LMFDB - Embedded newform 2340.2.bi.d.161.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2340,2,Mod(161,2340)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2340, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 0, 3])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2340.161"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2340.bi (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$18.6849940730$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	12.0.58498535041007616.52
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 12x^{9} + 72x^{6} - 324x^{3} + 729 $ x^12 - 12x^9 + 72x^6 - 324*x^3 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			161.5
Root			$-0.779723 + 1.54662i$ of defining polynomial
Character	$\chi$	$=$	2340.161
Dual form			2340.2.bi.d.1061.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 + 0.707107i) q^{5} +(-0.766897 - 0.766897i) q^{7} +(0.912129 - 0.912129i) q^{11} +(3.52305 + 0.766897i) q^{13} +2.49877 q^{17} +(2.00000 - 2.00000i) q^{19} -2.90881 q^{23} +1.00000i q^{25} +3.23847i q^{29} +(-1.53379 + 1.53379i) q^{31} -1.08456i q^{35} +(1.52305 + 1.52305i) q^{37} +(-1.25698 - 1.25698i) q^{41} -9.64748i q^{43} +(2.90881 - 2.90881i) q^{47} -5.82374i q^{49} +6.80660i q^{53} +1.28995 q^{55} +11.4036 q^{61} +(1.94889 + 3.03345i) q^{65} +(5.04610 - 5.04610i) q^{67} +(10.2175 + 10.2175i) q^{71} +(0.756152 + 0.756152i) q^{73} -1.39902 q^{77} +0.688564 q^{79} +(3.91298 + 3.91298i) q^{83} +(1.76690 + 1.76690i) q^{85} +(-2.98566 + 2.98566i) q^{89} +(-2.11368 - 3.28995i) q^{91} +2.82843 q^{95} +(7.05684 - 7.05684i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{13} + 24 q^{19} - 12 q^{37} - 24 q^{55} - 12 q^{73} + 24 q^{79} + 12 q^{85} + 72 q^{91} + 36 q^{97}+O(q^{100}) $ 12 * q + 12 * q^13 + 24 * q^19 - 12 * q^37 - 24 * q^55 - 12 * q^73 + 24 * q^79 + 12 * q^85 + 72 * q^91 + 36 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times$.

$n$	$937$	$1081$	$1171$	$2081$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i
$6$		0			0
$7$	−0.766897	−	0.766897i	−0.289860	−	0.289860i	0.547165	−	0.837025i	$-0.315707\pi$
$7$	−0.766897	−	0.766897i	−0.289860	−	0.289860i	−0.837025	+	0.547165i	$0.815707\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.912129	−	0.912129i	0.275017	−	0.275017i	−0.556099	−	0.831116i	$-0.687702\pi$
$11$	0.912129	−	0.912129i	0.275017	−	0.275017i	0.831116	+	0.556099i	$0.187702\pi$
$12$		0			0
$13$	3.52305	+	0.766897i	0.977118	+	0.212699i
$13$	3.52305	+	0.766897i	0.977118	+	0.212699i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.49877			0.606041			0.303020	−	0.952984i	$-0.402005\pi$
$17$	2.49877			0.606041			0.303020	+	0.952984i	$0.402005\pi$
$18$		0			0
$19$	2.00000	−	2.00000i	0.458831	−	0.458831i	−0.439440	−	0.898272i	$-0.644823\pi$
$19$	2.00000	−	2.00000i	0.458831	−	0.458831i	0.898272	+	0.439440i	$0.144823\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.90881			−0.606530			−0.303265	−	0.952906i	$-0.598077\pi$
$23$	−2.90881			−0.606530			−0.303265	+	0.952906i	$0.598077\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			3.23847i			0.601369i	0.953724	+	0.300685i	$0.0972151\pi$
$29$			3.23847i			0.601369i	−0.953724	+	0.300685i	$0.902785\pi$
$30$		0			0
$31$	−1.53379	+	1.53379i	−0.275477	+	0.275477i	−0.831301	−	0.555823i	$-0.812403\pi$
$31$	−1.53379	+	1.53379i	−0.275477	+	0.275477i	0.555823	+	0.831301i	$0.312403\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	1.08456i		−	0.183323i
$36$		0			0
$37$	1.52305	+	1.52305i	0.250388	+	0.250388i	0.821130	−	0.570742i	$-0.193344\pi$
$37$	1.52305	+	1.52305i	0.250388	+	0.250388i	−0.570742	+	0.821130i	$0.693344\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.25698	−	1.25698i	−0.196308	−	0.196308i	0.602107	−	0.798415i	$-0.294328\pi$
$41$	−1.25698	−	1.25698i	−0.196308	−	0.196308i	−0.798415	+	0.602107i	$0.794328\pi$
$42$		0			0
$43$		−	9.64748i		−	1.47123i	−0.677402	−	0.735613i	$-0.736894\pi$
$43$		−	9.64748i		−	1.47123i	0.677402	−	0.735613i	$-0.263106\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.90881	−	2.90881i	0.424294	−	0.424294i	−0.462385	−	0.886679i	$-0.653006\pi$
$47$	2.90881	−	2.90881i	0.424294	−	0.424294i	0.886679	+	0.462385i	$0.153006\pi$
$48$		0			0
$49$		−	5.82374i		−	0.831963i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			6.80660i			0.934958i	0.884004	+	0.467479i	$0.154838\pi$
$53$			6.80660i			0.934958i	−0.884004	+	0.467479i	$0.845162\pi$
$54$		0			0
$55$	1.28995			0.173936
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$59$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$60$		0			0
$61$	11.4036			1.46009			0.730043	−	0.683402i	$-0.239500\pi$
$61$	11.4036			1.46009			0.730043	+	0.683402i	$0.239500\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.94889	+	3.03345i	0.241730	+	0.376253i
$66$		0			0
$67$	5.04610	−	5.04610i	0.616479	−	0.616479i	−0.328148	−	0.944626i	$-0.606424\pi$
$67$	5.04610	−	5.04610i	0.616479	−	0.616479i	0.944626	+	0.328148i	$0.106424\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	10.2175	+	10.2175i	1.21259	+	1.21259i	0.970171	+	0.242423i	$0.0779423\pi$
$71$	10.2175	+	10.2175i	1.21259	+	1.21259i	0.242423	+	0.970171i	$0.422058\pi$
$72$		0			0
$73$	0.756152	+	0.756152i	0.0885008	+	0.0885008i	0.749971	−	0.661470i	$-0.230067\pi$
$73$	0.756152	+	0.756152i	0.0885008	+	0.0885008i	−0.661470	+	0.749971i	$0.730067\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.39902			−0.159433
$78$		0			0
$79$	0.688564			0.0774695			0.0387348	−	0.999250i	$-0.487667\pi$
$79$	0.688564			0.0774695			0.0387348	+	0.999250i	$0.487667\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.91298	+	3.91298i	0.429506	+	0.429506i	0.888460	−	0.458954i	$-0.151776\pi$
$83$	3.91298	+	3.91298i	0.429506	+	0.429506i	−0.458954	+	0.888460i	$0.651776\pi$
$84$		0			0
$85$	1.76690	+	1.76690i	0.191647	+	0.191647i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.98566	+	2.98566i	−0.316479	+	0.316479i	−0.847413	−	0.530934i	$-0.821841\pi$
$89$	−2.98566	+	2.98566i	−0.316479	+	0.316479i	0.530934	+	0.847413i	$0.321841\pi$
$90$		0			0
$91$	−2.11368	−	3.28995i	−0.221574	−	0.344880i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.82843			0.290191
$96$		0			0
$97$	7.05684	−	7.05684i	0.716514	−	0.716514i	−0.251376	−	0.967890i	$-0.580883\pi$
$97$	7.05684	−	7.05684i	0.716514	−	0.716514i	0.967890	+	0.251376i	$0.0808830\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	17.8862			1.77975			0.889873	−	0.456208i	$-0.150793\pi$
$101$	17.8862			1.77975			0.889873	+	0.456208i	$0.150793\pi$
$102$		0			0
$103$		−	7.62599i		−	0.751411i	−0.926739	−	0.375705i	$-0.877400\pi$
$103$		−	7.62599i		−	0.751411i	0.926739	−	0.375705i	$-0.122600\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			5.73724i			0.554640i	0.960778	+	0.277320i	$0.0894462\pi$
$107$			5.73724i			0.554640i	−0.960778	+	0.277320i	$0.910554\pi$
$108$		0			0
$109$	6.86984	−	6.86984i	0.658011	−	0.658011i	−0.296898	−	0.954909i	$-0.595952\pi$
$109$	6.86984	−	6.86984i	0.658011	−	0.658011i	0.954909	+	0.296898i	$0.0959522\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			0.249271i			0.0234495i	0.999931	+	0.0117247i	$0.00373218\pi$
$113$			0.249271i			0.0234495i	−0.999931	+	0.0117247i	$0.996268\pi$
$114$		0			0
$115$	−2.05684	−	2.05684i	−0.191801	−	0.191801i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.91630	−	1.91630i	−0.175667	−	0.175667i
$120$		0			0
$121$			9.33604i			0.848731i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	+	0.707107i	−0.0632456	+	0.0632456i
$126$		0			0
$127$		−	13.0461i		−	1.15765i	−0.815450	−	0.578827i	$-0.803511\pi$
$127$		−	13.0461i		−	1.15765i	0.815450	−	0.578827i	$-0.196489\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	1.50980i		−	0.131912i	−0.997823	−	0.0659558i	$-0.978990\pi$
$131$		−	1.50980i		−	0.131912i	0.997823	−	0.0659558i	$-0.0210096\pi$
$132$		0			0
$133$	−3.06759			−0.265994
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−12.5590	+	12.5590i	−1.07299	+	1.07299i	−0.0758735	+	0.997117i	$0.524175\pi$
$137$	−12.5590	+	12.5590i	−1.07299	+	1.07299i	−0.997117	+	0.0758735i	$0.975825\pi$
$138$		0			0
$139$	−3.31144			−0.280872			−0.140436	−	0.990090i	$-0.544850\pi$
$139$	−3.31144			−0.280872			−0.140436	+	0.990090i	$0.544850\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3.91298	−	2.51397i	0.327220	−	0.210228i
$144$		0			0
$145$	−2.28995	+	2.28995i	−0.190170	+	0.190170i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−0.567275	−	0.567275i	−0.0464730	−	0.0464730i	0.683488	−	0.729961i	$-0.260462\pi$
$149$	−0.567275	−	0.567275i	−0.0464730	−	0.0464730i	−0.729961	+	0.683488i	$0.760462\pi$
$150$		0			0
$151$	3.86984	+	3.86984i	0.314923	+	0.314923i	0.846813	−	0.531890i	$-0.178518\pi$
$151$	3.86984	+	3.86984i	0.314923	+	0.314923i	−0.531890	+	0.846813i	$0.678518\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.16911			−0.174227
$156$		0			0
$157$	−2.13517			−0.170405			−0.0852027	−	0.996364i	$-0.527154\pi$
$157$	−2.13517			−0.170405			−0.0852027	+	0.996364i	$0.527154\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2.23076	+	2.23076i	0.175809	+	0.175809i
$162$		0			0
$163$	9.83448	+	9.83448i	0.770296	+	0.770296i	0.978158	−	0.207862i	$-0.0666505\pi$
$163$	9.83448	+	9.83448i	0.770296	+	0.770296i	−0.207862	+	0.978158i	$0.566650\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−5.73724	+	5.73724i	−0.443961	+	0.443961i	−0.893341	−	0.449380i	$-0.851645\pi$
$167$	−5.73724	+	5.73724i	−0.443961	+	0.443961i	0.449380	+	0.893341i	$0.351645\pi$
$168$		0			0
$169$	11.8237	+	5.40363i	0.909518	+	0.415664i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−3.23847			−0.246216			−0.123108	−	0.992393i	$-0.539286\pi$
$173$	−3.23847			−0.246216			−0.123108	+	0.992393i	$0.539286\pi$
$174$		0			0
$175$	0.766897	−	0.766897i	0.0579719	−	0.0579719i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	10.3095			0.770571			0.385286	−	0.922797i	$-0.374103\pi$
$179$	10.3095			0.770571			0.385286	+	0.922797i	$0.374103\pi$
$180$		0			0
$181$			0.200867i			0.0149304i	0.999972	+	0.00746518i	$0.00237626\pi$
$181$			0.200867i			0.0149304i	−0.999972	+	0.00746518i	$0.997624\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			2.15392i			0.158359i
$186$		0			0
$187$	2.27920	−	2.27920i	0.166672	−	0.166672i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			3.14289i			0.227411i	0.993514	+	0.113706i	$0.0362721\pi$
$191$			3.14289i			0.227411i	−0.993514	+	0.113706i	$0.963728\pi$
$192$		0			0
$193$	−3.54454	−	3.54454i	−0.255141	−	0.255141i	0.567933	−	0.823075i	$-0.307743\pi$
$193$	−3.54454	−	3.54454i	−0.255141	−	0.255141i	−0.823075	+	0.567933i	$0.807743\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−4.16225	−	4.16225i	−0.296548	−	0.296548i	0.543112	−	0.839660i	$-0.317246\pi$
$197$	−4.16225	−	4.16225i	−0.296548	−	0.296548i	−0.839660	+	0.543112i	$0.817246\pi$
$198$		0			0
$199$		−	10.5799i		−	0.749989i	−0.927027	−	0.374994i	$-0.877645\pi$
$199$		−	10.5799i		−	0.749989i	0.927027	−	0.374994i	$-0.122355\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.48357	−	2.48357i	0.174313	−	0.174313i
$204$		0			0
$205$		−	1.77764i		−	0.124156i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	3.64852i		−	0.252373i
$210$		0			0
$211$	−4.44472			−0.305987			−0.152993	−	0.988227i	$-0.548891\pi$
$211$	−4.44472			−0.305987			−0.152993	+	0.988227i	$0.548891\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	6.82180	−	6.82180i	0.465243	−	0.465243i
$216$		0			0
$217$	2.35252			0.159700
$218$		0			0
$219$		0			0
$220$		0			0
$221$	8.80329	+	1.91630i	0.592173	+	0.128904i
$222$		0			0
$223$	−2.11368	+	2.11368i	−0.141543	+	0.141543i	−0.774328	−	0.632785i	$-0.781912\pi$
$223$	−2.11368	+	2.11368i	−0.141543	+	0.141543i	0.632785	+	0.774328i	$0.281912\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−17.7907	−	17.7907i	−1.18081	−	1.18081i	−0.979535	−	0.201272i	$-0.935493\pi$
$227$	−17.7907	−	17.7907i	−1.18081	−	1.18081i	−0.201272	−	0.979535i	$-0.564507\pi$
$228$		0			0
$229$	2.53379	+	2.53379i	0.167438	+	0.167438i	0.785852	−	0.618414i	$-0.212225\pi$
$229$	2.53379	+	2.53379i	0.167438	+	0.167438i	−0.618414	+	0.785852i	$0.712225\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	2.65954			0.174232			0.0871162	−	0.996198i	$-0.472235\pi$
$233$	2.65954			0.174232			0.0871162	+	0.996198i	$0.472235\pi$
$234$		0			0
$235$	4.11368			0.268347
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−5.90967	−	5.90967i	−0.382265	−	0.382265i	0.489653	−	0.871917i	$-0.337124\pi$
$239$	−5.90967	−	5.90967i	−0.382265	−	0.382265i	−0.871917	+	0.489653i	$0.837124\pi$
$240$		0			0
$241$	12.2899	+	12.2899i	0.791665	+	0.791665i	0.981765	−	0.190100i	$-0.0608812\pi$
$241$	12.2899	+	12.2899i	0.791665	+	0.791665i	−0.190100	+	0.981765i	$0.560881\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.11801	−	4.11801i	0.263090	−	0.263090i
$246$		0			0
$247$	8.57989	−	5.51230i	0.545925	−	0.350739i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	23.1027			1.45823			0.729113	−	0.684393i	$-0.239933\pi$
$251$	23.1027			1.45823			0.729113	+	0.684393i	$0.239933\pi$
$252$		0			0
$253$	−2.65321	+	2.65321i	−0.166806	+	0.166806i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−19.0208			−1.18648			−0.593242	−	0.805024i	$-0.702152\pi$
$257$	−19.0208			−1.18648			−0.593242	+	0.805024i	$0.702152\pi$
$258$		0			0
$259$		−	2.33604i		−	0.145155i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			1.50980i			0.0930980i	0.998916	+	0.0465490i	$0.0148224\pi$
$263$			1.50980i			0.0930980i	−0.998916	+	0.0465490i	$0.985178\pi$
$264$		0			0
$265$	−4.81299	+	4.81299i	−0.295660	+	0.295660i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			22.8838i			1.39525i	0.716464	+	0.697624i	$0.245759\pi$
$269$			22.8838i			1.39525i	−0.716464	+	0.697624i	$0.754241\pi$
$270$		0			0
$271$	−5.28995	−	5.28995i	−0.321341	−	0.321341i	0.527940	−	0.849282i	$-0.322964\pi$
$271$	−5.28995	−	5.28995i	−0.321341	−	0.321341i	−0.849282	+	0.527940i	$0.822964\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.912129	+	0.912129i	0.0550034	+	0.0550034i
$276$		0			0
$277$		−	7.27346i		−	0.437020i	−0.975835	−	0.218510i	$-0.929880\pi$
$277$		−	7.27346i		−	0.437020i	0.975835	−	0.218510i	$-0.0701196\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	9.48945	−	9.48945i	0.566093	−	0.566093i	−0.364938	−	0.931032i	$-0.618910\pi$
$281$	9.48945	−	9.48945i	0.566093	−	0.566093i	0.931032	+	0.364938i	$0.118910\pi$
$282$		0			0
$283$		−	11.8863i		−	0.706568i	−0.935516	−	0.353284i	$-0.885065\pi$
$283$		−	11.8863i		−	0.706568i	0.935516	−	0.353284i	$-0.114935\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			1.92795i			0.113803i
$288$		0			0
$289$	−10.7562			−0.632715
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−5.34239	+	5.34239i	−0.312106	+	0.312106i	−0.845725	−	0.533619i	$-0.820832\pi$
$293$	−5.34239	+	5.34239i	−0.312106	+	0.312106i	0.533619	+	0.845725i	$0.320832\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−10.2479	−	2.23076i	−0.592651	−	0.129008i
$300$		0			0
$301$	−7.39862	+	7.39862i	−0.426449	+	0.426449i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	8.06358	+	8.06358i	0.461719	+	0.461719i
$306$		0			0
$307$	−19.9267	−	19.9267i	−1.13728	−	1.13728i	−0.988937	−	0.148339i	$-0.952607\pi$
$307$	−19.9267	−	19.9267i	−1.13728	−	1.13728i	−0.148339	−	0.988937i	$-0.547393\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	9.14460			0.518543			0.259271	−	0.965805i	$-0.416518\pi$
$311$	9.14460			0.518543			0.259271	+	0.965805i	$0.416518\pi$
$312$		0			0
$313$	20.2274			1.14332			0.571659	−	0.820491i	$-0.306300\pi$
$313$	20.2274			1.14332			0.571659	+	0.820491i	$0.306300\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−12.0382	−	12.0382i	−0.676133	−	0.676133i	0.282990	−	0.959123i	$-0.408674\pi$
$317$	−12.0382	−	12.0382i	−0.676133	−	0.676133i	−0.959123	+	0.282990i	$0.908674\pi$
$318$		0			0
$319$	2.95390	+	2.95390i	0.165387	+	0.165387i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	4.99754	−	4.99754i	0.278070	−	0.278070i
$324$		0			0
$325$	−0.766897	+	3.52305i	−0.0425398	+	0.195424i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.46152			−0.245972
$330$		0			0
$331$	−16.0922	+	16.0922i	−0.884507	+	0.884507i	−0.993989	−	0.109482i	$-0.965081\pi$
$331$	−16.0922	+	16.0922i	−0.884507	+	0.884507i	0.109482	+	0.993989i	$0.465081\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	7.13626			0.389895
$336$		0			0
$337$		−	1.42011i		−	0.0773583i	−0.999252	−	0.0386792i	$-0.987685\pi$
$337$		−	1.42011i		−	0.0773583i	0.999252	−	0.0386792i	$-0.0123150\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			2.79803i			0.151522i
$342$		0			0
$343$	−9.83448	+	9.83448i	−0.531012	+	0.531012i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	4.19705i		−	0.225310i	−0.993634	−	0.112655i	$-0.964065\pi$
$347$		−	4.19705i		−	0.225310i	0.993634	−	0.112655i	$-0.0359354\pi$
$348$		0			0
$349$	−3.46621	−	3.46621i	−0.185542	−	0.185542i	0.608224	−	0.793766i	$-0.291882\pi$
$349$	−3.46621	−	3.46621i	−0.185542	−	0.185542i	−0.793766	+	0.608224i	$0.791882\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−19.1897	−	19.1897i	−1.02136	−	1.02136i	−0.999767	−	0.0215962i	$-0.993125\pi$
$353$	−19.1897	−	19.1897i	−1.02136	−	1.02136i	−0.0215962	−	0.999767i	$-0.506875\pi$
$354$		0			0
$355$			14.4497i			0.766912i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	4.17745	−	4.17745i	0.220477	−	0.220477i	−0.588222	−	0.808699i	$-0.700172\pi$
$359$	4.17745	−	4.17745i	0.220477	−	0.220477i	0.808699	+	0.588222i	$0.200172\pi$
$360$		0			0
$361$			11.0000i			0.578947i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			1.06936i			0.0559728i
$366$		0			0
$367$	−10.9539			−0.571789			−0.285895	−	0.958261i	$-0.592291\pi$
$367$	−10.9539			−0.571789			−0.285895	+	0.958261i	$0.592291\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	5.21996	−	5.21996i	0.271007	−	0.271007i
$372$		0			0
$373$	15.6690			0.811308			0.405654	−	0.914027i	$-0.367044\pi$
$373$	15.6690			0.811308			0.405654	+	0.914027i	$0.367044\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2.48357	+	11.4093i	−0.127911	+	0.587608i
$378$		0			0
$379$	−13.6690	+	13.6690i	−0.702128	+	0.702128i	−0.964867	−	0.262739i	$-0.915374\pi$
$379$	−13.6690	+	13.6690i	−0.702128	+	0.702128i	0.262739	+	0.964867i	$0.415374\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.90464	+	1.90464i	0.0973228	+	0.0973228i	0.754092	−	0.656769i	$-0.228077\pi$
$383$	1.90464	+	1.90464i	0.0973228	+	0.0973228i	−0.656769	+	0.754092i	$0.728077\pi$
$384$		0			0
$385$	−0.989255	−	0.989255i	−0.0504171	−	0.0504171i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−33.8526			−1.71640			−0.858198	−	0.513319i	$-0.828416\pi$
$389$	−33.8526			−1.71640			−0.858198	+	0.513319i	$0.828416\pi$
$390$		0			0
$391$	−7.26845			−0.367582
$392$		0			0
$393$		0			0
$394$		0			0
$395$	0.486889	+	0.486889i	0.0244980	+	0.0244980i
$396$		0			0
$397$	14.1244	+	14.1244i	0.708885	+	0.708885i	0.966301	−	0.257416i	$-0.0828710\pi$
$397$	14.1244	+	14.1244i	0.708885	+	0.708885i	−0.257416	+	0.966301i	$0.582871\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	0.410044	−	0.410044i	0.0204766	−	0.0204766i	−0.696794	−	0.717271i	$-0.745391\pi$
$401$	0.410044	−	0.410044i	0.0204766	−	0.0204766i	0.717271	+	0.696794i	$0.245391\pi$
$402$		0			0
$403$	−6.57989	+	4.22737i	−0.327768	+	0.210580i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	2.77843			0.137722
$408$		0			0
$409$	−3.68856	+	3.68856i	−0.182388	+	0.182388i	−0.792395	−	0.610008i	$-0.791166\pi$
$409$	−3.68856	+	3.68856i	−0.182388	+	0.182388i	0.610008	+	0.792395i	$0.291166\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$			5.53379i			0.271643i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	34.2323i		−	1.67236i	−0.548458	−	0.836178i	$-0.684785\pi$
$419$		−	34.2323i		−	1.67236i	0.548458	−	0.836178i	$-0.315215\pi$
$420$		0			0
$421$	−23.7858	+	23.7858i	−1.15925	+	1.15925i	−0.174610	+	0.984638i	$0.555866\pi$
$421$	−23.7858	+	23.7858i	−1.15925	+	1.15925i	−0.984638	+	0.174610i	$0.944134\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			2.49877i			0.121208i
$426$		0			0
$427$	−8.74541	−	8.74541i	−0.423220	−	0.423220i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	7.45072	+	7.45072i	0.358889	+	0.358889i	0.863403	−	0.504515i	$-0.168329\pi$
$431$	7.45072	+	7.45072i	0.358889	+	0.358889i	−0.504515	+	0.863403i	$0.668329\pi$
$432$		0			0
$433$			22.4332i			1.07807i	0.842282	+	0.539036i	$0.181212\pi$
$433$			22.4332i			1.07807i	−0.842282	+	0.539036i	$0.818788\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−5.81763	+	5.81763i	−0.278295	+	0.278295i
$438$		0			0
$439$		−	24.2438i		−	1.15710i	−0.815648	−	0.578548i	$-0.803620\pi$
$439$		−	24.2438i		−	1.15710i	0.815648	−	0.578548i	$-0.196380\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	14.8818i		−	0.707057i	−0.935424	−	0.353529i	$-0.884982\pi$
$443$		−	14.8818i		−	0.707057i	0.935424	−	0.353529i	$-0.115018\pi$
$444$		0			0
$445$	−4.22236			−0.200159
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−7.95280	+	7.95280i	−0.375316	+	0.375316i	−0.869409	−	0.494093i	$-0.835500\pi$
$449$	−7.95280	+	7.95280i	−0.375316	+	0.375316i	0.494093	+	0.869409i	$0.335500\pi$
$450$		0			0
$451$	−2.29306			−0.107976
$452$		0			0
$453$		0			0
$454$		0			0
$455$	0.831742	−	3.82094i	0.0389927	−	0.179129i
$456$		0			0
$457$	−10.9432	+	10.9432i	−0.511899	+	0.511899i	−0.915108	−	0.403209i	$-0.867895\pi$
$457$	−10.9432	+	10.9432i	−0.511899	+	0.511899i	0.403209	+	0.915108i	$0.367895\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−19.9633	−	19.9633i	−0.929784	−	0.929784i	0.0679078	−	0.997692i	$-0.478368\pi$
$461$	−19.9633	−	19.9633i	−0.929784	−	0.929784i	−0.997692	+	0.0679078i	$0.978368\pi$
$462$		0			0
$463$	−27.0865	−	27.0865i	−1.25881	−	1.25881i	−0.951660	−	0.307155i	$-0.900623\pi$
$463$	−27.0865	−	27.0865i	−1.25881	−	1.25881i	−0.307155	−	0.951660i	$-0.599377\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−32.1739			−1.48883			−0.744416	−	0.667716i	$-0.767272\pi$
$467$	−32.1739			−1.48883			−0.744416	+	0.667716i	$0.767272\pi$
$468$		0			0
$469$	−7.73967			−0.357385
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−8.79974	−	8.79974i	−0.404613	−	0.404613i
$474$		0			0
$475$	2.00000	+	2.00000i	0.0917663	+	0.0917663i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−10.5927	+	10.5927i	−0.483995	+	0.483995i	−0.906405	−	0.422410i	$-0.861184\pi$
$479$	−10.5927	+	10.5927i	−0.483995	+	0.483995i	0.422410	+	0.906405i	$0.361184\pi$
$480$		0			0
$481$	4.19775	+	6.53379i	0.191401	+	0.297915i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	9.97988			0.453163
$486$		0			0
$487$	−5.36828	+	5.36828i	−0.243260	+	0.243260i	−0.818197	−	0.574937i	$-0.805026\pi$
$487$	−5.36828	+	5.36828i	−0.243260	+	0.243260i	0.574937	+	0.818197i	$0.305026\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	35.2061			1.58883			0.794414	−	0.607377i	$-0.207778\pi$
$491$	35.2061			1.58883			0.794414	+	0.607377i	$0.207778\pi$
$492$		0			0
$493$			8.09219i			0.364454i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	15.6715i		−	0.702964i
$498$		0			0
$499$	−6.20087	+	6.20087i	−0.277589	+	0.277589i	−0.832146	−	0.554557i	$-0.812888\pi$
$499$	−6.20087	+	6.20087i	−0.277589	+	0.277589i	0.554557	+	0.832146i	$0.312888\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			30.4230i			1.35649i	0.734834	+	0.678247i	$0.237260\pi$
$503$			30.4230i			1.35649i	−0.734834	+	0.678247i	$0.762740\pi$
$504$		0			0
$505$	12.6475	+	12.6475i	0.562805	+	0.562805i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−13.6097	−	13.6097i	−0.603238	−	0.603238i	0.337933	−	0.941170i	$-0.390272\pi$
$509$	−13.6097	−	13.6097i	−0.603238	−	0.603238i	−0.941170	+	0.337933i	$0.890272\pi$
$510$		0			0
$511$		−	1.15978i		−	0.0513057i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	5.39239	−	5.39239i	0.237617	−	0.237617i
$516$		0			0
$517$		−	5.30643i		−	0.233376i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			29.8593i			1.30816i	0.756426	+	0.654079i	$0.226944\pi$
$521$			29.8593i			1.30816i	−0.756426	+	0.654079i	$0.773056\pi$
$522$		0			0
$523$	−25.9785			−1.13596			−0.567980	−	0.823042i	$-0.692275\pi$
$523$	−25.9785			−1.13596			−0.567980	+	0.823042i	$0.692275\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.83260	+	3.83260i	−0.166951	+	0.166951i
$528$		0			0
$529$	−14.5388			−0.632122
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3.46443	−	5.39239i	−0.150061	−	0.233570i
$534$		0			0
$535$	−4.05684	+	4.05684i	−0.175393	+	0.175393i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−5.31200	−	5.31200i	−0.228804	−	0.228804i
$540$		0			0
$541$	12.0511	+	12.0511i	0.518118	+	0.518118i	0.917001	−	0.398884i	$-0.130602\pi$
$541$	12.0511	+	12.0511i	0.518118	+	0.518118i	−0.398884	+	0.917001i	$0.630602\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	9.71541			0.416163
$546$		0			0
$547$	−17.8319			−0.762435			−0.381218	−	0.924485i	$-0.624495\pi$
$547$	−17.8319			−0.762435			−0.381218	+	0.924485i	$0.624495\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	6.47694	+	6.47694i	0.275927	+	0.275927i
$552$		0			0
$553$	−0.528058	−	0.528058i	−0.0224553	−	0.0224553i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2.48357	+	2.48357i	−0.105232	+	0.105232i	−0.757763	−	0.652530i	$-0.773708\pi$
$557$	−2.48357	+	2.48357i	−0.105232	+	0.105232i	0.652530	+	0.757763i	$0.273708\pi$
$558$		0			0
$559$	7.39862	−	33.9885i	0.312928	−	1.43756i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−15.7323			−0.663038			−0.331519	−	0.943449i	$-0.607561\pi$
$563$	−15.7323			−0.663038			−0.331519	+	0.943449i	$0.607561\pi$
$564$		0			0
$565$	−0.176261	+	0.176261i	−0.00741537	+	0.00741537i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−17.0124			−0.713199			−0.356599	−	0.934257i	$-0.616064\pi$
$569$	−17.0124			−0.713199			−0.356599	+	0.934257i	$0.616064\pi$
$570$		0			0
$571$			46.3461i			1.93952i	0.244056	+	0.969761i	$0.421522\pi$
$571$			46.3461i			1.93952i	−0.244056	+	0.969761i	$0.578478\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	2.90881i		−	0.121306i
$576$		0			0
$577$	6.59064	−	6.59064i	0.274372	−	0.274372i	−0.556486	−	0.830857i	$-0.687851\pi$
$577$	6.59064	−	6.59064i	0.274372	−	0.274372i	0.830857	+	0.556486i	$0.187851\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	6.00171i		−	0.248993i
$582$		0			0
$583$	6.20850	+	6.20850i	0.257130	+	0.257130i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−30.3193	−	30.3193i	−1.25141	−	1.25141i	−0.955088	−	0.296324i	$-0.904239\pi$
$587$	−30.3193	−	30.3193i	−1.25141	−	1.25141i	−0.296324	−	0.955088i	$-0.595761\pi$
$588$		0			0
$589$			6.13517i			0.252795i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	13.5480	−	13.5480i	0.556350	−	0.556350i	−0.371916	−	0.928266i	$-0.621299\pi$
$593$	13.5480	−	13.5480i	0.556350	−	0.556350i	0.928266	+	0.371916i	$0.121299\pi$
$594$		0			0
$595$		−	2.71005i		−	0.111101i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	0.813004i		−	0.0332184i	−0.999862	−	0.0166092i	$-0.994713\pi$
$599$		−	0.813004i		−	0.0332184i	0.999862	−	0.0166092i	$-0.00528712\pi$
$600$		0			0
$601$	−11.5288			−0.470269			−0.235134	−	0.971963i	$-0.575553\pi$
$601$	−11.5288			−0.470269			−0.235134	+	0.971963i	$0.575553\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−6.60158	+	6.60158i	−0.268392	+	0.268392i
$606$		0			0
$607$	21.1598			0.858849			0.429424	−	0.903103i	$-0.358716\pi$
$607$	21.1598			0.858849			0.429424	+	0.903103i	$0.358716\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	12.4787	−	8.01713i	0.504832	−	0.324338i
$612$		0			0
$613$	−6.43397	+	6.43397i	−0.259866	+	0.259866i	−0.824999	−	0.565134i	$-0.808825\pi$
$613$	−6.43397	+	6.43397i	−0.259866	+	0.259866i	0.565134	+	0.824999i	$0.308825\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−10.9385	−	10.9385i	−0.440366	−	0.440366i	0.451769	−	0.892135i	$-0.350793\pi$
$617$	−10.9385	−	10.9385i	−0.440366	−	0.440366i	−0.892135	+	0.451769i	$0.850793\pi$
$618$		0			0
$619$	2.35753	+	2.35753i	0.0947572	+	0.0947572i	0.752896	−	0.658139i	$-0.228656\pi$
$619$	2.35753	+	2.35753i	0.0947572	+	0.0947572i	−0.658139	+	0.752896i	$0.728656\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	4.57938			0.183469
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	3.80575	+	3.80575i	0.151745	+	0.151745i
$630$		0			0
$631$	16.1137	+	16.1137i	0.641476	+	0.641476i	0.950918	−	0.309443i	$-0.100142\pi$
$631$	16.1137	+	16.1137i	0.641476	+	0.641476i	−0.309443	+	0.950918i	$0.600142\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	9.22498	−	9.22498i	0.366082	−	0.366082i
$636$		0			0
$637$	4.46621	−	20.5173i	0.176958	−	0.812926i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	4.52671			0.178794			0.0893971	−	0.995996i	$-0.471506\pi$
$641$	4.52671			0.178794			0.0893971	+	0.995996i	$0.471506\pi$
$642$		0			0
$643$	−22.4359	+	22.4359i	−0.884784	+	0.884784i	−0.994016	−	0.109232i	$-0.965161\pi$
$643$	−22.4359	+	22.4359i	−0.884784	+	0.884784i	0.109232	+	0.994016i	$0.465161\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	26.3867			1.03737			0.518684	−	0.854966i	$-0.326422\pi$
$647$	26.3867			1.03737			0.518684	+	0.854966i	$0.326422\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	13.0424i		−	0.510388i	−0.966890	−	0.255194i	$-0.917861\pi$
$653$		−	13.0424i		−	0.510388i	0.966890	−	0.255194i	$-0.0821393\pi$
$654$		0			0
$655$	1.06759	−	1.06759i	0.0417141	−	0.0417141i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	27.8857i		−	1.08627i	−0.839644	−	0.543137i	$-0.817237\pi$
$659$		−	27.8857i		−	1.08627i	0.839644	−	0.543137i	$-0.182763\pi$
$660$		0			0
$661$	7.67208	+	7.67208i	0.298409	+	0.298409i	0.840391	−	0.541981i	$-0.182326\pi$
$661$	7.67208	+	7.67208i	0.298409	+	0.298409i	−0.541981	+	0.840391i	$0.682326\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2.16911	−	2.16911i	−0.0841145	−	0.0841145i
$666$		0			0
$667$		−	9.42011i		−	0.364748i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	10.4016	−	10.4016i	0.401549	−	0.401549i
$672$		0			0
$673$		−	10.8187i		−	0.417031i	−0.978019	−	0.208516i	$-0.933137\pi$
$673$		−	10.8187i		−	0.417031i	0.978019	−	0.208516i	$-0.0668632\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			21.1703i			0.813640i	0.913508	+	0.406820i	$0.133362\pi$
$677$			21.1703i			0.813640i	−0.913508	+	0.406820i	$0.866638\pi$
$678$		0			0
$679$	−10.8237			−0.415377
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−21.0443	+	21.0443i	−0.805238	+	0.805238i	−0.983909	−	0.178670i	$-0.942820\pi$
$683$	−21.0443	+	21.0443i	−0.805238	+	0.805238i	0.178670	+	0.983909i	$0.442820\pi$
$684$		0			0
$685$	−17.7612			−0.678619
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−5.21996	+	23.9800i	−0.198865	+	0.913565i
$690$		0			0
$691$	−27.2899	+	27.2899i	−1.03816	+	1.03816i	−0.0389160	+	0.999242i	$0.512390\pi$
$691$	−27.2899	+	27.2899i	−1.03816	+	1.03816i	−0.999242	+	0.0389160i	$0.987610\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−2.34154	−	2.34154i	−0.0888196	−	0.0888196i
$696$		0			0
$697$	−3.14091	−	3.14091i	−0.118970	−	0.118970i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	34.1438			1.28959			0.644796	−	0.764354i	$-0.276942\pi$
$701$	34.1438			1.28959			0.644796	+	0.764354i	$0.276942\pi$
$702$		0			0
$703$	6.09219			0.229771
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−13.7169	−	13.7169i	−0.515877	−	0.515877i
$708$		0			0
$709$	18.0511	+	18.0511i	0.677924	+	0.677924i	0.959530	−	0.281606i	$-0.0908673\pi$
$709$	18.0511	+	18.0511i	0.677924	+	0.677924i	−0.281606	+	0.959530i	$0.590867\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	4.46152	−	4.46152i	0.167085	−	0.167085i
$714$		0			0
$715$	4.54454	+	0.989255i	0.169956	+	0.0369960i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	9.65022			0.359893			0.179946	−	0.983676i	$-0.442408\pi$
$719$	9.65022			0.359893			0.179946	+	0.983676i	$0.442408\pi$
$720$		0			0
$721$	−5.84834	+	5.84834i	−0.217804	+	0.217804i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3.23847			−0.120274
$726$		0			0
$727$		−	0.672083i		−	0.0249262i	−0.999922	−	0.0124631i	$-0.996033\pi$
$727$		−	0.672083i		−	0.0249262i	0.999922	−	0.0124631i	$-0.00396723\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	24.1068i		−	0.891623i
$732$		0			0
$733$	5.14904	−	5.14904i	0.190184	−	0.190184i	−0.605592	−	0.795776i	$-0.707063\pi$
$733$	5.14904	−	5.14904i	0.190184	−	0.190184i	0.795776	+	0.605592i	$0.207063\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	9.20538i		−	0.339085i
$738$		0			0
$739$	2.60138	+	2.60138i	0.0956933	+	0.0956933i	0.753333	−	0.657639i	$-0.228445\pi$
$739$	2.60138	+	2.60138i	0.0956933	+	0.0956933i	−0.657639	+	0.753333i	$0.728445\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−25.8078	−	25.8078i	−0.946796	−	0.946796i	0.0518587	−	0.998654i	$-0.483485\pi$
$743$	−25.8078	−	25.8078i	−0.946796	−	0.946796i	−0.998654	+	0.0518587i	$0.983485\pi$
$744$		0			0
$745$		−	0.802248i		−	0.0293921i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4.39987	−	4.39987i	0.160768	−	0.160768i
$750$		0			0
$751$		−	35.0511i		−	1.27903i	−0.768777	−	0.639517i	$-0.779135\pi$
$751$		−	35.0511i		−	1.27903i	0.768777	−	0.639517i	$-0.220865\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			5.47277i			0.199175i
$756$		0			0
$757$	−23.6690			−0.860263			−0.430132	−	0.902766i	$-0.641533\pi$
$757$	−23.6690			−0.860263			−0.430132	+	0.902766i	$0.641533\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−6.00171	+	6.00171i	−0.217562	+	0.217562i	−0.807470	−	0.589908i	$-0.799164\pi$
$761$	−6.00171	+	6.00171i	−0.217562	+	0.217562i	0.589908	+	0.807470i	$0.299164\pi$
$762$		0			0
$763$	−10.5369			−0.381462
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−15.3525	+	15.3525i	−0.553626	+	0.553626i	−0.927485	−	0.373859i	$-0.878034\pi$
$769$	−15.3525	+	15.3525i	−0.553626	+	0.553626i	0.373859	+	0.927485i	$0.378034\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	2.24241	+	2.24241i	0.0806540	+	0.0806540i	0.746283	−	0.665629i	$-0.231837\pi$
$773$	2.24241	+	2.24241i	0.0806540	+	0.0806540i	−0.665629	+	0.746283i	$0.731837\pi$
$774$		0			0
$775$	−1.53379	−	1.53379i	−0.0550955	−	0.0550955i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−5.02793			−0.180144
$780$		0			0
$781$	18.6394			0.666968
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.50980	−	1.50980i	−0.0538869	−	0.0538869i
$786$		0			0
$787$	−0.715064	−	0.715064i	−0.0254893	−	0.0254893i	0.694247	−	0.719737i	$-0.255737\pi$
$787$	−0.715064	−	0.715064i	−0.0254893	−	0.0254893i	−0.719737	+	0.694247i	$0.755737\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	0.191165	−	0.191165i	0.00679705	−	0.00679705i
$792$		0			0
$793$	40.1755	+	8.74541i	1.42668	+	0.310558i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−38.9609			−1.38007			−0.690034	−	0.723777i	$-0.742404\pi$
$797$	−38.9609			−1.38007			−0.690034	+	0.723777i	$0.742404\pi$
$798$		0			0
$799$	7.26845	−	7.26845i	0.257139	−	0.257139i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.37942			0.0486785
$804$		0			0
$805$			3.15477i			0.111191i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1.69828i			0.0597085i	0.999554	+	0.0298542i	$0.00950431\pi$
$809$			1.69828i			0.0597085i	−0.999554	+	0.0298542i	$0.990496\pi$
$810$		0			0
$811$	−3.97851	+	3.97851i	−0.139704	+	0.139704i	−0.773500	−	0.633796i	$-0.781496\pi$
$811$	−3.97851	+	3.97851i	−0.139704	+	0.139704i	0.633796	+	0.773500i	$0.281496\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			13.9081i			0.487178i
$816$		0			0
$817$	−19.2950	−	19.2950i	−0.675045	−	0.675045i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	25.6853	+	25.6853i	0.896425	+	0.896425i	0.995118	−	0.0986930i	$-0.0314662\pi$
$821$	25.6853	+	25.6853i	0.896425	+	0.896425i	−0.0986930	+	0.995118i	$0.531466\pi$
$822$		0			0
$823$		−	37.2735i		−	1.29927i	−0.760246	−	0.649636i	$-0.774921\pi$
$823$		−	37.2735i		−	1.29927i	0.760246	−	0.649636i	$-0.225079\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−26.5171	+	26.5171i	−0.922090	+	0.922090i	−0.997177	−	0.0750873i	$-0.976076\pi$
$827$	−26.5171	+	26.5171i	−0.922090	+	0.922090i	0.0750873	+	0.997177i	$0.476076\pi$
$828$		0			0
$829$		−	20.5799i		−	0.714769i	−0.933957	−	0.357385i	$-0.883669\pi$
$829$		−	20.5799i		−	0.714769i	0.933957	−	0.357385i	$-0.116331\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	14.5522i		−	0.504203i
$834$		0			0
$835$	−8.11368			−0.280786
$836$		0			0
$837$		0			0
$838$		0			0
$839$	11.5432	−	11.5432i	0.398516	−	0.398516i	−0.479193	−	0.877709i	$-0.659071\pi$
$839$	11.5432	−	11.5432i	0.398516	−	0.398516i	0.877709	+	0.479193i	$0.159071\pi$
$840$		0			0
$841$	18.5123			0.638355
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4.53970	+	12.1816i	0.156171	+	0.419059i
$846$		0			0
$847$	7.15978	−	7.15978i	0.246013	−	0.246013i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−4.43026	−	4.43026i	−0.151867	−	0.151867i
$852$		0			0
$853$	−10.5906	−	10.5906i	−0.362616	−	0.362616i	0.502159	−	0.864775i	$-0.332539\pi$
$853$	−10.5906	−	10.5906i	−0.362616	−	0.362616i	−0.864775	+	0.502159i	$0.832539\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−19.4389			−0.664022			−0.332011	−	0.943276i	$-0.607727\pi$
$857$	−19.4389			−0.664022			−0.332011	+	0.943276i	$0.607727\pi$
$858$		0			0
$859$	28.6986			0.979183			0.489592	−	0.871952i	$-0.337146\pi$
$859$	28.6986			0.979183			0.489592	+	0.871952i	$0.337146\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	34.6075	+	34.6075i	1.17805	+	1.17805i	0.980239	+	0.197814i	$0.0633843\pi$
$863$	34.6075	+	34.6075i	1.17805	+	1.17805i	0.197814	+	0.980239i	$0.436616\pi$
$864$		0			0
$865$	−2.28995	−	2.28995i	−0.0778605	−	0.0778605i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	0.628059	−	0.628059i	0.0213055	−	0.0213055i
$870$		0			0
$871$	21.6475	−	13.9078i	0.733497	−	0.471248i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.08456			0.0366647
$876$		0			0
$877$	−3.47122	+	3.47122i	−0.117215	+	0.117215i	−0.763281	−	0.646066i	$-0.776413\pi$
$877$	−3.47122	+	3.47122i	−0.117215	+	0.117215i	0.646066	+	0.763281i	$0.276413\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	45.3200			1.52687			0.763435	−	0.645884i	$-0.223511\pi$
$881$	45.3200			1.52687			0.763435	+	0.645884i	$0.223511\pi$
$882$		0			0
$883$		−	4.78577i		−	0.161054i	−0.996752	−	0.0805269i	$-0.974340\pi$
$883$		−	4.78577i		−	0.161054i	0.996752	−	0.0805269i	$-0.0256603\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	18.4000i		−	0.617811i	−0.951093	−	0.308905i	$-0.900037\pi$
$887$		−	18.4000i		−	0.617811i	0.951093	−	0.308905i	$-0.0999626\pi$
$888$		0			0
$889$	−10.0050	+	10.0050i	−0.335557	+	0.335557i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	11.6353i		−	0.389359i
$894$		0			0
$895$	7.28995	+	7.28995i	0.243676	+	0.243676i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−4.96715	−	4.96715i	−0.165664	−	0.165664i
$900$		0			0
$901$			17.0081i			0.566623i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.142035	+	0.142035i	−0.00472139	+	0.00472139i
$906$		0			0
$907$		−	31.5830i		−	1.04870i	−0.851504	−	0.524348i	$-0.824309\pi$
$907$		−	31.5830i		−	1.04870i	0.851504	−	0.524348i	$-0.175691\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			0.0233077i			0.000772218i	1.00000		0.000386109i	$0.000122902\pi$
$911$			0.0233077i			0.000772218i	−1.00000		0.000386109i	$0.999877\pi$
$912$		0			0
$913$	7.13829			0.236243
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.15786	+	1.15786i	−0.0382358	+	0.0382358i
$918$		0			0
$919$	40.8730			1.34827			0.674137	−	0.738606i	$-0.264516\pi$
$919$	40.8730			1.34827			0.674137	+	0.738606i	$0.264516\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	28.1610	+	43.8325i	0.926930	+	1.44276i
$924$		0			0
$925$	−1.52305	+	1.52305i	−0.0500775	+	0.0500775i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	20.0285	+	20.0285i	0.657114	+	0.657114i	0.954696	−	0.297582i	$-0.0961803\pi$
$929$	20.0285	+	20.0285i	0.657114	+	0.657114i	−0.297582	+	0.954696i	$0.596180\pi$
$930$		0			0
$931$	−11.6475	−	11.6475i	−0.381731	−	0.381731i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	3.22328			0.105412
$936$		0			0
$937$	36.9209			1.20615			0.603077	−	0.797683i	$-0.293941\pi$
$937$	36.9209			1.20615			0.603077	+	0.797683i	$0.293941\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−3.89424	−	3.89424i	−0.126949	−	0.126949i	0.640778	−	0.767726i	$-0.278612\pi$
$941$	−3.89424	−	3.89424i	−0.126949	−	0.126949i	−0.767726	+	0.640778i	$0.778612\pi$
$942$		0			0
$943$	3.65633	+	3.65633i	0.119066	+	0.119066i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−20.0705	+	20.0705i	−0.652205	+	0.652205i	−0.953524	−	0.301319i	$-0.902573\pi$
$947$	−20.0705	+	20.0705i	−0.652205	+	0.652205i	0.301319	+	0.953524i	$0.402573\pi$
$948$		0			0
$949$	2.08407	+	3.24385i	0.0676517	+	0.105300i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−16.3335			−0.529095			−0.264547	−	0.964373i	$-0.585223\pi$
$953$	−16.3335			−0.529095			−0.264547	+	0.964373i	$0.585223\pi$
$954$		0			0
$955$	−2.22236	+	2.22236i	−0.0719138	+	0.0719138i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	19.2630			0.622034
$960$		0			0
$961$			26.2950i			0.848224i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	5.01273i		−	0.161366i
$966$		0			0
$967$	3.97851	−	3.97851i	0.127940	−	0.127940i	−0.640237	−	0.768177i	$-0.721164\pi$
$967$	3.97851	−	3.97851i	0.127940	−	0.127940i	0.768177	+	0.640237i	$0.221164\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			12.4179i			0.398508i	0.979948	+	0.199254i	$0.0638519\pi$
$971$			12.4179i			0.398508i	−0.979948	+	0.199254i	$0.936148\pi$
$972$		0			0
$973$	2.53953	+	2.53953i	0.0814136	+	0.0814136i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.83945	+	1.83945i	0.0588493	+	0.0588493i	0.735919	−	0.677070i	$-0.236750\pi$
$977$	1.83945	+	1.83945i	0.0588493	+	0.0588493i	−0.677070	+	0.735919i	$0.736750\pi$
$978$		0			0
$979$			5.44661i			0.174074i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−17.7603	+	17.7603i	−0.566464	+	0.566464i	−0.931136	−	0.364672i	$-0.881181\pi$
$983$	−17.7603	+	17.7603i	−0.566464	+	0.566464i	0.364672	+	0.931136i	$0.381181\pi$
$984$		0			0
$985$		−	5.88632i		−	0.187554i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			28.0627i			0.892342i
$990$		0			0
$991$	−33.2293			−1.05556			−0.527781	−	0.849380i	$-0.676976\pi$
$991$	−33.2293			−1.05556			−0.527781	+	0.849380i	$0.676976\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	7.48111	−	7.48111i	0.237167	−	0.237167i
$996$		0			0
$997$	46.3411			1.46764			0.733818	−	0.679346i	$-0.237737\pi$
$997$	46.3411			1.46764			0.733818	+	0.679346i	$0.237737\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2340.2.bi.d.161.5	yes	12
3.2	odd	2	inner	2340.2.bi.d.161.2	✓	12
13.8	odd	4	inner	2340.2.bi.d.1061.2	yes	12
39.8	even	4	inner	2340.2.bi.d.1061.5	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2340.2.bi.d.161.2	✓	12	3.2	odd	2	inner
2340.2.bi.d.161.5	yes	12	1.1	even	1	trivial
2340.2.bi.d.1061.2	yes	12	13.8	odd	4	inner
2340.2.bi.d.1061.5	yes	12	39.8	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2340.bi (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 + 0.707107i) q^{5} +(-0.766897 - 0.766897i) q^{7} +(0.912129 - 0.912129i) q^{11} +(3.52305 + 0.766897i) q^{13} +2.49877 q^{17} +(2.00000 - 2.00000i) q^{19} -2.90881 q^{23} +1.00000i q^{25} +3.23847i q^{29} +(-1.53379 + 1.53379i) q^{31} -1.08456i q^{35} +(1.52305 + 1.52305i) q^{37} +(-1.25698 - 1.25698i) q^{41} -9.64748i q^{43} +(2.90881 - 2.90881i) q^{47} -5.82374i q^{49} +6.80660i q^{53} +1.28995 q^{55} +11.4036 q^{61} +(1.94889 + 3.03345i) q^{65} +(5.04610 - 5.04610i) q^{67} +(10.2175 + 10.2175i) q^{71} +(0.756152 + 0.756152i) q^{73} -1.39902 q^{77} +0.688564 q^{79} +(3.91298 + 3.91298i) q^{83} +(1.76690 + 1.76690i) q^{85} +(-2.98566 + 2.98566i) q^{89} +(-2.11368 - 3.28995i) q^{91} +2.82843 q^{95} +(7.05684 - 7.05684i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{13} + 24 q^{19} - 12 q^{37} - 24 q^{55} - 12 q^{73} + 24 q^{79} + 12 q^{85} + 72 q^{91} + 36 q^{97}+O(q^{100}) \) 12 * q + 12 * q^13 + 24 * q^19 - 12 * q^37 - 24 * q^55 - 12 * q^73 + 24 * q^79 + 12 * q^85 + 72 * q^91 + 36 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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