Embedded newform 308.2.n.c.219.15

• Label: 308.2.n.c.219.15
• Level: $308$
• Weight: $2$
• Character: 308.219
• Analytic conductor: $2.459$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	308.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.45939238226\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(40\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			219.15
Character	\(\chi\)	\(=\)	308.219
Dual form			308.2.n.c.263.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.533373 - 1.30978i) q^{2} +(-0.581000 + 0.335441i) q^{3} +(-1.43103 + 1.39720i) q^{4} +(-0.251848 + 0.436213i) q^{5} +(0.749242 + 0.582066i) q^{6} +(1.57544 - 2.12556i) q^{7} +(2.59329 + 1.12910i) q^{8} +(-1.27496 + 2.20829i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 6 q^{4} - 4 q^{5} + 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/308\mathbb{Z}\right)^\times\).

\(n\)	\(45\)	\(57\)	\(155\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.533373	−	1.30978i	−0.377151	−	0.926152i
							\(3\)	−0.581000	+	0.335441i	−0.335441	+	0.193667i	−0.658254	−	0.752796i	\(-0.728705\pi\)
0.322813	+	0.946463i	\(0.395371\pi\)
\(5\)	−0.251848	+	0.436213i	−0.112630	+	0.195080i	−0.916830	−	0.399278i	\(-0.869261\pi\)
							0.804200	+	0.594359i	\(0.202594\pi\)
\(7\)	1.57544	−	2.12556i	0.595459	−	0.803385i
							\(11\)	3.29805	−	0.350529i	0.994399	−	0.105689i
\(13\)			2.90055i			0.804468i								0.915537	+	0.402234i	\(0.131766\pi\)
							−0.915537	+	0.402234i	\(0.868234\pi\)
\(17\)	5.10761	−	2.94888i	1.23878	−	0.715209i	0.269934	−	0.962879i	\(-0.412998\pi\)
							0.968844	+	0.247670i	\(0.0796648\pi\)
\(19\)	2.84959	−	4.93563i	0.653741	−	1.13231i	−0.328467	−	0.944515i	\(-0.606532\pi\)
							0.982208	−	0.187797i	\(-0.0601346\pi\)
\(23\)	2.31877	+	1.33874i	0.483497	+	0.279147i	0.721873	−	0.692026i	\(-0.243282\pi\)
							−0.238376	+	0.971173i	\(0.576615\pi\)
\(29\)		−	3.72196i		−	0.691150i	−0.938391	−	0.345575i	\(-0.887684\pi\)
							0.938391	−	0.345575i	\(-0.112316\pi\)
\(31\)	1.43943	−	0.831056i	0.258529	−	0.149262i	−0.365134	−	0.930955i	\(-0.618977\pi\)
							0.623664	+	0.781693i	\(0.285644\pi\)
\(37\)	−1.57751	+	2.73233i	−0.259341	+	0.449192i	−0.966066	−	0.258297i	\(-0.916839\pi\)
							0.706724	+	0.707489i	\(0.250172\pi\)
\(41\)			4.54779i			0.710245i	0.934820	+	0.355123i	\(0.115561\pi\)
							−0.934820	+	0.355123i	\(0.884439\pi\)
\(43\)	−9.19460			−1.40216			−0.701082	−	0.713081i	\(-0.747299\pi\)
							−0.701082	+	0.713081i	\(0.747299\pi\)
\(47\)	6.64561	+	3.83684i	0.969362	+	0.559661i	0.899042	−	0.437863i	\(-0.144265\pi\)
							0.0703202	+	0.997524i	\(0.477598\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	308.2.n.c.219.15	yes	80
4.3	odd	2	inner	308.2.n.c.219.2	✓	80
7.4	even	3	inner	308.2.n.c.263.39	yes	80
11.10	odd	2	inner	308.2.n.c.219.26	yes	80
28.11	odd	6	inner	308.2.n.c.263.26	yes	80
44.43	even	2	inner	308.2.n.c.219.39	yes	80
77.32	odd	6	inner	308.2.n.c.263.2	yes	80
308.263	even	6	inner	308.2.n.c.263.15	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
308.2.n.c.219.2	✓	80	4.3	odd	2	inner
308.2.n.c.219.15	yes	80	1.1	even	1	trivial
308.2.n.c.219.26	yes	80	11.10	odd	2	inner
308.2.n.c.219.39	yes	80	44.43	even	2	inner
308.2.n.c.263.2	yes	80	77.32	odd	6	inner
308.2.n.c.263.15	yes	80	308.263	even	6	inner
308.2.n.c.263.26	yes	80	28.11	odd	6	inner
308.2.n.c.263.39	yes	80	7.4	even	3	inner