Embedded newform 35.5.l.a.2.6

• Label: 35.5.l.a.2.6
• 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			2.6
Character	\(\chi\)	\(=\)	35.2
Dual form			35.5.l.a.18.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.93554 + 0.518627i) q^{2} +(0.952096 + 0.255113i) q^{3} +(-10.3791 + 5.99235i) q^{4} +(3.12714 - 24.8036i) q^{5} -1.97513 q^{6} +(-47.7288 - 11.0890i) q^{7} +(39.6520 - 39.6520i) q^{8} +(-69.3067 - 40.0142i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 2 q^{2} - 2 q^{3} + 16 q^{5} - 144 q^{6} + 46 q^{7} + 108 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.93554	+	0.518627i	−0.483886	+	0.129657i	−0.492511	−	0.870306i	\(-0.663921\pi\)
							0.00862561	+	0.999963i	\(0.497254\pi\)
\(3\)	0.952096	+	0.255113i	0.105788	+	0.0283459i	0.311325	−	0.950304i	\(-0.399227\pi\)
							−0.205536	+	0.978649i	\(0.565894\pi\)
\(5\)	3.12714	−	24.8036i	0.125086	−	0.992146i
							\(7\)	−47.7288	−	11.0890i	−0.974056	−	0.226305i
\(11\)	−46.1603	−	79.9519i	−0.381490	−	0.660760i								0.609786	−	0.792566i	\(-0.291256\pi\)
							−0.991275	+	0.131807i	\(0.957922\pi\)
\(13\)	−64.8218	+	64.8218i	−0.383561	+	0.383561i	−0.872383	−	0.488822i	\(-0.837427\pi\)
							0.488822	+	0.872383i	\(0.337427\pi\)
\(17\)	−17.2363	+	64.3266i	−0.0596410	+	0.222583i	−0.989314	−	0.145804i	\(-0.953423\pi\)
							0.929673	+	0.368387i	\(0.120090\pi\)
\(19\)	270.008	+	155.889i	0.747944	+	0.431826i	0.824951	−	0.565205i	\(-0.191203\pi\)
							−0.0770064	+	0.997031i	\(0.524536\pi\)
\(23\)	225.082	+	840.019i	0.425486	+	1.58794i	0.762859	+	0.646565i	\(0.223795\pi\)
							−0.337372	+	0.941371i	\(0.609538\pi\)
\(29\)		−	1133.18i		−	1.34742i	−0.738997	−	0.673708i	\(-0.764700\pi\)
							0.738997	−	0.673708i	\(-0.235300\pi\)
\(31\)	−858.076	−	1486.23i	−0.892899	−	1.54655i	−0.836383	−	0.548145i	\(-0.815334\pi\)
							−0.0565160	−	0.998402i	\(-0.517999\pi\)
\(37\)	−370.889	+	99.3793i	−0.270919	+	0.0725926i	−0.391721	−	0.920084i	\(-0.628120\pi\)
							0.120802	+	0.992677i	\(0.461453\pi\)
\(41\)	−1219.29			−0.725336			−0.362668	−	0.931918i	\(-0.618134\pi\)
							−0.362668	+	0.931918i	\(0.618134\pi\)
\(43\)	1625.68	−	1625.68i	0.879221	−	0.879221i	−0.114233	−	0.993454i	\(-0.536441\pi\)
							0.993454	+	0.114233i	\(0.0364411\pi\)
\(47\)	−272.196	+	72.9348i	−0.123222	+	0.0330171i	−0.319903	−	0.947450i	\(-0.603650\pi\)
							0.196681	+	0.980467i	\(0.436984\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	35.5.l.a.2.6	✓	56
5.3	odd	4	inner	35.5.l.a.23.9	yes	56
7.4	even	3	inner	35.5.l.a.32.9	yes	56
35.18	odd	12	inner	35.5.l.a.18.6	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.5.l.a.2.6	✓	56	1.1	even	1	trivial
35.5.l.a.18.6	yes	56	35.18	odd	12	inner
35.5.l.a.23.9	yes	56	5.3	odd	4	inner
35.5.l.a.32.9	yes	56	7.4	even	3	inner