Embedded newform 91.10.i.a.34.12

• Label: 91.10.i.a.34.12
• Level: $91$
• Weight: $10$
• Character: 91.34
• Analytic conductor: $46.868$
• Analytic rank: $0$
• Dimension: $164$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	91.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(46.8682610909\)
Analytic rank:	\(0\)
Dimension:	\(164\)
Relative dimension:	\(82\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			34.12
Character	\(\chi\)	\(=\)	91.34
Dual form			91.10.i.a.83.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-25.7334 - 25.7334i) q^{2} +103.954i q^{3} +812.416i q^{4} +(-1902.22 + 1902.22i) q^{5} +(2675.09 - 2675.09i) q^{6} +(4287.59 - 4687.23i) q^{7} +(7730.72 - 7730.72i) q^{8} +8876.58 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 164 q - 4 q^{2} - 1142 q^{7} + 2044 q^{8} - 1023524 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−25.7334	−	25.7334i	−1.13727	−	1.13727i	−0.988938	−	0.148328i	\(-0.952611\pi\)
							−0.148328	−	0.988938i	\(-0.547389\pi\)
\(3\)			103.954i			0.740961i	0.928840	+	0.370480i	\(0.120807\pi\)
							−0.928840	+	0.370480i	\(0.879193\pi\)
\(5\)	−1902.22	+	1902.22i	−1.36111	+	1.36111i	−0.488615	+	0.872499i	\(0.662498\pi\)
							−0.872499	+	0.488615i	\(0.837502\pi\)
\(7\)	4287.59	−	4687.23i	0.674951	−	0.737862i
							\(11\)	−20047.5	+	20047.5i	−0.412852	+	0.412852i	−0.882731	−	0.469879i	\(-0.844298\pi\)
0.469879	+	0.882731i	\(0.344298\pi\)
\(13\)	−70990.7	−	74597.7i	−0.689376	−	0.724404i
							\(17\)	−422804.			−1.22778			−0.613888	−	0.789393i	\(-0.710395\pi\)
−0.613888	+	0.789393i	\(0.710395\pi\)
\(19\)	378822.	−	378822.i	0.666874	−	0.666874i	−0.290117	−	0.956991i	\(-0.593694\pi\)
							0.956991	+	0.290117i	\(0.0936943\pi\)
\(23\)		−	223737.i		−	0.166710i	−0.996520	−	0.0833551i	\(-0.973436\pi\)
							0.996520	−	0.0833551i	\(-0.0265636\pi\)
\(29\)	−4.28981e6			−1.12628			−0.563141	−	0.826361i	\(-0.690407\pi\)
							−0.563141	+	0.826361i	\(0.690407\pi\)
\(31\)	−3.07836e6	+	3.07836e6i	−0.598675	+	0.598675i	−0.939960	−	0.341285i	\(-0.889138\pi\)
							0.341285	+	0.939960i	\(0.389138\pi\)
\(37\)	1.44645e7	−	1.44645e7i	1.26881	−	1.26881i	0.322105	−	0.946704i	\(-0.395610\pi\)
							0.946704	−	0.322105i	\(-0.104390\pi\)
\(41\)	5.72077e6	−	5.72077e6i	0.316175	−	0.316175i	−0.531121	−	0.847296i	\(-0.678229\pi\)
							0.847296	+	0.531121i	\(0.178229\pi\)
\(43\)			1.73547e7i			0.774121i	0.922054	+	0.387060i	\(0.126510\pi\)
							−0.922054	+	0.387060i	\(0.873490\pi\)
\(47\)	2.30456e7	+	2.30456e7i	0.688887	+	0.688887i	0.961986	−	0.273099i	\(-0.0880486\pi\)
							−0.273099	+	0.961986i	\(0.588049\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.i.a.34.12	yes	164
7.6	odd	2	inner	91.10.i.a.34.11	✓	164
13.5	odd	4	inner	91.10.i.a.83.12	yes	164
91.83	even	4	inner	91.10.i.a.83.11	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.i.a.34.11	✓	164	7.6	odd	2	inner
91.10.i.a.34.12	yes	164	1.1	even	1	trivial
91.10.i.a.83.11	yes	164	91.83	even	4	inner
91.10.i.a.83.12	yes	164	13.5	odd	4	inner