Newform orbit 735.2.bt.a

• Label: 735.2.bt.a
• Level: $735$
• Weight: $2$
• Character orbit: 735.bt
• Analytic conductor: $5.869$
• Analytic rank: $0$
• Dimension: $2592$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	735.bt (of order \(84\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.86900454856\)
Analytic rank:	\(0\)
Dimension:	\(2592\)
Relative dimension:	\(108\) over \(\Q(\zeta_{84})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{84}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 2592 q - 26 q^{3} - 60 q^{6} - 44 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1	−2.47342	+	1.30724i	0.804554	+	1.53385i	3.28230	−	4.81424i	−1.23201	−	1.86605i	−3.99511	−	2.74211i	−2.08765	−	1.62533i	−1.19865	+	10.6383i	−1.70539	+	2.46813i	5.48666	+	3.00501i
2.2	−2.41713	+	1.27749i	1.62626	−	0.596048i	3.08391	−	4.52326i	−1.48100	+	1.67531i	−3.16944	+	3.51826i	0.757349	+	2.53504i	−1.06357	+	9.43945i	2.28945	−	1.93866i	1.43957	−	5.94140i
2.3	−2.40929	+	1.27335i	−1.59757	−	0.669165i	3.05664	−	4.48327i	2.18084	+	0.493901i	4.70109	−	0.422045i	−0.273808	+	2.63155i	−1.04535	+	9.27778i	2.10444	+	2.13807i	−5.88319	+	1.58702i
2.4	−2.37410	+	1.25475i	0.844125	−	1.51243i	2.93532	−	4.30532i	1.92805	−	1.13254i	−0.106315	+	4.64983i	−2.60051	−	0.487200i	−0.965329	+	8.56754i	−1.57491	−	2.55336i	−3.15633	+	5.10796i
2.5	−2.32462	+	1.22860i	1.20134	+	1.24771i	2.76778	−	4.05959i	1.19650	+	1.88901i	−4.32560	−	1.42450i	2.16504	−	1.52073i	−0.857658	+	7.61192i	−0.113575	+	2.99785i	−5.10227	−	2.92122i
2.6	−2.23465	+	1.18105i	−0.0918273	−	1.72961i	2.47214	−	3.62596i	0.358215	+	2.20719i	2.24796	+	3.75663i	0.481517	−	2.60157i	−0.675940	+	5.99914i	−2.98314	+	0.317652i	−3.40728	−	4.50922i
2.7	−2.21152	+	1.16882i	−0.621090	+	1.61686i	2.39804	−	3.51728i	1.83855	−	1.27268i	−0.516274	−	4.30167i	2.51475	+	0.822208i	−0.632106	+	5.61009i	−2.22849	−	2.00843i	−2.57846	+	4.96351i
2.8	−2.20580	+	1.16580i	0.621271	−	1.61679i	2.37982	−	3.49055i	−1.76904	−	1.36767i	0.514457	+	4.29060i	2.63127	+	0.276405i	−0.621433	+	5.51537i	−2.22804	−	2.00893i	5.49656	+	0.954455i
2.9	−2.19175	+	1.15838i	−0.787652	+	1.54260i	2.33531	−	3.42527i	0.134992	+	2.23199i	−0.0605679	−	4.29339i	−2.52629	+	0.786055i	−0.595546	+	5.28561i	−1.75921	−	2.43006i	−2.88135	−	4.73560i
2.10	−2.15911	+	1.14112i	−0.967723	−	1.43649i	2.23296	−	3.27515i	−2.23489	−	0.0725309i	3.72864	+	1.99726i	−2.40435	−	1.10413i	−0.536996	+	4.76597i	−1.12702	+	2.78026i	4.90815	−	2.39368i
2.11	−2.12871	+	1.12505i	−1.70111	+	0.325914i	2.13901	−	3.13734i	−1.49573	−	1.66217i	3.25450	−	2.60762i	−1.08181	+	2.41447i	−0.484472	+	4.29981i	2.78756	−	1.10883i	5.05399	+	1.85549i
2.12	−2.00465	+	1.05949i	1.69998	−	0.331757i	1.76946	−	2.59532i	1.45463	−	1.69825i	−3.05637	+	2.46617i	1.69991	−	2.02739i	−0.289697	+	2.57113i	2.77987	−	1.12796i	−1.11675	+	4.94555i
2.13	−2.00177	+	1.05797i	−1.58145	+	0.706417i	1.76116	−	2.58315i	0.212009	−	2.22599i	2.41833	−	3.08721i	−0.213018	−	2.63716i	−0.285546	+	2.53429i	2.00195	−	2.23432i	1.93064	+	4.68024i
2.14	−1.97132	+	1.04187i	−1.73186	+	0.0257939i	1.67395	−	2.45523i	2.07339	+	0.837291i	3.38717	−	1.85522i	−0.772356	−	2.53051i	−0.242552	+	2.15271i	2.99867	−	0.0893428i	−4.95965	+	0.509639i
2.15	−1.92422	+	1.01698i	1.70518	+	0.303886i	1.54172	−	2.26130i	1.79669	+	1.33112i	−3.59019	+	1.14939i	−2.64572	−	0.0129655i	−0.179559	+	1.59363i	2.81531	+	1.03636i	−4.81095	−	0.734169i
2.16	−1.92375	+	1.01673i	−1.42833	−	0.979726i	1.54042	−	2.25938i	−1.28197	+	1.83209i	3.74387	+	0.432516i	2.24981	+	1.39224i	−0.178952	+	1.58824i	1.08027	+	2.79875i	0.603455	−	4.82789i
2.17	−1.87913	+	0.993150i	0.885664	+	1.48849i	1.41815	−	2.08004i	−1.66321	+	1.49457i	−3.14257	−	1.91747i	0.980909	+	2.45720i	−0.123147	+	1.09296i	−1.43120	+	2.63660i	1.64107	−	4.46030i
2.18	−1.80971	+	0.956459i	−0.239317	−	1.71544i	1.23359	−	1.80935i	0.560641	−	2.16464i	2.07384	+	2.87555i	0.0994689	+	2.64388i	−0.0435126	+	0.386185i	−2.88545	+	0.821068i	1.05580	+	4.45360i
2.19	−1.79081	+	0.946469i	1.73198	+	0.0161946i	1.18454	−	1.73741i	−1.49157	−	1.66590i	−3.11696	+	1.61026i	−2.02796	+	1.69923i	−0.0233116	+	0.206896i	2.99948	+	0.0560974i	4.24783	+	1.57158i
2.20	−1.76189	+	0.931187i	1.29564	+	1.14948i	1.11051	−	1.62882i	−1.60285	−	1.55912i	−3.35317	−	0.818774i	2.64480	−	0.0710896i	0.00638796	−	0.0566947i	0.357387	+	2.97864i	4.27589	+	1.25445i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.c	odd	4	1	inner
15.e	even	4	1	inner
49.g	even	21	1	inner
147.n	odd	42	1	inner
245.w	odd	84	1	inner
735.bt	even	84	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	735.2.bt.a	✓	2592
3.b	odd	2	1	inner	735.2.bt.a	✓	2592
5.c	odd	4	1	inner	735.2.bt.a	✓	2592
15.e	even	4	1	inner	735.2.bt.a	✓	2592
49.g	even	21	1	inner	735.2.bt.a	✓	2592
147.n	odd	42	1	inner	735.2.bt.a	✓	2592
245.w	odd	84	1	inner	735.2.bt.a	✓	2592
735.bt	even	84	1	inner	735.2.bt.a	✓	2592

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
735.2.bt.a	✓	2592	1.a	even	1	1	trivial
735.2.bt.a	✓	2592	3.b	odd	2	1	inner
735.2.bt.a	✓	2592	5.c	odd	4	1	inner
735.2.bt.a	✓	2592	15.e	even	4	1	inner
735.2.bt.a	✓	2592	49.g	even	21	1	inner
735.2.bt.a	✓	2592	147.n	odd	42	1	inner
735.2.bt.a	✓	2592	245.w	odd	84	1	inner
735.2.bt.a	✓	2592	735.bt	even	84	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(735, [\chi])\).