Embedded newform 124.5.l.a.35.13

• Label: 124.5.l.a.35.13
• 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			35.13
Character	\(\chi\)	\(=\)	124.35
Dual form			124.5.l.a.39.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.12327 - 2.49904i) q^{2} +(9.51100 + 3.09031i) q^{3} +(3.50959 + 15.6103i) q^{4} +25.8175 q^{5} +(-21.9826 - 33.4202i) q^{6} +(52.0808 + 71.6831i) q^{7} +(28.0495 - 57.5259i) q^{8} +(15.3787 + 11.1732i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 248 q - 3 q^{2} - 31 q^{4} - 16 q^{5} - 10 q^{6} - 237 q^{8} + 1560 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.12327	−	2.49904i	−0.780817	−	0.624760i
							\(3\)	9.51100	+	3.09031i	1.05678	+	0.343368i	0.785325	−	0.619084i	\(-0.212496\pi\)
0.271453	+	0.962452i	\(0.412496\pi\)
\(5\)	25.8175			1.03270			0.516350	−	0.856378i	\(-0.327290\pi\)
							0.516350	+	0.856378i	\(0.327290\pi\)
\(7\)	52.0808	+	71.6831i	1.06287	+	1.46292i	0.877087	+	0.480331i	\(0.159483\pi\)
							0.185787	+	0.982590i	\(0.440517\pi\)
\(11\)	3.62203	+	4.98529i	0.0299341	+	0.0412008i	0.823722	−	0.566994i	\(-0.191894\pi\)
							−0.793788	+	0.608195i	\(0.791894\pi\)
\(13\)	−58.1297	+	178.905i	−0.343963	+	1.05861i	0.618174	+	0.786041i	\(0.287873\pi\)
							−0.962137	+	0.272567i	\(0.912127\pi\)
\(17\)	−59.0006	−	42.8664i	−0.204154	−	0.148327i	0.481011	−	0.876715i	\(-0.340270\pi\)
							−0.685165	+	0.728388i	\(0.740270\pi\)
\(19\)	−498.162	+	161.863i	−1.37995	+	0.448373i	−0.902653	−	0.430369i	\(-0.858383\pi\)
							−0.477297	+	0.878742i	\(0.658383\pi\)
\(23\)	243.368	−	334.967i	0.460053	−	0.633209i	−0.514467	−	0.857510i	\(-0.672010\pi\)
							0.974520	+	0.224302i	\(0.0720101\pi\)
\(29\)	184.841	+	568.881i	0.219787	+	0.676434i	0.998779	+	0.0494004i	\(0.0157310\pi\)
							−0.778992	+	0.627033i	\(0.784269\pi\)
\(31\)	885.114	−	374.292i	0.921034	−	0.389482i
							\(37\)	2140.62			1.56364			0.781819	−	0.623505i	\(-0.214292\pi\)
0.781819	+	0.623505i	\(0.214292\pi\)
\(41\)	−734.540	−	2260.68i	−0.436966	−	1.34484i	−0.891058	−	0.453889i	\(-0.850036\pi\)
							0.454092	−	0.890955i	\(-0.349964\pi\)
\(43\)	2467.24	−	801.655i	1.33436	−	0.433561i	0.446960	−	0.894554i	\(-0.352506\pi\)
							0.887404	+	0.460992i	\(0.152506\pi\)
\(47\)	2710.12	+	880.571i	1.22685	+	0.398629i	0.849574	−	0.527470i	\(-0.176859\pi\)
							0.377280	+	0.926099i	\(0.376859\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.5.l.a.35.13	✓	248
4.3	odd	2	inner	124.5.l.a.35.38	yes	248
31.8	even	5	inner	124.5.l.a.39.38	yes	248
124.39	odd	10	inner	124.5.l.a.39.13	yes	248

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.5.l.a.35.13	✓	248	1.1	even	1	trivial
124.5.l.a.35.38	yes	248	4.3	odd	2	inner
124.5.l.a.39.13	yes	248	124.39	odd	10	inner
124.5.l.a.39.38	yes	248	31.8	even	5	inner