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Properties

Label 210.4.d.b
Level $210$
Weight $4$
Character orbit 210.d
Analytic conductor $12.390$
Analytic rank $0$
Dimension $24$
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] = [210,4,Mod(209,210)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(210, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("210.209");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 210.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: \(12.3904011012\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 48 q^{2} + 96 q^{4} + 192 q^{8} + 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 48 q^{2} + 96 q^{4} + 192 q^{8} + 30 q^{9} + 138 q^{15} + 384 q^{16} + 60 q^{18} - 50 q^{21} + 144 q^{23} - 24 q^{25} + 276 q^{30} + 768 q^{32} + 168 q^{35} + 120 q^{36} + 246 q^{39} - 100 q^{42} + 288 q^{46} - 192 q^{49} - 48 q^{50} - 850 q^{51} - 456 q^{53} - 184 q^{57} + 552 q^{60} + 32 q^{63} + 1536 q^{64} + 432 q^{65} + 336 q^{70} + 240 q^{72} - 2328 q^{77} + 492 q^{78} - 684 q^{79} - 1454 q^{81} - 200 q^{84} - 12 q^{85} - 2148 q^{91} + 576 q^{92} - 2144 q^{93} - 1512 q^{95} - 384 q^{98} - 2542 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1 2.00000 −5.12775 − 0.840332i 4.00000 7.01878 + 8.70268i −10.2555 − 1.68066i −15.0645 − 10.7732i 8.00000 25.5877 + 8.61803i 14.0376 + 17.4054i
209.2 2.00000 −5.12775 + 0.840332i 4.00000 7.01878 − 8.70268i −10.2555 + 1.68066i −15.0645 + 10.7732i 8.00000 25.5877 − 8.61803i 14.0376 − 17.4054i
209.3 2.00000 −4.67505 − 2.26802i 4.00000 −7.81790 + 7.99253i −9.35010 − 4.53604i 15.4771 − 10.1715i 8.00000 16.7122 + 21.2062i −15.6358 + 15.9851i
209.4 2.00000 −4.67505 + 2.26802i 4.00000 −7.81790 − 7.99253i −9.35010 + 4.53604i 15.4771 + 10.1715i 8.00000 16.7122 − 21.2062i −15.6358 − 15.9851i
209.5 2.00000 −4.37834 − 2.79824i 4.00000 −10.8308 − 2.77380i −8.75667 − 5.59649i −3.07039 + 18.2640i 8.00000 11.3397 + 24.5033i −21.6616 − 5.54761i
209.6 2.00000 −4.37834 + 2.79824i 4.00000 −10.8308 + 2.77380i −8.75667 + 5.59649i −3.07039 − 18.2640i 8.00000 11.3397 − 24.5033i −21.6616 + 5.54761i
209.7 2.00000 −3.37351 − 3.95214i 4.00000 11.1048 − 1.29762i −6.74702 − 7.90428i 18.4981 − 0.906354i 8.00000 −4.23883 + 26.6652i 22.2096 − 2.59524i
209.8 2.00000 −3.37351 + 3.95214i 4.00000 11.1048 + 1.29762i −6.74702 + 7.90428i 18.4981 + 0.906354i 8.00000 −4.23883 − 26.6652i 22.2096 + 2.59524i
209.9 2.00000 −2.33332 − 4.64280i 4.00000 −2.31759 − 10.9375i −4.66665 − 9.28560i −13.6656 − 12.5000i 8.00000 −16.1112 + 21.6663i −4.63517 − 21.8750i
209.10 2.00000 −2.33332 + 4.64280i 4.00000 −2.31759 + 10.9375i −4.66665 + 9.28560i −13.6656 + 12.5000i 8.00000 −16.1112 − 21.6663i −4.63517 + 21.8750i
209.11 2.00000 −0.777981 − 5.13758i 4.00000 −3.95270 + 10.4583i −1.55596 − 10.2752i 0.407437 + 18.5158i 8.00000 −25.7895 + 7.99388i −7.90539 + 20.9166i
209.12 2.00000 −0.777981 + 5.13758i 4.00000 −3.95270 − 10.4583i −1.55596 + 10.2752i 0.407437 − 18.5158i 8.00000 −25.7895 − 7.99388i −7.90539 − 20.9166i
209.13 2.00000 0.777981 − 5.13758i 4.00000 3.95270 + 10.4583i 1.55596 − 10.2752i −0.407437 − 18.5158i 8.00000 −25.7895 − 7.99388i 7.90539 + 20.9166i
209.14 2.00000 0.777981 + 5.13758i 4.00000 3.95270 − 10.4583i 1.55596 + 10.2752i −0.407437 + 18.5158i 8.00000 −25.7895 + 7.99388i 7.90539 − 20.9166i
209.15 2.00000 2.33332 − 4.64280i 4.00000 2.31759 − 10.9375i 4.66665 − 9.28560i 13.6656 + 12.5000i 8.00000 −16.1112 − 21.6663i 4.63517 − 21.8750i
209.16 2.00000 2.33332 + 4.64280i 4.00000 2.31759 + 10.9375i 4.66665 + 9.28560i 13.6656 − 12.5000i 8.00000 −16.1112 + 21.6663i 4.63517 + 21.8750i
209.17 2.00000 3.37351 − 3.95214i 4.00000 −11.1048 − 1.29762i 6.74702 − 7.90428i −18.4981 + 0.906354i 8.00000 −4.23883 − 26.6652i −22.2096 − 2.59524i
209.18 2.00000 3.37351 + 3.95214i 4.00000 −11.1048 + 1.29762i 6.74702 + 7.90428i −18.4981 − 0.906354i 8.00000 −4.23883 + 26.6652i −22.2096 + 2.59524i
209.19 2.00000 4.37834 − 2.79824i 4.00000 10.8308 − 2.77380i 8.75667 − 5.59649i 3.07039 − 18.2640i 8.00000 11.3397 − 24.5033i 21.6616 − 5.54761i
209.20 2.00000 4.37834 + 2.79824i 4.00000 10.8308 + 2.77380i 8.75667 + 5.59649i 3.07039 + 18.2640i 8.00000 11.3397 + 24.5033i 21.6616 + 5.54761i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
15.d odd 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.4.d.b yes 24
3.b odd 2 1 210.4.d.a ✓ 24
5.b even 2 1 210.4.d.a ✓ 24
7.b odd 2 1 inner 210.4.d.b yes 24
15.d odd 2 1 inner 210.4.d.b yes 24
21.c even 2 1 210.4.d.a ✓ 24
35.c odd 2 1 210.4.d.a ✓ 24
105.g even 2 1 inner 210.4.d.b yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.4.d.a ✓ 24 3.b odd 2 1
210.4.d.a ✓ 24 5.b even 2 1
210.4.d.a ✓ 24 21.c even 2 1
210.4.d.a ✓ 24 35.c odd 2 1
210.4.d.b yes 24 1.a even 1 1 trivial
210.4.d.b yes 24 7.b odd 2 1 inner
210.4.d.b yes 24 15.d odd 2 1 inner
210.4.d.b yes 24 105.g even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23}^{6} - 36T_{23}^{5} - 29508T_{23}^{4} + 1832864T_{23}^{3} + 123342048T_{23}^{2} - 9321215616T_{23} + 105074233344 \) acting on \(S_{4}^{\mathrm{new}}(210, [\chi])\). Copy content Toggle raw display