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Properties

Label 735.2.m.c
Level $735$
Weight $2$
Character orbit 735.m
Analytic conductor $5.869$
Analytic rank $0$
Dimension $32$
Inner twists $4$

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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(97,735)] 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("735.97"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(735, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == 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: \(16\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 24 q^{8} + 16 q^{11} - 8 q^{15} - 8 q^{22} + 16 q^{23} - 24 q^{25} - 16 q^{30} - 48 q^{32} - 32 q^{36} - 8 q^{37} + 40 q^{43} + 80 q^{46} + 72 q^{50} + 16 q^{51} + 48 q^{53} + 16 q^{57} - 8 q^{58}+ \cdots + 24 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1 −1.59147 − 1.59147i −0.707107 − 0.707107i 3.06556i 1.90623 + 1.16889i 2.25068i 0 1.69581 − 1.69581i 1.00000i −1.17345 − 4.89396i
97.2 −1.59147 − 1.59147i 0.707107 + 0.707107i 3.06556i −1.90623 − 1.16889i − 2.25068i 0 1.69581 − 1.69581i 1.00000i 1.17345 + 4.89396i
97.3 −1.25995 − 1.25995i −0.707107 − 0.707107i 1.17495i −0.0279443 − 2.23589i 1.78184i 0 −1.03952 + 1.03952i 1.00000i −2.78191 + 2.85232i
97.4 −1.25995 − 1.25995i 0.707107 + 0.707107i 1.17495i 0.0279443 + 2.23589i − 1.78184i 0 −1.03952 + 1.03952i 1.00000i 2.78191 − 2.85232i
97.5 −1.09556 − 1.09556i −0.707107 − 0.707107i 0.400511i −0.531984 + 2.17186i 1.54936i 0 −1.75234 + 1.75234i 1.00000i 2.96223 − 1.79659i
97.6 −1.09556 − 1.09556i 0.707107 + 0.707107i 0.400511i 0.531984 − 2.17186i − 1.54936i 0 −1.75234 + 1.75234i 1.00000i −2.96223 + 1.79659i
97.7 −0.474562 − 0.474562i −0.707107 − 0.707107i − 1.54958i −0.484013 − 2.18306i 0.671132i 0 −1.68450 + 1.68450i 1.00000i −0.806302 + 1.26569i
97.8 −0.474562 − 0.474562i 0.707107 + 0.707107i − 1.54958i 0.484013 + 2.18306i − 0.671132i 0 −1.68450 + 1.68450i 1.00000i 0.806302 − 1.26569i
97.9 0.288785 + 0.288785i −0.707107 − 0.707107i − 1.83321i 1.48985 + 1.66743i − 0.408404i 0 1.10697 − 1.10697i 1.00000i −0.0512822 + 0.911777i
97.10 0.288785 + 0.288785i 0.707107 + 0.707107i − 1.83321i −1.48985 − 1.66743i 0.408404i 0 1.10697 − 1.10697i 1.00000i 0.0512822 − 0.911777i
97.11 0.709756 + 0.709756i −0.707107 − 0.707107i − 0.992492i 2.20896 − 0.347130i − 1.00375i 0 2.12394 − 2.12394i 1.00000i 1.81420 + 1.32145i
97.12 0.709756 + 0.709756i 0.707107 + 0.707107i − 0.992492i −2.20896 + 0.347130i 1.00375i 0 2.12394 − 2.12394i 1.00000i −1.81420 − 1.32145i
97.13 1.64576 + 1.64576i −0.707107 − 0.707107i 3.41703i −0.941924 − 2.02800i − 2.32745i 0 −2.33208 + 2.33208i 1.00000i 1.78741 − 4.88777i
97.14 1.64576 + 1.64576i 0.707107 + 0.707107i 3.41703i 0.941924 + 2.02800i 2.32745i 0 −2.33208 + 2.33208i 1.00000i −1.78741 + 4.88777i
97.15 1.77725 + 1.77725i −0.707107 − 0.707107i 4.31723i −2.20496 + 0.371678i − 2.51341i 0 −4.11829 + 4.11829i 1.00000i −4.57933 − 3.25820i
97.16 1.77725 + 1.77725i 0.707107 + 0.707107i 4.31723i 2.20496 − 0.371678i 2.51341i 0 −4.11829 + 4.11829i 1.00000i 4.57933 + 3.25820i
538.1 −1.59147 + 1.59147i −0.707107 + 0.707107i − 3.06556i 1.90623 − 1.16889i − 2.25068i 0 1.69581 + 1.69581i − 1.00000i −1.17345 + 4.89396i
538.2 −1.59147 + 1.59147i 0.707107 − 0.707107i − 3.06556i −1.90623 + 1.16889i 2.25068i 0 1.69581 + 1.69581i − 1.00000i 1.17345 − 4.89396i
538.3 −1.25995 + 1.25995i −0.707107 + 0.707107i − 1.17495i −0.0279443 + 2.23589i − 1.78184i 0 −1.03952 − 1.03952i − 1.00000i −2.78191 − 2.85232i
538.4 −1.25995 + 1.25995i 0.707107 − 0.707107i − 1.17495i 0.0279443 − 2.23589i 1.78184i 0 −1.03952 − 1.03952i − 1.00000i 2.78191 + 2.85232i
See all 32 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 97.16
Significant digits:
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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
35.f 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.m.c 32
5.c odd 4 1 inner 735.2.m.c 32
7.b odd 2 1 inner 735.2.m.c 32
7.c even 3 1 105.2.u.a ✓ 32
7.c even 3 1 735.2.v.b 32
7.d odd 6 1 105.2.u.a ✓ 32
7.d odd 6 1 735.2.v.b 32
21.g even 6 1 315.2.bz.d 32
21.h odd 6 1 315.2.bz.d 32
35.f even 4 1 inner 735.2.m.c 32
35.i odd 6 1 525.2.bc.e 32
35.j even 6 1 525.2.bc.e 32
35.k even 12 1 105.2.u.a ✓ 32
35.k even 12 1 525.2.bc.e 32
35.k even 12 1 735.2.v.b 32
35.l odd 12 1 105.2.u.a ✓ 32
35.l odd 12 1 525.2.bc.e 32
35.l odd 12 1 735.2.v.b 32
105.w odd 12 1 315.2.bz.d 32
105.x even 12 1 315.2.bz.d 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.u.a ✓ 32 7.c even 3 1
105.2.u.a ✓ 32 7.d odd 6 1
105.2.u.a ✓ 32 35.k even 12 1
105.2.u.a ✓ 32 35.l odd 12 1
315.2.bz.d 32 21.g even 6 1
315.2.bz.d 32 21.h odd 6 1
315.2.bz.d 32 105.w odd 12 1
315.2.bz.d 32 105.x even 12 1
525.2.bc.e 32 35.i odd 6 1
525.2.bc.e 32 35.j even 6 1
525.2.bc.e 32 35.k even 12 1
525.2.bc.e 32 35.l odd 12 1
735.2.m.c 32 1.a even 1 1 trivial
735.2.m.c 32 5.c odd 4 1 inner
735.2.m.c 32 7.b odd 2 1 inner
735.2.m.c 32 35.f even 4 1 inner
735.2.v.b 32 7.c even 3 1
735.2.v.b 32 7.d odd 6 1
735.2.v.b 32 35.k even 12 1
735.2.v.b 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 \(S_{2}^{\mathrm{new}}(735, [\chi])\):

\( T_{2}^{16} + 4 T_{2}^{13} + 56 T_{2}^{12} + 24 T_{2}^{11} + 8 T_{2}^{10} + 96 T_{2}^{9} + 852 T_{2}^{8} + \cdots + 100 \) Copy content Toggle raw display
\( T_{13}^{32} + 3752 T_{13}^{28} + 4933436 T_{13}^{24} + 2838285760 T_{13}^{20} + 768155582310 T_{13}^{16} + \cdots + 9926437890625 \) Copy content Toggle raw display