Embedded newform 40.6.k.a.27.6

• Label: 40.6.k.a.27.6
• Level: $40$
• Weight: $6$
• Character: 40.27
• Analytic conductor: $6.415$
• Analytic rank: $0$
• Dimension: $56$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 40 = 2^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	40.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.41535279252\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(28\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			27.6
Character	\(\chi\)	\(=\)	40.27
Dual form			40.6.k.a.3.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.60651 + 3.28330i) q^{2} +(11.5932 - 11.5932i) q^{3} +(10.4398 - 30.2491i) q^{4} +(-48.9682 - 26.9650i) q^{5} +(-15.3402 + 91.4681i) q^{6} +(-75.8554 + 75.8554i) q^{7} +(51.2259 + 173.620i) q^{8} -25.8041i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 2 q^{2} - 4 q^{3} + 176 q^{6} + 244 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).

\(n\)	\(17\)	\(21\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.60651	+	3.28330i	−0.814323	+	0.580412i
							\(3\)	11.5932	−	11.5932i	0.743703	−	0.743703i	−0.229585	−	0.973289i	\(-0.573737\pi\)
0.973289	+	0.229585i	\(0.0737369\pi\)
\(5\)	−48.9682	−	26.9650i	−0.875970	−	0.482365i
							\(7\)	−75.8554	+	75.8554i	−0.585115	+	0.585115i	−0.936305	−	0.351189i	\(-0.885777\pi\)
0.351189	+	0.936305i	\(0.385777\pi\)
\(11\)	−570.121			−1.42064			−0.710322	−	0.703877i	\(-0.751451\pi\)
							−0.710322	+	0.703877i	\(0.751451\pi\)
\(13\)	−373.587	−	373.587i	−0.613104	−	0.613104i	0.330650	−	0.943753i	\(-0.392732\pi\)
							−0.943753	+	0.330650i	\(0.892732\pi\)
\(17\)	120.205	+	120.205i	0.100879	+	0.100879i	0.755745	−	0.654866i	\(-0.227275\pi\)
							−0.654866	+	0.755745i	\(0.727275\pi\)
\(19\)		−	1018.64i		−	0.647349i	−0.946169	−	0.323674i	\(-0.895082\pi\)
							0.946169	−	0.323674i	\(-0.104918\pi\)
\(23\)	−785.825	−	785.825i	−0.309746	−	0.309746i	0.535065	−	0.844811i	\(-0.320287\pi\)
							−0.844811	+	0.535065i	\(0.820287\pi\)
\(29\)	−6337.75			−1.39939			−0.699697	−	0.714440i	\(-0.746682\pi\)
							−0.699697	+	0.714440i	\(0.746682\pi\)
\(31\)		−	2427.71i		−	0.453725i	−0.973927	−	0.226863i	\(-0.927153\pi\)
							0.973927	−	0.226863i	\(-0.0728468\pi\)
\(37\)	6307.89	−	6307.89i	0.757494	−	0.757494i	−0.218371	−	0.975866i	\(-0.570074\pi\)
							0.975866	+	0.218371i	\(0.0700744\pi\)
\(41\)	−18431.2			−1.71235			−0.856176	−	0.516685i	\(-0.827166\pi\)
							−0.856176	+	0.516685i	\(0.827166\pi\)
\(43\)	15967.9	−	15967.9i	1.31697	−	1.31697i	0.400815	−	0.916159i	\(-0.368727\pi\)
							0.916159	−	0.400815i	\(-0.131273\pi\)
\(47\)	1022.27	−	1022.27i	0.0675028	−	0.0675028i	−0.672549	−	0.740052i	\(-0.734801\pi\)
							0.740052	+	0.672549i	\(0.234801\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	40.6.k.a.27.6	yes	56
4.3	odd	2		160.6.o.a.47.6		56
5.3	odd	4	inner	40.6.k.a.3.21	yes	56
8.3	odd	2	inner	40.6.k.a.27.21	yes	56
8.5	even	2		160.6.o.a.47.5		56
20.3	even	4		160.6.o.a.143.5		56
40.3	even	4	inner	40.6.k.a.3.6	✓	56
40.13	odd	4		160.6.o.a.143.6		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.k.a.3.6	✓	56	40.3	even	4	inner
40.6.k.a.3.21	yes	56	5.3	odd	4	inner
40.6.k.a.27.6	yes	56	1.1	even	1	trivial
40.6.k.a.27.21	yes	56	8.3	odd	2	inner
160.6.o.a.47.5		56	8.5	even	2
160.6.o.a.47.6		56	4.3	odd	2
160.6.o.a.143.5		56	20.3	even	4
160.6.o.a.143.6		56	40.13	odd	4