Embedded newform 912.2.bv.c.539.93

• Label: 912.2.bv.c.539.93
• Level: $912$
• Weight: $2$
• Character: 912.539
• Analytic conductor: $7.282$
• Analytic rank: $0$
• Dimension: $608$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	912.bv (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.28235666434\)
Analytic rank:	\(0\)
Dimension:	\(608\)
Relative dimension:	\(152\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			539.93
Character	\(\chi\)	\(=\)	912.539
Dual form			912.2.bv.c.467.93

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.455088 + 1.33899i) q^{2} +(1.26606 + 1.18198i) q^{3} +(-1.58579 + 1.21872i) q^{4} +(-0.660998 - 0.177114i) q^{5} +(-1.00649 + 2.23315i) q^{6} +4.00908 q^{7} +(-2.35352 - 1.56873i) q^{8} +(0.205829 + 2.99293i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 608 q - 4 q^{4} - 2 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\)	\(97\)	\(229\)	\(305\)	\(799\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.455088	+	1.33899i	0.321796	+	0.946809i
							\(3\)	1.26606	+	1.18198i	0.730962	+	0.682419i
\(5\)	−0.660998	−	0.177114i	−0.295607	−	0.0792077i								0.107967	−	0.994154i	\(-0.465566\pi\)
							−0.403575	+	0.914947i	\(0.632232\pi\)
\(7\)	4.00908			1.51529			0.757645	−	0.652666i	\(-0.226350\pi\)
							0.757645	+	0.652666i	\(0.226350\pi\)
\(11\)	0.806273	+	0.806273i	0.243100	+	0.243100i	0.818132	−	0.575031i	\(-0.195010\pi\)
							−0.575031	+	0.818132i	\(0.695010\pi\)
\(13\)	1.05243	+	3.92772i	0.291891	+	1.08935i	0.943655	+	0.330931i	\(0.107363\pi\)
							−0.651763	+	0.758422i	\(0.725971\pi\)
\(17\)	5.44955	−	3.14630i	1.32171	−	0.763090i	0.337709	−	0.941251i	\(-0.390348\pi\)
							0.984001	+	0.178161i	\(0.0570148\pi\)
\(19\)	−2.47626	+	3.58722i	−0.568093	+	0.822964i
							\(23\)	−1.80276	−	1.04083i	−0.375902	−	0.217027i	0.300132	−	0.953898i	\(-0.402969\pi\)
−0.676034	+	0.736871i	\(0.736303\pi\)
\(29\)	−3.58990	+	0.961912i	−0.666629	+	0.178623i	−0.576236	−	0.817284i	\(-0.695479\pi\)
							−0.0903929	+	0.995906i	\(0.528812\pi\)
\(31\)		−	4.59424i		−	0.825150i	−0.910924	−	0.412575i	\(-0.864630\pi\)
							0.910924	−	0.412575i	\(-0.135370\pi\)
\(37\)	8.24631	+	8.24631i	1.35568	+	1.35568i	0.879163	+	0.476522i	\(0.158103\pi\)
							0.476522	+	0.879163i	\(0.341897\pi\)
\(41\)	−3.33285	−	5.77267i	−0.520504	−	0.901540i	−0.999716	−	0.0238403i	\(-0.992411\pi\)
							0.479212	−	0.877699i	\(-0.340923\pi\)
\(43\)	−5.17787	−	1.38741i	−0.789618	−	0.211578i	−0.158597	−	0.987343i	\(-0.550697\pi\)
							−0.631021	+	0.775766i	\(0.717364\pi\)
\(47\)	−3.77759	+	6.54298i	−0.551018	+	0.954392i	0.447183	+	0.894442i	\(0.352427\pi\)
							−0.998201	+	0.0599493i	\(0.980906\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.bv.c.539.93	yes	608
3.2	odd	2	inner	912.2.bv.c.539.60	yes	608
16.3	odd	4	inner	912.2.bv.c.83.43	yes	608
19.11	even	3	inner	912.2.bv.c.11.110	yes	608
48.35	even	4	inner	912.2.bv.c.83.110	yes	608
57.11	odd	6	inner	912.2.bv.c.11.43	✓	608
304.163	odd	12	inner	912.2.bv.c.467.60	yes	608
912.467	even	12	inner	912.2.bv.c.467.93	yes	608

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.2.bv.c.11.43	✓	608	57.11	odd	6	inner
912.2.bv.c.11.110	yes	608	19.11	even	3	inner
912.2.bv.c.83.43	yes	608	16.3	odd	4	inner
912.2.bv.c.83.110	yes	608	48.35	even	4	inner
912.2.bv.c.467.60	yes	608	304.163	odd	12	inner
912.2.bv.c.467.93	yes	608	912.467	even	12	inner
912.2.bv.c.539.60	yes	608	3.2	odd	2	inner
912.2.bv.c.539.93	yes	608	1.1	even	1	trivial