Embedded newform 123.5.p.a.7.14

• Label: 123.5.p.a.7.14
• Level: $123$
• Weight: $5$
• Character: 123.7
• Analytic conductor: $12.715$
• Analytic rank: $0$
• Dimension: $448$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 123 = 3 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	123.p (of order \(40\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.7145054593\)
Analytic rank:	\(0\)
Dimension:	\(448\)
Relative dimension:	\(28\) over \(\Q(\zeta_{40})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{40}]$

Embedding invariants

Embedding label			7.14
Character	\(\chi\)	\(=\)	123.7
Dual form			123.5.p.a.88.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.140677 - 0.276094i) q^{2} +(-1.98848 + 4.80062i) q^{3} +(9.34813 - 12.8666i) q^{4} +(-0.000529835 - 0.00334525i) q^{5} +(1.60515 - 0.126328i) q^{6} +(32.6540 + 2.56992i) q^{7} +(-9.76428 - 1.54651i) q^{8} +(-19.0919 - 19.0919i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 448 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/123\mathbb{Z}\right)^\times\).

\(n\)	\(83\)	\(88\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{39}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.140677	−	0.276094i	−0.0351692	−	0.0690234i	0.872753	−	0.488162i	\(-0.162332\pi\)
							−0.907922	+	0.419138i	\(0.862332\pi\)
\(3\)	−1.98848	+	4.80062i	−0.220942	+	0.533402i
							\(5\)	−0.000529835	−	0.00334525i	−2.11934e−5	−	0.000133810i	0.987678	−	0.156501i	\(-0.0500216\pi\)
−0.987699	+	0.156368i	\(0.950022\pi\)
\(7\)	32.6540	+	2.56992i	0.666407	+	0.0524474i	0.407151	−	0.913361i	\(-0.366522\pi\)
							0.259257	+	0.965808i	\(0.416522\pi\)
\(11\)	38.7016	−	9.29143i	0.319848	−	0.0767887i	−0.0703425	−	0.997523i	\(-0.522409\pi\)
							0.390190	+	0.920734i	\(0.372409\pi\)
\(13\)	−123.370	−	105.368i	−0.730001	−	0.623480i	0.204737	−	0.978817i	\(-0.434366\pi\)
							−0.934738	+	0.355337i	\(0.884366\pi\)
\(17\)	218.820	+	134.093i	0.757161	+	0.463989i	0.846751	−	0.531990i	\(-0.178556\pi\)
							−0.0895895	+	0.995979i	\(0.528556\pi\)
\(19\)	405.629	−	346.440i	1.12363	−	0.959668i	0.124158	−	0.992262i	\(-0.460377\pi\)
							0.999469	+	0.0325948i	\(0.0103771\pi\)
\(23\)	−367.452	−	119.392i	−0.694616	−	0.225694i	−0.0596326	−	0.998220i	\(-0.518993\pi\)
							−0.634983	+	0.772526i	\(0.718993\pi\)
\(29\)	1351.58	−	828.250i	1.60711	−	0.984840i	0.628835	−	0.777539i	\(-0.283532\pi\)
							0.978277	−	0.207301i	\(-0.0664679\pi\)
\(31\)	326.631	+	449.569i	0.339887	+	0.467814i	0.944408	−	0.328775i	\(-0.106636\pi\)
							−0.604522	+	0.796589i	\(0.706636\pi\)
\(37\)	−102.723	−	74.6325i	−0.0750349	−	0.0545160i	0.549635	−	0.835405i	\(-0.314767\pi\)
							−0.624670	+	0.780889i	\(0.714767\pi\)
\(41\)	1426.08	+	889.983i	0.848349	+	0.529437i
							\(43\)	−1582.72	−	3106.27i	−0.855989	−	1.67997i	−0.725173	−	0.688566i	\(-0.758240\pi\)
−0.130816	−	0.991407i	\(-0.541760\pi\)
\(47\)	−938.322	+	73.8475i	−0.424772	+	0.0334303i	−0.289044	−	0.957316i	\(-0.593337\pi\)
							−0.135728	+	0.990746i	\(0.543337\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	123.5.p.a.7.14	✓	448
41.6	odd	40	inner	123.5.p.a.88.14	yes	448

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
123.5.p.a.7.14	✓	448	1.1	even	1	trivial
123.5.p.a.88.14	yes	448	41.6	odd	40	inner