LMFDB - Embedded newform 280.2.s.b.13.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [280,2,Mod(13,280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(280, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 3, 2]))

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

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

chi := DirichletCharacter("280.13");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 280 = 2^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	280.s (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:	$2.23581125660$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.40282095616.8
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 8x^{4} - 36x^{2} + 81 $ x^8 - 4x^6 + 8x^4 - 36*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			13.1
Root			$-1.03179 + 1.39119i$ of defining polynomial
Character	$\chi$	$=$	280.13
Dual form			280.2.s.b.237.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.00000i) q^{2} +(-2.42298 - 2.42298i) q^{3} -2.00000i q^{4} +(1.75060 - 1.39119i) q^{5} -4.84596 q^{6} +(1.87083 - 1.87083i) q^{7} +(-2.00000 - 2.00000i) q^{8} +8.74166i q^{9} +O(q^{10})$	\(q+(1.00000 - 1.00000i) q^{2} +(-2.42298 - 2.42298i) q^{3} -2.00000i q^{4} +(1.75060 - 1.39119i) q^{5} -4.84596 q^{6} +(1.87083 - 1.87083i) q^{7} +(-2.00000 - 2.00000i) q^{8} +8.74166i q^{9} +(0.359404 - 3.14179i) q^{10} +(-4.84596 + 4.84596i) q^{12} +(2.42298 + 2.42298i) q^{13} -3.74166i q^{14} +(-7.61249 - 0.870829i) q^{15} -4.00000 q^{16} +(8.74166 + 8.74166i) q^{18} +4.22000i q^{19} +(-2.78238 - 3.50119i) q^{20} -9.06596 q^{21} +(0.741657 + 0.741657i) q^{23} +9.69192i q^{24} +(1.12917 - 4.87083i) q^{25} +4.84596 q^{26} +(13.9119 - 13.9119i) q^{27} +(-3.74166 - 3.74166i) q^{28} +(-8.48331 + 6.74166i) q^{30} +(-4.00000 + 4.00000i) q^{32} +(0.672384 - 5.87775i) q^{35} +17.4833 q^{36} +(4.22000 + 4.22000i) q^{38} -11.7417i q^{39} +(-6.28357 - 0.718808i) q^{40} +(-9.06596 + 9.06596i) q^{42} +(12.1613 + 15.3031i) q^{45} +1.48331 q^{46} +(9.69192 + 9.69192i) q^{48} -7.00000i q^{49} +(-3.74166 - 6.00000i) q^{50} +(4.84596 - 4.84596i) q^{52} -27.8238i q^{54} -7.48331 q^{56} +(10.2250 - 10.2250i) q^{57} -0.625959i q^{59} +(-1.74166 + 15.2250i) q^{60} +5.47192 q^{61} +(16.3541 + 16.3541i) q^{63} +8.00000i q^{64} +(7.61249 + 0.870829i) q^{65} -3.59404i q^{69} +(-5.20536 - 6.55013i) q^{70} -7.22497 q^{71} +(17.4833 - 17.4833i) q^{72} +(-14.5379 + 9.06596i) q^{75} +8.44000 q^{76} +(-11.7417 - 11.7417i) q^{78} +15.7417i q^{79} +(-7.00238 + 5.56477i) q^{80} -41.1916 q^{81} +(-11.4889 - 11.4889i) q^{83} +18.1319i q^{84} +(27.4644 + 3.14179i) q^{90} +9.06596 q^{91} +(1.48331 - 1.48331i) q^{92} +(5.87083 + 7.38751i) q^{95} +19.3838 q^{96} +(-7.00000 - 7.00000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} - 16 q^{8}+O(q^{10}) $ 8 * q + 8 * q^2 - 16 * q^8	$ 8 q + 8 q^{2} - 16 q^{8} - 16 q^{15} - 32 q^{16} + 40 q^{18} - 24 q^{23} + 24 q^{25} - 8 q^{30} - 32 q^{32} + 80 q^{36} - 48 q^{46} - 8 q^{57} + 16 q^{60} + 56 q^{63} + 16 q^{65} + 32 q^{71} + 80 q^{72} - 64 q^{78} - 120 q^{81} - 48 q^{92} + 32 q^{95} - 56 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 - 16 * q^8 - 16 * q^15 - 32 * q^16 + 40 * q^18 - 24 * q^23 + 24 * q^25 - 8 * q^30 - 32 * q^32 + 80 * q^36 - 48 * q^46 - 8 * q^57 + 16 * q^60 + 56 * q^63 + 16 * q^65 + 32 * q^71 + 80 * q^72 - 64 * q^78 - 120 * q^81 - 48 * q^92 + 32 * q^95 - 56 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/280\mathbb{Z}\right)^\times$.

$n$	$57$	$71$	$141$	$241$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$	1.00000	−	1.00000i	0.707107	−	0.707107i
$2$	1.00000	−	1.00000i	0.707107	−	0.707107i
$3$	−2.42298	−	2.42298i	−1.39891	−	1.39891i	−0.803205	−	0.595703i	$-0.796874\pi$
$3$	−2.42298	−	2.42298i	−1.39891	−	1.39891i	−0.595703	−	0.803205i	$-0.703126\pi$
$4$		−	2.00000i		−	1.00000i
$5$	1.75060	−	1.39119i	0.782890	−	0.622160i
$5$	1.75060	−	1.39119i	0.782890	−	0.622160i
$6$	−4.84596			−1.97835
$7$	1.87083	−	1.87083i	0.707107	−	0.707107i
$7$	1.87083	−	1.87083i	0.707107	−	0.707107i
$8$	−2.00000	−	2.00000i	−0.707107	−	0.707107i
$9$			8.74166i			2.91389i
$10$	0.359404	−	3.14179i	0.113654	−	0.993520i
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$	−4.84596	+	4.84596i	−1.39891	+	1.39891i
$13$	2.42298	+	2.42298i	0.672014	+	0.672014i	0.958180	−	0.286166i	$-0.0923810\pi$
$13$	2.42298	+	2.42298i	0.672014	+	0.672014i	−0.286166	+	0.958180i	$0.592381\pi$
$14$		−	3.74166i		−	1.00000i
$15$	−7.61249	−	0.870829i	−1.96554	−	0.224847i
$16$	−4.00000			−1.00000
$17$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$17$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$18$	8.74166	+	8.74166i	2.06043	+	2.06043i
$19$			4.22000i			0.968134i	0.875031	+	0.484067i	$0.160841\pi$
$19$			4.22000i			0.968134i	−0.875031	+	0.484067i	$0.839159\pi$
$20$	−2.78238	−	3.50119i	−0.622160	−	0.782890i
$21$	−9.06596			−1.97835
$22$		0			0
$23$	0.741657	+	0.741657i	0.154646	+	0.154646i	0.780189	−	0.625543i	$-0.215123\pi$
$23$	0.741657	+	0.741657i	0.154646	+	0.154646i	−0.625543	+	0.780189i	$0.715123\pi$
$24$			9.69192i			1.97835i
$25$	1.12917	−	4.87083i	0.225834	−	0.974166i
$26$	4.84596			0.950371
$27$	13.9119	−	13.9119i	2.67735	−	2.67735i
$28$	−3.74166	−	3.74166i	−0.707107	−	0.707107i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$	−8.48331	+	6.74166i	−1.54883	+	1.23085i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−4.00000	+	4.00000i	−0.707107	+	0.707107i
$33$		0			0
$34$		0			0
$35$	0.672384	−	5.87775i	0.113654	−	0.993520i
$36$	17.4833			2.91389
$37$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$37$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$38$	4.22000	+	4.22000i	0.684574	+	0.684574i
$39$		−	11.7417i		−	1.88017i
$40$	−6.28357	−	0.718808i	−0.993520	−	0.113654i
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$	−9.06596	+	9.06596i	−1.39891	+	1.39891i
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$	12.1613	+	15.3031i	1.81290	+	2.28125i
$46$	1.48331			0.218703
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$	9.69192	+	9.69192i	1.39891	+	1.39891i
$49$		−	7.00000i		−	1.00000i
$50$	−3.74166	−	6.00000i	−0.529150	−	0.848528i
$51$		0			0
$52$	4.84596	−	4.84596i	0.672014	−	0.672014i
$53$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$53$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$54$		−	27.8238i		−	3.78634i
$55$		0			0
$56$	−7.48331			−1.00000
$57$	10.2250	−	10.2250i	1.35433	−	1.35433i
$58$		0			0
$59$		−	0.625959i		−	0.0814929i	−0.999170	−	0.0407464i	$-0.987026\pi$
$59$		−	0.625959i		−	0.0814929i	0.999170	−	0.0407464i	$-0.0129736\pi$
$60$	−1.74166	+	15.2250i	−0.224847	+	1.96554i
$61$	5.47192			0.700607			0.350304	−	0.936636i	$-0.386078\pi$
$61$	5.47192			0.700607			0.350304	+	0.936636i	$0.386078\pi$
$62$		0			0
$63$	16.3541	+	16.3541i	2.06043	+	2.06043i
$64$			8.00000i			1.00000i
$65$	7.61249	+	0.870829i	0.944213	+	0.108013i
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$		0			0
$69$		−	3.59404i		−	0.432672i
$70$	−5.20536	−	6.55013i	−0.622160	−	0.782890i
$71$	−7.22497			−0.857446			−0.428723	−	0.903436i	$-0.641036\pi$
$71$	−7.22497			−0.857446			−0.428723	+	0.903436i	$0.641036\pi$
$72$	17.4833	−	17.4833i	2.06043	−	2.06043i
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$		0			0
$75$	−14.5379	+	9.06596i	−1.67869	+	1.04685i
$76$	8.44000			0.968134
$77$		0			0
$78$	−11.7417	−	11.7417i	−1.32948	−	1.32948i
$79$			15.7417i			1.77107i	0.464568	+	0.885537i	$0.346210\pi$
$79$			15.7417i			1.77107i	−0.464568	+	0.885537i	$0.653790\pi$
$80$	−7.00238	+	5.56477i	−0.782890	+	0.622160i
$81$	−41.1916			−4.57684
$82$		0			0
$83$	−11.4889	−	11.4889i	−1.26107	−	1.26107i	−0.950573	−	0.310502i	$-0.899503\pi$
$83$	−11.4889	−	11.4889i	−1.26107	−	1.26107i	−0.310502	−	0.950573i	$-0.600497\pi$
$84$			18.1319i			1.97835i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	27.4644	+	3.14179i	2.89501	+	0.331173i
$91$	9.06596			0.950371
$92$	1.48331	−	1.48331i	0.154646	−	0.154646i
$93$		0			0
$94$		0			0
$95$	5.87083	+	7.38751i	0.602334	+	0.757943i
$96$	19.3838			1.97835
$97$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$97$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$98$	−7.00000	−	7.00000i	−0.707107	−	0.707107i
$99$		0			0
$100$	−9.74166	−	2.25834i	−0.974166	−	0.225834i
$101$	−15.1638			−1.50886			−0.754429	−	0.656382i	$-0.772086\pi$
$101$	−15.1638			−1.50886			−0.754429	+	0.656382i	$0.772086\pi$
$102$		0			0
$103$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$103$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$104$		−	9.69192i		−	0.950371i
$105$	−15.8708	+	12.6125i	−1.54883	+	1.23085i
$106$		0			0
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$	−27.8238	−	27.8238i	−2.67735	−	2.67735i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$	−7.48331	+	7.48331i	−0.707107	+	0.707107i
$113$	11.2250	+	11.2250i	1.05596	+	1.05596i	0.998339	+	0.0576178i	$0.0183505\pi$
$113$	11.2250	+	11.2250i	1.05596	+	1.05596i	0.0576178	+	0.998339i	$0.481650\pi$
$114$		−	20.4499i		−	1.91531i
$115$	2.33013	+	0.266555i	0.217286	+	0.0248564i
$116$		0			0
$117$	−21.1809	+	21.1809i	−1.95817	+	1.95817i
$118$	−0.625959	−	0.625959i	−0.0576242	−	0.0576242i
$119$		0			0
$120$	13.4833	+	16.9666i	1.23085	+	1.54883i
$121$	11.0000			1.00000
$122$	5.47192	−	5.47192i	0.495404	−	0.495404i
$123$		0			0
$124$		0			0
$125$	−4.79953	−	10.0977i	−0.429283	−	0.903170i
$126$	32.7083			2.91389
$127$	−12.2250	+	12.2250i	−1.08479	+	1.08479i	−0.0887357	+	0.996055i	$0.528283\pi$
$127$	−12.2250	+	12.2250i	−1.08479	+	1.08479i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$	8.00000	+	8.00000i	0.707107	+	0.707107i
$129$		0			0
$130$	8.48331	−	6.74166i	0.744036	−	0.591282i
$131$	22.3519			1.95290			0.976448	−	0.215753i	$-0.0692205\pi$
$131$	22.3519			1.95290			0.976448	+	0.215753i	$0.0692205\pi$
$132$		0			0
$133$	7.89490	+	7.89490i	0.684574	+	0.684574i
$134$		0			0
$135$	5.00000	−	43.7083i	0.430331	−	3.76181i
$136$		0			0
$137$	16.4833	−	16.4833i	1.40826	−	1.40826i	0.639343	−	0.768922i	$-0.279207\pi$
$137$	16.4833	−	16.4833i	1.40826	−	1.40826i	0.768922	−	0.639343i	$-0.220793\pi$
$138$	−3.59404	−	3.59404i	−0.305945	−	0.305945i
$139$			20.0098i			1.69721i	0.529028	+	0.848604i	$0.322557\pi$
$139$			20.0098i			1.69721i	−0.529028	+	0.848604i	$0.677443\pi$
$140$	−11.7555	−	1.34477i	−0.993520	−	0.113654i
$141$		0			0
$142$	−7.22497	+	7.22497i	−0.606306	+	0.606306i
$143$		0			0
$144$		−	34.9666i		−	2.91389i
$145$		0			0
$146$		0			0
$147$	−16.9609	+	16.9609i	−1.39891	+	1.39891i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$	−5.47192	+	23.6038i	−0.446780	+	1.92724i
$151$	22.4499			1.82695			0.913475	−	0.406894i	$-0.133388\pi$
$151$	22.4499			1.82695			0.913475	+	0.406894i	$0.133388\pi$
$152$	8.44000	−	8.44000i	0.684574	−	0.684574i
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−23.4833			−1.88017
$157$	−6.64298	+	6.64298i	−0.530167	+	0.530167i	−0.920622	−	0.390455i	$-0.872318\pi$
$157$	−6.64298	+	6.64298i	−0.530167	+	0.530167i	0.390455	+	0.920622i	$0.372318\pi$
$158$	15.7417	+	15.7417i	1.25234	+	1.25234i
$159$		0			0
$160$	−1.43762	+	12.5671i	−0.113654	+	0.993520i
$161$	2.77503			0.218703
$162$	−41.1916	+	41.1916i	−3.23632	+	3.23632i
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$		0			0
$165$		0			0
$166$	−22.9779			−1.78343
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$	18.1319	+	18.1319i	1.39891	+	1.39891i
$169$		−	1.25834i		−	0.0967956i
$170$		0			0
$171$	−36.8898			−2.82103
$172$		0			0
$173$	−12.7409	−	12.7409i	−0.968669	−	0.968669i	0.0308546	−	0.999524i	$-0.490177\pi$
$173$	−12.7409	−	12.7409i	−0.968669	−	0.968669i	−0.999524	+	0.0308546i	$0.990177\pi$
$174$		0			0
$175$	−7.00000	−	11.2250i	−0.529150	−	0.848528i
$176$		0			0
$177$	−1.51669	+	1.51669i	−0.114001	+	0.114001i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	30.6062	−	24.3226i	2.28125	−	1.81290i
$181$	18.7579			1.39426			0.697131	−	0.716944i	$-0.254460\pi$
$181$	18.7579			1.39426			0.697131	+	0.716944i	$0.254460\pi$
$182$	9.06596	−	9.06596i	0.672014	−	0.672014i
$183$	−13.2583	−	13.2583i	−0.980085	−	0.980085i
$184$		−	2.96663i		−	0.218703i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		−	52.0536i		−	3.78634i
$190$	13.2583	+	1.51669i	0.961861	+	0.110032i
$191$	−27.2250			−1.96993			−0.984965	−	0.172754i	$-0.944733\pi$
$191$	−27.2250			−1.96993			−0.984965	+	0.172754i	$0.944733\pi$
$192$	19.3838	−	19.3838i	1.39891	−	1.39891i
$193$	6.00000	+	6.00000i	0.431889	+	0.431889i	0.889271	−	0.457381i	$-0.151213\pi$
$193$	6.00000	+	6.00000i	0.431889	+	0.431889i	−0.457381	+	0.889271i	$0.651213\pi$
$194$		0			0
$195$	−16.3349	−	20.5549i	−1.16977	−	1.47197i
$196$	−14.0000			−1.00000
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	−12.0000	+	7.48331i	−0.848528	+	0.529150i
$201$		0			0
$202$	−15.1638	+	15.1638i	−1.06692	+	1.06692i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−6.48331	+	6.48331i	−0.450622	+	0.450622i
$208$	−9.69192	−	9.69192i	−0.672014	−	0.672014i
$209$		0			0
$210$	−3.25834	+	28.4833i	−0.224847	+	1.96554i
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$	17.5060	+	17.5060i	1.19949	+	1.19949i
$214$		0			0
$215$		0			0
$216$	−55.6477			−3.78634
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$223$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$224$			14.9666i			1.00000i
$225$	42.5791	+	9.87083i	2.83861	+	0.658055i
$226$	22.4499			1.49335
$227$	3.04894	−	3.04894i	0.202365	−	0.202365i	−0.598647	−	0.801013i	$-0.704295\pi$
$227$	3.04894	−	3.04894i	0.202365	−	0.202365i	0.801013	+	0.598647i	$0.204295\pi$
$228$	−20.4499	−	20.4499i	−1.35433	−	1.35433i
$229$			23.6038i			1.55979i	0.625913	+	0.779893i	$0.284726\pi$
$229$			23.6038i			1.55979i	−0.625913	+	0.779893i	$0.715274\pi$
$230$	2.59668	−	2.06358i	0.171220	−	0.136068i
$231$		0			0
$232$		0			0
$233$	−17.9666	−	17.9666i	−1.17703	−	1.17703i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	−17.9666	−	17.9666i	−1.17703	−	1.17703i	−0.196537	−	0.980497i	$-0.562969\pi$
$234$			42.3617i			2.76927i
$235$		0			0
$236$	−1.25192			−0.0814929
$237$	38.1417	−	38.1417i	2.47757	−	2.47757i
$238$		0			0
$239$		−	7.48331i		−	0.484055i	−0.970269	−	0.242028i	$-0.922188\pi$
$239$		−	7.48331i		−	0.484055i	0.970269	−	0.242028i	$-0.0778125\pi$
$240$	30.4499	+	3.48331i	1.96554	+	0.224847i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	11.0000	−	11.0000i	0.707107	−	0.707107i
$243$	58.0706	+	58.0706i	3.72523	+	3.72523i
$244$		−	10.9438i		−	0.700607i
$245$	−9.73834	−	12.2542i	−0.622160	−	0.782890i
$246$		0			0
$247$	−10.2250	+	10.2250i	−0.650599	+	0.650599i
$248$		0			0
$249$			55.6749i			3.52825i
$250$	−14.8973	−	5.29821i	−0.942187	−	0.335088i
$251$	29.7017			1.87476			0.937378	−	0.348315i	$-0.113246\pi$
$251$	29.7017			1.87476			0.937378	+	0.348315i	$0.113246\pi$
$252$	32.7083	−	32.7083i	2.06043	−	2.06043i
$253$		0			0
$254$			24.4499i			1.53413i
$255$		0			0
$256$	16.0000			1.00000
$257$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$257$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$258$		0			0
$259$		0			0
$260$	1.74166	−	15.2250i	0.108013	−	0.944213i
$261$		0			0
$262$	22.3519	−	22.3519i	1.38091	−	1.38091i
$263$	11.2250	+	11.2250i	0.692161	+	0.692161i	0.962707	−	0.270546i	$-0.0872041\pi$
$263$	11.2250	+	11.2250i	0.692161	+	0.692161i	−0.270546	+	0.962707i	$0.587204\pi$
$264$		0			0
$265$		0			0
$266$	15.7898			0.968134
$267$		0			0
$268$		0			0
$269$			7.81404i			0.476430i	0.971212	+	0.238215i	$0.0765624\pi$
$269$			7.81404i			0.476430i	−0.971212	+	0.238215i	$0.923438\pi$
$270$	−38.7083	−	48.7083i	−2.35571	−	2.96429i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$	−21.9666	−	21.9666i	−1.32948	−	1.32948i
$274$		−	32.9666i		−	1.99159i
$275$		0			0
$276$	−7.18808			−0.432672
$277$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$277$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$278$	20.0098	+	20.0098i	1.20011	+	1.20011i
$279$		0			0
$280$	−13.1003	+	10.4107i	−0.782890	+	0.622160i
$281$	−14.9666			−0.892834			−0.446417	−	0.894825i	$-0.647300\pi$
$281$	−14.9666			−0.892834			−0.446417	+	0.894825i	$0.647300\pi$
$282$		0			0
$283$	−15.0830	−	15.0830i	−0.896590	−	0.896590i	0.0985428	−	0.995133i	$-0.468582\pi$
$283$	−15.0830	−	15.0830i	−0.896590	−	0.896590i	−0.995133	+	0.0985428i	$0.968582\pi$
$284$			14.4499i			0.857446i
$285$	3.67490	−	32.1247i	0.217682	−	1.90290i
$286$		0			0
$287$		0			0
$288$	−34.9666	−	34.9666i	−2.06043	−	2.06043i
$289$		−	17.0000i		−	1.00000i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−16.3349	−	16.3349i	−0.954295	−	0.954295i	0.0447054	−	0.999000i	$-0.485765\pi$
$293$	−16.3349	−	16.3349i	−0.954295	−	0.954295i	−0.999000	+	0.0447054i	$0.985765\pi$
$294$			33.9217i			1.97835i
$295$	−0.870829	−	1.09580i	−0.0507016	−	0.0638000i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			3.59404i			0.207849i
$300$	18.1319	+	29.0758i	1.04685	+	1.67869i
$301$		0			0
$302$	22.4499	−	22.4499i	1.29185	−	1.29185i
$303$	36.7417	+	36.7417i	2.11075	+	2.11075i
$304$		−	16.8800i		−	0.968134i
$305$	9.57912	−	7.61249i	0.548499	−	0.435890i
$306$		0			0
$307$	−24.7749	+	24.7749i	−1.41398	+	1.41398i	−0.694083	+	0.719895i	$0.744190\pi$
$307$	−24.7749	+	24.7749i	−1.41398	+	1.41398i	−0.719895	+	0.694083i	$0.755810\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$	−23.4833	+	23.4833i	−1.32948	+	1.32948i
$313$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$313$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$314$			13.2860i			0.749770i
$315$	51.3812	+	5.87775i	2.89501	+	0.331173i
$316$	31.4833			1.77107
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$		0			0
$320$	11.1295	+	14.0048i	0.622160	+	0.782890i
$321$		0			0
$322$	2.77503	−	2.77503i	0.154646	−	0.154646i
$323$		0			0
$324$			82.3832i			4.57684i
$325$	14.5379	−	9.06596i	0.806416	−	0.502889i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$	−22.9779	+	22.9779i	−1.26107	+	1.26107i
$333$		0			0
$334$		0			0
$335$		0			0
$336$	36.2638			1.97835
$337$	−18.0000	+	18.0000i	−0.980522	+	0.980522i	−0.999814	−	0.0192914i	$-0.993859\pi$
$337$	−18.0000	+	18.0000i	−0.980522	+	0.980522i	0.0192914	+	0.999814i	$0.493859\pi$
$338$	−1.25834	−	1.25834i	−0.0684448	−	0.0684448i
$339$		−	54.3958i		−	2.95437i
$340$		0			0
$341$		0			0
$342$	−36.8898	+	36.8898i	−1.99477	+	1.99477i
$343$	−13.0958	−	13.0958i	−0.707107	−	0.707107i
$344$		0			0
$345$	−5.00000	−	6.29171i	−0.269191	−	0.338734i
$346$	−25.4817			−1.36991
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$			35.6379i			1.90765i	0.300360	+	0.953826i	$0.402893\pi$
$349$			35.6379i			1.90765i	−0.300360	+	0.953826i	$0.597107\pi$
$350$	−18.2250	−	4.22497i	−0.974166	−	0.225834i
$351$	67.4166			3.59843
$352$		0			0
$353$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$353$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$354$			3.03337i			0.161222i
$355$	−12.6480	+	10.0513i	−0.671286	+	0.533469i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			6.00000i			0.316668i	0.987386	+	0.158334i	$0.0506123\pi$
$359$			6.00000i			0.316668i	−0.987386	+	0.158334i	$0.949388\pi$
$360$	6.28357	−	54.9289i	0.331173	−	2.89501i
$361$	1.19160			0.0627159
$362$	18.7579	−	18.7579i	0.985892	−	0.985892i
$363$	−26.6528	−	26.6528i	−1.39891	−	1.39891i
$364$		−	18.1319i		−	0.950371i
$365$		0			0
$366$	−26.5167			−1.38605
$367$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$367$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$368$	−2.96663	−	2.96663i	−0.154646	−	0.154646i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$373$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$374$		0			0
$375$	−12.8375	+	36.0958i	−0.662923	+	1.86398i
$376$		0			0
$377$		0			0
$378$	−52.0536	−	52.0536i	−2.67735	−	2.67735i
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$	14.7750	−	11.7417i	0.757943	−	0.602334i
$381$	59.2417			3.03504
$382$	−27.2250	+	27.2250i	−1.39295	+	1.39295i
$383$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$		−	38.7677i		−	1.97835i
$385$		0			0
$386$	12.0000			0.610784
$387$		0			0
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$	−36.8898	−	4.22000i	−1.86799	−	0.213688i
$391$		0			0
$392$	−14.0000	+	14.0000i	−0.707107	+	0.707107i
$393$	−54.1582	−	54.1582i	−2.73192	−	2.73192i
$394$		0			0
$395$	21.8997	+	27.5573i	1.10189	+	1.38656i
$396$		0			0
$397$	21.1809	−	21.1809i	1.06304	−	1.06304i	0.0651619	−	0.997875i	$-0.479244\pi$
$397$	21.1809	−	21.1809i	1.06304	−	1.06304i	0.997875	−	0.0651619i	$-0.0207564\pi$
$398$		0			0
$399$		−	38.2583i		−	1.91531i
$400$	−4.51669	+	19.4833i	−0.225834	+	0.974166i
$401$	−37.2250			−1.85893			−0.929463	−	0.368915i	$-0.879729\pi$
$401$	−37.2250			−1.85893			−0.929463	+	0.368915i	$0.879729\pi$
$402$		0			0
$403$		0			0
$404$			30.3277i			1.50886i
$405$	−72.1098	+	57.3054i	−3.58317	+	2.84753i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$	−79.8775			−3.94006
$412$		0			0
$413$	−1.17106	−	1.17106i	−0.0576242	−	0.0576242i
$414$			12.9666i			0.637275i
$415$	−36.0958	−	4.12917i	−1.77187	−	0.202693i
$416$	−19.3838			−0.950371
$417$	48.4833	−	48.4833i	2.37424	−	2.37424i
$418$		0			0
$419$		−	2.96808i		−	0.145000i	−0.997368	−	0.0725002i	$-0.976902\pi$
$419$		−	2.96808i		−	0.145000i	0.997368	−	0.0725002i	$-0.0230978\pi$
$420$	25.2250	+	31.7417i	1.23085	+	1.54883i
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$	35.0119			1.69633
$427$	10.2370	−	10.2370i	0.495404	−	0.495404i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−18.0000			−0.867029			−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000			−0.867029			−0.433515	+	0.901146i	$0.642727\pi$
$432$	−55.6477	+	55.6477i	−2.67735	+	2.67735i
$433$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$433$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3.12979	+	3.12979i	−0.149718	+	0.149718i
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	61.1916			2.91389
$442$		0			0
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	14.9666	+	14.9666i	0.707107	+	0.707107i
$449$			29.9333i			1.41264i	0.707894	+	0.706319i	$0.249646\pi$
$449$			29.9333i			1.41264i	−0.707894	+	0.706319i	$0.750354\pi$
$450$	52.4499	−	32.7083i	2.47251	−	1.54188i
$451$		0			0
$452$	22.4499	−	22.4499i	1.05596	−	1.05596i
$453$	−54.3958	−	54.3958i	−2.55574	−	2.55574i
$454$		−	6.09788i		−	0.286188i
$455$	15.8708	−	12.6125i	0.744036	−	0.591282i
$456$	−40.8999			−1.91531
$457$	3.74166	−	3.74166i	0.175027	−	0.175027i	−0.614157	−	0.789184i	$-0.710504\pi$
$457$	3.74166	−	3.74166i	0.175027	−	0.175027i	0.789184	+	0.614157i	$0.210504\pi$
$458$	23.6038	+	23.6038i	1.10293	+	1.10293i
$459$		0			0
$460$	0.533109	−	4.66026i	0.0248564	−	0.217286i
$461$	−36.8898			−1.71813			−0.859064	−	0.511867i	$-0.828954\pi$
$461$	−36.8898			−1.71813			−0.859064	+	0.511867i	$0.828954\pi$
$462$		0			0
$463$	−24.0000	−	24.0000i	−1.11537	−	1.11537i	−0.992411	−	0.122963i	$-0.960760\pi$
$463$	−24.0000	−	24.0000i	−1.11537	−	1.11537i	−0.122963	−	0.992411i	$-0.539240\pi$
$464$		0			0
$465$		0			0
$466$	−35.9333			−1.66458
$467$	3.67490	−	3.67490i	0.170054	−	0.170054i	−0.616949	−	0.787003i	$-0.711632\pi$
$467$	3.67490	−	3.67490i	0.170054	−	0.170054i	0.787003	+	0.616949i	$0.211632\pi$
$468$	42.3617	+	42.3617i	1.95817	+	1.95817i
$469$		0			0
$470$		0			0
$471$	32.1916			1.48331
$472$	−1.25192	+	1.25192i	−0.0576242	+	0.0576242i
$473$		0			0
$474$		−	76.2834i		−	3.50381i
$475$	20.5549	+	4.76510i	0.943123	+	0.218638i
$476$		0			0
$477$		0			0
$478$	−7.48331	−	7.48331i	−0.342279	−	0.342279i
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$	33.9333	−	26.9666i	1.54883	−	1.23085i
$481$		0			0
$482$		0			0
$483$	−6.72384	−	6.72384i	−0.305945	−	0.305945i
$484$		−	22.0000i		−	1.00000i
$485$		0			0
$486$	116.141			5.26828
$487$	7.77503	−	7.77503i	0.352320	−	0.352320i	−0.508652	−	0.860972i	$-0.669856\pi$
$487$	7.77503	−	7.77503i	0.352320	−	0.352320i	0.860972	+	0.508652i	$0.169856\pi$
$488$	−10.9438	−	10.9438i	−0.495404	−	0.495404i
$489$		0			0
$490$	−21.9925	−	2.51583i	−0.993520	−	0.113654i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$			20.4499i			0.920086i
$495$		0			0
$496$		0			0
$497$	−13.5167	+	13.5167i	−0.606306	+	0.606306i
$498$	55.6749	+	55.6749i	2.49485	+	2.49485i
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	−20.1955	+	9.59907i	−0.903170	+	0.429283i
$501$		0			0
$502$	29.7017	−	29.7017i	1.32565	−	1.32565i
$503$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$503$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$504$		−	65.4166i		−	2.91389i
$505$	−26.5457	+	21.0958i	−1.18127	+	0.938751i
$506$		0			0
$507$	−3.04894	+	3.04894i	−0.135408	+	0.135408i
$508$	24.4499	+	24.4499i	1.08479	+	1.08479i
$509$			44.2396i			1.96089i	0.196805	+	0.980443i	$0.436943\pi$
$509$			44.2396i			1.96089i	−0.196805	+	0.980443i	$0.563057\pi$
$510$		0			0
$511$		0			0
$512$	16.0000	−	16.0000i	0.707107	−	0.707107i
$513$	58.7083	+	58.7083i	2.59203	+	2.59203i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$			61.7417i			2.71016i
$520$	−13.4833	−	16.9666i	−0.591282	−	0.744036i
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−6.01702	−	6.01702i	−0.263106	−	0.263106i	0.563209	−	0.826315i	$-0.309567\pi$
$523$	−6.01702	−	6.01702i	−0.263106	−	0.263106i	−0.826315	+	0.563209i	$0.809567\pi$
$524$		−	44.7038i		−	1.95290i
$525$	−10.2370	+	44.1587i	−0.446780	+	1.92724i
$526$	22.4499			0.978864
$527$		0			0
$528$		0			0
$529$		−	21.8999i		−	0.952169i
$530$		0			0
$531$	5.47192			0.237461
$532$	15.7898	−	15.7898i	0.684574	−	0.684574i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	7.81404	+	7.81404i	0.336887	+	0.336887i
$539$		0			0
$540$	−87.4166	−	10.0000i	−3.76181	−	0.430331i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	−45.4499	−	45.4499i	−1.95044	−	1.95044i
$544$		0			0
$545$		0			0
$546$	−43.9333			−1.88017
$547$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$547$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$548$	−32.9666	−	32.9666i	−1.40826	−	1.40826i
$549$			47.8336i			2.04149i
$550$		0			0
$551$		0			0
$552$	−7.18808	+	7.18808i	−0.305945	+	0.305945i
$553$	29.4499	+	29.4499i	1.25234	+	1.25234i
$554$		0			0
$555$		0			0
$556$	40.0196			1.69721
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$		0			0
$559$		0			0
$560$	−2.68953	+	23.5110i	−0.113654	+	0.993520i
$561$		0			0
$562$	−14.9666	+	14.9666i	−0.631329	+	0.631329i
$563$	18.2128	+	18.2128i	0.767577	+	0.767577i	0.977679	−	0.210103i	$-0.0673798\pi$
$563$	18.2128	+	18.2128i	0.767577	+	0.767577i	−0.210103	+	0.977679i	$0.567380\pi$
$564$		0			0
$565$	35.2665	+	4.03430i	1.48367	+	0.169724i
$566$	−30.1660			−1.26797
$567$	−77.0624	+	77.0624i	−3.23632	+	3.23632i
$568$	14.4499	+	14.4499i	0.606306	+	0.606306i
$569$		−	31.6749i		−	1.32788i	−0.747785	−	0.663941i	$-0.768883\pi$
$569$		−	31.6749i		−	1.32788i	0.747785	−	0.663941i	$-0.231117\pi$
$570$	−28.4498	−	35.7996i	−1.19163	−	1.49948i
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$	65.9655	+	65.9655i	2.75575	+	2.75575i
$574$		0			0
$575$	4.44994	−	2.77503i	0.185576	−	0.115727i
$576$	−69.9333			−2.91389
$577$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$577$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$578$	−17.0000	−	17.0000i	−0.707107	−	0.707107i
$579$		−	29.0758i		−	1.20835i
$580$		0			0
$581$	−42.9877			−1.78343
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−7.61249	+	66.5457i	−0.314738	+	2.75133i
$586$	−32.6698			−1.34958
$587$	10.8630	−	10.8630i	0.448363	−	0.448363i	−0.446447	−	0.894810i	$-0.647311\pi$
$587$	10.8630	−	10.8630i	0.448363	−	0.448363i	0.894810	+	0.446447i	$0.147311\pi$
$588$	33.9217	+	33.9217i	1.39891	+	1.39891i
$589$		0			0
$590$	−1.96663	−	0.224972i	−0.0809649	−	0.00926196i
$591$		0			0
$592$		0			0
$593$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$593$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	3.59404	+	3.59404i	0.146971	+	0.146971i
$599$		−	41.6749i		−	1.70279i	−0.524524	−	0.851395i	$-0.675757\pi$
$599$		−	41.6749i		−	1.70279i	0.524524	−	0.851395i	$-0.324243\pi$
$600$	47.2077	+	10.9438i	1.92724	+	0.446780i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		−	44.8999i		−	1.82695i
$605$	19.2566	−	15.3031i	0.782890	−	0.622160i
$606$	73.4833			2.98506
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$	−16.8800	−	16.8800i	−0.684574	−	0.684574i
$609$		0			0
$610$	1.96663	−	17.1916i	0.0796265	−	0.696068i
$611$		0			0
$612$		0			0
$613$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$613$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$614$			49.5498i			1.99967i
$615$		0			0
$616$		0			0
$617$	−33.6749	+	33.6749i	−1.35570	+	1.35570i	−0.476558	+	0.879143i	$0.658116\pi$
$617$	−33.6749	+	33.6749i	−1.35570	+	1.35570i	−0.879143	+	0.476558i	$0.841884\pi$
$618$		0			0
$619$		−	49.0855i		−	1.97291i	−0.164018	−	0.986457i	$-0.552446\pi$
$619$		−	49.0855i		−	1.97291i	0.164018	−	0.986457i	$-0.447554\pi$
$620$		0			0
$621$	20.6358			0.828084
$622$		0			0
$623$		0			0
$624$			46.9666i			1.88017i
$625$	−22.4499	−	11.0000i	−0.897998	−	0.440000i
$626$		0			0
$627$		0			0
$628$	13.2860	+	13.2860i	0.530167	+	0.530167i
$629$		0			0
$630$	57.2590	−	45.5035i	2.28125	−	1.81290i
$631$	50.1916			1.99810			0.999048	−	0.0436231i	$-0.0138901\pi$
$631$	50.1916			1.99810			0.999048	+	0.0436231i	$0.0138901\pi$
$632$	31.4833	−	31.4833i	1.25234	−	1.25234i
$633$		0			0
$634$		0			0
$635$	−4.39371	+	38.4083i	−0.174359	+	1.52419i
$636$		0			0
$637$	16.9609	−	16.9609i	0.672014	−	0.672014i
$638$		0			0
$639$		−	63.1582i		−	2.49850i
$640$	25.1343	+	2.87523i	0.993520	+	0.113654i
$641$	22.7750			0.899560			0.449780	−	0.893140i	$-0.351502\pi$
$641$	22.7750			0.899560			0.449780	+	0.893140i	$0.351502\pi$
$642$		0			0
$643$	−23.0587	−	23.0587i	−0.909348	−	0.909348i	0.0868719	−	0.996219i	$-0.472313\pi$
$643$	−23.0587	−	23.0587i	−0.909348	−	0.909348i	−0.996219	+	0.0868719i	$0.972313\pi$
$644$		−	5.55006i		−	0.218703i
$645$		0			0
$646$		0			0
$647$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$647$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$648$	82.3832	+	82.3832i	3.23632	+	3.23632i
$649$		0			0
$650$	5.47192	−	23.6038i	0.214626	−	0.925819i
$651$		0			0
$652$		0			0
$653$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$653$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$654$		0			0
$655$	39.1292	−	31.0958i	1.52890	−	1.21501i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−44.0779			−1.71443			−0.857215	−	0.514958i	$-0.827807\pi$
$661$	−44.0779			−1.71443			−0.857215	+	0.514958i	$0.827807\pi$
$662$		0			0
$663$		0			0
$664$			45.9558i			1.78343i
$665$	24.8041	+	2.83746i	0.961861	+	0.110032i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	36.2638	−	36.2638i	1.39891	−	1.39891i
$673$	9.44994	+	9.44994i	0.364269	+	0.364269i	0.865382	−	0.501113i	$-0.167076\pi$
$673$	9.44994	+	9.44994i	0.364269	+	0.364269i	−0.501113	+	0.865382i	$0.667076\pi$
$674$			36.0000i			1.38667i
$675$	−52.0536	−	83.4715i	−2.00355	−	3.21282i
$676$	−2.51669			−0.0967956
$677$	−30.8728	+	30.8728i	−1.18654	+	1.18654i	−0.208519	+	0.978018i	$0.566864\pi$
$677$	−30.8728	+	30.8728i	−1.18654	+	1.18654i	−0.978018	+	0.208519i	$0.933136\pi$
$678$	−54.3958	−	54.3958i	−2.08906	−	2.08906i
$679$		0			0
$680$		0			0
$681$	−14.7750			−0.566180
$682$		0			0
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$			73.7796i			2.82103i
$685$	5.92417	−	51.7871i	0.226351	−	1.97868i
$686$	−26.1916			−1.00000
$687$	57.1916	−	57.1916i	2.18200	−	2.18200i
$688$		0			0
$689$		0			0
$690$	−11.2917	−	1.29171i	−0.429868	−	0.0491747i
$691$	50.1758			1.90878			0.954388	−	0.298570i	$-0.0965096\pi$
$691$	50.1758			1.90878			0.954388	+	0.298570i	$0.0965096\pi$
$692$	−25.4817	+	25.4817i	−0.968669	+	0.968669i
$693$		0			0
$694$		0			0
$695$	27.8375	+	35.0291i	1.05593	+	1.32873i
$696$		0			0
$697$		0			0
$698$	35.6379	+	35.6379i	1.34891	+	1.34891i
$699$			87.0655i			3.29312i
$700$	−22.4499	+	14.0000i	−0.848528	+	0.529150i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$	67.4166	−	67.4166i	2.54447	−	2.54447i
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−28.3689	+	28.3689i	−1.06692	+	1.06692i
$708$	3.03337	+	3.03337i	0.114001	+	0.114001i
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$	−2.59668	+	22.6993i	−0.0974518	+	0.851891i
$711$	−137.608			−5.16071
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−18.1319	+	18.1319i	−0.677149	+	0.677149i
$718$	6.00000	+	6.00000i	0.223918	+	0.223918i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	−48.6453	−	61.2124i	−1.81290	−	2.28125i
$721$		0			0
$722$	1.19160	−	1.19160i	0.0443468	−	0.0443468i
$723$		0			0
$724$		−	37.5158i		−	1.39426i
$725$		0			0
$726$	−53.3055			−1.97835
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$	−18.1319	−	18.1319i	−0.672014	−	0.672014i
$729$		−	157.833i		−	5.84567i
$730$		0			0
$731$		0			0
$732$	−26.5167	+	26.5167i	−0.980085	+	0.980085i
$733$	−36.9706	−	36.9706i	−1.36554	−	1.36554i	−0.866683	−	0.498859i	$-0.833753\pi$
$733$	−36.9706	−	36.9706i	−1.36554	−	1.36554i	−0.498859	−	0.866683i	$-0.666247\pi$
$734$		0			0
$735$	−6.09580	+	53.2874i	−0.224847	+	1.96554i
$736$	−5.93326			−0.218703
$737$		0			0
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$	49.5498			1.82026
$742$		0			0
$743$	30.7417	+	30.7417i	1.12780	+	1.12780i	0.990534	+	0.137268i	$0.0438322\pi$
$743$	30.7417	+	30.7417i	1.12780	+	1.12780i	0.137268	+	0.990534i	$0.456168\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	100.432	−	100.432i	3.67463	−	3.67463i
$748$		0			0
$749$		0			0
$750$	23.2583	+	48.9333i	0.849275	+	1.78679i
$751$	22.4499			0.819210			0.409605	−	0.912263i	$-0.365667\pi$
$751$	22.4499			0.819210			0.409605	+	0.912263i	$0.365667\pi$
$752$		0			0
$753$	−71.9666	−	71.9666i	−2.62261	−	2.62261i
$754$		0			0
$755$	39.3008	−	31.2322i	1.43030	−	1.13666i
$756$	−104.107			−3.78634
$757$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$757$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$758$		0			0
$759$		0			0
$760$	3.03337	−	26.5167i	0.110032	−	0.961861i
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$	59.2417	−	59.2417i	2.14610	−	2.14610i
$763$		0			0
$764$			54.4499i			1.96993i
$765$		0			0
$766$		0			0
$767$	1.51669	−	1.51669i	0.0547643	−	0.0547643i
$768$	−38.7677	−	38.7677i	−1.39891	−	1.39891i
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$	12.0000	−	12.0000i	0.431889	−	0.431889i
$773$	30.2468	+	30.2468i	1.08790	+	1.08790i	0.995744	+	0.0921578i	$0.0293764\pi$
$773$	30.2468	+	30.2468i	1.08790	+	1.08790i	0.0921578	+	0.995744i	$0.470624\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	−41.1098	+	32.6698i	−1.47197	+	1.16977i
$781$		0			0
$782$		0			0
$783$		0			0
$784$			28.0000i			1.00000i
$785$	−2.38751	+	20.8708i	−0.0852140	+	0.744912i
$786$	−108.316			−3.86352
$787$	−20.5549	+	20.5549i	−0.732703	+	0.732703i	−0.971154	−	0.238451i	$-0.923360\pi$
$787$	−20.5549	+	20.5549i	−0.732703	+	0.732703i	0.238451	+	0.971154i	$0.423360\pi$
$788$		0			0
$789$		−	54.3958i		−	1.93654i
$790$	49.4569	+	5.65762i	1.75960	+	0.201289i
$791$	42.0000			1.49335
$792$		0			0
$793$	13.2583	+	13.2583i	0.470818	+	0.470818i
$794$		−	42.3617i		−	1.50336i
$795$		0			0
$796$		0			0
$797$	−38.6868	+	38.6868i	−1.37036	+	1.37036i	−0.510449	+	0.859908i	$0.670521\pi$
$797$	−38.6868	+	38.6868i	−1.37036	+	1.37036i	−0.859908	+	0.510449i	$0.829479\pi$
$798$	−38.2583	−	38.2583i	−1.35433	−	1.35433i
$799$		0			0
$800$	14.9666	+	24.0000i	0.529150	+	0.848528i
$801$		0			0
$802$	−37.2250	+	37.2250i	−1.31446	+	1.31446i
$803$		0			0
$804$		0			0
$805$	4.85795	−	3.86060i	0.171220	−	0.136068i
$806$		0			0
$807$	18.9333	−	18.9333i	0.666482	−	0.666482i
$808$	30.3277	+	30.3277i	1.06692	+	1.06692i
$809$		−	11.6749i		−	0.410468i	−0.978713	−	0.205234i	$-0.934204\pi$
$809$		−	11.6749i		−	0.410468i	0.978713	−	0.205234i	$-0.0657956\pi$
$810$	−14.8044	+	129.415i	−0.520175	+	4.54719i
$811$	−33.2958			−1.16917			−0.584586	−	0.811332i	$-0.698743\pi$
$811$	−33.2958			−1.16917			−0.584586	+	0.811332i	$0.698743\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$			79.2515i			2.76927i
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$	−79.8775	+	79.8775i	−2.78605	+	2.78605i
$823$	36.0000	+	36.0000i	1.25488	+	1.25488i	0.953506	+	0.301376i	$0.0974458\pi$
$823$	36.0000	+	36.0000i	1.25488	+	1.25488i	0.301376	+	0.953506i	$0.402554\pi$
$824$		0			0
$825$		0			0
$826$	−2.34212			−0.0814929
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$	12.9666	+	12.9666i	0.450622	+	0.450622i
$829$		−	52.6796i		−	1.82964i	−0.403864	−	0.914819i	$-0.632333\pi$
$829$		−	52.6796i		−	1.82964i	0.403864	−	0.914819i	$-0.367667\pi$
$830$	−40.2250	+	31.9666i	−1.39623	+	1.10958i
$831$		0			0
$832$	−19.3838	+	19.3838i	−0.672014	+	0.672014i
$833$		0			0
$834$		−	96.9666i		−	3.35768i
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−2.96808	−	2.96808i	−0.102531	−	0.102531i
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$	56.9666	+	6.51669i	1.96554	+	0.224847i
$841$	−29.0000			−1.00000
$842$		0			0
$843$	36.2638	+	36.2638i	1.24899	+	1.24899i
$844$		0			0
$845$	−1.75060	−	2.20285i	−0.0602223	−	0.0757803i
$846$		0			0
$847$	20.5791	−	20.5791i	0.707107	−	0.707107i
$848$		0			0
$849$			73.0915i			2.50849i
$850$		0			0
$851$		0			0
$852$	35.0119	−	35.0119i	1.19949	−	1.19949i
$853$	35.7187	+	35.7187i	1.22299	+	1.22299i	0.966565	+	0.256421i	$0.0825433\pi$
$853$	35.7187	+	35.7187i	1.22299	+	1.22299i	0.256421	+	0.966565i	$0.417457\pi$
$854$		−	20.4740i		−	0.700607i
$855$	−64.5791	+	51.3208i	−2.20856	+	1.75513i
$856$		0			0
$857$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$857$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$858$		0			0
$859$		−	58.6158i		−	1.99994i	−0.00751074	−	0.999972i	$-0.502391\pi$
$859$		−	58.6158i		−	1.99994i	0.00751074	−	0.999972i	$-0.497609\pi$
$860$		0			0
$861$		0			0
$862$	−18.0000	+	18.0000i	−0.613082	+	0.613082i
$863$	11.2250	+	11.2250i	0.382102	+	0.382102i	0.871859	−	0.489757i	$-0.162914\pi$
$863$	11.2250	+	11.2250i	0.382102	+	0.382102i	−0.489757	+	0.871859i	$0.662914\pi$
$864$			111.295i			3.78634i
$865$	−40.0291	−	4.57912i	−1.36103	−	0.155695i
$866$		0			0
$867$	−41.1906	+	41.1906i	−1.39891	+	1.39891i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$			6.25959i			0.211734i
$875$	−27.8703	−	9.91205i	−0.942187	−	0.335088i
$876$		0			0
$877$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$877$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$878$		0			0
$879$			79.1582i			2.66994i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	61.1916	−	61.1916i	2.06043	−	2.06043i
$883$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$883$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$884$		0			0
$885$	−0.545103	+	4.76510i	−0.0183234	+	0.160177i
$886$		0			0
$887$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$887$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$888$		0			0
$889$			45.7417i			1.53413i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	29.9333			1.00000
$897$	8.70829	−	8.70829i	0.290761	−	0.290761i
$898$	29.9333	+	29.9333i	0.998886	+	0.998886i
$899$		0			0
$900$	19.7417	−	85.1582i	0.658055	−	2.83861i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		−	44.8999i		−	1.49335i
$905$	32.8375	−	26.0958i	1.09155	−	0.867454i
$906$	−108.792			−3.61436
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$	−6.09788	−	6.09788i	−0.202365	−	0.202365i
$909$		−	132.557i		−	4.39664i
$910$	3.25834	−	28.4833i	0.108013	−	0.944213i
$911$	−52.3832			−1.73553			−0.867766	−	0.496972i	$-0.834445\pi$
$911$	−52.3832			−1.73553			−0.867766	+	0.496972i	$0.834445\pi$
$912$	−40.8999	+	40.8999i	−1.35433	+	1.35433i
$913$		0			0
$914$		−	7.48331i		−	0.247526i
$915$	−41.6549	−	4.76510i	−1.37707	−	0.157529i
$916$	47.2077			1.55979
$917$	41.8166	−	41.8166i	1.38091	−	1.38091i
$918$		0			0
$919$			53.1582i			1.75353i	0.480921	+	0.876764i	$0.340303\pi$
$919$			53.1582i			1.75353i	−0.480921	+	0.876764i	$0.659697\pi$
$920$	−4.12715	−	5.19337i	−0.136068	−	0.171220i
$921$	120.058			3.95605
$922$	−36.8898	+	36.8898i	−1.21490	+	1.21490i
$923$	−17.5060	−	17.5060i	−0.576216	−	0.576216i
$924$		0			0
$925$		0			0
$926$	−48.0000			−1.57738
$927$		0			0
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	29.5400			0.968134
$932$	−35.9333	+	35.9333i	−1.17703	+	1.17703i
$933$		0			0
$934$		−	7.34979i		−	0.240493i
$935$		0			0
$936$	84.7234			2.76927
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	46.5817			1.51852			0.759260	−	0.650787i	$-0.225561\pi$
$941$	46.5817			1.51852			0.759260	+	0.650787i	$0.225561\pi$
$942$	32.1916	−	32.1916i	1.04886	−	1.04886i
$943$		0			0
$944$			2.50384i			0.0814929i
$945$	−72.4166	−	91.1249i	−2.35571	−	2.96429i
$946$		0			0
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$	−76.2834	−	76.2834i	−2.47757	−	2.47757i
$949$		0			0
$950$	25.3200	−	15.7898i	0.821489	−	0.512289i
$951$		0			0
$952$		0			0
$953$	12.0334	+	12.0334i	0.389799	+	0.389799i	0.874616	−	0.484817i	$-0.161114\pi$
$953$	12.0334	+	12.0334i	0.389799	+	0.389799i	−0.484817	+	0.874616i	$0.661114\pi$
$954$		0			0
$955$	−47.6599	+	37.8752i	−1.54224	+	1.22561i
$956$	−14.9666			−0.484055
$957$		0			0
$958$		0			0
$959$		−	61.6749i		−	1.99159i
$960$	6.96663	−	60.8999i	0.224847	−	1.96554i
$961$	31.0000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	18.8507	+	2.15642i	0.606826	+	0.0694178i
$966$	−13.4477			−0.432672
$967$	17.7750	−	17.7750i	0.571606	−	0.571606i	−0.360971	−	0.932577i	$-0.617555\pi$
$967$	17.7750	−	17.7750i	0.571606	−	0.571606i	0.932577	+	0.360971i	$0.117555\pi$
$968$	−22.0000	−	22.0000i	−0.707107	−	0.707107i
$969$		0			0
$970$		0			0
$971$	57.5255			1.84608			0.923041	−	0.384701i	$-0.125695\pi$
$971$	57.5255			1.84608			0.923041	+	0.384701i	$0.125695\pi$
$972$	116.141	−	116.141i	3.72523	−	3.72523i
$973$	37.4349	+	37.4349i	1.20011	+	1.20011i
$974$		−	15.5501i		−	0.498256i
$975$	−57.1916	−	13.2583i	−1.83160	−	0.424607i
$976$	−21.8877			−0.700607
$977$	−20.9333	+	20.9333i	−0.669714	+	0.669714i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$	−20.9333	+	20.9333i	−0.669714	+	0.669714i	0.287936	+	0.957650i	$0.407031\pi$
$978$		0			0
$979$		0			0
$980$	−24.5083	+	19.4767i	−0.782890	+	0.622160i
$981$		0			0
$982$		0			0
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	20.4499	+	20.4499i	0.650599	+	0.650599i
$989$		0			0
$990$		0			0
$991$	−9.80840			−0.311574			−0.155787	−	0.987791i	$-0.549791\pi$
$991$	−9.80840			−0.311574			−0.155787	+	0.987791i	$0.549791\pi$
$992$		0			0
$993$		0			0
$994$			27.0334i			0.857446i
$995$		0			0
$996$	111.350			3.52825
$997$	−42.2809	+	42.2809i	−1.33905	+	1.33905i	−0.442065	+	0.896983i	$0.645754\pi$
$997$	−42.2809	+	42.2809i	−1.33905	+	1.33905i	−0.896983	+	0.442065i	$0.854246\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.2.s.b.13.1	✓	8
4.3	odd	2		1120.2.w.b.433.4		8
5.2	odd	4	inner	280.2.s.b.237.1	yes	8
7.6	odd	2	inner	280.2.s.b.13.4	yes	8
8.3	odd	2		1120.2.w.b.433.1		8
8.5	even	2	inner	280.2.s.b.13.4	yes	8
20.7	even	4		1120.2.w.b.657.4		8
28.27	even	2		1120.2.w.b.433.1		8
35.27	even	4	inner	280.2.s.b.237.4	yes	8
40.27	even	4		1120.2.w.b.657.1		8
40.37	odd	4	inner	280.2.s.b.237.4	yes	8
56.13	odd	2	CM	280.2.s.b.13.1	✓	8
56.27	even	2		1120.2.w.b.433.4		8
140.27	odd	4		1120.2.w.b.657.1		8
280.27	odd	4		1120.2.w.b.657.4		8
280.237	even	4	inner	280.2.s.b.237.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.s.b.13.1	✓	8	1.1	even	1	trivial
280.2.s.b.13.1	✓	8	56.13	odd	2	CM
280.2.s.b.13.4	yes	8	7.6	odd	2	inner
280.2.s.b.13.4	yes	8	8.5	even	2	inner
280.2.s.b.237.1	yes	8	5.2	odd	4	inner
280.2.s.b.237.1	yes	8	280.237	even	4	inner
280.2.s.b.237.4	yes	8	35.27	even	4	inner
280.2.s.b.237.4	yes	8	40.37	odd	4	inner
1120.2.w.b.433.1		8	8.3	odd	2
1120.2.w.b.433.1		8	28.27	even	2
1120.2.w.b.433.4		8	4.3	odd	2
1120.2.w.b.433.4		8	56.27	even	2
1120.2.w.b.657.1		8	40.27	even	4
1120.2.w.b.657.1		8	140.27	odd	4
1120.2.w.b.657.4		8	20.7	even	4
1120.2.w.b.657.4		8	280.27	odd	4

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:	\( N \)	\(=\)	\( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	280.s (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.00000i) q^{2} +(-2.42298 - 2.42298i) q^{3} -2.00000i q^{4} +(1.75060 - 1.39119i) q^{5} -4.84596 q^{6} +(1.87083 - 1.87083i) q^{7} +(-2.00000 - 2.00000i) q^{8} +8.74166i q^{9} +O(q^{10})\)	\(q+(1.00000 - 1.00000i) q^{2} +(-2.42298 - 2.42298i) q^{3} -2.00000i q^{4} +(1.75060 - 1.39119i) q^{5} -4.84596 q^{6} +(1.87083 - 1.87083i) q^{7} +(-2.00000 - 2.00000i) q^{8} +8.74166i q^{9} +(0.359404 - 3.14179i) q^{10} +(-4.84596 + 4.84596i) q^{12} +(2.42298 + 2.42298i) q^{13} -3.74166i q^{14} +(-7.61249 - 0.870829i) q^{15} -4.00000 q^{16} +(8.74166 + 8.74166i) q^{18} +4.22000i q^{19} +(-2.78238 - 3.50119i) q^{20} -9.06596 q^{21} +(0.741657 + 0.741657i) q^{23} +9.69192i q^{24} +(1.12917 - 4.87083i) q^{25} +4.84596 q^{26} +(13.9119 - 13.9119i) q^{27} +(-3.74166 - 3.74166i) q^{28} +(-8.48331 + 6.74166i) q^{30} +(-4.00000 + 4.00000i) q^{32} +(0.672384 - 5.87775i) q^{35} +17.4833 q^{36} +(4.22000 + 4.22000i) q^{38} -11.7417i q^{39} +(-6.28357 - 0.718808i) q^{40} +(-9.06596 + 9.06596i) q^{42} +(12.1613 + 15.3031i) q^{45} +1.48331 q^{46} +(9.69192 + 9.69192i) q^{48} -7.00000i q^{49} +(-3.74166 - 6.00000i) q^{50} +(4.84596 - 4.84596i) q^{52} -27.8238i q^{54} -7.48331 q^{56} +(10.2250 - 10.2250i) q^{57} -0.625959i q^{59} +(-1.74166 + 15.2250i) q^{60} +5.47192 q^{61} +(16.3541 + 16.3541i) q^{63} +8.00000i q^{64} +(7.61249 + 0.870829i) q^{65} -3.59404i q^{69} +(-5.20536 - 6.55013i) q^{70} -7.22497 q^{71} +(17.4833 - 17.4833i) q^{72} +(-14.5379 + 9.06596i) q^{75} +8.44000 q^{76} +(-11.7417 - 11.7417i) q^{78} +15.7417i q^{79} +(-7.00238 + 5.56477i) q^{80} -41.1916 q^{81} +(-11.4889 - 11.4889i) q^{83} +18.1319i q^{84} +(27.4644 + 3.14179i) q^{90} +9.06596 q^{91} +(1.48331 - 1.48331i) q^{92} +(5.87083 + 7.38751i) q^{95} +19.3838 q^{96} +(-7.00000 - 7.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 16 q^{8}+O(q^{10}) \) 8 * q + 8 * q^2 - 16 * q^8	\( 8 q + 8 q^{2} - 16 q^{8} - 16 q^{15} - 32 q^{16} + 40 q^{18} - 24 q^{23} + 24 q^{25} - 8 q^{30} - 32 q^{32} + 80 q^{36} - 48 q^{46} - 8 q^{57} + 16 q^{60} + 56 q^{63} + 16 q^{65} + 32 q^{71} + 80 q^{72} - 64 q^{78} - 120 q^{81} - 48 q^{92} + 32 q^{95} - 56 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 - 16 * q^8 - 16 * q^15 - 32 * q^16 + 40 * q^18 - 24 * q^23 + 24 * q^25 - 8 * q^30 - 32 * q^32 + 80 * q^36 - 48 * q^46 - 8 * q^57 + 16 * q^60 + 56 * q^63 + 16 * q^65 + 32 * q^71 + 80 * q^72 - 64 * q^78 - 120 * q^81 - 48 * q^92 + 32 * q^95 - 56 * q^98

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