Embedded newform 123.4.g.a.10.2

• Label: 123.4.g.a.10.2
• Level: $123$
• Weight: $4$
• Character: 123.10
• Analytic conductor: $7.257$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 123 = 3 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	123.g (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.25723493071\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(10\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			10.2
Character	\(\chi\)	\(=\)	123.10
Dual form			123.4.g.a.37.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.01879 - 2.91982i) q^{2} -3.00000 q^{3} +(5.15318 + 15.8598i) q^{4} +(-5.25396 - 16.1700i) q^{5} +(12.0564 + 8.75946i) q^{6} +(20.7309 - 15.0619i) q^{7} +(13.3181 - 40.9888i) q^{8} +9.00000 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 2 q^{2} - 120 q^{3} - 40 q^{4} - 2 q^{5} + 6 q^{6} - 14 q^{7} - 12 q^{8} + 360 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(83\)	\(88\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.01879	−	2.91982i	−1.42086	−	1.03231i	−0.991629	−	0.129116i	\(-0.958786\pi\)
							−0.429227	−	0.903197i	\(-0.641214\pi\)
\(3\)	−3.00000			−0.577350
							\(5\)	−5.25396	−	16.1700i	−0.469929	−	1.44629i	−0.852677	−	0.522439i	\(-0.825022\pi\)
0.382748	−	0.923853i	\(-0.374978\pi\)
\(7\)	20.7309	−	15.0619i	1.11936	−	0.813265i	0.135251	−	0.990811i	\(-0.456816\pi\)
							0.984112	+	0.177547i	\(0.0568160\pi\)
\(11\)	20.6566	−	63.5745i	0.566200	−	1.74258i	−0.0981591	−	0.995171i	\(-0.531295\pi\)
							0.664359	−	0.747414i	\(-0.268705\pi\)
\(13\)	−29.6162	−	21.5174i	−0.631851	−	0.459066i	0.225190	−	0.974315i	\(-0.427700\pi\)
							−0.857041	+	0.515248i	\(0.827700\pi\)
\(17\)	−24.6272	+	75.7948i	−0.351352	+	1.08135i	0.606743	+	0.794898i	\(0.292476\pi\)
							−0.958095	+	0.286451i	\(0.907524\pi\)
\(19\)	75.7056	−	55.0034i	0.914109	−	0.664139i	−0.0279420	−	0.999610i	\(-0.508895\pi\)
							0.942051	+	0.335471i	\(0.108895\pi\)
\(23\)	−32.3520	−	23.5051i	−0.293298	−	0.213094i	0.431399	−	0.902161i	\(-0.358020\pi\)
							−0.724697	+	0.689068i	\(0.758020\pi\)
\(29\)	38.1852	+	117.522i	0.244511	+	0.752526i	0.995717	+	0.0924584i	\(0.0294725\pi\)
							−0.751206	+	0.660068i	\(0.770527\pi\)
\(31\)	−78.5845	+	241.858i	−0.455297	+	1.40126i	0.415490	+	0.909598i	\(0.363610\pi\)
							−0.870786	+	0.491662i	\(0.836390\pi\)
\(37\)	−100.511	−	309.341i	−0.446591	−	1.37447i	−0.880729	−	0.473620i	\(-0.842947\pi\)
							0.434138	−	0.900847i	\(-0.357053\pi\)
\(41\)	92.5939	−	245.657i	0.352701	−	0.935736i
							\(43\)	−146.818	−	106.669i	−0.520686	−	0.378301i	0.296176	−	0.955133i	\(-0.404288\pi\)
−0.816862	+	0.576833i	\(0.804288\pi\)
\(47\)	−9.27001	−	6.73506i	−0.0287696	−	0.0209023i	0.573308	−	0.819340i	\(-0.305660\pi\)
							−0.602077	+	0.798438i	\(0.705660\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	123.4.g.a.10.2	✓	40
41.37	even	5	inner	123.4.g.a.37.2	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
123.4.g.a.10.2	✓	40	1.1	even	1	trivial
123.4.g.a.37.2	yes	40	41.37	even	5	inner