Embedded newform 135.4.q.a.113.41

• Label: 135.4.q.a.113.41
• 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			113.41
Character	\(\chi\)	\(=\)	135.113
Dual form			135.4.q.a.92.41

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.36719 + 0.294591i) q^{2} +(0.172148 - 5.19330i) q^{3} +(3.37270 + 0.594698i) q^{4} +(10.8769 - 2.58705i) q^{5} +(2.10955 - 17.4361i) q^{6} +(23.9650 + 16.7805i) q^{7} +(-14.9377 - 4.00254i) q^{8} +(-26.9407 - 1.78803i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 12 q^{2} - 12 q^{3} - 12 q^{5} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.36719	+	0.294591i	1.19048	+	0.104154i	0.665127	−	0.746731i	\(-0.268378\pi\)
							0.525353	+	0.850884i	\(0.323933\pi\)
\(3\)	0.172148	−	5.19330i	0.0331299	−	0.999451i
							\(5\)	10.8769	−	2.58705i	0.972860	−	0.231393i
\(7\)	23.9650	+	16.7805i	1.29399	+	0.906062i								0.998781	−	0.0493700i	\(-0.0157214\pi\)
							0.295210	+	0.955432i	\(0.404610\pi\)
\(11\)	22.4411	−	61.6565i	0.615114	−	1.69001i	−0.103523	−	0.994627i	\(-0.533011\pi\)
							0.718637	−	0.695385i	\(-0.244766\pi\)
\(13\)	1.51980	+	17.3714i	0.0324244	+	0.370613i	0.995020	+	0.0996717i	\(0.0317793\pi\)
							−0.962596	+	0.270941i	\(0.912665\pi\)
\(17\)	−63.5513	+	17.0285i	−0.906674	+	0.242943i	−0.681880	−	0.731464i	\(-0.738837\pi\)
							−0.224794	+	0.974406i	\(0.572171\pi\)
\(19\)	56.0791	−	32.3773i	0.677128	−	0.390940i	−0.121644	−	0.992574i	\(-0.538817\pi\)
							0.798772	+	0.601634i	\(0.205483\pi\)
\(23\)	33.9745	+	48.5206i	0.308008	+	0.439880i	0.943077	−	0.332575i	\(-0.107917\pi\)
							−0.635069	+	0.772455i	\(0.719028\pi\)
\(29\)	−97.4542	+	81.7737i	−0.624027	+	0.523621i	−0.899066	−	0.437812i	\(-0.855754\pi\)
							0.275040	+	0.961433i	\(0.411309\pi\)
\(31\)	−44.9547	+	254.951i	−0.260455	+	1.47711i	0.521217	+	0.853424i	\(0.325478\pi\)
							−0.781672	+	0.623690i	\(0.785633\pi\)
\(37\)	65.3707	+	243.967i	0.290456	+	1.08400i	0.944759	+	0.327765i	\(0.106295\pi\)
							−0.654303	+	0.756232i	\(0.727038\pi\)
\(41\)	−130.910	+	156.012i	−0.498650	+	0.594268i	−0.955395	−	0.295329i	\(-0.904571\pi\)
							0.456745	+	0.889597i	\(0.349015\pi\)
\(43\)	−10.4162	−	22.3376i	−0.0369407	−	0.0792197i	0.886971	−	0.461824i	\(-0.152805\pi\)
							−0.923912	+	0.382605i	\(0.875027\pi\)
\(47\)	−63.9236	+	91.2923i	−0.198387	+	0.283327i	−0.906003	−	0.423271i	\(-0.860882\pi\)
							0.707616	+	0.706597i	\(0.249771\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.q.a.113.41	yes	624
5.2	odd	4	inner	135.4.q.a.32.12	✓	624
27.11	odd	18	inner	135.4.q.a.38.12	yes	624
135.92	even	36	inner	135.4.q.a.92.41	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.q.a.32.12	✓	624	5.2	odd	4	inner
135.4.q.a.38.12	yes	624	27.11	odd	18	inner
135.4.q.a.92.41	yes	624	135.92	even	36	inner
135.4.q.a.113.41	yes	624	1.1	even	1	trivial