Embedded newform 91.10.u.a.30.6

• Label: 91.10.u.a.30.6
• Level: $91$
• Weight: $10$
• Character: 91.30
• Analytic conductor: $46.868$
• Analytic rank: $0$
• Dimension: $164$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	91.u (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(46.8682610909\)
Analytic rank:	\(0\)
Dimension:	\(164\)
Relative dimension:	\(82\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			30.6
Character	\(\chi\)	\(=\)	91.30
Dual form			91.10.u.a.88.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-35.0393 + 20.2299i) q^{2} +269.124 q^{3} +(562.501 - 974.280i) q^{4} +(-63.4069 - 36.6080i) q^{5} +(-9429.90 + 5444.36i) q^{6} +(5725.53 + 2751.71i) q^{7} +24802.0i q^{8} +52744.6 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 164 q - 3 q^{2} + 322 q^{3} + 20481 q^{4} - 6 q^{6} + 1023 q^{7} + 1021818 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−35.0393	+	20.2299i	−1.54853	+	0.894045i	−0.550278	+	0.834982i	\(0.685478\pi\)
							−0.998254	+	0.0590636i	\(0.981189\pi\)
\(3\)	269.124			1.91825			0.959127	−	0.282974i	\(-0.0913212\pi\)
							0.959127	+	0.282974i	\(0.0913212\pi\)
\(5\)	−63.4069	−	36.6080i	−0.0453703	−	0.0261946i	0.477143	−	0.878826i	\(-0.341672\pi\)
							−0.522514	+	0.852631i	\(0.675006\pi\)
\(7\)	5725.53	+	2751.71i	0.901311	+	0.433173i
							\(11\)			14608.9i			0.300850i	0.988621	+	0.150425i	\(0.0480642\pi\)
−0.988621	+	0.150425i	\(0.951936\pi\)
\(13\)	101990.	+	14233.7i	0.990402	+	0.138220i
							\(17\)	−173258.	+	300091.i	−0.503121	+	0.871431i	0.496872	+	0.867824i	\(0.334482\pi\)
−0.999993	+	0.00360780i	\(0.998852\pi\)
\(19\)			784200.i			1.38050i	0.723573	+	0.690248i	\(0.242499\pi\)
							−0.723573	+	0.690248i	\(0.757501\pi\)
\(23\)	−570795.	−	988646.i	−0.425309	−	0.736658i	0.571140	−	0.820853i	\(-0.306501\pi\)
							−0.996449	+	0.0841952i	\(0.973168\pi\)
\(29\)	−1.04968e6	+	1.81810e6i	−0.275592	+	0.477340i	−0.970284	−	0.241967i	\(-0.922207\pi\)
							0.694692	+	0.719307i	\(0.255541\pi\)
\(31\)	−3.14632e6	+	1.81653e6i	−0.611893	+	0.353276i	−0.773706	−	0.633545i	\(-0.781599\pi\)
							0.161813	+	0.986821i	\(0.448266\pi\)
\(37\)	9.38233e6	−	5.41689e6i	0.823006	−	0.475163i	−0.0284459	−	0.999595i	\(-0.509056\pi\)
							0.851452	+	0.524433i	\(0.175722\pi\)
\(41\)	2.15267e7	+	1.24285e7i	1.18974	+	0.686894i	0.958247	−	0.285943i	\(-0.0923069\pi\)
							0.231489	+	0.972837i	\(0.425640\pi\)
\(43\)	−2.94056e6	−	5.09320e6i	−0.131166	−	0.227186i	0.792960	−	0.609273i	\(-0.208539\pi\)
							−0.924126	+	0.382087i	\(0.875205\pi\)
\(47\)	2.91842e7	+	1.68495e7i	0.872383	+	0.503671i	0.868139	−	0.496320i	\(-0.165316\pi\)
							0.00424366	+	0.999991i	\(0.498649\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.u.a.30.6	yes	164
7.4	even	3		91.10.k.a.4.6	✓	164
13.10	even	6		91.10.k.a.23.77	yes	164
91.88	even	6	inner	91.10.u.a.88.6	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.k.a.4.6	✓	164	7.4	even	3
91.10.k.a.23.77	yes	164	13.10	even	6
91.10.u.a.30.6	yes	164	1.1	even	1	trivial
91.10.u.a.88.6	yes	164	91.88	even	6	inner