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Properties

Label 735.2.v.a
Level $735$
Weight $2$
Character orbit 735.v
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(178,735)]
 
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([0, 9, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("735.178");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.v (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: no (minimal twist has level 105)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 48 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 48 q^{8} + 16 q^{11} + 16 q^{15} + 48 q^{16} - 32 q^{22} + 40 q^{23} + 8 q^{30} - 48 q^{32} - 32 q^{36} - 32 q^{37} - 32 q^{43} - 64 q^{46} - 144 q^{50} + 16 q^{51} - 24 q^{53} + 16 q^{57} - 32 q^{58} - 40 q^{60} - 40 q^{65} + 32 q^{67} + 128 q^{71} - 24 q^{72} - 16 q^{78} + 16 q^{81} + 96 q^{85} - 64 q^{86} + 64 q^{88} - 80 q^{92} - 24 q^{93} + 72 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
178.1 −2.03317 − 0.544785i −0.258819 − 0.965926i 2.10492 + 1.21528i −2.22675 − 0.203934i 2.10489i 0 −0.640825 − 0.640825i −0.866025 + 0.500000i 4.41625 + 1.62773i
178.2 −2.03317 − 0.544785i 0.258819 + 0.965926i 2.10492 + 1.21528i 2.22675 + 0.203934i − 2.10489i 0 −0.640825 − 0.640825i −0.866025 + 0.500000i −4.41625 − 1.62773i
178.3 −0.737849 − 0.197706i −0.258819 − 0.965926i −1.22672 − 0.708245i 1.19764 − 1.88830i 0.763878i 0 1.84539 + 1.84539i −0.866025 + 0.500000i −1.25700 + 1.15650i
178.4 −0.737849 − 0.197706i 0.258819 + 0.965926i −1.22672 − 0.708245i −1.19764 + 1.88830i − 0.763878i 0 1.84539 + 1.84539i −0.866025 + 0.500000i 1.25700 − 1.15650i
178.5 0.228203 + 0.0611467i −0.258819 − 0.965926i −1.68371 − 0.972092i 1.18965 + 1.89334i − 0.236253i 0 −0.658899 − 0.658899i −0.866025 + 0.500000i 0.155711 + 0.504808i
178.6 0.228203 + 0.0611467i 0.258819 + 0.965926i −1.68371 − 0.972092i −1.18965 − 1.89334i 0.236253i 0 −0.658899 − 0.658899i −0.866025 + 0.500000i −0.155711 − 0.504808i
178.7 2.54281 + 0.681344i −0.258819 − 0.965926i 4.26961 + 2.46506i −0.678180 + 2.13074i − 2.63251i 0 5.45433 + 5.45433i −0.866025 + 0.500000i −3.17625 + 4.95601i
178.8 2.54281 + 0.681344i 0.258819 + 0.965926i 4.26961 + 2.46506i 0.678180 − 2.13074i 2.63251i 0 5.45433 + 5.45433i −0.866025 + 0.500000i 3.17625 − 4.95601i
313.1 −0.681344 − 2.54281i −0.965926 − 0.258819i −4.26961 + 2.46506i −2.18437 + 0.478051i 2.63251i 0 5.45433 + 5.45433i 0.866025 + 0.500000i 2.70390 + 5.22872i
313.2 −0.681344 − 2.54281i 0.965926 + 0.258819i −4.26961 + 2.46506i 2.18437 − 0.478051i − 2.63251i 0 5.45433 + 5.45433i 0.866025 + 0.500000i −2.70390 − 5.22872i
313.3 −0.0611467 − 0.228203i −0.965926 − 0.258819i 1.68371 − 0.972092i −1.04485 + 1.97694i 0.236253i 0 −0.658899 − 0.658899i 0.866025 + 0.500000i 0.515032 + 0.117555i
313.4 −0.0611467 − 0.228203i 0.965926 + 0.258819i 1.68371 − 0.972092i 1.04485 − 1.97694i − 0.236253i 0 −0.658899 − 0.658899i 0.866025 + 0.500000i −0.515032 − 0.117555i
313.5 0.197706 + 0.737849i −0.965926 − 0.258819i 1.22672 − 0.708245i 2.23413 + 0.0930365i − 0.763878i 0 1.84539 + 1.84539i 0.866025 + 0.500000i 0.373055 + 1.66685i
313.6 0.197706 + 0.737849i 0.965926 + 0.258819i 1.22672 − 0.708245i −2.23413 − 0.0930365i 0.763878i 0 1.84539 + 1.84539i 0.866025 + 0.500000i −0.373055 − 1.66685i
313.7 0.544785 + 2.03317i −0.965926 − 0.258819i −2.10492 + 1.21528i −0.936763 − 2.03039i − 2.10489i 0 −0.640825 − 0.640825i 0.866025 + 0.500000i 3.61778 − 3.01072i
313.8 0.544785 + 2.03317i 0.965926 + 0.258819i −2.10492 + 1.21528i 0.936763 + 2.03039i 2.10489i 0 −0.640825 − 0.640825i 0.866025 + 0.500000i −3.61778 + 3.01072i
472.1 −0.681344 + 2.54281i −0.965926 + 0.258819i −4.26961 − 2.46506i −2.18437 − 0.478051i − 2.63251i 0 5.45433 − 5.45433i 0.866025 − 0.500000i 2.70390 − 5.22872i
472.2 −0.681344 + 2.54281i 0.965926 − 0.258819i −4.26961 − 2.46506i 2.18437 + 0.478051i 2.63251i 0 5.45433 − 5.45433i 0.866025 − 0.500000i −2.70390 + 5.22872i
472.3 −0.0611467 + 0.228203i −0.965926 + 0.258819i 1.68371 + 0.972092i −1.04485 − 1.97694i − 0.236253i 0 −0.658899 + 0.658899i 0.866025 − 0.500000i 0.515032 − 0.117555i
472.4 −0.0611467 + 0.228203i 0.965926 − 0.258819i 1.68371 + 0.972092i 1.04485 + 1.97694i 0.236253i 0 −0.658899 + 0.658899i 0.866025 − 0.500000i −0.515032 + 0.117555i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 178.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
35.f even 4 1 inner
35.k even 12 1 inner
35.l odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.v.a 32
5.c odd 4 1 inner 735.2.v.a 32
7.b odd 2 1 inner 735.2.v.a 32
7.c even 3 1 105.2.m.a ✓ 16
7.c even 3 1 inner 735.2.v.a 32
7.d odd 6 1 105.2.m.a ✓ 16
7.d odd 6 1 inner 735.2.v.a 32
21.g even 6 1 315.2.p.e 16
21.h odd 6 1 315.2.p.e 16
28.f even 6 1 1680.2.cz.d 16
28.g odd 6 1 1680.2.cz.d 16
35.f even 4 1 inner 735.2.v.a 32
35.i odd 6 1 525.2.m.b 16
35.j even 6 1 525.2.m.b 16
35.k even 12 1 105.2.m.a ✓ 16
35.k even 12 1 525.2.m.b 16
35.k even 12 1 inner 735.2.v.a 32
35.l odd 12 1 105.2.m.a ✓ 16
35.l odd 12 1 525.2.m.b 16
35.l odd 12 1 inner 735.2.v.a 32
105.w odd 12 1 315.2.p.e 16
105.x even 12 1 315.2.p.e 16
140.w even 12 1 1680.2.cz.d 16
140.x odd 12 1 1680.2.cz.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.m.a ✓ 16 7.c even 3 1
105.2.m.a ✓ 16 7.d odd 6 1
105.2.m.a ✓ 16 35.k even 12 1
105.2.m.a ✓ 16 35.l odd 12 1
315.2.p.e 16 21.g even 6 1
315.2.p.e 16 21.h odd 6 1
315.2.p.e 16 105.w odd 12 1
315.2.p.e 16 105.x even 12 1
525.2.m.b 16 35.i odd 6 1
525.2.m.b 16 35.j even 6 1
525.2.m.b 16 35.k even 12 1
525.2.m.b 16 35.l odd 12 1
735.2.v.a 32 1.a even 1 1 trivial
735.2.v.a 32 5.c odd 4 1 inner
735.2.v.a 32 7.b odd 2 1 inner
735.2.v.a 32 7.c even 3 1 inner
735.2.v.a 32 7.d odd 6 1 inner
735.2.v.a 32 35.f even 4 1 inner
735.2.v.a 32 35.k even 12 1 inner
735.2.v.a 32 35.l odd 12 1 inner
1680.2.cz.d 16 28.f even 6 1
1680.2.cz.d 16 28.g odd 6 1
1680.2.cz.d 16 140.w even 12 1
1680.2.cz.d 16 140.x odd 12 1

Hecke kernels

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

\( T_{2}^{16} - 8 T_{2}^{13} - 34 T_{2}^{12} + 24 T_{2}^{11} + 32 T_{2}^{10} + 132 T_{2}^{9} + 1059 T_{2}^{8} + \cdots + 1 \) Copy content Toggle raw display
\( T_{13}^{16} + 736T_{13}^{12} + 8576T_{13}^{8} + 18432T_{13}^{4} + 4096 \) Copy content Toggle raw display