Embedded newform 336.2.bu.a.275.7

• Label: 336.2.bu.a.275.7
• Level: $336$
• Weight: $2$
• Character: 336.275
• Analytic conductor: $2.683$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	336.bu (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.68297350792\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(60\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			275.7
Character	\(\chi\)	\(=\)	336.275
Dual form			336.2.bu.a.11.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.33024 + 0.480058i) q^{2} +(-0.332434 + 1.69985i) q^{3} +(1.53909 - 1.27719i) q^{4} +(0.419196 - 0.112323i) q^{5} +(-0.373809 - 2.42080i) q^{6} +(1.95186 + 1.78613i) q^{7} +(-1.43423 + 2.43782i) q^{8} +(-2.77897 - 1.13018i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 2 q^{3} - 4 q^{4} - 8 q^{6} - 16 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\)	\(85\)	\(113\)	\(127\)	\(241\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.33024	+	0.480058i	−0.940623	+	0.339453i
							\(3\)	−0.332434	+	1.69985i	−0.191931	+	0.981408i
\(5\)	0.419196	−	0.112323i	0.187470	−	0.0502324i								−0.163863	−	0.986483i	\(-0.552395\pi\)
							0.351333	+	0.936251i	\(0.385729\pi\)
\(7\)	1.95186	+	1.78613i	0.737733	+	0.675092i
							\(11\)	2.16054	+	0.578915i	0.651427	+	0.174549i	0.569374	−	0.822079i	\(-0.307186\pi\)
0.0820532	+	0.996628i	\(0.473852\pi\)
\(13\)	1.27333	+	1.27333i	0.353157	+	0.353157i	0.861283	−	0.508126i	\(-0.169662\pi\)
							−0.508126	+	0.861283i	\(0.669662\pi\)
\(17\)	2.45410	−	1.41688i	0.595208	−	0.343643i	−0.171946	−	0.985106i	\(-0.555006\pi\)
							0.767154	+	0.641463i	\(0.221672\pi\)
\(19\)	1.13523	+	4.23675i	0.260440	+	0.971976i	0.964983	+	0.262314i	\(0.0844856\pi\)
							−0.704542	+	0.709662i	\(0.748848\pi\)
\(23\)	0.224843	+	0.129813i	0.0468830	+	0.0270679i	0.523258	−	0.852174i	\(-0.324716\pi\)
							−0.476375	+	0.879242i	\(0.658050\pi\)
\(29\)	−5.96729	+	5.96729i	−1.10810	+	1.10810i	−0.114697	+	0.993400i	\(0.536590\pi\)
							−0.993400	+	0.114697i	\(0.963410\pi\)
\(31\)	−4.40724	+	2.54452i	−0.791563	+	0.457009i	−0.840513	−	0.541792i	\(-0.817746\pi\)
							0.0489494	+	0.998801i	\(0.484413\pi\)
\(37\)	−1.23825	−	4.62122i	−0.203567	−	0.759724i	−0.989881	−	0.141897i	\(-0.954680\pi\)
							0.786314	−	0.617827i	\(-0.211987\pi\)
\(41\)	−7.14247			−1.11547			−0.557733	−	0.830020i	\(-0.688329\pi\)
							−0.557733	+	0.830020i	\(0.688329\pi\)
\(43\)	5.18463	+	5.18463i	0.790648	+	0.790648i	0.981599	−	0.190951i	\(-0.0611573\pi\)
							−0.190951	+	0.981599i	\(0.561157\pi\)
\(47\)	5.44958	−	9.43895i	0.794903	−	1.37681i	−0.127998	−	0.991774i	\(-0.540855\pi\)
							0.922901	−	0.385037i	\(-0.125811\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.bu.a.275.7	yes	240
3.2	odd	2	inner	336.2.bu.a.275.54	yes	240
7.4	even	3	inner	336.2.bu.a.179.35	yes	240
16.11	odd	4	inner	336.2.bu.a.107.26	yes	240
21.11	odd	6	inner	336.2.bu.a.179.26	yes	240
48.11	even	4	inner	336.2.bu.a.107.35	yes	240
112.11	odd	12	inner	336.2.bu.a.11.54	yes	240
336.11	even	12	inner	336.2.bu.a.11.7	✓	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.bu.a.11.7	✓	240	336.11	even	12	inner
336.2.bu.a.11.54	yes	240	112.11	odd	12	inner
336.2.bu.a.107.26	yes	240	16.11	odd	4	inner
336.2.bu.a.107.35	yes	240	48.11	even	4	inner
336.2.bu.a.179.26	yes	240	21.11	odd	6	inner
336.2.bu.a.179.35	yes	240	7.4	even	3	inner
336.2.bu.a.275.7	yes	240	1.1	even	1	trivial
336.2.bu.a.275.54	yes	240	3.2	odd	2	inner