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Properties

Label 1000.4.a.d
Level $1000$
Weight $4$
Character orbit 1000.a
Self dual yes
Analytic conductor $59.002$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1000,4,Mod(1,1000)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1000.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1000, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1000.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,4,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(59.0019100057\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 56x^{6} - 44x^{5} + 924x^{4} + 832x^{3} - 5656x^{2} - 3540x + 11255 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 5^{3} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 1) q^{3} + (\beta_{7} + \beta_{3} - 1) q^{7} + (\beta_{7} - \beta_{6} + 2 \beta_{4} + \cdots + 8) q^{9} + (\beta_{7} + \beta_{5} - 2 \beta_{4} + \cdots - 1) q^{11} + (\beta_{7} + \beta_{5} + \beta_{4} + \cdots + 6) q^{13}+ \cdots + ( - 18 \beta_{7} + 15 \beta_{6} + \cdots - 116) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} - 8 q^{7} + 60 q^{9} - 16 q^{11} + 40 q^{13} + 112 q^{17} + 16 q^{19} - 256 q^{21} + 312 q^{23} + 268 q^{27} - 188 q^{29} + 464 q^{31} + 144 q^{33} - 72 q^{37} + 672 q^{39} - 380 q^{41} - 44 q^{43}+ \cdots - 832 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 56x^{6} - 44x^{5} + 924x^{4} + 832x^{3} - 5656x^{2} - 3540x + 11255 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3799 \nu^{7} - 171007 \nu^{6} + 30789 \nu^{5} + 7058790 \nu^{4} + 2517243 \nu^{3} + \cdots + 110853336 ) / 2575866 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5242 \nu^{7} + 97303 \nu^{6} + 347388 \nu^{5} - 4620453 \nu^{4} - 11370492 \nu^{3} + \cdots - 155529615 ) / 2575866 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 50056 \nu^{7} - 157503 \nu^{6} - 2293816 \nu^{5} + 5075970 \nu^{4} + 29824038 \nu^{3} + \cdots + 183581212 ) / 2575866 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 62391 \nu^{7} - 196873 \nu^{6} - 2838209 \nu^{5} + 6149841 \nu^{4} + 36224223 \nu^{3} + \cdots + 184865465 ) / 2575866 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1720 \nu^{7} - 5320 \nu^{6} - 77430 \nu^{5} + 157725 \nu^{4} + 988200 \nu^{3} - 1508600 \nu^{2} + \cdots + 4892763 ) / 62826 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 31716 \nu^{7} - 101751 \nu^{6} - 1287754 \nu^{5} + 2575617 \nu^{4} + 13585974 \nu^{3} + \cdots + 21853727 ) / 858622 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 142391 \nu^{7} + 667128 \nu^{6} + 6055343 \nu^{5} - 23454036 \nu^{4} - 77586105 \nu^{3} + \cdots - 742071440 ) / 2575866 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} - 2\beta_{5} + 4\beta_{4} - 7\beta_{3} - 3\beta_{2} - 3\beta _1 + 2 ) / 20 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 3\beta_{6} - 9\beta_{5} + 6\beta_{4} + 2\beta_{3} - 4\beta_{2} - \beta _1 + 141 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -13\beta_{7} + 47\beta_{6} - 80\beta_{5} + 42\beta_{4} - 71\beta_{3} - 101\beta_{2} - 73\beta _1 + 378 ) / 20 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6\beta_{7} + 70\beta_{6} - 210\beta_{5} + 101\beta_{4} + 47\beta_{3} - 105\beta_{2} - 53\beta _1 + 1644 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 305 \beta_{7} + 1993 \beta_{6} - 3906 \beta_{5} + 930 \beta_{4} - 505 \beta_{3} - 3589 \beta_{2} + \cdots + 21988 ) / 20 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 13 \beta_{7} + 5937 \beta_{6} - 16529 \beta_{5} + 6388 \beta_{4} + 3516 \beta_{3} - 9356 \beta_{2} + \cdots + 103959 ) / 10 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 8717 \beta_{7} + 81119 \beta_{6} - 174332 \beta_{5} + 39088 \beta_{4} + 11451 \beta_{3} + \cdots + 1007368 ) / 20 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−4.49163
−2.49981
6.36248
−3.58691
1.46068
2.55736
−3.33153
3.52936
0 −7.75314 0 0 0 18.5615 0 33.1111 0
1.2 0 −6.64241 0 0 0 8.68836 0 17.1216 0
1.3 0 −2.87484 0 0 0 −1.16117 0 −18.7353 0
1.4 0 −1.38332 0 0 0 −11.7456 0 −25.0864 0
1.5 0 1.43173 0 0 0 −34.6352 0 −24.9502 0
1.6 0 5.10051 0 0 0 24.3225 0 −0.984757 0
1.7 0 6.72411 0 0 0 13.2349 0 18.2137 0
1.8 0 9.39735 0 0 0 −25.2652 0 61.3103 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( -1 \)

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 1000.4.a.d yes 8
4.b odd 2 1 2000.4.a.o 8
5.b even 2 1 1000.4.a.a ✓ 8
5.c odd 4 2 1000.4.c.a 16
20.d odd 2 1 2000.4.a.t 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1000.4.a.a ✓ 8 5.b even 2 1
1000.4.a.d yes 8 1.a even 1 1 trivial
1000.4.c.a 16 5.c odd 4 2
2000.4.a.o 8 4.b odd 2 1
2000.4.a.t 8 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 4T_{3}^{7} - 130T_{3}^{6} + 380T_{3}^{5} + 5035T_{3}^{4} - 9364T_{3}^{3} - 56814T_{3}^{2} + 18960T_{3} + 94505 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1000))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{7} + \cdots + 94505 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 8 T^{7} + \cdots + 619569409 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 505089997056 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 714023149824 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots - 4284166752000 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots - 136866037155584 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 222235110426769 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 20\!\cdots\!01 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots - 22\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 27\!\cdots\!49 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 70\!\cdots\!09 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 20\!\cdots\!21 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots - 20\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots - 18\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots - 69\!\cdots\!75 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots - 97\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 14\!\cdots\!20 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots - 95\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots - 20\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots - 59\!\cdots\!31 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 26\!\cdots\!21 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots - 47\!\cdots\!04 \) Copy content Toggle raw display
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