Newform orbit 128.4.k.a

• Label: 128.4.k.a
• Level: $128$
• Weight: $4$
• Character orbit: 128.k
• Analytic conductor: $7.552$
• Analytic rank: $0$
• Dimension: $752$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	128.k (of order \(32\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.55224448073\)
Analytic rank:	\(0\)
Dimension:	\(752\)
Relative dimension:	\(47\) over \(\Q(\zeta_{32})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{32}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 752 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1	−2.82809	+	0.0437786i	3.90803	+	1.18549i	7.99617	−	0.247619i	−3.72183	−	0.366569i	−11.1042	−	3.18158i	−11.6508	−	17.4367i	−22.6030	+	1.05035i	−8.58237	−	5.73455i	10.5417	+	0.873752i
5.2	−2.82184	+	0.192933i	−9.75553	−	2.95931i	7.92555	−	1.08885i	−0.385980	−	0.0380157i	28.0995	+	6.46853i	8.69824	+	13.0178i	−22.1546	+	4.60167i	63.9633	+	42.7389i	1.09651	+	0.0328059i
5.3	−2.73119	+	0.735238i	−2.99224	−	0.907685i	6.91885	−	4.01616i	20.1386	+	1.98348i	8.83975	+	0.279059i	−4.42679	−	6.62516i	−15.9439	+	16.0559i	−14.3201	−	9.56838i	−56.4608	+	9.38941i
5.4	−2.72237	−	0.767264i	−3.00383	−	0.911201i	6.82261	+	4.17756i	1.47948	+	0.145716i	7.47840	+	4.78536i	3.74690	+	5.60763i	−15.3684	−	16.6076i	−14.2570	−	9.52622i	−3.91588	−	1.53184i
5.5	−2.71781	+	0.783272i	−1.31785	−	0.399764i	6.77297	−	4.25757i	−18.9129	−	1.86275i	3.89478	+	0.0542507i	16.8694	+	25.2468i	−15.0728	+	16.8763i	−20.8728	−	13.9467i	52.8606	−	9.75131i
5.6	−2.60432	−	1.10341i	9.60431	+	2.91344i	5.56498	+	5.74726i	−17.8404	−	1.75713i	−21.7980	−	18.1850i	10.3895	+	15.5490i	−8.15141	−	21.1082i	61.3050	+	40.9627i	44.5233	+	24.2614i
5.7	−2.53887	+	1.24665i	7.58094	+	2.29965i	4.89175	−	6.33015i	9.48044	+	0.933742i	−22.1139	+	3.61223i	6.95727	+	10.4123i	−4.52806	+	22.1697i	29.7326	+	19.8667i	−25.2337	+	9.44810i
5.8	−2.50388	−	1.31551i	3.59854	+	1.09161i	4.53886	+	6.58777i	17.1492	+	1.68905i	−7.57431	−	7.46718i	15.6405	+	23.4076i	−2.69848	−	22.4659i	−10.6918	−	7.14401i	−40.7177	−	26.7892i
5.9	−2.42367	−	1.45802i	−4.84264	−	1.46900i	3.74833	+	7.06753i	−16.3353	−	1.60888i	9.59512	+	10.6211i	−11.5654	−	17.3089i	1.21991	−	22.5945i	−1.15646	−	0.772719i	37.2455	+	27.7166i
5.10	−2.36116	+	1.55721i	−6.77444	−	2.05500i	3.15018	−	7.35366i	−5.70543	−	0.561936i	19.1956	−	5.69705i	−18.2713	−	27.3450i	4.01313	+	22.2687i	19.2204	+	12.8426i	14.3465	−	7.55774i
5.11	−2.23791	−	1.72967i	7.71088	+	2.33907i	2.01649	+	7.74169i	13.5785	+	1.33737i	−13.2105	−	18.5719i	−18.9453	−	28.3537i	8.87784	−	20.8131i	31.5368	+	21.0722i	−28.0744	−	26.4793i
5.12	−2.14725	+	1.84101i	3.92169	+	1.18963i	1.22134	−	7.90622i	−5.31043	−	0.523032i	−10.6110	+	4.66546i	−0.627511	−	0.939136i	11.9329	+	19.2251i	−8.48522	−	5.66964i	12.3657	−	8.65350i
5.13	−1.88791	+	2.10613i	−3.74897	−	1.13724i	−0.871580	−	7.95238i	4.16267	+	0.409988i	9.47289	−	5.74882i	10.4814	+	15.6865i	18.3942	+	13.1777i	−9.68822	−	6.47346i	−8.72225	+	7.99312i
5.14	−1.83459	−	2.15274i	−7.96376	−	2.41578i	−1.26855	+	7.89878i	9.56707	+	0.942274i	9.40970	+	21.5759i	2.63589	+	3.94489i	19.3313	−	11.7602i	35.1358	+	23.4770i	−15.5232	−	22.3241i
5.15	−1.79489	−	2.18595i	2.18691	+	0.663393i	−1.55674	+	7.84707i	−5.13671	−	0.505922i	−2.47513	−	5.97120i	4.12327	+	6.17091i	19.9475	−	10.6817i	−18.1072	−	12.0988i	8.11391	+	12.1367i
5.16	−1.35278	+	2.48395i	6.26012	+	1.89899i	−4.33998	−	6.72046i	−18.2007	−	1.79261i	−13.1855	+	12.9809i	−7.34552	−	10.9934i	22.5643	−	1.68898i	13.1332	+	8.77534i	29.0742	−	42.7844i
5.17	−1.13660	−	2.59001i	−2.41332	−	0.732073i	−5.41626	+	5.88762i	13.6673	+	1.34611i	0.846916	+	7.08259i	−16.0547	−	24.0276i	21.4051	+	7.33627i	−17.1615	−	11.4669i	−12.0478	−	36.9283i
5.18	−1.11449	+	2.59960i	0.572589	+	0.173693i	−5.51584	−	5.79444i	8.36768	+	0.824145i	−1.08968	+	1.29492i	−6.62562	−	9.91594i	21.2105	−	7.88116i	−22.1520	−	14.8015i	−11.4681	+	20.8341i
5.19	−0.935141	−	2.66937i	3.85090	+	1.16816i	−6.25102	+	4.99247i	−11.1272	−	1.09593i	−0.482895	−	11.3718i	−0.831707	−	1.24474i	19.1723	+	12.0176i	−8.98487	−	6.00350i	7.48005	+	30.7274i
5.20	−0.668572	+	2.74827i	−6.55625	−	1.98882i	−7.10602	−	3.67484i	−14.4435	−	1.42256i	9.84915	−	16.6887i	0.534510	+	0.799951i	14.8504	−	17.0724i	16.5794	+	11.0780i	13.5661	−	38.7437i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
128.k	even	32	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	128.4.k.a	✓	752
128.k	even	32	1	inner	128.4.k.a	✓	752

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
128.4.k.a	✓	752	1.a	even	1	1	trivial
128.4.k.a	✓	752	128.k	even	32	1	inner

Hecke kernels

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