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Properties

Label 5292.2.bm.b
Level $5292$
Weight $2$
Character orbit 5292.bm
Analytic conductor $42.257$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5292,2,Mod(2285,5292)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5292, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5292.2285");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5292.bm (of order \(6\), degree \(2\), 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: \(42.2568327497\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3x^{14} - 9x^{12} - 9x^{10} + 225x^{8} - 81x^{6} - 729x^{4} - 2187x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{37}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{15} - \beta_1) q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{15} - \beta_1) q^{5} - \beta_{4} q^{11} + ( - \beta_{13} + \beta_{12} + \cdots - \beta_{3}) q^{13}+ \cdots + (\beta_{15} + 2 \beta_{12} + \cdots + \beta_{3}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{25} + 12 q^{29} - 2 q^{37} + 4 q^{43} + 36 q^{53} - 24 q^{65} + 14 q^{67} + 20 q^{79} + 6 q^{85} - 60 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3x^{14} - 9x^{12} - 9x^{10} + 225x^{8} - 81x^{6} - 729x^{4} - 2187x^{2} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{15} + 15\nu^{13} + 72\nu^{11} + 153\nu^{9} - 423\nu^{7} - 891\nu^{5} + 1944\nu^{3} + 17496\nu ) / 15309 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{14} - 12\nu^{12} + 18\nu^{10} + 369\nu^{8} - 153\nu^{6} - 1782\nu^{4} - 4617\nu^{2} + 9477 ) / 5103 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{15} - 9\nu^{13} + 45\nu^{11} - 117\nu^{9} + 90\nu^{7} - 864\nu^{5} + 5184\nu^{3} - 9477\nu ) / 5103 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{14} + 3\nu^{12} + 9\nu^{10} + 9\nu^{8} - 225\nu^{6} + 81\nu^{4} + 2187 ) / 729 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{14} + 6\nu^{12} - 36\nu^{10} - 90\nu^{8} + 144\nu^{6} + 1215\nu^{4} - 729\nu^{2} - 4374 ) / 729 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13\nu^{14} + 6\nu^{12} - 9\nu^{10} - 279\nu^{8} - 396\nu^{6} + 324\nu^{4} + 11664\nu^{2} + 13122 ) / 5103 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{15} - 12\nu^{13} + 18\nu^{11} + 369\nu^{9} - 153\nu^{7} - 1782\nu^{5} - 4617\nu^{3} + 4374\nu ) / 5103 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -19\nu^{14} + 30\nu^{12} + 144\nu^{10} + 495\nu^{8} - 1602\nu^{6} - 648\nu^{4} - 4617\nu^{2} + 14580 ) / 5103 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 23\nu^{14} - 33\nu^{12} - 234\nu^{10} - 1017\nu^{8} + 3879\nu^{6} + 8424\nu^{4} + 2187\nu^{2} - 82377 ) / 5103 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -3\nu^{15} + 4\nu^{13} + 15\nu^{11} + 45\nu^{9} - 306\nu^{7} + 90\nu^{5} + 405\nu^{3} + 1944\nu ) / 1701 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -10\nu^{15} - 24\nu^{13} + 36\nu^{11} + 738\nu^{9} - 306\nu^{7} - 3564\nu^{5} - 9234\nu^{3} + 24057\nu ) / 5103 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -11\nu^{15} + 3\nu^{13} + 27\nu^{11} + 207\nu^{9} - 261\nu^{7} - 27\nu^{5} - 2673\nu^{3} - 3645\nu ) / 5103 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 34\nu^{15} + 69\nu^{13} - 576\nu^{11} - 1602\nu^{9} + 3195\nu^{7} + 16767\nu^{5} - 3645\nu^{3} - 78732\nu ) / 15309 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -47\nu^{14} - 12\nu^{12} + 396\nu^{10} + 1314\nu^{8} - 3366\nu^{6} - 8019\nu^{4} - 2916\nu^{2} + 34992 ) / 5103 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 62\nu^{15} - 15\nu^{13} - 639\nu^{11} - 2421\nu^{9} + 7794\nu^{7} + 17901\nu^{5} + 3159\nu^{3} - 139968\nu ) / 15309 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - 2\beta_{7} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{14} + 2\beta_{6} + \beta_{5} - 2\beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} + 2\beta_{3} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{8} + \beta_{6} + \beta_{4} + 2\beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{15} - 3\beta_{13} - \beta_{12} + 5\beta_{10} - 3\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{8} - 6\beta_{4} - 3\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3\beta_{13} + 6\beta_{12} + 6\beta_{11} - 6\beta_{10} - 9\beta_{7} + 6\beta_{3} + 15\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3\beta_{9} + 12\beta_{8} + 15\beta_{6} + 3\beta_{5} - 9\beta_{4} + 24\beta_{2} - 24 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 9\beta_{15} + 9\beta_{13} - 18\beta_{12} + 6\beta_{11} + 27\beta_{10} + 33\beta_{7} + 18\beta_{3} + 54\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -21\beta_{14} + 18\beta_{9} + 45\beta_{8} - 15\beta_{6} - 39\beta_{5} + 24\beta_{4} - 18\beta_{2} + 72 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 54\beta_{15} - 117\beta_{13} + 54\beta_{12} - 72\beta_{11} + 9\beta_{10} + 18\beta_{7} - 126\beta_{3} + 9\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -36\beta_{14} - 63\beta_{9} + 126\beta_{8} + 18\beta_{5} - 180\beta_{4} - 153\beta_{2} - 198 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 189 \beta_{15} + 270 \beta_{13} + 198 \beta_{12} - 27 \beta_{11} - 342 \beta_{10} + 27 \beta_{7} + \cdots + 567 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -297\beta_{14} + 81\beta_{9} + 297\beta_{8} + 81\beta_{6} - 270\beta_{5} + 297\beta_{4} + 432\beta_{2} - 1026 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 243 \beta_{15} - 351 \beta_{13} - 513 \beta_{12} - 567 \beta_{11} + 324 \beta_{10} + 1107 \beta_{7} + \cdots + 351 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5292\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(2647\)
\(\chi(n)\) \(1 + \beta_{2}\) \(-\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2285.1
1.71965 + 0.206851i
−1.69483 + 0.357142i
−0.744857 + 1.56371i
0.604587 + 1.62311i
−0.604587 − 1.62311i
0.744857 − 1.56371i
1.69483 − 0.357142i
−1.71965 − 0.206851i
1.71965 − 0.206851i
−1.69483 − 0.357142i
−0.744857 − 1.56371i
0.604587 − 1.62311i
−0.604587 + 1.62311i
0.744857 + 1.56371i
1.69483 + 0.357142i
−1.71965 + 0.206851i
0 0 0 −4.18671 0 0 0 0 0
2285.2 0 0 0 −2.42488 0 0 0 0 0
2285.3 0 0 0 −0.553827 0 0 0 0 0
2285.4 0 0 0 −0.533560 0 0 0 0 0
2285.5 0 0 0 0.533560 0 0 0 0 0
2285.6 0 0 0 0.553827 0 0 0 0 0
2285.7 0 0 0 2.42488 0 0 0 0 0
2285.8 0 0 0 4.18671 0 0 0 0 0
4625.1 0 0 0 −4.18671 0 0 0 0 0
4625.2 0 0 0 −2.42488 0 0 0 0 0
4625.3 0 0 0 −0.553827 0 0 0 0 0
4625.4 0 0 0 −0.533560 0 0 0 0 0
4625.5 0 0 0 0.533560 0 0 0 0 0
4625.6 0 0 0 0.553827 0 0 0 0 0
4625.7 0 0 0 2.42488 0 0 0 0 0
4625.8 0 0 0 4.18671 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2285.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
63.n odd 6 1 inner
63.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5292.2.bm.b 16
3.b odd 2 1 1764.2.bm.b 16
7.b odd 2 1 inner 5292.2.bm.b 16
7.c even 3 1 756.2.x.a 16
7.c even 3 1 5292.2.w.a 16
7.d odd 6 1 756.2.x.a 16
7.d odd 6 1 5292.2.w.a 16
9.c even 3 1 1764.2.w.a 16
9.d odd 6 1 5292.2.w.a 16
21.c even 2 1 1764.2.bm.b 16
21.g even 6 1 252.2.x.a ✓ 16
21.g even 6 1 1764.2.w.a 16
21.h odd 6 1 252.2.x.a ✓ 16
21.h odd 6 1 1764.2.w.a 16
28.f even 6 1 3024.2.cc.c 16
28.g odd 6 1 3024.2.cc.c 16
63.g even 3 1 1764.2.bm.b 16
63.g even 3 1 2268.2.f.b 16
63.h even 3 1 252.2.x.a ✓ 16
63.i even 6 1 756.2.x.a 16
63.j odd 6 1 756.2.x.a 16
63.k odd 6 1 1764.2.bm.b 16
63.k odd 6 1 2268.2.f.b 16
63.l odd 6 1 1764.2.w.a 16
63.n odd 6 1 2268.2.f.b 16
63.n odd 6 1 inner 5292.2.bm.b 16
63.o even 6 1 5292.2.w.a 16
63.s even 6 1 2268.2.f.b 16
63.s even 6 1 inner 5292.2.bm.b 16
63.t odd 6 1 252.2.x.a ✓ 16
84.j odd 6 1 1008.2.cc.c 16
84.n even 6 1 1008.2.cc.c 16
252.r odd 6 1 3024.2.cc.c 16
252.u odd 6 1 1008.2.cc.c 16
252.bb even 6 1 3024.2.cc.c 16
252.bj even 6 1 1008.2.cc.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.x.a ✓ 16 21.g even 6 1
252.2.x.a ✓ 16 21.h odd 6 1
252.2.x.a ✓ 16 63.h even 3 1
252.2.x.a ✓ 16 63.t odd 6 1
756.2.x.a 16 7.c even 3 1
756.2.x.a 16 7.d odd 6 1
756.2.x.a 16 63.i even 6 1
756.2.x.a 16 63.j odd 6 1
1008.2.cc.c 16 84.j odd 6 1
1008.2.cc.c 16 84.n even 6 1
1008.2.cc.c 16 252.u odd 6 1
1008.2.cc.c 16 252.bj even 6 1
1764.2.w.a 16 9.c even 3 1
1764.2.w.a 16 21.g even 6 1
1764.2.w.a 16 21.h odd 6 1
1764.2.w.a 16 63.l odd 6 1
1764.2.bm.b 16 3.b odd 2 1
1764.2.bm.b 16 21.c even 2 1
1764.2.bm.b 16 63.g even 3 1
1764.2.bm.b 16 63.k odd 6 1
2268.2.f.b 16 63.g even 3 1
2268.2.f.b 16 63.k odd 6 1
2268.2.f.b 16 63.n odd 6 1
2268.2.f.b 16 63.s even 6 1
3024.2.cc.c 16 28.f even 6 1
3024.2.cc.c 16 28.g odd 6 1
3024.2.cc.c 16 252.r odd 6 1
3024.2.cc.c 16 252.bb even 6 1
5292.2.w.a 16 7.c even 3 1
5292.2.w.a 16 7.d odd 6 1
5292.2.w.a 16 9.d odd 6 1
5292.2.w.a 16 63.o even 6 1
5292.2.bm.b 16 1.a even 1 1 trivial
5292.2.bm.b 16 7.b odd 2 1 inner
5292.2.bm.b 16 63.n odd 6 1 inner
5292.2.bm.b 16 63.s even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 24T_{5}^{6} + 117T_{5}^{4} - 63T_{5}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(5292, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 24 T^{6} + 117 T^{4} + \cdots + 9)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + 45 T^{6} + \cdots + 3969)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} - 48 T^{14} + \cdots + 81 \) Copy content Toggle raw display
$17$ \( T^{16} + 78 T^{14} + \cdots + 810000 \) Copy content Toggle raw display
$19$ \( T^{16} - 75 T^{14} + \cdots + 810000 \) Copy content Toggle raw display
$23$ \( (T^{8} + 81 T^{6} + \cdots + 50625)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 6 T^{7} + \cdots + 245025)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} - 72 T^{14} + \cdots + 531441 \) Copy content Toggle raw display
$37$ \( (T^{8} + T^{7} + \cdots + 372100)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 331869318561 \) Copy content Toggle raw display
$43$ \( (T^{8} - 2 T^{7} + \cdots + 461041)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 24685405970481 \) Copy content Toggle raw display
$53$ \( (T^{8} - 18 T^{7} + \cdots + 41990400)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + 96 T^{14} + \cdots + 194481 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 12\!\cdots\!01 \) Copy content Toggle raw display
$67$ \( (T^{8} - 7 T^{7} + \cdots + 3940225)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 207 T^{6} + \cdots + 15876)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 5802782976 \) Copy content Toggle raw display
$79$ \( (T^{8} - 10 T^{7} + \cdots + 319225)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 30237384321 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 44\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 83955602727441 \) Copy content Toggle raw display
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