Newform orbit 128.6.g.a

• Label: 128.6.g.a
• Level: $128$
• Weight: $6$
• Character orbit: 128.g
• Analytic conductor: $20.529$
• Analytic rank: $0$
• Dimension: $76$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	128.g (of order \(8\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(20.5291289361\)
Analytic rank:	\(0\)
Dimension:	\(76\)
Relative dimension:	\(19\) over \(\Q(\zeta_{8})\)
Twist minimal:	no (minimal twist has level 32)
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 76 q + 4 q^{3} - 4 q^{5} + 4 q^{7} - 4 q^{9}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
17.1		0		−10.7913	+	26.0525i		0		−26.5895	+	11.0137i		0		−131.646	+	131.646i		0		−390.452	−	390.452i		0
17.2		0		−10.1367	+	24.4721i		0		−58.4422	+	24.2075i		0		163.765	−	163.765i		0		−324.305	−	324.305i		0
17.3		0		−9.04699	+	21.8414i		0		83.7809	−	34.7032i		0		34.8236	−	34.8236i		0		−223.371	−	223.371i		0
17.4		0		−7.44181	+	17.9661i		0		29.7710	−	12.3316i		0		−32.2015	+	32.2015i		0		−95.5738	−	95.5738i		0
17.5		0		−6.19399	+	14.9536i		0		−2.09949	+	0.869637i		0		63.4847	−	63.4847i		0		−13.4180	−	13.4180i		0
17.6		0		−4.90532	+	11.8425i		0		56.4155	−	23.3681i		0		−77.5935	+	77.5935i		0		55.6443	+	55.6443i		0
17.7		0		−4.68811	+	11.3181i		0		−98.5667	+	40.8277i		0		−104.908	+	104.908i		0		65.7060	+	65.7060i		0
17.8		0		−1.59693	+	3.85534i		0		−32.3386	+	13.3951i		0		71.7275	−	71.7275i		0		159.514	+	159.514i		0
17.9		0		−0.399918	+	0.965487i		0		−41.3553	+	17.1299i		0		−125.552	+	125.552i		0		171.055	+	171.055i		0
17.10		0		−0.355593	+	0.858478i		0		−47.1761	+	19.5410i		0		64.1149	−	64.1149i		0		171.216	+	171.216i		0
17.11		0		0.512391	−	1.23702i		0		50.7130	−	21.0060i		0		40.9895	−	40.9895i		0		170.559	+	170.559i		0
17.12		0		3.34607	−	8.07812i		0		90.7112	−	37.5738i		0		136.759	−	136.759i		0		117.767	+	117.767i		0
17.13		0		4.24511	−	10.2486i		0		36.3338	−	15.0500i		0		−153.750	+	153.750i		0		84.8142	+	84.8142i		0
17.14		0		5.15883	−	12.4545i		0		50.0856	−	20.7461i		0		−106.306	+	106.306i		0		43.3257	+	43.3257i		0
17.15		0		6.55463	−	15.8243i		0		−38.0662	+	15.7675i		0		−29.7916	+	29.7916i		0		−35.6173	−	35.6173i		0
17.16		0		7.16078	−	17.2876i		0		−8.84628	+	3.66425i		0		158.442	−	158.442i		0		−75.7590	−	75.7590i		0
17.17		0		7.87640	−	19.0153i		0		−91.3601	+	37.8426i		0		−18.5982	+	18.5982i		0		−127.718	−	127.718i		0
17.18		0		9.61436	−	23.2111i		0		−17.5778	+	7.28095i		0		10.1181	−	10.1181i		0		−274.493	−	274.493i		0
17.19		0		11.3810	−	27.4761i		0		62.9000	−	26.0541i		0		−32.1730	+	32.1730i		0		−453.581	−	453.581i		0
49.1		0		−26.1289	+	10.8229i		0		−1.85266	+	4.47271i		0		−18.4008	−	18.4008i		0		393.755	−	393.755i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
32.g	even	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	128.6.g.a		76
4.b	odd	2	1		32.6.g.a	✓	76
32.g	even	8	1	inner	128.6.g.a		76
32.h	odd	8	1		32.6.g.a	✓	76

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
32.6.g.a	✓	76	4.b	odd	2	1
32.6.g.a	✓	76	32.h	odd	8	1
128.6.g.a		76	1.a	even	1	1	trivial
128.6.g.a		76	32.g	even	8	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(128, [\chi])\).