Embedded newform 15.8.e.a.8.1

• Label: 15.8.e.a.8.1
• Level: $15$
• Weight: $8$
• Character: 15.8
• Analytic conductor: $4.686$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 15 = 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	15.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.68577538226\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			8.1
Character	\(\chi\)	\(=\)	15.8
Dual form			15.8.e.a.2.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-14.4708 - 14.4708i) q^{2} +(-45.2107 + 11.9578i) q^{3} +290.808i q^{4} +(-140.740 - 241.490i) q^{5} +(827.274 + 481.197i) q^{6} +(196.040 - 196.040i) q^{7} +(2355.96 - 2355.96i) q^{8} +(1901.02 - 1081.24i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{3} + 924 q^{6} + 1344 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(7\)	\(11\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−14.4708	−	14.4708i	−1.27905	−	1.27905i	−0.941200	−	0.337850i	\(-0.890300\pi\)
							−0.337850	−	0.941200i	\(-0.609700\pi\)
\(3\)	−45.2107	+	11.9578i	−0.966757	+	0.255697i
							\(5\)	−140.740	−	241.490i	−0.503527	−	0.863980i
\(7\)	196.040	−	196.040i	0.216024	−	0.216024i								−0.590797	−	0.806821i	\(-0.701186\pi\)
							0.806821	+	0.590797i	\(0.201186\pi\)
\(11\)			7416.50i			1.68006i	0.542540	+	0.840030i	\(0.317463\pi\)
							−0.542540	+	0.840030i	\(0.682537\pi\)
\(13\)	−8.05172	−	8.05172i	−0.00101645	−	0.00101645i	0.706598	−	0.707615i	\(-0.250229\pi\)
							−0.707615	+	0.706598i	\(0.750229\pi\)
\(17\)	5935.29	+	5935.29i	0.293002	+	0.293002i	0.838265	−	0.545263i	\(-0.183570\pi\)
							−0.545263	+	0.838265i	\(0.683570\pi\)
\(19\)			8611.20i			0.288022i	0.989576	+	0.144011i	\(0.0460001\pi\)
							−0.989576	+	0.144011i	\(0.954000\pi\)
\(23\)	−12814.5	+	12814.5i	−0.219611	+	0.219611i	−0.808335	−	0.588723i	\(-0.799631\pi\)
							0.588723	+	0.808335i	\(0.299631\pi\)
\(29\)	75559.3			0.575301			0.287650	−	0.957735i	\(-0.407126\pi\)
							0.287650	+	0.957735i	\(0.407126\pi\)
\(31\)	88519.2			0.533669			0.266834	−	0.963742i	\(-0.414022\pi\)
							0.266834	+	0.963742i	\(0.414022\pi\)
\(37\)	−227492.	+	227492.i	−0.738346	+	0.738346i	−0.972258	−	0.233912i	\(-0.924847\pi\)
							0.233912	+	0.972258i	\(0.424847\pi\)
\(41\)		−	362186.i		−	0.820706i	−0.911927	−	0.410353i	\(-0.865406\pi\)
							0.911927	−	0.410353i	\(-0.134594\pi\)
\(43\)	354357.	+	354357.i	0.679676	+	0.679676i	0.959927	−	0.280251i	\(-0.0904177\pi\)
							−0.280251	+	0.959927i	\(0.590418\pi\)
\(47\)	818248.	+	818248.i	1.14959	+	1.14959i	0.986634	+	0.162954i	\(0.0521023\pi\)
							0.162954	+	0.986634i	\(0.447898\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.8.e.a.8.1	yes	24
3.2	odd	2	inner	15.8.e.a.8.12	yes	24
5.2	odd	4	inner	15.8.e.a.2.12	yes	24
5.3	odd	4		75.8.e.d.32.1		24
5.4	even	2		75.8.e.d.68.12		24
15.2	even	4	inner	15.8.e.a.2.1	✓	24
15.8	even	4		75.8.e.d.32.12		24
15.14	odd	2		75.8.e.d.68.1		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.8.e.a.2.1	✓	24	15.2	even	4	inner
15.8.e.a.2.12	yes	24	5.2	odd	4	inner
15.8.e.a.8.1	yes	24	1.1	even	1	trivial
15.8.e.a.8.12	yes	24	3.2	odd	2	inner
75.8.e.d.32.1		24	5.3	odd	4
75.8.e.d.32.12		24	15.8	even	4
75.8.e.d.68.1		24	15.14	odd	2
75.8.e.d.68.12		24	5.4	even	2