Embedded newform 124.5.l.a.35.17

• Label: 124.5.l.a.35.17
• Level: $124$
• Weight: $5$
• Character: 124.35
• Analytic conductor: $12.818$
• Analytic rank: $0$
• Dimension: $248$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 124 = 2^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	124.l (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.8178754224\)
Analytic rank:	\(0\)
Dimension:	\(248\)
Relative dimension:	\(62\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			35.17
Character	\(\chi\)	\(=\)	124.35
Dual form			124.5.l.a.39.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.55746 + 3.07561i) q^{2} +(-1.74347 - 0.566487i) q^{3} +(-2.91878 - 15.7315i) q^{4} +42.7016 q^{5} +(6.20114 - 3.91346i) q^{6} +(8.91114 + 12.2651i) q^{7} +(55.8487 + 31.2557i) q^{8} +(-62.8116 - 45.6353i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 248 q - 3 q^{2} - 31 q^{4} - 16 q^{5} - 10 q^{6} - 237 q^{8} + 1560 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/124\mathbb{Z}\right)^\times\).

\(n\)	\(63\)	\(65\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.55746	+	3.07561i	−0.639366	+	0.768903i
							\(3\)	−1.74347	−	0.566487i	−0.193719	−	0.0629430i	0.210551	−	0.977583i	\(-0.432474\pi\)
−0.404270	+	0.914640i	\(0.632474\pi\)
\(5\)	42.7016			1.70807			0.854033	−	0.520220i	\(-0.174150\pi\)
							0.854033	+	0.520220i	\(0.174150\pi\)
\(7\)	8.91114	+	12.2651i	0.181860	+	0.250309i	0.890208	−	0.455555i	\(-0.150559\pi\)
							−0.708348	+	0.705864i	\(0.750559\pi\)
\(11\)	−67.7297	−	93.2220i	−0.559750	−	0.770429i	0.431545	−	0.902091i	\(-0.357969\pi\)
							−0.991295	+	0.131662i	\(0.957969\pi\)
\(13\)	44.0291	−	135.508i	0.260527	−	0.801821i	−0.732163	−	0.681130i	\(-0.761489\pi\)
							0.992690	−	0.120691i	\(-0.0385111\pi\)
\(17\)	−181.955	−	132.198i	−0.629604	−	0.457434i	0.226659	−	0.973974i	\(-0.427220\pi\)
							−0.856263	+	0.516540i	\(0.827220\pi\)
\(19\)	316.117	−	102.713i	0.875670	−	0.284522i	0.163512	−	0.986541i	\(-0.447718\pi\)
							0.712158	+	0.702019i	\(0.247718\pi\)
\(23\)	154.314	−	212.394i	0.291708	−	0.401502i	−0.637860	−	0.770152i	\(-0.720180\pi\)
							0.929568	+	0.368651i	\(0.120180\pi\)
\(29\)	213.879	+	658.252i	0.254315	+	0.782702i	0.993964	+	0.109708i	\(0.0349917\pi\)
							−0.739649	+	0.672993i	\(0.765008\pi\)
\(31\)	−794.961	−	539.960i	−0.827223	−	0.561874i
							\(37\)	895.420			0.654069			0.327035	−	0.945012i	\(-0.393951\pi\)
0.327035	+	0.945012i	\(0.393951\pi\)
\(41\)	−714.416	−	2198.75i	−0.424995	−	1.30800i	−0.902999	−	0.429642i	\(-0.858640\pi\)
							0.478005	−	0.878357i	\(-0.341360\pi\)
\(43\)	3382.30	−	1098.98i	1.82926	−	0.594362i	0.829925	−	0.557876i	\(-0.188383\pi\)
							0.999334	−	0.0364866i	\(-0.0116166\pi\)
\(47\)	834.174	+	271.040i	0.377625	+	0.122698i	0.491679	−	0.870777i	\(-0.336383\pi\)
							−0.114053	+	0.993475i	\(0.536383\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.5.l.a.35.17	✓	248
4.3	odd	2	inner	124.5.l.a.35.58	yes	248
31.8	even	5	inner	124.5.l.a.39.58	yes	248
124.39	odd	10	inner	124.5.l.a.39.17	yes	248

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.5.l.a.35.17	✓	248	1.1	even	1	trivial
124.5.l.a.35.58	yes	248	4.3	odd	2	inner
124.5.l.a.39.17	yes	248	124.39	odd	10	inner
124.5.l.a.39.58	yes	248	31.8	even	5	inner