LMFDB - Embedded newform 2760.2.k.e.2209.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2760,2,Mod(2209,2760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2760, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2760.2209");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2760.k (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:	$22.0387109579$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 2 x^{14} + 141 x^{12} - 298 x^{11} + 314 x^{10} + 144 x^{9} + 4788 x^{8} + \cdots + 20736 $ x^16 - 2x^15 + 2x^14 + 141x^12 - 298x^11 + 314x^10 + 144x^9 + 4788x^8 - 10568x^7 + 11688x^6 + 8928x^5 + 8320x^4 - 18624x^3 + 28800x^2 + 34560x + 20736
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2209.2
Root			$1.83968 + 1.83968i$ of defining polynomial
Character	$\chi$	$=$	2760.2209
Dual form			2760.2.k.e.2209.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(-1.83968 - 1.27105i) q^{5} +0.440442i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +(-1.83968 - 1.27105i) q^{5} +0.440442i q^{7} -1.00000 q^{9} +4.37833 q^{11} +1.54184i q^{13} +(-1.27105 + 1.83968i) q^{15} +1.98228i q^{17} -2.13752 q^{19} +0.440442 q^{21} +1.00000i q^{23} +(1.76884 + 4.67666i) q^{25} +1.00000i q^{27} -8.68220 q^{29} -4.07541 q^{31} -4.37833i q^{33} +(0.559826 - 0.810273i) q^{35} +8.30387i q^{37} +1.54184 q^{39} -3.39578 q^{41} -4.06053i q^{43} +(1.83968 + 1.27105i) q^{45} +7.95277i q^{47} +6.80601 q^{49} +1.98228 q^{51} -4.68348i q^{53} +(-8.05473 - 5.56510i) q^{55} +2.13752i q^{57} +0.331363 q^{59} -6.44524 q^{61} -0.440442i q^{63} +(1.95976 - 2.83649i) q^{65} +10.6175i q^{67} +1.00000 q^{69} -0.114463 q^{71} +5.83786i q^{73} +(4.67666 - 1.76884i) q^{75} +1.92840i q^{77} +9.06492 q^{79} +1.00000 q^{81} +0.983785i q^{83} +(2.51959 - 3.64676i) q^{85} +8.68220i q^{87} +16.0850 q^{89} -0.679091 q^{91} +4.07541i q^{93} +(3.93235 + 2.71690i) q^{95} +6.11195i q^{97} -4.37833 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{5} - 16 q^{9}+O(q^{10}) $ 16 * q - 2 * q^5 - 16 * q^9	$ 16 q - 2 q^{5} - 16 q^{9} - 2 q^{15} - 8 q^{19} + 6 q^{21} + 22 q^{29} + 30 q^{31} + 2 q^{35} - 4 q^{39} - 22 q^{41} + 2 q^{45} - 38 q^{49} + 2 q^{51} + 12 q^{55} - 6 q^{59} - 4 q^{61} + 20 q^{65} + 16 q^{69} - 10 q^{71} - 4 q^{75} - 84 q^{79} + 16 q^{81} + 22 q^{85} + 16 q^{89} + 36 q^{91} + 76 q^{95}+O(q^{100}) $ 16 * q - 2 * q^5 - 16 * q^9 - 2 * q^15 - 8 * q^19 + 6 * q^21 + 22 * q^29 + 30 * q^31 + 2 * q^35 - 4 * q^39 - 22 * q^41 + 2 * q^45 - 38 * q^49 + 2 * q^51 + 12 * q^55 - 6 * q^59 - 4 * q^61 + 20 * q^65 + 16 * q^69 - 10 * q^71 - 4 * q^75 - 84 * q^79 + 16 * q^81 + 22 * q^85 + 16 * q^89 + 36 * q^91 + 76 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2760\mathbb{Z}\right)^\times$.

$n$	$1201$	$1381$	$1657$	$1841$	$2071$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	−1.83968	−	1.27105i	−0.822730	−	0.568433i
$5$	−1.83968	−	1.27105i	−0.822730	−	0.568433i
$6$		0			0
$7$			0.440442i			0.166471i	0.996530	+	0.0832357i	$0.0265254\pi$
$7$			0.440442i			0.166471i	−0.996530	+	0.0832357i	$0.973475\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	4.37833			1.32012			0.660058	−	0.751214i	$-0.270531\pi$
$11$	4.37833			1.32012			0.660058	+	0.751214i	$0.270531\pi$
$12$		0			0
$13$			1.54184i			0.427630i	0.976874	+	0.213815i	$0.0685889\pi$
$13$			1.54184i			0.427630i	−0.976874	+	0.213815i	$0.931411\pi$
$14$		0			0
$15$	−1.27105	+	1.83968i	−0.328185	+	0.475003i
$16$		0			0
$17$			1.98228i			0.480774i	0.970677	+	0.240387i	$0.0772744\pi$
$17$			1.98228i			0.480774i	−0.970677	+	0.240387i	$0.922726\pi$
$18$		0			0
$19$	−2.13752			−0.490381			−0.245190	−	0.969475i	$-0.578850\pi$
$19$	−2.13752			−0.490381			−0.245190	+	0.969475i	$0.578850\pi$
$20$		0			0
$21$	0.440442			0.0961124
$22$		0			0
$23$			1.00000i			0.208514i
$23$			1.00000i			0.208514i
$24$		0			0
$25$	1.76884	+	4.67666i	0.353769	+	0.935333i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	−8.68220			−1.61224			−0.806122	−	0.591749i	$-0.798438\pi$
$29$	−8.68220			−1.61224			−0.806122	+	0.591749i	$0.798438\pi$
$30$		0			0
$31$	−4.07541			−0.731965			−0.365983	−	0.930622i	$-0.619267\pi$
$31$	−4.07541			−0.731965			−0.365983	+	0.930622i	$0.619267\pi$
$32$		0			0
$33$		−	4.37833i		−	0.762170i
$34$		0			0
$35$	0.559826	−	0.810273i	0.0946278	−	0.136961i
$36$		0			0
$37$			8.30387i			1.36515i	0.730816	+	0.682574i	$0.239140\pi$
$37$			8.30387i			1.36515i	−0.730816	+	0.682574i	$0.760860\pi$
$38$		0			0
$39$	1.54184			0.246892
$40$		0			0
$41$	−3.39578			−0.530332			−0.265166	−	0.964203i	$-0.585427\pi$
$41$	−3.39578			−0.530332			−0.265166	+	0.964203i	$0.585427\pi$
$42$		0			0
$43$		−	4.06053i		−	0.619225i	−0.950863	−	0.309613i	$-0.899801\pi$
$43$		−	4.06053i		−	0.619225i	0.950863	−	0.309613i	$-0.100199\pi$
$44$		0			0
$45$	1.83968	+	1.27105i	0.274243	+	0.189478i
$46$		0			0
$47$			7.95277i			1.16003i	0.814606	+	0.580015i	$0.196953\pi$
$47$			7.95277i			1.16003i	−0.814606	+	0.580015i	$0.803047\pi$
$48$		0			0
$49$	6.80601			0.972287
$50$		0			0
$51$	1.98228			0.277575
$52$		0			0
$53$		−	4.68348i		−	0.643325i	−0.946854	−	0.321662i	$-0.895758\pi$
$53$		−	4.68348i		−	0.643325i	0.946854	−	0.321662i	$-0.104242\pi$
$54$		0			0
$55$	−8.05473	−	5.56510i	−1.08610	−	0.750398i
$56$		0			0
$57$			2.13752i			0.283121i
$58$		0			0
$59$	0.331363			0.0431398			0.0215699	−	0.999767i	$-0.493134\pi$
$59$	0.331363			0.0431398			0.0215699	+	0.999767i	$0.493134\pi$
$60$		0			0
$61$	−6.44524			−0.825228			−0.412614	−	0.910906i	$-0.635384\pi$
$61$	−6.44524			−0.825228			−0.412614	+	0.910906i	$0.635384\pi$
$62$		0			0
$63$		−	0.440442i		−	0.0554905i
$64$		0			0
$65$	1.95976	−	2.83649i	0.243079	−	0.351824i
$66$		0			0
$67$			10.6175i			1.29714i	0.761157	+	0.648568i	$0.224632\pi$
$67$			10.6175i			1.29714i	−0.761157	+	0.648568i	$0.775368\pi$
$68$		0			0
$69$	1.00000			0.120386
$70$		0			0
$71$	−0.114463			−0.0135843			−0.00679214	−	0.999977i	$-0.502162\pi$
$71$	−0.114463			−0.0135843			−0.00679214	+	0.999977i	$0.502162\pi$
$72$		0			0
$73$			5.83786i			0.683270i	0.939833	+	0.341635i	$0.110981\pi$
$73$			5.83786i			0.683270i	−0.939833	+	0.341635i	$0.889019\pi$
$74$		0			0
$75$	4.67666	−	1.76884i	0.540015	−	0.204248i
$76$		0			0
$77$			1.92840i			0.219762i
$78$		0			0
$79$	9.06492			1.01988			0.509942	−	0.860209i	$-0.329667\pi$
$79$	9.06492			1.01988			0.509942	+	0.860209i	$0.329667\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			0.983785i			0.107984i	0.998541	+	0.0539922i	$0.0171946\pi$
$83$			0.983785i			0.107984i	−0.998541	+	0.0539922i	$0.982805\pi$
$84$		0			0
$85$	2.51959	−	3.64676i	0.273288	−	0.395547i
$86$		0			0
$87$			8.68220i			0.930830i
$88$		0			0
$89$	16.0850			1.70501			0.852504	−	0.522721i	$-0.175083\pi$
$89$	16.0850			1.70501			0.852504	+	0.522721i	$0.175083\pi$
$90$		0			0
$91$	−0.679091			−0.0711881
$92$		0			0
$93$			4.07541i			0.422600i
$94$		0			0
$95$	3.93235	+	2.71690i	0.403451	+	0.278748i
$96$		0			0
$97$			6.11195i			0.620574i	0.950643	+	0.310287i	$0.100425\pi$
$97$			6.11195i			0.620574i	−0.950643	+	0.310287i	$0.899575\pi$
$98$		0			0
$99$	−4.37833			−0.440039
$100$		0			0
$101$	18.6484			1.85559			0.927793	−	0.373094i	$-0.121703\pi$
$101$	18.6484			1.85559			0.927793	+	0.373094i	$0.121703\pi$
$102$		0			0
$103$			18.9534i			1.86753i	0.357882	+	0.933767i	$0.383499\pi$
$103$			18.9534i			1.86753i	−0.357882	+	0.933767i	$0.616501\pi$
$104$		0			0
$105$	−0.810273	−	0.559826i	−0.0790745	−	0.0546334i
$106$		0			0
$107$		−	0.505129i		−	0.0488327i	−0.999702	−	0.0244163i	$-0.992227\pi$
$107$		−	0.505129i		−	0.0488327i	0.999702	−	0.0244163i	$-0.00777273\pi$
$108$		0			0
$109$	−7.32612			−0.701715			−0.350858	−	0.936429i	$-0.614110\pi$
$109$	−7.32612			−0.701715			−0.350858	+	0.936429i	$0.614110\pi$
$110$		0			0
$111$	8.30387			0.788169
$112$		0			0
$113$			1.35717i			0.127672i	0.997960	+	0.0638361i	$0.0203335\pi$
$113$			1.35717i			0.127672i	−0.997960	+	0.0638361i	$0.979667\pi$
$114$		0			0
$115$	1.27105	−	1.83968i	0.118526	−	0.171551i
$116$		0			0
$117$		−	1.54184i		−	0.142543i
$118$		0			0
$119$	−0.873081			−0.0800352
$120$		0			0
$121$	8.16980			0.742709
$122$		0			0
$123$			3.39578i			0.306187i
$124$		0			0
$125$	2.69019	−	10.8519i	0.240618	−	0.970620i
$126$		0			0
$127$			9.74517i			0.864744i	0.901695	+	0.432372i	$0.142323\pi$
$127$			9.74517i			0.864744i	−0.901695	+	0.432372i	$0.857677\pi$
$128$		0			0
$129$	−4.06053			−0.357510
$130$		0			0
$131$	6.33863			0.553809			0.276904	−	0.960897i	$-0.410691\pi$
$131$	6.33863			0.553809			0.276904	+	0.960897i	$0.410691\pi$
$132$		0			0
$133$		−	0.941454i		−	0.0816344i
$134$		0			0
$135$	1.27105	−	1.83968i	0.109395	−	0.158334i
$136$		0			0
$137$		−	6.53737i		−	0.558525i	−0.960215	−	0.279262i	$-0.909910\pi$
$137$		−	6.53737i		−	0.558525i	0.960215	−	0.279262i	$-0.0900900\pi$
$138$		0			0
$139$	−14.0805			−1.19429			−0.597147	−	0.802132i	$-0.703699\pi$
$139$	−14.0805			−1.19429			−0.597147	+	0.802132i	$0.703699\pi$
$140$		0			0
$141$	7.95277			0.669744
$142$		0			0
$143$			6.75069i			0.564521i
$144$		0			0
$145$	15.9725	+	11.0356i	1.32644	+	0.916453i
$146$		0			0
$147$		−	6.80601i		−	0.561350i
$148$		0			0
$149$	13.2390			1.08459			0.542293	−	0.840190i	$-0.317556\pi$
$149$	13.2390			1.08459			0.542293	+	0.840190i	$0.317556\pi$
$150$		0			0
$151$	−17.5984			−1.43214			−0.716068	−	0.698031i	$-0.754060\pi$
$151$	−17.5984			−1.43214			−0.716068	+	0.698031i	$0.754060\pi$
$152$		0			0
$153$		−	1.98228i		−	0.160258i
$154$		0			0
$155$	7.49745	+	5.18007i	0.602210	+	0.416073i
$156$		0			0
$157$			6.07530i			0.484862i	0.970169	+	0.242431i	$0.0779447\pi$
$157$			6.07530i			0.484862i	−0.970169	+	0.242431i	$0.922055\pi$
$158$		0			0
$159$	−4.68348			−0.371424
$160$		0			0
$161$	−0.440442			−0.0347117
$162$		0			0
$163$			19.6560i			1.53958i	0.638300	+	0.769788i	$0.279638\pi$
$163$			19.6560i			1.53958i	−0.638300	+	0.769788i	$0.720362\pi$
$164$		0			0
$165$	−5.56510	+	8.05473i	−0.433242	+	0.627060i
$166$		0			0
$167$			12.5885i			0.974131i	0.873365	+	0.487066i	$0.161933\pi$
$167$			12.5885i			0.974131i	−0.873365	+	0.487066i	$0.838067\pi$
$168$		0			0
$169$	10.6227			0.817133
$170$		0			0
$171$	2.13752			0.163460
$172$		0			0
$173$		−	20.8105i		−	1.58220i	−0.611689	−	0.791098i	$-0.709510\pi$
$173$		−	20.8105i		−	1.58220i	0.611689	−	0.791098i	$-0.290490\pi$
$174$		0			0
$175$	−2.05980	+	0.779073i	−0.155706	+	0.0588924i
$176$		0			0
$177$		−	0.331363i		−	0.0249068i
$178$		0			0
$179$	8.63239			0.645215			0.322608	−	0.946533i	$-0.395441\pi$
$179$	8.63239			0.645215			0.322608	+	0.946533i	$0.395441\pi$
$180$		0			0
$181$	11.6263			0.864176			0.432088	−	0.901831i	$-0.357777\pi$
$181$	11.6263			0.864176			0.432088	+	0.901831i	$0.357777\pi$
$182$		0			0
$183$			6.44524i			0.476446i
$184$		0			0
$185$	10.5547	−	15.2765i	0.775995	−	1.12315i
$186$		0			0
$187$			8.67909i			0.634678i
$188$		0			0
$189$	−0.440442			−0.0320375
$190$		0			0
$191$	15.9114			1.15131			0.575655	−	0.817693i	$-0.304747\pi$
$191$	15.9114			1.15131			0.575655	+	0.817693i	$0.304747\pi$
$192$		0			0
$193$		−	19.7365i		−	1.42066i	−0.703868	−	0.710331i	$-0.748545\pi$
$193$		−	19.7365i		−	1.42066i	0.703868	−	0.710331i	$-0.251455\pi$
$194$		0			0
$195$	−2.83649	−	1.95976i	−0.203125	−	0.140341i
$196$		0			0
$197$			16.0851i			1.14602i	0.819550	+	0.573008i	$0.194224\pi$
$197$			16.0851i			1.14602i	−0.819550	+	0.573008i	$0.805776\pi$
$198$		0			0
$199$	−16.8660			−1.19560			−0.597801	−	0.801645i	$-0.703959\pi$
$199$	−16.8660			−1.19560			−0.597801	+	0.801645i	$0.703959\pi$
$200$		0			0
$201$	10.6175			0.748902
$202$		0			0
$203$		−	3.82401i		−	0.268393i
$204$		0			0
$205$	6.24715	+	4.31622i	0.436320	+	0.301458i
$206$		0			0
$207$		−	1.00000i		−	0.0695048i
$208$		0			0
$209$	−9.35877			−0.647360
$210$		0			0
$211$	−1.43733			−0.0989501			−0.0494751	−	0.998775i	$-0.515755\pi$
$211$	−1.43733			−0.0989501			−0.0494751	+	0.998775i	$0.515755\pi$
$212$		0			0
$213$			0.114463i			0.00784289i
$214$		0			0
$215$	−5.16116	+	7.47008i	−0.351988	+	0.509455i
$216$		0			0
$217$		−	1.79498i		−	0.121851i
$218$		0			0
$219$	5.83786			0.394486
$220$		0			0
$221$	−3.05636			−0.205593
$222$		0			0
$223$		−	7.88799i		−	0.528219i	−0.964493	−	0.264109i	$-0.914922\pi$
$223$		−	7.88799i		−	0.528219i	0.964493	−	0.264109i	$-0.0850780\pi$
$224$		0			0
$225$	−1.76884	−	4.67666i	−0.117923	−	0.311778i
$226$		0			0
$227$		−	11.2384i		−	0.745916i	−0.927848	−	0.372958i	$-0.878344\pi$
$227$		−	11.2384i		−	0.745916i	0.927848	−	0.372958i	$-0.121656\pi$
$228$		0			0
$229$	−27.3251			−1.80569			−0.902847	−	0.429961i	$-0.858527\pi$
$229$	−27.3251			−1.80569			−0.902847	+	0.429961i	$0.858527\pi$
$230$		0			0
$231$	1.92840			0.126880
$232$		0			0
$233$			7.77622i			0.509437i	0.967015	+	0.254718i	$0.0819828\pi$
$233$			7.77622i			0.509437i	−0.967015	+	0.254718i	$0.918017\pi$
$234$		0			0
$235$	10.1084	−	14.6305i	0.659399	−	0.954392i
$236$		0			0
$237$		−	9.06492i		−	0.588830i
$238$		0			0
$239$	3.23833			0.209470			0.104735	−	0.994500i	$-0.466601\pi$
$239$	3.23833			0.209470			0.104735	+	0.994500i	$0.466601\pi$
$240$		0			0
$241$	−9.74677			−0.627844			−0.313922	−	0.949449i	$-0.601643\pi$
$241$	−9.74677			−0.627844			−0.313922	+	0.949449i	$0.601643\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	−12.5209	−	8.65081i	−0.799930	−	0.552680i
$246$		0			0
$247$		−	3.29571i		−	0.209701i
$248$		0			0
$249$	0.983785			0.0623448
$250$		0			0
$251$	−19.7066			−1.24387			−0.621934	−	0.783070i	$-0.713653\pi$
$251$	−19.7066			−1.24387			−0.621934	+	0.783070i	$0.713653\pi$
$252$		0			0
$253$			4.37833i			0.275263i
$254$		0			0
$255$	−3.64676	−	2.51959i	−0.228369	−	0.157783i
$256$		0			0
$257$			6.57303i			0.410014i	0.978760	+	0.205007i	$0.0657217\pi$
$257$			6.57303i			0.410014i	−0.978760	+	0.205007i	$0.934278\pi$
$258$		0			0
$259$	−3.65738			−0.227258
$260$		0			0
$261$	8.68220			0.537415
$262$		0			0
$263$			16.0540i			0.989929i	0.868913	+	0.494965i	$0.164819\pi$
$263$			16.0540i			0.989929i	−0.868913	+	0.494965i	$0.835181\pi$
$264$		0			0
$265$	−5.95295	+	8.61610i	−0.365687	+	0.529283i
$266$		0			0
$267$		−	16.0850i		−	0.984387i
$268$		0			0
$269$	3.78234			0.230614			0.115307	−	0.993330i	$-0.463215\pi$
$269$	3.78234			0.230614			0.115307	+	0.993330i	$0.463215\pi$
$270$		0			0
$271$	−26.1008			−1.58551			−0.792755	−	0.609541i	$-0.791354\pi$
$271$	−26.1008			−1.58551			−0.792755	+	0.609541i	$0.791354\pi$
$272$		0			0
$273$			0.679091i			0.0411005i
$274$		0			0
$275$	7.74458	+	20.4760i	0.467016	+	1.23475i
$276$		0			0
$277$		−	28.9682i		−	1.74053i	−0.492584	−	0.870265i	$-0.663948\pi$
$277$		−	28.9682i		−	1.74053i	0.492584	−	0.870265i	$-0.336052\pi$
$278$		0			0
$279$	4.07541			0.243988
$280$		0			0
$281$	2.01319			0.120097			0.0600484	−	0.998195i	$-0.480875\pi$
$281$	2.01319			0.120097			0.0600484	+	0.998195i	$0.480875\pi$
$282$		0			0
$283$		−	6.53160i		−	0.388263i	−0.980975	−	0.194132i	$-0.937811\pi$
$283$		−	6.53160i		−	0.388263i	0.980975	−	0.194132i	$-0.0621889\pi$
$284$		0			0
$285$	2.71690	−	3.93235i	0.160935	−	0.232932i
$286$		0			0
$287$		−	1.49565i		−	0.0882852i
$288$		0			0
$289$	13.0706			0.768856
$290$		0			0
$291$	6.11195			0.358289
$292$		0			0
$293$		−	27.2819i		−	1.59383i	−0.604094	−	0.796913i	$-0.706465\pi$
$293$		−	27.2819i		−	1.59383i	0.604094	−	0.796913i	$-0.293535\pi$
$294$		0			0
$295$	−0.609602	−	0.421181i	−0.0354924	−	0.0245221i
$296$		0			0
$297$			4.37833i			0.254057i
$298$		0			0
$299$	−1.54184			−0.0891669
$300$		0			0
$301$	1.78843			0.103083
$302$		0			0
$303$		−	18.6484i		−	1.07132i
$304$		0			0
$305$	11.8572	+	8.19225i	0.678940	+	0.469087i
$306$		0			0
$307$			6.38858i			0.364616i	0.983242	+	0.182308i	$0.0583567\pi$
$307$			6.38858i			0.364616i	−0.983242	+	0.182308i	$0.941643\pi$
$308$		0			0
$309$	18.9534			1.07822
$310$		0			0
$311$	−1.42344			−0.0807160			−0.0403580	−	0.999185i	$-0.512850\pi$
$311$	−1.42344			−0.0807160			−0.0403580	+	0.999185i	$0.512850\pi$
$312$		0			0
$313$		−	5.54673i		−	0.313520i	−0.987637	−	0.156760i	$-0.949895\pi$
$313$		−	5.54673i		−	0.313520i	0.987637	−	0.156760i	$-0.0501049\pi$
$314$		0			0
$315$	−0.559826	+	0.810273i	−0.0315426	+	0.0456537i
$316$		0			0
$317$			19.7357i			1.10847i	0.832360	+	0.554235i	$0.186989\pi$
$317$			19.7357i			1.10847i	−0.832360	+	0.554235i	$0.813011\pi$
$318$		0			0
$319$	−38.0136			−2.12835
$320$		0			0
$321$	−0.505129			−0.0281936
$322$		0			0
$323$		−	4.23717i		−	0.235762i
$324$		0			0
$325$	−7.21067	+	2.72727i	−0.399976	+	0.151282i
$326$		0			0
$327$			7.32612i			0.405136i
$328$		0			0
$329$	−3.50273			−0.193112
$330$		0			0
$331$	25.2951			1.39035			0.695173	−	0.718842i	$-0.255328\pi$
$331$	25.2951			1.39035			0.695173	+	0.718842i	$0.255328\pi$
$332$		0			0
$333$		−	8.30387i		−	0.455049i
$334$		0			0
$335$	13.4954	−	19.5328i	0.737335	−	1.06719i
$336$		0			0
$337$			33.9240i			1.84796i	0.382445	+	0.923978i	$0.375082\pi$
$337$			33.9240i			1.84796i	−0.382445	+	0.923978i	$0.624918\pi$
$338$		0			0
$339$	1.35717			0.0737116
$340$		0			0
$341$	−17.8435			−0.966280
$342$		0			0
$343$			6.08075i			0.328330i
$344$		0			0
$345$	−1.83968	−	1.27105i	−0.0990450	−	0.0684313i
$346$		0			0
$347$			7.05791i			0.378888i	0.981891	+	0.189444i	$0.0606686\pi$
$347$			7.05791i			0.378888i	−0.981891	+	0.189444i	$0.939331\pi$
$348$		0			0
$349$	9.65204			0.516662			0.258331	−	0.966057i	$-0.416828\pi$
$349$	9.65204			0.516662			0.258331	+	0.966057i	$0.416828\pi$
$350$		0			0
$351$	−1.54184			−0.0822973
$352$		0			0
$353$			0.678918i			0.0361352i	0.999837	+	0.0180676i	$0.00575140\pi$
$353$			0.678918i			0.0361352i	−0.999837	+	0.0180676i	$0.994249\pi$
$354$		0			0
$355$	0.210576	+	0.145489i	0.0111762	+	0.00772175i
$356$		0			0
$357$			0.873081i			0.0462083i
$358$		0			0
$359$	−25.5392			−1.34791			−0.673953	−	0.738774i	$-0.735405\pi$
$359$	−25.5392			−1.34791			−0.673953	+	0.738774i	$0.735405\pi$
$360$		0			0
$361$	−14.4310			−0.759527
$362$		0			0
$363$		−	8.16980i		−	0.428803i
$364$		0			0
$365$	7.42023	−	10.7398i	0.388393	−	0.562146i
$366$		0			0
$367$			16.7580i			0.874763i	0.899276	+	0.437381i	$0.144094\pi$
$367$			16.7580i			0.874763i	−0.899276	+	0.437381i	$0.855906\pi$
$368$		0			0
$369$	3.39578			0.176777
$370$		0			0
$371$	2.06280			0.107095
$372$		0			0
$373$			16.0137i			0.829158i	0.910013	+	0.414579i	$0.136071\pi$
$373$			16.0137i			0.829158i	−0.910013	+	0.414579i	$0.863929\pi$
$374$		0			0
$375$	−10.8519	−	2.69019i	−0.560388	−	0.138921i
$376$		0			0
$377$		−	13.3866i		−	0.689444i
$378$		0			0
$379$	12.0022			0.616510			0.308255	−	0.951304i	$-0.400255\pi$
$379$	12.0022			0.616510			0.308255	+	0.951304i	$0.400255\pi$
$380$		0			0
$381$	9.74517			0.499260
$382$		0			0
$383$			28.1638i			1.43910i	0.694440	+	0.719551i	$0.255652\pi$
$383$			28.1638i			1.43910i	−0.694440	+	0.719551i	$0.744348\pi$
$384$		0			0
$385$	2.45110	−	3.54764i	0.124920	−	0.180805i
$386$		0			0
$387$			4.06053i			0.206408i
$388$		0			0
$389$	10.4438			0.529522			0.264761	−	0.964314i	$-0.414707\pi$
$389$	10.4438			0.529522			0.264761	+	0.964314i	$0.414707\pi$
$390$		0			0
$391$	−1.98228			−0.100248
$392$		0			0
$393$		−	6.33863i		−	0.319742i
$394$		0			0
$395$	−16.6766	−	11.5220i	−0.839088	−	0.579735i
$396$		0			0
$397$		−	2.87091i		−	0.144087i	−0.997401	−	0.0720434i	$-0.977048\pi$
$397$		−	2.87091i		−	0.144087i	0.997401	−	0.0720434i	$-0.0229520\pi$
$398$		0			0
$399$	−0.941454			−0.0471316
$400$		0			0
$401$	−22.5894			−1.12806			−0.564030	−	0.825754i	$-0.690750\pi$
$401$	−22.5894			−1.12806			−0.564030	+	0.825754i	$0.690750\pi$
$402$		0			0
$403$		−	6.28363i		−	0.313010i
$404$		0			0
$405$	−1.83968	−	1.27105i	−0.0914144	−	0.0631592i
$406$		0			0
$407$			36.3571i			1.80216i
$408$		0			0
$409$	9.68315			0.478801			0.239401	−	0.970921i	$-0.423049\pi$
$409$	9.68315			0.478801			0.239401	+	0.970921i	$0.423049\pi$
$410$		0			0
$411$	−6.53737			−0.322465
$412$		0			0
$413$			0.145946i			0.00718155i
$414$		0			0
$415$	1.25044	−	1.80985i	0.0613819	−	0.0888420i
$416$		0			0
$417$			14.0805i			0.689526i
$418$		0			0
$419$	−10.5876			−0.517240			−0.258620	−	0.965979i	$-0.583268\pi$
$419$	−10.5876			−0.517240			−0.258620	+	0.965979i	$0.583268\pi$
$420$		0			0
$421$	22.8744			1.11483			0.557414	−	0.830235i	$-0.311794\pi$
$421$	22.8744			1.11483			0.557414	+	0.830235i	$0.311794\pi$
$422$		0			0
$423$		−	7.95277i		−	0.386677i
$424$		0			0
$425$	−9.27047	+	3.50635i	−0.449684	+	0.170083i
$426$		0			0
$427$		−	2.83876i		−	0.137377i
$428$		0			0
$429$	6.75069			0.325926
$430$		0			0
$431$	−11.5601			−0.556829			−0.278415	−	0.960461i	$-0.589809\pi$
$431$	−11.5601			−0.556829			−0.278415	+	0.960461i	$0.589809\pi$
$432$		0			0
$433$			22.9689i			1.10382i	0.833905	+	0.551908i	$0.186100\pi$
$433$			22.9689i			1.10382i	−0.833905	+	0.551908i	$0.813900\pi$
$434$		0			0
$435$	11.0356	−	15.9725i	0.529114	−	0.765822i
$436$		0			0
$437$		−	2.13752i		−	0.102251i
$438$		0			0
$439$	−29.8244			−1.42344			−0.711719	−	0.702464i	$-0.752083\pi$
$439$	−29.8244			−1.42344			−0.711719	+	0.702464i	$0.752083\pi$
$440$		0			0
$441$	−6.80601			−0.324096
$442$		0			0
$443$		−	0.280731i		−	0.0133379i	−0.999978	−	0.00666897i	$-0.997877\pi$
$443$		−	0.280731i		−	0.0133379i	0.999978	−	0.00666897i	$-0.00212281\pi$
$444$		0			0
$445$	−29.5913	−	20.4449i	−1.40276	−	0.969182i
$446$		0			0
$447$		−	13.2390i		−	0.626186i
$448$		0			0
$449$	11.2864			0.532637			0.266319	−	0.963885i	$-0.414193\pi$
$449$	11.2864			0.532637			0.266319	+	0.963885i	$0.414193\pi$
$450$		0			0
$451$	−14.8679			−0.700100
$452$		0			0
$453$			17.5984i			0.826844i
$454$		0			0
$455$	1.24931	+	0.863162i	0.0585686	+	0.0404657i
$456$		0			0
$457$		−	26.1999i		−	1.22558i	−0.790245	−	0.612791i	$-0.790047\pi$
$457$		−	26.1999i		−	1.22558i	0.790245	−	0.612791i	$-0.209953\pi$
$458$		0			0
$459$	−1.98228			−0.0925250
$460$		0			0
$461$	−7.50427			−0.349509			−0.174754	−	0.984612i	$-0.555913\pi$
$461$	−7.50427			−0.349509			−0.174754	+	0.984612i	$0.555913\pi$
$462$		0			0
$463$		−	19.0294i		−	0.884371i	−0.896924	−	0.442186i	$-0.854203\pi$
$463$		−	19.0294i		−	0.884371i	0.896924	−	0.442186i	$-0.145797\pi$
$464$		0			0
$465$	5.18007	−	7.49745i	0.240220	−	0.347686i
$466$		0			0
$467$		−	24.2683i		−	1.12300i	−0.827476	−	0.561501i	$-0.810224\pi$
$467$		−	24.2683i		−	1.12300i	0.827476	−	0.561501i	$-0.189776\pi$
$468$		0			0
$469$	−4.67640			−0.215936
$470$		0			0
$471$	6.07530			0.279935
$472$		0			0
$473$		−	17.7784i		−	0.817450i
$474$		0			0
$475$	−3.78094	−	9.99646i	−0.173481	−	0.458669i
$476$		0			0
$477$			4.68348i			0.214442i
$478$		0			0
$479$	−40.0270			−1.82888			−0.914439	−	0.404724i	$-0.867368\pi$
$479$	−40.0270			−1.82888			−0.914439	+	0.404724i	$0.867368\pi$
$480$		0			0
$481$	−12.8032			−0.583778
$482$		0			0
$483$			0.440442i			0.0200408i
$484$		0			0
$485$	7.76862	−	11.2440i	0.352755	−	0.510565i
$486$		0			0
$487$			35.7242i			1.61882i	0.587245	+	0.809409i	$0.300212\pi$
$487$			35.7242i			1.61882i	−0.587245	+	0.809409i	$0.699788\pi$
$488$		0			0
$489$	19.6560			0.888875
$490$		0			0
$491$	22.3336			1.00790			0.503950	−	0.863733i	$-0.331880\pi$
$491$	22.3336			1.00790			0.503950	+	0.863733i	$0.331880\pi$
$492$		0			0
$493$		−	17.2106i		−	0.775126i
$494$		0			0
$495$	8.05473	+	5.56510i	0.362033	+	0.250133i
$496$		0			0
$497$		−	0.0504144i		−	0.00226140i
$498$		0			0
$499$	−36.7479			−1.64506			−0.822530	−	0.568722i	$-0.807438\pi$
$499$	−36.7479			−1.64506			−0.822530	+	0.568722i	$0.807438\pi$
$500$		0			0
$501$	12.5885			0.562415
$502$		0			0
$503$			5.92619i			0.264236i	0.991234	+	0.132118i	$0.0421777\pi$
$503$			5.92619i			0.264236i	−0.991234	+	0.132118i	$0.957822\pi$
$504$		0			0
$505$	−34.3071	−	23.7031i	−1.52665	−	1.05478i
$506$		0			0
$507$		−	10.6227i		−	0.471772i
$508$		0			0
$509$	24.9290			1.10496			0.552479	−	0.833527i	$-0.313682\pi$
$509$	24.9290			1.10496			0.552479	+	0.833527i	$0.313682\pi$
$510$		0			0
$511$	−2.57124			−0.113745
$512$		0			0
$513$		−	2.13752i		−	0.0943738i
$514$		0			0
$515$	24.0908	−	34.8682i	1.06157	−	1.53648i
$516$		0			0
$517$			34.8199i			1.53138i
$518$		0			0
$519$	−20.8105			−0.913481
$520$		0			0
$521$	−25.3540			−1.11078			−0.555390	−	0.831590i	$-0.687431\pi$
$521$	−25.3540			−1.11078			−0.555390	+	0.831590i	$0.687431\pi$
$522$		0			0
$523$		−	13.3009i		−	0.581608i	−0.956783	−	0.290804i	$-0.906077\pi$
$523$		−	13.3009i		−	0.581608i	0.956783	−	0.290804i	$-0.0939228\pi$
$524$		0			0
$525$	0.779073	+	2.05980i	0.0340015	+	0.0898971i
$526$		0			0
$527$		−	8.07861i		−	0.351910i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	−0.331363			−0.0143799
$532$		0			0
$533$		−	5.23575i		−	0.226786i
$534$		0			0
$535$	−0.642046	+	0.929276i	−0.0277581	+	0.0401761i
$536$		0			0
$537$		−	8.63239i		−	0.372515i
$538$		0			0
$539$	29.7990			1.28353
$540$		0			0
$541$	0.967409			0.0415922			0.0207961	−	0.999784i	$-0.493380\pi$
$541$	0.967409			0.0415922			0.0207961	+	0.999784i	$0.493380\pi$
$542$		0			0
$543$		−	11.6263i		−	0.498932i
$544$		0			0
$545$	13.4777	+	9.31190i	0.577322	+	0.398878i
$546$		0			0
$547$		−	7.24001i		−	0.309561i	−0.987949	−	0.154780i	$-0.950533\pi$
$547$		−	7.24001i		−	0.309561i	0.987949	−	0.154780i	$-0.0494670\pi$
$548$		0			0
$549$	6.44524			0.275076
$550$		0			0
$551$	18.5584			0.790614
$552$		0			0
$553$			3.99257i			0.169782i
$554$		0			0
$555$	−15.2765	−	10.5547i	−0.648450	−	0.448021i
$556$		0			0
$557$		−	1.32889i		−	0.0563068i	−0.999604	−	0.0281534i	$-0.991037\pi$
$557$		−	1.32889i		−	0.0563068i	0.999604	−	0.0281534i	$-0.00896269\pi$
$558$		0			0
$559$	6.26069			0.264799
$560$		0			0
$561$	8.67909			0.366432
$562$		0			0
$563$			1.89875i			0.0800226i	0.999199	+	0.0400113i	$0.0127394\pi$
$563$			1.89875i			0.0800226i	−0.999199	+	0.0400113i	$0.987261\pi$
$564$		0			0
$565$	1.72504	−	2.49677i	0.0725731	−	0.105040i
$566$		0			0
$567$			0.440442i			0.0184968i
$568$		0			0
$569$	20.1644			0.845336			0.422668	−	0.906285i	$-0.361094\pi$
$569$	20.1644			0.845336			0.422668	+	0.906285i	$0.361094\pi$
$570$		0			0
$571$	3.64458			0.152521			0.0762604	−	0.997088i	$-0.475702\pi$
$571$	3.64458			0.152521			0.0762604	+	0.997088i	$0.475702\pi$
$572$		0			0
$573$		−	15.9114i		−	0.664709i
$574$		0			0
$575$	−4.67666	+	1.76884i	−0.195030	+	0.0737658i
$576$		0			0
$577$			25.9250i			1.07927i	0.841898	+	0.539636i	$0.181438\pi$
$577$			25.9250i			1.07927i	−0.841898	+	0.539636i	$0.818562\pi$
$578$		0			0
$579$	−19.7365			−0.820219
$580$		0			0
$581$	−0.433300			−0.0179763
$582$		0			0
$583$		−	20.5058i		−	0.849264i
$584$		0			0
$585$	−1.95976	+	2.83649i	−0.0810262	+	0.117275i
$586$		0			0
$587$			3.25203i			0.134226i	0.997745	+	0.0671128i	$0.0213787\pi$
$587$			3.25203i			0.134226i	−0.997745	+	0.0671128i	$0.978621\pi$
$588$		0			0
$589$	8.71127			0.358941
$590$		0			0
$591$	16.0851			0.661652
$592$		0			0
$593$		−	8.77771i		−	0.360457i	−0.983625	−	0.180229i	$-0.942316\pi$
$593$		−	8.77771i		−	0.360457i	0.983625	−	0.180229i	$-0.0576838\pi$
$594$		0			0
$595$	1.60619	+	1.10973i	0.0658473	+	0.0454946i
$596$		0			0
$597$			16.8660i			0.690281i
$598$		0			0
$599$	−0.419404			−0.0171364			−0.00856818	−	0.999963i	$-0.502727\pi$
$599$	−0.419404			−0.0171364			−0.00856818	+	0.999963i	$0.502727\pi$
$600$		0			0
$601$	30.4638			1.24265			0.621323	−	0.783555i	$-0.286596\pi$
$601$	30.4638			1.24265			0.621323	+	0.783555i	$0.286596\pi$
$602$		0			0
$603$		−	10.6175i		−	0.432379i
$604$		0			0
$605$	−15.0298	−	10.3843i	−0.611049	−	0.422180i
$606$		0			0
$607$			10.2907i			0.417686i	0.977949	+	0.208843i	$0.0669697\pi$
$607$			10.2907i			0.417686i	−0.977949	+	0.208843i	$0.933030\pi$
$608$		0			0
$609$	−3.82401			−0.154957
$610$		0			0
$611$	−12.2619			−0.496063
$612$		0			0
$613$		−	3.68868i		−	0.148984i	−0.997222	−	0.0744922i	$-0.976266\pi$
$613$		−	3.68868i		−	0.148984i	0.997222	−	0.0744922i	$-0.0237336\pi$
$614$		0			0
$615$	4.31622	−	6.24715i	0.174047	−	0.251909i
$616$		0			0
$617$			18.0596i			0.727052i	0.931584	+	0.363526i	$0.118427\pi$
$617$			18.0596i			0.727052i	−0.931584	+	0.363526i	$0.881573\pi$
$618$		0			0
$619$	18.7179			0.752335			0.376168	−	0.926552i	$-0.377242\pi$
$619$	18.7179			0.752335			0.376168	+	0.926552i	$0.377242\pi$
$620$		0			0
$621$	−1.00000			−0.0401286
$622$		0			0
$623$			7.08452i			0.283835i
$624$		0			0
$625$	−18.7424	+	16.5446i	−0.749696	+	0.661783i
$626$		0			0
$627$			9.35877i			0.373753i
$628$		0			0
$629$	−16.4606			−0.656328
$630$		0			0
$631$	9.24365			0.367984			0.183992	−	0.982928i	$-0.441098\pi$
$631$	9.24365			0.367984			0.183992	+	0.982928i	$0.441098\pi$
$632$		0			0
$633$			1.43733i			0.0571289i
$634$		0			0
$635$	12.3866	−	17.9280i	0.491549	−	0.711451i
$636$		0			0
$637$			10.4938i			0.415779i
$638$		0			0
$639$	0.114463			0.00452809
$640$		0			0
$641$	−46.7528			−1.84662			−0.923312	−	0.384051i	$-0.874529\pi$
$641$	−46.7528			−1.84662			−0.923312	+	0.384051i	$0.874529\pi$
$642$		0			0
$643$		−	8.78044i		−	0.346267i	−0.984898	−	0.173133i	$-0.944611\pi$
$643$		−	8.78044i		−	0.346267i	0.984898	−	0.173133i	$-0.0553892\pi$
$644$		0			0
$645$	7.47008	+	5.16116i	0.294134	+	0.203220i
$646$		0			0
$647$		−	14.8980i		−	0.585701i	−0.956158	−	0.292850i	$-0.905396\pi$
$647$		−	14.8980i		−	0.585701i	0.956158	−	0.292850i	$-0.0946038\pi$
$648$		0			0
$649$	1.45082			0.0569496
$650$		0			0
$651$	−1.79498			−0.0703509
$652$		0			0
$653$		−	47.3611i		−	1.85338i	−0.375825	−	0.926691i	$-0.622641\pi$
$653$		−	47.3611i		−	1.85338i	0.375825	−	0.926691i	$-0.377359\pi$
$654$		0			0
$655$	−11.6611	−	8.05674i	−0.455635	−	0.314803i
$656$		0			0
$657$		−	5.83786i		−	0.227757i
$658$		0			0
$659$	−3.65260			−0.142285			−0.0711425	−	0.997466i	$-0.522664\pi$
$659$	−3.65260			−0.142285			−0.0711425	+	0.997466i	$0.522664\pi$
$660$		0			0
$661$	−49.5117			−1.92578			−0.962891	−	0.269892i	$-0.913012\pi$
$661$	−49.5117			−1.92578			−0.962891	+	0.269892i	$0.913012\pi$
$662$		0			0
$663$			3.05636i			0.118699i
$664$		0			0
$665$	−1.19664	+	1.73197i	−0.0464036	+	0.0671630i
$666$		0			0
$667$		−	8.68220i		−	0.336176i
$668$		0			0
$669$	−7.88799			−0.304967
$670$		0			0
$671$	−28.2194			−1.08940
$672$		0			0
$673$		−	12.1519i		−	0.468423i	−0.972186	−	0.234212i	$-0.924749\pi$
$673$		−	12.1519i		−	0.468423i	0.972186	−	0.234212i	$-0.0752508\pi$
$674$		0			0
$675$	−4.67666	+	1.76884i	−0.180005	+	0.0680828i
$676$		0			0
$677$		−	1.80948i		−	0.0695438i	−0.999395	−	0.0347719i	$-0.988930\pi$
$677$		−	1.80948i		−	0.0695438i	0.999395	−	0.0347719i	$-0.0110705\pi$
$678$		0			0
$679$	−2.69196			−0.103308
$680$		0			0
$681$	−11.2384			−0.430655
$682$		0			0
$683$		−	11.6965i		−	0.447554i	−0.974640	−	0.223777i	$-0.928161\pi$
$683$		−	11.6965i		−	0.447554i	0.974640	−	0.223777i	$-0.0718388\pi$
$684$		0			0
$685$	−8.30935	+	12.0267i	−0.317484	+	0.459515i
$686$		0			0
$687$			27.3251i			1.04252i
$688$		0			0
$689$	7.22117			0.275105
$690$		0			0
$691$	−6.73929			−0.256375			−0.128187	−	0.991750i	$-0.540916\pi$
$691$	−6.73929			−0.256375			−0.128187	+	0.991750i	$0.540916\pi$
$692$		0			0
$693$		−	1.92840i		−	0.0732539i
$694$		0			0
$695$	25.9036	+	17.8971i	0.982581	+	0.678875i
$696$		0			0
$697$		−	6.73140i		−	0.254970i
$698$		0			0
$699$	7.77622			0.294124
$700$		0			0
$701$	−19.3116			−0.729388			−0.364694	−	0.931127i	$-0.618826\pi$
$701$	−19.3116			−0.729388			−0.364694	+	0.931127i	$0.618826\pi$
$702$		0			0
$703$		−	17.7497i		−	0.669442i
$704$		0			0
$705$	−14.6305	−	10.1084i	−0.551018	−	0.380704i
$706$		0			0
$707$			8.21355i			0.308902i
$708$		0			0
$709$	−17.5481			−0.659034			−0.329517	−	0.944150i	$-0.606886\pi$
$709$	−17.5481			−0.659034			−0.329517	+	0.944150i	$0.606886\pi$
$710$		0			0
$711$	−9.06492			−0.339961
$712$		0			0
$713$		−	4.07541i		−	0.152625i
$714$		0			0
$715$	8.58049	−	12.4191i	0.320892	−	0.464448i
$716$		0			0
$717$		−	3.23833i		−	0.120938i
$718$		0			0
$719$	11.5933			0.432356			0.216178	−	0.976354i	$-0.430641\pi$
$719$	11.5933			0.432356			0.216178	+	0.976354i	$0.430641\pi$
$720$		0			0
$721$	−8.34788			−0.310891
$722$		0			0
$723$			9.74677i			0.362486i
$724$		0			0
$725$	−15.3575	−	40.6038i	−0.570362	−	1.50799i
$726$		0			0
$727$		−	5.68704i		−	0.210921i	−0.994424	−	0.105460i	$-0.966368\pi$
$727$		−	5.68704i		−	0.210921i	0.994424	−	0.105460i	$-0.0336316\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	8.04912			0.297708
$732$		0			0
$733$			4.70829i			0.173905i	0.996212	+	0.0869524i	$0.0277128\pi$
$733$			4.70829i			0.173905i	−0.996212	+	0.0869524i	$0.972287\pi$
$734$		0			0
$735$	−8.65081	+	12.5209i	−0.319090	+	0.461840i
$736$		0			0
$737$			46.4870i			1.71237i
$738$		0			0
$739$	3.31899			0.122091			0.0610455	−	0.998135i	$-0.480557\pi$
$739$	3.31899			0.122091			0.0610455	+	0.998135i	$0.480557\pi$
$740$		0			0
$741$	−3.29571			−0.121071
$742$		0			0
$743$		−	52.7027i		−	1.93347i	−0.255772	−	0.966737i	$-0.582330\pi$
$743$		−	52.7027i		−	1.93347i	0.255772	−	0.966737i	$-0.417670\pi$
$744$		0			0
$745$	−24.3556	−	16.8275i	−0.892321	−	0.616514i
$746$		0			0
$747$		−	0.983785i		−	0.0359948i
$748$		0			0
$749$	0.222480			0.00812925
$750$		0			0
$751$	46.1568			1.68428			0.842142	−	0.539256i	$-0.181294\pi$
$751$	46.1568			1.68428			0.842142	+	0.539256i	$0.181294\pi$
$752$		0			0
$753$			19.7066i			0.718148i
$754$		0			0
$755$	32.3754	+	22.3685i	1.17826	+	0.814072i
$756$		0			0
$757$			15.2118i			0.552882i	0.961031	+	0.276441i	$0.0891550\pi$
$757$			15.2118i			0.552882i	−0.961031	+	0.276441i	$0.910845\pi$
$758$		0			0
$759$	4.37833			0.158923
$760$		0			0
$761$	51.3859			1.86274			0.931370	−	0.364075i	$-0.118615\pi$
$761$	51.3859			1.86274			0.931370	+	0.364075i	$0.118615\pi$
$762$		0			0
$763$		−	3.22673i		−	0.116816i
$764$		0			0
$765$	−2.51959	+	3.64676i	−0.0910959	+	0.131849i
$766$		0			0
$767$			0.510909i			0.0184479i
$768$		0			0
$769$	−5.82921			−0.210207			−0.105103	−	0.994461i	$-0.533517\pi$
$769$	−5.82921			−0.210207			−0.105103	+	0.994461i	$0.533517\pi$
$770$		0			0
$771$	6.57303			0.236722
$772$		0			0
$773$		−	19.3010i		−	0.694207i	−0.937827	−	0.347104i	$-0.887165\pi$
$773$		−	19.3010i		−	0.694207i	0.937827	−	0.347104i	$-0.112835\pi$
$774$		0			0
$775$	−7.20876	−	19.0593i	−0.258946	−	0.684631i
$776$		0			0
$777$			3.65738i			0.131208i
$778$		0			0
$779$	7.25855			0.260065
$780$		0			0
$781$	−0.501158			−0.0179328
$782$		0			0
$783$		−	8.68220i		−	0.310277i
$784$		0			0
$785$	7.72203	−	11.1766i	0.275611	−	0.398910i
$786$		0			0
$787$			32.6018i			1.16213i	0.813857	+	0.581065i	$0.197364\pi$
$787$			32.6018i			1.16213i	−0.813857	+	0.581065i	$0.802636\pi$
$788$		0			0
$789$	16.0540			0.571536
$790$		0			0
$791$	−0.597757			−0.0212538
$792$		0			0
$793$		−	9.93753i		−	0.352892i
$794$		0			0
$795$	8.61610	+	5.95295i	0.305581	+	0.211129i
$796$		0			0
$797$		−	27.9847i		−	0.991270i	−0.868531	−	0.495635i	$-0.834935\pi$
$797$		−	27.9847i		−	0.991270i	0.868531	−	0.495635i	$-0.165065\pi$
$798$		0			0
$799$	−15.7646			−0.557713
$800$		0			0
$801$	−16.0850			−0.568336
$802$		0			0
$803$			25.5601i			0.901996i
$804$		0			0
$805$	0.810273	+	0.559826i	0.0285584	+	0.0197313i
$806$		0			0
$807$		−	3.78234i		−	0.133145i
$808$		0			0
$809$	0.910124			0.0319983			0.0159991	−	0.999872i	$-0.494907\pi$
$809$	0.910124			0.0319983			0.0159991	+	0.999872i	$0.494907\pi$
$810$		0			0
$811$	−31.0612			−1.09071			−0.545354	−	0.838206i	$-0.683605\pi$
$811$	−31.0612			−1.09071			−0.545354	+	0.838206i	$0.683605\pi$
$812$		0			0
$813$			26.1008i			0.915395i
$814$		0			0
$815$	24.9838	−	36.1607i	0.875145	−	1.26665i
$816$		0			0
$817$			8.67947i			0.303656i
$818$		0			0
$819$	0.679091			0.0237294
$820$		0			0
$821$	−5.96427			−0.208155			−0.104077	−	0.994569i	$-0.533189\pi$
$821$	−5.96427			−0.208155			−0.104077	+	0.994569i	$0.533189\pi$
$822$		0			0
$823$			2.06165i			0.0718644i	0.999354	+	0.0359322i	$0.0114400\pi$
$823$			2.06165i			0.0718644i	−0.999354	+	0.0359322i	$0.988560\pi$
$824$		0			0
$825$	20.4760	−	7.74458i	0.712883	−	0.269632i
$826$		0			0
$827$			28.3767i			0.986755i	0.869815	+	0.493377i	$0.164238\pi$
$827$			28.3767i			0.986755i	−0.869815	+	0.493377i	$0.835762\pi$
$828$		0			0
$829$	3.52893			0.122565			0.0612824	−	0.998120i	$-0.480481\pi$
$829$	3.52893			0.122565			0.0612824	+	0.998120i	$0.480481\pi$
$830$		0			0
$831$	−28.9682			−1.00490
$832$		0			0
$833$			13.4914i			0.467451i
$834$		0			0
$835$	16.0007	−	23.1589i	0.553728	−	0.801447i
$836$		0			0
$837$		−	4.07541i		−	0.140867i
$838$		0			0
$839$	6.09369			0.210377			0.105189	−	0.994452i	$-0.466455\pi$
$839$	6.09369			0.210377			0.105189	+	0.994452i	$0.466455\pi$
$840$		0			0
$841$	46.3807			1.59933
$842$		0			0
$843$		−	2.01319i		−	0.0693379i
$844$		0			0
$845$	−19.5424	−	13.5021i	−0.672280	−	0.464485i
$846$		0			0
$847$			3.59832i			0.123640i
$848$		0			0
$849$	−6.53160			−0.224164
$850$		0			0
$851$	−8.30387			−0.284653
$852$		0			0
$853$		−	6.06875i		−	0.207790i	−0.994588	−	0.103895i	$-0.966869\pi$
$853$		−	6.06875i		−	0.207790i	0.994588	−	0.103895i	$-0.0331306\pi$
$854$		0			0
$855$	−3.93235	−	2.71690i	−0.134484	−	0.0929161i
$856$		0			0
$857$		−	47.8622i		−	1.63494i	−0.575971	−	0.817470i	$-0.695376\pi$
$857$		−	47.8622i		−	1.63494i	0.575971	−	0.817470i	$-0.304624\pi$
$858$		0			0
$859$	11.4023			0.389042			0.194521	−	0.980898i	$-0.437685\pi$
$859$	11.4023			0.389042			0.194521	+	0.980898i	$0.437685\pi$
$860$		0			0
$861$	−1.49565			−0.0509715
$862$		0			0
$863$			13.5553i			0.461429i	0.973021	+	0.230715i	$0.0741064\pi$
$863$			13.5553i			0.461429i	−0.973021	+	0.230715i	$0.925894\pi$
$864$		0			0
$865$	−26.4513	+	38.2847i	−0.899372	+	1.30172i
$866$		0			0
$867$		−	13.0706i		−	0.443899i
$868$		0			0
$869$	39.6892			1.34637
$870$		0			0
$871$	−16.3705			−0.554694
$872$		0			0
$873$		−	6.11195i		−	0.206858i
$874$		0			0
$875$	4.77962	+	1.18487i	0.161581	+	0.0400560i
$876$		0			0
$877$			51.0505i			1.72385i	0.507033	+	0.861927i	$0.330742\pi$
$877$			51.0505i			1.72385i	−0.507033	+	0.861927i	$0.669258\pi$
$878$		0			0
$879$	−27.2819			−0.920196
$880$		0			0
$881$	−34.0651			−1.14768			−0.573841	−	0.818966i	$-0.694547\pi$
$881$	−34.0651			−1.14768			−0.573841	+	0.818966i	$0.694547\pi$
$882$		0			0
$883$		−	25.4229i		−	0.855549i	−0.903885	−	0.427775i	$-0.859298\pi$
$883$		−	25.4229i		−	0.855549i	0.903885	−	0.427775i	$-0.140702\pi$
$884$		0			0
$885$	−0.421181	+	0.609602i	−0.0141578	+	0.0204916i
$886$		0			0
$887$		−	41.7418i		−	1.40155i	−0.713382	−	0.700776i	$-0.752837\pi$
$887$		−	41.7418i		−	1.40155i	0.713382	−	0.700776i	$-0.247163\pi$
$888$		0			0
$889$	−4.29218			−0.143955
$890$		0			0
$891$	4.37833			0.146680
$892$		0			0
$893$		−	16.9992i		−	0.568856i
$894$		0			0
$895$	−15.8808	−	10.9722i	−0.530838	−	0.366762i
$896$		0			0
$897$			1.54184i			0.0514805i
$898$		0			0
$899$	35.3835			1.18011
$900$		0			0
$901$	9.28397			0.309294
$902$		0			0
$903$		−	1.78843i		−	0.0595152i
$904$		0			0
$905$	−21.3887	−	14.7777i	−0.710983	−	0.491226i
$906$		0			0
$907$			24.8662i			0.825670i	0.910806	+	0.412835i	$0.135461\pi$
$907$			24.8662i			0.825670i	−0.910806	+	0.412835i	$0.864539\pi$
$908$		0			0
$909$	−18.6484			−0.618529
$910$		0			0
$911$	−0.430252			−0.0142549			−0.00712744	−	0.999975i	$-0.502269\pi$
$911$	−0.430252			−0.0142549			−0.00712744	+	0.999975i	$0.502269\pi$
$912$		0			0
$913$			4.30734i			0.142552i
$914$		0			0
$915$	8.19225	−	11.8572i	0.270827	−	0.391986i
$916$		0			0
$917$			2.79180i			0.0921934i
$918$		0			0
$919$	−31.1930			−1.02896			−0.514481	−	0.857502i	$-0.672015\pi$
$919$	−31.1930			−1.02896			−0.514481	+	0.857502i	$0.672015\pi$
$920$		0			0
$921$	6.38858			0.210511
$922$		0			0
$923$		−	0.176484i		−	0.00580904i
$924$		0			0
$925$	−38.8344	+	14.6882i	−1.27687	+	0.482947i
$926$		0			0
$927$		−	18.9534i		−	0.622511i
$928$		0			0
$929$	−29.9373			−0.982210			−0.491105	−	0.871100i	$-0.663407\pi$
$929$	−29.9373			−0.982210			−0.491105	+	0.871100i	$0.663407\pi$
$930$		0			0
$931$	−14.5480			−0.476791
$932$		0			0
$933$			1.42344i			0.0466014i
$934$		0			0
$935$	11.0316	−	15.9667i	0.360772	−	0.522169i
$936$		0			0
$937$		−	26.4873i		−	0.865301i	−0.901562	−	0.432650i	$-0.857578\pi$
$937$		−	26.4873i		−	0.865301i	0.901562	−	0.432650i	$-0.142422\pi$
$938$		0			0
$939$	−5.54673			−0.181011
$940$		0			0
$941$	−29.6599			−0.966885			−0.483443	−	0.875376i	$-0.660614\pi$
$941$	−29.6599			−0.966885			−0.483443	+	0.875376i	$0.660614\pi$
$942$		0			0
$943$		−	3.39578i		−	0.110582i
$944$		0			0
$945$	0.810273	+	0.559826i	0.0263582	+	0.0182111i
$946$		0			0
$947$			9.45905i			0.307378i	0.988119	+	0.153689i	$0.0491154\pi$
$947$			9.45905i			0.307378i	−0.988119	+	0.153689i	$0.950885\pi$
$948$		0			0
$949$	−9.00104			−0.292186
$950$		0			0
$951$	19.7357			0.639975
$952$		0			0
$953$		−	24.1695i		−	0.782926i	−0.920194	−	0.391463i	$-0.871969\pi$
$953$		−	24.1695i		−	0.782926i	0.920194	−	0.391463i	$-0.128031\pi$
$954$		0			0
$955$	−29.2719	−	20.2243i	−0.947216	−	0.654442i
$956$		0			0
$957$			38.0136i			1.22880i
$958$		0			0
$959$	2.87933			0.0929785
$960$		0			0
$961$	−14.3910			−0.464227
$962$		0			0
$963$			0.505129i			0.0162776i
$964$		0			0
$965$	−25.0861	+	36.3088i	−0.807551	+	1.16882i
$966$		0			0
$967$		−	2.60086i		−	0.0836382i	−0.999125	−	0.0418191i	$-0.986685\pi$
$967$		−	2.60086i		−	0.0836382i	0.999125	−	0.0418191i	$-0.0133153\pi$
$968$		0			0
$969$	−4.23717			−0.136117
$970$		0			0
$971$	−56.2190			−1.80415			−0.902076	−	0.431577i	$-0.857957\pi$
$971$	−56.2190			−1.80415			−0.902076	+	0.431577i	$0.857957\pi$
$972$		0			0
$973$		−	6.20165i		−	0.198816i
$974$		0			0
$975$	2.72727	+	7.21067i	0.0873426	+	0.230926i
$976$		0			0
$977$			9.57871i			0.306450i	0.988191	+	0.153225i	$0.0489659\pi$
$977$			9.57871i			0.306450i	−0.988191	+	0.153225i	$0.951034\pi$
$978$		0			0
$979$	70.4255			2.25081
$980$		0			0
$981$	7.32612			0.233905
$982$		0			0
$983$		−	4.45103i		−	0.141966i	−0.997478	−	0.0709829i	$-0.977386\pi$
$983$		−	4.45103i		−	0.141966i	0.997478	−	0.0709829i	$-0.0226136\pi$
$984$		0			0
$985$	20.4450	−	29.5914i	0.651433	−	0.942861i
$986$		0			0
$987$			3.50273i			0.111493i
$988$		0			0
$989$	4.06053			0.129117
$990$		0			0
$991$	47.1285			1.49708			0.748542	−	0.663087i	$-0.230754\pi$
$991$	47.1285			1.49708			0.748542	+	0.663087i	$0.230754\pi$
$992$		0			0
$993$		−	25.2951i		−	0.802717i
$994$		0			0
$995$	31.0281	+	21.4376i	0.983657	+	0.679619i
$996$		0			0
$997$		−	51.2950i		−	1.62453i	−0.583290	−	0.812264i	$-0.698235\pi$
$997$		−	51.2950i		−	1.62453i	0.583290	−	0.812264i	$-0.301765\pi$
$998$		0			0
$999$	−8.30387			−0.262723

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2760.2.k.e.2209.2	✓	16
5.4	even	2	inner	2760.2.k.e.2209.10	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.k.e.2209.2	✓	16	1.1	even	1	trivial
2760.2.k.e.2209.10	yes	16	5.4	even	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:	\( N \)	\(=\)	\( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2760.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(-1.83968 - 1.27105i) q^{5} +0.440442i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(-1.83968 - 1.27105i) q^{5} +0.440442i q^{7} -1.00000 q^{9} +4.37833 q^{11} +1.54184i q^{13} +(-1.27105 + 1.83968i) q^{15} +1.98228i q^{17} -2.13752 q^{19} +0.440442 q^{21} +1.00000i q^{23} +(1.76884 + 4.67666i) q^{25} +1.00000i q^{27} -8.68220 q^{29} -4.07541 q^{31} -4.37833i q^{33} +(0.559826 - 0.810273i) q^{35} +8.30387i q^{37} +1.54184 q^{39} -3.39578 q^{41} -4.06053i q^{43} +(1.83968 + 1.27105i) q^{45} +7.95277i q^{47} +6.80601 q^{49} +1.98228 q^{51} -4.68348i q^{53} +(-8.05473 - 5.56510i) q^{55} +2.13752i q^{57} +0.331363 q^{59} -6.44524 q^{61} -0.440442i q^{63} +(1.95976 - 2.83649i) q^{65} +10.6175i q^{67} +1.00000 q^{69} -0.114463 q^{71} +5.83786i q^{73} +(4.67666 - 1.76884i) q^{75} +1.92840i q^{77} +9.06492 q^{79} +1.00000 q^{81} +0.983785i q^{83} +(2.51959 - 3.64676i) q^{85} +8.68220i q^{87} +16.0850 q^{89} -0.679091 q^{91} +4.07541i q^{93} +(3.93235 + 2.71690i) q^{95} +6.11195i q^{97} -4.37833 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{5} - 16 q^{9}+O(q^{10}) \) 16 * q - 2 * q^5 - 16 * q^9	\( 16 q - 2 q^{5} - 16 q^{9} - 2 q^{15} - 8 q^{19} + 6 q^{21} + 22 q^{29} + 30 q^{31} + 2 q^{35} - 4 q^{39} - 22 q^{41} + 2 q^{45} - 38 q^{49} + 2 q^{51} + 12 q^{55} - 6 q^{59} - 4 q^{61} + 20 q^{65} + 16 q^{69} - 10 q^{71} - 4 q^{75} - 84 q^{79} + 16 q^{81} + 22 q^{85} + 16 q^{89} + 36 q^{91} + 76 q^{95}+O(q^{100}) \) 16 * q - 2 * q^5 - 16 * q^9 - 2 * q^15 - 8 * q^19 + 6 * q^21 + 22 * q^29 + 30 * q^31 + 2 * q^35 - 4 * q^39 - 22 * q^41 + 2 * q^45 - 38 * q^49 + 2 * q^51 + 12 * q^55 - 6 * q^59 - 4 * q^61 + 20 * q^65 + 16 * q^69 - 10 * q^71 - 4 * q^75 - 84 * q^79 + 16 * q^81 + 22 * q^85 + 16 * q^89 + 36 * q^91 + 76 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more