Embedded newform 950.2.bc.a.111.5

• Label: 950.2.bc.a.111.5
• Level: $950$
• Weight: $2$
• Character: 950.111
• Analytic conductor: $7.586$
• Analytic rank: $0$
• Dimension: $600$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	950.bc (of order \(45\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.58578819202\)
Analytic rank:	\(0\)
Dimension:	\(600\)
Relative dimension:	\(25\) over \(\Q(\zeta_{45})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{45}]$

Embedding invariants

Embedding label			111.5
Character	\(\chi\)	\(=\)	950.111
Dual form			950.2.bc.a.291.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.997564 - 0.0697565i) q^{2} +(-0.0816741 - 2.33884i) q^{3} +(0.990268 + 0.139173i) q^{4} +(1.44922 - 1.70287i) q^{5} +(-0.0816741 + 2.33884i) q^{6} +(1.27710 + 2.21200i) q^{7} +(-0.978148 - 0.207912i) q^{8} +(-2.47082 + 0.172776i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 600 q - 42 q^{7} + 75 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{2}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.997564	−	0.0697565i	−0.705384	−	0.0493253i
							\(3\)	−0.0816741	−	2.33884i	−0.0471546	−	1.35033i	−0.760418	−	0.649434i	\(-0.775006\pi\)
0.713263	−	0.700896i	\(-0.247217\pi\)
\(5\)	1.44922	−	1.70287i	0.648111	−	0.761546i
							\(7\)	1.27710	+	2.21200i	0.482699	+	0.836058i	0.999803	−	0.0198641i	\(-0.00632337\pi\)
−0.517104	+	0.855922i	\(0.672990\pi\)
\(11\)	0.542997	+	5.16627i	0.163720	+	1.55769i	0.700305	+	0.713844i	\(0.253047\pi\)
							−0.536585	+	0.843846i	\(0.680286\pi\)
\(13\)	−3.07783	−	4.56307i	−0.853637	−	1.26557i	−0.962437	−	0.271505i	\(-0.912479\pi\)
							0.108800	−	0.994064i	\(-0.465299\pi\)
\(17\)	1.61983	−	4.00922i	0.392867	−	0.972379i	−0.593053	−	0.805163i	\(-0.702078\pi\)
							0.985920	−	0.167216i	\(-0.0534778\pi\)
\(19\)	−4.30945	−	0.654697i	−0.988656	−	0.150198i
							\(23\)	−1.09952	−	1.06179i	−0.229266	−	0.221400i	0.571148	−	0.820847i	\(-0.306498\pi\)
−0.800414	+	0.599447i	\(0.795387\pi\)
\(29\)	−2.20404	−	5.45518i	−0.409279	−	1.01300i	−0.981252	−	0.192729i	\(-0.938266\pi\)
							0.571973	−	0.820272i	\(-0.306178\pi\)
\(31\)	5.04697	−	5.60522i	0.906462	−	1.00673i	−0.0934766	−	0.995621i	\(-0.529798\pi\)
							0.999938	−	0.0111062i	\(-0.00353530\pi\)
\(37\)	−0.980072	−	0.712064i	−0.161123	−	0.117063i	0.504302	−	0.863527i	\(-0.331750\pi\)
							−0.665425	+	0.746465i	\(0.731750\pi\)
\(41\)	−1.10337	−	0.316386i	−0.172317	−	0.0494112i	0.188369	−	0.982098i	\(-0.439680\pi\)
							−0.360687	+	0.932687i	\(0.617458\pi\)
\(43\)	0.197461	−	1.11986i	0.0301125	−	0.170776i	−0.966043	−	0.258382i	\(-0.916811\pi\)
							0.996155	+	0.0876060i	\(0.0279216\pi\)
\(47\)	−2.19475	−	5.43219i	−0.320137	−	0.792366i	−0.998346	−	0.0574972i	\(-0.981688\pi\)
							0.678209	−	0.734869i	\(-0.262756\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bc.a.111.5	✓	600
19.6	even	9	inner	950.2.bc.a.861.21	yes	600
25.16	even	5	inner	950.2.bc.a.491.21	yes	600
475.291	even	45	inner	950.2.bc.a.291.5	yes	600

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bc.a.111.5	✓	600	1.1	even	1	trivial
950.2.bc.a.291.5	yes	600	475.291	even	45	inner
950.2.bc.a.491.21	yes	600	25.16	even	5	inner
950.2.bc.a.861.21	yes	600	19.6	even	9	inner