Embedded newform 950.2.bi.a.3.9

• Label: 950.2.bi.a.3.9
• Level: $950$
• Weight: $2$
• Character: 950.3
• 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			3.9
Character	\(\chi\)	\(=\)	950.3
Dual form			950.2.bi.a.317.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.974370 - 0.224951i) q^{2} +(-0.957340 + 0.530662i) q^{3} +(0.898794 + 0.438371i) q^{4} +(1.21737 + 1.87564i) q^{5} +(1.05218 - 0.301707i) q^{6} +(1.58359 + 0.424321i) q^{7} +(-0.777146 - 0.629320i) q^{8} +(-0.954860 + 1.52810i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2400 q + 12 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{7}{20}\right)\)	\(e\left(\frac{13}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.974370	−	0.224951i	−0.688984	−	0.159064i
							\(3\)	−0.957340	+	0.530662i	−0.552721	+	0.306378i	−0.736332	−	0.676621i	\(-0.763444\pi\)
0.183611	+	0.982999i	\(0.441221\pi\)
\(5\)	1.21737	+	1.87564i	0.544423	+	0.838811i
							\(7\)	1.58359	+	0.424321i	0.598540	+	0.160378i	0.545353	−	0.838206i	\(-0.316396\pi\)
0.0531866	+	0.998585i	\(0.483062\pi\)
\(11\)	0.124758	+	0.138558i	0.0376159	+	0.0417767i	0.761659	−	0.647978i	\(-0.224385\pi\)
							−0.724043	+	0.689755i	\(0.757718\pi\)
\(13\)	3.59236	+	3.85233i	0.996340	+	1.06844i	0.997655	+	0.0684452i	\(0.0218038\pi\)
							−0.00131445	+	0.999999i	\(0.500418\pi\)
\(17\)	−2.46571	+	2.14341i	−0.598023	+	0.519853i	−0.901383	−	0.433023i	\(-0.857447\pi\)
							0.303360	+	0.952876i	\(0.401891\pi\)
\(19\)	4.35863	+	0.0483187i	0.999939	+	0.0110851i
							\(23\)	−3.82814	−	5.08012i	−0.798223	−	1.05928i	−0.996858	−	0.0792044i	\(-0.974762\pi\)
0.198635	−	0.980073i	\(-0.436349\pi\)
\(29\)	4.28625	−	0.299724i	0.795937	−	0.0556573i	0.334003	−	0.942572i	\(-0.391600\pi\)
							0.461934	+	0.886915i	\(0.347156\pi\)
\(31\)	−0.382957	+	0.860135i	−0.0687811	+	0.154485i	−0.944665	−	0.328038i	\(-0.893613\pi\)
							0.875884	+	0.482523i	\(0.160279\pi\)
\(37\)	−4.70326	+	9.23067i	−0.773212	+	1.51751i	0.0804924	+	0.996755i	\(0.474351\pi\)
							−0.853704	+	0.520758i	\(0.825649\pi\)
\(41\)	8.10743	−	6.33422i	1.26617	−	0.989239i	0.266491	−	0.963837i	\(-0.414136\pi\)
							0.999677	−	0.0254016i	\(-0.00808647\pi\)
\(43\)	3.14422	−	2.20161i	0.479489	−	0.335742i	−0.308716	−	0.951154i	\(-0.599899\pi\)
							0.788206	+	0.615412i	\(0.211010\pi\)
\(47\)	−5.41650	+	6.23097i	−0.790078	+	0.908880i	−0.997592	−	0.0693516i	\(-0.977907\pi\)
							0.207515	+	0.978232i	\(0.433463\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bi.a.3.9	✓	2400
19.13	odd	18	inner	950.2.bi.a.203.42	yes	2400
25.17	odd	20	inner	950.2.bi.a.117.42	yes	2400
475.317	even	180	inner	950.2.bi.a.317.9	yes	2400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bi.a.3.9	✓	2400	1.1	even	1	trivial
950.2.bi.a.117.42	yes	2400	25.17	odd	20	inner
950.2.bi.a.203.42	yes	2400	19.13	odd	18	inner
950.2.bi.a.317.9	yes	2400	475.317	even	180	inner