Embedded newform 35.5.l.a.2.4

• Label: 35.5.l.a.2.4
• 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			2.4
Character	\(\chi\)	\(=\)	35.2
Dual form			35.5.l.a.18.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.20038 + 1.39344i) q^{2} +(2.56504 + 0.687301i) q^{3} +(11.2458 - 6.49278i) q^{4} +(-19.5926 + 15.5283i) q^{5} -14.2969 q^{6} +(20.1640 - 44.6589i) q^{7} +(11.4758 - 11.4758i) q^{8} +(-64.0410 - 36.9741i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 2 q^{2} - 2 q^{3} + 16 q^{5} - 144 q^{6} + 46 q^{7} + 108 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.20038	+	1.39344i	−1.30009	+	0.348359i	−0.841486	−	0.540279i	\(-0.818319\pi\)
							−0.458608	+	0.888639i	\(0.651652\pi\)
\(3\)	2.56504	+	0.687301i	0.285005	+	0.0763668i	0.398489	−	0.917173i	\(-0.369535\pi\)
							−0.113485	+	0.993540i	\(0.536201\pi\)
\(5\)	−19.5926	+	15.5283i	−0.783706	+	0.621133i
							\(7\)	20.1640	−	44.6589i	0.411510	−	0.911405i
\(11\)	−91.3743	−	158.265i	−0.755159	−	1.30797i								−0.945295	−	0.326216i	\(-0.894226\pi\)
							0.190136	−	0.981758i	\(-0.439107\pi\)
\(13\)	141.894	−	141.894i	0.839607	−	0.839607i	−0.149200	−	0.988807i	\(-0.547670\pi\)
							0.988807	+	0.149200i	\(0.0476699\pi\)
\(17\)	−110.602	+	412.771i	−0.382705	+	1.42827i	0.459048	+	0.888412i	\(0.348191\pi\)
							−0.841753	+	0.539863i	\(0.818476\pi\)
\(19\)	−213.281	−	123.138i	−0.590805	−	0.341101i	0.174611	−	0.984638i	\(-0.444133\pi\)
							−0.765416	+	0.643536i	\(0.777467\pi\)
\(23\)	−132.073	−	492.904i	−0.249666	−	0.931766i	−0.970981	−	0.239158i	\(-0.923128\pi\)
							0.721315	−	0.692608i	\(-0.243538\pi\)
\(29\)			1081.57i			1.28605i	0.765844	+	0.643026i	\(0.222321\pi\)
							−0.765844	+	0.643026i	\(0.777679\pi\)
\(31\)	−227.280	−	393.660i	−0.236503	−	0.409636i	0.723205	−	0.690633i	\(-0.242668\pi\)
							−0.959709	+	0.280997i	\(0.909335\pi\)
\(37\)	72.2086	−	19.3482i	0.0527455	−	0.0141331i	−0.232350	−	0.972632i	\(-0.574641\pi\)
							0.285095	+	0.958499i	\(0.407975\pi\)
\(41\)	−1376.70			−0.818976			−0.409488	−	0.912315i	\(-0.634293\pi\)
							−0.409488	+	0.912315i	\(0.634293\pi\)
\(43\)	−493.766	+	493.766i	−0.267045	+	0.267045i	−0.827908	−	0.560864i	\(-0.810469\pi\)
							0.560864	+	0.827908i	\(0.310469\pi\)
\(47\)	826.377	−	221.427i	0.374095	−	0.100239i	−0.0668723	−	0.997762i	\(-0.521302\pi\)
							0.440968	+	0.897523i	\(0.354635\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	35.5.l.a.2.4	✓	56
5.3	odd	4	inner	35.5.l.a.23.11	yes	56
7.4	even	3	inner	35.5.l.a.32.11	yes	56
35.18	odd	12	inner	35.5.l.a.18.4	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.5.l.a.2.4	✓	56	1.1	even	1	trivial
35.5.l.a.18.4	yes	56	35.18	odd	12	inner
35.5.l.a.23.11	yes	56	5.3	odd	4	inner
35.5.l.a.32.11	yes	56	7.4	even	3	inner