LMFDB - Embedded newform 720.2.h.a.431.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,2,Mod(431,720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("720.431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	720.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$5.74922894553$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			431.8
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	720.431
Dual form			720.2.h.a.431.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{5} +2.44949i q^{7} +O(q^{10})$	$q+1.00000i q^{5} +2.44949i q^{7} +5.91359 q^{11} -6.24264 q^{13} +4.89898i q^{19} -1.43488 q^{23} -1.00000 q^{25} +6.00000i q^{29} +1.43488i q^{31} -2.44949 q^{35} -2.24264 q^{37} +4.24264i q^{41} +11.8272i q^{43} +11.8272 q^{47} +1.00000 q^{49} -8.48528i q^{53} +5.91359i q^{55} +5.91359 q^{59} -6.48528 q^{61} -6.24264i q^{65} -6.92820i q^{67} +2.86976 q^{71} +6.48528 q^{73} +14.4853i q^{77} -8.36308i q^{79} -14.6969 q^{83} -7.75736i q^{89} -15.2913i q^{91} -4.89898 q^{95} +6.48528 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 16 q^{13} - 8 q^{25} + 16 q^{37} + 8 q^{49} + 16 q^{61} - 16 q^{73} - 16 q^{97}+O(q^{100}) $ 8 * q - 16 * q^13 - 8 * q^25 + 16 * q^37 + 8 * q^49 + 16 * q^61 - 16 * q^73 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	2.44949i	0.925820i	0.886405	+	0.462910i	$0.153195\pi$
$7$	2.44949i	0.925820i	−0.886405	+	0.462910i	$0.846805\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.91359	1.78301	0.891507	−	0.453006i	$-0.149648\pi$
$11$	5.91359	1.78301	0.891507	+	0.453006i	$0.149648\pi$
$12$	0	0
$13$	−6.24264	−1.73140	−0.865699	−	0.500566i	$-0.833125\pi$
$13$	−6.24264	−1.73140	−0.865699	+	0.500566i	$0.833125\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	4.89898i	1.12390i	0.827170	+	0.561951i	$0.189949\pi$
$19$	4.89898i	1.12390i	−0.827170	+	0.561951i	$0.810051\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.43488	−0.299193	−0.149596	−	0.988747i	$-0.547797\pi$
$23$	−1.43488	−0.299193	−0.149596	+	0.988747i	$0.547797\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.00000i	1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$	6.00000i	1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$	0	0
$31$	1.43488i	0.257712i	0.991663	+	0.128856i	$0.0411304\pi$
$31$	1.43488i	0.257712i	−0.991663	+	0.128856i	$0.958870\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.44949	−0.414039
$36$	0	0
$37$	−2.24264	−0.368688	−0.184344	−	0.982862i	$-0.559016\pi$
$37$	−2.24264	−0.368688	−0.184344	+	0.982862i	$0.559016\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.24264i	0.662589i	0.943527	+	0.331295i	$0.107485\pi$
$41$	4.24264i	0.662589i	−0.943527	+	0.331295i	$0.892515\pi$
$42$	0	0
$43$	11.8272i	1.80363i	0.432124	+	0.901814i	$0.357764\pi$
$43$	11.8272i	1.80363i	−0.432124	+	0.901814i	$0.642236\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.8272	1.72517	0.862586	−	0.505911i	$-0.168843\pi$
$47$	11.8272	1.72517	0.862586	+	0.505911i	$0.168843\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 8.48528i	− 1.16554i	−0.812636	−	0.582772i	$-0.801968\pi$
$53$	− 8.48528i	− 1.16554i	0.812636	−	0.582772i	$-0.198032\pi$
$54$	0	0
$55$	5.91359i	0.797388i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.91359	0.769884	0.384942	−	0.922941i	$-0.374221\pi$
$59$	5.91359	0.769884	0.384942	+	0.922941i	$0.374221\pi$
$60$	0	0
$61$	−6.48528	−0.830355	−0.415178	−	0.909740i	$-0.636281\pi$
$61$	−6.48528	−0.830355	−0.415178	+	0.909740i	$0.636281\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 6.24264i	− 0.774304i
$66$	0	0
$67$	− 6.92820i	− 0.846415i	−0.906033	−	0.423207i	$-0.860904\pi$
$67$	− 6.92820i	− 0.846415i	0.906033	−	0.423207i	$-0.139096\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.86976	0.340577	0.170289	−	0.985394i	$-0.445530\pi$
$71$	2.86976	0.340577	0.170289	+	0.985394i	$0.445530\pi$
$72$	0	0
$73$	6.48528	0.759045	0.379522	−	0.925183i	$-0.376088\pi$
$73$	6.48528	0.759045	0.379522	+	0.925183i	$0.376088\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	14.4853i	1.65075i
$78$	0	0
$79$	− 8.36308i	− 0.940920i	−0.882421	−	0.470460i	$-0.844088\pi$
$79$	− 8.36308i	− 0.940920i	0.882421	−	0.470460i	$-0.155912\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−14.6969	−1.61320	−0.806599	−	0.591099i	$-0.798694\pi$
$83$	−14.6969	−1.61320	−0.806599	+	0.591099i	$0.798694\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 7.75736i	− 0.822278i	−0.911573	−	0.411139i	$-0.865131\pi$
$89$	− 7.75736i	− 0.822278i	0.911573	−	0.411139i	$-0.134869\pi$
$90$	0	0
$91$	− 15.2913i	− 1.60296i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.89898	−0.502625
$96$	0	0
$97$	6.48528	0.658481	0.329240	−	0.944246i	$-0.393207\pi$
$97$	6.48528	0.658481	0.329240	+	0.944246i	$0.393207\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	14.4853i	1.44134i	0.693279	+	0.720670i	$0.256166\pi$
$101$	14.4853i	1.44134i	−0.693279	+	0.720670i	$0.743834\pi$
$102$	0	0
$103$	11.4069i	1.12396i	0.827152	+	0.561978i	$0.189960\pi$
$103$	11.4069i	1.12396i	−0.827152	+	0.561978i	$0.810040\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.95743	−0.865947	−0.432974	−	0.901407i	$-0.642536\pi$
$107$	−8.95743	−0.865947	−0.432974	+	0.901407i	$0.642536\pi$
$108$	0	0
$109$	−10.4853	−1.00431	−0.502154	−	0.864778i	$-0.667459\pi$
$109$	−10.4853	−1.00431	−0.502154	+	0.864778i	$0.667459\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 8.48528i	− 0.798228i	−0.916901	−	0.399114i	$-0.869318\pi$
$113$	− 8.48528i	− 0.798228i	0.916901	−	0.399114i	$-0.130682\pi$
$114$	0	0
$115$	− 1.43488i	− 0.133803i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	23.9706	2.17914
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	− 7.34847i	− 0.652071i	−0.945357	−	0.326036i	$-0.894287\pi$
$127$	− 7.34847i	− 0.652071i	0.945357	−	0.326036i	$-0.105713\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	20.6105	1.80075	0.900375	−	0.435114i	$-0.143292\pi$
$131$	20.6105	1.80075	0.900375	+	0.435114i	$0.143292\pi$
$132$	0	0
$133$	−12.0000	−1.04053
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 20.4853i	− 1.75018i	−0.483964	−	0.875088i	$-0.660803\pi$
$137$	− 20.4853i	− 1.75018i	0.483964	−	0.875088i	$-0.339197\pi$
$138$	0	0
$139$	− 6.33386i	− 0.537231i	−0.963248	−	0.268615i	$-0.913434\pi$
$139$	− 6.33386i	− 0.537231i	0.963248	−	0.268615i	$-0.0865661\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−36.9164	−3.08711
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 14.4853i	− 1.18668i	−0.804952	−	0.593340i	$-0.797809\pi$
$149$	− 14.4853i	− 1.18668i	0.804952	−	0.593340i	$-0.202191\pi$
$150$	0	0
$151$	− 20.1903i	− 1.64306i	−0.570165	−	0.821530i	$-0.693121\pi$
$151$	− 20.1903i	− 1.64306i	0.570165	−	0.821530i	$-0.306879\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.43488	−0.115252
$156$	0	0
$157$	9.75736	0.778722	0.389361	−	0.921085i	$-0.372696\pi$
$157$	9.75736	0.778722	0.389361	+	0.921085i	$0.372696\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 3.51472i	− 0.276999i
$162$	0	0
$163$	− 7.76874i	− 0.608494i	−0.952593	−	0.304247i	$-0.901595\pi$
$163$	− 7.76874i	− 0.608494i	0.952593	−	0.304247i	$-0.0984049\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.43488	0.111034	0.0555171	−	0.998458i	$-0.482319\pi$
$167$	1.43488	0.111034	0.0555171	+	0.998458i	$0.482319\pi$
$168$	0	0
$169$	25.9706	1.99774
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.51472i	0.267219i	0.991034	+	0.133610i	$0.0426568\pi$
$173$	3.51472i	0.267219i	−0.991034	+	0.133610i	$0.957343\pi$
$174$	0	0
$175$	− 2.44949i	− 0.185164i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−14.8710	−1.11151	−0.555756	−	0.831345i	$-0.687571\pi$
$179$	−14.8710	−1.11151	−0.555756	+	0.831345i	$0.687571\pi$
$180$	0	0
$181$	18.4853	1.37400	0.687000	−	0.726657i	$-0.258927\pi$
$181$	18.4853	1.37400	0.687000	+	0.726657i	$0.258927\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 2.24264i	− 0.164882i
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	14.6969	1.06343	0.531717	−	0.846922i	$-0.321547\pi$
$191$	14.6969	1.06343	0.531717	+	0.846922i	$0.321547\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.9706i	0.781620i	0.920471	+	0.390810i	$0.127805\pi$
$197$	10.9706i	0.781620i	−0.920471	+	0.390810i	$0.872195\pi$
$198$	0	0
$199$	− 3.46410i	− 0.245564i	−0.992434	−	0.122782i	$-0.960818\pi$
$199$	− 3.46410i	− 0.245564i	0.992434	−	0.122782i	$-0.0391815\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−14.6969	−1.03152
$204$	0	0
$205$	−4.24264	−0.296319
$206$	0	0
$207$	0	0
$208$	0	0
$209$	28.9706i	2.00394i
$210$	0	0
$211$	− 3.46410i	− 0.238479i	−0.992866	−	0.119239i	$-0.961954\pi$
$211$	− 3.46410i	− 0.238479i	0.992866	−	0.119239i	$-0.0380456\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−11.8272	−0.806607
$216$	0	0
$217$	−3.51472	−0.238595
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	4.47871i	0.299917i	0.988692	+	0.149958i	$0.0479140\pi$
$223$	4.47871i	0.299917i	−0.988692	+	0.149958i	$0.952086\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	23.6544	1.57000	0.784998	−	0.619499i	$-0.212664\pi$
$227$	23.6544	1.57000	0.784998	+	0.619499i	$0.212664\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 24.0000i	− 1.57229i	−0.618041	−	0.786146i	$-0.712073\pi$
$233$	− 24.0000i	− 1.57229i	0.618041	−	0.786146i	$-0.287927\pi$
$234$	0	0
$235$	11.8272i	0.771520i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	11.8272	0.765037	0.382518	−	0.923948i	$-0.375057\pi$
$239$	11.8272	0.765037	0.382518	+	0.923948i	$0.375057\pi$
$240$	0	0
$241$	−12.9706	−0.835507	−0.417754	−	0.908560i	$-0.637183\pi$
$241$	−12.9706	−0.835507	−0.417754	+	0.908560i	$0.637183\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.00000i	0.0638877i
$246$	0	0
$247$	− 30.5826i	− 1.94592i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−14.8710	−0.938650	−0.469325	−	0.883025i	$-0.655503\pi$
$251$	−14.8710	−0.938650	−0.469325	+	0.883025i	$0.655503\pi$
$252$	0	0
$253$	−8.48528	−0.533465
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.51472i	0.219242i	0.993973	+	0.109621i	$0.0349637\pi$
$257$	3.51472i	0.219242i	−0.993973	+	0.109621i	$0.965036\pi$
$258$	0	0
$259$	− 5.49333i	− 0.341339i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	19.3497	1.19316	0.596578	−	0.802555i	$-0.296527\pi$
$263$	19.3497	1.19316	0.596578	+	0.802555i	$0.296527\pi$
$264$	0	0
$265$	8.48528	0.521247
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 10.9706i	− 0.668887i	−0.942416	−	0.334444i	$-0.891452\pi$
$269$	− 10.9706i	− 0.668887i	0.942416	−	0.334444i	$-0.108548\pi$
$270$	0	0
$271$	6.33386i	0.384754i	0.981321	+	0.192377i	$0.0616197\pi$
$271$	6.33386i	0.384754i	−0.981321	+	0.192377i	$0.938380\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.91359	−0.356603
$276$	0	0
$277$	14.2426	0.855757	0.427879	−	0.903836i	$-0.359261\pi$
$277$	14.2426	0.855757	0.427879	+	0.903836i	$0.359261\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4.24264i	0.253095i	0.991961	+	0.126547i	$0.0403896\pi$
$281$	4.24264i	0.253095i	−0.991961	+	0.126547i	$0.959610\pi$
$282$	0	0
$283$	23.6544i	1.40611i	0.711137	+	0.703053i	$0.248180\pi$
$283$	23.6544i	1.40611i	−0.711137	+	0.703053i	$0.751820\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.3923	−0.613438
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 22.9706i	− 1.34195i	−0.741478	−	0.670977i	$-0.765875\pi$
$293$	− 22.9706i	− 1.34195i	0.741478	−	0.670977i	$-0.234125\pi$
$294$	0	0
$295$	5.91359i	0.344303i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8.95743	0.518021
$300$	0	0
$301$	−28.9706	−1.66984
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 6.48528i	− 0.371346i
$306$	0	0
$307$	23.6544i	1.35003i	0.737806	+	0.675013i	$0.235862\pi$
$307$	23.6544i	1.35003i	−0.737806	+	0.675013i	$0.764138\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8.95743	0.507929	0.253965	−	0.967214i	$-0.418265\pi$
$311$	8.95743	0.507929	0.253965	+	0.967214i	$0.418265\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	34.9706i	1.96414i	0.188510	+	0.982071i	$0.439634\pi$
$317$	34.9706i	1.96414i	−0.188510	+	0.982071i	$0.560366\pi$
$318$	0	0
$319$	35.4815i	1.98659i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	6.24264	0.346279
$326$	0	0
$327$	0	0
$328$	0	0
$329$	28.9706i	1.59720i
$330$	0	0
$331$	− 2.02922i	− 0.111536i	−0.998444	−	0.0557681i	$-0.982239\pi$
$331$	− 2.02922i	− 0.111536i	0.998444	−	0.0557681i	$-0.0177608\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6.92820	0.378528
$336$	0	0
$337$	−27.4558	−1.49562	−0.747808	−	0.663916i	$-0.768893\pi$
$337$	−27.4558	−1.49562	−0.747808	+	0.663916i	$0.768893\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	8.48528i	0.459504i
$342$	0	0
$343$	19.5959i	1.05808i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.7846	1.11578	0.557888	−	0.829916i	$-0.311612\pi$
$347$	20.7846	1.11578	0.557888	+	0.829916i	$0.311612\pi$
$348$	0	0
$349$	−18.4853	−0.989494	−0.494747	−	0.869037i	$-0.664739\pi$
$349$	−18.4853	−0.989494	−0.494747	+	0.869037i	$0.664739\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	12.0000i	0.638696i	0.947638	+	0.319348i	$0.103464\pi$
$353$	12.0000i	0.638696i	−0.947638	+	0.319348i	$0.896536\pi$
$354$	0	0
$355$	2.86976i	0.152311i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−8.95743	−0.472755	−0.236377	−	0.971661i	$-0.575960\pi$
$359$	−8.95743	−0.472755	−0.236377	+	0.971661i	$0.575960\pi$
$360$	0	0
$361$	−5.00000	−0.263158
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6.48528i	0.339455i
$366$	0	0
$367$	28.9736i	1.51241i	0.654335	+	0.756205i	$0.272949\pi$
$367$	28.9736i	1.51241i	−0.654335	+	0.756205i	$0.727051\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	20.7846	1.07908
$372$	0	0
$373$	2.72792	0.141246	0.0706232	−	0.997503i	$-0.477501\pi$
$373$	2.72792	0.141246	0.0706232	+	0.997503i	$0.477501\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 37.4558i	− 1.92907i
$378$	0	0
$379$	− 17.3205i	− 0.889695i	−0.895606	−	0.444847i	$-0.853258\pi$
$379$	− 17.3205i	− 0.889695i	0.895606	−	0.444847i	$-0.146742\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	11.8272	0.604341	0.302170	−	0.953254i	$-0.402289\pi$
$383$	11.8272	0.604341	0.302170	+	0.953254i	$0.402289\pi$
$384$	0	0
$385$	−14.4853	−0.738238
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 22.9706i	− 1.16465i	−0.812955	−	0.582327i	$-0.802142\pi$
$389$	− 22.9706i	− 1.16465i	0.812955	−	0.582327i	$-0.197858\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.36308	0.420792
$396$	0	0
$397$	−10.7279	−0.538419	−0.269209	−	0.963082i	$-0.586762\pi$
$397$	−10.7279	−0.538419	−0.269209	+	0.963082i	$0.586762\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	28.2426i	1.41037i	0.709023	+	0.705185i	$0.249136\pi$
$401$	28.2426i	1.41037i	−0.709023	+	0.705185i	$0.750864\pi$
$402$	0	0
$403$	− 8.95743i	− 0.446201i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−13.2621	−0.657376
$408$	0	0
$409$	−18.9706	−0.938034	−0.469017	−	0.883189i	$-0.655392\pi$
$409$	−18.9706	−0.938034	−0.469017	+	0.883189i	$0.655392\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	14.4853i	0.712774i
$414$	0	0
$415$	− 14.6969i	− 0.721444i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	3.04384	0.148701	0.0743506	−	0.997232i	$-0.476312\pi$
$419$	3.04384	0.148701	0.0743506	+	0.997232i	$0.476312\pi$
$420$	0	0
$421$	26.9706	1.31446	0.657232	−	0.753688i	$-0.271727\pi$
$421$	26.9706	1.31446	0.657232	+	0.753688i	$0.271727\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 15.8856i	− 0.768760i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	11.8272	0.569695	0.284848	−	0.958573i	$-0.408057\pi$
$431$	11.8272	0.569695	0.284848	+	0.958573i	$0.408057\pi$
$432$	0	0
$433$	−23.4558	−1.12722	−0.563608	−	0.826042i	$-0.690587\pi$
$433$	−23.4558	−1.12722	−0.563608	+	0.826042i	$0.690587\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 7.02944i	− 0.336264i
$438$	0	0
$439$	16.1318i	0.769930i	0.922931	+	0.384965i	$0.125786\pi$
$439$	16.1318i	0.769930i	−0.922931	+	0.384965i	$0.874214\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.86976	0.136346	0.0681731	−	0.997674i	$-0.478283\pi$
$443$	2.86976	0.136346	0.0681731	+	0.997674i	$0.478283\pi$
$444$	0	0
$445$	7.75736	0.367734
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0.727922i	0.0343528i	0.999852	+	0.0171764i	$0.00546768\pi$
$449$	0.727922i	0.0343528i	−0.999852	+	0.0171764i	$0.994532\pi$
$450$	0	0
$451$	25.0892i	1.18141i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	15.2913	0.716866
$456$	0	0
$457$	−17.5147	−0.819304	−0.409652	−	0.912242i	$-0.634350\pi$
$457$	−17.5147	−0.819304	−0.409652	+	0.912242i	$0.634350\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 19.4558i	− 0.906149i	−0.891473	−	0.453074i	$-0.850327\pi$
$461$	− 19.4558i	− 0.906149i	0.891473	−	0.453074i	$-0.149673\pi$
$462$	0	0
$463$	− 9.37769i	− 0.435818i	−0.975969	−	0.217909i	$-0.930076\pi$
$463$	− 9.37769i	− 0.435818i	0.975969	−	0.217909i	$-0.0699237\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−11.8272	−0.547297	−0.273648	−	0.961830i	$-0.588230\pi$
$467$	−11.8272	−0.547297	−0.273648	+	0.961830i	$0.588230\pi$
$468$	0	0
$469$	16.9706	0.783628
$470$	0	0
$471$	0	0
$472$	0	0
$473$	69.9411i	3.21590i
$474$	0	0
$475$	− 4.89898i	− 0.224781i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	20.7846	0.949673	0.474837	−	0.880074i	$-0.342507\pi$
$479$	20.7846	0.949673	0.474837	+	0.880074i	$0.342507\pi$
$480$	0	0
$481$	14.0000	0.638345
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6.48528i	0.294481i
$486$	0	0
$487$	35.0613i	1.58878i	0.607409	+	0.794389i	$0.292209\pi$
$487$	35.0613i	1.58878i	−0.607409	+	0.794389i	$0.707791\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−17.7408	−0.800630	−0.400315	−	0.916378i	$-0.631099\pi$
$491$	−17.7408	−0.800630	−0.400315	+	0.916378i	$0.631099\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.02944i	0.315313i
$498$	0	0
$499$	29.7420i	1.33144i	0.746203	+	0.665718i	$0.231875\pi$
$499$	29.7420i	1.33144i	−0.746203	+	0.665718i	$0.768125\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−6.08767	−0.271436	−0.135718	−	0.990748i	$-0.543334\pi$
$503$	−6.08767	−0.271436	−0.135718	+	0.990748i	$0.543334\pi$
$504$	0	0
$505$	−14.4853	−0.644587
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 14.4853i	− 0.642049i	−0.947071	−	0.321024i	$-0.895973\pi$
$509$	− 14.4853i	− 0.642049i	0.947071	−	0.321024i	$-0.104027\pi$
$510$	0	0
$511$	15.8856i	0.702739i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−11.4069	−0.502649
$516$	0	0
$517$	69.9411	3.07601
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 33.2132i	− 1.45510i	−0.686056	−	0.727548i	$-0.740660\pi$
$521$	− 33.2132i	− 1.45510i	0.686056	−	0.727548i	$-0.259340\pi$
$522$	0	0
$523$	− 15.8856i	− 0.694630i	−0.937749	−	0.347315i	$-0.887093\pi$
$523$	− 15.8856i	− 0.694630i	0.937749	−	0.347315i	$-0.112907\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−20.9411	−0.910484
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 26.4853i	− 1.14720i
$534$	0	0
$535$	− 8.95743i	− 0.387263i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.91359	0.254716
$540$	0	0
$541$	−5.51472	−0.237096	−0.118548	−	0.992948i	$-0.537824\pi$
$541$	−5.51472	−0.237096	−0.118548	+	0.992948i	$0.537824\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 10.4853i	− 0.449140i
$546$	0	0
$547$	− 10.6385i	− 0.454869i	−0.973793	−	0.227435i	$-0.926966\pi$
$547$	− 10.6385i	− 0.454869i	0.973793	−	0.227435i	$-0.0730338\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−29.3939	−1.25222
$552$	0	0
$553$	20.4853	0.871123
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8.48528i	0.359533i	0.983709	+	0.179766i	$0.0575342\pi$
$557$	8.48528i	0.359533i	−0.983709	+	0.179766i	$0.942466\pi$
$558$	0	0
$559$	− 73.8329i	− 3.12280i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	26.5241	1.11786	0.558929	−	0.829215i	$-0.311212\pi$
$563$	26.5241	1.11786	0.558929	+	0.829215i	$0.311212\pi$
$564$	0	0
$565$	8.48528	0.356978
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 28.2426i	− 1.18399i	−0.805941	−	0.591997i	$-0.798340\pi$
$569$	− 28.2426i	− 1.18399i	0.805941	−	0.591997i	$-0.201660\pi$
$570$	0	0
$571$	25.6836i	1.07482i	0.843320	+	0.537412i	$0.180598\pi$
$571$	25.6836i	1.07482i	−0.843320	+	0.537412i	$0.819402\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.43488	0.0598385
$576$	0	0
$577$	−30.4853	−1.26912	−0.634559	−	0.772874i	$-0.718818\pi$
$577$	−30.4853	−1.26912	−0.634559	+	0.772874i	$0.718818\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 36.0000i	− 1.49353i
$582$	0	0
$583$	− 50.1785i	− 2.07818i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14.6969	−0.606608	−0.303304	−	0.952894i	$-0.598090\pi$
$587$	−14.6969	−0.606608	−0.303304	+	0.952894i	$0.598090\pi$
$588$	0	0
$589$	−7.02944	−0.289643
$590$	0	0
$591$	0	0
$592$	0	0
$593$	20.4853i	0.841230i	0.907239	+	0.420615i	$0.138186\pi$
$593$	20.4853i	0.841230i	−0.907239	+	0.420615i	$0.861814\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−35.4815	−1.44974	−0.724868	−	0.688887i	$-0.758099\pi$
$599$	−35.4815	−1.44974	−0.724868	+	0.688887i	$0.758099\pi$
$600$	0	0
$601$	−28.0000	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	23.9706i	0.974542i
$606$	0	0
$607$	− 30.1623i	− 1.22425i	−0.790761	−	0.612125i	$-0.790315\pi$
$607$	− 30.1623i	− 1.22425i	0.790761	−	0.612125i	$-0.209685\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−73.8329	−2.98696
$612$	0	0
$613$	23.2132	0.937572	0.468786	−	0.883312i	$-0.344691\pi$
$613$	23.2132	0.937572	0.468786	+	0.883312i	$0.344691\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 21.9411i	− 0.883316i	−0.897183	−	0.441658i	$-0.854390\pi$
$617$	− 21.9411i	− 0.883316i	0.897183	−	0.441658i	$-0.145610\pi$
$618$	0	0
$619$	0.594346i	0.0238888i	0.999929	+	0.0119444i	$0.00380211\pi$
$619$	0.594346i	0.0238888i	−0.999929	+	0.0119444i	$0.996198\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	19.0016	0.761282
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	− 11.2328i	− 0.447172i	−0.974684	−	0.223586i	$-0.928224\pi$
$631$	− 11.2328i	− 0.447172i	0.974684	−	0.223586i	$-0.0717764\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	7.34847	0.291615
$636$	0	0
$637$	−6.24264	−0.247342
$638$	0	0
$639$	0	0
$640$	0	0
$641$	24.7279i	0.976694i	0.872649	+	0.488347i	$0.162400\pi$
$641$	24.7279i	0.976694i	−0.872649	+	0.488347i	$0.837600\pi$
$642$	0	0
$643$	− 14.6969i	− 0.579591i	−0.957089	−	0.289795i	$-0.906413\pi$
$643$	− 14.6969i	− 0.579591i	0.957089	−	0.289795i	$-0.0935872\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	11.8272	0.464975	0.232487	−	0.972599i	$-0.425314\pi$
$647$	11.8272	0.464975	0.232487	+	0.972599i	$0.425314\pi$
$648$	0	0
$649$	34.9706	1.37271
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 13.0294i	− 0.509881i	−0.966957	−	0.254941i	$-0.917944\pi$
$653$	− 13.0294i	− 0.509881i	0.966957	−	0.254941i	$-0.0820559\pi$
$654$	0	0
$655$	20.6105i	0.805320i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	11.6531	0.453940	0.226970	−	0.973902i	$-0.427118\pi$
$659$	11.6531	0.453940	0.226970	+	0.973902i	$0.427118\pi$
$660$	0	0
$661$	−13.5147	−0.525662	−0.262831	−	0.964842i	$-0.584656\pi$
$661$	−13.5147	−0.525662	−0.262831	+	0.964842i	$0.584656\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 12.0000i	− 0.465340i
$666$	0	0
$667$	− 8.60927i	− 0.333352i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−38.3513	−1.48054
$672$	0	0
$673$	34.4853	1.32931	0.664655	−	0.747150i	$-0.268579\pi$
$673$	34.4853	1.32931	0.664655	+	0.747150i	$0.268579\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	27.9411i	1.07386i	0.843625	+	0.536932i	$0.180417\pi$
$677$	27.9411i	1.07386i	−0.843625	+	0.536932i	$0.819583\pi$
$678$	0	0
$679$	15.8856i	0.609635i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	41.2211	1.57728	0.788640	−	0.614855i	$-0.210786\pi$
$683$	41.2211	1.57728	0.788640	+	0.614855i	$0.210786\pi$
$684$	0	0
$685$	20.4853	0.782702
$686$	0	0
$687$	0	0
$688$	0	0
$689$	52.9706i	2.01802i
$690$	0	0
$691$	− 14.6969i	− 0.559098i	−0.960131	−	0.279549i	$-0.909815\pi$
$691$	− 14.6969i	− 0.559098i	0.960131	−	0.279549i	$-0.0901849\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.33386	0.240257
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 19.4558i	− 0.734837i	−0.930056	−	0.367419i	$-0.880242\pi$
$701$	− 19.4558i	− 0.734837i	0.930056	−	0.367419i	$-0.119758\pi$
$702$	0	0
$703$	− 10.9867i	− 0.414369i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−35.4815	−1.33442
$708$	0	0
$709$	31.9411	1.19957	0.599787	−	0.800160i	$-0.295252\pi$
$709$	31.9411	1.19957	0.599787	+	0.800160i	$0.295252\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 2.05887i	− 0.0771055i
$714$	0	0
$715$	− 36.9164i	− 1.38060i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−38.3513	−1.43026	−0.715131	−	0.698990i	$-0.753633\pi$
$719$	−38.3513	−1.43026	−0.715131	+	0.698990i	$0.753633\pi$
$720$	0	0
$721$	−27.9411	−1.04058
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 6.00000i	− 0.222834i
$726$	0	0
$727$	10.5664i	0.391886i	0.980615	+	0.195943i	$0.0627767\pi$
$727$	10.5664i	0.391886i	−0.980615	+	0.195943i	$0.937223\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	17.7574	0.655883	0.327942	−	0.944698i	$-0.393645\pi$
$733$	17.7574	0.655883	0.327942	+	0.944698i	$0.393645\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 40.9706i	− 1.50917i
$738$	0	0
$739$	− 6.08767i	− 0.223939i	−0.993712	−	0.111969i	$-0.964284\pi$
$739$	− 6.08767i	− 0.223939i	0.993712	−	0.111969i	$-0.0357158\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	17.5667	0.644460	0.322230	−	0.946661i	$-0.395568\pi$
$743$	17.5667	0.644460	0.322230	+	0.946661i	$0.395568\pi$
$744$	0	0
$745$	14.4853	0.530700
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 21.9411i	− 0.801711i
$750$	0	0
$751$	− 16.9723i	− 0.619330i	−0.950846	−	0.309665i	$-0.899783\pi$
$751$	− 16.9723i	− 0.619330i	0.950846	−	0.309665i	$-0.100217\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	20.1903	0.734799
$756$	0	0
$757$	7.21320	0.262168	0.131084	−	0.991371i	$-0.458154\pi$
$757$	7.21320	0.262168	0.131084	+	0.991371i	$0.458154\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 17.6985i	− 0.641570i	−0.947152	−	0.320785i	$-0.896053\pi$
$761$	− 17.6985i	− 0.641570i	0.947152	−	0.320785i	$-0.103947\pi$
$762$	0	0
$763$	− 25.6836i	− 0.929808i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−36.9164	−1.33297
$768$	0	0
$769$	−16.0000	−0.576975	−0.288487	−	0.957484i	$-0.593152\pi$
$769$	−16.0000	−0.576975	−0.288487	+	0.957484i	$0.593152\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 22.9706i	− 0.826194i	−0.910687	−	0.413097i	$-0.864447\pi$
$773$	− 22.9706i	− 0.826194i	0.910687	−	0.413097i	$-0.135553\pi$
$774$	0	0
$775$	− 1.43488i	− 0.0515423i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−20.7846	−0.744686
$780$	0	0
$781$	16.9706	0.607254
$782$	0	0
$783$	0	0
$784$	0	0
$785$	9.75736i	0.348255i
$786$	0	0
$787$	− 36.6702i	− 1.30715i	−0.756861	−	0.653576i	$-0.773268\pi$
$787$	− 36.6702i	− 1.30715i	0.756861	−	0.653576i	$-0.226732\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	20.7846	0.739016
$792$	0	0
$793$	40.4853	1.43767
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 44.4853i	− 1.57575i	−0.615836	−	0.787875i	$-0.711181\pi$
$797$	− 44.4853i	− 1.57575i	0.615836	−	0.787875i	$-0.288819\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	38.3513	1.35339
$804$	0	0
$805$	3.51472	0.123878
$806$	0	0
$807$	0	0
$808$	0	0
$809$	41.6985i	1.46604i	0.680207	+	0.733020i	$0.261890\pi$
$809$	41.6985i	1.46604i	−0.680207	+	0.733020i	$0.738110\pi$
$810$	0	0
$811$	4.65279i	0.163382i	0.996658	+	0.0816908i	$0.0260320\pi$
$811$	4.65279i	0.163382i	−0.996658	+	0.0816908i	$0.973968\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	7.76874	0.272127
$816$	0	0
$817$	−57.9411	−2.02710
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 18.0000i	− 0.628204i	−0.949389	−	0.314102i	$-0.898297\pi$
$821$	− 18.0000i	− 0.628204i	0.949389	−	0.314102i	$-0.101703\pi$
$822$	0	0
$823$	− 14.2767i	− 0.497654i	−0.968548	−	0.248827i	$-0.919955\pi$
$823$	− 14.2767i	− 0.497654i	0.968548	−	0.248827i	$-0.0800450\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−26.5241	−0.922334	−0.461167	−	0.887313i	$-0.652569\pi$
$827$	−26.5241	−0.922334	−0.461167	+	0.887313i	$0.652569\pi$
$828$	0	0
$829$	−10.4853	−0.364169	−0.182084	−	0.983283i	$-0.558284\pi$
$829$	−10.4853	−0.364169	−0.182084	+	0.983283i	$0.558284\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1.43488i	0.0496560i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−8.95743	−0.309245	−0.154622	−	0.987974i	$-0.549416\pi$
$839$	−8.95743	−0.309245	−0.154622	+	0.987974i	$0.549416\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	0	0
$844$	0	0
$845$	25.9706i	0.893415i
$846$	0	0
$847$	58.7156i	2.01749i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.21792	0.110309
$852$	0	0
$853$	−17.7574	−0.608000	−0.304000	−	0.952672i	$-0.598322\pi$
$853$	−17.7574	−0.608000	−0.304000	+	0.952672i	$0.598322\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 4.97056i	− 0.169791i	−0.996390	−	0.0848956i	$-0.972944\pi$
$857$	− 4.97056i	− 0.169791i	0.996390	−	0.0848956i	$-0.0270557\pi$
$858$	0	0
$859$	43.8446i	1.49596i	0.663722	+	0.747980i	$0.268976\pi$
$859$	43.8446i	1.49596i	−0.663722	+	0.747980i	$0.731024\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	25.0892	0.854048	0.427024	−	0.904240i	$-0.359562\pi$
$863$	25.0892	0.854048	0.427024	+	0.904240i	$0.359562\pi$
$864$	0	0
$865$	−3.51472	−0.119504
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 49.4558i	− 1.67767i
$870$	0	0
$871$	43.2503i	1.46548i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.44949	0.0828079
$876$	0	0
$877$	30.2426	1.02122	0.510611	−	0.859812i	$-0.329419\pi$
$877$	30.2426	1.02122	0.510611	+	0.859812i	$0.329419\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	33.2132i	1.11898i	0.828837	+	0.559491i	$0.189003\pi$
$881$	33.2132i	1.11898i	−0.828837	+	0.559491i	$0.810997\pi$
$882$	0	0
$883$	− 25.6836i	− 0.864322i	−0.901797	−	0.432161i	$-0.857751\pi$
$883$	− 25.6836i	− 0.864322i	0.901797	−	0.432161i	$-0.142249\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	19.3497	0.649700	0.324850	−	0.945765i	$-0.394686\pi$
$887$	19.3497	0.649700	0.324850	+	0.945765i	$0.394686\pi$
$888$	0	0
$889$	18.0000	0.603701
$890$	0	0
$891$	0	0
$892$	0	0
$893$	57.9411i	1.93893i
$894$	0	0
$895$	− 14.8710i	− 0.497083i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−8.60927	−0.287135
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	18.4853i	0.614472i
$906$	0	0
$907$	52.2077i	1.73353i	0.498718	+	0.866764i	$0.333804\pi$
$907$	52.2077i	1.73353i	−0.498718	+	0.866764i	$0.666196\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−53.0482	−1.75757	−0.878783	−	0.477221i	$-0.841644\pi$
$911$	−53.0482	−1.75757	−0.878783	+	0.477221i	$0.841644\pi$
$912$	0	0
$913$	−86.9117	−2.87636
$914$	0	0
$915$	0	0
$916$	0	0
$917$	50.4853i	1.66717i
$918$	0	0
$919$	− 1.43488i	− 0.0473323i	−0.999720	−	0.0236661i	$-0.992466\pi$
$919$	− 1.43488i	− 0.0473323i	0.999720	−	0.0236661i	$-0.00753387\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−17.9149	−0.589675
$924$	0	0
$925$	2.24264	0.0737376
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0.727922i	0.0238823i	0.999929	+	0.0119412i	$0.00380108\pi$
$929$	0.727922i	0.0238823i	−0.999929	+	0.0119412i	$0.996199\pi$
$930$	0	0
$931$	4.89898i	0.160558i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−5.02944	−0.164305	−0.0821523	−	0.996620i	$-0.526179\pi$
$937$	−5.02944	−0.164305	−0.0821523	+	0.996620i	$0.526179\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 10.9706i	− 0.357630i	−0.983883	−	0.178815i	$-0.942774\pi$
$941$	− 10.9706i	− 0.357630i	0.983883	−	0.178815i	$-0.0572264\pi$
$942$	0	0
$943$	− 6.08767i	− 0.198242i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−20.7846	−0.675409	−0.337705	−	0.941252i	$-0.609650\pi$
$947$	−20.7846	−0.675409	−0.337705	+	0.941252i	$0.609650\pi$
$948$	0	0
$949$	−40.4853	−1.31421
$950$	0	0
$951$	0	0
$952$	0	0
$953$	54.4264i	1.76304i	0.472143	+	0.881522i	$0.343481\pi$
$953$	54.4264i	1.76304i	−0.472143	+	0.881522i	$0.656519\pi$
$954$	0	0
$955$	14.6969i	0.475582i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	50.1785	1.62035
$960$	0	0
$961$	28.9411	0.933585
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 10.0000i	− 0.321911i
$966$	0	0
$967$	− 17.4946i	− 0.562588i	−0.959622	−	0.281294i	$-0.909236\pi$
$967$	− 17.4946i	− 0.562588i	0.959622	−	0.281294i	$-0.0907636\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	32.7859	1.05215	0.526074	−	0.850439i	$-0.323663\pi$
$971$	32.7859	1.05215	0.526074	+	0.850439i	$0.323663\pi$
$972$	0	0
$973$	15.5147	0.497379
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 4.97056i	− 0.159022i	−0.996834	−	0.0795112i	$-0.974664\pi$
$977$	− 4.97056i	− 0.159022i	0.996834	−	0.0795112i	$-0.0253359\pi$
$978$	0	0
$979$	− 45.8739i	− 1.46613i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	11.8272	0.377229	0.188614	−	0.982051i	$-0.439600\pi$
$983$	11.8272	0.377229	0.188614	+	0.982051i	$0.439600\pi$
$984$	0	0
$985$	−10.9706	−0.349551
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 16.9706i	− 0.539633i
$990$	0	0
$991$	28.7995i	0.914847i	0.889249	+	0.457424i	$0.151228\pi$
$991$	28.7995i	0.914847i	−0.889249	+	0.457424i	$0.848772\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	3.46410	0.109819
$996$	0	0
$997$	4.78680	0.151599	0.0757997	−	0.997123i	$-0.475849\pi$
$997$	4.78680	0.151599	0.0757997	+	0.997123i	$0.475849\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.2.h.a.431.8	yes	8
3.2	odd	2	inner	720.2.h.a.431.3	yes	8
4.3	odd	2	inner	720.2.h.a.431.5	yes	8
5.2	odd	4		3600.2.o.c.3599.4		8
5.3	odd	4		3600.2.o.d.3599.7		8
5.4	even	2		3600.2.h.j.1151.4		8
8.3	odd	2		2880.2.h.f.1151.2		8
8.5	even	2		2880.2.h.f.1151.3		8
12.11	even	2	inner	720.2.h.a.431.2	✓	8
15.2	even	4		3600.2.o.d.3599.2		8
15.8	even	4		3600.2.o.c.3599.5		8
15.14	odd	2		3600.2.h.j.1151.1		8
20.3	even	4		3600.2.o.d.3599.1		8
20.7	even	4		3600.2.o.c.3599.6		8
20.19	odd	2		3600.2.h.j.1151.5		8
24.5	odd	2		2880.2.h.f.1151.8		8
24.11	even	2		2880.2.h.f.1151.5		8
60.23	odd	4		3600.2.o.c.3599.3		8
60.47	odd	4		3600.2.o.d.3599.8		8
60.59	even	2		3600.2.h.j.1151.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.h.a.431.2	✓	8	12.11	even	2	inner
720.2.h.a.431.3	yes	8	3.2	odd	2	inner
720.2.h.a.431.5	yes	8	4.3	odd	2	inner
720.2.h.a.431.8	yes	8	1.1	even	1	trivial
2880.2.h.f.1151.2		8	8.3	odd	2
2880.2.h.f.1151.3		8	8.5	even	2
2880.2.h.f.1151.5		8	24.11	even	2
2880.2.h.f.1151.8		8	24.5	odd	2
3600.2.h.j.1151.1		8	15.14	odd	2
3600.2.h.j.1151.4		8	5.4	even	2
3600.2.h.j.1151.5		8	20.19	odd	2
3600.2.h.j.1151.8		8	60.59	even	2
3600.2.o.c.3599.3		8	60.23	odd	4
3600.2.o.c.3599.4		8	5.2	odd	4
3600.2.o.c.3599.5		8	15.8	even	4
3600.2.o.c.3599.6		8	20.7	even	4
3600.2.o.d.3599.1		8	20.3	even	4
3600.2.o.d.3599.2		8	15.2	even	4
3600.2.o.d.3599.7		8	5.3	odd	4
3600.2.o.d.3599.8		8	60.47	odd	4

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:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	720.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{5} +2.44949i q^{7} +O(q^{10})\)	\(q+1.00000i q^{5} +2.44949i q^{7} +5.91359 q^{11} -6.24264 q^{13} +4.89898i q^{19} -1.43488 q^{23} -1.00000 q^{25} +6.00000i q^{29} +1.43488i q^{31} -2.44949 q^{35} -2.24264 q^{37} +4.24264i q^{41} +11.8272i q^{43} +11.8272 q^{47} +1.00000 q^{49} -8.48528i q^{53} +5.91359i q^{55} +5.91359 q^{59} -6.48528 q^{61} -6.24264i q^{65} -6.92820i q^{67} +2.86976 q^{71} +6.48528 q^{73} +14.4853i q^{77} -8.36308i q^{79} -14.6969 q^{83} -7.75736i q^{89} -15.2913i q^{91} -4.89898 q^{95} +6.48528 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 16 q^{13} - 8 q^{25} + 16 q^{37} + 8 q^{49} + 16 q^{61} - 16 q^{73} - 16 q^{97}+O(q^{100}) \) 8 * q - 16 * q^13 - 8 * q^25 + 16 * q^37 + 8 * q^49 + 16 * q^61 - 16 * q^73 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more