Properties

Label 2004.2.a.d
Level $2004$
Weight $2$
Character orbit 2004.a
Self dual yes
Analytic conductor $16.002$
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] = [2004,2,Mod(1,2004)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2004, 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("2004.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2004 = 2^{2} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2004.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: \(16.0020205651\)
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: \( x^{9} - 29x^{7} - 7x^{6} + 266x^{5} + 69x^{4} - 901x^{3} - 199x^{2} + 875x + 391 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + q^{3} + ( - \beta_1 + 1) q^{5} - \beta_{6} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + ( - \beta_1 + 1) q^{5} - \beta_{6} q^{7} + q^{9} + ( - \beta_{5} + 1) q^{11} + (\beta_{7} + \beta_{6} - \beta_{4} + 1) q^{13} + ( - \beta_1 + 1) q^{15} + (\beta_{8} + \beta_{3} - \beta_{2} + 1) q^{17} + ( - \beta_{7} - \beta_{2}) q^{19} - \beta_{6} q^{21} + ( - \beta_{8} + \beta_{5} + \beta_{2} + \beta_1 + 2) q^{23} + (\beta_{4} + 2 \beta_{2} - 2 \beta_1 + 3) q^{25} + q^{27} + ( - \beta_{8} + \beta_{7} + \beta_{5} - 2 \beta_{3} + \beta_1) q^{29} + ( - \beta_{7} - \beta_{5} + \beta_{4} - \beta_{2} + 1) q^{31} + ( - \beta_{5} + 1) q^{33} + ( - \beta_{7} - 3 \beta_{6} + \beta_{5} - 2 \beta_{3} + 2 \beta_{2}) q^{35} + (\beta_{8} - \beta_{7} + 2 \beta_{4} + \beta_{2} - \beta_1 + 2) q^{37} + (\beta_{7} + \beta_{6} - \beta_{4} + 1) q^{39} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 2) q^{41} + ( - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} - 2) q^{43} + ( - \beta_1 + 1) q^{45} + (\beta_{6} - \beta_{4} + 2 \beta_{3} + 2 \beta_1 + 4) q^{47} + ( - 2 \beta_{8} + \beta_{7} + 2 \beta_{6} - \beta_{4} + 2) q^{49} + (\beta_{8} + \beta_{3} - \beta_{2} + 1) q^{51} + (\beta_{8} - 2 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 2) q^{53} + (2 \beta_{8} - 2 \beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_1 + 1) q^{55} + ( - \beta_{7} - \beta_{2}) q^{57} + ( - \beta_{8} + 2 \beta_{7} + \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{59} + (\beta_{8} - \beta_{7} + 2 \beta_{6} - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 3) q^{61} - \beta_{6} q^{63} + (\beta_{7} + 3 \beta_{6} - 2 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 2) q^{65} + (\beta_{7} + \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1) q^{67} + ( - \beta_{8} + \beta_{5} + \beta_{2} + \beta_1 + 2) q^{69} + (\beta_{8} + \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{71} + (2 \beta_{7} + 2 \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} - 3) q^{73} + (\beta_{4} + 2 \beta_{2} - 2 \beta_1 + 3) q^{75} + (\beta_{7} + \beta_{6} - 3 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} + 2) q^{77} + ( - 2 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{79} + q^{81} + (\beta_{8} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_1 + 1) q^{83} + (2 \beta_{6} - \beta_{5} - 2 \beta_{4} - 2 \beta_{2} + 1) q^{85} + ( - \beta_{8} + \beta_{7} + \beta_{5} - 2 \beta_{3} + \beta_1) q^{87} + ( - \beta_{7} + 2 \beta_{4} + \beta_{2} + 4) q^{89} + (\beta_{8} - 2 \beta_{6} - 2 \beta_{4} + \beta_1 - 4) q^{91} + ( - \beta_{7} - \beta_{5} + \beta_{4} - \beta_{2} + 1) q^{93} + (\beta_{8} + \beta_{7} - \beta_{6} + \beta_{4} + 2 \beta_{2} + \beta_1 + 1) q^{95} + ( - \beta_{8} - \beta_{7} - \beta_{6} + 2 \beta_{4} - 2 \beta_{3} + 5 \beta_{2} - \beta_1 + 4) q^{97} + ( - \beta_{5} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{3} + 9 q^{5} + 2 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{3} + 9 q^{5} + 2 q^{7} + 9 q^{9} + 7 q^{11} + 6 q^{13} + 9 q^{15} + 7 q^{17} + 2 q^{19} + 2 q^{21} + 19 q^{23} + 22 q^{25} + 9 q^{27} + 13 q^{29} + 12 q^{31} + 7 q^{33} + 4 q^{35} + 15 q^{37} + 6 q^{39} + 18 q^{41} - 6 q^{43} + 9 q^{45} + 25 q^{47} + 19 q^{49} + 7 q^{51} + 17 q^{53} - 3 q^{55} + 2 q^{57} + 3 q^{59} + 14 q^{61} + 2 q^{63} + 14 q^{65} - 4 q^{67} + 19 q^{69} + 17 q^{71} - 20 q^{73} + 22 q^{75} + 14 q^{77} - 8 q^{79} + 9 q^{81} - q^{83} + 5 q^{85} + 13 q^{87} + 36 q^{89} - 41 q^{91} + 12 q^{93} + 5 q^{95} + 31 q^{97} + 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 29x^{7} - 7x^{6} + 266x^{5} + 69x^{4} - 901x^{3} - 199x^{2} + 875x + 391 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1006 \nu^{8} + 2871 \nu^{7} + 22263 \nu^{6} - 53775 \nu^{5} - 157887 \nu^{4} + 312589 \nu^{3} + 454514 \nu^{2} - 550699 \nu - 575149 ) / 51607 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1976 \nu^{8} - 9538 \nu^{7} - 25159 \nu^{6} + 160208 \nu^{5} - 1570 \nu^{4} - 711563 \nu^{3} + 508013 \nu^{2} + 557721 \nu - 255465 ) / 51607 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2012 \nu^{8} - 5742 \nu^{7} - 44526 \nu^{6} + 107550 \nu^{5} + 315774 \nu^{4} - 625178 \nu^{3} - 857421 \nu^{2} + 1101398 \nu + 789049 ) / 51607 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2980 \nu^{8} - 18354 \nu^{7} - 43479 \nu^{6} + 348793 \nu^{5} + 150155 \nu^{4} - 1846062 \nu^{3} - 203222 \nu^{2} + 2331119 \nu + 1064431 ) / 51607 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3832 \nu^{8} - 14527 \nu^{7} - 71773 \nu^{6} + 255110 \nu^{5} + 401245 \nu^{4} - 1246613 \nu^{3} - 814593 \nu^{2} + 1505084 \nu + 875617 ) / 51607 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3879 \nu^{8} - 3837 \nu^{7} - 87023 \nu^{6} + 28674 \nu^{5} + 546052 \nu^{4} + 57429 \nu^{3} - 948686 \nu^{2} - 418506 \nu + 172141 ) / 51607 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3886 \nu^{8} + 8833 \nu^{7} + 75020 \nu^{6} - 124516 \nu^{5} - 361191 \nu^{4} + 420341 \nu^{3} + 179180 \nu^{2} - 75695 \nu + 292783 ) / 51607 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + 2\beta_{2} + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + 2\beta_{7} - \beta_{6} + 2\beta_{5} - \beta_{3} + 4\beta_{2} + 10\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{8} + 4\beta_{7} + \beta_{5} + 14\beta_{4} + 2\beta_{3} + 31\beta_{2} + 5\beta _1 + 79 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 22\beta_{8} + 38\beta_{7} - 23\beta_{6} + 36\beta_{5} + 17\beta_{4} - 12\beta_{3} + 91\beta_{2} + 123\beta _1 + 91 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 115 \beta_{8} + 102 \beta_{7} - 21 \beta_{6} + 38 \beta_{5} + 207 \beta_{4} + 46 \beta_{3} + 486 \beta_{2} + 163 \beta _1 + 1059 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 431 \beta_{8} + 648 \beta_{7} - 397 \beta_{6} + 555 \beta_{5} + 464 \beta_{4} - 97 \beta_{3} + 1703 \beta_{2} + 1714 \beta _1 + 2087 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2125 \beta_{8} + 2069 \beta_{7} - 679 \beta_{6} + 965 \beta_{5} + 3251 \beta_{4} + 758 \beta_{3} + 7981 \beta_{2} + 3699 \beta _1 + 15652 \) 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
4.16840
2.69402
1.79204
1.61974
−0.529979
−0.907808
−2.42964
−3.19525
−3.21153
0 1.00000 0 −3.16840 0 −0.230890 0 1.00000 0
1.2 0 1.00000 0 −1.69402 0 −4.12928 0 1.00000 0
1.3 0 1.00000 0 −0.792043 0 3.80237 0 1.00000 0
1.4 0 1.00000 0 −0.619742 0 1.05844 0 1.00000 0
1.5 0 1.00000 0 1.52998 0 −1.05249 0 1.00000 0
1.6 0 1.00000 0 1.90781 0 2.81337 0 1.00000 0
1.7 0 1.00000 0 3.42964 0 3.44225 0 1.00000 0
1.8 0 1.00000 0 4.19525 0 1.43344 0 1.00000 0
1.9 0 1.00000 0 4.21153 0 −5.13720 0 1.00000 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
\(2\) \(-1\)
\(3\) \(-1\)
\(167\) \(-1\)

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 2004.2.a.d 9
3.b odd 2 1 6012.2.a.h 9
4.b odd 2 1 8016.2.a.bb 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2004.2.a.d 9 1.a even 1 1 trivial
6012.2.a.h 9 3.b odd 2 1
8016.2.a.bb 9 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{9} - 9T_{5}^{8} + 7T_{5}^{7} + 126T_{5}^{6} - 259T_{5}^{5} - 405T_{5}^{4} + 964T_{5}^{3} + 506T_{5}^{2} - 856T_{5} - 466 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2004))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( (T - 1)^{9} \) Copy content Toggle raw display
$5$ \( T^{9} - 9 T^{8} + 7 T^{7} + 126 T^{6} + \cdots - 466 \) Copy content Toggle raw display
$7$ \( T^{9} - 2 T^{8} - 39 T^{7} + 109 T^{6} + \cdots - 288 \) Copy content Toggle raw display
$11$ \( T^{9} - 7 T^{8} - 23 T^{7} + 168 T^{6} + \cdots + 36 \) Copy content Toggle raw display
$13$ \( T^{9} - 6 T^{8} - 45 T^{7} + \cdots - 5520 \) Copy content Toggle raw display
$17$ \( T^{9} - 7 T^{8} - 58 T^{7} + \cdots - 1562 \) Copy content Toggle raw display
$19$ \( T^{9} - 2 T^{8} - 69 T^{7} + 134 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( T^{9} - 19 T^{8} + 47 T^{7} + \cdots - 288512 \) Copy content Toggle raw display
$29$ \( T^{9} - 13 T^{8} - 107 T^{7} + \cdots - 272720 \) Copy content Toggle raw display
$31$ \( T^{9} - 12 T^{8} - 42 T^{7} + \cdots - 280000 \) Copy content Toggle raw display
$37$ \( T^{9} - 15 T^{8} - 91 T^{7} + \cdots - 5712 \) Copy content Toggle raw display
$41$ \( T^{9} - 18 T^{8} + 22 T^{7} + \cdots + 11178 \) Copy content Toggle raw display
$43$ \( T^{9} + 6 T^{8} - 290 T^{7} + \cdots - 8589994 \) Copy content Toggle raw display
$47$ \( T^{9} - 25 T^{8} + 25 T^{7} + \cdots + 51047340 \) Copy content Toggle raw display
$53$ \( T^{9} - 17 T^{8} - 159 T^{7} + \cdots + 12400550 \) Copy content Toggle raw display
$59$ \( T^{9} - 3 T^{8} - 283 T^{7} + \cdots + 881152 \) Copy content Toggle raw display
$61$ \( T^{9} - 14 T^{8} - 263 T^{7} + \cdots - 4149788 \) Copy content Toggle raw display
$67$ \( T^{9} + 4 T^{8} - 201 T^{7} + \cdots + 150554 \) Copy content Toggle raw display
$71$ \( T^{9} - 17 T^{8} + \cdots + 343184800 \) Copy content Toggle raw display
$73$ \( T^{9} + 20 T^{8} - 159 T^{7} + \cdots + 45115056 \) Copy content Toggle raw display
$79$ \( T^{9} + 8 T^{8} - 240 T^{7} + \cdots + 2054818 \) Copy content Toggle raw display
$83$ \( T^{9} + T^{8} - 259 T^{7} + \cdots - 3972576 \) Copy content Toggle raw display
$89$ \( T^{9} - 36 T^{8} + 425 T^{7} + \cdots - 93648 \) Copy content Toggle raw display
$97$ \( T^{9} - 31 T^{8} + \cdots - 481925792 \) Copy content Toggle raw display
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