LMFDB - Embedded newform 4320.2.m.c.2159.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4320,2,Mod(2159,4320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4320, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 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("4320.2159");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4320 = 2^{5} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4320.m (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:	$34.4953736732$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	no (minimal twist has level 1080)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2159.2
Character	$\chi$	$=$	4320.2159
Dual form			4320.2.m.c.2159.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.17257 - 0.529077i) q^{5} +1.59486 q^{7} -3.61186i q^{11} -2.18660 q^{13} -2.27828 q^{17} -5.81486 q^{19} -1.07910i q^{23} +(4.44016 + 2.29892i) q^{25} +2.55585 q^{29} +5.93344i q^{31} +(-3.46495 - 0.843804i) q^{35} +2.49682 q^{37} -10.3047i q^{41} +0.983745i q^{43} +10.6532i q^{47} -4.45642 q^{49} -8.26543i q^{53} +(-1.91095 + 7.84703i) q^{55} +2.81213i q^{59} +8.34242i q^{61} +(4.75054 + 1.15688i) q^{65} +6.00359i q^{67} +2.34652 q^{71} +5.96697i q^{73} -5.76041i q^{77} +11.2489i q^{79} -6.40570 q^{83} +(4.94973 + 1.20539i) q^{85} +6.53221i q^{89} -3.48732 q^{91} +(12.6332 + 3.07651i) q^{95} -16.1661i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 16 q^{19} + 48 q^{49}+O(q^{100}) $ 48 * q + 16 * q^19 + 48 * q^49

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4320\mathbb{Z}\right)^\times$.

$n$	$2081$	$2431$	$3457$	$3781$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

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
$5$	−2.17257	−	0.529077i	−0.971605	−	0.236610i
$5$	−2.17257	−	0.529077i	−0.971605	−	0.236610i
$6$		0			0
$7$	1.59486			0.602801			0.301400	−	0.953498i	$-0.402546\pi$
$7$	1.59486			0.602801			0.301400	+	0.953498i	$0.402546\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	3.61186i		−	1.08902i	−0.838756	−	0.544508i	$-0.816716\pi$
$11$		−	3.61186i		−	1.08902i	0.838756	−	0.544508i	$-0.183284\pi$
$12$		0			0
$13$	−2.18660			−0.606452			−0.303226	−	0.952919i	$-0.598064\pi$
$13$	−2.18660			−0.606452			−0.303226	+	0.952919i	$0.598064\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.27828			−0.552564			−0.276282	−	0.961077i	$-0.589102\pi$
$17$	−2.27828			−0.552564			−0.276282	+	0.961077i	$0.589102\pi$
$18$		0			0
$19$	−5.81486			−1.33402			−0.667010	−	0.745049i	$-0.732426\pi$
$19$	−5.81486			−1.33402			−0.667010	+	0.745049i	$0.732426\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.07910i		−	0.225009i	−0.993651	−	0.112504i	$-0.964113\pi$
$23$		−	1.07910i		−	0.225009i	0.993651	−	0.112504i	$-0.0358873\pi$
$24$		0			0
$25$	4.44016	+	2.29892i	0.888031	+	0.459783i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.55585			0.474609			0.237305	−	0.971435i	$-0.423736\pi$
$29$	2.55585			0.474609			0.237305	+	0.971435i	$0.423736\pi$
$30$		0			0
$31$			5.93344i			1.06568i	0.846217	+	0.532838i	$0.178875\pi$
$31$			5.93344i			1.06568i	−0.846217	+	0.532838i	$0.821125\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.46495	−	0.843804i	−0.585684	−	0.142629i
$36$		0			0
$37$	2.49682			0.410475			0.205237	−	0.978712i	$-0.434203\pi$
$37$	2.49682			0.410475			0.205237	+	0.978712i	$0.434203\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	10.3047i		−	1.60933i	−0.593731	−	0.804664i	$-0.702346\pi$
$41$		−	10.3047i		−	1.60933i	0.593731	−	0.804664i	$-0.297654\pi$
$42$		0			0
$43$			0.983745i			0.150020i	0.997183	+	0.0750098i	$0.0238988\pi$
$43$			0.983745i			0.150020i	−0.997183	+	0.0750098i	$0.976101\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			10.6532i			1.55394i	0.629541	+	0.776968i	$0.283243\pi$
$47$			10.6532i			1.55394i	−0.629541	+	0.776968i	$0.716757\pi$
$48$		0			0
$49$	−4.45642			−0.636631
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	8.26543i		−	1.13534i	−0.823255	−	0.567672i	$-0.807844\pi$
$53$		−	8.26543i		−	1.13534i	0.823255	−	0.567672i	$-0.192156\pi$
$54$		0			0
$55$	−1.91095	+	7.84703i	−0.257672	+	1.05809i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			2.81213i			0.366108i	0.983103	+	0.183054i	$0.0585983\pi$
$59$			2.81213i			0.366108i	−0.983103	+	0.183054i	$0.941402\pi$
$60$		0			0
$61$			8.34242i			1.06814i	0.845441	+	0.534069i	$0.179338\pi$
$61$			8.34242i			1.06814i	−0.845441	+	0.534069i	$0.820662\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.75054	+	1.15688i	0.589232	+	0.143493i
$66$		0			0
$67$			6.00359i			0.733455i	0.930328	+	0.366728i	$0.119522\pi$
$67$			6.00359i			0.733455i	−0.930328	+	0.366728i	$0.880478\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2.34652			0.278480			0.139240	−	0.990259i	$-0.455534\pi$
$71$	2.34652			0.278480			0.139240	+	0.990259i	$0.455534\pi$
$72$		0			0
$73$			5.96697i			0.698381i	0.937052	+	0.349191i	$0.113543\pi$
$73$			5.96697i			0.698381i	−0.937052	+	0.349191i	$0.886457\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	5.76041i		−	0.656460i
$78$		0			0
$79$			11.2489i			1.26559i	0.774317	+	0.632797i	$0.218093\pi$
$79$			11.2489i			1.26559i	−0.774317	+	0.632797i	$0.781907\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.40570			−0.703117			−0.351559	−	0.936166i	$-0.614348\pi$
$83$	−6.40570			−0.703117			−0.351559	+	0.936166i	$0.614348\pi$
$84$		0			0
$85$	4.94973	+	1.20539i	0.536874	+	0.130742i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			6.53221i			0.692413i	0.938158	+	0.346206i	$0.112530\pi$
$89$			6.53221i			0.692413i	−0.938158	+	0.346206i	$0.887470\pi$
$90$		0			0
$91$	−3.48732			−0.365570
$92$		0			0
$93$		0			0
$94$		0			0
$95$	12.6332	+	3.07651i	1.29614	+	0.315643i
$96$		0			0
$97$		−	16.1661i		−	1.64142i	−0.571348	−	0.820708i	$-0.693579\pi$
$97$		−	16.1661i		−	1.64142i	0.571348	−	0.820708i	$-0.306421\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	9.33401			0.928769			0.464385	−	0.885634i	$-0.346276\pi$
$101$	9.33401			0.928769			0.464385	+	0.885634i	$0.346276\pi$
$102$		0			0
$103$	19.4431			1.91579			0.957895	−	0.287120i	$-0.0926977\pi$
$103$	19.4431			1.91579			0.957895	+	0.287120i	$0.0926977\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−12.4680			−1.20532			−0.602662	−	0.797997i	$-0.705893\pi$
$107$	−12.4680			−1.20532			−0.602662	+	0.797997i	$0.705893\pi$
$108$		0			0
$109$			7.46156i			0.714688i	0.933973	+	0.357344i	$0.116318\pi$
$109$			7.46156i			0.714688i	−0.933973	+	0.357344i	$0.883682\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	13.1420			1.23629			0.618146	−	0.786063i	$-0.287884\pi$
$113$	13.1420			1.23629			0.618146	+	0.786063i	$0.287884\pi$
$114$		0			0
$115$	−0.570929	+	2.34443i	−0.0532394	+	0.218620i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.63354			−0.333086
$120$		0			0
$121$	−2.04551			−0.185955
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−8.43026	−	7.34375i	−0.754026	−	0.656845i
$126$		0			0
$127$	−17.9156			−1.58975			−0.794876	−	0.606772i	$-0.792464\pi$
$127$	−17.9156			−1.58975			−0.794876	+	0.606772i	$0.792464\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			13.8512i			1.21019i	0.796155	+	0.605093i	$0.206864\pi$
$131$			13.8512i			1.21019i	−0.796155	+	0.605093i	$0.793136\pi$
$132$		0			0
$133$	−9.27389			−0.804148
$134$		0			0
$135$		0			0
$136$		0			0
$137$	14.7529			1.26043			0.630214	−	0.776422i	$-0.282967\pi$
$137$	14.7529			1.26043			0.630214	+	0.776422i	$0.282967\pi$
$138$		0			0
$139$	−8.56738			−0.726676			−0.363338	−	0.931657i	$-0.618363\pi$
$139$	−8.56738			−0.726676			−0.363338	+	0.931657i	$0.618363\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			7.89767i			0.660436i
$144$		0			0
$145$	−5.55277	−	1.35224i	−0.461132	−	0.112297i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−17.2194			−1.41067			−0.705335	−	0.708874i	$-0.749203\pi$
$149$	−17.2194			−1.41067			−0.705335	+	0.708874i	$0.749203\pi$
$150$		0			0
$151$			6.38199i			0.519359i	0.965695	+	0.259680i	$0.0836170\pi$
$151$			6.38199i			0.519359i	−0.965695	+	0.259680i	$0.916383\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3.13924	−	12.8908i	0.252150	−	1.03542i
$156$		0			0
$157$	24.3034			1.93962			0.969811	−	0.243857i	$-0.0784129\pi$
$157$	24.3034			1.93962			0.969811	+	0.243857i	$0.0784129\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	1.72102i		−	0.135636i
$162$		0			0
$163$		−	19.9078i		−	1.55930i	−0.626215	−	0.779650i	$-0.715397\pi$
$163$		−	19.9078i		−	1.55930i	0.626215	−	0.779650i	$-0.284603\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			7.62227i			0.589829i	0.955524	+	0.294915i	$0.0952912\pi$
$167$			7.62227i			0.589829i	−0.955524	+	0.294915i	$0.904709\pi$
$168$		0			0
$169$	−8.21880			−0.632215
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	13.7524i		−	1.04558i	−0.852463	−	0.522788i	$-0.824892\pi$
$173$		−	13.7524i		−	1.04558i	0.852463	−	0.522788i	$-0.175108\pi$
$174$		0			0
$175$	7.08143	+	3.66645i	0.535306	+	0.277158i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			14.1889i			1.06053i	0.847832	+	0.530265i	$0.177908\pi$
$179$			14.1889i			1.06053i	−0.847832	+	0.530265i	$0.822092\pi$
$180$		0			0
$181$			19.4024i			1.44217i	0.692846	+	0.721086i	$0.256357\pi$
$181$			19.4024i			1.44217i	−0.692846	+	0.721086i	$0.743643\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−5.42453	−	1.32101i	−0.398819	−	0.0971226i
$186$		0			0
$187$			8.22882i			0.601751i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−24.8392			−1.79730			−0.898651	−	0.438664i	$-0.855452\pi$
$191$	−24.8392			−1.79730			−0.898651	+	0.438664i	$0.855452\pi$
$192$		0			0
$193$			6.04021i			0.434784i	0.976084	+	0.217392i	$0.0697549\pi$
$193$			6.04021i			0.434784i	−0.976084	+	0.217392i	$0.930245\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			17.5571i			1.25089i	0.780269	+	0.625444i	$0.215082\pi$
$197$			17.5571i			1.25089i	−0.780269	+	0.625444i	$0.784918\pi$
$198$		0			0
$199$			11.9068i			0.844048i	0.906585	+	0.422024i	$0.138680\pi$
$199$			11.9068i			0.844048i	−0.906585	+	0.422024i	$0.861320\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	4.07622			0.286095
$204$		0			0
$205$	−5.45199	+	22.3878i	−0.380784	+	1.56363i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			21.0024i			1.45277i
$210$		0			0
$211$	6.86570			0.472654			0.236327	−	0.971674i	$-0.424056\pi$
$211$	6.86570			0.472654			0.236327	+	0.971674i	$0.424056\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	0.520477	−	2.13726i	0.0354962	−	0.145760i
$216$		0			0
$217$			9.46301i			0.642391i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	4.98168			0.335104
$222$		0			0
$223$	4.99364			0.334399			0.167200	−	0.985923i	$-0.446528\pi$
$223$	4.99364			0.334399			0.167200	+	0.985923i	$0.446528\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	12.3395			0.819000			0.409500	−	0.912310i	$-0.365703\pi$
$227$	12.3395			0.819000			0.409500	+	0.912310i	$0.365703\pi$
$228$		0			0
$229$			25.0215i			1.65347i	0.562593	+	0.826734i	$0.309804\pi$
$229$			25.0215i			1.65347i	−0.562593	+	0.826734i	$0.690196\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6.88604			−0.451120			−0.225560	−	0.974229i	$-0.572421\pi$
$233$	−6.88604			−0.451120			−0.225560	+	0.974229i	$0.572421\pi$
$234$		0			0
$235$	5.63638	−	23.1450i	0.367677	−	1.50981i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	8.26310			0.534495			0.267248	−	0.963628i	$-0.413886\pi$
$239$	8.26310			0.534495			0.267248	+	0.963628i	$0.413886\pi$
$240$		0			0
$241$	−18.2659			−1.17661			−0.588304	−	0.808640i	$-0.700204\pi$
$241$	−18.2659			−1.17661			−0.588304	+	0.808640i	$0.700204\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	9.68190	+	2.35779i	0.618554	+	0.150634i
$246$		0			0
$247$	12.7147			0.809019
$248$		0			0
$249$		0			0
$250$		0			0
$251$			11.7200i			0.739758i	0.929080	+	0.369879i	$0.120601\pi$
$251$			11.7200i			0.739758i	−0.929080	+	0.369879i	$0.879399\pi$
$252$		0			0
$253$	−3.89757			−0.245038
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−14.0351			−0.875484			−0.437742	−	0.899101i	$-0.644222\pi$
$257$	−14.0351			−0.875484			−0.437742	+	0.899101i	$0.644222\pi$
$258$		0			0
$259$	3.98208			0.247435
$260$		0			0
$261$		0			0
$262$		0			0
$263$			2.15821i			0.133081i	0.997784	+	0.0665404i	$0.0211961\pi$
$263$			2.15821i			0.133081i	−0.997784	+	0.0665404i	$0.978804\pi$
$264$		0			0
$265$	−4.37305	+	17.9573i	−0.268634	+	1.10311i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−16.4601			−1.00359			−0.501794	−	0.864987i	$-0.667326\pi$
$269$	−16.4601			−1.00359			−0.501794	+	0.864987i	$0.667326\pi$
$270$		0			0
$271$			17.3557i			1.05429i	0.849777	+	0.527143i	$0.176737\pi$
$271$			17.3557i			1.05429i	−0.849777	+	0.527143i	$0.823263\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	8.30336	−	16.0372i	0.500711	−	0.967080i
$276$		0			0
$277$	4.12691			0.247962			0.123981	−	0.992285i	$-0.460434\pi$
$277$	4.12691			0.247962			0.123981	+	0.992285i	$0.460434\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	4.04370i		−	0.241227i	−0.992700	−	0.120613i	$-0.961514\pi$
$281$		−	4.04370i		−	0.241227i	0.992700	−	0.120613i	$-0.0384862\pi$
$282$		0			0
$283$			16.6792i			0.991473i	0.868473	+	0.495736i	$0.165102\pi$
$283$			16.6792i			0.991473i	−0.868473	+	0.495736i	$0.834898\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	16.4346i		−	0.970104i
$288$		0			0
$289$	−11.8094			−0.694673
$290$		0			0
$291$		0			0
$292$		0			0
$293$			30.6478i			1.79046i	0.445601	+	0.895232i	$0.352990\pi$
$293$			30.6478i			1.79046i	−0.445601	+	0.895232i	$0.647010\pi$
$294$		0			0
$295$	1.48783	−	6.10956i	0.0866249	−	0.355712i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			2.35957i			0.136457i
$300$		0			0
$301$			1.56894i			0.0904320i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	4.41378	−	18.1245i	0.252732	−	1.03781i
$306$		0			0
$307$			32.3125i			1.84417i	0.386988	+	0.922085i	$0.373515\pi$
$307$			32.3125i			1.84417i	−0.386988	+	0.922085i	$0.626485\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−5.05675			−0.286742			−0.143371	−	0.989669i	$-0.545794\pi$
$311$	−5.05675			−0.286742			−0.143371	+	0.989669i	$0.545794\pi$
$312$		0			0
$313$			10.5648i			0.597158i	0.954385	+	0.298579i	$0.0965127\pi$
$313$			10.5648i			0.597158i	−0.954385	+	0.298579i	$0.903487\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	4.83085i		−	0.271328i	−0.990755	−	0.135664i	$-0.956683\pi$
$317$		−	4.83085i		−	0.271328i	0.990755	−	0.135664i	$-0.0433167\pi$
$318$		0			0
$319$		−	9.23136i		−	0.516857i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	13.2479			0.737132
$324$		0			0
$325$	−9.70882	−	5.02680i	−0.538549	−	0.278837i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			16.9904i			0.936713i
$330$		0			0
$331$	−4.14240			−0.227687			−0.113843	−	0.993499i	$-0.536316\pi$
$331$	−4.14240			−0.227687			−0.113843	+	0.993499i	$0.536316\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	3.17636	−	13.0432i	0.173543	−	0.712629i
$336$		0			0
$337$		−	23.5875i		−	1.28489i	−0.766332	−	0.642445i	$-0.777920\pi$
$337$		−	23.5875i		−	1.28489i	0.766332	−	0.642445i	$-0.222080\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	21.4307			1.16054
$342$		0			0
$343$	−18.2714			−0.986563
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−10.0100			−0.537365			−0.268683	−	0.963229i	$-0.586588\pi$
$347$	−10.0100			−0.537365			−0.268683	+	0.963229i	$0.586588\pi$
$348$		0			0
$349$			0.434946i			0.0232821i	0.999932	+	0.0116411i	$0.00370555\pi$
$349$			0.434946i			0.0232821i	−0.999932	+	0.0116411i	$0.996294\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−23.0388			−1.22623			−0.613115	−	0.789994i	$-0.710084\pi$
$353$	−23.0388			−1.22623			−0.613115	+	0.789994i	$0.710084\pi$
$354$		0			0
$355$	−5.09798	−	1.24149i	−0.270573	−	0.0658913i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	12.9336			0.682611			0.341306	−	0.939952i	$-0.389131\pi$
$359$	12.9336			0.682611			0.341306	+	0.939952i	$0.389131\pi$
$360$		0			0
$361$	14.8126			0.779608
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3.15699	−	12.9637i	0.165244	−	0.678551i
$366$		0			0
$367$	−21.3144			−1.11260			−0.556301	−	0.830981i	$-0.687780\pi$
$367$	−21.3144			−1.11260			−0.556301	+	0.830981i	$0.687780\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	13.1822i		−	0.684386i
$372$		0			0
$373$	28.2669			1.46360			0.731802	−	0.681517i	$-0.238679\pi$
$373$	28.2669			1.46360			0.731802	+	0.681517i	$0.238679\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−5.58861			−0.287828
$378$		0			0
$379$	22.8586			1.17417			0.587084	−	0.809526i	$-0.300276\pi$
$379$	22.8586			1.17417			0.587084	+	0.809526i	$0.300276\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	26.5168i		−	1.35494i	−0.735549	−	0.677472i	$-0.763076\pi$
$383$		−	26.5168i		−	1.35494i	0.735549	−	0.677472i	$-0.236924\pi$
$384$		0			0
$385$	−3.04770	+	12.5149i	−0.155325	+	0.637819i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	16.2240			0.822589			0.411295	−	0.911502i	$-0.365077\pi$
$389$	16.2240			0.822589			0.411295	+	0.911502i	$0.365077\pi$
$390$		0			0
$391$			2.45850i			0.124332i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	5.95151	−	24.4390i	0.299453	−	1.22966i
$396$		0			0
$397$	−18.3726			−0.922093			−0.461046	−	0.887376i	$-0.652526\pi$
$397$	−18.3726			−0.922093			−0.461046	+	0.887376i	$0.652526\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	13.0115i		−	0.649761i	−0.945755	−	0.324881i	$-0.894676\pi$
$401$		−	13.0115i		−	0.649761i	0.945755	−	0.324881i	$-0.105324\pi$
$402$		0			0
$403$		−	12.9740i		−	0.646282i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	9.01816i		−	0.447014i
$408$		0			0
$409$	27.8630			1.37774			0.688869	−	0.724885i	$-0.258107\pi$
$409$	27.8630			1.37774			0.688869	+	0.724885i	$0.258107\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			4.48496i			0.220690i
$414$		0			0
$415$	13.9169	+	3.38911i	0.683152	+	0.166365i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			21.9663i			1.07312i	0.843861	+	0.536562i	$0.180277\pi$
$419$			21.9663i			1.07312i	−0.843861	+	0.536562i	$0.819723\pi$
$420$		0			0
$421$		−	10.0594i		−	0.490263i	−0.969490	−	0.245132i	$-0.921169\pi$
$421$		−	10.0594i		−	0.490263i	0.969490	−	0.245132i	$-0.0788312\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−10.1159	−	5.23758i	−0.490694	−	0.254060i
$426$		0			0
$427$			13.3050i			0.643874i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	9.80514			0.472297			0.236148	−	0.971717i	$-0.424115\pi$
$431$	9.80514			0.472297			0.236148	+	0.971717i	$0.424115\pi$
$432$		0			0
$433$		−	14.7117i		−	0.706997i	−0.935435	−	0.353499i	$-0.884992\pi$
$433$		−	14.7117i		−	0.706997i	0.935435	−	0.353499i	$-0.115008\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			6.27484i			0.300166i
$438$		0			0
$439$			20.5609i			0.981319i	0.871352	+	0.490659i	$0.163244\pi$
$439$			20.5609i			0.981319i	−0.871352	+	0.490659i	$0.836756\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−0.505156			−0.0240007			−0.0120003	−	0.999928i	$-0.503820\pi$
$443$	−0.505156			−0.0240007			−0.0120003	+	0.999928i	$0.503820\pi$
$444$		0			0
$445$	3.45604	−	14.1917i	0.163832	−	0.672751i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			11.5637i			0.545726i	0.962053	+	0.272863i	$0.0879706\pi$
$449$			11.5637i			0.545726i	−0.962053	+	0.272863i	$0.912029\pi$
$450$		0			0
$451$	−37.2192			−1.75258
$452$		0			0
$453$		0			0
$454$		0			0
$455$	7.57645	+	1.84506i	0.355190	+	0.0864976i
$456$		0			0
$457$		−	13.7815i		−	0.644670i	−0.946626	−	0.322335i	$-0.895532\pi$
$457$		−	13.7815i		−	0.644670i	0.946626	−	0.322335i	$-0.104468\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	11.6050			0.540501			0.270250	−	0.962790i	$-0.412894\pi$
$461$	11.6050			0.540501			0.270250	+	0.962790i	$0.412894\pi$
$462$		0			0
$463$	23.6187			1.09765			0.548827	−	0.835936i	$-0.315074\pi$
$463$	23.6187			1.09765			0.548827	+	0.835936i	$0.315074\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	25.1522			1.16390			0.581952	−	0.813223i	$-0.302289\pi$
$467$	25.1522			1.16390			0.581952	+	0.813223i	$0.302289\pi$
$468$		0			0
$469$			9.57489i			0.442128i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3.55314			0.163374
$474$		0			0
$475$	−25.8189	−	13.3679i	−1.18465	−	0.613360i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−18.6766			−0.853354			−0.426677	−	0.904404i	$-0.640316\pi$
$479$	−18.6766			−0.853354			−0.426677	+	0.904404i	$0.640316\pi$
$480$		0			0
$481$	−5.45954			−0.248933
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−8.55309	+	35.1220i	−0.388376	+	1.59481i
$486$		0			0
$487$	−21.6007			−0.978823			−0.489411	−	0.872053i	$-0.662788\pi$
$487$	−21.6007			−0.978823			−0.489411	+	0.872053i	$0.662788\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	30.9795i		−	1.39809i	−0.715079	−	0.699043i	$-0.753609\pi$
$491$		−	30.9795i		−	1.39809i	0.715079	−	0.699043i	$-0.246391\pi$
$492$		0			0
$493$	−5.82294			−0.262252
$494$		0			0
$495$		0			0
$496$		0			0
$497$	3.74237			0.167868
$498$		0			0
$499$	12.1240			0.542745			0.271372	−	0.962474i	$-0.412523\pi$
$499$	12.1240			0.542745			0.271372	+	0.962474i	$0.412523\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			7.93957i			0.354008i	0.984210	+	0.177004i	$0.0566405\pi$
$503$			7.93957i			0.354008i	−0.984210	+	0.177004i	$0.943359\pi$
$504$		0			0
$505$	−20.2788	−	4.93841i	−0.902396	−	0.219756i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−14.5479			−0.644826			−0.322413	−	0.946599i	$-0.604494\pi$
$509$	−14.5479			−0.644826			−0.322413	+	0.946599i	$0.604494\pi$
$510$		0			0
$511$			9.51650i			0.420985i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−42.2417	−	10.2869i	−1.86139	−	0.453296i
$516$		0			0
$517$	38.4780			1.69226
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	40.9736i		−	1.79508i	−0.440929	−	0.897542i	$-0.645351\pi$
$521$		−	40.9736i		−	1.79508i	0.440929	−	0.897542i	$-0.354649\pi$
$522$		0			0
$523$			0.689354i			0.0301434i	0.999886	+	0.0150717i	$0.00479765\pi$
$523$			0.689354i			0.0301434i	−0.999886	+	0.0150717i	$0.995202\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	13.5180i		−	0.588855i
$528$		0			0
$529$	21.8355			0.949371
$530$		0			0
$531$		0			0
$532$		0			0
$533$			22.5323i			0.975980i
$534$		0			0
$535$	27.0876	+	6.59651i	1.17110	+	0.285192i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			16.0959i			0.693301i
$540$		0			0
$541$		−	2.61438i		−	0.112401i	−0.998420	−	0.0562004i	$-0.982101\pi$
$541$		−	2.61438i		−	0.112401i	0.998420	−	0.0562004i	$-0.0178986\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	3.94774	−	16.2108i	0.169102	−	0.694394i
$546$		0			0
$547$		−	6.89720i		−	0.294903i	−0.989069	−	0.147452i	$-0.952893\pi$
$547$		−	6.89720i		−	0.294903i	0.989069	−	0.147452i	$-0.0471070\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−14.8619			−0.633138
$552$		0			0
$553$			17.9404i			0.762902i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	29.1931i		−	1.23695i	−0.785803	−	0.618477i	$-0.787750\pi$
$557$		−	29.1931i		−	1.23695i	0.785803	−	0.618477i	$-0.212250\pi$
$558$		0			0
$559$		−	2.15105i		−	0.0909798i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−31.4932			−1.32728			−0.663639	−	0.748053i	$-0.730989\pi$
$563$	−31.4932			−1.32728			−0.663639	+	0.748053i	$0.730989\pi$
$564$		0			0
$565$	−28.5519	−	6.95311i	−1.20119	−	0.292519i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			42.4551i			1.77981i	0.456145	+	0.889905i	$0.349230\pi$
$569$			42.4551i			1.77981i	−0.456145	+	0.889905i	$0.650770\pi$
$570$		0			0
$571$	5.84033			0.244410			0.122205	−	0.992505i	$-0.461003\pi$
$571$	5.84033			0.244410			0.122205	+	0.992505i	$0.461003\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.48077	−	4.79139i	0.103455	−	0.199815i
$576$		0			0
$577$		−	45.1309i		−	1.87882i	−0.342791	−	0.939412i	$-0.611372\pi$
$577$		−	45.1309i		−	1.87882i	0.342791	−	0.939412i	$-0.388628\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−10.2162			−0.423840
$582$		0			0
$583$	−29.8535			−1.23641
$584$		0			0
$585$		0			0
$586$		0			0
$587$	6.69464			0.276317			0.138159	−	0.990410i	$-0.455882\pi$
$587$	6.69464			0.276317			0.138159	+	0.990410i	$0.455882\pi$
$588$		0			0
$589$		−	34.5021i		−	1.42163i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.59700			−0.0655810			−0.0327905	−	0.999462i	$-0.510439\pi$
$593$	−1.59700			−0.0655810			−0.0327905	+	0.999462i	$0.510439\pi$
$594$		0			0
$595$	7.89414	+	1.92242i	0.323628	+	0.0788116i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−25.0749			−1.02453			−0.512266	−	0.858827i	$-0.671194\pi$
$599$	−25.0749			−1.02453			−0.512266	+	0.858827i	$0.671194\pi$
$600$		0			0
$601$	2.98059			0.121581			0.0607904	−	0.998151i	$-0.480638\pi$
$601$	2.98059			0.121581			0.0607904	+	0.998151i	$0.480638\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	4.44401	+	1.08223i	0.180675	+	0.0439989i
$606$		0			0
$607$	30.2075			1.22608			0.613042	−	0.790050i	$-0.289946\pi$
$607$	30.2075			1.22608			0.613042	+	0.790050i	$0.289946\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	23.2943i		−	0.942388i
$612$		0			0
$613$	−41.9630			−1.69487			−0.847434	−	0.530900i	$-0.821854\pi$
$613$	−41.9630			−1.69487			−0.847434	+	0.530900i	$0.821854\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−36.0698			−1.45211			−0.726057	−	0.687635i	$-0.758649\pi$
$617$	−36.0698			−1.45211			−0.726057	+	0.687635i	$0.758649\pi$
$618$		0			0
$619$	−18.8294			−0.756817			−0.378408	−	0.925639i	$-0.623529\pi$
$619$	−18.8294			−0.756817			−0.378408	+	0.925639i	$0.623529\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			10.4180i			0.417387i
$624$		0			0
$625$	14.4300	+	20.4151i	0.577198	+	0.816604i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−5.68846			−0.226814
$630$		0			0
$631$		−	48.3505i		−	1.92480i	−0.271629	−	0.962402i	$-0.587562\pi$
$631$		−	48.3505i		−	1.92480i	0.271629	−	0.962402i	$-0.412438\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	38.9230	+	9.47873i	1.54461	+	0.376152i
$636$		0			0
$637$	9.74438			0.386086
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	15.2377i		−	0.601853i	−0.953647	−	0.300927i	$-0.902704\pi$
$641$		−	15.2377i		−	0.601853i	0.953647	−	0.300927i	$-0.0972959\pi$
$642$		0			0
$643$		−	22.2897i		−	0.879019i	−0.898238	−	0.439509i	$-0.855152\pi$
$643$		−	22.2897i		−	0.879019i	0.898238	−	0.439509i	$-0.144848\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	11.8135i		−	0.464437i	−0.972664	−	0.232218i	$-0.925402\pi$
$647$		−	11.8135i		−	0.464437i	0.972664	−	0.232218i	$-0.0745984\pi$
$648$		0			0
$649$	10.1570			0.398697
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	5.33196i		−	0.208656i	−0.994543	−	0.104328i	$-0.966731\pi$
$653$		−	5.33196i		−	0.208656i	0.994543	−	0.104328i	$-0.0332691\pi$
$654$		0			0
$655$	7.32836	−	30.0928i	0.286343	−	1.17582i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			30.7464i			1.19771i	0.800858	+	0.598854i	$0.204377\pi$
$659$			30.7464i			1.19771i	−0.800858	+	0.598854i	$0.795623\pi$
$660$		0			0
$661$		−	40.7254i		−	1.58404i	−0.610498	−	0.792018i	$-0.709031\pi$
$661$		−	40.7254i		−	1.58404i	0.610498	−	0.792018i	$-0.290969\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	20.1482	+	4.90660i	0.781314	+	0.190270i
$666$		0			0
$667$		−	2.75803i		−	0.106791i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	30.1316			1.16322
$672$		0			0
$673$		−	3.14342i		−	0.121170i	−0.998163	−	0.0605850i	$-0.980703\pi$
$673$		−	3.14342i		−	0.121170i	0.998163	−	0.0605850i	$-0.0192966\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			35.1895i			1.35244i	0.736699	+	0.676221i	$0.236384\pi$
$677$			35.1895i			1.35244i	−0.736699	+	0.676221i	$0.763616\pi$
$678$		0			0
$679$		−	25.7826i		−	0.989447i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	29.6866			1.13593			0.567964	−	0.823054i	$-0.307731\pi$
$683$	29.6866			1.13593			0.567964	+	0.823054i	$0.307731\pi$
$684$		0			0
$685$	−32.0518	−	7.80543i	−1.22464	−	0.298230i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			18.0731i			0.688532i
$690$		0			0
$691$	33.5824			1.27753			0.638767	−	0.769400i	$-0.279445\pi$
$691$	33.5824			1.27753			0.638767	+	0.769400i	$0.279445\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	18.6133	+	4.53280i	0.706041	+	0.171939i
$696$		0			0
$697$			23.4771i			0.889257i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	6.08892			0.229975			0.114988	−	0.993367i	$-0.463317\pi$
$701$	6.08892			0.229975			0.114988	+	0.993367i	$0.463317\pi$
$702$		0			0
$703$	−14.5187			−0.547582
$704$		0			0
$705$		0			0
$706$		0			0
$707$	14.8865			0.559863
$708$		0			0
$709$		−	23.7395i		−	0.891556i	−0.895144	−	0.445778i	$-0.852927\pi$
$709$		−	23.7395i		−	0.891556i	0.895144	−	0.445778i	$-0.147073\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	6.40280			0.239787
$714$		0			0
$715$	4.17847	−	17.1583i	0.156266	−	0.641683i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−50.5874			−1.88659			−0.943295	−	0.331956i	$-0.892292\pi$
$719$	−50.5874			−1.88659			−0.943295	+	0.331956i	$0.892292\pi$
$720$		0			0
$721$	31.0091			1.15484
$722$		0			0
$723$		0			0
$724$		0			0
$725$	11.3484	+	5.87568i	0.421468	+	0.218217i
$726$		0			0
$727$	−26.3853			−0.978577			−0.489289	−	0.872122i	$-0.662744\pi$
$727$	−26.3853			−0.978577			−0.489289	+	0.872122i	$0.662744\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	2.24125i		−	0.0828955i
$732$		0			0
$733$	37.4800			1.38435			0.692177	−	0.721727i	$-0.256652\pi$
$733$	37.4800			1.38435			0.692177	+	0.721727i	$0.256652\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	21.6841			0.798744
$738$		0			0
$739$	−18.7497			−0.689718			−0.344859	−	0.938655i	$-0.612073\pi$
$739$	−18.7497			−0.689718			−0.344859	+	0.938655i	$0.612073\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	27.8320i		−	1.02106i	−0.859861	−	0.510529i	$-0.829450\pi$
$743$		−	27.8320i		−	1.02106i	0.859861	−	0.510529i	$-0.170550\pi$
$744$		0			0
$745$	37.4105	+	9.11040i	1.37061	+	0.333779i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−19.8847			−0.726570
$750$		0			0
$751$		−	8.78033i		−	0.320399i	−0.987085	−	0.160199i	$-0.948786\pi$
$751$		−	8.78033i		−	0.320399i	0.987085	−	0.160199i	$-0.0512137\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	3.37657	−	13.8654i	0.122886	−	0.504612i
$756$		0			0
$757$	−15.1104			−0.549197			−0.274598	−	0.961559i	$-0.588545\pi$
$757$	−15.1104			−0.549197			−0.274598	+	0.961559i	$0.588545\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			16.6305i			0.602854i	0.953489	+	0.301427i	$0.0974629\pi$
$761$			16.6305i			0.602854i	−0.953489	+	0.301427i	$0.902537\pi$
$762$		0			0
$763$			11.9001i			0.430814i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	6.14899i		−	0.222027i
$768$		0			0
$769$	−10.4045			−0.375195			−0.187598	−	0.982246i	$-0.560070\pi$
$769$	−10.4045			−0.375195			−0.187598	+	0.982246i	$0.560070\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	3.04874i		−	0.109656i	−0.998496	−	0.0548278i	$-0.982539\pi$
$773$		−	3.04874i		−	0.109656i	0.998496	−	0.0548278i	$-0.0174610\pi$
$774$		0			0
$775$	−13.6405	+	26.3454i	−0.489981	+	0.946354i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			59.9205i			2.14687i
$780$		0			0
$781$		−	8.47528i		−	0.303269i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−52.8009	−	12.8584i	−1.88455	−	0.458935i
$786$		0			0
$787$			29.9306i			1.06691i	0.845828	+	0.533455i	$0.179107\pi$
$787$			29.9306i			1.06691i	−0.845828	+	0.533455i	$0.820893\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	20.9596			0.745238
$792$		0			0
$793$		−	18.2415i		−	0.647775i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	34.7092i		−	1.22946i	−0.788736	−	0.614731i	$-0.789264\pi$
$797$		−	34.7092i		−	1.22946i	0.788736	−	0.614731i	$-0.210736\pi$
$798$		0			0
$799$		−	24.2711i		−	0.858649i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	21.5519			0.760548
$804$		0			0
$805$	−0.910553	+	3.73905i	−0.0320928	+	0.131784i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			37.9701i			1.33496i	0.744629	+	0.667478i	$0.232626\pi$
$809$			37.9701i			1.33496i	−0.744629	+	0.667478i	$0.767374\pi$
$810$		0			0
$811$	5.59781			0.196566			0.0982828	−	0.995159i	$-0.468665\pi$
$811$	5.59781			0.196566			0.0982828	+	0.995159i	$0.468665\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−10.5328	+	43.2512i	−0.368947	+	1.51502i
$816$		0			0
$817$		−	5.72033i		−	0.200129i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−14.5614			−0.508195			−0.254097	−	0.967179i	$-0.581778\pi$
$821$	−14.5614			−0.508195			−0.254097	+	0.967179i	$0.581778\pi$
$822$		0			0
$823$	4.09828			0.142857			0.0714285	−	0.997446i	$-0.477244\pi$
$823$	4.09828			0.142857			0.0714285	+	0.997446i	$0.477244\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−35.6559			−1.23988			−0.619938	−	0.784651i	$-0.712842\pi$
$827$	−35.6559			−1.23988			−0.619938	+	0.784651i	$0.712842\pi$
$828$		0			0
$829$		−	25.9129i		−	0.899991i	−0.893031	−	0.449995i	$-0.851426\pi$
$829$		−	25.9129i		−	0.899991i	0.893031	−	0.449995i	$-0.148574\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	10.1530			0.351780
$834$		0			0
$835$	4.03277	−	16.5599i	0.139560	−	0.573081i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−9.17285			−0.316682			−0.158341	−	0.987385i	$-0.550614\pi$
$839$	−9.17285			−0.316682			−0.158341	+	0.987385i	$0.550614\pi$
$840$		0			0
$841$	−22.4676			−0.774746
$842$		0			0
$843$		0			0
$844$		0			0
$845$	17.8560	+	4.34838i	0.614263	+	0.149589i
$846$		0			0
$847$	−3.26230			−0.112094
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	2.69433i		−	0.0923605i
$852$		0			0
$853$	16.1963			0.554550			0.277275	−	0.960791i	$-0.410569\pi$
$853$	16.1963			0.554550			0.277275	+	0.960791i	$0.410569\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	38.3773			1.31094			0.655471	−	0.755220i	$-0.272470\pi$
$857$	38.3773			1.31094			0.655471	+	0.755220i	$0.272470\pi$
$858$		0			0
$859$	−9.75306			−0.332770			−0.166385	−	0.986061i	$-0.553209\pi$
$859$	−9.75306			−0.332770			−0.166385	+	0.986061i	$0.553209\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			17.7874i			0.605489i	0.953072	+	0.302745i	$0.0979029\pi$
$863$			17.7874i			0.605489i	−0.953072	+	0.302745i	$0.902097\pi$
$864$		0			0
$865$	−7.27608	+	29.8781i	−0.247394	+	1.01589i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	40.6292			1.37825
$870$		0			0
$871$		−	13.1274i		−	0.444806i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−13.4451	−	11.7123i	−0.454527	−	0.395947i
$876$		0			0
$877$	−5.30899			−0.179272			−0.0896360	−	0.995975i	$-0.528570\pi$
$877$	−5.30899			−0.179272			−0.0896360	+	0.995975i	$0.528570\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			46.1938i			1.55631i	0.628073	+	0.778154i	$0.283844\pi$
$881$			46.1938i			1.55631i	−0.628073	+	0.778154i	$0.716156\pi$
$882$		0			0
$883$			35.4510i			1.19302i	0.802605	+	0.596510i	$0.203447\pi$
$883$			35.4510i			1.19302i	−0.802605	+	0.596510i	$0.796553\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			1.33801i			0.0449261i	0.999748	+	0.0224631i	$0.00715082\pi$
$887$			1.33801i			0.0449261i	−0.999748	+	0.0224631i	$0.992849\pi$
$888$		0			0
$889$	−28.5729			−0.958304
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	61.9471i		−	2.07298i
$894$		0			0
$895$	7.50704	−	30.8265i	0.250933	−	1.03042i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			15.1650i			0.505780i
$900$		0			0
$901$			18.8310i			0.627351i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	10.2654	−	42.1532i	0.341233	−	1.40122i
$906$		0			0
$907$			2.19143i			0.0727653i	0.999338	+	0.0363827i	$0.0115835\pi$
$907$			2.19143i			0.0727653i	−0.999338	+	0.0363827i	$0.988416\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	48.6279			1.61111			0.805557	−	0.592519i	$-0.201866\pi$
$911$	48.6279			1.61111			0.805557	+	0.592519i	$0.201866\pi$
$912$		0			0
$913$			23.1365i			0.765706i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			22.0908i			0.729502i
$918$		0			0
$919$		−	14.9150i		−	0.492002i	−0.969270	−	0.246001i	$-0.920883\pi$
$919$		−	14.9150i		−	0.492002i	0.969270	−	0.246001i	$-0.0791166\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−5.13088			−0.168885
$924$		0			0
$925$	11.0863	+	5.73998i	0.364514	+	0.188730i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			11.8899i			0.390094i	0.980794	+	0.195047i	$0.0624860\pi$
$929$			11.8899i			0.390094i	−0.980794	+	0.195047i	$0.937514\pi$
$930$		0			0
$931$	25.9134			0.849278
$932$		0			0
$933$		0			0
$934$		0			0
$935$	4.35368	−	17.8777i	0.142381	−	0.584664i
$936$		0			0
$937$		−	36.1612i		−	1.18133i	−0.806916	−	0.590667i	$-0.798865\pi$
$937$		−	36.1612i		−	1.18133i	0.806916	−	0.590667i	$-0.201135\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−37.1768			−1.21193			−0.605965	−	0.795491i	$-0.707213\pi$
$941$	−37.1768			−1.21193			−0.605965	+	0.795491i	$0.707213\pi$
$942$		0			0
$943$	−11.1199			−0.362113
$944$		0			0
$945$		0			0
$946$		0			0
$947$	31.9333			1.03769			0.518846	−	0.854868i	$-0.326362\pi$
$947$	31.9333			1.03769			0.518846	+	0.854868i	$0.326362\pi$
$948$		0			0
$949$		−	13.0474i		−	0.423535i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−5.34819			−0.173245			−0.0866224	−	0.996241i	$-0.527607\pi$
$953$	−5.34819			−0.173245			−0.0866224	+	0.996241i	$0.527607\pi$
$954$		0			0
$955$	53.9650	+	13.1419i	1.74627	+	0.425260i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	23.5289			0.759787
$960$		0			0
$961$	−4.20569			−0.135667
$962$		0			0
$963$		0			0
$964$		0			0
$965$	3.19573	−	13.1228i	0.102874	−	0.422438i
$966$		0			0
$967$	−30.4607			−0.979550			−0.489775	−	0.871849i	$-0.662921\pi$
$967$	−30.4607			−0.979550			−0.489775	+	0.871849i	$0.662921\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	4.26310i		−	0.136809i	−0.997658	−	0.0684046i	$-0.978209\pi$
$971$		−	4.26310i		−	0.136809i	0.997658	−	0.0684046i	$-0.0217909\pi$
$972$		0			0
$973$	−13.6638			−0.438041
$974$		0			0
$975$		0			0
$976$		0			0
$977$	14.4535			0.462410			0.231205	−	0.972905i	$-0.425733\pi$
$977$	14.4535			0.462410			0.231205	+	0.972905i	$0.425733\pi$
$978$		0			0
$979$	23.5934			0.754048
$980$		0			0
$981$		0			0
$982$		0			0
$983$			17.2658i			0.550694i	0.961345	+	0.275347i	$0.0887928\pi$
$983$			17.2658i			0.550694i	−0.961345	+	0.275347i	$0.911207\pi$
$984$		0			0
$985$	9.28903	−	38.1440i	0.295973	−	1.21537i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.06156			0.0337558
$990$		0			0
$991$		−	48.1671i		−	1.53008i	−0.643984	−	0.765039i	$-0.722720\pi$
$991$		−	48.1671i		−	1.53008i	0.643984	−	0.765039i	$-0.277280\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	6.29959	−	25.8683i	0.199710	−	0.820081i
$996$		0			0
$997$	−26.3097			−0.833235			−0.416617	−	0.909082i	$-0.636785\pi$
$997$	−26.3097			−0.833235			−0.416617	+	0.909082i	$0.636785\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4320.2.m.c.2159.2		48
3.2	odd	2	inner	4320.2.m.c.2159.48		48
4.3	odd	2		1080.2.m.c.539.7	yes	48
5.4	even	2	inner	4320.2.m.c.2159.3		48
8.3	odd	2	inner	4320.2.m.c.2159.47		48
8.5	even	2		1080.2.m.c.539.6	yes	48
12.11	even	2		1080.2.m.c.539.42	yes	48
15.14	odd	2	inner	4320.2.m.c.2159.45		48
20.19	odd	2		1080.2.m.c.539.41	yes	48
24.5	odd	2		1080.2.m.c.539.43	yes	48
24.11	even	2	inner	4320.2.m.c.2159.1		48
40.19	odd	2	inner	4320.2.m.c.2159.46		48
40.29	even	2		1080.2.m.c.539.44	yes	48
60.59	even	2		1080.2.m.c.539.8	yes	48
120.29	odd	2		1080.2.m.c.539.5	✓	48
120.59	even	2	inner	4320.2.m.c.2159.4		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.2.m.c.539.5	✓	48	120.29	odd	2
1080.2.m.c.539.6	yes	48	8.5	even	2
1080.2.m.c.539.7	yes	48	4.3	odd	2
1080.2.m.c.539.8	yes	48	60.59	even	2
1080.2.m.c.539.41	yes	48	20.19	odd	2
1080.2.m.c.539.42	yes	48	12.11	even	2
1080.2.m.c.539.43	yes	48	24.5	odd	2
1080.2.m.c.539.44	yes	48	40.29	even	2
4320.2.m.c.2159.1		48	24.11	even	2	inner
4320.2.m.c.2159.2		48	1.1	even	1	trivial
4320.2.m.c.2159.3		48	5.4	even	2	inner
4320.2.m.c.2159.4		48	120.59	even	2	inner
4320.2.m.c.2159.45		48	15.14	odd	2	inner
4320.2.m.c.2159.46		48	40.19	odd	2	inner
4320.2.m.c.2159.47		48	8.3	odd	2	inner
4320.2.m.c.2159.48		48	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4320 = 2^{5} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4320.m (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.17257 - 0.529077i) q^{5} +1.59486 q^{7} -3.61186i q^{11} -2.18660 q^{13} -2.27828 q^{17} -5.81486 q^{19} -1.07910i q^{23} +(4.44016 + 2.29892i) q^{25} +2.55585 q^{29} +5.93344i q^{31} +(-3.46495 - 0.843804i) q^{35} +2.49682 q^{37} -10.3047i q^{41} +0.983745i q^{43} +10.6532i q^{47} -4.45642 q^{49} -8.26543i q^{53} +(-1.91095 + 7.84703i) q^{55} +2.81213i q^{59} +8.34242i q^{61} +(4.75054 + 1.15688i) q^{65} +6.00359i q^{67} +2.34652 q^{71} +5.96697i q^{73} -5.76041i q^{77} +11.2489i q^{79} -6.40570 q^{83} +(4.94973 + 1.20539i) q^{85} +6.53221i q^{89} -3.48732 q^{91} +(12.6332 + 3.07651i) q^{95} -16.1661i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 16 q^{19} + 48 q^{49}+O(q^{100}) \) 48 * q + 16 * q^19 + 48 * q^49

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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