Newform orbit 128.4.g.a

• Label: 128.4.g.a
• Level: $128$
• Weight: $4$
• Character orbit: 128.g
• Analytic conductor: $7.552$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(7.55224448073\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(11\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 4 q^{3} - 4 q^{5} + 4 q^{7} - 4 q^{9}+O(q^{10}) \)

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		−3.54796	+	8.56554i		0		−7.55322	+	3.12865i		0		−7.16166	+	7.16166i		0		−41.6886	−	41.6886i		0
17.2		0		−2.92731	+	7.06715i		0		13.5234	−	5.60159i		0		23.0737	−	23.0737i		0		−22.2836	−	22.2836i		0
17.3		0		−1.94138	+	4.68690i		0		4.93338	−	2.04347i		0		−14.0755	+	14.0755i		0		0.893826	+	0.893826i		0
17.4		0		−1.64064	+	3.96085i		0		−11.8087	+	4.89132i		0		5.11236	−	5.11236i		0		6.09524	+	6.09524i		0
17.5		0		−0.729459	+	1.76107i		0		4.29822	−	1.78038i		0		−1.47807	+	1.47807i		0		16.5226	+	16.5226i		0
17.6		0		0.477813	−	1.15354i		0		−16.3468	+	6.77105i		0		18.0222	−	18.0222i		0		17.9895	+	17.9895i		0
17.7		0		0.998206	−	2.40988i		0		17.4005	−	7.20752i		0		−4.37099	+	4.37099i		0		14.2808	+	14.2808i		0
17.8		0		1.20185	−	2.90153i		0		−3.98512	+	1.65069i		0		−22.4050	+	22.4050i		0		12.1174	+	12.1174i		0
17.9		0		1.90169	−	4.59109i		0		0.188811	−	0.0782080i		0		11.4103	−	11.4103i		0		1.63019	+	1.63019i		0
17.10		0		3.21198	−	7.75440i		0		−13.6472	+	5.65283i		0		−9.07689	+	9.07689i		0		−30.7220	−	30.7220i		0
17.11		0		3.28810	−	7.93817i		0		11.2895	−	4.67626i		0		11.8490	−	11.8490i		0		−33.1111	−	33.1111i		0
49.1		0		−7.57022	+	3.13569i		0		−8.03744	+	19.4041i		0		1.85119	+	1.85119i		0		28.3838	−	28.3838i		0
49.2		0		−6.97998	+	2.89120i		0		1.57150	−	3.79394i		0		−15.6607	−	15.6607i		0		21.2692	−	21.2692i		0
49.3		0		−5.53310	+	2.29188i		0		4.22177	−	10.1923i		0		11.6451	+	11.6451i		0		6.27055	−	6.27055i		0
49.4		0		−4.56924	+	1.89264i		0		1.37033	−	3.30826i		0		6.14642	+	6.14642i		0		−1.79604	+	1.79604i		0
49.5		0		−0.143768	+	0.0595506i		0		−0.767542	+	1.85301i		0		5.47741	+	5.47741i		0		−19.0748	+	19.0748i		0
49.6		0		1.36212	−	0.564209i		0		6.58151	−	15.8892i		0		−14.5517	−	14.5517i		0		−17.5548	+	17.5548i		0
49.7		0		1.65706	−	0.686375i		0		−4.13953	+	9.99370i		0		−24.2273	−	24.2273i		0		−16.8172	+	16.8172i		0
49.8		0		2.66269	−	1.10292i		0		−5.50788	+	13.2972i		0		6.48055	+	6.48055i		0		−13.2184	+	13.2184i		0
49.9		0		5.56908	−	2.30679i		0		6.28381	−	15.1704i		0		16.6573	+	16.6573i		0		6.60151	−	6.60151i		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	128.4.g.a		44
4.b	odd	2	1		32.4.g.a	✓	44
8.b	even	2	1		256.4.g.a		44
8.d	odd	2	1		256.4.g.b		44
32.g	even	8	1	inner	128.4.g.a		44
32.g	even	8	1		256.4.g.a		44
32.h	odd	8	1		32.4.g.a	✓	44
32.h	odd	8	1		256.4.g.b		44

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
32.4.g.a	✓	44	4.b	odd	2	1
32.4.g.a	✓	44	32.h	odd	8	1
128.4.g.a		44	1.a	even	1	1	trivial
128.4.g.a		44	32.g	even	8	1	inner
256.4.g.a		44	8.b	even	2	1
256.4.g.a		44	32.g	even	8	1
256.4.g.b		44	8.d	odd	2	1
256.4.g.b		44	32.h	odd	8	1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(128, [\chi])\).