Embedded newform 124.3.l.a.35.1

• Label: 124.3.l.a.35.1
• Level: $124$
• Weight: $3$
• Character: 124.35
• Analytic conductor: $3.379$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 124 = 2^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	124.l (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.37875527807\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(30\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			35.1
Character	\(\chi\)	\(=\)	124.35
Dual form			124.3.l.a.39.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99963 - 0.0384241i) q^{2} +(2.41735 + 0.785446i) q^{3} +(3.99705 + 0.153668i) q^{4} +7.23273 q^{5} +(-4.80363 - 1.66349i) q^{6} +(0.574978 + 0.791389i) q^{7} +(-7.98671 - 0.460862i) q^{8} +(-2.05448 - 1.49267i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 3 q^{2} + q^{4} - 16 q^{5} - 10 q^{6} + 27 q^{8} + 72 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(63\)	\(65\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99963	−	0.0384241i	−0.999815	−	0.0192121i
							\(3\)	2.41735	+	0.785446i	0.805784	+	0.261815i	0.682811	−	0.730595i	\(-0.260757\pi\)
0.122973	+	0.992410i	\(0.460757\pi\)
\(5\)	7.23273			1.44655			0.723273	−	0.690563i	\(-0.242637\pi\)
							0.723273	+	0.690563i	\(0.242637\pi\)
\(7\)	0.574978	+	0.791389i	0.0821397	+	0.113056i	0.848110	−	0.529821i	\(-0.177741\pi\)
							−0.765970	+	0.642876i	\(0.777741\pi\)
\(11\)	1.68954	+	2.32545i	0.153594	+	0.211405i	0.878879	−	0.477044i	\(-0.158292\pi\)
							−0.725285	+	0.688449i	\(0.758292\pi\)
\(13\)	1.24994	−	3.84692i	0.0961491	−	0.295917i	−0.891402	−	0.453213i	\(-0.850278\pi\)
							0.987551	+	0.157296i	\(0.0502778\pi\)
\(17\)	12.2534	+	8.90264i	0.720790	+	0.523685i	0.886637	−	0.462467i	\(-0.153036\pi\)
							−0.165846	+	0.986152i	\(0.553036\pi\)
\(19\)	−8.30194	+	2.69746i	−0.436944	+	0.141972i	−0.519225	−	0.854637i	\(-0.673779\pi\)
							0.0822811	+	0.996609i	\(0.473779\pi\)
\(23\)	6.72934	−	9.26214i	0.292580	−	0.402702i	−0.637270	−	0.770641i	\(-0.719936\pi\)
							0.929850	+	0.367939i	\(0.119936\pi\)
\(29\)	16.9115	+	52.0483i	0.583156	+	1.79477i	0.606554	+	0.795043i	\(0.292552\pi\)
							−0.0233978	+	0.999726i	\(0.507448\pi\)
\(31\)	−8.81687	−	29.7197i	−0.284415	−	0.958701i
							\(37\)	−33.0371			−0.892894			−0.446447	−	0.894810i	\(-0.647311\pi\)
−0.446447	+	0.894810i	\(0.647311\pi\)
\(41\)	−0.725016	−	2.23137i	−0.0176833	−	0.0544237i	0.941825	−	0.336102i	\(-0.109109\pi\)
							−0.959509	+	0.281679i	\(0.909109\pi\)
\(43\)	−50.8134	+	16.5103i	−1.18171	+	0.383960i	−0.833000	−	0.553273i	\(-0.813379\pi\)
							−0.348706	+	0.937232i	\(0.613379\pi\)
\(47\)	−20.7881	−	6.75445i	−0.442299	−	0.143712i	0.0793969	−	0.996843i	\(-0.474701\pi\)
							−0.521696	+	0.853131i	\(0.674701\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.3.l.a.35.1	✓	120
4.3	odd	2	inner	124.3.l.a.35.24	yes	120
31.8	even	5	inner	124.3.l.a.39.24	yes	120
124.39	odd	10	inner	124.3.l.a.39.1	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.3.l.a.35.1	✓	120	1.1	even	1	trivial
124.3.l.a.35.24	yes	120	4.3	odd	2	inner
124.3.l.a.39.1	yes	120	124.39	odd	10	inner
124.3.l.a.39.24	yes	120	31.8	even	5	inner