Newform orbit 464.2.t.a

• Label: 464.2.t.a
• Level: $464$
• Weight: $2$
• Character orbit: 464.t
• Analytic conductor: $3.705$
• Analytic rank: $0$
• Dimension: $116$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 464 = 2^{4} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	464.t (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.70505865379\)
Analytic rank:	\(0\)
Dimension:	\(116\)
Relative dimension:	\(58\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 116 q - 4 q^{2} - 4 q^{3} - 8 q^{6} - 8 q^{7} - 10 q^{8} + 108 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
75.1	−1.41358	−	0.0423294i	2.68431			1.99642	+	0.119672i	−0.838918	+	0.838918i	−3.79449	−	0.113625i	1.67686			−2.81703	−	0.253673i	4.20551			1.22139	−	1.15037i
75.2	−1.38956	+	0.262926i	−2.19356			1.86174	−	0.730702i	1.27297	−	1.27297i	3.04808	−	0.576744i	−3.34983			−2.39487	+	1.50485i	1.81171			−1.43417	+	2.10356i
75.3	−1.37821	+	0.317080i	0.552615			1.79892	−	0.874004i	−0.781153	+	0.781153i	−0.761618	+	0.175223i	−0.637113			−2.20216	+	1.77496i	−2.69462			0.828904	−	1.32428i
75.4	−1.37217	+	0.342266i	1.98082			1.76571	−	0.939296i	2.68321	−	2.68321i	−2.71802	+	0.677966i	2.10378			−2.10136	+	1.89322i	0.923633			−2.76345	+	4.60020i
75.5	−1.36695	−	0.362573i	−0.495601			1.73708	+	0.991235i	−0.878643	+	0.878643i	0.677460	+	0.179692i	5.13464			−2.01510	−	1.98478i	−2.75438			1.51963	−	0.882485i
75.6	−1.35394	+	0.408481i	−0.689570			1.66629	−	1.10612i	−2.77352	+	2.77352i	0.933634	−	0.281677i	−1.49484			−1.80422	+	2.17826i	−2.52449			2.62224	−	4.88810i
75.7	−1.33315	−	0.471915i	0.956370			1.55459	+	1.25827i	0.193407	−	0.193407i	−1.27499	−	0.451326i	−4.00606			−1.47871	−	2.41110i	−2.08536			−0.349113	+	0.166570i
75.8	−1.33243	−	0.473963i	−3.30359			1.55072	+	1.26304i	−0.860765	+	0.860765i	4.40179	+	1.56578i	0.0940520			−1.46758	−	2.41789i	7.91371			1.55488	−	0.738935i
75.9	−1.32515	−	0.493953i	0.483780			1.51202	+	1.30912i	2.13346	−	2.13346i	−0.641079	−	0.238965i	−0.788169			−1.35700	−	2.48164i	−2.76596			−3.88098	+	1.77332i
75.10	−1.29541	+	0.567383i	−1.99994			1.35615	−	1.46998i	1.75731	−	1.75731i	2.59073	−	1.13473i	5.23781			−0.922727	+	2.67368i	0.999744			−1.27936	+	3.27350i
75.11	−1.14724	−	0.826943i	−1.24943			0.632331	+	1.89741i	−2.79557	+	2.79557i	1.43340	+	1.03321i	−2.41658			0.843610	−	2.69969i	−1.43893			5.51897	−	0.895420i
75.12	−1.13391	+	0.845131i	3.02225			0.571506	−	1.91661i	−2.21820	+	2.21820i	−3.42696	+	2.55420i	−4.61929			0.971748	+	2.65626i	6.13402			0.640570	−	4.38991i
75.13	−1.08113	+	0.911676i	−2.88181			0.337693	−	1.97128i	−1.46472	+	1.46472i	3.11562	−	2.62728i	1.66671			1.43208	+	2.43909i	5.30485			0.248206	−	2.91891i
75.14	−1.07952	−	0.913582i	3.27960			0.330735	+	1.97246i	1.30138	−	1.30138i	−3.54040	−	2.99618i	−2.27702			1.44497	−	2.43147i	7.75575			−2.59379	+	0.215950i
75.15	−1.07840	−	0.914912i	−2.48955			0.325872	+	1.97327i	3.12350	−	3.12350i	2.68471	+	2.27772i	1.10065			1.45395	−	2.42611i	3.19784			−6.22610	+	0.510639i
75.16	−0.966648	+	1.03227i	−0.0538569			−0.131183	−	1.99569i	0.695345	−	0.695345i	0.0520607	−	0.0555951i	−0.265456			2.18691	+	1.79372i	−2.99710			0.0456334	+	1.38994i
75.17	−0.951132	+	1.04659i	−0.691461			−0.190696	−	1.99089i	1.76409	−	1.76409i	0.657671	−	0.723675i	−2.67588			2.26502	+	1.69402i	−2.52188			0.168394	+	3.52417i
75.18	−0.949119	+	1.04841i	2.44381			−0.198348	−	1.99014i	−0.0104209	+	0.0104209i	−2.31946	+	2.56212i	3.06495			2.27475	+	1.68093i	2.97219			−0.00103475	−	0.0208160i
75.19	−0.896949	−	1.09338i	−1.13960			−0.390963	+	1.96141i	−0.141976	+	0.141976i	1.02216	+	1.24602i	1.14278			2.49525	−	1.33182i	−1.70132			0.282580	+	0.0278886i
75.20	−0.856001	−	1.12573i	2.20050			−0.534525	+	1.92725i	−1.50681	+	1.50681i	−1.88363	−	2.47717i	3.50430			2.62711	−	1.04800i	1.84221			2.98608	+	0.406427i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
464.t	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	464.2.t.a	yes	116
16.f	odd	4	1		464.2.j.a	✓	116
29.c	odd	4	1		464.2.j.a	✓	116
464.t	even	4	1	inner	464.2.t.a	yes	116

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
464.2.j.a	✓	116	16.f	odd	4	1
464.2.j.a	✓	116	29.c	odd	4	1
464.2.t.a	yes	116	1.a	even	1	1	trivial
464.2.t.a	yes	116	464.t	even	4	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(464, [\chi])\).