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Properties

Label 500.2.g.b
Level $500$
Weight $2$
Character orbit 500.g
Analytic conductor $3.993$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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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] = [500,2,Mod(101,500)] 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("500.101"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(500, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 500 = 2^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 500.g (of order \(5\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.99252010106\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + x^{14} - 4x^{12} - 49x^{10} + 11x^{8} + 395x^{6} + 900x^{4} + 1125x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: no (minimal twist has level 100)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{13} - \beta_{11} - \beta_1) q^{3} + ( - \beta_{15} + \beta_{13} + \cdots - \beta_1) q^{7} + ( - \beta_{9} - \beta_{8} + \cdots - \beta_{2}) q^{9} + ( - \beta_{9} - \beta_{8} + \cdots - \beta_{3}) q^{11}+ \cdots + ( - \beta_{9} + 4 \beta_{5} + \cdots - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{9} + 10 q^{11} + 16 q^{19} - 4 q^{21} + 16 q^{29} - 24 q^{31} - 44 q^{39} + 26 q^{41} - 28 q^{49} - 28 q^{51} - 18 q^{59} + 32 q^{61} + 28 q^{69} - 2 q^{71} + 48 q^{79} - 6 q^{81} + 74 q^{89}+ \cdots - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + x^{14} - 4x^{12} - 49x^{10} + 11x^{8} + 395x^{6} + 900x^{4} + 1125x^{2} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 22849 \nu^{15} - 2422021 \nu^{13} + 6573709 \nu^{11} - 1538146 \nu^{9} + 97097069 \nu^{7} + \cdots - 35093875 \nu ) / 858900625 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 948392 \nu^{14} - 2081693 \nu^{12} - 1547103 \nu^{10} - 35207443 \nu^{8} + 136185777 \nu^{6} + \cdots + 181387875 ) / 171780125 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 122987 \nu^{14} + 97172 \nu^{12} - 701513 \nu^{10} - 5799603 \nu^{8} + 3932417 \nu^{6} + \cdots + 52412250 ) / 15616375 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1379032 \nu^{14} - 312743 \nu^{12} - 4990978 \nu^{10} - 61900293 \nu^{8} + 94590252 \nu^{6} + \cdots + 563878625 ) / 171780125 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 281280 \nu^{14} + 69219 \nu^{12} + 939009 \nu^{10} + 12381889 \nu^{8} - 18275316 \nu^{6} + \cdots - 167096475 ) / 34356025 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1157122 \nu^{15} + 1428203 \nu^{13} + 8678488 \nu^{11} + 43919978 \nu^{9} + \cdots + 846395125 \nu ) / 858900625 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 441862 \nu^{14} + 108648 \nu^{12} + 1544473 \nu^{10} + 20140738 \nu^{8} - 29602957 \nu^{6} + \cdots - 215410900 ) / 34356025 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2817591 \nu^{14} + 2379174 \nu^{12} + 8884104 \nu^{10} + 118381374 \nu^{8} - 262226761 \nu^{6} + \cdots - 860602375 ) / 171780125 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 626638 \nu^{14} + 144621 \nu^{12} + 2783831 \nu^{10} + 27217946 \nu^{8} - 41997039 \nu^{6} + \cdots - 222107450 ) / 34356025 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 257689 \nu^{15} - 260409 \nu^{13} + 1184711 \nu^{11} + 13501191 \nu^{9} - 4440499 \nu^{7} + \cdots - 189940875 \nu ) / 78081875 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3711283 \nu^{15} + 2442853 \nu^{13} - 18411087 \nu^{11} - 175485672 \nu^{9} + \cdots + 1573606875 \nu ) / 858900625 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 11296137 \nu^{15} - 2500148 \nu^{13} - 48798533 \nu^{11} - 491452273 \nu^{9} + \cdots + 4625990250 \nu ) / 858900625 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1033909 \nu^{15} - 651386 \nu^{13} - 3517956 \nu^{11} - 44316636 \nu^{9} + 84515054 \nu^{7} + \cdots + 416073000 \nu ) / 78081875 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1468861 \nu^{15} - 201274 \nu^{13} - 5426879 \nu^{11} - 66369199 \nu^{9} + 90901286 \nu^{7} + \cdots + 722601875 \nu ) / 78081875 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 18299323 \nu^{15} - 8550017 \nu^{13} - 66698807 \nu^{11} - 789938892 \nu^{9} + \cdots + 7417196625 \nu ) / 858900625 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{15} - 5\beta_{14} + 4\beta_{13} + 4\beta_{12} - 5\beta_{10} + 5\beta_{6} + \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{9} + 2\beta_{8} + 3\beta_{7} + 2\beta_{5} + 11\beta_{4} - 4\beta_{3} - \beta_{2} + 4 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -6\beta_{15} + 14\beta_{13} - 6\beta_{12} + 5\beta_{11} - 9\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{9} - 13\beta_{8} + 3\beta_{7} + 7\beta_{5} + 6\beta_{4} - 19\beta_{3} - 26\beta_{2} + 14 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 39\beta_{15} - 20\beta_{14} - 6\beta_{13} - 36\beta_{12} - 30\beta_{10} - 29\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 23\beta_{9} - 23\beta_{8} + 63\beta_{7} + 52\beta_{5} + 156\beta_{4} + 11\beta_{3} - 11\beta_{2} + 144 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 26 \beta_{15} - 190 \beta_{14} + 184 \beta_{13} + 49 \beta_{12} + 190 \beta_{11} - 115 \beta_{10} + \cdots + 66 \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -57\beta_{9} - 18\beta_{8} + 363\beta_{7} - 208\beta_{5} + 381\beta_{4} - 114\beta_{3} - 96\beta_{2} + 39 ) / 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -\beta_{15} + 284\beta_{13} - 286\beta_{12} + 285\beta_{11} + 285\beta_{10} + 190\beta_{6} - 479\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 208\beta_{9} - 688\beta_{8} + 208\beta_{7} - 493\beta_{5} - 149\beta_{4} - 344\beta_{3} - 896\beta_{2} - 606 ) / 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2344 \beta_{15} - 290 \beta_{14} - 1651 \beta_{13} - 1846 \beta_{12} + 195 \beta_{11} - 290 \beta_{10} + \cdots - 624 \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 2208 \beta_{9} - 683 \beta_{8} + 2888 \beta_{7} - 683 \beta_{5} + 3956 \beta_{4} + 2891 \beta_{3} + \cdots + 1784 ) / 5 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 906 \beta_{15} - 4675 \beta_{14} + 2364 \beta_{13} + 3769 \beta_{12} + 7530 \beta_{11} + \cdots + 5166 \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 393 \beta_{9} + 1012 \beta_{8} + 10778 \beta_{7} - 18118 \beta_{5} - 619 \beta_{4} + 1631 \beta_{3} + \cdots - 18511 ) / 5 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1639\beta_{15} + 19130\beta_{14} - 12431\beta_{13} - 9336\beta_{12} + 30920\beta_{10} - 13429\beta_1 ) / 5 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(251\) \(377\)
\(\chi(n)\) \(1\) \(\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
−0.0566033 − 1.17421i
−0.644389 − 0.983224i
0.644389 + 0.983224i
0.0566033 + 1.17421i
−0.917186 + 1.66637i
−1.86824 + 0.357358i
1.86824 − 0.357358i
0.917186 − 1.66637i
−0.917186 − 1.66637i
−1.86824 − 0.357358i
1.86824 + 0.357358i
0.917186 + 1.66637i
−0.0566033 + 1.17421i
−0.644389 + 0.983224i
0.644389 − 0.983224i
0.0566033 − 1.17421i
0 −1.68703 − 1.22570i 0 0 0 −0.909715 0 0.416683 + 1.28242i 0
101.2 0 −0.148189 − 0.107666i 0 0 0 −1.35874 0 −0.916683 − 2.82126i 0
101.3 0 0.148189 + 0.107666i 0 0 0 1.35874 0 −0.916683 − 2.82126i 0
101.4 0 1.68703 + 1.22570i 0 0 0 0.909715 0 0.416683 + 1.28242i 0
201.1 0 −0.713605 − 2.19625i 0 0 0 4.21139 0 −1.88723 + 1.37116i 0
201.2 0 −0.350334 − 1.07822i 0 0 0 −0.768409 0 1.38723 − 1.00788i 0
201.3 0 0.350334 + 1.07822i 0 0 0 0.768409 0 1.38723 − 1.00788i 0
201.4 0 0.713605 + 2.19625i 0 0 0 −4.21139 0 −1.88723 + 1.37116i 0
301.1 0 −0.713605 + 2.19625i 0 0 0 4.21139 0 −1.88723 − 1.37116i 0
301.2 0 −0.350334 + 1.07822i 0 0 0 −0.768409 0 1.38723 + 1.00788i 0
301.3 0 0.350334 − 1.07822i 0 0 0 0.768409 0 1.38723 + 1.00788i 0
301.4 0 0.713605 − 2.19625i 0 0 0 −4.21139 0 −1.88723 − 1.37116i 0
401.1 0 −1.68703 + 1.22570i 0 0 0 −0.909715 0 0.416683 − 1.28242i 0
401.2 0 −0.148189 + 0.107666i 0 0 0 −1.35874 0 −0.916683 + 2.82126i 0
401.3 0 0.148189 − 0.107666i 0 0 0 1.35874 0 −0.916683 + 2.82126i 0
401.4 0 1.68703 − 1.22570i 0 0 0 0.909715 0 0.416683 − 1.28242i 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 101.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
25.d even 5 1 inner
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 500.2.g.b 16
5.b even 2 1 inner 500.2.g.b 16
5.c odd 4 1 100.2.i.a ✓ 8
5.c odd 4 1 500.2.i.a 8
15.e even 4 1 900.2.w.a 8
20.e even 4 1 400.2.y.b 8
25.d even 5 1 inner 500.2.g.b 16
25.d even 5 1 2500.2.a.f 8
25.e even 10 1 inner 500.2.g.b 16
25.e even 10 1 2500.2.a.f 8
25.f odd 20 1 100.2.i.a ✓ 8
25.f odd 20 1 500.2.i.a 8
25.f odd 20 2 2500.2.c.b 8
75.l even 20 1 900.2.w.a 8
100.h odd 10 1 10000.2.a.bi 8
100.j odd 10 1 10000.2.a.bi 8
100.l even 20 1 400.2.y.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.2.i.a ✓ 8 5.c odd 4 1
100.2.i.a ✓ 8 25.f odd 20 1
400.2.y.b 8 20.e even 4 1
400.2.y.b 8 100.l even 20 1
500.2.g.b 16 1.a even 1 1 trivial
500.2.g.b 16 5.b even 2 1 inner
500.2.g.b 16 25.d even 5 1 inner
500.2.g.b 16 25.e even 10 1 inner
500.2.i.a 8 5.c odd 4 1
500.2.i.a 8 25.f odd 20 1
900.2.w.a 8 15.e even 4 1
900.2.w.a 8 75.l even 20 1
2500.2.a.f 8 25.d even 5 1
2500.2.a.f 8 25.e even 10 1
2500.2.c.b 8 25.f odd 20 2
10000.2.a.bi 8 100.h odd 10 1
10000.2.a.bi 8 100.j odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 8T_{3}^{14} + 38T_{3}^{12} + 146T_{3}^{10} + 755T_{3}^{8} + 1246T_{3}^{6} + 863T_{3}^{4} - 17T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(500, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 8 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 21 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 5 T^{7} + \cdots + 400)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 38 T^{14} + \cdots + 14641 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 36136489216 \) Copy content Toggle raw display
$19$ \( (T^{8} - 8 T^{7} + \cdots + 121)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 132 T^{14} + \cdots + 62742241 \) Copy content Toggle raw display
$29$ \( (T^{8} - 8 T^{7} + 63 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 12 T^{7} + \cdots + 6241)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 3262808641 \) Copy content Toggle raw display
$41$ \( (T^{8} - 13 T^{7} + \cdots + 99856)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 125 T^{6} + \cdots + 400)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 5393580481 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 13521270961 \) Copy content Toggle raw display
$59$ \( (T^{8} + 9 T^{7} + \cdots + 167281)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 16 T^{7} + \cdots + 3721)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 1475789056 \) Copy content Toggle raw display
$71$ \( (T^{8} + T^{7} + 312 T^{6} + \cdots + 841)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 428216844946801 \) Copy content Toggle raw display
$79$ \( (T^{8} - 24 T^{7} + \cdots + 12453841)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 6904497224881 \) Copy content Toggle raw display
$89$ \( (T^{8} - 37 T^{7} + \cdots + 205176976)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 91284285426961 \) Copy content Toggle raw display
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