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Properties

Label 500.4.c.c
Level $500$
Weight $4$
Character orbit 500.c
Analytic conductor $29.501$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [500,4,Mod(249,500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(500, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("500.249");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 500 = 2^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 500.c (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(29.5009550029\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 132x^{10} + 6524x^{8} + 151506x^{6} + 1711604x^{4} + 8580972x^{2} + 13184161 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{3} - \beta_1) q^{7} + ( - \beta_{9} - \beta_{4} - 10) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + (\beta_{3} - \beta_1) q^{7} + ( - \beta_{9} - \beta_{4} - 10) q^{9} + (\beta_{8} - \beta_{4} - 2) q^{11} + ( - \beta_{5} + \beta_{2} + \beta_1) q^{13} + ( - \beta_{6} - 2 \beta_{3} + \cdots + \beta_1) q^{17}+ \cdots + (11 \beta_{11} + \beta_{10} + \cdots + 418) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 128 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 128 q^{9} - 30 q^{11} + 42 q^{19} - 368 q^{21} + 172 q^{29} + 492 q^{31} + 712 q^{39} - 938 q^{41} - 906 q^{49} + 224 q^{51} - 1526 q^{59} + 1044 q^{61} + 2016 q^{69} + 956 q^{71} - 2144 q^{79} + 868 q^{81} + 918 q^{89} + 1528 q^{91} + 5150 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 132x^{10} + 6524x^{8} + 151506x^{6} + 1711604x^{4} + 8580972x^{2} + 13184161 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1542287 \nu^{11} + 185877521 \nu^{9} + 7791699159 \nu^{7} + 130602268971 \nu^{5} + \cdots - 206270302637 \nu ) / 286095494880 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -125\nu^{11} - 9275\nu^{9} - 40725\nu^{7} + 9010575\nu^{5} + 173542025\nu^{3} + 707981975\nu ) / 20391696 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1008893 \nu^{11} - 137204699 \nu^{9} - 6940761021 \nu^{7} - 157886039889 \nu^{5} + \cdots - 4317862949137 \nu ) / 95365164960 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -157\nu^{10} - 16891\nu^{8} - 570069\nu^{6} - 5728401\nu^{4} + 7136281\nu^{2} + 144721975 ) / 15758496 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8521607 \nu^{11} + 1036938941 \nu^{9} + 44923274739 \nu^{7} + 831724450371 \nu^{5} + \cdots + 11426419630303 \nu ) / 143047747440 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1988521 \nu^{11} - 239355423 \nu^{9} - 10200769617 \nu^{7} - 184131999213 \nu^{5} + \cdots - 2741430806709 \nu ) / 31788388320 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 9212371 \nu^{11} - 1094951893 \nu^{9} - 45856789947 \nu^{7} - 809962764543 \nu^{5} + \cdots - 12021765369479 \nu ) / 95365164960 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2989531 \nu^{10} - 382380293 \nu^{8} - 17425066107 \nu^{6} - 333780906183 \nu^{4} + \cdots - 2189634786919 ) / 31788388320 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4639339 \nu^{10} + 586739542 \nu^{8} + 26447328783 \nu^{6} + 504446769927 \nu^{4} + \cdots + 7032081960971 ) / 35761936860 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 293875 \nu^{10} - 33820525 \nu^{8} - 1355688135 \nu^{6} - 22297256415 \nu^{4} + \cdots - 198614352899 ) / 733578192 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 48502241 \nu^{10} - 5798175623 \nu^{8} - 244978262577 \nu^{6} - 4389368531013 \nu^{4} + \cdots - 63436310602549 ) / 47682582480 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -5\beta_{7} + 5\beta_{6} - 5\beta_{5} - 5\beta_{3} + \beta_{2} + 15\beta_1 ) / 50 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - \beta_{10} + 17\beta_{9} + 17\beta_{8} - \beta_{4} - 1103 ) / 50 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 85\beta_{7} - 137\beta_{6} + 28\beta_{5} + 70\beta_{3} - 10\beta_{2} - 250\beta_1 ) / 25 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -96\beta_{11} + 133\beta_{10} - 915\beta_{9} - 771\beta_{8} - 183\beta_{4} + 36787 ) / 50 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -7200\beta_{7} + 12430\beta_{6} - 695\beta_{5} - 3925\beta_{3} + 1566\beta_{2} + 16975\beta_1 ) / 50 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2660\beta_{11} - 4284\beta_{10} + 20816\beta_{9} + 16488\beta_{8} + 16088\beta_{4} - 726253 ) / 25 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 318425\beta_{7} - 546193\beta_{6} + 8377\beta_{5} + 95585\beta_{3} - 108893\beta_{2} - 660795\beta_1 ) / 50 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -260733\beta_{11} + 475149\beta_{10} - 1805925\beta_{9} - 1428573\beta_{8} - 2515539\beta_{4} + 61578851 ) / 50 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 7133725 \beta_{7} + 11965789 \beta_{6} - 63256 \beta_{5} - 712210 \beta_{3} + 3330326 \beta_{2} + 14105970 \beta_1 ) / 25 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 12282400 \beta_{11} - 24907033 \beta_{10} + 77283887 \beta_{9} + 62862071 \beta_{8} + 155418251 \beta_{4} - 2697233351 ) / 50 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 643590680 \beta_{7} - 1053846670 \beta_{6} + 5986715 \beta_{5} - 42333895 \beta_{3} + \cdots - 1263613115 \beta_1 ) / 50 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/500\mathbb{Z}\right)^\times\).

\(n\) \(251\) \(377\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
249.1
4.55540i
3.71831i
− 1.62104i
6.84581i
− 3.12750i
− 6.17644i
6.17644i
3.12750i
− 6.84581i
1.62104i
− 3.71831i
− 4.55540i
0 − 9.26390i 0 0 0 − 34.7650i 0 −58.8198 0
249.2 0 − 7.96155i 0 0 0 − 0.0689265i 0 −36.3863 0
249.3 0 − 7.11328i 0 0 0 15.3866i 0 −23.5988 0
249.4 0 − 4.45839i 0 0 0 14.7973i 0 7.12278 0
249.5 0 − 2.26710i 0 0 0 − 24.0678i 0 21.8603 0
249.6 0 − 1.08545i 0 0 0 16.3532i 0 25.8218 0
249.7 0 1.08545i 0 0 0 − 16.3532i 0 25.8218 0
249.8 0 2.26710i 0 0 0 24.0678i 0 21.8603 0
249.9 0 4.45839i 0 0 0 − 14.7973i 0 7.12278 0
249.10 0 7.11328i 0 0 0 − 15.3866i 0 −23.5988 0
249.11 0 7.96155i 0 0 0 0.0689265i 0 −36.3863 0
249.12 0 9.26390i 0 0 0 34.7650i 0 −58.8198 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 249.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 500.4.c.c 12
5.b even 2 1 inner 500.4.c.c 12
5.c odd 4 1 500.4.a.b ✓ 6
5.c odd 4 1 500.4.a.c yes 6
20.e even 4 1 2000.4.a.l 6
20.e even 4 1 2000.4.a.m 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
500.4.a.b ✓ 6 5.c odd 4 1
500.4.a.c yes 6 5.c odd 4 1
500.4.c.c 12 1.a even 1 1 trivial
500.4.c.c 12 5.b even 2 1 inner
2000.4.a.l 6 20.e even 4 1
2000.4.a.m 6 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 226T_{3}^{10} + 18355T_{3}^{8} + 641930T_{3}^{6} + 8944105T_{3}^{4} + 37796716T_{3}^{2} + 33131536 \) acting on \(S_{4}^{\mathrm{new}}(500, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 226 T^{10} + \cdots + 33131536 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 46108543441 \) Copy content Toggle raw display
$11$ \( (T^{6} + 15 T^{5} + \cdots + 730959875)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 92\!\cdots\!41 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 27\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( (T^{6} - 21 T^{5} + \cdots - 1220233819)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 62\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 11561139063884)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 10595036078116)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 2501624923189)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 59\!\cdots\!21 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 15\!\cdots\!61 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 623930671754649)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 34\!\cdots\!76)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 23\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 870909015303524)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 12472751663244)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 49\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 30\!\cdots\!24)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
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