Embedded newform 135.4.q.a.113.42

• Label: 135.4.q.a.113.42
• 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			113.42
Character	\(\chi\)	\(=\)	135.113
Dual form			135.4.q.a.92.42

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.83816 + 0.335795i) q^{2} +(-3.82673 + 3.51513i) q^{3} +(6.74023 + 1.18848i) q^{4} +(-10.6316 + 3.45978i) q^{5} +(-15.8679 + 12.2066i) q^{6} +(-8.65544 - 6.06060i) q^{7} +(-4.30133 - 1.15254i) q^{8} +(2.28771 - 26.9029i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 12 q^{2} - 12 q^{3} - 12 q^{5} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.83816	+	0.335795i	1.35699	+	0.118722i	0.742312	−	0.670054i	\(-0.233729\pi\)
							0.614681	+	0.788776i	\(0.289285\pi\)
\(3\)	−3.82673	+	3.51513i	−0.736454	+	0.676487i
							\(5\)	−10.6316	+	3.45978i	−0.950915	+	0.309453i
\(7\)	−8.65544	−	6.06060i	−0.467350	−	0.327242i								0.316069	−	0.948736i	\(-0.397637\pi\)
							−0.783419	+	0.621495i	\(0.786526\pi\)
\(11\)	−18.9037	+	51.9374i	−0.518152	+	1.42361i	0.354403	+	0.935093i	\(0.384684\pi\)
							−0.872554	+	0.488517i	\(0.837538\pi\)
\(13\)	−0.436374	−	4.98778i	−0.00930987	−	0.106412i	0.990118	−	0.140238i	\(-0.0447866\pi\)
							−0.999428	+	0.0338252i	\(0.989231\pi\)
\(17\)	3.33191	−	0.892782i	0.0475357	−	0.0127371i	−0.234973	−	0.972002i	\(-0.575500\pi\)
							0.282509	+	0.959265i	\(0.408833\pi\)
\(19\)	−26.0625	+	15.0472i	−0.314692	+	0.181687i	−0.649024	−	0.760768i	\(-0.724823\pi\)
							0.334332	+	0.942455i	\(0.391489\pi\)
\(23\)	49.6315	+	70.8811i	0.449951	+	0.642597i	0.978486	−	0.206315i	\(-0.0661471\pi\)
							−0.528534	+	0.848912i	\(0.677258\pi\)
\(29\)	−143.711	+	120.588i	−0.920221	+	0.772157i	−0.974036	−	0.226394i	\(-0.927306\pi\)
							0.0538151	+	0.998551i	\(0.482862\pi\)
\(31\)	−2.89290	+	16.4065i	−0.0167607	+	0.0950544i	−0.992041	−	0.125919i	\(-0.959812\pi\)
							0.975280	+	0.220973i	\(0.0709233\pi\)
\(37\)	103.730	+	387.126i	0.460895	+	1.72008i	0.670152	+	0.742224i	\(0.266229\pi\)
							−0.209257	+	0.977861i	\(0.567104\pi\)
\(41\)	243.559	−	290.263i	0.927746	−	1.10564i	−0.0664206	−	0.997792i	\(-0.521158\pi\)
							0.994167	−	0.107853i	\(-0.0343977\pi\)
\(43\)	45.4976	+	97.5699i	0.161356	+	0.346030i	0.970318	−	0.241831i	\(-0.0777480\pi\)
							−0.808962	+	0.587861i	\(0.799970\pi\)
\(47\)	−24.7027	+	35.2792i	−0.0766652	+	0.109489i	−0.855653	−	0.517549i	\(-0.826844\pi\)
							0.778988	+	0.627039i	\(0.215733\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.q.a.113.42	yes	624
5.2	odd	4	inner	135.4.q.a.32.11	✓	624
27.11	odd	18	inner	135.4.q.a.38.11	yes	624
135.92	even	36	inner	135.4.q.a.92.42	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.q.a.32.11	✓	624	5.2	odd	4	inner
135.4.q.a.38.11	yes	624	27.11	odd	18	inner
135.4.q.a.92.42	yes	624	135.92	even	36	inner
135.4.q.a.113.42	yes	624	1.1	even	1	trivial