Embedded newform 135.4.q.a.113.22

• Label: 135.4.q.a.113.22
• 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			113.22
Character	\(\chi\)	\(=\)	135.113
Dual form			135.4.q.a.92.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.10849 - 0.0969803i) q^{2} +(-3.39868 + 3.93052i) q^{3} +(-6.65912 - 1.17418i) q^{4} +(11.1316 + 1.04333i) q^{5} +(4.14859 - 4.02734i) q^{6} +(-23.5869 - 16.5158i) q^{7} +(15.8662 + 4.25133i) q^{8} +(-3.89795 - 26.7171i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 12 q^{2} - 12 q^{3} - 12 q^{5} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.10849	−	0.0969803i	−0.391911	−	0.0342877i	−0.110502	−	0.993876i	\(-0.535246\pi\)
							−0.281408	+	0.959588i	\(0.590802\pi\)
\(3\)	−3.39868	+	3.93052i	−0.654076	+	0.756429i
							\(5\)	11.1316	+	1.04333i	0.995636	+	0.0933185i
\(7\)	−23.5869	−	16.5158i	−1.27358	−	0.891767i								−0.275986	−	0.961162i	\(-0.589004\pi\)
							−0.997589	+	0.0693948i	\(0.977893\pi\)
\(11\)	−2.37318	+	6.52025i	−0.0650491	+	0.178721i	−0.967959	−	0.251110i	\(-0.919204\pi\)
							0.902910	+	0.429831i	\(0.141427\pi\)
\(13\)	4.54279	+	51.9243i	0.0969187	+	1.10779i	0.876473	+	0.481451i	\(0.159890\pi\)
							−0.779554	+	0.626334i	\(0.784554\pi\)
\(17\)	93.1752	−	24.9662i	1.32931	−	0.356188i	0.476853	−	0.878983i	\(-0.341777\pi\)
							0.852458	+	0.522795i	\(0.175111\pi\)
\(19\)	107.354	−	61.9811i	1.29625	−	0.748391i	0.316497	−	0.948593i	\(-0.397493\pi\)
							0.979755	+	0.200202i	\(0.0641598\pi\)
\(23\)	−2.99639	−	4.27929i	−0.0271648	−	0.0387954i	0.805340	−	0.592813i	\(-0.201983\pi\)
							−0.832505	+	0.554018i	\(0.813094\pi\)
\(29\)	−48.2501	+	40.4867i	−0.308959	+	0.259248i	−0.784062	−	0.620683i	\(-0.786855\pi\)
							0.475102	+	0.879931i	\(0.342411\pi\)
\(31\)	41.3524	−	234.521i	0.239584	−	1.35875i	−0.593157	−	0.805087i	\(-0.702119\pi\)
							0.832741	−	0.553662i	\(-0.186770\pi\)
\(37\)	71.3678	+	266.348i	0.317103	+	1.18344i	0.922016	+	0.387152i	\(0.126541\pi\)
							−0.604913	+	0.796291i	\(0.706792\pi\)
\(41\)	166.426	−	198.339i	0.633937	−	0.755497i	−0.349462	−	0.936950i	\(-0.613636\pi\)
							0.983400	+	0.181454i	\(0.0580802\pi\)
\(43\)	−25.2690	−	54.1896i	−0.0896160	−	0.192182i	0.856391	−	0.516327i	\(-0.172701\pi\)
							−0.946007	+	0.324145i	\(0.894923\pi\)
\(47\)	−109.602	+	156.527i	−0.340150	+	0.485785i	−0.952388	−	0.304889i	\(-0.901381\pi\)
							0.612238	+	0.790674i	\(0.290270\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.q.a.113.22	yes	624
5.2	odd	4	inner	135.4.q.a.32.31	✓	624
27.11	odd	18	inner	135.4.q.a.38.31	yes	624
135.92	even	36	inner	135.4.q.a.92.22	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.q.a.32.31	✓	624	5.2	odd	4	inner
135.4.q.a.38.31	yes	624	27.11	odd	18	inner
135.4.q.a.92.22	yes	624	135.92	even	36	inner
135.4.q.a.113.22	yes	624	1.1	even	1	trivial