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Properties

Label 735.2.y.e
Level $735$
Weight $2$
Character orbit 735.y
Analytic conductor $5.869$
Analytic rank $0$
Dimension $32$
Inner twists $8$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [735,2,Mod(128,735)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
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

Copy content comment:select newform
 
Copy content sage:traces = [32,-4,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content 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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{2} - 56 q^{8} + 32 q^{9} + 8 q^{15} + 48 q^{16} + 28 q^{18} + 16 q^{22} + 16 q^{23} - 8 q^{25} + 64 q^{29} - 64 q^{30} + 52 q^{32} - 80 q^{36} + 24 q^{37} - 8 q^{39} + 48 q^{43} - 8 q^{44} - 88 q^{46}+ \cdots - 80 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
128.1 −2.57291 + 0.689408i −1.01575 + 1.40294i 4.41251 − 2.54756i −1.82584 − 1.29085i 1.64624 − 4.30991i 0 −5.82966 + 5.82966i −0.936492 − 2.85008i 5.58764 + 2.06250i
128.2 −2.57291 + 0.689408i 1.01575 − 1.40294i 4.41251 − 2.54756i 1.82584 + 1.29085i −1.64624 + 4.30991i 0 −5.82966 + 5.82966i −0.936492 − 2.85008i −5.58764 − 2.06250i
128.3 −0.632011 + 0.169347i −1.72286 + 0.178197i −1.36129 + 0.785942i 0.893095 + 2.04997i 1.05869 − 0.404383i 0 1.65258 − 1.65258i 2.93649 − 0.614017i −0.911602 − 1.14436i
128.4 −0.632011 + 0.169347i 1.72286 − 0.178197i −1.36129 + 0.785942i −0.893095 − 2.04997i −1.05869 + 0.404383i 0 1.65258 − 1.65258i 2.93649 − 0.614017i 0.911602 + 1.14436i
128.5 0.362630 − 0.0971664i −1.01575 + 1.40294i −1.60999 + 0.929529i 1.74522 − 1.39793i −0.232024 + 0.607446i 0 −1.02444 + 1.02444i −0.936492 − 2.85008i 0.497037 − 0.676508i
128.6 0.362630 − 0.0971664i 1.01575 − 1.40294i −1.60999 + 0.929529i −1.74522 + 1.39793i 0.232024 − 0.607446i 0 −1.02444 + 1.02444i −0.936492 − 2.85008i −0.497037 + 0.676508i
128.7 1.47626 − 0.395563i −1.72286 + 0.178197i 0.290825 − 0.167908i 0.768666 − 2.09980i −2.47290 + 0.944565i 0 −1.79848 + 1.79848i 2.93649 − 0.614017i 0.304149 − 3.40391i
128.8 1.47626 − 0.395563i 1.72286 − 0.178197i 0.290825 − 0.167908i −0.768666 + 2.09980i 2.47290 − 0.944565i 0 −1.79848 + 1.79848i 2.93649 − 0.614017i −0.304149 + 3.40391i
263.1 −0.395563 + 1.47626i −1.01575 − 1.40294i −0.290825 − 0.167908i −1.43415 − 1.71558i 2.47290 − 0.944565i 0 −1.79848 + 1.79848i −0.936492 + 2.85008i 3.09994 − 1.43855i
263.2 −0.395563 + 1.47626i 1.01575 + 1.40294i −0.290825 − 0.167908i 1.43415 + 1.71558i −2.47290 + 0.944565i 0 −1.79848 + 1.79848i −0.936492 + 2.85008i −3.09994 + 1.43855i
263.3 −0.0971664 + 0.362630i −1.72286 − 0.178197i 1.60999 + 0.929529i −0.338034 − 2.21037i 0.232024 − 0.607446i 0 −1.02444 + 1.02444i 2.93649 + 0.614017i 0.834392 + 0.0921922i
263.4 −0.0971664 + 0.362630i 1.72286 + 0.178197i 1.60999 + 0.929529i 0.338034 + 2.21037i −0.232024 + 0.607446i 0 −1.02444 + 1.02444i 2.93649 + 0.614017i −0.834392 − 0.0921922i
263.5 0.169347 − 0.632011i −1.01575 − 1.40294i 1.36129 + 0.785942i 2.22187 + 0.251543i −1.05869 + 0.404383i 0 1.65258 − 1.65258i −0.936492 + 2.85008i 0.535245 − 1.36165i
263.6 0.169347 − 0.632011i 1.01575 + 1.40294i 1.36129 + 0.785942i −2.22187 − 0.251543i 1.05869 − 0.404383i 0 1.65258 − 1.65258i −0.936492 + 2.85008i −0.535245 + 1.36165i
263.7 0.689408 − 2.57291i −1.72286 − 0.178197i −4.41251 − 2.54756i −2.03083 + 0.935797i −1.64624 + 4.30991i 0 −5.82966 + 5.82966i 2.93649 + 0.614017i 1.00765 + 5.87029i
263.8 0.689408 − 2.57291i 1.72286 + 0.178197i −4.41251 − 2.54756i 2.03083 − 0.935797i 1.64624 − 4.30991i 0 −5.82966 + 5.82966i 2.93649 + 0.614017i −1.00765 − 5.87029i
422.1 −0.395563 − 1.47626i −1.01575 + 1.40294i −0.290825 + 0.167908i −1.43415 + 1.71558i 2.47290 + 0.944565i 0 −1.79848 − 1.79848i −0.936492 − 2.85008i 3.09994 + 1.43855i
422.2 −0.395563 − 1.47626i 1.01575 − 1.40294i −0.290825 + 0.167908i 1.43415 − 1.71558i −2.47290 − 0.944565i 0 −1.79848 − 1.79848i −0.936492 − 2.85008i −3.09994 − 1.43855i
422.3 −0.0971664 − 0.362630i −1.72286 + 0.178197i 1.60999 − 0.929529i −0.338034 + 2.21037i 0.232024 + 0.607446i 0 −1.02444 − 1.02444i 2.93649 − 0.614017i 0.834392 − 0.0921922i
422.4 −0.0971664 − 0.362630i 1.72286 − 0.178197i 1.60999 − 0.929529i 0.338034 − 2.21037i −0.232024 − 0.607446i 0 −1.02444 − 1.02444i 2.93649 − 0.614017i −0.834392 + 0.0921922i
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.e 32
3.b odd 2 1 735.2.y.f 32
5.c odd 4 1 735.2.y.f 32
7.b odd 2 1 inner 735.2.y.e 32
7.c even 3 1 735.2.j.d yes 16
7.c even 3 1 inner 735.2.y.e 32
7.d odd 6 1 735.2.j.d yes 16
7.d odd 6 1 inner 735.2.y.e 32
15.e even 4 1 inner 735.2.y.e 32
21.c even 2 1 735.2.y.f 32
21.g even 6 1 735.2.j.c ✓ 16
21.g even 6 1 735.2.y.f 32
21.h odd 6 1 735.2.j.c ✓ 16
21.h odd 6 1 735.2.y.f 32
35.f even 4 1 735.2.y.f 32
35.k even 12 1 735.2.j.c ✓ 16
35.k even 12 1 735.2.y.f 32
35.l odd 12 1 735.2.j.c ✓ 16
35.l odd 12 1 735.2.y.f 32
105.k odd 4 1 inner 735.2.y.e 32
105.w odd 12 1 735.2.j.d yes 16
105.w odd 12 1 inner 735.2.y.e 32
105.x even 12 1 735.2.j.d yes 16
105.x even 12 1 inner 735.2.y.e 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
735.2.j.c ✓ 16 21.g even 6 1
735.2.j.c ✓ 16 21.h odd 6 1
735.2.j.c ✓ 16 35.k even 12 1
735.2.j.c ✓ 16 35.l odd 12 1
735.2.j.d yes 16 7.c even 3 1
735.2.j.d yes 16 7.d odd 6 1
735.2.j.d yes 16 105.w odd 12 1
735.2.j.d yes 16 105.x even 12 1
735.2.y.e 32 1.a even 1 1 trivial
735.2.y.e 32 7.b odd 2 1 inner
735.2.y.e 32 7.c even 3 1 inner
735.2.y.e 32 7.d odd 6 1 inner
735.2.y.e 32 15.e even 4 1 inner
735.2.y.e 32 105.k odd 4 1 inner
735.2.y.e 32 105.w odd 12 1 inner
735.2.y.e 32 105.x even 12 1 inner
735.2.y.f 32 3.b odd 2 1
735.2.y.f 32 5.c odd 4 1
735.2.y.f 32 21.c even 2 1
735.2.y.f 32 21.g even 6 1
735.2.y.f 32 21.h odd 6 1
735.2.y.f 32 35.f even 4 1
735.2.y.f 32 35.k even 12 1
735.2.y.f 32 35.l odd 12 1

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