Embedded newform 315.2.cb.a.13.20

• Label: 315.2.cb.a.13.20
• 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			13.20
Character	\(\chi\)	\(=\)	315.13
Dual form			315.2.cb.a.97.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.461159 - 0.123567i) q^{2} +(1.73179 + 0.0298722i) q^{3} +(-1.53465 - 0.886032i) q^{4} +(-1.48035 - 1.67588i) q^{5} +(-0.794941 - 0.227769i) q^{6} +(-2.64575 + 0.00158185i) q^{7} +(1.27342 + 1.27342i) q^{8} +(2.99822 + 0.103465i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 4 q^{2} - 2 q^{7} - 32 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.461159	−	0.123567i	−0.326089	−	0.0873752i	0.0920609	−	0.995753i	\(-0.470655\pi\)
							−0.418150	+	0.908378i	\(0.637321\pi\)
\(3\)	1.73179	+	0.0298722i	0.999851	+	0.0172467i
							\(5\)	−1.48035	−	1.67588i	−0.662030	−	0.749477i
\(7\)	−2.64575	+	0.00158185i	−1.00000	+	0.000597884i
							\(11\)	−1.25923	−	2.18105i	−0.379671	−	0.657610i	0.611343	−	0.791366i	\(-0.290630\pi\)
−0.991014	+	0.133756i	\(0.957296\pi\)
\(13\)	−0.779818	−	2.91032i	−0.216283	−	0.807178i	−0.985711	−	0.168445i	\(-0.946125\pi\)
							0.769428	−	0.638733i	\(-0.220541\pi\)
\(17\)	−3.82405	−	3.82405i	−0.927470	−	0.927470i	0.0700724	−	0.997542i	\(-0.477677\pi\)
							−0.997542	+	0.0700724i	\(0.977677\pi\)
\(19\)	−6.20625			−1.42381			−0.711906	−	0.702275i	\(-0.752168\pi\)
							−0.711906	+	0.702275i	\(0.752168\pi\)
\(23\)	3.58157	−	0.959678i	0.746809	−	0.200107i	0.134707	−	0.990886i	\(-0.456991\pi\)
							0.612102	+	0.790779i	\(0.290324\pi\)
\(29\)	8.36661	−	4.83047i	1.55364	−	0.896995i	0.555800	−	0.831316i	\(-0.312412\pi\)
							0.997841	−	0.0656794i	\(-0.0209214\pi\)
\(31\)	−2.24164	−	1.29421i	−0.402611	−	0.232448i	0.284999	−	0.958528i	\(-0.408007\pi\)
							−0.687610	+	0.726080i	\(0.741340\pi\)
\(37\)	−0.746045	+	0.746045i	−0.122649	+	0.122649i	−0.765767	−	0.643118i	\(-0.777640\pi\)
							0.643118	+	0.765767i	\(0.277640\pi\)
\(41\)	5.53616	+	3.19630i	0.864602	+	0.499178i	0.865551	−	0.500821i	\(-0.166969\pi\)
							−0.000948497		1.00000i	\(0.500302\pi\)
\(43\)	−2.46923	+	9.21529i	−0.376554	+	1.40532i	0.474507	+	0.880252i	\(0.342626\pi\)
							−0.851061	+	0.525067i	\(0.824040\pi\)
\(47\)	1.24571	−	4.64907i	0.181706	−	0.678136i	−0.813606	−	0.581417i	\(-0.802498\pi\)
							0.995312	−	0.0967192i	\(-0.0308349\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.cb.a.13.20	yes	176
3.2	odd	2		945.2.ce.a.118.26		176
5.2	odd	4	inner	315.2.cb.a.202.26	yes	176
7.6	odd	2	inner	315.2.cb.a.13.19	✓	176
9.2	odd	6		945.2.ce.a.748.19		176
9.7	even	3	inner	315.2.cb.a.223.25	yes	176
15.2	even	4		945.2.ce.a.307.20		176
21.20	even	2		945.2.ce.a.118.25		176
35.27	even	4	inner	315.2.cb.a.202.25	yes	176
45.2	even	12		945.2.ce.a.937.25		176
45.7	odd	12	inner	315.2.cb.a.97.19	yes	176
63.20	even	6		945.2.ce.a.748.20		176
63.34	odd	6	inner	315.2.cb.a.223.26	yes	176
105.62	odd	4		945.2.ce.a.307.19		176
315.97	even	12	inner	315.2.cb.a.97.20	yes	176
315.272	odd	12		945.2.ce.a.937.26		176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.cb.a.13.19	✓	176	7.6	odd	2	inner
315.2.cb.a.13.20	yes	176	1.1	even	1	trivial
315.2.cb.a.97.19	yes	176	45.7	odd	12	inner
315.2.cb.a.97.20	yes	176	315.97	even	12	inner
315.2.cb.a.202.25	yes	176	35.27	even	4	inner
315.2.cb.a.202.26	yes	176	5.2	odd	4	inner
315.2.cb.a.223.25	yes	176	9.7	even	3	inner
315.2.cb.a.223.26	yes	176	63.34	odd	6	inner
945.2.ce.a.118.25		176	21.20	even	2
945.2.ce.a.118.26		176	3.2	odd	2
945.2.ce.a.307.19		176	105.62	odd	4
945.2.ce.a.307.20		176	15.2	even	4
945.2.ce.a.748.19		176	9.2	odd	6
945.2.ce.a.748.20		176	63.20	even	6
945.2.ce.a.937.25		176	45.2	even	12
945.2.ce.a.937.26		176	315.272	odd	12