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Properties

Label 10000.2.a.bq
Level $10000$
Weight $2$
Character orbit 10000.a
Self dual yes
Analytic conductor $79.850$
Analytic rank $0$
Dimension $16$
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: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 26 x^{14} + 110 x^{13} + 250 x^{12} - 1154 x^{11} - 1074 x^{10} + 5784 x^{9} + \cdots + 80 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 5^{3} \)
Twist minimal: no (minimal twist has level 200)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{8} - 1) q^{7} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + (\beta_{8} - 1) q^{7} + (\beta_{2} + 1) q^{9} + (\beta_{5} + 1) q^{11} + ( - \beta_{11} + \beta_{10} + \beta_{9} + \cdots + 1) q^{13}+ \cdots + ( - \beta_{14} + 2 \beta_{13} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{3} - 8 q^{7} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{3} - 8 q^{7} + 20 q^{9} + 12 q^{11} + 10 q^{13} + 8 q^{17} + 12 q^{19} + 8 q^{21} - 12 q^{23} - 22 q^{27} + 16 q^{29} + 2 q^{31} + 24 q^{33} + 22 q^{37} + 4 q^{39} + 20 q^{41} - 26 q^{43} - 24 q^{47} + 30 q^{49} + 30 q^{51} + 16 q^{53} + 24 q^{57} + 40 q^{59} + 22 q^{61} - 52 q^{63} - 50 q^{67} + 22 q^{69} + 4 q^{71} + 32 q^{73} + 30 q^{77} + 8 q^{79} + 44 q^{81} - 30 q^{83} - 48 q^{87} + 28 q^{89} + 32 q^{91} + 40 q^{93} + 36 q^{97} + 46 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 26 x^{14} + 110 x^{13} + 250 x^{12} - 1154 x^{11} - 1074 x^{10} + 5784 x^{9} + \cdots + 80 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 177 \nu^{15} - 6961 \nu^{14} + 34464 \nu^{13} + 147458 \nu^{12} - 1048624 \nu^{11} - 886366 \nu^{10} + \cdots + 2503340 ) / 624200 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3320 \nu^{15} + 7068 \nu^{14} + 106035 \nu^{13} - 192200 \nu^{12} - 1354885 \nu^{11} + \cdots + 651760 ) / 312100 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16083 \nu^{15} - 40628 \nu^{14} - 491454 \nu^{13} + 1092422 \nu^{12} + 6003374 \nu^{11} + \cdots - 5172720 ) / 624200 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8147 \nu^{15} - 29268 \nu^{14} - 218890 \nu^{13} + 790135 \nu^{12} + 2228950 \nu^{11} + \cdots - 799260 ) / 312100 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 34985 \nu^{15} + 160082 \nu^{14} + 828954 \nu^{13} - 4334162 \nu^{12} - 6655254 \nu^{11} + \cdots - 11782400 ) / 1248400 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 9952 \nu^{15} + 47268 \nu^{14} + 222753 \nu^{13} - 1261704 \nu^{12} - 1520633 \nu^{11} + \cdots - 1490220 ) / 312100 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 19853 \nu^{15} - 110886 \nu^{14} - 371328 \nu^{13} + 2921034 \nu^{12} + 1061908 \nu^{11} + \cdots + 5150560 ) / 624200 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 45159 \nu^{15} + 195662 \nu^{14} + 1091690 \nu^{13} - 5247810 \nu^{12} - 9150790 \nu^{11} + \cdots + 1927440 ) / 1248400 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 48227 \nu^{15} - 237254 \nu^{14} - 1043610 \nu^{13} + 6309830 \nu^{12} + 6421510 \nu^{11} + \cdots + 13166720 ) / 1248400 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 17507 \nu^{15} + 57768 \nu^{14} + 498354 \nu^{13} - 1586982 \nu^{12} - 5557734 \nu^{11} + \cdots + 1137920 ) / 312100 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 70841 \nu^{15} + 216442 \nu^{14} + 2080246 \nu^{13} - 5960498 \nu^{12} - 24203026 \nu^{11} + \cdots + 10337280 ) / 1248400 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 47457 \nu^{15} + 211662 \nu^{14} + 1129108 \nu^{13} - 5692644 \nu^{12} - 9103068 \nu^{11} + \cdots - 4435440 ) / 624200 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 29696 \nu^{15} - 104425 \nu^{14} - 808595 \nu^{13} + 2827855 \nu^{12} + 8411315 \nu^{11} + \cdots - 2334800 ) / 312100 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} - \beta_{13} + \beta_{12} - \beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{2} + 7\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} + \beta_{14} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - 2 \beta_{7} + \cdots + 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10 \beta_{14} - 13 \beta_{13} + 12 \beta_{12} + 3 \beta_{11} - \beta_{9} - 7 \beta_{8} - 10 \beta_{7} + \cdots + 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 18 \beta_{15} + 15 \beta_{14} - \beta_{13} + 17 \beta_{12} + 18 \beta_{11} + 12 \beta_{10} - 19 \beta_{9} + \cdots + 239 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8 \beta_{15} + 90 \beta_{14} - 140 \beta_{13} + 128 \beta_{12} + 54 \beta_{11} + 2 \beta_{10} + \cdots + 258 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 238 \beta_{15} + 178 \beta_{14} - 36 \beta_{13} + 214 \beta_{12} + 256 \beta_{11} + 114 \beta_{10} + \cdots + 2068 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 212 \beta_{15} + 817 \beta_{14} - 1431 \beta_{13} + 1323 \beta_{12} + 754 \beta_{11} + 56 \beta_{10} + \cdots + 2674 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2885 \beta_{15} + 1975 \beta_{14} - 708 \beta_{13} + 2471 \beta_{12} + 3295 \beta_{11} + 1033 \beta_{10} + \cdots + 18415 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3776 \beta_{15} + 7650 \beta_{14} - 14421 \beta_{13} + 13548 \beta_{12} + 9679 \beta_{11} + 960 \beta_{10} + \cdots + 27222 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 33886 \beta_{15} + 21335 \beta_{14} - 10963 \beta_{13} + 27629 \beta_{12} + 40274 \beta_{11} + \cdots + 167249 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 56916 \beta_{15} + 74056 \beta_{14} - 145338 \beta_{13} + 138764 \beta_{12} + 119414 \beta_{11} + \cdots + 274612 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 392784 \beta_{15} + 227784 \beta_{14} - 150324 \beta_{13} + 304908 \beta_{12} + 478292 \beta_{11} + \cdots + 1541960 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 784500 \beta_{15} + 738565 \beta_{14} - 1473625 \beta_{13} + 1428221 \beta_{12} + 1439528 \beta_{11} + \cdots + 2758914 \) 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
3.36926
3.07129
2.93924
2.30769
1.62428
1.56513
0.720900
0.686530
−0.0595607
−0.631998
−0.764364
−0.853232
−2.02783
−2.24082
−2.71029
−2.99623
0 −3.36926 0 0 0 −0.794375 0 8.35194 0
1.2 0 −3.07129 0 0 0 −4.49756 0 6.43283 0
1.3 0 −2.93924 0 0 0 3.13612 0 5.63915 0
1.4 0 −2.30769 0 0 0 −4.74404 0 2.32542 0
1.5 0 −1.62428 0 0 0 −2.51754 0 −0.361706 0
1.6 0 −1.56513 0 0 0 −0.0338937 0 −0.550359 0
1.7 0 −0.720900 0 0 0 4.42421 0 −2.48030 0
1.8 0 −0.686530 0 0 0 3.54998 0 −2.52868 0
1.9 0 0.0595607 0 0 0 −1.71675 0 −2.99645 0
1.10 0 0.631998 0 0 0 2.09441 0 −2.60058 0
1.11 0 0.764364 0 0 0 −4.32704 0 −2.41575 0
1.12 0 0.853232 0 0 0 −1.47738 0 −2.27199 0
1.13 0 2.02783 0 0 0 0.498275 0 1.11208 0
1.14 0 2.24082 0 0 0 1.85909 0 2.02129 0
1.15 0 2.71029 0 0 0 0.760910 0 4.34569 0
1.16 0 2.99623 0 0 0 −4.21442 0 5.97742 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
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.bq 16
4.b odd 2 1 5000.2.a.r 16
5.b even 2 1 10000.2.a.br 16
20.d odd 2 1 5000.2.a.q 16
25.f odd 20 2 400.2.y.d 32
100.h odd 10 2 1000.2.m.e 32
100.j odd 10 2 1000.2.m.d 32
100.l even 20 2 200.2.q.a ✓ 32
100.l even 20 2 1000.2.q.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.q.a ✓ 32 100.l even 20 2
400.2.y.d 32 25.f odd 20 2
1000.2.m.d 32 100.j odd 10 2
1000.2.m.e 32 100.h odd 10 2
1000.2.q.c 32 100.l even 20 2
5000.2.a.q 16 20.d odd 2 1
5000.2.a.r 16 4.b odd 2 1
10000.2.a.bq 16 1.a even 1 1 trivial
10000.2.a.br 16 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(10000))\):

\( T_{3}^{16} + 4 T_{3}^{15} - 26 T_{3}^{14} - 110 T_{3}^{13} + 250 T_{3}^{12} + 1154 T_{3}^{11} + \cdots + 80 \) Copy content Toggle raw display
\( T_{7}^{16} + 8 T_{7}^{15} - 39 T_{7}^{14} - 416 T_{7}^{13} + 376 T_{7}^{12} + 8084 T_{7}^{11} + \cdots - 4864 \) Copy content Toggle raw display
\( T_{11}^{16} - 12 T_{11}^{15} - 31 T_{11}^{14} + 794 T_{11}^{13} - 756 T_{11}^{12} - 18090 T_{11}^{11} + \cdots - 112384 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 4 T^{15} + \cdots + 80 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 8 T^{15} + \cdots - 4864 \) Copy content Toggle raw display
$11$ \( T^{16} - 12 T^{15} + \cdots - 112384 \) Copy content Toggle raw display
$13$ \( T^{16} - 10 T^{15} + \cdots - 9030899 \) Copy content Toggle raw display
$17$ \( T^{16} - 8 T^{15} + \cdots + 6010000 \) Copy content Toggle raw display
$19$ \( T^{16} - 12 T^{15} + \cdots + 2808976 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 6486139136 \) Copy content Toggle raw display
$29$ \( T^{16} - 16 T^{15} + \cdots + 35290081 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 2080762000 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 421220021 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 48434041616 \) Copy content Toggle raw display
$43$ \( T^{16} + 26 T^{15} + \cdots + 10244096 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots - 2949551104 \) Copy content Toggle raw display
$53$ \( T^{16} - 16 T^{15} + \cdots + 91170496 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots - 2071241191424 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots - 57875079875 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 10874045696 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots - 17821854244720 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 467056954256 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots - 39769573120 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots - 5160481904 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots - 47559207352064 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots - 42316001699 \) Copy content Toggle raw display
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