Embedded newform 124.5.l.a.35.2

• Label: 124.5.l.a.35.2
• 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			35.2
Character	\(\chi\)	\(=\)	124.35
Dual form			124.5.l.a.39.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.99862 - 0.105051i) q^{2} +(7.35414 + 2.38950i) q^{3} +(15.9779 + 0.840119i) q^{4} -14.1850 q^{5} +(-29.1554 - 10.3273i) q^{6} +(12.6030 + 17.3466i) q^{7} +(-63.8014 - 5.03782i) q^{8} +(-17.1567 - 12.4651i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 248 q - 3 q^{2} - 31 q^{4} - 16 q^{5} - 10 q^{6} - 237 q^{8} + 1560 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.99862	−	0.105051i	−0.999655	−	0.0262628i
							\(3\)	7.35414	+	2.38950i	0.817127	+	0.265501i	0.687613	−	0.726077i	\(-0.258658\pi\)
0.129513	+	0.991578i	\(0.458658\pi\)
\(5\)	−14.1850			−0.567401			−0.283701	−	0.958913i	\(-0.591562\pi\)
							−0.283701	+	0.958913i	\(0.591562\pi\)
\(7\)	12.6030	+	17.3466i	0.257205	+	0.354012i	0.918018	−	0.396538i	\(-0.129789\pi\)
							−0.660814	+	0.750550i	\(0.729789\pi\)
\(11\)	−119.560	−	164.560i	−0.988098	−	1.36000i	−0.932351	−	0.361556i	\(-0.882246\pi\)
							−0.0557476	−	0.998445i	\(-0.517754\pi\)
\(13\)	−62.6022	+	192.670i	−0.370427	+	1.14006i	0.576085	+	0.817390i	\(0.304580\pi\)
							−0.946512	+	0.322668i	\(0.895420\pi\)
\(17\)	−141.989	−	103.161i	−0.491313	−	0.356960i	0.314376	−	0.949299i	\(-0.398205\pi\)
							−0.805689	+	0.592339i	\(0.798205\pi\)
\(19\)	389.761	−	126.641i	1.07967	−	0.350806i	0.285424	−	0.958401i	\(-0.407866\pi\)
							0.794247	+	0.607595i	\(0.207866\pi\)
\(23\)	−88.3133	+	121.553i	−0.166944	+	0.229778i	−0.884290	−	0.466939i	\(-0.845357\pi\)
							0.717346	+	0.696717i	\(0.245357\pi\)
\(29\)	−154.650	−	475.964i	−0.183888	−	0.565950i	0.816039	−	0.577996i	\(-0.196165\pi\)
							−0.999927	+	0.0120469i	\(0.996165\pi\)
\(31\)	−632.446	−	723.556i	−0.658113	−	0.752920i
							\(37\)	−1849.42			−1.35092			−0.675462	−	0.737395i	\(-0.736056\pi\)
−0.675462	+	0.737395i	\(0.736056\pi\)
\(41\)	−272.811	−	839.627i	−0.162291	−	0.499481i	0.836535	−	0.547913i	\(-0.184578\pi\)
							−0.998826	+	0.0484322i	\(0.984578\pi\)
\(43\)	−2152.29	+	699.321i	−1.16403	+	0.378216i	−0.826411	−	0.563067i	\(-0.809621\pi\)
							−0.337618	+	0.941283i	\(0.609621\pi\)
\(47\)	2449.63	+	795.932i	1.10893	+	0.360313i	0.805533	−	0.592551i	\(-0.201879\pi\)
							0.303397	+	0.952864i	\(0.401879\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.5.l.a.35.2	✓	248
4.3	odd	2	inner	124.5.l.a.35.50	yes	248
31.8	even	5	inner	124.5.l.a.39.50	yes	248
124.39	odd	10	inner	124.5.l.a.39.2	yes	248

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.5.l.a.35.2	✓	248	1.1	even	1	trivial
124.5.l.a.35.50	yes	248	4.3	odd	2	inner
124.5.l.a.39.2	yes	248	124.39	odd	10	inner
124.5.l.a.39.50	yes	248	31.8	even	5	inner