LMFDB - Embedded newform 2760.2.k.d.2209.13

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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} + 2 x^{12} + 112 x^{10} - 228 x^{9} + 232 x^{8} + 40 x^{7} + 1316 x^{6} - 2688 x^{5} + \cdots + 128 $ x^14 - 2x^13 + 2x^12 + 112x^10 - 228x^9 + 232x^8 + 40x^7 + 1316x^6 - 2688x^5 + 2752x^4 + 608x^3 + 256x^2 - 256x + 128
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2209.13
Root			$1.31001 + 1.31001i$ of defining polynomial
Character	$\chi$	$=$	2760.2209
Dual form			2760.2.k.d.2209.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(1.31001 + 1.81215i) q^{5} +2.34779i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(1.31001 + 1.81215i) q^{5} +2.34779i q^{7} -1.00000 q^{9} -5.12790 q^{11} +3.04538i q^{13} +(-1.81215 + 1.31001i) q^{15} -5.94899i q^{17} +1.42109 q^{19} -2.34779 q^{21} +1.00000i q^{23} +(-1.56775 + 4.74786i) q^{25} -1.00000i q^{27} -3.37811 q^{29} -9.68542 q^{31} -5.12790i q^{33} +(-4.25455 + 3.07563i) q^{35} -2.22974i q^{37} -3.04538 q^{39} +0.660506 q^{41} +7.61377i q^{43} +(-1.31001 - 1.81215i) q^{45} -1.56781i q^{47} +1.48787 q^{49} +5.94899 q^{51} -7.23139i q^{53} +(-6.71761 - 9.29251i) q^{55} +1.42109i q^{57} +6.22029 q^{59} -3.90988 q^{61} -2.34779i q^{63} +(-5.51867 + 3.98948i) q^{65} +2.05818i q^{67} -1.00000 q^{69} -9.61577 q^{71} +0.0524738i q^{73} +(-4.74786 - 1.56775i) q^{75} -12.0393i q^{77} -9.76701 q^{79} +1.00000 q^{81} -11.2737i q^{83} +(10.7804 - 7.79324i) q^{85} -3.37811i q^{87} +1.96261 q^{89} -7.14992 q^{91} -9.68542i q^{93} +(1.86164 + 2.57522i) q^{95} -6.67370i q^{97} +5.12790 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 2 q^{5} - 14 q^{9}+O(q^{10}) $ 14 * q + 2 * q^5 - 14 * q^9	$ 14 q + 2 q^{5} - 14 q^{9} + 8 q^{11} - 4 q^{15} + 8 q^{19} - 10 q^{21} - 10 q^{25} + 26 q^{29} - 18 q^{31} - 10 q^{35} + 12 q^{39} - 2 q^{41} - 2 q^{45} + 4 q^{49} - 6 q^{51} - 32 q^{55} - 10 q^{59} - 24 q^{61} - 36 q^{65} - 14 q^{69} - 38 q^{71} + 4 q^{79} + 14 q^{81} - 14 q^{85} - 16 q^{89} - 4 q^{91} - 8 q^{95} - 8 q^{99}+O(q^{100}) $ 14 * q + 2 * q^5 - 14 * q^9 + 8 * q^11 - 4 * q^15 + 8 * q^19 - 10 * q^21 - 10 * q^25 + 26 * q^29 - 18 * q^31 - 10 * q^35 + 12 * q^39 - 2 * q^41 - 2 * q^45 + 4 * q^49 - 6 * q^51 - 32 * q^55 - 10 * q^59 - 24 * q^61 - 36 * q^65 - 14 * q^69 - 38 * q^71 + 4 * q^79 + 14 * q^81 - 14 * q^85 - 16 * q^89 - 4 * q^91 - 8 * q^95 - 8 * q^99

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
$5$	1.31001	+	1.81215i	0.585854	+	0.810416i
$5$	1.31001	+	1.81215i	0.585854	+	0.810416i
$6$		0			0
$7$			2.34779i			0.887383i	0.896180	+	0.443691i	$0.146331\pi$
$7$			2.34779i			0.887383i	−0.896180	+	0.443691i	$0.853669\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−5.12790			−1.54612			−0.773061	−	0.634332i	$-0.781275\pi$
$11$	−5.12790			−1.54612			−0.773061	+	0.634332i	$0.781275\pi$
$12$		0			0
$13$			3.04538i			0.844636i	0.906448	+	0.422318i	$0.138784\pi$
$13$			3.04538i			0.844636i	−0.906448	+	0.422318i	$0.861216\pi$
$14$		0			0
$15$	−1.81215	+	1.31001i	−0.467894	+	0.338243i
$16$		0			0
$17$		−	5.94899i		−	1.44284i	−0.692496	−	0.721421i	$-0.743489\pi$
$17$		−	5.94899i		−	1.44284i	0.692496	−	0.721421i	$-0.256511\pi$
$18$		0			0
$19$	1.42109			0.326020			0.163010	−	0.986624i	$-0.447880\pi$
$19$	1.42109			0.326020			0.163010	+	0.986624i	$0.447880\pi$
$20$		0			0
$21$	−2.34779			−0.512331
$22$		0			0
$23$			1.00000i			0.208514i
$23$			1.00000i			0.208514i
$24$		0			0
$25$	−1.56775	+	4.74786i	−0.313550	+	0.949572i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−3.37811			−0.627300			−0.313650	−	0.949539i	$-0.601552\pi$
$29$	−3.37811			−0.627300			−0.313650	+	0.949539i	$0.601552\pi$
$30$		0			0
$31$	−9.68542			−1.73955			−0.869776	−	0.493446i	$-0.835737\pi$
$31$	−9.68542			−1.73955			−0.869776	+	0.493446i	$0.835737\pi$
$32$		0			0
$33$		−	5.12790i		−	0.892654i
$34$		0			0
$35$	−4.25455	+	3.07563i	−0.719149	+	0.519877i
$36$		0			0
$37$		−	2.22974i		−	0.366567i	−0.983060	−	0.183284i	$-0.941327\pi$
$37$		−	2.22974i		−	0.366567i	0.983060	−	0.183284i	$-0.0586727\pi$
$38$		0			0
$39$	−3.04538			−0.487651
$40$		0			0
$41$	0.660506			0.103154			0.0515768	−	0.998669i	$-0.483575\pi$
$41$	0.660506			0.103154			0.0515768	+	0.998669i	$0.483575\pi$
$42$		0			0
$43$			7.61377i			1.16109i	0.814228	+	0.580545i	$0.197160\pi$
$43$			7.61377i			1.16109i	−0.814228	+	0.580545i	$0.802840\pi$
$44$		0			0
$45$	−1.31001	−	1.81215i	−0.195285	−	0.270139i
$46$		0			0
$47$		−	1.56781i		−	0.228689i	−0.993441	−	0.114345i	$-0.963523\pi$
$47$		−	1.56781i		−	0.228689i	0.993441	−	0.114345i	$-0.0364768\pi$
$48$		0			0
$49$	1.48787			0.212552
$50$		0			0
$51$	5.94899			0.833026
$52$		0			0
$53$		−	7.23139i		−	0.993307i	−0.867949	−	0.496654i	$-0.834562\pi$
$53$		−	7.23139i		−	0.993307i	0.867949	−	0.496654i	$-0.165438\pi$
$54$		0			0
$55$	−6.71761	−	9.29251i	−0.905802	−	1.25300i
$56$		0			0
$57$			1.42109i			0.188228i
$58$		0			0
$59$	6.22029			0.809812			0.404906	−	0.914358i	$-0.367304\pi$
$59$	6.22029			0.809812			0.404906	+	0.914358i	$0.367304\pi$
$60$		0			0
$61$	−3.90988			−0.500609			−0.250305	−	0.968167i	$-0.580531\pi$
$61$	−3.90988			−0.500609			−0.250305	+	0.968167i	$0.580531\pi$
$62$		0			0
$63$		−	2.34779i		−	0.295794i
$64$		0			0
$65$	−5.51867	+	3.98948i	−0.684507	+	0.494834i
$66$		0			0
$67$			2.05818i			0.251446i	0.992065	+	0.125723i	$0.0401251\pi$
$67$			2.05818i			0.251446i	−0.992065	+	0.125723i	$0.959875\pi$
$68$		0			0
$69$	−1.00000			−0.120386
$70$		0			0
$71$	−9.61577			−1.14118			−0.570591	−	0.821234i	$-0.693286\pi$
$71$	−9.61577			−1.14118			−0.570591	+	0.821234i	$0.693286\pi$
$72$		0			0
$73$			0.0524738i			0.00614159i	0.999995	+	0.00307080i	$0.000977466\pi$
$73$			0.0524738i			0.00614159i	−0.999995	+	0.00307080i	$0.999023\pi$
$74$		0			0
$75$	−4.74786	−	1.56775i	−0.548236	−	0.181028i
$76$		0			0
$77$		−	12.0393i		−	1.37200i
$78$		0			0
$79$	−9.76701			−1.09887			−0.549437	−	0.835535i	$-0.685158\pi$
$79$	−9.76701			−1.09887			−0.549437	+	0.835535i	$0.685158\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	11.2737i		−	1.23745i	−0.785608	−	0.618725i	$-0.787650\pi$
$83$		−	11.2737i		−	1.23745i	0.785608	−	0.618725i	$-0.212350\pi$
$84$		0			0
$85$	10.7804	−	7.79324i	1.16930	−	0.845295i
$86$		0			0
$87$		−	3.37811i		−	0.362172i
$88$		0			0
$89$	1.96261			0.208036			0.104018	−	0.994575i	$-0.466830\pi$
$89$	1.96261			0.208036			0.104018	+	0.994575i	$0.466830\pi$
$90$		0			0
$91$	−7.14992			−0.749516
$92$		0			0
$93$		−	9.68542i		−	1.00433i
$94$		0			0
$95$	1.86164	+	2.57522i	0.191000	+	0.264212i
$96$		0			0
$97$		−	6.67370i		−	0.677611i	−0.940856	−	0.338806i	$-0.889977\pi$
$97$		−	6.67370i		−	0.677611i	0.940856	−	0.338806i	$-0.110023\pi$
$98$		0			0
$99$	5.12790			0.515374
$100$		0			0
$101$	14.5419			1.44697			0.723486	−	0.690339i	$-0.242539\pi$
$101$	14.5419			1.44697			0.723486	+	0.690339i	$0.242539\pi$
$102$		0			0
$103$		−	6.55181i		−	0.645569i	−0.946472	−	0.322785i	$-0.895381\pi$
$103$		−	6.55181i		−	0.645569i	0.946472	−	0.322785i	$-0.104619\pi$
$104$		0			0
$105$	−3.07563	−	4.25455i	−0.300151	−	0.415201i
$106$		0			0
$107$			18.7537i			1.81299i	0.422222	+	0.906493i	$0.361250\pi$
$107$			18.7537i			1.81299i	−0.422222	+	0.906493i	$0.638750\pi$
$108$		0			0
$109$	−12.3966			−1.18738			−0.593692	−	0.804693i	$-0.702330\pi$
$109$	−12.3966			−1.18738			−0.593692	+	0.804693i	$0.702330\pi$
$110$		0			0
$111$	2.22974			0.211638
$112$		0			0
$113$			17.8456i			1.67877i	0.543538	+	0.839384i	$0.317084\pi$
$113$			17.8456i			1.67877i	−0.543538	+	0.839384i	$0.682916\pi$
$114$		0			0
$115$	−1.81215	+	1.31001i	−0.168984	+	0.122159i
$116$		0			0
$117$		−	3.04538i		−	0.281545i
$118$		0			0
$119$	13.9670			1.28035
$120$		0			0
$121$	15.2954			1.39049
$122$		0			0
$123$			0.660506i			0.0595558i
$124$		0			0
$125$	−10.6576	+	3.37875i	−0.953243	+	0.302205i
$126$		0			0
$127$			2.24525i			0.199234i	0.995026	+	0.0996168i	$0.0317617\pi$
$127$			2.24525i			0.199234i	−0.995026	+	0.0996168i	$0.968238\pi$
$128$		0			0
$129$	−7.61377			−0.670355
$130$		0			0
$131$	15.6589			1.36812			0.684062	−	0.729423i	$-0.260212\pi$
$131$	15.6589			1.36812			0.684062	+	0.729423i	$0.260212\pi$
$132$		0			0
$133$			3.33642i			0.289304i
$134$		0			0
$135$	1.81215	−	1.31001i	0.155965	−	0.112748i
$136$		0			0
$137$			5.36091i			0.458014i	0.973425	+	0.229007i	$0.0735478\pi$
$137$			5.36091i			0.458014i	−0.973425	+	0.229007i	$0.926452\pi$
$138$		0			0
$139$	11.2880			0.957435			0.478718	−	0.877969i	$-0.341102\pi$
$139$	11.2880			0.957435			0.478718	+	0.877969i	$0.341102\pi$
$140$		0			0
$141$	1.56781			0.132034
$142$		0			0
$143$		−	15.6164i		−	1.30591i
$144$		0			0
$145$	−4.42536	−	6.12163i	−0.367506	−	0.508374i
$146$		0			0
$147$			1.48787i			0.122717i
$148$		0			0
$149$	1.78557			0.146279			0.0731397	−	0.997322i	$-0.476698\pi$
$149$	1.78557			0.146279			0.0731397	+	0.997322i	$0.476698\pi$
$150$		0			0
$151$	−14.1905			−1.15480			−0.577402	−	0.816460i	$-0.695933\pi$
$151$	−14.1905			−1.15480			−0.577402	+	0.816460i	$0.695933\pi$
$152$		0			0
$153$			5.94899i			0.480948i
$154$		0			0
$155$	−12.6880	−	17.5514i	−1.01912	−	1.40976i
$156$		0			0
$157$		−	2.63045i		−	0.209933i	−0.994476	−	0.104967i	$-0.966526\pi$
$157$		−	2.63045i		−	0.209933i	0.994476	−	0.104967i	$-0.0334736\pi$
$158$		0			0
$159$	7.23139			0.573486
$160$		0			0
$161$	−2.34779			−0.185032
$162$		0			0
$163$			2.92501i			0.229104i	0.993417	+	0.114552i	$0.0365433\pi$
$163$			2.92501i			0.229104i	−0.993417	+	0.114552i	$0.963457\pi$
$164$		0			0
$165$	9.29251	−	6.71761i	0.723421	−	0.522965i
$166$		0			0
$167$			9.90446i			0.766430i	0.923659	+	0.383215i	$0.125183\pi$
$167$			9.90446i			0.766430i	−0.923659	+	0.383215i	$0.874817\pi$
$168$		0			0
$169$	3.72566			0.286589
$170$		0			0
$171$	−1.42109			−0.108673
$172$		0			0
$173$		−	13.6303i		−	1.03629i	−0.855293	−	0.518145i	$-0.826623\pi$
$173$		−	13.6303i		−	1.03629i	0.855293	−	0.518145i	$-0.173377\pi$
$174$		0			0
$175$	−11.1470	−	3.68075i	−0.842633	−	0.278238i
$176$		0			0
$177$			6.22029i			0.467545i
$178$		0			0
$179$	−23.9064			−1.78684			−0.893422	−	0.449218i	$-0.851703\pi$
$179$	−23.9064			−1.78684			−0.893422	+	0.449218i	$0.851703\pi$
$180$		0			0
$181$	−6.16844			−0.458497			−0.229248	−	0.973368i	$-0.573627\pi$
$181$	−6.16844			−0.458497			−0.229248	+	0.973368i	$0.573627\pi$
$182$		0			0
$183$		−	3.90988i		−	0.289027i
$184$		0			0
$185$	4.04062	−	2.92098i	0.297072	−	0.214755i
$186$		0			0
$187$			30.5059i			2.23081i
$188$		0			0
$189$	2.34779			0.170777
$190$		0			0
$191$	7.09026			0.513033			0.256517	−	0.966540i	$-0.417425\pi$
$191$	7.09026			0.513033			0.256517	+	0.966540i	$0.417425\pi$
$192$		0			0
$193$		−	15.5694i		−	1.12071i	−0.828253	−	0.560355i	$-0.810665\pi$
$193$		−	15.5694i		−	1.12071i	0.828253	−	0.560355i	$-0.189335\pi$
$194$		0			0
$195$	−3.98948	−	5.51867i	−0.285692	−	0.395200i
$196$		0			0
$197$			19.4095i			1.38287i	0.722439	+	0.691435i	$0.243021\pi$
$197$			19.4095i			1.38287i	−0.722439	+	0.691435i	$0.756979\pi$
$198$		0			0
$199$	−11.9937			−0.850209			−0.425105	−	0.905144i	$-0.639763\pi$
$199$	−11.9937			−0.850209			−0.425105	+	0.905144i	$0.639763\pi$
$200$		0			0
$201$	−2.05818			−0.145172
$202$		0			0
$203$		−	7.93111i		−	0.556655i
$204$		0			0
$205$	0.865269	+	1.19693i	0.0604330	+	0.0835974i
$206$		0			0
$207$		−	1.00000i		−	0.0695048i
$208$		0			0
$209$	−7.28720			−0.504066
$210$		0			0
$211$	−21.2488			−1.46283			−0.731413	−	0.681935i	$-0.761139\pi$
$211$	−21.2488			−1.46283			−0.731413	+	0.681935i	$0.761139\pi$
$212$		0			0
$213$		−	9.61577i		−	0.658862i
$214$		0			0
$215$	−13.7973	+	9.97412i	−0.940966	+	0.680229i
$216$		0			0
$217$		−	22.7394i		−	1.54365i
$218$		0			0
$219$	−0.0524738			−0.00354585
$220$		0			0
$221$	18.1169			1.21868
$222$		0			0
$223$			6.96844i			0.466641i	0.972400	+	0.233321i	$0.0749592\pi$
$223$			6.96844i			0.466641i	−0.972400	+	0.233321i	$0.925041\pi$
$224$		0			0
$225$	1.56775	−	4.74786i	0.104517	−	0.316524i
$226$		0			0
$227$		−	22.5332i		−	1.49558i	−0.663935	−	0.747790i	$-0.731115\pi$
$227$		−	22.5332i		−	1.49558i	0.663935	−	0.747790i	$-0.268885\pi$
$228$		0			0
$229$	2.79145			0.184465			0.0922323	−	0.995738i	$-0.470600\pi$
$229$	2.79145			0.184465			0.0922323	+	0.995738i	$0.470600\pi$
$230$		0			0
$231$	12.0393			0.792125
$232$		0			0
$233$		−	20.1342i		−	1.31904i	−0.751688	−	0.659518i	$-0.770760\pi$
$233$		−	20.1342i		−	1.31904i	0.751688	−	0.659518i	$-0.229240\pi$
$234$		0			0
$235$	2.84111	−	2.05385i	0.185333	−	0.133978i
$236$		0			0
$237$		−	9.76701i		−	0.634436i
$238$		0			0
$239$	−23.5778			−1.52512			−0.762560	−	0.646917i	$-0.776058\pi$
$239$	−23.5778			−1.52512			−0.762560	+	0.646917i	$0.776058\pi$
$240$		0			0
$241$	−12.4333			−0.800899			−0.400449	−	0.916319i	$-0.631146\pi$
$241$	−12.4333			−0.800899			−0.400449	+	0.916319i	$0.631146\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	1.94912	+	2.69623i	0.124525	+	0.172256i
$246$		0			0
$247$			4.32775i			0.275368i
$248$		0			0
$249$	11.2737			0.714442
$250$		0			0
$251$	4.94900			0.312378			0.156189	−	0.987727i	$-0.450079\pi$
$251$	4.94900			0.312378			0.156189	+	0.987727i	$0.450079\pi$
$252$		0			0
$253$		−	5.12790i		−	0.322389i
$254$		0			0
$255$	7.79324	+	10.7804i	0.488032	+	0.675098i
$256$		0			0
$257$		−	6.81640i		−	0.425195i	−0.977140	−	0.212598i	$-0.931808\pi$
$257$		−	6.81640i		−	0.425195i	0.977140	−	0.212598i	$-0.0681924\pi$
$258$		0			0
$259$	5.23497			0.325285
$260$		0			0
$261$	3.37811			0.209100
$262$		0			0
$263$		−	0.723986i		−	0.0446429i	−0.999751	−	0.0223215i	$-0.992894\pi$
$263$		−	0.723986i		−	0.0446429i	0.999751	−	0.0223215i	$-0.00710573\pi$
$264$		0			0
$265$	13.1043	−	9.47319i	0.804993	−	0.581933i
$266$		0			0
$267$			1.96261i			0.120110i
$268$		0			0
$269$	−2.42725			−0.147992			−0.0739960	−	0.997259i	$-0.523575\pi$
$269$	−2.42725			−0.147992			−0.0739960	+	0.997259i	$0.523575\pi$
$270$		0			0
$271$	−22.4755			−1.36529			−0.682643	−	0.730752i	$-0.739170\pi$
$271$	−22.4755			−1.36529			−0.682643	+	0.730752i	$0.739170\pi$
$272$		0			0
$273$		−	7.14992i		−	0.432733i
$274$		0			0
$275$	8.03926	−	24.3466i	0.484786	−	1.46815i
$276$		0			0
$277$			19.7701i			1.18787i	0.804512	+	0.593936i	$0.202427\pi$
$277$			19.7701i			1.18787i	−0.804512	+	0.593936i	$0.797573\pi$
$278$		0			0
$279$	9.68542			0.579851
$280$		0			0
$281$	−13.8906			−0.828641			−0.414321	−	0.910131i	$-0.635981\pi$
$281$	−13.8906			−0.828641			−0.414321	+	0.910131i	$0.635981\pi$
$282$		0			0
$283$			32.7851i			1.94887i	0.224662	+	0.974437i	$0.427872\pi$
$283$			32.7851i			1.94887i	−0.224662	+	0.974437i	$0.572128\pi$
$284$		0			0
$285$	−2.57522	+	1.86164i	−0.152543	+	0.110274i
$286$		0			0
$287$			1.55073i			0.0915368i
$288$		0			0
$289$	−18.3905			−1.08179
$290$		0			0
$291$	6.67370			0.391219
$292$		0			0
$293$		−	21.6422i		−	1.26435i	−0.774826	−	0.632174i	$-0.782163\pi$
$293$		−	21.6422i		−	1.26435i	0.774826	−	0.632174i	$-0.217837\pi$
$294$		0			0
$295$	8.14864	+	11.2721i	0.474432	+	0.656285i
$296$		0			0
$297$			5.12790i			0.297551i
$298$		0			0
$299$	−3.04538			−0.176119
$300$		0			0
$301$	−17.8756			−1.03033
$302$		0			0
$303$			14.5419i			0.835410i
$304$		0			0
$305$	−5.12199	−	7.08528i	−0.293284	−	0.405702i
$306$		0			0
$307$			27.9277i			1.59392i	0.604032	+	0.796960i	$0.293560\pi$
$307$			27.9277i			1.59392i	−0.604032	+	0.796960i	$0.706440\pi$
$308$		0			0
$309$	6.55181			0.372720
$310$		0			0
$311$	−29.9204			−1.69663			−0.848314	−	0.529493i	$-0.822382\pi$
$311$	−29.9204			−1.69663			−0.848314	+	0.529493i	$0.822382\pi$
$312$		0			0
$313$			20.3987i			1.15300i	0.817096	+	0.576502i	$0.195583\pi$
$313$			20.3987i			1.15300i	−0.817096	+	0.576502i	$0.804417\pi$
$314$		0			0
$315$	4.25455	−	3.07563i	0.239716	−	0.173292i
$316$		0			0
$317$			18.5483i			1.04178i	0.853625	+	0.520888i	$0.174399\pi$
$317$			18.5483i			1.04178i	−0.853625	+	0.520888i	$0.825601\pi$
$318$		0			0
$319$	17.3226			0.969881
$320$		0			0
$321$	−18.7537			−1.04673
$322$		0			0
$323$		−	8.45404i		−	0.470395i
$324$		0			0
$325$	−14.4590	−	4.77439i	−0.802043	−	0.264835i
$326$		0			0
$327$		−	12.3966i		−	0.685536i
$328$		0			0
$329$	3.68090			0.202935
$330$		0			0
$331$	−1.29507			−0.0711837			−0.0355918	−	0.999366i	$-0.511332\pi$
$331$	−1.29507			−0.0711837			−0.0355918	+	0.999366i	$0.511332\pi$
$332$		0			0
$333$			2.22974i			0.122189i
$334$		0			0
$335$	−3.72971	+	2.69623i	−0.203776	+	0.147311i
$336$		0			0
$337$			24.9099i			1.35693i	0.734634	+	0.678464i	$0.237354\pi$
$337$			24.9099i			1.35693i	−0.734634	+	0.678464i	$0.762646\pi$
$338$		0			0
$339$	−17.8456			−0.969238
$340$		0			0
$341$	49.6659			2.68956
$342$		0			0
$343$			19.9278i			1.07600i
$344$		0			0
$345$	−1.31001	−	1.81215i	−0.0705286	−	0.0975627i
$346$		0			0
$347$			3.49215i			0.187469i	0.995597	+	0.0937343i	$0.0298804\pi$
$347$			3.49215i			0.187469i	−0.995597	+	0.0937343i	$0.970120\pi$
$348$		0			0
$349$	−9.97677			−0.534044			−0.267022	−	0.963690i	$-0.586040\pi$
$349$	−9.97677			−0.534044			−0.267022	+	0.963690i	$0.586040\pi$
$350$		0			0
$351$	3.04538			0.162550
$352$		0			0
$353$		−	11.8191i		−	0.629067i	−0.949246	−	0.314533i	$-0.898152\pi$
$353$		−	11.8191i		−	0.629067i	0.949246	−	0.314533i	$-0.101848\pi$
$354$		0			0
$355$	−12.5968	−	17.4252i	−0.668566	−	0.924833i
$356$		0			0
$357$			13.9670i			0.739212i
$358$		0			0
$359$	27.3226			1.44203			0.721017	−	0.692918i	$-0.243675\pi$
$359$	27.3226			1.44203			0.721017	+	0.692918i	$0.243675\pi$
$360$		0			0
$361$	−16.9805			−0.893711
$362$		0			0
$363$			15.2954i			0.802801i
$364$		0			0
$365$	−0.0950901	+	0.0687412i	−0.00497725	+	0.00359808i
$366$		0			0
$367$			29.2705i			1.52791i	0.645271	+	0.763954i	$0.276744\pi$
$367$			29.2705i			1.52791i	−0.645271	+	0.763954i	$0.723256\pi$
$368$		0			0
$369$	−0.660506			−0.0343846
$370$		0			0
$371$	16.9778			0.881444
$372$		0			0
$373$		−	15.3321i		−	0.793866i	−0.917848	−	0.396933i	$-0.870075\pi$
$373$		−	15.3321i		−	0.793866i	0.917848	−	0.396933i	$-0.129925\pi$
$374$		0			0
$375$	−3.37875	−	10.6576i	−0.174478	−	0.550355i
$376$		0			0
$377$		−	10.2876i		−	0.529840i
$378$		0			0
$379$	23.3102			1.19736			0.598682	−	0.800987i	$-0.295691\pi$
$379$	23.3102			1.19736			0.598682	+	0.800987i	$0.295691\pi$
$380$		0			0
$381$	−2.24525			−0.115028
$382$		0			0
$383$		−	5.16874i		−	0.264110i	−0.991242	−	0.132055i	$-0.957842\pi$
$383$		−	5.16874i		−	0.264110i	0.991242	−	0.132055i	$-0.0421576\pi$
$384$		0			0
$385$	21.8169	−	15.7716i	1.11189	−	0.803793i
$386$		0			0
$387$		−	7.61377i		−	0.387030i
$388$		0			0
$389$	6.65255			0.337298			0.168649	−	0.985676i	$-0.446060\pi$
$389$	6.65255			0.337298			0.168649	+	0.985676i	$0.446060\pi$
$390$		0			0
$391$	5.94899			0.300853
$392$		0			0
$393$			15.6589i			0.789887i
$394$		0			0
$395$	−12.7949	−	17.6993i	−0.643780	−	0.890546i
$396$		0			0
$397$			29.5882i			1.48499i	0.669852	+	0.742495i	$0.266358\pi$
$397$			29.5882i			1.48499i	−0.669852	+	0.742495i	$0.733642\pi$
$398$		0			0
$399$	−3.33642			−0.167030
$400$		0			0
$401$	34.7824			1.73695			0.868474	−	0.495735i	$-0.165101\pi$
$401$	34.7824			1.73695			0.868474	+	0.495735i	$0.165101\pi$
$402$		0			0
$403$		−	29.4958i		−	1.46929i
$404$		0			0
$405$	1.31001	+	1.81215i	0.0650949	+	0.0900463i
$406$		0			0
$407$			11.4339i			0.566757i
$408$		0			0
$409$	5.39469			0.266750			0.133375	−	0.991066i	$-0.457418\pi$
$409$	5.39469			0.266750			0.133375	+	0.991066i	$0.457418\pi$
$410$		0			0
$411$	−5.36091			−0.264434
$412$		0			0
$413$			14.6040i			0.718613i
$414$		0			0
$415$	20.4296	−	14.7687i	1.00285	−	0.724965i
$416$		0			0
$417$			11.2880i			0.552776i
$418$		0			0
$419$	23.1777			1.13231			0.566153	−	0.824300i	$-0.308431\pi$
$419$	23.1777			1.13231			0.566153	+	0.824300i	$0.308431\pi$
$420$		0			0
$421$	−26.3963			−1.28648			−0.643238	−	0.765666i	$-0.722409\pi$
$421$	−26.3963			−1.28648			−0.643238	+	0.765666i	$0.722409\pi$
$422$		0			0
$423$			1.56781i			0.0762297i
$424$		0			0
$425$	28.2450	+	9.32652i	1.37008	+	0.452403i
$426$		0			0
$427$		−	9.17960i		−	0.444232i
$428$		0			0
$429$	15.6164			0.753968
$430$		0			0
$431$	−21.3126			−1.02659			−0.513296	−	0.858211i	$-0.671576\pi$
$431$	−21.3126			−1.02659			−0.513296	+	0.858211i	$0.671576\pi$
$432$		0			0
$433$		−	7.16071i		−	0.344122i	−0.985086	−	0.172061i	$-0.944957\pi$
$433$		−	7.16071i		−	0.344122i	0.985086	−	0.172061i	$-0.0550426\pi$
$434$		0			0
$435$	6.12163	−	4.42536i	0.293510	−	0.212180i
$436$		0			0
$437$			1.42109i			0.0679798i
$438$		0			0
$439$	26.3125			1.25583			0.627914	−	0.778283i	$-0.283909\pi$
$439$	26.3125			1.25583			0.627914	+	0.778283i	$0.283909\pi$
$440$		0			0
$441$	−1.48787			−0.0708507
$442$		0			0
$443$			30.9196i			1.46903i	0.678590	+	0.734517i	$0.262591\pi$
$443$			30.9196i			1.46903i	−0.678590	+	0.734517i	$0.737409\pi$
$444$		0			0
$445$	2.57104	+	3.55654i	0.121879	+	0.168596i
$446$		0			0
$447$			1.78557i			0.0844544i
$448$		0			0
$449$	24.6521			1.16340			0.581702	−	0.813402i	$-0.302387\pi$
$449$	24.6521			1.16340			0.581702	+	0.813402i	$0.302387\pi$
$450$		0			0
$451$	−3.38701			−0.159488
$452$		0			0
$453$		−	14.1905i		−	0.666726i
$454$		0			0
$455$	−9.36647	−	12.9567i	−0.439107	−	0.607420i
$456$		0			0
$457$			25.7815i			1.20601i	0.797739	+	0.603003i	$0.206029\pi$
$457$			25.7815i			1.20601i	−0.797739	+	0.603003i	$0.793971\pi$
$458$		0			0
$459$	−5.94899			−0.277675
$460$		0			0
$461$	−11.3832			−0.530167			−0.265083	−	0.964225i	$-0.585399\pi$
$461$	−11.3832			−0.530167			−0.265083	+	0.964225i	$0.585399\pi$
$462$		0			0
$463$			25.8714i			1.20234i	0.799120	+	0.601172i	$0.205299\pi$
$463$			25.8714i			1.20234i	−0.799120	+	0.601172i	$0.794701\pi$
$464$		0			0
$465$	17.5514	−	12.6880i	0.813927	−	0.588392i
$466$		0			0
$467$		−	12.7023i		−	0.587793i	−0.955837	−	0.293897i	$-0.905048\pi$
$467$		−	12.7023i		−	0.587793i	0.955837	−	0.293897i	$-0.0949522\pi$
$468$		0			0
$469$	−4.83217			−0.223129
$470$		0			0
$471$	2.63045			0.121205
$472$		0			0
$473$		−	39.0427i		−	1.79518i
$474$		0			0
$475$	−2.22791	+	6.74712i	−0.102223	+	0.309579i
$476$		0			0
$477$			7.23139i			0.331102i
$478$		0			0
$479$	−12.5837			−0.574965			−0.287482	−	0.957786i	$-0.592818\pi$
$479$	−12.5837			−0.574965			−0.287482	+	0.957786i	$0.592818\pi$
$480$		0			0
$481$	6.79041			0.309616
$482$		0			0
$483$		−	2.34779i		−	0.106828i
$484$		0			0
$485$	12.0937	−	8.74261i	0.549147	−	0.396982i
$486$		0			0
$487$		−	1.28446i		−	0.0582044i	−0.999576	−	0.0291022i	$-0.990735\pi$
$487$		−	1.28446i		−	0.0582044i	0.999576	−	0.0291022i	$-0.00926483\pi$
$488$		0			0
$489$	−2.92501			−0.132274
$490$		0			0
$491$	−0.706233			−0.0318718			−0.0159359	−	0.999873i	$-0.505073\pi$
$491$	−0.706233			−0.0318718			−0.0159359	+	0.999873i	$0.505073\pi$
$492$		0			0
$493$			20.0964i			0.905095i
$494$		0			0
$495$	6.71761	+	9.29251i	0.301934	+	0.417667i
$496$		0			0
$497$		−	22.5758i		−	1.01266i
$498$		0			0
$499$	27.8463			1.24657			0.623287	−	0.781993i	$-0.285797\pi$
$499$	27.8463			1.24657			0.623287	+	0.781993i	$0.285797\pi$
$500$		0			0
$501$	−9.90446			−0.442499
$502$		0			0
$503$		−	7.00320i		−	0.312257i	−0.987737	−	0.156129i	$-0.950099\pi$
$503$		−	7.00320i		−	0.312257i	0.987737	−	0.156129i	$-0.0499014\pi$
$504$		0			0
$505$	19.0500	+	26.3520i	0.847715	+	1.17265i
$506$		0			0
$507$			3.72566i			0.165462i
$508$		0			0
$509$	17.7027			0.784658			0.392329	−	0.919825i	$-0.371669\pi$
$509$	17.7027			0.784658			0.392329	+	0.919825i	$0.371669\pi$
$510$		0			0
$511$	−0.123198			−0.00544994
$512$		0			0
$513$		−	1.42109i		−	0.0627426i
$514$		0			0
$515$	11.8728	−	8.58294i	0.523180	−	0.378210i
$516$		0			0
$517$			8.03960i			0.353581i
$518$		0			0
$519$	13.6303			0.598302
$520$		0			0
$521$	−9.20828			−0.403422			−0.201711	−	0.979445i	$-0.564650\pi$
$521$	−9.20828			−0.403422			−0.201711	+	0.979445i	$0.564650\pi$
$522$		0			0
$523$		−	20.8363i		−	0.911109i	−0.890208	−	0.455554i	$-0.849441\pi$
$523$		−	20.8363i		−	0.911109i	0.890208	−	0.455554i	$-0.150559\pi$
$524$		0			0
$525$	3.68075	−	11.1470i	0.160641	−	0.486495i
$526$		0			0
$527$			57.6185i			2.50990i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	−6.22029			−0.269937
$532$		0			0
$533$			2.01149i			0.0871273i
$534$		0			0
$535$	−33.9844	+	24.5675i	−1.46927	+	1.06215i
$536$		0			0
$537$		−	23.9064i		−	1.03164i
$538$		0			0
$539$	−7.62963			−0.328631
$540$		0			0
$541$	31.2983			1.34562			0.672809	−	0.739816i	$-0.265088\pi$
$541$	31.2983			1.34562			0.672809	+	0.739816i	$0.265088\pi$
$542$		0			0
$543$		−	6.16844i		−	0.264713i
$544$		0			0
$545$	−16.2397	−	22.4645i	−0.695634	−	0.962275i
$546$		0			0
$547$			2.09538i			0.0895919i	0.998996	+	0.0447960i	$0.0142638\pi$
$547$			2.09538i			0.0895919i	−0.998996	+	0.0447960i	$0.985736\pi$
$548$		0			0
$549$	3.90988			0.166870
$550$		0			0
$551$	−4.80059			−0.204512
$552$		0			0
$553$		−	22.9309i		−	0.975122i
$554$		0			0
$555$	2.92098	+	4.04062i	0.123989	+	0.171515i
$556$		0			0
$557$			27.9523i			1.18438i	0.805799	+	0.592189i	$0.201736\pi$
$557$			27.9523i			1.18438i	−0.805799	+	0.592189i	$0.798264\pi$
$558$		0			0
$559$	−23.1868			−0.980698
$560$		0			0
$561$	−30.5059			−1.28796
$562$		0			0
$563$			5.34755i			0.225372i	0.993631	+	0.112686i	$0.0359455\pi$
$563$			5.34755i			0.225372i	−0.993631	+	0.112686i	$0.964055\pi$
$564$		0			0
$565$	−32.3388	+	23.3779i	−1.36050	+	0.983514i
$566$		0			0
$567$			2.34779i			0.0985981i
$568$		0			0
$569$	−2.18158			−0.0914566			−0.0457283	−	0.998954i	$-0.514561\pi$
$569$	−2.18158			−0.0914566			−0.0457283	+	0.998954i	$0.514561\pi$
$570$		0			0
$571$	16.8120			0.703560			0.351780	−	0.936083i	$-0.385577\pi$
$571$	16.8120			0.703560			0.351780	+	0.936083i	$0.385577\pi$
$572$		0			0
$573$			7.09026i			0.296200i
$574$		0			0
$575$	−4.74786	−	1.56775i	−0.197999	−	0.0653796i
$576$		0			0
$577$			26.7283i			1.11271i	0.830944	+	0.556356i	$0.187801\pi$
$577$			26.7283i			1.11271i	−0.830944	+	0.556356i	$0.812199\pi$
$578$		0			0
$579$	15.5694			0.647042
$580$		0			0
$581$	26.4683			1.09809
$582$		0			0
$583$			37.0819i			1.53577i
$584$		0			0
$585$	5.51867	−	3.98948i	0.228169	−	0.164945i
$586$		0			0
$587$		−	19.2868i		−	0.796049i	−0.917375	−	0.398025i	$-0.869696\pi$
$587$		−	19.2868i		−	0.796049i	0.917375	−	0.398025i	$-0.130304\pi$
$588$		0			0
$589$	−13.7638			−0.567129
$590$		0			0
$591$	−19.4095			−0.798400
$592$		0			0
$593$			35.4233i			1.45466i	0.686288	+	0.727330i	$0.259239\pi$
$593$			35.4233i			1.45466i	−0.686288	+	0.727330i	$0.740761\pi$
$594$		0			0
$595$	18.2969	+	25.3103i	0.750100	+	1.03762i
$596$		0			0
$597$		−	11.9937i		−	0.490869i
$598$		0			0
$599$	11.4900			0.469470			0.234735	−	0.972059i	$-0.424578\pi$
$599$	11.4900			0.469470			0.234735	+	0.972059i	$0.424578\pi$
$600$		0			0
$601$	−3.88058			−0.158292			−0.0791460	−	0.996863i	$-0.525219\pi$
$601$	−3.88058			−0.158292			−0.0791460	+	0.996863i	$0.525219\pi$
$602$		0			0
$603$		−	2.05818i		−	0.0838154i
$604$		0			0
$605$	20.0371	+	27.7175i	0.814625	+	1.12688i
$606$		0			0
$607$			9.33306i			0.378817i	0.981898	+	0.189409i	$0.0606571\pi$
$607$			9.33306i			0.378817i	−0.981898	+	0.189409i	$0.939343\pi$
$608$		0			0
$609$	7.93111			0.321385
$610$		0			0
$611$	4.77459			0.193159
$612$		0			0
$613$		−	33.8177i		−	1.36588i	−0.730473	−	0.682942i	$-0.760700\pi$
$613$		−	33.8177i		−	1.36588i	0.730473	−	0.682942i	$-0.239300\pi$
$614$		0			0
$615$	−1.19693	+	0.865269i	−0.0482650	+	0.0348910i
$616$		0			0
$617$		−	4.80257i		−	0.193344i	−0.995316	−	0.0966721i	$-0.969180\pi$
$617$		−	4.80257i		−	0.193344i	0.995316	−	0.0966721i	$-0.0308198\pi$
$618$		0			0
$619$	16.2393			0.652714			0.326357	−	0.945247i	$-0.394179\pi$
$619$	16.2393			0.652714			0.326357	+	0.945247i	$0.394179\pi$
$620$		0			0
$621$	1.00000			0.0401286
$622$		0			0
$623$			4.60780i			0.184608i
$624$		0			0
$625$	−20.0843	−	14.8869i	−0.803373	−	0.595476i
$626$		0			0
$627$		−	7.28720i		−	0.291023i
$628$		0			0
$629$	−13.2647			−0.528899
$630$		0			0
$631$	10.0075			0.398392			0.199196	−	0.979960i	$-0.436167\pi$
$631$	10.0075			0.398392			0.199196	+	0.979960i	$0.436167\pi$
$632$		0			0
$633$		−	21.2488i		−	0.844563i
$634$		0			0
$635$	−4.06872	+	2.94130i	−0.161462	+	0.116722i
$636$		0			0
$637$			4.53112i			0.179529i
$638$		0			0
$639$	9.61577			0.380394
$640$		0			0
$641$	−44.7892			−1.76907			−0.884533	−	0.466477i	$-0.845523\pi$
$641$	−44.7892			−1.76907			−0.884533	+	0.466477i	$0.845523\pi$
$642$		0			0
$643$			11.5666i			0.456143i	0.973644	+	0.228071i	$0.0732419\pi$
$643$			11.5666i			0.456143i	−0.973644	+	0.228071i	$0.926758\pi$
$644$		0			0
$645$	−9.97412	−	13.7973i	−0.392730	−	0.543267i
$646$		0			0
$647$			17.1475i			0.674137i	0.941480	+	0.337068i	$0.109435\pi$
$647$			17.1475i			0.674137i	−0.941480	+	0.337068i	$0.890565\pi$
$648$		0			0
$649$	−31.8970			−1.25207
$650$		0			0
$651$	22.7394			0.891226
$652$		0			0
$653$			21.3924i			0.837149i	0.908182	+	0.418574i	$0.137470\pi$
$653$			21.3924i			0.837149i	−0.908182	+	0.418574i	$0.862530\pi$
$654$		0			0
$655$	20.5133	+	28.3762i	0.801522	+	1.10875i
$656$		0			0
$657$		−	0.0524738i		−	0.00204720i
$658$		0			0
$659$	−6.62300			−0.257996			−0.128998	−	0.991645i	$-0.541176\pi$
$659$	−6.62300			−0.257996			−0.128998	+	0.991645i	$0.541176\pi$
$660$		0			0
$661$	23.7129			0.922323			0.461162	−	0.887316i	$-0.347433\pi$
$661$	23.7129			0.922323			0.461162	+	0.887316i	$0.347433\pi$
$662$		0			0
$663$			18.1169i			0.703604i
$664$		0			0
$665$	−6.04608	+	4.37074i	−0.234457	+	0.169490i
$666$		0			0
$667$		−	3.37811i		−	0.130801i
$668$		0			0
$669$	−6.96844			−0.269416
$670$		0			0
$671$	20.0495			0.774003
$672$		0			0
$673$			23.6330i			0.910986i	0.890239	+	0.455493i	$0.150537\pi$
$673$			23.6330i			0.910986i	−0.890239	+	0.455493i	$0.849463\pi$
$674$		0			0
$675$	4.74786	+	1.56775i	0.182745	+	0.0603426i
$676$		0			0
$677$			1.32984i			0.0511097i	0.999673	+	0.0255549i	$0.00813525\pi$
$677$			1.32984i			0.0511097i	−0.999673	+	0.0255549i	$0.991865\pi$
$678$		0			0
$679$	15.6685			0.601301
$680$		0			0
$681$	22.5332			0.863474
$682$		0			0
$683$			24.3183i			0.930514i	0.885176	+	0.465257i	$0.154038\pi$
$683$			24.3183i			0.930514i	−0.885176	+	0.465257i	$0.845962\pi$
$684$		0			0
$685$	−9.71476	+	7.02285i	−0.371182	+	0.268329i
$686$		0			0
$687$			2.79145i			0.106501i
$688$		0			0
$689$	22.0223			0.838984
$690$		0			0
$691$	−1.72885			−0.0657687			−0.0328843	−	0.999459i	$-0.510469\pi$
$691$	−1.72885			−0.0657687			−0.0328843	+	0.999459i	$0.510469\pi$
$692$		0			0
$693$			12.0393i			0.457334i
$694$		0			0
$695$	14.7874	+	20.4555i	0.560918	+	0.775921i
$696$		0			0
$697$		−	3.92934i		−	0.148834i
$698$		0			0
$699$	20.1342			0.761546
$700$		0			0
$701$	47.8216			1.80620			0.903098	−	0.429435i	$-0.141287\pi$
$701$	47.8216			1.80620			0.903098	+	0.429435i	$0.141287\pi$
$702$		0			0
$703$		−	3.16866i		−	0.119508i
$704$		0			0
$705$	2.05385	+	2.84111i	0.0773525	+	0.107002i
$706$		0			0
$707$			34.1414i			1.28402i
$708$		0			0
$709$	−26.7220			−1.00357			−0.501783	−	0.864993i	$-0.667323\pi$
$709$	−26.7220			−1.00357			−0.501783	+	0.864993i	$0.667323\pi$
$710$		0			0
$711$	9.76701			0.366292
$712$		0			0
$713$		−	9.68542i		−	0.362722i
$714$		0			0
$715$	28.2992	−	20.4577i	1.05833	−	0.765073i
$716$		0			0
$717$		−	23.5778i		−	0.880528i
$718$		0			0
$719$	−42.1856			−1.57326			−0.786628	−	0.617427i	$-0.788175\pi$
$719$	−42.1856			−1.57326			−0.786628	+	0.617427i	$0.788175\pi$
$720$		0			0
$721$	15.3823			0.572867
$722$		0			0
$723$		−	12.4333i		−	0.462399i
$724$		0			0
$725$	5.29603	−	16.0388i	0.196690	−	0.595666i
$726$		0			0
$727$		−	38.6940i		−	1.43508i	−0.696516	−	0.717541i	$-0.745268\pi$
$727$		−	38.6940i		−	1.43508i	0.696516	−	0.717541i	$-0.254732\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	45.2943			1.67527
$732$		0			0
$733$		−	14.7327i		−	0.544165i	−0.962274	−	0.272083i	$-0.912288\pi$
$733$		−	14.7327i		−	0.544165i	0.962274	−	0.272083i	$-0.0877124\pi$
$734$		0			0
$735$	−2.69623	+	1.94912i	−0.0994519	+	0.0718943i
$736$		0			0
$737$		−	10.5541i		−	0.388766i
$738$		0			0
$739$	−46.3802			−1.70612			−0.853062	−	0.521810i	$-0.825257\pi$
$739$	−46.3802			−1.70612			−0.853062	+	0.521810i	$0.825257\pi$
$740$		0			0
$741$	−4.32775			−0.158984
$742$		0			0
$743$		−	14.4596i		−	0.530473i	−0.964183	−	0.265236i	$-0.914550\pi$
$743$		−	14.4596i		−	0.530473i	0.964183	−	0.265236i	$-0.0854500\pi$
$744$		0			0
$745$	2.33911	+	3.23571i	0.0856984	+	0.118547i
$746$		0			0
$747$			11.2737i			0.412483i
$748$		0			0
$749$	−44.0297			−1.60881
$750$		0			0
$751$	8.88999			0.324400			0.162200	−	0.986758i	$-0.448141\pi$
$751$	8.88999			0.324400			0.162200	+	0.986758i	$0.448141\pi$
$752$		0			0
$753$			4.94900i			0.180352i
$754$		0			0
$755$	−18.5896	−	25.7152i	−0.676546	−	0.935872i
$756$		0			0
$757$		−	17.5976i		−	0.639597i	−0.947486	−	0.319798i	$-0.896385\pi$
$757$		−	17.5976i		−	0.639597i	0.947486	−	0.319798i	$-0.103615\pi$
$758$		0			0
$759$	5.12790			0.186131
$760$		0			0
$761$	−24.7513			−0.897236			−0.448618	−	0.893724i	$-0.648084\pi$
$761$	−24.7513			−0.897236			−0.448618	+	0.893724i	$0.648084\pi$
$762$		0			0
$763$		−	29.1048i		−	1.05366i
$764$		0			0
$765$	−10.7804	+	7.79324i	−0.389768	+	0.281765i
$766$		0			0
$767$			18.9431i			0.683997i
$768$		0			0
$769$	12.3353			0.444822			0.222411	−	0.974953i	$-0.428607\pi$
$769$	12.3353			0.444822			0.222411	+	0.974953i	$0.428607\pi$
$770$		0			0
$771$	6.81640			0.245487
$772$		0			0
$773$		−	13.5916i		−	0.488857i	−0.969667	−	0.244429i	$-0.921400\pi$
$773$		−	13.5916i		−	0.488857i	0.969667	−	0.244429i	$-0.0786004\pi$
$774$		0			0
$775$	15.1843	−	45.9850i	0.545436	−	1.65183i
$776$		0			0
$777$			5.23497i			0.187804i
$778$		0			0
$779$	0.938636			0.0336301
$780$		0			0
$781$	49.3088			1.76441
$782$		0			0
$783$			3.37811i			0.120724i
$784$		0			0
$785$	4.76677	−	3.44592i	0.170133	−	0.122990i
$786$		0			0
$787$			13.5835i			0.484200i	0.970251	+	0.242100i	$0.0778362\pi$
$787$			13.5835i			0.484200i	−0.970251	+	0.242100i	$0.922164\pi$
$788$		0			0
$789$	0.723986			0.0257746
$790$		0			0
$791$	−41.8977			−1.48971
$792$		0			0
$793$		−	11.9071i		−	0.422833i
$794$		0			0
$795$	9.47319	+	13.1043i	0.335979	+	0.464763i
$796$		0			0
$797$			43.8025i			1.55157i	0.631000	+	0.775783i	$0.282645\pi$
$797$			43.8025i			1.55157i	−0.631000	+	0.775783i	$0.717355\pi$
$798$		0			0
$799$	−9.32691			−0.329962
$800$		0			0
$801$	−1.96261			−0.0693454
$802$		0			0
$803$		−	0.269081i		−	0.00949565i
$804$		0			0
$805$	−3.07563	−	4.25455i	−0.108402	−	0.149953i
$806$		0			0
$807$		−	2.42725i		−	0.0854432i
$808$		0			0
$809$	38.1383			1.34087			0.670436	−	0.741967i	$-0.266107\pi$
$809$	38.1383			1.34087			0.670436	+	0.741967i	$0.266107\pi$
$810$		0			0
$811$	55.1094			1.93515			0.967576	−	0.252580i	$-0.0812792\pi$
$811$	55.1094			1.93515			0.967576	+	0.252580i	$0.0812792\pi$
$812$		0			0
$813$		−	22.4755i		−	0.788249i
$814$		0			0
$815$	−5.30055	+	3.83179i	−0.185670	+	0.134222i
$816$		0			0
$817$			10.8198i			0.378538i
$818$		0			0
$819$	7.14992			0.249839
$820$		0			0
$821$	30.5189			1.06512			0.532559	−	0.846393i	$-0.321230\pi$
$821$	30.5189			1.06512			0.532559	+	0.846393i	$0.321230\pi$
$822$		0			0
$823$			9.70526i			0.338304i	0.985590	+	0.169152i	$0.0541029\pi$
$823$			9.70526i			0.338304i	−0.985590	+	0.169152i	$0.945897\pi$
$824$		0			0
$825$	24.3466	+	8.03926i	0.847639	+	0.279891i
$826$		0			0
$827$		−	37.0405i		−	1.28802i	−0.765016	−	0.644012i	$-0.777269\pi$
$827$		−	37.0405i		−	1.28802i	0.765016	−	0.644012i	$-0.222731\pi$
$828$		0			0
$829$	5.73431			0.199161			0.0995804	−	0.995030i	$-0.468250\pi$
$829$	5.73431			0.199161			0.0995804	+	0.995030i	$0.468250\pi$
$830$		0			0
$831$	−19.7701			−0.685818
$832$		0			0
$833$		−	8.85130i		−	0.306679i
$834$		0			0
$835$	−17.9483	+	12.9749i	−0.621128	+	0.449017i
$836$		0			0
$837$			9.68542i			0.334777i
$838$		0			0
$839$	−35.7441			−1.23402			−0.617012	−	0.786953i	$-0.711657\pi$
$839$	−35.7441			−1.23402			−0.617012	+	0.786953i	$0.711657\pi$
$840$		0			0
$841$	−17.5884			−0.606495
$842$		0			0
$843$		−	13.8906i		−	0.478416i
$844$		0			0
$845$	4.88065	+	6.75144i	0.167900	+	0.232257i
$846$		0			0
$847$			35.9105i			1.23390i
$848$		0			0
$849$	−32.7851			−1.12518
$850$		0			0
$851$	2.22974			0.0764345
$852$		0			0
$853$			2.31904i			0.0794025i	0.999212	+	0.0397012i	$0.0126406\pi$
$853$			2.31904i			0.0794025i	−0.999212	+	0.0397012i	$0.987359\pi$
$854$		0			0
$855$	−1.86164	−	2.57522i	−0.0636667	−	0.0880706i
$856$		0			0
$857$			46.8462i			1.60024i	0.599842	+	0.800118i	$0.295230\pi$
$857$			46.8462i			1.60024i	−0.599842	+	0.800118i	$0.704770\pi$
$858$		0			0
$859$	20.3627			0.694767			0.347383	−	0.937723i	$-0.387070\pi$
$859$	20.3627			0.694767			0.347383	+	0.937723i	$0.387070\pi$
$860$		0			0
$861$	−1.55073			−0.0528488
$862$		0			0
$863$		−	4.74687i		−	0.161585i	−0.996731	−	0.0807927i	$-0.974255\pi$
$863$		−	4.74687i		−	0.161585i	0.996731	−	0.0807927i	$-0.0257452\pi$
$864$		0			0
$865$	24.7000	−	17.8558i	0.839826	−	0.607114i
$866$		0			0
$867$		−	18.3905i		−	0.624575i
$868$		0			0
$869$	50.0843			1.69899
$870$		0			0
$871$	−6.26793			−0.212381
$872$		0			0
$873$			6.67370i			0.225870i
$874$		0			0
$875$	−7.93262	−	25.0218i	−0.268171	−	0.845891i
$876$		0			0
$877$		−	8.55753i		−	0.288967i	−0.989507	−	0.144484i	$-0.953848\pi$
$877$		−	8.55753i		−	0.288967i	0.989507	−	0.144484i	$-0.0461521\pi$
$878$		0			0
$879$	21.6422			0.729972
$880$		0			0
$881$	−16.8980			−0.569308			−0.284654	−	0.958630i	$-0.591879\pi$
$881$	−16.8980			−0.569308			−0.284654	+	0.958630i	$0.591879\pi$
$882$		0			0
$883$		−	43.6172i		−	1.46783i	−0.679239	−	0.733917i	$-0.737690\pi$
$883$		−	43.6172i		−	1.46783i	0.679239	−	0.733917i	$-0.262310\pi$
$884$		0			0
$885$	−11.2721	+	8.14864i	−0.378907	+	0.273913i
$886$		0			0
$887$		−	20.2349i		−	0.679420i	−0.940530	−	0.339710i	$-0.889671\pi$
$887$		−	20.2349i		−	0.679420i	0.940530	−	0.339710i	$-0.110329\pi$
$888$		0			0
$889$	−5.27138			−0.176796
$890$		0			0
$891$	−5.12790			−0.171791
$892$		0			0
$893$		−	2.22800i		−	0.0745572i
$894$		0			0
$895$	−31.3176	−	43.3218i	−1.04683	−	1.44809i
$896$		0			0
$897$		−	3.04538i		−	0.101682i
$898$		0			0
$899$	32.7184			1.09122
$900$		0			0
$901$	−43.0195			−1.43319
$902$		0			0
$903$		−	17.8756i		−	0.594861i
$904$		0			0
$905$	−8.08072	−	11.1781i	−0.268612	−	0.371573i
$906$		0			0
$907$		−	4.26287i		−	0.141546i	−0.997492	−	0.0707732i	$-0.977453\pi$
$907$		−	4.26287i		−	0.141546i	0.997492	−	0.0707732i	$-0.0225467\pi$
$908$		0			0
$909$	−14.5419			−0.482324
$910$		0			0
$911$	35.3973			1.17276			0.586382	−	0.810035i	$-0.300552\pi$
$911$	35.3973			1.17276			0.586382	+	0.810035i	$0.300552\pi$
$912$		0			0
$913$			57.8105i			1.91325i
$914$		0			0
$915$	7.08528	−	5.12199i	0.234232	−	0.169328i
$916$		0			0
$917$			36.7639i			1.21405i
$918$		0			0
$919$	47.7259			1.57433			0.787166	−	0.616741i	$-0.211547\pi$
$919$	47.7259			1.57433			0.787166	+	0.616741i	$0.211547\pi$
$920$		0			0
$921$	−27.9277			−0.920250
$922$		0			0
$923$		−	29.2837i		−	0.963884i
$924$		0			0
$925$	10.5865	+	3.49567i	0.348082	+	0.114937i
$926$		0			0
$927$			6.55181i			0.215190i
$928$		0			0
$929$	−33.4958			−1.09896			−0.549481	−	0.835506i	$-0.685175\pi$
$929$	−33.4958			−1.09896			−0.549481	+	0.835506i	$0.685175\pi$
$930$		0			0
$931$	2.11439			0.0692962
$932$		0			0
$933$		−	29.9204i		−	0.979549i
$934$		0			0
$935$	−55.2811	+	39.9630i	−1.80789	+	1.30693i
$936$		0			0
$937$		−	1.02343i		−	0.0334338i	−0.999860	−	0.0167169i	$-0.994679\pi$
$937$		−	1.02343i		−	0.0334338i	0.999860	−	0.0167169i	$-0.00532141\pi$
$938$		0			0
$939$	−20.3987			−0.665687
$940$		0			0
$941$	37.0543			1.20794			0.603969	−	0.797008i	$-0.293585\pi$
$941$	37.0543			1.20794			0.603969	+	0.797008i	$0.293585\pi$
$942$		0			0
$943$			0.660506i			0.0215090i
$944$		0			0
$945$	3.07563	+	4.25455i	0.100050	+	0.138400i
$946$		0			0
$947$		−	42.0657i		−	1.36695i	−0.729973	−	0.683476i	$-0.760467\pi$
$947$		−	42.0657i		−	1.36695i	0.729973	−	0.683476i	$-0.239533\pi$
$948$		0			0
$949$	−0.159803			−0.00518741
$950$		0			0
$951$	−18.5483			−0.601470
$952$		0			0
$953$			38.9701i			1.26237i	0.775634	+	0.631183i	$0.217430\pi$
$953$			38.9701i			1.26237i	−0.775634	+	0.631183i	$0.782570\pi$
$954$		0			0
$955$	9.28831	+	12.8486i	0.300563	+	0.415771i
$956$		0			0
$957$			17.3226i			0.559961i
$958$		0			0
$959$	−12.5863			−0.406433
$960$		0			0
$961$	62.8074			2.02604
$962$		0			0
$963$		−	18.7537i		−	0.604328i
$964$		0			0
$965$	28.2140	−	20.3961i	0.908241	−	0.656572i
$966$		0			0
$967$			37.6179i			1.20971i	0.796335	+	0.604856i	$0.206769\pi$
$967$			37.6179i			1.20971i	−0.796335	+	0.604856i	$0.793231\pi$
$968$		0			0
$969$	8.45404			0.271583
$970$		0			0
$971$	10.5396			0.338233			0.169117	−	0.985596i	$-0.445909\pi$
$971$	10.5396			0.338233			0.169117	+	0.985596i	$0.445909\pi$
$972$		0			0
$973$			26.5019i			0.849611i
$974$		0			0
$975$	4.77439	−	14.4590i	0.152903	−	0.463060i
$976$		0			0
$977$			53.8645i			1.72328i	0.507521	+	0.861639i	$0.330562\pi$
$977$			53.8645i			1.72328i	−0.507521	+	0.861639i	$0.669438\pi$
$978$		0			0
$979$	−10.0641			−0.321649
$980$		0			0
$981$	12.3966			0.395794
$982$		0			0
$983$		−	31.5212i		−	1.00537i	−0.864469	−	0.502685i	$-0.832345\pi$
$983$		−	31.5212i		−	1.00537i	0.864469	−	0.502685i	$-0.167655\pi$
$984$		0			0
$985$	−35.1728	+	25.4266i	−1.12070	+	0.810160i
$986$		0			0
$987$			3.68090i			0.117164i
$988$		0			0
$989$	−7.61377			−0.242104
$990$		0			0
$991$	−12.8851			−0.409309			−0.204654	−	0.978834i	$-0.565607\pi$
$991$	−12.8851			−0.409309			−0.204654	+	0.978834i	$0.565607\pi$
$992$		0			0
$993$		−	1.29507i		−	0.0410979i
$994$		0			0
$995$	−15.7118	−	21.7343i	−0.498099	−	0.689024i
$996$		0			0
$997$		−	8.54232i		−	0.270538i	−0.990809	−	0.135269i	$-0.956810\pi$
$997$		−	8.54232i		−	0.270538i	0.990809	−	0.135269i	$-0.0431899\pi$
$998$		0			0
$999$	−2.22974			−0.0705459

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2760.2.k.d.2209.13	yes	14
5.4	even	2	inner	2760.2.k.d.2209.6	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.k.d.2209.6	✓	14	5.4	even	2	inner
2760.2.k.d.2209.13	yes	14	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(1.31001 + 1.81215i) q^{5} +2.34779i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(1.31001 + 1.81215i) q^{5} +2.34779i q^{7} -1.00000 q^{9} -5.12790 q^{11} +3.04538i q^{13} +(-1.81215 + 1.31001i) q^{15} -5.94899i q^{17} +1.42109 q^{19} -2.34779 q^{21} +1.00000i q^{23} +(-1.56775 + 4.74786i) q^{25} -1.00000i q^{27} -3.37811 q^{29} -9.68542 q^{31} -5.12790i q^{33} +(-4.25455 + 3.07563i) q^{35} -2.22974i q^{37} -3.04538 q^{39} +0.660506 q^{41} +7.61377i q^{43} +(-1.31001 - 1.81215i) q^{45} -1.56781i q^{47} +1.48787 q^{49} +5.94899 q^{51} -7.23139i q^{53} +(-6.71761 - 9.29251i) q^{55} +1.42109i q^{57} +6.22029 q^{59} -3.90988 q^{61} -2.34779i q^{63} +(-5.51867 + 3.98948i) q^{65} +2.05818i q^{67} -1.00000 q^{69} -9.61577 q^{71} +0.0524738i q^{73} +(-4.74786 - 1.56775i) q^{75} -12.0393i q^{77} -9.76701 q^{79} +1.00000 q^{81} -11.2737i q^{83} +(10.7804 - 7.79324i) q^{85} -3.37811i q^{87} +1.96261 q^{89} -7.14992 q^{91} -9.68542i q^{93} +(1.86164 + 2.57522i) q^{95} -6.67370i q^{97} +5.12790 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 2 q^{5} - 14 q^{9}+O(q^{10}) \) 14 * q + 2 * q^5 - 14 * q^9	\( 14 q + 2 q^{5} - 14 q^{9} + 8 q^{11} - 4 q^{15} + 8 q^{19} - 10 q^{21} - 10 q^{25} + 26 q^{29} - 18 q^{31} - 10 q^{35} + 12 q^{39} - 2 q^{41} - 2 q^{45} + 4 q^{49} - 6 q^{51} - 32 q^{55} - 10 q^{59} - 24 q^{61} - 36 q^{65} - 14 q^{69} - 38 q^{71} + 4 q^{79} + 14 q^{81} - 14 q^{85} - 16 q^{89} - 4 q^{91} - 8 q^{95} - 8 q^{99}+O(q^{100}) \) 14 * q + 2 * q^5 - 14 * q^9 + 8 * q^11 - 4 * q^15 + 8 * q^19 - 10 * q^21 - 10 * q^25 + 26 * q^29 - 18 * q^31 - 10 * q^35 + 12 * q^39 - 2 * q^41 - 2 * q^45 + 4 * q^49 - 6 * q^51 - 32 * q^55 - 10 * q^59 - 24 * q^61 - 36 * q^65 - 14 * q^69 - 38 * q^71 + 4 * q^79 + 14 * q^81 - 14 * q^85 - 16 * q^89 - 4 * q^91 - 8 * q^95 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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