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Properties

Label 213.4.b.b
Level $213$
Weight $4$
Character orbit 213.b
Analytic conductor $12.567$
Analytic rank $0$
Dimension $56$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [213,4,Mod(212,213)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(213, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("213.212");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 213 = 3 \cdot 71 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 213.b (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: \(12.5674068312\)
Analytic rank: \(0\)
Dimension: \(56\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 2 q^{3} - 164 q^{4} - 16 q^{6} - 66 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 2 q^{3} - 164 q^{4} - 16 q^{6} - 66 q^{9} - 44 q^{10} - 292 q^{12} + 124 q^{15} - 12 q^{16} + 292 q^{18} + 20 q^{19} - 1708 q^{24} + 460 q^{25} + 242 q^{27} + 2456 q^{30} + 400 q^{36} - 148 q^{37} + 716 q^{40} - 1084 q^{43} + 236 q^{45} - 600 q^{48} - 7912 q^{49} + 564 q^{54} + 2188 q^{57} + 3520 q^{58} - 1000 q^{60} + 6752 q^{64} - 240 q^{72} + 716 q^{73} + 5684 q^{75} - 3132 q^{76} - 1852 q^{79} + 3278 q^{81} - 10630 q^{87} + 1220 q^{90} + 2472 q^{91} + 7204 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
212.1 − 5.10881i −2.19239 + 4.71099i −18.1000 − 5.28572i 24.0676 + 11.2005i − 30.7054i 51.5988i −17.3868 − 20.6567i −27.0038
212.2 − 5.10881i −2.19239 + 4.71099i −18.1000 − 5.28572i 24.0676 + 11.2005i 30.7054i 51.5988i −17.3868 − 20.6567i −27.0038
212.3 − 4.95987i 3.80703 + 3.53645i −16.6003 − 0.347034i 17.5403 − 18.8824i − 11.5438i 42.6564i 1.98701 + 26.9268i −1.72125
212.4 − 4.95987i 3.80703 + 3.53645i −16.6003 − 0.347034i 17.5403 − 18.8824i 11.5438i 42.6564i 1.98701 + 26.9268i −1.72125
212.5 − 4.55662i −3.00775 − 4.23715i −12.7628 − 10.7547i −19.3071 + 13.7052i − 26.6452i 21.7020i −8.90689 + 25.4886i −49.0049
212.6 − 4.55662i −3.00775 − 4.23715i −12.7628 − 10.7547i −19.3071 + 13.7052i 26.6452i 21.7020i −8.90689 + 25.4886i −49.0049
212.7 − 4.52632i 5.13706 − 0.781391i −12.4876 15.3155i −3.53682 − 23.2520i − 24.2862i 20.3121i 25.7789 − 8.02811i 69.3228
212.8 − 4.52632i 5.13706 − 0.781391i −12.4876 15.3155i −3.53682 − 23.2520i 24.2862i 20.3121i 25.7789 − 8.02811i 69.3228
212.9 − 3.80560i 3.08429 − 4.18177i −6.48261 − 3.26232i −15.9141 − 11.7376i − 14.5354i − 5.77459i −7.97433 − 25.7955i −12.4151
212.10 − 3.80560i 3.08429 − 4.18177i −6.48261 − 3.26232i −15.9141 − 11.7376i 14.5354i − 5.77459i −7.97433 − 25.7955i −12.4151
212.11 − 3.52239i −4.94156 + 1.60654i −4.40724 − 13.0782i 5.65886 + 17.4061i − 16.1366i − 12.6551i 21.8381 − 15.8776i −46.0667
212.12 − 3.52239i −4.94156 + 1.60654i −4.40724 − 13.0782i 5.65886 + 17.4061i 16.1366i − 12.6551i 21.8381 − 15.8776i −46.0667
212.13 − 3.27753i −1.07166 − 5.08444i −2.74220 16.9661i −16.6644 + 3.51240i − 26.0954i − 17.2326i −24.7031 + 10.8976i 55.6069
212.14 − 3.27753i −1.07166 − 5.08444i −2.74220 16.9661i −16.6644 + 3.51240i 26.0954i − 17.2326i −24.7031 + 10.8976i 55.6069
212.15 − 3.12894i 1.61109 + 4.94008i −1.79026 8.04412i 15.4572 − 5.04101i − 22.3665i − 19.4299i −21.8088 + 15.9179i 25.1696
212.16 − 3.12894i 1.61109 + 4.94008i −1.79026 8.04412i 15.4572 − 5.04101i 22.3665i − 19.4299i −21.8088 + 15.9179i 25.1696
212.17 − 2.41054i 5.16962 + 0.524392i 2.18931 − 6.99655i 1.26407 − 12.4616i − 24.3771i − 24.5617i 26.4500 + 5.42181i −16.8654
212.18 − 2.41054i 5.16962 + 0.524392i 2.18931 − 6.99655i 1.26407 − 12.4616i 24.3771i − 24.5617i 26.4500 + 5.42181i −16.8654
212.19 − 2.13157i −4.65571 − 2.30746i 3.45643 5.01171i −4.91851 + 9.92394i − 5.72485i − 24.4201i 16.3512 + 21.4858i 10.6828
212.20 − 2.13157i −4.65571 − 2.30746i 3.45643 5.01171i −4.91851 + 9.92394i 5.72485i − 24.4201i 16.3512 + 21.4858i 10.6828
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 212.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
71.b odd 2 1 inner
213.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 213.4.b.b ✓ 56
3.b odd 2 1 inner 213.4.b.b ✓ 56
71.b odd 2 1 inner 213.4.b.b ✓ 56
213.b even 2 1 inner 213.4.b.b ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
213.4.b.b ✓ 56 1.a even 1 1 trivial
213.4.b.b ✓ 56 3.b odd 2 1 inner
213.4.b.b ✓ 56 71.b odd 2 1 inner
213.4.b.b ✓ 56 213.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} + 153 T_{2}^{26} + 10318 T_{2}^{24} + 404011 T_{2}^{22} + 10184478 T_{2}^{20} + \cdots + 118816768 \) acting on \(S_{4}^{\mathrm{new}}(213, [\chi])\). Copy content Toggle raw display