Properties

Label 4005.2.a.n
Level $4005$
Weight $2$
Character orbit 4005.a
Self dual yes
Analytic conductor $31.980$
Analytic rank $0$
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] = [4005,2,Mod(1,4005)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4005, 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("4005.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4005.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: \(31.9800860095\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.10407557.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 8x^{4} + 2x^{3} + 18x^{2} + 7x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1335)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{2} + (\beta_{3} + \beta_{2} + 2) q^{4} + q^{5} - \beta_{4} q^{7} + ( - \beta_{5} + \beta_{3} + 2 \beta_{2} + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{2} + (\beta_{3} + \beta_{2} + 2) q^{4} + q^{5} - \beta_{4} q^{7} + ( - \beta_{5} + \beta_{3} + 2 \beta_{2} + 2) q^{8} + (\beta_{2} + 1) q^{10} + \beta_1 q^{13} + (\beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{14} + ( - 2 \beta_{5} + \beta_{4} + \beta_{3} + 3 \beta_{2} - \beta_1 + 2) q^{16} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 3) q^{17} + 2 q^{19} + (\beta_{3} + \beta_{2} + 2) q^{20} + ( - \beta_{4} + \beta_{3} - \beta_{2} + 3) q^{23} + q^{25} + \beta_{4} q^{26} + (2 \beta_{5} - 3 \beta_{4} + \beta_1) q^{28} + ( - \beta_{5} + \beta_1) q^{29} + (2 \beta_{4} + 2) q^{31} + ( - 2 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 3) q^{32} + (\beta_{5} - 2 \beta_{4} - \beta_{3} + 3 \beta_{2} + \beta_1 + 1) q^{34} - \beta_{4} q^{35} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{37} + (2 \beta_{2} + 2) q^{38} + ( - \beta_{5} + \beta_{3} + 2 \beta_{2} + 2) q^{40} + ( - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{41} + (\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{43} + ( - 2 \beta_{4} + 4 \beta_{2}) q^{46} + (\beta_{4} - 2 \beta_{3} + 2) q^{47} + (2 \beta_{5} - 2 \beta_{4} + \beta_1 + 1) q^{49} + (\beta_{2} + 1) q^{50} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{52} + (2 \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 4) q^{53} + (3 \beta_{5} - 3 \beta_{4} - \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 3) q^{56} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 1) q^{58} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 1) q^{59} + (2 \beta_{5} - 2 \beta_{4} + 2) q^{61} + ( - 2 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + 4 \beta_{2}) q^{62} + ( - 3 \beta_{5} + 4 \beta_{4} + 6 \beta_{2} + 1) q^{64} + \beta_1 q^{65} + ( - 2 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{67} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 8) q^{68} + (\beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{70} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 2) q^{71} + ( - 2 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + \beta_{2} - 4 \beta_1 - 1) q^{73} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{74} + (2 \beta_{3} + 2 \beta_{2} + 4) q^{76} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{79} + ( - 2 \beta_{5} + \beta_{4} + \beta_{3} + 3 \beta_{2} - \beta_1 + 2) q^{80} + ( - 3 \beta_{5} + 4 \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{82} + (\beta_{4} + \beta_{3} - \beta_{2} + 3) q^{83} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 3) q^{85} + (\beta_{5} - \beta_{3} - 3 \beta_{2} + \beta_1 + 3) q^{86} + q^{89} + ( - \beta_{5} + \beta_{3} - \beta_{2} + \beta_1 - 3) q^{91} + (2 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} + 8) q^{92} + (\beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} + 3) q^{94} + 2 q^{95} + (\beta_{4} - \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 5) q^{97} + (4 \beta_{5} - 5 \beta_{4} - 3 \beta_{2} + 2 \beta_1 + 5) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{2} + 8 q^{4} + 6 q^{5} + q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{2} + 8 q^{4} + 6 q^{5} + q^{7} + 9 q^{8} + 4 q^{10} - 3 q^{13} + 5 q^{14} + 12 q^{16} + 13 q^{17} + 12 q^{19} + 8 q^{20} + 19 q^{23} + 6 q^{25} - q^{26} - 6 q^{28} + 10 q^{31} + 17 q^{32} - 2 q^{34} + q^{35} - q^{37} + 8 q^{38} + 9 q^{40} + 4 q^{41} - 7 q^{43} - 6 q^{46} + 15 q^{47} - q^{49} + 4 q^{50} + q^{52} + 27 q^{53} + 14 q^{56} - 6 q^{58} + 4 q^{59} + 8 q^{61} - 2 q^{62} - q^{64} - 3 q^{65} - 11 q^{67} + 47 q^{68} + 5 q^{70} + 16 q^{71} + q^{73} + 10 q^{74} + 16 q^{76} + 12 q^{80} + q^{82} + 17 q^{83} + 13 q^{85} + 20 q^{86} + 6 q^{89} - 18 q^{91} + 36 q^{92} + 17 q^{94} + 12 q^{95} - 29 q^{97} + 23 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 8x^{4} + 2x^{3} + 18x^{2} + 7x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 2\nu^{3} - 4\nu^{2} + 5\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{5} - 2\nu^{4} - 6\nu^{3} + 7\nu^{2} + 10\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\nu^{5} - 4\nu^{4} - 11\nu^{3} + 14\nu^{2} + 18\nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - 2\beta_{4} - \beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 2\beta_{4} + 3\beta_{2} - \beta _1 + 12 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{5} - 6\beta_{4} - \beta_{2} - \beta _1 + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11\beta_{5} - 22\beta_{4} + 4\beta_{3} + 13\beta_{2} - 3\beta _1 + 60 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 41\beta_{5} - 78\beta_{4} + 8\beta_{3} + 3\beta_{2} - \beta _1 + 108 ) / 4 \) 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
0.191235
−0.763968
2.09854
−1.43605
2.63450
−1.72426
−2.15466 0 2.64258 1.00000 0 −2.12396 −1.38454 0 −2.15466
1.2 −0.652386 0 −1.57439 1.00000 0 1.82035 2.33188 0 −0.652386
1.3 0.305330 0 −1.90677 1.00000 0 1.72660 −1.19286 0 0.305330
1.4 1.49828 0 0.244835 1.00000 0 −3.23110 −2.62972 0 1.49828
1.5 2.30610 0 3.31809 1.00000 0 4.21574 3.03965 0 2.30610
1.6 2.69734 0 5.27566 1.00000 0 −1.40763 8.83558 0 2.69734
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(89\) \(-1\)

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 4005.2.a.n 6
3.b odd 2 1 1335.2.a.g 6
15.d odd 2 1 6675.2.a.u 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1335.2.a.g 6 3.b odd 2 1
4005.2.a.n 6 1.a even 1 1 trivial
6675.2.a.u 6 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4005))\):

\( T_{2}^{6} - 4T_{2}^{5} - 2T_{2}^{4} + 21T_{2}^{3} - 13T_{2}^{2} - 11T_{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{6} - T_{7}^{5} - 20T_{7}^{4} + 7T_{7}^{3} + 96T_{7}^{2} - 16T_{7} - 128 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 4 T^{5} - 2 T^{4} + 21 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T - 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - T^{5} - 20 T^{4} + 7 T^{3} + \cdots - 128 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 3 T^{5} - 20 T^{4} - 17 T^{3} + \cdots - 32 \) Copy content Toggle raw display
$17$ \( T^{6} - 13 T^{5} + 22 T^{4} + \cdots - 128 \) Copy content Toggle raw display
$19$ \( (T - 2)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} - 19 T^{5} + 114 T^{4} + \cdots - 2048 \) Copy content Toggle raw display
$29$ \( T^{6} - 55 T^{4} - 157 T^{3} + \cdots + 652 \) Copy content Toggle raw display
$31$ \( T^{6} - 10 T^{5} - 40 T^{4} + \cdots - 3904 \) Copy content Toggle raw display
$37$ \( T^{6} + T^{5} - 48 T^{4} - 79 T^{3} + \cdots - 32 \) Copy content Toggle raw display
$41$ \( T^{6} - 4 T^{5} - 117 T^{4} + \cdots - 4964 \) Copy content Toggle raw display
$43$ \( T^{6} + 7 T^{5} - 72 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$47$ \( T^{6} - 15 T^{5} + 26 T^{4} + \cdots + 512 \) Copy content Toggle raw display
$53$ \( T^{6} - 27 T^{5} + 174 T^{4} + \cdots - 34376 \) Copy content Toggle raw display
$59$ \( T^{6} - 4 T^{5} - 39 T^{4} + 147 T^{3} + \cdots - 4 \) Copy content Toggle raw display
$61$ \( T^{6} - 8 T^{5} - 92 T^{4} + \cdots - 1024 \) Copy content Toggle raw display
$67$ \( T^{6} + 11 T^{5} - 204 T^{4} + \cdots + 53888 \) Copy content Toggle raw display
$71$ \( T^{6} - 16 T^{5} - 68 T^{4} + \cdots + 22016 \) Copy content Toggle raw display
$73$ \( T^{6} - T^{5} - 442 T^{4} + \cdots - 2343136 \) Copy content Toggle raw display
$79$ \( T^{6} - 121 T^{4} + 39 T^{3} + \cdots - 10048 \) Copy content Toggle raw display
$83$ \( T^{6} - 17 T^{5} + 52 T^{4} + \cdots + 512 \) Copy content Toggle raw display
$89$ \( (T - 1)^{6} \) Copy content Toggle raw display
$97$ \( T^{6} + 29 T^{5} + 134 T^{4} + \cdots - 30112 \) Copy content Toggle raw display
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