LMFDB - Embedded newform 2740.1.cd.a.423.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2740,1,Mod(3,2740)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2740, base_ring=CyclotomicField(136))

chi = DirichletCharacter(H, H._module([68, 102, 1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2740.3");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2740 = 2^{2} \cdot 5 \cdot 137 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2740.cd (of order $136$, degree $64$, 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:	$1.36743813461$
Analytic rank:	$0$
Dimension:	$64$
Coefficient field:	$\Q(\zeta_{136})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{64} - x^{60} + x^{56} - x^{52} + x^{48} - x^{44} + x^{40} - x^{36} + x^{32} - x^{28} + x^{24} + \cdots + 1 $ x^64 - x^60 + x^56 - x^52 + x^48 - x^44 + x^40 - x^36 + x^32 - x^28 + x^24 - x^20 + x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{136}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{136} - \cdots)$

Embedding invariants

Embedding label			423.1
Root			$0.317791 - 0.948161i$ of defining polynomial
Character	$\chi$	$=$	2740.423
Dual form			2740.1.cd.a.1587.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.602635 + 0.798017i) q^{2} +(-0.273663 + 0.961826i) q^{4} +(0.914794 + 0.403921i) q^{5} +(-0.932472 + 0.361242i) q^{8} +(-0.824997 + 0.565136i) q^{9} +O(q^{10})$	\(q+(0.602635 + 0.798017i) q^{2} +(-0.273663 + 0.961826i) q^{4} +(0.914794 + 0.403921i) q^{5} +(-0.932472 + 0.361242i) q^{8} +(-0.824997 + 0.565136i) q^{9} +(0.228951 + 0.973438i) q^{10} +(0.216788 + 0.0782748i) q^{13} +(-0.850217 - 0.526432i) q^{16} +(-0.291826 + 0.753290i) q^{17} +(-0.948161 - 0.317791i) q^{18} +(-0.638847 + 0.769334i) q^{20} +(0.673696 + 0.739009i) q^{25} +(0.0681792 + 0.220171i) q^{26} +(-0.136828 + 0.838799i) q^{29} +(-0.0922684 - 0.995734i) q^{32} +(-0.777003 + 0.221076i) q^{34} +(-0.317791 - 0.948161i) q^{36} +0.276313i q^{37} +(-0.998933 - 0.0461835i) q^{40} +(-0.0529974 - 0.127947i) q^{41} +(-0.982973 + 0.183750i) q^{45} +(0.0922684 + 0.995734i) q^{49} +(-0.183750 + 0.982973i) q^{50} +(-0.134613 + 0.187091i) q^{52} +(-0.0956383 - 1.37786i) q^{53} +(-0.751834 + 0.396298i) q^{58} +(0.595012 - 0.868610i) q^{61} +(0.739009 - 0.673696i) q^{64} +(0.166699 + 0.159170i) q^{65} +(-0.644672 - 0.486834i) q^{68} +(0.565136 - 0.824997i) q^{72} +(1.64823 + 0.0762025i) q^{73} +(-0.220502 + 0.166516i) q^{74} +(-0.565136 - 0.824997i) q^{80} +(0.361242 - 0.932472i) q^{81} +(0.0701658 - 0.119398i) q^{82} +(-0.571231 + 0.571231i) q^{85} +(0.299742 + 0.510058i) q^{89} +(-0.739009 - 0.673696i) q^{90} +(-0.212941 - 1.83545i) q^{97} +(-0.739009 + 0.673696i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 64 q + 4 q^{2} - 4 q^{4} + 4 q^{8}+O(q^{10}) $ 64 * q + 4 * q^2 - 4 * q^4 + 4 * q^8	$ 64 q + 4 q^{2} - 4 q^{4} + 4 q^{8} - 4 q^{13} - 4 q^{16} + 4 q^{26} + 4 q^{32} - 4 q^{45} - 4 q^{49} - 4 q^{52} - 4 q^{64} - 4 q^{65} + 4 q^{73} + 4 q^{85} + 4 q^{90} + 4 q^{97} + 4 q^{98}+O(q^{100}) $ 64 * q + 4 * q^2 - 4 * q^4 + 4 * q^8 - 4 * q^13 - 4 * q^16 + 4 * q^26 + 4 * q^32 - 4 * q^45 - 4 * q^49 - 4 * q^52 - 4 * q^64 - 4 * q^65 + 4 * q^73 + 4 * q^85 + 4 * q^90 + 4 * q^97 + 4 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2740\mathbb{Z}\right)^\times$.

$n$	$1097$	$1371$	$1921$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$	$e\left(\frac{21}{136}\right)$

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 $

$2$	0.602635	+	0.798017i	0.602635	+	0.798017i
$2$	0.602635	+	0.798017i	0.602635	+	0.798017i
$3$		0			0		−0.295806	−	0.955248i	$-0.595588\pi$
$3$		0			0		0.295806	+	0.955248i	$0.404412\pi$
$4$	−0.273663	+	0.961826i	−0.273663	+	0.961826i
$5$	0.914794	+	0.403921i	0.914794	+	0.403921i
$5$	0.914794	+	0.403921i	0.914794	+	0.403921i
$6$		0			0
$7$		0			0		−0.739009	−	0.673696i	$-0.764706\pi$
$7$		0			0		0.739009	+	0.673696i	$0.235294\pi$
$8$	−0.932472	+	0.361242i	−0.932472	+	0.361242i
$9$	−0.824997	+	0.565136i	−0.824997	+	0.565136i
$10$	0.228951	+	0.973438i	0.228951	+	0.973438i
$11$		0			0		0.873622	−	0.486604i	$-0.161765\pi$
$11$		0			0		−0.873622	+	0.486604i	$0.838235\pi$
$12$		0			0
$13$	0.216788	+	0.0782748i	0.216788	+	0.0782748i	0.445738	−	0.895163i	$-0.352941\pi$
$13$	0.216788	+	0.0782748i	0.216788	+	0.0782748i	−0.228951	+	0.973438i	$0.573529\pi$
$14$		0			0
$15$		0			0
$16$	−0.850217	−	0.526432i	−0.850217	−	0.526432i
$17$	−0.291826	+	0.753290i	−0.291826	+	0.753290i	0.707107	+	0.707107i	$0.250000\pi$
$17$	−0.291826	+	0.753290i	−0.291826	+	0.753290i	−0.998933	+	0.0461835i	$0.985294\pi$
$18$	−0.948161	−	0.317791i	−0.948161	−	0.317791i
$19$		0			0		0.317791	−	0.948161i	$-0.397059\pi$
$19$		0			0		−0.317791	+	0.948161i	$0.602941\pi$
$20$	−0.638847	+	0.769334i	−0.638847	+	0.769334i
$21$		0			0
$22$		0			0
$23$		0			0		0.160996	−	0.986955i	$-0.448529\pi$
$23$		0			0		−0.160996	+	0.986955i	$0.551471\pi$
$24$		0			0
$25$	0.673696	+	0.739009i	0.673696	+	0.739009i
$26$	0.0681792	+	0.220171i	0.0681792	+	0.220171i
$27$		0			0
$28$		0			0
$29$	−0.136828	+	0.838799i	−0.136828	+	0.838799i	0.824997	+	0.565136i	$0.191176\pi$
$29$	−0.136828	+	0.838799i	−0.136828	+	0.838799i	−0.961826	+	0.273663i	$0.911765\pi$
$30$		0			0
$31$		0			0		−0.656446	−	0.754373i	$-0.727941\pi$
$31$		0			0		0.656446	+	0.754373i	$0.272059\pi$
$32$	−0.0922684	−	0.995734i	−0.0922684	−	0.995734i
$33$		0			0
$34$	−0.777003	+	0.221076i	−0.777003	+	0.221076i
$35$		0			0
$36$	−0.317791	−	0.948161i	−0.317791	−	0.948161i
$37$			0.276313i			0.276313i	0.990410	+	0.138156i	$0.0441176\pi$
$37$			0.276313i			0.276313i	−0.990410	+	0.138156i	$0.955882\pi$
$38$		0			0
$39$		0			0
$40$	−0.998933	−	0.0461835i	−0.998933	−	0.0461835i
$41$	−0.0529974	−	0.127947i	−0.0529974	−	0.127947i	0.895163	−	0.445738i	$-0.147059\pi$
$41$	−0.0529974	−	0.127947i	−0.0529974	−	0.127947i	−0.948161	+	0.317791i	$0.897059\pi$
$42$		0			0
$43$		0			0		−0.754373	−	0.656446i	$-0.772059\pi$
$43$		0			0		0.754373	+	0.656446i	$0.227941\pi$
$44$		0			0
$45$	−0.982973	+	0.183750i	−0.982973	+	0.183750i
$46$		0			0
$47$		0			0		0.545930	−	0.837831i	$-0.316176\pi$
$47$		0			0		−0.545930	+	0.837831i	$0.683824\pi$
$48$		0			0
$49$	0.0922684	+	0.995734i	0.0922684	+	0.995734i
$50$	−0.183750	+	0.982973i	−0.183750	+	0.982973i
$51$		0			0
$52$	−0.134613	+	0.187091i	−0.134613	+	0.187091i
$53$	−0.0956383	−	1.37786i	−0.0956383	−	1.37786i	−0.769334	−	0.638847i	$-0.779412\pi$
$53$	−0.0956383	−	1.37786i	−0.0956383	−	1.37786i	0.673696	−	0.739009i	$-0.264706\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−0.751834	+	0.396298i	−0.751834	+	0.396298i
$59$		0			0		0.183750	−	0.982973i	$-0.441176\pi$
$59$		0			0		−0.183750	+	0.982973i	$0.558824\pi$
$60$		0			0
$61$	0.595012	−	0.868610i	0.595012	−	0.868610i	−0.403921	−	0.914794i	$-0.632353\pi$
$61$	0.595012	−	0.868610i	0.595012	−	0.868610i	0.998933	+	0.0461835i	$0.0147059\pi$
$62$		0			0
$63$		0			0
$64$	0.739009	−	0.673696i	0.739009	−	0.673696i
$65$	0.166699	+	0.159170i	0.166699	+	0.159170i
$66$		0			0
$67$		0			0		0.905220	−	0.424943i	$-0.139706\pi$
$67$		0			0		−0.905220	+	0.424943i	$0.860294\pi$
$68$	−0.644672	−	0.486834i	−0.644672	−	0.486834i
$69$		0			0
$70$		0			0
$71$		0			0		−0.993337	−	0.115243i	$-0.963235\pi$
$71$		0			0		0.993337	+	0.115243i	$0.0367647\pi$
$72$	0.565136	−	0.824997i	0.565136	−	0.824997i
$73$	1.64823	+	0.0762025i	1.64823	+	0.0762025i	0.850217	−	0.526432i	$-0.176471\pi$
$73$	1.64823	+	0.0762025i	1.64823	+	0.0762025i	0.798017	+	0.602635i	$0.205882\pi$
$74$	−0.220502	+	0.166516i	−0.220502	+	0.166516i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.295806	−	0.955248i	$-0.404412\pi$
$79$		0			0		−0.295806	+	0.955248i	$0.595588\pi$
$80$	−0.565136	−	0.824997i	−0.565136	−	0.824997i
$81$	0.361242	−	0.932472i	0.361242	−	0.932472i
$82$	0.0701658	−	0.119398i	0.0701658	−	0.119398i
$83$		0			0		0.723251	−	0.690585i	$-0.242647\pi$
$83$		0			0		−0.723251	+	0.690585i	$0.757353\pi$
$84$		0			0
$85$	−0.571231	+	0.571231i	−0.571231	+	0.571231i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.299742	+	0.510058i	0.299742	+	0.510058i	0.973438	−	0.228951i	$-0.0735294\pi$
$89$	0.299742	+	0.510058i	0.299742	+	0.510058i	−0.673696	+	0.739009i	$0.735294\pi$
$90$	−0.739009	−	0.673696i	−0.739009	−	0.673696i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.212941	−	1.83545i	−0.212941	−	1.83545i	−0.486604	−	0.873622i	$-0.661765\pi$
$97$	−0.212941	−	1.83545i	−0.212941	−	1.83545i	0.273663	−	0.961826i	$-0.411765\pi$
$98$	−0.739009	+	0.673696i	−0.739009	+	0.673696i
$99$		0			0
$100$	−0.895163	+	0.445738i	−0.895163	+	0.445738i
$101$	0.174970	−	0.0586442i	0.174970	−	0.0586442i	−0.228951	−	0.973438i	$-0.573529\pi$
$101$	0.174970	−	0.0586442i	0.174970	−	0.0586442i	0.403921	+	0.914794i	$0.367647\pi$
$102$		0			0
$103$		0			0		−0.873622	−	0.486604i	$-0.838235\pi$
$103$		0			0		0.873622	+	0.486604i	$0.161765\pi$
$104$	−0.230425	+	0.00532375i	−0.230425	+	0.00532375i
$105$		0			0
$106$	1.04192	−	0.906665i	1.04192	−	0.906665i
$107$		0			0		0.0922684	−	0.995734i	$-0.470588\pi$
$107$		0			0		−0.0922684	+	0.995734i	$0.529412\pi$
$108$		0			0
$109$	−0.352279	+	1.49780i	−0.352279	+	1.49780i	0.445738	+	0.895163i	$0.352941\pi$
$109$	−0.352279	+	1.49780i	−0.352279	+	1.49780i	−0.798017	+	0.602635i	$0.794118\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.0475735	+	0.410060i	−0.0475735	+	0.410060i	0.948161	+	0.317791i	$0.102941\pi$
$113$	−0.0475735	+	0.410060i	−0.0475735	+	0.410060i	−0.995734	+	0.0922684i	$0.970588\pi$
$114$		0			0
$115$		0			0
$116$	−0.769334	−	0.361153i	−0.769334	−	0.361153i
$117$	−0.223085	+	0.0579382i	−0.223085	+	0.0579382i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.526432	−	0.850217i	0.526432	−	0.850217i
$122$	1.05174	−	0.0486249i	1.05174	−	0.0486249i
$123$		0			0
$124$		0			0
$125$	0.317791	+	0.948161i	0.317791	+	0.948161i
$126$		0			0
$127$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$127$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$128$	0.982973	+	0.183750i	0.982973	+	0.183750i
$129$		0			0
$130$	−0.0265619	+	0.228951i	−0.0265619	+	0.228951i
$131$		0			0		−0.811724	−	0.584041i	$-0.801471\pi$
$131$		0			0		0.811724	+	0.584041i	$0.198529\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		−	0.807842i		−	0.807842i
$137$	−0.673696	−	0.739009i	−0.673696	−	0.739009i
$137$	−0.673696	−	0.739009i	−0.673696	−	0.739009i
$138$		0			0
$139$		0			0		−0.990410	−	0.138156i	$-0.955882\pi$
$139$		0			0		0.990410	+	0.138156i	$0.0441176\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.998933	−	0.0461835i	0.998933	−	0.0461835i
$145$	−0.463978	+	0.712061i	−0.463978	+	0.712061i
$146$	0.932472	+	1.36124i	0.932472	+	1.36124i
$147$		0			0
$148$	−0.265765	−	0.0756166i	−0.265765	−	0.0756166i
$149$	0.347190	−	0.363613i	0.347190	−	0.363613i	−0.526432	−	0.850217i	$-0.676471\pi$
$149$	0.347190	−	0.363613i	0.347190	−	0.363613i	0.873622	+	0.486604i	$0.161765\pi$
$150$		0			0
$151$		0			0		−0.995734	−	0.0922684i	$-0.970588\pi$
$151$		0			0		0.995734	+	0.0922684i	$0.0294118\pi$
$152$		0			0
$153$	−0.184956	−	0.786384i	−0.184956	−	0.786384i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.254719	+	0.980770i	0.254719	+	0.980770i	0.961826	+	0.273663i	$0.0882353\pi$
$157$	0.254719	+	0.980770i	0.254719	+	0.980770i	−0.707107	+	0.707107i	$0.750000\pi$
$158$		0			0
$159$		0			0
$160$	0.317791	−	0.948161i	0.317791	−	0.948161i
$161$		0			0
$162$	0.961826	−	0.273663i	0.961826	−	0.273663i
$163$		0			0		0.466296	−	0.884629i	$-0.345588\pi$
$163$		0			0		−0.466296	+	0.884629i	$0.654412\pi$
$164$	0.137566	−	0.0159599i	0.137566	−	0.0159599i
$165$		0			0
$166$		0			0
$167$		0			0		−0.995734	−	0.0922684i	$-0.970588\pi$
$167$		0			0		0.995734	+	0.0922684i	$0.0294118\pi$
$168$		0			0
$169$	−0.728464	−	0.604909i	−0.728464	−	0.604909i
$170$	−0.800095	−	0.111609i	−0.800095	−	0.111609i
$171$		0			0
$172$		0			0
$173$	0.658809	−	1.32307i	0.658809	−	1.32307i	−0.273663	−	0.961826i	$-0.588235\pi$
$173$	0.658809	−	1.32307i	0.658809	−	1.32307i	0.932472	−	0.361242i	$-0.117647\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−0.226400	+	0.546578i	−0.226400	+	0.546578i
$179$		0			0		0.545930	−	0.837831i	$-0.316176\pi$
$179$		0			0		−0.545930	+	0.837831i	$0.683824\pi$
$180$	0.0922684	−	0.995734i	0.0922684	−	0.995734i
$181$	0.0899135	−	0.0211475i	0.0899135	−	0.0211475i	−0.183750	−	0.982973i	$-0.558824\pi$
$181$	0.0899135	−	0.0211475i	0.0899135	−	0.0211475i	0.273663	+	0.961826i	$0.411765\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.111609	+	0.252769i	−0.111609	+	0.252769i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.999733	−	0.0230979i	$-0.992647\pi$
$191$		0			0		0.999733	+	0.0230979i	$0.00735294\pi$
$192$		0			0
$193$	−0.148441	−	0.336186i	−0.148441	−	0.336186i	0.824997	−	0.565136i	$-0.191176\pi$
$193$	−0.148441	−	0.336186i	−0.148441	−	0.336186i	−0.973438	+	0.228951i	$0.926471\pi$
$194$	1.33639	−	1.27604i	1.33639	−	1.27604i
$195$		0			0
$196$	−0.982973	−	0.183750i	−0.982973	−	0.183750i
$197$	0.455948	−	1.93857i	0.455948	−	1.93857i	0.138156	−	0.990410i	$-0.455882\pi$
$197$	0.455948	−	1.93857i	0.455948	−	1.93857i	0.317791	−	0.948161i	$-0.397059\pi$
$198$		0			0
$199$		0			0		−0.545930	−	0.837831i	$-0.683824\pi$
$199$		0			0		0.545930	+	0.837831i	$0.316176\pi$
$200$	−0.895163	−	0.445738i	−0.895163	−	0.445738i
$201$		0			0
$202$	0.152242	+	0.104288i	0.152242	+	0.104288i
$203$		0			0
$204$		0			0
$205$	0.00319880	−	0.138452i	0.00319880	−	0.138452i
$206$		0			0
$207$		0			0
$208$	−0.143110	−	0.180675i	−0.143110	−	0.180675i
$209$		0			0
$210$		0			0
$211$		0			0		−0.602635	−	0.798017i	$-0.705882\pi$
$211$		0			0		0.602635	+	0.798017i	$0.294118\pi$
$212$	1.35143	+	0.285081i	1.35143	+	0.285081i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−1.40756	+	0.621500i	−1.40756	+	0.621500i
$219$		0			0
$220$		0			0
$221$	−0.122228	+	0.140462i	−0.122228	+	0.140462i
$222$		0			0
$223$		0			0		0.754373	−	0.656446i	$-0.227941\pi$
$223$		0			0		−0.754373	+	0.656446i	$0.772059\pi$
$224$		0			0
$225$	−0.973438	−	0.228951i	−0.973438	−	0.228951i
$226$	−0.355904	+	0.209152i	−0.355904	+	0.209152i
$227$		0			0		−0.837831	−	0.545930i	$-0.816176\pi$
$227$		0			0		0.837831	+	0.545930i	$0.183824\pi$
$228$		0			0
$229$	1.41908	−	0.512381i	1.41908	−	0.512381i	0.486604	−	0.873622i	$-0.338235\pi$
$229$	1.41908	−	0.512381i	1.41908	−	0.512381i	0.932472	+	0.361242i	$0.117647\pi$
$230$		0			0
$231$		0			0
$232$	−0.175421	−	0.831585i	−0.175421	−	0.831585i
$233$	0.0426793	+	0.0176784i	0.0426793	+	0.0176784i	0.403921	−	0.914794i	$-0.367647\pi$
$233$	0.0426793	+	0.0176784i	0.0426793	+	0.0176784i	−0.361242	+	0.932472i	$0.617647\pi$
$234$	−0.180675	−	0.143110i	−0.180675	−	0.143110i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.978467	−	0.206405i	$-0.933824\pi$
$239$		0			0		0.978467	+	0.206405i	$0.0661765\pi$
$240$		0			0
$241$	0.0387043	+	1.67521i	0.0387043	+	1.67521i	0.565136	+	0.824997i	$0.308824\pi$
$241$	0.0387043	+	1.67521i	0.0387043	+	1.67521i	−0.526432	+	0.850217i	$0.676471\pi$
$242$	0.995734	−	0.0922684i	0.995734	−	0.0922684i
$243$		0			0
$244$	0.672619	+	0.810004i	0.672619	+	0.810004i
$245$	−0.317791	+	0.948161i	−0.317791	+	0.948161i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−0.565136	+	0.824997i	−0.565136	+	0.824997i
$251$		0			0		0.811724	−	0.584041i	$-0.198529\pi$
$251$		0			0		−0.811724	+	0.584041i	$0.801471\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.445738	+	0.895163i	0.445738	+	0.895163i
$257$	−0.907490	−	0.351564i	−0.907490	−	0.351564i	−0.138156	−	0.990410i	$-0.544118\pi$
$257$	−0.907490	−	0.351564i	−0.907490	−	0.351564i	−0.769334	+	0.638847i	$0.779412\pi$
$258$		0			0
$259$		0			0
$260$	−0.198714	+	0.116777i	−0.198714	+	0.116777i
$261$	−0.361153	−	0.769334i	−0.361153	−	0.769334i
$262$		0			0
$263$		0			0		0.273663	−	0.961826i	$-0.411765\pi$
$263$		0			0		−0.273663	+	0.961826i	$0.588235\pi$
$264$		0			0
$265$	0.469056	−	1.29908i	0.469056	−	1.29908i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.491922	+	1.89410i	−0.491922	+	1.89410i	−0.0461835	+	0.998933i	$0.514706\pi$
$269$	−0.491922	+	1.89410i	−0.491922	+	1.89410i	−0.445738	+	0.895163i	$0.647059\pi$
$270$		0			0
$271$		0			0		0.466296	−	0.884629i	$-0.345588\pi$
$271$		0			0		−0.466296	+	0.884629i	$0.654412\pi$
$272$	0.644672	−	0.486834i	0.644672	−	0.486834i
$273$		0			0
$274$	0.183750	−	0.982973i	0.183750	−	0.982973i
$275$		0			0
$276$		0			0
$277$	0.731115	−	0.226400i	0.731115	−	0.226400i	0.0922684	−	0.995734i	$-0.470588\pi$
$277$	0.731115	−	0.226400i	0.731115	−	0.226400i	0.638847	+	0.769334i	$0.279412\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.0914812	+	1.97871i	0.0914812	+	1.97871i	0.183750	+	0.982973i	$0.441176\pi$
$281$	0.0914812	+	1.97871i	0.0914812	+	1.97871i	−0.0922684	+	0.995734i	$0.529412\pi$
$282$		0			0
$283$		0			0		0.183750	−	0.982973i	$-0.441176\pi$
$283$		0			0		−0.183750	+	0.982973i	$0.558824\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.638847	+	0.769334i	0.638847	+	0.769334i
$289$	0.256725	+	0.234036i	0.256725	+	0.234036i
$290$	−0.847846	+	0.0588498i	−0.847846	+	0.0588498i
$291$		0			0
$292$	−0.524354	+	1.56446i	−0.524354	+	1.56446i
$293$	−0.778814	−	1.56407i	−0.778814	−	1.56407i	−0.824997	−	0.565136i	$-0.808824\pi$
$293$	−0.778814	−	1.56407i	−0.778814	−	1.56407i	0.0461835	−	0.998933i	$-0.485294\pi$
$294$		0			0
$295$		0			0
$296$	−0.0998157	−	0.257654i	−0.0998157	−	0.257654i
$297$		0			0
$298$	0.499398	+	0.0579382i	0.499398	+	0.0579382i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.895163	−	0.554262i	0.895163	−	0.554262i
$306$	0.516087	−	0.621500i	0.516087	−	0.621500i
$307$		0			0		0.690585	−	0.723251i	$-0.257353\pi$
$307$		0			0		−0.690585	+	0.723251i	$0.742647\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$311$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$312$		0			0
$313$	0.308486	+	0.338393i	0.308486	+	0.338393i	0.873622	−	0.486604i	$-0.161765\pi$
$313$	0.308486	+	0.338393i	0.308486	+	0.338393i	−0.565136	+	0.824997i	$0.691176\pi$
$314$	−0.629169	+	0.794316i	−0.629169	+	0.794316i
$315$		0			0
$316$		0			0
$317$	1.30974	−	1.50512i	1.30974	−	1.50512i	0.602635	−	0.798017i	$-0.294118\pi$
$317$	1.30974	−	1.50512i	1.30974	−	1.50512i	0.707107	−	0.707107i	$-0.250000\pi$
$318$		0			0
$319$		0			0
$320$	0.948161	−	0.317791i	0.948161	−	0.317791i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.798017	+	0.602635i	0.798017	+	0.602635i
$325$	0.0882033	+	0.212941i	0.0882033	+	0.212941i
$326$		0			0
$327$		0			0
$328$	0.0956383	+	0.100162i	0.0956383	+	0.100162i
$329$		0			0
$330$		0			0
$331$		0			0		−0.690585	−	0.723251i	$-0.742647\pi$
$331$		0			0		0.690585	+	0.723251i	$0.257353\pi$
$332$		0			0
$333$	−0.156154	−	0.227957i	−0.156154	−	0.227957i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.0756166	−	0.542077i	0.0756166	−	0.542077i	−0.914794	−	0.403921i	$-0.867647\pi$
$337$	0.0756166	−	0.542077i	0.0756166	−	0.542077i	0.990410	−	0.138156i	$-0.0441176\pi$
$338$	0.0437300	−	0.945866i	0.0437300	−	0.945866i
$339$		0			0
$340$	−0.393100	−	0.705749i	−0.393100	−	0.705749i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	1.45285	−	0.271585i	1.45285	−	0.271585i
$347$		0			0		0.0461835	−	0.998933i	$-0.485294\pi$
$347$		0			0		−0.0461835	+	0.998933i	$0.514706\pi$
$348$		0			0
$349$	1.21192	−	0.789689i	1.21192	−	0.789689i	0.228951	−	0.973438i	$-0.426471\pi$
$349$	1.21192	−	0.789689i	1.21192	−	0.789689i	0.982973	+	0.183750i	$0.0588235\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.509130	+	0.965891i	0.509130	+	0.965891i	0.995734	+	0.0922684i	$0.0294118\pi$
$353$	0.509130	+	0.965891i	0.509130	+	0.965891i	−0.486604	+	0.873622i	$0.661765\pi$
$354$		0			0
$355$		0			0
$356$	−0.572616	+	0.148716i	−0.572616	+	0.148716i
$357$		0			0
$358$		0			0
$359$		0			0		0.986955	−	0.160996i	$-0.0514706\pi$
$359$		0			0		−0.986955	+	0.160996i	$0.948529\pi$
$360$	0.850217	−	0.526432i	0.850217	−	0.526432i
$361$	−0.798017	−	0.602635i	−0.798017	−	0.602635i
$362$	0.0710610	+	0.0590083i	0.0710610	+	0.0590083i
$363$		0			0
$364$		0			0
$365$	1.47701	+	0.735466i	1.47701	+	0.735466i
$366$		0			0
$367$		0			0		−0.850217	−	0.526432i	$-0.823529\pi$
$367$		0			0		0.850217	+	0.526432i	$0.176471\pi$
$368$		0			0
$369$	0.116030	+	0.0756052i	0.116030	+	0.0756052i
$370$	−0.268973	+	0.0632619i	−0.268973	+	0.0632619i
$371$		0			0
$372$		0			0
$373$	−0.611320	−	1.82393i	−0.611320	−	1.82393i	−0.565136	−	0.824997i	$-0.691176\pi$
$373$	−0.611320	−	1.82393i	−0.611320	−	1.82393i	−0.0461835	−	0.998933i	$-0.514706\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.0953195	+	0.171131i	−0.0953195	+	0.171131i
$378$		0			0
$379$		0			0		0.638847	−	0.769334i	$-0.279412\pi$
$379$		0			0		−0.638847	+	0.769334i	$0.720588\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.526432	−	0.850217i	$-0.676471\pi$
$383$		0			0		0.526432	+	0.850217i	$0.323529\pi$
$384$		0			0
$385$		0			0
$386$	0.178827	−	0.321055i	0.178827	−	0.321055i
$387$		0			0
$388$	1.82366	+	0.297482i	1.82366	+	0.297482i
$389$	0.408302	+	1.05395i	0.408302	+	1.05395i	0.973438	+	0.228951i	$0.0735294\pi$
$389$	0.408302	+	1.05395i	0.408302	+	1.05395i	−0.565136	+	0.824997i	$0.691176\pi$
$390$		0			0
$391$		0			0
$392$	−0.445738	−	0.895163i	−0.445738	−	0.895163i
$393$		0			0
$394$	1.82178	−	0.804396i	1.82178	−	0.804396i
$395$		0			0
$396$		0			0
$397$	1.01962	+	1.22788i	1.01962	+	1.22788i	0.973438	+	0.228951i	$0.0735294\pi$
$397$	1.01962	+	1.22788i	1.01962	+	1.22788i	0.0461835	+	0.998933i	$0.485294\pi$
$398$		0			0
$399$		0			0
$400$	−0.183750	−	0.982973i	−0.183750	−	0.982973i
$401$	−0.259924	+	0.627512i	−0.259924	+	0.627512i	−0.998933	−	0.0461835i	$-0.985294\pi$
$401$	−0.259924	+	0.627512i	−0.259924	+	0.627512i	0.739009	+	0.673696i	$0.235294\pi$
$402$		0			0
$403$		0			0
$404$	0.00852254	+	0.184340i	0.00852254	+	0.184340i
$405$	0.707107	−	0.707107i	0.707107	−	0.707107i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.247345	−	1.77316i	0.247345	−	1.77316i	−0.317791	−	0.948161i	$-0.602941\pi$
$409$	0.247345	−	1.77316i	0.247345	−	1.77316i	0.565136	−	0.824997i	$-0.308824\pi$
$410$	0.112415	−	0.0808832i	0.112415	−	0.0808832i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	0.0579382	−	0.223085i	0.0579382	−	0.223085i
$417$		0			0
$418$		0			0
$419$		0			0		−0.914794	−	0.403921i	$-0.867647\pi$
$419$		0			0		0.914794	+	0.403921i	$0.132353\pi$
$420$		0			0
$421$	1.63458	+	0.677066i	1.63458	+	0.677066i	0.995734	−	0.0922684i	$-0.0294118\pi$
$421$	1.63458	+	0.677066i	1.63458	+	0.677066i	0.638847	+	0.769334i	$0.279412\pi$
$422$		0			0
$423$		0			0
$424$	0.586919	+	1.25026i	0.586919	+	1.25026i
$425$	−0.753290	+	0.291826i	−0.753290	+	0.291826i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.506653	−	0.862150i	$-0.330882\pi$
$431$		0			0		−0.506653	+	0.862150i	$0.669118\pi$
$432$		0			0
$433$	0.516087	−	1.16883i	0.516087	−	1.16883i	−0.445738	−	0.895163i	$-0.647059\pi$
$433$	0.516087	−	1.16883i	0.516087	−	1.16883i	0.961826	−	0.273663i	$-0.0882353\pi$
$434$		0			0
$435$		0			0
$436$	−1.34421	−	0.748723i	−1.34421	−	0.748723i
$437$		0			0
$438$		0			0
$439$		0			0		−0.973438	−	0.228951i	$-0.926471\pi$
$439$		0			0		0.973438	+	0.228951i	$0.0735294\pi$
$440$		0			0
$441$	−0.638847	−	0.769334i	−0.638847	−	0.769334i
$442$	−0.185750	−	0.0128931i	−0.185750	−	0.0128931i
$443$		0			0		0.995734	−	0.0922684i	$-0.0294118\pi$
$443$		0			0		−0.995734	+	0.0922684i	$0.970588\pi$
$444$		0			0
$445$	0.0681792	+	0.587671i	0.0681792	+	0.587671i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.78842	−	0.890525i	−1.78842	−	0.890525i	−0.914794	−	0.403921i	$-0.867647\pi$
$449$	−1.78842	−	0.890525i	−1.78842	−	0.890525i	−0.873622	−	0.486604i	$-0.838235\pi$
$450$	−0.403921	−	0.914794i	−0.403921	−	0.914794i
$451$		0			0
$452$	−0.381387	−	0.157976i	−0.381387	−	0.157976i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.275866	−	0.890856i	0.275866	−	0.890856i	−0.707107	−	0.707107i	$-0.750000\pi$
$457$	0.275866	−	0.890856i	0.275866	−	0.890856i	0.982973	−	0.183750i	$-0.0588235\pi$
$458$	1.26407	+	0.823669i	1.26407	+	0.823669i
$459$		0			0
$460$		0			0
$461$	−1.18475	+	1.56886i	−1.18475	+	1.56886i	−0.445738	+	0.895163i	$0.647059\pi$
$461$	−1.18475	+	1.56886i	−1.18475	+	1.56886i	−0.739009	+	0.673696i	$0.764706\pi$
$462$		0			0
$463$		0			0		−0.811724	−	0.584041i	$-0.801471\pi$
$463$		0			0		0.811724	+	0.584041i	$0.198529\pi$
$464$	0.557905	−	0.641131i	0.557905	−	0.641131i
$465$		0			0
$466$	0.0116124	+	0.0447124i	0.0116124	+	0.0447124i
$467$		0			0		0.914794	−	0.403921i	$-0.132353\pi$
$467$		0			0		−0.914794	+	0.403921i	$0.867647\pi$
$468$	0.00532375	−	0.230425i	0.00532375	−	0.230425i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.857578	+	1.08268i	0.857578	+	1.08268i
$478$		0			0
$479$		0			0		−0.138156	−	0.990410i	$-0.544118\pi$
$479$		0			0		0.138156	+	0.990410i	$0.455882\pi$
$480$		0			0
$481$	−0.0216283	+	0.0599012i	−0.0216283	+	0.0599012i
$482$	−1.31353	+	1.04043i	−1.31353	+	1.04043i
$483$		0			0
$484$	0.673696	+	0.739009i	0.673696	+	0.739009i
$485$	0.546578	−	1.76507i	0.546578	−	1.76507i
$486$		0			0
$487$		0			0		0.982973	−	0.183750i	$-0.0588235\pi$
$487$		0			0		−0.982973	+	0.183750i	$0.941176\pi$
$488$	−0.241054	+	1.02490i	−0.241054	+	1.02490i
$489$		0			0
$490$	−0.948161	+	0.317791i	−0.948161	+	0.317791i
$491$		0			0		0.723251	−	0.690585i	$-0.242647\pi$
$491$		0			0		−0.723251	+	0.690585i	$0.757353\pi$
$492$		0			0
$493$	−0.591929	−	0.347855i	−0.591929	−	0.347855i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.995734	−	0.0922684i	$-0.0294118\pi$
$499$		0			0		−0.995734	+	0.0922684i	$0.970588\pi$
$500$	−0.998933	+	0.0461835i	−0.998933	+	0.0461835i
$501$		0			0
$502$		0			0
$503$		0			0		0.424943	−	0.905220i	$-0.360294\pi$
$503$		0			0		−0.424943	+	0.905220i	$0.639706\pi$
$504$		0			0
$505$	0.183750	+	0.0170269i	0.183750	+	0.0170269i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.20398	−	0.0556635i	−1.20398	−	0.0556635i	−0.565136	−	0.824997i	$-0.691176\pi$
$509$	−1.20398	−	0.0556635i	−1.20398	−	0.0556635i	−0.638847	+	0.769334i	$0.720588\pi$
$510$		0			0
$511$		0			0
$512$	−0.445738	+	0.895163i	−0.445738	+	0.895163i
$513$		0			0
$514$	−0.266331	−	0.936057i	−0.266331	−	0.936057i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−0.212941	−	0.0882033i	−0.212941	−	0.0882033i
$521$	−1.86860	+	0.216788i	−1.86860	+	0.216788i	−0.973438	−	0.228951i	$-0.926471\pi$
$521$	−1.86860	+	0.216788i	−1.86860	+	0.216788i	−0.895163	+	0.445738i	$0.852941\pi$
$522$	0.396298	−	0.751834i	0.396298	−	0.751834i
$523$		0			0		0.961826	−	0.273663i	$-0.0882353\pi$
$523$		0			0		−0.961826	+	0.273663i	$0.911765\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.948161	−	0.317791i	−0.948161	−	0.317791i
$530$	1.31936	−	0.408559i	1.31936	−	0.408559i
$531$		0			0
$532$		0			0
$533$	−0.00147416	−	0.0318857i	−0.00147416	−	0.0318857i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−1.80797	+	0.748886i	−1.80797	+	0.748886i
$539$		0			0
$540$		0			0
$541$	0.890525	−	0.0411715i	0.890525	−	0.0411715i	0.403921	−	0.914794i	$-0.367647\pi$
$541$	0.890525	−	0.0411715i	0.890525	−	0.0411715i	0.486604	+	0.873622i	$0.338235\pi$
$542$		0			0
$543$		0			0
$544$	0.777003	+	0.221076i	0.777003	+	0.221076i
$545$	−0.927255	+	1.22788i	−0.927255	+	1.22788i
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$	0.895163	−	0.445738i	0.895163	−	0.445738i
$549$			1.05286i			1.05286i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	0.621267	+	0.447006i	0.621267	+	0.447006i
$555$		0			0
$556$		0			0
$557$	−0.710182	−	0.132756i	−0.710182	−	0.132756i	−0.183750	−	0.982973i	$-0.558824\pi$
$557$	−0.710182	−	0.132756i	−0.710182	−	0.132756i	−0.526432	+	0.850217i	$0.676471\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−1.52391	+	1.26544i	−1.52391	+	1.26544i
$563$		0			0		0.998933	−	0.0461835i	$-0.0147059\pi$
$563$		0			0		−0.998933	+	0.0461835i	$0.985294\pi$
$564$		0			0
$565$	−0.209152	+	0.355904i	−0.209152	+	0.355904i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1.18846	+	0.557905i	1.18846	+	0.557905i	0.914794	−	0.403921i	$-0.132353\pi$
$569$	1.18846	+	0.557905i	1.18846	+	0.557905i	0.273663	+	0.961826i	$0.411765\pi$
$570$		0			0
$571$		0			0		−0.584041	−	0.811724i	$-0.698529\pi$
$571$		0			0		0.584041	+	0.811724i	$0.301471\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.228951	+	0.973438i	−0.228951	+	0.973438i
$577$	−1.57132	+	1.13058i	−1.57132	+	1.13058i	−0.638847	+	0.769334i	$0.720588\pi$
$577$	−1.57132	+	1.13058i	−1.57132	+	1.13058i	−0.932472	+	0.361242i	$0.882353\pi$
$578$	−0.0320532	+	0.345909i	−0.0320532	+	0.345909i
$579$		0			0
$580$	−0.557905	−	0.641131i	−0.557905	−	0.641131i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−1.56446	+	0.524354i	−1.56446	+	0.524354i
$585$	−0.227480	−	0.0371073i	−0.227480	−	0.0371073i
$586$	0.778814	−	1.56407i	0.778814	−	1.56407i
$587$		0			0		0.739009	−	0.673696i	$-0.235294\pi$
$587$		0			0		−0.739009	+	0.673696i	$0.764706\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.145460	−	0.234926i	0.145460	−	0.234926i
$593$	−1.63885	−	0.769334i	−1.63885	−	0.769334i	−0.638847	−	0.769334i	$-0.720588\pi$
$593$	−1.63885	−	0.769334i	−1.63885	−	0.769334i	−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$	0.254719	+	0.433444i	0.254719	+	0.433444i
$597$		0			0
$598$		0			0
$599$		0			0		0.0692444	−	0.997600i	$-0.477941\pi$
$599$		0			0		−0.0692444	+	0.997600i	$0.522059\pi$
$600$		0			0
$601$	−0.703522	−	0.612196i	−0.703522	−	0.612196i	0.228951	−	0.973438i	$-0.426471\pi$
$601$	−0.703522	−	0.612196i	−0.703522	−	0.612196i	−0.932472	+	0.361242i	$0.882353\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.824997	−	0.565136i	0.824997	−	0.565136i
$606$		0			0
$607$		0			0		0.824997	−	0.565136i	$-0.191176\pi$
$607$		0			0		−0.824997	+	0.565136i	$0.808824\pi$
$608$		0			0
$609$		0			0
$610$	0.981767	+	0.380338i	0.981767	+	0.380338i
$611$		0			0
$612$	0.806980	+	0.0373089i	0.806980	+	0.0373089i
$613$	0.359191	−	0.524354i	0.359191	−	0.524354i	−0.602635	−	0.798017i	$-0.705882\pi$
$613$	0.359191	−	0.524354i	0.359191	−	0.524354i	0.961826	+	0.273663i	$0.0882353\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.48826	−	1.12388i	−1.48826	−	1.12388i	−0.961826	−	0.273663i	$-0.911765\pi$
$617$	−1.48826	−	1.12388i	−1.48826	−	1.12388i	−0.526432	−	0.850217i	$-0.676471\pi$
$618$		0			0
$619$		0			0		0.115243	−	0.993337i	$-0.463235\pi$
$619$		0			0		−0.115243	+	0.993337i	$0.536765\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.0922684	+	0.995734i	−0.0922684	+	0.995734i
$626$	−0.0841391	+	0.450104i	−0.0841391	+	0.450104i
$627$		0			0
$628$	−1.01304	−	0.0234053i	−1.01304	−	0.0234053i
$629$	−0.208144	−	0.0806353i	−0.208144	−	0.0806353i
$630$		0			0
$631$		0			0		0.0230979	−	0.999733i	$-0.492647\pi$
$631$		0			0		−0.0230979	+	0.999733i	$0.507353\pi$
$632$		0			0
$633$		0			0
$634$	1.99041	+	0.138156i	1.99041	+	0.138156i
$635$		0			0
$636$		0			0
$637$	−0.0579382	+	0.223085i	−0.0579382	+	0.223085i
$638$		0			0
$639$		0			0
$640$	0.824997	+	0.565136i	0.824997	+	0.565136i
$641$	0.418885	+	1.78099i	0.418885	+	1.78099i	0.602635	+	0.798017i	$0.294118\pi$
$641$	0.418885	+	1.78099i	0.418885	+	1.78099i	−0.183750	+	0.982973i	$0.558824\pi$
$642$		0			0
$643$		0			0		0.978467	−	0.206405i	$-0.0661765\pi$
$643$		0			0		−0.978467	+	0.206405i	$0.933824\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.948161	−	0.317791i	$-0.102941\pi$
$647$		0			0		−0.948161	+	0.317791i	$0.897059\pi$
$648$			1.00000i			1.00000i
$649$		0			0
$650$	−0.116777	+	0.198714i	−0.116777	+	0.198714i
$651$		0			0
$652$		0			0
$653$	0.104288	+	1.12545i	0.104288	+	1.12545i	0.873622	+	0.486604i	$0.161765\pi$
$653$	0.104288	+	1.12545i	0.104288	+	1.12545i	−0.769334	+	0.638847i	$0.779412\pi$
$654$		0			0
$655$		0			0
$656$	−0.0222961	+	0.136682i	−0.0222961	+	0.136682i
$657$	−1.40285	+	0.868610i	−1.40285	+	0.868610i
$658$		0			0
$659$		0			0		−0.295806	−	0.955248i	$-0.595588\pi$
$659$		0			0		0.295806	+	0.955248i	$0.404412\pi$
$660$		0			0
$661$	1.12411	+	1.41918i	1.12411	+	1.41918i	0.895163	+	0.445738i	$0.147059\pi$
$661$	1.12411	+	1.41918i	1.12411	+	1.41918i	0.228951	+	0.973438i	$0.426471\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	0.0878098	−	0.261989i	0.0878098	−	0.261989i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.44612	+	0.0334111i	−1.44612	+	0.0334111i	−0.739009	−	0.673696i	$-0.764706\pi$
$673$	−1.44612	+	0.0334111i	−1.44612	+	0.0334111i	−0.707107	+	0.707107i	$0.750000\pi$
$674$	0.478156	−	0.266331i	0.478156	−	0.266331i
$675$		0			0
$676$	0.781170	−	0.535114i	0.781170	−	0.535114i
$677$	−1.12388	+	0.435393i	−1.12388	+	0.435393i	−0.850217	−	0.526432i	$-0.823529\pi$
$677$	−1.12388	+	0.435393i	−1.12388	+	0.435393i	−0.273663	+	0.961826i	$0.588235\pi$
$678$		0			0
$679$		0			0
$680$	0.326304	−	0.739009i	0.326304	−	0.739009i
$681$		0			0
$682$		0			0
$683$		0			0		−0.602635	−	0.798017i	$-0.705882\pi$
$683$		0			0		0.602635	+	0.798017i	$0.294118\pi$
$684$		0			0
$685$	−0.317791	−	0.948161i	−0.317791	−	0.948161i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.0871181	−	0.306188i	0.0871181	−	0.306188i
$690$		0			0
$691$		0			0		0.584041	−	0.811724i	$-0.301471\pi$
$691$		0			0		−0.584041	+	0.811724i	$0.698529\pi$
$692$	1.09227	+	0.995734i	1.09227	+	0.995734i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	0.111847	−	0.00258412i	0.111847	−	0.00258412i
$698$	1.36053	+	0.491242i	1.36053	+	0.491242i
$699$		0			0
$700$		0			0
$701$	0.465346	+	0.288130i	0.465346	+	0.288130i	0.739009	−	0.673696i	$-0.235294\pi$
$701$	0.465346	+	0.288130i	0.465346	+	0.288130i	−0.273663	+	0.961826i	$0.588235\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−0.463978	+	0.988373i	−0.463978	+	0.988373i
$707$		0			0
$708$		0			0
$709$	−0.677943	−	0.855892i	−0.677943	−	0.855892i	0.317791	−	0.948161i	$-0.397059\pi$
$709$	−0.677943	−	0.855892i	−0.677943	−	0.855892i	−0.995734	+	0.0922684i	$0.970588\pi$
$710$		0			0
$711$		0			0
$712$	−0.463756	−	0.367336i	−0.463756	−	0.367336i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.961826	−	0.273663i	$-0.0882353\pi$
$719$		0			0		−0.961826	+	0.273663i	$0.911765\pi$
$720$	0.932472	+	0.361242i	0.932472	+	0.361242i
$721$		0			0
$722$		−	1.00000i		−	1.00000i
$723$		0			0
$724$	−0.00426582	+	0.0922684i	−0.00426582	+	0.0922684i
$725$	−0.712061	+	0.463978i	−0.712061	+	0.463978i
$726$		0			0
$727$		0			0		0.978467	−	0.206405i	$-0.0661765\pi$
$727$		0			0		−0.978467	+	0.206405i	$0.933824\pi$
$728$		0			0
$729$	0.228951	+	0.973438i	0.228951	+	0.973438i
$730$	0.303186	+	1.62190i	0.303186	+	1.62190i
$731$		0			0
$732$		0			0
$733$	−0.312159	+	1.20194i	−0.312159	+	1.20194i	0.602635	+	0.798017i	$0.294118\pi$
$733$	−0.312159	+	1.20194i	−0.312159	+	1.20194i	−0.914794	+	0.403921i	$0.867647\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	0.00958957	+	0.138156i	0.00958957	+	0.138156i
$739$		0			0		0.0230979	−	0.999733i	$-0.492647\pi$
$739$		0			0		−0.0230979	+	0.999733i	$0.507353\pi$
$740$	−0.212577	−	0.176521i	−0.212577	−	0.176521i
$741$		0			0
$742$		0			0
$743$		0			0		0.884629	−	0.466296i	$-0.154412\pi$
$743$		0			0		−0.884629	+	0.466296i	$0.845588\pi$
$744$		0			0
$745$	0.464478	−	0.192393i	0.464478	−	0.192393i
$746$	1.08713	−	1.58701i	1.08713	−	1.58701i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.115243	−	0.993337i	$-0.463235\pi$
$751$		0			0		−0.115243	+	0.993337i	$0.536765\pi$
$752$		0			0
$753$		0			0
$754$	−0.194009	+	0.0270630i	−0.194009	+	0.0270630i
$755$		0			0
$756$		0			0
$757$	0.681142	−	0.994344i	0.681142	−	0.994344i	−0.317791	−	0.948161i	$-0.602941\pi$
$757$	0.681142	−	0.994344i	0.681142	−	0.994344i	0.998933	−	0.0461835i	$-0.0147059\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.524354	−	0.359191i	−0.524354	−	0.359191i	0.273663	−	0.961826i	$-0.411765\pi$
$761$	−0.524354	−	0.359191i	−0.524354	−	0.359191i	−0.798017	+	0.602635i	$0.794118\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	0.148441	−	0.794087i	0.148441	−	0.794087i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−1.49869	−	1.30415i	−1.49869	−	1.30415i	−0.824997	−	0.565136i	$-0.808824\pi$
$769$	−1.49869	−	1.30415i	−1.49869	−	1.30415i	−0.673696	−	0.739009i	$-0.735294\pi$
$770$		0			0
$771$		0			0
$772$	0.363975	−	0.0507723i	0.363975	−	0.0507723i
$773$	−0.406040	+	0.488975i	−0.406040	+	0.488975i	−0.932472	−	0.361242i	$-0.882353\pi$
$773$	−0.406040	+	0.488975i	−0.406040	+	0.488975i	0.526432	+	0.850217i	$0.323529\pi$
$774$		0			0
$775$		0			0
$776$	0.861602	+	1.63458i	0.861602	+	1.63458i
$777$		0			0
$778$	−0.595012	+	0.960977i	−0.595012	+	0.960977i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.445738	−	0.895163i	0.445738	−	0.895163i
$785$	−0.163138	+	1.00009i	−0.163138	+	1.00009i
$786$		0			0
$787$		0			0		−0.545930	−	0.837831i	$-0.683824\pi$
$787$		0			0		0.545930	+	0.837831i	$0.316176\pi$
$788$	1.73979	+	0.969057i	1.73979	+	0.969057i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	0.196982	−	0.141730i	0.196982	−	0.141730i
$794$	−0.365413	+	1.55364i	−0.365413	+	1.55364i
$795$		0			0
$796$		0			0
$797$	−0.243964	−	0.857445i	−0.243964	−	0.857445i	−0.982973	−	0.183750i	$-0.941176\pi$
$797$	−0.243964	−	0.857445i	−0.243964	−	0.857445i	0.739009	−	0.673696i	$-0.235294\pi$
$798$		0			0
$799$		0			0
$800$	0.673696	−	0.739009i	0.673696	−	0.739009i
$801$	−0.535539	−	0.251402i	−0.535539	−	0.251402i
$802$	−0.657405	+	0.170737i	−0.657405	+	0.170737i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−0.141970	+	0.117891i	−0.141970	+	0.117891i
$809$	−1.80609	+	0.847846i	−1.80609	+	0.847846i	−0.873622	+	0.486604i	$0.838235\pi$
$809$	−1.80609	+	0.847846i	−1.80609	+	0.847846i	−0.932472	+	0.361242i	$0.882353\pi$
$810$	0.990410	+	0.138156i	0.990410	+	0.138156i
$811$		0			0		−0.486604	−	0.873622i	$-0.661765\pi$
$811$		0			0		0.486604	+	0.873622i	$0.338235\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	1.56407	−	0.871181i	1.56407	−	0.871181i
$819$		0			0
$820$	0.132291	+	0.0409658i	0.132291	+	0.0409658i
$821$		−	1.74724i		−	1.74724i	−0.486604	−	0.873622i	$-0.661765\pi$
$821$		−	1.74724i		−	1.74724i	0.486604	−	0.873622i	$-0.338235\pi$
$822$		0			0
$823$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$823$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.506653	−	0.862150i	$-0.669118\pi$
$827$		0			0		0.506653	+	0.862150i	$0.330882\pi$
$828$		0			0
$829$	−0.457413	+	0.0211475i	−0.457413	+	0.0211475i	−0.273663	−	0.961826i	$-0.588235\pi$
$829$	−0.457413	+	0.0211475i	−0.457413	+	0.0211475i	−0.183750	+	0.982973i	$0.558824\pi$
$830$		0			0
$831$		0			0
$832$	0.212941	−	0.0882033i	0.212941	−	0.0882033i
$833$	−0.777003	−	0.221076i	−0.777003	−	0.221076i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.403921	−	0.914794i	$-0.632353\pi$
$839$		0			0		0.403921	+	0.914794i	$0.367647\pi$
$840$		0			0
$841$	0.263298	+	0.0882486i	0.263298	+	0.0882486i
$842$	0.444745	+	1.71245i	0.444745	+	1.71245i
$843$		0			0
$844$		0			0
$845$	−0.422059	−	0.847609i	−0.422059	−	0.847609i
$846$		0			0
$847$		0			0
$848$	−0.644034	+	1.22182i	−0.644034	+	1.22182i
$849$		0			0
$850$	−0.686841	−	0.425274i	−0.686841	−	0.425274i
$851$		0			0
$852$		0			0
$853$	0.122511	−	1.76501i	0.122511	−	1.76501i	−0.403921	−	0.914794i	$-0.632353\pi$
$853$	0.122511	−	1.76501i	0.122511	−	1.76501i	0.526432	−	0.850217i	$-0.323529\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−0.569067	+	0.370803i	−0.569067	+	0.370803i	−0.798017	−	0.602635i	$-0.794118\pi$
$857$	−0.569067	+	0.370803i	−0.569067	+	0.370803i	0.228951	+	0.973438i	$0.426471\pi$
$858$		0			0
$859$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$859$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$863$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$864$		0			0
$865$	1.13709	−	0.944227i	1.13709	−	0.944227i
$866$	1.24376	−	0.292529i	1.24376	−	0.292529i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−0.212577	−	1.52391i	−0.212577	−	1.52391i
$873$	1.21295	+	1.39390i	1.21295	+	1.39390i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1.68717	−	0.991487i	−1.68717	−	0.991487i	−0.948161	−	0.317791i	$-0.897059\pi$
$877$	−1.68717	−	0.991487i	−1.68717	−	0.991487i	−0.739009	−	0.673696i	$-0.764706\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−0.876298	−	0.163808i	−0.876298	−	0.163808i	−0.273663	−	0.961826i	$-0.588235\pi$
$881$	−0.876298	−	0.163808i	−0.876298	−	0.163808i	−0.602635	+	0.798017i	$0.705882\pi$
$882$	0.228951	−	0.973438i	0.228951	−	0.973438i
$883$		0			0		0.982973	−	0.183750i	$-0.0588235\pi$
$883$		0			0		−0.982973	+	0.183750i	$0.941176\pi$
$884$	−0.101650	−	0.156001i	−0.101650	−	0.156001i
$885$		0			0
$886$		0			0
$887$		0			0		−0.824997	−	0.565136i	$-0.808824\pi$
$887$		0			0		0.824997	+	0.565136i	$0.191176\pi$
$888$		0			0
$889$		0			0
$890$	−0.427884	+	0.408559i	−0.427884	+	0.408559i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−0.367107	−	1.96385i	−0.367107	−	1.96385i
$899$		0			0
$900$	0.486604	−	0.873622i	0.486604	−	0.873622i
$901$	1.06584	+	0.330051i	1.06584	+	0.330051i
$902$		0			0
$903$		0			0
$904$	−0.103770	−	0.399555i	−0.103770	−	0.399555i
$905$	0.0907942	+	0.0169724i	0.0907942	+	0.0169724i
$906$		0			0
$907$		0			0		−0.811724	−	0.584041i	$-0.801471\pi$
$907$		0			0		0.811724	+	0.584041i	$0.198529\pi$
$908$		0			0
$909$	−0.111208	+	0.147263i	−0.111208	+	0.147263i
$910$		0			0
$911$		0			0		0.862150	−	0.506653i	$-0.169118\pi$
$911$		0			0		−0.862150	+	0.506653i	$0.830882\pi$
$912$		0			0
$913$		0			0
$914$	0.877165	−	0.316715i	0.877165	−	0.316715i
$915$		0			0
$916$	0.104472	+	1.50512i	0.104472	+	1.50512i
$917$		0			0
$918$		0			0
$919$		0			0		−0.783885	−	0.620906i	$-0.786765\pi$
$919$		0			0		0.783885	+	0.620906i	$0.213235\pi$
$920$		0			0
$921$		0			0
$922$	−1.96595			−1.96595
$923$		0			0
$924$		0			0
$925$	−0.204198	+	0.186151i	−0.204198	+	0.186151i
$926$		0			0
$927$		0			0
$928$	0.847846	+	0.0588498i	0.847846	+	0.0588498i
$929$	−0.621731	−	0.748723i	−0.621731	−	0.748723i	0.361242	−	0.932472i	$-0.382353\pi$
$929$	−0.621731	−	0.748723i	−0.621731	−	0.748723i	−0.982973	+	0.183750i	$0.941176\pi$
$930$		0			0
$931$		0			0
$932$	−0.0286833	+	0.0362122i	−0.0286833	+	0.0362122i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	0.187091	−	0.134613i	0.187091	−	0.134613i
$937$	−0.544240	+	1.23259i	−0.544240	+	1.23259i	0.403921	+	0.914794i	$0.367647\pi$
$937$	−0.544240	+	1.23259i	−0.544240	+	1.23259i	−0.948161	+	0.317791i	$0.897059\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	0.815517	+	1.63778i	0.815517	+	1.63778i	0.769334	+	0.638847i	$0.220588\pi$
$941$	0.815517	+	1.63778i	0.815517	+	1.63778i	0.0461835	+	0.998933i	$0.485294\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.0230979	−	0.999733i	$-0.507353\pi$
$947$		0			0		0.0230979	+	0.999733i	$0.492647\pi$
$948$		0			0
$949$	0.351352	+	0.145535i	0.351352	+	0.145535i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	0.311653	+	1.91053i	0.311653	+	1.91053i	0.403921	+	0.914794i	$0.367647\pi$
$953$	0.311653	+	1.91053i	0.311653	+	1.91053i	−0.0922684	+	0.995734i	$0.529412\pi$
$954$	−0.347190	+	1.33682i	−0.347190	+	1.33682i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.138156	+	0.990410i	−0.138156	+	0.990410i
$962$	−0.0608362	+	0.0188388i	−0.0608362	+	0.0188388i
$963$		0			0
$964$	−1.62186	−	0.421217i	−1.62186	−	0.421217i
$965$		−	0.367499i		−	0.367499i
$966$		0			0
$967$		0			0		−0.361242	−	0.932472i	$-0.617647\pi$
$967$		0			0		0.361242	+	0.932472i	$0.382353\pi$
$968$	−0.183750	+	0.982973i	−0.183750	+	0.982973i
$969$		0			0
$970$	1.73794	−	0.627512i	1.73794	−	0.627512i
$971$		0			0		−0.723251	−	0.690585i	$-0.757353\pi$
$971$		0			0		0.723251	+	0.690585i	$0.242647\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−0.963154	+	0.425274i	−0.963154	+	0.425274i
$977$	−0.428189	+	1.27754i	−0.428189	+	1.27754i	0.486604	+	0.873622i	$0.338235\pi$
$977$	−0.428189	+	1.27754i	−0.428189	+	1.27754i	−0.914794	+	0.403921i	$0.867647\pi$
$978$		0			0
$979$		0			0
$980$	−0.824997	−	0.565136i	−0.824997	−	0.565136i
$981$	−0.555831	−	1.43477i	−0.555831	−	1.43477i
$982$		0			0
$983$		0			0		−0.993337	−	0.115243i	$-0.963235\pi$
$983$		0			0		0.993337	+	0.115243i	$0.0367647\pi$
$984$		0			0
$985$	1.20013	−	1.58923i	1.20013	−	1.58923i
$986$	−0.0791229	−	0.681999i	−0.0791229	−	0.681999i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.638847	−	0.769334i	$-0.279412\pi$
$991$		0			0		−0.638847	+	0.769334i	$0.720588\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.592663	−	1.76827i	−0.592663	−	1.76827i	−0.638847	−	0.769334i	$-0.720588\pi$
$997$	−0.592663	−	1.76827i	−0.592663	−	1.76827i	0.0461835	−	0.998933i	$-0.485294\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2740.1.cd.a.423.1	yes	64
4.3	odd	2	CM	2740.1.cd.a.423.1	yes	64
5.2	odd	4		2740.1.bw.a.2067.1	✓	64
20.7	even	4		2740.1.bw.a.2067.1	✓	64
137.80	odd	136		2740.1.bw.a.2683.1	yes	64
548.491	even	136		2740.1.bw.a.2683.1	yes	64
685.217	even	136	inner	2740.1.cd.a.1587.1	yes	64
2740.1587	odd	136	inner	2740.1.cd.a.1587.1	yes	64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2740.1.bw.a.2067.1	✓	64	5.2	odd	4
2740.1.bw.a.2067.1	✓	64	20.7	even	4
2740.1.bw.a.2683.1	yes	64	137.80	odd	136
2740.1.bw.a.2683.1	yes	64	548.491	even	136
2740.1.cd.a.423.1	yes	64	1.1	even	1	trivial
2740.1.cd.a.423.1	yes	64	4.3	odd	2	CM
2740.1.cd.a.1587.1	yes	64	685.217	even	136	inner
2740.1.cd.a.1587.1	yes	64	2740.1587	odd	136	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:	\( N \)	\(=\)	\( 2740 = 2^{2} \cdot 5 \cdot 137 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2740.cd (of order \(136\), degree \(64\), minimal)

\(f(q)\)	\(=\)	\(q+(0.602635 + 0.798017i) q^{2} +(-0.273663 + 0.961826i) q^{4} +(0.914794 + 0.403921i) q^{5} +(-0.932472 + 0.361242i) q^{8} +(-0.824997 + 0.565136i) q^{9} +O(q^{10})\)	\(q+(0.602635 + 0.798017i) q^{2} +(-0.273663 + 0.961826i) q^{4} +(0.914794 + 0.403921i) q^{5} +(-0.932472 + 0.361242i) q^{8} +(-0.824997 + 0.565136i) q^{9} +(0.228951 + 0.973438i) q^{10} +(0.216788 + 0.0782748i) q^{13} +(-0.850217 - 0.526432i) q^{16} +(-0.291826 + 0.753290i) q^{17} +(-0.948161 - 0.317791i) q^{18} +(-0.638847 + 0.769334i) q^{20} +(0.673696 + 0.739009i) q^{25} +(0.0681792 + 0.220171i) q^{26} +(-0.136828 + 0.838799i) q^{29} +(-0.0922684 - 0.995734i) q^{32} +(-0.777003 + 0.221076i) q^{34} +(-0.317791 - 0.948161i) q^{36} +0.276313i q^{37} +(-0.998933 - 0.0461835i) q^{40} +(-0.0529974 - 0.127947i) q^{41} +(-0.982973 + 0.183750i) q^{45} +(0.0922684 + 0.995734i) q^{49} +(-0.183750 + 0.982973i) q^{50} +(-0.134613 + 0.187091i) q^{52} +(-0.0956383 - 1.37786i) q^{53} +(-0.751834 + 0.396298i) q^{58} +(0.595012 - 0.868610i) q^{61} +(0.739009 - 0.673696i) q^{64} +(0.166699 + 0.159170i) q^{65} +(-0.644672 - 0.486834i) q^{68} +(0.565136 - 0.824997i) q^{72} +(1.64823 + 0.0762025i) q^{73} +(-0.220502 + 0.166516i) q^{74} +(-0.565136 - 0.824997i) q^{80} +(0.361242 - 0.932472i) q^{81} +(0.0701658 - 0.119398i) q^{82} +(-0.571231 + 0.571231i) q^{85} +(0.299742 + 0.510058i) q^{89} +(-0.739009 - 0.673696i) q^{90} +(-0.212941 - 1.83545i) q^{97} +(-0.739009 + 0.673696i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 4 q^{2} - 4 q^{4} + 4 q^{8}+O(q^{10}) \) 64 * q + 4 * q^2 - 4 * q^4 + 4 * q^8	\( 64 q + 4 q^{2} - 4 q^{4} + 4 q^{8} - 4 q^{13} - 4 q^{16} + 4 q^{26} + 4 q^{32} - 4 q^{45} - 4 q^{49} - 4 q^{52} - 4 q^{64} - 4 q^{65} + 4 q^{73} + 4 q^{85} + 4 q^{90} + 4 q^{97} + 4 q^{98}+O(q^{100}) \) 64 * q + 4 * q^2 - 4 * q^4 + 4 * q^8 - 4 * q^13 - 4 * q^16 + 4 * q^26 + 4 * q^32 - 4 * q^45 - 4 * q^49 - 4 * q^52 - 4 * q^64 - 4 * q^65 + 4 * q^73 + 4 * q^85 + 4 * q^90 + 4 * q^97 + 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more