Embedded newform 135.4.q.a.113.52

• Label: 135.4.q.a.113.52
• 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			113.52
Character	\(\chi\)	\(=\)	135.113
Dual form			135.4.q.a.92.52

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.58023 + 0.488207i) q^{2} +(2.83566 + 4.35420i) q^{3} +(23.0222 + 4.05944i) q^{4} +(-10.9372 + 2.31889i) q^{5} +(13.6979 + 25.6818i) q^{6} +(-14.3762 - 10.0663i) q^{7} +(83.2020 + 22.2939i) q^{8} +(-10.9181 + 24.6940i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 12 q^{2} - 12 q^{3} - 12 q^{5} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.58023	+	0.488207i	1.97291	+	0.172607i	0.998613	−	0.0526563i	\(-0.0167688\pi\)
							0.974298	+	0.225264i	\(0.0723243\pi\)
\(3\)	2.83566	+	4.35420i	0.545722	+	0.837966i
							\(5\)	−10.9372	+	2.31889i	−0.978255	+	0.207408i
\(7\)	−14.3762	−	10.0663i	−0.776243	−	0.543531i								0.116986	−	0.993134i	\(-0.462677\pi\)
							−0.893229	+	0.449602i	\(0.851566\pi\)
\(11\)	12.3044	−	33.8060i	0.337264	−	0.926626i	−0.648903	−	0.760871i	\(-0.724772\pi\)
							0.986167	−	0.165755i	\(-0.0530060\pi\)
\(13\)	−2.69873	−	30.8466i	−0.0575764	−	0.658101i	−0.969070	−	0.246786i	\(-0.920625\pi\)
							0.911494	−	0.411314i	\(-0.134930\pi\)
\(17\)	12.4779	−	3.34345i	0.178020	−	0.0477004i	−0.168708	−	0.985666i	\(-0.553959\pi\)
							0.346728	+	0.937966i	\(0.387293\pi\)
\(19\)	−9.93741	+	5.73737i	−0.119989	+	0.0692759i	−0.558793	−	0.829307i	\(-0.688736\pi\)
							0.438804	+	0.898583i	\(0.355402\pi\)
\(23\)	−80.0201	−	114.281i	−0.725450	−	1.03605i	−0.997280	−	0.0737049i	\(-0.976518\pi\)
							0.271830	−	0.962345i	\(-0.412371\pi\)
\(29\)	−30.1487	+	25.2978i	−0.193051	+	0.161989i	−0.734190	−	0.678944i	\(-0.762438\pi\)
							0.541139	+	0.840933i	\(0.317993\pi\)
\(31\)	−45.3142	+	256.990i	−0.262538	+	1.48893i	0.513417	+	0.858139i	\(0.328379\pi\)
							−0.775955	+	0.630788i	\(0.782732\pi\)
\(37\)	−23.9876	−	89.5228i	−0.106582	−	0.397769i	0.891938	−	0.452158i	\(-0.149346\pi\)
							−0.998520	+	0.0543887i	\(0.982679\pi\)
\(41\)	12.8739	−	15.3425i	0.0490382	−	0.0584415i	−0.740968	−	0.671541i	\(-0.765633\pi\)
							0.790006	+	0.613099i	\(0.210077\pi\)
\(43\)	87.5391	+	187.728i	0.310456	+	0.665774i	0.998187	−	0.0601913i	\(-0.0191711\pi\)
							−0.687731	+	0.725965i	\(0.741393\pi\)
\(47\)	19.0543	−	27.2123i	0.0591351	−	0.0844537i	−0.788495	−	0.615041i	\(-0.789140\pi\)
							0.847631	+	0.530587i	\(0.178028\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.q.a.113.52	yes	624
5.2	odd	4	inner	135.4.q.a.32.1	✓	624
27.11	odd	18	inner	135.4.q.a.38.1	yes	624
135.92	even	36	inner	135.4.q.a.92.52	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.q.a.32.1	✓	624	5.2	odd	4	inner
135.4.q.a.38.1	yes	624	27.11	odd	18	inner
135.4.q.a.92.52	yes	624	135.92	even	36	inner
135.4.q.a.113.52	yes	624	1.1	even	1	trivial