Embedded newform 124.3.l.a.35.11

• Label: 124.3.l.a.35.11
• 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			35.11
Character	\(\chi\)	\(=\)	124.35
Dual form			124.3.l.a.39.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.904691 - 1.78369i) q^{2} +(-1.26908 - 0.412350i) q^{3} +(-2.36307 + 3.22737i) q^{4} -3.39555 q^{5} +(0.412626 + 2.63670i) q^{6} +(4.74424 + 6.52988i) q^{7} +(7.89446 + 1.29520i) q^{8} +(-5.84061 - 4.24345i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 3 q^{2} + q^{4} - 16 q^{5} - 10 q^{6} + 27 q^{8} + 72 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.904691	−	1.78369i	−0.452346	−	0.891843i
							\(3\)	−1.26908	−	0.412350i	−0.423028	−	0.137450i	0.0897638	−	0.995963i	\(-0.471389\pi\)
−0.512792	+	0.858513i	\(0.671389\pi\)
\(5\)	−3.39555			−0.679110			−0.339555	−	0.940586i	\(-0.610277\pi\)
							−0.339555	+	0.940586i	\(0.610277\pi\)
\(7\)	4.74424	+	6.52988i	0.677748	+	0.932840i	0.999904	−	0.0138462i	\(-0.00440751\pi\)
							−0.322156	+	0.946687i	\(0.604408\pi\)
\(11\)	11.3616	+	15.6380i	1.03288	+	1.42163i	0.902766	+	0.430132i	\(0.141533\pi\)
							0.130110	+	0.991500i	\(0.458467\pi\)
\(13\)	−1.03750	+	3.19309i	−0.0798076	+	0.245623i	−0.982998	−	0.183618i	\(-0.941219\pi\)
							0.903190	+	0.429241i	\(0.141219\pi\)
\(17\)	16.9607	+	12.3227i	0.997688	+	0.724863i	0.961591	−	0.274485i	\(-0.0885074\pi\)
							0.0360971	+	0.999348i	\(0.488507\pi\)
\(19\)	−17.3206	+	5.62780i	−0.911610	+	0.296200i	−0.727021	−	0.686616i	\(-0.759096\pi\)
							−0.184590	+	0.982816i	\(0.559096\pi\)
\(23\)	5.79544	−	7.97674i	0.251976	−	0.346815i	−0.664226	−	0.747532i	\(-0.731239\pi\)
							0.916202	+	0.400717i	\(0.131239\pi\)
\(29\)	11.5876	+	35.6630i	0.399573	+	1.22976i	0.925343	+	0.379132i	\(0.123777\pi\)
							−0.525770	+	0.850627i	\(0.676223\pi\)
\(31\)	−8.73751	+	29.7432i	−0.281855	+	0.959457i
							\(37\)	−2.06373			−0.0557766			−0.0278883	−	0.999611i	\(-0.508878\pi\)
−0.0278883	+	0.999611i	\(0.508878\pi\)
\(41\)	−15.8322	−	48.7264i	−0.386151	−	1.18845i	−0.935642	−	0.352950i	\(-0.885178\pi\)
							0.549491	−	0.835499i	\(-0.314822\pi\)
\(43\)	−75.3810	+	24.4928i	−1.75305	+	0.569599i	−0.996443	−	0.0842740i	\(-0.973143\pi\)
							−0.756604	+	0.653873i	\(0.773143\pi\)
\(47\)	19.5843	+	6.36333i	0.416688	+	0.135390i	0.509855	−	0.860260i	\(-0.329699\pi\)
							−0.0931671	+	0.995650i	\(0.529699\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.3.l.a.35.11	✓	120
4.3	odd	2	inner	124.3.l.a.35.15	yes	120
31.8	even	5	inner	124.3.l.a.39.15	yes	120
124.39	odd	10	inner	124.3.l.a.39.11	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.3.l.a.35.11	✓	120	1.1	even	1	trivial
124.3.l.a.35.15	yes	120	4.3	odd	2	inner
124.3.l.a.39.11	yes	120	124.39	odd	10	inner
124.3.l.a.39.15	yes	120	31.8	even	5	inner