Embedded newform 315.2.cb.a.13.4

• Label: 315.2.cb.a.13.4
• Level: $315$
• Weight: $2$
• Character: 315.13
• Analytic conductor: $2.515$
• Analytic rank: $0$
• Dimension: $176$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.cb (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.51528766367\)
Analytic rank:	\(0\)
Dimension:	\(176\)
Relative dimension:	\(44\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			13.4
Character	\(\chi\)	\(=\)	315.13
Dual form			315.2.cb.a.97.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.51063 - 0.672721i) q^{2} +(1.39463 + 1.02713i) q^{3} +(4.11865 + 2.37791i) q^{4} +(-0.00805112 + 2.23605i) q^{5} +(-2.81044 - 3.51694i) q^{6} +(2.11812 + 1.58542i) q^{7} +(-5.06493 - 5.06493i) q^{8} +(0.890010 + 2.86494i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 4 q^{2} - 2 q^{7} - 32 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(136\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.51063	−	0.672721i	−1.77528	−	0.475686i	−0.785572	−	0.618770i	\(-0.787631\pi\)
							−0.989710	+	0.143084i	\(0.954298\pi\)
\(3\)	1.39463	+	1.02713i	0.805193	+	0.593013i
							\(5\)	−0.00805112	+	2.23605i	−0.00360057	+	0.999994i
\(7\)	2.11812	+	1.58542i	0.800574	+	0.599234i
							\(11\)	1.91949	+	3.32465i	0.578747	+	1.00242i	0.995623	+	0.0934560i	\(0.0297915\pi\)
−0.416876	+	0.908963i	\(0.636875\pi\)
\(13\)	−0.479455	−	1.78935i	−0.132977	−	0.496277i	0.867021	−	0.498271i	\(-0.166032\pi\)
							−0.999998	+	0.00199480i	\(0.999365\pi\)
\(17\)	−4.95480	−	4.95480i	−1.20172	−	1.20172i	−0.973645	−	0.228071i	\(-0.926758\pi\)
							−0.228071	−	0.973645i	\(-0.573242\pi\)
\(19\)	−2.83221			−0.649754			−0.324877	−	0.945756i	\(-0.605323\pi\)
							−0.324877	+	0.945756i	\(0.605323\pi\)
\(23\)	−1.72515	+	0.462252i	−0.359718	+	0.0963862i	−0.434152	−	0.900840i	\(-0.642952\pi\)
							0.0744336	+	0.997226i	\(0.476285\pi\)
\(29\)	7.98382	−	4.60946i	1.48256	−	0.855955i	0.482754	−	0.875756i	\(-0.339636\pi\)
							0.999804	+	0.0198004i	\(0.00630306\pi\)
\(31\)	1.77653	+	1.02568i	0.319075	+	0.184218i	0.650980	−	0.759095i	\(-0.274358\pi\)
							−0.331905	+	0.943313i	\(0.607691\pi\)
\(37\)	2.06919	−	2.06919i	0.340172	−	0.340172i	−0.516260	−	0.856432i	\(-0.672676\pi\)
							0.856432	+	0.516260i	\(0.172676\pi\)
\(41\)	4.47513	+	2.58372i	0.698899	+	0.403509i	0.806937	−	0.590637i	\(-0.201124\pi\)
							−0.108038	+	0.994147i	\(0.534457\pi\)
\(43\)	−0.491018	+	1.83250i	−0.0748795	+	0.279454i	−0.993206	−	0.116370i	\(-0.962874\pi\)
							0.918326	+	0.395824i	\(0.129541\pi\)
\(47\)	−0.378577	+	1.41287i	−0.0552211	+	0.206088i	−0.988024	−	0.154298i	\(-0.950688\pi\)
							0.932803	+	0.360386i	\(0.117355\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.cb.a.13.4	yes	176
3.2	odd	2		945.2.ce.a.118.42		176
5.2	odd	4	inner	315.2.cb.a.202.42	yes	176
7.6	odd	2	inner	315.2.cb.a.13.3	✓	176
9.2	odd	6		945.2.ce.a.748.4		176
9.7	even	3	inner	315.2.cb.a.223.41	yes	176
15.2	even	4		945.2.ce.a.307.3		176
21.20	even	2		945.2.ce.a.118.41		176
35.27	even	4	inner	315.2.cb.a.202.41	yes	176
45.2	even	12		945.2.ce.a.937.41		176
45.7	odd	12	inner	315.2.cb.a.97.3	yes	176
63.20	even	6		945.2.ce.a.748.3		176
63.34	odd	6	inner	315.2.cb.a.223.42	yes	176
105.62	odd	4		945.2.ce.a.307.4		176
315.97	even	12	inner	315.2.cb.a.97.4	yes	176
315.272	odd	12		945.2.ce.a.937.42		176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.cb.a.13.3	✓	176	7.6	odd	2	inner
315.2.cb.a.13.4	yes	176	1.1	even	1	trivial
315.2.cb.a.97.3	yes	176	45.7	odd	12	inner
315.2.cb.a.97.4	yes	176	315.97	even	12	inner
315.2.cb.a.202.41	yes	176	35.27	even	4	inner
315.2.cb.a.202.42	yes	176	5.2	odd	4	inner
315.2.cb.a.223.41	yes	176	9.7	even	3	inner
315.2.cb.a.223.42	yes	176	63.34	odd	6	inner
945.2.ce.a.118.41		176	21.20	even	2
945.2.ce.a.118.42		176	3.2	odd	2
945.2.ce.a.307.3		176	15.2	even	4
945.2.ce.a.307.4		176	105.62	odd	4
945.2.ce.a.748.3		176	63.20	even	6
945.2.ce.a.748.4		176	9.2	odd	6
945.2.ce.a.937.41		176	45.2	even	12
945.2.ce.a.937.42		176	315.272	odd	12