Embedded newform 75.9.j.a.11.13

• Label: 75.9.j.a.11.13
• Level: $75$
• Weight: $9$
• Character: 75.11
• Analytic conductor: $30.553$
• Analytic rank: $0$
• Dimension: $312$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	75.j (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(30.5533957546\)
Analytic rank:	\(0\)
Dimension:	\(312\)
Relative dimension:	\(78\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			11.13
Character	\(\chi\)	\(=\)	75.11
Dual form			75.9.j.a.41.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-13.7113 - 18.8720i) q^{2} +(-65.6121 + 47.4980i) q^{3} +(-89.0441 + 274.050i) q^{4} +(-427.105 + 456.297i) q^{5} +(1796.01 + 586.972i) q^{6} +3153.01 q^{7} +(713.322 - 231.773i) q^{8} +(2048.89 - 6232.88i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 73 q^{3} + 9722 q^{4} - 515 q^{6} - 176 q^{7} + 7907 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\)	\(26\)	\(52\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−13.7113	−	18.8720i	−0.856957	−	1.17950i	−0.982287	−	0.187385i	\(-0.939999\pi\)
							0.125329	−	0.992115i	\(-0.460001\pi\)
\(3\)	−65.6121	+	47.4980i	−0.810026	+	0.586395i
							\(5\)	−427.105	+	456.297i	−0.683367	+	0.730075i
\(7\)	3153.01			1.31321										0.656604	−	0.754236i	\(-0.271992\pi\)
							0.656604	+	0.754236i	\(0.271992\pi\)
\(11\)	−8172.01	−	11247.8i	−0.558159	−	0.768240i	0.432932	−	0.901427i	\(-0.357479\pi\)
							−0.991091	+	0.133186i	\(0.957479\pi\)
\(13\)	−7113.52	−	5168.27i	−0.249064	−	0.180956i	0.456248	−	0.889853i	\(-0.349193\pi\)
							−0.705312	+	0.708897i	\(0.749193\pi\)
\(17\)	34586.7	−	11237.9i	0.414107	−	0.134552i	−0.0945511	−	0.995520i	\(-0.530142\pi\)
							0.508659	+	0.860968i	\(0.330142\pi\)
\(19\)	14757.7	+	45419.4i	0.113241	+	0.348519i	0.991576	−	0.129526i	\(-0.0413455\pi\)
							−0.878335	+	0.478045i	\(0.841345\pi\)
\(23\)	−160126.	−	220394.i	−0.572202	−	0.787569i	0.420611	−	0.907241i	\(-0.361816\pi\)
							−0.992813	+	0.119672i	\(0.961816\pi\)
\(29\)	424824.	+	138034.i	0.600644	+	0.195161i	0.593528	−	0.804814i	\(-0.297735\pi\)
							0.00711636	+	0.999975i	\(0.497735\pi\)
\(31\)	−94976.5	−	292308.i	−0.102842	−	0.316514i	0.886376	−	0.462966i	\(-0.153215\pi\)
							−0.989218	+	0.146451i	\(0.953215\pi\)
\(37\)	−1.55260e6	−	1.12803e6i	−0.828425	−	0.601886i	0.0906886	−	0.995879i	\(-0.471093\pi\)
							−0.919113	+	0.393994i	\(0.871093\pi\)
\(41\)	−2.02318e6	+	2.78467e6i	−0.715977	+	0.985457i	0.283671	+	0.958922i	\(0.408448\pi\)
							−0.999648	+	0.0265357i	\(0.991552\pi\)
\(43\)	5.20557e6			1.52263			0.761315	−	0.648382i	\(-0.224554\pi\)
							0.761315	+	0.648382i	\(0.224554\pi\)
\(47\)	6.50129e6	+	2.11240e6i	1.33232	+	0.432896i	0.886708	−	0.462331i	\(-0.152987\pi\)
							0.445611	+	0.895227i	\(0.352987\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.9.j.a.11.13	✓	312
3.2	odd	2	inner	75.9.j.a.11.66	yes	312
25.16	even	5	inner	75.9.j.a.41.66	yes	312
75.41	odd	10	inner	75.9.j.a.41.13	yes	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.9.j.a.11.13	✓	312	1.1	even	1	trivial
75.9.j.a.11.66	yes	312	3.2	odd	2	inner
75.9.j.a.41.13	yes	312	75.41	odd	10	inner
75.9.j.a.41.66	yes	312	25.16	even	5	inner