Embedded newform 456.2.bu.c.131.10

• Label: 456.2.bu.c.131.10
• Level: $456$
• Weight: $2$
• Character: 456.131
• Analytic conductor: $3.641$
• Analytic rank: $0$
• Dimension: $432$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.64117833217\)
Analytic rank:	\(0\)
Dimension:	\(432\)
Relative dimension:	\(72\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			131.10
Character	\(\chi\)	\(=\)	456.131
Dual form			456.2.bu.c.275.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.23959 + 0.680745i) q^{2} +(-0.975855 + 1.43098i) q^{3} +(1.07317 - 1.68769i) q^{4} +(0.356822 - 0.129873i) q^{5} +(0.235529 - 2.43814i) q^{6} +(-1.66467 + 0.961100i) q^{7} +(-0.181410 + 2.82260i) q^{8} +(-1.09541 - 2.79286i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 18 q^{3} - 6 q^{4} + 6 q^{6} - 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(229\)	\(305\)	\(343\)
\(\chi(n)\)	\(e\left(\frac{5}{9}\right)\)	\(-1\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.23959	+	0.680745i	−0.876523	+	0.481359i
							\(3\)	−0.975855	+	1.43098i	−0.563410	+	0.826177i
\(5\)	0.356822	−	0.129873i	0.159576	−	0.0580808i								−0.260997	−	0.965340i	\(-0.584051\pi\)
							0.420573	+	0.907259i	\(0.361829\pi\)
\(7\)	−1.66467	+	0.961100i	−0.629187	+	0.363262i	−0.780437	−	0.625234i	\(-0.785004\pi\)
							0.151250	+	0.988496i	\(0.451670\pi\)
\(11\)	0.285778	+	0.164994i	0.0861654	+	0.0497476i	0.542464	−	0.840079i	\(-0.317492\pi\)
							−0.456298	+	0.889827i	\(0.650825\pi\)
\(13\)	−2.20788	−	2.63125i	−0.612356	−	0.729777i	0.367380	−	0.930071i	\(-0.380255\pi\)
							−0.979736	+	0.200294i	\(0.935810\pi\)
\(17\)	−4.48165	−	0.790236i	−1.08696	−	0.191660i	−0.398669	−	0.917095i	\(-0.630528\pi\)
							−0.688291	+	0.725435i	\(0.741639\pi\)
\(19\)	3.91742	+	1.91150i	0.898718	+	0.438527i
							\(23\)	−6.59560	−	2.40060i	−1.37528	−	0.500560i	−0.454534	−	0.890729i	\(-0.650194\pi\)
−0.920743	+	0.390169i	\(0.872417\pi\)
\(29\)	−1.57245	−	8.91779i	−0.291996	−	1.65599i	−0.679166	−	0.733985i	\(-0.737658\pi\)
							0.387170	−	0.922008i	\(-0.373453\pi\)
\(31\)	2.02049	−	1.16653i	0.362890	−	0.209515i	−0.307458	−	0.951562i	\(-0.599478\pi\)
							0.670348	+	0.742047i	\(0.266145\pi\)
\(37\)		−	4.18527i		−	0.688055i	−0.938960	−	0.344027i	\(-0.888209\pi\)
							0.938960	−	0.344027i	\(-0.111791\pi\)
\(41\)	6.03778	−	7.19555i	0.942943	−	1.12376i	−0.0492173	−	0.998788i	\(-0.515673\pi\)
							0.992161	−	0.124968i	\(-0.0398829\pi\)
\(43\)	1.50204	−	0.546697i	0.229059	−	0.0833705i	−0.224941	−	0.974372i	\(-0.572219\pi\)
							0.454000	+	0.891002i	\(0.349997\pi\)
\(47\)	−1.36351	−	7.73282i	−0.198888	−	1.12795i	−0.906772	−	0.421620i	\(-0.861462\pi\)
							0.707885	−	0.706328i	\(-0.249650\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	456.2.bu.c.131.10	✓	432
3.2	odd	2	inner	456.2.bu.c.131.63	yes	432
8.3	odd	2	inner	456.2.bu.c.131.19	yes	432
19.9	even	9	inner	456.2.bu.c.275.54	yes	432
24.11	even	2	inner	456.2.bu.c.131.54	yes	432
57.47	odd	18	inner	456.2.bu.c.275.19	yes	432
152.123	odd	18	inner	456.2.bu.c.275.63	yes	432
456.275	even	18	inner	456.2.bu.c.275.10	yes	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.bu.c.131.10	✓	432	1.1	even	1	trivial
456.2.bu.c.131.19	yes	432	8.3	odd	2	inner
456.2.bu.c.131.54	yes	432	24.11	even	2	inner
456.2.bu.c.131.63	yes	432	3.2	odd	2	inner
456.2.bu.c.275.10	yes	432	456.275	even	18	inner
456.2.bu.c.275.19	yes	432	57.47	odd	18	inner
456.2.bu.c.275.54	yes	432	19.9	even	9	inner
456.2.bu.c.275.63	yes	432	152.123	odd	18	inner