Embedded newform 51.6.h.a.25.8

• Label: 51.6.h.a.25.8
• Level: $51$
• Weight: $6$
• Character: 51.25
• Analytic conductor: $8.180$
• Analytic rank: $0$
• Dimension: $56$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 51 = 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	51.h (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.17957481046\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(14\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			25.8
Character	\(\chi\)	\(=\)	51.25
Dual form			51.6.h.a.49.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.434767 + 0.434767i) q^{2} +(-3.44415 + 8.31492i) q^{3} +31.6220i q^{4} +(-22.4402 - 9.29502i) q^{5} +(-2.11765 - 5.11245i) q^{6} +(-28.5998 + 11.8464i) q^{7} +(-27.6607 - 27.6607i) q^{8} +(-57.2756 - 57.2756i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 88 q^{5} + 360 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/51\mathbb{Z}\right)^\times\).

\(n\)	\(35\)	\(37\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.434767	+	0.434767i	−0.0768567	+	0.0768567i	−0.744490	−	0.667633i	\(-0.767307\pi\)
							0.667633	+	0.744490i	\(0.267307\pi\)
\(3\)	−3.44415	+	8.31492i	−0.220942	+	0.533402i
							\(5\)	−22.4402	−	9.29502i	−0.401422	−	0.166274i	0.172833	−	0.984951i	\(-0.444708\pi\)
−0.574255	+	0.818677i	\(0.694708\pi\)
\(7\)	−28.5998	+	11.8464i	−0.220606	+	0.0913782i	−0.490249	−	0.871582i	\(-0.663094\pi\)
							0.269643	+	0.962960i	\(0.413094\pi\)
\(11\)	−160.452	−	387.366i	−0.399820	−	0.965251i	−0.987708	−	0.156309i	\(-0.950041\pi\)
							0.587888	−	0.808942i	\(-0.299959\pi\)
\(13\)		−	58.8613i		−	0.0965987i	−0.998833	−	0.0482993i	\(-0.984620\pi\)
							0.998833	−	0.0482993i	\(-0.0153801\pi\)
\(17\)	−1190.94	−	38.8426i	−0.999469	−	0.0325977i
							\(19\)	−921.811	+	921.811i	−0.585812	+	0.585812i	−0.936494	−	0.350683i	\(-0.885950\pi\)
0.350683	+	0.936494i	\(0.385950\pi\)
\(23\)	100.894	+	243.580i	0.0397691	+	0.0960111i	0.942516	−	0.334161i	\(-0.108453\pi\)
							−0.902747	+	0.430172i	\(0.858453\pi\)
\(29\)	5409.11	+	2240.53i	1.19435	+	0.494715i	0.889168	−	0.457580i	\(-0.151284\pi\)
							0.305179	+	0.952295i	\(0.401284\pi\)
\(31\)	−1051.98	+	2539.70i	−0.196609	+	0.474655i	−0.991181	−	0.132514i	\(-0.957695\pi\)
							0.794572	+	0.607169i	\(0.207695\pi\)
\(37\)	−5099.42	+	12311.1i	−0.612373	+	1.47840i	0.248013	+	0.968757i	\(0.420222\pi\)
							−0.860386	+	0.509643i	\(0.829778\pi\)
\(41\)	5014.37	−	2077.02i	0.465862	−	0.192966i	−0.137390	−	0.990517i	\(-0.543871\pi\)
							0.603252	+	0.797551i	\(0.293871\pi\)
\(43\)	6514.75	+	6514.75i	0.537312	+	0.537312i	0.922739	−	0.385426i	\(-0.125946\pi\)
							−0.385426	+	0.922739i	\(0.625946\pi\)
\(47\)		−	5141.34i		−	0.339494i	−0.985488	−	0.169747i	\(-0.945705\pi\)
							0.985488	−	0.169747i	\(-0.0542950\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	51.6.h.a.25.8	✓	56
17.7	odd	16		867.6.a.u.1.14		28
17.10	odd	16		867.6.a.t.1.14		28
17.15	even	8	inner	51.6.h.a.49.8	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.6.h.a.25.8	✓	56	1.1	even	1	trivial
51.6.h.a.49.8	yes	56	17.15	even	8	inner
867.6.a.t.1.14		28	17.10	odd	16
867.6.a.u.1.14		28	17.7	odd	16