Embedded newform 252.2.bj.b.103.6

• Label: 252.2.bj.b.103.6
• Level: $252$
• Weight: $2$
• Character: 252.103
• Analytic conductor: $2.012$
• Analytic rank: $0$
• Dimension: $84$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	252.bj (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.01223013094\)
Analytic rank:	\(0\)
Dimension:	\(84\)
Relative dimension:	\(42\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			103.6
Character	\(\chi\)	\(=\)	252.103
Dual form			252.2.bj.b.115.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.33100 + 0.477940i) q^{2} +(-1.43852 - 0.964710i) q^{3} +(1.54315 - 1.27228i) q^{4} +(3.01564 - 1.74108i) q^{5} +(2.37575 + 0.596507i) q^{6} +(-1.90425 - 1.83680i) q^{7} +(-1.44586 + 2.43094i) q^{8} +(1.13867 + 2.77551i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q + 2 q^{2} - 2 q^{4} + 6 q^{5} - 16 q^{8} + 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(73\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.33100	+	0.477940i	−0.941162	+	0.337955i
							\(3\)	−1.43852	−	0.964710i	−0.830529	−	0.556975i
\(5\)	3.01564	−	1.74108i	1.34863	−	0.778634i								0.360577	−	0.932729i	\(-0.382580\pi\)
							0.988056	+	0.154096i	\(0.0492464\pi\)
\(7\)	−1.90425	−	1.83680i	−0.719740	−	0.694244i
							\(11\)	1.71090	+	0.987787i	0.515855	+	0.297829i	0.735237	−	0.677810i	\(-0.237071\pi\)
−0.219382	+	0.975639i	\(0.570404\pi\)
\(13\)	−5.52258	−	3.18847i	−1.53169	−	0.884321i	−0.999284	−	0.0378315i	\(-0.987955\pi\)
							−0.532405	−	0.846490i	\(-0.678712\pi\)
\(17\)	1.04054	−	0.600755i	0.252367	−	0.145704i	−0.368480	−	0.929636i	\(-0.620122\pi\)
							0.620848	+	0.783931i	\(0.286788\pi\)
\(19\)	2.25453	−	3.90496i	0.517224	−	0.895858i	−0.482576	−	0.875854i	\(-0.660299\pi\)
							0.999800	−	0.0200042i	\(-0.00636795\pi\)
\(23\)	−0.511185	+	0.295133i	−0.106589	+	0.0615394i	−0.552347	−	0.833614i	\(-0.686268\pi\)
							0.445758	+	0.895154i	\(0.352934\pi\)
\(29\)	−2.00631	−	3.47503i	−0.372562	−	0.645296i	0.617397	−	0.786652i	\(-0.288187\pi\)
							−0.989959	+	0.141356i	\(0.954854\pi\)
\(31\)	−5.27982			−0.948283			−0.474142	−	0.880449i	\(-0.657241\pi\)
							−0.474142	+	0.880449i	\(0.657241\pi\)
\(37\)	0.506783	−	0.877774i	0.0833146	−	0.144305i	−0.821357	−	0.570414i	\(-0.806783\pi\)
							0.904672	+	0.426109i	\(0.140116\pi\)
\(41\)	−4.54294	−	2.62287i	−0.709488	−	0.409623i	0.101384	−	0.994847i	\(-0.467673\pi\)
							−0.810871	+	0.585225i	\(0.801006\pi\)
\(43\)	2.66552	−	1.53894i	0.406488	−	0.234686i	−0.282792	−	0.959181i	\(-0.591261\pi\)
							0.689280	+	0.724495i	\(0.257927\pi\)
\(47\)	0.349732			0.0510136			0.0255068	−	0.999675i	\(-0.491880\pi\)
							0.0255068	+	0.999675i	\(0.491880\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.2.bj.b.103.6	yes	84
3.2	odd	2		756.2.bj.b.523.37		84
4.3	odd	2	inner	252.2.bj.b.103.5	yes	84
7.3	odd	6		252.2.n.b.31.23	✓	84
9.2	odd	6		756.2.n.b.19.10		84
9.7	even	3		252.2.n.b.187.33	yes	84
12.11	even	2		756.2.bj.b.523.38		84
21.17	even	6		756.2.n.b.199.20		84
28.3	even	6		252.2.n.b.31.33	yes	84
36.7	odd	6		252.2.n.b.187.23	yes	84
36.11	even	6		756.2.n.b.19.20		84
63.38	even	6		756.2.bj.b.451.37		84
63.52	odd	6	inner	252.2.bj.b.115.6	yes	84
84.59	odd	6		756.2.n.b.199.10		84
252.115	even	6	inner	252.2.bj.b.115.5	yes	84
252.227	odd	6		756.2.bj.b.451.38		84

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.n.b.31.23	✓	84	7.3	odd	6
252.2.n.b.31.33	yes	84	28.3	even	6
252.2.n.b.187.23	yes	84	36.7	odd	6
252.2.n.b.187.33	yes	84	9.7	even	3
252.2.bj.b.103.5	yes	84	4.3	odd	2	inner
252.2.bj.b.103.6	yes	84	1.1	even	1	trivial
252.2.bj.b.115.5	yes	84	252.115	even	6	inner
252.2.bj.b.115.6	yes	84	63.52	odd	6	inner
756.2.n.b.19.10		84	9.2	odd	6
756.2.n.b.19.20		84	36.11	even	6
756.2.n.b.199.10		84	84.59	odd	6
756.2.n.b.199.20		84	21.17	even	6
756.2.bj.b.451.37		84	63.38	even	6
756.2.bj.b.451.38		84	252.227	odd	6
756.2.bj.b.523.37		84	3.2	odd	2
756.2.bj.b.523.38		84	12.11	even	2