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Properties

Label 840.2.bt.a
Level $840$
Weight $2$
Character orbit 840.bt
Analytic conductor $6.707$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [840,2,Mod(97,840)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(840, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 2])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("840.97"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.bt (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.70743376979\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{7} - 8 q^{11} + 16 q^{13} + 4 q^{15} + 20 q^{17} + 8 q^{19} + 24 q^{23} - 4 q^{25} + 4 q^{37} - 16 q^{43} - 4 q^{45} - 24 q^{47} + 36 q^{49} + 16 q^{53} + 28 q^{55} + 4 q^{57} + 8 q^{59} + 24 q^{65}+ \cdots - 28 q^{97}+O(q^{100}) \) 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 0 −0.707107 − 0.707107i 0 −1.98667 − 1.02624i 0 −2.56958 − 0.630291i 0 1.00000i 0
97.2 0 −0.707107 − 0.707107i 0 −1.35806 + 1.77642i 0 −0.498068 + 2.59845i 0 1.00000i 0
97.3 0 −0.707107 − 0.707107i 0 −0.793022 − 2.09072i 0 2.38904 − 1.13687i 0 1.00000i 0
97.4 0 −0.707107 − 0.707107i 0 −0.503511 + 2.17864i 0 0.918874 − 2.48106i 0 1.00000i 0
97.5 0 −0.707107 − 0.707107i 0 1.75776 + 1.38213i 0 2.02918 + 1.69778i 0 1.00000i 0
97.6 0 −0.707107 − 0.707107i 0 2.17640 − 0.513127i 0 −2.56234 + 0.659108i 0 1.00000i 0
97.7 0 0.707107 + 0.707107i 0 −2.19074 − 0.447937i 0 1.88862 − 1.85286i 0 1.00000i 0
97.8 0 0.707107 + 0.707107i 0 −1.43773 − 1.71258i 0 −2.23600 + 1.41433i 0 1.00000i 0
97.9 0 0.707107 + 0.707107i 0 −0.0188126 + 2.23599i 0 −2.30215 − 1.30388i 0 1.00000i 0
97.10 0 0.707107 + 0.707107i 0 0.600524 + 2.15392i 0 2.62016 − 0.367100i 0 1.00000i 0
97.11 0 0.707107 + 0.707107i 0 1.54016 − 1.62109i 0 −2.38404 − 1.14733i 0 1.00000i 0
97.12 0 0.707107 + 0.707107i 0 2.21371 − 0.315413i 0 0.706295 + 2.54973i 0 1.00000i 0
433.1 0 −0.707107 + 0.707107i 0 −1.98667 + 1.02624i 0 −2.56958 + 0.630291i 0 − 1.00000i 0
433.2 0 −0.707107 + 0.707107i 0 −1.35806 − 1.77642i 0 −0.498068 − 2.59845i 0 − 1.00000i 0
433.3 0 −0.707107 + 0.707107i 0 −0.793022 + 2.09072i 0 2.38904 + 1.13687i 0 − 1.00000i 0
433.4 0 −0.707107 + 0.707107i 0 −0.503511 − 2.17864i 0 0.918874 + 2.48106i 0 − 1.00000i 0
433.5 0 −0.707107 + 0.707107i 0 1.75776 − 1.38213i 0 2.02918 − 1.69778i 0 − 1.00000i 0
433.6 0 −0.707107 + 0.707107i 0 2.17640 + 0.513127i 0 −2.56234 − 0.659108i 0 − 1.00000i 0
433.7 0 0.707107 − 0.707107i 0 −2.19074 + 0.447937i 0 1.88862 + 1.85286i 0 − 1.00000i 0
433.8 0 0.707107 − 0.707107i 0 −1.43773 + 1.71258i 0 −2.23600 − 1.41433i 0 − 1.00000i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 840.2.bt.a ✓ 24
4.b odd 2 1 1680.2.cz.f 24
5.c odd 4 1 840.2.bt.b yes 24
7.b odd 2 1 840.2.bt.b yes 24
20.e even 4 1 1680.2.cz.e 24
28.d even 2 1 1680.2.cz.e 24
35.f even 4 1 inner 840.2.bt.a ✓ 24
140.j odd 4 1 1680.2.cz.f 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.bt.a ✓ 24 1.a even 1 1 trivial
840.2.bt.a ✓ 24 35.f even 4 1 inner
840.2.bt.b yes 24 5.c odd 4 1
840.2.bt.b yes 24 7.b odd 2 1
1680.2.cz.e 24 20.e even 4 1
1680.2.cz.e 24 28.d even 2 1
1680.2.cz.f 24 4.b odd 2 1
1680.2.cz.f 24 140.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{24} - 16 T_{13}^{23} + 128 T_{13}^{22} - 544 T_{13}^{21} + 3456 T_{13}^{20} - 34688 T_{13}^{19} + \cdots + 2316304384 \) acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\). Copy content Toggle raw display