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Properties

Label 735.2.y.f
Level $735$
Weight $2$
Character orbit 735.y
Analytic conductor $5.869$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [735,2,Mod(128,735)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 9, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("735.128");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.y (of order \(12\), degree \(4\), not 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: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{2} + 56 q^{8} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{2} + 56 q^{8} - 32 q^{9} - 56 q^{15} + 48 q^{16} - 12 q^{18} + 16 q^{22} - 16 q^{23} - 8 q^{25} - 64 q^{29} - 8 q^{30} - 52 q^{32} - 80 q^{36} + 24 q^{37} + 8 q^{39} + 48 q^{43} + 8 q^{44} - 88 q^{46} + 104 q^{50} - 40 q^{51} + 8 q^{53} + 32 q^{57} - 8 q^{58} + 116 q^{60} + 112 q^{65} - 16 q^{67} - 116 q^{72} + 8 q^{74} + 128 q^{78} + 16 q^{81} + 48 q^{85} + 80 q^{88} - 160 q^{92} - 80 q^{93} - 72 q^{95} + 80 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
128.1 −1.47626 + 0.395563i −1.40294 − 1.01575i 0.290825 − 0.167908i 0.768666 − 2.09980i 2.47290 + 0.944565i 0 1.79848 − 1.79848i 0.936492 + 2.85008i −0.304149 + 3.40391i
128.2 −1.47626 + 0.395563i 1.40294 + 1.01575i 0.290825 − 0.167908i −0.768666 + 2.09980i −2.47290 − 0.944565i 0 1.79848 − 1.79848i 0.936492 + 2.85008i 0.304149 − 3.40391i
128.3 −0.362630 + 0.0971664i −0.178197 − 1.72286i −1.60999 + 0.929529i 1.74522 − 1.39793i 0.232024 + 0.607446i 0 1.02444 − 1.02444i −2.93649 + 0.614017i −0.497037 + 0.676508i
128.4 −0.362630 + 0.0971664i 0.178197 + 1.72286i −1.60999 + 0.929529i −1.74522 + 1.39793i −0.232024 − 0.607446i 0 1.02444 − 1.02444i −2.93649 + 0.614017i 0.497037 − 0.676508i
128.5 0.632011 − 0.169347i −1.40294 − 1.01575i −1.36129 + 0.785942i 0.893095 + 2.04997i −1.05869 − 0.404383i 0 −1.65258 + 1.65258i 0.936492 + 2.85008i 0.911602 + 1.14436i
128.6 0.632011 − 0.169347i 1.40294 + 1.01575i −1.36129 + 0.785942i −0.893095 − 2.04997i 1.05869 + 0.404383i 0 −1.65258 + 1.65258i 0.936492 + 2.85008i −0.911602 − 1.14436i
128.7 2.57291 − 0.689408i −0.178197 − 1.72286i 4.41251 − 2.54756i −1.82584 − 1.29085i −1.64624 − 4.30991i 0 5.82966 − 5.82966i −2.93649 + 0.614017i −5.58764 − 2.06250i
128.8 2.57291 − 0.689408i 0.178197 + 1.72286i 4.41251 − 2.54756i 1.82584 + 1.29085i 1.64624 + 4.30991i 0 5.82966 − 5.82966i −2.93649 + 0.614017i 5.58764 + 2.06250i
263.1 −0.689408 + 2.57291i −1.40294 + 1.01575i −4.41251 − 2.54756i 2.03083 − 0.935797i −1.64624 − 4.30991i 0 5.82966 − 5.82966i 0.936492 − 2.85008i 1.00765 + 5.87029i
263.2 −0.689408 + 2.57291i 1.40294 − 1.01575i −4.41251 − 2.54756i −2.03083 + 0.935797i 1.64624 + 4.30991i 0 5.82966 − 5.82966i 0.936492 − 2.85008i −1.00765 − 5.87029i
263.3 −0.169347 + 0.632011i −0.178197 + 1.72286i 1.36129 + 0.785942i −2.22187 − 0.251543i −1.05869 − 0.404383i 0 −1.65258 + 1.65258i −2.93649 − 0.614017i 0.535245 − 1.36165i
263.4 −0.169347 + 0.632011i 0.178197 − 1.72286i 1.36129 + 0.785942i 2.22187 + 0.251543i 1.05869 + 0.404383i 0 −1.65258 + 1.65258i −2.93649 − 0.614017i −0.535245 + 1.36165i
263.5 0.0971664 − 0.362630i −1.40294 + 1.01575i 1.60999 + 0.929529i 0.338034 + 2.21037i 0.232024 + 0.607446i 0 1.02444 − 1.02444i 0.936492 − 2.85008i 0.834392 + 0.0921922i
263.6 0.0971664 − 0.362630i 1.40294 − 1.01575i 1.60999 + 0.929529i −0.338034 − 2.21037i −0.232024 − 0.607446i 0 1.02444 − 1.02444i 0.936492 − 2.85008i −0.834392 − 0.0921922i
263.7 0.395563 − 1.47626i −0.178197 + 1.72286i −0.290825 − 0.167908i 1.43415 + 1.71558i 2.47290 + 0.944565i 0 1.79848 − 1.79848i −2.93649 − 0.614017i 3.09994 − 1.43855i
263.8 0.395563 − 1.47626i 0.178197 − 1.72286i −0.290825 − 0.167908i −1.43415 − 1.71558i −2.47290 − 0.944565i 0 1.79848 − 1.79848i −2.93649 − 0.614017i −3.09994 + 1.43855i
422.1 −0.689408 − 2.57291i −1.40294 − 1.01575i −4.41251 + 2.54756i 2.03083 + 0.935797i −1.64624 + 4.30991i 0 5.82966 + 5.82966i 0.936492 + 2.85008i 1.00765 − 5.87029i
422.2 −0.689408 − 2.57291i 1.40294 + 1.01575i −4.41251 + 2.54756i −2.03083 − 0.935797i 1.64624 − 4.30991i 0 5.82966 + 5.82966i 0.936492 + 2.85008i −1.00765 + 5.87029i
422.3 −0.169347 − 0.632011i −0.178197 − 1.72286i 1.36129 − 0.785942i −2.22187 + 0.251543i −1.05869 + 0.404383i 0 −1.65258 − 1.65258i −2.93649 + 0.614017i 0.535245 + 1.36165i
422.4 −0.169347 − 0.632011i 0.178197 + 1.72286i 1.36129 − 0.785942i 2.22187 − 0.251543i 1.05869 − 0.404383i 0 −1.65258 − 1.65258i −2.93649 + 0.614017i −0.535245 − 1.36165i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 128.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
15.e even 4 1 inner
105.k odd 4 1 inner
105.w odd 12 1 inner
105.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.y.f 32
3.b odd 2 1 735.2.y.e 32
5.c odd 4 1 735.2.y.e 32
7.b odd 2 1 inner 735.2.y.f 32
7.c even 3 1 735.2.j.c ✓ 16
7.c even 3 1 inner 735.2.y.f 32
7.d odd 6 1 735.2.j.c ✓ 16
7.d odd 6 1 inner 735.2.y.f 32
15.e even 4 1 inner 735.2.y.f 32
21.c even 2 1 735.2.y.e 32
21.g even 6 1 735.2.j.d yes 16
21.g even 6 1 735.2.y.e 32
21.h odd 6 1 735.2.j.d yes 16
21.h odd 6 1 735.2.y.e 32
35.f even 4 1 735.2.y.e 32
35.k even 12 1 735.2.j.d yes 16
35.k even 12 1 735.2.y.e 32
35.l odd 12 1 735.2.j.d yes 16
35.l odd 12 1 735.2.y.e 32
105.k odd 4 1 inner 735.2.y.f 32
105.w odd 12 1 735.2.j.c ✓ 16
105.w odd 12 1 inner 735.2.y.f 32
105.x even 12 1 735.2.j.c ✓ 16
105.x even 12 1 inner 735.2.y.f 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
735.2.j.c ✓ 16 7.c even 3 1
735.2.j.c ✓ 16 7.d odd 6 1
735.2.j.c ✓ 16 105.w odd 12 1
735.2.j.c ✓ 16 105.x even 12 1
735.2.j.d yes 16 21.g even 6 1
735.2.j.d yes 16 21.h odd 6 1
735.2.j.d yes 16 35.k even 12 1
735.2.j.d yes 16 35.l odd 12 1
735.2.y.e 32 3.b odd 2 1
735.2.y.e 32 5.c odd 4 1
735.2.y.e 32 21.c even 2 1
735.2.y.e 32 21.g even 6 1
735.2.y.e 32 21.h odd 6 1
735.2.y.e 32 35.f even 4 1
735.2.y.e 32 35.k even 12 1
735.2.y.e 32 35.l odd 12 1
735.2.y.f 32 1.a even 1 1 trivial
735.2.y.f 32 7.b odd 2 1 inner
735.2.y.f 32 7.c even 3 1 inner
735.2.y.f 32 7.d odd 6 1 inner
735.2.y.f 32 15.e even 4 1 inner
735.2.y.f 32 105.k odd 4 1 inner
735.2.y.f 32 105.w odd 12 1 inner
735.2.y.f 32 105.x even 12 1 inner

Hecke kernels

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

\( T_{2}^{16} - 2 T_{2}^{15} + 2 T_{2}^{14} - 16 T_{2}^{13} + 2 T_{2}^{12} + 62 T_{2}^{11} + 106 T_{2}^{9} + \cdots + 1 \) Copy content Toggle raw display
\( T_{11}^{16} - 44 T_{11}^{14} + 1560 T_{11}^{12} - 15296 T_{11}^{10} + 113904 T_{11}^{8} - 233216 T_{11}^{6} + \cdots + 256 \) Copy content Toggle raw display
\( T_{13}^{16} + 2584T_{13}^{12} + 1444656T_{13}^{8} + 227214464T_{13}^{4} + 3102044416 \) Copy content Toggle raw display
\( T_{17}^{32} - 1064 T_{17}^{28} + 818320 T_{17}^{24} - 298001536 T_{17}^{20} + 79143496704 T_{17}^{16} + \cdots + 55\!\cdots\!76 \) Copy content Toggle raw display