Embedded newform 672.2.ch.a.11.12

• Label: 672.2.ch.a.11.12
• Level: $672$
• Weight: $2$
• Character: 672.11
• Analytic conductor: $5.366$
• Analytic rank: $0$
• Dimension: $992$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.ch (of order \(24\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.36594701583\)
Analytic rank:	\(0\)
Dimension:	\(992\)
Relative dimension:	\(124\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

Embedding invariants

Embedding label			11.12
Character	\(\chi\)	\(=\)	672.11
Dual form			672.2.ch.a.611.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.34744 + 0.429412i) q^{2} +(0.649335 - 1.60573i) q^{3} +(1.63121 - 1.15722i) q^{4} +(0.323918 + 2.46040i) q^{5} +(-0.185422 + 2.44246i) q^{6} +(1.39830 - 2.24605i) q^{7} +(-1.70104 + 2.25975i) q^{8} +(-2.15673 - 2.08531i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 4 q^{3} - 8 q^{4} - 16 q^{6} - 16 q^{7} - 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(421\)	\(449\)	\(577\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{8}\right)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.34744	+	0.429412i	−0.952787	+	0.303640i
							\(3\)	0.649335	−	1.60573i	0.374894	−	0.927068i
\(5\)	0.323918	+	2.46040i	0.144861	+	1.10033i								0.897223	+	0.441577i	\(0.145581\pi\)
							−0.752363	+	0.658749i	\(0.771086\pi\)
\(7\)	1.39830	−	2.24605i	0.528508	−	0.848928i
							\(11\)	−1.69356	+	1.29951i	−0.510627	+	0.391818i	−0.831690	−	0.555240i	\(-0.812626\pi\)
0.321063	+	0.947058i	\(0.395960\pi\)
\(13\)	−1.32330	−	3.19473i	−0.367017	−	0.886057i	−0.994236	−	0.107214i	\(-0.965807\pi\)
							0.627219	−	0.778843i	\(-0.284193\pi\)
\(17\)	3.12001	−	5.40401i	0.756713	−	1.31066i	−0.187806	−	0.982206i	\(-0.560138\pi\)
							0.944518	−	0.328459i	\(-0.106529\pi\)
\(19\)	−0.976565	−	0.749345i	−0.224039	−	0.171912i	0.490643	−	0.871361i	\(-0.336762\pi\)
							−0.714682	+	0.699449i	\(0.753429\pi\)
\(23\)	1.32828	+	0.355911i	0.276965	+	0.0742125i	0.394628	−	0.918841i	\(-0.370874\pi\)
							−0.117663	+	0.993054i	\(0.537540\pi\)
\(29\)	−2.91333	−	7.03339i	−0.540991	−	1.30607i	−0.924024	−	0.382334i	\(-0.875120\pi\)
							0.383033	−	0.923735i	\(-0.374880\pi\)
\(31\)	4.96272	+	2.86523i	0.891331	+	0.514610i	0.874378	−	0.485246i	\(-0.161270\pi\)
							0.0169534	+	0.999856i	\(0.494603\pi\)
\(37\)	8.68803	−	1.14380i	1.42830	−	0.188040i	0.623612	−	0.781734i	\(-0.285665\pi\)
							0.804691	+	0.593694i	\(0.202331\pi\)
\(41\)	−1.81436	−	1.81436i	−0.283355	−	0.283355i	0.551091	−	0.834445i	\(-0.314212\pi\)
							−0.834445	+	0.551091i	\(0.814212\pi\)
\(43\)	−3.33702	+	8.05627i	−0.508890	+	1.22857i	0.435632	+	0.900125i	\(0.356525\pi\)
							−0.944523	+	0.328446i	\(0.893475\pi\)
\(47\)	5.86631	−	3.38692i	0.855689	−	0.494032i	−0.00687705	−	0.999976i	\(-0.502189\pi\)
							0.862566	+	0.505944i	\(0.168856\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.ch.a.11.12	✓	992
3.2	odd	2	inner	672.2.ch.a.11.113	yes	992
7.2	even	3	inner	672.2.ch.a.107.96	yes	992
21.2	odd	6	inner	672.2.ch.a.107.29	yes	992
32.3	odd	8	inner	672.2.ch.a.515.29	yes	992
96.35	even	8	inner	672.2.ch.a.515.96	yes	992
224.163	odd	24	inner	672.2.ch.a.611.113	yes	992
672.611	even	24	inner	672.2.ch.a.611.12	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.ch.a.11.12	✓	992	1.1	even	1	trivial
672.2.ch.a.11.113	yes	992	3.2	odd	2	inner
672.2.ch.a.107.29	yes	992	21.2	odd	6	inner
672.2.ch.a.107.96	yes	992	7.2	even	3	inner
672.2.ch.a.515.29	yes	992	32.3	odd	8	inner
672.2.ch.a.515.96	yes	992	96.35	even	8	inner
672.2.ch.a.611.12	yes	992	672.611	even	24	inner
672.2.ch.a.611.113	yes	992	224.163	odd	24	inner