Properties

Label 8037.2.a.i
Level $8037$
Weight $2$
Character orbit 8037.a
Self dual yes
Analytic conductor $64.176$
Analytic rank $1$
Dimension $6$
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] = [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: \(6\)
Coefficient field: 6.6.5476681.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 11x^{4} + 19x^{3} + 32x^{2} - 44x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2679)
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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - 1) q^{2} + (2 \beta_{4} + \beta_{2} + 1) q^{4} + ( - \beta_{5} - \beta_{4}) q^{5} + \beta_1 q^{7} + ( - \beta_{4} - 2 \beta_{2} - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} - 1) q^{2} + (2 \beta_{4} + \beta_{2} + 1) q^{4} + ( - \beta_{5} - \beta_{4}) q^{5} + \beta_1 q^{7} + ( - \beta_{4} - 2 \beta_{2} - 2) q^{8} + (\beta_{5} - \beta_1 + 1) q^{10} + (\beta_{4} + 2) q^{11} + ( - \beta_{5} - 2 \beta_{4} - 2 \beta_{2} - 3) q^{13} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_1 + 1) q^{14} + ( - \beta_{4} - \beta_{2}) q^{16} + (\beta_{5} - \beta_{3} + 1) q^{17} - q^{19} + (\beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{20} + ( - 3 \beta_{4} - \beta_{2} - 4) q^{22} + (2 \beta_{4} + 2 \beta_{2} + \beta_1 + 1) q^{23} + ( - \beta_{5} - 3 \beta_{4} - 3 \beta_{2} - \beta_1 - 2) q^{25} + (\beta_{5} + 4 \beta_{4} + \beta_{2} - \beta_1 + 4) q^{26} + ( - 2 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{28} + (\beta_{2} + \beta_1) q^{29} + (\beta_{5} + 2 \beta_{2} - \beta_1 + 1) q^{31} + (3 \beta_{4} + 5 \beta_{2} + 5) q^{32} + ( - \beta_{5} - \beta_{4} - 1) q^{34} + (3 \beta_{4} + \beta_{3} + \beta_{2}) q^{35} + (\beta_{5} + \beta_{3} + 3 \beta_{2} - \beta_1) q^{37} + (\beta_{4} + 1) q^{38} + (\beta_{4} + 2 \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{40} + (2 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - 2 \beta_1 - 1) q^{41} + ( - 2 \beta_{5} - \beta_{3} + 4 \beta_{2} + \beta_1 + 2) q^{43} + (5 \beta_{4} + 3 \beta_{2} + 5) q^{44} + (\beta_{5} - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{46} + q^{47} + (\beta_{5} + \beta_{2} - 2) q^{49} + (3 \beta_{4} - \beta_{3} + 2 \beta_{2} + 3) q^{50} + ( - 4 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 - 5) q^{52} + (\beta_{5} + 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 2) q^{53} + ( - 2 \beta_{5} - \beta_{4} + \beta_1 - 1) q^{55} + (\beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{56} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_1 + 2) q^{58} + (\beta_{5} + \beta_{4} - 3 \beta_{3} - 2) q^{59} + (3 \beta_{5} + 2 \beta_{4} - \beta_{2} - 2 \beta_1 + 2) q^{61} + ( - 2 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{62} + ( - 6 \beta_{4} - \beta_{2} - 6) q^{64} + ( - \beta_{4} + 2 \beta_{3} - 5 \beta_{2} - 2 \beta_1 + 2) q^{65} + ( - \beta_{5} + 4 \beta_{4} - \beta_{3} + \beta_1 - 1) q^{67} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_1) q^{68} + ( - 2 \beta_{4} - 2 \beta_{2} + \beta_1 - 4) q^{70} + (\beta_{5} + 5 \beta_1 - 2) q^{71} + (\beta_{5} - \beta_{3} + 6 \beta_{2} - 2 \beta_1 + 2) q^{73} + ( - 2 \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 4) q^{74} + ( - 2 \beta_{4} - \beta_{2} - 1) q^{76} + ( - \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_1 - 1) q^{77} + (4 \beta_{5} + 2 \beta_{4} - 4 \beta_{2} - 2) q^{79} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{80} + ( - 4 \beta_{5} - 2 \beta_{3} + \beta_{2} + 6 \beta_1 - 3) q^{82} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} + 5 \beta_{2} + 3 \beta_1) q^{83} + (2 \beta_{4} - \beta_{3} + 7 \beta_{2} - 1) q^{85} + (3 \beta_{5} - 4 \beta_{4} + \beta_{3} - 3 \beta_{2} - 4 \beta_1) q^{86} + ( - 4 \beta_{4} - 3 \beta_{2} - 4) q^{88} + (\beta_{5} - \beta_{3} - 5 \beta_{2} - 2 \beta_1 + 5) q^{89} + (\beta_{5} + 2 \beta_{4} - \beta_{2} - 3 \beta_1 + 1) q^{91} + ( - 2 \beta_{5} + \beta_{4} - \beta_{3} + \beta_1 + 3) q^{92} + ( - \beta_{4} - 1) q^{94} + (\beta_{5} + \beta_{4}) q^{95} + (\beta_{5} - 5 \beta_{4} - 2 \beta_{3} - 5 \beta_{2} - \beta_1 - 8) q^{97} + ( - \beta_{5} + 3 \beta_{4} + \beta_{2} + \beta_1 + 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{2} + 4 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{2} + 4 q^{5} + 2 q^{7} - 6 q^{8} + 2 q^{10} + 10 q^{11} - 8 q^{13} + q^{14} + 4 q^{16} + 3 q^{17} - 6 q^{19} - 7 q^{20} - 16 q^{22} + 10 q^{26} - 7 q^{28} - 2 q^{31} + 14 q^{32} - 2 q^{34} - 7 q^{35} - 9 q^{37} + 4 q^{38} - 4 q^{40} - 18 q^{41} + 9 q^{43} + 14 q^{44} - 7 q^{46} + 6 q^{47} - 16 q^{49} + 7 q^{50} - 21 q^{52} - 20 q^{53} + 2 q^{55} + 5 q^{56} + 7 q^{58} - 19 q^{59} + 13 q^{62} - 22 q^{64} + 22 q^{65} - 11 q^{67} - 14 q^{70} - 4 q^{71} - 7 q^{73} + 27 q^{74} + q^{77} - 16 q^{79} + 5 q^{80} - 2 q^{82} - 7 q^{83} - 25 q^{85} + q^{86} - 10 q^{88} + 33 q^{89} - 4 q^{91} + 21 q^{92} - 4 q^{94} - 4 q^{95} - 34 q^{97} + 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 11x^{4} + 19x^{3} + 32x^{2} - 44x - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 2\nu^{3} - 5\nu^{2} + 9\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 3\nu^{3} + 7\nu^{2} - 12\nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 4\nu^{4} - 3\nu^{3} + 21\nu^{2} - 10\nu - 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{4} + 2\nu^{3} + 7\nu^{2} - 9\nu - 8 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - \beta_{4} + \beta_{3} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{5} - 2\beta_{4} + 2\beta_{3} + 7\beta_{2} + \beta _1 + 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10\beta_{5} - 7\beta_{4} + 11\beta_{3} + 7\beta_{2} + 29\beta _1 + 18 \) 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
−2.20947
1.40754
2.41126
−0.164281
2.84087
−2.28591
−2.24698 0 3.04892 −1.57380 0 −2.20947 −2.35690 0 3.53629
1.2 −2.24698 0 3.04892 1.32682 0 1.40754 −2.35690 0 −2.98134
1.3 −0.554958 0 −1.69202 −2.17107 0 2.41126 2.04892 0 1.20486
1.4 −0.554958 0 −1.69202 3.61612 0 −0.164281 2.04892 0 −2.00679
1.5 0.801938 0 −1.35690 −0.0216037 0 2.84087 −2.69202 0 −0.0173248
1.6 0.801938 0 −1.35690 2.82354 0 −2.28591 −2.69202 0 2.26430
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.i 6
3.b odd 2 1 2679.2.a.i 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2679.2.a.i 6 3.b odd 2 1
8037.2.a.i 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8037))\):

\( T_{2}^{3} + 2T_{2}^{2} - T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{6} - 4T_{5}^{5} - 7T_{5}^{4} + 30T_{5}^{3} + 14T_{5}^{2} - 46T_{5} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{3} + 2 T^{2} - T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 4 T^{5} - 7 T^{4} + 30 T^{3} + \cdots - 1 \) Copy content Toggle raw display
$7$ \( T^{6} - 2 T^{5} - 11 T^{4} + 19 T^{3} + \cdots - 8 \) Copy content Toggle raw display
$11$ \( (T^{3} - 5 T^{2} + 6 T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 8 T^{5} - T^{4} - 152 T^{3} + \cdots - 169 \) Copy content Toggle raw display
$17$ \( T^{6} - 3 T^{5} - 27 T^{4} + 59 T^{3} + \cdots - 13 \) Copy content Toggle raw display
$19$ \( (T + 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} - 22 T^{4} - 7 T^{3} + 110 T^{2} + \cdots - 41 \) Copy content Toggle raw display
$29$ \( T^{6} - 15 T^{4} - 14 T^{3} + 26 T^{2} + \cdots - 13 \) Copy content Toggle raw display
$31$ \( T^{6} + 2 T^{5} - 42 T^{4} + \cdots - 1625 \) Copy content Toggle raw display
$37$ \( T^{6} + 9 T^{5} - 54 T^{4} + \cdots - 7384 \) Copy content Toggle raw display
$41$ \( T^{6} + 18 T^{5} + 36 T^{4} + \cdots - 2743 \) Copy content Toggle raw display
$43$ \( T^{6} - 9 T^{5} - 59 T^{4} + \cdots - 4073 \) Copy content Toggle raw display
$47$ \( (T - 1)^{6} \) Copy content Toggle raw display
$53$ \( T^{6} + 20 T^{5} + 13 T^{4} + \cdots + 116213 \) Copy content Toggle raw display
$59$ \( T^{6} + 19 T^{5} - 33 T^{4} + \cdots - 2344 \) Copy content Toggle raw display
$61$ \( T^{6} - 112 T^{4} - 287 T^{3} + \cdots + 8869 \) Copy content Toggle raw display
$67$ \( T^{6} + 11 T^{5} - 77 T^{4} + \cdots + 1987 \) Copy content Toggle raw display
$71$ \( T^{6} + 4 T^{5} - 377 T^{4} + \cdots - 482488 \) Copy content Toggle raw display
$73$ \( T^{6} + 7 T^{5} - 253 T^{4} + \cdots - 53423 \) Copy content Toggle raw display
$79$ \( T^{6} + 16 T^{5} - 168 T^{4} + \cdots + 171968 \) Copy content Toggle raw display
$83$ \( T^{6} + 7 T^{5} - 102 T^{4} + \cdots + 8632 \) Copy content Toggle raw display
$89$ \( T^{6} - 33 T^{5} + 283 T^{4} + \cdots - 195539 \) Copy content Toggle raw display
$97$ \( T^{6} + 34 T^{5} + 298 T^{4} + \cdots + 43597 \) Copy content Toggle raw display
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