Embedded newform 751.2.l.a.8.19

• Label: 751.2.l.a.8.19
• 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			8.19
Character	\(\chi\)	\(=\)	751.8
Dual form			751.2.l.a.94.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.988352 - 0.838777i) q^{2} +(1.47190 - 2.59946i) q^{3} +(-0.0519812 - 0.315359i) q^{4} +(0.871274 - 1.16805i) q^{5} +(-3.63513 + 1.33459i) q^{6} +(-2.70677 - 2.95599i) q^{7} +(-1.54687 + 2.57850i) q^{8} +(-3.04739 - 5.07973i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6100 q - 75 q^{2} - 100 q^{3} - 75 q^{4} - 100 q^{5} - 100 q^{6} - 100 q^{7} - 75 q^{8} - 100 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.988352	−	0.838777i	−0.698870	−	0.593105i	0.225663	−	0.974205i	\(-0.427545\pi\)
							−0.924534	+	0.381101i	\(0.875545\pi\)
\(3\)	1.47190	−	2.59946i	0.849804	−	1.50080i	−0.0156879	−	0.999877i	\(-0.504994\pi\)
							0.865492	−	0.500923i	\(-0.167006\pi\)
\(5\)	0.871274	−	1.16805i	0.389646	−	0.522369i	−0.562998	−	0.826458i	\(-0.690352\pi\)
							0.952644	+	0.304089i	\(0.0983521\pi\)
\(7\)	−2.70677	−	2.95599i	−1.02306	−	1.11726i	−0.993048	−	0.117709i	\(-0.962445\pi\)
							−0.0300135	−	0.999549i	\(-0.509555\pi\)
\(11\)	3.47612	−	3.26429i	1.04809	−	0.984221i	0.0481957	−	0.998838i	\(-0.484653\pi\)
							0.999893	+	0.0146170i	\(0.00465290\pi\)
\(13\)	−0.844673	−	1.25985i	−0.234270	−	0.349421i	0.694878	−	0.719127i	\(-0.255458\pi\)
							−0.929149	+	0.369707i	\(0.879458\pi\)
\(17\)	1.09349	+	7.86017i	0.265209	+	1.90637i	0.401772	+	0.915740i	\(0.368394\pi\)
							−0.136563	+	0.990631i	\(0.543606\pi\)
\(19\)	6.73040	−	1.02268i	1.54406	−	0.234620i	0.678189	−	0.734887i	\(-0.262765\pi\)
							0.865871	+	0.500267i	\(0.166765\pi\)
\(23\)	−1.01830	−	1.00559i	−0.212331	−	0.209679i	0.588619	−	0.808410i	\(-0.299672\pi\)
							−0.800950	+	0.598731i	\(0.795672\pi\)
\(29\)	−3.43412	−	1.46053i	−0.637699	−	0.271213i	0.0482801	−	0.998834i	\(-0.484626\pi\)
							−0.685979	+	0.727621i	\(0.740626\pi\)
\(31\)	8.80796	−	2.02700i	1.58196	−	0.364060i	0.659973	−	0.751289i	\(-0.270568\pi\)
							0.921984	+	0.387229i	\(0.126568\pi\)
\(37\)	1.19051	−	3.37312i	0.195719	−	0.554538i	−0.803447	−	0.595376i	\(-0.797003\pi\)
							0.999166	+	0.0408386i	\(0.0130030\pi\)
\(41\)	−0.678070	+	0.636750i	−0.105897	+	0.0994436i	−0.735687	−	0.677321i	\(-0.763141\pi\)
							0.629791	+	0.776765i	\(0.283141\pi\)
\(43\)	−3.27478	+	4.62856i	−0.499399	+	0.705849i	−0.985807	−	0.167883i	\(-0.946307\pi\)
							0.486408	+	0.873732i	\(0.338307\pi\)
\(47\)	10.8137	+	3.66445i	1.57734	+	0.534515i	0.966375	−	0.257139i	\(-0.0827797\pi\)
							0.610970	+	0.791654i	\(0.290780\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	751.2.l.a.8.19	✓	6100
751.94	even	125	inner	751.2.l.a.94.19	yes	6100

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
751.2.l.a.8.19	✓	6100	1.1	even	1	trivial
751.2.l.a.94.19	yes	6100	751.94	even	125	inner