Embedded newform 135.4.q.a.32.35

• Label: 135.4.q.a.32.35
• Level: $135$
• Weight: $4$
• Character: 135.32
• 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			32.35
Character	\(\chi\)	\(=\)	135.32
Dual form			135.4.q.a.38.35

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.220818 - 2.52396i) q^{2} +(-3.69802 - 3.65030i) q^{3} +(1.55687 + 0.274518i) q^{4} +(-6.66483 - 8.97664i) q^{5} +(-10.0298 + 8.52758i) q^{6} +(-20.7042 + 29.5687i) q^{7} +(6.28260 - 23.4470i) q^{8} +(0.350671 + 26.9977i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 12 q^{2} - 12 q^{3} - 12 q^{5} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(56\)	\(82\)
\(\chi(n)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.220818	−	2.52396i	0.0780708	−	0.892353i	−0.851656	−	0.524102i	\(-0.824401\pi\)
							0.929726	−	0.368251i	\(-0.120043\pi\)
\(3\)	−3.69802	−	3.65030i	−0.711684	−	0.702500i
							\(5\)	−6.66483	−	8.97664i	−0.596120	−	0.802895i
\(7\)	−20.7042	+	29.5687i	−1.11792	+	1.59656i								−0.375222	+	0.926935i	\(0.622434\pi\)
							−0.742700	+	0.669624i	\(0.766455\pi\)
\(11\)	4.71115	−	12.9438i	0.129133	−	0.354790i	−0.858230	−	0.513266i	\(-0.828436\pi\)
							0.987363	+	0.158475i	\(0.0506578\pi\)
\(13\)	−37.6002	+	3.28959i	−0.802186	+	0.0701822i	−0.480872	−	0.876791i	\(-0.659680\pi\)
							−0.321314	+	0.946973i	\(0.604124\pi\)
\(17\)	11.4833	+	42.8563i	0.163830	+	0.611422i	0.998187	+	0.0601969i	\(0.0191729\pi\)
							−0.834356	+	0.551225i	\(0.814160\pi\)
\(19\)	59.3367	−	34.2581i	0.716462	−	0.413649i	−0.0969873	−	0.995286i	\(-0.530921\pi\)
							0.813449	+	0.581636i	\(0.197587\pi\)
\(23\)	−97.9177	+	68.5627i	−0.887706	+	0.621579i	−0.925932	−	0.377690i	\(-0.876718\pi\)
							0.0382256	+	0.999269i	\(0.487829\pi\)
\(29\)	−142.883	+	119.893i	−0.914924	+	0.767712i	−0.973049	−	0.230597i	\(-0.925932\pi\)
							0.0581258	+	0.998309i	\(0.481488\pi\)
\(31\)	16.8172	−	95.3752i	0.0974343	−	0.552577i	−0.896540	−	0.442963i	\(-0.853927\pi\)
							0.993974	−	0.109614i	\(-0.0349616\pi\)
\(37\)	−199.396	+	53.4280i	−0.885959	+	0.237392i	−0.672976	−	0.739664i	\(-0.734984\pi\)
							−0.212983	+	0.977056i	\(0.568318\pi\)
\(41\)	34.3923	−	40.9872i	0.131004	−	0.156125i	−0.696554	−	0.717504i	\(-0.745285\pi\)
							0.827559	+	0.561379i	\(0.189729\pi\)
\(43\)	−156.153	+	72.8155i	−0.553794	+	0.258239i	−0.679308	−	0.733853i	\(-0.737720\pi\)
							0.125514	+	0.992092i	\(0.459942\pi\)
\(47\)	−257.964	−	180.628i	−0.800593	−	0.560581i	0.100131	−	0.994974i	\(-0.468074\pi\)
							−0.900723	+	0.434393i	\(0.856963\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.q.a.32.35	✓	624
5.3	odd	4	inner	135.4.q.a.113.18	yes	624
27.11	odd	18	inner	135.4.q.a.92.18	yes	624
135.38	even	36	inner	135.4.q.a.38.35	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.q.a.32.35	✓	624	1.1	even	1	trivial
135.4.q.a.38.35	yes	624	135.38	even	36	inner
135.4.q.a.92.18	yes	624	27.11	odd	18	inner
135.4.q.a.113.18	yes	624	5.3	odd	4	inner