LMFDB - Embedded newform 1350.2.f.d.107.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1350,2,Mod(107,1350)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-8,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)

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			107.1
Root			$0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	1350.107
Dual form			1350.2.f.d.593.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

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

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	0.436698	−	0.899608i	$-0.356148\pi$
$7$	−1.22474	−	1.22474i	−0.462910	−	0.462910i	−0.899608	+	0.436698i	$0.856148\pi$
$8$	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	$0.509367\pi$
$17$	−4.24264	+	4.24264i	−1.02899	+	1.02899i	−0.999567	+	0.0294245i	$0.990633\pi$
$18$		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	0.994003	−	0.109351i	$-0.0348774\pi$
$23$	4.24264	+	4.24264i	0.884652	+	0.884652i	−0.109351	+	0.994003i	$0.534877\pi$
$24$		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	$-0.997849\pi$
$37$	−6.12372	−	6.12372i	−1.00673	−	1.00673i	−0.00675691	−	0.999977i	$-0.502151\pi$
$38$	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	−0.607527	−	0.794299i	$-0.707838\pi$
$43$	1.22474	−	1.22474i	0.186772	−	0.186772i	0.794299	+	0.607527i	$0.207838\pi$
$44$		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	$0.0583542\pi$
$53$	8.48528	+	8.48528i	1.16554	+	1.16554i	0.182300	+	0.983243i	$0.441646\pi$
$54$		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	0.995238	−	0.0974792i	$-0.0310779\pi$
$67$	7.34847	+	7.34847i	0.897758	+	0.897758i	−0.0974792	+	0.995238i	$0.531078\pi$
$68$	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	$-0.498910\pi$
$73$	8.57321	−	8.57321i	1.00342	−	1.00342i	0.999994	−	0.00342468i	$-0.00109011\pi$
$74$	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	0.900515	−	0.434825i	$-0.143190\pi$
$83$	4.24264	+	4.24264i	0.465690	+	0.465690i	−0.434825	+	0.900515i	$0.643190\pi$
$84$		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	0.324214	−	0.945984i	$-0.394900\pi$
$97$	−6.12372	−	6.12372i	−0.621770	−	0.621770i	−0.945984	+	0.324214i	$0.894900\pi$
$98$	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	−0.337822	−	0.941210i	$-0.609690\pi$
$103$	6.12372	−	6.12372i	0.603388	−	0.603388i	0.941210	+	0.337822i	$0.109690\pi$
$104$		0			0
$105$		0			0
$106$	−12.0000			−1.16554
$107$	−12.7279	+	12.7279i	−1.23045	+	1.23045i	−0.266666	+	0.963789i	$0.585922\pi$
$107$	−12.7279	+	12.7279i	−1.23045	+	1.23045i	−0.963789	+	0.266666i	$0.914078\pi$
$108$		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	0.953491	−	0.301420i	$-0.0974607\pi$
$127$	7.34847	+	7.34847i	0.652071	+	0.652071i	−0.301420	+	0.953491i	$0.597461\pi$
$128$	−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	−0.244662	−	0.969608i	$-0.578677\pi$
$137$	8.48528	−	8.48528i	0.724947	−	0.724947i	0.969608	+	0.244662i	$0.0786770\pi$
$138$		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	0.907904	−	0.419178i	$-0.137682\pi$
$157$	6.12372	+	6.12372i	0.488726	+	0.488726i	−0.419178	+	0.907904i	$0.637682\pi$
$158$	−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	−0.358105	−	0.933681i	$-0.616577\pi$
$163$	7.34847	−	7.34847i	0.575577	−	0.575577i	0.933681	+	0.358105i	$0.116577\pi$
$164$		0			0
$165$		0			0
$166$	−6.00000			−0.465690
$167$	4.24264	−	4.24264i	0.328305	−	0.328305i	−0.523636	−	0.851942i	$-0.675425\pi$
$167$	4.24264	−	4.24264i	0.328305	−	0.328305i	0.851942	+	0.523636i	$0.175425\pi$
$168$		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	0.527187	−	0.849749i	$-0.323247\pi$
$173$	−4.24264	−	4.24264i	−0.322562	−	0.322562i	−0.849749	+	0.527187i	$0.823247\pi$
$174$		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	−0.661652	−	0.749811i	$-0.730144\pi$
$193$	1.22474	−	1.22474i	0.0881591	−	0.0881591i	0.749811	+	0.661652i	$0.230144\pi$
$194$	8.66025			0.621770
$195$		0			0
$196$	−4.00000			−0.285714
$197$	−8.48528	+	8.48528i	−0.604551	+	0.604551i	−0.941517	−	0.336966i	$-0.890599\pi$
$197$	−8.48528	+	8.48528i	−0.604551	+	0.604551i	0.336966	+	0.941517i	$0.390599\pi$
$198$		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	−0.933273	−	0.359168i	$-0.883060\pi$
$223$	−8.57321	+	8.57321i	−0.574105	+	0.574105i	0.359168	+	0.933273i	$0.383060\pi$
$224$	−1.73205			−0.115728
$225$		0			0
$226$		0			0
$227$	16.9706	−	16.9706i	1.12638	−	1.12638i	0.135614	−	0.990762i	$-0.456699\pi$
$227$	16.9706	−	16.9706i	1.12638	−	1.12638i	0.990762	−	0.135614i	$-0.0433007\pi$
$228$		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	0.154205	−	0.988039i	$-0.450718\pi$
$233$	−12.7279	−	12.7279i	−0.833834	−	0.833834i	−0.988039	+	0.154205i	$0.950718\pi$
$234$		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	−0.391066	−	0.920363i	$-0.627893\pi$
$257$	8.48528	−	8.48528i	0.529297	−	0.529297i	0.920363	+	0.391066i	$0.127893\pi$
$258$		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	0.195805	−	0.980643i	$-0.437268\pi$
$263$	−12.7279	−	12.7279i	−0.784837	−	0.784837i	−0.980643	+	0.195805i	$0.937268\pi$
$264$		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	0.175087	−	0.984553i	$-0.443979\pi$
$277$	−13.4722	−	13.4722i	−0.809466	−	0.809466i	−0.984553	+	0.175087i	$0.943979\pi$
$278$	−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	$-0.410758\pi$
$283$	20.8207	−	20.8207i	1.23766	−	1.23766i	0.960955	−	0.276705i	$-0.0892425\pi$
$284$	−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	0.572233	−	0.820091i	$-0.306077\pi$
$293$	−4.24264	−	4.24264i	−0.247858	−	0.247858i	−0.820091	+	0.572233i	$0.806077\pi$
$294$		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	$0.0675939\pi$
$307$	20.8207	+	20.8207i	1.18830	+	1.18830i	0.210760	+	0.977538i	$0.432406\pi$
$308$		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	−0.987615	−	0.156895i	$-0.949852\pi$
$313$	−14.6969	+	14.6969i	−0.830720	+	0.830720i	0.156895	+	0.987615i	$0.449852\pi$
$314$	−8.66025			−0.488726
$315$		0			0
$316$	13.0000			0.731307
$317$	4.24264	−	4.24264i	0.238290	−	0.238290i	−0.577851	−	0.816142i	$-0.696109\pi$
$317$	4.24264	−	4.24264i	0.238290	−	0.238290i	0.816142	+	0.577851i	$0.196109\pi$
$318$		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	0.182595	−	0.983188i	$-0.441550\pi$
$337$	−14.6969	−	14.6969i	−0.800593	−	0.800593i	−0.983188	+	0.182595i	$0.941550\pi$
$338$	−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	−0.277465	−	0.960736i	$-0.589494\pi$
$347$	12.7279	−	12.7279i	0.683271	−	0.683271i	0.960736	+	0.277465i	$0.0894943\pi$
$348$		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	0.488806	−	0.872393i	$-0.337433\pi$
$367$	−7.34847	−	7.34847i	−0.383587	−	0.383587i	−0.872393	+	0.488806i	$0.837433\pi$
$368$	−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	−0.893322	−	0.449418i	$-0.851632\pi$
$373$	−8.57321	+	8.57321i	−0.443904	+	0.443904i	0.449418	+	0.893322i	$0.351632\pi$
$374$		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	0.590355	−	0.807144i	$-0.298988\pi$
$383$	−4.24264	−	4.24264i	−0.216789	−	0.216789i	−0.807144	+	0.590355i	$0.798988\pi$
$384$		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	0.536536	−	0.843877i	$-0.319732\pi$
$397$	−6.12372	−	6.12372i	−0.307341	−	0.307341i	−0.843877	+	0.536536i	$0.819732\pi$
$398$	−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	−0.304939	−	0.952372i	$-0.598636\pi$
$433$	13.4722	−	13.4722i	0.647432	−	0.647432i	0.952372	+	0.304939i	$0.0986362\pi$
$434$	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	0.599100	−	0.800674i	$-0.295525\pi$
$443$	−4.24264	−	4.24264i	−0.201574	−	0.201574i	−0.800674	+	0.599100i	$0.795525\pi$
$444$		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	0.961677	−	0.274184i	$-0.0884076\pi$
$457$	14.6969	+	14.6969i	0.687494	+	0.687494i	−0.274184	+	0.961677i	$0.588408\pi$
$458$	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	−0.479248	−	0.877679i	$-0.659091\pi$
$463$	8.57321	−	8.57321i	0.398431	−	0.398431i	0.877679	+	0.479248i	$0.159091\pi$
$464$	−10.3923			−0.482451
$465$		0			0
$466$	18.0000			0.833834
$467$	4.24264	−	4.24264i	0.196326	−	0.196326i	−0.602097	−	0.798423i	$-0.705668\pi$
$467$	4.24264	−	4.24264i	0.196326	−	0.196326i	0.798423	+	0.602097i	$0.205668\pi$
$468$		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	0.999999	−	0.00102669i	$-0.000326807\pi$
$487$	22.0454	+	22.0454i	0.998973	+	0.998973i	−0.00102669	+	0.999999i	$0.500327\pi$
$488$	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	0.795337	−	0.606167i	$-0.207294\pi$
$503$	4.24264	+	4.24264i	0.189170	+	0.189170i	−0.606167	+	0.795337i	$0.707294\pi$
$504$		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	−0.963590	−	0.267384i	$-0.913841\pi$
$523$	−15.9217	+	15.9217i	−0.696207	+	0.696207i	0.267384	+	0.963590i	$0.413841\pi$
$524$	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	0.680439	−	0.732805i	$-0.261789\pi$
$547$	−1.22474	−	1.22474i	−0.0523663	−	0.0523663i	−0.732805	+	0.680439i	$0.761789\pi$
$548$	−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	−0.384024	−	0.923323i	$-0.625462\pi$
$557$	12.7279	−	12.7279i	0.539299	−	0.539299i	0.923323	+	0.384024i	$0.125462\pi$
$558$		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	$0.0241210\pi$
$563$	25.4558	+	25.4558i	1.07284	+	1.07284i	0.0757057	+	0.997130i	$0.475879\pi$
$564$		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	$-0.938784\pi$
$577$	−28.1691	−	28.1691i	−1.17270	−	1.17270i	−0.191132	−	0.981564i	$-0.561216\pi$
$578$	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	$0.583803\pi$
$587$	−29.6985	+	29.6985i	−1.22579	+	1.22579i	−0.965543	+	0.260245i	$0.916197\pi$
$588$		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	$0.0810186\pi$
$593$	29.6985	+	29.6985i	1.21957	+	1.21957i	0.251788	+	0.967782i	$0.418981\pi$
$594$		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	0.998539	−	0.0540328i	$-0.0172076\pi$
$607$	23.2702	+	23.2702i	0.944506	+	0.944506i	−0.0540328	+	0.998539i	$0.517208\pi$
$608$	−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	−0.858718	−	0.512449i	$-0.828738\pi$
$613$	−8.57321	+	8.57321i	−0.346269	+	0.346269i	0.512449	+	0.858718i	$0.328738\pi$
$614$	−29.4449			−1.18830
$615$		0			0
$616$		0			0
$617$	−16.9706	+	16.9706i	−0.683209	+	0.683209i	−0.960722	−	0.277513i	$-0.910490\pi$
$617$	−16.9706	+	16.9706i	−0.683209	+	0.683209i	0.277513	+	0.960722i	$0.410490\pi$
$618$		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	−0.992404	−	0.123018i	$-0.960743\pi$
$643$	−22.0454	+	22.0454i	−0.869386	+	0.869386i	0.123018	+	0.992404i	$0.460743\pi$
$644$	−10.3923			−0.409514
$645$		0			0
$646$	−6.00000			−0.236067
$647$	16.9706	−	16.9706i	0.667182	−	0.667182i	−0.289881	−	0.957063i	$-0.593616\pi$
$647$	16.9706	−	16.9706i	0.667182	−	0.667182i	0.957063	+	0.289881i	$0.0936157\pi$
$648$		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	0.853366	−	0.521312i	$-0.174557\pi$
$653$	8.48528	+	8.48528i	0.332055	+	0.332055i	−0.521312	+	0.853366i	$0.674557\pi$
$654$		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	−0.330182	−	0.943917i	$-0.607110\pi$
$673$	15.9217	−	15.9217i	0.613736	−	0.613736i	0.943917	+	0.330182i	$0.107110\pi$
$674$	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	0.621262	−	0.783603i	$-0.286620\pi$
$683$	−4.24264	−	4.24264i	−0.162340	−	0.162340i	−0.783603	+	0.621262i	$0.786620\pi$
$684$		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	0.811487	−	0.584371i	$-0.198659\pi$
$727$	6.12372	+	6.12372i	0.227116	+	0.227116i	−0.584371	+	0.811487i	$0.698659\pi$
$728$		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	−0.924362	−	0.381518i	$-0.875402\pi$
$733$	−14.6969	+	14.6969i	−0.542844	+	0.542844i	0.381518	+	0.924362i	$0.375402\pi$
$734$	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	0.355857	−	0.934540i	$-0.384189\pi$
$757$	−15.9217	−	15.9217i	−0.578683	−	0.578683i	−0.934540	+	0.355857i	$0.884189\pi$
$758$	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	0.779277	−	0.626680i	$-0.215587\pi$
$773$	4.24264	+	4.24264i	0.152597	+	0.152597i	−0.626680	+	0.779277i	$0.715587\pi$
$774$		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	0.684941	−	0.728598i	$-0.259828\pi$
$787$	−1.22474	−	1.22474i	−0.0436574	−	0.0436574i	−0.728598	+	0.684941i	$0.759828\pi$
$788$	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	−0.841235	−	0.540670i	$-0.818171\pi$
$797$	−8.48528	+	8.48528i	−0.300564	+	0.300564i	0.540670	+	0.841235i	$0.318171\pi$
$798$		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	−0.209380	−	0.977834i	$-0.567144\pi$
$823$	22.0454	−	22.0454i	0.768455	−	0.768455i	0.977834	+	0.209380i	$0.0671445\pi$
$824$	8.66025			0.301694
$825$		0			0
$826$	18.0000			0.626300
$827$	8.48528	−	8.48528i	0.295062	−	0.295062i	−0.544014	−	0.839076i	$-0.683096\pi$
$827$	8.48528	−	8.48528i	0.295062	−	0.295062i	0.839076	+	0.544014i	$0.183096\pi$
$828$		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	−0.912435	−	0.409221i	$-0.865800\pi$
$853$	−14.6969	+	14.6969i	−0.503214	+	0.503214i	0.409221	+	0.912435i	$0.365800\pi$
$854$	8.66025			0.296348
$855$		0			0
$856$	−18.0000			−0.615227
$857$	−16.9706	+	16.9706i	−0.579703	+	0.579703i	−0.934821	−	0.355118i	$-0.884441\pi$
$857$	−16.9706	+	16.9706i	−0.579703	+	0.579703i	0.355118	+	0.934821i	$0.384441\pi$
$858$		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	0.631199	−	0.775621i	$-0.282563\pi$
$863$	−4.24264	−	4.24264i	−0.144421	−	0.144421i	−0.775621	+	0.631199i	$0.782563\pi$
$864$		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	0.802899	−	0.596115i	$-0.203290\pi$
$877$	6.12372	+	6.12372i	0.206783	+	0.206783i	−0.596115	+	0.802899i	$0.703290\pi$
$878$	−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	−0.263883	−	0.964555i	$-0.585003\pi$
$883$	20.8207	−	20.8207i	0.700671	−	0.700671i	0.964555	+	0.263883i	$0.0850034\pi$
$884$		0			0
$885$		0			0
$886$	6.00000			0.201574
$887$	8.48528	−	8.48528i	0.284908	−	0.284908i	−0.550155	−	0.835063i	$-0.685431\pi$
$887$	8.48528	−	8.48528i	0.284908	−	0.284908i	0.835063	+	0.550155i	$0.185431\pi$
$888$		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	0.598092	−	0.801427i	$-0.295926\pi$
$907$	−6.12372	−	6.12372i	−0.203335	−	0.203335i	−0.801427	+	0.598092i	$0.795926\pi$
$908$	−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	$-0.999915\pi$
$937$	−30.6186	−	30.6186i	−1.00027	−	1.00027i	−0.000266809	−	1.00000i	$-0.500085\pi$
$938$	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	−0.999410	−	0.0343392i	$-0.989067\pi$
$947$	−29.6985	+	29.6985i	−0.965071	+	0.965071i	0.0343392	+	0.999410i	$0.489067\pi$
$948$		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	0.162168	−	0.986763i	$-0.448151\pi$
$953$	−25.4558	−	25.4558i	−0.824596	−	0.824596i	−0.986763	+	0.162168i	$0.948151\pi$
$954$		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	0.915141	−	0.403134i	$-0.132079\pi$
$967$	15.9217	+	15.9217i	0.512007	+	0.512007i	−0.403134	+	0.915141i	$0.632079\pi$
$968$	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	−0.880762	−	0.473560i	$-0.842969\pi$
$977$	−12.7279	+	12.7279i	−0.407202	+	0.407202i	0.473560	+	0.880762i	$0.342969\pi$
$978$		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	$0.156153\pi$
$983$	42.4264	+	42.4264i	1.35319	+	1.35319i	0.471127	+	0.882066i	$0.343847\pi$
$984$		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.107.1	✓	8
3.2	odd	2	inner	1350.2.f.d.107.3	yes	8
5.2	odd	4	inner	1350.2.f.d.593.2	yes	8
5.3	odd	4	inner	1350.2.f.d.593.3	yes	8
5.4	even	2	inner	1350.2.f.d.107.4	yes	8
15.2	even	4	inner	1350.2.f.d.593.4	yes	8
15.8	even	4	inner	1350.2.f.d.593.1	yes	8
15.14	odd	2	inner	1350.2.f.d.107.2	yes	8

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

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

Introduction

L-functions

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