Embedded newform 75.9.j.a.11.19

• Label: 75.9.j.a.11.19
• 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			11.19
Character	\(\chi\)	\(=\)	75.11
Dual form			75.9.j.a.41.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-11.3177 - 15.5774i) q^{2} +(-6.75329 - 80.7180i) q^{3} +(-35.4584 + 109.130i) q^{4} +(-344.132 + 521.726i) q^{5} +(-1180.95 + 1018.74i) q^{6} -4430.09 q^{7} +(-2586.70 + 840.469i) q^{8} +(-6469.79 + 1090.22i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 73 q^{3} + 9722 q^{4} - 515 q^{6} - 176 q^{7} + 7907 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−11.3177	−	15.5774i	−0.707354	−	0.973589i	−0.999850	−	0.0173225i	\(-0.994486\pi\)
							0.292496	−	0.956267i	\(-0.405514\pi\)
\(3\)	−6.75329	−	80.7180i	−0.0833740	−	0.996518i
							\(5\)	−344.132	+	521.726i	−0.550612	+	0.834762i
\(7\)	−4430.09			−1.84510										−0.922552	−	0.385873i	\(-0.873900\pi\)
							−0.922552	+	0.385873i	\(0.873900\pi\)
\(11\)	−9456.74	−	13016.1i	−0.645908	−	0.889016i	0.353005	−	0.935621i	\(-0.385160\pi\)
							−0.998913	+	0.0466052i	\(0.985160\pi\)
\(13\)	−26460.8	−	19224.9i	−0.926466	−	0.673117i	0.0186594	−	0.999826i	\(-0.494060\pi\)
							−0.945125	+	0.326709i	\(0.894060\pi\)
\(17\)	−14290.5	+	4643.26i	−0.171101	+	0.0555940i	−0.393315	−	0.919404i	\(-0.628672\pi\)
							0.222214	+	0.974998i	\(0.428672\pi\)
\(19\)	−52083.5	−	160297.i	−0.399656	−	1.23001i	−0.925276	−	0.379294i	\(-0.876167\pi\)
							0.525620	−	0.850719i	\(-0.323833\pi\)
\(23\)	13945.1	+	19193.8i	0.0498323	+	0.0685883i	0.833206	−	0.552963i	\(-0.186503\pi\)
							−0.783374	+	0.621551i	\(0.786503\pi\)
\(29\)	838951.	+	272592.i	1.18616	+	0.385408i	0.834653	−	0.550776i	\(-0.185668\pi\)
							0.351511	+	0.936184i	\(0.385668\pi\)
\(31\)	−439635.	−	1.35306e6i	−0.476042	−	1.46511i	−0.844548	−	0.535480i	\(-0.820131\pi\)
							0.368506	−	0.929625i	\(-0.379869\pi\)
\(37\)	1.15247e6	+	837320.i	0.614927	+	0.446770i	0.851146	−	0.524929i	\(-0.175908\pi\)
							−0.236219	+	0.971700i	\(0.575908\pi\)
\(41\)	222547.	−	306310.i	0.0787564	−	0.108399i	−0.767821	−	0.640664i	\(-0.778659\pi\)
							0.846578	+	0.532265i	\(0.178659\pi\)
\(43\)	−135471.			−0.0396253			−0.0198127	−	0.999804i	\(-0.506307\pi\)
							−0.0198127	+	0.999804i	\(0.506307\pi\)
\(47\)	−5.96498e6	−	1.93814e6i	−1.22241	−	0.397186i	−0.374452	−	0.927246i	\(-0.622169\pi\)
							−0.847960	+	0.530061i	\(0.822169\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.9.j.a.11.19	✓	312
3.2	odd	2	inner	75.9.j.a.11.60	yes	312
25.16	even	5	inner	75.9.j.a.41.60	yes	312
75.41	odd	10	inner	75.9.j.a.41.19	yes	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.9.j.a.11.19	✓	312	1.1	even	1	trivial
75.9.j.a.11.60	yes	312	3.2	odd	2	inner
75.9.j.a.41.19	yes	312	75.41	odd	10	inner
75.9.j.a.41.60	yes	312	25.16	even	5	inner