Properties

Label 4004.2.a.h
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: \( x^{9} - 3x^{8} - 19x^{7} + 51x^{6} + 116x^{5} - 247x^{4} - 249x^{3} + 288x^{2} + 189x - 14 \) 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{4} q^{5} + q^{7} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{4} q^{5} + q^{7} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 2) q^{9} + q^{11} - q^{13} + (\beta_{7} - \beta_{6} - \beta_{4} + \beta_1 - 2) q^{15} + (\beta_{8} - \beta_{3} - \beta_{2}) q^{17} + (\beta_{4} - \beta_{2} + 1) q^{19} + \beta_1 q^{21} + (\beta_{8} + \beta_{5} - \beta_1 + 1) q^{23} + (\beta_{6} + \beta_{3} + 1) q^{25} + (\beta_{7} + \beta_{6} + 2 \beta_{3} + 2 \beta_1 + 3) q^{27} + ( - \beta_{7} - \beta_{6} - \beta_{4} + \beta_1 + 1) q^{29} + ( - 2 \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{2} + 2) q^{31} + \beta_1 q^{33} - \beta_{4} q^{35} + ( - \beta_{8} - \beta_{4} - \beta_{3}) q^{37} - \beta_1 q^{39} + ( - \beta_{7} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{41} + (\beta_{8} + \beta_{7} - \beta_{5} - \beta_{3} + \beta_{2} + \beta_1) q^{43} + (\beta_{8} + \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 4 \beta_{4} - \beta_{2} - 1) q^{45} + ( - \beta_{8} + \beta_{5} - 2 \beta_{4} + \beta_1 + 1) q^{47} + q^{49} + ( - \beta_{8} - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{51} + ( - \beta_{8} - \beta_{7} + \beta_{6} - 3 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 3) q^{53} - \beta_{4} q^{55} + ( - \beta_{8} - \beta_{7} + 2 \beta_{6} - \beta_{5} + 3 \beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{57} + ( - 2 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{2} - \beta_1 + 3) q^{59} + (\beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{2} - 2 \beta_1 + 5) q^{61} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 2) q^{63} + \beta_{4} q^{65} + (\beta_{7} + \beta_{6} + 2 \beta_{4} - \beta_{3} - \beta_{2} + 1) q^{67} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} + 2 \beta_{2} - 2) q^{69} + (\beta_{8} + \beta_{5} + \beta_{4} + \beta_{2} + \beta_1) q^{71} + ( - \beta_{8} + 3 \beta_{7} - \beta_{5} + \beta_{3} - \beta_{2} + 2 \beta_1) q^{73} + (\beta_{6} + 2 \beta_{4} + 3 \beta_{3} - \beta_{2} + \beta_1 + 4) q^{75} + q^{77} + ( - \beta_{8} - 2 \beta_{7} - \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 4) q^{79} + (\beta_{8} + \beta_{6} + 2 \beta_{5} + 4 \beta_{3} - \beta_{2} + 2 \beta_1 + 8) q^{81} + (\beta_{7} + 2 \beta_{5} + \beta_{4} + \beta_{3} - \beta_1 + 2) q^{83} + ( - \beta_{8} - 2 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{2} - \beta_1 + 1) q^{85} + (\beta_{8} + \beta_{7} - \beta_{4} + \beta_{2} + 3 \beta_1 + 3) q^{87} + (\beta_{8} - 2 \beta_{5} + \beta_{4} + 2 \beta_{2}) q^{89} - q^{91} + ( - 2 \beta_{7} + 2 \beta_{6} + \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \cdots + 1) q^{93}+ \cdots + ( - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 3 q^{3} + 9 q^{7} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 3 q^{3} + 9 q^{7} + 20 q^{9} + 9 q^{11} - 9 q^{13} - 9 q^{15} + 5 q^{17} + 10 q^{19} + 3 q^{21} + 8 q^{23} + 3 q^{25} + 27 q^{27} + 14 q^{29} + 11 q^{31} + 3 q^{33} - 3 q^{39} + 14 q^{41} + 8 q^{43} + 4 q^{45} + 10 q^{47} + 9 q^{49} - 15 q^{51} + 21 q^{53} - 8 q^{57} + 23 q^{59} + 34 q^{61} + 20 q^{63} + 10 q^{67} - 16 q^{69} + 4 q^{71} + 9 q^{73} + 30 q^{75} + 9 q^{77} - 34 q^{79} + 69 q^{81} + 15 q^{83} + 5 q^{85} + 39 q^{87} - 9 q^{91} + 3 q^{93} - 64 q^{95} + 15 q^{97} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 19x^{7} + 51x^{6} + 116x^{5} - 247x^{4} - 249x^{3} + 288x^{2} + 189x - 14 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11 \nu^{8} - 1526 \nu^{7} + 1071 \nu^{6} + 27349 \nu^{5} - 11387 \nu^{4} - 132924 \nu^{3} + 26935 \nu^{2} + 141008 \nu + 36543 ) / 15311 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 38 \nu^{8} - 296 \nu^{7} - 916 \nu^{6} + 7131 \nu^{5} + 7323 \nu^{4} - 46071 \nu^{3} - 19277 \nu^{2} + 55726 \nu - 10711 ) / 15311 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 49 \nu^{8} + 1230 \nu^{7} - 1987 \nu^{6} - 20218 \nu^{5} + 18710 \nu^{4} + 86853 \nu^{3} - 61523 \nu^{2} - 69971 \nu + 29301 ) / 15311 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 95 \nu^{8} - 740 \nu^{7} - 2290 \nu^{6} + 10172 \nu^{5} + 25963 \nu^{4} - 30967 \nu^{3} - 117092 \nu^{2} + 16827 \nu + 103366 ) / 15311 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 626 \nu^{8} - 847 \nu^{7} - 9449 \nu^{6} + 9491 \nu^{5} + 32800 \nu^{4} - 14361 \nu^{3} + 7997 \nu^{2} - 64309 \nu - 41874 ) / 15311 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 702 \nu^{8} + 1439 \nu^{7} + 11281 \nu^{6} - 23753 \nu^{5} - 47446 \nu^{4} + 121814 \nu^{3} + 30557 \nu^{2} - 169631 \nu + 17363 ) / 15311 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 979 \nu^{8} + 1985 \nu^{7} + 18764 \nu^{6} - 31010 \nu^{5} - 110094 \nu^{4} + 127655 \nu^{3} + 192431 \nu^{2} - 81863 \nu - 70160 ) / 15311 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + 2\beta_{3} + 8\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} + \beta_{6} + 2\beta_{5} - 9\beta_{4} + 13\beta_{3} - 10\beta_{2} + 11\beta _1 + 44 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{8} + 11\beta_{7} + 12\beta_{6} + 31\beta_{3} - \beta_{2} + 74\beta _1 + 49 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 19\beta_{8} + 21\beta_{6} + 29\beta_{5} - 81\beta_{4} + 148\beta_{3} - 95\beta_{2} + 124\beta _1 + 434 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 22\beta_{8} + 110\beta_{7} + 133\beta_{6} + 3\beta_{5} - 2\beta_{4} + 394\beta_{3} - 31\beta_{2} + 734\beta _1 + 673 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 249 \beta_{8} + 5 \beta_{7} + 310 \beta_{6} + 337 \beta_{5} - 741 \beta_{4} + 1649 \beta_{3} - 924 \beta_{2} + 1440 \beta _1 + 4485 \) 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
−2.98175
−2.56319
−1.21819
−0.648104
0.0675528
1.32285
2.29815
3.32951
3.39317
0 −2.98175 0 2.28847 0 1.00000 0 5.89080 0
1.2 0 −2.56319 0 −1.31026 0 1.00000 0 3.56996 0
1.3 0 −1.21819 0 3.23173 0 1.00000 0 −1.51602 0
1.4 0 −0.648104 0 −1.99671 0 1.00000 0 −2.57996 0
1.5 0 0.0675528 0 −1.58844 0 1.00000 0 −2.99544 0
1.6 0 1.32285 0 −0.265081 0 1.00000 0 −1.25008 0
1.7 0 2.29815 0 0.952431 0 1.00000 0 2.28151 0
1.8 0 3.32951 0 2.67963 0 1.00000 0 8.08561 0
1.9 0 3.39317 0 −3.99177 0 1.00000 0 8.51361 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.h 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4004.2.a.h 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{9} - 3T_{3}^{8} - 19T_{3}^{7} + 51T_{3}^{6} + 116T_{3}^{5} - 247T_{3}^{4} - 249T_{3}^{3} + 288T_{3}^{2} + 189T_{3} - 14 \) 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
$3$ \( T^{9} - 3 T^{8} - 19 T^{7} + 51 T^{6} + \cdots - 14 \) Copy content Toggle raw display
$5$ \( T^{9} - 24 T^{7} + 4 T^{6} + 171 T^{5} + \cdots + 83 \) 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
$17$ \( T^{9} - 5 T^{8} - 95 T^{7} + \cdots + 61022 \) Copy content Toggle raw display
$19$ \( T^{9} - 10 T^{8} - 42 T^{7} + \cdots - 19801 \) Copy content Toggle raw display
$23$ \( T^{9} - 8 T^{8} - 61 T^{7} + \cdots - 11536 \) Copy content Toggle raw display
$29$ \( T^{9} - 14 T^{8} - 47 T^{7} + \cdots - 147280 \) Copy content Toggle raw display
$31$ \( T^{9} - 11 T^{8} - 186 T^{7} + \cdots - 15554608 \) Copy content Toggle raw display
$37$ \( T^{9} - 188 T^{7} - 175 T^{6} + \cdots + 118400 \) Copy content Toggle raw display
$41$ \( T^{9} - 14 T^{8} - 86 T^{7} + \cdots - 1040000 \) Copy content Toggle raw display
$43$ \( T^{9} - 8 T^{8} - 211 T^{7} + \cdots + 1770695 \) Copy content Toggle raw display
$47$ \( T^{9} - 10 T^{8} - 241 T^{7} + \cdots - 11326672 \) Copy content Toggle raw display
$53$ \( T^{9} - 21 T^{8} + \cdots - 199204765 \) Copy content Toggle raw display
$59$ \( T^{9} - 23 T^{8} - 13 T^{7} + \cdots - 775936 \) Copy content Toggle raw display
$61$ \( T^{9} - 34 T^{8} + 311 T^{7} + \cdots + 262558 \) Copy content Toggle raw display
$67$ \( T^{9} - 10 T^{8} - 253 T^{7} + \cdots + 235526 \) Copy content Toggle raw display
$71$ \( T^{9} - 4 T^{8} - 253 T^{7} + \cdots - 19670560 \) Copy content Toggle raw display
$73$ \( T^{9} - 9 T^{8} - 464 T^{7} + \cdots - 41371456 \) Copy content Toggle raw display
$79$ \( T^{9} + 34 T^{8} + 243 T^{7} + \cdots + 3373045 \) Copy content Toggle raw display
$83$ \( T^{9} - 15 T^{8} - 261 T^{7} + \cdots + 743353 \) Copy content Toggle raw display
$89$ \( T^{9} - 538 T^{7} + \cdots - 318322999 \) Copy content Toggle raw display
$97$ \( T^{9} - 15 T^{8} - 90 T^{7} + \cdots + 336832 \) Copy content Toggle raw display
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