LMFDB - Embedded newform 55.2.e.b.32.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [55,2,Mod(32,55)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(55, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([1, 2]))

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

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

chi := DirichletCharacter("55.32");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 55 = 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	55.e (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:	$0.439177211117$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 9 $ x^4 - 5*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			32.1
Root			$-1.65831 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	55.32
Dual form			55.2.e.b.43.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.15831 + 2.15831i) q^{3} +2.00000i q^{4} +(1.65831 - 1.50000i) q^{5} -6.31662i q^{9} +3.31662 q^{11} +(-4.31662 - 4.31662i) q^{12} +(-0.341688 + 6.81662i) q^{15} -4.00000 q^{16} +(3.00000 + 3.31662i) q^{20} +(2.84169 - 2.84169i) q^{23} +(0.500000 - 4.97494i) q^{25} +(7.15831 + 7.15831i) q^{27} -9.94987 q^{31} +(-7.15831 + 7.15831i) q^{33} +12.6332 q^{36} +(1.47494 + 1.47494i) q^{37} +6.63325i q^{44} +(-9.47494 - 10.4749i) q^{45} +(-9.31662 - 9.31662i) q^{47} +(8.63325 - 8.63325i) q^{48} +7.00000i q^{49} +(3.63325 - 3.63325i) q^{53} +(5.50000 - 4.97494i) q^{55} -3.31662i q^{59} +(-13.6332 - 0.683375i) q^{60} -8.00000i q^{64} +(11.4749 + 11.4749i) q^{67} +12.2665i q^{69} -3.00000 q^{71} +(9.65831 + 11.8166i) q^{75} +(-6.63325 + 6.00000i) q^{80} -11.9499 q^{81} +9.00000i q^{89} +(5.68338 + 5.68338i) q^{92} +(21.4749 - 21.4749i) q^{93} +(-3.52506 - 3.52506i) q^{97} -20.9499i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 4 q^{12} - 8 q^{15} - 16 q^{16} + 12 q^{20} + 18 q^{23} + 2 q^{25} + 22 q^{27} - 22 q^{33} + 24 q^{36} - 14 q^{37} - 18 q^{45} - 24 q^{47} + 8 q^{48} - 12 q^{53} + 22 q^{55} - 28 q^{60}+ \cdots - 34 q^{97}+O(q^{100}) $ 4 * q - 2 * q^3 - 4 * q^12 - 8 * q^15 - 16 * q^16 + 12 * q^20 + 18 * q^23 + 2 * q^25 + 22 * q^27 - 22 * q^33 + 24 * q^36 - 14 * q^37 - 18 * q^45 - 24 * q^47 + 8 * q^48 - 12 * q^53 + 22 * q^55 - 28 * q^60 + 26 * q^67 - 12 * q^71 + 32 * q^75 - 8 * q^81 + 36 * q^92 + 66 * q^93 - 34 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/55\mathbb{Z}\right)^\times$.

$n$	$12$	$46$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$

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		−0.707107	−	0.707107i	$-0.750000\pi$
$2$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$3$	−2.15831	+	2.15831i	−1.24610	+	1.24610i	−0.288675	+	0.957427i	$0.593215\pi$
$3$	−2.15831	+	2.15831i	−1.24610	+	1.24610i	−0.957427	+	0.288675i	$0.906785\pi$
$4$			2.00000i			1.00000i
$5$	1.65831	−	1.50000i	0.741620	−	0.670820i
$5$	1.65831	−	1.50000i	0.741620	−	0.670820i
$6$		0			0
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$		0			0
$9$		−	6.31662i		−	2.10554i
$10$		0			0
$11$	3.31662			1.00000
$11$	3.31662			1.00000
$12$	−4.31662	−	4.31662i	−1.24610	−	1.24610i
$13$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$13$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$14$		0			0
$15$	−0.341688	+	6.81662i	−0.0882234	+	1.76004i
$16$	−4.00000			−1.00000
$17$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$17$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	3.00000	+	3.31662i	0.670820	+	0.741620i
$21$		0			0
$22$		0			0
$23$	2.84169	−	2.84169i	0.592533	−	0.592533i	−0.345782	−	0.938315i	$-0.612386\pi$
$23$	2.84169	−	2.84169i	0.592533	−	0.592533i	0.938315	+	0.345782i	$0.112386\pi$
$24$		0			0
$25$	0.500000	−	4.97494i	0.100000	−	0.994987i
$26$		0			0
$27$	7.15831	+	7.15831i	1.37762	+	1.37762i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−9.94987			−1.78705			−0.893525	−	0.449013i	$-0.851776\pi$
$31$	−9.94987			−1.78705			−0.893525	+	0.449013i	$0.851776\pi$
$32$		0			0
$33$	−7.15831	+	7.15831i	−1.24610	+	1.24610i
$34$		0			0
$35$		0			0
$36$	12.6332			2.10554
$37$	1.47494	+	1.47494i	0.242478	+	0.242478i	0.817875	−	0.575396i	$-0.195152\pi$
$37$	1.47494	+	1.47494i	0.242478	+	0.242478i	−0.575396	+	0.817875i	$0.695152\pi$
$38$		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	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$			6.63325i			1.00000i
$45$	−9.47494	−	10.4749i	−1.41244	−	1.56151i
$46$		0			0
$47$	−9.31662	−	9.31662i	−1.35897	−	1.35897i	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−9.31662	−	9.31662i	−1.35897	−	1.35897i	−0.483779	−	0.875190i	$-0.660736\pi$
$48$	8.63325	−	8.63325i	1.24610	−	1.24610i
$49$			7.00000i			1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.63325	−	3.63325i	0.499065	−	0.499065i	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	3.63325	−	3.63325i	0.499065	−	0.499065i	0.911147	+	0.412082i	$0.135198\pi$
$54$		0			0
$55$	5.50000	−	4.97494i	0.741620	−	0.670820i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	3.31662i		−	0.431788i	−0.976417	−	0.215894i	$-0.930733\pi$
$59$		−	3.31662i		−	0.431788i	0.976417	−	0.215894i	$-0.0692665\pi$
$60$	−13.6332	−	0.683375i	−1.76004	−	0.0882234i
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$		−	8.00000i		−	1.00000i
$65$		0			0
$66$		0			0
$67$	11.4749	+	11.4749i	1.40189	+	1.40189i	0.794101	+	0.607785i	$0.207942\pi$
$67$	11.4749	+	11.4749i	1.40189	+	1.40189i	0.607785	+	0.794101i	$0.292058\pi$
$68$		0			0
$69$			12.2665i			1.47671i
$70$		0			0
$71$	−3.00000			−0.356034			−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000			−0.356034			−0.178017	+	0.984027i	$0.556968\pi$
$72$		0			0
$73$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$73$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$74$		0			0
$75$	9.65831	+	11.8166i	1.11525	+	1.36447i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	−6.63325	+	6.00000i	−0.741620	+	0.670820i
$81$	−11.9499			−1.32776
$82$		0			0
$83$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$83$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			9.00000i			0.953998i	0.878904	+	0.476999i	$0.158275\pi$
$89$			9.00000i			0.953998i	−0.878904	+	0.476999i	$0.841725\pi$
$90$		0			0
$91$		0			0
$92$	5.68338	+	5.68338i	0.592533	+	0.592533i
$93$	21.4749	−	21.4749i	2.22685	−	2.22685i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−3.52506	−	3.52506i	−0.357916	−	0.357916i	0.505128	−	0.863044i	$-0.331445\pi$
$97$	−3.52506	−	3.52506i	−0.357916	−	0.357916i	−0.863044	+	0.505128i	$0.831445\pi$
$98$		0			0
$99$		−	20.9499i		−	2.10554i
$100$	9.94987	+	1.00000i	0.994987	+	0.100000i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	−7.94987	+	7.94987i	−0.783324	+	0.783324i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	−7.94987	+	7.94987i	−0.783324	+	0.783324i	0.197066	+	0.980390i	$0.436859\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$	−14.3166	+	14.3166i	−1.37762	+	1.37762i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$	−6.36675			−0.604305
$112$		0			0
$113$	−12.1583	+	12.1583i	−1.14376	+	1.14376i	−0.156001	+	0.987757i	$0.549860\pi$
$113$	−12.1583	+	12.1583i	−1.14376	+	1.14376i	−0.987757	+	0.156001i	$0.950140\pi$
$114$		0			0
$115$	0.449874	−	8.97494i	0.0419510	−	0.836917i
$116$		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
$124$		−	19.8997i		−	1.78705i
$125$	−6.63325	−	9.00000i	−0.593296	−	0.804984i
$126$		0			0
$127$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$127$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$	−14.3166	−	14.3166i	−1.24610	−	1.24610i
$133$		0			0
$134$		0			0
$135$	22.6082	+	1.13325i	1.94580	+	0.0975346i
$136$		0			0
$137$	−10.1082	−	10.1082i	−0.863601	−	0.863601i	0.128154	−	0.991754i	$-0.459095\pi$
$137$	−10.1082	−	10.1082i	−0.863601	−	0.863601i	−0.991754	+	0.128154i	$0.959095\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	40.2164			3.38683
$142$		0			0
$143$		0			0
$144$			25.2665i			2.10554i
$145$		0			0
$146$		0			0
$147$	−15.1082	−	15.1082i	−1.24610	−	1.24610i
$148$	−2.94987	+	2.94987i	−0.242478	+	0.242478i
$149$		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
$155$	−16.5000	+	14.9248i	−1.32531	+	1.19879i
$156$		0			0
$157$	16.4749	+	16.4749i	1.31484	+	1.31484i	0.917800	+	0.397043i	$0.129964\pi$
$157$	16.4749	+	16.4749i	1.31484	+	1.31484i	0.397043	+	0.917800i	$0.370036\pi$
$158$		0			0
$159$			15.6834i			1.24377i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−17.9499	+	17.9499i	−1.40594	+	1.40594i	−0.626608	+	0.779334i	$0.715557\pi$
$163$	−17.9499	+	17.9499i	−1.40594	+	1.40594i	−0.779334	+	0.626608i	$0.784443\pi$
$164$		0			0
$165$	−1.13325	+	22.6082i	−0.0882234	+	1.76004i
$166$		0			0
$167$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$167$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$168$		0			0
$169$		−	13.0000i		−	1.00000i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$173$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$174$		0			0
$175$		0			0
$176$	−13.2665			−1.00000
$177$	7.15831	+	7.15831i	0.538052	+	0.538052i
$178$		0			0
$179$		−	21.0000i		−	1.56961i	−0.619740	−	0.784807i	$-0.712762\pi$
$179$		−	21.0000i		−	1.56961i	0.619740	−	0.784807i	$-0.287238\pi$
$180$	20.9499	−	18.9499i	1.56151	−	1.41244i
$181$	−9.94987			−0.739568			−0.369784	−	0.929118i	$-0.620568\pi$
$181$	−9.94987			−0.739568			−0.369784	+	0.929118i	$0.620568\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	4.65831	+	0.233501i	0.342486	+	0.0171673i
$186$		0			0
$187$		0			0
$188$	18.6332	−	18.6332i	1.35897	−	1.35897i
$189$		0			0
$190$		0			0
$191$	23.2164			1.67988			0.839939	−	0.542681i	$-0.182591\pi$
$191$	23.2164			1.67988			0.839939	+	0.542681i	$0.182591\pi$
$192$	17.2665	+	17.2665i	1.24610	+	1.24610i
$193$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$193$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$194$		0			0
$195$		0			0
$196$	−14.0000			−1.00000
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$		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
$201$	−49.5330			−3.49379
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−17.9499	−	17.9499i	−1.24760	−	1.24760i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	7.26650	+	7.26650i	0.499065	+	0.499065i
$213$	6.47494	−	6.47494i	0.443655	−	0.443655i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	9.94987	+	11.0000i	0.670820	+	0.741620i
$221$		0			0
$222$		0			0
$223$	14.4248	−	14.4248i	0.965957	−	0.965957i	−0.0334825	−	0.999439i	$-0.510660\pi$
$223$	14.4248	−	14.4248i	0.965957	−	0.965957i	0.999439	+	0.0334825i	$0.0106598\pi$
$224$		0			0
$225$	−31.4248	−	3.15831i	−2.09499	−	0.210554i
$226$		0			0
$227$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$		0			0
$229$			29.8496i			1.97252i	0.165205	+	0.986259i	$0.447172\pi$
$229$			29.8496i			1.97252i	−0.165205	+	0.986259i	$0.552828\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$233$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$234$		0			0
$235$	−29.4248	−	1.47494i	−1.91946	−	0.0962143i
$236$	6.63325			0.431788
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$	1.36675	−	27.2665i	0.0882234	−	1.76004i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	4.31662	−	4.31662i	0.276912	−	0.276912i
$244$		0			0
$245$	10.5000	+	11.6082i	0.670820	+	0.741620i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	27.0000			1.70422			0.852112	−	0.523359i	$-0.175321\pi$
$251$	27.0000			1.70422			0.852112	+	0.523359i	$0.175321\pi$
$252$		0			0
$253$	9.42481	−	9.42481i	0.592533	−	0.592533i
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$	22.2665	+	22.2665i	1.38895	+	1.38895i	0.827541	+	0.561405i	$0.189739\pi$
$257$	22.2665	+	22.2665i	1.38895	+	1.38895i	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	$-0.250000\pi$
$263$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$264$		0			0
$265$	0.575188	−	11.4749i	0.0353335	−	0.704900i
$266$		0			0
$267$	−19.4248	−	19.4248i	−1.18878	−	1.18878i
$268$	−22.9499	+	22.9499i	−1.40189	+	1.40189i
$269$			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
$275$	1.65831	−	16.5000i	0.100000	−	0.994987i
$276$	−24.5330			−1.47671
$277$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$277$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$278$		0			0
$279$			62.8496i			3.76271i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$283$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$284$		−	6.00000i		−	0.356034i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$			17.0000i			1.00000i
$290$		0			0
$291$	15.2164			0.892000
$292$		0			0
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$		0			0
$295$	−4.97494	−	5.50000i	−0.289652	−	0.320222i
$296$		0			0
$297$	23.7414	+	23.7414i	1.37762	+	1.37762i
$298$		0			0
$299$		0			0
$300$	−23.6332	+	19.3166i	−1.36447	+	1.11525i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		−	34.3166i		−	1.95220i
$310$		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
$313$	24.4248	−	24.4248i	1.38057	−	1.38057i	0.536972	−	0.843600i	$-0.319568\pi$
$313$	24.4248	−	24.4248i	1.38057	−	1.38057i	0.843600	−	0.536972i	$-0.180432\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−25.1082	−	25.1082i	−1.41022	−	1.41022i	−0.758236	−	0.651981i	$-0.773938\pi$
$317$	−25.1082	−	25.1082i	−1.41022	−	1.41022i	−0.651981	−	0.758236i	$-0.726062\pi$
$318$		0			0
$319$		0			0
$320$	−12.0000	−	13.2665i	−0.670820	−	0.741620i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		−	23.8997i		−	1.32776i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−9.94987			−0.546895			−0.273447	−	0.961887i	$-0.588164\pi$
$331$	−9.94987			−0.546895			−0.273447	+	0.961887i	$0.588164\pi$
$332$		0			0
$333$	9.31662	−	9.31662i	0.510548	−	0.510548i
$334$		0			0
$335$	36.2414	+	1.81662i	1.98008	+	0.0992528i
$336$		0			0
$337$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$337$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$338$		0			0
$339$		−	52.4829i		−	2.85048i
$340$		0			0
$341$	−33.0000			−1.78705
$342$		0			0
$343$		0			0
$344$		0			0
$345$	18.3997	+	20.3417i	0.990609	+	1.09516i
$346$		0			0
$347$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$347$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$348$		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
$353$	−13.7414	+	13.7414i	−0.731383	+	0.731383i	−0.970894	−	0.239511i	$-0.923013\pi$
$353$	−13.7414	+	13.7414i	−0.731383	+	0.731383i	0.239511	+	0.970894i	$0.423013\pi$
$354$		0			0
$355$	−4.97494	+	4.50000i	−0.264042	+	0.238835i
$356$	−18.0000			−0.953998
$357$		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
$363$	−23.7414	+	23.7414i	−1.24610	+	1.24610i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−13.5251	−	13.5251i	−0.706003	−	0.706003i	0.259690	−	0.965692i	$-0.416380\pi$
$367$	−13.5251	−	13.5251i	−0.706003	−	0.706003i	−0.965692	+	0.259690i	$0.916380\pi$
$368$	−11.3668	+	11.3668i	−0.592533	+	0.592533i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	42.9499	+	42.9499i	2.22685	+	2.22685i
$373$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$373$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$374$		0			0
$375$	33.7414	+	5.10819i	1.74240	+	0.263786i
$376$		0			0
$377$		0			0
$378$		0			0
$379$			29.8496i			1.53327i	0.642082	+	0.766636i	$0.278071\pi$
$379$			29.8496i			1.53327i	−0.642082	+	0.766636i	$0.721929\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	17.8417	−	17.8417i	0.911668	−	0.911668i	−0.0847358	−	0.996403i	$-0.527005\pi$
$383$	17.8417	−	17.8417i	0.911668	−	0.911668i	0.996403	+	0.0847358i	$0.0270046\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	7.05013	−	7.05013i	0.357916	−	0.357916i
$389$		−	36.4829i		−	1.84976i	−0.380265	−	0.924878i	$-0.624167\pi$
$389$		−	36.4829i		−	1.84976i	0.380265	−	0.924878i	$-0.375833\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	41.8997			2.10554
$397$	−20.8997	−	20.8997i	−1.04893	−	1.04893i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	−20.8997	−	20.8997i	−1.04893	−	1.04893i	−0.0501886	−	0.998740i	$-0.515982\pi$
$398$		0			0
$399$		0			0
$400$	−2.00000	+	19.8997i	−0.100000	+	0.994987i
$401$	−26.5330			−1.32499			−0.662497	−	0.749064i	$-0.730503\pi$
$401$	−26.5330			−1.32499			−0.662497	+	0.749064i	$0.730503\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−19.8166	+	17.9248i	−0.984696	+	0.890691i
$406$		0			0
$407$	4.89181	+	4.89181i	0.242478	+	0.242478i
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$	43.6332			2.15227
$412$	−15.8997	−	15.8997i	−0.783324	−	0.783324i
$413$		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	$0.199392\pi$
$419$			24.0000i			1.17248i	−0.810139	+	0.586238i	$0.800608\pi$
$420$		0			0
$421$	39.7995			1.93971			0.969854	−	0.243685i	$-0.0783563\pi$
$421$	39.7995			1.93971			0.969854	+	0.243685i	$0.0783563\pi$
$422$		0			0
$423$	−58.8496	+	58.8496i	−2.86137	+	2.86137i
$424$		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			$\pi$
$432$	−28.6332	−	28.6332i	−1.37762	−	1.37762i
$433$	29.4248	−	29.4248i	1.41407	−	1.41407i	0.696826	−	0.717241i	$-0.254595\pi$
$433$	29.4248	−	29.4248i	1.41407	−	1.41407i	0.717241	−	0.696826i	$-0.245405\pi$
$434$		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
$441$	44.2164			2.10554
$442$		0			0
$443$	−28.7414	+	28.7414i	−1.36555	+	1.36555i	−0.498870	+	0.866677i	$0.666252\pi$
$443$	−28.7414	+	28.7414i	−1.36555	+	1.36555i	−0.866677	+	0.498870i	$0.833748\pi$
$444$		−	12.7335i		−	0.604305i
$445$	13.5000	+	14.9248i	0.639961	+	0.707504i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			39.0000i			1.84052i	0.391303	+	0.920262i	$0.372024\pi$
$449$			39.0000i			1.84052i	−0.391303	+	0.920262i	$0.627976\pi$
$450$		0			0
$451$		0			0
$452$	−24.3166	−	24.3166i	−1.14376	−	1.14376i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$457$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$458$		0			0
$459$		0			0
$460$	17.9499	+	0.899749i	0.836917	+	0.0419510i
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−0.575188	+	0.575188i	−0.0267313	+	0.0267313i	−0.720346	−	0.693615i	$-0.756017\pi$
$463$	−0.575188	+	0.575188i	−0.0267313	+	0.0267313i	0.693615	+	0.720346i	$0.256017\pi$
$464$		0			0
$465$	3.39975	−	67.8246i	0.157660	−	3.14529i
$466$		0			0
$467$	23.0581	+	23.0581i	1.06700	+	1.06700i	0.997588	+	0.0694117i	$0.0221122\pi$
$467$	23.0581	+	23.0581i	1.06700	+	1.06700i	0.0694117	+	0.997588i	$0.477888\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−71.1161			−3.27686
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−22.9499	−	22.9499i	−1.05080	−	1.05080i
$478$		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
$484$			22.0000i			1.00000i
$485$	−11.1332	−	0.558061i	−0.505535	−	0.0253403i
$486$		0			0
$487$	26.4749	+	26.4749i	1.19969	+	1.19969i	0.974258	+	0.225436i	$0.0723806\pi$
$487$	26.4749	+	26.4749i	1.19969	+	1.19969i	0.225436	+	0.974258i	$0.427619\pi$
$488$		0			0
$489$		−	77.4829i		−	3.50390i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−31.4248	−	34.7414i	−1.41244	−	1.56151i
$496$	39.7995			1.78705
$497$		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	$-0.146945\pi$
$500$	18.0000	−	13.2665i	0.804984	−	0.593296i
$501$		0			0
$502$		0			0
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	28.0581	+	28.0581i	1.24610	+	1.24610i
$508$		0			0
$509$		−	3.31662i		−	0.147007i	−0.997295	−	0.0735034i	$-0.976582\pi$
$509$		−	3.31662i		−	0.147007i	0.997295	−	0.0735034i	$-0.0234180\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.25856	+	25.1082i	−0.0554589	+	1.10640i
$516$		0			0
$517$	−30.8997	−	30.8997i	−1.35897	−	1.35897i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−43.1161			−1.88895			−0.944476	−	0.328581i	$-0.893430\pi$
$521$	−43.1161			−1.88895			−0.944476	+	0.328581i	$0.893430\pi$
$522$		0			0
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	28.6332	−	28.6332i	1.24610	−	1.24610i
$529$			6.84962i			0.297810i
$530$		0			0
$531$	−20.9499			−0.909147
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	45.3246	+	45.3246i	1.95590	+	1.95590i
$538$		0			0
$539$			23.2164i			1.00000i
$540$	−2.26650	+	45.2164i	−0.0975346	+	1.94580i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	21.4749	−	21.4749i	0.921578	−	0.921578i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$	20.2164	−	20.2164i	0.863601	−	0.863601i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−10.5581	+	9.55013i	−0.448165	+	0.405380i
$556$		0			0
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$563$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$564$			80.4327i			3.38683i
$565$	−1.92481	+	38.3997i	−0.0809774	+	1.61549i
$566$		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
$573$	−50.1082	+	50.1082i	−2.09330	+	2.09330i
$574$		0			0
$575$	−12.7164	−	15.5581i	−0.530309	−	0.648816i
$576$	−50.5330			−2.10554
$577$	−18.5251	−	18.5251i	−0.771208	−	0.771208i	0.207109	−	0.978318i	$-0.433594\pi$
$577$	−18.5251	−	18.5251i	−0.771208	−	0.771208i	−0.978318	+	0.207109i	$0.933594\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	12.0501	−	12.0501i	0.499065	−	0.499065i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	20.6834	+	20.6834i	0.853694	+	0.853694i	0.990586	−	0.136892i	$-0.0437113\pi$
$587$	20.6834	+	20.6834i	0.853694	+	0.853694i	−0.136892	+	0.990586i	$0.543711\pi$
$588$	30.2164	−	30.2164i	1.24610	−	1.24610i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−5.89975	−	5.89975i	−0.242478	−	0.242478i
$593$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$593$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	42.9499	+	42.9499i	1.75782	+	1.75782i
$598$		0			0
$599$		−	36.0000i		−	1.47092i	−0.677568	−	0.735460i	$-0.736966\pi$
$599$		−	36.0000i		−	1.47092i	0.677568	−	0.735460i	$-0.263034\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$	72.4829	−	72.4829i	2.95173	−	2.95173i
$604$		0			0
$605$	18.2414	−	16.5000i	0.741620	−	0.670820i
$606$		0			0
$607$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$607$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$613$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7.73350	−	7.73350i	−0.311339	−	0.311339i	0.534089	−	0.845428i	$-0.320655\pi$
$617$	−7.73350	−	7.73350i	−0.311339	−	0.311339i	−0.845428	+	0.534089i	$0.820655\pi$
$618$		0			0
$619$		−	1.00000i		−	0.0401934i	−0.999798	−	0.0200967i	$-0.993603\pi$
$619$		−	1.00000i		−	0.0401934i	0.999798	−	0.0200967i	$-0.00639741\pi$
$620$	−29.8496	−	33.0000i	−1.19879	−	1.32531i
$621$	40.6834			1.63257
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−24.5000	−	4.97494i	−0.980000	−	0.198997i
$626$		0			0
$627$		0			0
$628$	−32.9499	+	32.9499i	−1.31484	+	1.31484i
$629$		0			0
$630$		0			0
$631$	7.00000			0.278666			0.139333	−	0.990246i	$-0.455504\pi$
$631$	7.00000			0.278666			0.139333	+	0.990246i	$0.455504\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−31.3668			−1.24377
$637$		0			0
$638$		0			0
$639$			18.9499i			0.749645i
$640$		0			0
$641$	23.2164			0.916992			0.458496	−	0.888697i	$-0.348388\pi$
$641$	23.2164			0.916992			0.458496	+	0.888697i	$0.348388\pi$
$642$		0			0
$643$	−5.57519	+	5.57519i	−0.219864	+	0.219864i	−0.808441	−	0.588577i	$-0.799688\pi$
$643$	−5.57519	+	5.57519i	−0.219864	+	0.219864i	0.588577	+	0.808441i	$0.299688\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	8.05806	+	8.05806i	0.316795	+	0.316795i	0.847535	−	0.530740i	$-0.178086\pi$
$647$	8.05806	+	8.05806i	0.316795	+	0.316795i	−0.530740	+	0.847535i	$0.678086\pi$
$648$		0			0
$649$		−	11.0000i		−	0.431788i
$650$		0			0
$651$		0			0
$652$	−35.8997	−	35.8997i	−1.40594	−	1.40594i
$653$	−27.1583	+	27.1583i	−1.06279	+	1.06279i	−0.0648948	+	0.997892i	$0.520671\pi$
$653$	−27.1583	+	27.1583i	−1.06279	+	1.06279i	−0.997892	+	0.0648948i	$0.979329\pi$
$654$		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	$0.5\pi$
$660$	−45.2164	−	2.26650i	−1.76004	−	0.0882234i
$661$	−13.0000			−0.505641			−0.252821	−	0.967513i	$-0.581358\pi$
$661$	−13.0000			−0.505641			−0.252821	+	0.967513i	$0.581358\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$			62.2665i			2.40736i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$673$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$674$		0			0
$675$	39.1913	−	32.0330i	1.50847	−	1.23295i
$676$	26.0000			1.00000
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	35.2164	−	35.2164i	1.34752	−	1.34752i	0.459167	−	0.888350i	$-0.348148\pi$
$683$	35.2164	−	35.2164i	1.34752	−	1.34752i	0.888350	−	0.459167i	$-0.151852\pi$
$684$		0			0
$685$	−31.9248	−	1.60025i	−1.21978	−	0.0611425i
$686$		0			0
$687$	−64.4248	−	64.4248i	−2.45796	−	2.45796i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	17.0000			0.646710			0.323355	−	0.946278i	$-0.395189\pi$
$691$	17.0000			0.646710			0.323355	+	0.946278i	$0.395189\pi$
$692$		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
$704$		−	26.5330i		−	1.00000i
$705$	66.6913	−	60.3246i	2.51174	−	2.27195i
$706$		0			0
$707$		0			0
$708$	−14.3166	+	14.3166i	−0.538052	+	0.538052i
$709$			19.0000i			0.713560i	0.934188	+	0.356780i	$0.116125\pi$
$709$			19.0000i			0.713560i	−0.934188	+	0.356780i	$0.883875\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−28.2744	+	28.2744i	−1.05889	+	1.05889i
$714$		0			0
$715$		0			0
$716$	42.0000			1.56961
$717$		0			0
$718$		0			0
$719$		−	51.0000i		−	1.90198i	−0.309223	−	0.950990i	$-0.600069\pi$
$719$		−	51.0000i		−	1.90198i	0.309223	−	0.950990i	$-0.399931\pi$
$720$	37.8997	+	41.8997i	1.41244	+	1.56151i
$721$		0			0
$722$		0			0
$723$		0			0
$724$		−	19.8997i		−	0.739568i
$725$		0			0
$726$		0			0
$727$	31.4749	+	31.4749i	1.16734	+	1.16734i	0.982831	+	0.184510i	$0.0590699\pi$
$727$	31.4749	+	31.4749i	1.16734	+	1.16734i	0.184510	+	0.982831i	$0.440930\pi$
$728$		0			0
$729$		−	17.2164i		−	0.637643i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$733$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$734$		0			0
$735$	−47.7164	−	2.39181i	−1.76004	−	0.0882234i
$736$		0			0
$737$	38.0581	+	38.0581i	1.40189	+	1.40189i
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	−0.467002	+	9.31662i	−0.0171673	+	0.342486i
$741$		0			0
$742$		0			0
$743$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−23.0000			−0.839282			−0.419641	−	0.907690i	$-0.637844\pi$
$751$	−23.0000			−0.839282			−0.419641	+	0.907690i	$0.637844\pi$
$752$	37.2665	+	37.2665i	1.35897	+	1.35897i
$753$	−58.2744	+	58.2744i	−2.12364	+	2.12364i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.899749	−	0.899749i	−0.0327019	−	0.0327019i	0.690567	−	0.723269i	$-0.257361\pi$
$757$	−0.899749	−	0.899749i	−0.0327019	−	0.0327019i	−0.723269	+	0.690567i	$0.757361\pi$
$758$		0			0
$759$			40.6834i			1.47671i
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$			46.4327i			1.67988i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−34.5330	+	34.5330i	−1.24610	+	1.24610i
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$	−96.1161			−3.46154
$772$		0			0
$773$	33.6332	−	33.6332i	1.20970	−	1.20970i	0.238581	−	0.971123i	$-0.423318\pi$
$773$	33.6332	−	33.6332i	1.20970	−	1.20970i	0.971123	−	0.238581i	$-0.0766824\pi$
$774$		0			0
$775$	−4.97494	+	49.5000i	−0.178705	+	1.77809i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−9.94987			−0.356034
$782$		0			0
$783$		0			0
$784$		−	28.0000i		−	1.00000i
$785$	52.0330	+	2.60819i	1.85714	+	0.0930902i
$786$		0			0
$787$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$787$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	23.5251	+	26.0079i	0.834348	+	0.922406i
$796$	39.7995			1.41066
$797$	−26.6913	−	26.6913i	−0.945455	−	0.945455i	0.0531327	−	0.998587i	$-0.483079\pi$
$797$	−26.6913	−	26.6913i	−0.945455	−	0.945455i	−0.998587	+	0.0531327i	$0.983079\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	56.8496			2.00868
$802$		0			0
$803$		0			0
$804$		−	99.0660i		−	3.49379i
$805$		0			0
$806$		0			0
$807$	−28.6332	−	28.6332i	−1.00794	−	1.00794i
$808$		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
$815$	−2.84169	+	56.6913i	−0.0995400	+	1.98581i
$816$		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
$823$	39.4248	−	39.4248i	1.37426	−	1.37426i	0.520246	−	0.854016i	$-0.325840\pi$
$823$	39.4248	−	39.4248i	1.37426	−	1.37426i	0.854016	−	0.520246i	$-0.174160\pi$
$824$		0			0
$825$	32.0330	+	39.1913i	1.11525	+	1.36447i
$826$		0			0
$827$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$827$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$828$	35.8997	−	35.8997i	1.24760	−	1.24760i
$829$			29.0000i			1.00721i	0.863934	+	0.503606i	$0.167994\pi$
$829$			29.0000i			1.00721i	−0.863934	+	0.503606i	$0.832006\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−71.2243	−	71.2243i	−2.46187	−	2.46187i
$838$		0			0
$839$		−	36.4829i		−	1.25953i	−0.776786	−	0.629764i	$-0.783151\pi$
$839$		−	36.4829i		−	1.25953i	0.776786	−	0.629764i	$-0.216849\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−19.5000	−	21.5581i	−0.670820	−	0.741620i
$846$		0			0
$847$		0			0
$848$	−14.5330	+	14.5330i	−0.499065	+	0.499065i
$849$		0			0
$850$		0			0
$851$	8.38262			0.287353
$852$	12.9499	+	12.9499i	0.443655	+	0.443655i
$853$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$853$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$857$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$858$		0			0
$859$		−	31.0000i		−	1.05771i	−0.848713	−	0.528853i	$-0.822622\pi$
$859$		−	31.0000i		−	1.05771i	0.848713	−	0.528853i	$-0.177378\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	5.21637	−	5.21637i	0.177567	−	0.177567i	−0.612727	−	0.790295i	$-0.709928\pi$
$863$	5.21637	−	5.21637i	0.177567	−	0.177567i	0.790295	+	0.612727i	$0.209928\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−36.6913	−	36.6913i	−1.24610	−	1.24610i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−22.2665	+	22.2665i	−0.753607	+	0.753607i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$877$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$878$		0			0
$879$		0			0
$880$	−22.0000	+	19.8997i	−0.741620	+	0.670820i
$881$	57.0000			1.92038			0.960189	−	0.279350i	$-0.0901189\pi$
$881$	57.0000			1.92038			0.960189	+	0.279350i	$0.0901189\pi$
$882$		0			0
$883$	−37.9499	+	37.9499i	−1.27711	+	1.27711i	−0.334840	+	0.942275i	$0.608682\pi$
$883$	−37.9499	+	37.9499i	−1.27711	+	1.27711i	−0.942275	+	0.334840i	$0.891318\pi$
$884$		0			0
$885$	22.6082	+	1.13325i	0.759966	+	0.0380938i
$886$		0			0
$887$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$887$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−39.6332			−1.32776
$892$	28.8496	+	28.8496i	0.965957	+	0.965957i
$893$		0			0
$894$		0			0
$895$	−31.5000	−	34.8246i	−1.05293	−	1.16406i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	6.31662	−	62.8496i	0.210554	−	2.09499i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−16.5000	+	14.9248i	−0.548479	+	0.496118i
$906$		0			0
$907$	33.8496	+	33.8496i	1.12396	+	1.12396i	0.991140	+	0.132818i	$0.0424025\pi$
$907$	33.8496	+	33.8496i	1.12396	+	1.12396i	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	$-0.464952\pi$
$911$	6.63325			0.219769			0.109885	+	0.993944i	$0.464952\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−59.6992			−1.97252
$917$		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
$925$	8.07519	−	6.60025i	0.265511	−	0.217015i
$926$		0			0
$927$	50.2164	+	50.2164i	1.64932	+	1.64932i
$928$		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
$933$	−25.8997	+	25.8997i	−0.847920	+	0.847920i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$937$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$938$		0			0
$939$			105.433i			3.44067i
$940$	2.94987	−	58.8496i	0.0962143	−	1.91946i
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$			13.2665i			0.431788i
$945$		0			0
$946$		0			0
$947$	−40.1082	−	40.1082i	−1.30334	−	1.30334i	−0.926126	−	0.377215i	$-0.876882\pi$
$947$	−40.1082	−	40.1082i	−1.30334	−	1.30334i	−0.377215	−	0.926126i	$-0.623118\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	108.383			3.51455
$952$		0			0
$953$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$953$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$954$		0			0
$955$	38.5000	−	34.8246i	1.24583	−	1.12690i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	54.5330	+	2.73350i	1.76004	+	0.0882234i
$961$	68.0000			2.19355
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−43.1161			−1.38366			−0.691831	−	0.722059i	$-0.743196\pi$
$971$	−43.1161			−1.38366			−0.691831	+	0.722059i	$0.743196\pi$
$972$	8.63325	+	8.63325i	0.276912	+	0.276912i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−41.6913	−	41.6913i	−1.33382	−	1.33382i	−0.901920	−	0.431903i	$-0.857842\pi$
$977$	−41.6913	−	41.6913i	−1.33382	−	1.33382i	−0.431903	−	0.901920i	$-0.642158\pi$
$978$		0			0
$979$			29.8496i			0.953998i
$980$	−23.2164	+	21.0000i	−0.741620	+	0.670820i
$981$		0			0
$982$		0			0
$983$	−43.7414	+	43.7414i	−1.39514	+	1.39514i	−0.581811	+	0.813324i	$0.697656\pi$
$983$	−43.7414	+	43.7414i	−1.39514	+	1.39514i	−0.813324	+	0.581811i	$0.802344\pi$
$984$		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	$-0.897103\pi$
$991$	−59.6992			−1.89641			−0.948205	+	0.317660i	$0.897103\pi$
$992$		0			0
$993$	21.4749	−	21.4749i	0.681487	−	0.681487i
$994$		0			0
$995$	−29.8496	−	33.0000i	−0.946297	−	1.04617i
$996$		0			0
$997$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$997$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$998$		0			0
$999$			21.1161i			0.668085i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	55.2.e.b.32.1	✓	4
3.2	odd	2		495.2.k.a.307.1		4
4.3	odd	2		880.2.bd.d.417.2		4
5.2	odd	4		275.2.e.a.43.2		4
5.3	odd	4	inner	55.2.e.b.43.1	yes	4
5.4	even	2		275.2.e.a.32.2		4
11.2	odd	10		605.2.m.a.282.2		16
11.3	even	5		605.2.m.a.112.2		16
11.4	even	5		605.2.m.a.457.2		16
11.5	even	5		605.2.m.a.602.1		16
11.6	odd	10		605.2.m.a.602.1		16
11.7	odd	10		605.2.m.a.457.2		16
11.8	odd	10		605.2.m.a.112.2		16
11.9	even	5		605.2.m.a.282.2		16
11.10	odd	2	CM	55.2.e.b.32.1	✓	4
15.8	even	4		495.2.k.a.208.1		4
20.3	even	4		880.2.bd.d.593.2		4
33.32	even	2		495.2.k.a.307.1		4
44.43	even	2		880.2.bd.d.417.2		4
55.3	odd	20		605.2.m.a.233.2		16
55.8	even	20		605.2.m.a.233.2		16
55.13	even	20		605.2.m.a.403.1		16
55.18	even	20		605.2.m.a.578.2		16
55.28	even	20		605.2.m.a.118.2		16
55.32	even	4		275.2.e.a.43.2		4
55.38	odd	20		605.2.m.a.118.2		16
55.43	even	4	inner	55.2.e.b.43.1	yes	4
55.48	odd	20		605.2.m.a.578.2		16
55.53	odd	20		605.2.m.a.403.1		16
55.54	odd	2		275.2.e.a.32.2		4
165.98	odd	4		495.2.k.a.208.1		4
220.43	odd	4		880.2.bd.d.593.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.e.b.32.1	✓	4	1.1	even	1	trivial
55.2.e.b.32.1	✓	4	11.10	odd	2	CM
55.2.e.b.43.1	yes	4	5.3	odd	4	inner
55.2.e.b.43.1	yes	4	55.43	even	4	inner
275.2.e.a.32.2		4	5.4	even	2
275.2.e.a.32.2		4	55.54	odd	2
275.2.e.a.43.2		4	5.2	odd	4
275.2.e.a.43.2		4	55.32	even	4
495.2.k.a.208.1		4	15.8	even	4
495.2.k.a.208.1		4	165.98	odd	4
495.2.k.a.307.1		4	3.2	odd	2
495.2.k.a.307.1		4	33.32	even	2
605.2.m.a.112.2		16	11.3	even	5
605.2.m.a.112.2		16	11.8	odd	10
605.2.m.a.118.2		16	55.28	even	20
605.2.m.a.118.2		16	55.38	odd	20
605.2.m.a.233.2		16	55.3	odd	20
605.2.m.a.233.2		16	55.8	even	20
605.2.m.a.282.2		16	11.2	odd	10
605.2.m.a.282.2		16	11.9	even	5
605.2.m.a.403.1		16	55.13	even	20
605.2.m.a.403.1		16	55.53	odd	20
605.2.m.a.457.2		16	11.4	even	5
605.2.m.a.457.2		16	11.7	odd	10
605.2.m.a.578.2		16	55.18	even	20
605.2.m.a.578.2		16	55.48	odd	20
605.2.m.a.602.1		16	11.5	even	5
605.2.m.a.602.1		16	11.6	odd	10
880.2.bd.d.417.2		4	4.3	odd	2
880.2.bd.d.417.2		4	44.43	even	2
880.2.bd.d.593.2		4	20.3	even	4
880.2.bd.d.593.2		4	220.43	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 55 = 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	55.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.15831 + 2.15831i) q^{3} +2.00000i q^{4} +(1.65831 - 1.50000i) q^{5} -6.31662i q^{9} +3.31662 q^{11} +(-4.31662 - 4.31662i) q^{12} +(-0.341688 + 6.81662i) q^{15} -4.00000 q^{16} +(3.00000 + 3.31662i) q^{20} +(2.84169 - 2.84169i) q^{23} +(0.500000 - 4.97494i) q^{25} +(7.15831 + 7.15831i) q^{27} -9.94987 q^{31} +(-7.15831 + 7.15831i) q^{33} +12.6332 q^{36} +(1.47494 + 1.47494i) q^{37} +6.63325i q^{44} +(-9.47494 - 10.4749i) q^{45} +(-9.31662 - 9.31662i) q^{47} +(8.63325 - 8.63325i) q^{48} +7.00000i q^{49} +(3.63325 - 3.63325i) q^{53} +(5.50000 - 4.97494i) q^{55} -3.31662i q^{59} +(-13.6332 - 0.683375i) q^{60} -8.00000i q^{64} +(11.4749 + 11.4749i) q^{67} +12.2665i q^{69} -3.00000 q^{71} +(9.65831 + 11.8166i) q^{75} +(-6.63325 + 6.00000i) q^{80} -11.9499 q^{81} +9.00000i q^{89} +(5.68338 + 5.68338i) q^{92} +(21.4749 - 21.4749i) q^{93} +(-3.52506 - 3.52506i) q^{97} -20.9499i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 4 q^{12} - 8 q^{15} - 16 q^{16} + 12 q^{20} + 18 q^{23} + 2 q^{25} + 22 q^{27} - 22 q^{33} + 24 q^{36} - 14 q^{37} - 18 q^{45} - 24 q^{47} + 8 q^{48} - 12 q^{53} + 22 q^{55} - 28 q^{60}+ \cdots - 34 q^{97}+O(q^{100}) \) 4 * q - 2 * q^3 - 4 * q^12 - 8 * q^15 - 16 * q^16 + 12 * q^20 + 18 * q^23 + 2 * q^25 + 22 * q^27 - 22 * q^33 + 24 * q^36 - 14 * q^37 - 18 * q^45 - 24 * q^47 + 8 * q^48 - 12 * q^53 + 22 * q^55 - 28 * q^60 + 26 * q^67 - 12 * q^71 + 32 * q^75 - 8 * q^81 + 36 * q^92 + 66 * q^93 - 34 * q^97

Introduction

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