Embedded newform 35.5.l.a.2.12

• Label: 35.5.l.a.2.12
• Level: $35$
• Weight: $5$
• Character: 35.2
• Analytic conductor: $3.618$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.61794870793\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(14\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			2.12
Character	\(\chi\)	\(=\)	35.2
Dual form			35.5.l.a.18.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.58177 - 1.49563i) q^{2} +(11.2270 + 3.00826i) q^{3} +(15.0628 - 8.69652i) q^{4} +(-22.0427 - 11.7949i) q^{5} +67.1656 q^{6} +(-38.6780 + 30.0834i) q^{7} +(5.69217 - 5.69217i) q^{8} +(46.8474 + 27.0474i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 2 q^{2} - 2 q^{3} + 16 q^{5} - 144 q^{6} + 46 q^{7} + 108 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.58177	−	1.49563i	1.39544	−	0.373908i	0.518736	−	0.854934i	\(-0.326403\pi\)
							0.876706	+	0.481027i	\(0.159736\pi\)
\(3\)	11.2270	+	3.00826i	1.24744	+	0.334251i	0.821348	−	0.570428i	\(-0.193222\pi\)
							0.426094	+	0.904679i	\(0.359889\pi\)
\(5\)	−22.0427	−	11.7949i	−0.881708	−	0.471795i
							\(7\)	−38.6780	+	30.0834i	−0.789348	+	0.613946i
\(11\)	−7.11215	−	12.3186i	−0.0587781	−	0.101807i								0.835139	−	0.550039i	\(-0.185387\pi\)
							−0.893917	+	0.448232i	\(0.852054\pi\)
\(13\)	195.546	−	195.546i	1.15708	−	1.15708i	0.171975	−	0.985101i	\(-0.444985\pi\)
							0.985101	−	0.171975i	\(-0.0550148\pi\)
\(17\)	−10.4712	+	39.0789i	−0.0362324	+	0.135221i	−0.981673	−	0.190574i	\(-0.938965\pi\)
							0.945440	+	0.325795i	\(0.105632\pi\)
\(19\)	222.686	+	128.568i	0.616859	+	0.356144i	0.775645	−	0.631169i	\(-0.217425\pi\)
							−0.158786	+	0.987313i	\(0.550758\pi\)
\(23\)	−22.4812	−	83.9010i	−0.0424975	−	0.158603i	0.941416	−	0.337247i	\(-0.109496\pi\)
							−0.983914	+	0.178644i	\(0.942829\pi\)
\(29\)			1116.77i			1.32791i	0.747774	+	0.663953i	\(0.231123\pi\)
							−0.747774	+	0.663953i	\(0.768877\pi\)
\(31\)	−890.180	−	1541.84i	−0.926306	−	1.60441i	−0.789447	−	0.613819i	\(-0.789632\pi\)
							−0.136860	−	0.990590i	\(-0.543701\pi\)
\(37\)	805.400	−	215.806i	0.588313	−	0.157638i	0.0476309	−	0.998865i	\(-0.484833\pi\)
							0.540682	+	0.841227i	\(0.318166\pi\)
\(41\)	2737.91			1.62874			0.814370	−	0.580346i	\(-0.197083\pi\)
							0.814370	+	0.580346i	\(0.197083\pi\)
\(43\)	−209.828	+	209.828i	−0.113482	+	0.113482i	−0.761568	−	0.648086i	\(-0.775570\pi\)
							0.648086	+	0.761568i	\(0.275570\pi\)
\(47\)	−523.197	+	140.190i	−0.236848	+	0.0634632i	−0.375290	−	0.926907i	\(-0.622457\pi\)
							0.138443	+	0.990370i	\(0.455790\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	35.5.l.a.2.12	✓	56
5.3	odd	4	inner	35.5.l.a.23.3	yes	56
7.4	even	3	inner	35.5.l.a.32.3	yes	56
35.18	odd	12	inner	35.5.l.a.18.12	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.5.l.a.2.12	✓	56	1.1	even	1	trivial
35.5.l.a.18.12	yes	56	35.18	odd	12	inner
35.5.l.a.23.3	yes	56	5.3	odd	4	inner
35.5.l.a.32.3	yes	56	7.4	even	3	inner