Embedded newform 353.2.f.a.36.5

• Label: 353.2.f.a.36.5
• Level: $353$
• Weight: $2$
• Character: 353.36
• Analytic conductor: $2.819$
• Analytic rank: $0$
• Dimension: $232$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 353 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	353.f (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.81871919135\)
Analytic rank:	\(0\)
Dimension:	\(232\)
Relative dimension:	\(29\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			36.5
Character	\(\chi\)	\(=\)	353.36
Dual form			353.2.f.a.304.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.54451 + 1.54451i) q^{2} +(-1.53117 + 0.304569i) q^{3} -2.77104i q^{4} +(2.16346 - 3.23785i) q^{5} +(1.89451 - 2.83533i) q^{6} +(-0.326993 + 0.218490i) q^{7} +(1.19088 + 1.19088i) q^{8} +(-0.519907 + 0.215353i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 232 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} + 8 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/353\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{15}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.54451	+	1.54451i	−1.09214	+	1.09214i	−0.0968349	+	0.995300i	\(0.530872\pi\)
							−0.995300	+	0.0968349i	\(0.969128\pi\)
\(3\)	−1.53117	+	0.304569i	−0.884024	+	0.175843i	−0.616167	−	0.787615i	\(-0.711315\pi\)
							−0.267857	+	0.963459i	\(0.586315\pi\)
\(5\)	2.16346	−	3.23785i	0.967529	−	1.44801i	0.0749132	−	0.997190i	\(-0.476132\pi\)
							0.892615	−	0.450819i	\(-0.148868\pi\)
\(7\)	−0.326993	+	0.218490i	−0.123592	+	0.0825814i	−0.615830	−	0.787879i	\(-0.711179\pi\)
							0.492238	+	0.870461i	\(0.336179\pi\)
\(11\)	0.677469	+	0.677469i	0.204265	+	0.204265i	0.801824	−	0.597560i	\(-0.203863\pi\)
							−0.597560	+	0.801824i	\(0.703863\pi\)
\(13\)	−5.17912	+	1.03019i	−1.43643	+	0.285724i	−0.851071	−	0.525050i	\(-0.824047\pi\)
							−0.585359	+	0.810774i	\(0.699047\pi\)
\(17\)	−2.83100	+	2.83100i	−0.686619	+	0.686619i	−0.961483	−	0.274864i	\(-0.911367\pi\)
							0.274864	+	0.961483i	\(0.411367\pi\)
\(19\)	−2.11326	+	5.10186i	−0.484815	+	1.17045i	0.472482	+	0.881340i	\(0.343358\pi\)
							−0.957297	+	0.289106i	\(0.906642\pi\)
\(23\)	−2.39132	+	5.77317i	−0.498626	+	1.20379i	0.451598	+	0.892221i	\(0.350854\pi\)
							−0.950224	+	0.311568i	\(0.899146\pi\)
\(29\)	4.87562	+	4.87562i	0.905381	+	0.905381i	0.995895	−	0.0905146i	\(-0.0288512\pi\)
							−0.0905146	+	0.995895i	\(0.528851\pi\)
\(31\)	−2.59807	−	1.73598i	−0.466628	−	0.311791i	0.299936	−	0.953959i	\(-0.403035\pi\)
							−0.766563	+	0.642169i	\(0.778035\pi\)
\(37\)	−5.64156	−	8.44319i	−0.927467	−	1.38805i	−0.921630	−	0.388070i	\(-0.873142\pi\)
							−0.00583713	−	0.999983i	\(-0.501858\pi\)
\(41\)	−3.25587	−	1.34863i	−0.508482	−	0.210620i	0.113667	−	0.993519i	\(-0.463740\pi\)
							−0.622149	+	0.782899i	\(0.713740\pi\)
\(43\)	0.652205	−	1.57456i	0.0994604	−	0.240119i	−0.866315	−	0.499498i	\(-0.833518\pi\)
							0.965776	+	0.259379i	\(0.0835179\pi\)
\(47\)	2.60108	+	6.27957i	0.379407	+	0.915969i	0.992077	+	0.125630i	\(0.0400953\pi\)
							−0.612670	+	0.790339i	\(0.709905\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	353.2.f.a.36.5	✓	232
353.304	even	16	inner	353.2.f.a.304.5	yes	232

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
353.2.f.a.36.5	✓	232	1.1	even	1	trivial
353.2.f.a.304.5	yes	232	353.304	even	16	inner