			0
$715$		−	0.0613829i		−	0.00229559i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			10.8209i			0.403550i	0.979432	+	0.201775i	$0.0646709\pi$
$719$			10.8209i			0.403550i	−0.979432	+	0.201775i	$0.935329\pi$
$720$		0			0
$721$		−	41.8883i		−	1.56000i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	33.7446			1.25324
$726$		0			0
$727$	−13.7022			−0.508186			−0.254093	−	0.967180i	$-0.581777\pi$
$727$	−13.7022			−0.508186			−0.254093	+	0.967180i	$0.581777\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	3.13834i		−	0.116076i
$732$		0			0
$733$	−19.6540			−0.725938			−0.362969	−	0.931801i	$-0.618237\pi$
$733$	−19.6540			−0.725938			−0.362969	+	0.931801i	$0.618237\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2.72074			0.100220
$738$		0			0
$739$	−40.2070			−1.47904			−0.739519	−	0.673136i	$-0.764947\pi$
$739$	−40.2070			−1.47904			−0.739519	+	0.673136i	$0.764947\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−43.9263			−1.61150			−0.805750	−	0.592255i	$-0.798238\pi$
$743$	−43.9263			−1.61150			−0.805750	+	0.592255i	$0.798238\pi$
$744$		0			0
$745$	0.297811			0.0109110
$746$		0			0
$747$		0			0
$748$		0			0
$749$	13.2301			0.483418
$750$		0			0
$751$			32.9996i			1.20417i	0.798432	+	0.602086i	$0.205663\pi$
$751$			32.9996i			1.20417i	−0.798432	+	0.602086i	$0.794337\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.82447			0.0663993
$756$		0			0
$757$	39.0052			1.41767			0.708834	−	0.705376i	$-0.249222\pi$
$757$	39.0052			1.41767			0.708834	+	0.705376i	$0.249222\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			51.4302i			1.86434i	0.362019	+	0.932171i	$0.382088\pi$
$761$			51.4302i			1.86434i	−0.362019	+	0.932171i	$0.617912\pi$
$762$		0			0
$763$			21.5246i			0.779243i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	11.5874i		−	0.418397i
$768$		0			0
$769$	32.4496			1.17016			0.585081	−	0.810975i	$-0.301063\pi$
$769$	32.4496			1.17016			0.585081	+	0.810975i	$0.301063\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	4.25872			0.153175			0.0765877	−	0.997063i	$-0.475597\pi$
$773$	4.25872			0.153175			0.0765877	+	0.997063i	$0.475597\pi$
$774$		0			0
$775$			45.9281i			1.64979i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−39.9570	+	20.0200i	−1.43161	+	0.717289i
$780$		0			0
$781$		−	3.44196i		−	0.123163i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	1.28844i		−	0.0459864i
$786$		0			0
$787$		−	16.7056i		−	0.595489i	−0.954646	−	0.297745i	$-0.903766\pi$
$787$		−	16.7056i		−	0.595489i	0.954646	−	0.297745i	$-0.0962344\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	44.2757			1.57426
$792$		0			0
$793$			7.47860i			0.265573i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−18.9058			−0.669677			−0.334839	−	0.942276i	$-0.608682\pi$
$797$	−18.9058			−0.669677			−0.334839	+	0.942276i	$0.608682\pi$
$798$		0			0
$799$	−23.6493			−0.836651
$800$		0			0
$801$		0			0
$802$		0			0
$803$			1.11537i			0.0393607i
$804$		0			0
$805$	−2.02053			−0.0712144
$806$		0			0
$807$		0			0
$808$		0			0
$809$			25.2054i			0.886173i	0.896479	+	0.443087i	$0.146117\pi$
$809$			25.2054i			0.886173i	−0.896479	+	0.443087i	$0.853883\pi$
$810$		0			0
$811$			22.9592i			0.806206i	0.915155	+	0.403103i	$0.132068\pi$
$811$			22.9592i			0.806206i	−0.915155	+	0.403103i	$0.867932\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	0.312017i		−	0.0109295i
$816$		0			0
$817$	−3.53198	+	1.76965i	−0.123568	+	0.0619123i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			10.8369i			0.378209i	0.981957	+	0.189105i	$0.0605585\pi$
$821$			10.8369i			0.378209i	−0.981957	+	0.189105i	$0.939441\pi$
$822$		0			0
$823$	−13.8195			−0.481717			−0.240859	−	0.970560i	$-0.577429\pi$
$823$	−13.8195			−0.481717			−0.240859	+	0.970560i	$0.577429\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.69087			−0.0587972			−0.0293986	−	0.999568i	$-0.509359\pi$
$827$	−1.69087			−0.0587972			−0.0293986	+	0.999568i	$0.509359\pi$
$828$		0			0
$829$			30.0696i			1.04436i	0.852835	+	0.522181i	$0.174881\pi$
$829$			30.0696i			1.04436i	−0.852835	+	0.522181i	$0.825119\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	11.8741i		−	0.411413i
$834$		0			0
$835$			0.572808i			0.0198228i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	12.6741			0.437560			0.218780	−	0.975774i	$-0.429792\pi$
$839$	12.6741			0.437560			0.218780	+	0.975774i	$0.429792\pi$
$840$		0			0
$841$	16.7163			0.576424
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	1.14999i		−	0.0395607i
$846$		0			0
$847$	34.2240			1.17595
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−71.4180			−2.44818
$852$		0			0
$853$	−42.3248			−1.44917			−0.724587	−	0.689184i	$-0.757969\pi$
$853$	−42.3248			−1.44917			−0.724587	+	0.689184i	$0.757969\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−33.1395			−1.13202			−0.566012	−	0.824397i	$-0.691514\pi$
$857$	−33.1395			−1.13202			−0.566012	+	0.824397i	$0.691514\pi$
$858$		0			0
$859$	1.45407			0.0496124			0.0248062	−	0.999692i	$-0.492103\pi$
$859$	1.45407			0.0496124			0.0248062	+	0.999692i	$0.492103\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	20.9799			0.714164			0.357082	−	0.934073i	$-0.383772\pi$
$863$	20.9799			0.714164			0.357082	+	0.934073i	$0.383772\pi$
$864$		0			0
$865$		−	0.453943i		−	0.0154345i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	6.70526			0.227460
$870$		0			0
$871$	−4.32479			−0.146540
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	3.09628i		−	0.104673i
$876$		0			0
$877$			21.0550i			0.710976i	0.934681	+	0.355488i	$0.115685\pi$
$877$			21.0550i			0.710976i	−0.934681	+	0.355488i	$0.884315\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	39.7276i		−	1.33846i	−0.743056	−	0.669229i	$-0.766625\pi$
$881$		−	39.7276i		−	1.33846i	0.743056	−	0.669229i	$-0.233375\pi$
$882$		0			0
$883$	18.5714			0.624976			0.312488	−	0.949922i	$-0.398838\pi$
$883$	18.5714			0.624976			0.312488	+	0.949922i	$0.398838\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	24.8332			0.833816			0.416908	−	0.908949i	$-0.363114\pi$
$887$	24.8332			0.833816			0.416908	+	0.908949i	$0.363114\pi$
$888$		0			0
$889$		−	21.5246i		−	0.721912i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	13.3354	+	26.6155i	0.446252	+	0.890655i
$894$		0			0
$895$		−	1.73369i		−	0.0579508i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			62.2221i			2.07522i
$900$		0			0
$901$			14.5000i			0.483063i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.676470			−0.0224866
$906$		0			0
$907$		−	53.2421i		−	1.76788i	−0.467604	−	0.883938i	$-0.654883\pi$
$907$		−	53.2421i		−	1.76788i	0.467604	−	0.883938i	$-0.345117\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−27.2481			−0.902771			−0.451385	−	0.892329i	$-0.649070\pi$
$911$	−27.2481			−0.902771			−0.451385	+	0.892329i	$0.649070\pi$
$912$		0			0
$913$	2.29811			0.0760565
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	45.0557i		−	1.48787i
$918$		0			0
$919$	28.5782			0.942709			0.471354	−	0.881944i	$-0.343765\pi$
$919$	28.5782			0.942709			0.471354	+	0.881944i	$0.343765\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			5.47122i			0.180087i
$924$		0			0
$925$		−	54.6703i		−	1.79755i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			32.6638i			1.07166i	0.844325	+	0.535832i	$0.180002\pi$
$929$			32.6638i			1.07166i	−0.844325	+	0.535832i	$0.819998\pi$
$930$		0			0
$931$	−13.3634	+	6.69558i	−0.437969	+	0.219439i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			0.210798i			0.00689384i
$936$		0			0
$937$	19.8834			0.649563			0.324782	−	0.945789i	$-0.394709\pi$
$937$	19.8834			0.649563			0.324782	+	0.945789i	$0.394709\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	3.94622			0.128643			0.0643216	−	0.997929i	$-0.479512\pi$
$941$	3.94622			0.128643			0.0643216	+	0.997929i	$0.479512\pi$
$942$		0			0
$943$		−	66.8462i		−	2.17681i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	16.5600i		−	0.538129i	−0.963122	−	0.269064i	$-0.913286\pi$
$947$		−	16.5600i		−	0.538129i	0.963122	−	0.269064i	$-0.0867145\pi$
$948$		0			0
$949$		−	1.77296i		−	0.0575528i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−42.5655			−1.37883			−0.689416	−	0.724366i	$-0.742133\pi$
$953$	−42.5655			−1.37883			−0.689416	+	0.724366i	$0.742133\pi$
$954$		0			0
$955$	−0.880491			−0.0284920
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	62.1481i		−	2.00687i
$960$		0			0
$961$	−53.6873			−1.73185
$962$		0			0
$963$		0			0
$964$		0			0
$965$	0.243502			0.00783862
$966$		0			0
$967$	−18.5494			−0.596510			−0.298255	−	0.954486i	$-0.596405\pi$
$967$	−18.5494			−0.596510			−0.298255	+	0.954486i	$0.596405\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	58.7529			1.88547			0.942735	−	0.333543i	$-0.108244\pi$
$971$	58.7529			1.88547			0.942735	+	0.333543i	$0.108244\pi$
$972$		0			0
$973$	31.5997			1.01304
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−14.6646			−0.469163			−0.234581	−	0.972096i	$-0.575372\pi$
$977$	−14.6646			−0.469163			−0.234581	+	0.972096i	$0.575372\pi$
$978$		0			0
$979$			2.09603i			0.0669895i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−16.0944			−0.513332			−0.256666	−	0.966500i	$-0.582624\pi$
$983$	−16.0944			−0.513332			−0.256666	+	0.966500i	$0.582624\pi$
$984$		0			0
$985$	1.41732			0.0451596
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	5.90883i		−	0.187890i
$990$		0			0
$991$			3.01572i			0.0957974i	0.998852	+	0.0478987i	$0.0152525\pi$
$991$			3.01572i			0.0957974i	−0.998852	+	0.0478987i	$0.984748\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			0.296889i			0.00941202i
$996$		0			0
$997$	−13.4640			−0.426409			−0.213205	−	0.977008i	$-0.568390\pi$
$997$	−13.4640			−0.426409			−0.213205	+	0.977008i	$0.568390\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.f.h.1025.4		8
3.2	odd	2		2736.2.f.g.1025.5		8
4.3	odd	2		1368.2.f.d.1025.4	yes	8
12.11	even	2		1368.2.f.c.1025.5	yes	8
19.18	odd	2		2736.2.f.g.1025.4		8
57.56	even	2	inner	2736.2.f.h.1025.5		8
76.75	even	2		1368.2.f.c.1025.4	✓	8
228.227	odd	2		1368.2.f.d.1025.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1368.2.f.c.1025.4	✓	8	76.75	even	2
1368.2.f.c.1025.5	yes	8	12.11	even	2
1368.2.f.d.1025.4	yes	8	4.3	odd	2
1368.2.f.d.1025.5	yes	8	228.227	odd	2
2736.2.f.g.1025.4		8	19.18	odd	2
2736.2.f.g.1025.5		8	3.2	odd	2
2736.2.f.h.1025.4		8	1.1	even	1	trivial
2736.2.f.h.1025.5		8	57.56	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.0959660i q^{5} +3.22941 q^{7} +O(q^{10})\)	\(q-0.0959660i q^{5} +3.22941 q^{7} -0.634345i q^{11} +1.00833i q^{13} -3.46277i q^{17} +(-3.89710 + 1.95259i) q^{19} -6.51966i q^{23} +4.99079 q^{25} +6.76138 q^{29} +9.20257i q^{31} -0.309913i q^{35} -10.9542i q^{37} +10.2530 q^{41} +0.906309 q^{43} -6.82958i q^{47} +3.42907 q^{49} -4.18738 q^{53} -0.0608756 q^{55} -11.4916 q^{59} +7.41679 q^{61} +0.0967658 q^{65} +4.28905i q^{67} +5.42600 q^{71} -1.75831 q^{73} -2.04856i q^{77} +10.5704i q^{79} +3.62281i q^{83} -0.332308 q^{85} -3.30425 q^{89} +3.25632i q^{91} +(0.187383 + 0.373989i) q^{95} -6.30572i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 4 q^{19} - 8 q^{25} + 32 q^{29} - 24 q^{41} + 28 q^{43} + 4 q^{49} + 8 q^{53} - 12 q^{55} - 8 q^{59} - 8 q^{61} - 24 q^{65} + 24 q^{71} + 4 q^{73} - 4 q^{85} + 16 q^{89} - 40 q^{95}+O(q^{100}) \) 8 * q - 4 * q^19 - 8 * q^25 + 32 * q^29 - 24 * q^41 + 28 * q^43 + 4 * q^49 + 8 * q^53 - 12 * q^55 - 8 * q^59 - 8 * q^61 - 24 * q^65 + 24 * q^71 + 4 * q^73 - 4 * q^85 + 16 * q^89 - 40 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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