Embedded newform 456.2.bk.a.253.7

• Label: 456.2.bk.a.253.7
• Level: $456$
• Weight: $2$
• Character: 456.253
• Analytic conductor: $3.641$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.64117833217\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(40\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			253.7
Character	\(\chi\)	\(=\)	456.253
Dual form			456.2.bk.a.301.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.11709 + 0.867238i) q^{2} +(0.342020 - 0.939693i) q^{3} +(0.495798 - 1.93757i) q^{4} +(-0.520571 + 0.0917907i) q^{5} +(0.432868 + 1.34634i) q^{6} +(-1.60418 + 2.77852i) q^{7} +(1.12648 + 2.59442i) q^{8} +(-0.766044 - 0.642788i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 6 q^{4} + 6 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(229\)	\(305\)	\(343\)
\(\chi(n)\)	\(e\left(\frac{7}{9}\right)\)	\(-1\)	\(1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.11709	+	0.867238i	−0.789905	+	0.613230i
							\(3\)	0.342020	−	0.939693i	0.197465	−	0.542532i
\(5\)	−0.520571	+	0.0917907i	−0.232806	+	0.0410501i								−0.288834	−	0.957379i	\(-0.593268\pi\)
							0.0560277	+	0.998429i	\(0.482156\pi\)
\(7\)	−1.60418	+	2.77852i	−0.606322	+	1.05018i	0.385519	+	0.922700i	\(0.374023\pi\)
							−0.991841	+	0.127481i	\(0.959311\pi\)
\(11\)	−2.78308	+	1.60681i	−0.839129	+	0.484471i	−0.856968	−	0.515369i	\(-0.827655\pi\)
							0.0178390	+	0.999841i	\(0.494321\pi\)
\(13\)	−0.857041	−	2.35470i	−0.237700	−	0.653076i	−0.999983	−	0.00583210i	\(-0.998144\pi\)
							0.762283	−	0.647244i	\(-0.224079\pi\)
\(17\)	−0.870216	+	0.730198i	−0.211058	+	0.177099i	−0.742188	−	0.670191i	\(-0.766212\pi\)
							0.531130	+	0.847290i	\(0.321768\pi\)
\(19\)	3.90669	−	1.93334i	0.896255	−	0.443539i
							\(23\)	−1.50449	+	8.53241i	−0.313709	+	1.77913i	0.265662	+	0.964066i	\(0.414409\pi\)
−0.579371	+	0.815064i	\(0.696702\pi\)
\(29\)	−6.85130	+	8.16506i	−1.27225	+	1.51621i	−0.526137	+	0.850400i	\(0.676360\pi\)
							−0.746118	+	0.665814i	\(0.768084\pi\)
\(31\)	−2.73785	+	4.74210i	−0.491733	+	0.851706i	−0.999955	−	0.00951982i	\(-0.996970\pi\)
							0.508222	+	0.861226i	\(0.330303\pi\)
\(37\)		−	4.11790i		−	0.676979i	−0.940970	−	0.338489i	\(-0.890084\pi\)
							0.940970	−	0.338489i	\(-0.109916\pi\)
\(41\)	−8.03640	−	2.92501i	−1.25507	−	0.456810i	−0.372961	−	0.927847i	\(-0.621657\pi\)
							−0.882113	+	0.471037i	\(0.843880\pi\)
\(43\)	9.80241	−	1.72843i	1.49485	−	0.263583i	0.634358	−	0.773040i	\(-0.281265\pi\)
							0.860496	+	0.509457i	\(0.170154\pi\)
\(47\)	1.26671	+	1.06290i	0.184769	+	0.155039i	0.730480	−	0.682934i	\(-0.239296\pi\)
							−0.545712	+	0.837973i	\(0.683741\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	456.2.bk.a.253.7	✓	240
8.5	even	2	inner	456.2.bk.a.253.11	yes	240
19.16	even	9	inner	456.2.bk.a.301.11	yes	240
152.149	even	18	inner	456.2.bk.a.301.7	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.bk.a.253.7	✓	240	1.1	even	1	trivial
456.2.bk.a.253.11	yes	240	8.5	even	2	inner
456.2.bk.a.301.7	yes	240	152.149	even	18	inner
456.2.bk.a.301.11	yes	240	19.16	even	9	inner