LMFDB - Embedded newform 2240.2.bd.b.561.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(561,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("2240.561");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.bd (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$52$
Relative dimension:	$26$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 560)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			561.13
Character	$\chi$	$=$	2240.561
Dual form			2240.2.bd.b.1681.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.280441 + 0.280441i) q^{3} +(0.707107 + 0.707107i) q^{5} +1.00000i q^{7} +2.84271i q^{9} +O(q^{10})$	\(q+(-0.280441 + 0.280441i) q^{3} +(0.707107 + 0.707107i) q^{5} +1.00000i q^{7} +2.84271i q^{9} +(-0.538137 - 0.538137i) q^{11} +(1.81644 - 1.81644i) q^{13} -0.396603 q^{15} -6.12274 q^{17} +(-2.27187 + 2.27187i) q^{19} +(-0.280441 - 0.280441i) q^{21} +3.11586i q^{23} +1.00000i q^{25} +(-1.63853 - 1.63853i) q^{27} +(-1.87483 + 1.87483i) q^{29} -6.37128 q^{31} +0.301831 q^{33} +(-0.707107 + 0.707107i) q^{35} +(1.86176 + 1.86176i) q^{37} +1.01881i q^{39} -9.14711i q^{41} +(5.29325 + 5.29325i) q^{43} +(-2.01010 + 2.01010i) q^{45} +1.91208 q^{47} -1.00000 q^{49} +(1.71707 - 1.71707i) q^{51} +(-5.44648 - 5.44648i) q^{53} -0.761040i q^{55} -1.27425i q^{57} +(3.78946 + 3.78946i) q^{59} +(-8.50125 + 8.50125i) q^{61} -2.84271 q^{63} +2.56883 q^{65} +(2.02955 - 2.02955i) q^{67} +(-0.873814 - 0.873814i) q^{69} -5.82741i q^{71} -7.24382i q^{73} +(-0.280441 - 0.280441i) q^{75} +(0.538137 - 0.538137i) q^{77} -9.61104 q^{79} -7.60909 q^{81} +(-5.03556 + 5.03556i) q^{83} +(-4.32943 - 4.32943i) q^{85} -1.05156i q^{87} -8.76458i q^{89} +(1.81644 + 1.81644i) q^{91} +(1.78677 - 1.78677i) q^{93} -3.21291 q^{95} +5.00295 q^{97} +(1.52976 - 1.52976i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 52 q+O(q^{10}) $ 52 * q	$ 52 q + 4 q^{11} - 8 q^{15} - 8 q^{19} + 24 q^{27} + 4 q^{29} - 12 q^{37} - 36 q^{43} - 52 q^{49} + 8 q^{51} - 4 q^{53} - 24 q^{59} - 16 q^{61} + 68 q^{63} + 40 q^{65} + 12 q^{67} - 72 q^{69} - 4 q^{77} + 16 q^{79} - 116 q^{81} + 16 q^{85} + 8 q^{93} + 32 q^{95} + 52 q^{99}+O(q^{100}) $ 52 * q + 4 * q^11 - 8 * q^15 - 8 * q^19 + 24 * q^27 + 4 * q^29 - 12 * q^37 - 36 * q^43 - 52 * q^49 + 8 * q^51 - 4 * q^53 - 24 * q^59 - 16 * q^61 + 68 * q^63 + 40 * q^65 + 12 * q^67 - 72 * q^69 - 4 * q^77 + 16 * q^79 - 116 * q^81 + 16 * q^85 + 8 * q^93 + 32 * q^95 + 52 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$.

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−0.280441	+	0.280441i	−0.161913	+	0.161913i	−0.783413	−	0.621501i	$-0.786523\pi$
$3$	−0.280441	+	0.280441i	−0.161913	+	0.161913i	0.621501	+	0.783413i	$0.286523\pi$
$4$		0			0
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$			2.84271i			0.947569i
$10$		0			0
$11$	−0.538137	−	0.538137i	−0.162254	−	0.162254i	0.621310	−	0.783565i	$-0.286601\pi$
$11$	−0.538137	−	0.538137i	−0.162254	−	0.162254i	−0.783565	+	0.621310i	$0.786601\pi$
$12$		0			0
$13$	1.81644	−	1.81644i	0.503790	−	0.503790i	−0.408824	−	0.912613i	$-0.634061\pi$
$13$	1.81644	−	1.81644i	0.503790	−	0.503790i	0.912613	+	0.408824i	$0.134061\pi$
$14$		0			0
$15$	−0.396603			−0.102403
$16$		0			0
$17$	−6.12274			−1.48498			−0.742492	−	0.669855i	$-0.766356\pi$
$17$	−6.12274			−1.48498			−0.742492	+	0.669855i	$0.766356\pi$
$18$		0			0
$19$	−2.27187	+	2.27187i	−0.521202	+	0.521202i	−0.917934	−	0.396732i	$-0.870144\pi$
$19$	−2.27187	+	2.27187i	−0.521202	+	0.521202i	0.396732	+	0.917934i	$0.370144\pi$
$20$		0			0
$21$	−0.280441	−	0.280441i	−0.0611972	−	0.0611972i
$22$		0			0
$23$			3.11586i			0.649701i	0.945765	+	0.324851i	$0.105314\pi$
$23$			3.11586i			0.649701i	−0.945765	+	0.324851i	$0.894686\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$	−1.63853	−	1.63853i	−0.315336	−	0.315336i
$28$		0			0
$29$	−1.87483	+	1.87483i	−0.348147	+	0.348147i	−0.859419	−	0.511272i	$-0.829175\pi$
$29$	−1.87483	+	1.87483i	−0.348147	+	0.348147i	0.511272	+	0.859419i	$0.329175\pi$
$30$		0			0
$31$	−6.37128			−1.14432			−0.572158	−	0.820144i	$-0.693893\pi$
$31$	−6.37128			−1.14432			−0.572158	+	0.820144i	$0.693893\pi$
$32$		0			0
$33$	0.301831			0.0525420
$34$		0			0
$35$	−0.707107	+	0.707107i	−0.119523	+	0.119523i
$36$		0			0
$37$	1.86176	+	1.86176i	0.306072	+	0.306072i	0.843384	−	0.537312i	$-0.180560\pi$
$37$	1.86176	+	1.86176i	0.306072	+	0.306072i	−0.537312	+	0.843384i	$0.680560\pi$
$38$		0			0
$39$			1.01881i			0.163140i
$40$		0			0
$41$		−	9.14711i		−	1.42854i	−0.699871	−	0.714269i	$-0.746759\pi$
$41$		−	9.14711i		−	1.42854i	0.699871	−	0.714269i	$-0.253241\pi$
$42$		0			0
$43$	5.29325	+	5.29325i	0.807214	+	0.807214i	0.984211	−	0.176998i	$-0.0566385\pi$
$43$	5.29325	+	5.29325i	0.807214	+	0.807214i	−0.176998	+	0.984211i	$0.556639\pi$
$44$		0			0
$45$	−2.01010	+	2.01010i	−0.299648	+	0.299648i
$46$		0			0
$47$	1.91208			0.278905			0.139452	−	0.990229i	$-0.455466\pi$
$47$	1.91208			0.278905			0.139452	+	0.990229i	$0.455466\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	1.71707	−	1.71707i	0.240438	−	0.240438i
$52$		0			0
$53$	−5.44648	−	5.44648i	−0.748132	−	0.748132i	0.225996	−	0.974128i	$-0.427436\pi$
$53$	−5.44648	−	5.44648i	−0.748132	−	0.748132i	−0.974128	+	0.225996i	$0.927436\pi$
$54$		0			0
$55$		−	0.761040i		−	0.102619i
$56$		0			0
$57$		−	1.27425i		−	0.168779i
$58$		0			0
$59$	3.78946	+	3.78946i	0.493346	+	0.493346i	0.909359	−	0.416013i	$-0.136573\pi$
$59$	3.78946	+	3.78946i	0.493346	+	0.493346i	−0.416013	+	0.909359i	$0.636573\pi$
$60$		0			0
$61$	−8.50125	+	8.50125i	−1.08847	+	1.08847i	−0.0927872	+	0.995686i	$0.529578\pi$
$61$	−8.50125	+	8.50125i	−1.08847	+	1.08847i	−0.995686	+	0.0927872i	$0.970422\pi$
$62$		0			0
$63$	−2.84271			−0.358147
$64$		0			0
$65$	2.56883			0.318625
$66$		0			0
$67$	2.02955	−	2.02955i	0.247949	−	0.247949i	−0.572180	−	0.820128i	$-0.693902\pi$
$67$	2.02955	−	2.02955i	0.247949	−	0.247949i	0.820128	+	0.572180i	$0.193902\pi$
$68$		0			0
$69$	−0.873814	−	0.873814i	−0.105195	−	0.105195i
$70$		0			0
$71$		−	5.82741i		−	0.691586i	−0.938311	−	0.345793i	$-0.887610\pi$
$71$		−	5.82741i		−	0.691586i	0.938311	−	0.345793i	$-0.112390\pi$
$72$		0			0
$73$		−	7.24382i		−	0.847825i	−0.905703	−	0.423912i	$-0.860656\pi$
$73$		−	7.24382i		−	0.847825i	0.905703	−	0.423912i	$-0.139344\pi$
$74$		0			0
$75$	−0.280441	−	0.280441i	−0.0323825	−	0.0323825i
$76$		0			0
$77$	0.538137	−	0.538137i	0.0613264	−	0.0613264i
$78$		0			0
$79$	−9.61104			−1.08133			−0.540663	−	0.841239i	$-0.681827\pi$
$79$	−9.61104			−1.08133			−0.540663	+	0.841239i	$0.681827\pi$
$80$		0			0
$81$	−7.60909			−0.845455
$82$		0			0
$83$	−5.03556	+	5.03556i	−0.552724	+	0.552724i	−0.927226	−	0.374502i	$-0.877814\pi$
$83$	−5.03556	+	5.03556i	−0.552724	+	0.552724i	0.374502	+	0.927226i	$0.377814\pi$
$84$		0			0
$85$	−4.32943	−	4.32943i	−0.469593	−	0.469593i
$86$		0			0
$87$		−	1.05156i		−	0.112739i
$88$		0			0
$89$		−	8.76458i		−	0.929044i	−0.885562	−	0.464522i	$-0.846226\pi$
$89$		−	8.76458i		−	0.929044i	0.885562	−	0.464522i	$-0.153774\pi$
$90$		0			0
$91$	1.81644	+	1.81644i	0.190415	+	0.190415i
$92$		0			0
$93$	1.78677	−	1.78677i	0.185279	−	0.185279i
$94$		0			0
$95$	−3.21291			−0.329637
$96$		0			0
$97$	5.00295			0.507972			0.253986	−	0.967208i	$-0.418258\pi$
$97$	5.00295			0.507972			0.253986	+	0.967208i	$0.418258\pi$
$98$		0			0
$99$	1.52976	−	1.52976i	0.153747	−	0.153747i
$100$		0			0
$101$	−0.387033	−	0.387033i	−0.0385113	−	0.0385113i	0.687589	−	0.726100i	$-0.258669\pi$
$101$	−0.387033	−	0.387033i	−0.0385113	−	0.0385113i	−0.726100	+	0.687589i	$0.758669\pi$
$102$		0			0
$103$			7.23262i			0.712652i	0.934362	+	0.356326i	$0.115971\pi$
$103$			7.23262i			0.712652i	−0.934362	+	0.356326i	$0.884029\pi$
$104$		0			0
$105$		−	0.396603i		−	0.0387045i
$106$		0			0
$107$	1.27803	+	1.27803i	0.123552	+	0.123552i	0.766179	−	0.642627i	$-0.222155\pi$
$107$	1.27803	+	1.27803i	0.123552	+	0.123552i	−0.642627	+	0.766179i	$0.722155\pi$
$108$		0			0
$109$	−12.3875	+	12.3875i	−1.18651	+	1.18651i	−0.208486	+	0.978025i	$0.566853\pi$
$109$	−12.3875	+	12.3875i	−1.18651	+	1.18651i	−0.978025	+	0.208486i	$0.933147\pi$
$110$		0			0
$111$	−1.04423			−0.0991138
$112$		0			0
$113$	−13.2156			−1.24322			−0.621608	−	0.783328i	$-0.713520\pi$
$113$	−13.2156			−1.24322			−0.621608	+	0.783328i	$0.713520\pi$
$114$		0			0
$115$	−2.20324	+	2.20324i	−0.205454	+	0.205454i
$116$		0			0
$117$	5.16360	+	5.16360i	0.477375	+	0.477375i
$118$		0			0
$119$		−	6.12274i		−	0.561271i
$120$		0			0
$121$		−	10.4208i		−	0.947347i
$122$		0			0
$123$	2.56522	+	2.56522i	0.231298	+	0.231298i
$124$		0			0
$125$	−0.707107	+	0.707107i	−0.0632456	+	0.0632456i
$126$		0			0
$127$	−3.33945			−0.296328			−0.148164	−	0.988963i	$-0.547336\pi$
$127$	−3.33945			−0.296328			−0.148164	+	0.988963i	$0.547336\pi$
$128$		0			0
$129$	−2.96889			−0.261396
$130$		0			0
$131$	−0.369805	+	0.369805i	−0.0323100	+	0.0323100i	−0.723077	−	0.690767i	$-0.757273\pi$
$131$	−0.369805	+	0.369805i	−0.0323100	+	0.0323100i	0.690767	+	0.723077i	$0.257273\pi$
$132$		0			0
$133$	−2.27187	−	2.27187i	−0.196996	−	0.196996i
$134$		0			0
$135$		−	2.31724i		−	0.199436i
$136$		0			0
$137$			18.0983i			1.54624i	0.634258	+	0.773122i	$0.281306\pi$
$137$			18.0983i			1.54624i	−0.634258	+	0.773122i	$0.718694\pi$
$138$		0			0
$139$	3.13269	+	3.13269i	0.265711	+	0.265711i	0.827369	−	0.561658i	$-0.189836\pi$
$139$	3.13269	+	3.13269i	0.265711	+	0.265711i	−0.561658	+	0.827369i	$0.689836\pi$
$140$		0			0
$141$	−0.536224	+	0.536224i	−0.0451582	+	0.0451582i
$142$		0			0
$143$	−1.95499			−0.163484
$144$		0			0
$145$	−2.65141			−0.220188
$146$		0			0
$147$	0.280441	−	0.280441i	0.0231304	−	0.0231304i
$148$		0			0
$149$	4.44990	+	4.44990i	0.364550	+	0.364550i	0.865485	−	0.500935i	$-0.167010\pi$
$149$	4.44990	+	4.44990i	0.364550	+	0.364550i	−0.500935	+	0.865485i	$0.667010\pi$
$150$		0			0
$151$			6.91123i			0.562428i	0.959645	+	0.281214i	$0.0907371\pi$
$151$			6.91123i			0.562428i	−0.959645	+	0.281214i	$0.909263\pi$
$152$		0			0
$153$		−	17.4052i		−	1.40712i
$154$		0			0
$155$	−4.50517	−	4.50517i	−0.361864	−	0.361864i
$156$		0			0
$157$	−4.89790	+	4.89790i	−0.390895	+	0.390895i	−0.875006	−	0.484111i	$-0.839143\pi$
$157$	−4.89790	+	4.89790i	−0.390895	+	0.390895i	0.484111	+	0.875006i	$0.339143\pi$
$158$		0			0
$159$	3.05483			0.242264
$160$		0			0
$161$	−3.11586			−0.245564
$162$		0			0
$163$	−16.9053	+	16.9053i	−1.32413	+	1.32413i	−0.413728	+	0.910401i	$0.635773\pi$
$163$	−16.9053	+	16.9053i	−1.32413	+	1.32413i	−0.910401	+	0.413728i	$0.864227\pi$
$164$		0			0
$165$	0.213427	+	0.213427i	0.0166153	+	0.0166153i
$166$		0			0
$167$			15.1283i			1.17067i	0.810793	+	0.585333i	$0.199036\pi$
$167$			15.1283i			1.17067i	−0.810793	+	0.585333i	$0.800964\pi$
$168$		0			0
$169$			6.40109i			0.492392i
$170$		0			0
$171$	−6.45825	−	6.45825i	−0.493875	−	0.493875i
$172$		0			0
$173$	−7.29280	+	7.29280i	−0.554461	+	0.554461i	−0.927725	−	0.373264i	$-0.878239\pi$
$173$	−7.29280	+	7.29280i	−0.554461	+	0.554461i	0.373264	+	0.927725i	$0.378239\pi$
$174$		0			0
$175$	−1.00000			−0.0755929
$176$		0			0
$177$	−2.12544			−0.159758
$178$		0			0
$179$	13.9726	−	13.9726i	1.04436	−	1.04436i	0.0453906	−	0.998969i	$-0.485547\pi$
$179$	13.9726	−	13.9726i	1.04436	−	1.04436i	0.998969	−	0.0453906i	$-0.0144532\pi$
$180$		0			0
$181$	5.66754	+	5.66754i	0.421265	+	0.421265i	0.885639	−	0.464374i	$-0.153721\pi$
$181$	5.66754	+	5.66754i	0.421265	+	0.421265i	−0.464374	+	0.885639i	$0.653721\pi$
$182$		0			0
$183$		−	4.76819i		−	0.352475i
$184$		0			0
$185$			2.63293i			0.193577i
$186$		0			0
$187$	3.29487	+	3.29487i	0.240945	+	0.240945i
$188$		0			0
$189$	1.63853	−	1.63853i	0.119186	−	0.119186i
$190$		0			0
$191$	−11.7887			−0.853003			−0.426501	−	0.904487i	$-0.640254\pi$
$191$	−11.7887			−0.853003			−0.426501	+	0.904487i	$0.640254\pi$
$192$		0			0
$193$	27.5669			1.98431			0.992154	−	0.125024i	$-0.0399009\pi$
$193$	27.5669			1.98431			0.992154	+	0.125024i	$0.0399009\pi$
$194$		0			0
$195$	−0.720406	+	0.720406i	−0.0515893	+	0.0515893i
$196$		0			0
$197$	9.60554	+	9.60554i	0.684366	+	0.684366i	0.960981	−	0.276615i	$-0.0892125\pi$
$197$	9.60554	+	9.60554i	0.684366	+	0.684366i	−0.276615	+	0.960981i	$0.589213\pi$
$198$		0			0
$199$		−	22.8687i		−	1.62112i	−0.585657	−	0.810559i	$-0.699163\pi$
$199$		−	22.8687i		−	1.62112i	0.585657	−	0.810559i	$-0.300837\pi$
$200$		0			0
$201$			1.13834i			0.0802921i
$202$		0			0
$203$	−1.87483	−	1.87483i	−0.131587	−	0.131587i
$204$		0			0
$205$	6.46798	−	6.46798i	0.451744	−	0.451744i
$206$		0			0
$207$	−8.85747			−0.615637
$208$		0			0
$209$	2.44515			0.169135
$210$		0			0
$211$	11.7547	−	11.7547i	0.809224	−	0.809224i	−0.175293	−	0.984516i	$-0.556087\pi$
$211$	11.7547	−	11.7547i	0.809224	−	0.809224i	0.984516	+	0.175293i	$0.0560871\pi$
$212$		0			0
$213$	1.63424	+	1.63424i	0.111977	+	0.111977i
$214$		0			0
$215$			7.48579i			0.510527i
$216$		0			0
$217$		−	6.37128i		−	0.432511i
$218$		0			0
$219$	2.03146	+	2.03146i	0.137273	+	0.137273i
$220$		0			0
$221$	−11.1216	+	11.1216i	−0.748120	+	0.748120i
$222$		0			0
$223$	−18.9818			−1.27111			−0.635557	−	0.772054i	$-0.719230\pi$
$223$	−18.9818			−1.27111			−0.635557	+	0.772054i	$0.719230\pi$
$224$		0			0
$225$	−2.84271			−0.189514
$226$		0			0
$227$	18.0055	−	18.0055i	1.19507	−	1.19507i	0.219442	−	0.975625i	$-0.429576\pi$
$227$	18.0055	−	18.0055i	1.19507	−	1.19507i	0.975625	−	0.219442i	$-0.0704238\pi$
$228$		0			0
$229$	11.5882	+	11.5882i	0.765769	+	0.765769i	0.977359	−	0.211589i	$-0.0678640\pi$
$229$	11.5882	+	11.5882i	0.765769	+	0.765769i	−0.211589	+	0.977359i	$0.567864\pi$
$230$		0			0
$231$			0.301831i			0.0198590i
$232$		0			0
$233$		−	14.3065i		−	0.937248i	−0.883398	−	0.468624i	$-0.844750\pi$
$233$		−	14.3065i		−	0.937248i	0.883398	−	0.468624i	$-0.155250\pi$
$234$		0			0
$235$	1.35204	+	1.35204i	0.0881975	+	0.0881975i
$236$		0			0
$237$	2.69533	−	2.69533i	0.175080	−	0.175080i
$238$		0			0
$239$	−25.7832			−1.66777			−0.833887	−	0.551935i	$-0.813890\pi$
$239$	−25.7832			−1.66777			−0.833887	+	0.551935i	$0.813890\pi$
$240$		0			0
$241$	−20.5848			−1.32598			−0.662991	−	0.748627i	$-0.730713\pi$
$241$	−20.5848			−1.32598			−0.662991	+	0.748627i	$0.730713\pi$
$242$		0			0
$243$	7.04950	−	7.04950i	0.452226	−	0.452226i
$244$		0			0
$245$	−0.707107	−	0.707107i	−0.0451754	−	0.0451754i
$246$		0			0
$247$			8.25342i			0.525153i
$248$		0			0
$249$		−	2.82435i		−	0.178986i
$250$		0			0
$251$	−11.6736	−	11.6736i	−0.736832	−	0.736832i	0.235131	−	0.971964i	$-0.424448\pi$
$251$	−11.6736	−	11.6736i	−0.736832	−	0.736832i	−0.971964	+	0.235131i	$0.924448\pi$
$252$		0			0
$253$	1.67676	−	1.67676i	0.105417	−	0.105417i
$254$		0			0
$255$	2.42830			0.152066
$256$		0			0
$257$	−8.94910			−0.558229			−0.279115	−	0.960258i	$-0.590041\pi$
$257$	−8.94910			−0.558229			−0.279115	+	0.960258i	$0.590041\pi$
$258$		0			0
$259$	−1.86176	+	1.86176i	−0.115684	+	0.115684i
$260$		0			0
$261$	−5.32959	−	5.32959i	−0.329894	−	0.329894i
$262$		0			0
$263$			9.54669i			0.588674i	0.955702	+	0.294337i	$0.0950989\pi$
$263$			9.54669i			0.588674i	−0.955702	+	0.294337i	$0.904901\pi$
$264$		0			0
$265$		−	7.70249i		−	0.473160i
$266$		0			0
$267$	2.45795	+	2.45795i	0.150424	+	0.150424i
$268$		0			0
$269$	−1.67064	+	1.67064i	−0.101860	+	0.101860i	−0.756200	−	0.654340i	$-0.772947\pi$
$269$	−1.67064	+	1.67064i	−0.101860	+	0.101860i	0.654340	+	0.756200i	$0.272947\pi$
$270$		0			0
$271$	−22.7151			−1.37984			−0.689922	−	0.723883i	$-0.742355\pi$
$271$	−22.7151			−1.37984			−0.689922	+	0.723883i	$0.742355\pi$
$272$		0			0
$273$	−1.01881			−0.0616611
$274$		0			0
$275$	0.538137	−	0.538137i	0.0324509	−	0.0324509i
$276$		0			0
$277$	11.0399	+	11.0399i	0.663324	+	0.663324i	0.956162	−	0.292838i	$-0.0945996\pi$
$277$	11.0399	+	11.0399i	0.663324	+	0.663324i	−0.292838	+	0.956162i	$0.594600\pi$
$278$		0			0
$279$		−	18.1117i		−	1.08432i
$280$		0			0
$281$			12.6943i			0.757279i	0.925544	+	0.378640i	$0.123608\pi$
$281$			12.6943i			0.757279i	−0.925544	+	0.378640i	$0.876392\pi$
$282$		0			0
$283$	9.51217	+	9.51217i	0.565440	+	0.565440i	0.930848	−	0.365408i	$-0.119070\pi$
$283$	9.51217	+	9.51217i	0.565440	+	0.565440i	−0.365408	+	0.930848i	$0.619070\pi$
$284$		0			0
$285$	0.901031	−	0.901031i	0.0533724	−	0.0533724i
$286$		0			0
$287$	9.14711			0.539937
$288$		0			0
$289$	20.4880			1.20518
$290$		0			0
$291$	−1.40303	+	1.40303i	−0.0822471	+	0.0822471i
$292$		0			0
$293$	2.94317	+	2.94317i	0.171942	+	0.171942i	0.787832	−	0.615890i	$-0.211203\pi$
$293$	2.94317	+	2.94317i	0.171942	+	0.171942i	−0.615890	+	0.787832i	$0.711203\pi$
$294$		0			0
$295$			5.35911i			0.312020i
$296$		0			0
$297$			1.76351i			0.102329i
$298$		0			0
$299$	5.65977	+	5.65977i	0.327313	+	0.327313i
$300$		0			0
$301$	−5.29325	+	5.29325i	−0.305098	+	0.305098i
$302$		0			0
$303$	0.217080			0.0124709
$304$		0			0
$305$	−12.0226			−0.688411
$306$		0			0
$307$	−2.93576	+	2.93576i	−0.167553	+	0.167553i	−0.785903	−	0.618350i	$-0.787801\pi$
$307$	−2.93576	+	2.93576i	−0.167553	+	0.167553i	0.618350	+	0.785903i	$0.287801\pi$
$308$		0			0
$309$	−2.02832	−	2.02832i	−0.115387	−	0.115387i
$310$		0			0
$311$		−	29.0817i		−	1.64907i	−0.565811	−	0.824535i	$-0.691437\pi$
$311$		−	29.0817i		−	1.64907i	0.565811	−	0.824535i	$-0.308563\pi$
$312$		0			0
$313$			1.96859i			0.111271i	0.998451	+	0.0556355i	$0.0177185\pi$
$313$			1.96859i			0.111271i	−0.998451	+	0.0556355i	$0.982282\pi$
$314$		0			0
$315$	−2.01010	−	2.01010i	−0.113256	−	0.113256i
$316$		0			0
$317$	−2.79818	+	2.79818i	−0.157162	+	0.157162i	−0.781308	−	0.624146i	$-0.785447\pi$
$317$	−2.79818	+	2.79818i	−0.157162	+	0.157162i	0.624146	+	0.781308i	$0.285447\pi$
$318$		0			0
$319$	2.01783			0.112977
$320$		0			0
$321$	−0.716826			−0.0400093
$322$		0			0
$323$	13.9101	−	13.9101i	0.773977	−	0.773977i
$324$		0			0
$325$	1.81644	+	1.81644i	0.100758	+	0.100758i
$326$		0			0
$327$		−	6.94795i		−	0.384222i
$328$		0			0
$329$			1.91208i			0.105416i
$330$		0			0
$331$	−13.5439	−	13.5439i	−0.744439	−	0.744439i	0.228990	−	0.973429i	$-0.426458\pi$
$331$	−13.5439	−	13.5439i	−0.744439	−	0.744439i	−0.973429	+	0.228990i	$0.926458\pi$
$332$		0			0
$333$	−5.29245	+	5.29245i	−0.290024	+	0.290024i
$334$		0			0
$335$	2.87021			0.156817
$336$		0			0
$337$	17.1772			0.935703			0.467852	−	0.883807i	$-0.345028\pi$
$337$	17.1772			0.935703			0.467852	+	0.883807i	$0.345028\pi$
$338$		0			0
$339$	3.70619	−	3.70619i	0.201292	−	0.201292i
$340$		0			0
$341$	3.42862	+	3.42862i	0.185670	+	0.185670i
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$		−	1.23576i		−	0.0665311i
$346$		0			0
$347$	18.6176	+	18.6176i	0.999445	+	0.999445i	1.00000		0.000555063i	$-0.000176682\pi$
$347$	18.6176	+	18.6176i	0.999445	+	0.999445i	−0.000555063		1.00000i	$0.500177\pi$
$348$		0			0
$349$	5.04982	−	5.04982i	0.270311	−	0.270311i	−0.558915	−	0.829225i	$-0.688782\pi$
$349$	5.04982	−	5.04982i	0.270311	−	0.270311i	0.829225	+	0.558915i	$0.188782\pi$
$350$		0			0
$351$	−5.95260			−0.317726
$352$		0			0
$353$	20.8219			1.10824			0.554120	−	0.832437i	$-0.313055\pi$
$353$	20.8219			1.10824			0.554120	+	0.832437i	$0.313055\pi$
$354$		0			0
$355$	4.12060	−	4.12060i	0.218699	−	0.218699i
$356$		0			0
$357$	1.71707	+	1.71707i	0.0908769	+	0.0908769i
$358$		0			0
$359$			18.8326i			0.993948i	0.867766	+	0.496974i	$0.165556\pi$
$359$			18.8326i			0.993948i	−0.867766	+	0.496974i	$0.834444\pi$
$360$		0			0
$361$			8.67723i			0.456696i
$362$		0			0
$363$	2.92242	+	2.92242i	0.153387	+	0.153387i
$364$		0			0
$365$	5.12215	−	5.12215i	0.268106	−	0.268106i
$366$		0			0
$367$	32.9016			1.71745			0.858725	−	0.512436i	$-0.171257\pi$
$367$	32.9016			1.71745			0.858725	+	0.512436i	$0.171257\pi$
$368$		0			0
$369$	26.0025			1.35364
$370$		0			0
$371$	5.44648	−	5.44648i	0.282767	−	0.282767i
$372$		0			0
$373$	16.4641	+	16.4641i	0.852481	+	0.852481i	0.990438	−	0.137957i	$-0.0440536\pi$
$373$	16.4641	+	16.4641i	0.852481	+	0.852481i	−0.137957	+	0.990438i	$0.544054\pi$
$374$		0			0
$375$		−	0.396603i		−	0.0204805i
$376$		0			0
$377$			6.81104i			0.350786i
$378$		0			0
$379$	15.6190	+	15.6190i	0.802293	+	0.802293i	0.983454	−	0.181161i	$-0.0579854\pi$
$379$	15.6190	+	15.6190i	0.802293	+	0.802293i	−0.181161	+	0.983454i	$0.557985\pi$
$380$		0			0
$381$	0.936519	−	0.936519i	0.0479793	−	0.0479793i
$382$		0			0
$383$	28.4217			1.45228			0.726139	−	0.687548i	$-0.241313\pi$
$383$	28.4217			1.45228			0.726139	+	0.687548i	$0.241313\pi$
$384$		0			0
$385$	0.761040			0.0387862
$386$		0			0
$387$	−15.0472	+	15.0472i	−0.764890	+	0.764890i
$388$		0			0
$389$	8.97817	+	8.97817i	0.455211	+	0.455211i	0.897080	−	0.441868i	$-0.145684\pi$
$389$	8.97817	+	8.97817i	0.455211	+	0.455211i	−0.441868	+	0.897080i	$0.645684\pi$
$390$		0			0
$391$		−	19.0776i		−	0.964796i
$392$		0			0
$393$		−	0.207417i		−	0.0104628i
$394$		0			0
$395$	−6.79603	−	6.79603i	−0.341945	−	0.341945i
$396$		0			0
$397$	15.2501	−	15.2501i	0.765382	−	0.765382i	−0.211908	−	0.977290i	$-0.567968\pi$
$397$	15.2501	−	15.2501i	0.765382	−	0.765382i	0.977290	+	0.211908i	$0.0679677\pi$
$398$		0			0
$399$	1.27425			0.0637923
$400$		0			0
$401$	10.5556			0.527122			0.263561	−	0.964643i	$-0.415103\pi$
$401$	10.5556			0.527122			0.263561	+	0.964643i	$0.415103\pi$
$402$		0			0
$403$	−11.5730	+	11.5730i	−0.576494	+	0.576494i
$404$		0			0
$405$	−5.38044	−	5.38044i	−0.267356	−	0.267356i
$406$		0			0
$407$		−	2.00377i		−	0.0993230i
$408$		0			0
$409$		−	2.17136i		−	0.107367i	−0.998558	−	0.0536833i	$-0.982904\pi$
$409$		−	2.17136i		−	0.107367i	0.998558	−	0.0536833i	$-0.0170961\pi$
$410$		0			0
$411$	−5.07551	−	5.07551i	−0.250356	−	0.250356i
$412$		0			0
$413$	−3.78946	+	3.78946i	−0.186467	+	0.186467i
$414$		0			0
$415$	−7.12136			−0.349574
$416$		0			0
$417$	−1.75707			−0.0860441
$418$		0			0
$419$	23.9970	−	23.9970i	1.17233	−	1.17233i	0.190677	−	0.981653i	$-0.438932\pi$
$419$	23.9970	−	23.9970i	1.17233	−	1.17233i	0.981653	−	0.190677i	$-0.0610682\pi$
$420$		0			0
$421$	−22.4639	−	22.4639i	−1.09482	−	1.09482i	−0.995006	−	0.0998155i	$-0.968175\pi$
$421$	−22.4639	−	22.4639i	−1.09482	−	1.09482i	−0.0998155	−	0.995006i	$-0.531825\pi$
$422$		0			0
$423$			5.43547i			0.264281i
$424$		0			0
$425$		−	6.12274i		−	0.296997i
$426$		0			0
$427$	−8.50125	−	8.50125i	−0.411404	−	0.411404i
$428$		0			0
$429$	0.548258	−	0.548258i	0.0264701	−	0.0264701i
$430$		0			0
$431$	26.4487			1.27399			0.636995	−	0.770868i	$-0.280177\pi$
$431$	26.4487			1.27399			0.636995	+	0.770868i	$0.280177\pi$
$432$		0			0
$433$	2.56549			0.123289			0.0616447	−	0.998098i	$-0.480365\pi$
$433$	2.56549			0.123289			0.0616447	+	0.998098i	$0.480365\pi$
$434$		0			0
$435$	0.743564	−	0.743564i	0.0356512	−	0.0356512i
$436$		0			0
$437$	−7.07882	−	7.07882i	−0.338626	−	0.338626i
$438$		0			0
$439$			21.1679i			1.01029i	0.863036	+	0.505143i	$0.168560\pi$
$439$			21.1679i			1.01029i	−0.863036	+	0.505143i	$0.831440\pi$
$440$		0			0
$441$		−	2.84271i		−	0.135367i
$442$		0			0
$443$	3.78588	+	3.78588i	0.179873	+	0.179873i	0.791300	−	0.611428i	$-0.209404\pi$
$443$	3.78588	+	3.78588i	0.179873	+	0.179873i	−0.611428	+	0.791300i	$0.709404\pi$
$444$		0			0
$445$	6.19750	−	6.19750i	0.293790	−	0.293790i
$446$		0			0
$447$	−2.49587			−0.118050
$448$		0			0
$449$	0.750349			0.0354111			0.0177056	−	0.999843i	$-0.494364\pi$
$449$	0.750349			0.0354111			0.0177056	+	0.999843i	$0.494364\pi$
$450$		0			0
$451$	−4.92239	+	4.92239i	−0.231786	+	0.231786i
$452$		0			0
$453$	−1.93819	−	1.93819i	−0.0910642	−	0.0910642i
$454$		0			0
$455$			2.56883i			0.120429i
$456$		0			0
$457$			5.38872i			0.252074i	0.992026	+	0.126037i	$0.0402258\pi$
$457$			5.38872i			0.252074i	−0.992026	+	0.126037i	$0.959774\pi$
$458$		0			0
$459$	10.0323	+	10.0323i	0.468269	+	0.468269i
$460$		0			0
$461$	7.34944	−	7.34944i	0.342297	−	0.342297i	−0.514933	−	0.857230i	$-0.672183\pi$
$461$	7.34944	−	7.34944i	0.342297	−	0.342297i	0.857230	+	0.514933i	$0.172183\pi$
$462$		0			0
$463$	−21.5027			−0.999317			−0.499659	−	0.866222i	$-0.666541\pi$
$463$	−21.5027			−0.999317			−0.499659	+	0.866222i	$0.666541\pi$
$464$		0			0
$465$	2.52687			0.117181
$466$		0			0
$467$	−0.415352	+	0.415352i	−0.0192202	+	0.0192202i	−0.716652	−	0.697431i	$-0.754326\pi$
$467$	−0.415352	+	0.415352i	−0.0192202	+	0.0192202i	0.697431	+	0.716652i	$0.254326\pi$
$468$		0			0
$469$	2.02955	+	2.02955i	0.0937158	+	0.0937158i
$470$		0			0
$471$		−	2.74714i		−	0.126582i
$472$		0			0
$473$		−	5.69699i		−	0.261948i
$474$		0			0
$475$	−2.27187	−	2.27187i	−0.104240	−	0.104240i
$476$		0			0
$477$	15.4827	−	15.4827i	0.708906	−	0.708906i
$478$		0			0
$479$	−5.02367			−0.229538			−0.114769	−	0.993392i	$-0.536613\pi$
$479$	−5.02367			−0.229538			−0.114769	+	0.993392i	$0.536613\pi$
$480$		0			0
$481$	6.76356			0.308392
$482$		0			0
$483$	0.873814	−	0.873814i	0.0397599	−	0.0397599i
$484$		0			0
$485$	3.53762	+	3.53762i	0.160635	+	0.160635i
$486$		0			0
$487$			25.4422i			1.15290i	0.817134	+	0.576448i	$0.195562\pi$
$487$			25.4422i			1.15290i	−0.817134	+	0.576448i	$0.804438\pi$
$488$		0			0
$489$		−	9.48190i		−	0.428786i
$490$		0			0
$491$	4.76455	+	4.76455i	0.215021	+	0.215021i	0.806396	−	0.591375i	$-0.201415\pi$
$491$	4.76455	+	4.76455i	0.215021	+	0.215021i	−0.591375	+	0.806396i	$0.701415\pi$
$492$		0			0
$493$	11.4791	−	11.4791i	0.516993	−	0.516993i
$494$		0			0
$495$	2.16341			0.0972382
$496$		0			0
$497$	5.82741			0.261395
$498$		0			0
$499$	−20.4662	+	20.4662i	−0.916195	+	0.916195i	−0.996750	−	0.0805556i	$-0.974331\pi$
$499$	−20.4662	+	20.4662i	−0.916195	+	0.916195i	0.0805556	+	0.996750i	$0.474331\pi$
$500$		0			0
$501$	−4.24261	−	4.24261i	−0.189546	−	0.189546i
$502$		0			0
$503$		−	25.7819i		−	1.14956i	−0.818309	−	0.574778i	$-0.805088\pi$
$503$		−	25.7819i		−	1.14956i	0.818309	−	0.574778i	$-0.194912\pi$
$504$		0			0
$505$		−	0.547348i		−	0.0243567i
$506$		0			0
$507$	−1.79513	−	1.79513i	−0.0797244	−	0.0797244i
$508$		0			0
$509$	−30.6240	+	30.6240i	−1.35739	+	1.35739i	−0.480261	+	0.877126i	$0.659458\pi$
$509$	−30.6240	+	30.6240i	−1.35739	+	1.35739i	−0.877126	+	0.480261i	$0.840542\pi$
$510$		0			0
$511$	7.24382			0.320448
$512$		0			0
$513$	7.44507			0.328708
$514$		0			0
$515$	−5.11424	+	5.11424i	−0.225360	+	0.225360i
$516$		0			0
$517$	−1.02896	−	1.02896i	−0.0452535	−	0.0452535i
$518$		0			0
$519$		−	4.09040i		−	0.179549i
$520$		0			0
$521$		−	4.78332i		−	0.209561i	−0.994495	−	0.104780i	$-0.966586\pi$
$521$		−	4.78332i		−	0.209561i	0.994495	−	0.104780i	$-0.0334140\pi$
$522$		0			0
$523$	22.6776	+	22.6776i	0.991623	+	0.991623i	0.999965	−	0.00834240i	$-0.00265550\pi$
$523$	22.6776	+	22.6776i	0.991623	+	0.991623i	−0.00834240	+	0.999965i	$0.502655\pi$
$524$		0			0
$525$	0.280441	−	0.280441i	0.0122394	−	0.0122394i
$526$		0			0
$527$	39.0097			1.69929
$528$		0			0
$529$	13.2914			0.577888
$530$		0			0
$531$	−10.7723	+	10.7723i	−0.467479	+	0.467479i
$532$		0			0
$533$	−16.6152	−	16.6152i	−0.719683	−	0.719683i
$534$		0			0
$535$			1.80741i			0.0781412i
$536$		0			0
$537$			7.83697i			0.338190i
$538$		0			0
$539$	0.538137	+	0.538137i	0.0231792	+	0.0231792i
$540$		0			0
$541$	−20.7807	+	20.7807i	−0.893433	+	0.893433i	−0.994845	−	0.101412i	$-0.967664\pi$
$541$	−20.7807	+	20.7807i	−0.893433	+	0.893433i	0.101412	+	0.994845i	$0.467664\pi$
$542$		0			0
$543$	−3.17882			−0.136416
$544$		0			0
$545$	−17.5186			−0.750416
$546$		0			0
$547$	19.7510	−	19.7510i	0.844491	−	0.844491i	−0.144948	−	0.989439i	$-0.546301\pi$
$547$	19.7510	−	19.7510i	0.844491	−	0.844491i	0.989439	+	0.144948i	$0.0463014\pi$
$548$		0			0
$549$	−24.1665	−	24.1665i	−1.03140	−	1.03140i
$550$		0			0
$551$		−	8.51874i		−	0.362911i
$552$		0			0
$553$		−	9.61104i		−	0.408703i
$554$		0			0
$555$	−0.738382	−	0.738382i	−0.0313425	−	0.0313425i
$556$		0			0
$557$	28.6107	−	28.6107i	1.21227	−	1.21227i	0.241998	−	0.970277i	$-0.422197\pi$
$557$	28.6107	−	28.6107i	1.21227	−	1.21227i	0.970277	−	0.241998i	$-0.0778026\pi$
$558$		0			0
$559$	19.2298			0.813332
$560$		0			0
$561$	−1.84803			−0.0780241
$562$		0			0
$563$	22.1708	−	22.1708i	0.934387	−	0.934387i	−0.0635892	−	0.997976i	$-0.520255\pi$
$563$	22.1708	−	22.1708i	0.934387	−	0.934387i	0.997976	+	0.0635892i	$0.0202547\pi$
$564$		0			0
$565$	−9.34482	−	9.34482i	−0.393139	−	0.393139i
$566$		0			0
$567$		−	7.60909i		−	0.319552i
$568$		0			0
$569$		−	29.9798i		−	1.25682i	−0.777882	−	0.628410i	$-0.783706\pi$
$569$		−	29.9798i		−	1.25682i	0.777882	−	0.628410i	$-0.216294\pi$
$570$		0			0
$571$	28.8651	+	28.8651i	1.20797	+	1.20797i	0.971684	+	0.236284i	$0.0759296\pi$
$571$	28.8651	+	28.8651i	1.20797	+	1.20797i	0.236284	+	0.971684i	$0.424070\pi$
$572$		0			0
$573$	3.30604	−	3.30604i	0.138112	−	0.138112i
$574$		0			0
$575$	−3.11586			−0.129940
$576$		0			0
$577$	−20.5378			−0.855002			−0.427501	−	0.904015i	$-0.640606\pi$
$577$	−20.5378			−0.855002			−0.427501	+	0.904015i	$0.640606\pi$
$578$		0			0
$579$	−7.73088	+	7.73088i	−0.321284	+	0.321284i
$580$		0			0
$581$	−5.03556	−	5.03556i	−0.208910	−	0.208910i
$582$		0			0
$583$			5.86190i			0.242775i
$584$		0			0
$585$			7.30244i			0.301919i
$586$		0			0
$587$	−29.7578	−	29.7578i	−1.22824	−	1.22824i	−0.964630	−	0.263606i	$-0.915088\pi$
$587$	−29.7578	−	29.7578i	−1.22824	−	1.22824i	−0.263606	−	0.964630i	$-0.584912\pi$
$588$		0			0
$589$	14.4747	−	14.4747i	0.596420	−	0.596420i
$590$		0			0
$591$	−5.38757			−0.221615
$592$		0			0
$593$	−35.9198			−1.47505			−0.737525	−	0.675320i	$-0.764006\pi$
$593$	−35.9198			−1.47505			−0.737525	+	0.675320i	$0.764006\pi$
$594$		0			0
$595$	4.32943	−	4.32943i	0.177489	−	0.177489i
$596$		0			0
$597$	6.41332	+	6.41332i	0.262480	+	0.262480i
$598$		0			0
$599$			15.8766i			0.648699i	0.945937	+	0.324349i	$0.105145\pi$
$599$			15.8766i			0.648699i	−0.945937	+	0.324349i	$0.894855\pi$
$600$		0			0
$601$		−	32.0705i		−	1.30818i	−0.756415	−	0.654092i	$-0.773051\pi$
$601$		−	32.0705i		−	1.30818i	0.756415	−	0.654092i	$-0.226949\pi$
$602$		0			0
$603$	5.76941	+	5.76941i	0.234948	+	0.234948i
$604$		0			0
$605$	7.36863	−	7.36863i	0.299577	−	0.299577i
$606$		0			0
$607$	43.6666			1.77237			0.886187	−	0.463328i	$-0.153345\pi$
$607$	43.6666			1.77237			0.886187	+	0.463328i	$0.153345\pi$
$608$		0			0
$609$	1.05156			0.0426113
$610$		0			0
$611$	3.47317	−	3.47317i	0.140509	−	0.140509i
$612$		0			0
$613$	18.0335	+	18.0335i	0.728367	+	0.728367i	0.970294	−	0.241927i	$-0.0777796\pi$
$613$	18.0335	+	18.0335i	0.728367	+	0.728367i	−0.241927	+	0.970294i	$0.577780\pi$
$614$		0			0
$615$			3.62777i			0.146286i
$616$		0			0
$617$			14.2892i			0.575262i	0.957741	+	0.287631i	$0.0928676\pi$
$617$			14.2892i			0.575262i	−0.957741	+	0.287631i	$0.907132\pi$
$618$		0			0
$619$	32.0653	+	32.0653i	1.28881	+	1.28881i	0.935508	+	0.353307i	$0.114943\pi$
$619$	32.0653	+	32.0653i	1.28881	+	1.28881i	0.353307	+	0.935508i	$0.385057\pi$
$620$		0			0
$621$	5.10544	−	5.10544i	0.204874	−	0.204874i
$622$		0			0
$623$	8.76458			0.351146
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$	−0.685720	+	0.685720i	−0.0273850	+	0.0273850i
$628$		0			0
$629$	−11.3991	−	11.3991i	−0.454512	−	0.454512i
$630$		0			0
$631$			14.5737i			0.580169i	0.957001	+	0.290085i	$0.0936834\pi$
$631$			14.5737i			0.580169i	−0.957001	+	0.290085i	$0.906317\pi$
$632$		0			0
$633$			6.59297i			0.262047i
$634$		0			0
$635$	−2.36135	−	2.36135i	−0.0937072	−	0.0937072i
$636$		0			0
$637$	−1.81644	+	1.81644i	−0.0719700	+	0.0719700i
$638$		0			0
$639$	16.5656			0.655325
$640$		0			0
$641$	−14.6153			−0.577268			−0.288634	−	0.957440i	$-0.593201\pi$
$641$	−14.6153			−0.577268			−0.288634	+	0.957440i	$0.593201\pi$
$642$		0			0
$643$	5.01589	−	5.01589i	0.197807	−	0.197807i	−0.601252	−	0.799059i	$-0.705331\pi$
$643$	5.01589	−	5.01589i	0.197807	−	0.197807i	0.799059	+	0.601252i	$0.205331\pi$
$644$		0			0
$645$	−2.09932	−	2.09932i	−0.0826607	−	0.0826607i
$646$		0			0
$647$			18.5494i			0.729252i	0.931154	+	0.364626i	$0.118803\pi$
$647$			18.5494i			0.729252i	−0.931154	+	0.364626i	$0.881197\pi$
$648$		0			0
$649$		−	4.07850i		−	0.160095i
$650$		0			0
$651$	1.78677	+	1.78677i	0.0700289	+	0.0700289i
$652$		0			0
$653$	−0.459246	+	0.459246i	−0.0179717	+	0.0179717i	−0.716036	−	0.698064i	$-0.754045\pi$
$653$	−0.459246	+	0.459246i	−0.0179717	+	0.0179717i	0.698064	+	0.716036i	$0.254045\pi$
$654$		0			0
$655$	−0.522983			−0.0204346
$656$		0			0
$657$	20.5920			0.803372
$658$		0			0
$659$	−13.4361	+	13.4361i	−0.523395	+	0.523395i	−0.918595	−	0.395200i	$-0.870675\pi$
$659$	−13.4361	+	13.4361i	−0.523395	+	0.523395i	0.395200	+	0.918595i	$0.370675\pi$
$660$		0			0
$661$	−5.00769	−	5.00769i	−0.194777	−	0.194777i	0.602980	−	0.797756i	$-0.293980\pi$
$661$	−5.00769	−	5.00769i	−0.194777	−	0.194777i	−0.797756	+	0.602980i	$0.793980\pi$
$662$		0			0
$663$		−	6.23790i		−	0.242260i
$664$		0			0
$665$		−	3.21291i		−	0.124591i
$666$		0			0
$667$	−5.84171	−	5.84171i	−0.226192	−	0.226192i
$668$		0			0
$669$	5.32327	−	5.32327i	0.205810	−	0.205810i
$670$		0			0
$671$	9.14966			0.353219
$672$		0			0
$673$	9.29970			0.358477			0.179239	−	0.983806i	$-0.442637\pi$
$673$	9.29970			0.358477			0.179239	+	0.983806i	$0.442637\pi$
$674$		0			0
$675$	1.63853	−	1.63853i	0.0630672	−	0.0630672i
$676$		0			0
$677$	22.9866	+	22.9866i	0.883447	+	0.883447i	0.993883	−	0.110436i	$-0.0352248\pi$
$677$	22.9866	+	22.9866i	0.883447	+	0.883447i	−0.110436	+	0.993883i	$0.535225\pi$
$678$		0			0
$679$			5.00295i			0.191995i
$680$		0			0
$681$			10.0990i			0.386993i
$682$		0			0
$683$	−20.0382	−	20.0382i	−0.766740	−	0.766740i	0.210791	−	0.977531i	$-0.432396\pi$
$683$	−20.0382	−	20.0382i	−0.766740	−	0.766740i	−0.977531	+	0.210791i	$0.932396\pi$
$684$		0			0
$685$	−12.7974	+	12.7974i	−0.488965	+	0.488965i
$686$		0			0
$687$	−6.49961			−0.247975
$688$		0			0
$689$	−19.7864			−0.753802
$690$		0			0
$691$	−15.3051	+	15.3051i	−0.582233	+	0.582233i	−0.935516	−	0.353283i	$-0.885065\pi$
$691$	−15.3051	+	15.3051i	−0.582233	+	0.582233i	0.353283	+	0.935516i	$0.385065\pi$
$692$		0			0
$693$	1.52976	+	1.52976i	0.0581109	+	0.0581109i
$694$		0			0
$695$			4.43030i			0.168051i
$696$		0			0
$697$			56.0054i			2.12136i
$698$		0			0
$699$	4.01212	+	4.01212i	0.151752	+	0.151752i
$700$		0			0
$701$	21.6040	−	21.6040i	0.815973	−	0.815973i	−0.169549	−	0.985522i	$-0.554231\pi$
$701$	21.6040	−	21.6040i	0.815973	−	0.815973i	0.985522	+	0.169549i	$0.0542310\pi$
$702$		0			0
$703$	−8.45936			−0.319051
$704$		0			0
$705$	−0.758335			−0.0285606
$706$		0			0
$707$	0.387033	−	0.387033i	0.0145559	−	0.0145559i
$708$		0			0
$709$	13.1886	+	13.1886i	0.495310	+	0.495310i	0.909974	−	0.414664i	$-0.136101\pi$
$709$	13.1886	+	13.1886i	0.495310	+	0.495310i	−0.414664	+	0.909974i	$0.636101\pi$
$710$		0			0
$711$		−	27.3214i		−	1.02463i
$712$		0			0
$713$		−	19.8520i		−	0.743463i
$714$		0			0
$715$	−1.38238	−	1.38238i	−0.0516982	−	0.0516982i
$716$		0			0
$717$	7.23065	−	7.23065i	0.270034	−	0.270034i
$718$		0			0
$719$	13.0320			0.486011			0.243006	−	0.970025i	$-0.421867\pi$
$719$	13.0320			0.486011			0.243006	+	0.970025i	$0.421867\pi$
$720$		0			0
$721$	−7.23262			−0.269357
$722$		0			0
$723$	5.77282	−	5.77282i	0.214693	−	0.214693i
$724$		0			0
$725$	−1.87483	−	1.87483i	−0.0696295	−	0.0696295i
$726$		0			0
$727$			44.3601i			1.64522i	0.568603	+	0.822612i	$0.307484\pi$
$727$			44.3601i			1.64522i	−0.568603	+	0.822612i	$0.692516\pi$
$728$		0			0
$729$		−	18.8733i		−	0.699013i
$730$		0			0
$731$	−32.4092	−	32.4092i	−1.19870	−	1.19870i
$732$		0			0
$733$	−24.4991	+	24.4991i	−0.904894	+	0.904894i	−0.995854	−	0.0909607i	$-0.971006\pi$
$733$	−24.4991	+	24.4991i	−0.904894	+	0.904894i	0.0909607	+	0.995854i	$0.471006\pi$
$734$		0			0
$735$	0.396603			0.0146289
$736$		0			0
$737$	−2.18435			−0.0804615
$738$		0			0
$739$	−20.7191	+	20.7191i	−0.762164	+	0.762164i	−0.976713	−	0.214549i	$-0.931172\pi$
$739$	−20.7191	+	20.7191i	−0.762164	+	0.762164i	0.214549	+	0.976713i	$0.431172\pi$
$740$		0			0
$741$	−2.31460	−	2.31460i	−0.0850289	−	0.0850289i
$742$		0			0
$743$		−	14.9717i		−	0.549258i	−0.961550	−	0.274629i	$-0.911445\pi$
$743$		−	14.9717i		−	0.549258i	0.961550	−	0.274629i	$-0.0885550\pi$
$744$		0			0
$745$			6.29311i			0.230562i
$746$		0			0
$747$	−14.3146	−	14.3146i	−0.523744	−	0.523744i
$748$		0			0
$749$	−1.27803	+	1.27803i	−0.0466983	+	0.0466983i
$750$		0			0
$751$	−8.96690			−0.327207			−0.163603	−	0.986526i	$-0.552312\pi$
$751$	−8.96690			−0.327207			−0.163603	+	0.986526i	$0.552312\pi$
$752$		0			0
$753$	6.54752			0.238605
$754$		0			0
$755$	−4.88698	+	4.88698i	−0.177855	+	0.177855i
$756$		0			0
$757$	−1.12599	−	1.12599i	−0.0409248	−	0.0409248i	0.686348	−	0.727273i	$-0.259213\pi$
$757$	−1.12599	−	1.12599i	−0.0409248	−	0.0409248i	−0.727273	+	0.686348i	$0.759213\pi$
$758$		0			0
$759$			0.940463i			0.0341366i
$760$		0			0
$761$		−	22.3845i		−	0.811437i	−0.913998	−	0.405719i	$-0.867021\pi$
$761$		−	22.3845i		−	0.811437i	0.913998	−	0.405719i	$-0.132979\pi$
$762$		0			0
$763$	−12.3875	−	12.3875i	−0.448459	−	0.448459i
$764$		0			0
$765$	12.3073	−	12.3073i	0.444972	−	0.444972i
$766$		0			0
$767$	13.7667			0.497085
$768$		0			0
$769$	31.0269			1.11886			0.559429	−	0.828878i	$-0.311020\pi$
$769$	31.0269			1.11886			0.559429	+	0.828878i	$0.311020\pi$
$770$		0			0
$771$	2.50969	−	2.50969i	0.0903844	−	0.0903844i
$772$		0			0
$773$	4.39284	+	4.39284i	0.157999	+	0.157999i	0.781680	−	0.623680i	$-0.214363\pi$
$773$	4.39284	+	4.39284i	0.157999	+	0.157999i	−0.623680	+	0.781680i	$0.714363\pi$
$774$		0			0
$775$		−	6.37128i		−	0.228863i
$776$		0			0
$777$		−	1.04423i		−	0.0374615i
$778$		0			0
$779$	20.7810	+	20.7810i	0.744558	+	0.744558i
$780$		0			0
$781$	−3.13594	+	3.13594i	−0.112213	+	0.112213i
$782$		0			0
$783$	6.14395			0.219567
$784$		0			0
$785$	−6.92667			−0.247224
$786$		0			0
$787$	−10.0484	+	10.0484i	−0.358188	+	0.358188i	−0.863145	−	0.504957i	$-0.831508\pi$
$787$	−10.0484	+	10.0484i	−0.358188	+	0.358188i	0.504957	+	0.863145i	$0.331508\pi$
$788$		0			0
$789$	−2.67728	−	2.67728i	−0.0953138	−	0.0953138i
$790$		0			0
$791$		−	13.2156i		−	0.469892i
$792$		0			0
$793$			30.8840i			1.09672i
$794$		0			0
$795$	2.16009	+	2.16009i	0.0766106	+	0.0766106i
$796$		0			0
$797$	−7.24211	+	7.24211i	−0.256529	+	0.256529i	−0.823641	−	0.567112i	$-0.808061\pi$
$797$	−7.24211	+	7.24211i	−0.256529	+	0.256529i	0.567112	+	0.823641i	$0.308061\pi$
$798$		0			0
$799$	−11.7071			−0.414169
$800$		0			0
$801$	24.9151			0.880333
$802$		0			0
$803$	−3.89816	+	3.89816i	−0.137563	+	0.137563i
$804$		0			0
$805$	−2.20324	−	2.20324i	−0.0776542	−	0.0776542i
$806$		0			0
$807$		−	0.937029i		−	0.0329850i
$808$		0			0
$809$			45.1883i			1.58874i	0.607436	+	0.794368i	$0.292198\pi$
$809$			45.1883i			1.58874i	−0.607436	+	0.794368i	$0.707802\pi$
$810$		0			0
$811$	−7.86020	−	7.86020i	−0.276009	−	0.276009i	0.555505	−	0.831513i	$-0.312525\pi$
$811$	−7.86020	−	7.86020i	−0.276009	−	0.276009i	−0.831513	+	0.555505i	$0.812525\pi$
$812$		0			0
$813$	6.37025	−	6.37025i	0.223414	−	0.223414i
$814$		0			0
$815$	−23.9078			−0.837452
$816$		0			0
$817$	−24.0512			−0.841443
$818$		0			0
$819$	−5.16360	+	5.16360i	−0.180431	+	0.180431i
$820$		0			0
$821$	−22.0445	−	22.0445i	−0.769358	−	0.769358i	0.208635	−	0.977993i	$-0.433098\pi$
$821$	−22.0445	−	22.0445i	−0.769358	−	0.769358i	−0.977993	+	0.208635i	$0.933098\pi$
$822$		0			0
$823$			12.2385i			0.426608i	0.976986	+	0.213304i	$0.0684225\pi$
$823$			12.2385i			0.426608i	−0.976986	+	0.213304i	$0.931577\pi$
$824$		0			0
$825$			0.301831i			0.0105084i
$826$		0			0
$827$	−36.9246	−	36.9246i	−1.28399	−	1.28399i	−0.938375	−	0.345620i	$-0.887669\pi$
$827$	−36.9246	−	36.9246i	−1.28399	−	1.28399i	−0.345620	−	0.938375i	$-0.612331\pi$
$828$		0			0
$829$	3.79488	−	3.79488i	0.131801	−	0.131801i	−0.638128	−	0.769930i	$-0.720291\pi$
$829$	3.79488	−	3.79488i	0.131801	−	0.131801i	0.769930	+	0.638128i	$0.220291\pi$
$830$		0			0
$831$	−6.19209			−0.214801
$832$		0			0
$833$	6.12274			0.212141
$834$		0			0
$835$	−10.6974	+	10.6974i	−0.370197	+	0.370197i
$836$		0			0
$837$	10.4396	+	10.4396i	0.360844	+	0.360844i
$838$		0			0
$839$		−	7.66012i		−	0.264457i	−0.991219	−	0.132228i	$-0.957787\pi$
$839$		−	7.66012i		−	0.264457i	0.991219	−	0.132228i	$-0.0422132\pi$
$840$		0			0
$841$			21.9700i			0.757587i
$842$		0			0
$843$	−3.56000	−	3.56000i	−0.122613	−	0.122613i
$844$		0			0
$845$	−4.52626	+	4.52626i	−0.155708	+	0.155708i
$846$		0			0
$847$	10.4208			0.358064
$848$		0			0
$849$	−5.33520			−0.183104
$850$		0			0
$851$	−5.80099	+	5.80099i	−0.198855	+	0.198855i
$852$		0			0
$853$	−9.32297	−	9.32297i	−0.319212	−	0.319212i	0.529252	−	0.848465i	$-0.322473\pi$
$853$	−9.32297	−	9.32297i	−0.319212	−	0.319212i	−0.848465	+	0.529252i	$0.822473\pi$
$854$		0			0
$855$		−	9.13335i		−	0.312354i
$856$		0			0
$857$			10.3346i			0.353022i	0.984299	+	0.176511i	$0.0564812\pi$
$857$			10.3346i			0.353022i	−0.984299	+	0.176511i	$0.943519\pi$
$858$		0			0
$859$	−31.5542	−	31.5542i	−1.07662	−	1.07662i	−0.996810	−	0.0798052i	$-0.974570\pi$
$859$	−31.5542	−	31.5542i	−1.07662	−	1.07662i	−0.0798052	−	0.996810i	$-0.525430\pi$
$860$		0			0
$861$	−2.56522	+	2.56522i	−0.0874226	+	0.0874226i
$862$		0			0
$863$	7.29297			0.248256			0.124128	−	0.992266i	$-0.460387\pi$
$863$	7.29297			0.248256			0.124128	+	0.992266i	$0.460387\pi$
$864$		0			0
$865$	−10.3136			−0.350672
$866$		0			0
$867$	−5.74567	+	5.74567i	−0.195133	+	0.195133i
$868$		0			0
$869$	5.17205	+	5.17205i	0.175450	+	0.175450i
$870$		0			0
$871$		−	7.37310i		−	0.249828i
$872$		0			0
$873$			14.2219i			0.481338i
$874$		0			0
$875$	−0.707107	−	0.707107i	−0.0239046	−	0.0239046i
$876$		0			0
$877$	33.8672	−	33.8672i	1.14362	−	1.14362i	0.155832	−	0.987784i	$-0.450194\pi$
$877$	33.8672	−	33.8672i	1.14362	−	1.14362i	0.987784	−	0.155832i	$-0.0498059\pi$
$878$		0			0
$879$	−1.65077			−0.0556791
$880$		0			0
$881$	−34.4761			−1.16153			−0.580765	−	0.814071i	$-0.697246\pi$
$881$	−34.4761			−1.16153			−0.580765	+	0.814071i	$0.697246\pi$
$882$		0			0
$883$	−11.7629	+	11.7629i	−0.395853	+	0.395853i	−0.876767	−	0.480915i	$-0.840305\pi$
$883$	−11.7629	+	11.7629i	−0.395853	+	0.395853i	0.480915	+	0.876767i	$0.340305\pi$
$884$		0			0
$885$	−1.50291	−	1.50291i	−0.0505199	−	0.0505199i
$886$		0			0
$887$			24.7921i			0.832438i	0.909264	+	0.416219i	$0.136645\pi$
$887$			24.7921i			0.832438i	−0.909264	+	0.416219i	$0.863355\pi$
$888$		0			0
$889$		−	3.33945i		−	0.112002i
$890$		0			0
$891$	4.09473	+	4.09473i	0.137179	+	0.137179i
$892$		0			0
$893$	−4.34398	+	4.34398i	−0.145366	+	0.145366i
$894$		0			0
$895$	19.7602			0.660511
$896$		0			0
$897$	−3.17446			−0.105992
$898$		0			0
$899$	11.9451	−	11.9451i	0.398390	−	0.398390i
$900$		0			0
$901$	33.3474	+	33.3474i	1.11096	+	1.11096i
$902$		0			0
$903$		−	2.96889i		−	0.0987985i
$904$		0			0
$905$			8.01511i			0.266431i
$906$		0			0
$907$	4.41209	+	4.41209i	0.146501	+	0.146501i	0.776553	−	0.630052i	$-0.216966\pi$
$907$	4.41209	+	4.41209i	0.146501	+	0.146501i	−0.630052	+	0.776553i	$0.716966\pi$
$908$		0			0
$909$	1.10022	−	1.10022i	0.0364921	−	0.0364921i
$910$		0			0
$911$	−52.7897			−1.74900			−0.874500	−	0.485025i	$-0.838811\pi$
$911$	−52.7897			−1.74900			−0.874500	+	0.485025i	$0.838811\pi$
$912$		0			0
$913$	5.41964			0.179364
$914$		0			0
$915$	3.37162	−	3.37162i	0.111462	−	0.111462i
$916$		0			0
$917$	−0.369805	−	0.369805i	−0.0122120	−	0.0122120i
$918$		0			0
$919$		−	56.1208i		−	1.85126i	−0.378436	−	0.925628i	$-0.623538\pi$
$919$		−	56.1208i		−	1.85126i	0.378436	−	0.925628i	$-0.376462\pi$
$920$		0			0
$921$		−	1.64661i		−	0.0542577i
$922$		0			0
$923$	−10.5851	−	10.5851i	−0.348414	−	0.348414i
$924$		0			0
$925$	−1.86176	+	1.86176i	−0.0612144	+	0.0612144i
$926$		0			0
$927$	−20.5602			−0.675286
$928$		0			0
$929$	−12.2572			−0.402146			−0.201073	−	0.979576i	$-0.564443\pi$
$929$	−12.2572			−0.402146			−0.201073	+	0.979576i	$0.564443\pi$
$930$		0			0
$931$	2.27187	−	2.27187i	0.0744575	−	0.0744575i
$932$		0			0
$933$	8.15569	+	8.15569i	0.267005	+	0.267005i
$934$		0			0
$935$			4.65965i			0.152387i
$936$		0			0
$937$			40.6023i			1.32642i	0.748434	+	0.663209i	$0.230806\pi$
$937$			40.6023i			1.32642i	−0.748434	+	0.663209i	$0.769194\pi$
$938$		0			0
$939$	−0.552072	−	0.552072i	−0.0180162	−	0.0180162i
$940$		0			0
$941$	29.5220	−	29.5220i	0.962389	−	0.962389i	−0.0369292	−	0.999318i	$-0.511758\pi$
$941$	29.5220	−	29.5220i	0.962389	−	0.962389i	0.999318	+	0.0369292i	$0.0117576\pi$
$942$		0			0
$943$	28.5011			0.928123
$944$		0			0
$945$	2.31724			0.0753797
$946$		0			0
$947$	41.1792	−	41.1792i	1.33814	−	1.33814i	0.440286	−	0.897858i	$-0.354877\pi$
$947$	41.1792	−	41.1792i	1.33814	−	1.33814i	0.897858	−	0.440286i	$-0.145123\pi$
$948$		0			0
$949$	−13.1580	−	13.1580i	−0.427125	−	0.427125i
$950$		0			0
$951$		−	1.56945i		−	0.0508929i
$952$		0			0
$953$			22.8564i			0.740390i	0.928954	+	0.370195i	$0.120709\pi$
$953$			22.8564i			0.740390i	−0.928954	+	0.370195i	$0.879291\pi$
$954$		0			0
$955$	−8.33589	−	8.33589i	−0.269743	−	0.269743i
$956$		0			0
$957$	−0.565882	+	0.565882i	−0.0182924	+	0.0182924i
$958$		0			0
$959$	−18.0983			−0.584425
$960$		0			0
$961$	9.59319			0.309458
$962$		0			0
$963$	−3.63307	+	3.63307i	−0.117074	+	0.117074i
$964$		0			0
$965$	19.4927	+	19.4927i	0.627493	+	0.627493i
$966$		0			0
$967$		−	8.32781i		−	0.267804i	−0.990995	−	0.133902i	$-0.957249\pi$
$967$		−	8.32781i		−	0.267804i	0.990995	−	0.133902i	$-0.0427508\pi$
$968$		0			0
$969$			7.80191i			0.250633i
$970$		0			0
$971$	−23.7276	−	23.7276i	−0.761455	−	0.761455i	0.215131	−	0.976585i	$-0.430982\pi$
$971$	−23.7276	−	23.7276i	−0.761455	−	0.761455i	−0.976585	+	0.215131i	$0.930982\pi$
$972$		0			0
$973$	−3.13269	+	3.13269i	−0.100429	+	0.100429i
$974$		0			0
$975$	−1.01881			−0.0326280
$976$		0			0
$977$	−22.1968			−0.710137			−0.355069	−	0.934840i	$-0.615542\pi$
$977$	−22.1968			−0.710137			−0.355069	+	0.934840i	$0.615542\pi$
$978$		0			0
$979$	−4.71654	+	4.71654i	−0.150741	+	0.150741i
$980$		0			0
$981$	−35.2141	−	35.2141i	−1.12430	−	1.12430i
$982$		0			0
$983$		−	5.10478i		−	0.162817i	−0.996681	−	0.0814085i	$-0.974058\pi$
$983$		−	5.10478i		−	0.162817i	0.996681	−	0.0814085i	$-0.0259419\pi$
$984$		0			0
$985$			13.5843i			0.432831i
$986$		0			0
$987$	−0.536224	−	0.536224i	−0.0170682	−	0.0170682i
$988$		0			0
$989$	−16.4930	+	16.4930i	−0.524448	+	0.524448i
$990$		0			0
$991$	8.05644			0.255921			0.127961	−	0.991779i	$-0.459157\pi$
$991$	8.05644			0.255921			0.127961	+	0.991779i	$0.459157\pi$
$992$		0			0
$993$	7.59652			0.241068
$994$		0			0
$995$	16.1706	−	16.1706i	0.512643	−	0.512643i
$996$		0			0
$997$	−29.0178	−	29.0178i	−0.919002	−	0.919002i	0.0779549	−	0.996957i	$-0.475161\pi$
$997$	−29.0178	−	29.0178i	−0.919002	−	0.919002i	−0.996957	+	0.0779549i	$0.975161\pi$
$998$		0			0
$999$		−	6.10112i		−	0.193031i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.bd.b.561.13		52
4.3	odd	2		560.2.bd.b.421.24	yes	52
16.3	odd	4		560.2.bd.b.141.24	✓	52
16.13	even	4	inner	2240.2.bd.b.1681.13		52

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.bd.b.141.24	✓	52	16.3	odd	4
560.2.bd.b.421.24	yes	52	4.3	odd	2
2240.2.bd.b.561.13		52	1.1	even	1	trivial
2240.2.bd.b.1681.13		52	16.13	even	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.bd (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.280441 + 0.280441i) q^{3} +(0.707107 + 0.707107i) q^{5} +1.00000i q^{7} +2.84271i q^{9} +O(q^{10})\)	\(q+(-0.280441 + 0.280441i) q^{3} +(0.707107 + 0.707107i) q^{5} +1.00000i q^{7} +2.84271i q^{9} +(-0.538137 - 0.538137i) q^{11} +(1.81644 - 1.81644i) q^{13} -0.396603 q^{15} -6.12274 q^{17} +(-2.27187 + 2.27187i) q^{19} +(-0.280441 - 0.280441i) q^{21} +3.11586i q^{23} +1.00000i q^{25} +(-1.63853 - 1.63853i) q^{27} +(-1.87483 + 1.87483i) q^{29} -6.37128 q^{31} +0.301831 q^{33} +(-0.707107 + 0.707107i) q^{35} +(1.86176 + 1.86176i) q^{37} +1.01881i q^{39} -9.14711i q^{41} +(5.29325 + 5.29325i) q^{43} +(-2.01010 + 2.01010i) q^{45} +1.91208 q^{47} -1.00000 q^{49} +(1.71707 - 1.71707i) q^{51} +(-5.44648 - 5.44648i) q^{53} -0.761040i q^{55} -1.27425i q^{57} +(3.78946 + 3.78946i) q^{59} +(-8.50125 + 8.50125i) q^{61} -2.84271 q^{63} +2.56883 q^{65} +(2.02955 - 2.02955i) q^{67} +(-0.873814 - 0.873814i) q^{69} -5.82741i q^{71} -7.24382i q^{73} +(-0.280441 - 0.280441i) q^{75} +(0.538137 - 0.538137i) q^{77} -9.61104 q^{79} -7.60909 q^{81} +(-5.03556 + 5.03556i) q^{83} +(-4.32943 - 4.32943i) q^{85} -1.05156i q^{87} -8.76458i q^{89} +(1.81644 + 1.81644i) q^{91} +(1.78677 - 1.78677i) q^{93} -3.21291 q^{95} +5.00295 q^{97} +(1.52976 - 1.52976i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q+O(q^{10}) \) 52 * q	\( 52 q + 4 q^{11} - 8 q^{15} - 8 q^{19} + 24 q^{27} + 4 q^{29} - 12 q^{37} - 36 q^{43} - 52 q^{49} + 8 q^{51} - 4 q^{53} - 24 q^{59} - 16 q^{61} + 68 q^{63} + 40 q^{65} + 12 q^{67} - 72 q^{69} - 4 q^{77} + 16 q^{79} - 116 q^{81} + 16 q^{85} + 8 q^{93} + 32 q^{95} + 52 q^{99}+O(q^{100}) \) 52 * q + 4 * q^11 - 8 * q^15 - 8 * q^19 + 24 * q^27 + 4 * q^29 - 12 * q^37 - 36 * q^43 - 52 * q^49 + 8 * q^51 - 4 * q^53 - 24 * q^59 - 16 * q^61 + 68 * q^63 + 40 * q^65 + 12 * q^67 - 72 * q^69 - 4 * q^77 + 16 * q^79 - 116 * q^81 + 16 * q^85 + 8 * q^93 + 32 * q^95 + 52 * q^99

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