LMFDB - Embedded newform 121.2.c.c.3.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [121,2,Mod(3,121)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(121, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([8]))

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

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

chi := DirichletCharacter("121.3");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 121 = 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	121.c (of order $5$, degree $4$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$0.966189864457$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + x^{2} - x + 1 $ x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{5}]$

Embedding invariants

Embedding label			3.1
Root			$0.809017 + 0.587785i$ of defining polynomial
Character	$\chi$	$=$	121.3
Dual form			121.2.c.c.81.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.309017 - 0.951057i) q^{3} +(-0.618034 + 1.90211i) q^{4} +(2.42705 + 1.76336i) q^{5} +(1.61803 - 1.17557i) q^{9} +O(q^{10})$	$q+(-0.309017 - 0.951057i) q^{3} +(-0.618034 + 1.90211i) q^{4} +(2.42705 + 1.76336i) q^{5} +(1.61803 - 1.17557i) q^{9} +2.00000 q^{12} +(0.927051 - 2.85317i) q^{15} +(-3.23607 - 2.35114i) q^{16} +(-4.85410 + 3.52671i) q^{20} -9.00000 q^{23} +(1.23607 + 3.80423i) q^{25} +(-4.04508 - 2.93893i) q^{27} +(4.04508 - 2.93893i) q^{31} +(1.23607 + 3.80423i) q^{36} +(2.16312 - 6.65740i) q^{37} +6.00000 q^{45} +(-3.70820 - 11.4127i) q^{47} +(-1.23607 + 3.80423i) q^{48} +(5.66312 + 4.11450i) q^{49} +(-4.85410 + 3.52671i) q^{53} +(-4.63525 + 14.2658i) q^{59} +(4.85410 + 3.52671i) q^{60} +(6.47214 - 4.70228i) q^{64} +13.0000 q^{67} +(2.78115 + 8.55951i) q^{69} +(2.42705 + 1.76336i) q^{71} +(3.23607 - 2.35114i) q^{75} +(-3.70820 - 11.4127i) q^{80} +(0.309017 - 0.951057i) q^{81} -9.00000 q^{89} +(5.56231 - 17.1190i) q^{92} +(-4.04508 - 2.93893i) q^{93} +(-13.7533 + 9.99235i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} + 2 q^{4} + 3 q^{5} + 2 q^{9}+O(q^{10}) $ 4 * q + q^3 + 2 * q^4 + 3 * q^5 + 2 * q^9	$ 4 q + q^{3} + 2 q^{4} + 3 q^{5} + 2 q^{9} + 8 q^{12} - 3 q^{15} - 4 q^{16} - 6 q^{20} - 36 q^{23} - 4 q^{25} - 5 q^{27} + 5 q^{31} - 4 q^{36} - 7 q^{37} + 24 q^{45} + 12 q^{47} + 4 q^{48} + 7 q^{49} - 6 q^{53} + 15 q^{59} + 6 q^{60} + 8 q^{64} + 52 q^{67} - 9 q^{69} + 3 q^{71} + 4 q^{75} + 12 q^{80} - q^{81} - 36 q^{89} - 18 q^{92} - 5 q^{93} - 17 q^{97}+O(q^{100}) $ 4 * q + q^3 + 2 * q^4 + 3 * q^5 + 2 * q^9 + 8 * q^12 - 3 * q^15 - 4 * q^16 - 6 * q^20 - 36 * q^23 - 4 * q^25 - 5 * q^27 + 5 * q^31 - 4 * q^36 - 7 * q^37 + 24 * q^45 + 12 * q^47 + 4 * q^48 + 7 * q^49 - 6 * q^53 + 15 * q^59 + 6 * q^60 + 8 * q^64 + 52 * q^67 - 9 * q^69 + 3 * q^71 + 4 * q^75 + 12 * q^80 - q^81 - 36 * q^89 - 18 * q^92 - 5 * q^93 - 17 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/121\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$e\left(\frac{4}{5}\right)$

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		−0.587785	−	0.809017i	$-0.700000\pi$
$2$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$3$	−0.309017	−	0.951057i	−0.178411	−	0.549093i	0.821362	−	0.570408i	$-0.193215\pi$
$3$	−0.309017	−	0.951057i	−0.178411	−	0.549093i	−0.999773	+	0.0213149i	$0.993215\pi$
$4$	−0.618034	+	1.90211i	−0.309017	+	0.951057i
$5$	2.42705	+	1.76336i	1.08541	+	0.788597i	0.978618	−	0.205685i	$-0.0659421\pi$
$5$	2.42705	+	1.76336i	1.08541	+	0.788597i	0.106792	+	0.994281i	$0.465942\pi$
$6$		0			0
$7$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$7$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$8$		0			0
$9$	1.61803	−	1.17557i	0.539345	−	0.391857i
$10$		0			0
$11$		0			0
$11$		0			0
$12$	2.00000			0.577350
$13$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$13$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$14$		0			0
$15$	0.927051	−	2.85317i	0.239364	−	0.736685i
$16$	−3.23607	−	2.35114i	−0.809017	−	0.587785i
$17$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$17$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$18$		0			0
$19$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$19$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$20$	−4.85410	+	3.52671i	−1.08541	+	0.788597i
$21$		0			0
$22$		0			0
$23$	−9.00000			−1.87663			−0.938315	−	0.345782i	$-0.887614\pi$
$23$	−9.00000			−1.87663			−0.938315	+	0.345782i	$0.887614\pi$
$24$		0			0
$25$	1.23607	+	3.80423i	0.247214	+	0.760845i
$26$		0			0
$27$	−4.04508	−	2.93893i	−0.778477	−	0.565597i
$28$		0			0
$29$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$29$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$30$		0			0
$31$	4.04508	−	2.93893i	0.726519	−	0.527847i	−0.161942	−	0.986800i	$-0.551776\pi$
$31$	4.04508	−	2.93893i	0.726519	−	0.527847i	0.888460	+	0.458954i	$0.151776\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	1.23607	+	3.80423i	0.206011	+	0.634038i
$37$	2.16312	−	6.65740i	0.355615	−	1.09447i	−0.600038	−	0.799972i	$-0.704848\pi$
$37$	2.16312	−	6.65740i	0.355615	−	1.09447i	0.955652	−	0.294497i	$-0.0951522\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$41$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$	6.00000			0.894427
$46$		0			0
$47$	−3.70820	−	11.4127i	−0.540897	−	1.66471i	−0.730550	−	0.682859i	$-0.760736\pi$
$47$	−3.70820	−	11.4127i	−0.540897	−	1.66471i	0.189653	−	0.981851i	$-0.439264\pi$
$48$	−1.23607	+	3.80423i	−0.178411	+	0.549093i
$49$	5.66312	+	4.11450i	0.809017	+	0.587785i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.85410	+	3.52671i	−0.666762	+	0.484431i	−0.868940	−	0.494918i	$-0.835198\pi$
$53$	−4.85410	+	3.52671i	−0.666762	+	0.484431i	0.202178	+	0.979349i	$0.435198\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.63525	+	14.2658i	−0.603459	+	1.85726i	−0.0964021	+	0.995342i	$0.530733\pi$
$59$	−4.63525	+	14.2658i	−0.603459	+	1.85726i	−0.507057	+	0.861913i	$0.669267\pi$
$60$	4.85410	+	3.52671i	0.626662	+	0.455296i
$61$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$61$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$62$		0			0
$63$		0			0
$64$	6.47214	−	4.70228i	0.809017	−	0.587785i
$65$		0			0
$66$		0			0
$67$	13.0000			1.58820			0.794101	−	0.607785i	$-0.207942\pi$
$67$	13.0000			1.58820			0.794101	+	0.607785i	$0.207942\pi$
$68$		0			0
$69$	2.78115	+	8.55951i	0.334811	+	1.03044i
$70$		0			0
$71$	2.42705	+	1.76336i	0.288038	+	0.209272i	0.722416	−	0.691459i	$-0.243032\pi$
$71$	2.42705	+	1.76336i	0.288038	+	0.209272i	−0.434378	+	0.900731i	$0.643032\pi$
$72$		0			0
$73$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$73$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$74$		0			0
$75$	3.23607	−	2.35114i	0.373669	−	0.271486i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$79$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$80$	−3.70820	−	11.4127i	−0.414590	−	1.27598i
$81$	0.309017	−	0.951057i	0.0343352	−	0.105673i
$82$		0			0
$83$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$83$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−9.00000			−0.953998			−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000			−0.953998			−0.476999	+	0.878904i	$0.658275\pi$
$90$		0			0
$91$		0			0
$92$	5.56231	−	17.1190i	0.579910	−	1.78478i
$93$	−4.04508	−	2.93893i	−0.419456	−	0.304752i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−13.7533	+	9.99235i	−1.39643	+	1.01457i	−0.401310	+	0.915942i	$0.631445\pi$
$97$	−13.7533	+	9.99235i	−1.39643	+	1.01457i	−0.995124	+	0.0986273i	$0.968555\pi$
$98$		0			0
$99$		0			0
$100$	−8.00000			−0.800000
$101$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$101$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$102$		0			0
$103$	−1.23607	+	3.80423i	−0.121793	+	0.374842i	−0.993303	−	0.115536i	$-0.963141\pi$
$103$	−1.23607	+	3.80423i	−0.121793	+	0.374842i	0.871510	+	0.490378i	$0.163141\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$107$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$108$	8.09017	−	5.87785i	0.778477	−	0.565597i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$	−7.00000			−0.664411
$112$		0			0
$113$	6.48936	+	19.9722i	0.610467	+	1.87883i	0.453599	+	0.891206i	$0.350140\pi$
$113$	6.48936	+	19.9722i	0.610467	+	1.87883i	0.156868	+	0.987620i	$0.449860\pi$
$114$		0			0
$115$	−21.8435	−	15.8702i	−2.03691	−	1.47990i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$	3.09017	+	9.51057i	0.277505	+	0.854074i
$125$	0.927051	−	2.85317i	0.0829180	−	0.255195i
$126$		0			0
$127$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$127$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−4.63525	−	14.2658i	−0.398939	−	1.22781i
$136$		0			0
$137$	2.42705	+	1.76336i	0.207357	+	0.150654i	0.686617	−	0.727019i	$-0.259095\pi$
$137$	2.42705	+	1.76336i	0.207357	+	0.150654i	−0.479260	+	0.877673i	$0.659095\pi$
$138$		0			0
$139$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$139$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$140$		0			0
$141$	−9.70820	+	7.05342i	−0.817578	+	0.594005i
$142$		0			0
$143$		0			0
$144$	−8.00000			−0.666667
$145$		0			0
$146$		0			0
$147$	2.16312	−	6.65740i	0.178411	−	0.549093i
$148$	11.3262	+	8.22899i	0.931011	+	0.676419i
$149$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$149$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$150$		0			0
$151$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$151$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	15.0000			1.20483
$156$		0			0
$157$	−7.10739	−	21.8743i	−0.567232	−	1.74576i	−0.661226	−	0.750186i	$-0.729964\pi$
$157$	−7.10739	−	21.8743i	−0.567232	−	1.74576i	0.0939948	−	0.995573i	$-0.470036\pi$
$158$		0			0
$159$	4.85410	+	3.52671i	0.384955	+	0.279686i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	12.9443	−	9.40456i	1.01387	−	0.736622i	0.0488556	−	0.998806i	$-0.484443\pi$
$163$	12.9443	−	9.40456i	1.01387	−	0.736622i	0.965018	+	0.262184i	$0.0844426\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$167$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$168$		0			0
$169$	−4.01722	+	12.3637i	−0.309017	+	0.951057i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$173$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	15.0000			1.12747
$178$		0			0
$179$	6.48936	+	19.9722i	0.485037	+	1.49279i	0.831927	+	0.554885i	$0.187238\pi$
$179$	6.48936	+	19.9722i	0.485037	+	1.49279i	−0.346890	+	0.937906i	$0.612762\pi$
$180$	−3.70820	+	11.4127i	−0.276393	+	0.850651i
$181$	20.2254	+	14.6946i	1.50334	+	1.09224i	0.969026	+	0.246960i	$0.0794316\pi$
$181$	20.2254	+	14.6946i	1.50334	+	1.09224i	0.534318	+	0.845283i	$0.320568\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	16.9894	−	12.3435i	1.24908	−	0.907511i
$186$		0			0
$187$		0			0
$188$	24.0000			1.75038
$189$		0			0
$190$		0			0
$191$	−4.63525	+	14.2658i	−0.335395	+	1.03224i	0.631132	+	0.775676i	$0.282591\pi$
$191$	−4.63525	+	14.2658i	−0.335395	+	1.03224i	−0.966527	+	0.256565i	$0.917409\pi$
$192$	−6.47214	−	4.70228i	−0.467086	−	0.339358i
$193$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$193$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$194$		0			0
$195$		0			0
$196$	−11.3262	+	8.22899i	−0.809017	+	0.587785i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	−20.0000			−1.41776			−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000			−1.41776			−0.708881	+	0.705328i	$0.750800\pi$
$200$		0			0
$201$	−4.01722	−	12.3637i	−0.283353	−	0.872071i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−14.5623	+	10.5801i	−1.01215	+	0.735370i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$211$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$212$	−3.70820	−	11.4127i	−0.254680	−	0.783826i
$213$	0.927051	−	2.85317i	0.0635205	−	0.195496i
$214$		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
$223$	−0.309017	−	0.951057i	−0.0206933	−	0.0636875i	0.940177	−	0.340687i	$-0.110660\pi$
$223$	−0.309017	−	0.951057i	−0.0206933	−	0.0636875i	−0.960870	+	0.277000i	$0.910660\pi$
$224$		0			0
$225$	6.47214	+	4.70228i	0.431476	+	0.313485i
$226$		0			0
$227$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$227$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$228$		0			0
$229$	4.04508	−	2.93893i	0.267307	−	0.194210i	−0.446055	−	0.895005i	$-0.647172\pi$
$229$	4.04508	−	2.93893i	0.267307	−	0.194210i	0.713362	+	0.700796i	$0.247172\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$233$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$234$		0			0
$235$	11.1246	−	34.2380i	0.725690	−	2.23344i
$236$	−24.2705	−	17.6336i	−1.57988	−	1.14785i
$237$		0			0
$238$		0			0
$239$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$239$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$240$	−9.70820	+	7.05342i	−0.626662	+	0.455296i
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$		0			0
$243$	−16.0000			−1.02640
$244$		0			0
$245$	6.48936	+	19.9722i	0.414590	+	1.27598i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	21.8435	−	15.8702i	1.37875	−	1.00172i	0.381751	−	0.924265i	$-0.375321\pi$
$251$	21.8435	−	15.8702i	1.37875	−	1.00172i	0.996996	−	0.0774530i	$-0.0246788\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	4.94427	+	15.2169i	0.309017	+	0.951057i
$257$	5.56231	−	17.1190i	0.346967	−	1.06785i	−0.613555	−	0.789652i	$-0.710261\pi$
$257$	5.56231	−	17.1190i	0.346967	−	1.06785i	0.960522	−	0.278203i	$-0.0897388\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$	−18.0000			−1.10573
$266$		0			0
$267$	2.78115	+	8.55951i	0.170204	+	0.523833i
$268$	−8.03444	+	24.7275i	−0.490782	+	1.51047i
$269$	−24.2705	−	17.6336i	−1.47980	−	1.07514i	−0.977621	−	0.210373i	$-0.932532\pi$
$269$	−24.2705	−	17.6336i	−1.47980	−	1.07514i	−0.502178	−	0.864764i	$-0.667468\pi$
$270$		0			0
$271$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$271$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	−18.0000			−1.08347
$277$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$277$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$278$		0			0
$279$	3.09017	−	9.51057i	0.185004	−	0.569383i
$280$		0			0
$281$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$281$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$282$		0			0
$283$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$283$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$284$	−4.85410	+	3.52671i	−0.288038	+	0.209272i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−5.25329	−	16.1680i	−0.309017	−	0.951057i
$290$		0			0
$291$	13.7533	+	9.99235i	0.806232	+	0.585762i
$292$		0			0
$293$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$293$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$294$		0			0
$295$	−36.4058	+	26.4503i	−2.11963	+	1.54000i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	2.47214	+	7.60845i	0.142729	+	0.439274i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$	4.00000			0.227552
$310$		0			0
$311$	−3.70820	−	11.4127i	−0.210273	−	0.647154i	−0.999456	−	0.0329949i	$-0.989495\pi$
$311$	−3.70820	−	11.4127i	−0.210273	−	0.647154i	0.789183	−	0.614159i	$-0.210505\pi$
$312$		0			0
$313$	−15.3713	−	11.1679i	−0.868839	−	0.631248i	0.0614365	−	0.998111i	$-0.480432\pi$
$313$	−15.3713	−	11.1679i	−0.868839	−	0.631248i	−0.930275	+	0.366863i	$0.880432\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	21.8435	−	15.8702i	1.22685	−	0.891359i	0.230201	−	0.973143i	$-0.426062\pi$
$317$	21.8435	−	15.8702i	1.22685	−	0.891359i	0.996650	+	0.0817838i	$0.0260617\pi$
$318$		0			0
$319$		0			0
$320$	24.0000			1.34164
$321$		0			0
$322$		0			0
$323$		0			0
$324$	1.61803	+	1.17557i	0.0898908	+	0.0653095i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	35.0000			1.92377			0.961887	−	0.273447i	$-0.0881639\pi$
$331$	35.0000			1.92377			0.961887	+	0.273447i	$0.0881639\pi$
$332$		0			0
$333$	−4.32624	−	13.3148i	−0.237076	−	0.729646i
$334$		0			0
$335$	31.5517	+	22.9236i	1.72385	+	1.25245i
$336$		0			0
$337$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$337$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$338$		0			0
$339$	16.9894	−	12.3435i	0.922735	−	0.670406i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−8.34346	+	25.6785i	−0.449197	+	1.38249i
$346$		0			0
$347$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$347$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$348$		0			0
$349$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$349$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.00000			−0.479022			−0.239511	−	0.970894i	$-0.576987\pi$
$353$	−9.00000			−0.479022			−0.239511	+	0.970894i	$0.576987\pi$
$354$		0			0
$355$	2.78115	+	8.55951i	0.147608	+	0.454292i
$356$	5.56231	−	17.1190i	0.294802	−	0.907306i
$357$		0			0
$358$		0			0
$359$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$359$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$360$		0			0
$361$	15.3713	−	11.1679i	0.809017	−	0.587785i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−11.4336	+	35.1891i	−0.596831	+	1.83686i	−0.0514358	+	0.998676i	$0.516380\pi$
$367$	−11.4336	+	35.1891i	−0.596831	+	1.83686i	−0.545395	+	0.838179i	$0.683620\pi$
$368$	29.1246	+	21.1603i	1.51823	+	1.10306i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	8.09017	−	5.87785i	0.419456	−	0.304752i
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$	−3.00000			−0.154919
$376$		0			0
$377$		0			0
$378$		0			0
$379$	20.2254	+	14.6946i	1.03891	+	0.754813i	0.970073	−	0.242815i	$-0.0780709\pi$
$379$	20.2254	+	14.6946i	1.03891	+	0.754813i	0.0688378	+	0.997628i	$0.478071\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−31.5517	+	22.9236i	−1.61221	+	1.17134i	−0.756301	+	0.654224i	$0.772995\pi$
$383$	−31.5517	+	22.9236i	−1.61221	+	1.17134i	−0.855914	+	0.517119i	$0.827005\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−10.5066	−	32.3359i	−0.533391	−	1.64161i
$389$	−4.63525	+	14.2658i	−0.235017	+	0.723307i	0.762102	+	0.647456i	$0.224167\pi$
$389$	−4.63525	+	14.2658i	−0.235017	+	0.723307i	−0.997119	+	0.0758507i	$0.975833\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	2.00000			0.100377			0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000			0.100377			0.0501886	+	0.998740i	$0.484018\pi$
$398$		0			0
$399$		0			0
$400$	4.94427	−	15.2169i	0.247214	−	0.760845i
$401$	−24.2705	−	17.6336i	−1.21201	−	0.880578i	−0.216600	−	0.976261i	$-0.569497\pi$
$401$	−24.2705	−	17.6336i	−1.21201	−	0.880578i	−0.995412	+	0.0956827i	$0.969497\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	2.42705	−	1.76336i	0.120601	−	0.0876219i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$409$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$410$		0			0
$411$	0.927051	−	2.85317i	0.0457281	−	0.140736i
$412$	−6.47214	−	4.70228i	−0.318859	−	0.231665i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	24.0000			1.17248			0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000			1.17248			0.586238	+	0.810139i	$0.300608\pi$
$420$		0			0
$421$	3.09017	+	9.51057i	0.150606	+	0.463517i	0.997689	−	0.0679432i	$-0.0216437\pi$
$421$	3.09017	+	9.51057i	0.150606	+	0.463517i	−0.847084	+	0.531460i	$0.821644\pi$
$422$		0			0
$423$	−19.4164	−	14.1068i	−0.944058	−	0.685898i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$431$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$432$	6.18034	+	19.0211i	0.297352	+	0.915155i
$433$	8.96149	−	27.5806i	0.430662	−	1.32544i	−0.466805	−	0.884360i	$-0.654595\pi$
$433$	8.96149	−	27.5806i	0.430662	−	1.32544i	0.897467	−	0.441081i	$-0.145405\pi$
$434$		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
$441$	14.0000			0.666667
$442$		0			0
$443$	6.48936	+	19.9722i	0.308319	+	0.948907i	0.978418	+	0.206636i	$0.0662515\pi$
$443$	6.48936	+	19.9722i	0.308319	+	0.948907i	−0.670099	+	0.742271i	$0.733748\pi$
$444$	4.32624	−	13.3148i	0.205314	−	0.631892i
$445$	−21.8435	−	15.8702i	−1.03548	−	0.752320i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−31.5517	+	22.9236i	−1.48902	+	1.08183i	−0.514505	+	0.857487i	$0.672024\pi$
$449$	−31.5517	+	22.9236i	−1.48902	+	1.08183i	−0.974510	+	0.224346i	$0.927976\pi$
$450$		0			0
$451$		0			0
$452$	−42.0000			−1.97551
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$457$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$458$		0			0
$459$		0			0
$460$	43.6869	−	31.7404i	2.03691	−	1.47990i
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	−31.0000			−1.44069			−0.720346	−	0.693615i	$-0.756017\pi$
$463$	−31.0000			−1.44069			−0.720346	+	0.693615i	$0.756017\pi$
$464$		0			0
$465$	−4.63525	−	14.2658i	−0.214955	−	0.661563i
$466$		0			0
$467$	2.42705	+	1.76336i	0.112311	+	0.0815984i	0.642523	−	0.766267i	$-0.277888\pi$
$467$	2.42705	+	1.76336i	0.112311	+	0.0815984i	−0.530212	+	0.847865i	$0.677888\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−18.6074	+	13.5191i	−0.857383	+	0.622925i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−3.70820	+	11.4127i	−0.169787	+	0.522551i
$478$		0			0
$479$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$479$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−51.0000			−2.31579
$486$		0			0
$487$	13.2877	+	40.8954i	0.602125	+	1.85315i	0.515465	+	0.856911i	$0.327619\pi$
$487$	13.2877	+	40.8954i	0.602125	+	1.85315i	0.0866600	+	0.996238i	$0.472381\pi$
$488$		0			0
$489$	−12.9443	−	9.40456i	−0.585360	−	0.425289i
$490$		0			0
$491$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$491$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−20.0000			−0.898027
$497$		0			0
$498$		0			0
$499$	12.3607	−	38.0423i	0.553340	−	1.70301i	−0.146947	−	0.989144i	$-0.546945\pi$
$499$	12.3607	−	38.0423i	0.553340	−	1.70301i	0.700287	−	0.713861i	$-0.253055\pi$
$500$	4.85410	+	3.52671i	0.217082	+	0.157719i
$501$		0			0
$502$		0			0
$503$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$503$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	13.0000			0.577350
$508$		0			0
$509$	−13.9058	−	42.7975i	−0.616362	−	1.89697i	−0.378087	−	0.925770i	$-0.623418\pi$
$509$	−13.9058	−	42.7975i	−0.616362	−	1.89697i	−0.238275	−	0.971198i	$-0.576582\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−9.70820	+	7.05342i	−0.427795	+	0.310811i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−4.63525	+	14.2658i	−0.203074	+	0.624998i	0.796713	+	0.604358i	$0.206570\pi$
$521$	−4.63525	+	14.2658i	−0.203074	+	0.624998i	−0.999787	+	0.0206400i	$0.993430\pi$
$522$		0			0
$523$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$523$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	58.0000			2.52174
$530$		0			0
$531$	9.27051	+	28.5317i	0.402306	+	1.23817i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	16.9894	−	12.3435i	0.733145	−	0.532661i
$538$		0			0
$539$		0			0
$540$	30.0000			1.29099
$541$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$541$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$542$		0			0
$543$	7.72542	−	23.7764i	0.331530	−	1.02034i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$547$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$548$	−4.85410	+	3.52671i	−0.207357	+	0.150654i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−16.9894	−	12.3435i	−0.721158	−	0.523952i
$556$		0			0
$557$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$557$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$563$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$564$	−7.41641	−	22.8254i	−0.312287	−	0.961121i
$565$	−19.4681	+	59.9166i	−0.819028	+	2.52071i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$569$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$	15.0000			0.626634
$574$		0			0
$575$	−11.1246	−	34.2380i	−0.463928	−	1.42782i
$576$	4.94427	−	15.2169i	0.206011	−	0.634038i
$577$	38.0238	+	27.6259i	1.58295	+	1.15008i	0.913212	+	0.407486i	$0.133594\pi$
$577$	38.0238	+	27.6259i	1.58295	+	1.15008i	0.669740	+	0.742596i	$0.266406\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−14.8328	+	45.6507i	−0.612216	+	1.88421i	−0.175916	+	0.984405i	$0.556289\pi$
$587$	−14.8328	+	45.6507i	−0.612216	+	1.88421i	−0.436300	+	0.899801i	$0.643711\pi$
$588$	11.3262	+	8.22899i	0.467086	+	0.339358i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−22.6525	+	16.4580i	−0.931011	+	0.676419i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	6.18034	+	19.0211i	0.252944	+	0.778483i
$598$		0			0
$599$	29.1246	+	21.1603i	1.19000	+	0.864585i	0.993264	−	0.115872i	$-0.0369661\pi$
$599$	29.1246	+	21.1603i	1.19000	+	0.864585i	0.196735	+	0.980457i	$0.436966\pi$
$600$		0			0
$601$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$601$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$602$		0			0
$603$	21.0344	−	15.2824i	0.856589	−	0.622348i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$607$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$613$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−42.0000			−1.69086			−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000			−1.69086			−0.845428	+	0.534089i	$0.820655\pi$
$618$		0			0
$619$	−0.309017	−	0.951057i	−0.0124204	−	0.0382262i	0.944654	−	0.328068i	$-0.106397\pi$
$619$	−0.309017	−	0.951057i	−0.0124204	−	0.0382262i	−0.957075	+	0.289841i	$0.906397\pi$
$620$	−9.27051	+	28.5317i	−0.372313	+	1.14586i
$621$	36.4058	+	26.4503i	1.46091	+	1.06142i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	23.4615	−	17.0458i	0.938460	−	0.681831i
$626$		0			0
$627$		0			0
$628$	46.0000			1.83560
$629$		0			0
$630$		0			0
$631$	2.16312	−	6.65740i	0.0861124	−	0.265027i	−0.898723	−	0.438516i	$-0.855504\pi$
$631$	2.16312	−	6.65740i	0.0861124	−	0.265027i	0.984836	+	0.173489i	$0.0555042\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−9.70820	+	7.05342i	−0.384955	+	0.279686i
$637$		0			0
$638$		0			0
$639$	6.00000			0.237356
$640$		0			0
$641$	−13.9058	−	42.7975i	−0.549245	−	1.69040i	−0.710678	−	0.703518i	$-0.751612\pi$
$641$	−13.9058	−	42.7975i	−0.549245	−	1.69040i	0.161433	−	0.986884i	$-0.448388\pi$
$642$		0			0
$643$	−33.1697	−	24.0992i	−1.30809	−	0.950379i	−0.308086	−	0.951359i	$-0.599688\pi$
$643$	−33.1697	−	24.0992i	−1.30809	−	0.950379i	−1.00000		0.000979141i	$0.999688\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	21.8435	−	15.8702i	0.858755	−	0.623922i	−0.0687910	−	0.997631i	$-0.521914\pi$
$647$	21.8435	−	15.8702i	0.858755	−	0.623922i	0.927546	+	0.373709i	$0.121914\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	9.88854	+	30.4338i	0.387265	+	1.19188i
$653$	15.7599	−	48.5039i	0.616731	−	1.89810i	0.246647	−	0.969105i	$-0.420671\pi$
$653$	15.7599	−	48.5039i	0.616731	−	1.89810i	0.370084	−	0.928998i	$-0.379329\pi$
$654$		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
$661$	13.0000			0.505641			0.252821	−	0.967513i	$-0.418642\pi$
$661$	13.0000			0.505641			0.252821	+	0.967513i	$0.418642\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−0.809017	+	0.587785i	−0.0312784	+	0.0227251i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$673$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$674$		0			0
$675$	6.18034	−	19.0211i	0.237881	−	0.732124i
$676$	−21.0344	−	15.2824i	−0.809017	−	0.587785i
$677$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$677$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	24.0000			0.918334			0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000			0.918334			0.459167	+	0.888350i	$0.348148\pi$
$684$		0			0
$685$	2.78115	+	8.55951i	0.106262	+	0.327042i
$686$		0			0
$687$	−4.04508	−	2.93893i	−0.154330	−	0.112127i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−13.7533	+	9.99235i	−0.523200	+	0.380127i	−0.817808	−	0.575491i	$-0.804811\pi$
$691$	−13.7533	+	9.99235i	−0.523200	+	0.380127i	0.294608	+	0.955618i	$0.404811\pi$
$692$		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		0.951057	−	0.309017i	$-0.100000\pi$
$701$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−36.0000			−1.35584
$706$		0			0
$707$		0			0
$708$	−9.27051	+	28.5317i	−0.348407	+	1.07229i
$709$	−15.3713	−	11.1679i	−0.577282	−	0.419420i	0.260461	−	0.965484i	$-0.416125\pi$
$709$	−15.3713	−	11.1679i	−0.577282	−	0.419420i	−0.837743	+	0.546064i	$0.816125\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−36.4058	+	26.4503i	−1.36341	+	0.990573i
$714$		0			0
$715$		0			0
$716$	−42.0000			−1.56961
$717$		0			0
$718$		0			0
$719$	15.7599	−	48.5039i	0.587744	−	1.80889i	−0.000216702	−	1.00000i	$-0.500069\pi$
$719$	15.7599	−	48.5039i	0.587744	−	1.80889i	0.587961	−	0.808890i	$-0.299931\pi$
$720$	−19.4164	−	14.1068i	−0.723607	−	0.525731i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	−40.4508	+	29.3893i	−1.50334	+	1.09224i
$725$		0			0
$726$		0			0
$727$	−53.0000			−1.96566			−0.982831	−	0.184510i	$-0.940930\pi$
$727$	−53.0000			−1.96566			−0.982831	+	0.184510i	$0.940930\pi$
$728$		0			0
$729$	4.01722	+	12.3637i	0.148786	+	0.457916i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$733$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$734$		0			0
$735$	16.9894	−	12.3435i	0.626662	−	0.455296i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$739$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$740$	12.9787	+	39.9444i	0.477107	+	1.46838i
$741$		0			0
$742$		0			0
$743$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$743$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−7.10739	−	21.8743i	−0.259352	−	0.798205i	−0.992941	−	0.118611i	$-0.962156\pi$
$751$	−7.10739	−	21.8743i	−0.259352	−	0.798205i	0.733588	−	0.679594i	$-0.237844\pi$
$752$	−14.8328	+	45.6507i	−0.540897	+	1.66471i
$753$	−21.8435	−	15.8702i	−0.796020	−	0.578342i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	30.7426	−	22.3358i	1.11736	−	0.811810i	0.133554	−	0.991042i	$-0.457361\pi$
$757$	30.7426	−	22.3358i	1.11736	−	0.811810i	0.983807	+	0.179232i	$0.0573612\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$761$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$762$		0			0
$763$		0			0
$764$	−24.2705	−	17.6336i	−0.878076	−	0.637960i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	12.9443	−	9.40456i	0.467086	−	0.339358i
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$	−18.0000			−0.648254
$772$		0			0
$773$	16.6869	+	51.3571i	0.600187	+	1.84718i	0.526998	+	0.849867i	$0.323318\pi$
$773$	16.6869	+	51.3571i	0.600187	+	1.84718i	0.0731890	+	0.997318i	$0.476682\pi$
$774$		0			0
$775$	16.1803	+	11.7557i	0.581215	+	0.422277i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−8.65248	−	26.6296i	−0.309017	−	0.951057i
$785$	21.3222	−	65.6229i	0.761021	−	2.34218i
$786$		0			0
$787$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$787$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	5.56231	+	17.1190i	0.197275	+	0.607149i
$796$	12.3607	−	38.0423i	0.438113	−	1.34837i
$797$	2.42705	+	1.76336i	0.0859706	+	0.0624613i	0.629940	−	0.776644i	$-0.283079\pi$
$797$	2.42705	+	1.76336i	0.0859706	+	0.0624613i	−0.543970	+	0.839105i	$0.683079\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−14.5623	+	10.5801i	−0.514534	+	0.373831i
$802$		0			0
$803$		0			0
$804$	26.0000			0.916949
$805$		0			0
$806$		0			0
$807$	−9.27051	+	28.5317i	−0.326337	+	1.00436i
$808$		0			0
$809$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$809$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$810$		0			0
$811$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$811$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	48.0000			1.68137
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$821$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$822$		0			0
$823$	39.6418	−	28.8015i	1.38183	−	1.00396i	0.385121	−	0.922866i	$-0.374160\pi$
$823$	39.6418	−	28.8015i	1.38183	−	1.00396i	0.996707	−	0.0810902i	$-0.0258402\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$827$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$828$	−11.1246	−	34.2380i	−0.386607	−	1.18985i
$829$	8.96149	−	27.5806i	0.311246	−	0.957915i	−0.666027	−	0.745928i	$-0.732006\pi$
$829$	8.96149	−	27.5806i	0.311246	−	0.957915i	0.977272	−	0.211987i	$-0.0679936\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−25.0000			−0.864126
$838$		0			0
$839$	−13.9058	−	42.7975i	−0.480080	−	1.47754i	−0.838982	−	0.544160i	$-0.816849\pi$
$839$	−13.9058	−	42.7975i	−0.480080	−	1.47754i	0.358901	−	0.933376i	$-0.383151\pi$
$840$		0			0
$841$	23.4615	+	17.0458i	0.809017	+	0.587785i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−31.5517	+	22.9236i	−1.08541	+	0.788597i
$846$		0			0
$847$		0			0
$848$	24.0000			0.824163
$849$		0			0
$850$		0			0
$851$	−19.4681	+	59.9166i	−0.667357	+	2.05391i
$852$	4.85410	+	3.52671i	0.166299	+	0.120823i
$853$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$853$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$	−31.0000			−1.05771			−0.528853	−	0.848713i	$-0.677378\pi$
$859$	−31.0000			−1.05771			−0.528853	+	0.848713i	$0.677378\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	29.1246	+	21.1603i	0.991413	+	0.720304i	0.960230	−	0.279210i	$-0.0900725\pi$
$863$	29.1246	+	21.1603i	0.991413	+	0.720304i	0.0311832	+	0.999514i	$0.490072\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−13.7533	+	9.99235i	−0.467086	+	0.339358i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−10.5066	+	32.3359i	−0.355594	+	1.09441i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$877$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	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
$883$	−17.3050	−	53.2592i	−0.582358	−	1.79231i	−0.609631	−	0.792686i	$-0.708682\pi$
$883$	−17.3050	−	53.2592i	−0.582358	−	1.79231i	0.0272727	−	0.999628i	$-0.491318\pi$
$884$		0			0
$885$	36.4058	+	26.4503i	1.22377	+	0.889118i
$886$		0			0
$887$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$887$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	2.00000			0.0669650
$893$		0			0
$894$		0			0
$895$	−19.4681	+	59.9166i	−0.650746	+	2.00279i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−12.9443	+	9.40456i	−0.431476	+	0.313485i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	23.1763	+	71.3292i	0.770405	+	2.37106i
$906$		0			0
$907$	−6.47214	−	4.70228i	−0.214904	−	0.156137i	0.475126	−	0.879918i	$-0.342403\pi$
$907$	−6.47214	−	4.70228i	−0.214904	−	0.156137i	−0.690030	+	0.723781i	$0.742403\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	48.5410	−	35.2671i	1.60824	−	1.16845i	0.739529	−	0.673124i	$-0.235048\pi$
$911$	48.5410	−	35.2671i	1.60824	−	1.16845i	0.868706	−	0.495327i	$-0.164952\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	3.09017	+	9.51057i	0.102102	+	0.314238i
$917$		0			0
$918$		0			0
$919$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$919$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	28.0000			0.920634
$926$		0			0
$927$	2.47214	+	7.60845i	0.0811956	+	0.249894i
$928$		0			0
$929$	−24.2705	−	17.6336i	−0.796290	−	0.578538i	0.113534	−	0.993534i	$-0.463783\pi$
$929$	−24.2705	−	17.6336i	−0.796290	−	0.578538i	−0.909823	+	0.414996i	$0.863783\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−9.70820	+	7.05342i	−0.317832	+	0.230919i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$937$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$938$		0			0
$939$	−5.87132	+	18.0701i	−0.191603	+	0.589695i
$940$	58.2492	+	42.3205i	1.89988	+	1.38034i
$941$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$941$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$942$		0			0
$943$		0			0
$944$	48.5410	−	35.2671i	1.57988	−	1.14785i
$945$		0			0
$946$		0			0
$947$	57.0000			1.85225			0.926126	−	0.377215i	$-0.123118\pi$
$947$	57.0000			1.85225			0.926126	+	0.377215i	$0.123118\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−21.8435	−	15.8702i	−0.708323	−	0.514627i
$952$		0			0
$953$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$953$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$954$		0			0
$955$	−36.4058	+	26.4503i	−1.17806	+	0.855913i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−7.41641	−	22.8254i	−0.239364	−	0.736685i
$961$	−1.85410	+	5.70634i	−0.0598097	+	0.184075i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−13.9058	−	42.7975i	−0.446257	−	1.37344i	−0.881099	−	0.472932i	$-0.843196\pi$
$971$	−13.9058	−	42.7975i	−0.446257	−	1.37344i	0.434842	−	0.900507i	$-0.356804\pi$
$972$	9.88854	−	30.4338i	0.317175	−	0.976165i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	21.8435	−	15.8702i	0.698834	−	0.507733i	−0.180718	−	0.983535i	$-0.557842\pi$
$977$	21.8435	−	15.8702i	0.698834	−	0.507733i	0.879552	+	0.475802i	$0.157842\pi$
$978$		0			0
$979$		0			0
$980$	−42.0000			−1.34164
$981$		0			0
$982$		0			0
$983$	15.7599	−	48.5039i	0.502662	−	1.54703i	−0.302005	−	0.953306i	$-0.597656\pi$
$983$	15.7599	−	48.5039i	0.502662	−	1.54703i	0.804666	−	0.593727i	$-0.202344\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−20.0000			−0.635321			−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000			−0.635321			−0.317660	+	0.948205i	$0.602897\pi$
$992$		0			0
$993$	−10.8156	−	33.2870i	−0.343223	−	1.05633i
$994$		0			0
$995$	−48.5410	−	35.2671i	−1.53885	−	1.11804i
$996$		0			0
$997$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$997$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$998$		0			0
$999$	−28.3156	+	20.5725i	−0.895866	+	0.650885i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	121.2.c.c.3.1		4
11.2	odd	10		121.2.a.b.1.1	✓	1
11.3	even	5	inner	121.2.c.c.27.1		4
11.4	even	5	inner	121.2.c.c.81.1		4
11.5	even	5	inner	121.2.c.c.9.1		4
11.6	odd	10	inner	121.2.c.c.9.1		4
11.7	odd	10	inner	121.2.c.c.81.1		4
11.8	odd	10	inner	121.2.c.c.27.1		4
11.9	even	5		121.2.a.b.1.1	✓	1
11.10	odd	2	CM	121.2.c.c.3.1		4
33.2	even	10		1089.2.a.g.1.1		1
33.20	odd	10		1089.2.a.g.1.1		1
44.31	odd	10		1936.2.a.h.1.1		1
44.35	even	10		1936.2.a.h.1.1		1
55.9	even	10		3025.2.a.d.1.1		1
55.24	odd	10		3025.2.a.d.1.1		1
77.13	even	10		5929.2.a.e.1.1		1
77.20	odd	10		5929.2.a.e.1.1		1
88.13	odd	10		7744.2.a.bb.1.1		1
88.35	even	10		7744.2.a.n.1.1		1
88.53	even	10		7744.2.a.bb.1.1		1
88.75	odd	10		7744.2.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.2.a.b.1.1	✓	1	11.2	odd	10
121.2.a.b.1.1	✓	1	11.9	even	5
121.2.c.c.3.1		4	1.1	even	1	trivial
121.2.c.c.3.1		4	11.10	odd	2	CM
121.2.c.c.9.1		4	11.5	even	5	inner
121.2.c.c.9.1		4	11.6	odd	10	inner
121.2.c.c.27.1		4	11.3	even	5	inner
121.2.c.c.27.1		4	11.8	odd	10	inner
121.2.c.c.81.1		4	11.4	even	5	inner
121.2.c.c.81.1		4	11.7	odd	10	inner
1089.2.a.g.1.1		1	33.2	even	10
1089.2.a.g.1.1		1	33.20	odd	10
1936.2.a.h.1.1		1	44.31	odd	10
1936.2.a.h.1.1		1	44.35	even	10
3025.2.a.d.1.1		1	55.9	even	10
3025.2.a.d.1.1		1	55.24	odd	10
5929.2.a.e.1.1		1	77.13	even	10
5929.2.a.e.1.1		1	77.20	odd	10
7744.2.a.n.1.1		1	88.35	even	10
7744.2.a.n.1.1		1	88.75	odd	10
7744.2.a.bb.1.1		1	88.13	odd	10
7744.2.a.bb.1.1		1	88.53	even	10

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:	\( N \)	\(=\)	\( 121 = 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	121.c (of order \(5\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.309017 - 0.951057i) q^{3} +(-0.618034 + 1.90211i) q^{4} +(2.42705 + 1.76336i) q^{5} +(1.61803 - 1.17557i) q^{9} +O(q^{10})\)	\(q+(-0.309017 - 0.951057i) q^{3} +(-0.618034 + 1.90211i) q^{4} +(2.42705 + 1.76336i) q^{5} +(1.61803 - 1.17557i) q^{9} +2.00000 q^{12} +(0.927051 - 2.85317i) q^{15} +(-3.23607 - 2.35114i) q^{16} +(-4.85410 + 3.52671i) q^{20} -9.00000 q^{23} +(1.23607 + 3.80423i) q^{25} +(-4.04508 - 2.93893i) q^{27} +(4.04508 - 2.93893i) q^{31} +(1.23607 + 3.80423i) q^{36} +(2.16312 - 6.65740i) q^{37} +6.00000 q^{45} +(-3.70820 - 11.4127i) q^{47} +(-1.23607 + 3.80423i) q^{48} +(5.66312 + 4.11450i) q^{49} +(-4.85410 + 3.52671i) q^{53} +(-4.63525 + 14.2658i) q^{59} +(4.85410 + 3.52671i) q^{60} +(6.47214 - 4.70228i) q^{64} +13.0000 q^{67} +(2.78115 + 8.55951i) q^{69} +(2.42705 + 1.76336i) q^{71} +(3.23607 - 2.35114i) q^{75} +(-3.70820 - 11.4127i) q^{80} +(0.309017 - 0.951057i) q^{81} -9.00000 q^{89} +(5.56231 - 17.1190i) q^{92} +(-4.04508 - 2.93893i) q^{93} +(-13.7533 + 9.99235i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} + 2 q^{4} + 3 q^{5} + 2 q^{9}+O(q^{10}) \) 4 * q + q^3 + 2 * q^4 + 3 * q^5 + 2 * q^9	\( 4 q + q^{3} + 2 q^{4} + 3 q^{5} + 2 q^{9} + 8 q^{12} - 3 q^{15} - 4 q^{16} - 6 q^{20} - 36 q^{23} - 4 q^{25} - 5 q^{27} + 5 q^{31} - 4 q^{36} - 7 q^{37} + 24 q^{45} + 12 q^{47} + 4 q^{48} + 7 q^{49} - 6 q^{53} + 15 q^{59} + 6 q^{60} + 8 q^{64} + 52 q^{67} - 9 q^{69} + 3 q^{71} + 4 q^{75} + 12 q^{80} - q^{81} - 36 q^{89} - 18 q^{92} - 5 q^{93} - 17 q^{97}+O(q^{100}) \) 4 * q + q^3 + 2 * q^4 + 3 * q^5 + 2 * q^9 + 8 * q^12 - 3 * q^15 - 4 * q^16 - 6 * q^20 - 36 * q^23 - 4 * q^25 - 5 * q^27 + 5 * q^31 - 4 * q^36 - 7 * q^37 + 24 * q^45 + 12 * q^47 + 4 * q^48 + 7 * q^49 - 6 * q^53 + 15 * q^59 + 6 * q^60 + 8 * q^64 + 52 * q^67 - 9 * q^69 + 3 * q^71 + 4 * q^75 + 12 * q^80 - q^81 - 36 * q^89 - 18 * q^92 - 5 * q^93 - 17 * q^97

Introduction

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