Embedded newform 5.35.c.a.2.5

• Label: 5.35.c.a.2.5
• 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			2.5
Character	\(\chi\)	\(=\)	5.2
Dual form			5.35.c.a.3.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-85294.4 - 85294.4i) q^{2} +(-1.68126e8 + 1.68126e8i) q^{3} -2.62960e9i q^{4} +(-5.92285e11 + 4.80911e11i) q^{5} +2.86804e13 q^{6} +(2.04951e14 + 2.04951e14i) q^{7} +(-1.68964e15 + 1.68964e15i) q^{8} -3.98554e16i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−85294.4	−	85294.4i	−0.650745	−	0.650745i	0.302428	−	0.953172i	\(-0.402203\pi\)
							−0.953172	+	0.302428i	\(0.902203\pi\)
\(3\)	−1.68126e8	+	1.68126e8i	−1.30189	+	1.30189i	−0.374767	+	0.927119i	\(0.622277\pi\)
							−0.927119	+	0.374767i	\(0.877723\pi\)
\(5\)	−5.92285e11	+	4.80911e11i	−0.776320	+	0.630339i
							\(7\)	2.04951e14	+	2.04951e14i	0.881014	+	0.881014i	0.993638	−	0.112624i	\(-0.0359256\pi\)
−0.112624	+	0.993638i	\(0.535926\pi\)
\(11\)	−4.22981e17			−0.836845			−0.418422	−	0.908253i	\(-0.637417\pi\)
							−0.418422	+	0.908253i	\(0.637417\pi\)
\(13\)	8.97147e18	−	8.97147e18i	1.03711	−	1.03711i	0.0378303	−	0.999284i	\(-0.487955\pi\)
							0.999284	−	0.0378303i	\(-0.0120446\pi\)
\(17\)	−2.34375e20	−	2.34375e20i	−0.283322	−	0.283322i	0.551111	−	0.834432i	\(-0.314204\pi\)
							−0.834432	+	0.551111i	\(0.814204\pi\)
\(19\)		−	3.96974e21i		−	0.724355i	−0.932109	−	0.362177i	\(-0.882033\pi\)
							0.932109	−	0.362177i	\(-0.117967\pi\)
\(23\)	−1.87042e22	+	1.87042e22i	−0.132607	+	0.132607i	−0.770295	−	0.637688i	\(-0.779891\pi\)
							0.637688	+	0.770295i	\(0.279891\pi\)
\(29\)			3.22017e24i			0.443724i	0.975078	+	0.221862i	\(0.0712135\pi\)
							−0.975078	+	0.221862i	\(0.928787\pi\)
\(31\)	3.15358e25			1.39848			0.699238	−	0.714889i	\(-0.253523\pi\)
							0.699238	+	0.714889i	\(0.253523\pi\)
\(37\)	3.27113e26	+	3.27113e26i	0.716586	+	0.716586i	0.967904	−	0.251318i	\(-0.0808641\pi\)
							−0.251318	+	0.967904i	\(0.580864\pi\)
\(41\)	−9.81913e26			−0.375619			−0.187810	−	0.982205i	\(-0.560139\pi\)
							−0.187810	+	0.982205i	\(0.560139\pi\)
\(43\)	2.79636e27	−	2.79636e27i	0.476024	−	0.476024i	−0.427833	−	0.903858i	\(-0.640723\pi\)
							0.903858	+	0.427833i	\(0.140723\pi\)
\(47\)	−9.12764e27	−	9.12764e27i	−0.342527	−	0.342527i	0.514789	−	0.857317i	\(-0.327870\pi\)
							−0.857317	+	0.514789i	\(0.827870\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.35.c.a.2.5	✓	32
5.2	odd	4		25.35.c.b.18.12		32
5.3	odd	4	inner	5.35.c.a.3.5	yes	32
5.4	even	2		25.35.c.b.7.12		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.35.c.a.2.5	✓	32	1.1	even	1	trivial
5.35.c.a.3.5	yes	32	5.3	odd	4	inner
25.35.c.b.7.12		32	5.4	even	2
25.35.c.b.18.12		32	5.2	odd	4