Embedded newform 15.12.e.a.2.15

• Label: 15.12.e.a.2.15
• Level: $15$
• Weight: $12$
• Character: 15.2
• Analytic conductor: $11.525$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• 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			2.15
Character	\(\chi\)	\(=\)	15.2
Dual form			15.12.e.a.8.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(37.2586 - 37.2586i) q^{2} +(-407.704 - 104.521i) q^{3} -728.404i q^{4} +(1506.89 + 6823.30i) q^{5} +(-19084.8 + 11296.2i) q^{6} +(-5795.67 - 5795.67i) q^{7} +(49166.3 + 49166.3i) q^{8} +(155298. + 85226.8i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 504 q^{3} + 2940 q^{6} + 31504 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	37.2586	−	37.2586i	0.823306	−	0.823306i	−0.163275	−	0.986581i	\(-0.552206\pi\)
							0.986581	+	0.163275i	\(0.0522056\pi\)
\(3\)	−407.704	−	104.521i	−0.968675	−	0.248333i
							\(5\)	1506.89	+	6823.30i	0.215648	+	0.976471i
\(7\)	−5795.67	−	5795.67i	−0.130336	−	0.130336i								0.638929	−	0.769265i	\(-0.279378\pi\)
							−0.769265	+	0.638929i	\(0.779378\pi\)
\(11\)			364518.i			0.682432i	0.939985	+	0.341216i	\(0.110839\pi\)
							−0.939985	+	0.341216i	\(0.889161\pi\)
\(13\)	609210.	−	609210.i	0.455071	−	0.455071i	−0.441963	−	0.897033i	\(-0.645718\pi\)
							0.897033	+	0.441963i	\(0.145718\pi\)
\(17\)	−3.57407e6	+	3.57407e6i	−0.610513	+	0.610513i	−0.943080	−	0.332567i	\(-0.892085\pi\)
							0.332567	+	0.943080i	\(0.392085\pi\)
\(19\)			1.30082e7i			1.20524i	0.798028	+	0.602620i	\(0.205877\pi\)
							−0.798028	+	0.602620i	\(0.794123\pi\)
\(23\)	3.07670e7	+	3.07670e7i	0.996741	+	0.996741i	0.999995	−	0.00325393i	\(-0.00103576\pi\)
							−0.00325393	+	0.999995i	\(0.501036\pi\)
\(29\)	−9.62766e7			−0.871629			−0.435814	−	0.900037i	\(-0.643540\pi\)
							−0.435814	+	0.900037i	\(0.643540\pi\)
\(31\)	−1.54727e7			−0.0970684			−0.0485342	−	0.998822i	\(-0.515455\pi\)
							−0.0485342	+	0.998822i	\(0.515455\pi\)
\(37\)	−4.15338e8	−	4.15338e8i	−0.984673	−	0.984673i	0.0152109	−	0.999884i	\(-0.495158\pi\)
							−0.999884	+	0.0152109i	\(0.995158\pi\)
\(41\)			1.11979e9i			1.50948i	0.656024	+	0.754740i	\(0.272237\pi\)
							−0.656024	+	0.754740i	\(0.727763\pi\)
\(43\)	1.03152e9	−	1.03152e9i	1.07004	−	1.07004i	0.0726869	−	0.997355i	\(-0.476843\pi\)
							0.997355	−	0.0726869i	\(-0.0231574\pi\)
\(47\)	1.26480e9	−	1.26480e9i	0.804424	−	0.804424i	−0.179360	−	0.983784i	\(-0.557403\pi\)
							0.983784	+	0.179360i	\(0.0574025\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.12.e.a.2.15	yes	40
3.2	odd	2	inner	15.12.e.a.2.6	✓	40
5.2	odd	4		75.12.e.d.68.15		40
5.3	odd	4	inner	15.12.e.a.8.6	yes	40
5.4	even	2		75.12.e.d.32.6		40
15.2	even	4		75.12.e.d.68.6		40
15.8	even	4	inner	15.12.e.a.8.15	yes	40
15.14	odd	2		75.12.e.d.32.15		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.12.e.a.2.6	✓	40	3.2	odd	2	inner
15.12.e.a.2.15	yes	40	1.1	even	1	trivial
15.12.e.a.8.6	yes	40	5.3	odd	4	inner
15.12.e.a.8.15	yes	40	15.8	even	4	inner
75.12.e.d.32.6		40	5.4	even	2
75.12.e.d.32.15		40	15.14	odd	2
75.12.e.d.68.6		40	15.2	even	4
75.12.e.d.68.15		40	5.2	odd	4