LMFDB - Embedded newform 1350.2.f.d.107.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,2,Mod(107,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([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("1350.107");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1350.f (of order $4$, degree $2$, 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:	$10.7798042729$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			107.4
Root			$-0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	1350.107
Dual form			1350.2.f.d.593.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 - 0.707107i) q^{2} -1.00000i q^{4} +(1.22474 + 1.22474i) q^{7} +(-0.707107 - 0.707107i) q^{8} +O(q^{10})$	$q+(0.707107 - 0.707107i) q^{2} -1.00000i q^{4} +(1.22474 + 1.22474i) q^{7} +(-0.707107 - 0.707107i) q^{8} +1.73205 q^{14} -1.00000 q^{16} +(4.24264 - 4.24264i) q^{17} -1.00000i q^{19} +(-4.24264 - 4.24264i) q^{23} +(1.22474 - 1.22474i) q^{28} +10.3923 q^{29} +7.00000 q^{31} +(-0.707107 + 0.707107i) q^{32} -6.00000i q^{34} +(6.12372 + 6.12372i) q^{37} +(-0.707107 - 0.707107i) q^{38} +(-1.22474 + 1.22474i) q^{43} -6.00000 q^{46} -4.00000i q^{49} +(-8.48528 - 8.48528i) q^{53} -1.73205i q^{56} +(7.34847 - 7.34847i) q^{58} +10.3923 q^{59} +5.00000 q^{61} +(4.94975 - 4.94975i) q^{62} +1.00000i q^{64} +(-7.34847 - 7.34847i) q^{67} +(-4.24264 - 4.24264i) q^{68} -10.3923i q^{71} +(-8.57321 + 8.57321i) q^{73} +8.66025 q^{74} -1.00000 q^{76} +13.0000i q^{79} +(-4.24264 - 4.24264i) q^{83} +1.73205i q^{86} -10.3923 q^{89} +(-4.24264 + 4.24264i) q^{92} +(6.12372 + 6.12372i) q^{97} +(-2.82843 - 2.82843i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 8 q^{16} + 56 q^{31} - 48 q^{46} + 40 q^{61} - 8 q^{76}+O(q^{100}) $ 8 * q - 8 * q^16 + 56 * q^31 - 48 * q^46 + 40 * q^61 - 8 * q^76

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

$n$	$1001$	$1027$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	0.707107	−	0.707107i	0.500000	−	0.500000i
$2$	0.707107	−	0.707107i	0.500000	−	0.500000i
$3$		0			0
$3$		0			0
$4$		−	1.00000i		−	0.500000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.22474	+	1.22474i	0.462910	+	0.462910i	0.899608	−	0.436698i	$-0.143852\pi$
$7$	1.22474	+	1.22474i	0.462910	+	0.462910i	−0.436698	+	0.899608i	$0.643852\pi$
$8$	−0.707107	−	0.707107i	−0.250000	−	0.250000i
$9$		0			0
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$	1.73205			0.462910
$15$		0			0
$16$	−1.00000			−0.250000
$17$	4.24264	−	4.24264i	1.02899	−	1.02899i	0.0294245	−	0.999567i	$-0.490633\pi$
$17$	4.24264	−	4.24264i	1.02899	−	1.02899i	0.999567	−	0.0294245i	$-0.00936746\pi$
$18$		0			0
$19$		−	1.00000i		−	0.229416i	−0.993399	−	0.114708i	$-0.963407\pi$
$19$		−	1.00000i		−	0.229416i	0.993399	−	0.114708i	$-0.0365932\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.24264	−	4.24264i	−0.884652	−	0.884652i	0.109351	−	0.994003i	$-0.465123\pi$
$23$	−4.24264	−	4.24264i	−0.884652	−	0.884652i	−0.994003	+	0.109351i	$0.965123\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$	1.22474	−	1.22474i	0.231455	−	0.231455i
$29$	10.3923			1.92980			0.964901	−	0.262613i	$-0.0845842\pi$
$29$	10.3923			1.92980			0.964901	+	0.262613i	$0.0845842\pi$
$30$		0			0
$31$	7.00000			1.25724			0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000			1.25724			0.628619	+	0.777714i	$0.283621\pi$
$32$	−0.707107	+	0.707107i	−0.125000	+	0.125000i
$33$		0			0
$34$		−	6.00000i		−	1.02899i
$35$		0			0
$36$		0			0
$37$	6.12372	+	6.12372i	1.00673	+	1.00673i	0.999977	+	0.00675691i	$0.00215081\pi$
$37$	6.12372	+	6.12372i	1.00673	+	1.00673i	0.00675691	+	0.999977i	$0.497849\pi$
$38$	−0.707107	−	0.707107i	−0.114708	−	0.114708i
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	−1.22474	+	1.22474i	−0.186772	+	0.186772i	−0.794299	−	0.607527i	$-0.792162\pi$
$43$	−1.22474	+	1.22474i	−0.186772	+	0.186772i	0.607527	+	0.794299i	$0.292162\pi$
$44$		0			0
$45$		0			0
$46$	−6.00000			−0.884652
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$		0			0
$49$		−	4.00000i		−	0.571429i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−8.48528	−	8.48528i	−1.16554	−	1.16554i	−0.983243	−	0.182300i	$-0.941646\pi$
$53$	−8.48528	−	8.48528i	−1.16554	−	1.16554i	−0.182300	−	0.983243i	$-0.558354\pi$
$54$		0			0
$55$		0			0
$56$		−	1.73205i		−	0.231455i
$57$		0			0
$58$	7.34847	−	7.34847i	0.964901	−	0.964901i
$59$	10.3923			1.35296			0.676481	−	0.736460i	$-0.263504\pi$
$59$	10.3923			1.35296			0.676481	+	0.736460i	$0.263504\pi$
$60$		0			0
$61$	5.00000			0.640184			0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000			0.640184			0.320092	+	0.947386i	$0.396286\pi$
$62$	4.94975	−	4.94975i	0.628619	−	0.628619i
$63$		0			0
$64$			1.00000i			0.125000i
$65$		0			0
$66$		0			0
$67$	−7.34847	−	7.34847i	−0.897758	−	0.897758i	0.0974792	−	0.995238i	$-0.468922\pi$
$67$	−7.34847	−	7.34847i	−0.897758	−	0.897758i	−0.995238	+	0.0974792i	$0.968922\pi$
$68$	−4.24264	−	4.24264i	−0.514496	−	0.514496i
$69$		0			0
$70$		0			0
$71$		−	10.3923i		−	1.23334i	−0.787222	−	0.616670i	$-0.788481\pi$
$71$		−	10.3923i		−	1.23334i	0.787222	−	0.616670i	$-0.211519\pi$
$72$		0			0
$73$	−8.57321	+	8.57321i	−1.00342	+	1.00342i	−0.00342468	+	0.999994i	$0.501090\pi$
$73$	−8.57321	+	8.57321i	−1.00342	+	1.00342i	−0.999994	+	0.00342468i	$0.998910\pi$
$74$	8.66025			1.00673
$75$		0			0
$76$	−1.00000			−0.114708
$77$		0			0
$78$		0			0
$79$			13.0000i			1.46261i	0.682048	+	0.731307i	$0.261089\pi$
$79$			13.0000i			1.46261i	−0.682048	+	0.731307i	$0.738911\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.24264	−	4.24264i	−0.465690	−	0.465690i	0.434825	−	0.900515i	$-0.356810\pi$
$83$	−4.24264	−	4.24264i	−0.465690	−	0.465690i	−0.900515	+	0.434825i	$0.856810\pi$
$84$		0			0
$85$		0			0
$86$			1.73205i			0.186772i
$87$		0			0
$88$		0			0
$89$	−10.3923			−1.10158			−0.550791	−	0.834643i	$-0.685674\pi$
$89$	−10.3923			−1.10158			−0.550791	+	0.834643i	$0.685674\pi$
$90$		0			0
$91$		0			0
$92$	−4.24264	+	4.24264i	−0.442326	+	0.442326i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	6.12372	+	6.12372i	0.621770	+	0.621770i	0.945984	−	0.324214i	$-0.105100\pi$
$97$	6.12372	+	6.12372i	0.621770	+	0.621770i	−0.324214	+	0.945984i	$0.605100\pi$
$98$	−2.82843	−	2.82843i	−0.285714	−	0.285714i
$99$		0			0
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	−6.12372	+	6.12372i	−0.603388	+	0.603388i	−0.941210	−	0.337822i	$-0.890310\pi$
$103$	−6.12372	+	6.12372i	−0.603388	+	0.603388i	0.337822	+	0.941210i	$0.390310\pi$
$104$		0			0
$105$		0			0
$106$	−12.0000			−1.16554
$107$	12.7279	−	12.7279i	1.23045	−	1.23045i	0.266666	−	0.963789i	$-0.414078\pi$
$107$	12.7279	−	12.7279i	1.23045	−	1.23045i	0.963789	−	0.266666i	$-0.0859219\pi$
$108$		0			0
$109$			1.00000i			0.0957826i	0.998853	+	0.0478913i	$0.0152501\pi$
$109$			1.00000i			0.0957826i	−0.998853	+	0.0478913i	$0.984750\pi$
$110$		0			0
$111$		0			0
$112$	−1.22474	−	1.22474i	−0.115728	−	0.115728i
$113$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$113$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$114$		0			0
$115$		0			0
$116$		−	10.3923i		−	0.964901i
$117$		0			0
$118$	7.34847	−	7.34847i	0.676481	−	0.676481i
$119$	10.3923			0.952661
$120$		0			0
$121$	11.0000			1.00000
$122$	3.53553	−	3.53553i	0.320092	−	0.320092i
$123$		0			0
$124$		−	7.00000i		−	0.628619i
$125$		0			0
$126$		0			0
$127$	−7.34847	−	7.34847i	−0.652071	−	0.652071i	0.301420	−	0.953491i	$-0.402539\pi$
$127$	−7.34847	−	7.34847i	−0.652071	−	0.652071i	−0.953491	+	0.301420i	$0.902539\pi$
$128$	0.707107	+	0.707107i	0.0625000	+	0.0625000i
$129$		0			0
$130$		0			0
$131$			10.3923i			0.907980i	0.891007	+	0.453990i	$0.150000\pi$
$131$			10.3923i			0.907980i	−0.891007	+	0.453990i	$0.850000\pi$
$132$		0			0
$133$	1.22474	−	1.22474i	0.106199	−	0.106199i
$134$	−10.3923			−0.897758
$135$		0			0
$136$	−6.00000			−0.514496
$137$	−8.48528	+	8.48528i	−0.724947	+	0.724947i	−0.969608	−	0.244662i	$-0.921323\pi$
$137$	−8.48528	+	8.48528i	−0.724947	+	0.724947i	0.244662	+	0.969608i	$0.421323\pi$
$138$		0			0
$139$			11.0000i			0.933008i	0.884519	+	0.466504i	$0.154487\pi$
$139$			11.0000i			0.933008i	−0.884519	+	0.466504i	$0.845513\pi$
$140$		0			0
$141$		0			0
$142$	−7.34847	−	7.34847i	−0.616670	−	0.616670i
$143$		0			0
$144$		0			0
$145$		0			0
$146$			12.1244i			1.00342i
$147$		0			0
$148$	6.12372	−	6.12372i	0.503367	−	0.503367i
$149$	10.3923			0.851371			0.425685	−	0.904871i	$-0.360033\pi$
$149$	10.3923			0.851371			0.425685	+	0.904871i	$0.360033\pi$
$150$		0			0
$151$	−5.00000			−0.406894			−0.203447	−	0.979086i	$-0.565214\pi$
$151$	−5.00000			−0.406894			−0.203447	+	0.979086i	$0.565214\pi$
$152$	−0.707107	+	0.707107i	−0.0573539	+	0.0573539i
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−6.12372	−	6.12372i	−0.488726	−	0.488726i	0.419178	−	0.907904i	$-0.362318\pi$
$157$	−6.12372	−	6.12372i	−0.488726	−	0.488726i	−0.907904	+	0.419178i	$0.862318\pi$
$158$	9.19239	+	9.19239i	0.731307	+	0.731307i
$159$		0			0
$160$		0			0
$161$		−	10.3923i		−	0.819028i
$162$		0			0
$163$	−7.34847	+	7.34847i	−0.575577	+	0.575577i	−0.933681	−	0.358105i	$-0.883423\pi$
$163$	−7.34847	+	7.34847i	−0.575577	+	0.575577i	0.358105	+	0.933681i	$0.383423\pi$
$164$		0			0
$165$		0			0
$166$	−6.00000			−0.465690
$167$	−4.24264	+	4.24264i	−0.328305	+	0.328305i	−0.851942	−	0.523636i	$-0.824575\pi$
$167$	−4.24264	+	4.24264i	−0.328305	+	0.328305i	0.523636	+	0.851942i	$0.324575\pi$
$168$		0			0
$169$			13.0000i			1.00000i
$170$		0			0
$171$		0			0
$172$	1.22474	+	1.22474i	0.0933859	+	0.0933859i
$173$	4.24264	+	4.24264i	0.322562	+	0.322562i	0.849749	−	0.527187i	$-0.176753\pi$
$173$	4.24264	+	4.24264i	0.322562	+	0.322562i	−0.527187	+	0.849749i	$0.676753\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−7.34847	+	7.34847i	−0.550791	+	0.550791i
$179$	−20.7846			−1.55351			−0.776757	−	0.629800i	$-0.783137\pi$
$179$	−20.7846			−1.55351			−0.776757	+	0.629800i	$0.783137\pi$
$180$		0			0
$181$	−2.00000			−0.148659			−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000			−0.148659			−0.0743294	+	0.997234i	$0.523682\pi$
$182$		0			0
$183$		0			0
$184$			6.00000i			0.442326i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$			20.7846i			1.50392i	0.659208	+	0.751961i	$0.270892\pi$
$191$			20.7846i			1.50392i	−0.659208	+	0.751961i	$0.729108\pi$
$192$		0			0
$193$	−1.22474	+	1.22474i	−0.0881591	+	0.0881591i	−0.749811	−	0.661652i	$-0.769856\pi$
$193$	−1.22474	+	1.22474i	−0.0881591	+	0.0881591i	0.661652	+	0.749811i	$0.269856\pi$
$194$	8.66025			0.621770
$195$		0			0
$196$	−4.00000			−0.285714
$197$	8.48528	−	8.48528i	0.604551	−	0.604551i	−0.336966	−	0.941517i	$-0.609401\pi$
$197$	8.48528	−	8.48528i	0.604551	−	0.604551i	0.941517	+	0.336966i	$0.109401\pi$
$198$		0			0
$199$			16.0000i			1.13421i	0.823646	+	0.567105i	$0.191937\pi$
$199$			16.0000i			1.13421i	−0.823646	+	0.567105i	$0.808063\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	12.7279	+	12.7279i	0.893325	+	0.893325i
$204$		0			0
$205$		0			0
$206$			8.66025i			0.603388i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$	−8.48528	+	8.48528i	−0.582772	+	0.582772i
$213$		0			0
$214$		−	18.0000i		−	1.23045i
$215$		0			0
$216$		0			0
$217$	8.57321	+	8.57321i	0.581988	+	0.581988i
$218$	0.707107	+	0.707107i	0.0478913	+	0.0478913i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	8.57321	−	8.57321i	0.574105	−	0.574105i	−0.359168	−	0.933273i	$-0.616940\pi$
$223$	8.57321	−	8.57321i	0.574105	−	0.574105i	0.933273	+	0.359168i	$0.116940\pi$
$224$	−1.73205			−0.115728
$225$		0			0
$226$		0			0
$227$	−16.9706	+	16.9706i	−1.12638	+	1.12638i	−0.135614	+	0.990762i	$0.543301\pi$
$227$	−16.9706	+	16.9706i	−1.12638	+	1.12638i	−0.990762	+	0.135614i	$0.956699\pi$
$228$		0			0
$229$		−	25.0000i		−	1.65205i	−0.563636	−	0.826023i	$-0.690598\pi$
$229$		−	25.0000i		−	1.65205i	0.563636	−	0.826023i	$-0.309402\pi$
$230$		0			0
$231$		0			0
$232$	−7.34847	−	7.34847i	−0.482451	−	0.482451i
$233$	12.7279	+	12.7279i	0.833834	+	0.833834i	0.988039	−	0.154205i	$-0.0492816\pi$
$233$	12.7279	+	12.7279i	0.833834	+	0.833834i	−0.154205	+	0.988039i	$0.549282\pi$
$234$		0			0
$235$		0			0
$236$		−	10.3923i		−	0.676481i
$237$		0			0
$238$	7.34847	−	7.34847i	0.476331	−	0.476331i
$239$	−20.7846			−1.34444			−0.672222	−	0.740349i	$-0.734660\pi$
$239$	−20.7846			−1.34444			−0.672222	+	0.740349i	$0.734660\pi$
$240$		0			0
$241$	26.0000			1.67481			0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000			1.67481			0.837404	+	0.546585i	$0.184072\pi$
$242$	7.77817	−	7.77817i	0.500000	−	0.500000i
$243$		0			0
$244$		−	5.00000i		−	0.320092i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−4.94975	−	4.94975i	−0.314309	−	0.314309i
$249$		0			0
$250$		0			0
$251$		−	10.3923i		−	0.655956i	−0.944685	−	0.327978i	$-0.893633\pi$
$251$		−	10.3923i		−	0.655956i	0.944685	−	0.327978i	$-0.106367\pi$
$252$		0			0
$253$		0			0
$254$	−10.3923			−0.652071
$255$		0			0
$256$	1.00000			0.0625000
$257$	−8.48528	+	8.48528i	−0.529297	+	0.529297i	−0.920363	−	0.391066i	$-0.872107\pi$
$257$	−8.48528	+	8.48528i	−0.529297	+	0.529297i	0.391066	+	0.920363i	$0.372107\pi$
$258$		0			0
$259$			15.0000i			0.932055i
$260$		0			0
$261$		0			0
$262$	7.34847	+	7.34847i	0.453990	+	0.453990i
$263$	12.7279	+	12.7279i	0.784837	+	0.784837i	0.980643	−	0.195805i	$-0.0627321\pi$
$263$	12.7279	+	12.7279i	0.784837	+	0.784837i	−0.195805	+	0.980643i	$0.562732\pi$
$264$		0			0
$265$		0			0
$266$		−	1.73205i		−	0.106199i
$267$		0			0
$268$	−7.34847	+	7.34847i	−0.448879	+	0.448879i
$269$	−31.1769			−1.90089			−0.950445	−	0.310893i	$-0.899372\pi$
$269$	−31.1769			−1.90089			−0.950445	+	0.310893i	$0.899372\pi$
$270$		0			0
$271$	−1.00000			−0.0607457			−0.0303728	−	0.999539i	$-0.509669\pi$
$271$	−1.00000			−0.0607457			−0.0303728	+	0.999539i	$0.509669\pi$
$272$	−4.24264	+	4.24264i	−0.257248	+	0.257248i
$273$		0			0
$274$			12.0000i			0.724947i
$275$		0			0
$276$		0			0
$277$	13.4722	+	13.4722i	0.809466	+	0.809466i	0.984553	−	0.175087i	$-0.0560208\pi$
$277$	13.4722	+	13.4722i	0.809466	+	0.809466i	−0.175087	+	0.984553i	$0.556021\pi$
$278$	7.77817	+	7.77817i	0.466504	+	0.466504i
$279$		0			0
$280$		0			0
$281$			20.7846i			1.23991i	0.784639	+	0.619953i	$0.212848\pi$
$281$			20.7846i			1.23991i	−0.784639	+	0.619953i	$0.787152\pi$
$282$		0			0
$283$	−20.8207	+	20.8207i	−1.23766	+	1.23766i	−0.276705	+	0.960955i	$0.589242\pi$
$283$	−20.8207	+	20.8207i	−1.23766	+	1.23766i	−0.960955	+	0.276705i	$0.910758\pi$
$284$	−10.3923			−0.616670
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$		−	19.0000i		−	1.11765i
$290$		0			0
$291$		0			0
$292$	8.57321	+	8.57321i	0.501709	+	0.501709i
$293$	4.24264	+	4.24264i	0.247858	+	0.247858i	0.820091	−	0.572233i	$-0.193923\pi$
$293$	4.24264	+	4.24264i	0.247858	+	0.247858i	−0.572233	+	0.820091i	$0.693923\pi$
$294$		0			0
$295$		0			0
$296$		−	8.66025i		−	0.503367i
$297$		0			0
$298$	7.34847	−	7.34847i	0.425685	−	0.425685i
$299$		0			0
$300$		0			0
$301$	−3.00000			−0.172917
$302$	−3.53553	+	3.53553i	−0.203447	+	0.203447i
$303$		0			0
$304$			1.00000i			0.0573539i
$305$		0			0
$306$		0			0
$307$	−20.8207	−	20.8207i	−1.18830	−	1.18830i	−0.977538	−	0.210760i	$-0.932406\pi$
$307$	−20.8207	−	20.8207i	−1.18830	−	1.18830i	−0.210760	−	0.977538i	$-0.567594\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	20.7846i		−	1.17859i	−0.807919	−	0.589294i	$-0.799406\pi$
$311$		−	20.7846i		−	1.17859i	0.807919	−	0.589294i	$-0.200594\pi$
$312$		0			0
$313$	14.6969	−	14.6969i	0.830720	−	0.830720i	−0.156895	−	0.987615i	$-0.550148\pi$
$313$	14.6969	−	14.6969i	0.830720	−	0.830720i	0.987615	+	0.156895i	$0.0501485\pi$
$314$	−8.66025			−0.488726
$315$		0			0
$316$	13.0000			0.731307
$317$	−4.24264	+	4.24264i	−0.238290	+	0.238290i	−0.816142	−	0.577851i	$-0.803891\pi$
$317$	−4.24264	+	4.24264i	−0.238290	+	0.238290i	0.577851	+	0.816142i	$0.303891\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$	−7.34847	−	7.34847i	−0.409514	−	0.409514i
$323$	−4.24264	−	4.24264i	−0.236067	−	0.236067i
$324$		0			0
$325$		0			0
$326$			10.3923i			0.575577i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−13.0000			−0.714545			−0.357272	−	0.934000i	$-0.616293\pi$
$331$	−13.0000			−0.714545			−0.357272	+	0.934000i	$0.616293\pi$
$332$	−4.24264	+	4.24264i	−0.232845	+	0.232845i
$333$		0			0
$334$			6.00000i			0.328305i
$335$		0			0
$336$		0			0
$337$	14.6969	+	14.6969i	0.800593	+	0.800593i	0.983188	−	0.182595i	$-0.0584497\pi$
$337$	14.6969	+	14.6969i	0.800593	+	0.800593i	−0.182595	+	0.983188i	$0.558450\pi$
$338$	9.19239	+	9.19239i	0.500000	+	0.500000i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	13.4722	−	13.4722i	0.727430	−	0.727430i
$344$	1.73205			0.0933859
$345$		0			0
$346$	6.00000			0.322562
$347$	−12.7279	+	12.7279i	−0.683271	+	0.683271i	−0.960736	−	0.277465i	$-0.910506\pi$
$347$	−12.7279	+	12.7279i	−0.683271	+	0.683271i	0.277465	+	0.960736i	$0.410506\pi$
$348$		0			0
$349$		−	31.0000i		−	1.65939i	−0.558216	−	0.829696i	$-0.688514\pi$
$349$		−	31.0000i		−	1.65939i	0.558216	−	0.829696i	$-0.311486\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$353$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$354$		0			0
$355$		0			0
$356$			10.3923i			0.550791i
$357$		0			0
$358$	−14.6969	+	14.6969i	−0.776757	+	0.776757i
$359$	−10.3923			−0.548485			−0.274242	−	0.961661i	$-0.588427\pi$
$359$	−10.3923			−0.548485			−0.274242	+	0.961661i	$0.588427\pi$
$360$		0			0
$361$	18.0000			0.947368
$362$	−1.41421	+	1.41421i	−0.0743294	+	0.0743294i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	7.34847	+	7.34847i	0.383587	+	0.383587i	0.872393	−	0.488806i	$-0.162567\pi$
$367$	7.34847	+	7.34847i	0.383587	+	0.383587i	−0.488806	+	0.872393i	$0.662567\pi$
$368$	4.24264	+	4.24264i	0.221163	+	0.221163i
$369$		0			0
$370$		0			0
$371$		−	20.7846i		−	1.07908i
$372$		0			0
$373$	8.57321	−	8.57321i	0.443904	−	0.443904i	−0.449418	−	0.893322i	$-0.648368\pi$
$373$	8.57321	−	8.57321i	0.443904	−	0.443904i	0.893322	+	0.449418i	$0.148368\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	16.0000i		−	0.821865i	−0.911666	−	0.410932i	$-0.865203\pi$
$379$		−	16.0000i		−	0.821865i	0.911666	−	0.410932i	$-0.134797\pi$
$380$		0			0
$381$		0			0
$382$	14.6969	+	14.6969i	0.751961	+	0.751961i
$383$	4.24264	+	4.24264i	0.216789	+	0.216789i	0.807144	−	0.590355i	$-0.201012\pi$
$383$	4.24264	+	4.24264i	0.216789	+	0.216789i	−0.590355	+	0.807144i	$0.701012\pi$
$384$		0			0
$385$		0			0
$386$			1.73205i			0.0881591i
$387$		0			0
$388$	6.12372	−	6.12372i	0.310885	−	0.310885i
$389$	−20.7846			−1.05382			−0.526911	−	0.849921i	$-0.676650\pi$
$389$	−20.7846			−1.05382			−0.526911	+	0.849921i	$0.676650\pi$
$390$		0			0
$391$	−36.0000			−1.82060
$392$	−2.82843	+	2.82843i	−0.142857	+	0.142857i
$393$		0			0
$394$		−	12.0000i		−	0.604551i
$395$		0			0
$396$		0			0
$397$	6.12372	+	6.12372i	0.307341	+	0.307341i	0.843877	−	0.536536i	$-0.180268\pi$
$397$	6.12372	+	6.12372i	0.307341	+	0.307341i	−0.536536	+	0.843877i	$0.680268\pi$
$398$	11.3137	+	11.3137i	0.567105	+	0.567105i
$399$		0			0
$400$		0			0
$401$			20.7846i			1.03793i	0.854794	+	0.518967i	$0.173683\pi$
$401$			20.7846i			1.03793i	−0.854794	+	0.518967i	$0.826317\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	18.0000			0.893325
$407$		0			0
$408$		0			0
$409$		−	22.0000i		−	1.08783i	−0.839140	−	0.543915i	$-0.816941\pi$
$409$		−	22.0000i		−	1.08783i	0.839140	−	0.543915i	$-0.183059\pi$
$410$		0			0
$411$		0			0
$412$	6.12372	+	6.12372i	0.301694	+	0.301694i
$413$	12.7279	+	12.7279i	0.626300	+	0.626300i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	10.3923			0.507697			0.253849	−	0.967244i	$-0.418303\pi$
$419$	10.3923			0.507697			0.253849	+	0.967244i	$0.418303\pi$
$420$		0			0
$421$	13.0000			0.633581			0.316791	−	0.948495i	$-0.397395\pi$
$421$	13.0000			0.633581			0.316791	+	0.948495i	$0.397395\pi$
$422$	−2.82843	+	2.82843i	−0.137686	+	0.137686i
$423$		0			0
$424$			12.0000i			0.582772i
$425$		0			0
$426$		0			0
$427$	6.12372	+	6.12372i	0.296348	+	0.296348i
$428$	−12.7279	−	12.7279i	−0.615227	−	0.615227i
$429$		0			0
$430$		0			0
$431$			31.1769i			1.50174i	0.660451	+	0.750870i	$0.270365\pi$
$431$			31.1769i			1.50174i	−0.660451	+	0.750870i	$0.729635\pi$
$432$		0			0
$433$	−13.4722	+	13.4722i	−0.647432	+	0.647432i	−0.952372	−	0.304939i	$-0.901364\pi$
$433$	−13.4722	+	13.4722i	−0.647432	+	0.647432i	0.304939	+	0.952372i	$0.401364\pi$
$434$	12.1244			0.581988
$435$		0			0
$436$	1.00000			0.0478913
$437$	−4.24264	+	4.24264i	−0.202953	+	0.202953i
$438$		0			0
$439$			31.0000i			1.47955i	0.672855	+	0.739775i	$0.265068\pi$
$439$			31.0000i			1.47955i	−0.672855	+	0.739775i	$0.734932\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4.24264	+	4.24264i	0.201574	+	0.201574i	0.800674	−	0.599100i	$-0.204475\pi$
$443$	4.24264	+	4.24264i	0.201574	+	0.201574i	−0.599100	+	0.800674i	$0.704475\pi$
$444$		0			0
$445$		0			0
$446$		−	12.1244i		−	0.574105i
$447$		0			0
$448$	−1.22474	+	1.22474i	−0.0578638	+	0.0578638i
$449$	31.1769			1.47133			0.735665	−	0.677346i	$-0.236870\pi$
$449$	31.1769			1.47133			0.735665	+	0.677346i	$0.236870\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$			24.0000i			1.12638i
$455$		0			0
$456$		0			0
$457$	−14.6969	−	14.6969i	−0.687494	−	0.687494i	0.274184	−	0.961677i	$-0.411592\pi$
$457$	−14.6969	−	14.6969i	−0.687494	−	0.687494i	−0.961677	+	0.274184i	$0.911592\pi$
$458$	−17.6777	−	17.6777i	−0.826023	−	0.826023i
$459$		0			0
$460$		0			0
$461$		−	10.3923i		−	0.484018i	−0.970274	−	0.242009i	$-0.922194\pi$
$461$		−	10.3923i		−	0.484018i	0.970274	−	0.242009i	$-0.0778063\pi$
$462$		0			0
$463$	−8.57321	+	8.57321i	−0.398431	+	0.398431i	−0.877679	−	0.479248i	$-0.840909\pi$
$463$	−8.57321	+	8.57321i	−0.398431	+	0.398431i	0.479248	+	0.877679i	$0.340909\pi$
$464$	−10.3923			−0.482451
$465$		0			0
$466$	18.0000			0.833834
$467$	−4.24264	+	4.24264i	−0.196326	+	0.196326i	−0.798423	−	0.602097i	$-0.794332\pi$
$467$	−4.24264	+	4.24264i	−0.196326	+	0.196326i	0.602097	+	0.798423i	$0.294332\pi$
$468$		0			0
$469$		−	18.0000i		−	0.831163i
$470$		0			0
$471$		0			0
$472$	−7.34847	−	7.34847i	−0.338241	−	0.338241i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		−	10.3923i		−	0.476331i
$477$		0			0
$478$	−14.6969	+	14.6969i	−0.672222	+	0.672222i
$479$	10.3923			0.474837			0.237418	−	0.971408i	$-0.423699\pi$
$479$	10.3923			0.474837			0.237418	+	0.971408i	$0.423699\pi$
$480$		0			0
$481$		0			0
$482$	18.3848	−	18.3848i	0.837404	−	0.837404i
$483$		0			0
$484$		−	11.0000i		−	0.500000i
$485$		0			0
$486$		0			0
$487$	−22.0454	−	22.0454i	−0.998973	−	0.998973i	0.00102669	−	0.999999i	$-0.499673\pi$
$487$	−22.0454	−	22.0454i	−0.998973	−	0.998973i	−0.999999	+	0.00102669i	$0.999673\pi$
$488$	−3.53553	−	3.53553i	−0.160046	−	0.160046i
$489$		0			0
$490$		0			0
$491$			20.7846i			0.937996i	0.883199	+	0.468998i	$0.155385\pi$
$491$			20.7846i			0.937996i	−0.883199	+	0.468998i	$0.844615\pi$
$492$		0			0
$493$	44.0908	−	44.0908i	1.98575	−	1.98575i
$494$		0			0
$495$		0			0
$496$	−7.00000			−0.314309
$497$	12.7279	−	12.7279i	0.570925	−	0.570925i
$498$		0			0
$499$			7.00000i			0.313363i	0.987649	+	0.156682i	$0.0500796\pi$
$499$			7.00000i			0.313363i	−0.987649	+	0.156682i	$0.949920\pi$
$500$		0			0
$501$		0			0
$502$	−7.34847	−	7.34847i	−0.327978	−	0.327978i
$503$	−4.24264	−	4.24264i	−0.189170	−	0.189170i	0.606167	−	0.795337i	$-0.292706\pi$
$503$	−4.24264	−	4.24264i	−0.189170	−	0.189170i	−0.795337	+	0.606167i	$0.792706\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−7.34847	+	7.34847i	−0.326036	+	0.326036i
$509$	−20.7846			−0.921262			−0.460631	−	0.887592i	$-0.652377\pi$
$509$	−20.7846			−0.921262			−0.460631	+	0.887592i	$0.652377\pi$
$510$		0			0
$511$	−21.0000			−0.928985
$512$	0.707107	−	0.707107i	0.0312500	−	0.0312500i
$513$		0			0
$514$			12.0000i			0.529297i
$515$		0			0
$516$		0			0
$517$		0			0
$518$	10.6066	+	10.6066i	0.466027	+	0.466027i
$519$		0			0
$520$		0			0
$521$		−	20.7846i		−	0.910590i	−0.890341	−	0.455295i	$-0.849534\pi$
$521$		−	20.7846i		−	0.910590i	0.890341	−	0.455295i	$-0.150466\pi$
$522$		0			0
$523$	15.9217	−	15.9217i	0.696207	−	0.696207i	−0.267384	−	0.963590i	$-0.586159\pi$
$523$	15.9217	−	15.9217i	0.696207	−	0.696207i	0.963590	+	0.267384i	$0.0861591\pi$
$524$	10.3923			0.453990
$525$		0			0
$526$	18.0000			0.784837
$527$	29.6985	−	29.6985i	1.29369	−	1.29369i
$528$		0			0
$529$			13.0000i			0.565217i
$530$		0			0
$531$		0			0
$532$	−1.22474	−	1.22474i	−0.0530994	−	0.0530994i
$533$		0			0
$534$		0			0
$535$		0			0
$536$			10.3923i			0.448879i
$537$		0			0
$538$	−22.0454	+	22.0454i	−0.950445	+	0.950445i
$539$		0			0
$540$		0			0
$541$	−37.0000			−1.59075			−0.795377	−	0.606115i	$-0.792727\pi$
$541$	−37.0000			−1.59075			−0.795377	+	0.606115i	$0.792727\pi$
$542$	−0.707107	+	0.707107i	−0.0303728	+	0.0303728i
$543$		0			0
$544$			6.00000i			0.257248i
$545$		0			0
$546$		0			0
$547$	1.22474	+	1.22474i	0.0523663	+	0.0523663i	0.732805	−	0.680439i	$-0.238211\pi$
$547$	1.22474	+	1.22474i	0.0523663	+	0.0523663i	−0.680439	+	0.732805i	$0.738211\pi$
$548$	8.48528	+	8.48528i	0.362473	+	0.362473i
$549$		0			0
$550$		0			0
$551$		−	10.3923i		−	0.442727i
$552$		0			0
$553$	−15.9217	+	15.9217i	−0.677059	+	0.677059i
$554$	19.0526			0.809466
$555$		0			0
$556$	11.0000			0.466504
$557$	−12.7279	+	12.7279i	−0.539299	+	0.539299i	−0.923323	−	0.384024i	$-0.874538\pi$
$557$	−12.7279	+	12.7279i	−0.539299	+	0.539299i	0.384024	+	0.923323i	$0.374538\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	14.6969	+	14.6969i	0.619953	+	0.619953i
$563$	−25.4558	−	25.4558i	−1.07284	−	1.07284i	−0.997130	−	0.0757057i	$-0.975879\pi$
$563$	−25.4558	−	25.4558i	−1.07284	−	1.07284i	−0.0757057	−	0.997130i	$-0.524121\pi$
$564$		0			0
$565$		0			0
$566$			29.4449i			1.23766i
$567$		0			0
$568$	−7.34847	+	7.34847i	−0.308335	+	0.308335i
$569$	10.3923			0.435668			0.217834	−	0.975986i	$-0.430101\pi$
$569$	10.3923			0.435668			0.217834	+	0.975986i	$0.430101\pi$
$570$		0			0
$571$	−25.0000			−1.04622			−0.523109	−	0.852266i	$-0.675228\pi$
$571$	−25.0000			−1.04622			−0.523109	+	0.852266i	$0.675228\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	28.1691	+	28.1691i	1.17270	+	1.17270i	0.981564	+	0.191132i	$0.0612158\pi$
$577$	28.1691	+	28.1691i	1.17270	+	1.17270i	0.191132	+	0.981564i	$0.438784\pi$
$578$	−13.4350	−	13.4350i	−0.558824	−	0.558824i
$579$		0			0
$580$		0			0
$581$		−	10.3923i		−	0.431145i
$582$		0			0
$583$		0			0
$584$	12.1244			0.501709
$585$		0			0
$586$	6.00000			0.247858
$587$	29.6985	−	29.6985i	1.22579	−	1.22579i	0.260245	−	0.965543i	$-0.416197\pi$
$587$	29.6985	−	29.6985i	1.22579	−	1.22579i	0.965543	−	0.260245i	$-0.0838034\pi$
$588$		0			0
$589$		−	7.00000i		−	0.288430i
$590$		0			0
$591$		0			0
$592$	−6.12372	−	6.12372i	−0.251684	−	0.251684i
$593$	−29.6985	−	29.6985i	−1.21957	−	1.21957i	−0.967782	−	0.251788i	$-0.918981\pi$
$593$	−29.6985	−	29.6985i	−1.21957	−	1.21957i	−0.251788	−	0.967782i	$-0.581019\pi$
$594$		0			0
$595$		0			0
$596$		−	10.3923i		−	0.425685i
$597$		0			0
$598$		0			0
$599$	20.7846			0.849236			0.424618	−	0.905373i	$-0.360408\pi$
$599$	20.7846			0.849236			0.424618	+	0.905373i	$0.360408\pi$
$600$		0			0
$601$	13.0000			0.530281			0.265141	−	0.964210i	$-0.414582\pi$
$601$	13.0000			0.530281			0.265141	+	0.964210i	$0.414582\pi$
$602$	−2.12132	+	2.12132i	−0.0864586	+	0.0864586i
$603$		0			0
$604$			5.00000i			0.203447i
$605$		0			0
$606$		0			0
$607$	−23.2702	−	23.2702i	−0.944506	−	0.944506i	0.0540328	−	0.998539i	$-0.482792\pi$
$607$	−23.2702	−	23.2702i	−0.944506	−	0.944506i	−0.998539	+	0.0540328i	$0.982792\pi$
$608$	0.707107	+	0.707107i	0.0286770	+	0.0286770i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	8.57321	−	8.57321i	0.346269	−	0.346269i	−0.512449	−	0.858718i	$-0.671262\pi$
$613$	8.57321	−	8.57321i	0.346269	−	0.346269i	0.858718	+	0.512449i	$0.171262\pi$
$614$	−29.4449			−1.18830
$615$		0			0
$616$		0			0
$617$	16.9706	−	16.9706i	0.683209	−	0.683209i	−0.277513	−	0.960722i	$-0.589510\pi$
$617$	16.9706	−	16.9706i	0.683209	−	0.683209i	0.960722	+	0.277513i	$0.0895101\pi$
$618$		0			0
$619$			13.0000i			0.522514i	0.965269	+	0.261257i	$0.0841370\pi$
$619$			13.0000i			0.522514i	−0.965269	+	0.261257i	$0.915863\pi$
$620$		0			0
$621$		0			0
$622$	−14.6969	−	14.6969i	−0.589294	−	0.589294i
$623$	−12.7279	−	12.7279i	−0.509933	−	0.509933i
$624$		0			0
$625$		0			0
$626$		−	20.7846i		−	0.830720i
$627$		0			0
$628$	−6.12372	+	6.12372i	−0.244363	+	0.244363i
$629$	51.9615			2.07184
$630$		0			0
$631$	20.0000			0.796187			0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000			0.796187			0.398094	+	0.917345i	$0.369672\pi$
$632$	9.19239	−	9.19239i	0.365654	−	0.365654i
$633$		0			0
$634$			6.00000i			0.238290i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	31.1769i		−	1.23141i	−0.787975	−	0.615707i	$-0.788870\pi$
$641$		−	31.1769i		−	1.23141i	0.787975	−	0.615707i	$-0.211130\pi$
$642$		0			0
$643$	22.0454	−	22.0454i	0.869386	−	0.869386i	−0.123018	−	0.992404i	$-0.539257\pi$
$643$	22.0454	−	22.0454i	0.869386	−	0.869386i	0.992404	+	0.123018i	$0.0392574\pi$
$644$	−10.3923			−0.409514
$645$		0			0
$646$	−6.00000			−0.236067
$647$	−16.9706	+	16.9706i	−0.667182	+	0.667182i	−0.957063	−	0.289881i	$-0.906384\pi$
$647$	−16.9706	+	16.9706i	−0.667182	+	0.667182i	0.289881	+	0.957063i	$0.406384\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	7.34847	+	7.34847i	0.287788	+	0.287788i
$653$	−8.48528	−	8.48528i	−0.332055	−	0.332055i	0.521312	−	0.853366i	$-0.325443\pi$
$653$	−8.48528	−	8.48528i	−0.332055	−	0.332055i	−0.853366	+	0.521312i	$0.825443\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−10.3923			−0.404827			−0.202413	−	0.979300i	$-0.564878\pi$
$659$	−10.3923			−0.404827			−0.202413	+	0.979300i	$0.564878\pi$
$660$		0			0
$661$	−29.0000			−1.12797			−0.563985	−	0.825785i	$-0.690732\pi$
$661$	−29.0000			−1.12797			−0.563985	+	0.825785i	$0.690732\pi$
$662$	−9.19239	+	9.19239i	−0.357272	+	0.357272i
$663$		0			0
$664$			6.00000i			0.232845i
$665$		0			0
$666$		0			0
$667$	−44.0908	−	44.0908i	−1.70720	−	1.70720i
$668$	4.24264	+	4.24264i	0.164153	+	0.164153i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−15.9217	+	15.9217i	−0.613736	+	0.613736i	−0.943917	−	0.330182i	$-0.892890\pi$
$673$	−15.9217	+	15.9217i	−0.613736	+	0.613736i	0.330182	+	0.943917i	$0.392890\pi$
$674$	20.7846			0.800593
$675$		0			0
$676$	13.0000			0.500000
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$			15.0000i			0.575647i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	4.24264	+	4.24264i	0.162340	+	0.162340i	0.783603	−	0.621262i	$-0.213380\pi$
$683$	4.24264	+	4.24264i	0.162340	+	0.162340i	−0.621262	+	0.783603i	$0.713380\pi$
$684$		0			0
$685$		0			0
$686$		−	19.0526i		−	0.727430i
$687$		0			0
$688$	1.22474	−	1.22474i	0.0466930	−	0.0466930i
$689$		0			0
$690$		0			0
$691$	−44.0000			−1.67384			−0.836919	−	0.547326i	$-0.815646\pi$
$691$	−44.0000			−1.67384			−0.836919	+	0.547326i	$0.815646\pi$
$692$	4.24264	−	4.24264i	0.161281	−	0.161281i
$693$		0			0
$694$			18.0000i			0.683271i
$695$		0			0
$696$		0			0
$697$		0			0
$698$	−21.9203	−	21.9203i	−0.829696	−	0.829696i
$699$		0			0
$700$		0			0
$701$			31.1769i			1.17754i	0.808302	+	0.588768i	$0.200387\pi$
$701$			31.1769i			1.17754i	−0.808302	+	0.588768i	$0.799613\pi$
$702$		0			0
$703$	6.12372	−	6.12372i	0.230961	−	0.230961i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	13.0000i		−	0.488225i	−0.969747	−	0.244113i	$-0.921503\pi$
$709$		−	13.0000i		−	0.488225i	0.969747	−	0.244113i	$-0.0784967\pi$
$710$		0			0
$711$		0			0
$712$	7.34847	+	7.34847i	0.275396	+	0.275396i
$713$	−29.6985	−	29.6985i	−1.11222	−	1.11222i
$714$		0			0
$715$		0			0
$716$			20.7846i			0.776757i
$717$		0			0
$718$	−7.34847	+	7.34847i	−0.274242	+	0.274242i
$719$	51.9615			1.93784			0.968919	−	0.247378i	$-0.0795691\pi$
$719$	51.9615			1.93784			0.968919	+	0.247378i	$0.0795691\pi$
$720$		0			0
$721$	−15.0000			−0.558629
$722$	12.7279	−	12.7279i	0.473684	−	0.473684i
$723$		0			0
$724$			2.00000i			0.0743294i
$725$		0			0
$726$		0			0
$727$	−6.12372	−	6.12372i	−0.227116	−	0.227116i	0.584371	−	0.811487i	$-0.301341\pi$
$727$	−6.12372	−	6.12372i	−0.227116	−	0.227116i	−0.811487	+	0.584371i	$0.801341\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			10.3923i			0.384373i
$732$		0			0
$733$	14.6969	−	14.6969i	0.542844	−	0.542844i	−0.381518	−	0.924362i	$-0.624598\pi$
$733$	14.6969	−	14.6969i	0.542844	−	0.542844i	0.924362	+	0.381518i	$0.124598\pi$
$734$	10.3923			0.383587
$735$		0			0
$736$	6.00000			0.221163
$737$		0			0
$738$		0			0
$739$			20.0000i			0.735712i	0.929883	+	0.367856i	$0.119908\pi$
$739$			20.0000i			0.735712i	−0.929883	+	0.367856i	$0.880092\pi$
$740$		0			0
$741$		0			0
$742$	−14.6969	−	14.6969i	−0.539542	−	0.539542i
$743$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$		0			0
$746$		−	12.1244i		−	0.443904i
$747$		0			0
$748$		0			0
$749$	31.1769			1.13918
$750$		0			0
$751$	−11.0000			−0.401396			−0.200698	−	0.979653i	$-0.564321\pi$
$751$	−11.0000			−0.401396			−0.200698	+	0.979653i	$0.564321\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	15.9217	+	15.9217i	0.578683	+	0.578683i	0.934540	−	0.355857i	$-0.115811\pi$
$757$	15.9217	+	15.9217i	0.578683	+	0.578683i	−0.355857	+	0.934540i	$0.615811\pi$
$758$	−11.3137	−	11.3137i	−0.410932	−	0.410932i
$759$		0			0
$760$		0			0
$761$		−	41.5692i		−	1.50688i	−0.657515	−	0.753442i	$-0.728392\pi$
$761$		−	41.5692i		−	1.50688i	0.657515	−	0.753442i	$-0.271608\pi$
$762$		0			0
$763$	−1.22474	+	1.22474i	−0.0443387	+	0.0443387i
$764$	20.7846			0.751961
$765$		0			0
$766$	6.00000			0.216789
$767$		0			0
$768$		0			0
$769$		−	14.0000i		−	0.504853i	−0.967616	−	0.252426i	$-0.918771\pi$
$769$		−	14.0000i		−	0.504853i	0.967616	−	0.252426i	$-0.0812286\pi$
$770$		0			0
$771$		0			0
$772$	1.22474	+	1.22474i	0.0440795	+	0.0440795i
$773$	−4.24264	−	4.24264i	−0.152597	−	0.152597i	0.626680	−	0.779277i	$-0.284413\pi$
$773$	−4.24264	−	4.24264i	−0.152597	−	0.152597i	−0.779277	+	0.626680i	$0.784413\pi$
$774$		0			0
$775$		0			0
$776$		−	8.66025i		−	0.310885i
$777$		0			0
$778$	−14.6969	+	14.6969i	−0.526911	+	0.526911i
$779$		0			0
$780$		0			0
$781$		0			0
$782$	−25.4558	+	25.4558i	−0.910299	+	0.910299i
$783$		0			0
$784$			4.00000i			0.142857i
$785$		0			0
$786$		0			0
$787$	1.22474	+	1.22474i	0.0436574	+	0.0436574i	0.728598	−	0.684941i	$-0.240172\pi$
$787$	1.22474	+	1.22474i	0.0436574	+	0.0436574i	−0.684941	+	0.728598i	$0.740172\pi$
$788$	−8.48528	−	8.48528i	−0.302276	−	0.302276i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	8.66025			0.307341
$795$		0			0
$796$	16.0000			0.567105
$797$	8.48528	−	8.48528i	0.300564	−	0.300564i	−0.540670	−	0.841235i	$-0.681829\pi$
$797$	8.48528	−	8.48528i	0.300564	−	0.300564i	0.841235	+	0.540670i	$0.181829\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$	14.6969	+	14.6969i	0.518967	+	0.518967i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−10.3923			−0.365374			−0.182687	−	0.983171i	$-0.558479\pi$
$809$	−10.3923			−0.365374			−0.182687	+	0.983171i	$0.558479\pi$
$810$		0			0
$811$	19.0000			0.667180			0.333590	−	0.942718i	$-0.391740\pi$
$811$	19.0000			0.667180			0.333590	+	0.942718i	$0.391740\pi$
$812$	12.7279	−	12.7279i	0.446663	−	0.446663i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1.22474	+	1.22474i	0.0428484	+	0.0428484i
$818$	−15.5563	−	15.5563i	−0.543915	−	0.543915i
$819$		0			0
$820$		0			0
$821$			20.7846i			0.725388i	0.931908	+	0.362694i	$0.118143\pi$
$821$			20.7846i			0.725388i	−0.931908	+	0.362694i	$0.881857\pi$
$822$		0			0
$823$	−22.0454	+	22.0454i	−0.768455	+	0.768455i	−0.977834	−	0.209380i	$-0.932856\pi$
$823$	−22.0454	+	22.0454i	−0.768455	+	0.768455i	0.209380	+	0.977834i	$0.432856\pi$
$824$	8.66025			0.301694
$825$		0			0
$826$	18.0000			0.626300
$827$	−8.48528	+	8.48528i	−0.295062	+	0.295062i	−0.839076	−	0.544014i	$-0.816904\pi$
$827$	−8.48528	+	8.48528i	−0.295062	+	0.295062i	0.544014	+	0.839076i	$0.316904\pi$
$828$		0			0
$829$		−	37.0000i		−	1.28506i	−0.766259	−	0.642532i	$-0.777884\pi$
$829$		−	37.0000i		−	1.28506i	0.766259	−	0.642532i	$-0.222116\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−16.9706	−	16.9706i	−0.587995	−	0.587995i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	7.34847	−	7.34847i	0.253849	−	0.253849i
$839$	−51.9615			−1.79391			−0.896956	−	0.442121i	$-0.854226\pi$
$839$	−51.9615			−1.79391			−0.896956	+	0.442121i	$0.854226\pi$
$840$		0			0
$841$	79.0000			2.72414
$842$	9.19239	−	9.19239i	0.316791	−	0.316791i
$843$		0			0
$844$			4.00000i			0.137686i
$845$		0			0
$846$		0			0
$847$	13.4722	+	13.4722i	0.462910	+	0.462910i
$848$	8.48528	+	8.48528i	0.291386	+	0.291386i
$849$		0			0
$850$		0			0
$851$		−	51.9615i		−	1.78122i
$852$		0			0
$853$	14.6969	−	14.6969i	0.503214	−	0.503214i	−0.409221	−	0.912435i	$-0.634200\pi$
$853$	14.6969	−	14.6969i	0.503214	−	0.503214i	0.912435	+	0.409221i	$0.134200\pi$
$854$	8.66025			0.296348
$855$		0			0
$856$	−18.0000			−0.615227
$857$	16.9706	−	16.9706i	0.579703	−	0.579703i	−0.355118	−	0.934821i	$-0.615559\pi$
$857$	16.9706	−	16.9706i	0.579703	−	0.579703i	0.934821	+	0.355118i	$0.115559\pi$
$858$		0			0
$859$		−	11.0000i		−	0.375315i	−0.982235	−	0.187658i	$-0.939910\pi$
$859$		−	11.0000i		−	0.375315i	0.982235	−	0.187658i	$-0.0600895\pi$
$860$		0			0
$861$		0			0
$862$	22.0454	+	22.0454i	0.750870	+	0.750870i
$863$	4.24264	+	4.24264i	0.144421	+	0.144421i	0.775621	−	0.631199i	$-0.217437\pi$
$863$	4.24264	+	4.24264i	0.144421	+	0.144421i	−0.631199	+	0.775621i	$0.717437\pi$
$864$		0			0
$865$		0			0
$866$			19.0526i			0.647432i
$867$		0			0
$868$	8.57321	−	8.57321i	0.290994	−	0.290994i
$869$		0			0
$870$		0			0
$871$		0			0
$872$	0.707107	−	0.707107i	0.0239457	−	0.0239457i
$873$		0			0
$874$			6.00000i			0.202953i
$875$		0			0
$876$		0			0
$877$	−6.12372	−	6.12372i	−0.206783	−	0.206783i	0.596115	−	0.802899i	$-0.296710\pi$
$877$	−6.12372	−	6.12372i	−0.206783	−	0.206783i	−0.802899	+	0.596115i	$0.796710\pi$
$878$	21.9203	+	21.9203i	0.739775	+	0.739775i
$879$		0			0
$880$		0			0
$881$			51.9615i			1.75063i	0.483555	+	0.875314i	$0.339345\pi$
$881$			51.9615i			1.75063i	−0.483555	+	0.875314i	$0.660655\pi$
$882$		0			0
$883$	−20.8207	+	20.8207i	−0.700671	+	0.700671i	−0.964555	−	0.263883i	$-0.914997\pi$
$883$	−20.8207	+	20.8207i	−0.700671	+	0.700671i	0.263883	+	0.964555i	$0.414997\pi$
$884$		0			0
$885$		0			0
$886$	6.00000			0.201574
$887$	−8.48528	+	8.48528i	−0.284908	+	0.284908i	−0.835063	−	0.550155i	$-0.814569\pi$
$887$	−8.48528	+	8.48528i	−0.284908	+	0.284908i	0.550155	+	0.835063i	$0.314569\pi$
$888$		0			0
$889$		−	18.0000i		−	0.603701i
$890$		0			0
$891$		0			0
$892$	−8.57321	−	8.57321i	−0.287052	−	0.287052i
$893$		0			0
$894$		0			0
$895$		0			0
$896$			1.73205i			0.0578638i
$897$		0			0
$898$	22.0454	−	22.0454i	0.735665	−	0.735665i
$899$	72.7461			2.42622
$900$		0			0
$901$	−72.0000			−2.39867
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	6.12372	+	6.12372i	0.203335	+	0.203335i	0.801427	−	0.598092i	$-0.204074\pi$
$907$	6.12372	+	6.12372i	0.203335	+	0.203335i	−0.598092	+	0.801427i	$0.704074\pi$
$908$	16.9706	+	16.9706i	0.563188	+	0.563188i
$909$		0			0
$910$		0			0
$911$		−	51.9615i		−	1.72156i	−0.508975	−	0.860781i	$-0.669976\pi$
$911$		−	51.9615i		−	1.72156i	0.508975	−	0.860781i	$-0.330024\pi$
$912$		0			0
$913$		0			0
$914$	−20.7846			−0.687494
$915$		0			0
$916$	−25.0000			−0.826023
$917$	−12.7279	+	12.7279i	−0.420313	+	0.420313i
$918$		0			0
$919$		−	1.00000i		−	0.0329870i	−0.999864	−	0.0164935i	$-0.994750\pi$
$919$		−	1.00000i		−	0.0329870i	0.999864	−	0.0164935i	$-0.00525028\pi$
$920$		0			0
$921$		0			0
$922$	−7.34847	−	7.34847i	−0.242009	−	0.242009i
$923$		0			0
$924$		0			0
$925$		0			0
$926$			12.1244i			0.398431i
$927$		0			0
$928$	−7.34847	+	7.34847i	−0.241225	+	0.241225i
$929$	−31.1769			−1.02288			−0.511441	−	0.859319i	$-0.670888\pi$
$929$	−31.1769			−1.02288			−0.511441	+	0.859319i	$0.670888\pi$
$930$		0			0
$931$	−4.00000			−0.131095
$932$	12.7279	−	12.7279i	0.416917	−	0.416917i
$933$		0			0
$934$			6.00000i			0.196326i
$935$		0			0
$936$		0			0
$937$	30.6186	+	30.6186i	1.00027	+	1.00027i	1.00000		0.000266809i	$8.49279e-5\pi$
$937$	30.6186	+	30.6186i	1.00027	+	1.00027i	0.000266809		1.00000i	$0.499915\pi$
$938$	−12.7279	−	12.7279i	−0.415581	−	0.415581i
$939$		0			0
$940$		0			0
$941$			20.7846i			0.677559i	0.940866	+	0.338779i	$0.110014\pi$
$941$			20.7846i			0.677559i	−0.940866	+	0.338779i	$0.889986\pi$
$942$		0			0
$943$		0			0
$944$	−10.3923			−0.338241
$945$		0			0
$946$		0			0
$947$	29.6985	−	29.6985i	0.965071	−	0.965071i	−0.0343392	−	0.999410i	$-0.510933\pi$
$947$	29.6985	−	29.6985i	0.965071	−	0.965071i	0.999410	+	0.0343392i	$0.0109326\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$	−7.34847	−	7.34847i	−0.238165	−	0.238165i
$953$	25.4558	+	25.4558i	0.824596	+	0.824596i	0.986763	−	0.162168i	$-0.0518485\pi$
$953$	25.4558	+	25.4558i	0.824596	+	0.824596i	−0.162168	+	0.986763i	$0.551849\pi$
$954$		0			0
$955$		0			0
$956$			20.7846i			0.672222i
$957$		0			0
$958$	7.34847	−	7.34847i	0.237418	−	0.237418i
$959$	−20.7846			−0.671170
$960$		0			0
$961$	18.0000			0.580645
$962$		0			0
$963$		0			0
$964$		−	26.0000i		−	0.837404i
$965$		0			0
$966$		0			0
$967$	−15.9217	−	15.9217i	−0.512007	−	0.512007i	0.403134	−	0.915141i	$-0.367921\pi$
$967$	−15.9217	−	15.9217i	−0.512007	−	0.512007i	−0.915141	+	0.403134i	$0.867921\pi$
$968$	−7.77817	−	7.77817i	−0.250000	−	0.250000i
$969$		0			0
$970$		0			0
$971$			20.7846i			0.667010i	0.942748	+	0.333505i	$0.108231\pi$
$971$			20.7846i			0.667010i	−0.942748	+	0.333505i	$0.891769\pi$
$972$		0			0
$973$	−13.4722	+	13.4722i	−0.431899	+	0.431899i
$974$	−31.1769			−0.998973
$975$		0			0
$976$	−5.00000			−0.160046
$977$	12.7279	−	12.7279i	0.407202	−	0.407202i	−0.473560	−	0.880762i	$-0.657031\pi$
$977$	12.7279	−	12.7279i	0.407202	−	0.407202i	0.880762	+	0.473560i	$0.157031\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	14.6969	+	14.6969i	0.468998	+	0.468998i
$983$	−42.4264	−	42.4264i	−1.35319	−	1.35319i	−0.882066	−	0.471127i	$-0.843847\pi$
$983$	−42.4264	−	42.4264i	−1.35319	−	1.35319i	−0.471127	−	0.882066i	$-0.656153\pi$
$984$		0			0
$985$		0			0
$986$		−	62.3538i		−	1.98575i
$987$		0			0
$988$		0			0
$989$	10.3923			0.330456
$990$		0			0
$991$	35.0000			1.11181			0.555906	−	0.831245i	$-0.312372\pi$
$991$	35.0000			1.11181			0.555906	+	0.831245i	$0.312372\pi$
$992$	−4.94975	+	4.94975i	−0.157155	+	0.157155i
$993$		0			0
$994$		−	18.0000i		−	0.570925i
$995$		0			0
$996$		0			0
$997$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$997$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$998$	4.94975	+	4.94975i	0.156682	+	0.156682i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.2.f.d.107.4	yes	8
3.2	odd	2	inner	1350.2.f.d.107.2	yes	8
5.2	odd	4	inner	1350.2.f.d.593.3	yes	8
5.3	odd	4	inner	1350.2.f.d.593.2	yes	8
5.4	even	2	inner	1350.2.f.d.107.1	✓	8
15.2	even	4	inner	1350.2.f.d.593.1	yes	8
15.8	even	4	inner	1350.2.f.d.593.4	yes	8
15.14	odd	2	inner	1350.2.f.d.107.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1350.2.f.d.107.1	✓	8	5.4	even	2	inner
1350.2.f.d.107.2	yes	8	3.2	odd	2	inner
1350.2.f.d.107.3	yes	8	15.14	odd	2	inner
1350.2.f.d.107.4	yes	8	1.1	even	1	trivial
1350.2.f.d.593.1	yes	8	15.2	even	4	inner
1350.2.f.d.593.2	yes	8	5.3	odd	4	inner
1350.2.f.d.593.3	yes	8	5.2	odd	4	inner
1350.2.f.d.593.4	yes	8	15.8	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1350.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 - 0.707107i) q^{2} -1.00000i q^{4} +(1.22474 + 1.22474i) q^{7} +(-0.707107 - 0.707107i) q^{8} +O(q^{10})\)	\(q+(0.707107 - 0.707107i) q^{2} -1.00000i q^{4} +(1.22474 + 1.22474i) q^{7} +(-0.707107 - 0.707107i) q^{8} +1.73205 q^{14} -1.00000 q^{16} +(4.24264 - 4.24264i) q^{17} -1.00000i q^{19} +(-4.24264 - 4.24264i) q^{23} +(1.22474 - 1.22474i) q^{28} +10.3923 q^{29} +7.00000 q^{31} +(-0.707107 + 0.707107i) q^{32} -6.00000i q^{34} +(6.12372 + 6.12372i) q^{37} +(-0.707107 - 0.707107i) q^{38} +(-1.22474 + 1.22474i) q^{43} -6.00000 q^{46} -4.00000i q^{49} +(-8.48528 - 8.48528i) q^{53} -1.73205i q^{56} +(7.34847 - 7.34847i) q^{58} +10.3923 q^{59} +5.00000 q^{61} +(4.94975 - 4.94975i) q^{62} +1.00000i q^{64} +(-7.34847 - 7.34847i) q^{67} +(-4.24264 - 4.24264i) q^{68} -10.3923i q^{71} +(-8.57321 + 8.57321i) q^{73} +8.66025 q^{74} -1.00000 q^{76} +13.0000i q^{79} +(-4.24264 - 4.24264i) q^{83} +1.73205i q^{86} -10.3923 q^{89} +(-4.24264 + 4.24264i) q^{92} +(6.12372 + 6.12372i) q^{97} +(-2.82843 - 2.82843i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 8 q^{16} + 56 q^{31} - 48 q^{46} + 40 q^{61} - 8 q^{76}+O(q^{100}) \) 8 * q - 8 * q^16 + 56 * q^31 - 48 * q^46 + 40 * q^61 - 8 * q^76

Introduction

L-functions

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