Embedded newform 115.5.i.a.14.15

• Label: 115.5.i.a.14.15
• Level: $115$
• Weight: $5$
• Character: 115.14
• Analytic conductor: $11.888$
• Analytic rank: $0$
• Dimension: $460$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	115.i (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8875457546\)
Analytic rank:	\(0\)
Dimension:	\(460\)
Relative dimension:	\(46\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			14.15
Character	\(\chi\)	\(=\)	115.14
Dual form			115.5.i.a.74.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.944999 - 3.21837i) q^{2} +(6.93728 - 6.01119i) q^{3} +(3.99518 - 2.56754i) q^{4} +(17.7521 + 17.6030i) q^{5} +(-25.9019 - 16.6462i) q^{6} +(19.4786 - 42.6523i) q^{7} +(-52.5982 - 45.5766i) q^{8} +(0.463978 - 3.22704i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 460 q + 306 q^{4} - 11 q^{5} - 182 q^{6} + 944 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{21}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.944999	−	3.21837i	−0.236250	−	0.804592i	−0.989207	−	0.146523i	\(-0.953192\pi\)
							0.752958	−	0.658069i	\(-0.228627\pi\)
\(3\)	6.93728	−	6.01119i	0.770809	−	0.667910i	−0.177904	−	0.984048i	\(-0.556932\pi\)
							0.948713	+	0.316138i	\(0.102386\pi\)
\(5\)	17.7521	+	17.6030i	0.710083	+	0.704118i
							\(7\)	19.4786	−	42.6523i	0.397523	−	0.870455i	−0.599992	−	0.800006i	\(-0.704830\pi\)
0.997515	−	0.0704487i	\(-0.0224431\pi\)
\(11\)	−9.95524	+	33.9044i	−0.0822747	+	0.280202i	−0.990347	−	0.138612i	\(-0.955736\pi\)
							0.908072	+	0.418814i	\(0.137554\pi\)
\(13\)	277.513	−	126.736i	1.64209	−	0.749917i	0.642227	−	0.766514i	\(-0.278011\pi\)
							0.999862	+	0.0165971i	\(0.00528328\pi\)
\(17\)	−27.8798	−	17.9172i	−0.0964697	−	0.0619973i	0.491517	−	0.870868i	\(-0.336443\pi\)
							−0.587986	+	0.808871i	\(0.700079\pi\)
\(19\)	12.0707	+	18.7824i	0.0334368	+	0.0520287i	0.857570	−	0.514368i	\(-0.171973\pi\)
							−0.824133	+	0.566397i	\(0.808337\pi\)
\(23\)	−381.883	−	366.069i	−0.721896	−	0.692002i
							\(29\)	−877.150	−	563.710i	−1.04298	−	0.670285i	−0.0972611	−	0.995259i	\(-0.531008\pi\)
−0.945723	+	0.324974i	\(0.894645\pi\)
\(31\)	−594.808	+	686.445i	−0.618947	+	0.714303i	−0.975507	−	0.219969i	\(-0.929404\pi\)
							0.356559	+	0.934273i	\(0.383950\pi\)
\(37\)	200.047	−	1391.36i	0.146127	−	1.01633i	−0.776356	−	0.630294i	\(-0.782934\pi\)
							0.922483	−	0.386038i	\(-0.126157\pi\)
\(41\)	355.409	+	2471.92i	0.211427	+	1.47051i	0.768397	+	0.639974i	\(0.221055\pi\)
							−0.556970	+	0.830533i	\(0.688036\pi\)
\(43\)	88.6910	+	102.355i	0.0479670	+	0.0553569i	0.779226	−	0.626743i	\(-0.215612\pi\)
							−0.731259	+	0.682100i	\(0.761067\pi\)
\(47\)		−	898.358i		−	0.406681i	−0.979108	−	0.203340i	\(-0.934820\pi\)
							0.979108	−	0.203340i	\(-0.0651798\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.5.i.a.14.15	✓	460
5.4	even	2	inner	115.5.i.a.14.32	yes	460
23.5	odd	22	inner	115.5.i.a.74.32	yes	460
115.74	odd	22	inner	115.5.i.a.74.15	yes	460

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.5.i.a.14.15	✓	460	1.1	even	1	trivial
115.5.i.a.14.32	yes	460	5.4	even	2	inner
115.5.i.a.74.15	yes	460	115.74	odd	22	inner
115.5.i.a.74.32	yes	460	23.5	odd	22	inner