LMFDB - Embedded newform 480.2.bi.a.113.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [480,2,Mod(17,480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(480, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 2, 1]))

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

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

chi := DirichletCharacter("480.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	480.bi (of order $4$, degree $2$, 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:	$3.83281929702$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			113.1
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	480.113
Dual form			480.2.bi.a.17.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 - 1.22474i) q^{3} +(0.224745 - 2.22474i) q^{5} +(3.44949 - 3.44949i) q^{7} +3.00000i q^{9} -1.55051 q^{11} +(-3.00000 + 2.44949i) q^{15} -8.44949 q^{21} +(-4.89898 - 1.00000i) q^{25} +(3.67423 - 3.67423i) q^{27} -5.34847i q^{29} -4.89898 q^{31} +(1.89898 + 1.89898i) q^{33} +(-6.89898 - 8.44949i) q^{35} +(6.67423 + 0.674235i) q^{45} -16.7980i q^{49} +(2.44949 + 2.44949i) q^{53} +(-0.348469 + 3.44949i) q^{55} +15.3485i q^{59} +(10.3485 + 10.3485i) q^{63} +(11.8990 + 11.8990i) q^{73} +(4.77526 + 7.22474i) q^{75} +(-5.34847 + 5.34847i) q^{77} -14.6969i q^{79} -9.00000 q^{81} +(4.00000 + 4.00000i) q^{83} +(-6.55051 + 6.55051i) q^{87} +(6.00000 + 6.00000i) q^{93} +(8.79796 - 8.79796i) q^{97} -4.65153i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} + 4 q^{7} - 16 q^{11} - 12 q^{15} - 24 q^{21} - 12 q^{33} - 8 q^{35} + 12 q^{45} + 28 q^{55} + 12 q^{63} + 28 q^{73} + 24 q^{75} + 8 q^{77} - 36 q^{81} + 16 q^{83} - 36 q^{87} + 24 q^{93}+ \cdots - 4 q^{97}+O(q^{100}) $ 4 * q - 4 * q^5 + 4 * q^7 - 16 * q^11 - 12 * q^15 - 24 * q^21 - 12 * q^33 - 8 * q^35 + 12 * q^45 + 28 * q^55 + 12 * q^63 + 28 * q^73 + 24 * q^75 + 8 * q^77 - 36 * q^81 + 16 * q^83 - 36 * q^87 + 24 * q^93 - 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/480\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$421$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$-1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	−1.22474	−	1.22474i	−0.707107	−	0.707107i
$3$	−1.22474	−	1.22474i	−0.707107	−	0.707107i
$4$		0			0
$5$	0.224745	−	2.22474i	0.100509	−	0.994936i
$5$	0.224745	−	2.22474i	0.100509	−	0.994936i
$6$		0			0
$7$	3.44949	−	3.44949i	1.30378	−	1.30378i	0.377964	−	0.925820i	$-0.376624\pi$
$7$	3.44949	−	3.44949i	1.30378	−	1.30378i	0.925820	−	0.377964i	$-0.123376\pi$
$8$		0			0
$9$			3.00000i			1.00000i
$10$		0			0
$11$	−1.55051			−0.467496			−0.233748	−	0.972297i	$-0.575099\pi$
$11$	−1.55051			−0.467496			−0.233748	+	0.972297i	$0.575099\pi$
$12$		0			0
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$		0			0
$15$	−3.00000	+	2.44949i	−0.774597	+	0.632456i
$16$		0			0
$17$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$17$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$	−8.44949			−1.84383
$22$		0			0
$23$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$23$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$24$		0			0
$25$	−4.89898	−	1.00000i	−0.979796	−	0.200000i
$26$		0			0
$27$	3.67423	−	3.67423i	0.707107	−	0.707107i
$28$		0			0
$29$		−	5.34847i		−	0.993186i	−0.867984	−	0.496593i	$-0.834584\pi$
$29$		−	5.34847i		−	0.993186i	0.867984	−	0.496593i	$-0.165416\pi$
$30$		0			0
$31$	−4.89898			−0.879883			−0.439941	−	0.898027i	$-0.645001\pi$
$31$	−4.89898			−0.879883			−0.439941	+	0.898027i	$0.645001\pi$
$32$		0			0
$33$	1.89898	+	1.89898i	0.330570	+	0.330570i
$34$		0			0
$35$	−6.89898	−	8.44949i	−1.16614	−	1.42822i
$36$		0			0
$37$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$37$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$	6.67423	+	0.674235i	0.994936	+	0.100509i
$46$		0			0
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$		0			0
$49$		−	16.7980i		−	2.39971i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.44949	+	2.44949i	0.336463	+	0.336463i	0.855034	−	0.518571i	$-0.173536\pi$
$53$	2.44949	+	2.44949i	0.336463	+	0.336463i	−0.518571	+	0.855034i	$0.673536\pi$
$54$		0			0
$55$	−0.348469	+	3.44949i	−0.0469876	+	0.465129i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			15.3485i			1.99820i	0.0424110	+	0.999100i	$0.486496\pi$
$59$			15.3485i			1.99820i	−0.0424110	+	0.999100i	$0.513504\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$	10.3485	+	10.3485i	1.30378	+	1.30378i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	11.8990	+	11.8990i	1.39267	+	1.39267i	0.819288	+	0.573382i	$0.194369\pi$
$73$	11.8990	+	11.8990i	1.39267	+	1.39267i	0.573382	+	0.819288i	$0.305631\pi$
$74$		0			0
$75$	4.77526	+	7.22474i	0.551399	+	0.834242i
$76$		0			0
$77$	−5.34847	+	5.34847i	−0.609515	+	0.609515i
$78$		0			0
$79$		−	14.6969i		−	1.65353i	−0.562544	−	0.826767i	$-0.690177\pi$
$79$		−	14.6969i		−	1.65353i	0.562544	−	0.826767i	$-0.309823\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$	4.00000	+	4.00000i	0.439057	+	0.439057i	0.891695	−	0.452638i	$-0.149517\pi$
$83$	4.00000	+	4.00000i	0.439057	+	0.439057i	−0.452638	+	0.891695i	$0.649517\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−6.55051	+	6.55051i	−0.702288	+	0.702288i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	6.00000	+	6.00000i	0.622171	+	0.622171i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	8.79796	−	8.79796i	0.893297	−	0.893297i	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	8.79796	−	8.79796i	0.893297	−	0.893297i	0.994832	+	0.101535i	$0.0323753\pi$
$98$		0			0
$99$		−	4.65153i		−	0.467496i
$100$		0			0
$101$	11.5505			1.14932			0.574659	−	0.818393i	$-0.305135\pi$
$101$	11.5505			1.14932			0.574659	+	0.818393i	$0.305135\pi$
$102$		0			0
$103$	0.348469	+	0.348469i	0.0343357	+	0.0343357i	0.724066	−	0.689730i	$-0.242271\pi$
$103$	0.348469	+	0.348469i	0.0343357	+	0.0343357i	−0.689730	+	0.724066i	$0.742271\pi$
$104$		0			0
$105$	−1.89898	+	18.7980i	−0.185321	+	1.83449i
$106$		0			0
$107$	8.00000	−	8.00000i	0.773389	−	0.773389i	−0.205308	−	0.978697i	$-0.565820\pi$
$107$	8.00000	−	8.00000i	0.773389	−	0.773389i	0.978697	+	0.205308i	$0.0658197\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$113$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−8.59592			−0.781447
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−3.32577	+	10.6742i	−0.297465	+	0.954733i
$126$		0			0
$127$	13.4495	−	13.4495i	1.19345	−	1.19345i	0.217357	−	0.976092i	$-0.430256\pi$
$127$	13.4495	−	13.4495i	1.19345	−	1.19345i	0.976092	−	0.217357i	$-0.0697436\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	18.4495			1.61194			0.805970	−	0.591957i	$-0.201644\pi$
$131$	18.4495			1.61194			0.805970	+	0.591957i	$0.201644\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−7.34847	−	9.00000i	−0.632456	−	0.774597i
$136$		0			0
$137$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$137$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−11.8990	−	1.20204i	−0.988156	−	0.0998241i
$146$		0			0
$147$	−20.5732	+	20.5732i	−1.69685	+	1.69685i
$148$		0			0
$149$			19.1464i			1.56854i	0.620422	+	0.784268i	$0.286961\pi$
$149$			19.1464i			1.56854i	−0.620422	+	0.784268i	$0.713039\pi$
$150$		0			0
$151$	−2.00000			−0.162758			−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000			−0.162758			−0.0813788	+	0.996683i	$0.525932\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.10102	+	10.8990i	−0.0884361	+	0.875427i
$156$		0			0
$157$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$157$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$158$		0			0
$159$		−	6.00000i		−	0.475831i
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$		0			0
$165$	4.65153	−	3.79796i	0.362121	−	0.295671i
$166$		0			0
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$			13.0000i			1.00000i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−14.0000	−	14.0000i	−1.06440	−	1.06440i	−0.997778	−	0.0666220i	$-0.978778\pi$
$173$	−14.0000	−	14.0000i	−1.06440	−	1.06440i	−0.0666220	−	0.997778i	$-0.521222\pi$
$174$		0			0
$175$	−20.3485	+	13.4495i	−1.53820	+	1.01669i
$176$		0			0
$177$	18.7980	−	18.7980i	1.41294	−	1.41294i
$178$		0			0
$179$		−	9.14643i		−	0.683636i	−0.939766	−	0.341818i	$-0.888957\pi$
$179$		−	9.14643i		−	0.683636i	0.939766	−	0.341818i	$-0.111043\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		−	25.3485i		−	1.84383i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	−8.10102	−	8.10102i	−0.583124	−	0.583124i	0.352636	−	0.935760i	$-0.385285\pi$
$193$	−8.10102	−	8.10102i	−0.583124	−	0.583124i	−0.935760	+	0.352636i	$0.885285\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	17.1464	−	17.1464i	1.22163	−	1.22163i	0.254581	−	0.967051i	$-0.418062\pi$
$197$	17.1464	−	17.1464i	1.22163	−	1.22163i	0.967051	−	0.254581i	$-0.0819375\pi$
$198$		0			0
$199$			14.0000i			0.992434i	0.868199	+	0.496217i	$0.165278\pi$
$199$			14.0000i			0.992434i	−0.868199	+	0.496217i	$0.834722\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−18.4495	−	18.4495i	−1.29490	−	1.29490i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−16.8990	+	16.8990i	−1.14718	+	1.14718i
$218$		0			0
$219$		−	29.1464i		−	1.96953i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	20.3485	+	20.3485i	1.36263	+	1.36263i	0.870544	+	0.492090i	$0.163767\pi$
$223$	20.3485	+	20.3485i	1.36263	+	1.36263i	0.492090	+	0.870544i	$0.336233\pi$
$224$		0			0
$225$	3.00000	−	14.6969i	0.200000	−	0.979796i
$226$		0			0
$227$	7.34847	−	7.34847i	0.487735	−	0.487735i	−0.419856	−	0.907591i	$-0.637919\pi$
$227$	7.34847	−	7.34847i	0.487735	−	0.487735i	0.907591	+	0.419856i	$0.137919\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$	13.1010			0.861984
$232$		0			0
$233$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$233$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−18.0000	+	18.0000i	−1.16923	+	1.16923i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	29.3939			1.89343			0.946713	−	0.322078i	$-0.104381\pi$
$241$	29.3939			1.89343			0.946713	+	0.322078i	$0.104381\pi$
$242$		0			0
$243$	11.0227	+	11.0227i	0.707107	+	0.707107i
$244$		0			0
$245$	−37.3712	−	3.77526i	−2.38756	−	0.241192i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		−	9.79796i		−	0.620920i
$250$		0			0
$251$	−26.0454			−1.64397			−0.821986	−	0.569508i	$-0.807134\pi$
$251$	−26.0454			−1.64397			−0.821986	+	0.569508i	$0.807134\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$257$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	16.0454			0.993186
$262$		0			0
$263$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$263$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$264$		0			0
$265$	6.00000	−	4.89898i	0.368577	−	0.300942i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			14.6515i			0.893320i	0.894704	+	0.446660i	$0.147387\pi$
$269$			14.6515i			0.893320i	−0.894704	+	0.446660i	$0.852613\pi$
$270$		0			0
$271$	−22.0000			−1.33640			−0.668202	−	0.743980i	$-0.732936\pi$
$271$	−22.0000			−1.33640			−0.668202	+	0.743980i	$0.732936\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	7.59592	+	1.55051i	0.458051	+	0.0934993i
$276$		0			0
$277$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$277$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$278$		0			0
$279$		−	14.6969i		−	0.879883i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$283$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$		−	17.0000i		−	1.00000i
$290$		0			0
$291$	−21.5505			−1.26331
$292$		0			0
$293$	−22.0454	−	22.0454i	−1.28791	−	1.28791i	−0.936056	−	0.351850i	$-0.885553\pi$
$293$	−22.0454	−	22.0454i	−1.28791	−	1.28791i	−0.351850	−	0.936056i	$-0.614447\pi$
$294$		0			0
$295$	34.1464	+	3.44949i	1.98808	+	0.200837i
$296$		0			0
$297$	−5.69694	+	5.69694i	−0.330570	+	0.330570i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−14.1464	−	14.1464i	−0.812691	−	0.812691i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		0			0
$309$		−	0.853572i		−	0.0485580i
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	21.8990	+	21.8990i	1.23780	+	1.23780i	0.960897	+	0.276907i	$0.0893093\pi$
$313$	21.8990	+	21.8990i	1.23780	+	1.23780i	0.276907	+	0.960897i	$0.410691\pi$
$314$		0			0
$315$	25.3485	−	20.6969i	1.42822	−	1.16614i
$316$		0			0
$317$	22.0000	−	22.0000i	1.23564	−	1.23564i	0.273879	−	0.961764i	$-0.411693\pi$
$317$	22.0000	−	22.0000i	1.23564	−	1.23564i	0.961764	−	0.273879i	$-0.0883068\pi$
$318$		0			0
$319$			8.29286i			0.464311i
$320$		0			0
$321$	−19.5959			−1.09374
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−25.6969	+	25.6969i	−1.39980	+	1.39980i	−0.599208	+	0.800593i	$0.704518\pi$
$337$	−25.6969	+	25.6969i	−1.39980	+	1.39980i	−0.800593	+	0.599208i	$0.795482\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	7.59592			0.411342
$342$		0			0
$343$	−33.7980	−	33.7980i	−1.82492	−	1.82492i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−17.1464	+	17.1464i	−0.920468	+	0.920468i	−0.997062	−	0.0765939i	$-0.975596\pi$
$347$	−17.1464	+	17.1464i	−0.920468	+	0.920468i	0.0765939	+	0.997062i	$0.475596\pi$
$348$		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
$353$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$353$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$	10.5278	+	10.5278i	0.552567	+	0.552567i
$364$		0			0
$365$	29.1464	−	23.7980i	1.52559	−	1.24564i
$366$		0			0
$367$	−16.5505	+	16.5505i	−0.863930	+	0.863930i	−0.991792	−	0.127862i	$-0.959188\pi$
$367$	−16.5505	+	16.5505i	−0.863930	+	0.863930i	0.127862	+	0.991792i	$0.459188\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	16.8990			0.877351
$372$		0			0
$373$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$373$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$374$		0			0
$375$	17.1464	−	9.00000i	0.885438	−	0.464758i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$	−32.9444			−1.68779
$382$		0			0
$383$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$		0			0
$385$	10.6969	+	13.1010i	0.545166	+	0.667690i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			39.1464i			1.98480i	0.123043	+	0.992401i	$0.460735\pi$
$389$			39.1464i			1.98480i	−0.123043	+	0.992401i	$0.539265\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−22.5959	−	22.5959i	−1.13981	−	1.13981i
$394$		0			0
$395$	−32.6969	−	3.30306i	−1.64516	−	0.166195i
$396$		0			0
$397$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$397$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−2.02270	+	20.0227i	−0.100509	+	0.994936i
$406$		0			0
$407$		0			0
$408$		0			0
$409$			39.1918i			1.93791i	0.247234	+	0.968956i	$0.420478\pi$
$409$			39.1918i			1.93791i	−0.247234	+	0.968956i	$0.579522\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	52.9444	+	52.9444i	2.60522	+	2.60522i
$414$		0			0
$415$	9.79796	−	8.00000i	0.480963	−	0.392705i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			35.3485i			1.72689i	0.504447	+	0.863443i	$0.331697\pi$
$419$			35.3485i			1.72689i	−0.504447	+	0.863443i	$0.668303\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	−12.5959	−	12.5959i	−0.605321	−	0.605321i	0.336399	−	0.941720i	$-0.390791\pi$
$433$	−12.5959	−	12.5959i	−0.605321	−	0.605321i	−0.941720	+	0.336399i	$0.890791\pi$
$434$		0			0
$435$	13.1010	+	16.0454i	0.628146	+	0.769318i
$436$		0			0
$437$		0			0
$438$		0			0
$439$			34.0000i			1.62273i	0.584539	+	0.811366i	$0.301275\pi$
$439$			34.0000i			1.62273i	−0.584539	+	0.811366i	$0.698725\pi$
$440$		0			0
$441$	50.3939			2.39971
$442$		0			0
$443$	22.0454	+	22.0454i	1.04741	+	1.04741i	0.998819	+	0.0485901i	$0.0154728\pi$
$443$	22.0454	+	22.0454i	1.04741	+	1.04741i	0.0485901	+	0.998819i	$0.484527\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	23.4495	−	23.4495i	1.10912	−	1.10912i
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	2.44949	+	2.44949i	0.115087	+	0.115087i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	28.7980	−	28.7980i	1.34711	−	1.34711i	0.458329	−	0.888783i	$-0.348448\pi$
$457$	28.7980	−	28.7980i	1.34711	−	1.34711i	0.888783	−	0.458329i	$-0.151552\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−12.9444			−0.602880			−0.301440	−	0.953485i	$-0.597467\pi$
$461$	−12.9444			−0.602880			−0.301440	+	0.953485i	$0.597467\pi$
$462$		0			0
$463$	−4.14643	−	4.14643i	−0.192701	−	0.192701i	0.604161	−	0.796862i	$-0.293508\pi$
$463$	−4.14643	−	4.14643i	−0.192701	−	0.192701i	−0.796862	+	0.604161i	$0.793508\pi$
$464$		0			0
$465$	14.6969	−	12.0000i	0.681554	−	0.556487i
$466$		0			0
$467$	28.0000	−	28.0000i	1.29569	−	1.29569i	0.364471	−	0.931215i	$-0.381250\pi$
$467$	28.0000	−	28.0000i	1.29569	−	1.29569i	0.931215	−	0.364471i	$-0.118750\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−7.34847	+	7.34847i	−0.336463	+	0.336463i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−17.5959	−	21.5505i	−0.798989	−	0.978558i
$486$		0			0
$487$	−21.0454	+	21.0454i	−0.953658	+	0.953658i	−0.998973	−	0.0453143i	$-0.985571\pi$
$487$	−21.0454	+	21.0454i	−0.953658	+	0.953658i	0.0453143	+	0.998973i	$0.485571\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	42.9444			1.93805			0.969027	−	0.246957i	$-0.0794305\pi$
$491$	42.9444			1.93805			0.969027	+	0.246957i	$0.0794305\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−10.3485	−	1.04541i	−0.465129	−	0.0469876i
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$503$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$504$		0			0
$505$	2.59592	−	25.6969i	0.115517	−	1.14350i
$506$		0			0
$507$	15.9217	−	15.9217i	0.707107	−	0.707107i
$508$		0			0
$509$		−	29.8434i		−	1.32278i	−0.750040	−	0.661392i	$-0.769966\pi$
$509$		−	29.8434i		−	1.32278i	0.750040	−	0.661392i	$-0.230034\pi$
$510$		0			0
$511$	82.0908			3.63148
$512$		0			0
$513$		0			0
$514$		0			0
$515$	0.853572	−	0.696938i	0.0376129	−	0.0307108i
$516$		0			0
$517$		0			0
$518$		0			0
$519$			34.2929i			1.50529i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$523$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$524$		0			0
$525$	41.3939	+	8.44949i	1.80658	+	0.368766i
$526$		0			0
$527$		0			0
$528$		0			0
$529$			23.0000i			1.00000i
$530$		0			0
$531$	−46.0454			−1.99820
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−16.0000	−	19.5959i	−0.691740	−	0.847205i
$536$		0			0
$537$	−11.2020	+	11.2020i	−0.483404	+	0.483404i
$538$		0			0
$539$			26.0454i			1.12186i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$547$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−50.6969	−	50.6969i	−2.15585	−	2.15585i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−31.8434	+	31.8434i	−1.34925	+	1.34925i	−0.462767	+	0.886480i	$0.653143\pi$
$557$	−31.8434	+	31.8434i	−1.34925	+	1.34925i	−0.886480	+	0.462767i	$0.846857\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−26.9444	−	26.9444i	−1.13557	−	1.13557i	−0.989235	−	0.146336i	$-0.953252\pi$
$563$	−26.9444	−	26.9444i	−1.13557	−	1.13557i	−0.146336	−	0.989235i	$-0.546748\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−31.0454	+	31.0454i	−1.30378	+	1.30378i
$568$		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
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	4.30306	−	4.30306i	0.179139	−	0.179139i	−0.611842	−	0.790980i	$-0.709571\pi$
$577$	4.30306	−	4.30306i	0.179139	−	0.179139i	0.790980	+	0.611842i	$0.209571\pi$
$578$		0			0
$579$			19.8434i			0.824662i
$580$		0			0
$581$	27.5959			1.14487
$582$		0			0
$583$	−3.79796	−	3.79796i	−0.157295	−	0.157295i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−32.0000	+	32.0000i	−1.32078	+	1.32078i	−0.407638	+	0.913144i	$0.633647\pi$
$587$	−32.0000	+	32.0000i	−1.32078	+	1.32078i	−0.913144	+	0.407638i	$0.866353\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	−42.0000			−1.72765
$592$		0			0
$593$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$593$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	17.1464	−	17.1464i	0.701757	−	0.701757i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	2.00000			0.0815817			0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000			0.0815817			0.0407909	+	0.999168i	$0.487012\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.93189	+	19.1237i	−0.0785424	+	0.777490i
$606$		0			0
$607$	−11.0454	+	11.0454i	−0.448319	+	0.448319i	−0.894795	−	0.446476i	$-0.852679\pi$
$607$	−11.0454	+	11.0454i	−0.448319	+	0.448319i	0.446476	+	0.894795i	$0.352679\pi$
$608$		0			0
$609$			45.1918i			1.83127i
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$613$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$617$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	23.0000	+	9.79796i	0.920000	+	0.391918i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−4.89898			−0.195025			−0.0975126	−	0.995234i	$-0.531089\pi$
$631$	−4.89898			−0.195025			−0.0975126	+	0.995234i	$0.531089\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−26.8990	−	32.9444i	−1.06745	−	1.30736i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$647$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$648$		0			0
$649$		−	23.7980i		−	0.934152i
$650$		0			0
$651$	41.3939			1.62235
$652$		0			0
$653$	−34.0000	−	34.0000i	−1.33052	−	1.33052i	−0.904901	−	0.425622i	$-0.860055\pi$
$653$	−34.0000	−	34.0000i	−1.33052	−	1.33052i	−0.425622	−	0.904901i	$-0.639945\pi$
$654$		0			0
$655$	4.14643	−	41.0454i	0.162014	−	1.60378i
$656$		0			0
$657$	−35.6969	+	35.6969i	−1.39267	+	1.39267i
$658$		0			0
$659$		−	49.1464i		−	1.91447i	−0.289307	−	0.957237i	$-0.593425\pi$
$659$		−	49.1464i		−	1.91447i	0.289307	−	0.957237i	$-0.406575\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		−	49.8434i		−	1.92706i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−2.59592	−	2.59592i	−0.100065	−	0.100065i	0.655302	−	0.755367i	$-0.272541\pi$
$673$	−2.59592	−	2.59592i	−0.100065	−	0.100065i	−0.755367	+	0.655302i	$0.772541\pi$
$674$		0			0
$675$	−21.6742	+	14.3258i	−0.834242	+	0.551399i
$676$		0			0
$677$	2.00000	−	2.00000i	0.0768662	−	0.0768662i	−0.667628	−	0.744495i	$-0.732690\pi$
$677$	2.00000	−	2.00000i	0.0768662	−	0.0768662i	0.744495	+	0.667628i	$0.232690\pi$
$678$		0			0
$679$		−	60.6969i		−	2.32933i
$680$		0			0
$681$	−18.0000			−0.689761
$682$		0			0
$683$	4.00000	+	4.00000i	0.153056	+	0.153056i	0.779481	−	0.626426i	$-0.215483\pi$
$683$	4.00000	+	4.00000i	0.153056	+	0.153056i	−0.626426	+	0.779481i	$0.715483\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$	−16.0454	−	16.0454i	−0.609515	−	0.609515i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−52.9444			−1.99968			−0.999841	−	0.0178345i	$-0.994323\pi$
$701$	−52.9444			−1.99968			−0.999841	+	0.0178345i	$0.994323\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	39.8434	−	39.8434i	1.49846	−	1.49846i
$708$		0			0
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$		0			0
$711$	44.0908			1.65353
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	2.40408			0.0895327
$722$		0			0
$723$	−36.0000	−	36.0000i	−1.33885	−	1.33885i
$724$		0			0
$725$	−5.34847	+	26.2020i	−0.198637	+	0.973119i
$726$		0			0
$727$	27.9444	−	27.9444i	1.03640	−	1.03640i	0.0370879	−	0.999312i	$-0.488192\pi$
$727$	27.9444	−	27.9444i	1.03640	−	1.03640i	0.999312	−	0.0370879i	$-0.0118082\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$733$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$734$		0			0
$735$	41.1464	+	50.3939i	1.51771	+	1.85881i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$		0			0
$745$	42.5959	+	4.30306i	1.56059	+	0.157652i
$746$		0			0
$747$	−12.0000	+	12.0000i	−0.439057	+	0.439057i
$748$		0			0
$749$		−	55.1918i		−	2.01667i
$750$		0			0
$751$	−53.8888			−1.96643			−0.983215	−	0.182453i	$-0.941596\pi$
$751$	−53.8888			−1.96643			−0.983215	+	0.182453i	$0.941596\pi$
$752$		0			0
$753$	31.8990	+	31.8990i	1.16246	+	1.16246i
$754$		0			0
$755$	−0.449490	+	4.44949i	−0.0163586	+	0.161934i
$756$		0			0
$757$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$757$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$			26.0000i			0.937584i	0.883309	+	0.468792i	$0.155311\pi$
$769$			26.0000i			0.937584i	−0.883309	+	0.468792i	$0.844689\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−14.0000	−	14.0000i	−0.503545	−	0.503545i	0.408993	−	0.912538i	$-0.365880\pi$
$773$	−14.0000	−	14.0000i	−0.503545	−	0.503545i	−0.912538	+	0.408993i	$0.865880\pi$
$774$		0			0
$775$	24.0000	+	4.89898i	0.862105	+	0.175977i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−19.6515	−	19.6515i	−0.702288	−	0.702288i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−13.3485	−	1.34847i	−0.473421	−	0.0478253i
$796$		0			0
$797$	−38.0000	+	38.0000i	−1.34603	+	1.34603i	−0.456101	+	0.889928i	$0.650754\pi$
$797$	−38.0000	+	38.0000i	−1.34603	+	1.34603i	−0.889928	+	0.456101i	$0.849246\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−18.4495	−	18.4495i	−0.651068	−	0.651068i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	17.9444	−	17.9444i	0.631672	−	0.631672i
$808$		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
$813$	26.9444	+	26.9444i	0.944981	+	0.944981i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	56.0454			1.95600			0.977999	−	0.208609i	$-0.0668936\pi$
$821$	56.0454			1.95600			0.977999	+	0.208609i	$0.0668936\pi$
$822$		0			0
$823$	5.85357	+	5.85357i	0.204043	+	0.204043i	0.801730	−	0.597687i	$-0.203913\pi$
$823$	5.85357	+	5.85357i	0.204043	+	0.204043i	−0.597687	+	0.801730i	$0.703913\pi$
$824$		0			0
$825$	−7.40408	−	11.2020i	−0.257777	−	0.390005i
$826$		0			0
$827$	7.34847	−	7.34847i	0.255531	−	0.255531i	−0.567702	−	0.823234i	$-0.692168\pi$
$827$	7.34847	−	7.34847i	0.255531	−	0.255531i	0.823234	+	0.567702i	$0.192168\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−18.0000	+	18.0000i	−0.622171	+	0.622171i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	0.393877			0.0135820
$842$		0			0
$843$		0			0
$844$		0			0
$845$	28.9217	+	2.92168i	0.994936	+	0.100509i
$846$		0			0
$847$	−29.6515	+	29.6515i	−1.01884	+	1.01884i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$853$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$857$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$	−34.2929	+	28.0000i	−1.16599	+	0.952029i
$866$		0			0
$867$	−20.8207	+	20.8207i	−0.707107	+	0.707107i
$868$		0			0
$869$			22.7878i			0.773022i
$870$		0			0
$871$		0			0
$872$		0			0
$873$	26.3939	+	26.3939i	0.893297	+	0.893297i
$874$		0			0
$875$	25.3485	+	48.2929i	0.856935	+	1.63260i
$876$		0			0
$877$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$877$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$878$		0			0
$879$			54.0000i			1.82137i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$883$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$884$		0			0
$885$	−37.5959	−	46.0454i	−1.26377	−	1.54780i
$886$		0			0
$887$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$887$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$888$		0			0
$889$		−	92.7878i		−	3.11200i
$890$		0			0
$891$	13.9546			0.467496
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−20.3485	−	2.05561i	−0.680174	−	0.0687116i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			26.2020i			0.873887i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$			34.6515i			1.14932i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$	−6.20204	−	6.20204i	−0.205258	−	0.205258i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	63.6413	−	63.6413i	2.10162	−	2.10162i
$918$		0			0
$919$			34.2929i			1.13122i	0.824674	+	0.565608i	$0.191359\pi$
$919$			34.2929i			1.13122i	−0.824674	+	0.565608i	$0.808641\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−1.04541	+	1.04541i	−0.0343357	+	0.0343357i
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	38.7980	−	38.7980i	1.26747	−	1.26747i	0.320085	−	0.947389i	$-0.396288\pi$
$937$	38.7980	−	38.7980i	1.26747	−	1.26747i	0.947389	−	0.320085i	$-0.103712\pi$
$938$		0			0
$939$		−	53.6413i		−	1.75052i
$940$		0			0
$941$	7.05561			0.230006			0.115003	−	0.993365i	$-0.463312\pi$
$941$	7.05561			0.230006			0.115003	+	0.993365i	$0.463312\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	−56.3939	−	5.69694i	−1.83449	−	0.185321i
$946$		0			0
$947$	−17.1464	+	17.1464i	−0.557184	+	0.557184i	−0.928505	−	0.371321i	$-0.878905\pi$
$947$	−17.1464	+	17.1464i	−0.557184	+	0.557184i	0.371321	+	0.928505i	$0.378905\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−53.8888			−1.74746
$952$		0			0
$953$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$953$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	10.1566	−	10.1566i	0.328317	−	0.328317i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−7.00000			−0.225806
$962$		0			0
$963$	24.0000	+	24.0000i	0.773389	+	0.773389i
$964$		0			0
$965$	−19.8434	+	16.2020i	−0.638781	+	0.521562i
$966$		0			0
$967$	33.4495	−	33.4495i	1.07566	−	1.07566i	0.0787703	−	0.996893i	$-0.474901\pi$
$967$	33.4495	−	33.4495i	1.07566	−	1.07566i	0.996893	−	0.0787703i	$-0.0250994\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−41.5505			−1.33342			−0.666710	−	0.745318i	$-0.732298\pi$
$971$	−41.5505			−1.33342			−0.666710	+	0.745318i	$0.732298\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$977$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$		0			0
$985$	−34.2929	−	42.0000i	−1.09266	−	1.33823i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	58.0000			1.84243			0.921215	−	0.389053i	$-0.127198\pi$
$991$	58.0000			1.84243			0.921215	+	0.389053i	$0.127198\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	31.1464	+	3.14643i	0.987408	+	0.0997485i
$996$		0			0
$997$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$997$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.2.bi.a.113.1		4
3.2	odd	2		480.2.bi.b.113.2		4
4.3	odd	2		120.2.w.b.53.2	yes	4
5.2	odd	4	inner	480.2.bi.a.17.1		4
8.3	odd	2		120.2.w.a.53.1	✓	4
8.5	even	2		480.2.bi.b.113.2		4
12.11	even	2		120.2.w.a.53.1	✓	4
15.2	even	4		480.2.bi.b.17.2		4
20.3	even	4		600.2.w.b.557.1		4
20.7	even	4		120.2.w.b.77.2	yes	4
20.19	odd	2		600.2.w.b.293.1		4
24.5	odd	2	CM	480.2.bi.a.113.1		4
24.11	even	2		120.2.w.b.53.2	yes	4
40.3	even	4		600.2.w.h.557.2		4
40.19	odd	2		600.2.w.h.293.2		4
40.27	even	4		120.2.w.a.77.1	yes	4
40.37	odd	4		480.2.bi.b.17.2		4
60.23	odd	4		600.2.w.h.557.2		4
60.47	odd	4		120.2.w.a.77.1	yes	4
60.59	even	2		600.2.w.h.293.2		4
120.59	even	2		600.2.w.b.293.1		4
120.77	even	4	inner	480.2.bi.a.17.1		4
120.83	odd	4		600.2.w.b.557.1		4
120.107	odd	4		120.2.w.b.77.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.w.a.53.1	✓	4	8.3	odd	2
120.2.w.a.53.1	✓	4	12.11	even	2
120.2.w.a.77.1	yes	4	40.27	even	4
120.2.w.a.77.1	yes	4	60.47	odd	4
120.2.w.b.53.2	yes	4	4.3	odd	2
120.2.w.b.53.2	yes	4	24.11	even	2
120.2.w.b.77.2	yes	4	20.7	even	4
120.2.w.b.77.2	yes	4	120.107	odd	4
480.2.bi.a.17.1		4	5.2	odd	4	inner
480.2.bi.a.17.1		4	120.77	even	4	inner
480.2.bi.a.113.1		4	1.1	even	1	trivial
480.2.bi.a.113.1		4	24.5	odd	2	CM
480.2.bi.b.17.2		4	15.2	even	4
480.2.bi.b.17.2		4	40.37	odd	4
480.2.bi.b.113.2		4	3.2	odd	2
480.2.bi.b.113.2		4	8.5	even	2
600.2.w.b.293.1		4	20.19	odd	2
600.2.w.b.293.1		4	120.59	even	2
600.2.w.b.557.1		4	20.3	even	4
600.2.w.b.557.1		4	120.83	odd	4
600.2.w.h.293.2		4	40.19	odd	2
600.2.w.h.293.2		4	60.59	even	2
600.2.w.h.557.2		4	40.3	even	4
600.2.w.h.557.2		4	60.23	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:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	480.bi (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 - 1.22474i) q^{3} +(0.224745 - 2.22474i) q^{5} +(3.44949 - 3.44949i) q^{7} +3.00000i q^{9} -1.55051 q^{11} +(-3.00000 + 2.44949i) q^{15} -8.44949 q^{21} +(-4.89898 - 1.00000i) q^{25} +(3.67423 - 3.67423i) q^{27} -5.34847i q^{29} -4.89898 q^{31} +(1.89898 + 1.89898i) q^{33} +(-6.89898 - 8.44949i) q^{35} +(6.67423 + 0.674235i) q^{45} -16.7980i q^{49} +(2.44949 + 2.44949i) q^{53} +(-0.348469 + 3.44949i) q^{55} +15.3485i q^{59} +(10.3485 + 10.3485i) q^{63} +(11.8990 + 11.8990i) q^{73} +(4.77526 + 7.22474i) q^{75} +(-5.34847 + 5.34847i) q^{77} -14.6969i q^{79} -9.00000 q^{81} +(4.00000 + 4.00000i) q^{83} +(-6.55051 + 6.55051i) q^{87} +(6.00000 + 6.00000i) q^{93} +(8.79796 - 8.79796i) q^{97} -4.65153i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} + 4 q^{7} - 16 q^{11} - 12 q^{15} - 24 q^{21} - 12 q^{33} - 8 q^{35} + 12 q^{45} + 28 q^{55} + 12 q^{63} + 28 q^{73} + 24 q^{75} + 8 q^{77} - 36 q^{81} + 16 q^{83} - 36 q^{87} + 24 q^{93}+ \cdots - 4 q^{97}+O(q^{100}) \) 4 * q - 4 * q^5 + 4 * q^7 - 16 * q^11 - 12 * q^15 - 24 * q^21 - 12 * q^33 - 8 * q^35 + 12 * q^45 + 28 * q^55 + 12 * q^63 + 28 * q^73 + 24 * q^75 + 8 * q^77 - 36 * q^81 + 16 * q^83 - 36 * q^87 + 24 * q^93 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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