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Properties

Label 360.6.b.a
Level $360$
Weight $6$
Character orbit 360.b
Analytic conductor $57.738$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [360,6,Mod(251,360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(360, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("360.251");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 360.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: \(57.7381751327\)
Analytic rank: \(0\)
Dimension: \(40\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 2 q^{2} - 10 q^{4} + 1000 q^{5} - 62 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 2 q^{2} - 10 q^{4} + 1000 q^{5} - 62 q^{8} - 50 q^{10} - 132 q^{14} + 226 q^{16} - 2360 q^{19} - 250 q^{20} - 2864 q^{22} - 11272 q^{23} + 25000 q^{25} + 2488 q^{26} + 4568 q^{28} + 20678 q^{32} + 12492 q^{34} - 21820 q^{38} - 1550 q^{40} - 12732 q^{44} - 21208 q^{46} - 21368 q^{47} - 66776 q^{49} - 1250 q^{50} - 9236 q^{52} + 928 q^{53} + 47076 q^{56} + 34100 q^{58} + 106340 q^{62} + 53342 q^{64} + 24464 q^{67} + 22244 q^{68} - 3300 q^{70} + 268752 q^{71} - 105136 q^{73} - 10384 q^{74} - 223236 q^{76} + 5650 q^{80} + 179476 q^{82} + 169480 q^{86} + 171376 q^{88} - 200112 q^{91} - 146632 q^{92} - 151196 q^{94} - 59000 q^{95} + 147376 q^{97} - 279370 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1 −5.60016 − 0.798855i 0 30.7237 + 8.94744i 25.0000 0 34.4838i −164.910 − 74.6509i 0 −140.004 − 19.9714i
251.2 −5.60016 + 0.798855i 0 30.7237 − 8.94744i 25.0000 0 − 34.4838i −164.910 + 74.6509i 0 −140.004 + 19.9714i
251.3 −5.30246 − 1.97077i 0 24.2322 + 20.8998i 25.0000 0 187.110i −87.3014 − 158.576i 0 −132.561 − 49.2692i
251.4 −5.30246 + 1.97077i 0 24.2322 − 20.8998i 25.0000 0 − 187.110i −87.3014 + 158.576i 0 −132.561 + 49.2692i
251.5 −5.25640 − 2.09052i 0 23.2595 + 21.9772i 25.0000 0 − 143.749i −76.3174 − 164.145i 0 −131.410 − 52.2629i
251.6 −5.25640 + 2.09052i 0 23.2595 − 21.9772i 25.0000 0 143.749i −76.3174 + 164.145i 0 −131.410 + 52.2629i
251.7 −5.09972 − 2.44803i 0 20.0143 + 24.9685i 25.0000 0 − 167.537i −40.9440 − 176.328i 0 −127.493 − 61.2006i
251.8 −5.09972 + 2.44803i 0 20.0143 − 24.9685i 25.0000 0 167.537i −40.9440 + 176.328i 0 −127.493 + 61.2006i
251.9 −4.32495 − 3.64620i 0 5.41044 + 31.5393i 25.0000 0 114.576i 91.5987 − 156.134i 0 −108.124 − 91.1550i
251.10 −4.32495 + 3.64620i 0 5.41044 − 31.5393i 25.0000 0 − 114.576i 91.5987 + 156.134i 0 −108.124 + 91.1550i
251.11 −3.65140 − 4.32056i 0 −5.33454 + 31.5522i 25.0000 0 − 130.292i 155.802 − 92.1616i 0 −91.2850 − 108.014i
251.12 −3.65140 + 4.32056i 0 −5.33454 − 31.5522i 25.0000 0 130.292i 155.802 + 92.1616i 0 −91.2850 + 108.014i
251.13 −3.13493 − 4.70873i 0 −12.3444 + 29.5232i 25.0000 0 − 40.5004i 177.715 − 34.4268i 0 −78.3734 − 117.718i
251.14 −3.13493 + 4.70873i 0 −12.3444 − 29.5232i 25.0000 0 40.5004i 177.715 + 34.4268i 0 −78.3734 + 117.718i
251.15 −2.02309 − 5.28272i 0 −23.8142 + 21.3748i 25.0000 0 201.232i 161.096 + 82.5604i 0 −50.5773 − 132.068i
251.16 −2.02309 + 5.28272i 0 −23.8142 − 21.3748i 25.0000 0 − 201.232i 161.096 − 82.5604i 0 −50.5773 + 132.068i
251.17 −1.83939 − 5.34945i 0 −25.2333 + 19.6795i 25.0000 0 43.7960i 151.688 + 98.7860i 0 −45.9848 − 133.736i
251.18 −1.83939 + 5.34945i 0 −25.2333 − 19.6795i 25.0000 0 − 43.7960i 151.688 − 98.7860i 0 −45.9848 + 133.736i
251.19 0.243254 − 5.65162i 0 −31.8817 − 2.74956i 25.0000 0 − 37.5151i −23.2948 + 179.514i 0 6.08135 − 141.291i
251.20 0.243254 + 5.65162i 0 −31.8817 + 2.74956i 25.0000 0 37.5151i −23.2948 − 179.514i 0 6.08135 + 141.291i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.6.b.a ✓ 40
3.b odd 2 1 360.6.b.b yes 40
8.d odd 2 1 360.6.b.b yes 40
24.f even 2 1 inner 360.6.b.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.6.b.a ✓ 40 1.a even 1 1 trivial
360.6.b.a ✓ 40 24.f even 2 1 inner
360.6.b.b yes 40 3.b odd 2 1
360.6.b.b yes 40 8.d odd 2 1