Embedded newform 124.5.l.a.35.10

• Label: 124.5.l.a.35.10
• 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			35.10
Character	\(\chi\)	\(=\)	124.35
Dual form			124.5.l.a.39.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.63647 + 1.66615i) q^{2} +(2.24232 + 0.728573i) q^{3} +(10.4479 - 12.1179i) q^{4} +22.7008 q^{5} +(-9.36804 + 1.08661i) q^{6} +(-53.6590 - 73.8553i) q^{7} +(-17.8031 + 61.4740i) q^{8} +(-61.0332 - 44.3432i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 248 q - 3 q^{2} - 31 q^{4} - 16 q^{5} - 10 q^{6} - 237 q^{8} + 1560 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.63647	+	1.66615i	−0.909118	+	0.416539i
							\(3\)	2.24232	+	0.728573i	0.249146	+	0.0809526i	0.430928	−	0.902386i	\(-0.358186\pi\)
−0.181781	+	0.983339i	\(0.558186\pi\)
\(5\)	22.7008			0.908032			0.454016	−	0.890994i	\(-0.349991\pi\)
							0.454016	+	0.890994i	\(0.349991\pi\)
\(7\)	−53.6590	−	73.8553i	−1.09508	−	1.50725i	−0.841747	−	0.539873i	\(-0.818472\pi\)
							−0.253335	−	0.967379i	\(-0.581528\pi\)
\(11\)	29.1859	+	40.1710i	0.241206	+	0.331991i	0.912407	−	0.409284i	\(-0.134221\pi\)
							−0.671201	+	0.741275i	\(0.734221\pi\)
\(13\)	−57.6576	+	177.452i	−0.341169	+	1.05001i	0.622433	+	0.782673i	\(0.286144\pi\)
							−0.963603	+	0.267338i	\(0.913856\pi\)
\(17\)	109.869	+	79.8248i	0.380171	+	0.276211i	0.761416	−	0.648264i	\(-0.224505\pi\)
							−0.381245	+	0.924474i	\(0.624505\pi\)
\(19\)	−486.104	+	157.945i	−1.34655	+	0.437520i	−0.891530	−	0.452962i	\(-0.850367\pi\)
							−0.455018	+	0.890482i	\(0.650367\pi\)
\(23\)	−525.146	+	722.801i	−0.992714	+	1.36635i	−0.0630235	+	0.998012i	\(0.520074\pi\)
							−0.929691	+	0.368342i	\(0.879926\pi\)
\(29\)	−195.619	−	602.055i	−0.232603	−	0.715880i	−0.997430	−	0.0716438i	\(-0.977176\pi\)
							0.764827	−	0.644236i	\(-0.222824\pi\)
\(31\)	429.902	−	859.480i	0.447349	−	0.894360i
							\(37\)	−1832.49			−1.33856			−0.669281	−	0.743010i	\(-0.733398\pi\)
−0.669281	+	0.743010i	\(0.733398\pi\)
\(41\)	−119.062	−	366.435i	−0.0708280	−	0.217986i	0.909376	−	0.415974i	\(-0.136559\pi\)
							−0.980204	+	0.197988i	\(0.936559\pi\)
\(43\)	1718.55	−	558.391i	0.929449	−	0.301996i	0.195112	−	0.980781i	\(-0.437493\pi\)
							0.734337	+	0.678785i	\(0.237493\pi\)
\(47\)	702.629	+	228.298i	0.318076	+	0.103349i	0.463704	−	0.885990i	\(-0.346520\pi\)
							−0.145628	+	0.989339i	\(0.546520\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.5.l.a.35.10	✓	248
4.3	odd	2	inner	124.5.l.a.35.60	yes	248
31.8	even	5	inner	124.5.l.a.39.60	yes	248
124.39	odd	10	inner	124.5.l.a.39.10	yes	248

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.5.l.a.35.10	✓	248	1.1	even	1	trivial
124.5.l.a.35.60	yes	248	4.3	odd	2	inner
124.5.l.a.39.10	yes	248	124.39	odd	10	inner
124.5.l.a.39.60	yes	248	31.8	even	5	inner