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Properties

Label 735.2.j.g
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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [735,2,Mod(197,735)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("735.197");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
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.

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 197.12
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.g 24
3.b odd 2 1 inner 735.2.j.g 24
5.c odd 4 1 inner 735.2.j.g 24
7.b odd 2 1 735.2.j.e 24
7.c even 3 2 105.2.x.a ✓ 48
7.d odd 6 2 735.2.y.i 48
15.e even 4 1 inner 735.2.j.g 24
21.c even 2 1 735.2.j.e 24
21.g even 6 2 735.2.y.i 48
21.h odd 6 2 105.2.x.a ✓ 48
35.f even 4 1 735.2.j.e 24
35.j even 6 2 525.2.bf.f 48
35.k even 12 2 735.2.y.i 48
35.l odd 12 2 105.2.x.a ✓ 48
35.l odd 12 2 525.2.bf.f 48
105.k odd 4 1 735.2.j.e 24
105.o odd 6 2 525.2.bf.f 48
105.w odd 12 2 735.2.y.i 48
105.x even 12 2 105.2.x.a ✓ 48
105.x even 12 2 525.2.bf.f 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.x.a ✓ 48 7.c even 3 2
105.2.x.a ✓ 48 21.h odd 6 2
105.2.x.a ✓ 48 35.l odd 12 2
105.2.x.a ✓ 48 105.x even 12 2
525.2.bf.f 48 35.j even 6 2
525.2.bf.f 48 35.l odd 12 2
525.2.bf.f 48 105.o odd 6 2
525.2.bf.f 48 105.x even 12 2
735.2.j.e 24 7.b odd 2 1
735.2.j.e 24 21.c even 2 1
735.2.j.e 24 35.f even 4 1
735.2.j.e 24 105.k odd 4 1
735.2.j.g 24 1.a even 1 1 trivial
735.2.j.g 24 3.b odd 2 1 inner
735.2.j.g 24 5.c odd 4 1 inner
735.2.j.g 24 15.e even 4 1 inner
735.2.y.i 48 7.d odd 6 2
735.2.y.i 48 21.g even 6 2
735.2.y.i 48 35.k even 12 2
735.2.y.i 48 105.w odd 12 2

Hecke kernels

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

\( T_{2}^{24} + 70T_{2}^{20} + 1477T_{2}^{16} + 9508T_{2}^{12} + 20736T_{2}^{8} + 11180T_{2}^{4} + 100 \) Copy content Toggle raw display
\( T_{13}^{12} + 4 T_{13}^{11} + 8 T_{13}^{10} + 26 T_{13}^{9} + 334 T_{13}^{8} + 1504 T_{13}^{7} + \cdots + 30976 \) Copy content Toggle raw display
\( T_{17}^{24} + 3624 T_{17}^{20} + 2187636 T_{17}^{16} + 288386740 T_{17}^{12} + 12787676836 T_{17}^{8} + \cdots + 310204441600 \) Copy content Toggle raw display