Embedded newform 91.10.e.a.53.12

• Label: 91.10.e.a.53.12
• Level: $91$
• Weight: $10$
• Character: 91.53
• Analytic conductor: $46.868$
• Analytic rank: $0$
• Dimension: $70$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	91.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(46.8682610909\)
Analytic rank:	\(0\)
Dimension:	\(70\)
Relative dimension:	\(35\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			53.12
Character	\(\chi\)	\(=\)	91.53
Dual form			91.10.e.a.79.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-8.75534 - 15.1647i) q^{2} +(126.100 - 218.411i) q^{3} +(102.688 - 177.861i) q^{4} +(-351.181 - 608.263i) q^{5} -4416.19 q^{6} +(-5379.03 + 3379.30i) q^{7} -12561.7 q^{8} +(-21960.9 - 38037.3i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q + 65 q^{2} - 8533 q^{4} - 2118 q^{5} + 17358 q^{6} - 8764 q^{7} - 106980 q^{8} - 215597 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−8.75534	−	15.1647i	−0.386935	−	0.670191i	0.605100	−	0.796149i	\(-0.293133\pi\)
							−0.992036	+	0.125958i	\(0.959800\pi\)
\(3\)	126.100	−	218.411i	0.898812	−	1.55679i	0.0697979	−	0.997561i	\(-0.477765\pi\)
							0.829014	−	0.559227i	\(-0.188902\pi\)
\(5\)	−351.181	−	608.263i	−0.251284	−	0.435237i	0.712595	−	0.701575i	\(-0.247520\pi\)
							−0.963880	+	0.266338i	\(0.914186\pi\)
\(7\)	−5379.03	+	3379.30i	−0.846764	+	0.531968i
							\(11\)	20977.4	−	36333.9i	0.432001	−	0.748248i	−0.565045	−	0.825060i	\(-0.691141\pi\)
0.997046	+	0.0768128i	\(0.0244744\pi\)
\(13\)	−28561.0			−0.277350
							\(17\)	−106309.	+	184132.i	−0.308708	+	0.534698i	−0.978080	−	0.208229i	\(-0.933230\pi\)
0.669372	+	0.742927i	\(0.266563\pi\)
\(19\)	257040.	+	445207.i	0.452491	+	0.783737i	0.998540	−	0.0540158i	\(-0.0172021\pi\)
							−0.546049	+	0.837753i	\(0.683869\pi\)
\(23\)	−262670.	−	454958.i	−0.195720	−	0.338997i	0.751416	−	0.659828i	\(-0.229371\pi\)
							−0.947136	+	0.320832i	\(0.896038\pi\)
\(29\)	3.55557e6			0.933509			0.466755	−	0.884387i	\(-0.345423\pi\)
							0.466755	+	0.884387i	\(0.345423\pi\)
\(31\)	1.38648e6	−	2.40145e6i	0.269641	−	0.467032i	−0.699128	−	0.714996i	\(-0.746428\pi\)
							0.968769	+	0.247965i	\(0.0797616\pi\)
\(37\)	90962.0	+	157551.i	0.00797907	+	0.0138202i	0.869987	−	0.493074i	\(-0.164127\pi\)
							−0.862008	+	0.506894i	\(0.830793\pi\)
\(41\)	−2.84328e7			−1.57142			−0.785711	−	0.618594i	\(-0.787702\pi\)
							−0.785711	+	0.618594i	\(0.787702\pi\)
\(43\)	4.54441e6			0.202707			0.101354	−	0.994850i	\(-0.467683\pi\)
							0.101354	+	0.994850i	\(0.467683\pi\)
\(47\)	−1.08174e7	−	1.87363e7i	−0.323357	−	0.560071i	0.657821	−	0.753174i	\(-0.271478\pi\)
							−0.981179	+	0.193103i	\(0.938145\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.e.a.53.12	✓	70
7.2	even	3	inner	91.10.e.a.79.12	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.e.a.53.12	✓	70	1.1	even	1	trivial
91.10.e.a.79.12	yes	70	7.2	even	3	inner