LMFDB - Embedded newform 880.2.b.h.529.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [880,2,Mod(529,880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(880, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("880.529");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 880 = 2^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	880.b (of order $2$, degree $1$, 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:	$7.02683537787$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 55)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			529.4
Root			$-1.18614 - 1.26217i$ of defining polynomial
Character	$\chi$	$=$	880.529
Dual form			880.2.b.h.529.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.52434i q^{3} +(-2.18614 - 0.469882i) q^{5} -3.46410i q^{7} -3.37228 q^{9} +1.00000 q^{11} +(1.18614 - 5.51856i) q^{15} -5.04868i q^{17} +4.00000 q^{19} +8.74456 q^{21} -2.52434i q^{23} +(4.55842 + 2.05446i) q^{25} -0.939764i q^{27} +2.74456 q^{29} +2.37228 q^{31} +2.52434i q^{33} +(-1.62772 + 7.57301i) q^{35} -11.0371i q^{37} -2.74456 q^{41} +3.46410i q^{43} +(7.37228 + 1.58457i) q^{45} -6.63325i q^{47} -5.00000 q^{49} +12.7446 q^{51} +3.16915i q^{53} +(-2.18614 - 0.469882i) q^{55} +10.0974i q^{57} -1.62772 q^{59} +10.7446 q^{61} +11.6819i q^{63} +0.644810i q^{67} +6.37228 q^{69} -7.11684 q^{71} +6.92820i q^{73} +(-5.18614 + 11.5070i) q^{75} -3.46410i q^{77} +12.7446 q^{79} -7.74456 q^{81} -6.63325i q^{83} +(-2.37228 + 11.0371i) q^{85} +6.92820i q^{87} +4.37228 q^{89} +5.98844i q^{93} +(-8.74456 - 1.87953i) q^{95} -4.10891i q^{97} -3.37228 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{5} - 2 q^{9} + 4 q^{11} - q^{15} + 16 q^{19} + 12 q^{21} + q^{25} - 12 q^{29} - 2 q^{31} - 18 q^{35} + 12 q^{41} + 18 q^{45} - 20 q^{49} + 28 q^{51} - 3 q^{55} - 18 q^{59} + 20 q^{61} + 14 q^{69}+ \cdots - 2 q^{99}+O(q^{100}) $ 4 * q - 3 * q^5 - 2 * q^9 + 4 * q^11 - q^15 + 16 * q^19 + 12 * q^21 + q^25 - 12 * q^29 - 2 * q^31 - 18 * q^35 + 12 * q^41 + 18 * q^45 - 20 * q^49 + 28 * q^51 - 3 * q^55 - 18 * q^59 + 20 * q^61 + 14 * q^69 + 6 * q^71 - 15 * q^75 + 28 * q^79 - 8 * q^81 + 2 * q^85 + 6 * q^89 - 12 * q^95 - 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/880\mathbb{Z}\right)^\times$.

$n$	$111$	$177$	$321$	$661$
$\chi(n)$	$1$	$-1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$			2.52434i			1.45743i	0.684819	+	0.728714i	$0.259881\pi$
$3$			2.52434i			1.45743i	−0.684819	+	0.728714i	$0.740119\pi$
$4$		0			0
$5$	−2.18614	−	0.469882i	−0.977672	−	0.210138i
$5$	−2.18614	−	0.469882i	−0.977672	−	0.210138i
$6$		0			0
$7$		−	3.46410i		−	1.30931i	−0.755929	−	0.654654i	$-0.772814\pi$
$7$		−	3.46410i		−	1.30931i	0.755929	−	0.654654i	$-0.227186\pi$
$8$		0			0
$9$	−3.37228			−1.12409
$10$		0			0
$11$	1.00000			0.301511
$11$	1.00000			0.301511
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$	1.18614	−	5.51856i	0.306260	−	1.42489i
$16$		0			0
$17$		−	5.04868i		−	1.22448i	−0.790671	−	0.612242i	$-0.790268\pi$
$17$		−	5.04868i		−	1.22448i	0.790671	−	0.612242i	$-0.209732\pi$
$18$		0			0
$19$	4.00000			0.917663			0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000			0.917663			0.458831	+	0.888523i	$0.348268\pi$
$20$		0			0
$21$	8.74456			1.90822
$22$		0			0
$23$		−	2.52434i		−	0.526361i	−0.964747	−	0.263180i	$-0.915229\pi$
$23$		−	2.52434i		−	0.526361i	0.964747	−	0.263180i	$-0.0847714\pi$
$24$		0			0
$25$	4.55842	+	2.05446i	0.911684	+	0.410891i
$26$		0			0
$27$		−	0.939764i		−	0.180858i
$28$		0			0
$29$	2.74456			0.509652			0.254826	−	0.966987i	$-0.417982\pi$
$29$	2.74456			0.509652			0.254826	+	0.966987i	$0.417982\pi$
$30$		0			0
$31$	2.37228			0.426074			0.213037	−	0.977044i	$-0.431664\pi$
$31$	2.37228			0.426074			0.213037	+	0.977044i	$0.431664\pi$
$32$		0			0
$33$			2.52434i			0.439431i
$34$		0			0
$35$	−1.62772	+	7.57301i	−0.275135	+	1.28007i
$36$		0			0
$37$		−	11.0371i		−	1.81449i	−0.420602	−	0.907245i	$-0.638181\pi$
$37$		−	11.0371i		−	1.81449i	0.420602	−	0.907245i	$-0.361819\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.74456			−0.428629			−0.214314	−	0.976765i	$-0.568752\pi$
$41$	−2.74456			−0.428629			−0.214314	+	0.976765i	$0.568752\pi$
$42$		0			0
$43$			3.46410i			0.528271i	0.964486	+	0.264135i	$0.0850865\pi$
$43$			3.46410i			0.528271i	−0.964486	+	0.264135i	$0.914913\pi$
$44$		0			0
$45$	7.37228	+	1.58457i	1.09899	+	0.236214i
$46$		0			0
$47$		−	6.63325i		−	0.967559i	−0.875190	−	0.483779i	$-0.839264\pi$
$47$		−	6.63325i		−	0.967559i	0.875190	−	0.483779i	$-0.160736\pi$
$48$		0			0
$49$	−5.00000			−0.714286
$50$		0			0
$51$	12.7446			1.78460
$52$		0			0
$53$			3.16915i			0.435316i	0.976025	+	0.217658i	$0.0698417\pi$
$53$			3.16915i			0.435316i	−0.976025	+	0.217658i	$0.930158\pi$
$54$		0			0
$55$	−2.18614	−	0.469882i	−0.294779	−	0.0633589i
$56$		0			0
$57$			10.0974i			1.33743i
$58$		0			0
$59$	−1.62772			−0.211911			−0.105955	−	0.994371i	$-0.533790\pi$
$59$	−1.62772			−0.211911			−0.105955	+	0.994371i	$0.533790\pi$
$60$		0			0
$61$	10.7446			1.37570			0.687850	−	0.725853i	$-0.258555\pi$
$61$	10.7446			1.37570			0.687850	+	0.725853i	$0.258555\pi$
$62$		0			0
$63$			11.6819i			1.47178i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			0.644810i			0.0787761i	0.999224	+	0.0393880i	$0.0125408\pi$
$67$			0.644810i			0.0787761i	−0.999224	+	0.0393880i	$0.987459\pi$
$68$		0			0
$69$	6.37228			0.767133
$70$		0			0
$71$	−7.11684			−0.844614			−0.422307	−	0.906453i	$-0.638780\pi$
$71$	−7.11684			−0.844614			−0.422307	+	0.906453i	$0.638780\pi$
$72$		0			0
$73$			6.92820i			0.810885i	0.914121	+	0.405442i	$0.132883\pi$
$73$			6.92820i			0.810885i	−0.914121	+	0.405442i	$0.867117\pi$
$74$		0			0
$75$	−5.18614	+	11.5070i	−0.598844	+	1.32871i
$76$		0			0
$77$		−	3.46410i		−	0.394771i
$78$		0			0
$79$	12.7446			1.43388			0.716938	−	0.697137i	$-0.245543\pi$
$79$	12.7446			1.43388			0.716938	+	0.697137i	$0.245543\pi$
$80$		0			0
$81$	−7.74456			−0.860507
$82$		0			0
$83$		−	6.63325i		−	0.728094i	−0.931381	−	0.364047i	$-0.881395\pi$
$83$		−	6.63325i		−	0.728094i	0.931381	−	0.364047i	$-0.118605\pi$
$84$		0			0
$85$	−2.37228	+	11.0371i	−0.257310	+	1.19714i
$86$		0			0
$87$			6.92820i			0.742781i
$88$		0			0
$89$	4.37228			0.463461			0.231730	−	0.972780i	$-0.425561\pi$
$89$	4.37228			0.463461			0.231730	+	0.972780i	$0.425561\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$			5.98844i			0.620972i
$94$		0			0
$95$	−8.74456	−	1.87953i	−0.897173	−	0.192835i
$96$		0			0
$97$		−	4.10891i		−	0.417197i	−0.978001	−	0.208598i	$-0.933110\pi$
$97$		−	4.10891i		−	0.417197i	0.978001	−	0.208598i	$-0.0668902\pi$
$98$		0			0
$99$	−3.37228			−0.338927
$100$		0			0
$101$	6.00000			0.597022			0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000			0.597022			0.298511	+	0.954406i	$0.403510\pi$
$102$		0			0
$103$		−	10.3923i		−	1.02398i	−0.858990	−	0.511992i	$-0.828908\pi$
$103$		−	10.3923i		−	1.02398i	0.858990	−	0.511992i	$-0.171092\pi$
$104$		0			0
$105$	−19.1168	−	4.10891i	−1.86561	−	0.400989i
$106$		0			0
$107$		−	6.63325i		−	0.641260i	−0.947204	−	0.320630i	$-0.896105\pi$
$107$		−	6.63325i		−	0.641260i	0.947204	−	0.320630i	$-0.103895\pi$
$108$		0			0
$109$	−10.0000			−0.957826			−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000			−0.957826			−0.478913	+	0.877862i	$0.658969\pi$
$110$		0			0
$111$	27.8614			2.64449
$112$		0			0
$113$		−	16.0858i		−	1.51322i	−0.653864	−	0.756612i	$-0.726853\pi$
$113$		−	16.0858i		−	1.51322i	0.653864	−	0.756612i	$-0.273147\pi$
$114$		0			0
$115$	−1.18614	+	5.51856i	−0.110608	+	0.514608i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−17.4891			−1.60323
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$		−	6.92820i		−	0.624695i
$124$		0			0
$125$	−9.00000	−	6.63325i	−0.804984	−	0.593296i
$126$		0			0
$127$		−	11.6819i		−	1.03660i	−0.855198	−	0.518302i	$-0.826564\pi$
$127$		−	11.6819i		−	1.03660i	0.855198	−	0.518302i	$-0.173436\pi$
$128$		0			0
$129$	−8.74456			−0.769916
$130$		0			0
$131$	−8.74456			−0.764016			−0.382008	−	0.924159i	$-0.624767\pi$
$131$	−8.74456			−0.764016			−0.382008	+	0.924159i	$0.624767\pi$
$132$		0			0
$133$		−	13.8564i		−	1.20150i
$134$		0			0
$135$	−0.441578	+	2.05446i	−0.0380050	+	0.176819i
$136$		0			0
$137$		−	2.22938i		−	0.190469i	−0.995455	−	0.0952346i	$-0.969640\pi$
$137$		−	2.22938i		−	0.190469i	0.995455	−	0.0952346i	$-0.0303601\pi$
$138$		0			0
$139$	18.2337			1.54656			0.773281	−	0.634064i	$-0.218614\pi$
$139$	18.2337			1.54656			0.773281	+	0.634064i	$0.218614\pi$
$140$		0			0
$141$	16.7446			1.41015
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−6.00000	−	1.28962i	−0.498273	−	0.107097i
$146$		0			0
$147$		−	12.6217i		−	1.04102i
$148$		0			0
$149$	−11.4891			−0.941226			−0.470613	−	0.882340i	$-0.655967\pi$
$149$	−11.4891			−0.941226			−0.470613	+	0.882340i	$0.655967\pi$
$150$		0			0
$151$	−22.2337			−1.80935			−0.904676	−	0.426100i	$-0.859887\pi$
$151$	−22.2337			−1.80935			−0.904676	+	0.426100i	$0.859887\pi$
$152$		0			0
$153$			17.0256i			1.37643i
$154$		0			0
$155$	−5.18614	−	1.11469i	−0.416561	−	0.0895342i
$156$		0			0
$157$			5.39853i			0.430850i	0.976520	+	0.215425i	$0.0691136\pi$
$157$			5.39853i			0.430850i	−0.976520	+	0.215425i	$0.930886\pi$
$158$		0			0
$159$	−8.00000			−0.634441
$160$		0			0
$161$	−8.74456			−0.689168
$162$		0			0
$163$		−	3.46410i		−	0.271329i	−0.990755	−	0.135665i	$-0.956683\pi$
$163$		−	3.46410i		−	0.271329i	0.990755	−	0.135665i	$-0.0433170\pi$
$164$		0			0
$165$	1.18614	−	5.51856i	0.0923409	−	0.429619i
$166$		0			0
$167$			22.3692i			1.73098i	0.500927	+	0.865490i	$0.332993\pi$
$167$			22.3692i			1.73098i	−0.500927	+	0.865490i	$0.667007\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$	−13.4891			−1.03154
$172$		0			0
$173$			1.87953i			0.142898i	0.997444	+	0.0714489i	$0.0227623\pi$
$173$			1.87953i			0.142898i	−0.997444	+	0.0714489i	$0.977238\pi$
$174$		0			0
$175$	7.11684	−	15.7908i	0.537983	−	1.19368i
$176$		0			0
$177$		−	4.10891i		−	0.308845i
$178$		0			0
$179$	−15.8614			−1.18554			−0.592769	−	0.805373i	$-0.701965\pi$
$179$	−15.8614			−1.18554			−0.592769	+	0.805373i	$0.701965\pi$
$180$		0			0
$181$	6.88316			0.511621			0.255810	−	0.966727i	$-0.417658\pi$
$181$	6.88316			0.511621			0.255810	+	0.966727i	$0.417658\pi$
$182$		0			0
$183$			27.1229i			2.00498i
$184$		0			0
$185$	−5.18614	+	24.1287i	−0.381293	+	1.77398i
$186$		0			0
$187$		−	5.04868i		−	0.369196i
$188$		0			0
$189$	−3.25544			−0.236798
$190$		0			0
$191$	13.6277			0.986067			0.493034	−	0.870010i	$-0.335888\pi$
$191$	13.6277			0.986067			0.493034	+	0.870010i	$0.335888\pi$
$192$		0			0
$193$		−	23.3639i		−	1.68177i	−0.541216	−	0.840883i	$-0.682036\pi$
$193$		−	23.3639i		−	1.68177i	0.541216	−	0.840883i	$-0.317964\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			1.87953i			0.133911i	0.997756	+	0.0669554i	$0.0213285\pi$
$197$			1.87953i			0.133911i	−0.997756	+	0.0669554i	$0.978671\pi$
$198$		0			0
$199$	−8.00000			−0.567105			−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000			−0.567105			−0.283552	+	0.958957i	$0.591513\pi$
$200$		0			0
$201$	−1.62772			−0.114810
$202$		0			0
$203$		−	9.50744i		−	0.667292i
$204$		0			0
$205$	6.00000	+	1.28962i	0.419058	+	0.0900710i
$206$		0			0
$207$			8.51278i			0.591679i
$208$		0			0
$209$	4.00000			0.276686
$210$		0			0
$211$	21.4891			1.47937			0.739686	−	0.672952i	$-0.234974\pi$
$211$	21.4891			1.47937			0.739686	+	0.672952i	$0.234974\pi$
$212$		0			0
$213$		−	17.9653i		−	1.23096i
$214$		0			0
$215$	1.62772	−	7.57301i	0.111009	−	0.516475i
$216$		0			0
$217$		−	8.21782i		−	0.557862i
$218$		0			0
$219$	−17.4891			−1.18181
$220$		0			0
$221$		0			0
$222$		0			0
$223$			7.57301i			0.507126i	0.967319	+	0.253563i	$0.0816026\pi$
$223$			7.57301i			0.507126i	−0.967319	+	0.253563i	$0.918397\pi$
$224$		0			0
$225$	−15.3723	−	6.92820i	−1.02482	−	0.461880i
$226$		0			0
$227$			9.80240i			0.650608i	0.945609	+	0.325304i	$0.105467\pi$
$227$			9.80240i			0.650608i	−0.945609	+	0.325304i	$0.894533\pi$
$228$		0			0
$229$	−20.3723			−1.34624			−0.673119	−	0.739534i	$-0.735046\pi$
$229$	−20.3723			−1.34624			−0.673119	+	0.739534i	$0.735046\pi$
$230$		0			0
$231$	8.74456			0.575350
$232$		0			0
$233$			17.0256i			1.11538i	0.830049	+	0.557691i	$0.188312\pi$
$233$			17.0256i			1.11538i	−0.830049	+	0.557691i	$0.811688\pi$
$234$		0			0
$235$	−3.11684	+	14.5012i	−0.203320	+	0.945955i
$236$		0			0
$237$			32.1716i			2.08977i
$238$		0			0
$239$	−3.25544			−0.210577			−0.105288	−	0.994442i	$-0.533577\pi$
$239$	−3.25544			−0.210577			−0.105288	+	0.994442i	$0.533577\pi$
$240$		0			0
$241$	5.25544			0.338532			0.169266	−	0.985570i	$-0.445860\pi$
$241$	5.25544			0.338532			0.169266	+	0.985570i	$0.445860\pi$
$242$		0			0
$243$		−	22.3692i		−	1.43498i
$244$		0			0
$245$	10.9307	+	2.34941i	0.698337	+	0.150098i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	16.7446			1.06114
$250$		0			0
$251$	4.88316			0.308222			0.154111	−	0.988054i	$-0.450749\pi$
$251$	4.88316			0.308222			0.154111	+	0.988054i	$0.450749\pi$
$252$		0			0
$253$		−	2.52434i		−	0.158704i
$254$		0			0
$255$	−27.8614	−	5.98844i	−1.74475	−	0.375011i
$256$		0			0
$257$			23.9538i			1.49419i	0.664715	+	0.747097i	$0.268553\pi$
$257$			23.9538i			1.49419i	−0.664715	+	0.747097i	$0.731447\pi$
$258$		0			0
$259$	−38.2337			−2.37573
$260$		0			0
$261$	−9.25544			−0.572897
$262$		0			0
$263$			14.1514i			0.872610i	0.899799	+	0.436305i	$0.143713\pi$
$263$			14.1514i			0.872610i	−0.899799	+	0.436305i	$0.856287\pi$
$264$		0			0
$265$	1.48913	−	6.92820i	0.0914762	−	0.425596i
$266$		0			0
$267$			11.0371i			0.675460i
$268$		0			0
$269$	−11.4891			−0.700504			−0.350252	−	0.936655i	$-0.613904\pi$
$269$	−11.4891			−0.700504			−0.350252	+	0.936655i	$0.613904\pi$
$270$		0			0
$271$	9.48913			0.576423			0.288212	−	0.957567i	$-0.406939\pi$
$271$	9.48913			0.576423			0.288212	+	0.957567i	$0.406939\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.55842	+	2.05446i	0.274883	+	0.123888i
$276$		0			0
$277$			8.21782i			0.493761i	0.969046	+	0.246881i	$0.0794055\pi$
$277$			8.21782i			0.493761i	−0.969046	+	0.246881i	$0.920594\pi$
$278$		0			0
$279$	−8.00000			−0.478947
$280$		0			0
$281$	−23.4891			−1.40124			−0.700622	−	0.713533i	$-0.747094\pi$
$281$	−23.4891			−1.40124			−0.700622	+	0.713533i	$0.747094\pi$
$282$		0			0
$283$		−	4.75372i		−	0.282579i	−0.989968	−	0.141290i	$-0.954875\pi$
$283$		−	4.75372i		−	0.282579i	0.989968	−	0.141290i	$-0.0451249\pi$
$284$		0			0
$285$	4.74456	−	22.0742i	0.281044	−	1.30756i
$286$		0			0
$287$			9.50744i			0.561207i
$288$		0			0
$289$	−8.48913			−0.499360
$290$		0			0
$291$	10.3723			0.608034
$292$		0			0
$293$			10.0974i			0.589894i	0.955514	+	0.294947i	$0.0953019\pi$
$293$			10.0974i			0.589894i	−0.955514	+	0.294947i	$0.904698\pi$
$294$		0			0
$295$	3.55842	+	0.764836i	0.207179	+	0.0445304i
$296$		0			0
$297$		−	0.939764i		−	0.0545306i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	12.0000			0.691669
$302$		0			0
$303$			15.1460i			0.870117i
$304$		0			0
$305$	−23.4891	−	5.04868i	−1.34498	−	0.289086i
$306$		0			0
$307$			28.1176i			1.60475i	0.596817	+	0.802377i	$0.296432\pi$
$307$			28.1176i			1.60475i	−0.596817	+	0.802377i	$0.703568\pi$
$308$		0			0
$309$	26.2337			1.49238
$310$		0			0
$311$	17.4891			0.991717			0.495859	−	0.868403i	$-0.334853\pi$
$311$	17.4891			0.991717			0.495859	+	0.868403i	$0.334853\pi$
$312$		0			0
$313$			31.8217i			1.79867i	0.437260	+	0.899335i	$0.355949\pi$
$313$			31.8217i			1.79867i	−0.437260	+	0.899335i	$0.644051\pi$
$314$		0			0
$315$	5.48913	−	25.5383i	0.309277	−	1.43892i
$316$		0			0
$317$		−	3.51900i		−	0.197647i	−0.995105	−	0.0988235i	$-0.968492\pi$
$317$		−	3.51900i		−	0.197647i	0.995105	−	0.0988235i	$-0.0315079\pi$
$318$		0			0
$319$	2.74456			0.153666
$320$		0			0
$321$	16.7446			0.934590
$322$		0			0
$323$		−	20.1947i		−	1.12366i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		−	25.2434i		−	1.39596i
$328$		0			0
$329$	−22.9783			−1.26683
$330$		0			0
$331$	−3.11684			−0.171317			−0.0856586	−	0.996325i	$-0.527299\pi$
$331$	−3.11684			−0.171317			−0.0856586	+	0.996325i	$0.527299\pi$
$332$		0			0
$333$			37.2203i			2.03966i
$334$		0			0
$335$	0.302985	−	1.40965i	0.0165538	−	0.0770172i
$336$		0			0
$337$		−	12.5668i		−	0.684556i	−0.939599	−	0.342278i	$-0.888801\pi$
$337$		−	12.5668i		−	0.684556i	0.939599	−	0.342278i	$-0.111199\pi$
$338$		0			0
$339$	40.6060			2.20541
$340$		0			0
$341$	2.37228			0.128466
$342$		0			0
$343$		−	6.92820i		−	0.374088i
$344$		0			0
$345$	−13.9307	−	2.99422i	−0.750004	−	0.161203i
$346$		0			0
$347$			29.2974i			1.57277i	0.617739	+	0.786383i	$0.288049\pi$
$347$			29.2974i			1.57277i	−0.617739	+	0.786383i	$0.711951\pi$
$348$		0			0
$349$	7.48913			0.400884			0.200442	−	0.979706i	$-0.435762\pi$
$349$	7.48913			0.400884			0.200442	+	0.979706i	$0.435762\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	21.7244i		−	1.15627i	−0.815941	−	0.578136i	$-0.803780\pi$
$353$		−	21.7244i		−	1.15627i	0.815941	−	0.578136i	$-0.196220\pi$
$354$		0			0
$355$	15.5584	+	3.34408i	0.825755	+	0.177485i
$356$		0			0
$357$		−	44.1485i		−	2.33658i
$358$		0			0
$359$	6.51087			0.343631			0.171815	−	0.985129i	$-0.445037\pi$
$359$	6.51087			0.343631			0.171815	+	0.985129i	$0.445037\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$			2.52434i			0.132493i
$364$		0			0
$365$	3.25544	−	15.1460i	0.170397	−	0.792779i
$366$		0			0
$367$			24.0087i			1.25324i	0.779324	+	0.626621i	$0.215563\pi$
$367$			24.0087i			1.25324i	−0.779324	+	0.626621i	$0.784437\pi$
$368$		0			0
$369$	9.25544			0.481819
$370$		0			0
$371$	10.9783			0.569962
$372$		0			0
$373$		−	8.21782i		−	0.425503i	−0.977106	−	0.212751i	$-0.931758\pi$
$373$		−	8.21782i		−	0.425503i	0.977106	−	0.212751i	$-0.0682424\pi$
$374$		0			0
$375$	16.7446	−	22.7190i	0.864685	−	1.17321i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−6.37228			−0.327322			−0.163661	−	0.986517i	$-0.552330\pi$
$379$	−6.37228			−0.327322			−0.163661	+	0.986517i	$0.552330\pi$
$380$		0			0
$381$	29.4891			1.51077
$382$		0			0
$383$			5.69349i			0.290924i	0.989364	+	0.145462i	$0.0464668\pi$
$383$			5.69349i			0.290924i	−0.989364	+	0.145462i	$0.953533\pi$
$384$		0			0
$385$	−1.62772	+	7.57301i	−0.0829562	+	0.385957i
$386$		0			0
$387$		−	11.6819i		−	0.593826i
$388$		0			0
$389$	9.86141			0.499993			0.249997	−	0.968247i	$-0.419571\pi$
$389$	9.86141			0.499993			0.249997	+	0.968247i	$0.419571\pi$
$390$		0			0
$391$	−12.7446			−0.644520
$392$		0			0
$393$		−	22.0742i		−	1.11350i
$394$		0			0
$395$	−27.8614	−	5.98844i	−1.40186	−	0.301311i
$396$		0			0
$397$			23.3639i			1.17260i	0.810095	+	0.586299i	$0.199416\pi$
$397$			23.3639i			1.17260i	−0.810095	+	0.586299i	$0.800584\pi$
$398$		0			0
$399$	34.9783			1.75110
$400$		0			0
$401$	11.4891			0.573740			0.286870	−	0.957970i	$-0.407385\pi$
$401$	11.4891			0.573740			0.286870	+	0.957970i	$0.407385\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	16.9307	+	3.63903i	0.841293	+	0.180825i
$406$		0			0
$407$		−	11.0371i		−	0.547089i
$408$		0			0
$409$	−4.51087			−0.223048			−0.111524	−	0.993762i	$-0.535573\pi$
$409$	−4.51087			−0.223048			−0.111524	+	0.993762i	$0.535573\pi$
$410$		0			0
$411$	5.62772			0.277595
$412$		0			0
$413$			5.63858i			0.277457i
$414$		0			0
$415$	−3.11684	+	14.5012i	−0.153000	+	0.711837i
$416$		0			0
$417$			46.0280i			2.25400i
$418$		0			0
$419$	22.9783			1.12256			0.561280	−	0.827626i	$-0.310309\pi$
$419$	22.9783			1.12256			0.561280	+	0.827626i	$0.310309\pi$
$420$		0			0
$421$	31.4891			1.53469			0.767343	−	0.641237i	$-0.221578\pi$
$421$	31.4891			1.53469			0.767343	+	0.641237i	$0.221578\pi$
$422$		0			0
$423$			22.3692i			1.08763i
$424$		0			0
$425$	10.3723	−	23.0140i	0.503130	−	1.11634i
$426$		0			0
$427$		−	37.2203i		−	1.80121i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−31.7228			−1.52803			−0.764017	−	0.645196i	$-0.776776\pi$
$431$	−31.7228			−1.52803			−0.764017	+	0.645196i	$0.776776\pi$
$432$		0			0
$433$			20.5446i			0.987308i	0.869658	+	0.493654i	$0.164339\pi$
$433$			20.5446i			0.987308i	−0.869658	+	0.493654i	$0.835661\pi$
$434$		0			0
$435$	3.25544	−	15.1460i	0.156086	−	0.726196i
$436$		0			0
$437$		−	10.0974i		−	0.483022i
$438$		0			0
$439$	−1.48913			−0.0710721			−0.0355360	−	0.999368i	$-0.511314\pi$
$439$	−1.48913			−0.0710721			−0.0355360	+	0.999368i	$0.511314\pi$
$440$		0			0
$441$	16.8614			0.802924
$442$		0			0
$443$		−	15.0911i		−	0.717001i	−0.933530	−	0.358500i	$-0.883288\pi$
$443$		−	15.0911i		−	0.717001i	0.933530	−	0.358500i	$-0.116712\pi$
$444$		0			0
$445$	−9.55842	−	2.05446i	−0.453113	−	0.0973905i
$446$		0			0
$447$		−	29.0024i		−	1.37177i
$448$		0			0
$449$	21.8614			1.03170			0.515852	−	0.856678i	$-0.327476\pi$
$449$	21.8614			1.03170			0.515852	+	0.856678i	$0.327476\pi$
$450$		0			0
$451$	−2.74456			−0.129236
$452$		0			0
$453$		−	56.1253i		−	2.63700i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	20.7846i		−	0.972263i	−0.873886	−	0.486132i	$-0.838408\pi$
$457$		−	20.7846i		−	0.972263i	0.873886	−	0.486132i	$-0.161592\pi$
$458$		0			0
$459$	−4.74456			−0.221457
$460$		0			0
$461$	−32.2337			−1.50127			−0.750636	−	0.660716i	$-0.770253\pi$
$461$	−32.2337			−1.50127			−0.750636	+	0.660716i	$0.770253\pi$
$462$		0			0
$463$		−	20.1398i		−	0.935976i	−0.883735	−	0.467988i	$-0.844979\pi$
$463$		−	20.1398i		−	0.935976i	0.883735	−	0.467988i	$-0.155021\pi$
$464$		0			0
$465$	2.81386	−	13.0916i	0.130490	−	0.607107i
$466$		0			0
$467$			4.40387i			0.203787i	0.994795	+	0.101893i	$0.0324900\pi$
$467$			4.40387i			0.203787i	−0.994795	+	0.101893i	$0.967510\pi$
$468$		0			0
$469$	2.23369			0.103142
$470$		0			0
$471$	−13.6277			−0.627932
$472$		0			0
$473$			3.46410i			0.159280i
$474$		0			0
$475$	18.2337	+	8.21782i	0.836619	+	0.377060i
$476$		0			0
$477$		−	10.6873i		−	0.489336i
$478$		0			0
$479$	17.4891			0.799099			0.399549	−	0.916712i	$-0.369167\pi$
$479$	17.4891			0.799099			0.399549	+	0.916712i	$0.369167\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		−	22.0742i		−	1.00441i
$484$		0			0
$485$	−1.93070	+	8.98266i	−0.0876687	+	0.407882i
$486$		0			0
$487$		−	22.7190i		−	1.02950i	−0.857341	−	0.514749i	$-0.827885\pi$
$487$		−	22.7190i		−	1.02950i	0.857341	−	0.514749i	$-0.172115\pi$
$488$		0			0
$489$	8.74456			0.395443
$490$		0			0
$491$	−29.4891			−1.33083			−0.665413	−	0.746476i	$-0.731744\pi$
$491$	−29.4891			−1.33083			−0.665413	+	0.746476i	$0.731744\pi$
$492$		0			0
$493$		−	13.8564i		−	0.624061i
$494$		0			0
$495$	7.37228	+	1.58457i	0.331359	+	0.0712213i
$496$		0			0
$497$			24.6535i			1.10586i
$498$		0			0
$499$	−20.0000			−0.895323			−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000			−0.895323			−0.447661	+	0.894203i	$0.647743\pi$
$500$		0			0
$501$	−56.4674			−2.52278
$502$		0			0
$503$			0.294954i			0.0131513i	0.999978	+	0.00657567i	$0.00209311\pi$
$503$			0.294954i			0.0131513i	−0.999978	+	0.00657567i	$0.997907\pi$
$504$		0			0
$505$	−13.1168	−	2.81929i	−0.583692	−	0.125457i
$506$		0			0
$507$			32.8164i			1.45743i
$508$		0			0
$509$	−28.3723			−1.25758			−0.628790	−	0.777575i	$-0.716449\pi$
$509$	−28.3723			−1.25758			−0.628790	+	0.777575i	$0.716449\pi$
$510$		0			0
$511$	24.0000			1.06170
$512$		0			0
$513$		−	3.75906i		−	0.165966i
$514$		0			0
$515$	−4.88316	+	22.7190i	−0.215178	+	1.00112i
$516$		0			0
$517$		−	6.63325i		−	0.291730i
$518$		0			0
$519$	−4.74456			−0.208263
$520$		0			0
$521$	18.6060			0.815142			0.407571	−	0.913173i	$-0.366376\pi$
$521$	18.6060			0.815142			0.407571	+	0.913173i	$0.366376\pi$
$522$		0			0
$523$			9.10268i			0.398033i	0.979996	+	0.199016i	$0.0637747\pi$
$523$			9.10268i			0.398033i	−0.979996	+	0.199016i	$0.936225\pi$
$524$		0			0
$525$	39.8614	+	17.9653i	1.73969	+	0.784071i
$526$		0			0
$527$		−	11.9769i		−	0.521721i
$528$		0			0
$529$	16.6277			0.722944
$530$		0			0
$531$	5.48913			0.238208
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−3.11684	+	14.5012i	−0.134753	+	0.626942i
$536$		0			0
$537$		−	40.0395i		−	1.72783i
$538$		0			0
$539$	−5.00000			−0.215365
$540$		0			0
$541$	−0.233688			−0.0100470			−0.00502351	−	0.999987i	$-0.501599\pi$
$541$	−0.233688			−0.0100470			−0.00502351	+	0.999987i	$0.501599\pi$
$542$		0			0
$543$			17.3754i			0.745650i
$544$		0			0
$545$	21.8614	+	4.69882i	0.936440	+	0.201275i
$546$		0			0
$547$			9.10268i			0.389203i	0.980882	+	0.194601i	$0.0623413\pi$
$547$			9.10268i			0.389203i	−0.980882	+	0.194601i	$0.937659\pi$
$548$		0			0
$549$	−36.2337			−1.54642
$550$		0			0
$551$	10.9783			0.467689
$552$		0			0
$553$		−	44.1485i		−	1.87738i
$554$		0			0
$555$	−60.9090	−	13.0916i	−2.58544	−	0.555706i
$556$		0			0
$557$			32.1716i			1.36315i	0.731747	+	0.681577i	$0.238705\pi$
$557$			32.1716i			1.36315i	−0.731747	+	0.681577i	$0.761295\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	12.7446			0.538076
$562$		0			0
$563$		−	12.2718i		−	0.517196i	−0.965985	−	0.258598i	$-0.916739\pi$
$563$		−	12.2718i		−	0.517196i	0.965985	−	0.258598i	$-0.0832605\pi$
$564$		0			0
$565$	−7.55842	+	35.1658i	−0.317985	+	1.47944i
$566$		0			0
$567$			26.8280i			1.12667i
$568$		0			0
$569$	38.7446			1.62426			0.812128	−	0.583479i	$-0.198309\pi$
$569$	38.7446			1.62426			0.812128	+	0.583479i	$0.198309\pi$
$570$		0			0
$571$	21.4891			0.899292			0.449646	−	0.893207i	$-0.351550\pi$
$571$	21.4891			0.899292			0.449646	+	0.893207i	$0.351550\pi$
$572$		0			0
$573$			34.4010i			1.43712i
$574$		0			0
$575$	5.18614	−	11.5070i	0.216277	−	0.479875i
$576$		0			0
$577$			31.8217i			1.32476i	0.749170	+	0.662378i	$0.230453\pi$
$577$			31.8217i			1.32476i	−0.749170	+	0.662378i	$0.769547\pi$
$578$		0			0
$579$	58.9783			2.45105
$580$		0			0
$581$	−22.9783			−0.953298
$582$		0			0
$583$			3.16915i			0.131253i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			28.0078i			1.15600i	0.816035	+	0.578002i	$0.196167\pi$
$587$			28.0078i			1.15600i	−0.816035	+	0.578002i	$0.803833\pi$
$588$		0			0
$589$	9.48913			0.390993
$590$		0			0
$591$	−4.74456			−0.195165
$592$		0			0
$593$		−	43.5586i		−	1.78874i	−0.447333	−	0.894368i	$-0.647626\pi$
$593$		−	43.5586i		−	1.78874i	0.447333	−	0.894368i	$-0.352374\pi$
$594$		0			0
$595$	38.2337	+	8.21782i	1.56743	+	0.336898i
$596$		0			0
$597$		−	20.1947i		−	0.826514i
$598$		0			0
$599$	−34.9783			−1.42917			−0.714586	−	0.699547i	$-0.753385\pi$
$599$	−34.9783			−1.42917			−0.714586	+	0.699547i	$0.753385\pi$
$600$		0			0
$601$	30.4674			1.24279			0.621395	−	0.783497i	$-0.286566\pi$
$601$	30.4674			1.24279			0.621395	+	0.783497i	$0.286566\pi$
$602$		0			0
$603$		−	2.17448i		−	0.0885517i
$604$		0			0
$605$	−2.18614	−	0.469882i	−0.0888793	−	0.0191034i
$606$		0			0
$607$		−	3.46410i		−	0.140604i	−0.997526	−	0.0703018i	$-0.977604\pi$
$607$		−	3.46410i		−	0.140604i	0.997526	−	0.0703018i	$-0.0223962\pi$
$608$		0			0
$609$	24.0000			0.972529
$610$		0			0
$611$		0			0
$612$		0			0
$613$			44.1485i			1.78314i	0.452883	+	0.891570i	$0.350395\pi$
$613$			44.1485i			1.78314i	−0.452883	+	0.891570i	$0.649605\pi$
$614$		0			0
$615$	−3.25544	+	15.1460i	−0.131272	+	0.610747i
$616$		0			0
$617$		−	3.75906i		−	0.151334i	−0.997133	−	0.0756669i	$-0.975891\pi$
$617$		−	3.75906i		−	0.151334i	0.997133	−	0.0756669i	$-0.0241086\pi$
$618$		0			0
$619$	−3.11684			−0.125277			−0.0626383	−	0.998036i	$-0.519951\pi$
$619$	−3.11684			−0.125277			−0.0626383	+	0.998036i	$0.519951\pi$
$620$		0			0
$621$	−2.37228			−0.0951964
$622$		0			0
$623$		−	15.1460i		−	0.606813i
$624$		0			0
$625$	16.5584	+	18.7302i	0.662337	+	0.749206i
$626$		0			0
$627$			10.0974i			0.403249i
$628$		0			0
$629$	−55.7228			−2.22181
$630$		0			0
$631$	16.6060			0.661073			0.330537	−	0.943793i	$-0.392770\pi$
$631$	16.6060			0.661073			0.330537	+	0.943793i	$0.392770\pi$
$632$		0			0
$633$			54.2458i			2.15608i
$634$		0			0
$635$	−5.48913	+	25.5383i	−0.217829	+	1.01346i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	24.0000			0.949425
$640$		0			0
$641$	−19.6277			−0.775248			−0.387624	−	0.921818i	$-0.626704\pi$
$641$	−19.6277			−0.775248			−0.387624	+	0.921818i	$0.626704\pi$
$642$		0			0
$643$			39.1547i			1.54411i	0.635556	+	0.772055i	$0.280771\pi$
$643$			39.1547i			1.54411i	−0.635556	+	0.772055i	$0.719229\pi$
$644$		0			0
$645$	19.1168	+	4.10891i	0.752725	+	0.161788i
$646$		0			0
$647$		−	41.0342i		−	1.61322i	−0.591083	−	0.806611i	$-0.701299\pi$
$647$		−	41.0342i		−	1.61322i	0.591083	−	0.806611i	$-0.298701\pi$
$648$		0			0
$649$	−1.62772			−0.0638935
$650$		0			0
$651$	20.7446			0.813044
$652$		0			0
$653$		−	25.5932i		−	1.00154i	−0.865580	−	0.500770i	$-0.833050\pi$
$653$		−	25.5932i		−	1.00154i	0.865580	−	0.500770i	$-0.166950\pi$
$654$		0			0
$655$	19.1168	+	4.10891i	0.746957	+	0.160548i
$656$		0			0
$657$		−	23.3639i		−	0.911511i
$658$		0			0
$659$	32.7446			1.27555			0.637774	−	0.770224i	$-0.279856\pi$
$659$	32.7446			1.27555			0.637774	+	0.770224i	$0.279856\pi$
$660$		0			0
$661$	35.3505			1.37498			0.687488	−	0.726196i	$-0.258713\pi$
$661$	35.3505			1.37498			0.687488	+	0.726196i	$0.258713\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6.51087	+	30.2921i	−0.252481	+	1.17468i
$666$		0			0
$667$		−	6.92820i		−	0.268261i
$668$		0			0
$669$	−19.1168			−0.739100
$670$		0			0
$671$	10.7446			0.414789
$672$		0			0
$673$			1.28962i			0.0497112i	0.999691	+	0.0248556i	$0.00791260\pi$
$673$			1.28962i			0.0497112i	−0.999691	+	0.0248556i	$0.992087\pi$
$674$		0			0
$675$	1.93070	−	4.28384i	0.0743128	−	0.164885i
$676$		0			0
$677$			43.4487i			1.66987i	0.550348	+	0.834935i	$0.314495\pi$
$677$			43.4487i			1.66987i	−0.550348	+	0.834935i	$0.685505\pi$
$678$		0			0
$679$	−14.2337			−0.546239
$680$		0			0
$681$	−24.7446			−0.948214
$682$		0			0
$683$			44.4434i			1.70058i	0.526314	+	0.850290i	$0.323573\pi$
$683$			44.4434i			1.70058i	−0.526314	+	0.850290i	$0.676427\pi$
$684$		0			0
$685$	−1.04755	+	4.87375i	−0.0400247	+	0.186216i
$686$		0			0
$687$		−	51.4265i		−	1.96204i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−16.1386			−0.613941			−0.306971	−	0.951719i	$-0.599315\pi$
$691$	−16.1386			−0.613941			−0.306971	+	0.951719i	$0.599315\pi$
$692$		0			0
$693$			11.6819i			0.443760i
$694$		0			0
$695$	−39.8614	−	8.56768i	−1.51203	−	0.324991i
$696$		0			0
$697$			13.8564i			0.524849i
$698$		0			0
$699$	−42.9783			−1.62559
$700$		0			0
$701$	−35.4891			−1.34041			−0.670203	−	0.742178i	$-0.733793\pi$
$701$	−35.4891			−1.34041			−0.670203	+	0.742178i	$0.733793\pi$
$702$		0			0
$703$		−	44.1485i		−	1.66509i
$704$		0			0
$705$	−36.6060	−	7.86797i	−1.37866	−	0.296325i
$706$		0			0
$707$		−	20.7846i		−	0.781686i
$708$		0			0
$709$	−41.1168			−1.54418			−0.772088	−	0.635516i	$-0.780787\pi$
$709$	−41.1168			−1.54418			−0.772088	+	0.635516i	$0.780787\pi$
$710$		0			0
$711$	−42.9783			−1.61181
$712$		0			0
$713$		−	5.98844i		−	0.224269i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	8.21782i		−	0.306900i
$718$		0			0
$719$	21.3505			0.796240			0.398120	−	0.917333i	$-0.369663\pi$
$719$	21.3505			0.796240			0.398120	+	0.917333i	$0.369663\pi$
$720$		0			0
$721$	−36.0000			−1.34071
$722$		0			0
$723$			13.2665i			0.493386i
$724$		0			0
$725$	12.5109	+	5.63858i	0.464642	+	0.209412i
$726$		0			0
$727$			15.7908i			0.585650i	0.956166	+	0.292825i	$0.0945953\pi$
$727$			15.7908i			0.585650i	−0.956166	+	0.292825i	$0.905405\pi$
$728$		0			0
$729$	33.2337			1.23088
$730$		0			0
$731$	17.4891			0.646859
$732$		0			0
$733$		−	30.2921i		−	1.11886i	−0.828877	−	0.559431i	$-0.811020\pi$
$733$		−	30.2921i		−	1.11886i	0.828877	−	0.559431i	$-0.188980\pi$
$734$		0			0
$735$	−5.93070	+	27.5928i	−0.218757	+	1.01778i
$736$		0			0
$737$			0.644810i			0.0237519i
$738$		0			0
$739$	0.744563			0.0273892			0.0136946	−	0.999906i	$-0.495641\pi$
$739$	0.744563			0.0273892			0.0136946	+	0.999906i	$0.495641\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	21.7793i		−	0.799004i	−0.916732	−	0.399502i	$-0.869183\pi$
$743$		−	21.7793i		−	0.799004i	0.916732	−	0.399502i	$-0.130817\pi$
$744$		0			0
$745$	25.1168	+	5.39853i	0.920210	+	0.197787i
$746$		0			0
$747$			22.3692i			0.818446i
$748$		0			0
$749$	−22.9783			−0.839607
$750$		0			0
$751$	−21.6277			−0.789207			−0.394603	−	0.918852i	$-0.629118\pi$
$751$	−21.6277			−0.789207			−0.394603	+	0.918852i	$0.629118\pi$
$752$		0			0
$753$			12.3267i			0.449211i
$754$		0			0
$755$	48.6060	+	10.4472i	1.76895	+	0.380213i
$756$		0			0
$757$		−	39.7995i		−	1.44654i	−0.690567	−	0.723269i	$-0.742639\pi$
$757$		−	39.7995i		−	1.44654i	0.690567	−	0.723269i	$-0.257361\pi$
$758$		0			0
$759$	6.37228			0.231299
$760$		0			0
$761$	21.2554			0.770509			0.385255	−	0.922810i	$-0.374114\pi$
$761$	21.2554			0.770509			0.385255	+	0.922810i	$0.374114\pi$
$762$		0			0
$763$			34.6410i			1.25409i
$764$		0			0
$765$	8.00000	−	37.2203i	0.289241	−	1.34570i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	51.2119			1.84675			0.923375	−	0.383900i	$-0.125419\pi$
$769$	51.2119			1.84675			0.923375	+	0.383900i	$0.125419\pi$
$770$		0			0
$771$	−60.4674			−2.17768
$772$		0			0
$773$			30.8820i			1.11075i	0.831601	+	0.555373i	$0.187425\pi$
$773$			30.8820i			1.11075i	−0.831601	+	0.555373i	$0.812575\pi$
$774$		0			0
$775$	10.8139	+	4.87375i	0.388445	+	0.175070i
$776$		0			0
$777$		−	96.5147i		−	3.46245i
$778$		0			0
$779$	−10.9783			−0.393337
$780$		0			0
$781$	−7.11684			−0.254661
$782$		0			0
$783$		−	2.57924i		−	0.0921745i
$784$		0			0
$785$	2.53667	−	11.8020i	0.0905377	−	0.421230i
$786$		0			0
$787$		−	4.75372i		−	0.169452i	−0.996404	−	0.0847259i	$-0.972999\pi$
$787$		−	4.75372i		−	0.169452i	0.996404	−	0.0847259i	$-0.0270015\pi$
$788$		0			0
$789$	−35.7228			−1.27177
$790$		0			0
$791$	−55.7228			−1.98128
$792$		0			0
$793$		0			0
$794$		0			0
$795$	17.4891	+	3.75906i	0.620275	+	0.133320i
$796$		0			0
$797$		−	31.2318i		−	1.10629i	−0.833086	−	0.553144i	$-0.813428\pi$
$797$		−	31.2318i		−	1.10629i	0.833086	−	0.553144i	$-0.186572\pi$
$798$		0			0
$799$	−33.4891			−1.18476
$800$		0			0
$801$	−14.7446			−0.520974
$802$		0			0
$803$			6.92820i			0.244491i
$804$		0			0
$805$	19.1168	+	4.10891i	0.673780	+	0.144820i
$806$		0			0
$807$		−	29.0024i		−	1.02093i
$808$		0			0
$809$	21.2554			0.747301			0.373651	−	0.927569i	$-0.378106\pi$
$809$	21.2554			0.747301			0.373651	+	0.927569i	$0.378106\pi$
$810$		0			0
$811$	−34.2337			−1.20211			−0.601054	−	0.799209i	$-0.705252\pi$
$811$	−34.2337			−1.20211			−0.601054	+	0.799209i	$0.705252\pi$
$812$		0			0
$813$			23.9538i			0.840095i
$814$		0			0
$815$	−1.62772	+	7.57301i	−0.0570165	+	0.265271i
$816$		0			0
$817$			13.8564i			0.484774i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−18.0000			−0.628204			−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000			−0.628204			−0.314102	+	0.949389i	$0.601703\pi$
$822$		0			0
$823$		−	33.5161i		−	1.16830i	−0.811646	−	0.584149i	$-0.801428\pi$
$823$		−	33.5161i		−	1.16830i	0.811646	−	0.584149i	$-0.198572\pi$
$824$		0			0
$825$	−5.18614	+	11.5070i	−0.180558	+	0.400622i
$826$		0			0
$827$			18.0202i			0.626624i	0.949650	+	0.313312i	$0.101439\pi$
$827$			18.0202i			0.626624i	−0.949650	+	0.313312i	$0.898561\pi$
$828$		0			0
$829$	−31.3505			−1.08885			−0.544424	−	0.838810i	$-0.683252\pi$
$829$	−31.3505			−1.08885			−0.544424	+	0.838810i	$0.683252\pi$
$830$		0			0
$831$	−20.7446			−0.719621
$832$		0			0
$833$			25.2434i			0.874631i
$834$		0			0
$835$	10.5109	−	48.9022i	0.363744	−	1.69233i
$836$		0			0
$837$		−	2.22938i		−	0.0770588i
$838$		0			0
$839$	7.11684			0.245701			0.122850	−	0.992425i	$-0.460796\pi$
$839$	7.11684			0.245701			0.122850	+	0.992425i	$0.460796\pi$
$840$		0			0
$841$	−21.4674			−0.740254
$842$		0			0
$843$		−	59.2945i		−	2.04221i
$844$		0			0
$845$	−28.4198	−	6.10846i	−0.977672	−	0.210138i
$846$		0			0
$847$		−	3.46410i		−	0.119028i
$848$		0			0
$849$	12.0000			0.411839
$850$		0			0
$851$	−27.8614			−0.955077
$852$		0			0
$853$			24.6535i			0.844119i	0.906568	+	0.422059i	$0.138693\pi$
$853$			24.6535i			0.844119i	−0.906568	+	0.422059i	$0.861307\pi$
$854$		0			0
$855$	29.4891	+	6.33830i	1.00851	+	0.216765i
$856$		0			0
$857$		−	10.6873i		−	0.365070i	−0.983199	−	0.182535i	$-0.941570\pi$
$857$		−	10.6873i		−	0.365070i	0.983199	−	0.182535i	$-0.0584303\pi$
$858$		0			0
$859$	11.1168			0.379302			0.189651	−	0.981852i	$-0.439264\pi$
$859$	11.1168			0.379302			0.189651	+	0.981852i	$0.439264\pi$
$860$		0			0
$861$	−24.0000			−0.817918
$862$		0			0
$863$			23.6588i			0.805355i	0.915342	+	0.402678i	$0.131920\pi$
$863$			23.6588i			0.805355i	−0.915342	+	0.402678i	$0.868080\pi$
$864$		0			0
$865$	0.883156	−	4.10891i	0.0300282	−	0.139707i
$866$		0			0
$867$		−	21.4294i		−	0.727781i
$868$		0			0
$869$	12.7446			0.432330
$870$		0			0
$871$		0			0
$872$		0			0
$873$			13.8564i			0.468968i
$874$		0			0
$875$	−22.9783	+	31.1769i	−0.776807	+	1.05397i
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$	−25.4891			−0.859727
$880$		0			0
$881$	21.8614			0.736530			0.368265	−	0.929721i	$-0.379952\pi$
$881$	21.8614			0.736530			0.368265	+	0.929721i	$0.379952\pi$
$882$		0			0
$883$			24.2487i			0.816034i	0.912974	+	0.408017i	$0.133780\pi$
$883$			24.2487i			0.816034i	−0.912974	+	0.408017i	$0.866220\pi$
$884$		0			0
$885$	−1.93070	+	8.98266i	−0.0648999	+	0.301949i
$886$		0			0
$887$			14.1514i			0.475156i	0.971368	+	0.237578i	$0.0763536\pi$
$887$			14.1514i			0.475156i	−0.971368	+	0.237578i	$0.923646\pi$
$888$		0			0
$889$	−40.4674			−1.35723
$890$		0			0
$891$	−7.74456			−0.259453
$892$		0			0
$893$		−	26.5330i		−	0.887893i
$894$		0			0
$895$	34.6753	+	7.45299i	1.15907	+	0.249126i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	6.51087			0.217150
$900$		0			0
$901$	16.0000			0.533037
$902$		0			0
$903$			30.2921i			1.00806i
$904$		0			0
$905$	−15.0475	−	3.23427i	−0.500197	−	0.107511i
$906$		0			0
$907$		−	19.8997i		−	0.660760i	−0.943848	−	0.330380i	$-0.892823\pi$
$907$		−	19.8997i		−	0.660760i	0.943848	−	0.330380i	$-0.107177\pi$
$908$		0			0
$909$	−20.2337			−0.671109
$910$		0			0
$911$	30.5109			1.01087			0.505435	−	0.862865i	$-0.331332\pi$
$911$	30.5109			1.01087			0.505435	+	0.862865i	$0.331332\pi$
$912$		0			0
$913$		−	6.63325i		−	0.219529i
$914$		0			0
$915$	12.7446	−	59.2945i	0.421322	−	1.96022i
$916$		0			0
$917$			30.2921i			1.00033i
$918$		0			0
$919$	6.23369			0.205630			0.102815	−	0.994700i	$-0.467215\pi$
$919$	6.23369			0.205630			0.102815	+	0.994700i	$0.467215\pi$
$920$		0			0
$921$	−70.9783			−2.33881
$922$		0			0
$923$		0			0
$924$		0			0
$925$	22.6753	−	50.3118i	0.745558	−	1.65424i
$926$		0			0
$927$			35.0458i			1.15105i
$928$		0			0
$929$	52.9783			1.73816			0.869080	−	0.494672i	$-0.164712\pi$
$929$	52.9783			1.73816			0.869080	+	0.494672i	$0.164712\pi$
$930$		0			0
$931$	−20.0000			−0.655474
$932$		0			0
$933$			44.1485i			1.44536i
$934$		0			0
$935$	−2.37228	+	11.0371i	−0.0775819	+	0.360952i
$936$		0			0
$937$		−	25.9431i		−	0.847524i	−0.905774	−	0.423762i	$-0.860709\pi$
$937$		−	25.9431i		−	0.847524i	0.905774	−	0.423762i	$-0.139291\pi$
$938$		0			0
$939$	−80.3288			−2.62143
$940$		0			0
$941$	10.4674			0.341227			0.170613	−	0.985338i	$-0.445425\pi$
$941$	10.4674			0.341227			0.170613	+	0.985338i	$0.445425\pi$
$942$		0			0
$943$			6.92820i			0.225613i
$944$		0			0
$945$	7.11684	+	1.52967i	0.231511	+	0.0497602i
$946$		0			0
$947$		−	56.1802i		−	1.82561i	−0.408393	−	0.912806i	$-0.633911\pi$
$947$		−	56.1802i		−	1.82561i	0.408393	−	0.912806i	$-0.366089\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	8.88316			0.288056
$952$		0			0
$953$			41.6790i			1.35012i	0.737765	+	0.675058i	$0.235881\pi$
$953$			41.6790i			1.35012i	−0.737765	+	0.675058i	$0.764119\pi$
$954$		0			0
$955$	−29.7921	−	6.40342i	−0.964050	−	0.207210i
$956$		0			0
$957$			6.92820i			0.223957i
$958$		0			0
$959$	−7.72281			−0.249383
$960$		0			0
$961$	−25.3723			−0.818461
$962$		0			0
$963$			22.3692i			0.720837i
$964$		0			0
$965$	−10.9783	+	51.0767i	−0.353402	+	1.64422i
$966$		0			0
$967$			46.3229i			1.48965i	0.667262	+	0.744823i	$0.267466\pi$
$967$			46.3229i			1.48965i	−0.667262	+	0.744823i	$0.732534\pi$
$968$		0			0
$969$	50.9783			1.63766
$970$		0			0
$971$	−54.0951			−1.73599			−0.867997	−	0.496569i	$-0.834593\pi$
$971$	−54.0951			−1.73599			−0.867997	+	0.496569i	$0.834593\pi$
$972$		0			0
$973$		−	63.1633i		−	2.02492i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	27.3630i		−	0.875419i	−0.899117	−	0.437709i	$-0.855790\pi$
$977$		−	27.3630i		−	0.875419i	0.899117	−	0.437709i	$-0.144210\pi$
$978$		0			0
$979$	4.37228			0.139739
$980$		0			0
$981$	33.7228			1.07669
$982$		0			0
$983$		−	8.16292i		−	0.260357i	−0.991491	−	0.130178i	$-0.958445\pi$
$983$		−	8.16292i		−	0.260357i	0.991491	−	0.130178i	$-0.0415550\pi$
$984$		0			0
$985$	0.883156	−	4.10891i	0.0281397	−	0.130921i
$986$		0			0
$987$		−	58.0049i		−	1.84632i
$988$		0			0
$989$	8.74456			0.278061
$990$		0			0
$991$	26.9783			0.856992			0.428496	−	0.903544i	$-0.359044\pi$
$991$	26.9783			0.856992			0.428496	+	0.903544i	$0.359044\pi$
$992$		0			0
$993$		−	7.86797i		−	0.249682i
$994$		0			0
$995$	17.4891	+	3.75906i	0.554443	+	0.119170i
$996$		0			0
$997$			22.0742i			0.699098i	0.936918	+	0.349549i	$0.113665\pi$
$997$			22.0742i			0.699098i	−0.936918	+	0.349549i	$0.886335\pi$
$998$		0			0
$999$	−10.3723			−0.328164

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	880.2.b.h.529.4		4
4.3	odd	2		55.2.b.a.34.3	yes	4
5.2	odd	4		4400.2.a.cc.1.4		4
5.3	odd	4		4400.2.a.cc.1.1		4
5.4	even	2	inner	880.2.b.h.529.1		4
12.11	even	2		495.2.c.a.199.2		4
20.3	even	4		275.2.a.h.1.3		4
20.7	even	4		275.2.a.h.1.2		4
20.19	odd	2		55.2.b.a.34.2	✓	4
44.3	odd	10		605.2.j.i.9.3		16
44.7	even	10		605.2.j.j.269.3		16
44.15	odd	10		605.2.j.i.269.2		16
44.19	even	10		605.2.j.j.9.2		16
44.27	odd	10		605.2.j.i.124.2		16
44.31	odd	10		605.2.j.i.444.3		16
44.35	even	10		605.2.j.j.444.2		16
44.39	even	10		605.2.j.j.124.3		16
44.43	even	2		605.2.b.c.364.2		4
60.23	odd	4		2475.2.a.bi.1.2		4
60.47	odd	4		2475.2.a.bi.1.3		4
60.59	even	2		495.2.c.a.199.3		4
220.19	even	10		605.2.j.j.9.3		16
220.39	even	10		605.2.j.j.124.2		16
220.43	odd	4		3025.2.a.ba.1.2		4
220.59	odd	10		605.2.j.i.269.3		16
220.79	even	10		605.2.j.j.444.3		16
220.87	odd	4		3025.2.a.ba.1.3		4
220.119	odd	10		605.2.j.i.444.2		16
220.139	even	10		605.2.j.j.269.2		16
220.159	odd	10		605.2.j.i.124.3		16
220.179	odd	10		605.2.j.i.9.2		16
220.219	even	2		605.2.b.c.364.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.b.a.34.2	✓	4	20.19	odd	2
55.2.b.a.34.3	yes	4	4.3	odd	2
275.2.a.h.1.2		4	20.7	even	4
275.2.a.h.1.3		4	20.3	even	4
495.2.c.a.199.2		4	12.11	even	2
495.2.c.a.199.3		4	60.59	even	2
605.2.b.c.364.2		4	44.43	even	2
605.2.b.c.364.3		4	220.219	even	2
605.2.j.i.9.2		16	220.179	odd	10
605.2.j.i.9.3		16	44.3	odd	10
605.2.j.i.124.2		16	44.27	odd	10
605.2.j.i.124.3		16	220.159	odd	10
605.2.j.i.269.2		16	44.15	odd	10
605.2.j.i.269.3		16	220.59	odd	10
605.2.j.i.444.2		16	220.119	odd	10
605.2.j.i.444.3		16	44.31	odd	10
605.2.j.j.9.2		16	44.19	even	10
605.2.j.j.9.3		16	220.19	even	10
605.2.j.j.124.2		16	220.39	even	10
605.2.j.j.124.3		16	44.39	even	10
605.2.j.j.269.2		16	220.139	even	10
605.2.j.j.269.3		16	44.7	even	10
605.2.j.j.444.2		16	44.35	even	10
605.2.j.j.444.3		16	220.79	even	10
880.2.b.h.529.1		4	5.4	even	2	inner
880.2.b.h.529.4		4	1.1	even	1	trivial
2475.2.a.bi.1.2		4	60.23	odd	4
2475.2.a.bi.1.3		4	60.47	odd	4
3025.2.a.ba.1.2		4	220.43	odd	4
3025.2.a.ba.1.3		4	220.87	odd	4
4400.2.a.cc.1.1		4	5.3	odd	4
4400.2.a.cc.1.4		4	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	880.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.52434i q^{3} +(-2.18614 - 0.469882i) q^{5} -3.46410i q^{7} -3.37228 q^{9} +1.00000 q^{11} +(1.18614 - 5.51856i) q^{15} -5.04868i q^{17} +4.00000 q^{19} +8.74456 q^{21} -2.52434i q^{23} +(4.55842 + 2.05446i) q^{25} -0.939764i q^{27} +2.74456 q^{29} +2.37228 q^{31} +2.52434i q^{33} +(-1.62772 + 7.57301i) q^{35} -11.0371i q^{37} -2.74456 q^{41} +3.46410i q^{43} +(7.37228 + 1.58457i) q^{45} -6.63325i q^{47} -5.00000 q^{49} +12.7446 q^{51} +3.16915i q^{53} +(-2.18614 - 0.469882i) q^{55} +10.0974i q^{57} -1.62772 q^{59} +10.7446 q^{61} +11.6819i q^{63} +0.644810i q^{67} +6.37228 q^{69} -7.11684 q^{71} +6.92820i q^{73} +(-5.18614 + 11.5070i) q^{75} -3.46410i q^{77} +12.7446 q^{79} -7.74456 q^{81} -6.63325i q^{83} +(-2.37228 + 11.0371i) q^{85} +6.92820i q^{87} +4.37228 q^{89} +5.98844i q^{93} +(-8.74456 - 1.87953i) q^{95} -4.10891i q^{97} -3.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{5} - 2 q^{9} + 4 q^{11} - q^{15} + 16 q^{19} + 12 q^{21} + q^{25} - 12 q^{29} - 2 q^{31} - 18 q^{35} + 12 q^{41} + 18 q^{45} - 20 q^{49} + 28 q^{51} - 3 q^{55} - 18 q^{59} + 20 q^{61} + 14 q^{69}+ \cdots - 2 q^{99}+O(q^{100}) \) 4 * q - 3 * q^5 - 2 * q^9 + 4 * q^11 - q^15 + 16 * q^19 + 12 * q^21 + q^25 - 12 * q^29 - 2 * q^31 - 18 * q^35 + 12 * q^41 + 18 * q^45 - 20 * q^49 + 28 * q^51 - 3 * q^55 - 18 * q^59 + 20 * q^61 + 14 * q^69 + 6 * q^71 - 15 * q^75 + 28 * q^79 - 8 * q^81 + 2 * q^85 + 6 * q^89 - 12 * q^95 - 2 * q^99

Introduction

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Modular forms

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Database

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