Embedded newform 950.2.bi.a.117.17

• Label: 950.2.bi.a.117.17
• Level: $950$
• Weight: $2$
• Character: 950.117
• Analytic conductor: $7.586$
• Analytic rank: $0$
• Dimension: $2400$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	950.bi (of order \(180\), degree \(48\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.58578819202\)
Analytic rank:	\(0\)
Dimension:	\(2400\)
Relative dimension:	\(50\) over \(\Q(\zeta_{180})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{180}]$

Embedding invariants

Embedding label			117.17
Character	\(\chi\)	\(=\)	950.117
Dual form			950.2.bi.a.203.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.390731 - 0.920505i) q^{2} +(0.753708 - 0.146506i) q^{3} +(-0.694658 + 0.719340i) q^{4} +(2.20873 - 0.348588i) q^{5} +(-0.429357 - 0.636548i) q^{6} +(-1.29466 + 4.83172i) q^{7} +(0.933580 + 0.358368i) q^{8} +(-2.23494 + 0.902974i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2400 q + 12 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{13}{20}\right)\)	\(e\left(\frac{13}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.390731	−	0.920505i	−0.276289	−	0.650895i
							\(3\)	0.753708	−	0.146506i	0.435154	−	0.0845853i	0.0315139	−	0.999503i	\(-0.489967\pi\)
0.403640	+	0.914918i	\(0.367745\pi\)
\(5\)	2.20873	−	0.348588i	0.987774	−	0.155893i
							\(7\)	−1.29466	+	4.83172i	−0.489334	+	1.82622i	0.0703635	+	0.997521i	\(0.477584\pi\)
−0.559697	+	0.828697i	\(0.689083\pi\)
\(11\)	1.30782	+	0.277985i	0.394321	+	0.0838156i	0.400805	−	0.916163i	\(-0.368730\pi\)
							−0.00648362	+	0.999979i	\(0.502064\pi\)
\(13\)	−1.27372	+	0.156393i	−0.353266	+	0.0433756i	−0.296557	−	0.955015i	\(-0.595838\pi\)
							−0.0567090	+	0.998391i	\(0.518061\pi\)
\(17\)	1.86407	−	3.10233i	0.452102	−	0.752425i	−0.544144	−	0.838992i	\(-0.683145\pi\)
							0.996246	+	0.0865673i	\(0.0275898\pi\)
\(19\)	−4.34903	+	0.293154i	−0.997736	+	0.0672541i
							\(23\)	−2.61883	+	8.56580i	−0.546064	+	1.78609i	0.0690009	+	0.997617i	\(0.478019\pi\)
−0.615064	+	0.788477i	\(0.710870\pi\)
\(29\)	−1.72150	+	6.90455i	−0.319674	+	1.28214i	0.568518	+	0.822671i	\(0.307517\pi\)
							−0.888192	+	0.459472i	\(0.848039\pi\)
\(31\)	−1.24075	−	0.130408i	−0.222845	−	0.0234220i	−0.00755228	−	0.999971i	\(-0.502404\pi\)
							−0.215293	+	0.976549i	\(0.569071\pi\)
\(37\)	2.97237	+	5.83361i	0.488655	+	0.959039i	0.995296	+	0.0968823i	\(0.0308870\pi\)
							−0.506641	+	0.862157i	\(0.669113\pi\)
\(41\)	7.53857	−	0.263253i	1.17733	−	0.0411132i	0.560461	−	0.828181i	\(-0.310624\pi\)
							0.616866	+	0.787068i	\(0.288402\pi\)
\(43\)	−2.58231	−	3.68792i	−0.393798	−	0.562402i	0.572475	−	0.819922i	\(-0.305983\pi\)
							−0.966273	+	0.257520i	\(0.917095\pi\)
\(47\)	−2.74727	+	1.65073i	−0.400730	+	0.240783i	−0.701135	−	0.713029i	\(-0.747323\pi\)
							0.300404	+	0.953812i	\(0.402878\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bi.a.117.17	yes	2400
19.13	odd	18	inner	950.2.bi.a.317.34	yes	2400
25.3	odd	20	inner	950.2.bi.a.3.34	✓	2400
475.203	even	180	inner	950.2.bi.a.203.17	yes	2400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bi.a.3.34	✓	2400	25.3	odd	20	inner
950.2.bi.a.117.17	yes	2400	1.1	even	1	trivial
950.2.bi.a.203.17	yes	2400	475.203	even	180	inner
950.2.bi.a.317.34	yes	2400	19.13	odd	18	inner