Embedded newform 35.8.k.a.12.3

• Label: 35.8.k.a.12.3
• Level: $35$
• Weight: $8$
• Character: 35.12
• Analytic conductor: $10.933$
• Analytic rank: $0$
• Dimension: $104$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 35 = 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	35.k (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.9334758919\)
Analytic rank:	\(0\)
Dimension:	\(104\)
Relative dimension:	\(26\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			12.3
Character	\(\chi\)	\(=\)	35.12
Dual form			35.8.k.a.3.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-19.2890 + 5.16849i) q^{2} +(1.60132 - 5.97622i) q^{3} +(234.503 - 135.390i) q^{4} +(61.4870 - 272.662i) q^{5} +123.552i q^{6} +(-419.741 + 804.587i) q^{7} +(-2016.14 + 2016.14i) q^{8} +(1860.85 + 1074.36i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q - 2 q^{2} - 6 q^{3} - 1746 q^{7} - 2524 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\)	\(22\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−19.2890	+	5.16849i	−1.70493	+	0.456834i	−0.974172	−	0.225806i	\(-0.927498\pi\)
							−0.730755	+	0.682640i	\(0.760832\pi\)
\(3\)	1.60132	−	5.97622i	0.0342416	−	0.127791i	−0.946689	−	0.322148i	\(-0.895595\pi\)
							0.980931	+	0.194356i	\(0.0622618\pi\)
\(5\)	61.4870	−	272.662i	0.219983	−	0.975504i
							\(7\)	−419.741	+	804.587i	−0.462529	+	0.886604i
\(11\)	−665.320	−	1152.37i	−0.150715	−	0.261046i								0.780776	−	0.624812i	\(-0.214824\pi\)
							−0.931490	+	0.363766i	\(0.881491\pi\)
\(13\)	−3109.23	−	3109.23i	−0.392510	−	0.392510i	0.483071	−	0.875581i	\(-0.339521\pi\)
							−0.875581	+	0.483071i	\(0.839521\pi\)
\(17\)	−1503.71	−	402.917i	−0.0742321	−	0.0198904i	0.221512	−	0.975158i	\(-0.428901\pi\)
							−0.295744	+	0.955267i	\(0.595568\pi\)
\(19\)	17349.2	−	30049.7i	0.580286	−	1.00509i	−0.415159	−	0.909749i	\(-0.636274\pi\)
							0.995445	−	0.0953363i	\(-0.0303927\pi\)
\(23\)	−3046.08	−	11368.1i	−0.0522028	−	0.194823i	0.934900	−	0.354911i	\(-0.115489\pi\)
							−0.987103	+	0.160088i	\(0.948822\pi\)
\(29\)		−	211193.i		−	1.60800i	−0.594627	−	0.804002i	\(-0.702700\pi\)
							0.594627	−	0.804002i	\(-0.297300\pi\)
\(31\)	−185973.	+	107371.i	−1.12120	+	0.647325i	−0.941707	−	0.336434i	\(-0.890779\pi\)
							−0.179493	+	0.983759i	\(0.557446\pi\)
\(37\)	−572299.	+	153347.i	−1.85745	+	0.497702i	−0.999862	−	0.0166375i	\(-0.994704\pi\)
							−0.857587	+	0.514339i	\(0.828037\pi\)
\(41\)		−	658942.i		−	1.49315i	−0.665301	−	0.746575i	\(-0.731697\pi\)
							0.665301	−	0.746575i	\(-0.268303\pi\)
\(43\)	63394.0	−	63394.0i	0.121593	−	0.121593i	−0.643692	−	0.765285i	\(-0.722598\pi\)
							0.765285	+	0.643692i	\(0.222598\pi\)
\(47\)	−253743.	−	946980.i	−0.356493	−	1.33045i	−0.878596	−	0.477567i	\(-0.841519\pi\)
							0.522103	−	0.852883i	\(-0.325148\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	35.8.k.a.12.3	yes	104
5.3	odd	4	inner	35.8.k.a.33.24	yes	104
7.3	odd	6	inner	35.8.k.a.17.24	yes	104
35.3	even	12	inner	35.8.k.a.3.3	✓	104

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.8.k.a.3.3	✓	104	35.3	even	12	inner
35.8.k.a.12.3	yes	104	1.1	even	1	trivial
35.8.k.a.17.24	yes	104	7.3	odd	6	inner
35.8.k.a.33.24	yes	104	5.3	odd	4	inner