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Properties

Label 213.3.d.a
Level $213$
Weight $3$
Character orbit 213.d
Analytic conductor $5.804$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [213,3,Mod(70,213)]
 
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([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("213.70");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 213 = 3 \cdot 71 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 213.d (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: \(5.80382963087\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{2} + 36 q^{4} + 8 q^{8} + 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{2} + 36 q^{4} + 8 q^{8} + 72 q^{9} - 28 q^{10} - 24 q^{12} + 12 q^{15} - 36 q^{16} + 12 q^{18} - 8 q^{19} + 164 q^{20} + 60 q^{25} - 76 q^{29} + 24 q^{30} + 56 q^{32} + 108 q^{36} + 76 q^{37} - 16 q^{38} - 180 q^{40} + 52 q^{43} - 144 q^{48} - 392 q^{49} + 48 q^{50} - 96 q^{57} - 200 q^{58} - 84 q^{60} - 216 q^{64} + 208 q^{71} + 24 q^{72} - 136 q^{73} - 572 q^{74} + 264 q^{75} - 20 q^{76} - 44 q^{77} - 44 q^{79} + 252 q^{80} + 216 q^{81} - 132 q^{83} - 660 q^{86} + 48 q^{87} + 420 q^{89} - 84 q^{90} + 684 q^{91} + 392 q^{95} + 300 q^{96} - 284 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
70.1 −3.47760 −1.73205 8.09373 3.58564 6.02339 − 10.3328i −14.2364 3.00000 −12.4694
70.2 −3.47760 −1.73205 8.09373 3.58564 6.02339 10.3328i −14.2364 3.00000 −12.4694
70.3 −2.89502 1.73205 4.38115 6.80856 −5.01433 − 7.72681i −1.10345 3.00000 −19.7109
70.4 −2.89502 1.73205 4.38115 6.80856 −5.01433 7.72681i −1.10345 3.00000 −19.7109
70.5 −2.19883 1.73205 0.834864 −6.08826 −3.80849 − 9.25572i 6.95960 3.00000 13.3871
70.6 −2.19883 1.73205 0.834864 −6.08826 −3.80849 9.25572i 6.95960 3.00000 13.3871
70.7 −1.99074 −1.73205 −0.0369651 0.788225 3.44806 − 5.43426i 8.03654 3.00000 −1.56915
70.8 −1.99074 −1.73205 −0.0369651 0.788225 3.44806 5.43426i 8.03654 3.00000 −1.56915
70.9 −0.699472 −1.73205 −3.51074 −6.49867 1.21152 − 4.64737i 5.25355 3.00000 4.54564
70.10 −0.699472 −1.73205 −3.51074 −6.49867 1.21152 4.64737i 5.25355 3.00000 4.54564
70.11 −0.404027 1.73205 −3.83676 0.805979 −0.699795 − 4.88474i 3.16626 3.00000 −0.325637
70.12 −0.404027 1.73205 −3.83676 0.805979 −0.699795 4.88474i 3.16626 3.00000 −0.325637
70.13 1.13729 −1.73205 −2.70658 3.69868 −1.96984 − 3.52282i −7.62730 3.00000 4.20645
70.14 1.13729 −1.73205 −2.70658 3.69868 −1.96984 3.52282i −7.62730 3.00000 4.20645
70.15 1.35919 1.73205 −2.15260 −7.36312 2.35419 − 7.99392i −8.36256 3.00000 −10.0079
70.16 1.35919 1.73205 −2.15260 −7.36312 2.35419 7.99392i −8.36256 3.00000 −10.0079
70.17 1.82565 1.73205 −0.667016 8.03724 3.16211 − 13.1891i −8.52032 3.00000 14.6732
70.18 1.82565 1.73205 −0.667016 8.03724 3.16211 13.1891i −8.52032 3.00000 14.6732
70.19 2.54568 −1.73205 2.48049 −6.75391 −4.40925 − 9.31625i −3.86819 3.00000 −17.1933
70.20 2.54568 −1.73205 2.48049 −6.75391 −4.40925 9.31625i −3.86819 3.00000 −17.1933
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 70.24
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 213.3.d.a ✓ 24
3.b odd 2 1 639.3.d.c 24
71.b odd 2 1 inner 213.3.d.a ✓ 24
213.b even 2 1 639.3.d.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
213.3.d.a ✓ 24 1.a even 1 1 trivial
213.3.d.a ✓ 24 71.b odd 2 1 inner
639.3.d.c 24 3.b odd 2 1
639.3.d.c 24 213.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(213, [\chi])\).