LMFDB - Embedded newform 460.2.g.a.459.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [460,2,Mod(459,460)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(460, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("460.459");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 460 = 2^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	460.g (of order $2$, degree $1$, 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:	$3.67311849298$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 9 $ x^4 + 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			459.1
Root			$-0.707107 + 1.58114i$ of defining polynomial
Character	$\chi$	$=$	460.459
Dual form			460.2.g.a.459.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} -1.41421 q^{3} +2.00000 q^{4} -2.23607i q^{5} +2.00000 q^{6} -3.16228i q^{7} -2.82843 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.41421 q^{2} -1.41421 q^{3} +2.00000 q^{4} -2.23607i q^{5} +2.00000 q^{6} -3.16228i q^{7} -2.82843 q^{8} -1.00000 q^{9} +3.16228i q^{10} -2.82843 q^{12} +4.47214i q^{14} +3.16228i q^{15} +4.00000 q^{16} +1.41421 q^{18} -4.47214i q^{20} +4.47214i q^{21} +(-0.707107 + 4.74342i) q^{23} +4.00000 q^{24} -5.00000 q^{25} +5.65685 q^{27} -6.32456i q^{28} -6.00000 q^{29} -4.47214i q^{30} -5.65685 q^{32} -7.07107 q^{35} -2.00000 q^{36} +6.32456i q^{40} -12.0000 q^{41} -6.32456i q^{42} -3.16228i q^{43} +2.23607i q^{45} +(1.00000 - 6.70820i) q^{46} -9.89949 q^{47} -5.65685 q^{48} -3.00000 q^{49} +7.07107 q^{50} -8.00000 q^{54} +8.94427i q^{56} +8.48528 q^{58} +6.32456i q^{60} +13.4164i q^{61} +3.16228i q^{63} +8.00000 q^{64} -15.8114i q^{67} +(1.00000 - 6.70820i) q^{69} +10.0000 q^{70} +2.82843 q^{72} +7.07107 q^{75} -8.94427i q^{80} -5.00000 q^{81} +16.9706 q^{82} +9.48683i q^{83} +8.94427i q^{84} +4.47214i q^{86} +8.48528 q^{87} -17.8885i q^{89} -3.16228i q^{90} +(-1.41421 + 9.48683i) q^{92} +14.0000 q^{94} +8.00000 q^{96} +4.24264 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4} + 8 q^{6} - 4 q^{9}+O(q^{10}) $ 4 * q + 8 * q^4 + 8 * q^6 - 4 * q^9	$ 4 q + 8 q^{4} + 8 q^{6} - 4 q^{9} + 16 q^{16} + 16 q^{24} - 20 q^{25} - 24 q^{29} - 8 q^{36} - 48 q^{41} + 4 q^{46} - 12 q^{49} - 32 q^{54} + 32 q^{64} + 4 q^{69} + 40 q^{70} - 20 q^{81} + 56 q^{94} + 32 q^{96}+O(q^{100}) $ 4 * q + 8 * q^4 + 8 * q^6 - 4 * q^9 + 16 * q^16 + 16 * q^24 - 20 * q^25 - 24 * q^29 - 8 * q^36 - 48 * q^41 + 4 * q^46 - 12 * q^49 - 32 * q^54 + 32 * q^64 + 4 * q^69 + 40 * q^70 - 20 * q^81 + 56 * q^94 + 32 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/460\mathbb{Z}\right)^\times$.

$n$	$231$	$277$	$281$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	−1.41421			−1.00000
$2$	−1.41421			−1.00000
$3$	−1.41421			−0.816497			−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421			−0.816497			−0.408248	+	0.912871i	$0.633860\pi$
$4$	2.00000			1.00000
$5$		−	2.23607i		−	1.00000i
$5$		−	2.23607i		−	1.00000i
$6$	2.00000			0.816497
$7$		−	3.16228i		−	1.19523i	−0.801784	−	0.597614i	$-0.796115\pi$
$7$		−	3.16228i		−	1.19523i	0.801784	−	0.597614i	$-0.203885\pi$
$8$	−2.82843			−1.00000
$9$	−1.00000			−0.333333
$10$			3.16228i			1.00000i
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	−2.82843			−0.816497
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$			4.47214i			1.19523i
$15$			3.16228i			0.816497i
$16$	4.00000			1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	1.41421			0.333333
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		−	4.47214i		−	1.00000i
$21$			4.47214i			0.975900i
$22$		0			0
$23$	−0.707107	+	4.74342i	−0.147442	+	0.989071i
$23$	−0.707107	+	4.74342i	−0.147442	+	0.989071i
$24$	4.00000			0.816497
$25$	−5.00000			−1.00000
$26$		0			0
$27$	5.65685			1.08866
$28$		−	6.32456i		−	1.19523i
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		−	4.47214i		−	0.816497i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−5.65685			−1.00000
$33$		0			0
$34$		0			0
$35$	−7.07107			−1.19523
$36$	−2.00000			−0.333333
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$		0			0
$40$			6.32456i			1.00000i
$41$	−12.0000			−1.87409			−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000			−1.87409			−0.937043	+	0.349215i	$0.886448\pi$
$42$		−	6.32456i		−	0.975900i
$43$		−	3.16228i		−	0.482243i	−0.970495	−	0.241121i	$-0.922485\pi$
$43$		−	3.16228i		−	0.482243i	0.970495	−	0.241121i	$-0.0775152\pi$
$44$		0			0
$45$			2.23607i			0.333333i
$46$	1.00000	−	6.70820i	0.147442	−	0.989071i
$47$	−9.89949			−1.44399			−0.721995	−	0.691898i	$-0.756775\pi$
$47$	−9.89949			−1.44399			−0.721995	+	0.691898i	$0.756775\pi$
$48$	−5.65685			−0.816497
$49$	−3.00000			−0.428571
$50$	7.07107			1.00000
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	−8.00000			−1.08866
$55$		0			0
$56$			8.94427i			1.19523i
$57$		0			0
$58$	8.48528			1.11417
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$			6.32456i			0.816497i
$61$			13.4164i			1.71780i	0.512148	+	0.858898i	$0.328850\pi$
$61$			13.4164i			1.71780i	−0.512148	+	0.858898i	$0.671150\pi$
$62$		0			0
$63$			3.16228i			0.398410i
$64$	8.00000			1.00000
$65$		0			0
$66$		0			0
$67$		−	15.8114i		−	1.93167i	−0.259161	−	0.965834i	$-0.583446\pi$
$67$		−	15.8114i		−	1.93167i	0.259161	−	0.965834i	$-0.416554\pi$
$68$		0			0
$69$	1.00000	−	6.70820i	0.120386	−	0.807573i
$70$	10.0000			1.19523
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	2.82843			0.333333
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	7.07107			0.816497
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		−	8.94427i		−	1.00000i
$81$	−5.00000			−0.555556
$82$	16.9706			1.87409
$83$			9.48683i			1.04132i	0.853766	+	0.520658i	$0.174313\pi$
$83$			9.48683i			1.04132i	−0.853766	+	0.520658i	$0.825687\pi$
$84$			8.94427i			0.975900i
$85$		0			0
$86$			4.47214i			0.482243i
$87$	8.48528			0.909718
$88$		0			0
$89$		−	17.8885i		−	1.89618i	−0.317999	−	0.948091i	$-0.603011\pi$
$89$		−	17.8885i		−	1.89618i	0.317999	−	0.948091i	$-0.396989\pi$
$90$		−	3.16228i		−	0.333333i
$91$		0			0
$92$	−1.41421	+	9.48683i	−0.147442	+	0.989071i
$93$		0			0
$94$	14.0000			1.44399
$95$		0			0
$96$	8.00000			0.816497
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$	4.24264			0.428571
$99$		0			0
$100$	−10.0000			−1.00000
$101$	18.0000			1.79107			0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000			1.79107			0.895533	+	0.444994i	$0.146794\pi$
$102$		0			0
$103$			15.8114i			1.55794i	0.627060	+	0.778971i	$0.284258\pi$
$103$			15.8114i			1.55794i	−0.627060	+	0.778971i	$0.715742\pi$
$104$		0			0
$105$	10.0000			0.975900
$106$		0			0
$107$		−	9.48683i		−	0.917127i	−0.888662	−	0.458563i	$-0.848364\pi$
$107$		−	9.48683i		−	0.917127i	0.888662	−	0.458563i	$-0.151636\pi$
$108$	11.3137			1.08866
$109$		−	13.4164i		−	1.28506i	−0.766261	−	0.642529i	$-0.777885\pi$
$109$		−	13.4164i		−	1.28506i	0.766261	−	0.642529i	$-0.222115\pi$
$110$		0			0
$111$		0			0
$112$		−	12.6491i		−	1.19523i
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$	10.6066	+	1.58114i	0.989071	+	0.147442i
$116$	−12.0000			−1.11417
$117$		0			0
$118$		0			0
$119$		0			0
$120$		−	8.94427i		−	0.816497i
$121$	−11.0000			−1.00000
$122$		−	18.9737i		−	1.71780i
$123$	16.9706			1.53018
$124$		0			0
$125$			11.1803i			1.00000i
$126$		−	4.47214i		−	0.398410i
$127$	−4.24264			−0.376473			−0.188237	−	0.982124i	$-0.560277\pi$
$127$	−4.24264			−0.376473			−0.188237	+	0.982124i	$0.560277\pi$
$128$	−11.3137			−1.00000
$129$			4.47214i			0.393750i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$			22.3607i			1.93167i
$135$		−	12.6491i		−	1.08866i
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$	−1.41421	+	9.48683i	−0.120386	+	0.807573i
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$	−14.1421			−1.19523
$141$	14.0000			1.17901
$142$		0			0
$143$		0			0
$144$	−4.00000			−0.333333
$145$			13.4164i			1.11417i
$146$		0			0
$147$	4.24264			0.349927
$148$		0			0
$149$			4.47214i			0.366372i	0.983078	+	0.183186i	$0.0586410\pi$
$149$			4.47214i			0.366372i	−0.983078	+	0.183186i	$0.941359\pi$
$150$	−10.0000			−0.816497
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$		0			0
$159$		0			0
$160$			12.6491i			1.00000i
$161$	15.0000	+	2.23607i	1.18217	+	0.176227i
$162$	7.07107			0.555556
$163$	12.7279			0.996928			0.498464	−	0.866910i	$-0.333898\pi$
$163$	12.7279			0.996928			0.498464	+	0.866910i	$0.333898\pi$
$164$	−24.0000			−1.87409
$165$		0			0
$166$		−	13.4164i		−	1.04132i
$167$	−24.0416			−1.86040			−0.930199	−	0.367057i	$-0.880366\pi$
$167$	−24.0416			−1.86040			−0.930199	+	0.367057i	$0.880366\pi$
$168$		−	12.6491i		−	0.975900i
$169$	13.0000			1.00000
$170$		0			0
$171$		0			0
$172$		−	6.32456i		−	0.482243i
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$	−12.0000			−0.909718
$175$			15.8114i			1.19523i
$176$		0			0
$177$		0			0
$178$			25.2982i			1.89618i
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$			4.47214i			0.333333i
$181$		−	26.8328i		−	1.99447i	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$		−	26.8328i		−	1.99447i	0.0743294	−	0.997234i	$-0.476318\pi$
$182$		0			0
$183$		−	18.9737i		−	1.40257i
$184$	2.00000	−	13.4164i	0.147442	−	0.989071i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−19.7990			−1.44399
$189$		−	17.8885i		−	1.30120i
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	−11.3137			−0.816497
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$	−6.00000			−0.428571
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	14.1421			1.00000
$201$			22.3607i			1.57720i
$202$	−25.4558			−1.79107
$203$			18.9737i			1.33169i
$204$		0			0
$205$			26.8328i			1.87409i
$206$		−	22.3607i		−	1.55794i
$207$	0.707107	−	4.74342i	0.0491473	−	0.329690i
$208$		0			0
$209$		0			0
$210$	−14.1421			−0.975900
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$			13.4164i			0.917127i
$215$	−7.07107			−0.482243
$216$	−16.0000			−1.08866
$217$		0			0
$218$			18.9737i			1.28506i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−29.6985			−1.98876			−0.994379	−	0.105881i	$-0.966234\pi$
$223$	−29.6985			−1.98876			−0.994379	+	0.105881i	$0.966234\pi$
$224$			17.8885i			1.19523i
$225$	5.00000			0.333333
$226$		0			0
$227$		−	28.4605i		−	1.88899i	−0.328526	−	0.944495i	$-0.606552\pi$
$227$		−	28.4605i		−	1.88899i	0.328526	−	0.944495i	$-0.393448\pi$
$228$		0			0
$229$		−	26.8328i		−	1.77316i	−0.462573	−	0.886581i	$-0.653074\pi$
$229$		−	26.8328i		−	1.77316i	0.462573	−	0.886581i	$-0.346926\pi$
$230$	−15.0000	−	2.23607i	−0.989071	−	0.147442i
$231$		0			0
$232$	16.9706			1.11417
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$			22.1359i			1.44399i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$			12.6491i			0.816497i
$241$			13.4164i			0.864227i	0.901819	+	0.432113i	$0.142232\pi$
$241$			13.4164i			0.864227i	−0.901819	+	0.432113i	$0.857768\pi$
$242$	15.5563			1.00000
$243$	−9.89949			−0.635053
$244$			26.8328i			1.71780i
$245$			6.70820i			0.428571i
$246$	−24.0000			−1.53018
$247$		0			0
$248$		0			0
$249$		−	13.4164i		−	0.850230i
$250$		−	15.8114i		−	1.00000i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$			6.32456i			0.398410i
$253$		0			0
$254$	6.00000			0.376473
$255$		0			0
$256$	16.0000			1.00000
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		−	6.32456i		−	0.393750i
$259$		0			0
$260$		0			0
$261$	6.00000			0.371391
$262$		0			0
$263$		−	28.4605i		−	1.75495i	−0.479623	−	0.877475i	$-0.659226\pi$
$263$		−	28.4605i		−	1.75495i	0.479623	−	0.877475i	$-0.340774\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			25.2982i			1.54823i
$268$		−	31.6228i		−	1.93167i
$269$	24.0000			1.46331			0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000			1.46331			0.731653	+	0.681677i	$0.238749\pi$
$270$			17.8885i			1.08866i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	2.00000	−	13.4164i	0.120386	−	0.807573i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$	20.0000			1.19523
$281$		−	31.3050i		−	1.86750i	−0.357930	−	0.933748i	$-0.616517\pi$
$281$		−	31.3050i		−	1.86750i	0.357930	−	0.933748i	$-0.383483\pi$
$282$	−19.7990			−1.17901
$283$		−	15.8114i		−	0.939889i	−0.882696	−	0.469945i	$-0.844274\pi$
$283$		−	15.8114i		−	0.939889i	0.882696	−	0.469945i	$-0.155726\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			37.9473i			2.23996i
$288$	5.65685			0.333333
$289$	−17.0000			−1.00000
$290$		−	18.9737i		−	1.11417i
$291$		0			0
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	−6.00000			−0.349927
$295$		0			0
$296$		0			0
$297$		0			0
$298$		−	6.32456i		−	0.366372i
$299$		0			0
$300$	14.1421			0.816497
$301$	−10.0000			−0.576390
$302$		0			0
$303$	−25.4558			−1.46240
$304$		0			0
$305$	30.0000			1.71780
$306$		0			0
$307$	4.24264			0.242140			0.121070	−	0.992644i	$-0.461367\pi$
$307$	4.24264			0.242140			0.121070	+	0.992644i	$0.461367\pi$
$308$		0			0
$309$		−	22.3607i		−	1.27205i
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$		0			0
$315$	7.07107			0.398410
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$		−	17.8885i		−	1.00000i
$321$			13.4164i			0.748831i
$322$	−21.2132	−	3.16228i	−1.18217	−	0.176227i
$323$		0			0
$324$	−10.0000			−0.555556
$325$		0			0
$326$	−18.0000			−0.996928
$327$			18.9737i			1.04925i
$328$	33.9411			1.87409
$329$			31.3050i			1.72590i
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$			18.9737i			1.04132i
$333$		0			0
$334$	34.0000			1.86040
$335$	−35.3553			−1.93167
$336$			17.8885i			0.975900i
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$	−18.3848			−1.00000
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	12.6491i		−	0.682988i
$344$			8.94427i			0.482243i
$345$	−15.0000	−	2.23607i	−0.807573	−	0.120386i
$346$		0			0
$347$	24.0416			1.29062			0.645311	−	0.763920i	$-0.276728\pi$
$347$	24.0416			1.29062			0.645311	+	0.763920i	$0.276728\pi$
$348$	16.9706			0.909718
$349$	−26.0000			−1.39175			−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000			−1.39175			−0.695874	+	0.718164i	$0.744983\pi$
$350$		−	22.3607i		−	1.19523i
$351$		0			0
$352$		0			0
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		−	35.7771i		−	1.89618i
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		−	6.32456i		−	0.333333i
$361$	−19.0000			−1.00000
$362$			37.9473i			1.99447i
$363$	15.5563			0.816497
$364$		0			0
$365$		0			0
$366$			26.8328i			1.40257i
$367$			3.16228i			0.165070i	0.996588	+	0.0825348i	$0.0263016\pi$
$367$			3.16228i			0.165070i	−0.996588	+	0.0825348i	$0.973698\pi$
$368$	−2.82843	+	18.9737i	−0.147442	+	0.989071i
$369$	12.0000			0.624695
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		−	15.8114i		−	0.816497i
$376$	28.0000			1.44399
$377$		0			0
$378$			25.2982i			1.30120i
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$	6.00000			0.307389
$382$		0			0
$383$		−	28.4605i		−	1.45426i	−0.686498	−	0.727132i	$-0.740853\pi$
$383$		−	28.4605i		−	1.45426i	0.686498	−	0.727132i	$-0.259147\pi$
$384$	16.0000			0.816497
$385$		0			0
$386$		0			0
$387$			3.16228i			0.160748i
$388$		0			0
$389$			31.3050i			1.58722i	0.608424	+	0.793612i	$0.291802\pi$
$389$			31.3050i			1.58722i	−0.608424	+	0.793612i	$0.708198\pi$
$390$		0			0
$391$		0			0
$392$	8.48528			0.428571
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	−20.0000			−1.00000
$401$		−	35.7771i		−	1.78662i	−0.449439	−	0.893311i	$-0.648376\pi$
$401$		−	35.7771i		−	1.78662i	0.449439	−	0.893311i	$-0.351624\pi$
$402$		−	31.6228i		−	1.57720i
$403$		0			0
$404$	36.0000			1.79107
$405$			11.1803i			0.555556i
$406$		−	26.8328i		−	1.33169i
$407$		0			0
$408$		0			0
$409$	−4.00000			−0.197787			−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000			−0.197787			−0.0988936	+	0.995098i	$0.531530\pi$
$410$		−	37.9473i		−	1.87409i
$411$		0			0
$412$			31.6228i			1.55794i
$413$		0			0
$414$	−1.00000	+	6.70820i	−0.0491473	+	0.329690i
$415$	21.2132			1.04132
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$	20.0000			0.975900
$421$			40.2492i			1.96163i	0.194948	+	0.980814i	$0.437546\pi$
$421$			40.2492i			1.96163i	−0.194948	+	0.980814i	$0.562454\pi$
$422$		0			0
$423$	9.89949			0.481330
$424$		0			0
$425$		0			0
$426$		0			0
$427$	42.4264			2.05316
$428$		−	18.9737i		−	0.917127i
$429$		0			0
$430$	10.0000			0.482243
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	22.6274			1.08866
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		−	18.9737i		−	0.909718i
$436$		−	26.8328i		−	1.28506i
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	3.00000			0.142857
$442$		0			0
$443$	41.0122			1.94855			0.974274	−	0.225367i	$-0.0723580\pi$
$443$	41.0122			1.94855			0.974274	+	0.225367i	$0.0723580\pi$
$444$		0			0
$445$	−40.0000			−1.89618
$446$	42.0000			1.98876
$447$		−	6.32456i		−	0.299141i
$448$		−	25.2982i		−	1.19523i
$449$	36.0000			1.69895			0.849473	−	0.527633i	$-0.176920\pi$
$449$	36.0000			1.69895			0.849473	+	0.527633i	$0.176920\pi$
$450$	−7.07107			−0.333333
$451$		0			0
$452$		0			0
$453$		0			0
$454$			40.2492i			1.88899i
$455$		0			0
$456$		0			0
$457$		0			0			−	1.00000i	$-0.5\pi$
$457$		0			0				1.00000i	$0.5\pi$
$458$			37.9473i			1.77316i
$459$		0			0
$460$	21.2132	+	3.16228i	0.989071	+	0.147442i
$461$	−42.0000			−1.95614			−0.978068	−	0.208288i	$-0.933211\pi$
$461$	−42.0000			−1.95614			−0.978068	+	0.208288i	$0.933211\pi$
$462$		0			0
$463$	12.7279			0.591517			0.295758	−	0.955263i	$-0.404428\pi$
$463$	12.7279			0.591517			0.295758	+	0.955263i	$0.404428\pi$
$464$	−24.0000			−1.11417
$465$		0			0
$466$		0			0
$467$			28.4605i			1.31699i	0.752583	+	0.658497i	$0.228808\pi$
$467$			28.4605i			1.31699i	−0.752583	+	0.658497i	$0.771192\pi$
$468$		0			0
$469$	−50.0000			−2.30879
$470$		−	31.3050i		−	1.44399i
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		−	17.8885i		−	0.816497i
$481$		0			0
$482$		−	18.9737i		−	0.864227i
$483$	−21.2132	−	3.16228i	−0.965234	−	0.143889i
$484$	−22.0000			−1.00000
$485$		0			0
$486$	14.0000			0.635053
$487$	38.1838			1.73027			0.865136	−	0.501538i	$-0.167232\pi$
$487$	38.1838			1.73027			0.865136	+	0.501538i	$0.167232\pi$
$488$		−	37.9473i		−	1.71780i
$489$	−18.0000			−0.813988
$490$		−	9.48683i		−	0.428571i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$	33.9411			1.53018
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$			18.9737i			0.850230i
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$			22.3607i			1.00000i
$501$	34.0000			1.51901
$502$		0			0
$503$			9.48683i			0.422997i	0.977378	+	0.211498i	$0.0678343\pi$
$503$			9.48683i			0.422997i	−0.977378	+	0.211498i	$0.932166\pi$
$504$		−	8.94427i		−	0.398410i
$505$		−	40.2492i		−	1.79107i
$506$		0			0
$507$	−18.3848			−0.816497
$508$	−8.48528			−0.376473
$509$	−6.00000			−0.265945			−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000			−0.265945			−0.132973	+	0.991120i	$0.542452\pi$
$510$		0			0
$511$		0			0
$512$	−22.6274			−1.00000
$513$		0			0
$514$		0			0
$515$	35.3553			1.55794
$516$			8.94427i			0.393750i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			17.8885i			0.783711i	0.920027	+	0.391856i	$0.128167\pi$
$521$			17.8885i			0.783711i	−0.920027	+	0.391856i	$0.871833\pi$
$522$	−8.48528			−0.371391
$523$		−	34.7851i		−	1.52104i	−0.649312	−	0.760522i	$-0.724943\pi$
$523$		−	34.7851i		−	1.52104i	0.649312	−	0.760522i	$-0.275057\pi$
$524$		0			0
$525$		−	22.3607i		−	0.975900i
$526$			40.2492i			1.75495i
$527$		0			0
$528$		0			0
$529$	−22.0000	−	6.70820i	−0.956522	−	0.291661i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		−	35.7771i		−	1.54823i
$535$	−21.2132			−0.917127
$536$			44.7214i			1.93167i
$537$		0			0
$538$	−33.9411			−1.46331
$539$		0			0
$540$		−	25.2982i		−	1.08866i
$541$	−38.0000			−1.63375			−0.816874	−	0.576816i	$-0.804295\pi$
$541$	−38.0000			−1.63375			−0.816874	+	0.576816i	$0.804295\pi$
$542$		0			0
$543$			37.9473i			1.62848i
$544$		0			0
$545$	−30.0000			−1.28506
$546$		0			0
$547$	−46.6690			−1.99542			−0.997712	−	0.0676046i	$-0.978464\pi$
$547$	−46.6690			−1.99542			−0.997712	+	0.0676046i	$0.978464\pi$
$548$		0			0
$549$		−	13.4164i		−	0.572598i
$550$		0			0
$551$		0			0
$552$	−2.82843	+	18.9737i	−0.120386	+	0.807573i
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$	−28.2843			−1.19523
$561$		0			0
$562$			44.2719i			1.86750i
$563$			47.4342i			1.99911i	0.0298010	+	0.999556i	$0.490513\pi$
$563$			47.4342i			1.99911i	−0.0298010	+	0.999556i	$0.509487\pi$
$564$	28.0000			1.17901
$565$		0			0
$566$			22.3607i			0.939889i
$567$			15.8114i			0.664016i
$568$		0			0
$569$			31.3050i			1.31237i	0.754599	+	0.656186i	$0.227831\pi$
$569$			31.3050i			1.31237i	−0.754599	+	0.656186i	$0.772169\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$		0			0
$574$		−	53.6656i		−	2.23996i
$575$	3.53553	−	23.7171i	0.147442	−	0.989071i
$576$	−8.00000			−0.333333
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	24.0416			1.00000
$579$		0			0
$580$			26.8328i			1.11417i
$581$	30.0000			1.24461
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−9.89949			−0.408596			−0.204298	−	0.978909i	$-0.565491\pi$
$587$	−9.89949			−0.408596			−0.204298	+	0.978909i	$0.565491\pi$
$588$	8.48528			0.349927
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$			8.94427i			0.366372i
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$	−20.0000			−0.816497
$601$	−28.0000			−1.14214			−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000			−1.14214			−0.571072	+	0.820900i	$0.693472\pi$
$602$	14.1421			0.576390
$603$			15.8114i			0.643890i
$604$		0			0
$605$			24.5967i			1.00000i
$606$	36.0000			1.46240
$607$	−46.6690			−1.89424			−0.947119	−	0.320882i	$-0.896021\pi$
$607$	−46.6690			−1.89424			−0.947119	+	0.320882i	$0.896021\pi$
$608$		0			0
$609$		−	26.8328i		−	1.08732i
$610$	−42.4264			−1.71780
$611$		0			0
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$	−6.00000			−0.242140
$615$		−	37.9473i		−	1.53018i
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$			31.6228i			1.27205i
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$	−4.00000	+	26.8328i	−0.160514	+	1.07676i
$622$		0			0
$623$	−56.5685			−2.26637
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$	−10.0000			−0.398410
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			9.48683i			0.376473i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$			25.2982i			1.00000i
$641$		−	49.1935i		−	1.94303i	−0.236986	−	0.971513i	$-0.576159\pi$
$641$		−	49.1935i		−	1.94303i	0.236986	−	0.971513i	$-0.423841\pi$
$642$		−	18.9737i		−	0.748831i
$643$		−	41.1096i		−	1.62120i	−0.585597	−	0.810602i	$-0.699140\pi$
$643$		−	41.1096i		−	1.62120i	0.585597	−	0.810602i	$-0.300860\pi$
$644$	30.0000	+	4.47214i	1.18217	+	0.176227i
$645$	10.0000			0.393750
$646$		0			0
$647$	−18.3848			−0.722780			−0.361390	−	0.932415i	$-0.617698\pi$
$647$	−18.3848			−0.722780			−0.361390	+	0.932415i	$0.617698\pi$
$648$	14.1421			0.555556
$649$		0			0
$650$		0			0
$651$		0			0
$652$	25.4558			0.996928
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		−	26.8328i		−	1.04925i
$655$		0			0
$656$	−48.0000			−1.87409
$657$		0			0
$658$		−	44.2719i		−	1.72590i
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$			40.2492i			1.56551i	0.622328	+	0.782757i	$0.286187\pi$
$661$			40.2492i			1.56551i	−0.622328	+	0.782757i	$0.713813\pi$
$662$		0			0
$663$		0			0
$664$		−	26.8328i		−	1.04132i
$665$		0			0
$666$		0			0
$667$	4.24264	−	28.4605i	0.164276	−	1.10199i
$668$	−48.0833			−1.86040
$669$	42.0000			1.62381
$670$	50.0000			1.93167
$671$		0			0
$672$		−	25.2982i		−	0.975900i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	−28.2843			−1.08866
$676$	26.0000			1.00000
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$			40.2492i			1.54235i
$682$		0			0
$683$	43.8406			1.67751			0.838757	−	0.544505i	$-0.183283\pi$
$683$	43.8406			1.67751			0.838757	+	0.544505i	$0.183283\pi$
$684$		0			0
$685$		0			0
$686$			17.8885i			0.682988i
$687$			37.9473i			1.44778i
$688$		−	12.6491i		−	0.482243i
$689$		0			0
$690$	21.2132	+	3.16228i	0.807573	+	0.120386i
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$	−34.0000			−1.29062
$695$		0			0
$696$	−24.0000			−0.909718
$697$		0			0
$698$	36.7696			1.39175
$699$		0			0
$700$			31.6228i			1.19523i
$701$		−	22.3607i		−	0.844551i	−0.906467	−	0.422276i	$-0.861231\pi$
$701$		−	22.3607i		−	0.844551i	0.906467	−	0.422276i	$-0.138769\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		−	31.3050i		−	1.17901i
$706$		0			0
$707$		−	56.9210i		−	2.14073i
$708$		0			0
$709$		−	26.8328i		−	1.00773i	−0.863783	−	0.503864i	$-0.831911\pi$
$709$		−	26.8328i		−	1.00773i	0.863783	−	0.503864i	$-0.168089\pi$
$710$		0			0
$711$		0			0
$712$			50.5964i			1.89618i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$			8.94427i			0.333333i
$721$	50.0000			1.86210
$722$	26.8701			1.00000
$723$		−	18.9737i		−	0.705638i
$724$		−	53.6656i		−	1.99447i
$725$	30.0000			1.11417
$726$	−22.0000			−0.816497
$727$			53.7587i			1.99380i	0.0786754	+	0.996900i	$0.474931\pi$
$727$			53.7587i			1.99380i	−0.0786754	+	0.996900i	$0.525069\pi$
$728$		0			0
$729$	29.0000			1.07407
$730$		0			0
$731$		0			0
$732$		−	37.9473i		−	1.40257i
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		−	4.47214i		−	0.165070i
$735$		−	9.48683i		−	0.349927i
$736$	4.00000	−	26.8328i	0.147442	−	0.989071i
$737$		0			0
$738$	−16.9706			−0.624695
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			47.4342i			1.74019i	0.492883	+	0.870095i	$0.335943\pi$
$743$			47.4342i			1.74019i	−0.492883	+	0.870095i	$0.664057\pi$
$744$		0			0
$745$	10.0000			0.366372
$746$		0			0
$747$		−	9.48683i		−	0.347105i
$748$		0			0
$749$	−30.0000			−1.09618
$750$			22.3607i			0.816497i
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$	−39.5980			−1.44399
$753$		0			0
$754$		0			0
$755$		0			0
$756$		−	35.7771i		−	1.30120i
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	42.0000			1.52250			0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000			1.52250			0.761249	+	0.648459i	$0.224586\pi$
$762$	−8.48528			−0.307389
$763$	−42.4264			−1.53594
$764$		0			0
$765$		0			0
$766$			40.2492i			1.45426i
$767$		0			0
$768$	−22.6274			−0.816497
$769$			53.6656i			1.93523i	0.252426	+	0.967616i	$0.418771\pi$
$769$			53.6656i			1.93523i	−0.252426	+	0.967616i	$0.581229\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		−	4.47214i		−	0.160748i
$775$		0			0
$776$		0			0
$777$		0			0
$778$		−	44.2719i		−	1.58722i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−33.9411			−1.21296
$784$	−12.0000			−0.428571
$785$		0			0
$786$		0			0
$787$			41.1096i			1.46540i	0.680552	+	0.732700i	$0.261740\pi$
$787$			41.1096i			1.46540i	−0.680552	+	0.732700i	$0.738260\pi$
$788$		0			0
$789$			40.2492i			1.43291i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$	28.2843			1.00000
$801$			17.8885i			0.632061i
$802$			50.5964i			1.78662i
$803$		0			0
$804$			44.7214i			1.57720i
$805$	5.00000	−	33.5410i	0.176227	−	1.18217i
$806$		0			0
$807$	−33.9411			−1.19478
$808$	−50.9117			−1.79107
$809$	54.0000			1.89854			0.949269	−	0.314464i	$-0.101825\pi$
$809$	54.0000			1.89854			0.949269	+	0.314464i	$0.101825\pi$
$810$		−	15.8114i		−	0.555556i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$			37.9473i			1.33169i
$813$		0			0
$814$		0			0
$815$		−	28.4605i		−	0.996928i
$816$		0			0
$817$		0			0
$818$	5.65685			0.197787
$819$		0			0
$820$			53.6656i			1.87409i
$821$	−48.0000			−1.67521			−0.837606	−	0.546275i	$-0.816045\pi$
$821$	−48.0000			−1.67521			−0.837606	+	0.546275i	$0.816045\pi$
$822$		0			0
$823$	55.1543			1.92256			0.961280	−	0.275575i	$-0.0888683\pi$
$823$	55.1543			1.92256			0.961280	+	0.275575i	$0.0888683\pi$
$824$		−	44.7214i		−	1.55794i
$825$		0			0
$826$		0			0
$827$		−	47.4342i		−	1.64945i	−0.565536	−	0.824724i	$-0.691331\pi$
$827$		−	47.4342i		−	1.64945i	0.565536	−	0.824724i	$-0.308669\pi$
$828$	1.41421	−	9.48683i	0.0491473	−	0.329690i
$829$	56.0000			1.94496			0.972480	−	0.232986i	$-0.0748495\pi$
$829$	56.0000			1.94496			0.972480	+	0.232986i	$0.0748495\pi$
$830$	−30.0000			−1.04132
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$			53.7587i			1.86040i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$	−28.2843			−0.975900
$841$	7.00000			0.241379
$842$		−	56.9210i		−	1.96163i
$843$			44.2719i			1.52480i
$844$		0			0
$845$		−	29.0689i		−	1.00000i
$846$	−14.0000			−0.481330
$847$			34.7851i			1.19523i
$848$		0			0
$849$			22.3607i			0.767417i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$	−60.0000			−2.05316
$855$		0			0
$856$			26.8328i			0.917127i
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$	−14.1421			−0.482243
$861$		−	53.6656i		−	1.82892i
$862$		0			0
$863$	57.9828			1.97376			0.986878	−	0.161468i	$-0.0516228\pi$
$863$	57.9828			1.97376			0.986878	+	0.161468i	$0.0516228\pi$
$864$	−32.0000			−1.08866
$865$		0			0
$866$		0			0
$867$	24.0416			0.816497
$868$		0			0
$869$		0			0
$870$			26.8328i			0.909718i
$871$		0			0
$872$			37.9473i			1.28506i
$873$		0			0
$874$		0			0
$875$	35.3553			1.19523
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	58.1378i		−	1.95871i	−0.202145	−	0.979356i	$-0.564791\pi$
$881$		−	58.1378i		−	1.95871i	0.202145	−	0.979356i	$-0.435209\pi$
$882$	−4.24264			−0.142857
$883$	−55.1543			−1.85609			−0.928045	−	0.372467i	$-0.878512\pi$
$883$	−55.1543			−1.85609			−0.928045	+	0.372467i	$0.878512\pi$
$884$		0			0
$885$		0			0
$886$	−58.0000			−1.94855
$887$	52.3259			1.75693			0.878466	−	0.477805i	$-0.158567\pi$
$887$	52.3259			1.75693			0.878466	+	0.477805i	$0.158567\pi$
$888$		0			0
$889$			13.4164i			0.449972i
$890$	56.5685			1.89618
$891$		0			0
$892$	−59.3970			−1.98876
$893$		0			0
$894$			8.94427i			0.299141i
$895$		0			0
$896$			35.7771i			1.19523i
$897$		0			0
$898$	−50.9117			−1.69895
$899$		0			0
$900$	10.0000			0.333333
$901$		0			0
$902$		0			0
$903$	14.1421			0.470621
$904$		0			0
$905$	−60.0000			−1.99447
$906$		0			0
$907$		−	60.0833i		−	1.99503i	−0.0704373	−	0.997516i	$-0.522439\pi$
$907$		−	60.0833i		−	1.99503i	0.0704373	−	0.997516i	$-0.477561\pi$
$908$		−	56.9210i		−	1.88899i
$909$	−18.0000			−0.597022
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	−42.4264			−1.40257
$916$		−	53.6656i		−	1.77316i
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$	−30.0000	−	4.47214i	−0.989071	−	0.147442i
$921$	−6.00000			−0.197707
$922$	59.3970			1.95614
$923$		0			0
$924$		0			0
$925$		0			0
$926$	−18.0000			−0.591517
$927$		−	15.8114i		−	0.519314i
$928$	33.9411			1.11417
$929$	−36.0000			−1.18112			−0.590561	−	0.806993i	$-0.701093\pi$
$929$	−36.0000			−1.18112			−0.590561	+	0.806993i	$0.701093\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		−	40.2492i		−	1.31699i
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$	70.7107			2.30879
$939$		0			0
$940$			44.2719i			1.44399i
$941$		−	44.7214i		−	1.45787i	−0.684580	−	0.728937i	$-0.740015\pi$
$941$		−	44.7214i		−	1.45787i	0.684580	−	0.728937i	$-0.259985\pi$
$942$		0			0
$943$	8.48528	−	56.9210i	0.276319	−	1.85360i
$944$		0			0
$945$	−40.0000			−1.30120
$946$		0			0
$947$	60.8112			1.97610			0.988049	−	0.154140i	$-0.0492608\pi$
$947$	60.8112			1.97610			0.988049	+	0.154140i	$0.0492608\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$			25.2982i			0.816497i
$961$	31.0000			1.00000
$962$		0			0
$963$			9.48683i			0.305709i
$964$			26.8328i			0.864227i
$965$		0			0
$966$	30.0000	+	4.47214i	0.965234	+	0.143889i
$967$	46.6690			1.50078			0.750388	−	0.660998i	$-0.229867\pi$
$967$	46.6690			1.50078			0.750388	+	0.660998i	$0.229867\pi$
$968$	31.1127			1.00000
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	−19.7990			−0.635053
$973$		0			0
$974$	−54.0000			−1.73027
$975$		0			0
$976$			53.6656i			1.71780i
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$	25.4558			0.813988
$979$		0			0
$980$			13.4164i			0.428571i
$981$			13.4164i			0.428353i
$982$		0			0
$983$		−	47.4342i		−	1.51291i	−0.654043	−	0.756457i	$-0.726928\pi$
$983$		−	47.4342i		−	1.51291i	0.654043	−	0.756457i	$-0.273072\pi$
$984$	−48.0000			−1.53018
$985$		0			0
$986$		0			0
$987$		−	44.2719i		−	1.40919i
$988$		0			0
$989$	15.0000	+	2.23607i	0.476972	+	0.0711028i
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		−	26.8328i		−	0.850230i
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	460.2.g.a.459.1	✓	4
4.3	odd	2	inner	460.2.g.a.459.3	yes	4
5.4	even	2	inner	460.2.g.a.459.3	yes	4
20.19	odd	2	CM	460.2.g.a.459.1	✓	4
23.22	odd	2	inner	460.2.g.a.459.2	yes	4
92.91	even	2	inner	460.2.g.a.459.4	yes	4
115.114	odd	2	inner	460.2.g.a.459.4	yes	4
460.459	even	2	inner	460.2.g.a.459.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.g.a.459.1	✓	4	1.1	even	1	trivial
460.2.g.a.459.1	✓	4	20.19	odd	2	CM
460.2.g.a.459.2	yes	4	23.22	odd	2	inner
460.2.g.a.459.2	yes	4	460.459	even	2	inner
460.2.g.a.459.3	yes	4	4.3	odd	2	inner
460.2.g.a.459.3	yes	4	5.4	even	2	inner
460.2.g.a.459.4	yes	4	92.91	even	2	inner
460.2.g.a.459.4	yes	4	115.114	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	460.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} -1.41421 q^{3} +2.00000 q^{4} -2.23607i q^{5} +2.00000 q^{6} -3.16228i q^{7} -2.82843 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{2} -1.41421 q^{3} +2.00000 q^{4} -2.23607i q^{5} +2.00000 q^{6} -3.16228i q^{7} -2.82843 q^{8} -1.00000 q^{9} +3.16228i q^{10} -2.82843 q^{12} +4.47214i q^{14} +3.16228i q^{15} +4.00000 q^{16} +1.41421 q^{18} -4.47214i q^{20} +4.47214i q^{21} +(-0.707107 + 4.74342i) q^{23} +4.00000 q^{24} -5.00000 q^{25} +5.65685 q^{27} -6.32456i q^{28} -6.00000 q^{29} -4.47214i q^{30} -5.65685 q^{32} -7.07107 q^{35} -2.00000 q^{36} +6.32456i q^{40} -12.0000 q^{41} -6.32456i q^{42} -3.16228i q^{43} +2.23607i q^{45} +(1.00000 - 6.70820i) q^{46} -9.89949 q^{47} -5.65685 q^{48} -3.00000 q^{49} +7.07107 q^{50} -8.00000 q^{54} +8.94427i q^{56} +8.48528 q^{58} +6.32456i q^{60} +13.4164i q^{61} +3.16228i q^{63} +8.00000 q^{64} -15.8114i q^{67} +(1.00000 - 6.70820i) q^{69} +10.0000 q^{70} +2.82843 q^{72} +7.07107 q^{75} -8.94427i q^{80} -5.00000 q^{81} +16.9706 q^{82} +9.48683i q^{83} +8.94427i q^{84} +4.47214i q^{86} +8.48528 q^{87} -17.8885i q^{89} -3.16228i q^{90} +(-1.41421 + 9.48683i) q^{92} +14.0000 q^{94} +8.00000 q^{96} +4.24264 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4} + 8 q^{6} - 4 q^{9}+O(q^{10}) \) 4 * q + 8 * q^4 + 8 * q^6 - 4 * q^9	\( 4 q + 8 q^{4} + 8 q^{6} - 4 q^{9} + 16 q^{16} + 16 q^{24} - 20 q^{25} - 24 q^{29} - 8 q^{36} - 48 q^{41} + 4 q^{46} - 12 q^{49} - 32 q^{54} + 32 q^{64} + 4 q^{69} + 40 q^{70} - 20 q^{81} + 56 q^{94} + 32 q^{96}+O(q^{100}) \) 4 * q + 8 * q^4 + 8 * q^6 - 4 * q^9 + 16 * q^16 + 16 * q^24 - 20 * q^25 - 24 * q^29 - 8 * q^36 - 48 * q^41 + 4 * q^46 - 12 * q^49 - 32 * q^54 + 32 * q^64 + 4 * q^69 + 40 * q^70 - 20 * q^81 + 56 * q^94 + 32 * q^96

Introduction

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