Embedded newform 751.2.o.a.2.2

• Label: 751.2.o.a.2.2
• Level: $751$
• Weight: $2$
• Character: 751.2
• Analytic conductor: $5.997$
• Analytic rank: $0$
• Dimension: $12400$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	751.o (of order \(375\), degree \(200\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.99676519180\)
Analytic rank:	\(0\)
Dimension:	\(12400\)
Relative dimension:	\(62\) over \(\Q(\zeta_{375})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{375}]$

Embedding invariants

Embedding label			2.2
Character	\(\chi\)	\(=\)	751.2
Dual form			751.2.o.a.376.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.88888 + 1.99464i) q^{2} +(0.335543 - 1.93453i) q^{3} +(-0.301878 - 5.53822i) q^{4} +(-0.210427 - 0.191072i) q^{5} +(3.22490 + 4.32338i) q^{6} +(-3.37950 - 0.958852i) q^{7} +(7.42798 + 6.30384i) q^{8} +(-0.805069 - 0.287939i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12400 q - 225 q^{2} - 200 q^{3} - 225 q^{4} - 200 q^{5} - 200 q^{6} - 200 q^{7} - 150 q^{8} - 200 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{208}{375}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.88888	+	1.99464i	−1.33564	+	1.41042i	−0.494930	+	0.868933i	\(0.664806\pi\)
							−0.840707	+	0.541490i	\(0.817860\pi\)
\(3\)	0.335543	−	1.93453i	0.193726	−	1.11690i	−0.714879	−	0.699248i	\(-0.753518\pi\)
							0.908605	−	0.417656i	\(-0.137148\pi\)
\(5\)	−0.210427	−	0.191072i	−0.0941060	−	0.0854499i	0.623826	−	0.781564i	\(-0.285578\pi\)
							−0.717931	+	0.696114i	\(0.754911\pi\)
\(7\)	−3.37950	−	0.958852i	−1.27733	−	0.362412i	−0.434541	−	0.900652i	\(-0.643089\pi\)
							−0.842792	+	0.538240i	\(0.819089\pi\)
\(11\)	−1.36747	+	4.89773i	−0.412308	+	1.47672i	0.412316	+	0.911041i	\(0.364720\pi\)
							−0.824624	+	0.565681i	\(0.808613\pi\)
\(13\)	0.312631	−	0.274459i	0.0867081	−	0.0761213i	−0.615287	−	0.788304i	\(-0.710960\pi\)
							0.701995	+	0.712182i	\(0.252293\pi\)
\(17\)	0.322632	+	6.99711i	0.0782498	+	1.69705i	0.566273	+	0.824218i	\(0.308385\pi\)
							−0.488023	+	0.872831i	\(0.662282\pi\)
\(19\)	1.90152	−	3.71271i	0.436239	−	0.851753i	−0.563433	−	0.826162i	\(-0.690520\pi\)
							0.999672	−	0.0255915i	\(-0.00814693\pi\)
\(23\)	2.22695	+	0.591715i	0.464352	+	0.123381i	0.481451	−	0.876473i	\(-0.340110\pi\)
							−0.0170992	+	0.999854i	\(0.505443\pi\)
\(29\)	−7.37158	−	0.994056i	−1.36887	−	0.184592i	−0.587784	−	0.809018i	\(-0.699999\pi\)
							−0.781084	+	0.624426i	\(0.785333\pi\)
\(31\)	5.66660	+	8.30070i	1.01775	+	1.49085i	0.864482	+	0.502665i	\(0.167647\pi\)
							0.153269	+	0.988184i	\(0.451020\pi\)
\(37\)	5.40469	−	2.35227i	0.888526	−	0.386711i	0.0951643	−	0.995462i	\(-0.469662\pi\)
							0.793362	+	0.608751i	\(0.208329\pi\)
\(41\)	0.934098	−	0.239836i	0.145882	−	0.0374560i	−0.175043	−	0.984561i	\(-0.556006\pi\)
							0.320925	+	0.947105i	\(0.396006\pi\)
\(43\)	3.69270	+	3.29702i	0.563132	+	0.502791i	0.898309	−	0.439364i	\(-0.144796\pi\)
							−0.335177	+	0.942155i	\(0.608796\pi\)
\(47\)	12.8917	+	1.40960i	1.88045	+	0.205611i	0.975515	−	0.219932i	\(-0.0705835\pi\)
							0.904938	+	0.425543i	\(0.139917\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	751.2.o.a.2.2	✓	12400
751.376	even	375	inner	751.2.o.a.376.2	yes	12400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
751.2.o.a.2.2	✓	12400	1.1	even	1	trivial
751.2.o.a.376.2	yes	12400	751.376	even	375	inner