Properties

Label 735.2.j.e
Level 735
Weight 2
Character orbit 735.j
Analytic conductor 5.869
Analytic rank 0
Dimension 24
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 735.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Coefficient ring index: multiple of None
Twist minimal: no (minimal twist has level 105)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24q - 2q^{3} + 12q^{6} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q - 2q^{3} + 12q^{6} - 8q^{10} - 10q^{12} + 8q^{13} + 2q^{15} + 8q^{16} - 14q^{18} - 4q^{22} - 4q^{25} - 20q^{27} - 40q^{30} - 24q^{31} - 4q^{33} + 4q^{36} - 4q^{37} - 16q^{40} + 8q^{43} + 40q^{45} + 32q^{46} - 22q^{48} - 8q^{51} + 36q^{52} + 20q^{55} - 44q^{57} - 56q^{58} + 50q^{60} - 8q^{61} + 76q^{66} - 12q^{67} + 34q^{72} + 52q^{73} + 6q^{75} - 32q^{76} - 60q^{78} - 20q^{81} + 104q^{82} - 12q^{85} - 46q^{87} + 42q^{90} + 44q^{93} + 12q^{96} + 60q^{97} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1 −1.72500 + 1.72500i −1.44593 0.953569i 3.95128i −1.96293 1.07094i 4.13914 0.849321i 0 3.36596 + 3.36596i 1.18141 + 2.75758i 5.23344 1.53868i
197.2 −1.59037 + 1.59037i 1.36228 1.06967i 3.05858i 0.812581 + 2.08320i −0.465359 + 3.86771i 0 1.68355 + 1.68355i 0.711613 2.91438i −4.60537 2.02076i
197.3 −1.06891 + 1.06891i −1.73199 + 0.0150256i 0.285117i 2.03205 + 0.933160i 1.83527 1.86739i 0 −1.83305 1.83305i 2.99955 0.0520482i −3.16952 + 1.17461i
197.4 −0.929340 + 0.929340i 1.05286 + 1.37531i 0.272655i −0.980304 + 2.00973i −2.25660 0.299670i 0 −2.11207 2.11207i −0.782976 + 2.89602i −0.956684 2.77876i
197.5 −0.664190 + 0.664190i −0.578521 + 1.63258i 1.11770i 0.459812 2.18828i −0.700094 1.46859i 0 −2.07075 2.07075i −2.33063 1.88896i 1.14803 + 1.75884i
197.6 −0.218381 + 0.218381i −0.354425 1.69540i 1.90462i −2.16448 + 0.561256i 0.447643 + 0.292843i 0 −0.852694 0.852694i −2.74877 + 1.20179i 0.350114 0.595249i
197.7 0.218381 0.218381i 1.69540 + 0.354425i 1.90462i 2.16448 0.561256i 0.447643 0.292843i 0 0.852694 + 0.852694i 2.74877 + 1.20179i 0.350114 0.595249i
197.8 0.664190 0.664190i −1.63258 + 0.578521i 1.11770i −0.459812 + 2.18828i −0.700094 + 1.46859i 0 2.07075 + 2.07075i 2.33063 1.88896i 1.14803 + 1.75884i
197.9 0.929340 0.929340i −1.37531 1.05286i 0.272655i 0.980304 2.00973i −2.25660 + 0.299670i 0 2.11207 + 2.11207i 0.782976 + 2.89602i −0.956684 2.77876i
197.10 1.06891 1.06891i −0.0150256 + 1.73199i 0.285117i −2.03205 0.933160i 1.83527 + 1.86739i 0 1.83305 + 1.83305i −2.99955 0.0520482i −3.16952 + 1.17461i
197.11 1.59037 1.59037i 1.06967 1.36228i 3.05858i −0.812581 2.08320i −0.465359 3.86771i 0 −1.68355 1.68355i −0.711613 2.91438i −4.60537 2.02076i
197.12 1.72500 1.72500i 0.953569 + 1.44593i 3.95128i 1.96293 + 1.07094i 4.13914 + 0.849321i 0 −3.36596 3.36596i −1.18141 + 2.75758i 5.23344 1.53868i
638.1 −1.72500 1.72500i −1.44593 + 0.953569i 3.95128i −1.96293 + 1.07094i 4.13914 + 0.849321i 0 3.36596 3.36596i 1.18141 2.75758i 5.23344 + 1.53868i
638.2 −1.59037 1.59037i 1.36228 + 1.06967i 3.05858i 0.812581 2.08320i −0.465359 3.86771i 0 1.68355 1.68355i 0.711613 + 2.91438i −4.60537 + 2.02076i
638.3 −1.06891 1.06891i −1.73199 0.0150256i 0.285117i 2.03205 0.933160i 1.83527 + 1.86739i 0 −1.83305 + 1.83305i 2.99955 + 0.0520482i −3.16952 1.17461i
638.4 −0.929340 0.929340i 1.05286 1.37531i 0.272655i −0.980304 2.00973i −2.25660 + 0.299670i 0 −2.11207 + 2.11207i −0.782976 2.89602i −0.956684 + 2.77876i
638.5 −0.664190 0.664190i −0.578521 1.63258i 1.11770i 0.459812 + 2.18828i −0.700094 + 1.46859i 0 −2.07075 + 2.07075i −2.33063 + 1.88896i 1.14803 1.75884i
638.6 −0.218381 0.218381i −0.354425 + 1.69540i 1.90462i −2.16448 0.561256i 0.447643 0.292843i 0 −0.852694 + 0.852694i −2.74877 1.20179i 0.350114 + 0.595249i
638.7 0.218381 + 0.218381i 1.69540 0.354425i 1.90462i 2.16448 + 0.561256i 0.447643 + 0.292843i 0 0.852694 0.852694i 2.74877 1.20179i 0.350114 + 0.595249i
638.8 0.664190 + 0.664190i −1.63258 0.578521i 1.11770i −0.459812 2.18828i −0.700094 1.46859i 0 2.07075 2.07075i 2.33063 + 1.88896i 1.14803 1.75884i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 638.12
Significant digits:
Format:

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.j.e 24
3.b odd 2 1 inner 735.2.j.e 24
5.c odd 4 1 inner 735.2.j.e 24
7.b odd 2 1 735.2.j.g 24
7.c even 3 2 735.2.y.i 48
7.d odd 6 2 105.2.x.a 48
15.e even 4 1 inner 735.2.j.e 24
21.c even 2 1 735.2.j.g 24
21.g even 6 2 105.2.x.a 48
21.h odd 6 2 735.2.y.i 48
35.f even 4 1 735.2.j.g 24
35.i odd 6 2 525.2.bf.f 48
35.k even 12 2 105.2.x.a 48
35.k even 12 2 525.2.bf.f 48
35.l odd 12 2 735.2.y.i 48
105.k odd 4 1 735.2.j.g 24
105.p even 6 2 525.2.bf.f 48
105.w odd 12 2 105.2.x.a 48
105.w odd 12 2 525.2.bf.f 48
105.x even 12 2 735.2.y.i 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.x.a 48 7.d odd 6 2
105.2.x.a 48 21.g even 6 2
105.2.x.a 48 35.k even 12 2
105.2.x.a 48 105.w odd 12 2
525.2.bf.f 48 35.i odd 6 2
525.2.bf.f 48 35.k even 12 2
525.2.bf.f 48 105.p even 6 2
525.2.bf.f 48 105.w odd 12 2
735.2.j.e 24 1.a even 1 1 trivial
735.2.j.e 24 3.b odd 2 1 inner
735.2.j.e 24 5.c odd 4 1 inner
735.2.j.e 24 15.e even 4 1 inner
735.2.j.g 24 7.b odd 2 1
735.2.j.g 24 21.c even 2 1
735.2.j.g 24 35.f even 4 1
735.2.j.g 24 105.k odd 4 1
735.2.y.i 48 7.c even 3 2
735.2.y.i 48 21.h odd 6 2
735.2.y.i 48 35.l odd 12 2
735.2.y.i 48 105.x even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(735, [\chi])\):

\( T_{2}^{24} + 70 T_{2}^{20} + 1477 T_{2}^{16} + 9508 T_{2}^{12} + 20736 T_{2}^{8} + 11180 T_{2}^{4} + 100 \)
\(T_{13}^{12} - \cdots\)
\( T_{17}^{24} + 3624 T_{17}^{20} + 2187636 T_{17}^{16} + 288386740 T_{17}^{12} + 12787676836 T_{17}^{8} + 173525086080 T_{17}^{4} + 310204441600 \)

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database