Embedded newform 5.35.c.a.2.3

• Label: 5.35.c.a.2.3
• 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			2.3
Character	\(\chi\)	\(=\)	5.2
Dual form			5.35.c.a.3.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-134834. - 134834. i) q^{2} +(1.19387e8 - 1.19387e8i) q^{3} +1.91805e10i q^{4} +(6.75906e11 - 3.53875e11i) q^{5} -3.21948e13 q^{6} +(2.58783e14 + 2.58783e14i) q^{7} +(2.69752e14 - 2.69752e14i) q^{8} -1.18293e16i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−134834.	−	134834.i	−1.02870	−	1.02870i	−0.999576	−	0.0291252i	\(-0.990728\pi\)
							−0.0291252	−	0.999576i	\(-0.509272\pi\)
\(3\)	1.19387e8	−	1.19387e8i	0.924476	−	0.924476i	−0.0728660	−	0.997342i	\(-0.523215\pi\)
							0.997342	+	0.0728660i	\(0.0232145\pi\)
\(5\)	6.75906e11	−	3.53875e11i	0.885924	−	0.463831i
							\(7\)	2.58783e14	+	2.58783e14i	1.11242	+	1.11242i	0.992822	+	0.119599i	\(0.0381608\pi\)
0.119599	+	0.992822i	\(0.461839\pi\)
\(11\)	−4.31991e17			−0.854672			−0.427336	−	0.904093i	\(-0.640548\pi\)
							−0.427336	+	0.904093i	\(0.640548\pi\)
\(13\)	1.72740e18	−	1.72740e18i	0.199689	−	0.199689i	−0.600177	−	0.799867i	\(-0.704903\pi\)
							0.799867	+	0.600177i	\(0.204903\pi\)
\(17\)	8.56895e20	+	8.56895e20i	1.03585	+	1.03585i	0.999333	+	0.0365150i	\(0.0116257\pi\)
							0.0365150	+	0.999333i	\(0.488374\pi\)
\(19\)		−	1.87411e21i		−	0.341967i	−0.985274	−	0.170984i	\(-0.945305\pi\)
							0.985274	−	0.170984i	\(-0.0546945\pi\)
\(23\)	6.09239e22	−	6.09239e22i	0.431931	−	0.431931i	−0.457354	−	0.889285i	\(-0.651203\pi\)
							0.889285	+	0.457354i	\(0.151203\pi\)
\(29\)		−	1.33115e25i		−	1.83426i	−0.398592	−	0.917129i	\(-0.630501\pi\)
							0.398592	−	0.917129i	\(-0.369499\pi\)
\(31\)	2.39855e25			1.06365			0.531827	−	0.846853i	\(-0.321506\pi\)
							0.531827	+	0.846853i	\(0.321506\pi\)
\(37\)	−1.49049e26	−	1.49049e26i	−0.326513	−	0.326513i	0.524746	−	0.851259i	\(-0.324160\pi\)
							−0.851259	+	0.524746i	\(0.824160\pi\)
\(41\)	−2.29441e27			−0.877700			−0.438850	−	0.898560i	\(-0.644614\pi\)
							−0.438850	+	0.898560i	\(0.644614\pi\)
\(43\)	2.19310e27	−	2.19310e27i	0.373332	−	0.373332i	−0.495357	−	0.868689i	\(-0.664963\pi\)
							0.868689	+	0.495357i	\(0.164963\pi\)
\(47\)	−1.32061e28	−	1.32061e28i	−0.495577	−	0.495577i	0.414481	−	0.910058i	\(-0.363963\pi\)
							−0.910058	+	0.414481i	\(0.863963\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.35.c.a.2.3	✓	32
5.2	odd	4		25.35.c.b.18.14		32
5.3	odd	4	inner	5.35.c.a.3.3	yes	32
5.4	even	2		25.35.c.b.7.14		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.35.c.a.2.3	✓	32	1.1	even	1	trivial
5.35.c.a.3.3	yes	32	5.3	odd	4	inner
25.35.c.b.7.14		32	5.4	even	2
25.35.c.b.18.14		32	5.2	odd	4