Embedded newform 15.12.e.a.2.10

• Label: 15.12.e.a.2.10
• Level: $15$
• Weight: $12$
• Character: 15.2
• Analytic conductor: $11.525$
• Analytic rank: $0$
• Dimension: $40$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 15 = 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	15.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.5251477084\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(20\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2.10
Character	\(\chi\)	\(=\)	15.2
Dual form			15.12.e.a.8.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.50709 + 1.50709i) q^{2} +(357.838 + 221.583i) q^{3} +2043.46i q^{4} +(-3168.43 + 6228.10i) q^{5} +(-873.238 + 205.348i) q^{6} +(-31114.0 - 31114.0i) q^{7} +(-6166.19 - 6166.19i) q^{8} +(78948.9 + 158582. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 504 q^{3} + 2940 q^{6} + 31504 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).

\(n\)	\(7\)	\(11\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.50709	+	1.50709i	−0.0333022	+	0.0333022i	−0.723562	−	0.690260i	\(-0.757496\pi\)
							0.690260	+	0.723562i	\(0.257496\pi\)
\(3\)	357.838	+	221.583i	0.850197	+	0.526465i
							\(5\)	−3168.43	+	6228.10i	−0.453428	+	0.891293i
\(7\)	−31114.0	−	31114.0i	−0.699707	−	0.699707i								0.264640	−	0.964347i	\(-0.414747\pi\)
							−0.964347	+	0.264640i	\(0.914747\pi\)
\(11\)		−	709142.i		−	1.32762i	−0.747902	−	0.663810i	\(-0.768939\pi\)
							0.747902	−	0.663810i	\(-0.231061\pi\)
\(13\)	−1.08129e6	+	1.08129e6i	−0.807707	+	0.807707i	−0.984286	−	0.176579i	\(-0.943497\pi\)
							0.176579	+	0.984286i	\(0.443497\pi\)
\(17\)	361717.	−	361717.i	0.0617874	−	0.0617874i	−0.675538	−	0.737325i	\(-0.736089\pi\)
							0.737325	+	0.675538i	\(0.236089\pi\)
\(19\)			1.54311e7i			1.42972i	0.699268	+	0.714860i	\(0.253510\pi\)
							−0.699268	+	0.714860i	\(0.746490\pi\)
\(23\)	2.97410e7	+	2.97410e7i	0.963503	+	0.963503i	0.999357	−	0.0358540i	\(-0.0114151\pi\)
							−0.0358540	+	0.999357i	\(0.511415\pi\)
\(29\)	2.75125e7			0.249081			0.124540	−	0.992215i	\(-0.460254\pi\)
							0.124540	+	0.992215i	\(0.460254\pi\)
\(31\)	1.35112e8			0.847625			0.423813	−	0.905750i	\(-0.360692\pi\)
							0.423813	+	0.905750i	\(0.360692\pi\)
\(37\)	−1.65731e7	−	1.65731e7i	−0.0392912	−	0.0392912i	0.687188	−	0.726479i	\(-0.258845\pi\)
							−0.726479	+	0.687188i	\(0.758845\pi\)
\(41\)			3.86111e8i			0.520476i	0.965544	+	0.260238i	\(0.0838011\pi\)
							−0.965544	+	0.260238i	\(0.916199\pi\)
\(43\)	−1.00218e9	+	1.00218e9i	−1.03960	+	1.03960i	−0.0404204	+	0.999183i	\(0.512870\pi\)
							−0.999183	+	0.0404204i	\(0.987130\pi\)
\(47\)	2.83280e8	−	2.83280e8i	0.180168	−	0.180168i	−0.611261	−	0.791429i	\(-0.709337\pi\)
							0.791429	+	0.611261i	\(0.209337\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.12.e.a.2.10	✓	40
3.2	odd	2	inner	15.12.e.a.2.11	yes	40
5.2	odd	4		75.12.e.d.68.10		40
5.3	odd	4	inner	15.12.e.a.8.11	yes	40
5.4	even	2		75.12.e.d.32.11		40
15.2	even	4		75.12.e.d.68.11		40
15.8	even	4	inner	15.12.e.a.8.10	yes	40
15.14	odd	2		75.12.e.d.32.10		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.12.e.a.2.10	✓	40	1.1	even	1	trivial
15.12.e.a.2.11	yes	40	3.2	odd	2	inner
15.12.e.a.8.10	yes	40	15.8	even	4	inner
15.12.e.a.8.11	yes	40	5.3	odd	4	inner
75.12.e.d.32.10		40	15.14	odd	2
75.12.e.d.32.11		40	5.4	even	2
75.12.e.d.68.10		40	5.2	odd	4
75.12.e.d.68.11		40	15.2	even	4