Properties

Label 6038.2.a.a
Level $6038$
Weight $2$
Character orbit 6038.a
Self dual yes
Analytic conductor $48.214$
Analytic rank $2$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6038,2,Mod(1,6038)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6038, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6038.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6038 = 2 \cdot 3019 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6038.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: \(48.2136727404\)
Analytic rank: \(2\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + ( - \beta - 1) q^{3} + q^{4} + (\beta - 3) q^{5} + ( - \beta - 1) q^{6} - 3 q^{7} + q^{8} + (3 \beta - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + ( - \beta - 1) q^{3} + q^{4} + (\beta - 3) q^{5} + ( - \beta - 1) q^{6} - 3 q^{7} + q^{8} + (3 \beta - 1) q^{9} + (\beta - 3) q^{10} + (\beta - 4) q^{11} + ( - \beta - 1) q^{12} - 3 q^{13} - 3 q^{14} + (\beta + 2) q^{15} + q^{16} + ( - 4 \beta - 1) q^{17} + (3 \beta - 1) q^{18} - 3 q^{19} + (\beta - 3) q^{20} + (3 \beta + 3) q^{21} + (\beta - 4) q^{22} + (4 \beta - 5) q^{23} + ( - \beta - 1) q^{24} + ( - 5 \beta + 5) q^{25} - 3 q^{26} + ( - 2 \beta + 1) q^{27} - 3 q^{28} + ( - 2 \beta - 6) q^{29} + (\beta + 2) q^{30} + ( - 5 \beta + 2) q^{31} + q^{32} + (2 \beta + 3) q^{33} + ( - 4 \beta - 1) q^{34} + ( - 3 \beta + 9) q^{35} + (3 \beta - 1) q^{36} + ( - 3 \beta + 3) q^{37} - 3 q^{38} + (3 \beta + 3) q^{39} + (\beta - 3) q^{40} + (10 \beta - 5) q^{41} + (3 \beta + 3) q^{42} + ( - 2 \beta - 6) q^{43} + (\beta - 4) q^{44} + ( - 7 \beta + 6) q^{45} + (4 \beta - 5) q^{46} + (9 \beta - 3) q^{47} + ( - \beta - 1) q^{48} + 2 q^{49} + ( - 5 \beta + 5) q^{50} + (9 \beta + 5) q^{51} - 3 q^{52} - 9 q^{53} + ( - 2 \beta + 1) q^{54} + ( - 6 \beta + 13) q^{55} - 3 q^{56} + (3 \beta + 3) q^{57} + ( - 2 \beta - 6) q^{58} + (2 \beta - 13) q^{59} + (\beta + 2) q^{60} + (\beta - 4) q^{61} + ( - 5 \beta + 2) q^{62} + ( - 9 \beta + 3) q^{63} + q^{64} + ( - 3 \beta + 9) q^{65} + (2 \beta + 3) q^{66} + (3 \beta - 3) q^{67} + ( - 4 \beta - 1) q^{68} + ( - 3 \beta + 1) q^{69} + ( - 3 \beta + 9) q^{70} + ( - 8 \beta + 1) q^{71} + (3 \beta - 1) q^{72} + (6 \beta - 9) q^{73} + ( - 3 \beta + 3) q^{74} + 5 \beta q^{75} - 3 q^{76} + ( - 3 \beta + 12) q^{77} + (3 \beta + 3) q^{78} + ( - 3 \beta + 3) q^{79} + (\beta - 3) q^{80} + ( - 6 \beta + 4) q^{81} + (10 \beta - 5) q^{82} + ( - 8 \beta + 11) q^{83} + (3 \beta + 3) q^{84} + (7 \beta - 1) q^{85} + ( - 2 \beta - 6) q^{86} + (10 \beta + 8) q^{87} + (\beta - 4) q^{88} + ( - 11 \beta + 1) q^{89} + ( - 7 \beta + 6) q^{90} + 9 q^{91} + (4 \beta - 5) q^{92} + (8 \beta + 3) q^{93} + (9 \beta - 3) q^{94} + ( - 3 \beta + 9) q^{95} + ( - \beta - 1) q^{96} + ( - 2 \beta + 17) q^{97} + 2 q^{98} + ( - 10 \beta + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 3 q^{3} + 2 q^{4} - 5 q^{5} - 3 q^{6} - 6 q^{7} + 2 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 3 q^{3} + 2 q^{4} - 5 q^{5} - 3 q^{6} - 6 q^{7} + 2 q^{8} + q^{9} - 5 q^{10} - 7 q^{11} - 3 q^{12} - 6 q^{13} - 6 q^{14} + 5 q^{15} + 2 q^{16} - 6 q^{17} + q^{18} - 6 q^{19} - 5 q^{20} + 9 q^{21} - 7 q^{22} - 6 q^{23} - 3 q^{24} + 5 q^{25} - 6 q^{26} - 6 q^{28} - 14 q^{29} + 5 q^{30} - q^{31} + 2 q^{32} + 8 q^{33} - 6 q^{34} + 15 q^{35} + q^{36} + 3 q^{37} - 6 q^{38} + 9 q^{39} - 5 q^{40} + 9 q^{42} - 14 q^{43} - 7 q^{44} + 5 q^{45} - 6 q^{46} + 3 q^{47} - 3 q^{48} + 4 q^{49} + 5 q^{50} + 19 q^{51} - 6 q^{52} - 18 q^{53} + 20 q^{55} - 6 q^{56} + 9 q^{57} - 14 q^{58} - 24 q^{59} + 5 q^{60} - 7 q^{61} - q^{62} - 3 q^{63} + 2 q^{64} + 15 q^{65} + 8 q^{66} - 3 q^{67} - 6 q^{68} - q^{69} + 15 q^{70} - 6 q^{71} + q^{72} - 12 q^{73} + 3 q^{74} + 5 q^{75} - 6 q^{76} + 21 q^{77} + 9 q^{78} + 3 q^{79} - 5 q^{80} + 2 q^{81} + 14 q^{83} + 9 q^{84} + 5 q^{85} - 14 q^{86} + 26 q^{87} - 7 q^{88} - 9 q^{89} + 5 q^{90} + 18 q^{91} - 6 q^{92} + 14 q^{93} + 3 q^{94} + 15 q^{95} - 3 q^{96} + 32 q^{97} + 4 q^{98} + 4 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.61803
−0.618034
1.00000 −2.61803 1.00000 −1.38197 −2.61803 −3.00000 1.00000 3.85410 −1.38197
1.2 1.00000 −0.381966 1.00000 −3.61803 −0.381966 −3.00000 1.00000 −2.85410 −3.61803
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3019\) \(1\)

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 6038.2.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6038.2.a.a 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 3T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6038))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + 5T + 5 \) Copy content Toggle raw display
$7$ \( (T + 3)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 7T + 11 \) Copy content Toggle raw display
$13$ \( (T + 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 6T - 11 \) Copy content Toggle raw display
$19$ \( (T + 3)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 6T - 11 \) Copy content Toggle raw display
$29$ \( T^{2} + 14T + 44 \) Copy content Toggle raw display
$31$ \( T^{2} + T - 31 \) Copy content Toggle raw display
$37$ \( T^{2} - 3T - 9 \) Copy content Toggle raw display
$41$ \( T^{2} - 125 \) Copy content Toggle raw display
$43$ \( T^{2} + 14T + 44 \) Copy content Toggle raw display
$47$ \( T^{2} - 3T - 99 \) Copy content Toggle raw display
$53$ \( (T + 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 24T + 139 \) Copy content Toggle raw display
$61$ \( T^{2} + 7T + 11 \) Copy content Toggle raw display
$67$ \( T^{2} + 3T - 9 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T - 71 \) Copy content Toggle raw display
$73$ \( T^{2} + 12T - 9 \) Copy content Toggle raw display
$79$ \( T^{2} - 3T - 9 \) Copy content Toggle raw display
$83$ \( T^{2} - 14T - 31 \) Copy content Toggle raw display
$89$ \( T^{2} + 9T - 131 \) Copy content Toggle raw display
$97$ \( T^{2} - 32T + 251 \) Copy content Toggle raw display
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