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Properties

Label 882.2.bl.a
Level $882$
Weight $2$
Character orbit 882.bl
Analytic conductor $7.043$
Analytic rank $0$
Dimension $240$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [882,2,Mod(17,882)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(882, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([21, 25]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.bl (of order \(42\), degree \(12\), 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: \(7.04280545828\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(20\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 20 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q - 20 q^{4} - 4 q^{7} - 12 q^{10} + 20 q^{16} + 24 q^{19} - 20 q^{22} + 24 q^{25} + 4 q^{28} + 12 q^{31} - 32 q^{37} + 44 q^{40} - 48 q^{43} + 204 q^{49} + 140 q^{55} + 136 q^{58} + 88 q^{61} + 40 q^{64} - 32 q^{67} - 16 q^{70} - 24 q^{73} - 28 q^{79} - 48 q^{82} - 112 q^{85} + 4 q^{88} + 80 q^{91} + 80 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −0.997204 + 0.0747301i 0 0.988831 − 0.149042i −2.75566 + 2.55688i 0 2.25212 − 1.38851i −0.974928 + 0.222521i 0 2.55688 − 2.75566i
17.2 −0.997204 + 0.0747301i 0 0.988831 − 0.149042i −2.52832 + 2.34594i 0 −2.16138 − 1.52592i −0.974928 + 0.222521i 0 2.34594 − 2.52832i
17.3 −0.997204 + 0.0747301i 0 0.988831 − 0.149042i −2.29299 + 2.12758i 0 −2.55204 + 0.697919i −0.974928 + 0.222521i 0 2.12758 − 2.29299i
17.4 −0.997204 + 0.0747301i 0 0.988831 − 0.149042i −0.904119 + 0.838900i 0 2.63744 + 0.209587i −0.974928 + 0.222521i 0 0.838900 − 0.904119i
17.5 −0.997204 + 0.0747301i 0 0.988831 − 0.149042i −0.485406 + 0.450391i 0 2.48033 + 0.920837i −0.974928 + 0.222521i 0 0.450391 − 0.485406i
17.6 −0.997204 + 0.0747301i 0 0.988831 − 0.149042i 0.149830 − 0.139022i 0 −0.801783 + 2.52134i −0.974928 + 0.222521i 0 −0.139022 + 0.149830i
17.7 −0.997204 + 0.0747301i 0 0.988831 − 0.149042i 0.847616 − 0.786472i 0 −1.96819 + 1.76811i −0.974928 + 0.222521i 0 −0.786472 + 0.847616i
17.8 −0.997204 + 0.0747301i 0 0.988831 − 0.149042i 1.36445 − 1.26602i 0 1.78586 − 1.95210i −0.974928 + 0.222521i 0 −1.26602 + 1.36445i
17.9 −0.997204 + 0.0747301i 0 0.988831 − 0.149042i 2.39946 − 2.22638i 0 1.13262 + 2.39106i −0.974928 + 0.222521i 0 −2.22638 + 2.39946i
17.10 −0.997204 + 0.0747301i 0 0.988831 − 0.149042i 2.55337 − 2.36918i 0 −1.33888 − 2.28197i −0.974928 + 0.222521i 0 −2.36918 + 2.55337i
17.11 0.997204 − 0.0747301i 0 0.988831 − 0.149042i −2.55337 + 2.36918i 0 −1.33888 − 2.28197i 0.974928 − 0.222521i 0 −2.36918 + 2.55337i
17.12 0.997204 − 0.0747301i 0 0.988831 − 0.149042i −2.39946 + 2.22638i 0 1.13262 + 2.39106i 0.974928 − 0.222521i 0 −2.22638 + 2.39946i
17.13 0.997204 − 0.0747301i 0 0.988831 − 0.149042i −1.36445 + 1.26602i 0 1.78586 − 1.95210i 0.974928 − 0.222521i 0 −1.26602 + 1.36445i
17.14 0.997204 − 0.0747301i 0 0.988831 − 0.149042i −0.847616 + 0.786472i 0 −1.96819 + 1.76811i 0.974928 − 0.222521i 0 −0.786472 + 0.847616i
17.15 0.997204 − 0.0747301i 0 0.988831 − 0.149042i −0.149830 + 0.139022i 0 −0.801783 + 2.52134i 0.974928 − 0.222521i 0 −0.139022 + 0.149830i
17.16 0.997204 − 0.0747301i 0 0.988831 − 0.149042i 0.485406 − 0.450391i 0 2.48033 + 0.920837i 0.974928 − 0.222521i 0 0.450391 − 0.485406i
17.17 0.997204 − 0.0747301i 0 0.988831 − 0.149042i 0.904119 − 0.838900i 0 2.63744 + 0.209587i 0.974928 − 0.222521i 0 0.838900 − 0.904119i
17.18 0.997204 − 0.0747301i 0 0.988831 − 0.149042i 2.29299 − 2.12758i 0 −2.55204 + 0.697919i 0.974928 − 0.222521i 0 2.12758 − 2.29299i
17.19 0.997204 − 0.0747301i 0 0.988831 − 0.149042i 2.52832 − 2.34594i 0 −2.16138 − 1.52592i 0.974928 − 0.222521i 0 2.34594 − 2.52832i
17.20 0.997204 − 0.0747301i 0 0.988831 − 0.149042i 2.75566 − 2.55688i 0 2.25212 − 1.38851i 0.974928 − 0.222521i 0 2.55688 − 2.75566i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
49.h odd 42 1 inner
147.o even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.2.bl.a ✓ 240
3.b odd 2 1 inner 882.2.bl.a ✓ 240
49.h odd 42 1 inner 882.2.bl.a ✓ 240
147.o even 42 1 inner 882.2.bl.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.2.bl.a ✓ 240 1.a even 1 1 trivial
882.2.bl.a ✓ 240 3.b odd 2 1 inner
882.2.bl.a ✓ 240 49.h odd 42 1 inner
882.2.bl.a ✓ 240 147.o even 42 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(882, [\chi])\).