Embedded newform 456.2.bj.a.29.48

• Label: 456.2.bj.a.29.48
• Level: $456$
• Weight: $2$
• Character: 456.29
• Analytic conductor: $3.641$
• Analytic rank: $0$
• Dimension: $456$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	456.bj (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.64117833217\)
Analytic rank:	\(0\)
Dimension:	\(456\)
Relative dimension:	\(76\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			29.48
Character	\(\chi\)	\(=\)	456.29
Dual form			456.2.bj.a.173.48

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.566068 - 1.29598i) q^{2} +(-1.48828 - 0.886022i) q^{3} +(-1.35913 - 1.46723i) q^{4} +(-1.49063 - 0.542546i) q^{5} +(-1.99073 + 1.42723i) q^{6} +(2.47083 - 4.27960i) q^{7} +(-2.67086 + 0.930859i) q^{8} +(1.42993 + 2.63729i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 6 q^{4} - 6 q^{6} - 12 q^{7} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/456\mathbb{Z}\right)^\times\).

\(n\)	\(97\)	\(229\)	\(305\)	\(343\)
\(\chi(n)\)	\(e\left(\frac{17}{18}\right)\)	\(-1\)	\(-1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.566068	−	1.29598i	0.400271	−	0.916397i
							\(3\)	−1.48828	−	0.886022i	−0.859257	−	0.511545i
\(5\)	−1.49063	−	0.542546i	−0.666632	−	0.242634i								−0.0135349	−	0.999908i	\(-0.504308\pi\)
							−0.653097	+	0.757274i	\(0.726531\pi\)
\(7\)	2.47083	−	4.27960i	0.933884	−	1.61754i	0.157273	−	0.987555i	\(-0.449730\pi\)
							0.776611	−	0.629980i	\(-0.216937\pi\)
\(11\)	−1.56878	−	2.71721i	−0.473006	−	0.819270i	0.526517	−	0.850165i	\(-0.323498\pi\)
							−0.999523	+	0.0308945i	\(0.990164\pi\)
\(13\)	0.796665	+	0.668481i	0.220955	+	0.185403i	0.746545	−	0.665334i	\(-0.231711\pi\)
							−0.525590	+	0.850738i	\(0.676155\pi\)
\(17\)	−4.93455	+	0.870094i	−1.19680	+	0.211029i	−0.736316	−	0.676638i	\(-0.763436\pi\)
							−0.460487	+	0.887666i	\(0.652325\pi\)
\(19\)	0.310232	+	4.34785i	0.0711720	+	0.997464i
							\(23\)	1.04161	+	2.86179i	0.217190	+	0.596724i	0.999663	−	0.0259603i	\(-0.00826435\pi\)
−0.782473	+	0.622684i	\(0.786042\pi\)
\(29\)	1.32946	+	0.234419i	0.246874	+	0.0435305i	0.295716	−	0.955276i	\(-0.404442\pi\)
							−0.0488417	+	0.998807i	\(0.515553\pi\)
\(31\)	4.61174	+	2.66259i	0.828293	+	0.478215i	0.853268	−	0.521473i	\(-0.174617\pi\)
							−0.0249746	+	0.999688i	\(0.507950\pi\)
\(37\)	−8.08676			−1.32945			−0.664727	−	0.747086i	\(-0.731452\pi\)
							−0.664727	+	0.747086i	\(0.731452\pi\)
\(41\)	8.91543	−	7.48094i	1.39236	−	1.16833i	0.427982	−	0.903787i	\(-0.359225\pi\)
							0.964375	−	0.264539i	\(-0.0852197\pi\)
\(43\)	2.81235	−	7.72687i	0.428879	−	1.17834i	−0.517614	−	0.855614i	\(-0.673180\pi\)
							0.946494	−	0.322722i	\(-0.104598\pi\)
\(47\)	−9.37853	−	1.65369i	−1.36800	−	0.241215i	−0.559069	−	0.829121i	\(-0.688841\pi\)
							−0.808929	+	0.587906i	\(0.799953\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	456.2.bj.a.29.48	yes	456
3.2	odd	2	inner	456.2.bj.a.29.29	yes	456
8.5	even	2	inner	456.2.bj.a.29.21	✓	456
19.2	odd	18	inner	456.2.bj.a.173.56	yes	456
24.5	odd	2	inner	456.2.bj.a.29.56	yes	456
57.2	even	18	inner	456.2.bj.a.173.21	yes	456
152.21	odd	18	inner	456.2.bj.a.173.29	yes	456
456.173	even	18	inner	456.2.bj.a.173.48	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.bj.a.29.21	✓	456	8.5	even	2	inner
456.2.bj.a.29.29	yes	456	3.2	odd	2	inner
456.2.bj.a.29.48	yes	456	1.1	even	1	trivial
456.2.bj.a.29.56	yes	456	24.5	odd	2	inner
456.2.bj.a.173.21	yes	456	57.2	even	18	inner
456.2.bj.a.173.29	yes	456	152.21	odd	18	inner
456.2.bj.a.173.48	yes	456	456.173	even	18	inner
456.2.bj.a.173.56	yes	456	19.2	odd	18	inner