Embedded newform 35.3.l.a.2.2

• Label: 35.3.l.a.2.2
• Level: $35$
• Weight: $3$
• Character: 35.2
• Analytic conductor: $0.954$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 35 = 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	35.l (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.953680925261\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(6\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			2.2
Character	\(\chi\)	\(=\)	35.2
Dual form			35.3.l.a.18.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.89206 + 0.506976i) q^{2} +(2.62470 + 0.703286i) q^{3} +(-0.141229 + 0.0815388i) q^{4} +(4.09808 + 2.86456i) q^{5} -5.32264 q^{6} +(2.16120 + 6.65802i) q^{7} +(5.76622 - 5.76622i) q^{8} +(-1.39979 - 0.808169i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 2 q^{2} - 2 q^{3} - 4 q^{5} - 6 q^{7} - 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.89206	+	0.506976i	−0.946031	+	0.253488i	−0.698677	−	0.715437i	\(-0.746228\pi\)
							−0.247354	+	0.968925i	\(0.579561\pi\)
\(3\)	2.62470	+	0.703286i	0.874900	+	0.234429i	0.668205	−	0.743977i	\(-0.267063\pi\)
							0.206695	+	0.978406i	\(0.433729\pi\)
\(5\)	4.09808	+	2.86456i	0.819617	+	0.572912i
							\(7\)	2.16120	+	6.65802i	0.308743	+	0.951146i
\(11\)	−9.62452	−	16.6702i	−0.874956	−	1.51547i								−0.856809	−	0.515634i	\(-0.827557\pi\)
							−0.0181472	−	0.999835i	\(-0.505777\pi\)
\(13\)	0.957657	−	0.957657i	0.0736659	−	0.0736659i	−0.669314	−	0.742980i	\(-0.733412\pi\)
							0.742980	+	0.669314i	\(0.233412\pi\)
\(17\)	3.31668	−	12.3780i	0.195099	−	0.728118i	−0.797143	−	0.603791i	\(-0.793656\pi\)
							0.992241	−	0.124327i	\(-0.0396773\pi\)
\(19\)	4.15867	+	2.40101i	0.218877	+	0.126369i	0.605430	−	0.795898i	\(-0.293001\pi\)
							−0.386553	+	0.922267i	\(0.626334\pi\)
\(23\)	2.74430	+	10.2419i	0.119317	+	0.445299i	0.999574	−	0.0292001i	\(-0.00929600\pi\)
							−0.880256	+	0.474499i	\(0.842629\pi\)
\(29\)		−	13.7098i		−	0.472752i	−0.971662	−	0.236376i	\(-0.924040\pi\)
							0.971662	−	0.236376i	\(-0.0759596\pi\)
\(31\)	−9.56292	−	16.5635i	−0.308481	−	0.534305i	0.669549	−	0.742768i	\(-0.266487\pi\)
							−0.978030	+	0.208462i	\(0.933154\pi\)
\(37\)	4.21549	−	1.12954i	0.113932	−	0.0305281i	−0.201402	−	0.979509i	\(-0.564550\pi\)
							0.315335	+	0.948981i	\(0.397883\pi\)
\(41\)	50.7411			1.23759			0.618794	−	0.785554i	\(-0.287622\pi\)
							0.618794	+	0.785554i	\(0.287622\pi\)
\(43\)	−31.8953	+	31.8953i	−0.741750	+	0.741750i	−0.972915	−	0.231165i	\(-0.925746\pi\)
							0.231165	+	0.972915i	\(0.425746\pi\)
\(47\)	−43.3605	+	11.6184i	−0.922564	+	0.247200i	−0.688681	−	0.725064i	\(-0.741810\pi\)
							−0.233883	+	0.972265i	\(0.575143\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	35.3.l.a.2.2	✓	24
3.2	odd	2		315.3.ca.a.37.5		24
5.2	odd	4		175.3.p.c.93.2		24
5.3	odd	4	inner	35.3.l.a.23.5	yes	24
5.4	even	2		175.3.p.c.107.5		24
7.2	even	3		245.3.g.c.197.5		12
7.3	odd	6		245.3.m.b.67.5		24
7.4	even	3	inner	35.3.l.a.32.5	yes	24
7.5	odd	6		245.3.g.b.197.5		12
7.6	odd	2		245.3.m.b.177.2		24
15.8	even	4		315.3.ca.a.163.2		24
21.11	odd	6		315.3.ca.a.172.2		24
35.3	even	12		245.3.m.b.18.2		24
35.4	even	6		175.3.p.c.32.2		24
35.13	even	4		245.3.m.b.128.5		24
35.18	odd	12	inner	35.3.l.a.18.2	yes	24
35.23	odd	12		245.3.g.c.148.5		12
35.32	odd	12		175.3.p.c.18.5		24
35.33	even	12		245.3.g.b.148.5		12
105.53	even	12		315.3.ca.a.298.5		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.3.l.a.2.2	✓	24	1.1	even	1	trivial
35.3.l.a.18.2	yes	24	35.18	odd	12	inner
35.3.l.a.23.5	yes	24	5.3	odd	4	inner
35.3.l.a.32.5	yes	24	7.4	even	3	inner
175.3.p.c.18.5		24	35.32	odd	12
175.3.p.c.32.2		24	35.4	even	6
175.3.p.c.93.2		24	5.2	odd	4
175.3.p.c.107.5		24	5.4	even	2
245.3.g.b.148.5		12	35.33	even	12
245.3.g.b.197.5		12	7.5	odd	6
245.3.g.c.148.5		12	35.23	odd	12
245.3.g.c.197.5		12	7.2	even	3
245.3.m.b.18.2		24	35.3	even	12
245.3.m.b.67.5		24	7.3	odd	6
245.3.m.b.128.5		24	35.13	even	4
245.3.m.b.177.2		24	7.6	odd	2
315.3.ca.a.37.5		24	3.2	odd	2
315.3.ca.a.163.2		24	15.8	even	4
315.3.ca.a.172.2		24	21.11	odd	6
315.3.ca.a.298.5		24	105.53	even	12