Embedded newform 15.12.e.a.2.19

• Label: 15.12.e.a.2.19
• 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			2.19
Character	\(\chi\)	\(=\)	15.2
Dual form			15.12.e.a.8.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(57.5586 - 57.5586i) q^{2} +(-239.801 - 345.894i) q^{3} -4577.97i q^{4} +(-1295.31 - 6866.61i) q^{5} +(-33711.7 - 6106.58i) q^{6} +(53787.8 + 53787.8i) q^{7} +(-145622. - 145622. i) q^{8} +(-62138.3 + 165891. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 504 q^{3} + 2940 q^{6} + 31504 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	57.5586	−	57.5586i	1.27188	−	1.27188i	0.326774	−	0.945103i	\(-0.394038\pi\)
							0.945103	−	0.326774i	\(-0.105962\pi\)
\(3\)	−239.801	−	345.894i	−0.569749	−	0.821819i
							\(5\)	−1295.31	−	6866.61i	−0.185370	−	0.982669i
\(7\)	53787.8	+	53787.8i	1.20961	+	1.20961i								0.971154	+	0.238452i	\(0.0766401\pi\)
							0.238452	+	0.971154i	\(0.423360\pi\)
\(11\)		−	299004.i		−	0.559779i	−0.960032	−	0.279890i	\(-0.909702\pi\)
							0.960032	−	0.279890i	\(-0.0902979\pi\)
\(13\)	257611.	−	257611.i	0.192432	−	0.192432i	−0.604314	−	0.796746i	\(-0.706553\pi\)
							0.796746	+	0.604314i	\(0.206553\pi\)
\(17\)	−1.91096e6	+	1.91096e6i	−0.326424	+	0.326424i	−0.851225	−	0.524801i	\(-0.824140\pi\)
							0.524801	+	0.851225i	\(0.324140\pi\)
\(19\)		−	1.22718e7i		−	1.13701i	−0.822680	−	0.568505i	\(-0.807522\pi\)
							0.822680	−	0.568505i	\(-0.192478\pi\)
\(23\)	16960.0	+	16960.0i	0.000549442	+	0.000549442i	0.707381	−	0.706832i	\(-0.249876\pi\)
							−0.706832	+	0.707381i	\(0.749876\pi\)
\(29\)	8.47515e7			0.767288			0.383644	−	0.923481i	\(-0.374669\pi\)
							0.383644	+	0.923481i	\(0.374669\pi\)
\(31\)	5.68233e7			0.356481			0.178241	−	0.983987i	\(-0.442959\pi\)
							0.178241	+	0.983987i	\(0.442959\pi\)
\(37\)	−2.28933e8	−	2.28933e8i	−0.542748	−	0.542748i	0.381586	−	0.924334i	\(-0.375378\pi\)
							−0.924334	+	0.381586i	\(0.875378\pi\)
\(41\)			4.14106e8i			0.558214i	0.960260	+	0.279107i	\(0.0900384\pi\)
							−0.960260	+	0.279107i	\(0.909962\pi\)
\(43\)	5.52861e8	−	5.52861e8i	0.573508	−	0.573508i	−0.359599	−	0.933107i	\(-0.617086\pi\)
							0.933107	+	0.359599i	\(0.117086\pi\)
\(47\)	3.42474e8	−	3.42474e8i	0.217816	−	0.217816i	−0.589762	−	0.807577i	\(-0.700778\pi\)
							0.807577	+	0.589762i	\(0.200778\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.12.e.a.2.19	yes	40
3.2	odd	2	inner	15.12.e.a.2.2	✓	40
5.2	odd	4		75.12.e.d.68.19		40
5.3	odd	4	inner	15.12.e.a.8.2	yes	40
5.4	even	2		75.12.e.d.32.2		40
15.2	even	4		75.12.e.d.68.2		40
15.8	even	4	inner	15.12.e.a.8.19	yes	40
15.14	odd	2		75.12.e.d.32.19		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.12.e.a.2.2	✓	40	3.2	odd	2	inner
15.12.e.a.2.19	yes	40	1.1	even	1	trivial
15.12.e.a.8.2	yes	40	5.3	odd	4	inner
15.12.e.a.8.19	yes	40	15.8	even	4	inner
75.12.e.d.32.2		40	5.4	even	2
75.12.e.d.32.19		40	15.14	odd	2
75.12.e.d.68.2		40	15.2	even	4
75.12.e.d.68.19		40	5.2	odd	4