Embedded newform 35.5.l.a.2.3

• Label: 35.5.l.a.2.3
• 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			2.3
Character	\(\chi\)	\(=\)	35.2
Dual form			35.5.l.a.18.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.55079 + 1.48733i) q^{2} +(15.0503 + 4.03273i) q^{3} +(14.7427 - 8.51169i) q^{4} +(-2.87000 - 24.8347i) q^{5} -89.5393 q^{6} +(40.0414 + 28.2434i) q^{7} +(-4.15850 + 4.15850i) q^{8} +(140.102 + 80.8879i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 2 q^{2} - 2 q^{3} + 16 q^{5} - 144 q^{6} + 46 q^{7} + 108 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.55079	+	1.48733i	−1.38770	+	0.371832i	−0.873911	−	0.486086i	\(-0.838424\pi\)
							−0.513786	+	0.857918i	\(0.671757\pi\)
\(3\)	15.0503	+	4.03273i	1.67226	+	0.448081i	0.965719	−	0.259590i	\(-0.0835876\pi\)
							0.706542	+	0.707671i	\(0.250254\pi\)
\(5\)	−2.87000	−	24.8347i	−0.114800	−	0.993389i
							\(7\)	40.0414	+	28.2434i	0.817171	+	0.576395i
\(11\)	48.2961	+	83.6513i	0.399142	+	0.691333i								0.993620	−	0.112778i	\(-0.0359749\pi\)
							−0.594479	+	0.804111i	\(0.702642\pi\)
\(13\)	−48.6898	+	48.6898i	−0.288106	+	0.288106i	−0.836331	−	0.548225i	\(-0.815304\pi\)
							0.548225	+	0.836331i	\(0.315304\pi\)
\(17\)	25.3859	−	94.7415i	0.0878405	−	0.327825i	−0.907996	−	0.418978i	\(-0.862388\pi\)
							0.995837	+	0.0911527i	\(0.0290551\pi\)
\(19\)	217.882	+	125.794i	0.603552	+	0.348461i	0.770438	−	0.637516i	\(-0.220038\pi\)
							−0.166886	+	0.985976i	\(0.553371\pi\)
\(23\)	−247.264	−	922.803i	−0.467418	−	1.74443i	−0.648745	−	0.761006i	\(-0.724706\pi\)
							0.181326	−	0.983423i	\(-0.441961\pi\)
\(29\)			7.20762i			0.00857029i	0.999991	+	0.00428515i	\(0.00136401\pi\)
							−0.999991	+	0.00428515i	\(0.998636\pi\)
\(31\)	−157.749	−	273.229i	−0.164151	−	0.284318i	0.772203	−	0.635376i	\(-0.219155\pi\)
							−0.936353	+	0.351059i	\(0.885822\pi\)
\(37\)	−1460.18	+	391.255i	−1.06661	+	0.285796i	−0.749098	−	0.662459i	\(-0.769513\pi\)
							−0.317508	+	0.948256i	\(0.602846\pi\)
\(41\)	−2207.36			−1.31313			−0.656563	−	0.754272i	\(-0.727990\pi\)
							−0.656563	+	0.754272i	\(0.727990\pi\)
\(43\)	1414.89	−	1414.89i	0.765217	−	0.765217i	−0.212043	−	0.977260i	\(-0.568012\pi\)
							0.977260	+	0.212043i	\(0.0680118\pi\)
\(47\)	1154.98	−	309.475i	0.522851	−	0.140097i	0.0122659	−	0.999925i	\(-0.496096\pi\)
							0.510585	+	0.859827i	\(0.329429\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	35.5.l.a.2.3	✓	56
5.3	odd	4	inner	35.5.l.a.23.12	yes	56
7.4	even	3	inner	35.5.l.a.32.12	yes	56
35.18	odd	12	inner	35.5.l.a.18.3	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.5.l.a.2.3	✓	56	1.1	even	1	trivial
35.5.l.a.18.3	yes	56	35.18	odd	12	inner
35.5.l.a.23.12	yes	56	5.3	odd	4	inner
35.5.l.a.32.12	yes	56	7.4	even	3	inner