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Properties

Label 10000.2.a.be
Level $10000$
Weight $2$
Character orbit 10000.a
Self dual yes
Analytic conductor $79.850$
Analytic rank $1$
Dimension $8$
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] = [10000,2,Mod(1,10000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10000, 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("10000.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10000 = 2^{4} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 10000.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: \(79.8504020213\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.6152203125.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} - 8x^{6} + 20x^{5} + 26x^{4} - 35x^{3} - 27x^{2} + 16x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 625)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} - 1) q^{3} + (\beta_{7} - \beta_{5} + \beta_{3} + \cdots - 1) q^{7}+ \cdots + (2 \beta_{6} - \beta_{5} - \beta_{4} + \cdots + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - 1) q^{3} + (\beta_{7} - \beta_{5} + \beta_{3} + \cdots - 1) q^{7}+ \cdots + ( - 7 \beta_{7} - 4 \beta_{6} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 5 q^{3} - 10 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 5 q^{3} - 10 q^{7} + 9 q^{9} - q^{11} + 10 q^{13} + 15 q^{17} + 10 q^{19} - 14 q^{21} - 30 q^{23} - 20 q^{27} + 10 q^{29} + 9 q^{31} + 5 q^{33} - 10 q^{37} - 8 q^{39} - 4 q^{41} - 30 q^{47} - 4 q^{49} + 14 q^{51} + 10 q^{53} - 10 q^{57} + 5 q^{59} + 6 q^{61} - 10 q^{67} + 3 q^{69} + 9 q^{71} + 5 q^{77} + 20 q^{79} + 8 q^{81} - 40 q^{83} - 40 q^{87} - 5 q^{89} - 6 q^{91} - 40 q^{93} + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} - 8x^{6} + 20x^{5} + 26x^{4} - 35x^{3} - 27x^{2} + 16x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - 4\nu^{5} - 2\nu^{4} + 17\nu^{3} - \nu^{2} - 15\nu + 1 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 4\nu^{6} - 2\nu^{5} + 17\nu^{4} - \nu^{3} - 15\nu^{2} + 4\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 4\nu^{5} + 5\nu^{4} - 26\nu^{3} - 5\nu^{2} + 36\nu - 4 ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{7} + 8\nu^{6} + 7\nu^{5} - 43\nu^{4} - 4\nu^{3} + 51\nu^{2} - 8\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{7} + 8\nu^{6} + 7\nu^{5} - 43\nu^{4} - 7\nu^{3} + 57\nu^{2} + \nu - 6 ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} + 2\beta_{2} + 5\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{7} + 3\beta_{6} + \beta_{5} + \beta_{3} + 8\beta_{2} + 10\beta _1 + 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -11\beta_{7} + 12\beta_{6} + 3\beta_{5} + 2\beta_{4} + 3\beta_{3} + 21\beta_{2} + 33\beta _1 + 44 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -33\beta_{7} + 37\beta_{6} + 14\beta_{5} + 8\beta_{4} + 17\beta_{3} + 67\beta_{2} + 83\beta _1 + 148 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -104\beta_{7} + 122\beta_{6} + 45\beta_{5} + 39\beta_{4} + 57\beta_{3} + 191\beta_{2} + 244\beta _1 + 406 \) 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
−1.32610
−0.0573749
2.68341
1.32675
−1.47435
3.01367
−1.66501
0.499011
0 −3.09326 0 0 0 −0.0237879 0 6.56824 0
1.2 0 −3.02566 0 0 0 −0.369971 0 6.15465 0
1.3 0 −2.11675 0 0 0 0.973070 0 1.48063 0
1.4 0 −0.759083 0 0 0 2.04213 0 −2.42379 0
1.5 0 −0.710340 0 0 0 −4.59110 0 −2.49542 0
1.6 0 0.687404 0 0 0 −1.01199 0 −2.52748 0
1.7 0 1.71538 0 0 0 −3.42409 0 −0.0574791 0
1.8 0 2.30231 0 0 0 −3.59425 0 2.30065 0
\(n\): e.g. 2-40 or 990-1000
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 10000.2.a.be 8
4.b odd 2 1 625.2.a.g yes 8
5.b even 2 1 10000.2.a.bn 8
12.b even 2 1 5625.2.a.s 8
20.d odd 2 1 625.2.a.e ✓ 8
20.e even 4 2 625.2.b.d 16
60.h even 2 1 5625.2.a.be 8
100.h odd 10 2 625.2.d.p 16
100.h odd 10 2 625.2.d.q 16
100.j odd 10 2 625.2.d.m 16
100.j odd 10 2 625.2.d.n 16
100.l even 20 4 625.2.e.j 32
100.l even 20 4 625.2.e.k 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
625.2.a.e ✓ 8 20.d odd 2 1
625.2.a.g yes 8 4.b odd 2 1
625.2.b.d 16 20.e even 4 2
625.2.d.m 16 100.j odd 10 2
625.2.d.n 16 100.j odd 10 2
625.2.d.p 16 100.h odd 10 2
625.2.d.q 16 100.h odd 10 2
625.2.e.j 32 100.l even 20 4
625.2.e.k 32 100.l even 20 4
5625.2.a.s 8 12.b even 2 1
5625.2.a.be 8 60.h even 2 1
10000.2.a.be 8 1.a even 1 1 trivial
10000.2.a.bn 8 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(10000))\):

\( T_{3}^{8} + 5T_{3}^{7} - 4T_{3}^{6} - 45T_{3}^{5} - 19T_{3}^{4} + 105T_{3}^{3} + 71T_{3}^{2} - 40T_{3} - 29 \) Copy content Toggle raw display
\( T_{7}^{8} + 10T_{7}^{7} + 24T_{7}^{6} - 35T_{7}^{5} - 154T_{7}^{4} - 25T_{7}^{3} + 124T_{7}^{2} + 45T_{7} + 1 \) Copy content Toggle raw display
\( T_{11}^{8} + T_{11}^{7} - 58T_{11}^{6} - 82T_{11}^{5} + 970T_{11}^{4} + 1617T_{11}^{3} - 3588T_{11}^{2} - 3591T_{11} + 2421 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 5 T^{7} + \cdots - 29 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 10 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{8} + T^{7} + \cdots + 2421 \) Copy content Toggle raw display
$13$ \( T^{8} - 10 T^{7} + \cdots + 361 \) Copy content Toggle raw display
$17$ \( T^{8} - 15 T^{7} + \cdots + 1611 \) Copy content Toggle raw display
$19$ \( T^{8} - 10 T^{7} + \cdots + 10525 \) Copy content Toggle raw display
$23$ \( T^{8} + 30 T^{7} + \cdots - 46089 \) Copy content Toggle raw display
$29$ \( T^{8} - 10 T^{7} + \cdots - 60975 \) Copy content Toggle raw display
$31$ \( T^{8} - 9 T^{7} + \cdots + 24001 \) Copy content Toggle raw display
$37$ \( T^{8} + 10 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$41$ \( T^{8} + 4 T^{7} + \cdots - 487629 \) Copy content Toggle raw display
$43$ \( T^{8} - 99 T^{6} + \cdots - 1949 \) Copy content Toggle raw display
$47$ \( T^{8} + 30 T^{7} + \cdots + 56961 \) Copy content Toggle raw display
$53$ \( T^{8} - 10 T^{7} + \cdots - 1899 \) Copy content Toggle raw display
$59$ \( T^{8} - 5 T^{7} + \cdots - 225 \) Copy content Toggle raw display
$61$ \( T^{8} - 6 T^{7} + \cdots - 103529 \) Copy content Toggle raw display
$67$ \( T^{8} + 10 T^{7} + \cdots - 1746299 \) Copy content Toggle raw display
$71$ \( T^{8} - 9 T^{7} + \cdots - 16749 \) Copy content Toggle raw display
$73$ \( T^{8} - 314 T^{6} + \cdots + 237091 \) Copy content Toggle raw display
$79$ \( T^{8} - 20 T^{7} + \cdots + 249525 \) Copy content Toggle raw display
$83$ \( T^{8} + 40 T^{7} + \cdots + 12262851 \) Copy content Toggle raw display
$89$ \( T^{8} + 5 T^{7} + \cdots + 1849275 \) Copy content Toggle raw display
$97$ \( T^{8} - 321 T^{6} + \cdots + 972421 \) Copy content Toggle raw display
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