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Properties

Label 4004.2.a.i
Level $4004$
Weight $2$
Character orbit 4004.a
Self dual yes
Analytic conductor $31.972$
Analytic rank $0$
Dimension $9$
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] = [4004,2,Mod(1,4004)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4004, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4004.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4004.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: \(31.9721009693\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 4x^{8} - 12x^{7} + 60x^{6} + 15x^{5} - 233x^{4} + 74x^{3} + 271x^{2} - 67x - 87 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{7} q^{5} + q^{7} + (\beta_{4} + \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{7} q^{5} + q^{7} + (\beta_{4} + \beta_{3} + 1) q^{9} - q^{11} + q^{13} + (2 \beta_{8} + \beta_{6} - \beta_{5} + \cdots - 1) q^{15}+ \cdots + ( - \beta_{4} - \beta_{3} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 4 q^{3} + 4 q^{5} + 9 q^{7} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 4 q^{3} + 4 q^{5} + 9 q^{7} + 13 q^{9} - 9 q^{11} + 9 q^{13} - 5 q^{17} + 17 q^{19} + 4 q^{21} - 8 q^{23} + 35 q^{25} + 4 q^{27} - 3 q^{29} + 9 q^{31} - 4 q^{33} + 4 q^{35} + 7 q^{37} + 4 q^{39} + 14 q^{41} + 21 q^{43} + 43 q^{45} + q^{47} + 9 q^{49} + 25 q^{51} - 8 q^{53} - 4 q^{55} + 8 q^{57} - 4 q^{59} + 30 q^{61} + 13 q^{63} + 4 q^{65} + 15 q^{67} + 9 q^{69} - q^{71} + 18 q^{73} - 32 q^{75} - 9 q^{77} + 13 q^{79} + 13 q^{81} + 2 q^{83} + 45 q^{85} + 21 q^{87} + 9 q^{91} - 2 q^{93} + 11 q^{95} + 29 q^{97} - 13 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 4x^{8} - 12x^{7} + 60x^{6} + 15x^{5} - 233x^{4} + 74x^{3} + 271x^{2} - 67x - 87 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 10 \nu^{8} - 189 \nu^{7} + 627 \nu^{6} + 2658 \nu^{5} - 6837 \nu^{4} - 8299 \nu^{3} + 17667 \nu^{2} + \cdots - 5712 ) / 681 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -46\nu^{8} + 84\nu^{7} + 705\nu^{6} - 1257\nu^{5} - 2712\nu^{4} + 5000\nu^{3} + 2817\nu^{2} - 5365\nu - 1623 ) / 681 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 46\nu^{8} - 84\nu^{7} - 705\nu^{6} + 1257\nu^{5} + 2712\nu^{4} - 5000\nu^{3} - 2136\nu^{2} + 5365\nu - 1101 ) / 681 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 77 \nu^{8} + 111 \nu^{7} + 1491 \nu^{6} - 1734 \nu^{5} - 9129 \nu^{4} + 7126 \nu^{3} + 20334 \nu^{2} + \cdots - 11703 ) / 681 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 176 \nu^{8} - 351 \nu^{7} - 2727 \nu^{6} + 4839 \nu^{5} + 11235 \nu^{4} - 14926 \nu^{3} - 15249 \nu^{2} + \cdots + 6417 ) / 681 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 298 \nu^{8} - 633 \nu^{7} - 4656 \nu^{6} + 8913 \nu^{5} + 19464 \nu^{4} - 29312 \nu^{3} - 26214 \nu^{2} + \cdots + 9774 ) / 681 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 143 \nu^{8} + 271 \nu^{7} + 2315 \nu^{6} - 3804 \nu^{5} - 10533 \nu^{4} + 12553 \nu^{3} + \cdots - 7044 ) / 227 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + 2\beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 8\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{8} - 3\beta_{7} + 2\beta_{6} - \beta_{5} + 11\beta_{4} + 10\beta_{3} + \beta_{2} + 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13\beta_{8} - 2\beta_{7} + 27\beta_{6} - 15\beta_{5} + 17\beta_{4} + 11\beta_{3} + \beta_{2} + 74\beta _1 - 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -13\beta_{8} - 46\beta_{7} + 35\beta_{6} - 15\beta_{5} + 116\beta_{4} + 95\beta_{3} + 15\beta_{2} + 5\beta _1 + 209 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 139 \beta_{8} - 43 \beta_{7} + 311 \beta_{6} - 171 \beta_{5} + 211 \beta_{4} + 109 \beta_{3} + \cdots - 212 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 133 \beta_{8} - 552 \beta_{7} + 466 \beta_{6} - 182 \beta_{5} + 1220 \beta_{4} + 920 \beta_{3} + \cdots + 1876 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−3.11788
−1.90247
−0.980714
−0.554873
0.848270
1.89826
2.10249
2.42810
3.27882
0 −3.11788 0 4.40203 0 1.00000 0 6.72118 0
1.2 0 −1.90247 0 −3.63968 0 1.00000 0 0.619386 0
1.3 0 −0.980714 0 −1.42373 0 1.00000 0 −2.03820 0
1.4 0 −0.554873 0 3.11431 0 1.00000 0 −2.69212 0
1.5 0 0.848270 0 −0.844151 0 1.00000 0 −2.28044 0
1.6 0 1.89826 0 3.07273 0 1.00000 0 0.603374 0
1.7 0 2.10249 0 −4.18458 0 1.00000 0 1.42045 0
1.8 0 2.42810 0 0.790216 0 1.00000 0 2.89567 0
1.9 0 3.27882 0 2.71285 0 1.00000 0 7.75069 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(11\) \(1\)
\(13\) \(-1\)

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 4004.2.a.i ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4004.2.a.i ✓ 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{9} - 4T_{3}^{8} - 12T_{3}^{7} + 60T_{3}^{6} + 15T_{3}^{5} - 233T_{3}^{4} + 74T_{3}^{3} + 271T_{3}^{2} - 67T_{3} - 87 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4004))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( T^{9} - 4 T^{8} + \cdots - 87 \) Copy content Toggle raw display
$5$ \( T^{9} - 4 T^{8} + \cdots - 1653 \) Copy content Toggle raw display
$7$ \( (T - 1)^{9} \) Copy content Toggle raw display
$11$ \( (T + 1)^{9} \) Copy content Toggle raw display
$13$ \( (T - 1)^{9} \) Copy content Toggle raw display
$17$ \( T^{9} + 5 T^{8} + \cdots - 2916 \) Copy content Toggle raw display
$19$ \( T^{9} - 17 T^{8} + \cdots - 23112 \) Copy content Toggle raw display
$23$ \( T^{9} + 8 T^{8} + \cdots + 2736 \) Copy content Toggle raw display
$29$ \( T^{9} + 3 T^{8} + \cdots - 53184 \) Copy content Toggle raw display
$31$ \( T^{9} - 9 T^{8} + \cdots - 598608 \) Copy content Toggle raw display
$37$ \( T^{9} - 7 T^{8} + \cdots + 891328 \) Copy content Toggle raw display
$41$ \( T^{9} - 14 T^{8} + \cdots - 1251072 \) Copy content Toggle raw display
$43$ \( T^{9} - 21 T^{8} + \cdots - 31422 \) Copy content Toggle raw display
$47$ \( T^{9} - T^{8} + \cdots - 2123136 \) Copy content Toggle raw display
$53$ \( T^{9} + 8 T^{8} + \cdots + 145422 \) Copy content Toggle raw display
$59$ \( T^{9} + 4 T^{8} + \cdots + 7749264 \) Copy content Toggle raw display
$61$ \( T^{9} - 30 T^{8} + \cdots + 6189536 \) Copy content Toggle raw display
$67$ \( T^{9} - 15 T^{8} + \cdots - 254075797 \) Copy content Toggle raw display
$71$ \( T^{9} + T^{8} + \cdots - 17857872 \) Copy content Toggle raw display
$73$ \( T^{9} - 18 T^{8} + \cdots - 3281792 \) Copy content Toggle raw display
$79$ \( T^{9} - 13 T^{8} + \cdots + 43551736 \) Copy content Toggle raw display
$83$ \( T^{9} - 2 T^{8} + \cdots + 32568006 \) Copy content Toggle raw display
$89$ \( T^{9} - 342 T^{7} + \cdots + 133893 \) Copy content Toggle raw display
$97$ \( T^{9} - 29 T^{8} + \cdots + 869184 \) Copy content Toggle raw display
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