LMFDB - Embedded newform 136.2.s.a.11.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [136,2,Mod(3,136)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(136, base_ring=CyclotomicField(16))

chi = DirichletCharacter(H, H._module([8, 8, 1]))

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

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

chi := DirichletCharacter("136.3");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 136 = 2^{3} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	136.s (of order $16$, degree $8$, 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:	$1.08596546749$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{16}]$

Embedding invariants

Embedding label			11.1
Root			$-0.923880 + 0.382683i$ of defining polynomial
Character	$\chi$	$=$	136.11
Dual form			136.2.s.a.99.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.30656 + 0.541196i) q^{2} +(-0.242031 - 1.21677i) q^{3} +(1.41421 + 1.41421i) q^{4} +(0.342284 - 1.72078i) q^{6} +(1.08239 + 2.61313i) q^{8} +(1.34968 - 0.559056i) q^{9} +O(q^{10})$	\(q+(1.30656 + 0.541196i) q^{2} +(-0.242031 - 1.21677i) q^{3} +(1.41421 + 1.41421i) q^{4} +(0.342284 - 1.72078i) q^{6} +(1.08239 + 2.61313i) q^{8} +(1.34968 - 0.559056i) q^{9} +(-3.86883 - 0.769558i) q^{11} +(1.37849 - 2.06306i) q^{12} +4.00000i q^{16} +(-3.85403 + 1.46508i) q^{17} +2.06600 q^{18} +(-4.23671 - 1.75490i) q^{19} +(-4.63838 - 3.09927i) q^{22} +(2.91761 - 1.94948i) q^{24} +(1.91342 + 4.61940i) q^{25} +(-3.07465 - 4.60154i) q^{27} +(-2.16478 + 5.22625i) q^{32} +4.89374i q^{33} +(-5.82843 - 0.171573i) q^{34} +(2.69936 + 1.11811i) q^{36} +(-4.58579 - 4.58579i) q^{38} +(3.05774 - 2.04312i) q^{41} +(11.5355 - 4.77817i) q^{43} +(-4.38303 - 6.55967i) q^{44} +(4.86709 - 0.968125i) q^{48} +(2.67878 - 6.46716i) q^{49} +7.07107i q^{50} +(2.71546 + 4.33489i) q^{51} +(-1.52689 - 7.67619i) q^{54} +(-1.10990 + 5.57986i) q^{57} +(0.0502525 + 0.121320i) q^{59} +(-5.65685 + 5.65685i) q^{64} +(-2.64847 + 6.39398i) q^{66} +13.1969i q^{67} +(-7.52235 - 3.37849i) q^{68} +(2.92177 + 2.92177i) q^{72} +(14.1638 + 9.46395i) q^{73} +(5.15765 - 3.44623i) q^{75} +(-3.50981 - 8.47343i) q^{76} +(-1.75586 + 1.75586i) q^{81} +(5.10086 - 1.01462i) q^{82} +(-1.63604 + 3.94975i) q^{83} +17.6578 q^{86} +(-2.17664 - 10.9427i) q^{88} +(-13.2898 - 13.2898i) q^{89} +(6.88311 + 1.36913i) q^{96} +(9.85750 - 14.7528i) q^{97} +(7.00000 - 7.00000i) q^{98} +(-5.65191 + 1.12424i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} + 8 q^{6} + 16 q^{9}+O(q^{10}) $ 8 * q - 8 * q^3 + 8 * q^6 + 16 * q^9	$ 8 q - 8 q^{3} + 8 q^{6} + 16 q^{9} - 16 q^{12} - 24 q^{22} + 32 q^{24} - 32 q^{27} - 24 q^{34} + 32 q^{36} - 48 q^{38} - 32 q^{41} + 64 q^{43} - 16 q^{44} + 32 q^{48} - 40 q^{51} + 40 q^{54} - 40 q^{57} + 40 q^{59} + 72 q^{66} - 48 q^{72} + 48 q^{73} + 56 q^{81} - 64 q^{83} - 32 q^{96} + 56 q^{98} - 48 q^{99}+O(q^{100}) $ 8 * q - 8 * q^3 + 8 * q^6 + 16 * q^9 - 16 * q^12 - 24 * q^22 + 32 * q^24 - 32 * q^27 - 24 * q^34 + 32 * q^36 - 48 * q^38 - 32 * q^41 + 64 * q^43 - 16 * q^44 + 32 * q^48 - 40 * q^51 + 40 * q^54 - 40 * q^57 + 40 * q^59 + 72 * q^66 - 48 * q^72 + 48 * q^73 + 56 * q^81 - 64 * q^83 - 32 * q^96 + 56 * q^98 - 48 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/136\mathbb{Z}\right)^\times$.

$n$	$69$	$103$	$105$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{7}{16}\right)$

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 $

$2$	1.30656	+	0.541196i	0.923880	+	0.382683i
$2$	1.30656	+	0.541196i	0.923880	+	0.382683i
$3$	−0.242031	−	1.21677i	−0.139737	−	0.702504i	−0.985599	−	0.169102i	$-0.945913\pi$
$3$	−0.242031	−	1.21677i	−0.139737	−	0.702504i	0.845862	−	0.533402i	$-0.179087\pi$
$4$	1.41421	+	1.41421i	0.707107	+	0.707107i
$5$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$5$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$6$	0.342284	−	1.72078i	0.139737	−	0.702504i
$7$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$7$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$8$	1.08239	+	2.61313i	0.382683	+	0.923880i
$9$	1.34968	−	0.559056i	0.449894	−	0.186352i
$10$		0			0
$11$	−3.86883	−	0.769558i	−1.16650	−	0.232030i	−0.426401	−	0.904534i	$-0.640219\pi$
$11$	−3.86883	−	0.769558i	−1.16650	−	0.232030i	−0.740094	+	0.672504i	$0.765219\pi$
$12$	1.37849	−	2.06306i	0.397937	−	0.595554i
$13$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$13$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$14$		0			0
$15$		0			0
$16$			4.00000i			1.00000i
$17$	−3.85403	+	1.46508i	−0.934740	+	0.355333i
$17$	−3.85403	+	1.46508i	−0.934740	+	0.355333i
$18$	2.06600			0.486962
$19$	−4.23671	−	1.75490i	−0.971969	−	0.402603i	−0.160524	−	0.987032i	$-0.551318\pi$
$19$	−4.23671	−	1.75490i	−0.971969	−	0.402603i	−0.811445	+	0.584429i	$0.801318\pi$
$20$		0			0
$21$		0			0
$22$	−4.63838	−	3.09927i	−0.988907	−	0.660766i
$23$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$23$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$24$	2.91761	−	1.94948i	0.595554	−	0.397937i
$25$	1.91342	+	4.61940i	0.382683	+	0.923880i
$26$		0			0
$27$	−3.07465	−	4.60154i	−0.591716	−	0.885566i
$28$		0			0
$29$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$29$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$30$		0			0
$31$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$31$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$32$	−2.16478	+	5.22625i	−0.382683	+	0.923880i
$33$			4.89374i			0.851891i
$34$	−5.82843	−	0.171573i	−0.999567	−	0.0294245i
$35$		0			0
$36$	2.69936	+	1.11811i	0.449894	+	0.186352i
$37$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$37$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$38$	−4.58579	−	4.58579i	−0.743913	−	0.743913i
$39$		0			0
$40$		0			0
$41$	3.05774	−	2.04312i	0.477539	−	0.319082i	−0.293400	−	0.955990i	$-0.594787\pi$
$41$	3.05774	−	2.04312i	0.477539	−	0.319082i	0.770940	+	0.636908i	$0.219787\pi$
$42$		0			0
$43$	11.5355	−	4.77817i	1.75915	−	0.728665i	0.762493	−	0.646997i	$-0.223975\pi$
$43$	11.5355	−	4.77817i	1.75915	−	0.728665i	0.996660	−	0.0816682i	$-0.0260248\pi$
$44$	−4.38303	−	6.55967i	−0.660766	−	0.988907i
$45$		0			0
$46$		0			0
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$	4.86709	−	0.968125i	0.702504	−	0.139737i
$49$	2.67878	−	6.46716i	0.382683	−	0.923880i
$50$			7.07107i			1.00000i
$51$	2.71546	+	4.33489i	0.380240	+	0.607005i
$52$		0			0
$53$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$53$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$54$	−1.52689	−	7.67619i	−0.207783	−	1.04460i
$55$		0			0
$56$		0			0
$57$	−1.10990	+	5.57986i	−0.147010	+	0.739071i
$58$		0			0
$59$	0.0502525	+	0.121320i	0.00654232	+	0.0157946i	0.927117	−	0.374772i	$-0.122279\pi$
$59$	0.0502525	+	0.121320i	0.00654232	+	0.0157946i	−0.920575	+	0.390567i	$0.872279\pi$
$60$		0			0
$61$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$61$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$62$		0			0
$63$		0			0
$64$	−5.65685	+	5.65685i	−0.707107	+	0.707107i
$65$		0			0
$66$	−2.64847	+	6.39398i	−0.326004	+	0.787044i
$67$			13.1969i			1.61226i	0.591736	+	0.806132i	$0.298443\pi$
$67$			13.1969i			1.61226i	−0.591736	+	0.806132i	$0.701557\pi$
$68$	−7.52235	−	3.37849i	−0.912219	−	0.409702i
$69$		0			0
$70$		0			0
$71$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$71$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$72$	2.92177	+	2.92177i	0.344334	+	0.344334i
$73$	14.1638	+	9.46395i	1.65775	+	1.10767i	0.872740	+	0.488185i	$0.162341\pi$
$73$	14.1638	+	9.46395i	1.65775	+	1.10767i	0.785007	+	0.619486i	$0.212659\pi$
$74$		0			0
$75$	5.15765	−	3.44623i	0.595554	−	0.397937i
$76$	−3.50981	−	8.47343i	−0.402603	−	0.971969i
$77$		0			0
$78$		0			0
$79$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$79$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$80$		0			0
$81$	−1.75586	+	1.75586i	−0.195095	+	0.195095i
$82$	5.10086	−	1.01462i	0.563296	−	0.112047i
$83$	−1.63604	+	3.94975i	−0.179579	+	0.433541i	−0.987878	−	0.155230i	$-0.950388\pi$
$83$	−1.63604	+	3.94975i	−0.179579	+	0.433541i	0.808300	+	0.588771i	$0.200388\pi$
$84$		0			0
$85$		0			0
$86$	17.6578			1.90409
$87$		0			0
$88$	−2.17664	−	10.9427i	−0.232030	−	1.16650i
$89$	−13.2898	−	13.2898i	−1.40872	−	1.40872i	−0.766646	−	0.642070i	$-0.778076\pi$
$89$	−13.2898	−	13.2898i	−1.40872	−	1.40872i	−0.642070	−	0.766646i	$-0.721924\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	6.88311	+	1.36913i	0.702504	+	0.139737i
$97$	9.85750	−	14.7528i	1.00088	−	1.49792i	0.139328	−	0.990246i	$-0.455506\pi$
$97$	9.85750	−	14.7528i	1.00088	−	1.49792i	0.861550	−	0.507673i	$-0.169494\pi$
$98$	7.00000	−	7.00000i	0.707107	−	0.707107i
$99$	−5.65191	+	1.12424i	−0.568038	+	0.112990i
$100$	−3.82683	+	9.23880i	−0.382683	+	0.923880i
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$	1.20190	+	7.13340i	0.119005	+	0.706312i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	13.2940	+	8.88276i	1.28518	+	0.858729i	0.995158	−	0.0982914i	$-0.0313377\pi$
$107$	13.2940	+	8.88276i	1.28518	+	0.858729i	0.290021	+	0.957020i	$0.406338\pi$
$108$	2.15935	−	10.8558i	0.207783	−	1.04460i
$109$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$109$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−20.8437	−	4.14606i	−1.96081	−	0.390029i	−0.985847	−	0.167646i	$-0.946383\pi$
$113$	−20.8437	−	4.14606i	−1.96081	−	0.390029i	−0.974959	−	0.222383i	$-0.928617\pi$
$114$	−4.46996	+	6.68976i	−0.418650	+	0.626554i
$115$		0			0
$116$		0			0
$117$		0			0
$118$			0.185709i			0.0170959i
$119$		0			0
$120$		0			0
$121$	4.21293	+	1.74505i	0.382993	+	0.158641i
$122$		0			0
$123$	−3.22608	−	3.22608i	−0.290886	−	0.290886i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$127$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$128$	−10.4525	+	4.32957i	−0.923880	+	0.382683i
$129$	−8.60591	−	12.8797i	−0.757708	−	1.13399i
$130$		0			0
$131$	−11.3409	+	16.9729i	−0.990861	+	1.48293i	−0.119170	+	0.992874i	$0.538023\pi$
$131$	−11.3409	+	16.9729i	−0.990861	+	1.48293i	−0.871692	+	0.490055i	$0.836977\pi$
$132$	−6.92079	+	6.92079i	−0.602378	+	0.602378i
$133$		0			0
$134$	−7.14214	+	17.2426i	−0.616987	+	1.48954i
$135$		0			0
$136$	−8.00000	−	8.48528i	−0.685994	−	0.727607i
$137$	−20.2426			−1.72945			−0.864723	−	0.502249i	$-0.832506\pi$
$137$	−20.2426			−1.72945			−0.864723	+	0.502249i	$0.832506\pi$
$138$		0			0
$139$	−4.48835	−	22.5644i	−0.380697	−	1.91389i	−0.405279	−	0.914193i	$-0.632826\pi$
$139$	−4.48835	−	22.5644i	−0.380697	−	1.91389i	0.0245830	−	0.999698i	$-0.492174\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	2.23623	+	5.39873i	0.186352	+	0.449894i
$145$		0			0
$146$	13.3840	+	20.0306i	1.10767	+	1.65775i
$147$	−8.51741	−	1.69422i	−0.702504	−	0.139737i
$148$		0			0
$149$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$149$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$150$	8.60388	−	1.71142i	0.702504	−	0.139737i
$151$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$151$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$152$		−	12.9706i		−	1.05205i
$153$	−4.38265	+	4.13201i	−0.354317	+	0.334053i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$157$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−3.24440	+	1.34388i	−0.254905	+	0.105585i
$163$	6.93328	+	10.3764i	0.543056	+	0.812741i	0.996928	−	0.0783260i	$-0.0249575\pi$
$163$	6.93328	+	10.3764i	0.543056	+	0.812741i	−0.453872	+	0.891067i	$0.649958\pi$
$164$	7.21371	+	1.43490i	0.563296	+	0.112047i
$165$		0			0
$166$	−4.27518	+	4.27518i	−0.331818	+	0.331818i
$167$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$167$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$168$		0			0
$169$		−	13.0000i		−	1.00000i
$170$		0			0
$171$	−6.69931			−0.512309
$172$	23.0711	+	9.55635i	1.75915	+	0.728665i
$173$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$173$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$174$		0			0
$175$		0			0
$176$	3.07823	−	15.4753i	0.232030	−	1.16650i
$177$	0.135457	−	0.0905092i	0.0101815	−	0.00680309i
$178$	−10.1716	−	24.5563i	−0.762392	−	1.84058i
$179$	24.6934	−	10.2283i	1.84567	−	0.764501i	0.904642	−	0.426172i	$-0.140138\pi$
$179$	24.6934	−	10.2283i	1.84567	−	0.764501i	0.941028	−	0.338330i	$-0.109862\pi$
$180$		0			0
$181$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$181$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	16.0380	−	2.70223i	1.17282	−	0.197606i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$191$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$192$	8.25224	+	5.51397i	0.595554	+	0.397937i
$193$	−5.13732	+	25.8271i	−0.369793	+	1.85907i	0.127996	+	0.991775i	$0.459146\pi$
$193$	−5.13732	+	25.8271i	−0.369793	+	1.85907i	−0.497788	+	0.867298i	$0.665854\pi$
$194$	20.8636	−	13.9406i	1.49792	−	1.00088i
$195$		0			0
$196$	12.9343	−	5.35757i	0.923880	−	0.382683i
$197$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$197$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$198$	−7.99301	−	1.58991i	−0.568038	−	0.112990i
$199$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$199$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$200$	−10.0000	+	10.0000i	−0.707107	+	0.707107i
$201$	16.0577	−	3.19407i	1.13262	−	0.225292i
$202$		0			0
$203$		0			0
$204$	−2.29021	+	9.97069i	−0.160347	+	0.698088i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	15.0406	+	10.0498i	1.04038	+	0.695160i
$210$		0			0
$211$	−18.4881	+	12.3534i	−1.27277	+	0.850440i	−0.993943	−	0.109900i	$-0.964947\pi$
$211$	−18.4881	+	12.3534i	−1.27277	+	0.850440i	−0.278831	+	0.960340i	$0.589947\pi$
$212$		0			0
$213$		0			0
$214$	12.5621	+	18.8005i	0.858729	+	1.28518i
$215$		0			0
$216$	8.69642	−	13.0151i	0.591716	−	0.885566i
$217$		0			0
$218$		0			0
$219$	8.08739	−	19.5247i	0.546495	−	1.31936i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$223$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$224$		0			0
$225$	5.16501	+	5.16501i	0.344334	+	0.344334i
$226$	−24.9897	−	16.6976i	−1.66229	−	1.11071i
$227$	−5.17291	+	26.0060i	−0.343338	+	1.72608i	0.294274	+	0.955721i	$0.404922\pi$
$227$	−5.17291	+	26.0060i	−0.343338	+	1.72608i	−0.637613	+	0.770357i	$0.720078\pi$
$228$	−9.46075	+	6.32147i	−0.626554	+	0.418650i
$229$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$229$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	11.8729	−	17.7690i	0.777818	−	1.16409i	−0.204865	−	0.978790i	$-0.565675\pi$
$233$	11.8729	−	17.7690i	0.777818	−	1.16409i	0.982683	−	0.185296i	$-0.0593245\pi$
$234$		0			0
$235$		0			0
$236$	−0.100505	+	0.242641i	−0.00654232	+	0.0157946i
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	4.32898	+	21.7633i	0.278854	+	1.40189i	0.825439	+	0.564491i	$0.190928\pi$
$241$	4.32898	+	21.7633i	0.278854	+	1.40189i	−0.546585	+	0.837404i	$0.684072\pi$
$242$	4.56004	+	4.56004i	0.293130	+	0.293130i
$243$	−11.2432	−	7.51244i	−0.721249	−	0.481923i
$244$		0			0
$245$		0			0
$246$	−2.46914	−	5.96102i	−0.157426	−	0.380061i
$247$		0			0
$248$		0			0
$249$	5.20192	+	1.03473i	0.329658	+	0.0655731i
$250$		0			0
$251$	21.9489	−	21.9489i	1.38540	−	1.38540i	0.550704	−	0.834700i	$-0.314359\pi$
$251$	21.9489	−	21.9489i	1.38540	−	1.38540i	0.834700	−	0.550704i	$-0.185641\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−16.0000			−1.00000
$257$	20.2635	+	8.39340i	1.26400	+	0.523566i	0.911135	−	0.412108i	$-0.135208\pi$
$257$	20.2635	+	8.39340i	1.26400	+	0.523566i	0.352865	+	0.935674i	$0.385208\pi$
$258$	−4.27375	−	21.4856i	−0.266072	−	1.33763i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−24.0033	+	16.0385i	−1.48293	+	0.990861i
$263$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$263$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$264$	−12.7880	+	5.29695i	−0.787044	+	0.326004i
$265$		0			0
$266$		0			0
$267$	−12.9541	+	19.3872i	−0.792779	+	1.18648i
$268$	−18.6633	+	18.6633i	−1.14004	+	1.14004i
$269$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$269$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$	−5.86030	−	15.4161i	−0.355333	−	0.934740i
$273$		0			0
$274$	−26.4483	−	10.9552i	−1.59780	−	0.661830i
$275$	−3.84779	−	19.3441i	−0.232030	−	1.16650i
$276$		0			0
$277$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$277$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$278$	6.34748	−	31.9109i	0.380697	−	1.91389i
$279$		0			0
$280$		0			0
$281$	−6.71852	+	2.78290i	−0.400794	+	0.166014i	−0.573969	−	0.818877i	$-0.694597\pi$
$281$	−6.71852	+	2.78290i	−0.400794	+	0.166014i	0.173176	+	0.984891i	$0.444597\pi$
$282$		0			0
$283$	−16.2919	−	3.24066i	−0.968454	−	0.192637i	−0.314571	−	0.949234i	$-0.601861\pi$
$283$	−16.2919	−	3.24066i	−0.968454	−	0.192637i	−0.653882	+	0.756596i	$0.726861\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$			8.26401i			0.486962i
$289$	12.7071	−	11.2929i	0.747477	−	0.664288i
$290$		0			0
$291$	−20.3366	−	8.42370i	−1.19215	−	0.493806i
$292$	6.64659	+	33.4147i	0.388963	+	1.95545i
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$	−10.2116	−	6.82319i	−0.595554	−	0.397937i
$295$		0			0
$296$		0			0
$297$	8.35414	+	20.1687i	0.484756	+	1.17030i
$298$		0			0
$299$		0			0
$300$	12.1677	+	2.42031i	0.702504	+	0.139737i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	7.01962	−	16.9469i	0.402603	−	0.971969i
$305$		0			0
$306$	−7.96244	+	3.02685i	−0.455182	+	0.173034i
$307$	−8.48528			−0.484281			−0.242140	−	0.970241i	$-0.577849\pi$
$307$	−8.48528			−0.484281			−0.242140	+	0.970241i	$0.577849\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$311$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$312$		0			0
$313$	−28.0843	+	18.7653i	−1.58742	+	1.06068i	−0.628186	+	0.778063i	$0.716202\pi$
$313$	−28.0843	+	18.7653i	−1.58742	+	1.06068i	−0.959233	+	0.282617i	$0.908798\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$317$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	7.59074	−	18.3257i	0.423674	−	1.02284i
$322$		0			0
$323$	18.8995	+	0.556349i	1.05160	+	0.0309561i
$324$	−4.96632			−0.275907
$325$		0			0
$326$	3.44311	+	17.3097i	0.190696	+	0.958693i
$327$		0			0
$328$	8.64860	+	5.77881i	0.477539	+	0.319082i
$329$		0			0
$330$		0			0
$331$	−5.19239	−	12.5355i	−0.285399	−	0.689015i	0.714545	−	0.699590i	$-0.246634\pi$
$331$	−5.19239	−	12.5355i	−0.285399	−	0.689015i	−0.999944	+	0.0105746i	$0.996634\pi$
$332$	−7.89949	+	3.27208i	−0.433541	+	0.179579i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	7.07745	−	1.40779i	0.385533	−	0.0766873i	0.00148149	−	0.999999i	$-0.499528\pi$
$337$	7.07745	−	1.40779i	0.385533	−	0.0766873i	0.384052	+	0.923312i	$0.374528\pi$
$338$	7.03555	−	16.9853i	0.382683	−	0.923880i
$339$			26.3655i			1.43198i
$340$		0			0
$341$		0			0
$342$	−8.75307	−	3.62564i	−0.473312	−	0.196052i
$343$		0			0
$344$	24.9719	+	24.9719i	1.34640	+	1.34640i
$345$		0			0
$346$		0			0
$347$	10.8574	−	7.25469i	0.582856	−	0.389452i	−0.228899	−	0.973450i	$-0.573512\pi$
$347$	10.8574	−	7.25469i	0.582856	−	0.389452i	0.811755	+	0.583998i	$0.198512\pi$
$348$		0			0
$349$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$349$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$350$		0			0
$351$		0			0
$352$	12.3971	−	18.5535i	0.660766	−	0.988907i
$353$	−13.4755	+	13.4755i	−0.717229	+	0.717229i	−0.968037	−	0.250808i	$-0.919304\pi$
$353$	−13.4755	+	13.4755i	−0.717229	+	0.717229i	0.250808	+	0.968037i	$0.419304\pi$
$354$	0.225966	−	0.0449474i	0.0120099	−	0.00238893i
$355$		0			0
$356$		−	37.5892i		−	1.99223i
$357$		0			0
$358$	37.7990			1.99774
$359$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$359$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$360$		0			0
$361$	1.43503	+	1.43503i	0.0755278	+	0.0755278i
$362$		0			0
$363$	1.10367	−	5.54853i	0.0579277	−	0.291222i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$367$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$368$		0			0
$369$	2.98476	−	4.46701i	0.155380	−	0.232543i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$	22.4171	+	5.14910i	1.15916	+	0.266253i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−32.3652	−	21.6257i	−1.66249	−	1.11084i	−0.844211	−	0.536011i	$-0.819930\pi$
$379$	−32.3652	−	21.6257i	−1.66249	−	1.11084i	−0.818275	−	0.574826i	$-0.805070\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$383$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$384$	7.79793	+	11.6704i	0.397937	+	0.595554i
$385$		0			0
$386$	−20.6897	+	30.9644i	−1.05308	+	1.57605i
$387$	12.8980	−	12.8980i	0.655644	−	0.655644i
$388$	34.8042	−	6.92299i	1.76692	−	0.351462i
$389$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$389$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$390$		0			0
$391$		0			0
$392$	19.7990			1.00000
$393$	23.3970	+	9.69136i	1.18022	+	0.488864i
$394$		0			0
$395$		0			0
$396$	−9.58292	−	6.40310i	−0.481560	−	0.321768i
$397$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$397$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$398$		0			0
$399$		0			0
$400$	−18.4776	+	7.65367i	−0.923880	+	0.382683i
$401$	−20.1438	−	30.1474i	−1.00593	−	1.50549i	−0.856122	−	0.516774i	$-0.827133\pi$
$401$	−20.1438	−	30.1474i	−1.00593	−	1.50549i	−0.149813	−	0.988714i	$-0.547867\pi$
$402$	22.7090	+	4.51710i	1.13262	+	0.225292i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−8.38841	+	11.7879i	−0.415288	+	0.583587i
$409$	−39.7765			−1.96682			−0.983412	−	0.181387i	$-0.941941\pi$
$409$	−39.7765			−1.96682			−0.983412	+	0.181387i	$0.941941\pi$
$410$		0			0
$411$	4.89935	+	24.6307i	0.241667	+	1.21494i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−26.3695	+	10.9226i	−1.29132	+	0.534882i
$418$	14.2126	+	21.2706i	0.695160	+	1.04038i
$419$	5.35667	+	1.06551i	0.261691	+	0.0520535i	0.324192	−	0.945991i	$-0.394908\pi$
$419$	5.35667	+	1.06551i	0.261691	+	0.0520535i	−0.0625011	+	0.998045i	$0.519908\pi$
$420$		0			0
$421$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$421$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$422$	−30.8415	+	6.13475i	−1.50134	+	0.298635i
$423$		0			0
$424$		0			0
$425$	−14.1421	−	15.0000i	−0.685994	−	0.727607i
$426$		0			0
$427$		0			0
$428$	6.23842	+	31.3627i	0.301545	+	1.51597i
$429$		0			0
$430$		0			0
$431$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$431$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$432$	18.4061	−	12.2986i	0.885566	−	0.591716i
$433$	5.69052	+	13.7381i	0.273469	+	0.660213i	0.999627	−	0.0273152i	$-0.00869578\pi$
$433$	5.69052	+	13.7381i	0.273469	+	0.660213i	−0.726158	+	0.687528i	$0.758696\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	21.1334	−	21.1334i	1.00979	−	1.00979i
$439$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$439$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$440$		0			0
$441$		−	10.2262i		−	0.486962i
$442$		0			0
$443$	37.7205			1.79216			0.896079	−	0.443895i	$-0.146404\pi$
$443$	37.7205			1.79216			0.896079	+	0.443895i	$0.146404\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	26.4232	−	17.6554i	1.24699	−	0.833212i	0.255938	−	0.966693i	$-0.417616\pi$
$449$	26.4232	−	17.6554i	1.24699	−	0.833212i	0.991051	+	0.133482i	$0.0426157\pi$
$450$	3.95313	+	9.54369i	0.186352	+	0.449894i
$451$	−13.4022	+	5.55136i	−0.631084	+	0.261403i
$452$	−23.6140	−	35.3408i	−1.11071	−	1.66229i
$453$		0			0
$454$	−20.8331	+	31.1789i	−0.977745	+	1.46330i
$455$		0			0
$456$	−15.7822	+	3.13928i	−0.739071	+	0.147010i
$457$	16.2200	−	39.1584i	0.758737	−	1.83175i	0.258031	−	0.966137i	$-0.416926\pi$
$457$	16.2200	−	39.1584i	0.758737	−	1.83175i	0.500706	−	0.865617i	$-0.333074\pi$
$458$		0			0
$459$	18.5914	+	13.2299i	0.867772	+	0.617518i
$460$		0			0
$461$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$461$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		0			0
$465$		0			0
$466$	25.1292	−	16.7908i	1.16409	−	0.777818i
$467$	0.301094	+	0.726905i	0.0139330	+	0.0336372i	0.930693	−	0.365801i	$-0.119205\pi$
$467$	0.301094	+	0.726905i	0.0139330	+	0.0336372i	−0.916760	+	0.399439i	$0.869205\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−0.262632	+	0.262632i	−0.0120886	+	0.0120886i
$473$	−48.3061	+	9.60867i	−2.22112	+	0.441807i
$474$		0			0
$475$		−	22.9289i		−	1.05205i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$479$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$480$		0			0
$481$		0			0
$482$	−6.12210	+	30.7779i	−0.278854	+	1.40189i
$483$		0			0
$484$	3.49010	+	8.42585i	0.158641	+	0.382993i
$485$		0			0
$486$	−10.6242	−	15.9002i	−0.481923	−	0.721249i
$487$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$487$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$488$		0			0
$489$	10.9476	−	10.9476i	0.495069	−	0.495069i
$490$		0			0
$491$	−15.1920	+	36.6766i	−0.685603	+	1.65519i	0.0678537	+	0.997695i	$0.478385\pi$
$491$	−15.1920	+	36.6766i	−0.685603	+	1.65519i	−0.753457	+	0.657497i	$0.771615\pi$
$492$		−	9.12473i		−	0.411375i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	6.23664	+	4.16719i	0.279471	+	0.186736i
$499$	−1.06403	+	5.34922i	−0.0476323	+	0.239464i	−0.997266	−	0.0738993i	$-0.976456\pi$
$499$	−1.06403	+	5.34922i	−0.0476323	+	0.239464i	0.949633	+	0.313363i	$0.101456\pi$
$500$		0			0
$501$		0			0
$502$	40.5563	−	16.7990i	1.81012	−	0.749776i
$503$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$503$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−15.8180	+	3.14640i	−0.702504	+	0.139737i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	−20.9050	−	8.65914i	−0.923880	−	0.382683i
$513$	4.95115	+	24.8911i	0.218599	+	1.09897i
$514$	21.9330	+	21.9330i	0.967423	+	0.967423i
$515$		0			0
$516$	6.04399	−	30.3852i	0.266072	−	1.33763i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	42.3844	+	8.43078i	1.85689	+	0.369359i	0.991325	−	0.131432i	$-0.0419576\pi$
$521$	42.3844	+	8.43078i	1.85689	+	0.369359i	0.865568	+	0.500791i	$0.166958\pi$
$522$		0			0
$523$	17.9364	−	17.9364i	0.784304	−	0.784304i	−0.196250	−	0.980554i	$-0.562876\pi$
$523$	17.9364	−	17.9364i	0.784304	−	0.784304i	0.980554	+	0.196250i	$0.0628764\pi$
$524$	−40.0418	+	7.96481i	−1.74923	+	0.347944i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−19.5750			−0.851891
$529$	−21.2492	−	8.80172i	−0.923880	−	0.382683i
$530$		0			0
$531$	0.135650	+	0.135650i	0.00588670	+	0.00588670i
$532$		0			0
$533$		0			0
$534$	−27.4177	+	18.3199i	−1.18648	+	0.792779i
$535$		0			0
$536$	−34.4853	+	14.2843i	−1.48954	+	0.616987i
$537$	−18.4221	−	27.5707i	−0.794973	−	1.18976i
$538$		0			0
$539$	−15.3406	+	22.9588i	−0.660766	+	0.988907i
$540$		0			0
$541$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$541$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$542$		0			0
$543$		0			0
$544$	0.686292	−	23.3137i	0.0294245	−	0.999567i
$545$		0			0
$546$		0			0
$547$	6.54205	+	32.8891i	0.279718	+	1.40624i	0.823646	+	0.567104i	$0.191936\pi$
$547$	6.54205	+	32.8891i	0.279718	+	1.40624i	−0.543928	+	0.839132i	$0.683064\pi$
$548$	−28.6274	−	28.6274i	−1.22290	−	1.22290i
$549$		0			0
$550$	5.44159	−	27.3567i	0.232030	−	1.16650i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	25.5635	−	38.2584i	1.08413	−	1.62252i
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−7.16970	−	18.8606i	−0.302705	−	0.796296i
$562$	−10.2843			−0.433816
$563$	−20.7782	−	8.60660i	−0.875696	−	0.362725i	−0.100870	−	0.994900i	$-0.532163\pi$
$563$	−20.7782	−	8.60660i	−0.875696	−	0.362725i	−0.774826	+	0.632175i	$0.782163\pi$
$564$		0			0
$565$		0			0
$566$	−19.5326	−	13.0512i	−0.821015	−	0.548585i
$567$		0			0
$568$		0			0
$569$	1.84924	+	4.46447i	0.0775243	+	0.187160i	0.957890	−	0.287135i	$-0.0927029\pi$
$569$	1.84924	+	4.46447i	0.0775243	+	0.187160i	−0.880366	+	0.474295i	$0.842703\pi$
$570$		0			0
$571$	25.5024	+	38.1671i	1.06724	+	1.59724i	0.765020	+	0.644007i	$0.222729\pi$
$571$	25.5024	+	38.1671i	1.06724	+	1.59724i	0.302224	+	0.953237i	$0.402271\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−4.47245	+	10.7975i	−0.186352	+	0.449894i
$577$		−	33.9411i		−	1.41299i	−0.707719	−	0.706494i	$-0.750276\pi$
$577$		−	33.9411i		−	1.41299i	0.707719	−	0.706494i	$-0.249724\pi$
$578$	22.7143	−	7.87784i	0.944791	−	0.327675i
$579$	32.6691			1.35768
$580$		0			0
$581$		0			0
$582$	−22.0122	−	22.0122i	−0.912435	−	0.912435i
$583$		0			0
$584$	−9.39970	+	47.2555i	−0.388963	+	1.95545i
$585$		0			0
$586$		0			0
$587$	−5.54328	+	2.29610i	−0.228796	+	0.0947702i	−0.494136	−	0.869385i	$-0.664516\pi$
$587$	−5.54328	+	2.29610i	−0.228796	+	0.0947702i	0.265341	+	0.964155i	$0.414516\pi$
$588$	−9.64945	−	14.4414i	−0.397937	−	0.595554i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−18.6360	+	44.9914i	−0.765290	+	1.84757i	−0.369586	+	0.929197i	$0.620500\pi$
$593$	−18.6360	+	44.9914i	−0.765290	+	1.84757i	−0.395705	+	0.918378i	$0.629500\pi$
$594$			30.8729i			1.26673i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$599$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$600$	14.5880	+	9.74742i	0.595554	+	0.397937i
$601$	2.23295	−	11.2258i	0.0910839	−	0.457910i	−0.908145	−	0.418655i	$-0.862502\pi$
$601$	2.23295	−	11.2258i	0.0910839	−	0.457910i	0.999229	−	0.0392547i	$-0.0124984\pi$
$602$		0			0
$603$	7.37784	+	17.8117i	0.300449	+	0.725348i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$607$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$608$	18.3431	−	18.3431i	0.743913	−	0.743913i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−12.0416	−	0.354470i	−0.486751	−	0.0143286i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	−11.0866	−	4.59220i	−0.447417	−	0.185326i
$615$		0			0
$616$		0			0
$617$	30.8880	+	20.6387i	1.24351	+	0.830884i	0.990624	−	0.136613i	$-0.0436217\pi$
$617$	30.8880	+	20.6387i	1.24351	+	0.830884i	0.252882	+	0.967497i	$0.418622\pi$
$618$		0			0
$619$	1.62356	−	1.08483i	0.0652564	−	0.0436029i	−0.522514	−	0.852631i	$-0.675006\pi$
$619$	1.62356	−	1.08483i	0.0652564	−	0.0436029i	0.587770	+	0.809028i	$0.300006\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−17.6777	+	17.6777i	−0.707107	+	0.707107i
$626$	−46.8497	+	9.31898i	−1.87249	+	0.372461i
$627$	8.58805	−	20.7334i	0.342974	−	0.828011i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$631$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$632$		0			0
$633$	19.5059	+	19.5059i	0.775291	+	0.775291i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	10.8597	−	16.2527i	0.428934	−	0.641945i	−0.552552	−	0.833478i	$-0.686346\pi$
$641$	10.8597	−	16.2527i	0.428934	−	0.641945i	0.981486	+	0.191534i	$0.0613461\pi$
$642$	19.8356	−	19.8356i	0.782847	−	0.782847i
$643$	−1.40476	+	0.279423i	−0.0553982	+	0.0110194i	−0.222712	−	0.974884i	$-0.571491\pi$
$643$	−1.40476	+	0.279423i	−0.0553982	+	0.0110194i	0.167313	+	0.985904i	$0.446491\pi$
$644$		0			0
$645$		0			0
$646$	24.3923	+	10.9552i	0.959702	+	0.431028i
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	−6.48881	−	2.68775i	−0.254905	−	0.105585i
$649$	−0.101055	−	0.508040i	−0.00396677	−	0.0199423i
$650$		0			0
$651$		0			0
$652$	−4.86929	+	24.4796i	−0.190696	+	0.958693i
$653$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$653$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$654$		0			0
$655$		0			0
$656$	8.17248	+	12.2310i	0.319082	+	0.477539i
$657$	24.4075	+	4.85495i	0.952228	+	0.189410i
$658$		0			0
$659$	12.7279	−	12.7279i	0.495809	−	0.495809i	−0.414321	−	0.910131i	$-0.635981\pi$
$659$	12.7279	−	12.7279i	0.495809	−	0.495809i	0.910131	+	0.414321i	$0.135981\pi$
$660$		0			0
$661$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$661$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$662$		−	19.1886i		−	0.745785i
$663$		0			0
$664$	−12.0920			−0.469262
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.0691823	+	0.103539i	0.00266678	+	0.00399112i	0.832801	−	0.553573i	$-0.186736\pi$
$673$	0.0691823	+	0.103539i	0.00266678	+	0.00399112i	−0.830134	+	0.557564i	$0.811736\pi$
$674$	10.0090	+	1.99092i	0.385533	+	0.0766873i
$675$	15.3732	−	23.0077i	0.591716	−	0.885566i
$676$	18.3848	−	18.3848i	0.707107	−	0.707107i
$677$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$677$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$678$	−14.2689	+	34.4482i	−0.547994	+	1.32297i
$679$		0			0
$680$		0			0
$681$	32.8954			1.26055
$682$		0			0
$683$	−6.85220	−	34.4483i	−0.262192	−	1.31813i	−0.857439	−	0.514585i	$-0.827946\pi$
$683$	−6.85220	−	34.4483i	−0.262192	−	1.31813i	0.595247	−	0.803543i	$-0.297054\pi$
$684$	−9.47425	−	9.47425i	−0.362257	−	0.362257i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	19.1127	+	46.1421i	0.728665	+	1.75915i
$689$		0			0
$690$		0			0
$691$	49.1200	+	9.77057i	1.86861	+	0.371690i	0.993651	−	0.112503i	$-0.0358867\pi$
$691$	49.1200	+	9.77057i	1.86861	+	0.371690i	0.874961	+	0.484193i	$0.160887\pi$
$692$		0			0
$693$		0			0
$694$	18.1121	−	3.60272i	0.687526	−	0.136757i
$695$		0			0
$696$		0			0
$697$	−8.79131	+	12.3541i	−0.332995	+	0.467944i
$698$		0			0
$699$	−24.4945	−	10.1459i	−0.926465	−	0.383755i
$700$		0			0
$701$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$701$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$702$		0			0
$703$		0			0
$704$	26.2387	−	17.5321i	0.988907	−	0.660766i
$705$		0			0
$706$	−24.8995	+	10.3137i	−0.937105	+	0.388162i
$707$		0			0
$708$	0.319564	+	0.0635652i	0.0120099	+	0.00238893i
$709$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$709$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$710$		0			0
$711$		0			0
$712$	20.3431	−	49.1127i	0.762392	−	1.84058i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	49.3868	+	20.4567i	1.84567	+	0.764501i
$717$		0			0
$718$		0			0
$719$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$719$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$720$		0			0
$721$		0			0
$722$	1.09832	+	2.65159i	0.0408754	+	0.0986819i
$723$	25.4332	−	10.5348i	0.945871	−	0.391792i
$724$		0			0
$725$		0			0
$726$	4.44486	−	6.65220i	0.164964	−	0.246886i
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$	−9.27053	+	22.3810i	−0.343353	+	0.828927i
$730$		0			0
$731$	−37.4579	+	35.3157i	−1.38543	+	1.30620i
$732$		0			0
$733$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$733$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	10.1558	−	51.0567i	0.374094	−	1.88070i
$738$	6.31731	−	4.22109i	0.232543	−	0.155380i
$739$	−16.2359	−	39.1969i	−0.597247	−	1.44188i	−0.876376	−	0.481627i	$-0.840046\pi$
$739$	−16.2359	−	39.1969i	−0.597247	−	1.44188i	0.279129	−	0.960253i	$-0.409954\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$743$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			6.24554i			0.228512i
$748$	26.5027	+	18.8597i	0.969036	+	0.689578i
$749$		0			0
$750$		0			0
$751$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$751$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$752$		0			0
$753$	−32.0192	−	21.3945i	−1.16684	−	0.779660i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$757$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$758$	−30.5834	−	45.7712i	−1.11084	−	1.66249i
$759$		0			0
$760$		0			0
$761$	−32.6569	+	32.6569i	−1.18381	+	1.18381i	−0.205061	+	0.978749i	$0.565739\pi$
$761$	−32.6569	+	32.6569i	−1.18381	+	1.18381i	−0.978749	+	0.205061i	$0.934261\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	3.87250	+	19.4684i	0.139737	+	0.702504i
$769$	36.4558	+	36.4558i	1.31463	+	1.31463i	0.917961	+	0.396670i	$0.129834\pi$
$769$	36.4558	+	36.4558i	1.31463	+	1.31463i	0.396670	+	0.917961i	$0.370166\pi$
$770$		0			0
$771$	5.30847	−	26.6875i	0.191180	−	0.961126i
$772$	−43.7903	+	29.2597i	−1.57605	+	1.05308i
$773$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$773$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$774$	23.8325	−	9.87172i	0.856640	−	0.354832i
$775$		0			0
$776$	49.2206	+	9.79059i	1.76692	+	0.351462i
$777$		0			0
$778$		0			0
$779$	−16.5403	+	3.29006i	−0.592616	+	0.117879i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	25.8686	+	10.7151i	0.923880	+	0.382683i
$785$		0			0
$786$	25.3247	+	25.3247i	0.903304	+	0.903304i
$787$	−40.6957	−	27.1920i	−1.45064	−	0.969289i	−0.996943	−	0.0781357i	$-0.975103\pi$
$787$	−40.6957	−	27.1920i	−1.45064	−	0.969289i	−0.453701	−	0.891154i	$-0.649897\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−9.05535	−	13.5523i	−0.321768	−	0.481560i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$797$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$798$		0			0
$799$		0			0
$800$	−28.2843			−1.00000
$801$	−25.3668	−	10.5073i	−0.896290	−	0.371255i
$802$	−10.0035	−	50.2912i	−0.353237	−	1.77584i
$803$	−47.5142	−	47.5142i	−1.67674	−	1.67674i
$804$	27.2261	+	18.1919i	0.960190	+	0.641579i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	24.2793	+	36.3366i	0.853615	+	1.27753i	0.959089	+	0.283103i	$0.0913639\pi$
$809$	24.2793	+	36.3366i	0.853615	+	1.27753i	−0.105474	+	0.994422i	$0.533636\pi$
$810$		0			0
$811$	−31.3275	+	46.8849i	−1.10006	+	1.64635i	−0.432876	+	0.901454i	$0.642501\pi$
$811$	−31.3275	+	46.8849i	−1.10006	+	1.64635i	−0.667180	+	0.744896i	$0.732499\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−17.3395	+	10.8618i	−0.607005	+	0.380240i
$817$	−57.2580			−2.00320
$818$	−51.9706	−	21.5269i	−1.81711	−	0.752671i
$819$		0			0
$820$		0			0
$821$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$821$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$822$	−6.92873	+	34.8331i	−0.241667	+	1.21494i
$823$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$823$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$824$		0			0
$825$	−22.6061	+	9.36377i	−0.787044	+	0.326004i
$826$		0			0
$827$	−45.5703	−	9.06449i	−1.58463	−	0.315203i	−0.677330	−	0.735679i	$-0.736863\pi$
$827$	−45.5703	−	9.06449i	−1.58463	−	0.315203i	−0.907304	+	0.420476i	$0.861863\pi$
$828$		0			0
$829$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$829$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.849242	+	28.8492i	−0.0294245	+	0.999567i
$834$	−40.3646			−1.39771
$835$		0			0
$836$	7.05805	+	35.4832i	0.244108	+	1.22721i
$837$		0			0
$838$	6.42218	+	4.29116i	0.221851	+	0.148236i
$839$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$839$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$840$		0			0
$841$	−11.0978	−	26.7925i	−0.382683	−	0.923880i
$842$		0			0
$843$	5.01225	+	7.50137i	0.172631	+	0.258361i
$844$	−43.6164	−	8.67584i	−1.50134	−	0.298635i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$			20.6079i			0.707261i
$850$	−10.3596	−	27.2521i	−0.355333	−	0.934740i
$851$		0			0
$852$		0			0
$853$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$853$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$854$		0			0
$855$		0			0
$856$	−8.82246	+	44.3535i	−0.301545	+	1.51597i
$857$	9.69307	−	6.47670i	0.331109	−	0.221240i	−0.378892	−	0.925441i	$-0.623695\pi$
$857$	9.69307	−	6.47670i	0.331109	−	0.221240i	0.710001	+	0.704201i	$0.248695\pi$
$858$		0			0
$859$	−46.5061	+	19.2635i	−1.58677	+	0.657261i	−0.989467	−	0.144757i	$-0.953760\pi$
$859$	−46.5061	+	19.2635i	−1.58677	+	0.657261i	−0.597300	+	0.802018i	$0.703760\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$863$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$864$	30.7047	−	6.10755i	1.04460	−	0.207783i
$865$		0			0
$866$			21.0294i			0.714609i
$867$	−16.8164	−	12.7284i	−0.571115	−	0.432280i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	5.05685	−	25.4225i	0.171148	−	0.860421i
$874$		0			0
$875$		0			0
$876$	39.0494	−	16.1748i	1.31936	−	0.546495i
$877$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$877$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	57.8597	−	11.5090i	1.94934	−	0.387748i	0.952921	−	0.303218i	$-0.0980609\pi$
$881$	57.8597	−	11.5090i	1.94934	−	0.387748i	0.996421	−	0.0845306i	$-0.0269391\pi$
$882$	5.53438	−	13.3612i	0.186352	−	0.449894i
$883$		−	24.5780i		−	0.827115i	−0.910478	−	0.413558i	$-0.864286\pi$
$883$		−	24.5780i		−	0.827115i	0.910478	−	0.413558i	$-0.135714\pi$
$884$		0			0
$885$		0			0
$886$	49.2843	+	20.4142i	1.65574	+	0.685829i
$887$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$887$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	8.14435	−	5.44188i	0.272846	−	0.182310i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	44.0787	−	8.76779i	1.47092	−	0.292585i
$899$		0			0
$900$			14.6088i			0.486962i
$901$		0			0
$902$	−20.5152			−0.683080
$903$		0			0
$904$	−11.7268	−	58.9548i	−0.390029	−	1.96081i
$905$		0			0
$906$		0			0
$907$	10.9845	−	55.2229i	0.364735	−	1.83365i	−0.166022	−	0.986122i	$-0.553092\pi$
$907$	10.9845	−	55.2229i	0.364735	−	1.83365i	0.530757	−	0.847524i	$-0.321908\pi$
$908$	−44.0936	+	29.4624i	−1.46330	+	0.977745i
$909$		0			0
$910$		0			0
$911$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$911$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$912$	−22.3194	−	4.43961i	−0.739071	−	0.147010i
$913$	9.36911	−	14.0219i	0.310072	−	0.464056i
$914$	42.3848	−	42.3848i	1.40196	−	1.40196i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	17.1309	+	27.3473i	0.565403	+	0.902594i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	2.05370	+	10.3247i	0.0676718	+	0.340209i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−6.35871	−	1.26483i	−0.208622	−	0.0414976i	0.0896726	−	0.995971i	$-0.471418\pi$
$929$	−6.35871	−	1.26483i	−0.208622	−	0.0414976i	−0.298295	+	0.954474i	$0.596418\pi$
$930$		0			0
$931$	−22.6985	+	22.6985i	−0.743913	+	0.743913i
$932$	41.9200	−	8.33840i	1.37313	−	0.273133i
$933$		0			0
$934$			1.11270i			0.0364086i
$935$		0			0
$936$		0			0
$937$	47.0363	+	19.4831i	1.53661	+	0.636484i	0.980833	−	0.194849i	$-0.0624217\pi$
$937$	47.0363	+	19.4831i	1.53661	+	0.636484i	0.555775	+	0.831333i	$0.312422\pi$
$938$		0			0
$939$	29.6304	+	29.6304i	0.966953	+	0.966953i
$940$		0			0
$941$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$941$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$942$		0			0
$943$		0			0
$944$	−0.485281	+	0.201010i	−0.0157946	+	0.00654232i
$945$		0			0
$946$	−68.3151	−	13.5887i	−2.22112	−	0.441807i
$947$	34.0844	−	51.0110i	1.10760	−	1.65763i	0.487435	−	0.873160i	$-0.337933\pi$
$947$	34.0844	−	51.0110i	1.10760	−	1.65763i	0.620161	−	0.784474i	$-0.287067\pi$
$948$		0			0
$949$		0			0
$950$	12.4090	−	29.9581i	0.402603	−	0.971969i
$951$		0			0
$952$		0			0
$953$	56.1212			1.81794			0.908972	−	0.416857i	$-0.136868\pi$
$953$	56.1212			1.81794			0.908972	+	0.416857i	$0.136868\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	28.6403	−	11.8632i	0.923880	−	0.382683i
$962$		0			0
$963$	22.9086	+	4.55681i	0.738220	+	0.146841i
$964$	−24.6558	+	36.9000i	−0.794110	+	1.18847i
$965$		0			0
$966$		0			0
$967$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$967$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$968$			12.8977i			0.414549i
$969$	−3.89732	−	23.1310i	−0.125200	−	0.743076i
$970$		0			0
$971$	35.0919	+	14.5355i	1.12615	+	0.466467i	0.866471	−	0.499227i	$-0.166383\pi$
$971$	35.0919	+	14.5355i	1.12615	+	0.466467i	0.259681	+	0.965694i	$0.416383\pi$
$972$	−5.27603	−	26.5244i	−0.169229	−	0.850771i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−6.99138	−	16.8787i	−0.223674	−	0.539997i	0.771709	−	0.635975i	$-0.219402\pi$
$977$	−6.99138	−	16.8787i	−0.223674	−	0.539997i	−0.995383	+	0.0959785i	$0.969402\pi$
$978$	20.2286	−	8.37895i	0.646839	−	0.267929i
$979$	41.1887	+	61.6432i	1.31640	+	1.97013i
$980$		0			0
$981$		0			0
$982$	−39.6985	+	39.6985i	−1.26683	+	1.26683i
$983$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$983$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$984$	4.93827	−	11.9220i	0.157426	−	0.380061i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$991$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$992$		0			0
$993$	−13.9962	+	9.35195i	−0.444155	+	0.296775i
$994$		0			0
$995$		0			0
$996$	5.89330	+	8.81995i	0.186736	+	0.279471i
$997$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$997$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$998$	−4.28519	+	6.41324i	−0.135645	+	0.203008i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	136.2.s.a.11.1	✓	8
4.3	odd	2		544.2.cc.b.79.1		8
8.3	odd	2	CM	136.2.s.a.11.1	✓	8
8.5	even	2		544.2.cc.b.79.1		8
17.14	odd	16	inner	136.2.s.a.99.1	yes	8
68.31	even	16		544.2.cc.b.303.1		8
136.99	even	16	inner	136.2.s.a.99.1	yes	8
136.133	odd	16		544.2.cc.b.303.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.2.s.a.11.1	✓	8	1.1	even	1	trivial
136.2.s.a.11.1	✓	8	8.3	odd	2	CM
136.2.s.a.99.1	yes	8	17.14	odd	16	inner
136.2.s.a.99.1	yes	8	136.99	even	16	inner
544.2.cc.b.79.1		8	4.3	odd	2
544.2.cc.b.79.1		8	8.5	even	2
544.2.cc.b.303.1		8	68.31	even	16
544.2.cc.b.303.1		8	136.133	odd	16

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:	\( N \)	\(=\)	\( 136 = 2^{3} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	136.s (of order \(16\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(1.30656 + 0.541196i) q^{2} +(-0.242031 - 1.21677i) q^{3} +(1.41421 + 1.41421i) q^{4} +(0.342284 - 1.72078i) q^{6} +(1.08239 + 2.61313i) q^{8} +(1.34968 - 0.559056i) q^{9} +O(q^{10})\)	\(q+(1.30656 + 0.541196i) q^{2} +(-0.242031 - 1.21677i) q^{3} +(1.41421 + 1.41421i) q^{4} +(0.342284 - 1.72078i) q^{6} +(1.08239 + 2.61313i) q^{8} +(1.34968 - 0.559056i) q^{9} +(-3.86883 - 0.769558i) q^{11} +(1.37849 - 2.06306i) q^{12} +4.00000i q^{16} +(-3.85403 + 1.46508i) q^{17} +2.06600 q^{18} +(-4.23671 - 1.75490i) q^{19} +(-4.63838 - 3.09927i) q^{22} +(2.91761 - 1.94948i) q^{24} +(1.91342 + 4.61940i) q^{25} +(-3.07465 - 4.60154i) q^{27} +(-2.16478 + 5.22625i) q^{32} +4.89374i q^{33} +(-5.82843 - 0.171573i) q^{34} +(2.69936 + 1.11811i) q^{36} +(-4.58579 - 4.58579i) q^{38} +(3.05774 - 2.04312i) q^{41} +(11.5355 - 4.77817i) q^{43} +(-4.38303 - 6.55967i) q^{44} +(4.86709 - 0.968125i) q^{48} +(2.67878 - 6.46716i) q^{49} +7.07107i q^{50} +(2.71546 + 4.33489i) q^{51} +(-1.52689 - 7.67619i) q^{54} +(-1.10990 + 5.57986i) q^{57} +(0.0502525 + 0.121320i) q^{59} +(-5.65685 + 5.65685i) q^{64} +(-2.64847 + 6.39398i) q^{66} +13.1969i q^{67} +(-7.52235 - 3.37849i) q^{68} +(2.92177 + 2.92177i) q^{72} +(14.1638 + 9.46395i) q^{73} +(5.15765 - 3.44623i) q^{75} +(-3.50981 - 8.47343i) q^{76} +(-1.75586 + 1.75586i) q^{81} +(5.10086 - 1.01462i) q^{82} +(-1.63604 + 3.94975i) q^{83} +17.6578 q^{86} +(-2.17664 - 10.9427i) q^{88} +(-13.2898 - 13.2898i) q^{89} +(6.88311 + 1.36913i) q^{96} +(9.85750 - 14.7528i) q^{97} +(7.00000 - 7.00000i) q^{98} +(-5.65191 + 1.12424i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} + 8 q^{6} + 16 q^{9}+O(q^{10}) \) 8 * q - 8 * q^3 + 8 * q^6 + 16 * q^9	\( 8 q - 8 q^{3} + 8 q^{6} + 16 q^{9} - 16 q^{12} - 24 q^{22} + 32 q^{24} - 32 q^{27} - 24 q^{34} + 32 q^{36} - 48 q^{38} - 32 q^{41} + 64 q^{43} - 16 q^{44} + 32 q^{48} - 40 q^{51} + 40 q^{54} - 40 q^{57} + 40 q^{59} + 72 q^{66} - 48 q^{72} + 48 q^{73} + 56 q^{81} - 64 q^{83} - 32 q^{96} + 56 q^{98} - 48 q^{99}+O(q^{100}) \) 8 * q - 8 * q^3 + 8 * q^6 + 16 * q^9 - 16 * q^12 - 24 * q^22 + 32 * q^24 - 32 * q^27 - 24 * q^34 + 32 * q^36 - 48 * q^38 - 32 * q^41 + 64 * q^43 - 16 * q^44 + 32 * q^48 - 40 * q^51 + 40 * q^54 - 40 * q^57 + 40 * q^59 + 72 * q^66 - 48 * q^72 + 48 * q^73 + 56 * q^81 - 64 * q^83 - 32 * q^96 + 56 * q^98 - 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more