Embedded newform 135.4.q.a.113.17

• Label: 135.4.q.a.113.17
• Level: $135$
• Weight: $4$
• Character: 135.113
• Analytic conductor: $7.965$
• Analytic rank: $0$
• Dimension: $624$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	135.q (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.96525785077\)
Analytic rank:	\(0\)
Dimension:	\(624\)
Relative dimension:	\(52\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			113.17
Character	\(\chi\)	\(=\)	135.113
Dual form			135.4.q.a.92.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.61874 - 0.229110i) q^{2} +(-5.08676 - 1.06061i) q^{3} +(-1.07314 - 0.189223i) q^{4} +(8.45829 - 7.31146i) q^{5} +(13.0779 + 3.94288i) q^{6} +(5.62520 + 3.93881i) q^{7} +(23.0803 + 6.18435i) q^{8} +(24.7502 + 10.7901i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 12 q^{2} - 12 q^{3} - 12 q^{5} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).

\(n\)	\(56\)	\(82\)
\(\chi(n)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{3}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.61874	−	0.229110i	−0.925866	−	0.0810028i	−0.385759	−	0.922599i	\(-0.626060\pi\)
							−0.540106	+	0.841597i	\(0.681616\pi\)
\(3\)	−5.08676	−	1.06061i	−0.978947	−	0.204114i
							\(5\)	8.45829	−	7.31146i	0.756532	−	0.653956i
\(7\)	5.62520	+	3.93881i	0.303732	+	0.212676i								0.715499	−	0.698614i	\(-0.246199\pi\)
							−0.411767	+	0.911289i	\(0.635088\pi\)
\(11\)	−5.34374	+	14.6818i	−0.146473	+	0.402430i	−0.991133	−	0.132872i	\(-0.957580\pi\)
							0.844661	+	0.535302i	\(0.179802\pi\)
\(13\)	−4.80156	−	54.8821i	−0.102440	−	1.17089i	−0.857234	−	0.514928i	\(-0.827819\pi\)
							0.754794	−	0.655962i	\(-0.227737\pi\)
\(17\)	−78.7375	+	21.0976i	−1.12333	+	0.300996i	−0.772231	−	0.635342i	\(-0.780859\pi\)
							−0.351101	+	0.936338i	\(0.614193\pi\)
\(19\)	112.291	−	64.8310i	1.35585	−	0.782802i	0.366792	−	0.930303i	\(-0.380456\pi\)
							0.989062	+	0.147501i	\(0.0471228\pi\)
\(23\)	−88.6441	−	126.597i	−0.803634	−	1.14771i	−0.986369	−	0.164546i	\(-0.947384\pi\)
							0.182736	−	0.983162i	\(-0.441505\pi\)
\(29\)	−114.349	+	95.9501i	−0.732209	+	0.614396i	−0.930733	−	0.365700i	\(-0.880830\pi\)
							0.198524	+	0.980096i	\(0.436385\pi\)
\(31\)	−49.0838	+	278.368i	−0.284378	+	1.61279i	0.423122	+	0.906073i	\(0.360934\pi\)
							−0.707500	+	0.706713i	\(0.750177\pi\)
\(37\)	−76.5889	−	285.834i	−0.340301	−	1.27002i	−0.898007	−	0.439982i	\(-0.854985\pi\)
							0.557706	−	0.830039i	\(-0.311682\pi\)
\(41\)	25.9242	−	30.8952i	0.0987482	−	0.117684i	−0.714407	−	0.699730i	\(-0.753303\pi\)
							0.813155	+	0.582047i	\(0.197748\pi\)
\(43\)	−78.0518	−	167.383i	−0.276809	−	0.593619i	0.717924	−	0.696122i	\(-0.245093\pi\)
							−0.994733	+	0.102503i	\(0.967315\pi\)
\(47\)	−186.459	+	266.290i	−0.578676	+	0.826435i	−0.996630	−	0.0820331i	\(-0.973859\pi\)
							0.417953	+	0.908468i	\(0.362748\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.q.a.113.17	yes	624
5.2	odd	4	inner	135.4.q.a.32.36	✓	624
27.11	odd	18	inner	135.4.q.a.38.36	yes	624
135.92	even	36	inner	135.4.q.a.92.17	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.q.a.32.36	✓	624	5.2	odd	4	inner
135.4.q.a.38.36	yes	624	27.11	odd	18	inner
135.4.q.a.92.17	yes	624	135.92	even	36	inner
135.4.q.a.113.17	yes	624	1.1	even	1	trivial