Embedded newform 91.10.q.a.36.1

• Label: 91.10.q.a.36.1
• Level: $91$
• Weight: $10$
• Character: 91.36
• Analytic conductor: $46.868$
• Analytic rank: $0$
• Dimension: $124$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(46.8682610909\)
Analytic rank:	\(0\)
Dimension:	\(124\)
Relative dimension:	\(62\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			36.1
Character	\(\chi\)	\(=\)	91.36
Dual form			91.10.q.a.43.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-38.9200 + 22.4705i) q^{2} +(-55.4073 - 95.9683i) q^{3} +(753.843 - 1305.69i) q^{4} +740.456i q^{5} +(4312.90 + 2490.06i) q^{6} +(-2079.33 - 1200.50i) q^{7} +44747.0i q^{8} +(3701.56 - 6411.29i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 124 q + 15360 q^{4} + 6570 q^{6} - 381386 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−38.9200	+	22.4705i	−1.72004	+	0.993063i	−0.801227	+	0.598361i	\(0.795819\pi\)
							−0.918809	+	0.394702i	\(0.870848\pi\)
\(3\)	−55.4073	−	95.9683i	−0.394931	−	0.684041i	0.598161	−	0.801376i	\(-0.295898\pi\)
							−0.993092	+	0.117335i	\(0.962565\pi\)
\(5\)			740.456i			0.529827i	0.964272	+	0.264913i	\(0.0853434\pi\)
							−0.964272	+	0.264913i	\(0.914657\pi\)
\(7\)	−2079.33	−	1200.50i	−0.327327	−	0.188982i
							\(11\)	−63355.0	+	36578.0i	−1.30471	+	0.753274i	−0.981208	−	0.192954i	\(-0.938193\pi\)
−0.323501	+	0.946228i	\(0.604860\pi\)
\(13\)	102950.	−	2421.57i	0.999723	−	0.0235154i
							\(17\)	159656.	−	276532.i	0.463623	−	0.803019i	−0.535515	−	0.844526i	\(-0.679882\pi\)
0.999138	+	0.0415068i	\(0.0132158\pi\)
\(19\)	−634228.	−	366172.i	−1.11649	−	0.644605i	−0.175987	−	0.984393i	\(-0.556312\pi\)
							−0.940502	+	0.339787i	\(0.889645\pi\)
\(23\)	63585.4	+	110133.i	0.0473786	+	0.0820622i	0.888742	−	0.458407i	\(-0.151580\pi\)
							−0.841364	+	0.540470i	\(0.818247\pi\)
\(29\)	1.94837e6	+	3.37468e6i	0.511541	+	0.886015i	0.999911	+	0.0133779i	\(0.00425846\pi\)
							−0.488370	+	0.872637i	\(0.662408\pi\)
\(31\)		−	1.26388e6i		−	0.245798i	−0.992419	−	0.122899i	\(-0.960781\pi\)
							0.992419	−	0.122899i	\(-0.0392192\pi\)
\(37\)	−1.38471e6	+	799464.i	−0.121465	+	0.0701279i	−0.559502	−	0.828829i	\(-0.689007\pi\)
							0.438036	+	0.898957i	\(0.355674\pi\)
\(41\)	1.93496e7	−	1.11715e7i	1.06941	−	0.617426i	0.141392	−	0.989954i	\(-0.454842\pi\)
							0.928021	+	0.372528i	\(0.121509\pi\)
\(43\)	8.55249e6	−	1.48133e7i	0.381491	−	0.660762i	−0.609784	−	0.792567i	\(-0.708744\pi\)
							0.991276	+	0.131805i	\(0.0420773\pi\)
\(47\)		−	3.39812e7i		−	1.01578i	−0.861422	−	0.507889i	\(-0.830426\pi\)
							0.861422	−	0.507889i	\(-0.169574\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.q.a.36.1	✓	124
13.4	even	6	inner	91.10.q.a.43.1	yes	124

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.q.a.36.1	✓	124	1.1	even	1	trivial
91.10.q.a.43.1	yes	124	13.4	even	6	inner