↑

Properties

Label 8037.2.a.s
Level $8037$
Weight $2$
Character orbit 8037.a
Self dual yes
Analytic conductor $64.176$
Analytic rank $1$
Dimension $24$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8037,2,Mod(1,8037)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8037, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8037.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8037.a (trivial)

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: yes
Analytic conductor: \(64.1757681045\)
Analytic rank: \(1\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 2679)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{2} + 32 q^{4} - 10 q^{5} + 6 q^{7} - 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{2} + 32 q^{4} - 10 q^{5} + 6 q^{7} - 15 q^{8} + 5 q^{10} - 15 q^{11} - 3 q^{13} - 27 q^{14} + 48 q^{16} - 22 q^{17} - 24 q^{19} - 29 q^{20} - 9 q^{22} - 15 q^{23} + 58 q^{25} - q^{26} + 18 q^{28} - 36 q^{29} - 16 q^{31} - 26 q^{32} + 10 q^{34} - 37 q^{35} + 18 q^{37} + 6 q^{38} + 7 q^{40} - 24 q^{41} + 2 q^{43} - 57 q^{44} - 23 q^{46} + 24 q^{47} + 64 q^{49} - 4 q^{50} - 40 q^{52} - 42 q^{53} + 8 q^{55} - 38 q^{56} - 5 q^{58} - 39 q^{59} + 6 q^{61} + 14 q^{62} + 53 q^{64} - 47 q^{65} - 26 q^{67} - 80 q^{68} - 46 q^{70} - 6 q^{71} + 3 q^{73} - 62 q^{74} - 32 q^{76} - 13 q^{77} + 10 q^{79} - 17 q^{80} + 15 q^{82} - 44 q^{83} + 21 q^{85} + 22 q^{86} - 88 q^{88} - 32 q^{89} - 17 q^{91} + 37 q^{92} - 6 q^{94} + 10 q^{95} + 28 q^{97} - 44 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.74619 0 5.54156 1.39198 0 3.03199 −9.72581 0 −3.82263
1.2 −2.73887 0 5.50141 1.01615 0 0.640714 −9.58990 0 −2.78309
1.3 −2.62778 0 4.90524 −2.66749 0 −4.09726 −7.63435 0 7.00959
1.4 −2.30982 0 3.33525 −0.261175 0 1.60168 −3.08417 0 0.603266
1.5 −2.26800 0 3.14381 −4.22801 0 5.05999 −2.59416 0 9.58911
1.6 −2.20954 0 2.88208 −2.70483 0 −0.952275 −1.94899 0 5.97643
1.7 −1.92175 0 1.69311 −0.940482 0 3.86468 0.589769 0 1.80737
1.8 −1.72798 0 0.985905 4.09118 0 3.75957 1.75233 0 −7.06946
1.9 −1.39021 0 −0.0673298 −3.64442 0 −1.02491 2.87401 0 5.06649
1.10 −0.992877 0 −1.01419 4.04955 0 −4.64503 2.99273 0 −4.02071
1.11 −0.543143 0 −1.70500 0.446531 0 −4.73017 2.01234 0 −0.242530
1.12 −0.456458 0 −1.79165 1.12773 0 −0.162029 1.73073 0 −0.514761
1.13 −0.165573 0 −1.97259 −4.22825 0 2.94294 0.657754 0 0.700086
1.14 −0.0666738 0 −1.99555 −2.45418 0 3.25184 0.266399 0 0.163629
1.15 0.0589819 0 −1.99652 2.29389 0 2.75379 −0.235722 0 0.135298
1.16 0.743704 0 −1.44690 −0.245686 0 1.64092 −2.56348 0 −0.182717
1.17 1.10355 0 −0.782179 −3.33604 0 −4.44205 −3.07027 0 −3.68148
1.18 1.17388 0 −0.622012 3.35404 0 −1.78420 −3.07792 0 3.93723
1.19 1.62396 0 0.637238 2.60823 0 −0.198208 −2.21307 0 4.23566
1.20 1.96663 0 1.86764 −1.40576 0 4.82031 −0.260313 0 −2.76460
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.24
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(19\) \(1\)
\(47\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8037.2.a.s 24
3.b odd 2 1 2679.2.a.p ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2679.2.a.p ✓ 24 3.b odd 2 1
8037.2.a.s 24 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8037))\):

\( T_{2}^{24} + 6 T_{2}^{23} - 22 T_{2}^{22} - 191 T_{2}^{21} + 104 T_{2}^{20} + 2544 T_{2}^{19} + \cdots + 16 \) Copy content Toggle raw display
\( T_{5}^{24} + 10 T_{5}^{23} - 39 T_{5}^{22} - 670 T_{5}^{21} - 107 T_{5}^{20} + 18359 T_{5}^{19} + \cdots + 48574 \) Copy content Toggle raw display