Embedded newform 5.35.c.a.2.12

• Label: 5.35.c.a.2.12
• Level: $5$
• Weight: $35$
• Character: 5.2
• Analytic conductor: $36.613$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 5 \)
Weight:	\( k \)	\(=\)	\( 35 \)
Character orbit:	\([\chi]\)	\(=\)	5.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(36.6128270213\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2.12
Character	\(\chi\)	\(=\)	5.2
Dual form			5.35.c.a.3.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(109228. + 109228. i) q^{2} +(1.53588e7 - 1.53588e7i) q^{3} +6.68164e9i q^{4} +(1.05031e11 - 7.55675e11i) q^{5} +3.35523e12 q^{6} +(5.89338e13 + 5.89338e13i) q^{7} +(1.14670e15 - 1.14670e15i) q^{8} +1.62054e16i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	32 * q + 131070 * q^2 - 78988260 * q^3 - 730721770860 * q^5 - 44043049959816 * q^6 + 5337954235100 * q^7 - 4536119823236220 * q^8

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	109228.	+	109228.i	0.833344	+	0.833344i	0.987973	−	0.154629i	\(-0.0494182\pi\)
							−0.154629	+	0.987973i	\(0.549418\pi\)
\(3\)	1.53588e7	−	1.53588e7i	0.118932	−	0.118932i	−0.645136	−	0.764068i	\(-0.723199\pi\)
							0.764068	+	0.645136i	\(0.223199\pi\)
\(5\)	1.05031e11	−	7.55675e11i	0.137666	−	0.990479i
							\(7\)	5.89338e13	+	5.89338e13i	0.253337	+	0.253337i	0.822337	−	0.569001i	\(-0.192670\pi\)
−0.569001	+	0.822337i	\(0.692670\pi\)
\(11\)	−8.40927e17			−1.66373			−0.831865	−	0.554978i	\(-0.812727\pi\)
							−0.831865	+	0.554978i	\(0.812727\pi\)
\(13\)	4.68922e18	−	4.68922e18i	0.542081	−	0.542081i	−0.382058	−	0.924138i	\(-0.624784\pi\)
							0.924138	+	0.382058i	\(0.124784\pi\)
\(17\)	−9.55163e20	−	9.55163e20i	−1.15464	−	1.15464i	−0.985612	−	0.169026i	\(-0.945938\pi\)
							−0.169026	−	0.985612i	\(-0.554062\pi\)
\(19\)		−	9.79104e21i		−	1.78656i	−0.449500	−	0.893280i	\(-0.648398\pi\)
							0.449500	−	0.893280i	\(-0.351602\pi\)
\(23\)	7.80719e22	−	7.80719e22i	0.553505	−	0.553505i	−0.373946	−	0.927451i	\(-0.621995\pi\)
							0.927451	+	0.373946i	\(0.121995\pi\)
\(29\)			8.16669e24i			1.12533i	0.826685	+	0.562665i	\(0.190224\pi\)
							−0.826685	+	0.562665i	\(0.809776\pi\)
\(31\)	8.59328e24			0.381075			0.190537	−	0.981680i	\(-0.438977\pi\)
							0.190537	+	0.981680i	\(0.438977\pi\)
\(37\)	−3.42933e26	−	3.42933e26i	−0.751242	−	0.751242i	0.223469	−	0.974711i	\(-0.428262\pi\)
							−0.974711	+	0.223469i	\(0.928262\pi\)
\(41\)	1.78189e27			0.681641			0.340820	−	0.940128i	\(-0.389295\pi\)
							0.340820	+	0.940128i	\(0.389295\pi\)
\(43\)	2.85740e27	−	2.85740e27i	0.486416	−	0.486416i	−0.420757	−	0.907173i	\(-0.638236\pi\)
							0.907173	+	0.420757i	\(0.138236\pi\)
\(47\)	−2.17350e28	−	2.17350e28i	−0.815635	−	0.815635i	0.169837	−	0.985472i	\(-0.445676\pi\)
							−0.985472	+	0.169837i	\(0.945676\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.35.c.a.2.12	✓	32
5.2	odd	4		25.35.c.b.18.5		32
5.3	odd	4	inner	5.35.c.a.3.12	yes	32
5.4	even	2		25.35.c.b.7.5		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.35.c.a.2.12	✓	32	1.1	even	1	trivial
5.35.c.a.3.12	yes	32	5.3	odd	4	inner
25.35.c.b.7.5		32	5.4	even	2
25.35.c.b.18.5		32	5.2	odd	4