Embedded newform 751.2.l.a.8.7

• Label: 751.2.l.a.8.7
• Level: $751$
• Weight: $2$
• Character: 751.8
• Analytic conductor: $5.997$
• Analytic rank: $0$
• Dimension: $6100$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	751.l (of order \(125\), degree \(100\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.99676519180\)
Analytic rank:	\(0\)
Dimension:	\(6100\)
Relative dimension:	\(61\) over \(\Q(\zeta_{125})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{125}]$

Embedding invariants

Embedding label			8.7
Character	\(\chi\)	\(=\)	751.8
Dual form			751.2.l.a.94.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.72580 - 1.46462i) q^{2} +(-1.33315 + 2.35442i) q^{3} +(0.507996 + 3.08191i) q^{4} +(0.954651 - 1.27983i) q^{5} +(5.74908 - 2.11069i) q^{6} +(-2.22702 - 2.43207i) q^{7} +(1.30824 - 2.18072i) q^{8} +(-2.22268 - 3.70501i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6100 q - 75 q^{2} - 100 q^{3} - 75 q^{4} - 100 q^{5} - 100 q^{6} - 100 q^{7} - 75 q^{8} - 100 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/751\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{83}{125}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.72580	−	1.46462i	−1.22032	−	1.03564i	−0.998160	−	0.0606332i	\(-0.980688\pi\)
							−0.222164	−	0.975009i	\(-0.571312\pi\)
\(3\)	−1.33315	+	2.35442i	−0.769695	+	1.35932i	0.158548	+	0.987351i	\(0.449319\pi\)
							−0.928244	+	0.371973i	\(0.878681\pi\)
\(5\)	0.954651	−	1.27983i	0.426933	−	0.572357i	−0.535273	−	0.844679i	\(-0.679791\pi\)
							0.962206	+	0.272321i	\(0.0877914\pi\)
\(7\)	−2.22702	−	2.43207i	−0.841733	−	0.919235i	0.155915	−	0.987770i	\(-0.450167\pi\)
							−0.997649	+	0.0685356i	\(0.978167\pi\)
\(11\)	−0.206572	+	0.193984i	−0.0622837	+	0.0584883i	−0.715064	−	0.699059i	\(-0.753602\pi\)
							0.652780	+	0.757547i	\(0.273602\pi\)
\(13\)	2.43544	+	3.63252i	0.675469	+	1.00748i	0.998140	+	0.0609696i	\(0.0194193\pi\)
							−0.322671	+	0.946511i	\(0.604581\pi\)
\(17\)	−0.407599	−	2.92989i	−0.0988572	−	0.710603i	−0.973969	−	0.226682i	\(-0.927212\pi\)
							0.875112	−	0.483921i	\(-0.160788\pi\)
\(19\)	0.657118	−	0.0998491i	0.150753	−	0.0229070i	−0.0744117	−	0.997228i	\(-0.523708\pi\)
							0.225165	+	0.974321i	\(0.427708\pi\)
\(23\)	0.112731	+	0.111324i	0.0235061	+	0.0232126i	0.714307	−	0.699832i	\(-0.246742\pi\)
							−0.690801	+	0.723045i	\(0.742742\pi\)
\(29\)	1.28714	+	0.547419i	0.239016	+	0.101653i	0.507567	−	0.861612i	\(-0.330545\pi\)
							−0.268551	+	0.963265i	\(0.586545\pi\)
\(31\)	−1.85382	+	0.426626i	−0.332957	+	0.0766243i	−0.387452	−	0.921890i	\(-0.626645\pi\)
							0.0544957	+	0.998514i	\(0.482645\pi\)
\(37\)	0.669401	−	1.89664i	0.110049	−	0.311806i	−0.874989	−	0.484142i	\(-0.839132\pi\)
							0.985038	+	0.172337i	\(0.0551317\pi\)
\(41\)	−1.74594	+	1.63955i	−0.272670	+	0.256055i	−0.808804	−	0.588079i	\(-0.799884\pi\)
							0.536133	+	0.844133i	\(0.319884\pi\)
\(43\)	−4.34783	+	6.14521i	−0.663038	+	0.937136i	−0.999990	−	0.00438608i	\(-0.998604\pi\)
							0.336952	+	0.941522i	\(0.390604\pi\)
\(47\)	−7.25386	−	2.45812i	−1.05808	−	0.358553i	−0.262838	−	0.964840i	\(-0.584658\pi\)
							−0.795246	+	0.606287i	\(0.792658\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	751.2.l.a.8.7	✓	6100
751.94	even	125	inner	751.2.l.a.94.7	yes	6100

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
751.2.l.a.8.7	✓	6100	1.1	even	1	trivial
751.2.l.a.94.7	yes	6100	751.94	even	125	inner