Embedded newform 950.2.bc.b.61.9

• Label: 950.2.bc.b.61.9
• Level: $950$
• Weight: $2$
• Character: 950.61
• 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			61.9
Character	\(\chi\)	\(=\)	950.61
Dual form			950.2.bc.b.841.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.961262 + 0.275637i) q^{2} +(-0.963527 - 0.135415i) q^{3} +(0.848048 - 0.529919i) q^{4} +(-2.13219 - 0.673618i) q^{5} +(0.963527 - 0.135415i) q^{6} +(0.212802 - 0.368583i) q^{7} +(-0.669131 + 0.743145i) q^{8} +(-1.97374 - 0.565960i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 600 q + 54 q^{7} - 75 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{1}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.961262	+	0.275637i	−0.679715	+	0.194905i
							\(3\)	−0.963527	−	0.135415i	−0.556293	−	0.0781818i	−0.144576	−	0.989494i	\(-0.546182\pi\)
−0.411717	+	0.911312i	\(0.635071\pi\)
\(5\)	−2.13219	−	0.673618i	−0.953545	−	0.301251i
							\(7\)	0.212802	−	0.368583i	0.0804314	−	0.139311i	−0.823004	−	0.568036i	\(-0.807704\pi\)
0.903435	+	0.428724i	\(0.141037\pi\)
\(11\)	−3.07951	−	1.37109i	−0.928507	−	0.413398i	−0.113953	−	0.993486i	\(-0.536351\pi\)
							−0.814554	+	0.580088i	\(0.803018\pi\)
\(13\)	−3.16881	+	3.06009i	−0.878871	+	0.848715i	−0.989409	−	0.145156i	\(-0.953631\pi\)
							0.110538	+	0.993872i	\(0.464743\pi\)
\(17\)	−0.0548052	+	1.56941i	−0.0132922	+	0.380639i	0.974456	+	0.224580i	\(0.0721011\pi\)
							−0.987748	+	0.156059i	\(0.950121\pi\)
\(19\)	3.61031	+	2.44246i	0.828263	+	0.560340i
							\(23\)	4.51082	+	0.315427i	0.940571	+	0.0657711i	0.531804	−	0.846868i	\(-0.321514\pi\)
0.408767	+	0.912639i	\(0.365959\pi\)
\(29\)	−0.358078	−	10.2540i	−0.0664934	−	1.90412i	−0.337158	−	0.941448i	\(-0.609466\pi\)
							0.270664	−	0.962674i	\(-0.412757\pi\)
\(31\)	−5.52323	−	1.17400i	−0.992001	−	0.210856i	−0.316802	−	0.948492i	\(-0.602609\pi\)
							−0.675199	+	0.737636i	\(0.735942\pi\)
\(37\)	−1.29338	−	0.939698i	−0.212631	−	0.154485i	0.476372	−	0.879244i	\(-0.341951\pi\)
							−0.689003	+	0.724758i	\(0.741951\pi\)
\(41\)	−2.36670	+	4.85246i	−0.369617	+	0.757827i	−0.999827	−	0.0185976i	\(-0.994080\pi\)
							0.630210	+	0.776425i	\(0.282969\pi\)
\(43\)	5.55872	−	4.66432i	0.847697	−	0.711303i	−0.111584	−	0.993755i	\(-0.535592\pi\)
							0.959281	+	0.282452i	\(0.0911479\pi\)
\(47\)	0.127076	+	3.63897i	0.0185359	+	0.530798i	0.972775	+	0.231753i	\(0.0744460\pi\)
							−0.954239	+	0.299045i	\(0.903332\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bc.b.61.9	✓	600
19.5	even	9	inner	950.2.bc.b.461.17	yes	600
25.16	even	5	inner	950.2.bc.b.441.17	yes	600
475.366	even	45	inner	950.2.bc.b.841.9	yes	600

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bc.b.61.9	✓	600	1.1	even	1	trivial
950.2.bc.b.441.17	yes	600	25.16	even	5	inner
950.2.bc.b.461.17	yes	600	19.5	even	9	inner
950.2.bc.b.841.9	yes	600	475.366	even	45	inner