LMFDB - Embedded newform 55.2.e.b.32.2

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			32.2
Root			$1.65831 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	55.32
Dual form			55.2.e.b.43.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.15831 - 1.15831i) q^{3} +2.00000i q^{4} +(-1.65831 - 1.50000i) q^{5} +0.316625i q^{9} +O(q^{10})$	$q+(1.15831 - 1.15831i) q^{3} +2.00000i q^{4} +(-1.65831 - 1.50000i) q^{5} +0.316625i q^{9} -3.31662 q^{11} +(2.31662 + 2.31662i) q^{12} +(-3.65831 + 0.183375i) q^{15} -4.00000 q^{16} +(3.00000 - 3.31662i) q^{20} +(6.15831 - 6.15831i) q^{23} +(0.500000 + 4.97494i) q^{25} +(3.84169 + 3.84169i) q^{27} +9.94987 q^{31} +(-3.84169 + 3.84169i) q^{33} -0.633250 q^{36} +(-8.47494 - 8.47494i) q^{37} -6.63325i q^{44} +(0.474937 - 0.525063i) q^{45} +(-2.68338 - 2.68338i) q^{47} +(-4.63325 + 4.63325i) q^{48} +7.00000i q^{49} +(-9.63325 + 9.63325i) q^{53} +(5.50000 + 4.97494i) q^{55} +3.31662i q^{59} +(-0.366750 - 7.31662i) q^{60} -8.00000i q^{64} +(1.52506 + 1.52506i) q^{67} -14.2665i q^{69} -3.00000 q^{71} +(6.34169 + 5.18338i) q^{75} +(6.63325 + 6.00000i) q^{80} +7.94987 q^{81} +9.00000i q^{89} +(12.3166 + 12.3166i) q^{92} +(11.5251 - 11.5251i) q^{93} +(-13.4749 - 13.4749i) q^{97} -1.05013i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3}+O(q^{10}) $ 4 * q - 2 * q^3	$ 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}+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

Display 100 coefficients 10 coefficients

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	$0.250000\pi$
$3$	1.15831	−	1.15831i	0.668752	−	0.668752i	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	1.15831	−	1.15831i	0.668752	−	0.668752i	0.957427	+	0.288675i	$0.0932147\pi$
$4$			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
$9$			0.316625i			0.105542i
$10$		0			0
$11$	−3.31662			−1.00000
$11$	−3.31662			−1.00000
$12$	2.31662	+	2.31662i	0.668752	+	0.668752i
$13$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$13$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$14$		0			0
$15$	−3.65831	+	0.183375i	−0.944572	+	0.0473473i
$16$	−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
$23$	6.15831	−	6.15831i	1.28410	−	1.28410i	0.345782	−	0.938315i	$-0.387614\pi$
$23$	6.15831	−	6.15831i	1.28410	−	1.28410i	0.938315	−	0.345782i	$-0.112386\pi$
$24$		0			0
$25$	0.500000	+	4.97494i	0.100000	+	0.994987i
$26$		0			0
$27$	3.84169	+	3.84169i	0.739333	+	0.739333i
$28$		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	$-0.148224\pi$
$31$	9.94987			1.78705			0.893525	+	0.449013i	$0.148224\pi$
$32$		0			0
$33$	−3.84169	+	3.84169i	−0.668752	+	0.668752i
$34$		0			0
$35$		0			0
$36$	−0.633250			−0.105542
$37$	−8.47494	−	8.47494i	−1.39327	−	1.39327i	−0.817875	−	0.575396i	$-0.804848\pi$
$37$	−8.47494	−	8.47494i	−1.39327	−	1.39327i	−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
$45$	0.474937	−	0.525063i	0.0707995	−	0.0782717i
$46$		0			0
$47$	−2.68338	−	2.68338i	−0.391411	−	0.391411i	0.483779	−	0.875190i	$-0.339264\pi$
$47$	−2.68338	−	2.68338i	−0.391411	−	0.391411i	−0.875190	+	0.483779i	$0.839264\pi$
$48$	−4.63325	+	4.63325i	−0.668752	+	0.668752i
$49$			7.00000i			1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−9.63325	+	9.63325i	−1.32323	+	1.32323i	−0.412082	+	0.911147i	$0.635198\pi$
$53$	−9.63325	+	9.63325i	−1.32323	+	1.32323i	−0.911147	+	0.412082i	$0.864802\pi$
$54$		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	$0.0692665\pi$
$59$			3.31662i			0.431788i	−0.976417	+	0.215894i	$0.930733\pi$
$60$	−0.366750	−	7.31662i	−0.0473473	−	0.944572i
$61$		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
$67$	1.52506	+	1.52506i	0.186316	+	0.186316i	0.794101	−	0.607785i	$-0.207942\pi$
$67$	1.52506	+	1.52506i	0.186316	+	0.186316i	−0.607785	+	0.794101i	$0.707942\pi$
$68$		0			0
$69$		−	14.2665i		−	1.71748i
$70$		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
$75$	6.34169	+	5.18338i	0.732275	+	0.598525i
$76$		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
$81$	7.94987			0.883319
$82$		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
$92$	12.3166	+	12.3166i	1.28410	+	1.28410i
$93$	11.5251	−	11.5251i	1.19509	−	1.19509i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−13.4749	−	13.4749i	−1.36817	−	1.36817i	−0.863044	−	0.505128i	$-0.831445\pi$
$97$	−13.4749	−	13.4749i	−1.36817	−	1.36817i	−0.505128	−	0.863044i	$-0.668555\pi$
$98$		0			0
$99$		−	1.05013i		−	0.105542i
$100$	−9.94987	+	1.00000i	−0.994987	+	0.100000i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	11.9499	−	11.9499i	1.17746	−	1.17746i	0.197066	−	0.980390i	$-0.436859\pi$
$103$	11.9499	−	11.9499i	1.17746	−	1.17746i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$		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	$0.250000\pi$
$108$	−7.68338	+	7.68338i	−0.739333	+	0.739333i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$	−19.6332			−1.86351
$112$		0			0
$113$	−8.84169	+	8.84169i	−0.831756	+	0.831756i	−0.987757	−	0.156001i	$-0.950140\pi$
$113$	−8.84169	+	8.84169i	−0.831756	+	0.831756i	0.156001	+	0.987757i	$0.450140\pi$
$114$		0			0
$115$	−19.4499	+	0.974937i	−1.81371	+	0.0909134i
$116$		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			$\pi$
$132$	−7.68338	−	7.68338i	−0.668752	−	0.668752i
$133$		0			0
$134$		0			0
$135$	−0.608187	−	12.1332i	−0.0523444	−	1.04426i
$136$		0			0
$137$	13.1082	+	13.1082i	1.11991	+	1.11991i	0.991754	+	0.128154i	$0.0409051\pi$
$137$	13.1082	+	13.1082i	1.11991	+	1.11991i	0.128154	+	0.991754i	$0.459095\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	−6.21637			−0.523513
$142$		0			0
$143$		0			0
$144$		−	1.26650i		−	0.105542i
$145$		0			0
$146$		0			0
$147$	8.10819	+	8.10819i	0.668752	+	0.668752i
$148$	16.9499	−	16.9499i	1.39327	−	1.39327i
$149$		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
$157$	6.52506	+	6.52506i	0.520757	+	0.520757i	0.917800	−	0.397043i	$-0.129964\pi$
$157$	6.52506	+	6.52506i	0.520757	+	0.520757i	−0.397043	+	0.917800i	$0.629964\pi$
$158$		0			0
$159$			22.3166i			1.76982i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1.94987	−	1.94987i	0.152726	−	0.152726i	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	1.94987	−	1.94987i	0.152726	−	0.152726i	0.779334	+	0.626608i	$0.215557\pi$
$164$		0			0
$165$	12.1332	−	0.608187i	0.944572	−	0.0473473i
$166$		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
$177$	3.84169	+	3.84169i	0.288759	+	0.288759i
$178$		0			0
$179$		−	21.0000i		−	1.56961i	−0.619740	−	0.784807i	$-0.712762\pi$
$179$		−	21.0000i		−	1.56961i	0.619740	−	0.784807i	$-0.287238\pi$
$180$	1.05013	+	0.949874i	0.0782717	+	0.0707995i
$181$	9.94987			0.739568			0.369784	−	0.929118i	$-0.379432\pi$
$181$	9.94987			0.739568			0.369784	+	0.929118i	$0.379432\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.34169	+	26.7665i	0.0986428	+	1.96791i
$186$		0			0
$187$		0			0
$188$	5.36675	−	5.36675i	0.391411	−	0.391411i
$189$		0			0
$190$		0			0
$191$	−23.2164			−1.67988			−0.839939	−	0.542681i	$-0.817409\pi$
$191$	−23.2164			−1.67988			−0.839939	+	0.542681i	$0.817409\pi$
$192$	−9.26650	−	9.26650i	−0.668752	−	0.668752i
$193$		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	$0.249200\pi$
$199$			19.8997i			1.41066i	−0.708881	+	0.705328i	$0.750800\pi$
$200$		0			0
$201$	3.53300			0.249198
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	1.94987	+	1.94987i	0.135526	+	0.135526i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	−19.2665	−	19.2665i	−1.32323	−	1.32323i
$213$	−3.47494	+	3.47494i	−0.238099	+	0.238099i
$214$		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
$223$	−15.4248	+	15.4248i	−1.03292	+	1.03292i	−0.0334825	+	0.999439i	$0.510660\pi$
$223$	−15.4248	+	15.4248i	−1.03292	+	1.03292i	−0.999439	+	0.0334825i	$0.989340\pi$
$224$		0			0
$225$	−1.57519	+	0.158312i	−0.105013	+	0.0105542i
$226$		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	$-0.552828\pi$
$229$		−	29.8496i		−	1.97252i	0.165205	−	0.986259i	$-0.447172\pi$
$230$		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
$235$	0.424812	+	8.47494i	0.0277117	+	0.552844i
$236$	−6.63325			−0.431788
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$	14.6332	−	0.733501i	0.944572	−	0.0473473i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	−2.31662	+	2.31662i	−0.148612	+	0.148612i
$244$		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
$253$	−20.4248	+	20.4248i	−1.28410	+	1.28410i
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$	−4.26650	−	4.26650i	−0.266137	−	0.266137i	0.561405	−	0.827541i	$-0.310261\pi$
$257$	−4.26650	−	4.26650i	−0.266137	−	0.266137i	−0.827541	+	0.561405i	$0.810261\pi$
$258$		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
$265$	30.4248	−	1.52506i	1.86898	−	0.0936839i
$266$		0			0
$267$	10.4248	+	10.4248i	0.637988	+	0.637988i
$268$	−3.05013	+	3.05013i	−0.186316	+	0.186316i
$269$		−	13.2665i		−	0.808873i	−0.914566	−	0.404436i	$-0.867468\pi$
$269$		−	13.2665i		−	0.808873i	0.914566	−	0.404436i	$-0.132532\pi$
$270$		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
$276$	28.5330			1.71748
$277$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$277$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$278$		0			0
$279$			3.15038i			0.188608i
$280$		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
$291$	−31.2164			−1.82994
$292$		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
$297$	−12.7414	−	12.7414i	−0.739333	−	0.739333i
$298$		0			0
$299$		0			0
$300$	−10.3668	+	12.6834i	−0.598525	+	0.732275i
$301$		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
$309$		−	27.6834i		−	1.57485i
$310$		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
$313$	−5.42481	+	5.42481i	−0.306628	+	0.306628i	−0.843600	−	0.536972i	$-0.819568\pi$
$313$	−5.42481	+	5.42481i	−0.306628	+	0.306628i	0.536972	+	0.843600i	$0.319568\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.89181	−	1.89181i	−0.106255	−	0.106255i	0.651981	−	0.758236i	$-0.273938\pi$
$317$	−1.89181	−	1.89181i	−0.106255	−	0.106255i	−0.758236	+	0.651981i	$0.773938\pi$
$318$		0			0
$319$		0			0
$320$	−12.0000	+	13.2665i	−0.670820	+	0.741620i
$321$		0			0
$322$		0			0
$323$		0			0
$324$			15.8997i			0.883319i
$325$		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	$-0.411836\pi$
$331$	9.94987			0.546895			0.273447	+	0.961887i	$0.411836\pi$
$332$		0			0
$333$	2.68338	−	2.68338i	0.147048	−	0.147048i
$334$		0			0
$335$	−0.241436	−	4.81662i	−0.0131911	−	0.263160i
$336$		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
$339$			20.4829i			1.11248i
$340$		0			0
$341$	−33.0000			−1.78705
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−21.3997	+	23.6583i	−1.15212	+	1.27372i
$346$		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
$353$	22.7414	−	22.7414i	1.21040	−	1.21040i	0.239511	−	0.970894i	$-0.423013\pi$
$353$	22.7414	−	22.7414i	1.21040	−	1.21040i	0.970894	−	0.239511i	$-0.0769871\pi$
$354$		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
$363$	12.7414	−	12.7414i	0.668752	−	0.668752i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−23.4749	−	23.4749i	−1.22538	−	1.22538i	−0.965692	−	0.259690i	$-0.916380\pi$
$367$	−23.4749	−	23.4749i	−1.22538	−	1.22538i	−0.259690	−	0.965692i	$-0.583620\pi$
$368$	−24.6332	+	24.6332i	−1.28410	+	1.28410i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	23.0501	+	23.0501i	1.19509	+	1.19509i
$373$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$373$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$374$		0			0
$375$	−2.74144	−	18.1082i	−0.141567	−	0.935103i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	29.8496i		−	1.53327i	−0.642082	−	0.766636i	$-0.721929\pi$
$379$		−	29.8496i		−	1.53327i	0.642082	−	0.766636i	$-0.278071\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	21.1583	−	21.1583i	1.08114	−	1.08114i	0.0847358	−	0.996403i	$-0.472995\pi$
$383$	21.1583	−	21.1583i	1.08114	−	1.08114i	0.996403	−	0.0847358i	$-0.0270046\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	26.9499	−	26.9499i	1.36817	−	1.36817i
$389$			36.4829i			1.84976i	0.380265	+	0.924878i	$0.375833\pi$
$389$			36.4829i			1.84976i	−0.380265	+	0.924878i	$0.624167\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	2.10025			0.105542
$397$	18.8997	+	18.8997i	0.948551	+	0.948551i	0.998740	−	0.0501886i	$-0.0159822\pi$
$397$	18.8997	+	18.8997i	0.948551	+	0.948551i	−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	$-0.269497\pi$
$401$	26.5330			1.32499			0.662497	+	0.749064i	$0.269497\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−13.1834	−	11.9248i	−0.655087	−	0.592549i
$406$		0			0
$407$	28.1082	+	28.1082i	1.39327	+	1.39327i
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$	30.3668			1.49788
$412$	23.8997	+	23.8997i	1.17746	+	1.17746i
$413$		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	$-0.921644\pi$
$421$	−39.7995			−1.93971			−0.969854	+	0.243685i	$0.921644\pi$
$422$		0			0
$423$	0.849623	−	0.849623i	0.0413101	−	0.0413101i
$424$		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			$\pi$
$432$	−15.3668	−	15.3668i	−0.739333	−	0.739333i
$433$	−0.424812	+	0.424812i	−0.0204151	+	0.0204151i	−0.717241	−	0.696826i	$-0.754595\pi$
$433$	−0.424812	+	0.424812i	−0.0204151	+	0.0204151i	0.696826	+	0.717241i	$0.254595\pi$
$434$		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
$441$	−2.21637			−0.105542
$442$		0			0
$443$	7.74144	−	7.74144i	0.367807	−	0.367807i	−0.498870	−	0.866677i	$-0.666252\pi$
$443$	7.74144	−	7.74144i	0.367807	−	0.367807i	0.866677	+	0.498870i	$0.166252\pi$
$444$		−	39.2665i		−	1.86351i
$445$	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
$452$	−17.6834	−	17.6834i	−0.831756	−	0.831756i
$453$		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
$460$	−1.94987	−	38.8997i	−0.0909134	−	1.81371i
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−30.4248	+	30.4248i	−1.41396	+	1.41396i	−0.693615	+	0.720346i	$0.743983\pi$
$463$	−30.4248	+	30.4248i	−1.41396	+	1.41396i	−0.720346	+	0.693615i	$0.756017\pi$
$464$		0			0
$465$	−36.3997	+	1.82456i	−1.68800	+	0.0846120i
$466$		0			0
$467$	−20.0581	−	20.0581i	−0.928176	−	0.928176i	0.0694117	−	0.997588i	$-0.477888\pi$
$467$	−20.0581	−	20.0581i	−0.928176	−	0.928176i	−0.997588	+	0.0694117i	$0.977888\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	15.1161			0.696514
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−3.05013	−	3.05013i	−0.139656	−	0.139656i
$478$		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
$485$	2.13325	+	42.5581i	0.0968659	+	1.93246i
$486$		0			0
$487$	16.5251	+	16.5251i	0.748822	+	0.748822i	0.974258	−	0.225436i	$-0.0723806\pi$
$487$	16.5251	+	16.5251i	0.748822	+	0.748822i	−0.225436	+	0.974258i	$0.572381\pi$
$488$		0			0
$489$		−	4.51713i		−	0.204272i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−1.57519	+	1.74144i	−0.0707995	+	0.0782717i
$496$	−39.7995			−1.78705
$497$		0			0
$498$		0			0
$499$			19.8997i			0.890835i	0.895323	+	0.445418i	$0.146945\pi$
$499$			19.8997i			0.890835i	−0.895323	+	0.445418i	$0.853055\pi$
$500$	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
$507$	−15.0581	−	15.0581i	−0.668752	−	0.668752i
$508$		0			0
$509$			3.31662i			0.147007i	0.997295	+	0.0735034i	$0.0234180\pi$
$509$			3.31662i			0.147007i	−0.997295	+	0.0735034i	$0.976582\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−37.7414	+	1.89181i	−1.66309	+	0.0833633i
$516$		0			0
$517$	8.89975	+	8.89975i	0.391411	+	0.391411i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	43.1161			1.88895			0.944476	−	0.328581i	$-0.106570\pi$
$521$	43.1161			1.88895			0.944476	+	0.328581i	$0.106570\pi$
$522$		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
$528$	15.3668	−	15.3668i	0.668752	−	0.668752i
$529$		−	52.8496i		−	2.29781i
$530$		0			0
$531$	−1.05013			−0.0455716
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−24.3246	−	24.3246i	−1.04968	−	1.04968i
$538$		0			0
$539$		−	23.2164i		−	1.00000i
$540$	24.2665	−	1.21637i	1.04426	−	0.0523444i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	11.5251	−	11.5251i	0.494588	−	0.494588i
$544$		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	$0.250000\pi$
$548$	−26.2164	+	26.2164i	−1.11991	+	1.11991i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	32.5581	+	29.4499i	1.38201	+	1.25008i
$556$		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	$0.750000\pi$
$564$		−	12.4327i		−	0.523513i
$565$	27.9248	−	1.39975i	1.17481	−	0.0588879i
$566$		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
$573$	−26.8918	+	26.8918i	−1.12342	+	1.12342i
$574$		0			0
$575$	33.7164	+	27.5581i	1.40607	+	1.14925i
$576$	2.53300			0.105542
$577$	−28.4749	−	28.4749i	−1.18543	−	1.18543i	−0.978318	−	0.207109i	$-0.933594\pi$
$577$	−28.4749	−	28.4749i	−1.18543	−	1.18543i	−0.207109	−	0.978318i	$-0.566406\pi$
$578$		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	0.990586	+	0.136892i	$0.0437113\pi$
$587$	27.3166	+	27.3166i	1.12748	+	1.12748i	0.136892	+	0.990586i	$0.456289\pi$
$588$	−16.2164	+	16.2164i	−0.668752	+	0.668752i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	33.8997	+	33.8997i	1.39327	+	1.39327i
$593$		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
$597$	23.0501	+	23.0501i	0.943379	+	0.943379i
$598$		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
$603$	−0.482873	+	0.482873i	−0.0196641	+	0.0196641i
$604$		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
$617$	−34.2665	−	34.2665i	−1.37952	−	1.37952i	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−34.2665	−	34.2665i	−1.37952	−	1.37952i	−0.534089	−	0.845428i	$-0.679345\pi$
$618$		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
$621$	47.3166			1.89875
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−24.5000	+	4.97494i	−0.980000	+	0.198997i
$626$		0			0
$627$		0			0
$628$	−13.0501	+	13.0501i	−0.520757	+	0.520757i
$629$		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
$636$	−44.6332			−1.76982
$637$		0			0
$638$		0			0
$639$		−	0.949874i		−	0.0375764i
$640$		0			0
$641$	−23.2164			−0.916992			−0.458496	−	0.888697i	$-0.651612\pi$
$641$	−23.2164			−0.916992			−0.458496	+	0.888697i	$0.651612\pi$
$642$		0			0
$643$	−35.4248	+	35.4248i	−1.39702	+	1.39702i	−0.588577	+	0.808441i	$0.700312\pi$
$643$	−35.4248	+	35.4248i	−1.39702	+	1.39702i	−0.808441	+	0.588577i	$0.799688\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−35.0581	−	35.0581i	−1.37827	−	1.37827i	−0.847535	−	0.530740i	$-0.821914\pi$
$647$	−35.0581	−	35.0581i	−1.37827	−	1.37827i	−0.530740	−	0.847535i	$-0.678086\pi$
$648$		0			0
$649$		−	11.0000i		−	0.431788i
$650$		0			0
$651$		0			0
$652$	3.89975	+	3.89975i	0.152726	+	0.152726i
$653$	−23.8417	+	23.8417i	−0.932997	+	0.932997i	−0.997892	−	0.0648948i	$-0.979329\pi$
$653$	−23.8417	+	23.8417i	−0.932997	+	0.932997i	0.0648948	+	0.997892i	$0.479329\pi$
$654$		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	$0.5\pi$
$660$	1.21637	+	24.2665i	0.0473473	+	0.944572i
$661$	−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
$669$			35.7335i			1.38154i
$670$		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
$675$	−17.1913	+	21.0330i	−0.661694	+	0.809560i
$676$	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
$683$	−11.2164	+	11.2164i	−0.429183	+	0.429183i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$	−11.2164	+	11.2164i	−0.429183	+	0.429183i	0.459167	+	0.888350i	$0.348148\pi$
$684$		0			0
$685$	−2.07519	−	41.3997i	−0.0792889	−	1.58180i
$686$		0			0
$687$	−34.5752	−	34.5752i	−1.31913	−	1.31913i
$688$		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
$705$	10.3087	+	9.32456i	0.388248	+	0.351183i
$706$		0			0
$707$		0			0
$708$	−7.68338	+	7.68338i	−0.288759	+	0.288759i
$709$			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
$713$	61.2744	−	61.2744i	2.29475	−	2.29475i
$714$		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	$-0.399931\pi$
$720$	−1.89975	+	2.10025i	−0.0707995	+	0.0782717i
$721$		0			0
$722$		0			0
$723$		0			0
$724$			19.8997i			0.739568i
$725$		0			0
$726$		0			0
$727$	21.5251	+	21.5251i	0.798320	+	0.798320i	0.982831	−	0.184510i	$-0.0590699\pi$
$727$	21.5251	+	21.5251i	0.798320	+	0.798320i	−0.184510	+	0.982831i	$0.559070\pi$
$728$		0			0
$729$			29.2164i			1.08209i
$730$		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
$735$	−1.28363	−	25.6082i	−0.0473473	−	0.944572i
$736$		0			0
$737$	−5.05806	−	5.05806i	−0.186316	−	0.186316i
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	−53.5330	+	2.68338i	−1.96791	+	0.0986428i
$741$		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	$0.637844\pi$
$752$	10.7335	+	10.7335i	0.391411	+	0.391411i
$753$	31.2744	−	31.2744i	1.13970	−	1.13970i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	38.8997	+	38.8997i	1.41384	+	1.41384i	0.723269	+	0.690567i	$0.242639\pi$
$757$	38.8997	+	38.8997i	1.41384	+	1.41384i	0.690567	+	0.723269i	$0.257361\pi$
$758$		0			0
$759$			47.3166i			1.71748i
$760$		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
$768$	18.5330	−	18.5330i	0.668752	−	0.668752i
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$	−9.88388			−0.355959
$772$		0			0
$773$	20.3668	−	20.3668i	0.732541	−	0.732541i	−0.238581	−	0.971123i	$-0.576682\pi$
$773$	20.3668	−	20.3668i	0.732541	−	0.732541i	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
$785$	−1.03300	−	20.6082i	−0.0368693	−	0.735538i
$786$		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
$795$	33.4749	−	37.0079i	1.18723	−	1.31254i
$796$	−39.7995			−1.41066
$797$	29.6913	+	29.6913i	1.05172	+	1.05172i	0.998587	+	0.0531327i	$0.0169206\pi$
$797$	29.6913	+	29.6913i	1.05172	+	1.05172i	0.0531327	+	0.998587i	$0.483079\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−2.84962			−0.100686
$802$		0			0
$803$		0			0
$804$			7.06600i			0.249198i
$805$		0			0
$806$		0			0
$807$	−15.3668	−	15.3668i	−0.540935	−	0.540935i
$808$		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
$815$	−6.15831	+	0.308689i	−0.215716	+	0.0108129i
$816$		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
$823$	9.57519	−	9.57519i	0.333770	−	0.333770i	−0.520246	−	0.854016i	$-0.674160\pi$
$823$	9.57519	−	9.57519i	0.333770	−	0.333770i	0.854016	+	0.520246i	$0.174160\pi$
$824$		0			0
$825$	−21.0330	−	17.1913i	−0.732275	−	0.598525i
$826$		0			0
$827$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$827$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$828$	−3.89975	+	3.89975i	−0.135526	+	0.135526i
$829$			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
$837$	38.2243	+	38.2243i	1.32123	+	1.32123i
$838$		0			0
$839$			36.4829i			1.25953i	0.776786	+	0.629764i	$0.216849\pi$
$839$			36.4829i			1.25953i	−0.776786	+	0.629764i	$0.783151\pi$
$840$		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
$848$	38.5330	−	38.5330i	1.32323	−	1.32323i
$849$		0			0
$850$		0			0
$851$	−104.383			−3.57819
$852$	−6.94987	−	6.94987i	−0.238099	−	0.238099i
$853$		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
$863$	−41.2164	+	41.2164i	−1.40302	+	1.40302i	−0.612727	+	0.790295i	$0.709928\pi$
$863$	−41.2164	+	41.2164i	−1.40302	+	1.40302i	−0.790295	+	0.612727i	$0.790072\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	19.6913	+	19.6913i	0.668752	+	0.668752i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	4.26650	−	4.26650i	0.144399	−	0.144399i
$874$		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
$883$	−18.0501	+	18.0501i	−0.607435	+	0.607435i	−0.942275	−	0.334840i	$-0.891318\pi$
$883$	−18.0501	+	18.0501i	−0.607435	+	0.607435i	0.334840	+	0.942275i	$0.391318\pi$
$884$		0			0
$885$	−0.608187	−	12.1332i	−0.0204440	−	0.407855i
$886$		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
$891$	−26.3668			−0.883319
$892$	−30.8496	−	30.8496i	−1.03292	−	1.03292i
$893$		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
$900$	−0.316625	−	3.15038i	−0.0105542	−	0.105013i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−16.5000	−	14.9248i	−0.548479	−	0.496118i
$906$		0			0
$907$	−25.8496	−	25.8496i	−0.858323	−	0.858323i	0.132818	−	0.991140i	$-0.457597\pi$
$907$	−25.8496	−	25.8496i	−0.858323	−	0.858323i	−0.991140	+	0.132818i	$0.957597\pi$
$908$		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
$916$	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
$925$	37.9248	−	46.3997i	1.24696	−	1.52561i
$926$		0			0
$927$	3.78363	+	3.78363i	0.124271	+	0.124271i
$928$		0			0
$929$			53.0660i			1.74104i	0.492134	+	0.870519i	$0.336217\pi$
$929$			53.0660i			1.74104i	−0.492134	+	0.870519i	$0.663783\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	13.8997	−	13.8997i	0.455057	−	0.455057i
$934$		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
$939$			12.5673i			0.410117i
$940$	−16.9499	+	0.849623i	−0.552844	+	0.0277117i
$941$		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
$947$	−16.8918	−	16.8918i	−0.548910	−	0.548910i	0.377215	−	0.926126i	$-0.376882\pi$
$947$	−16.8918	−	16.8918i	−0.548910	−	0.548910i	−0.926126	+	0.377215i	$0.876882\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−4.38262			−0.142116
$952$		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
$960$	1.46700	+	29.2665i	0.0473473	+	0.944572i
$961$	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	$-0.256804\pi$
$971$	43.1161			1.38366			0.691831	+	0.722059i	$0.256804\pi$
$972$	−4.63325	−	4.63325i	−0.148612	−	0.148612i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	14.6913	+	14.6913i	0.470017	+	0.470017i	0.901920	−	0.431903i	$-0.142158\pi$
$977$	14.6913	+	14.6913i	0.470017	+	0.470017i	−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
$983$	−7.25856	+	7.25856i	−0.231512	+	0.231512i	−0.813324	−	0.581811i	$-0.802344\pi$
$983$	−7.25856	+	7.25856i	−0.231512	+	0.231512i	0.581811	+	0.813324i	$0.302344\pi$
$984$		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
$993$	11.5251	−	11.5251i	0.365737	−	0.365737i
$994$		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
$999$		−	65.1161i		−	2.06018i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	55.2.e.b.32.2	✓	4
3.2	odd	2		495.2.k.a.307.2		4
4.3	odd	2		880.2.bd.d.417.1		4
5.2	odd	4		275.2.e.a.43.1		4
5.3	odd	4	inner	55.2.e.b.43.2	yes	4
5.4	even	2		275.2.e.a.32.1		4
11.2	odd	10		605.2.m.a.282.1		16
11.3	even	5		605.2.m.a.112.1		16
11.4	even	5		605.2.m.a.457.1		16
11.5	even	5		605.2.m.a.602.2		16
11.6	odd	10		605.2.m.a.602.2		16
11.7	odd	10		605.2.m.a.457.1		16
11.8	odd	10		605.2.m.a.112.1		16
11.9	even	5		605.2.m.a.282.1		16
11.10	odd	2	CM	55.2.e.b.32.2	✓	4
15.8	even	4		495.2.k.a.208.2		4
20.3	even	4		880.2.bd.d.593.1		4
33.32	even	2		495.2.k.a.307.2		4
44.43	even	2		880.2.bd.d.417.1		4
55.3	odd	20		605.2.m.a.233.1		16
55.8	even	20		605.2.m.a.233.1		16
55.13	even	20		605.2.m.a.403.2		16
55.18	even	20		605.2.m.a.578.1		16
55.28	even	20		605.2.m.a.118.1		16
55.32	even	4		275.2.e.a.43.1		4
55.38	odd	20		605.2.m.a.118.1		16
55.43	even	4	inner	55.2.e.b.43.2	yes	4
55.48	odd	20		605.2.m.a.578.1		16
55.53	odd	20		605.2.m.a.403.2		16
55.54	odd	2		275.2.e.a.32.1		4
165.98	odd	4		495.2.k.a.208.2		4
220.43	odd	4		880.2.bd.d.593.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.e.b.32.2	✓	4	1.1	even	1	trivial
55.2.e.b.32.2	✓	4	11.10	odd	2	CM
55.2.e.b.43.2	yes	4	5.3	odd	4	inner
55.2.e.b.43.2	yes	4	55.43	even	4	inner
275.2.e.a.32.1		4	5.4	even	2
275.2.e.a.32.1		4	55.54	odd	2
275.2.e.a.43.1		4	5.2	odd	4
275.2.e.a.43.1		4	55.32	even	4
495.2.k.a.208.2		4	15.8	even	4
495.2.k.a.208.2		4	165.98	odd	4
495.2.k.a.307.2		4	3.2	odd	2
495.2.k.a.307.2		4	33.32	even	2
605.2.m.a.112.1		16	11.3	even	5
605.2.m.a.112.1		16	11.8	odd	10
605.2.m.a.118.1		16	55.28	even	20
605.2.m.a.118.1		16	55.38	odd	20
605.2.m.a.233.1		16	55.3	odd	20
605.2.m.a.233.1		16	55.8	even	20
605.2.m.a.282.1		16	11.2	odd	10
605.2.m.a.282.1		16	11.9	even	5
605.2.m.a.403.2		16	55.13	even	20
605.2.m.a.403.2		16	55.53	odd	20
605.2.m.a.457.1		16	11.4	even	5
605.2.m.a.457.1		16	11.7	odd	10
605.2.m.a.578.1		16	55.18	even	20
605.2.m.a.578.1		16	55.48	odd	20
605.2.m.a.602.2		16	11.5	even	5
605.2.m.a.602.2		16	11.6	odd	10
880.2.bd.d.417.1		4	4.3	odd	2
880.2.bd.d.417.1		4	44.43	even	2
880.2.bd.d.593.1		4	20.3	even	4
880.2.bd.d.593.1		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(1.15831 - 1.15831i) q^{3} +2.00000i q^{4} +(-1.65831 - 1.50000i) q^{5} +0.316625i q^{9} +O(q^{10})\)	\(q+(1.15831 - 1.15831i) q^{3} +2.00000i q^{4} +(-1.65831 - 1.50000i) q^{5} +0.316625i q^{9} -3.31662 q^{11} +(2.31662 + 2.31662i) q^{12} +(-3.65831 + 0.183375i) q^{15} -4.00000 q^{16} +(3.00000 - 3.31662i) q^{20} +(6.15831 - 6.15831i) q^{23} +(0.500000 + 4.97494i) q^{25} +(3.84169 + 3.84169i) q^{27} +9.94987 q^{31} +(-3.84169 + 3.84169i) q^{33} -0.633250 q^{36} +(-8.47494 - 8.47494i) q^{37} -6.63325i q^{44} +(0.474937 - 0.525063i) q^{45} +(-2.68338 - 2.68338i) q^{47} +(-4.63325 + 4.63325i) q^{48} +7.00000i q^{49} +(-9.63325 + 9.63325i) q^{53} +(5.50000 + 4.97494i) q^{55} +3.31662i q^{59} +(-0.366750 - 7.31662i) q^{60} -8.00000i q^{64} +(1.52506 + 1.52506i) q^{67} -14.2665i q^{69} -3.00000 q^{71} +(6.34169 + 5.18338i) q^{75} +(6.63325 + 6.00000i) q^{80} +7.94987 q^{81} +9.00000i q^{89} +(12.3166 + 12.3166i) q^{92} +(11.5251 - 11.5251i) q^{93} +(-13.4749 - 13.4749i) q^{97} -1.05013i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3}+O(q^{10}) \) 4 * q - 2 * q^3	\( 4 q - 2 q^{3} - 4 q^{12} - 8 q^{15} - 16 q^{16} + 12 q^{20} + 18 q^{23} + 2 q^{25} + 22 q^{27} - 22 q^{33} + 24 q^{36} - 14 q^{37} - 18 q^{45} - 24 q^{47} + 8 q^{48} - 12 q^{53} + 22 q^{55} - 28 q^{60} + 26 q^{67} - 12 q^{71} + 32 q^{75} - 8 q^{81} + 36 q^{92} + 66 q^{93} - 34 q^{97}+O(q^{100}) \) 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

L-functions

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Database

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