↑

Properties

Label 840.2.cp.b
Level $840$
Weight $2$
Character orbit 840.cp
Analytic conductor $6.707$
Analytic rank $0$
Dimension $32$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [840,2,Mod(521,840)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(840, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 0, 1])) 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("840.521"); 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.cp (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,0,16] 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: \(6.70743376979\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{5} - 2 q^{7} + 8 q^{9} + 6 q^{19} - 14 q^{21} + 24 q^{23} - 16 q^{25} + 24 q^{27} + 42 q^{31} + 18 q^{33} + 2 q^{35} + 6 q^{37} + 12 q^{39} + 44 q^{41} - 20 q^{43} + 10 q^{45} + 4 q^{47} + 16 q^{49}+ \cdots - 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
521.1 0 −1.71210 − 0.262156i 0 0.500000 + 0.866025i 0 2.06354 − 1.65584i 0 2.86255 + 0.897672i 0
521.2 0 −1.68390 − 0.405545i 0 0.500000 + 0.866025i 0 2.40193 + 1.10938i 0 2.67107 + 1.36580i 0
521.3 0 −1.62344 + 0.603694i 0 0.500000 + 0.866025i 0 −1.29532 + 2.30698i 0 2.27111 − 1.96012i 0
521.4 0 −1.15951 + 1.28667i 0 0.500000 + 0.866025i 0 −2.64426 − 0.0889087i 0 −0.311064 − 2.98383i 0
521.5 0 −0.950788 − 1.44776i 0 0.500000 + 0.866025i 0 −0.887076 − 2.49261i 0 −1.19201 + 2.75302i 0
521.6 0 −0.801214 + 1.53560i 0 0.500000 + 0.866025i 0 1.33762 − 2.28271i 0 −1.71611 − 2.46068i 0
521.7 0 −0.789474 − 1.54167i 0 0.500000 + 0.866025i 0 −2.25949 − 1.37648i 0 −1.75346 + 2.43421i 0
521.8 0 −0.645495 + 1.60728i 0 0.500000 + 0.866025i 0 2.09890 + 1.61078i 0 −2.16667 − 2.07498i 0
521.9 0 0.0468355 − 1.73142i 0 0.500000 + 0.866025i 0 0.719245 + 2.54611i 0 −2.99561 − 0.162184i 0
521.10 0 0.382234 + 1.68935i 0 0.500000 + 0.866025i 0 −2.62618 − 0.321191i 0 −2.70779 + 1.29145i 0
521.11 0 1.10659 − 1.33247i 0 0.500000 + 0.866025i 0 2.52792 − 0.780793i 0 −0.550930 − 2.94898i 0
521.12 0 1.46285 − 0.927396i 0 0.500000 + 0.866025i 0 −1.49899 − 2.18014i 0 1.27987 − 2.71329i 0
521.13 0 1.51560 − 0.838429i 0 0.500000 + 0.866025i 0 −1.32384 + 2.29073i 0 1.59407 − 2.54144i 0
521.14 0 1.56554 + 0.741007i 0 0.500000 + 0.866025i 0 2.50460 + 0.852628i 0 1.90182 + 2.32015i 0
521.15 0 1.58492 + 0.698582i 0 0.500000 + 0.866025i 0 −2.26230 + 1.37186i 0 2.02397 + 2.21440i 0
521.16 0 1.70135 + 0.324654i 0 0.500000 + 0.866025i 0 0.143717 − 2.64185i 0 2.78920 + 1.10470i 0
761.1 0 −1.71210 + 0.262156i 0 0.500000 − 0.866025i 0 2.06354 + 1.65584i 0 2.86255 − 0.897672i 0
761.2 0 −1.68390 + 0.405545i 0 0.500000 − 0.866025i 0 2.40193 − 1.10938i 0 2.67107 − 1.36580i 0
761.3 0 −1.62344 − 0.603694i 0 0.500000 − 0.866025i 0 −1.29532 − 2.30698i 0 2.27111 + 1.96012i 0
761.4 0 −1.15951 − 1.28667i 0 0.500000 − 0.866025i 0 −2.64426 + 0.0889087i 0 −0.311064 + 2.98383i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 521.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 840.2.cp.b yes 32
3.b odd 2 1 840.2.cp.a ✓ 32
7.d odd 6 1 840.2.cp.a ✓ 32
21.g even 6 1 inner 840.2.cp.b yes 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.cp.a ✓ 32 3.b odd 2 1
840.2.cp.a ✓ 32 7.d odd 6 1
840.2.cp.b yes 32 1.a even 1 1 trivial
840.2.cp.b yes 32 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{32} - 102 T_{11}^{30} + 6555 T_{11}^{28} + 2100 T_{11}^{27} - 255278 T_{11}^{26} + \cdots + 306047942656 \) acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\). Copy content Toggle raw display