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Properties

Label 712.2.f.a
Level $712$
Weight $2$
Character orbit 712.f
Analytic conductor $5.685$
Analytic rank $0$
Dimension $88$
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] = [712,2,Mod(533,712)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(712, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("712.533");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 712 = 2^{3} \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 712.f (of order \(2\), degree \(1\), 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: \(5.68534862392\)
Analytic rank: \(0\)
Dimension: \(88\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 88 q - 2 q^{2} - 2 q^{4} - 2 q^{8} + 80 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q - 2 q^{2} - 2 q^{4} - 2 q^{8} + 80 q^{9} - 6 q^{10} - 2 q^{16} - 8 q^{17} - 16 q^{18} - 26 q^{20} + 12 q^{22} - 88 q^{25} + 18 q^{32} - 18 q^{34} - 16 q^{36} + 22 q^{40} + 52 q^{42} + 36 q^{44} + 16 q^{47} - 80 q^{49} + 32 q^{50} - 32 q^{55} - 24 q^{57} - 2 q^{64} - 6 q^{68} - 56 q^{72} + 16 q^{73} - 48 q^{78} + 16 q^{79} - 26 q^{80} + 64 q^{81} + 112 q^{84} - 48 q^{87} + 64 q^{88} + 16 q^{89} + 28 q^{90} - 68 q^{94} - 8 q^{97} - 34 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
533.1 −1.38587 − 0.281698i −2.38863 1.84129 + 0.780796i − 4.14694i 3.31033 + 0.672872i 2.30888i −2.33185 − 1.60077i 2.70553 −1.16819 + 5.74713i
533.2 −1.38587 − 0.281698i 2.38863 1.84129 + 0.780796i − 4.14694i −3.31033 − 0.672872i − 2.30888i −2.33185 − 1.60077i 2.70553 −1.16819 + 5.74713i
533.3 −1.38587 + 0.281698i −2.38863 1.84129 − 0.780796i 4.14694i 3.31033 − 0.672872i − 2.30888i −2.33185 + 1.60077i 2.70553 −1.16819 − 5.74713i
533.4 −1.38587 + 0.281698i 2.38863 1.84129 − 0.780796i 4.14694i −3.31033 + 0.672872i 2.30888i −2.33185 + 1.60077i 2.70553 −1.16819 − 5.74713i
533.5 −1.38297 − 0.295610i −0.484188 1.82523 + 0.817641i 2.52387i 0.669619 + 0.143131i 2.70992i −2.28254 − 1.67033i −2.76556 0.746082 − 3.49045i
533.6 −1.38297 − 0.295610i 0.484188 1.82523 + 0.817641i 2.52387i −0.669619 − 0.143131i − 2.70992i −2.28254 − 1.67033i −2.76556 0.746082 − 3.49045i
533.7 −1.38297 + 0.295610i −0.484188 1.82523 − 0.817641i − 2.52387i 0.669619 − 0.143131i − 2.70992i −2.28254 + 1.67033i −2.76556 0.746082 + 3.49045i
533.8 −1.38297 + 0.295610i 0.484188 1.82523 − 0.817641i − 2.52387i −0.669619 + 0.143131i 2.70992i −2.28254 + 1.67033i −2.76556 0.746082 + 3.49045i
533.9 −1.30958 − 0.533862i −1.88807 1.42998 + 1.39827i 0.614635i 2.47257 + 1.00797i 2.73276i −1.12619 − 2.59455i 0.564802 0.328130 − 0.804912i
533.10 −1.30958 − 0.533862i 1.88807 1.42998 + 1.39827i 0.614635i −2.47257 − 1.00797i − 2.73276i −1.12619 − 2.59455i 0.564802 0.328130 − 0.804912i
533.11 −1.30958 + 0.533862i −1.88807 1.42998 − 1.39827i − 0.614635i 2.47257 − 1.00797i − 2.73276i −1.12619 + 2.59455i 0.564802 0.328130 + 0.804912i
533.12 −1.30958 + 0.533862i 1.88807 1.42998 − 1.39827i − 0.614635i −2.47257 + 1.00797i 2.73276i −1.12619 + 2.59455i 0.564802 0.328130 + 0.804912i
533.13 −1.29291 − 0.573058i −2.39520 1.34321 + 1.48182i − 0.685546i 3.09677 + 1.37259i − 4.54203i −0.887473 − 2.68559i 2.73700 −0.392858 + 0.886346i
533.14 −1.29291 − 0.573058i 2.39520 1.34321 + 1.48182i − 0.685546i −3.09677 − 1.37259i 4.54203i −0.887473 − 2.68559i 2.73700 −0.392858 + 0.886346i
533.15 −1.29291 + 0.573058i −2.39520 1.34321 − 1.48182i 0.685546i 3.09677 − 1.37259i 4.54203i −0.887473 + 2.68559i 2.73700 −0.392858 − 0.886346i
533.16 −1.29291 + 0.573058i 2.39520 1.34321 − 1.48182i 0.685546i −3.09677 + 1.37259i − 4.54203i −0.887473 + 2.68559i 2.73700 −0.392858 − 0.886346i
533.17 −1.10931 − 0.877173i −3.33714 0.461137 + 1.94611i 3.17960i 3.70192 + 2.92725i 1.86632i 1.19553 − 2.56334i 8.13651 2.78906 − 3.52716i
533.18 −1.10931 − 0.877173i 3.33714 0.461137 + 1.94611i 3.17960i −3.70192 − 2.92725i − 1.86632i 1.19553 − 2.56334i 8.13651 2.78906 − 3.52716i
533.19 −1.10931 + 0.877173i −3.33714 0.461137 − 1.94611i − 3.17960i 3.70192 − 2.92725i − 1.86632i 1.19553 + 2.56334i 8.13651 2.78906 + 3.52716i
533.20 −1.10931 + 0.877173i 3.33714 0.461137 − 1.94611i − 3.17960i −3.70192 + 2.92725i 1.86632i 1.19553 + 2.56334i 8.13651 2.78906 + 3.52716i
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 533.88
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
89.b even 2 1 inner
712.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 712.2.f.a ✓ 88
4.b odd 2 1 2848.2.f.a 88
8.b even 2 1 inner 712.2.f.a ✓ 88
8.d odd 2 1 2848.2.f.a 88
89.b even 2 1 inner 712.2.f.a ✓ 88
356.d odd 2 1 2848.2.f.a 88
712.c odd 2 1 2848.2.f.a 88
712.f even 2 1 inner 712.2.f.a ✓ 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
712.2.f.a ✓ 88 1.a even 1 1 trivial
712.2.f.a ✓ 88 8.b even 2 1 inner
712.2.f.a ✓ 88 89.b even 2 1 inner
712.2.f.a ✓ 88 712.f even 2 1 inner
2848.2.f.a 88 4.b odd 2 1
2848.2.f.a 88 8.d odd 2 1
2848.2.f.a 88 356.d odd 2 1
2848.2.f.a 88 712.c odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(712, [\chi])\).