LMFDB - Embedded newform 220.2.k.b.153.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [220,2,Mod(153,220)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(220, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 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("220.153");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 220 = 2^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	220.k (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:	$1.75670884447$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.303595776.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 $ x^8 + 5x^6 + 16x^4 + 45*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			153.1
Root			$-1.26217 - 1.18614i$ of defining polynomial
Character	$\chi$	$=$	220.153
Dual form			220.2.k.b.197.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.44831 - 2.44831i) q^{3} +(-0.469882 - 2.18614i) q^{5} +8.98844i q^{9} -3.31662 q^{11} +(-4.20193 + 6.50277i) q^{15} +(-5.54013 - 5.54013i) q^{23} +(-4.55842 + 2.05446i) q^{25} +(14.6616 - 14.6616i) q^{27} -0.644810 q^{31} +(8.12012 + 8.12012i) q^{33} +(2.96014 - 2.96014i) q^{37} +(19.6500 - 4.22351i) q^{45} +(-2.68338 + 2.68338i) q^{47} -7.00000i q^{49} +(-9.63325 - 9.63325i) q^{53} +(1.55842 + 7.25061i) q^{55} -11.3321i q^{59} +(-10.7001 + 10.7001i) q^{67} +27.1279i q^{69} +15.8614 q^{71} +(16.1904 + 6.13048i) q^{75} -44.8267 q^{81} -9.86141i q^{89} +(1.57869 + 1.57869i) q^{93} +(3.68467 - 3.68467i) q^{97} -29.8113i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{3} - 16 q^{15} - 18 q^{23} - 2 q^{25} + 26 q^{27} + 22 q^{33} + 14 q^{37} + 18 q^{45} - 48 q^{47} - 24 q^{53} - 22 q^{55} - 26 q^{67} + 12 q^{71} + 64 q^{75} - 100 q^{81} + 18 q^{93} + 34 q^{97}+O(q^{100}) $ 8 * q + 2 * q^3 - 16 * q^15 - 18 * q^23 - 2 * q^25 + 26 * q^27 + 22 * q^33 + 14 * q^37 + 18 * q^45 - 48 * q^47 - 24 * q^53 - 22 * q^55 - 26 * q^67 + 12 * q^71 + 64 * q^75 - 100 * q^81 + 18 * q^93 + 34 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/220\mathbb{Z}\right)^\times$.

$n$	$101$	$111$	$177$
$\chi(n)$	$-1$	$1$	$e\left(\frac{3}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−2.44831	−	2.44831i	−1.41353	−	1.41353i	−0.728714	−	0.684819i	$-0.759881\pi$
$3$	−2.44831	−	2.44831i	−1.41353	−	1.41353i	−0.684819	−	0.728714i	$-0.740119\pi$
$4$		0			0
$5$	−0.469882	−	2.18614i	−0.210138	−	0.977672i
$5$	−0.469882	−	2.18614i	−0.210138	−	0.977672i
$6$		0			0
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$		0			0
$9$			8.98844i			2.99615i
$10$		0			0
$11$	−3.31662			−1.00000
$11$	−3.31662			−1.00000
$12$		0			0
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$		0			0
$15$	−4.20193	+	6.50277i	−1.08493	+	1.67901i
$16$		0			0
$17$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$17$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−5.54013	−	5.54013i	−1.15520	−	1.15520i	−0.985496	−	0.169701i	$-0.945720\pi$
$23$	−5.54013	−	5.54013i	−1.15520	−	1.15520i	−0.169701	−	0.985496i	$-0.554280\pi$
$24$		0			0
$25$	−4.55842	+	2.05446i	−0.911684	+	0.410891i
$26$		0			0
$27$	14.6616	−	14.6616i	2.82162	−	2.82162i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−0.644810			−0.115811			−0.0579057	−	0.998322i	$-0.518442\pi$
$31$	−0.644810			−0.115811			−0.0579057	+	0.998322i	$0.518442\pi$
$32$		0			0
$33$	8.12012	+	8.12012i	1.41353	+	1.41353i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2.96014	−	2.96014i	0.486643	−	0.486643i	−0.420602	−	0.907245i	$-0.638181\pi$
$37$	2.96014	−	2.96014i	0.486643	−	0.486643i	0.907245	+	0.420602i	$0.138181\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$	19.6500	−	4.22351i	2.92925	−	0.629603i
$46$		0			0
$47$	−2.68338	+	2.68338i	−0.391411	+	0.391411i	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−2.68338	+	2.68338i	−0.391411	+	0.391411i	0.483779	+	0.875190i	$0.339264\pi$
$48$		0			0
$49$		−	7.00000i		−	1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−9.63325	−	9.63325i	−1.32323	−	1.32323i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	−9.63325	−	9.63325i	−1.32323	−	1.32323i	−0.412082	−	0.911147i	$-0.635198\pi$
$54$		0			0
$55$	1.55842	+	7.25061i	0.210138	+	0.977672i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	11.3321i		−	1.47531i	−0.675178	−	0.737655i	$-0.735933\pi$
$59$		−	11.3321i		−	1.47531i	0.675178	−	0.737655i	$-0.264067\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−10.7001	+	10.7001i	−1.30723	+	1.30723i	−0.383819	+	0.923408i	$0.625391\pi$
$67$	−10.7001	+	10.7001i	−1.30723	+	1.30723i	−0.923408	+	0.383819i	$0.874609\pi$
$68$		0			0
$69$			27.1279i			3.26582i
$70$		0			0
$71$	15.8614			1.88240			0.941201	−	0.337846i	$-0.109698\pi$
$71$	15.8614			1.88240			0.941201	+	0.337846i	$0.109698\pi$
$72$		0			0
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$		0			0
$75$	16.1904	+	6.13048i	1.86950	+	0.707887i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	−44.8267			−4.98075
$82$		0			0
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	9.86141i		−	1.04531i	−0.852545	−	0.522654i	$-0.824942\pi$
$89$		−	9.86141i		−	1.04531i	0.852545	−	0.522654i	$-0.175058\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	1.57869	+	1.57869i	0.163703	+	0.163703i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	3.68467	−	3.68467i	0.374122	−	0.374122i	−0.494854	−	0.868976i	$-0.664778\pi$
$97$	3.68467	−	3.68467i	0.374122	−	0.374122i	0.868976	+	0.494854i	$0.164778\pi$
$98$		0			0
$99$		−	29.8113i		−	2.99615i
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	11.9499	+	11.9499i	1.17746	+	1.17746i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	11.9499	+	11.9499i	1.17746	+	1.17746i	0.197066	+	0.980390i	$0.436859\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$	−14.4947			−1.37577
$112$		0			0
$113$	−6.10856	−	6.10856i	−0.574645	−	0.574645i	0.358778	−	0.933423i	$-0.383194\pi$
$113$	−6.10856	−	6.10856i	−0.574645	−	0.574645i	−0.933423	+	0.358778i	$0.883194\pi$
$114$		0			0
$115$	−9.50830	+	14.7147i	−0.886653	+	1.37215i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$	6.63325	+	9.00000i	0.593296	+	0.804984i
$126$		0			0
$127$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−38.9414	−	25.1630i	−3.35154	−	2.16569i
$136$		0			0
$137$	2.19985	−	2.19985i	0.187946	−	0.187946i	−0.606861	−	0.794808i	$-0.707572\pi$
$137$	2.19985	−	2.19985i	0.187946	−	0.187946i	0.794808	+	0.606861i	$0.207572\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	13.1395			1.10654
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−17.1382	+	17.1382i	−1.41353	+	1.41353i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.302985	+	1.40965i	0.0243363	+	0.113225i
$156$		0			0
$157$	11.0052	−	11.0052i	0.878309	−	0.878309i	−0.115050	−	0.993360i	$-0.536703\pi$
$157$	11.0052	−	11.0052i	0.878309	−	0.878309i	0.993360	+	0.115050i	$0.0367030\pi$
$158$		0			0
$159$			47.1704i			3.74085i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1.94987	+	1.94987i	0.152726	+	0.152726i	0.779334	−	0.626608i	$-0.215557\pi$
$163$	1.94987	+	1.94987i	0.152726	+	0.152726i	−0.626608	+	0.779334i	$0.715557\pi$
$164$		0			0
$165$	13.9362	−	21.5672i	1.08493	−	1.67901i
$166$		0			0
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$			13.0000i			1.00000i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$173$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−27.7444	+	27.7444i	−2.08540	+	2.08540i
$178$		0			0
$179$			3.86141i			0.288615i	0.989533	+	0.144308i	$0.0460955\pi$
$179$			3.86141i			0.288615i	−0.989533	+	0.144308i	$0.953905\pi$
$180$		0			0
$181$	16.6757			1.23949			0.619747	−	0.784801i	$-0.287235\pi$
$181$	16.6757			1.23949			0.619747	+	0.784801i	$0.287235\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−7.86219	−	5.08036i	−0.578039	−	0.373515i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	24.5986			1.77989			0.889945	−	0.456068i	$-0.150743\pi$
$191$	24.5986			1.77989			0.889945	+	0.456068i	$0.150743\pi$
$192$		0			0
$193$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$193$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$		−	19.8997i		−	1.41066i	−0.708881	−	0.705328i	$-0.750800\pi$
$199$		−	19.8997i		−	1.41066i	0.708881	−	0.705328i	$-0.249200\pi$
$200$		0			0
$201$	52.3944			3.69562
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	49.7971	−	49.7971i	3.46114	−	3.46114i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$	−38.8336	−	38.8336i	−2.66084	−	2.66084i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−4.77985	−	4.77985i	−0.320082	−	0.320082i	0.528716	−	0.848799i	$-0.322674\pi$
$223$	−4.77985	−	4.77985i	−0.320082	−	0.320082i	−0.848799	+	0.528716i	$0.822674\pi$
$224$		0			0
$225$	−18.4664	−	40.9731i	−1.23109	−	2.73154i
$226$		0			0
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$		−	19.2549i		−	1.27240i	−0.771523	−	0.636201i	$-0.780505\pi$
$229$		−	19.2549i		−	1.27240i	0.771523	−	0.636201i	$-0.219495\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$233$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$234$		0			0
$235$	7.12711	+	4.60537i	0.464921	+	0.300421i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	65.7650	+	65.7650i	4.21883	+	4.21883i
$244$		0			0
$245$	−15.3030	+	3.28917i	−0.977672	+	0.210138i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−27.8614			−1.75860			−0.879298	−	0.476272i	$-0.841988\pi$
$251$	−27.8614			−1.75860			−0.879298	+	0.476272i	$0.841988\pi$
$252$		0			0
$253$	18.3745	+	18.3745i	1.15520	+	1.15520i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.26650	+	4.26650i	−0.266137	+	0.266137i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	−4.26650	+	4.26650i	−0.266137	+	0.266137i	0.561405	+	0.827541i	$0.310261\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$263$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$264$		0			0
$265$	−16.5331	+	25.5861i	−1.01562	+	1.57174i
$266$		0			0
$267$	−24.1438	+	24.1438i	−1.47758	+	1.47758i
$268$		0			0
$269$			13.2665i			0.808873i	0.914566	+	0.404436i	$0.132532\pi$
$269$			13.2665i			0.808873i	−0.914566	+	0.404436i	$0.867468\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	15.1186	−	6.81386i	0.911684	−	0.410891i
$276$		0			0
$277$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$277$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$278$		0			0
$279$		−	5.79584i		−	0.346988i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$283$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$		−	17.0000i		−	1.00000i
$290$		0			0
$291$	−18.0424			−1.05767
$292$		0			0
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$		0			0
$295$	−24.7735	+	5.32473i	−1.44237	+	0.310018i
$296$		0			0
$297$	−48.6269	+	48.6269i	−2.82162	+	2.82162i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		0			0
$309$		−	58.5140i		−	3.32874i
$310$		0			0
$311$	12.0000			0.680458			0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000			0.680458			0.340229	+	0.940343i	$0.389495\pi$
$312$		0			0
$313$	−18.4401	−	18.4401i	−1.04230	−	1.04230i	−0.999065	−	0.0432311i	$-0.986235\pi$
$313$	−18.4401	−	18.4401i	−1.04230	−	1.04230i	−0.0432311	−	0.999065i	$-0.513765\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−20.7984	+	20.7984i	−1.16816	+	1.16816i	−0.185514	+	0.982642i	$0.559395\pi$
$317$	−20.7984	+	20.7984i	−1.16816	+	1.16816i	−0.982642	+	0.185514i	$0.940605\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−35.2858			−1.93948			−0.969742	−	0.244131i	$-0.921497\pi$
$331$	−35.2858			−1.93948			−0.969742	+	0.244131i	$0.921497\pi$
$332$		0			0
$333$	26.6070	+	26.6070i	1.45805	+	1.45805i
$334$		0			0
$335$	28.4198	+	18.3642i	1.55274	+	1.00334i
$336$		0			0
$337$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$337$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$338$		0			0
$339$			29.9113i			1.62456i
$340$		0			0
$341$	2.13859			0.115811
$342$		0			0
$343$		0			0
$344$		0			0
$345$	59.3054	−	12.7469i	3.19290	−	0.686271i
$346$		0			0
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−23.2712	−	23.2712i	−1.23860	−	1.23860i	−0.960574	−	0.278024i	$-0.910320\pi$
$353$	−23.2712	−	23.2712i	−1.23860	−	1.23860i	−0.278024	−	0.960574i	$-0.589680\pi$
$354$		0			0
$355$	−7.45299	−	34.6753i	−0.395564	−	1.84037i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$	−26.9314	−	26.9314i	−1.41353	−	1.41353i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	0.0244206	−	0.0244206i	0.00127475	−	0.00127475i	−0.706469	−	0.707744i	$-0.749713\pi$
$367$	0.0244206	−	0.0244206i	0.00127475	−	0.00127475i	0.707744	+	0.706469i	$0.249713\pi$
$368$		0			0
$369$		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
$375$	5.79454	−	38.2750i	0.299229	−	1.97651i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	36.5754i		−	1.87875i	−0.342885	−	0.939377i	$-0.611404\pi$
$379$		−	36.5754i		−	1.87875i	0.342885	−	0.939377i	$-0.388596\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4.87220	+	4.87220i	0.248958	+	0.248958i	0.820543	−	0.571585i	$-0.193671\pi$
$383$	4.87220	+	4.87220i	0.248958	+	0.248958i	−0.571585	+	0.820543i	$0.693671\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$			31.2318i			1.58352i	0.610835	+	0.791758i	$0.290834\pi$
$389$			31.2318i			1.58352i	−0.610835	+	0.791758i	$0.709166\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	18.8997	−	18.8997i	0.948551	−	0.948551i	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	18.8997	−	18.8997i	0.948551	−	0.948551i	0.998740	+	0.0501886i	$0.0159822\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	26.5330			1.32499			0.662497	−	0.749064i	$-0.269497\pi$
$401$	26.5330			1.32499			0.662497	+	0.749064i	$0.269497\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	21.0633	+	97.9975i	1.04664	+	4.86954i
$406$		0			0
$407$	−9.81766	+	9.81766i	−0.486643	+	0.486643i
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$	−10.7718			−0.531336
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	24.0000i		−	1.17248i	−0.810139	−	0.586238i	$-0.800608\pi$
$419$		−	24.0000i		−	1.17248i	0.810139	−	0.586238i	$-0.199392\pi$
$420$		0			0
$421$	−39.7995			−1.93971			−0.969854	−	0.243685i	$-0.921644\pi$
$421$	−39.7995			−1.93971			−0.969854	+	0.243685i	$0.921644\pi$
$422$		0			0
$423$	−24.1194	−	24.1194i	−1.17272	−	1.17272i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	25.6950	+	25.6950i	1.23482	+	1.23482i	0.962089	+	0.272736i	$0.0879285\pi$
$433$	25.6950	+	25.6950i	1.23482	+	1.23482i	0.272736	+	0.962089i	$0.412071\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	62.9191			2.99615
$442$		0			0
$443$	21.0201	+	21.0201i	0.998695	+	0.998695i	0.999999	−	0.00130426i	$-0.000415158\pi$
$443$	21.0201	+	21.0201i	0.998695	+	0.998695i	−0.00130426	+	0.999999i	$0.500415\pi$
$444$		0			0
$445$	−21.5584	+	4.63370i	−1.02197	+	0.219658i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			33.8614i			1.59802i	0.601319	+	0.799009i	$0.294642\pi$
$449$			33.8614i			1.59802i	−0.601319	+	0.799009i	$0.705358\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$457$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	14.7143	+	14.7143i	0.683830	+	0.683830i	0.960861	−	0.277031i	$-0.0893503\pi$
$463$	14.7143	+	14.7143i	0.683830	+	0.683830i	−0.277031	+	0.960861i	$0.589350\pi$
$464$		0			0
$465$	2.70945	−	4.19305i	0.125648	−	0.194448i
$466$		0			0
$467$	−9.93984	+	9.93984i	−0.459961	+	0.459961i	−0.898642	−	0.438682i	$-0.855446\pi$
$467$	−9.93984	+	9.93984i	−0.459961	+	0.459961i	0.438682	+	0.898642i	$0.355446\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−53.8882			−2.48304
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	86.5879	−	86.5879i	3.96459	−	3.96459i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−9.78658	−	6.32386i	−0.444386	−	0.287151i
$486$		0			0
$487$	14.6654	−	14.6654i	0.664554	−	0.664554i	−0.291896	−	0.956450i	$-0.594286\pi$
$487$	14.6654	−	14.6654i	0.664554	−	0.664554i	0.956450	+	0.291896i	$0.0942860\pi$
$488$		0			0
$489$		−	9.54779i		−	0.431766i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−65.1717	+	14.0078i	−2.92925	+	0.629603i
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	19.8997i		−	0.890835i	−0.895323	−	0.445418i	$-0.853055\pi$
$499$		−	19.8997i		−	0.890835i	0.895323	−	0.445418i	$-0.146945\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$503$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	31.8280	−	31.8280i	1.41353	−	1.41353i
$508$		0			0
$509$			40.6295i			1.80087i	0.434992	+	0.900434i	$0.356751\pi$
$509$			40.6295i			1.80087i	−0.434992	+	0.900434i	$0.643249\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	20.5091	−	31.7391i	0.903738	−	1.39859i
$516$		0			0
$517$	8.89975	−	8.89975i	0.391411	−	0.391411i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−8.56768			−0.375357			−0.187678	−	0.982231i	$-0.560096\pi$
$521$	−8.56768			−0.375357			−0.187678	+	0.982231i	$0.560096\pi$
$522$		0			0
$523$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$523$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$			38.3861i			1.66896i
$530$		0			0
$531$	101.858			4.42024
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	9.45392	−	9.45392i	0.407967	−	0.407967i
$538$		0			0
$539$			23.2164i			1.00000i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	−40.8273	−	40.8273i	−1.75207	−	1.75207i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$547$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	6.81078	+	31.6874i	0.289101	+	1.34505i
$556$		0			0
$557$		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
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$		0			0
$565$	−10.4839	+	16.2245i	−0.441060	+	0.682569i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$	−60.2249	−	60.2249i	−2.51593	−	2.51593i
$574$		0			0
$575$	36.6362	+	13.8723i	1.52784	+	0.578515i
$576$		0			0
$577$	30.2806	−	30.2806i	1.26060	−	1.26060i	0.309797	−	0.950803i	$-0.399739\pi$
$577$	30.2806	−	30.2806i	1.26060	−	1.26060i	0.950803	−	0.309797i	$-0.100261\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	31.9499	+	31.9499i	1.32323	+	1.32323i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	27.3166	−	27.3166i	1.12748	−	1.12748i	0.136892	−	0.990586i	$-0.456289\pi$
$587$	27.3166	−	27.3166i	1.12748	−	1.12748i	0.990586	−	0.136892i	$-0.0437113\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		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
$597$	−48.7207	+	48.7207i	−1.99401	+	1.99401i
$598$		0			0
$599$			36.0000i			1.47092i	0.677568	+	0.735460i	$0.263034\pi$
$599$			36.0000i			1.47092i	−0.677568	+	0.735460i	$0.736966\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$	−96.1774	−	96.1774i	−3.91665	−	3.91665i
$604$		0			0
$605$	−5.16870	−	24.0475i	−0.210138	−	0.977672i
$606$		0			0
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$613$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−34.2665	+	34.2665i	−1.37952	+	1.37952i	−0.534089	+	0.845428i	$0.679345\pi$
$617$	−34.2665	+	34.2665i	−1.37952	+	1.37952i	−0.845428	+	0.534089i	$0.820655\pi$
$618$		0			0
$619$		−	43.5842i		−	1.75180i	−0.482495	−	0.875899i	$-0.660269\pi$
$619$		−	43.5842i		−	1.75180i	0.482495	−	0.875899i	$-0.339731\pi$
$620$		0			0
$621$	−162.454			−6.51905
$622$		0			0
$623$		0			0
$624$		0			0
$625$	16.5584	−	18.7302i	0.662337	−	0.749206i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	39.5842			1.57582			0.787911	−	0.615789i	$-0.211162\pi$
$631$	39.5842			1.57582			0.787911	+	0.615789i	$0.211162\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$			142.569i			5.63995i
$640$		0			0
$641$	−27.3630			−1.08077			−0.540386	−	0.841417i	$-0.681722\pi$
$641$	−27.3630			−1.08077			−0.540386	+	0.841417i	$0.681722\pi$
$642$		0			0
$643$	22.5407	+	22.5407i	0.888917	+	0.888917i	0.994419	−	0.105502i	$-0.0336450\pi$
$643$	22.5407	+	22.5407i	0.888917	+	0.888917i	−0.105502	+	0.994419i	$0.533645\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	24.5075	−	24.5075i	0.963490	−	0.963490i	−0.0358667	−	0.999357i	$-0.511419\pi$
$647$	24.5075	−	24.5075i	0.963490	−	0.963490i	0.999357	+	0.0358667i	$0.0114192\pi$
$648$		0			0
$649$			37.5842i			1.47531i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	35.4406	+	35.4406i	1.38690	+	1.38690i	0.831753	+	0.555147i	$0.187338\pi$
$653$	35.4406	+	35.4406i	1.38690	+	1.38690i	0.555147	+	0.831753i	$0.312662\pi$
$654$		0			0
$655$		0			0
$656$		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
$661$	49.5842			1.92860			0.964301	−	0.264807i	$-0.0853084\pi$
$661$	49.5842			1.92860			0.964301	+	0.264807i	$0.0853084\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$			23.4051i			0.904893i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$673$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$674$		0			0
$675$	−36.7120	+	96.9551i	−1.41305	+	3.73180i
$676$		0			0
$677$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$677$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−11.2164	−	11.2164i	−0.429183	−	0.429183i	0.459167	−	0.888350i	$-0.348148\pi$
$683$	−11.2164	−	11.2164i	−0.429183	−	0.429183i	−0.888350	+	0.459167i	$0.848148\pi$
$684$		0			0
$685$	−5.84286	−	3.77552i	−0.223244	−	0.144255i
$686$		0			0
$687$	−47.1421	+	47.1421i	−1.79858	+	1.79858i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−51.5842			−1.96236			−0.981178	−	0.193105i	$-0.938144\pi$
$691$	−51.5842			−1.96236			−0.981178	+	0.193105i	$0.938144\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−6.17400	−	28.7247i	−0.232526	−	1.08184i
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	33.5842i		−	1.26128i	−0.776075	−	0.630641i	$-0.782792\pi$
$709$		−	33.5842i		−	1.26128i	0.776075	−	0.630641i	$-0.217208\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	3.57233	+	3.57233i	0.133785	+	0.133785i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	39.8614i		−	1.48658i	−0.668970	−	0.743290i	$-0.733264\pi$
$719$		−	39.8614i		−	1.48658i	0.668970	−	0.743290i	$-0.266736\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−38.0206	+	38.0206i	−1.41011	+	1.41011i	−0.651206	+	0.758901i	$0.725737\pi$
$727$	−38.0206	+	38.0206i	−1.41011	+	1.41011i	−0.758901	+	0.651206i	$0.774263\pi$
$728$		0			0
$729$		−	187.546i		−	6.94615i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$733$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$734$		0			0
$735$	45.5194	+	29.4135i	1.67901	+	1.08493i
$736$		0			0
$737$	35.4883	−	35.4883i	1.30723	−	1.30723i
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−31.5842			−1.15252			−0.576262	−	0.817265i	$-0.695489\pi$
$751$	−31.5842			−1.15252			−0.576262	+	0.817265i	$0.695489\pi$
$752$		0			0
$753$	68.2133	+	68.2133i	2.48583	+	2.48583i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	38.8997	−	38.8997i	1.41384	−	1.41384i	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.8997	−	38.8997i	1.41384	−	1.41384i	0.723269	−	0.690567i	$-0.242639\pi$
$758$		0			0
$759$		−	89.9731i		−	3.26582i
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$	20.8914			0.752386
$772$		0			0
$773$	20.3668	+	20.3668i	0.732541	+	0.732541i	0.971123	−	0.238581i	$-0.0766824\pi$
$773$	20.3668	+	20.3668i	0.732541	+	0.732541i	−0.238581	+	0.971123i	$0.576682\pi$
$774$		0			0
$775$	2.93932	−	1.32473i	0.105583	−	0.0475859i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−52.6063			−1.88240
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−29.2300	−	18.8877i	−1.04326	−	0.674132i
$786$		0			0
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	103.121	−	22.1645i	3.65733	−	0.786094i
$796$		0			0
$797$	−37.9610	+	37.9610i	−1.34465	+	1.34465i	−0.453279	+	0.891368i	$0.649746\pi$
$797$	−37.9610	+	37.9610i	−1.34465	+	1.34465i	−0.891368	+	0.453279i	$0.850254\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	88.6387			3.13189
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	32.4805	−	32.4805i	1.14337	−	1.14337i
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	3.34649	−	5.17891i	0.117222	−	0.181409i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	29.3553	+	29.3553i	1.02326	+	1.02326i	0.999723	+	0.0235383i	$0.00749316\pi$
$823$	29.3553	+	29.3553i	1.02326	+	1.02326i	0.0235383	+	0.999723i	$0.492507\pi$
$824$		0			0
$825$	−53.6974	−	20.3325i	−1.86950	−	0.707887i
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$			57.5842i			1.99998i	0.00416865	+	0.999991i	$0.498673\pi$
$829$			57.5842i			1.99998i	−0.00416865	+	0.999991i	$0.501327\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−9.45392	+	9.45392i	−0.326775	+	0.326775i
$838$		0			0
$839$		−	20.7297i		−	0.715669i	−0.933785	−	0.357834i	$-0.883515\pi$
$839$		−	20.7297i		−	0.715669i	0.933785	−	0.357834i	$-0.116485\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	28.4198	−	6.10846i	0.977672	−	0.210138i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−32.7991			−1.12434
$852$		0			0
$853$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$853$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$854$		0			0
$855$		0			0
$856$		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
$859$			27.5842i			0.941161i	0.882357	+	0.470581i	$0.155956\pi$
$859$			27.5842i			0.941161i	−0.882357	+	0.470581i	$0.844044\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−41.2164	−	41.2164i	−1.40302	−	1.40302i	−0.790295	−	0.612727i	$-0.790072\pi$
$863$	−41.2164	−	41.2164i	−1.40302	−	1.40302i	−0.612727	−	0.790295i	$-0.709928\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−41.6213	+	41.6213i	−1.41353	+	1.41353i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	33.1195	+	33.1195i	1.12092	+	1.12092i
$874$		0			0
$875$		0			0
$876$		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
$879$		0			0
$880$		0			0
$881$	−14.1386			−0.476341			−0.238171	−	0.971223i	$-0.576548\pi$
$881$	−14.1386			−0.476341			−0.238171	+	0.971223i	$0.576548\pi$
$882$		0			0
$883$	−18.0501	−	18.0501i	−0.607435	−	0.607435i	0.334840	−	0.942275i	$-0.391318\pi$
$883$	−18.0501	−	18.0501i	−0.607435	−	0.607435i	−0.942275	+	0.334840i	$0.891318\pi$
$884$		0			0
$885$	73.6898	+	47.6166i	2.47706	+	1.60061i
$886$		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
$889$		0			0
$890$		0			0
$891$	148.673			4.98075
$892$		0			0
$893$		0			0
$894$		0			0
$895$	8.44158	−	1.81441i	0.282171	−	0.0606489i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−7.83561	−	36.4554i	−0.260464	−	1.21182i
$906$		0			0
$907$	−25.8496	+	25.8496i	−0.858323	+	0.858323i	−0.991140	−	0.132818i	$-0.957597\pi$
$907$	−25.8496	+	25.8496i	−0.858323	+	0.858323i	0.132818	+	0.991140i	$0.457597\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−6.63325			−0.219769			−0.109885	−	0.993944i	$-0.535048\pi$
$911$	−6.63325			−0.219769			−0.109885	+	0.993944i	$0.535048\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−7.41208	+	19.5750i	−0.243708	+	0.643623i
$926$		0			0
$927$	−107.411	+	107.411i	−3.52783	+	3.52783i
$928$		0			0
$929$		−	53.0660i		−	1.74104i	−0.492134	−	0.870519i	$-0.663783\pi$
$929$		−	53.0660i		−	1.74104i	0.492134	−	0.870519i	$-0.336217\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−29.3797	−	29.3797i	−0.961849	−	0.961849i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$		0			0
$939$			90.2942i			2.94664i
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	43.1806	−	43.1806i	1.40318	−	1.40318i	0.613441	−	0.789741i	$-0.289785\pi$
$947$	43.1806	−	43.1806i	1.40318	−	1.40318i	0.789741	−	0.613441i	$-0.210215\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	101.842			3.30245
$952$		0			0
$953$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$953$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$954$		0			0
$955$	−11.5584	−	53.7759i	−0.374022	−	1.74015i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−30.5842			−0.986588
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−60.5292			−1.94247			−0.971237	−	0.238114i	$-0.923471\pi$
$971$	−60.5292			−1.94247			−0.971237	+	0.238114i	$0.923471\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	28.7601	−	28.7601i	0.920116	−	0.920116i	−0.0769208	−	0.997037i	$-0.524509\pi$
$977$	28.7601	−	28.7601i	0.920116	−	0.920116i	0.997037	+	0.0769208i	$0.0245089\pi$
$978$		0			0
$979$			32.7066i			1.04531i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−34.2519	−	34.2519i	−1.09247	−	1.09247i	−0.995265	−	0.0972017i	$-0.969011\pi$
$983$	−34.2519	−	34.2519i	−1.09247	−	1.09247i	−0.0972017	−	0.995265i	$-0.530989\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	59.6992			1.89641			0.948205	−	0.317660i	$-0.102897\pi$
$991$	59.6992			1.89641			0.948205	+	0.317660i	$0.102897\pi$
$992$		0			0
$993$	86.3906	+	86.3906i	2.74152	+	2.74152i
$994$		0			0
$995$	−43.5036	+	9.35053i	−1.37916	+	0.296432i
$996$		0			0
$997$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$997$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$998$		0			0
$999$		−	86.8004i		−	2.74624i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	220.2.k.b.153.1	✓	8
3.2	odd	2		1980.2.y.b.1693.3		8
4.3	odd	2		880.2.bd.h.593.4		8
5.2	odd	4	inner	220.2.k.b.197.1	yes	8
5.3	odd	4		1100.2.k.b.857.4		8
5.4	even	2		1100.2.k.b.593.4		8
11.10	odd	2	CM	220.2.k.b.153.1	✓	8
15.2	even	4		1980.2.y.b.1297.3		8
20.7	even	4		880.2.bd.h.417.4		8
33.32	even	2		1980.2.y.b.1693.3		8
44.43	even	2		880.2.bd.h.593.4		8
55.32	even	4	inner	220.2.k.b.197.1	yes	8
55.43	even	4		1100.2.k.b.857.4		8
55.54	odd	2		1100.2.k.b.593.4		8
165.32	odd	4		1980.2.y.b.1297.3		8
220.87	odd	4		880.2.bd.h.417.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.2.k.b.153.1	✓	8	1.1	even	1	trivial
220.2.k.b.153.1	✓	8	11.10	odd	2	CM
220.2.k.b.197.1	yes	8	5.2	odd	4	inner
220.2.k.b.197.1	yes	8	55.32	even	4	inner
880.2.bd.h.417.4		8	20.7	even	4
880.2.bd.h.417.4		8	220.87	odd	4
880.2.bd.h.593.4		8	4.3	odd	2
880.2.bd.h.593.4		8	44.43	even	2
1100.2.k.b.593.4		8	5.4	even	2
1100.2.k.b.593.4		8	55.54	odd	2
1100.2.k.b.857.4		8	5.3	odd	4
1100.2.k.b.857.4		8	55.43	even	4
1980.2.y.b.1297.3		8	15.2	even	4
1980.2.y.b.1297.3		8	165.32	odd	4
1980.2.y.b.1693.3		8	3.2	odd	2
1980.2.y.b.1693.3		8	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	220.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.44831 - 2.44831i) q^{3} +(-0.469882 - 2.18614i) q^{5} +8.98844i q^{9} -3.31662 q^{11} +(-4.20193 + 6.50277i) q^{15} +(-5.54013 - 5.54013i) q^{23} +(-4.55842 + 2.05446i) q^{25} +(14.6616 - 14.6616i) q^{27} -0.644810 q^{31} +(8.12012 + 8.12012i) q^{33} +(2.96014 - 2.96014i) q^{37} +(19.6500 - 4.22351i) q^{45} +(-2.68338 + 2.68338i) q^{47} -7.00000i q^{49} +(-9.63325 - 9.63325i) q^{53} +(1.55842 + 7.25061i) q^{55} -11.3321i q^{59} +(-10.7001 + 10.7001i) q^{67} +27.1279i q^{69} +15.8614 q^{71} +(16.1904 + 6.13048i) q^{75} -44.8267 q^{81} -9.86141i q^{89} +(1.57869 + 1.57869i) q^{93} +(3.68467 - 3.68467i) q^{97} -29.8113i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{3} - 16 q^{15} - 18 q^{23} - 2 q^{25} + 26 q^{27} + 22 q^{33} + 14 q^{37} + 18 q^{45} - 48 q^{47} - 24 q^{53} - 22 q^{55} - 26 q^{67} + 12 q^{71} + 64 q^{75} - 100 q^{81} + 18 q^{93} + 34 q^{97}+O(q^{100}) \) 8 * q + 2 * q^3 - 16 * q^15 - 18 * q^23 - 2 * q^25 + 26 * q^27 + 22 * q^33 + 14 * q^37 + 18 * q^45 - 48 * q^47 - 24 * q^53 - 22 * q^55 - 26 * q^67 + 12 * q^71 + 64 * q^75 - 100 * q^81 + 18 * q^93 + 34 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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