Embedded newform 124.3.n.a.7.5

• Label: 124.3.n.a.7.5
• 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			7.5
Character	\(\chi\)	\(=\)	124.7
Dual form			124.3.n.a.71.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.76073 - 0.948596i) q^{2} +(-1.13434 - 5.33667i) q^{3} +(2.20033 + 3.34044i) q^{4} +(-3.56746 + 6.17902i) q^{5} +(-3.06507 + 10.4725i) q^{6} +(-0.864598 + 1.94192i) q^{7} +(-0.705452 - 7.96884i) q^{8} +(-18.9714 + 8.44661i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 6 q^{2} - 10 q^{4} - 8 q^{5} - 5 q^{6} - 27 q^{8} - 96 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.76073	−	0.948596i	−0.880364	−	0.474298i
							\(3\)	−1.13434	−	5.33667i	−0.378115	−	1.77889i	−0.596071	−	0.802932i	\(-0.703272\pi\)
0.217956	−	0.975959i	\(-0.430061\pi\)
\(5\)	−3.56746	+	6.17902i	−0.713492	+	1.23580i	0.250047	+	0.968234i	\(0.419554\pi\)
							−0.963538	+	0.267570i	\(0.913779\pi\)
\(7\)	−0.864598	+	1.94192i	−0.123514	+	0.277417i	−0.964711	−	0.263312i	\(-0.915185\pi\)
							0.841197	+	0.540729i	\(0.181852\pi\)
\(11\)	−5.86436	+	0.616369i	−0.533124	+	0.0560335i	−0.367266	−	0.930116i	\(-0.619706\pi\)
							−0.165858	+	0.986150i	\(0.553039\pi\)
\(13\)	13.3128	+	14.7854i	1.02406	+	1.13734i	0.990446	+	0.137904i	\(0.0440364\pi\)
							0.0336184	+	0.999435i	\(0.489297\pi\)
\(17\)	1.27665	−	12.1466i	0.0750973	−	0.714503i	−0.890592	−	0.454803i	\(-0.849710\pi\)
							0.965689	−	0.259700i	\(-0.0836237\pi\)
\(19\)	−6.48904	−	5.84276i	−0.341529	−	0.307514i	0.480461	−	0.877016i	\(-0.340469\pi\)
							−0.821990	+	0.569502i	\(0.807136\pi\)
\(23\)	−21.2352	+	29.2278i	−0.923271	+	1.27077i	0.0391560	+	0.999233i	\(0.487533\pi\)
							−0.962427	+	0.271540i	\(0.912467\pi\)
\(29\)	2.92488	+	9.00187i	0.100858	+	0.310409i	0.988736	−	0.149670i	\(-0.0478211\pi\)
							−0.887878	+	0.460079i	\(0.847821\pi\)
\(31\)	−29.8924	+	8.21263i	−0.964269	+	0.264923i
							\(37\)	9.28424	+	16.0808i	0.250926	+	0.434616i	0.963781	−	0.266695i	\(-0.0859317\pi\)
−0.712855	+	0.701311i	\(0.752598\pi\)
\(41\)	−74.6000	−	15.8567i	−1.81951	−	0.386749i	−0.833379	−	0.552702i	\(-0.813597\pi\)
							−0.986134	+	0.165953i	\(0.946930\pi\)
\(43\)	−2.70500	−	2.43559i	−0.0629070	−	0.0566417i	0.637078	−	0.770800i	\(-0.280143\pi\)
							−0.699984	+	0.714158i	\(0.746810\pi\)
\(47\)	−11.7812	−	3.82794i	−0.250664	−	0.0814455i	0.180990	−	0.983485i	\(-0.442070\pi\)
							−0.431654	+	0.902039i	\(0.642070\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.3.n.a.7.5	✓	240
4.3	odd	2	inner	124.3.n.a.7.21	yes	240
31.9	even	15	inner	124.3.n.a.71.21	yes	240
124.71	odd	30	inner	124.3.n.a.71.5	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.3.n.a.7.5	✓	240	1.1	even	1	trivial
124.3.n.a.7.21	yes	240	4.3	odd	2	inner
124.3.n.a.71.5	yes	240	124.71	odd	30	inner
124.3.n.a.71.21	yes	240	31.9	even	15	inner