Embedded newform 128.2.k.a.13.1

• Label: 128.2.k.a.13.1
• Level: $128$
• Weight: $2$
• Character: 128.13
• Analytic conductor: $1.022$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	128.k (of order \(32\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.02208514587\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(15\) over \(\Q(\zeta_{32})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label			13.1
Character	\(\chi\)	\(=\)	128.13
Dual form			128.2.k.a.69.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41410 + 0.0181162i) q^{2} +(0.180024 + 0.593459i) q^{3} +(1.99934 - 0.0512363i) q^{4} +(-0.372310 - 3.78013i) q^{5} +(-0.265323 - 0.835948i) q^{6} +(-0.885307 + 1.32496i) q^{7} +(-2.82634 + 0.108674i) q^{8} +(2.17462 - 1.45304i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{15}{32}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41410	+	0.0181162i	−0.999918	+	0.0128101i
							\(3\)	0.180024	+	0.593459i	0.103937	+	0.342634i	0.993449	−	0.114278i	\(-0.0364554\pi\)
−0.889512	+	0.456912i	\(0.848955\pi\)
\(5\)	−0.372310	−	3.78013i	−0.166502	−	1.69052i	−0.608537	−	0.793526i	\(-0.708243\pi\)
							0.442035	−	0.896998i	\(-0.354257\pi\)
\(7\)	−0.885307	+	1.32496i	−0.334615	+	0.500786i	−0.960179	−	0.279387i	\(-0.909869\pi\)
							0.625564	+	0.780173i	\(0.284869\pi\)
\(11\)	4.66841	−	2.49532i	1.40758	−	0.752367i	0.419711	−	0.907658i	\(-0.362131\pi\)
							0.987869	+	0.155291i	\(0.0496315\pi\)
\(13\)	−1.45318	−	0.143126i	−0.403041	−	0.0396960i	−0.105534	−	0.994416i	\(-0.533655\pi\)
							−0.297507	+	0.954720i	\(0.596155\pi\)
\(17\)	0.266237	+	0.110279i	0.0645721	+	0.0267466i	0.414736	−	0.909942i	\(-0.363874\pi\)
							−0.350164	+	0.936689i	\(0.613874\pi\)
\(19\)	0.978812	−	0.803290i	0.224555	−	0.184287i	−0.515388	−	0.856957i	\(-0.672352\pi\)
							0.739943	+	0.672669i	\(0.234852\pi\)
\(23\)	−0.210003	+	1.05575i	−0.0437886	+	0.220140i	−0.996484	−	0.0837878i	\(-0.973298\pi\)
							0.952695	+	0.303928i	\(0.0982982\pi\)
\(29\)	−2.93892	+	5.49833i	−0.545744	+	1.02101i	0.446272	+	0.894898i	\(0.352752\pi\)
							−0.992015	+	0.126117i	\(0.959748\pi\)
\(31\)	−5.49180	+	5.49180i	−0.986356	+	0.986356i	−0.999908	−	0.0135524i	\(-0.995686\pi\)
							0.0135524	+	0.999908i	\(0.495686\pi\)
\(37\)	6.67782	−	8.13695i	1.09783	−	1.33771i	0.161382	−	0.986892i	\(-0.448405\pi\)
							0.936445	−	0.350815i	\(-0.114095\pi\)
\(41\)	−6.40426	−	1.27389i	−1.00018	−	0.198948i	−0.332264	−	0.943186i	\(-0.607812\pi\)
							−0.667914	+	0.744239i	\(0.732812\pi\)
\(43\)	−2.63020	+	8.67062i	−0.401102	+	1.32226i	0.491290	+	0.870996i	\(0.336525\pi\)
							−0.892392	+	0.451260i	\(0.850975\pi\)
\(47\)	0.458439	−	1.10677i	0.0668702	−	0.161439i	−0.886911	−	0.461940i	\(-0.847154\pi\)
							0.953782	+	0.300500i	\(0.0971537\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.2.k.a.13.1	✓	240
4.3	odd	2		512.2.k.a.401.7		240
128.59	odd	32		512.2.k.a.369.7		240
128.69	even	32	inner	128.2.k.a.69.1	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.2.k.a.13.1	✓	240	1.1	even	1	trivial
128.2.k.a.69.1	yes	240	128.69	even	32	inner
512.2.k.a.369.7		240	128.59	odd	32
512.2.k.a.401.7		240	4.3	odd	2