Embedded newform 35.5.l.a.2.9

• Label: 35.5.l.a.2.9
• 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			2.9
Character	\(\chi\)	\(=\)	35.2
Dual form			35.5.l.a.18.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.34674 - 0.628807i) q^{2} +(-11.5823 - 3.10347i) q^{3} +(-8.74462 + 5.04871i) q^{4} +(11.4928 + 22.2017i) q^{5} -29.1322 q^{6} +(-36.0021 + 33.2392i) q^{7} +(-44.8336 + 44.8336i) q^{8} +(54.3705 + 31.3908i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 2 q^{2} - 2 q^{3} + 16 q^{5} - 144 q^{6} + 46 q^{7} + 108 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.34674	−	0.628807i	0.586685	−	0.157202i	0.0467472	−	0.998907i	\(-0.485114\pi\)
							0.539938	+	0.841705i	\(0.318448\pi\)
\(3\)	−11.5823	−	3.10347i	−1.28692	−	0.344830i	−0.450433	−	0.892810i	\(-0.648730\pi\)
							−0.836491	+	0.547980i	\(0.815397\pi\)
\(5\)	11.4928	+	22.2017i	0.459711	+	0.888069i
							\(7\)	−36.0021	+	33.2392i	−0.734737	+	0.678352i
\(11\)	−90.4787	−	156.714i	−0.747758	−	1.29515i								−0.948895	−	0.315591i	\(-0.897797\pi\)
							0.201138	−	0.979563i	\(-0.435536\pi\)
\(13\)	89.6472	−	89.6472i	0.530457	−	0.530457i	−0.390251	−	0.920708i	\(-0.627612\pi\)
							0.920708	+	0.390251i	\(0.127612\pi\)
\(17\)	−146.843	+	548.027i	−0.508109	+	1.89629i	−0.0695669	+	0.997577i	\(0.522162\pi\)
							−0.438542	+	0.898711i	\(0.644505\pi\)
\(19\)	102.687	+	59.2862i	0.284451	+	0.164228i	0.635437	−	0.772153i	\(-0.280820\pi\)
							−0.350986	+	0.936381i	\(0.614153\pi\)
\(23\)	−11.0640	−	41.2914i	−0.0209149	−	0.0780555i	0.954680	−	0.297635i	\(-0.0961980\pi\)
							−0.975595	+	0.219580i	\(0.929531\pi\)
\(29\)			15.5584i			0.0184998i	0.999957	+	0.00924992i	\(0.00294438\pi\)
							−0.999957	+	0.00924992i	\(0.997056\pi\)
\(31\)	132.134	+	228.862i	0.137496	+	0.238150i	0.926548	−	0.376176i	\(-0.122761\pi\)
							−0.789052	+	0.614326i	\(0.789428\pi\)
\(37\)	−639.930	+	171.469i	−0.467443	+	0.125251i	−0.484850	−	0.874597i	\(-0.661126\pi\)
							0.0174066	+	0.999848i	\(0.494459\pi\)
\(41\)	880.560			0.523831			0.261916	−	0.965091i	\(-0.415646\pi\)
							0.261916	+	0.965091i	\(0.415646\pi\)
\(43\)	−1623.67	+	1623.67i	−0.878133	+	0.878133i	−0.993341	−	0.115209i	\(-0.963246\pi\)
							0.115209	+	0.993341i	\(0.463246\pi\)
\(47\)	−2616.45	+	701.077i	−1.18445	+	0.317373i	−0.796691	−	0.604387i	\(-0.793418\pi\)
							−0.387761	+	0.921760i	\(0.626751\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	35.5.l.a.2.9	✓	56
5.3	odd	4	inner	35.5.l.a.23.6	yes	56
7.4	even	3	inner	35.5.l.a.32.6	yes	56
35.18	odd	12	inner	35.5.l.a.18.9	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.5.l.a.2.9	✓	56	1.1	even	1	trivial
35.5.l.a.18.9	yes	56	35.18	odd	12	inner
35.5.l.a.23.6	yes	56	5.3	odd	4	inner
35.5.l.a.32.6	yes	56	7.4	even	3	inner