LMFDB - Embedded newform 2448.2.be.v.1585.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2448,2,Mod(1441,2448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2448, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("2448.1441");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2448 = 2^{4} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2448.be (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:	$19.5473784148$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(i)$
Coefficient field:	6.0.269485056.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 8x^{3} + 81x^{2} - 72x + 32 $ x^6 - 8x^3 + 81x^2 - 72*x + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1585.1
Root			$1.84885 - 1.84885i$ of defining polynomial
Character	$\chi$	$=$	2448.1585
Dual form			2448.2.be.v.1441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.84885 + 1.84885i) q^{5} +(1.00000 + 1.00000i) q^{7} +(-0.151150 - 0.151150i) q^{11} +0.836493 q^{13} +(-3.68534 - 1.84885i) q^{17} +6.23189i q^{19} +(-1.68534 - 1.68534i) q^{23} -1.83649i q^{25} +(-1.83649 + 1.83649i) q^{31} -3.69770 q^{35} +(2.83649 - 2.83649i) q^{37} +(0.0123570 + 0.0123570i) q^{41} +10.2319i q^{43} -7.06839 q^{47} -5.00000i q^{49} -7.06839i q^{53} +0.558907 q^{55} +0.302300i q^{59} +(-6.23189 - 6.23189i) q^{61} +(-1.54655 + 1.54655i) q^{65} +2.32701 q^{67} +(-5.86121 + 5.86121i) q^{71} +(5.23189 - 5.23189i) q^{73} -0.302300i q^{77} +(-4.39540 - 4.39540i) q^{79} +3.39540i q^{83} +(10.2319 - 3.39540i) q^{85} -15.6977 q^{89} +(0.836493 + 0.836493i) q^{91} +(-11.5218 - 11.5218i) q^{95} +(-7.55891 + 7.55891i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{7} - 12 q^{11} - 6 q^{17} + 6 q^{23} - 6 q^{31} + 12 q^{37} - 6 q^{41} + 12 q^{47} - 36 q^{55} + 12 q^{61} + 24 q^{65} + 24 q^{67} - 18 q^{71} - 18 q^{73} + 18 q^{79} + 12 q^{85} - 72 q^{89}+ \cdots - 6 q^{97}+O(q^{100}) $ 6 * q + 6 * q^7 - 12 * q^11 - 6 * q^17 + 6 * q^23 - 6 * q^31 + 12 * q^37 - 6 * q^41 + 12 * q^47 - 36 * q^55 + 12 * q^61 + 24 * q^65 + 24 * q^67 - 18 * q^71 - 18 * q^73 + 18 * q^79 + 12 * q^85 - 72 * q^89 - 48 * q^95 - 6 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2448\mathbb{Z}\right)^\times$.

$n$	$613$	$1361$	$1873$	$2143$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$	$1$

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
$5$	−1.84885	+	1.84885i	−0.826831	+	0.826831i	−0.987077	−	0.160246i	$-0.948771\pi$
$5$	−1.84885	+	1.84885i	−0.826831	+	0.826831i	0.160246	+	0.987077i	$0.448771\pi$
$6$		0			0
$7$	1.00000	+	1.00000i	0.377964	+	0.377964i	0.870367	−	0.492403i	$-0.163881\pi$
$7$	1.00000	+	1.00000i	0.377964	+	0.377964i	−0.492403	+	0.870367i	$0.663881\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.151150	−	0.151150i	−0.0455734	−	0.0455734i	0.683953	−	0.729526i	$-0.260259\pi$
$11$	−0.151150	−	0.151150i	−0.0455734	−	0.0455734i	−0.729526	+	0.683953i	$0.760259\pi$
$12$		0			0
$13$	0.836493			0.232001			0.116001	−	0.993249i	$-0.462993\pi$
$13$	0.836493			0.232001			0.116001	+	0.993249i	$0.462993\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.68534	−	1.84885i	−0.893827	−	0.448412i
$17$	−3.68534	−	1.84885i	−0.893827	−	0.448412i
$18$		0			0
$19$			6.23189i			1.42969i	0.699281	+	0.714847i	$0.253504\pi$
$19$			6.23189i			1.42969i	−0.699281	+	0.714847i	$0.746496\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.68534	−	1.68534i	−0.351418	−	0.351418i	0.509219	−	0.860637i	$-0.329934\pi$
$23$	−1.68534	−	1.68534i	−0.351418	−	0.351418i	−0.860637	+	0.509219i	$0.829934\pi$
$24$		0			0
$25$		−	1.83649i		−	0.367299i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$29$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$30$		0			0
$31$	−1.83649	+	1.83649i	−0.329844	+	0.329844i	−0.852527	−	0.522683i	$-0.824931\pi$
$31$	−1.83649	+	1.83649i	−0.329844	+	0.329844i	0.522683	+	0.852527i	$0.324931\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.69770			−0.625025
$36$		0			0
$37$	2.83649	−	2.83649i	0.466317	−	0.466317i	−0.434402	−	0.900719i	$-0.643040\pi$
$37$	2.83649	−	2.83649i	0.466317	−	0.466317i	0.900719	+	0.434402i	$0.143040\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.0123570	+	0.0123570i	0.00192984	+	0.00192984i	0.708071	−	0.706141i	$-0.249566\pi$
$41$	0.0123570	+	0.0123570i	0.00192984	+	0.00192984i	−0.706141	+	0.708071i	$0.749566\pi$
$42$		0			0
$43$			10.2319i			1.56035i	0.625562	+	0.780175i	$0.284870\pi$
$43$			10.2319i			1.56035i	−0.625562	+	0.780175i	$0.715130\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−7.06839			−1.03103			−0.515515	−	0.856881i	$-0.672399\pi$
$47$	−7.06839			−1.03103			−0.515515	+	0.856881i	$0.672399\pi$
$48$		0			0
$49$		−	5.00000i		−	0.714286i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	7.06839i		−	0.970918i	−0.874260	−	0.485459i	$-0.838653\pi$
$53$		−	7.06839i		−	0.970918i	0.874260	−	0.485459i	$-0.161347\pi$
$54$		0			0
$55$	0.558907			0.0753630
$56$		0			0
$57$		0			0
$58$		0			0
$59$			0.302300i			0.0393561i	0.999806	+	0.0196780i	$0.00626412\pi$
$59$			0.302300i			0.0393561i	−0.999806	+	0.0196780i	$0.993736\pi$
$60$		0			0
$61$	−6.23189	−	6.23189i	−0.797912	−	0.797912i	0.184854	−	0.982766i	$-0.440819\pi$
$61$	−6.23189	−	6.23189i	−0.797912	−	0.797912i	−0.982766	+	0.184854i	$0.940819\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.54655	+	1.54655i	−0.191826	+	0.191826i
$66$		0			0
$67$	2.32701			0.284290			0.142145	−	0.989846i	$-0.454600\pi$
$67$	2.32701			0.284290			0.142145	+	0.989846i	$0.454600\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−5.86121	+	5.86121i	−0.695597	+	0.695597i	−0.963458	−	0.267860i	$-0.913683\pi$
$71$	−5.86121	+	5.86121i	−0.695597	+	0.695597i	0.267860	+	0.963458i	$0.413683\pi$
$72$		0			0
$73$	5.23189	−	5.23189i	0.612347	−	0.612347i	−0.331210	−	0.943557i	$-0.607457\pi$
$73$	5.23189	−	5.23189i	0.612347	−	0.612347i	0.943557	+	0.331210i	$0.107457\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	0.302300i		−	0.0344503i
$78$		0			0
$79$	−4.39540	−	4.39540i	−0.494521	−	0.494521i	0.415206	−	0.909727i	$-0.363709\pi$
$79$	−4.39540	−	4.39540i	−0.494521	−	0.494521i	−0.909727	+	0.415206i	$0.863709\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			3.39540i			0.372694i	0.982484	+	0.186347i	$0.0596648\pi$
$83$			3.39540i			0.372694i	−0.982484	+	0.186347i	$0.940335\pi$
$84$		0			0
$85$	10.2319	−	3.39540i	1.10980	−	0.368283i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−15.6977			−1.66395			−0.831976	−	0.554811i	$-0.812791\pi$
$89$	−15.6977			−1.66395			−0.831976	+	0.554811i	$0.812791\pi$
$90$		0			0
$91$	0.836493	+	0.836493i	0.0876883	+	0.0876883i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−11.5218	−	11.5218i	−1.18212	−	1.18212i
$96$		0			0
$97$	−7.55891	+	7.55891i	−0.767491	+	0.767491i	−0.977664	−	0.210173i	$-0.932597\pi$
$97$	−7.55891	+	7.55891i	−0.767491	+	0.767491i	0.210173	+	0.977664i	$0.432597\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.67299			−0.365476			−0.182738	−	0.983162i	$-0.558496\pi$
$101$	−3.67299			−0.365476			−0.182738	+	0.983162i	$0.558496\pi$
$102$		0			0
$103$	−16.2319			−1.59938			−0.799688	−	0.600416i	$-0.795002\pi$
$103$	−16.2319			−1.59938			−0.799688	+	0.600416i	$0.795002\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.84885	−	7.84885i	0.758777	−	0.758777i	−0.217323	−	0.976100i	$-0.569732\pi$
$107$	7.84885	−	7.84885i	0.758777	−	0.758777i	0.976100	+	0.217323i	$0.0697324\pi$
$108$		0			0
$109$	−6.00000	−	6.00000i	−0.574696	−	0.574696i	0.358741	−	0.933437i	$-0.383206\pi$
$109$	−6.00000	−	6.00000i	−0.574696	−	0.574696i	−0.933437	+	0.358741i	$0.883206\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.31466	+	4.31466i	0.405889	+	0.405889i	0.880302	−	0.474413i	$-0.157340\pi$
$113$	4.31466	+	4.31466i	0.405889	+	0.405889i	−0.474413	+	0.880302i	$0.657340\pi$
$114$		0			0
$115$	6.23189			0.581127
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.83649	−	5.53419i	−0.168351	−	0.507319i
$120$		0			0
$121$		−	10.9543i		−	0.995846i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−5.84885	−	5.84885i	−0.523137	−	0.523137i
$126$		0			0
$127$		−	6.23189i		−	0.552991i	−0.961015	−	0.276496i	$-0.910827\pi$
$127$		−	6.23189i		−	0.552991i	0.961015	−	0.276496i	$-0.0891731\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−0.151150	+	0.151150i	−0.0132060	+	0.0132060i	−0.713679	−	0.700473i	$-0.752972\pi$
$131$	−0.151150	+	0.151150i	−0.0132060	+	0.0132060i	0.700473	+	0.713679i	$0.252972\pi$
$132$		0			0
$133$	−6.23189	+	6.23189i	−0.540374	+	0.540374i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−10.7661			−0.919809			−0.459904	−	0.887968i	$-0.652116\pi$
$137$	−10.7661			−0.919809			−0.459904	+	0.887968i	$0.652116\pi$
$138$		0			0
$139$	0.836493	−	0.836493i	0.0709504	−	0.0709504i	−0.670741	−	0.741692i	$-0.734024\pi$
$139$	0.836493	−	0.836493i	0.0709504	−	0.0709504i	0.741692	+	0.670741i	$0.234024\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.126436	−	0.126436i	−0.0105731	−	0.0105731i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−10.7661			−0.881992			−0.440996	−	0.897509i	$-0.645375\pi$
$149$	−10.7661			−0.881992			−0.440996	+	0.897509i	$0.645375\pi$
$150$		0			0
$151$		−	2.32701i		−	0.189370i	−0.995507	−	0.0946849i	$-0.969816\pi$
$151$		−	2.32701i		−	0.189370i	0.995507	−	0.0946849i	$-0.0301843\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	6.79080i		−	0.545450i
$156$		0			0
$157$	−0.836493			−0.0667594			−0.0333797	−	0.999443i	$-0.510627\pi$
$157$	−0.836493			−0.0667594			−0.0333797	+	0.999443i	$0.510627\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	3.37069i		−	0.265647i
$162$		0			0
$163$	2.32701	+	2.32701i	0.182266	+	0.182266i	0.792342	−	0.610077i	$-0.208861\pi$
$163$	2.32701	+	2.32701i	0.182266	+	0.182266i	−0.610077	+	0.792342i	$0.708861\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−12.7784	+	12.7784i	−0.988826	+	0.988826i	−0.999938	−	0.0111126i	$-0.996463\pi$
$167$	−12.7784	+	12.7784i	−0.988826	+	0.988826i	0.0111126	+	0.999938i	$0.496463\pi$
$168$		0			0
$169$	−12.3003			−0.946175
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.54655	+	1.54655i	−0.117582	+	0.117582i	−0.763450	−	0.645867i	$-0.776496\pi$
$173$	−1.54655	+	1.54655i	−0.117582	+	0.117582i	0.645867	+	0.763450i	$0.276496\pi$
$174$		0			0
$175$	1.83649	−	1.83649i	0.138826	−	0.138826i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			15.3954i			1.15071i	0.817905	+	0.575353i	$0.195135\pi$
$179$			15.3954i			1.15071i	−0.817905	+	0.575353i	$0.804865\pi$
$180$		0			0
$181$	8.55891	+	8.55891i	0.636179	+	0.636179i	0.949611	−	0.313432i	$-0.101479\pi$
$181$	8.55891	+	8.55891i	0.636179	+	0.636179i	−0.313432	+	0.949611i	$0.601479\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			10.4885i			0.771130i
$186$		0			0
$187$	0.277586	+	0.836493i	0.0202991	+	0.0611704i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−6.76609			−0.489577			−0.244788	−	0.969577i	$-0.578718\pi$
$191$	−6.76609			−0.489577			−0.244788	+	0.969577i	$0.578718\pi$
$192$		0			0
$193$	18.3003	+	18.3003i	1.31728	+	1.31728i	0.915918	+	0.401365i	$0.131464\pi$
$193$	18.3003	+	18.3003i	1.31728	+	1.31728i	0.401365	+	0.915918i	$0.368536\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.47816	+	2.47816i	0.176562	+	0.176562i	0.789855	−	0.613293i	$-0.210156\pi$
$197$	2.47816	+	2.47816i	0.176562	+	0.176562i	−0.613293	+	0.789855i	$0.710156\pi$
$198$		0			0
$199$	11.2319	−	11.2319i	0.796208	−	0.796208i	−0.186287	−	0.982495i	$-0.559646\pi$
$199$	11.2319	−	11.2319i	0.796208	−	0.796208i	0.982495	+	0.186287i	$0.0596456\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−0.0456926			−0.00319131
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.941950	−	0.941950i	0.0651561	−	0.0651561i
$210$		0			0
$211$	−6.00000	−	6.00000i	−0.413057	−	0.413057i	0.469745	−	0.882802i	$-0.344346\pi$
$211$	−6.00000	−	6.00000i	−0.413057	−	0.413057i	−0.882802	+	0.469745i	$0.844346\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−18.9172	−	18.9172i	−1.29014	−	1.29014i
$216$		0			0
$217$	−3.67299			−0.249339
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3.08276	−	1.54655i	−0.207369	−	0.104032i
$222$		0			0
$223$			2.55891i			0.171357i	0.996323	+	0.0856785i	$0.0273058\pi$
$223$			2.55891i			0.171357i	−0.996323	+	0.0856785i	$0.972694\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	11.5218	+	11.5218i	0.764731	+	0.764731i	0.977174	−	0.212442i	$-0.0681418\pi$
$227$	11.5218	+	11.5218i	0.764731	+	0.764731i	−0.212442	+	0.977174i	$0.568142\pi$
$228$		0			0
$229$			22.4638i			1.48445i	0.670151	+	0.742224i	$0.266229\pi$
$229$			22.4638i			1.48445i	−0.670151	+	0.742224i	$0.733771\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	11.3830	−	11.3830i	0.745728	−	0.745728i	−0.227946	−	0.973674i	$-0.573201\pi$
$233$	11.3830	−	11.3830i	0.745728	−	0.745728i	0.973674	+	0.227946i	$0.0732009\pi$
$234$		0			0
$235$	13.0684	−	13.0684i	0.852488	−	0.852488i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−22.4638			−1.45306			−0.726531	−	0.687134i	$-0.758869\pi$
$239$	−22.4638			−1.45306			−0.726531	+	0.687134i	$0.758869\pi$
$240$		0			0
$241$	−8.06839	+	8.06839i	−0.519730	+	0.519730i	−0.917490	−	0.397759i	$-0.869788\pi$
$241$	−8.06839	+	8.06839i	−0.519730	+	0.519730i	0.397759	+	0.917490i	$0.369788\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	9.24425	+	9.24425i	0.590593	+	0.590593i
$246$		0			0
$247$			5.21294i			0.331691i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	22.4391			1.41634			0.708171	−	0.706041i	$-0.249521\pi$
$251$	22.4391			1.41634			0.708171	+	0.706041i	$0.249521\pi$
$252$		0			0
$253$			0.509479i			0.0320307i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	8.35173i		−	0.520966i	−0.965478	−	0.260483i	$-0.916118\pi$
$257$		−	8.35173i		−	0.520966i	0.965478	−	0.260483i	$-0.0838819\pi$
$258$		0			0
$259$	5.67299			0.352502
$260$		0			0
$261$		0			0
$262$		0			0
$263$			11.4201i			0.704194i	0.935963	+	0.352097i	$0.114531\pi$
$263$			11.4201i			0.704194i	−0.935963	+	0.352097i	$0.885469\pi$
$264$		0			0
$265$	13.0684	+	13.0684i	0.802785	+	0.802785i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	6.15115	−	6.15115i	0.375042	−	0.375042i	−0.494268	−	0.869310i	$-0.664564\pi$
$269$	6.15115	−	6.15115i	0.375042	−	0.375042i	0.869310	+	0.494268i	$0.164564\pi$
$270$		0			0
$271$	17.5779			1.06778			0.533890	−	0.845554i	$-0.320730\pi$
$271$	17.5779			1.06778			0.533890	+	0.845554i	$0.320730\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.277586	+	0.277586i	−0.0167391	+	0.0167391i
$276$		0			0
$277$	−7.06839	+	7.06839i	−0.424698	+	0.424698i	−0.886818	−	0.462119i	$-0.847089\pi$
$277$	−7.06839	+	7.06839i	−0.424698	+	0.424698i	0.462119	+	0.886818i	$0.347089\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	15.6977i		−	0.936446i	−0.883610	−	0.468223i	$-0.844894\pi$
$281$		−	15.6977i		−	0.936446i	0.883610	−	0.468223i	$-0.155106\pi$
$282$		0			0
$283$	−13.9049	−	13.9049i	−0.826559	−	0.826559i	0.160480	−	0.987039i	$-0.448696\pi$
$283$	−13.9049	−	13.9049i	−0.826559	−	0.826559i	−0.987039	+	0.160480i	$0.948696\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			0.0247141i			0.00145883i
$288$		0			0
$289$	10.1635	+	13.6273i	0.597853	+	0.801605i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−30.7661			−1.79737			−0.898687	−	0.438591i	$-0.855478\pi$
$293$	−30.7661			−1.79737			−0.898687	+	0.438591i	$0.855478\pi$
$294$		0			0
$295$	−0.558907	−	0.558907i	−0.0325408	−	0.0325408i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.40978	−	1.40978i	−0.0815295	−	0.0815295i
$300$		0			0
$301$	−10.2319	+	10.2319i	−0.589757	+	0.589757i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	23.0437			1.31948
$306$		0			0
$307$	−21.1178			−1.20526			−0.602629	−	0.798022i	$-0.705880\pi$
$307$	−21.1178			−1.20526			−0.602629	+	0.798022i	$0.705880\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	17.8612	−	17.8612i	1.01282	−	1.01282i	0.0128993	−	0.999917i	$-0.495894\pi$
$311$	17.8612	−	17.8612i	1.01282	−	1.01282i	0.999917	−	0.0128993i	$-0.00410610\pi$
$312$		0			0
$313$	−5.23189	−	5.23189i	−0.295724	−	0.295724i	0.543612	−	0.839336i	$-0.317056\pi$
$313$	−5.23189	−	5.23189i	−0.295724	−	0.295724i	−0.839336	+	0.543612i	$0.817056\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	22.4638	+	22.4638i	1.26169	+	1.26169i	0.950273	+	0.311419i	$0.100804\pi$
$317$	22.4638	+	22.4638i	1.26169	+	1.26169i	0.311419	+	0.950273i	$0.399196\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	11.5218	−	22.9667i	0.641092	−	1.27790i
$324$		0			0
$325$		−	1.53621i		−	0.0852138i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−7.06839	−	7.06839i	−0.389693	−	0.389693i
$330$		0			0
$331$		−	0.558907i		−	0.0307203i	−0.999882	−	0.0153602i	$-0.995111\pi$
$331$		−	0.558907i		−	0.0307203i	0.999882	−	0.0153602i	$-0.00488948\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−4.30230	+	4.30230i	−0.235060	+	0.235060i
$336$		0			0
$337$	4.39540	−	4.39540i	0.239433	−	0.239433i	−0.577182	−	0.816615i	$-0.695848\pi$
$337$	4.39540	−	4.39540i	0.239433	−	0.239433i	0.816615	+	0.577182i	$0.195848\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	0.555172			0.0300642
$342$		0			0
$343$	12.0000	−	12.0000i	0.647939	−	0.647939i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	21.8345	+	21.8345i	1.17214	+	1.17214i	0.981700	+	0.190436i	$0.0609903\pi$
$347$	21.8345	+	21.8345i	1.17214	+	1.17214i	0.190436	+	0.981700i	$0.439010\pi$
$348$		0			0
$349$			4.37271i			0.234066i	0.993128	+	0.117033i	$0.0373383\pi$
$349$			4.37271i			0.234066i	−0.993128	+	0.117033i	$0.962662\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−8.62931			−0.459292			−0.229646	−	0.973274i	$-0.573757\pi$
$353$	−8.62931			−0.459292			−0.229646	+	0.973274i	$0.573757\pi$
$354$		0			0
$355$		−	21.6730i		−	1.15028i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	13.8345i		−	0.730156i	−0.930977	−	0.365078i	$-0.881042\pi$
$359$		−	13.8345i		−	0.730156i	0.930977	−	0.365078i	$-0.118958\pi$
$360$		0			0
$361$	−19.8365			−1.04403
$362$		0			0
$363$		0			0
$364$		0			0
$365$			19.3460i			1.01261i
$366$		0			0
$367$	15.7414	+	15.7414i	0.821693	+	0.821693i	0.986351	−	0.164658i	$-0.0526520\pi$
$367$	15.7414	+	15.7414i	0.821693	+	0.821693i	−0.164658	+	0.986351i	$0.552652\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	7.06839	−	7.06839i	0.366972	−	0.366972i
$372$		0			0
$373$	29.8098			1.54349			0.771745	−	0.635932i	$-0.219384\pi$
$373$	29.8098			1.54349			0.771745	+	0.635932i	$0.219384\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−2.83649	+	2.83649i	−0.145701	+	0.145701i	−0.776194	−	0.630494i	$-0.782853\pi$
$379$	−2.83649	+	2.83649i	−0.145701	+	0.145701i	0.630494	+	0.776194i	$0.282853\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			16.9316i			0.865165i	0.901594	+	0.432583i	$0.142398\pi$
$383$			16.9316i			0.865165i	−0.901594	+	0.432583i	$0.857602\pi$
$384$		0			0
$385$	0.558907	+	0.558907i	0.0284845	+	0.0284845i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			1.23391i			0.0625619i	0.999511	+	0.0312810i	$0.00995866\pi$
$389$			1.23391i			0.0625619i	−0.999511	+	0.0312810i	$0.990041\pi$
$390$		0			0
$391$	3.09512	+	9.32701i	0.156527	+	0.471687i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	16.2529			0.817771
$396$		0			0
$397$	−17.5322	−	17.5322i	−0.879915	−	0.879915i	0.113611	−	0.993525i	$-0.463758\pi$
$397$	−17.5322	−	17.5322i	−0.879915	−	0.879915i	−0.993525	+	0.113611i	$0.963758\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	22.7537	+	22.7537i	1.13627	+	1.13627i	0.989114	+	0.147153i	$0.0470111\pi$
$401$	22.7537	+	22.7537i	1.13627	+	1.13627i	0.147153	+	0.989114i	$0.452989\pi$
$402$		0			0
$403$	−1.53621	+	1.53621i	−0.0765242	+	0.0765242i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−0.857472			−0.0425033
$408$		0			0
$409$	−17.1635			−0.848681			−0.424340	−	0.905503i	$-0.639494\pi$
$409$	−17.1635			−0.848681			−0.424340	+	0.905503i	$0.639494\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−0.302300	+	0.302300i	−0.0148752	+	0.0148752i
$414$		0			0
$415$	−6.27759	−	6.27759i	−0.308155	−	0.308155i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	25.2299	+	25.2299i	1.23256	+	1.23256i	0.962977	+	0.269583i	$0.0868858\pi$
$419$	25.2299	+	25.2299i	1.23256	+	1.23256i	0.269583	+	0.962977i	$0.413114\pi$
$420$		0			0
$421$	−19.6273			−0.956576			−0.478288	−	0.878203i	$-0.658742\pi$
$421$	−19.6273			−0.956576			−0.478288	+	0.878203i	$0.658742\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.39540	+	6.76811i	−0.164701	+	0.328301i
$426$		0			0
$427$		−	12.4638i		−	0.603165i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	2.46581	+	2.46581i	0.118774	+	0.118774i	0.763995	−	0.645222i	$-0.223235\pi$
$431$	2.46581	+	2.46581i	0.118774	+	0.118774i	−0.645222	+	0.763995i	$0.723235\pi$
$432$		0			0
$433$			0.0456926i			0.00219585i	0.999999	+	0.00109792i	$0.000349480\pi$
$433$			0.0456926i			0.00219585i	−0.999999	+	0.00109792i	$0.999651\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	10.5029	−	10.5029i	0.502421	−	0.502421i
$438$		0			0
$439$	−1.83649	+	1.83649i	−0.0876510	+	0.0876510i	−0.749573	−	0.661922i	$-0.769741\pi$
$439$	−1.83649	+	1.83649i	−0.0876510	+	0.0876510i	0.661922	+	0.749573i	$0.269741\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	10.7414			0.510338			0.255169	−	0.966896i	$-0.417869\pi$
$443$	10.7414			0.510338			0.255169	+	0.966896i	$0.417869\pi$
$444$		0			0
$445$	29.0227	−	29.0227i	1.37581	−	1.37581i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	19.6710	+	19.6710i	0.928330	+	0.928330i	0.997598	−	0.0692678i	$-0.0220663\pi$
$449$	19.6710	+	19.6710i	0.928330	+	0.928330i	−0.0692678	+	0.997598i	$0.522066\pi$
$450$		0			0
$451$		−	0.00373553i		−	0.000175899i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3.09310			−0.145007
$456$		0			0
$457$		−	14.8365i		−	0.694022i	−0.937861	−	0.347011i	$-0.887197\pi$
$457$		−	14.8365i		−	0.694022i	0.937861	−	0.347011i	$-0.112803\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	30.2109i		−	1.40706i	−0.710664	−	0.703531i	$-0.751605\pi$
$461$		−	30.2109i		−	1.40706i	0.710664	−	0.703531i	$-0.248395\pi$
$462$		0			0
$463$	−38.9276			−1.80912			−0.904559	−	0.426349i	$-0.859800\pi$
$463$	−38.9276			−1.80912			−0.904559	+	0.426349i	$0.859800\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	27.1425i		−	1.25601i	−0.778211	−	0.628003i	$-0.783872\pi$
$467$		−	27.1425i		−	1.25601i	0.778211	−	0.628003i	$-0.216128\pi$
$468$		0			0
$469$	2.32701	+	2.32701i	0.107452	+	0.107452i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.54655	−	1.54655i	0.0711105	−	0.0711105i
$474$		0			0
$475$	11.4448			0.525125
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−23.2422	+	23.2422i	−1.06196	+	1.06196i	−0.0640157	+	0.997949i	$0.520391\pi$
$479$	−23.2422	+	23.2422i	−1.06196	+	1.06196i	−0.997949	+	0.0640157i	$0.979609\pi$
$480$		0			0
$481$	2.37271	−	2.37271i	0.108186	−	0.108186i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	27.9506i		−	1.26917i
$486$		0			0
$487$	12.5322	+	12.5322i	0.567887	+	0.567887i	0.931536	−	0.363649i	$-0.118469\pi$
$487$	12.5322	+	12.5322i	0.567887	+	0.567887i	−0.363649	+	0.931536i	$0.618469\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			19.3460i			0.873071i	0.899687	+	0.436536i	$0.143795\pi$
$491$			19.3460i			0.873071i	−0.899687	+	0.436536i	$0.856205\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−11.7224			−0.525822
$498$		0			0
$499$	−16.7414	−	16.7414i	−0.749447	−	0.749447i	0.224929	−	0.974375i	$-0.427785\pi$
$499$	−16.7414	−	16.7414i	−0.749447	−	0.749447i	−0.974375	+	0.224929i	$0.927785\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−21.1055	−	21.1055i	−0.941046	−	0.941046i	0.0573106	−	0.998356i	$-0.481747\pi$
$503$	−21.1055	−	21.1055i	−0.941046	−	0.941046i	−0.998356	+	0.0573106i	$0.981747\pi$
$504$		0			0
$505$	6.79080	−	6.79080i	0.302187	−	0.302187i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	41.2546			1.82858			0.914289	−	0.405063i	$-0.132750\pi$
$509$	41.2546			1.82858			0.914289	+	0.405063i	$0.132750\pi$
$510$		0			0
$511$	10.4638			0.462891
$512$		0			0
$513$		0			0
$514$		0			0
$515$	30.0103	−	30.0103i	1.32241	−	1.32241i
$516$		0			0
$517$	1.06839	+	1.06839i	0.0469876	+	0.0469876i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	4.91926	+	4.91926i	0.215517	+	0.215517i	0.806606	−	0.591089i	$-0.201302\pi$
$521$	4.91926	+	4.91926i	0.215517	+	0.215517i	−0.591089	+	0.806606i	$0.701302\pi$
$522$		0			0
$523$	−30.6006			−1.33807			−0.669035	−	0.743231i	$-0.733292\pi$
$523$	−30.6006			−1.33807			−0.669035	+	0.743231i	$0.733292\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	10.1635	−	3.37271i	0.442729	−	0.146917i
$528$		0			0
$529$		−	17.3192i		−	0.753010i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	0.0103366	+	0.0103366i	0.000447727	+	0.000447727i
$534$		0			0
$535$			29.0227i			1.25476i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.755750	+	0.755750i	−0.0325524	+	0.0325524i
$540$		0			0
$541$	0.509479	−	0.509479i	0.0219042	−	0.0219042i	−0.696070	−	0.717974i	$-0.745070\pi$
$541$	0.509479	−	0.509479i	0.0219042	−	0.0219042i	0.717974	+	0.696070i	$0.245070\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	22.1862			0.950352
$546$		0			0
$547$	5.49052	−	5.49052i	0.234758	−	0.234758i	−0.579917	−	0.814675i	$-0.696915\pi$
$547$	5.49052	−	5.49052i	0.234758	−	0.234758i	0.814675	+	0.579917i	$0.196915\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		−	8.79080i		−	0.373823i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	12.9029			0.546712			0.273356	−	0.961913i	$-0.411866\pi$
$557$	12.9029			0.546712			0.273356	+	0.961913i	$0.411866\pi$
$558$		0			0
$559$			8.55891i			0.362003i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			36.8781i			1.55423i	0.629360	+	0.777114i	$0.283317\pi$
$563$			36.8781i			1.55423i	−0.629360	+	0.777114i	$0.716683\pi$
$564$		0			0
$565$	−15.9543			−0.671203
$566$		0			0
$567$		0			0
$568$		0			0
$569$			13.8592i			0.581007i	0.956874	+	0.290504i	$0.0938228\pi$
$569$			13.8592i			0.581007i	−0.956874	+	0.290504i	$0.906177\pi$
$570$		0			0
$571$	15.3954	+	15.3954i	0.644277	+	0.644277i	0.951604	−	0.307327i	$-0.0994344\pi$
$571$	15.3954	+	15.3954i	0.644277	+	0.644277i	−0.307327	+	0.951604i	$0.599434\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3.09512	+	3.09512i	−0.129075	+	0.129075i
$576$		0			0
$577$	6.83649			0.284607			0.142303	−	0.989823i	$-0.454549\pi$
$577$	6.83649			0.284607			0.142303	+	0.989823i	$0.454549\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3.39540	+	3.39540i	−0.140865	+	0.140865i
$582$		0			0
$583$	−1.06839	+	1.06839i	−0.0442480	+	0.0442480i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			46.7414i			1.92922i	0.263675	+	0.964611i	$0.415065\pi$
$587$			46.7414i			1.92922i	−0.263675	+	0.964611i	$0.584935\pi$
$588$		0			0
$589$	−11.4448	−	11.4448i	−0.471576	−	0.471576i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	8.32701i		−	0.341949i	−0.985275	−	0.170975i	$-0.945308\pi$
$593$		−	8.32701i		−	0.341949i	0.985275	−	0.170975i	$-0.0546917\pi$
$594$		0			0
$595$	13.6273	+	6.83649i	0.558665	+	0.280269i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	27.0931			1.10699			0.553497	−	0.832851i	$-0.313293\pi$
$599$	27.0931			1.10699			0.553497	+	0.832851i	$0.313293\pi$
$600$		0			0
$601$	−17.5095	−	17.5095i	−0.714227	−	0.714227i	0.253190	−	0.967417i	$-0.418520\pi$
$601$	−17.5095	−	17.5095i	−0.714227	−	0.714227i	−0.967417	+	0.253190i	$0.918520\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	20.2529	+	20.2529i	0.823396	+	0.823396i
$606$		0			0
$607$	−22.8098	+	22.8098i	−0.925820	+	0.925820i	−0.997432	−	0.0716130i	$-0.977185\pi$
$607$	−22.8098	+	22.8098i	−0.925820	+	0.925820i	0.0716130	+	0.997432i	$0.477185\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−5.91266			−0.239200
$612$		0			0
$613$	−3.81753			−0.154189			−0.0770944	−	0.997024i	$-0.524564\pi$
$613$	−3.81753			−0.154189			−0.0770944	+	0.997024i	$0.524564\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−25.8365	+	25.8365i	−1.04014	+	1.04014i	−0.0409780	+	0.999160i	$0.513047\pi$
$617$	−25.8365	+	25.8365i	−1.04014	+	1.04014i	−0.999160	+	0.0409780i	$0.986953\pi$
$618$		0			0
$619$	5.57787	+	5.57787i	0.224193	+	0.224193i	0.810262	−	0.586068i	$-0.199325\pi$
$619$	5.57787	+	5.57787i	0.224193	+	0.224193i	−0.586068	+	0.810262i	$0.699325\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−15.6977	−	15.6977i	−0.628915	−	0.628915i
$624$		0			0
$625$	30.8098			1.23239
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−15.6977	+	5.20920i	−0.625908	+	0.207704i
$630$		0			0
$631$			4.78706i			0.190570i	0.995450	+	0.0952850i	$0.0303762\pi$
$631$			4.78706i			0.190570i	−0.995450	+	0.0952850i	$0.969624\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	11.5218	+	11.5218i	0.457230	+	0.457230i
$636$		0			0
$637$		−	4.18247i		−	0.165715i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	22.7537	−	22.7537i	0.898718	−	0.898718i	−0.0966046	−	0.995323i	$-0.530798\pi$
$641$	22.7537	−	22.7537i	0.898718	−	0.898718i	0.995323	+	0.0966046i	$0.0307982\pi$
$642$		0			0
$643$	−22.1862	+	22.1862i	−0.874938	+	0.874938i	−0.993006	−	0.118067i	$-0.962330\pi$
$643$	−22.1862	+	22.1862i	−0.874938	+	0.874938i	0.118067	+	0.993006i	$0.462330\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−22.7661			−0.895027			−0.447514	−	0.894277i	$-0.647690\pi$
$647$	−22.7661			−0.895027			−0.447514	+	0.894277i	$0.647690\pi$
$648$		0			0
$649$	0.0456926	−	0.0456926i	0.00179359	−	0.00179359i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	14.4534	+	14.4534i	0.565607	+	0.565607i	0.930895	−	0.365288i	$-0.119029\pi$
$653$	14.4534	+	14.4534i	0.565607	+	0.565607i	−0.365288	+	0.930895i	$0.619029\pi$
$654$		0			0
$655$		−	0.558907i		−	0.0218383i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	43.0931			1.67867			0.839334	−	0.543615i	$-0.182945\pi$
$659$	43.0931			1.67867			0.839334	+	0.543615i	$0.182945\pi$
$660$		0			0
$661$		−	20.6463i		−	0.803046i	−0.915849	−	0.401523i	$-0.868481\pi$
$661$		−	20.6463i		−	0.803046i	0.915849	−	0.401523i	$-0.131519\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	23.0437i		−	0.893595i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$			1.88390i			0.0727272i
$672$		0			0
$673$	26.9049	+	26.9049i	1.03711	+	1.03711i	0.999284	+	0.0378224i	$0.0120421\pi$
$673$	26.9049	+	26.9049i	1.03711	+	1.03711i	0.0378224	+	0.999284i	$0.487958\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−12.9172	+	12.9172i	−0.496450	+	0.496450i	−0.910331	−	0.413881i	$-0.864173\pi$
$677$	−12.9172	+	12.9172i	−0.496450	+	0.496450i	0.413881	+	0.910331i	$0.364173\pi$
$678$		0			0
$679$	−15.1178			−0.580168
$680$		0			0
$681$		0			0
$682$		0			0
$683$	29.6586	−	29.6586i	1.13486	−	1.13486i	0.145496	−	0.989359i	$-0.453522\pi$
$683$	29.6586	−	29.6586i	1.13486	−	1.13486i	0.989359	−	0.145496i	$-0.0464779\pi$
$684$		0			0
$685$	19.9049	−	19.9049i	0.760526	−	0.760526i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	5.91266i		−	0.225254i
$690$		0			0
$691$	32.1368	+	32.1368i	1.22254	+	1.22254i	0.966726	+	0.255815i	$0.0823437\pi$
$691$	32.1368	+	32.1368i	1.22254	+	1.22254i	0.255815	+	0.966726i	$0.417656\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			3.09310i			0.117328i
$696$		0			0
$697$	−0.0226936	−	0.0683862i	−0.000859582	−	0.00259031i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−4.37644			−0.165296			−0.0826480	−	0.996579i	$-0.526338\pi$
$701$	−4.37644			−0.165296			−0.0826480	+	0.996579i	$0.526338\pi$
$702$		0			0
$703$	17.6767	+	17.6767i	0.666690	+	0.666690i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−3.67299	−	3.67299i	−0.138137	−	0.138137i
$708$		0			0
$709$	9.39540	−	9.39540i	0.352852	−	0.352852i	−0.508318	−	0.861170i	$-0.669732\pi$
$709$	9.39540	−	9.39540i	0.352852	−	0.352852i	0.861170	+	0.508318i	$0.169732\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	6.19024			0.231826
$714$		0			0
$715$	0.467522			0.0174843
$716$		0			0
$717$		0			0
$718$		0			0
$719$	24.4514	−	24.4514i	0.911884	−	0.911884i	−0.0845360	−	0.996420i	$-0.526941\pi$
$719$	24.4514	−	24.4514i	0.911884	−	0.911884i	0.996420	+	0.0845360i	$0.0269408\pi$
$720$		0			0
$721$	−16.2319	−	16.2319i	−0.604507	−	0.604507i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	18.1368			0.672656			0.336328	−	0.941745i	$-0.390815\pi$
$727$	18.1368			0.672656			0.336328	+	0.941745i	$0.390815\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	18.9172	−	37.7080i	0.699679	−	1.39468i
$732$		0			0
$733$			10.4638i			0.386489i	0.981151	+	0.193244i	$0.0619010\pi$
$733$			10.4638i			0.386489i	−0.981151	+	0.193244i	$0.938099\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−0.351728	−	0.351728i	−0.0129561	−	0.0129561i
$738$		0			0
$739$			50.6957i			1.86487i	0.361337	+	0.932435i	$0.382320\pi$
$739$			50.6957i			1.86487i	−0.361337	+	0.932435i	$0.617680\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−6.44109	+	6.44109i	−0.236301	+	0.236301i	−0.815316	−	0.579016i	$-0.803437\pi$
$743$	−6.44109	+	6.44109i	−0.236301	+	0.236301i	0.579016	+	0.815316i	$0.303437\pi$
$744$		0			0
$745$	19.9049	−	19.9049i	0.729258	−	0.729258i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	15.6977			0.573581
$750$		0			0
$751$	10.1635	−	10.1635i	0.370872	−	0.370872i	−0.496923	−	0.867795i	$-0.665537\pi$
$751$	10.1635	−	10.1635i	0.370872	−	0.370872i	0.867795	+	0.496923i	$0.165537\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4.30230	+	4.30230i	0.156577	+	0.156577i
$756$		0			0
$757$			38.4181i			1.39633i	0.715937	+	0.698165i	$0.246000\pi$
$757$			38.4181i			1.39633i	−0.715937	+	0.698165i	$0.754000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−12.2776			−0.445062			−0.222531	−	0.974926i	$-0.571432\pi$
$761$	−12.2776			−0.445062			−0.222531	+	0.974926i	$0.571432\pi$
$762$		0			0
$763$		−	12.0000i		−	0.434429i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			0.252872i			0.00913067i
$768$		0			0
$769$	20.9733			0.756315			0.378158	−	0.925741i	$-0.376558\pi$
$769$	20.9733			0.756315			0.378158	+	0.925741i	$0.376558\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	5.80976i		−	0.208962i	−0.994527	−	0.104481i	$-0.966682\pi$
$773$		−	5.80976i		−	0.208962i	0.994527	−	0.104481i	$-0.0333182\pi$
$774$		0			0
$775$	3.37271	+	3.37271i	0.121151	+	0.121151i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−0.0770077	+	0.0770077i	−0.00275909	+	0.00275909i
$780$		0			0
$781$	1.77184			0.0634015
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1.54655	−	1.54655i	0.0551987	−	0.0551987i
$786$		0			0
$787$	16.9316	−	16.9316i	0.603547	−	0.603547i	−0.337705	−	0.941252i	$-0.609651\pi$
$787$	16.9316	−	16.9316i	0.603547	−	0.603547i	0.941252	+	0.337705i	$0.109651\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			8.62931i			0.306823i
$792$		0			0
$793$	−5.21294	−	5.21294i	−0.185117	−	0.185117i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			26.4144i			0.935645i	0.883823	+	0.467822i	$0.154961\pi$
$797$			26.4144i			0.935645i	−0.883823	+	0.467822i	$0.845039\pi$
$798$		0			0
$799$	26.0494	+	13.0684i	0.921563	+	0.462326i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.58160			−0.0558135
$804$		0			0
$805$	6.23189	+	6.23189i	0.219645	+	0.219645i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−5.82212	−	5.82212i	−0.204695	−	0.204695i	0.597313	−	0.802008i	$-0.296235\pi$
$809$	−5.82212	−	5.82212i	−0.204695	−	0.204695i	−0.802008	+	0.597313i	$0.796235\pi$
$810$		0			0
$811$	−38.6500	+	38.6500i	−1.35718	+	1.35718i	−0.479815	+	0.877370i	$0.659296\pi$
$811$	−38.6500	+	38.6500i	−1.35718	+	1.35718i	−0.877370	+	0.479815i	$0.840704\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−8.60460			−0.301406
$816$		0			0
$817$	−63.7641			−2.23082
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−6.78046	+	6.78046i	−0.236640	+	0.236640i	−0.815457	−	0.578817i	$-0.803514\pi$
$821$	−6.78046	+	6.78046i	−0.236640	+	0.236640i	0.578817	+	0.815457i	$0.303514\pi$
$822$		0			0
$823$	−30.5322	−	30.5322i	−1.06428	−	1.06428i	−0.997787	−	0.0664981i	$-0.978817\pi$
$823$	−30.5322	−	30.5322i	−1.06428	−	1.06428i	−0.0664981	−	0.997787i	$-0.521183\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−4.45345	−	4.45345i	−0.154862	−	0.154862i	0.625424	−	0.780285i	$-0.284926\pi$
$827$	−4.45345	−	4.45345i	−0.154862	−	0.154862i	−0.780285	+	0.625424i	$0.784926\pi$
$828$		0			0
$829$	−11.0190			−0.382704			−0.191352	−	0.981521i	$-0.561287\pi$
$829$	−11.0190			−0.382704			−0.191352	+	0.981521i	$0.561287\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−9.24425	+	18.4267i	−0.320294	+	0.638448i
$834$		0			0
$835$		−	47.2509i		−	1.63518i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	24.5009	+	24.5009i	0.845864	+	0.845864i	0.989614	−	0.143750i	$-0.0459163\pi$
$839$	24.5009	+	24.5009i	0.845864	+	0.845864i	−0.143750	+	0.989614i	$0.545916\pi$
$840$		0			0
$841$			29.0000i			1.00000i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	22.7414	−	22.7414i	0.782327	−	0.782327i
$846$		0			0
$847$	10.9543	−	10.9543i	0.376394	−	0.376394i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−9.56093			−0.327744
$852$		0			0
$853$	27.1635	−	27.1635i	0.930061	−	0.930061i	−0.0676485	−	0.997709i	$-0.521550\pi$
$853$	27.1635	−	27.1635i	0.930061	−	0.930061i	0.997709	+	0.0676485i	$0.0215496\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−9.18247	−	9.18247i	−0.313667	−	0.313667i	0.532661	−	0.846328i	$-0.321192\pi$
$857$	−9.18247	−	9.18247i	−0.313667	−	0.313667i	−0.846328	+	0.532661i	$0.821192\pi$
$858$		0			0
$859$			28.4638i			0.971172i	0.874189	+	0.485586i	$0.161394\pi$
$859$			28.4638i			0.971172i	−0.874189	+	0.485586i	$0.838606\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−3.69770			−0.125871			−0.0629356	−	0.998018i	$-0.520046\pi$
$863$	−3.69770			−0.125871			−0.0629356	+	0.998018i	$0.520046\pi$
$864$		0			0
$865$		−	5.71868i		−	0.194441i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			1.32873i			0.0450740i
$870$		0			0
$871$	1.94653			0.0659557
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	11.6977i		−	0.395454i
$876$		0			0
$877$	21.9049	+	21.9049i	0.739675	+	0.739675i	0.972515	−	0.232840i	$-0.0748018\pi$
$877$	21.9049	+	21.9049i	0.739675	+	0.739675i	−0.232840	+	0.972515i	$0.574802\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	27.6710	−	27.6710i	0.932259	−	0.932259i	−0.0655883	−	0.997847i	$-0.520892\pi$
$881$	27.6710	−	27.6710i	0.932259	−	0.932259i	0.997847	+	0.0655883i	$0.0208924\pi$
$882$		0			0
$883$	−23.8135			−0.801388			−0.400694	−	0.916212i	$-0.631231\pi$
$883$	−23.8135			−0.801388			−0.400694	+	0.916212i	$0.631231\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−22.0124	+	22.0124i	−0.739103	+	0.739103i	−0.972404	−	0.233302i	$-0.925047\pi$
$887$	−22.0124	+	22.0124i	−0.739103	+	0.739103i	0.233302	+	0.972404i	$0.425047\pi$
$888$		0			0
$889$	6.23189	−	6.23189i	0.209011	−	0.209011i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	44.0494i		−	1.47406i
$894$		0			0
$895$	−28.4638	−	28.4638i	−0.951439	−	0.951439i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−13.0684	+	26.0494i	−0.435371	+	0.867832i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−31.6483			−1.05202
$906$		0			0
$907$	1.30028	+	1.30028i	0.0431751	+	0.0431751i	0.728365	−	0.685190i	$-0.240281\pi$
$907$	1.30028	+	1.30028i	0.0431751	+	0.0431751i	−0.685190	+	0.728365i	$0.740281\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−36.1491	−	36.1491i	−1.19767	−	1.19767i	−0.974861	−	0.222813i	$-0.928476\pi$
$911$	−36.1491	−	36.1491i	−1.19767	−	1.19767i	−0.222813	−	0.974861i	$-0.571524\pi$
$912$		0			0
$913$	0.513215	−	0.513215i	0.0169849	−	0.0169849i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−0.302300			−0.00998282
$918$		0			0
$919$	−41.5779			−1.37153			−0.685764	−	0.727824i	$-0.740532\pi$
$919$	−41.5779			−1.37153			−0.685764	+	0.727824i	$0.740532\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−4.90286	+	4.90286i	−0.161380	+	0.161380i
$924$		0			0
$925$	−5.20920	−	5.20920i	−0.171277	−	0.171277i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	33.8468	+	33.8468i	1.11048	+	1.11048i	0.993086	+	0.117393i	$0.0374537\pi$
$929$	33.8468	+	33.8468i	1.11048	+	1.11048i	0.117393	+	0.993086i	$0.462546\pi$
$930$		0			0
$931$	31.1595			1.02121
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−2.05976	−	1.03334i	−0.0673615	−	0.0337937i
$936$		0			0
$937$			30.2735i			0.988994i	0.869179	+	0.494497i	$0.164648\pi$
$937$			30.2735i			0.988994i	−0.869179	+	0.494497i	$0.835352\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−4.93161	−	4.93161i	−0.160766	−	0.160766i	0.622140	−	0.782906i	$-0.286263\pi$
$941$	−4.93161	−	4.93161i	−0.160766	−	0.160766i	−0.782906	+	0.622140i	$0.786263\pi$
$942$		0			0
$943$		−	0.0416517i		−	0.00135637i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−34.8155	+	34.8155i	−1.13135	+	1.13135i	−0.141400	+	0.989953i	$0.545160\pi$
$947$	−34.8155	+	34.8155i	−1.13135	+	1.13135i	−0.989953	+	0.141400i	$0.954840\pi$
$948$		0			0
$949$	4.37644	−	4.37644i	0.142065	−	0.142065i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−22.1862			−0.718682			−0.359341	−	0.933206i	$-0.616998\pi$
$953$	−22.1862			−0.718682			−0.359341	+	0.933206i	$0.616998\pi$
$954$		0			0
$955$	12.5095	−	12.5095i	0.404797	−	0.404797i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−10.7661	−	10.7661i	−0.347655	−	0.347655i
$960$		0			0
$961$			24.2546i			0.782406i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−67.6689			−2.17834
$966$		0			0
$967$		−	8.69568i		−	0.279634i	−0.990177	−	0.139817i	$-0.955349\pi$
$967$		−	8.69568i		−	0.279634i	0.990177	−	0.139817i	$-0.0446515\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			0.703456i			0.0225750i	0.999936	+	0.0112875i	$0.00359300\pi$
$971$			0.703456i			0.0225750i	−0.999936	+	0.0112875i	$0.996407\pi$
$972$		0			0
$973$	1.67299			0.0536335
$974$		0			0
$975$		0			0
$976$		0			0
$977$			22.1615i			0.709009i	0.935054	+	0.354504i	$0.115350\pi$
$977$			22.1615i			0.709009i	−0.935054	+	0.354504i	$0.884650\pi$
$978$		0			0
$979$	2.37271	+	2.37271i	0.0758320	+	0.0758320i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−34.9152	+	34.9152i	−1.11362	+	1.11362i	−0.120966	+	0.992657i	$0.538599\pi$
$983$	−34.9152	+	34.9152i	−1.11362	+	1.11362i	−0.992657	+	0.120966i	$0.961401\pi$
$984$		0			0
$985$	−9.16351			−0.291974
$986$		0			0
$987$		0			0
$988$		0			0
$989$	17.2443	−	17.2443i	0.548335	−	0.548335i
$990$		0			0
$991$	37.3687	−	37.3687i	1.18705	−	1.18705i	0.209177	−	0.977878i	$-0.432922\pi$
$991$	37.3687	−	37.3687i	1.18705	−	1.18705i	0.977878	−	0.209177i	$-0.0670785\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			41.5322i			1.31666i
$996$		0			0
$997$	−28.9733	−	28.9733i	−0.917593	−	0.917593i	0.0792613	−	0.996854i	$-0.474744\pi$
$997$	−28.9733	−	28.9733i	−0.917593	−	0.917593i	−0.996854	+	0.0792613i	$0.974744\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2448.2.be.v.1585.1		6
3.2	odd	2		2448.2.be.w.1585.3		6
4.3	odd	2		1224.2.w.j.361.1	yes	6
12.11	even	2		1224.2.w.i.361.3	yes	6
17.13	even	4	inner	2448.2.be.v.1441.1		6
51.47	odd	4		2448.2.be.w.1441.3		6
68.47	odd	4		1224.2.w.j.217.1	yes	6
204.47	even	4		1224.2.w.i.217.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1224.2.w.i.217.3	✓	6	204.47	even	4
1224.2.w.i.361.3	yes	6	12.11	even	2
1224.2.w.j.217.1	yes	6	68.47	odd	4
1224.2.w.j.361.1	yes	6	4.3	odd	2
2448.2.be.v.1441.1		6	17.13	even	4	inner
2448.2.be.v.1585.1		6	1.1	even	1	trivial
2448.2.be.w.1441.3		6	51.47	odd	4
2448.2.be.w.1585.3		6	3.2	odd	2

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:	\( N \)	\(=\)	\( 2448 = 2^{4} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2448.be (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.84885 + 1.84885i) q^{5} +(1.00000 + 1.00000i) q^{7} +(-0.151150 - 0.151150i) q^{11} +0.836493 q^{13} +(-3.68534 - 1.84885i) q^{17} +6.23189i q^{19} +(-1.68534 - 1.68534i) q^{23} -1.83649i q^{25} +(-1.83649 + 1.83649i) q^{31} -3.69770 q^{35} +(2.83649 - 2.83649i) q^{37} +(0.0123570 + 0.0123570i) q^{41} +10.2319i q^{43} -7.06839 q^{47} -5.00000i q^{49} -7.06839i q^{53} +0.558907 q^{55} +0.302300i q^{59} +(-6.23189 - 6.23189i) q^{61} +(-1.54655 + 1.54655i) q^{65} +2.32701 q^{67} +(-5.86121 + 5.86121i) q^{71} +(5.23189 - 5.23189i) q^{73} -0.302300i q^{77} +(-4.39540 - 4.39540i) q^{79} +3.39540i q^{83} +(10.2319 - 3.39540i) q^{85} -15.6977 q^{89} +(0.836493 + 0.836493i) q^{91} +(-11.5218 - 11.5218i) q^{95} +(-7.55891 + 7.55891i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{7} - 12 q^{11} - 6 q^{17} + 6 q^{23} - 6 q^{31} + 12 q^{37} - 6 q^{41} + 12 q^{47} - 36 q^{55} + 12 q^{61} + 24 q^{65} + 24 q^{67} - 18 q^{71} - 18 q^{73} + 18 q^{79} + 12 q^{85} - 72 q^{89}+ \cdots - 6 q^{97}+O(q^{100}) \) 6 * q + 6 * q^7 - 12 * q^11 - 6 * q^17 + 6 * q^23 - 6 * q^31 + 12 * q^37 - 6 * q^41 + 12 * q^47 - 36 * q^55 + 12 * q^61 + 24 * q^65 + 24 * q^67 - 18 * q^71 - 18 * q^73 + 18 * q^79 + 12 * q^85 - 72 * q^89 - 48 * q^95 - 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more