LMFDB - Embedded newform 1440.2.bj.a.17.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,2,Mod(17,1440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1440, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 2, 1]))

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

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

chi := DirichletCharacter("1440.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1440.bj (of order $4$, degree $2$, 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:	$11.4984578911$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			17.15
Character	$\chi$	$=$	1440.17
Dual form			1440.2.bj.a.593.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.29263 + 1.82458i) q^{5} +(0.306649 + 0.306649i) q^{7} +O(q^{10})$	$q+(1.29263 + 1.82458i) q^{5} +(0.306649 + 0.306649i) q^{7} -4.06731 q^{11} +(0.625224 + 0.625224i) q^{13} +(3.57491 - 3.57491i) q^{17} +6.82524 q^{19} +(1.58940 + 1.58940i) q^{23} +(-1.65821 + 4.71703i) q^{25} +8.50925i q^{29} +2.56730 q^{31} +(-0.163123 + 0.955891i) q^{35} +(-1.69023 + 1.69023i) q^{37} +5.19315i q^{41} +(4.18889 + 4.18889i) q^{43} +(-4.58429 + 4.58429i) q^{47} -6.81193i q^{49} +(-7.41934 + 7.41934i) q^{53} +(-5.25753 - 7.42115i) q^{55} +8.79560i q^{59} +6.08412i q^{61} +(-0.332590 + 1.94896i) q^{65} +(6.18046 - 6.18046i) q^{67} -14.7516i q^{71} +(3.05881 - 3.05881i) q^{73} +(-1.24724 - 1.24724i) q^{77} -8.56106i q^{79} +(-5.13122 + 5.13122i) q^{83} +(11.1438 + 1.90169i) q^{85} -9.88328 q^{89} +0.383449i q^{91} +(8.82252 + 12.4532i) q^{95} +(1.75624 + 1.75624i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q - 32 q^{31} - 32 q^{97}+O(q^{100}) $ 48 * q - 32 * q^31 - 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$991$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$-1$	$1$

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
$5$	1.29263	+	1.82458i	0.578082	+	0.815979i
$5$	1.29263	+	1.82458i	0.578082	+	0.815979i
$6$		0			0
$7$	0.306649	+	0.306649i	0.115902	+	0.115902i	0.762679	−	0.646777i	$-0.223883\pi$
$7$	0.306649	+	0.306649i	0.115902	+	0.115902i	−0.646777	+	0.762679i	$0.723883\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.06731			−1.22634			−0.613170	−	0.789951i	$-0.710106\pi$
$11$	−4.06731			−1.22634			−0.613170	+	0.789951i	$0.710106\pi$
$12$		0			0
$13$	0.625224	+	0.625224i	0.173406	+	0.173406i	0.788474	−	0.615068i	$-0.210871\pi$
$13$	0.625224	+	0.625224i	0.173406	+	0.173406i	−0.615068	+	0.788474i	$0.710871\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.57491	−	3.57491i	0.867044	−	0.867044i	−0.125100	−	0.992144i	$-0.539925\pi$
$17$	3.57491	−	3.57491i	0.867044	−	0.867044i	0.992144	+	0.125100i	$0.0399253\pi$
$18$		0			0
$19$	6.82524			1.56582			0.782909	−	0.622136i	$-0.213735\pi$
$19$	6.82524			1.56582			0.782909	+	0.622136i	$0.213735\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.58940	+	1.58940i	0.331413	+	0.331413i	0.853123	−	0.521710i	$-0.174706\pi$
$23$	1.58940	+	1.58940i	0.331413	+	0.331413i	−0.521710	+	0.853123i	$0.674706\pi$
$24$		0			0
$25$	−1.65821	+	4.71703i	−0.331643	+	0.943405i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			8.50925i			1.58013i	0.613025	+	0.790064i	$0.289953\pi$
$29$			8.50925i			1.58013i	−0.613025	+	0.790064i	$0.710047\pi$
$30$		0			0
$31$	2.56730			0.461101			0.230551	−	0.973060i	$-0.425947\pi$
$31$	2.56730			0.461101			0.230551	+	0.973060i	$0.425947\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.163123	+	0.955891i	−0.0275728	+	0.161575i
$36$		0			0
$37$	−1.69023	+	1.69023i	−0.277872	+	0.277872i	−0.832259	−	0.554387i	$-0.812953\pi$
$37$	−1.69023	+	1.69023i	−0.277872	+	0.277872i	0.554387	+	0.832259i	$0.312953\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			5.19315i			0.811033i	0.914088	+	0.405517i	$0.132908\pi$
$41$			5.19315i			0.811033i	−0.914088	+	0.405517i	$0.867092\pi$
$42$		0			0
$43$	4.18889	+	4.18889i	0.638800	+	0.638800i	0.950259	−	0.311460i	$-0.100818\pi$
$43$	4.18889	+	4.18889i	0.638800	+	0.638800i	−0.311460	+	0.950259i	$0.600818\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.58429	+	4.58429i	−0.668688	+	0.668688i	−0.957412	−	0.288724i	$-0.906769\pi$
$47$	−4.58429	+	4.58429i	−0.668688	+	0.668688i	0.288724	+	0.957412i	$0.406769\pi$
$48$		0			0
$49$		−	6.81193i		−	0.973133i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−7.41934	+	7.41934i	−1.01912	+	1.01912i	−0.0193111	+	0.999814i	$0.506147\pi$
$53$	−7.41934	+	7.41934i	−1.01912	+	1.01912i	−0.999814	+	0.0193111i	$0.993853\pi$
$54$		0			0
$55$	−5.25753	−	7.42115i	−0.708925	−	1.00067i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			8.79560i			1.14509i	0.819874	+	0.572545i	$0.194044\pi$
$59$			8.79560i			1.14509i	−0.819874	+	0.572545i	$0.805956\pi$
$60$		0			0
$61$			6.08412i			0.778992i	0.921028	+	0.389496i	$0.127351\pi$
$61$			6.08412i			0.778992i	−0.921028	+	0.389496i	$0.872649\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.332590	+	1.94896i	−0.0412527	+	0.241738i
$66$		0			0
$67$	6.18046	−	6.18046i	0.755063	−	0.755063i	−0.220356	−	0.975419i	$-0.570722\pi$
$67$	6.18046	−	6.18046i	0.755063	−	0.755063i	0.975419	+	0.220356i	$0.0707219\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	14.7516i		−	1.75069i	−0.483500	−	0.875344i	$-0.660635\pi$
$71$		−	14.7516i		−	1.75069i	0.483500	−	0.875344i	$-0.339365\pi$
$72$		0			0
$73$	3.05881	−	3.05881i	0.358007	−	0.358007i	−0.505071	−	0.863078i	$-0.668534\pi$
$73$	3.05881	−	3.05881i	0.358007	−	0.358007i	0.863078	+	0.505071i	$0.168534\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.24724	−	1.24724i	−0.142136	−	0.142136i
$78$		0			0
$79$		−	8.56106i		−	0.963194i	−0.876393	−	0.481597i	$-0.840057\pi$
$79$		−	8.56106i		−	0.963194i	0.876393	−	0.481597i	$-0.159943\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.13122	+	5.13122i	−0.563224	+	0.563224i	−0.930222	−	0.366998i	$-0.880386\pi$
$83$	−5.13122	+	5.13122i	−0.563224	+	0.563224i	0.366998	+	0.930222i	$0.380386\pi$
$84$		0			0
$85$	11.1438	+	1.90169i	1.20871	+	0.206267i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−9.88328			−1.04763			−0.523813	−	0.851833i	$-0.675491\pi$
$89$	−9.88328			−1.04763			−0.523813	+	0.851833i	$0.675491\pi$
$90$		0			0
$91$			0.383449i			0.0401964i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	8.82252	+	12.4532i	0.905171	+	1.27767i
$96$		0			0
$97$	1.75624	+	1.75624i	0.178319	+	0.178319i	0.790623	−	0.612303i	$-0.209757\pi$
$97$	1.75624	+	1.75624i	0.178319	+	0.178319i	−0.612303	+	0.790623i	$0.709757\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.88077			0.187143			0.0935716	−	0.995613i	$-0.470172\pi$
$101$	1.88077			0.187143			0.0935716	+	0.995613i	$0.470172\pi$
$102$		0			0
$103$	−10.7376	+	10.7376i	−1.05800	+	1.05800i	−0.0597943	+	0.998211i	$0.519044\pi$
$103$	−10.7376	+	10.7376i	−1.05800	+	1.05800i	−0.998211	+	0.0597943i	$0.980956\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	9.59725	+	9.59725i	0.927801	+	0.927801i	0.997564	−	0.0697628i	$-0.0222242\pi$
$107$	9.59725	+	9.59725i	0.927801	+	0.927801i	−0.0697628	+	0.997564i	$0.522224\pi$
$108$		0			0
$109$	3.12119			0.298956			0.149478	−	0.988765i	$-0.452241\pi$
$109$	3.12119			0.298956			0.149478	+	0.988765i	$0.452241\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	5.85646	+	5.85646i	0.550929	+	0.550929i	0.926709	−	0.375780i	$-0.122625\pi$
$113$	5.85646	+	5.85646i	0.550929	+	0.550929i	−0.375780	+	0.926709i	$0.622625\pi$
$114$		0			0
$115$	−0.845488	+	4.95451i	−0.0788421	+	0.462010i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.19249			0.200985
$120$		0			0
$121$	5.54302			0.503911
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−10.7501	+	3.07182i	−0.961515	+	0.274752i
$126$		0			0
$127$	14.7376	+	14.7376i	1.30775	+	1.30775i	0.923037	+	0.384712i	$0.125699\pi$
$127$	14.7376	+	14.7376i	1.30775	+	1.30775i	0.384712	+	0.923037i	$0.374301\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.86888			−0.512766			−0.256383	−	0.966575i	$-0.582531\pi$
$131$	−5.86888			−0.512766			−0.256383	+	0.966575i	$0.582531\pi$
$132$		0			0
$133$	2.09295	+	2.09295i	0.181482	+	0.181482i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	13.4582	−	13.4582i	1.14981	−	1.14981i	0.163222	−	0.986589i	$-0.447811\pi$
$137$	13.4582	−	13.4582i	1.14981	−	1.14981i	0.986589	−	0.163222i	$-0.0521887\pi$
$138$		0			0
$139$	−16.0990			−1.36550			−0.682752	−	0.730650i	$-0.739217\pi$
$139$	−16.0990			−1.36550			−0.682752	+	0.730650i	$0.739217\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.54298	−	2.54298i	−0.212655	−	0.212655i
$144$		0			0
$145$	−15.5258	+	10.9993i	−1.28935	+	0.913443i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	4.56103i		−	0.373654i	−0.982393	−	0.186827i	$-0.940180\pi$
$149$		−	4.56103i		−	0.373654i	0.982393	−	0.186827i	$-0.0598204\pi$
$150$		0			0
$151$	12.4433			1.01262			0.506309	−	0.862352i	$-0.331009\pi$
$151$	12.4433			1.01262			0.506309	+	0.862352i	$0.331009\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3.31857	+	4.68426i	0.266554	+	0.376249i
$156$		0			0
$157$	13.4700	−	13.4700i	1.07502	−	1.07502i	0.0780736	−	0.996948i	$-0.475123\pi$
$157$	13.4700	−	13.4700i	1.07502	−	1.07502i	0.996948	−	0.0780736i	$-0.0248769\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			0.974777i			0.0768232i
$162$		0			0
$163$	6.23902	+	6.23902i	0.488678	+	0.488678i	0.907889	−	0.419211i	$-0.137693\pi$
$163$	6.23902	+	6.23902i	0.488678	+	0.488678i	−0.419211	+	0.907889i	$0.637693\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4.26961	−	4.26961i	0.330392	−	0.330392i	−0.522343	−	0.852735i	$-0.674942\pi$
$167$	4.26961	−	4.26961i	0.330392	−	0.330392i	0.852735	+	0.522343i	$0.174942\pi$
$168$		0			0
$169$		−	12.2182i		−	0.939861i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	14.0168	−	14.0168i	1.06568	−	1.06568i	0.0679921	−	0.997686i	$-0.478341\pi$
$173$	14.0168	−	14.0168i	1.06568	−	1.06568i	0.997686	−	0.0679921i	$-0.0216593\pi$
$174$		0			0
$175$	−1.95496	+	0.937982i	−0.147781	+	0.0709048i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	16.8624i		−	1.26036i	−0.776451	−	0.630178i	$-0.782982\pi$
$179$		−	16.8624i		−	1.26036i	0.776451	−	0.630178i	$-0.217018\pi$
$180$		0			0
$181$		−	18.7043i		−	1.39028i	−0.718874	−	0.695140i	$-0.755342\pi$
$181$		−	18.7043i		−	1.39028i	0.718874	−	0.695140i	$-0.244658\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−5.26880	−	0.899123i	−0.387370	−	0.0661048i
$186$		0			0
$187$	−14.5403	+	14.5403i	−1.06329	+	1.06329i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	6.52419i		−	0.472074i	−0.971744	−	0.236037i	$-0.924151\pi$
$191$		−	6.52419i		−	0.472074i	0.971744	−	0.236037i	$-0.0758486\pi$
$192$		0			0
$193$	1.52917	−	1.52917i	0.110072	−	0.110072i	−0.649926	−	0.759998i	$-0.725200\pi$
$193$	1.52917	−	1.52917i	0.110072	−	0.110072i	0.759998	+	0.649926i	$0.225200\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−3.85488	−	3.85488i	−0.274648	−	0.274648i	0.556320	−	0.830968i	$-0.312213\pi$
$197$	−3.85488	−	3.85488i	−0.274648	−	0.274648i	−0.830968	+	0.556320i	$0.812213\pi$
$198$		0			0
$199$		−	14.3090i		−	1.01434i	−0.861847	−	0.507168i	$-0.830693\pi$
$199$		−	14.3090i		−	1.01434i	0.861847	−	0.507168i	$-0.169307\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2.60935	+	2.60935i	−0.183141	+	0.183141i
$204$		0			0
$205$	−9.47533	+	6.71282i	−0.661786	+	0.468844i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−27.7604			−1.92023
$210$		0			0
$211$		−	8.62279i		−	0.593618i	−0.954937	−	0.296809i	$-0.904078\pi$
$211$		−	8.62279i		−	0.593618i	0.954937	−	0.296809i	$-0.0959224\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2.22830	+	13.0577i	−0.151968	+	0.890526i
$216$		0			0
$217$	0.787261	+	0.787261i	0.0534427	+	0.0534427i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	4.47024			0.300701
$222$		0			0
$223$	−6.87395	+	6.87395i	−0.460314	+	0.460314i	−0.898758	−	0.438444i	$-0.855530\pi$
$223$	−6.87395	+	6.87395i	−0.460314	+	0.460314i	0.438444	+	0.898758i	$0.355530\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−9.19431	−	9.19431i	−0.610248	−	0.610248i	0.332763	−	0.943011i	$-0.392019\pi$
$227$	−9.19431	−	9.19431i	−0.610248	−	0.610248i	−0.943011	+	0.332763i	$0.892019\pi$
$228$		0			0
$229$	−2.31517			−0.152991			−0.0764954	−	0.997070i	$-0.524373\pi$
$229$	−2.31517			−0.152991			−0.0764954	+	0.997070i	$0.524373\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	13.1377	+	13.1377i	0.860682	+	0.860682i	0.991417	−	0.130735i	$-0.0417338\pi$
$233$	13.1377	+	13.1377i	0.860682	+	0.860682i	−0.130735	+	0.991417i	$0.541734\pi$
$234$		0			0
$235$	−14.2902	−	2.43863i	−0.932191	−	0.159079i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−12.3925			−0.801604			−0.400802	−	0.916165i	$-0.631268\pi$
$239$	−12.3925			−0.801604			−0.400802	+	0.916165i	$0.631268\pi$
$240$		0			0
$241$	18.0131			1.16033			0.580164	−	0.814499i	$-0.302988\pi$
$241$	18.0131			1.16033			0.580164	+	0.814499i	$0.302988\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	12.4289	−	8.80531i	0.794056	−	0.562551i
$246$		0			0
$247$	4.26731	+	4.26731i	0.271522	+	0.271522i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	12.9632			0.818228			0.409114	−	0.912483i	$-0.365838\pi$
$251$	12.9632			0.818228			0.409114	+	0.912483i	$0.365838\pi$
$252$		0			0
$253$	−6.46459	−	6.46459i	−0.406425	−	0.406425i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	8.26946	−	8.26946i	0.515835	−	0.515835i	−0.400473	−	0.916308i	$-0.631154\pi$
$257$	8.26946	−	8.26946i	0.515835	−	0.515835i	0.916308	+	0.400473i	$0.131154\pi$
$258$		0			0
$259$	−1.03661			−0.0644120
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2.53991	−	2.53991i	−0.156617	−	0.156617i	0.624448	−	0.781066i	$-0.285324\pi$
$263$	−2.53991	−	2.53991i	−0.156617	−	0.156617i	−0.781066	+	0.624448i	$0.785324\pi$
$264$		0			0
$265$	−23.1277	−	3.94674i	−1.42072	−	0.242447i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	1.90816i		−	0.116342i	−0.998307	−	0.0581712i	$-0.981473\pi$
$269$		−	1.90816i		−	0.116342i	0.998307	−	0.0581712i	$-0.0185269\pi$
$270$		0			0
$271$	−14.2812			−0.867524			−0.433762	−	0.901028i	$-0.642814\pi$
$271$	−14.2812			−0.867524			−0.433762	+	0.901028i	$0.642814\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	6.74447	−	19.1856i	0.406707	−	1.15694i
$276$		0			0
$277$	−20.5605	+	20.5605i	−1.23536	+	1.23536i	−0.273488	+	0.961876i	$0.588177\pi$
$277$	−20.5605	+	20.5605i	−1.23536	+	1.23536i	−0.961876	+	0.273488i	$0.911823\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	9.40066i		−	0.560796i	−0.959884	−	0.280398i	$-0.909533\pi$
$281$		−	9.40066i		−	0.560796i	0.959884	−	0.280398i	$-0.0904665\pi$
$282$		0			0
$283$	−18.4043	−	18.4043i	−1.09402	−	1.09402i	−0.995095	−	0.0989268i	$-0.968459\pi$
$283$	−18.4043	−	18.4043i	−1.09402	−	1.09402i	−0.0989268	−	0.995095i	$-0.531541\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.59247	+	1.59247i	−0.0940008	+	0.0940008i
$288$		0			0
$289$		−	8.56000i		−	0.503530i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	10.8096	−	10.8096i	0.631504	−	0.631504i	−0.316941	−	0.948445i	$-0.602656\pi$
$293$	10.8096	−	10.8096i	0.631504	−	0.631504i	0.948445	+	0.316941i	$0.102656\pi$
$294$		0			0
$295$	−16.0483	+	11.3695i	−0.934369	+	0.661955i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			1.98746i			0.114938i
$300$		0			0
$301$			2.56904i			0.148077i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−11.1010	+	7.86452i	−0.635641	+	0.450321i
$306$		0			0
$307$	23.8171	−	23.8171i	1.35931	−	1.35931i	0.484546	−	0.874766i	$-0.338985\pi$
$307$	23.8171	−	23.8171i	1.35931	−	1.35931i	0.874766	−	0.484546i	$-0.161015\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			0.743382i			0.0421533i	0.999778	+	0.0210767i	$0.00670941\pi$
$311$			0.743382i			0.0421533i	−0.999778	+	0.0210767i	$0.993291\pi$
$312$		0			0
$313$	−16.4757	+	16.4757i	−0.931262	+	0.931262i	−0.997785	−	0.0665231i	$-0.978809\pi$
$313$	−16.4757	+	16.4757i	−0.931262	+	0.931262i	0.0665231	+	0.997785i	$0.478809\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−12.0047	−	12.0047i	−0.674252	−	0.674252i	0.284442	−	0.958693i	$-0.408192\pi$
$317$	−12.0047	−	12.0047i	−0.674252	−	0.674252i	−0.958693	+	0.284442i	$0.908192\pi$
$318$		0			0
$319$		−	34.6098i		−	1.93777i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	24.3997	−	24.3997i	1.35763	−	1.35763i
$324$		0			0
$325$	−3.98595	+	1.91244i	−0.221101	+	0.106083i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.81154			−0.155005
$330$		0			0
$331$			20.3586i			1.11901i	0.828827	+	0.559505i	$0.189009\pi$
$331$			20.3586i			1.11901i	−0.828827	+	0.559505i	$0.810991\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	19.2658	+	3.28772i	1.05260	+	0.179627i
$336$		0			0
$337$	12.3690	+	12.3690i	0.673782	+	0.673782i	0.958586	−	0.284804i	$-0.0919284\pi$
$337$	12.3690	+	12.3690i	0.673782	+	0.673782i	−0.284804	+	0.958586i	$0.591928\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−10.4420			−0.565467
$342$		0			0
$343$	4.23542	−	4.23542i	0.228691	−	0.228691i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8.65625	−	8.65625i	−0.464692	−	0.464692i	0.435498	−	0.900190i	$-0.356572\pi$
$347$	−8.65625	−	8.65625i	−0.464692	−	0.464692i	−0.900190	+	0.435498i	$0.856572\pi$
$348$		0			0
$349$	−26.6201			−1.42494			−0.712469	−	0.701703i	$-0.752423\pi$
$349$	−26.6201			−1.42494			−0.712469	+	0.701703i	$0.752423\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2.06573	+	2.06573i	0.109948	+	0.109948i	0.759940	−	0.649993i	$-0.225228\pi$
$353$	2.06573	+	2.06573i	0.109948	+	0.109948i	−0.649993	+	0.759940i	$0.725228\pi$
$354$		0			0
$355$	26.9155	−	19.0683i	1.42852	−	1.01204i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−25.0314			−1.32110			−0.660552	−	0.750780i	$-0.729678\pi$
$359$	−25.0314			−1.32110			−0.660552	+	0.750780i	$0.729678\pi$
$360$		0			0
$361$	27.5840			1.45179
$362$		0			0
$363$		0			0
$364$		0			0
$365$	9.53497	+	1.62715i	0.499083	+	0.0851687i
$366$		0			0
$367$	17.6828	+	17.6828i	0.923033	+	0.923033i	0.997243	−	0.0742100i	$-0.0236435\pi$
$367$	17.6828	+	17.6828i	0.923033	+	0.923033i	−0.0742100	+	0.997243i	$0.523644\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−4.55027			−0.236238
$372$		0			0
$373$	−17.6344	−	17.6344i	−0.913073	−	0.913073i	0.0834395	−	0.996513i	$-0.473409\pi$
$373$	−17.6344	−	17.6344i	−0.913073	−	0.913073i	−0.996513	+	0.0834395i	$0.973409\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−5.32019	+	5.32019i	−0.274004	+	0.274004i
$378$		0			0
$379$	−17.1117			−0.878971			−0.439486	−	0.898250i	$-0.644839\pi$
$379$	−17.1117			−0.878971			−0.439486	+	0.898250i	$0.644839\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−7.26450	−	7.26450i	−0.371199	−	0.371199i	0.496715	−	0.867914i	$-0.334539\pi$
$383$	−7.26450	−	7.26450i	−0.371199	−	0.371199i	−0.867914	+	0.496715i	$0.834539\pi$
$384$		0			0
$385$	0.663472	−	3.88791i	0.0338137	−	0.198146i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	4.30166i		−	0.218103i	−0.994036	−	0.109051i	$-0.965219\pi$
$389$		−	4.30166i		−	0.218103i	0.994036	−	0.109051i	$-0.0347813\pi$
$390$		0			0
$391$	11.3639			0.574699
$392$		0			0
$393$		0			0
$394$		0			0
$395$	15.6204	−	11.0663i	0.785946	−	0.556805i
$396$		0			0
$397$	−9.05589	+	9.05589i	−0.454502	+	0.454502i	−0.896846	−	0.442344i	$-0.854147\pi$
$397$	−9.05589	+	9.05589i	−0.454502	+	0.454502i	0.442344	+	0.896846i	$0.354147\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	16.9061i		−	0.844249i	−0.906538	−	0.422124i	$-0.861285\pi$
$401$		−	16.9061i		−	0.844249i	0.906538	−	0.422124i	$-0.138715\pi$
$402$		0			0
$403$	1.60514	+	1.60514i	0.0799577	+	0.0799577i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	6.87468	−	6.87468i	0.340765	−	0.340765i
$408$		0			0
$409$			1.92157i			0.0950153i	0.998871	+	0.0475077i	$0.0151278\pi$
$409$			1.92157i			0.0950153i	−0.998871	+	0.0475077i	$0.984872\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2.69716	+	2.69716i	−0.132719	+	0.132719i
$414$		0			0
$415$	−15.9951	−	2.72957i	−0.785169	−	0.133989i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	8.33374i		−	0.407130i	−0.979061	−	0.203565i	$-0.934747\pi$
$419$		−	8.33374i		−	0.407130i	0.979061	−	0.203565i	$-0.0652528\pi$
$420$		0			0
$421$			11.0584i			0.538956i	0.963007	+	0.269478i	$0.0868511\pi$
$421$			11.0584i			0.538956i	−0.963007	+	0.269478i	$0.913149\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	10.9350	+	22.7909i	0.530425	+	1.10552i
$426$		0			0
$427$	−1.86569	+	1.86569i	−0.0902871	+	0.0902871i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	26.6362i		−	1.28302i	−0.767115	−	0.641509i	$-0.778309\pi$
$431$		−	26.6362i		−	1.28302i	0.767115	−	0.641509i	$-0.221691\pi$
$432$		0			0
$433$	−16.5454	+	16.5454i	−0.795123	+	0.795123i	−0.982322	−	0.187199i	$-0.940059\pi$
$433$	−16.5454	+	16.5454i	−0.795123	+	0.795123i	0.187199	+	0.982322i	$0.440059\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	10.8481	+	10.8481i	0.518933	+	0.518933i
$438$		0			0
$439$		−	5.35402i		−	0.255533i	−0.991804	−	0.127767i	$-0.959219\pi$
$439$		−	5.35402i		−	0.255533i	0.991804	−	0.127767i	$-0.0407808\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.73227	+	2.73227i	−0.129814	+	0.129814i	−0.769028	−	0.639215i	$-0.779260\pi$
$443$	−2.73227	+	2.73227i	−0.129814	+	0.129814i	0.639215	+	0.769028i	$0.279260\pi$
$444$		0			0
$445$	−12.7754	−	18.0329i	−0.605614	−	0.854841i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	28.9286			1.36522			0.682612	−	0.730781i	$-0.260844\pi$
$449$	28.9286			1.36522			0.682612	+	0.730781i	$0.260844\pi$
$450$		0			0
$451$		−	21.1222i		−	0.994603i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.699635	+	0.495658i	−0.0327994	+	0.0232368i
$456$		0			0
$457$	10.5512	+	10.5512i	0.493564	+	0.493564i	0.909427	−	0.415863i	$-0.136520\pi$
$457$	10.5512	+	10.5512i	0.493564	+	0.493564i	−0.415863	+	0.909427i	$0.636520\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	11.9624			0.557146			0.278573	−	0.960415i	$-0.410139\pi$
$461$	11.9624			0.557146			0.278573	+	0.960415i	$0.410139\pi$
$462$		0			0
$463$	−8.05265	+	8.05265i	−0.374238	+	0.374238i	−0.869018	−	0.494780i	$-0.835249\pi$
$463$	−8.05265	+	8.05265i	−0.374238	+	0.374238i	0.494780	+	0.869018i	$0.335249\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−13.8717	−	13.8717i	−0.641906	−	0.641906i	0.309118	−	0.951024i	$-0.399966\pi$
$467$	−13.8717	−	13.8717i	−0.641906	−	0.641906i	−0.951024	+	0.309118i	$0.899966\pi$
$468$		0			0
$469$	3.79046			0.175027
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−17.0375	−	17.0375i	−0.783386	−	0.783386i
$474$		0			0
$475$	−11.3177	+	32.1949i	−0.519292	+	1.47720i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5.00618			−0.228738			−0.114369	−	0.993438i	$-0.536485\pi$
$479$	−5.00618			−0.228738			−0.114369	+	0.993438i	$0.536485\pi$
$480$		0			0
$481$	−2.11354			−0.0963693
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−0.934240	+	5.47458i	−0.0424216	+	0.248588i
$486$		0			0
$487$	7.84638	+	7.84638i	0.355554	+	0.355554i	0.862171	−	0.506617i	$-0.169104\pi$
$487$	7.84638	+	7.84638i	0.355554	+	0.355554i	−0.506617	+	0.862171i	$0.669104\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.65566			−0.0747189			−0.0373594	−	0.999302i	$-0.511895\pi$
$491$	−1.65566			−0.0747189			−0.0373594	+	0.999302i	$0.511895\pi$
$492$		0			0
$493$	30.4198	+	30.4198i	1.37004	+	1.37004i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	4.52355	−	4.52355i	0.202909	−	0.202909i
$498$		0			0
$499$	28.0466			1.25554			0.627768	−	0.778400i	$-0.283969\pi$
$499$	28.0466			1.25554			0.627768	+	0.778400i	$0.283969\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	26.8464	+	26.8464i	1.19702	+	1.19702i	0.975055	+	0.221965i	$0.0712472\pi$
$503$	26.8464	+	26.8464i	1.19702	+	1.19702i	0.221965	+	0.975055i	$0.428753\pi$
$504$		0			0
$505$	2.43114	+	3.43162i	0.108184	+	0.152705i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	12.0740i		−	0.535170i	−0.963534	−	0.267585i	$-0.913774\pi$
$509$		−	12.0740i		−	0.535170i	0.963534	−	0.267585i	$-0.0862256\pi$
$510$		0			0
$511$	1.87596			0.0829878
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−33.4713	−	5.71189i	−1.47492	−	0.251696i
$516$		0			0
$517$	18.6457	−	18.6457i	0.820039	−	0.820039i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	20.1076i		−	0.880928i	−0.897770	−	0.440464i	$-0.854814\pi$
$521$		−	20.1076i		−	0.880928i	0.897770	−	0.440464i	$-0.145186\pi$
$522$		0			0
$523$	−5.72970	−	5.72970i	−0.250542	−	0.250542i	0.570651	−	0.821193i	$-0.306691\pi$
$523$	−5.72970	−	5.72970i	−0.250542	−	0.250542i	−0.821193	+	0.570651i	$0.806691\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	9.17788	−	9.17788i	0.399795	−	0.399795i
$528$		0			0
$529$		−	17.9476i		−	0.780331i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3.24688	+	3.24688i	−0.140638	+	0.140638i
$534$		0			0
$535$	−5.10529	+	29.9167i	−0.220721	+	1.29341i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			27.7063i			1.19339i
$540$		0			0
$541$		−	33.4356i		−	1.43751i	−0.695263	−	0.718755i	$-0.744712\pi$
$541$		−	33.4356i		−	1.43751i	0.695263	−	0.718755i	$-0.255288\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4.03454	+	5.69487i	0.172821	+	0.243941i
$546$		0			0
$547$	−9.08411	+	9.08411i	−0.388409	+	0.388409i	−0.874119	−	0.485711i	$-0.838561\pi$
$547$	−9.08411	+	9.08411i	−0.388409	+	0.388409i	0.485711	+	0.874119i	$0.338561\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			58.0777i			2.47419i
$552$		0			0
$553$	2.62524	−	2.62524i	0.111637	−	0.111637i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	5.55956	+	5.55956i	0.235566	+	0.235566i	0.815011	−	0.579445i	$-0.196731\pi$
$557$	5.55956	+	5.55956i	0.235566	+	0.235566i	−0.579445	+	0.815011i	$0.696731\pi$
$558$		0			0
$559$			5.23799i			0.221543i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6.38203	+	6.38203i	−0.268970	+	0.268970i	−0.828685	−	0.559715i	$-0.810911\pi$
$563$	−6.38203	+	6.38203i	−0.268970	+	0.268970i	0.559715	+	0.828685i	$0.310911\pi$
$564$		0			0
$565$	−3.11537	+	18.2558i	−0.131064	+	0.768029i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−24.2945			−1.01848			−0.509240	−	0.860625i	$-0.670073\pi$
$569$	−24.2945			−1.01848			−0.509240	+	0.860625i	$0.670073\pi$
$570$		0			0
$571$		−	3.35468i		−	0.140389i	−0.997533	−	0.0701945i	$-0.977638\pi$
$571$		−	3.35468i		−	0.140389i	0.997533	−	0.0701945i	$-0.0223620\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−10.1328	+	4.86168i	−0.422568	+	0.202746i
$576$		0			0
$577$	−16.6998	−	16.6998i	−0.695223	−	0.695223i	0.268153	−	0.963376i	$-0.413587\pi$
$577$	−16.6998	−	16.6998i	−0.695223	−	0.695223i	−0.963376	+	0.268153i	$0.913587\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3.14697			−0.130558
$582$		0			0
$583$	30.1768	−	30.1768i	1.24979	−	1.24979i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	31.8287	+	31.8287i	1.31371	+	1.31371i	0.918659	+	0.395050i	$0.129273\pi$
$587$	31.8287	+	31.8287i	1.31371	+	1.31371i	0.395050	+	0.918659i	$0.370727\pi$
$588$		0			0
$589$	17.5225			0.722000
$590$		0			0
$591$		0			0
$592$		0			0
$593$	8.80387	+	8.80387i	0.361532	+	0.361532i	0.864377	−	0.502845i	$-0.167713\pi$
$593$	8.80387	+	8.80387i	0.361532	+	0.361532i	−0.502845	+	0.864377i	$0.667713\pi$
$594$		0			0
$595$	2.83408	+	4.00038i	0.116186	+	0.163999i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−36.2251			−1.48012			−0.740060	−	0.672541i	$-0.765203\pi$
$599$	−36.2251			−1.48012			−0.740060	+	0.672541i	$0.765203\pi$
$600$		0			0
$601$	−5.92157			−0.241546			−0.120773	−	0.992680i	$-0.538537\pi$
$601$	−5.92157			−0.241546			−0.120773	+	0.992680i	$0.538537\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	7.16508	+	10.1137i	0.291302	+	0.411181i
$606$		0			0
$607$	−22.5127	−	22.5127i	−0.913763	−	0.913763i	0.0828026	−	0.996566i	$-0.473613\pi$
$607$	−22.5127	−	22.5127i	−0.913763	−	0.913763i	−0.996566	+	0.0828026i	$0.973613\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−5.73242			−0.231909
$612$		0			0
$613$	17.8351	+	17.8351i	0.720352	+	0.720352i	0.968677	−	0.248325i	$-0.0798800\pi$
$613$	17.8351	+	17.8351i	0.720352	+	0.720352i	−0.248325	+	0.968677i	$0.579880\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4.87456	+	4.87456i	−0.196242	+	0.196242i	−0.798387	−	0.602145i	$-0.794313\pi$
$617$	−4.87456	+	4.87456i	−0.196242	+	0.196242i	0.602145	+	0.798387i	$0.294313\pi$
$618$		0			0
$619$	−33.3372			−1.33994			−0.669968	−	0.742390i	$-0.733692\pi$
$619$	−33.3372			−1.33994			−0.669968	+	0.742390i	$0.733692\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−3.03070	−	3.03070i	−0.121422	−	0.121422i
$624$		0			0
$625$	−19.5007	−	15.6437i	−0.780026	−	0.625747i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			12.0848i			0.481854i
$630$		0			0
$631$	11.3469			0.451715			0.225857	−	0.974160i	$-0.427482\pi$
$631$	11.3469			0.451715			0.225857	+	0.974160i	$0.427482\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−7.83971	+	45.9402i	−0.311109	+	1.82308i
$636$		0			0
$637$	4.25899	−	4.25899i	0.168747	−	0.168747i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	0.578269i		−	0.0228403i	−0.999935	−	0.0114201i	$-0.996365\pi$
$641$		−	0.578269i		−	0.0228403i	0.999935	−	0.0114201i	$-0.00363522\pi$
$642$		0			0
$643$	23.4952	+	23.4952i	0.926559	+	0.926559i	0.997482	−	0.0709229i	$-0.0225944\pi$
$643$	23.4952	+	23.4952i	0.926559	+	0.926559i	−0.0709229	+	0.997482i	$0.522594\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	25.6996	−	25.6996i	1.01036	−	1.01036i	0.0104099	−	0.999946i	$-0.496686\pi$
$647$	25.6996	−	25.6996i	1.01036	−	1.01036i	0.999946	−	0.0104099i	$-0.00331364\pi$
$648$		0			0
$649$		−	35.7744i		−	1.40427i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−24.0153	+	24.0153i	−0.939791	+	0.939791i	−0.998288	−	0.0584965i	$-0.981369\pi$
$653$	−24.0153	+	24.0153i	−0.939791	+	0.939791i	0.0584965	+	0.998288i	$0.481369\pi$
$654$		0			0
$655$	−7.58629	−	10.7083i	−0.296421	−	0.418406i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	6.25348i		−	0.243601i	−0.992555	−	0.121800i	$-0.961133\pi$
$659$		−	6.25348i		−	0.243601i	0.992555	−	0.121800i	$-0.0388668\pi$
$660$		0			0
$661$		−	11.0800i		−	0.430960i	−0.976508	−	0.215480i	$-0.930868\pi$
$661$		−	11.0800i		−	0.430960i	0.976508	−	0.215480i	$-0.0691317\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.11335	+	6.52419i	−0.0431740	+	0.252997i
$666$		0			0
$667$	−13.5246	+	13.5246i	−0.523675	+	0.523675i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	24.7460i		−	0.955310i
$672$		0			0
$673$	5.39727	−	5.39727i	0.208049	−	0.208049i	−0.595389	−	0.803438i	$-0.703002\pi$
$673$	5.39727	−	5.39727i	0.208049	−	0.208049i	0.803438	+	0.595389i	$0.203002\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7.81449	+	7.81449i	0.300335	+	0.300335i	0.841145	−	0.540810i	$-0.181882\pi$
$677$	7.81449	+	7.81449i	0.300335	+	0.300335i	−0.540810	+	0.841145i	$0.681882\pi$
$678$		0			0
$679$			1.07710i			0.0413353i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−19.2280	+	19.2280i	−0.735740	+	0.735740i	−0.971750	−	0.236011i	$-0.924160\pi$
$683$	−19.2280	+	19.2280i	−0.735740	+	0.735740i	0.236011	+	0.971750i	$0.424160\pi$
$684$		0			0
$685$	41.9521	+	7.15914i	1.60291	+	0.273536i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−9.27750			−0.353445
$690$		0			0
$691$			23.0256i			0.875936i	0.898991	+	0.437968i	$0.144302\pi$
$691$			23.0256i			0.875936i	−0.898991	+	0.437968i	$0.855698\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−20.8101	−	29.3741i	−0.789373	−	1.11422i
$696$		0			0
$697$	18.5651	+	18.5651i	0.703201	+	0.703201i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−36.0487			−1.36154			−0.680771	−	0.732496i	$-0.738355\pi$
$701$	−36.0487			−1.36154			−0.680771	+	0.732496i	$0.738355\pi$
$702$		0			0
$703$	−11.5362	+	11.5362i	−0.435097	+	0.435097i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	0.576735	+	0.576735i	0.0216904	+	0.0216904i
$708$		0			0
$709$	10.2426			0.384667			0.192334	−	0.981330i	$-0.438394\pi$
$709$	10.2426			0.384667			0.192334	+	0.981330i	$0.438394\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	4.08047	+	4.08047i	0.152815	+	0.152815i
$714$		0			0
$715$	1.35275	−	7.92702i	0.0505899	−	0.296454i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	39.2886			1.46522			0.732609	−	0.680650i	$-0.238302\pi$
$719$	39.2886			1.46522			0.732609	+	0.680650i	$0.238302\pi$
$720$		0			0
$721$	−6.58534			−0.245251
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−40.1383	−	14.1101i	−1.49070	−	0.524038i
$726$		0			0
$727$	−15.2333	−	15.2333i	−0.564970	−	0.564970i	0.365745	−	0.930715i	$-0.380814\pi$
$727$	−15.2333	−	15.2333i	−0.564970	−	0.564970i	−0.930715	+	0.365745i	$0.880814\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	29.9498			1.10773
$732$		0			0
$733$	−12.3754	−	12.3754i	−0.457097	−	0.457097i	0.440604	−	0.897701i	$-0.354764\pi$
$733$	−12.3754	−	12.3754i	−0.457097	−	0.457097i	−0.897701	+	0.440604i	$0.854764\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−25.1379	+	25.1379i	−0.925965	+	0.925965i
$738$		0			0
$739$	25.8730			0.951753			0.475876	−	0.879512i	$-0.342131\pi$
$739$	25.8730			0.951753			0.475876	+	0.879512i	$0.342131\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	15.9812	+	15.9812i	0.586292	+	0.586292i	0.936625	−	0.350333i	$-0.113932\pi$
$743$	15.9812	+	15.9812i	0.586292	+	0.586292i	−0.350333	+	0.936625i	$0.613932\pi$
$744$		0			0
$745$	8.32198	−	5.89573i	0.304894	−	0.216003i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			5.88597i			0.215069i
$750$		0			0
$751$	33.3102			1.21551			0.607754	−	0.794125i	$-0.292071\pi$
$751$	33.3102			1.21551			0.607754	+	0.794125i	$0.292071\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	16.0845	+	22.7038i	0.585376	+	0.826275i
$756$		0			0
$757$	35.1121	−	35.1121i	1.27617	−	1.27617i	0.333378	−	0.942793i	$-0.391812\pi$
$757$	35.1121	−	35.1121i	1.27617	−	1.27617i	0.942793	−	0.333378i	$-0.108188\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			22.5518i			0.817502i	0.912646	+	0.408751i	$0.134036\pi$
$761$			22.5518i			0.817502i	−0.912646	+	0.408751i	$0.865964\pi$
$762$		0			0
$763$	0.957109	+	0.957109i	0.0346497	+	0.0346497i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−5.49922	+	5.49922i	−0.198565	+	0.198565i
$768$		0			0
$769$			16.8578i			0.607907i	0.952687	+	0.303953i	$0.0983067\pi$
$769$			16.8578i			0.607907i	−0.952687	+	0.303953i	$0.901693\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	2.59499	−	2.59499i	0.0933354	−	0.0933354i	−0.658897	−	0.752233i	$-0.728977\pi$
$773$	2.59499	−	2.59499i	0.0933354	−	0.0933354i	0.752233	+	0.658897i	$0.228977\pi$
$774$		0			0
$775$	−4.25713	+	12.1100i	−0.152921	+	0.435005i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			35.4445i			1.26993i
$780$		0			0
$781$			59.9992i			2.14694i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	41.9888	+	7.16540i	1.49864	+	0.255744i
$786$		0			0
$787$	25.1710	−	25.1710i	0.897248	−	0.897248i	−0.0979438	−	0.995192i	$-0.531227\pi$
$787$	25.1710	−	25.1710i	0.897248	−	0.897248i	0.995192	+	0.0979438i	$0.0312265\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			3.59176i			0.127708i
$792$		0			0
$793$	−3.80394	+	3.80394i	−0.135082	+	0.135082i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−0.184398	−	0.184398i	−0.00653173	−	0.00653173i	0.703833	−	0.710365i	$-0.251470\pi$
$797$	−0.184398	−	0.184398i	−0.00653173	−	0.00653173i	−0.710365	+	0.703833i	$0.751470\pi$
$798$		0			0
$799$			32.7769i			1.15956i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−12.4411	+	12.4411i	−0.439038	+	0.439038i
$804$		0			0
$805$	−1.77856	+	1.26003i	−0.0626861	+	0.0444101i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	23.3352			0.820422			0.410211	−	0.911991i	$-0.365455\pi$
$809$	23.3352			0.820422			0.410211	+	0.911991i	$0.365455\pi$
$810$		0			0
$811$			4.50503i			0.158193i	0.996867	+	0.0790964i	$0.0252035\pi$
$811$			4.50503i			0.158193i	−0.996867	+	0.0790964i	$0.974797\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3.31887	+	19.4484i	−0.116255	+	0.681247i
$816$		0			0
$817$	28.5902	+	28.5902i	1.00024	+	1.00024i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−13.4411			−0.469097			−0.234549	−	0.972104i	$-0.575361\pi$
$821$	−13.4411			−0.469097			−0.234549	+	0.972104i	$0.575361\pi$
$822$		0			0
$823$	−12.1118	+	12.1118i	−0.422190	+	0.422190i	−0.885957	−	0.463767i	$-0.846497\pi$
$823$	−12.1118	+	12.1118i	−0.422190	+	0.422190i	0.463767	+	0.885957i	$0.346497\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−3.92415	−	3.92415i	−0.136456	−	0.136456i	0.635579	−	0.772035i	$-0.280761\pi$
$827$	−3.92415	−	3.92415i	−0.136456	−	0.136456i	−0.772035	+	0.635579i	$0.780761\pi$
$828$		0			0
$829$	7.47321			0.259555			0.129778	−	0.991543i	$-0.458574\pi$
$829$	7.47321			0.259555			0.129778	+	0.991543i	$0.458574\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−24.3521	−	24.3521i	−0.843749	−	0.843749i
$834$		0			0
$835$	13.3093	+	2.27123i	0.460587	+	0.0785992i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	19.0880			0.658991			0.329495	−	0.944157i	$-0.393121\pi$
$839$	19.0880			0.658991			0.329495	+	0.944157i	$0.393121\pi$
$840$		0			0
$841$	−43.4073			−1.49680
$842$		0			0
$843$		0			0
$844$		0			0
$845$	22.2931	−	15.7936i	0.766906	−	0.543316i
$846$		0			0
$847$	1.69976	+	1.69976i	0.0584045	+	0.0584045i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−5.37290			−0.184181
$852$		0			0
$853$	0.0201924	+	0.0201924i	0.000691373	+	0.000691373i	0.707452	−	0.706761i	$-0.249844\pi$
$853$	0.0201924	+	0.0201924i	0.000691373	+	0.000691373i	−0.706761	+	0.707452i	$0.749844\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.32849	+	1.32849i	−0.0453805	+	0.0453805i	−0.729433	−	0.684052i	$-0.760216\pi$
$857$	−1.32849	+	1.32849i	−0.0453805	+	0.0453805i	0.684052	+	0.729433i	$0.260216\pi$
$858$		0			0
$859$	10.8364			0.369735			0.184867	−	0.982764i	$-0.440814\pi$
$859$	10.8364			0.369735			0.184867	+	0.982764i	$0.440814\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	8.57479	+	8.57479i	0.291889	+	0.291889i	0.837826	−	0.545937i	$-0.183826\pi$
$863$	8.57479	+	8.57479i	0.291889	+	0.291889i	−0.545937	+	0.837826i	$0.683826\pi$
$864$		0			0
$865$	43.6934	+	7.45629i	1.48562	+	0.253521i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			34.8205i			1.18120i
$870$		0			0
$871$	7.72835			0.261865
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−4.23847	−	2.35453i	−0.143286	−	0.0795975i
$876$		0			0
$877$	−6.82619	+	6.82619i	−0.230504	+	0.230504i	−0.812903	−	0.582399i	$-0.802114\pi$
$877$	−6.82619	+	6.82619i	−0.230504	+	0.230504i	0.582399	+	0.812903i	$0.302114\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	33.3297i		−	1.12290i	−0.827509	−	0.561452i	$-0.810243\pi$
$881$		−	33.3297i		−	1.12290i	0.827509	−	0.561452i	$-0.189757\pi$
$882$		0			0
$883$	−15.0182	−	15.0182i	−0.505403	−	0.505403i	0.407709	−	0.913112i	$-0.366328\pi$
$883$	−15.0182	−	15.0182i	−0.505403	−	0.505403i	−0.913112	+	0.407709i	$0.866328\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	37.6074	−	37.6074i	1.26273	−	1.26273i	0.312971	−	0.949763i	$-0.398676\pi$
$887$	37.6074	−	37.6074i	1.26273	−	1.26273i	0.949763	−	0.312971i	$-0.101324\pi$
$888$		0			0
$889$			9.03853i			0.303143i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−31.2889	+	31.2889i	−1.04704	+	1.04704i
$894$		0			0
$895$	30.7669	−	21.7969i	1.02842	−	0.728589i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			21.8458i			0.728598i
$900$		0			0
$901$			53.0470i			1.76725i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	34.1276	−	24.1778i	1.13444	−	0.803696i
$906$		0			0
$907$	−29.6844	+	29.6844i	−0.985654	+	0.985654i	−0.999899	−	0.0142450i	$-0.995466\pi$
$907$	−29.6844	+	29.6844i	−0.985654	+	0.985654i	0.0142450	+	0.999899i	$0.495466\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			9.37413i			0.310579i	0.987869	+	0.155289i	$0.0496310\pi$
$911$			9.37413i			0.310579i	−0.987869	+	0.155289i	$0.950369\pi$
$912$		0			0
$913$	20.8703	−	20.8703i	0.690705	−	0.690705i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.79969	−	1.79969i	−0.0594309	−	0.0594309i
$918$		0			0
$919$			12.2408i			0.403788i	0.979407	+	0.201894i	$0.0647097\pi$
$919$			12.2408i			0.403788i	−0.979407	+	0.201894i	$0.935290\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	9.22303	−	9.22303i	0.303580	−	0.303580i
$924$		0			0
$925$	−5.17009	−	10.7756i	−0.169991	−	0.354300i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	3.79903			0.124642			0.0623210	−	0.998056i	$-0.480150\pi$
$929$	3.79903			0.124642			0.0623210	+	0.998056i	$0.480150\pi$
$930$		0			0
$931$		−	46.4931i		−	1.52375i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−45.3252	−	7.73476i	−1.48229	−	0.252954i
$936$		0			0
$937$	−6.03853	−	6.03853i	−0.197270	−	0.197270i	0.601559	−	0.798829i	$-0.294547\pi$
$937$	−6.03853	−	6.03853i	−0.197270	−	0.197270i	−0.798829	+	0.601559i	$0.794547\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	29.9232			0.975469			0.487735	−	0.872992i	$-0.337823\pi$
$941$	29.9232			0.975469			0.487735	+	0.872992i	$0.337823\pi$
$942$		0			0
$943$	−8.25400	+	8.25400i	−0.268787	+	0.268787i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−24.1129	−	24.1129i	−0.783564	−	0.783564i	0.196867	−	0.980430i	$-0.436923\pi$
$947$	−24.1129	−	24.1129i	−0.783564	−	0.783564i	−0.980430	+	0.196867i	$0.936923\pi$
$948$		0			0
$949$	3.82489			0.124161
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−2.14110	−	2.14110i	−0.0693571	−	0.0693571i	0.671577	−	0.740934i	$-0.265617\pi$
$953$	−2.14110	−	2.14110i	−0.0693571	−	0.0693571i	−0.740934	+	0.671577i	$0.765617\pi$
$954$		0			0
$955$	11.9039	−	8.43336i	0.385202	−	0.272897i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	8.25389			0.266532
$960$		0			0
$961$	−24.4090			−0.787386
$962$		0			0
$963$		0			0
$964$		0			0
$965$	4.76674	+	0.813445i	0.153447	+	0.0261857i
$966$		0			0
$967$	−32.0860	−	32.0860i	−1.03182	−	1.03182i	−0.999477	−	0.0323385i	$-0.989705\pi$
$967$	−32.0860	−	32.0860i	−1.03182	−	1.03182i	−0.0323385	−	0.999477i	$-0.510295\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−43.3908			−1.39248			−0.696239	−	0.717810i	$-0.745144\pi$
$971$	−43.3908			−1.39248			−0.696239	+	0.717810i	$0.745144\pi$
$972$		0			0
$973$	−4.93676	−	4.93676i	−0.158265	−	0.158265i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−24.5104	+	24.5104i	−0.784156	+	0.784156i	−0.980529	−	0.196374i	$-0.937083\pi$
$977$	−24.5104	+	24.5104i	−0.784156	+	0.784156i	0.196374	+	0.980529i	$0.437083\pi$
$978$		0			0
$979$	40.1984			1.28475
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−20.3315	−	20.3315i	−0.648473	−	0.648473i	0.304151	−	0.952624i	$-0.401627\pi$
$983$	−20.3315	−	20.3315i	−0.648473	−	0.648473i	−0.952624	+	0.304151i	$0.901627\pi$
$984$		0			0
$985$	2.05061	−	12.0165i	0.0653380	−	0.382877i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			13.3157i			0.423413i
$990$		0			0
$991$	−3.08590			−0.0980269			−0.0490135	−	0.998798i	$-0.515608\pi$
$991$	−3.08590			−0.0980269			−0.0490135	+	0.998798i	$0.515608\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	26.1079	−	18.4962i	0.827676	−	0.586369i
$996$		0			0
$997$	−18.0364	+	18.0364i	−0.571217	+	0.571217i	−0.932468	−	0.361251i	$-0.882350\pi$
$997$	−18.0364	+	18.0364i	−0.571217	+	0.571217i	0.361251	+	0.932468i	$0.382350\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.2.bj.a.17.15		48
3.2	odd	2	inner	1440.2.bj.a.17.10		48
4.3	odd	2		360.2.x.a.197.19	yes	48
5.3	odd	4	inner	1440.2.bj.a.593.16		48
8.3	odd	2		360.2.x.a.197.18	yes	48
8.5	even	2	inner	1440.2.bj.a.17.9		48
12.11	even	2		360.2.x.a.197.6	yes	48
15.8	even	4	inner	1440.2.bj.a.593.9		48
20.3	even	4		360.2.x.a.53.7	yes	48
24.5	odd	2	inner	1440.2.bj.a.17.16		48
24.11	even	2		360.2.x.a.197.7	yes	48
40.3	even	4		360.2.x.a.53.6	✓	48
40.13	odd	4	inner	1440.2.bj.a.593.10		48
60.23	odd	4		360.2.x.a.53.18	yes	48
120.53	even	4	inner	1440.2.bj.a.593.15		48
120.83	odd	4		360.2.x.a.53.19	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.x.a.53.6	✓	48	40.3	even	4
360.2.x.a.53.7	yes	48	20.3	even	4
360.2.x.a.53.18	yes	48	60.23	odd	4
360.2.x.a.53.19	yes	48	120.83	odd	4
360.2.x.a.197.6	yes	48	12.11	even	2
360.2.x.a.197.7	yes	48	24.11	even	2
360.2.x.a.197.18	yes	48	8.3	odd	2
360.2.x.a.197.19	yes	48	4.3	odd	2
1440.2.bj.a.17.9		48	8.5	even	2	inner
1440.2.bj.a.17.10		48	3.2	odd	2	inner
1440.2.bj.a.17.15		48	1.1	even	1	trivial
1440.2.bj.a.17.16		48	24.5	odd	2	inner
1440.2.bj.a.593.9		48	15.8	even	4	inner
1440.2.bj.a.593.10		48	40.13	odd	4	inner
1440.2.bj.a.593.15		48	120.53	even	4	inner
1440.2.bj.a.593.16		48	5.3	odd	4	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:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1440.bj (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.29263 + 1.82458i) q^{5} +(0.306649 + 0.306649i) q^{7} +O(q^{10})\)	\(q+(1.29263 + 1.82458i) q^{5} +(0.306649 + 0.306649i) q^{7} -4.06731 q^{11} +(0.625224 + 0.625224i) q^{13} +(3.57491 - 3.57491i) q^{17} +6.82524 q^{19} +(1.58940 + 1.58940i) q^{23} +(-1.65821 + 4.71703i) q^{25} +8.50925i q^{29} +2.56730 q^{31} +(-0.163123 + 0.955891i) q^{35} +(-1.69023 + 1.69023i) q^{37} +5.19315i q^{41} +(4.18889 + 4.18889i) q^{43} +(-4.58429 + 4.58429i) q^{47} -6.81193i q^{49} +(-7.41934 + 7.41934i) q^{53} +(-5.25753 - 7.42115i) q^{55} +8.79560i q^{59} +6.08412i q^{61} +(-0.332590 + 1.94896i) q^{65} +(6.18046 - 6.18046i) q^{67} -14.7516i q^{71} +(3.05881 - 3.05881i) q^{73} +(-1.24724 - 1.24724i) q^{77} -8.56106i q^{79} +(-5.13122 + 5.13122i) q^{83} +(11.1438 + 1.90169i) q^{85} -9.88328 q^{89} +0.383449i q^{91} +(8.82252 + 12.4532i) q^{95} +(1.75624 + 1.75624i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q - 32 q^{31} - 32 q^{97}+O(q^{100}) \) 48 * q - 32 * q^31 - 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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