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Properties

Label 39.6.f.a
Level $39$
Weight $6$
Character orbit 39.f
Analytic conductor $6.255$
Analytic rank $0$
Dimension $44$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [39,6,Mod(5,39)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(39, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("39.5");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 39.f (of order \(4\), degree \(2\), 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: \(6.25496897271\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 4 q^{3} - 52 q^{6} + 148 q^{7} - 4 q^{9} + 2112 q^{13} + 2072 q^{15} - 8720 q^{16} - 2728 q^{18} + 5188 q^{19} - 1996 q^{21} + 2784 q^{22} + 6900 q^{24} + 12836 q^{27} - 15784 q^{28} + 12572 q^{31}+ \cdots + 584912 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −7.77041 − 7.77041i −4.70768 + 14.8606i 88.7585i −30.1421 − 30.1421i 152.054 − 78.8924i 60.1892 + 60.1892i 441.037 − 441.037i −198.675 − 139.918i 468.433i
5.2 −6.77656 − 6.77656i −0.111541 − 15.5881i 59.8434i 57.3759 + 57.3759i −104.877 + 106.389i 139.198 + 139.198i 188.683 − 188.683i −242.975 + 3.47741i − 777.622i
5.3 −6.46085 − 6.46085i 14.7948 − 4.91051i 51.4853i −18.6574 − 18.6574i −127.313 − 63.8612i −126.698 − 126.698i 125.892 − 125.892i 194.774 − 145.300i 241.085i
5.4 −6.01851 − 6.01851i −10.4416 − 11.5747i 40.4450i −55.1183 − 55.1183i −6.81986 + 132.505i −92.2834 − 92.2834i 50.8266 − 50.8266i −24.9476 + 241.716i 663.461i
5.5 −5.33357 − 5.33357i −15.0211 + 4.16738i 24.8940i 58.2277 + 58.2277i 102.343 + 57.8890i −64.3799 − 64.3799i −37.9005 + 37.9005i 208.266 − 125.197i − 621.123i
5.6 −4.70563 − 4.70563i 13.6569 + 7.51597i 12.2859i 2.98354 + 2.98354i −28.8968 − 99.6316i 130.405 + 130.405i −92.7671 + 92.7671i 130.020 + 205.289i − 28.0789i
5.7 −3.76705 − 3.76705i 2.33888 + 15.4120i − 3.61863i 14.1700 + 14.1700i 49.2471 − 66.8685i −76.3369 − 76.3369i −134.177 + 134.177i −232.059 + 72.0936i − 106.758i
5.8 −2.84235 − 2.84235i −14.9406 + 4.44742i − 15.8421i −43.3121 − 43.3121i 55.1074 + 29.8252i 113.155 + 113.155i −135.984 + 135.984i 203.441 − 132.894i 246.216i
5.9 −2.20001 − 2.20001i 7.79307 − 13.5007i − 22.3199i −32.7863 − 32.7863i −46.8465 + 12.5568i 82.1975 + 82.1975i −119.504 + 119.504i −121.536 − 210.423i 144.261i
5.10 −1.19509 − 1.19509i −8.76613 − 12.8901i − 29.1435i 15.3758 + 15.3758i −4.92854 + 25.8812i −44.9529 − 44.9529i −73.0722 + 73.0722i −89.3099 + 225.993i − 36.7511i
5.11 −0.778428 − 0.778428i 14.4049 − 5.95807i − 30.7881i 73.9451 + 73.9451i −15.8511 − 6.57526i −83.4924 − 83.4924i −48.8760 + 48.8760i 172.003 − 171.651i − 115.122i
5.12 0.778428 + 0.778428i 14.4049 + 5.95807i − 30.7881i −73.9451 − 73.9451i 6.57526 + 15.8511i −83.4924 − 83.4924i 48.8760 − 48.8760i 172.003 + 171.651i − 115.122i
5.13 1.19509 + 1.19509i −8.76613 + 12.8901i − 29.1435i −15.3758 − 15.3758i −25.8812 + 4.92854i −44.9529 − 44.9529i 73.0722 − 73.0722i −89.3099 − 225.993i − 36.7511i
5.14 2.20001 + 2.20001i 7.79307 + 13.5007i − 22.3199i 32.7863 + 32.7863i −12.5568 + 46.8465i 82.1975 + 82.1975i 119.504 − 119.504i −121.536 + 210.423i 144.261i
5.15 2.84235 + 2.84235i −14.9406 − 4.44742i − 15.8421i 43.3121 + 43.3121i −29.8252 − 55.1074i 113.155 + 113.155i 135.984 − 135.984i 203.441 + 132.894i 246.216i
5.16 3.76705 + 3.76705i 2.33888 − 15.4120i − 3.61863i −14.1700 − 14.1700i 66.8685 − 49.2471i −76.3369 − 76.3369i 134.177 − 134.177i −232.059 − 72.0936i − 106.758i
5.17 4.70563 + 4.70563i 13.6569 − 7.51597i 12.2859i −2.98354 − 2.98354i 99.6316 + 28.8968i 130.405 + 130.405i 92.7671 − 92.7671i 130.020 − 205.289i − 28.0789i
5.18 5.33357 + 5.33357i −15.0211 − 4.16738i 24.8940i −58.2277 − 58.2277i −57.8890 − 102.343i −64.3799 − 64.3799i 37.9005 − 37.9005i 208.266 + 125.197i − 621.123i
5.19 6.01851 + 6.01851i −10.4416 + 11.5747i 40.4450i 55.1183 + 55.1183i −132.505 + 6.81986i −92.2834 − 92.2834i −50.8266 + 50.8266i −24.9476 − 241.716i 663.461i
5.20 6.46085 + 6.46085i 14.7948 + 4.91051i 51.4853i 18.6574 + 18.6574i 63.8612 + 127.313i −126.698 − 126.698i −125.892 + 125.892i 194.774 + 145.300i 241.085i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.22
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.d odd 4 1 inner
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 39.6.f.a ✓ 44
3.b odd 2 1 inner 39.6.f.a ✓ 44
13.d odd 4 1 inner 39.6.f.a ✓ 44
39.f even 4 1 inner 39.6.f.a ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.6.f.a ✓ 44 1.a even 1 1 trivial
39.6.f.a ✓ 44 3.b odd 2 1 inner
39.6.f.a ✓ 44 13.d odd 4 1 inner
39.6.f.a ✓ 44 39.f even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(39, [\chi])\).