Embedded newform 91.10.u.a.30.3

• Label: 91.10.u.a.30.3
• 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			30.3
Character	\(\chi\)	\(=\)	91.30
Dual form			91.10.u.a.88.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-37.1580 + 21.4532i) q^{2} +7.72989 q^{3} +(664.476 - 1150.91i) q^{4} +(-1575.27 - 909.485i) q^{5} +(-287.227 + 165.831i) q^{6} +(1841.00 + 6079.83i) q^{7} +35052.4i q^{8} -19623.2 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 164 q - 3 q^{2} + 322 q^{3} + 20481 q^{4} - 6 q^{6} + 1023 q^{7} + 1021818 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−37.1580	+	21.4532i	−1.64217	+	0.948105i	−0.662104	+	0.749412i	\(0.730336\pi\)
							−0.980062	+	0.198693i	\(0.936330\pi\)
\(3\)	7.72989			0.0550970			0.0275485	−	0.999620i	\(-0.491230\pi\)
							0.0275485	+	0.999620i	\(0.491230\pi\)
\(5\)	−1575.27	−	909.485i	−1.12717	−	0.650774i	−0.183951	−	0.982935i	\(-0.558889\pi\)
							−0.943223	+	0.332161i	\(0.892222\pi\)
\(7\)	1841.00	+	6079.83i	0.289810	+	0.957084i
							\(11\)			26968.5i			0.555379i	0.960671	+	0.277689i	\(0.0895686\pi\)
−0.960671	+	0.277689i	\(0.910431\pi\)
\(13\)	−21210.2	−	100770.i	−0.205968	−	0.978559i
							\(17\)	296912.	−	514267.i	0.862199	−	1.49337i	−0.00760258	−	0.999971i	\(-0.502420\pi\)
0.869802	−	0.493402i	\(-0.164247\pi\)
\(19\)			518691.i			0.913099i	0.889698	+	0.456549i	\(0.150915\pi\)
							−0.889698	+	0.456549i	\(0.849085\pi\)
\(23\)	−903512.	−	1.56493e6i	−0.673223	−	1.16606i	−0.976985	−	0.213308i	\(-0.931576\pi\)
							0.303762	−	0.952748i	\(-0.401757\pi\)
\(29\)	1.13858e6	−	1.97207e6i	0.298931	−	0.517764i	−0.676960	−	0.736019i	\(-0.736703\pi\)
							0.975892	+	0.218255i	\(0.0700365\pi\)
\(31\)	2.87313e6	−	1.65880e6i	0.558763	−	0.322602i	−0.193886	−	0.981024i	\(-0.562109\pi\)
							0.752649	+	0.658422i	\(0.228776\pi\)
\(37\)	1.30120e7	−	7.51248e6i	1.14140	−	0.658985i	0.194620	−	0.980879i	\(-0.437653\pi\)
							0.946776	+	0.321894i	\(0.104319\pi\)
\(41\)	−1.32457e7	−	7.64743e6i	−0.732064	−	0.422657i	0.0871131	−	0.996198i	\(-0.472236\pi\)
							−0.819177	+	0.573541i	\(0.805569\pi\)
\(43\)	1.12094e7	+	1.94153e7i	0.500007	+	0.866038i	1.00000		8.28016e-6i	\(2.63566e-6\pi\)
							−0.499993	+	0.866030i	\(0.666664\pi\)
\(47\)	−3.29827e7	−	1.90425e7i	−0.985928	−	0.569226i	−0.0818736	−	0.996643i	\(-0.526090\pi\)
							−0.904055	+	0.427417i	\(0.859424\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.u.a.30.3	yes	164
7.4	even	3		91.10.k.a.4.3	✓	164
13.10	even	6		91.10.k.a.23.80	yes	164
91.88	even	6	inner	91.10.u.a.88.3	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.k.a.4.3	✓	164	7.4	even	3
91.10.k.a.23.80	yes	164	13.10	even	6
91.10.u.a.30.3	yes	164	1.1	even	1	trivial
91.10.u.a.88.3	yes	164	91.88	even	6	inner