Embedded newform 124.3.n.a.7.13

• Label: 124.3.n.a.7.13
• 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			7.13
Character	\(\chi\)	\(=\)	124.7
Dual form			124.3.n.a.71.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.561919 - 1.91944i) q^{2} +(0.736057 + 3.46287i) q^{3} +(-3.36849 + 2.15714i) q^{4} +(3.42633 - 5.93457i) q^{5} +(6.23317 - 3.35867i) q^{6} +(-2.34572 + 5.26857i) q^{7} +(6.03332 + 5.25348i) q^{8} +(-3.22781 + 1.43711i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 6 q^{2} - 10 q^{4} - 8 q^{5} - 5 q^{6} - 27 q^{8} - 96 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.561919	−	1.91944i	−0.280959	−	0.959720i
							\(3\)	0.736057	+	3.46287i	0.245352	+	1.15429i	0.912407	+	0.409283i	\(0.134221\pi\)
−0.667055	+	0.745008i	\(0.732446\pi\)
\(5\)	3.42633	−	5.93457i	0.685265	−	1.18691i	−0.288088	−	0.957604i	\(-0.593020\pi\)
							0.973353	−	0.229311i	\(-0.0736471\pi\)
\(7\)	−2.34572	+	5.26857i	−0.335102	+	0.752652i	0.664881	+	0.746949i	\(0.268482\pi\)
							−0.999984	+	0.00570327i	\(0.998185\pi\)
\(11\)	21.1181	−	2.21960i	1.91983	−	0.201782i	0.932577	−	0.360971i	\(-0.117555\pi\)
							0.987248	+	0.159189i	\(0.0508879\pi\)
\(13\)	11.1195	+	12.3495i	0.855349	+	0.949961i	0.999214	−	0.0396362i	\(-0.0126199\pi\)
							−0.143865	+	0.989597i	\(0.545953\pi\)
\(17\)	−0.527025	+	5.01431i	−0.0310015	+	0.294959i	0.968025	+	0.250852i	\(0.0807109\pi\)
							−0.999027	+	0.0441070i	\(0.985956\pi\)
\(19\)	−15.8066	−	14.2323i	−0.831928	−	0.749071i	0.138526	−	0.990359i	\(-0.455764\pi\)
							−0.970454	+	0.241288i	\(0.922430\pi\)
\(23\)	−4.49597	+	6.18817i	−0.195477	+	0.269051i	−0.895492	−	0.445077i	\(-0.853176\pi\)
							0.700016	+	0.714128i	\(0.253176\pi\)
\(29\)	−15.4208	−	47.4604i	−0.531752	−	1.63656i	−0.750564	−	0.660797i	\(-0.770218\pi\)
							0.218813	−	0.975767i	\(-0.429782\pi\)
\(31\)	−28.2018	+	12.8707i	−0.909737	+	0.415185i
							\(37\)	−2.79489	−	4.84089i	−0.0755375	−	0.130835i	0.825782	−	0.563989i	\(-0.190734\pi\)
−0.901320	+	0.433154i	\(0.857401\pi\)
\(41\)	7.97361	+	1.69484i	0.194478	+	0.0413377i	0.304121	−	0.952633i	\(-0.401637\pi\)
							−0.109643	+	0.993971i	\(0.534971\pi\)
\(43\)	24.3448	+	21.9201i	0.566158	+	0.509771i	0.901762	−	0.432232i	\(-0.142274\pi\)
							−0.335605	+	0.942003i	\(0.608941\pi\)
\(47\)	−72.1870	−	23.4550i	−1.53589	−	0.499042i	−0.585654	−	0.810561i	\(-0.699162\pi\)
							−0.950240	+	0.311519i	\(0.899162\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.3.n.a.7.13	yes	240
4.3	odd	2	inner	124.3.n.a.7.12	✓	240
31.9	even	15	inner	124.3.n.a.71.12	yes	240
124.71	odd	30	inner	124.3.n.a.71.13	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.3.n.a.7.12	✓	240	4.3	odd	2	inner
124.3.n.a.7.13	yes	240	1.1	even	1	trivial
124.3.n.a.71.12	yes	240	31.9	even	15	inner
124.3.n.a.71.13	yes	240	124.71	odd	30	inner