Embedded newform 124.3.n.a.7.14

• Label: 124.3.n.a.7.14
• Level: $124$
• Weight: $3$
• Character: 124.7
• Analytic conductor: $3.379$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 124 = 2^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	124.n (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.37875527807\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(30\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			7.14
Character	\(\chi\)	\(=\)	124.7
Dual form			124.3.n.a.71.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.329666 + 1.97264i) q^{2} +(-0.974786 - 4.58601i) q^{3} +(-3.78264 - 1.30063i) q^{4} +(-0.229111 + 0.396833i) q^{5} +(9.36791 - 0.411054i) q^{6} +(-4.30093 + 9.66006i) q^{7} +(3.81268 - 7.03303i) q^{8} +(-11.8594 + 5.28013i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 6 q^{2} - 10 q^{4} - 8 q^{5} - 5 q^{6} - 27 q^{8} - 96 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(63\)	\(65\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{14}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.329666	+	1.97264i	−0.164833	+	0.986321i
							\(3\)	−0.974786	−	4.58601i	−0.324929	−	1.52867i	−0.772846	−	0.634594i	\(-0.781167\pi\)
0.447917	−	0.894075i	\(-0.352166\pi\)
\(5\)	−0.229111	+	0.396833i	−0.0458223	+	0.0793666i	−0.888027	−	0.459792i	\(-0.847924\pi\)
							0.842205	+	0.539158i	\(0.181257\pi\)
\(7\)	−4.30093	+	9.66006i	−0.614419	+	1.38001i	0.291505	+	0.956569i	\(0.405844\pi\)
							−0.905925	+	0.423439i	\(0.860823\pi\)
\(11\)	−16.1795	+	1.70053i	−1.47086	+	0.154594i	−0.805798	−	0.592190i	\(-0.798263\pi\)
							−0.665066	+	0.746785i	\(0.731597\pi\)
\(13\)	−13.3445	−	14.8205i	−1.02650	−	1.14004i	−0.990052	−	0.140704i	\(-0.955063\pi\)
							−0.0364446	−	0.999336i	\(-0.511603\pi\)
\(17\)	−1.08199	+	10.2945i	−0.0636466	+	0.605557i	0.915488	+	0.402345i	\(0.131805\pi\)
							−0.979135	+	0.203212i	\(0.934862\pi\)
\(19\)	0.674206	+	0.607058i	0.0354845	+	0.0319504i	0.686682	−	0.726958i	\(-0.259066\pi\)
							−0.651198	+	0.758908i	\(0.725733\pi\)
\(23\)	−0.175172	+	0.241103i	−0.00761615	+	0.0104827i	−0.812808	−	0.582532i	\(-0.802062\pi\)
							0.805192	+	0.593014i	\(0.202062\pi\)
\(29\)	−5.90603	−	18.1769i	−0.203656	−	0.626790i	−0.999766	−	0.0216371i	\(-0.993112\pi\)
							0.796110	−	0.605152i	\(-0.206888\pi\)
\(31\)	−8.69128	−	29.7567i	−0.280364	−	0.959894i
							\(37\)	−28.6804	−	49.6760i	−0.775147	−	1.34259i	−0.934712	−	0.355406i	\(-0.884343\pi\)
0.159565	−	0.987187i	\(-0.448991\pi\)
\(41\)	27.7901	+	5.90697i	0.677807	+	0.144072i	0.533937	−	0.845524i	\(-0.320712\pi\)
							0.143870	+	0.989597i	\(0.454045\pi\)
\(43\)	1.39301	+	1.25427i	0.0323955	+	0.0291690i	0.685169	−	0.728384i	\(-0.259728\pi\)
							−0.652774	+	0.757553i	\(0.726395\pi\)
\(47\)	34.2996	+	11.1446i	0.729779	+	0.237119i	0.650258	−	0.759714i	\(-0.274661\pi\)
							0.0795209	+	0.996833i	\(0.474661\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.3.n.a.7.14	✓	240
4.3	odd	2	inner	124.3.n.a.7.23	yes	240
31.9	even	15	inner	124.3.n.a.71.23	yes	240
124.71	odd	30	inner	124.3.n.a.71.14	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.3.n.a.7.14	✓	240	1.1	even	1	trivial
124.3.n.a.7.23	yes	240	4.3	odd	2	inner
124.3.n.a.71.14	yes	240	124.71	odd	30	inner
124.3.n.a.71.23	yes	240	31.9	even	15	inner