Embedded newform 124.4.p.a.3.11

• Label: 124.4.p.a.3.11
• Level: $124$
• Weight: $4$
• Character: 124.3
• Analytic conductor: $7.316$
• Analytic rank: $0$
• Dimension: $368$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 124 = 2^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	124.p (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.31623684071\)
Analytic rank:	\(0\)
Dimension:	\(368\)
Relative dimension:	\(46\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			3.11
Character	\(\chi\)	\(=\)	124.3
Dual form			124.4.p.a.83.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.27279 + 1.68358i) q^{2} +(0.315000 - 2.99703i) q^{3} +(2.33112 - 7.65284i) q^{4} +(0.474237 - 0.821402i) q^{5} +(4.32981 + 7.34193i) q^{6} +(1.56190 + 7.34815i) q^{7} +(7.58603 + 21.3179i) q^{8} +(17.5270 + 3.72549i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 6 q^{2} + 6 q^{4} - 8 q^{5} - 9 q^{6} - 57 q^{8} + 360 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(63\)	\(65\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.27279	+	1.68358i	−0.803551	+	0.595235i
							\(3\)	0.315000	−	2.99703i	0.0606218	−	0.576778i	−0.921480	−	0.388426i	\(-0.873019\pi\)
0.982102	−	0.188352i	\(-0.0603146\pi\)
\(5\)	0.474237	−	0.821402i	0.0424170	−	0.0734685i	−0.844037	−	0.536284i	\(-0.819828\pi\)
							0.886454	+	0.462816i	\(0.153161\pi\)
\(7\)	1.56190	+	7.34815i	0.0843345	+	0.396763i	0.999986	−	0.00520943i	\(-0.00165822\pi\)
							−0.915652	+	0.401972i	\(0.868325\pi\)
\(11\)	−37.4294	−	41.5695i	−1.02594	−	1.13943i	−0.990142	−	0.140067i	\(-0.955268\pi\)
							−0.0358021	−	0.999359i	\(-0.511399\pi\)
\(13\)	−13.6973	−	30.7646i	−0.292227	−	0.656352i	0.706451	−	0.707762i	\(-0.250295\pi\)
							−0.998678	+	0.0514102i	\(0.983628\pi\)
\(17\)	6.58646	+	5.93047i	0.0939677	+	0.0846089i	0.714776	−	0.699354i	\(-0.246529\pi\)
							−0.620808	+	0.783963i	\(0.713195\pi\)
\(19\)	44.3439	−	99.5980i	0.535431	−	1.20260i	−0.420039	−	0.907506i	\(-0.637983\pi\)
							0.955469	−	0.295091i	\(-0.0953499\pi\)
\(23\)	−35.2006	−	108.336i	−0.319123	−	0.982159i	−0.974024	−	0.226445i	\(-0.927290\pi\)
							0.654901	−	0.755714i	\(-0.272710\pi\)
\(29\)	−119.421	−	164.369i	−0.764689	−	1.05250i	−0.996810	−	0.0798170i	\(-0.974566\pi\)
							0.232121	−	0.972687i	\(-0.425434\pi\)
\(31\)	−1.04340	−	172.598i	−0.00604516	−	0.999982i
							\(37\)	45.9963	−	26.5560i	0.204372	−	0.117994i	−0.394321	−	0.918973i	\(-0.629020\pi\)
0.598693	+	0.800979i	\(0.295687\pi\)
\(41\)	18.0795	+	172.015i	0.0688669	+	0.655224i	0.973446	+	0.228918i	\(0.0735188\pi\)
							−0.904579	+	0.426306i	\(0.859815\pi\)
\(43\)	−126.246	−	56.2083i	−0.447728	−	0.199341i	0.170471	−	0.985363i	\(-0.445471\pi\)
							−0.618199	+	0.786021i	\(0.712138\pi\)
\(47\)	−173.462	+	238.750i	−0.538340	+	0.740962i	−0.988373	−	0.152050i	\(-0.951412\pi\)
							0.450033	+	0.893012i	\(0.351412\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.4.p.a.3.11	yes	368
4.3	odd	2	inner	124.4.p.a.3.10	✓	368
31.21	odd	30	inner	124.4.p.a.83.10	yes	368
124.83	even	30	inner	124.4.p.a.83.11	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.4.p.a.3.10	✓	368	4.3	odd	2	inner
124.4.p.a.3.11	yes	368	1.1	even	1	trivial
124.4.p.a.83.10	yes	368	31.21	odd	30	inner
124.4.p.a.83.11	yes	368	124.83	even	30	inner