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Properties

Label 5292.2.bm.a
Level $5292$
Weight $2$
Character orbit 5292.bm
Analytic conductor $42.257$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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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: \( x^{16} - 2 x^{15} + 5 x^{14} - 17 x^{13} + 22 x^{12} - 31 x^{11} + 62 x^{10} - 52 x^{9} + 52 x^{8} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3^{6} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{11} - \beta_{9}) q^{5} + ( - \beta_{14} - \beta_{10} + \cdots + \beta_{2}) q^{11} + ( - \beta_{13} - \beta_{9} - \beta_1) q^{13} + (\beta_{9} + \beta_{8} + \cdots - 2 \beta_{2}) q^{17} + (\beta_{15} - 2 \beta_{12} - \beta_{11} + \cdots + 1) q^{19}+ \cdots + ( - 2 \beta_{15} - \beta_{14} + \cdots - 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 3 q^{13} - 9 q^{17} + 16 q^{25} - 6 q^{29} - 6 q^{31} + q^{37} + 6 q^{41} - 2 q^{43} - 18 q^{47} - 15 q^{59} - 3 q^{61} + 39 q^{65} - 7 q^{67} - q^{79} + 6 q^{85} - 21 q^{89} - 6 q^{95} - 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 5 x^{14} - 17 x^{13} + 22 x^{12} - 31 x^{11} + 62 x^{10} - 52 x^{9} + 52 x^{8} + \cdots + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 19 \nu^{15} + 139 \nu^{14} + 1928 \nu^{13} + 8221 \nu^{12} + 10009 \nu^{11} + 14762 \nu^{10} + \cdots + 4788801 ) / 103518 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1307 \nu^{15} - 5068 \nu^{14} - 824 \nu^{13} - 49267 \nu^{12} - 2716 \nu^{11} - 77018 \nu^{10} + \cdots - 19166868 ) / 621108 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3695 \nu^{15} + 20725 \nu^{14} - 51544 \nu^{13} + 99223 \nu^{12} - 215537 \nu^{11} + \cdots + 46300977 ) / 1242216 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 6677 \nu^{15} + 21577 \nu^{14} - 16612 \nu^{13} + 91129 \nu^{12} - 145673 \nu^{11} + \cdots - 5872095 ) / 1242216 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1669 \nu^{15} + 1930 \nu^{14} + 23477 \nu^{13} + 5248 \nu^{12} + 44857 \nu^{11} - 82192 \nu^{10} + \cdots - 4533651 ) / 310554 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2739 \nu^{15} - 9229 \nu^{14} + 1724 \nu^{13} - 88319 \nu^{12} - 8827 \nu^{11} - 147394 \nu^{10} + \cdots - 35170605 ) / 414072 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 8405 \nu^{15} - 16535 \nu^{14} - 52096 \nu^{13} - 27863 \nu^{12} - 20561 \nu^{11} + \cdots - 11234619 ) / 621108 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4934 \nu^{15} + 6616 \nu^{14} - 21253 \nu^{13} + 86401 \nu^{12} - 24581 \nu^{11} + \cdots + 34596153 ) / 310554 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13862 \nu^{15} - 1333 \nu^{14} + 45760 \nu^{13} - 164782 \nu^{12} - 15775 \nu^{11} + \cdots - 61443765 ) / 621108 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 27959 \nu^{15} + 45263 \nu^{14} + 129088 \nu^{13} - 58111 \nu^{12} - 95755 \nu^{11} + \cdots - 47062053 ) / 1242216 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 16952 \nu^{15} + 9175 \nu^{14} - 73804 \nu^{13} + 123904 \nu^{12} - 128807 \nu^{11} + \cdots + 23648031 ) / 621108 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 19793 \nu^{15} - 1430 \nu^{14} - 108688 \nu^{13} + 147133 \nu^{12} - 121322 \nu^{11} + \cdots + 66410442 ) / 621108 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 3656 \nu^{15} - 865 \nu^{14} + 15076 \nu^{13} - 31234 \nu^{12} + 13505 \nu^{11} - 75428 \nu^{10} + \cdots - 10890531 ) / 103518 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 27554 \nu^{15} + 18943 \nu^{14} - 80794 \nu^{13} + 318436 \nu^{12} - 60809 \nu^{11} + \cdots + 92446677 ) / 621108 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 23045 \nu^{15} + 19169 \nu^{14} - 116460 \nu^{13} + 283569 \nu^{12} - 185713 \nu^{11} + \cdots + 111152817 ) / 414072 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - \beta_{12} - \beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{7} - \beta_{6} + 2 \beta_{4} + \cdots + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} - \beta_{13} - 2\beta_{11} + \beta_{10} - \beta_{8} - \beta_{5} + \beta_{4} - 3\beta_{3} - \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 4 \beta_{15} + 4 \beta_{14} + \beta_{13} + \beta_{12} + \beta_{10} - 4 \beta_{9} + \beta_{8} + \cdots - 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2 \beta_{15} + 3 \beta_{14} + 4 \beta_{13} + 4 \beta_{12} - \beta_{11} + 5 \beta_{9} + 2 \beta_{8} + \cdots + 10 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6 \beta_{15} - 14 \beta_{14} - 2 \beta_{13} - \beta_{12} + \beta_{11} - 10 \beta_{10} - 5 \beta_{8} + \cdots - 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 5 \beta_{15} + 9 \beta_{14} - 10 \beta_{13} - 13 \beta_{12} - 13 \beta_{11} + 3 \beta_{10} - 6 \beta_{9} + \cdots + 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 12 \beta_{15} - 9 \beta_{14} + 2 \beta_{13} + 10 \beta_{12} - 15 \beta_{11} + 2 \beta_{10} - 15 \beta_{9} + \cdots + 60 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 61 \beta_{15} - 25 \beta_{14} + 25 \beta_{13} - 41 \beta_{12} + 6 \beta_{11} - 46 \beta_{10} + 54 \beta_{9} + \cdots + 15 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 56 \beta_{15} + 13 \beta_{14} - 67 \beta_{13} + 14 \beta_{12} + 7 \beta_{11} + 118 \beta_{10} + \cdots - 81 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 185 \beta_{15} + 207 \beta_{14} + 14 \beta_{13} + 99 \beta_{12} - 61 \beta_{11} + 112 \beta_{10} + \cdots - 632 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 7 \beta_{15} - 106 \beta_{14} + 234 \beta_{13} + 286 \beta_{12} + 274 \beta_{11} + 48 \beta_{10} + \cdots - 63 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 144 \beta_{15} - 439 \beta_{14} + 36 \beta_{13} - 147 \beta_{12} + 673 \beta_{11} - 220 \beta_{10} + \cdots - 502 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 373 \beta_{15} + 399 \beta_{14} - 651 \beta_{13} - 44 \beta_{12} - 632 \beta_{11} + 544 \beta_{10} + \cdots - 517 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 1073 \beta_{15} - 1398 \beta_{14} - 11 \beta_{13} + 363 \beta_{12} + 530 \beta_{11} - 367 \beta_{10} + \cdots - 1014 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1627 \beta_{15} - 544 \beta_{14} + 1472 \beta_{13} - 2368 \beta_{12} + 2547 \beta_{11} - 361 \beta_{10} + \cdots + 1741 ) / 3 \) Copy content Toggle raw display

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

Character values

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

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

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
−0.544978 − 1.64408i
1.68042 − 0.419752i
−0.268067 + 1.71118i
1.08696 − 1.34852i
1.68124 + 0.416458i
−1.61108 − 0.635951i
−0.811340 − 1.53027i
−0.213160 + 1.71888i
−0.544978 + 1.64408i
1.68042 + 0.419752i
−0.268067 − 1.71118i
1.08696 + 1.34852i
1.68124 − 0.416458i
−1.61108 + 0.635951i
−0.811340 + 1.53027i
−0.213160 − 1.71888i
0 0 0 −3.91482 0 0 0 0 0
2285.2 0 0 0 −2.96988 0 0 0 0 0
2285.3 0 0 0 −1.68574 0 0 0 0 0
2285.4 0 0 0 0.0764245 0 0 0 0 0
2285.5 0 0 0 0.699656 0 0 0 0 0
2285.6 0 0 0 2.18300 0 0 0 0 0
2285.7 0 0 0 2.74332 0 0 0 0 0
2285.8 0 0 0 2.86804 0 0 0 0 0
4625.1 0 0 0 −3.91482 0 0 0 0 0
4625.2 0 0 0 −2.96988 0 0 0 0 0
4625.3 0 0 0 −1.68574 0 0 0 0 0
4625.4 0 0 0 0.0764245 0 0 0 0 0
4625.5 0 0 0 0.699656 0 0 0 0 0
4625.6 0 0 0 2.18300 0 0 0 0 0
4625.7 0 0 0 2.74332 0 0 0 0 0
4625.8 0 0 0 2.86804 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
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.a 16
3.b odd 2 1 1764.2.bm.a 16
7.b odd 2 1 756.2.bm.a 16
7.c even 3 1 756.2.w.a 16
7.c even 3 1 5292.2.x.a 16
7.d odd 6 1 5292.2.w.b 16
7.d odd 6 1 5292.2.x.b 16
9.c even 3 1 1764.2.w.b 16
9.d odd 6 1 5292.2.w.b 16
21.c even 2 1 252.2.bm.a yes 16
21.g even 6 1 1764.2.w.b 16
21.g even 6 1 1764.2.x.b 16
21.h odd 6 1 252.2.w.a ✓ 16
21.h odd 6 1 1764.2.x.a 16
28.d even 2 1 3024.2.df.d 16
28.g odd 6 1 3024.2.ca.d 16
63.g even 3 1 252.2.bm.a yes 16
63.h even 3 1 1764.2.x.b 16
63.h even 3 1 2268.2.t.a 16
63.i even 6 1 5292.2.x.a 16
63.j odd 6 1 2268.2.t.b 16
63.j odd 6 1 5292.2.x.b 16
63.k odd 6 1 1764.2.bm.a 16
63.l odd 6 1 252.2.w.a ✓ 16
63.l odd 6 1 2268.2.t.b 16
63.n odd 6 1 756.2.bm.a 16
63.o even 6 1 756.2.w.a 16
63.o even 6 1 2268.2.t.a 16
63.s even 6 1 inner 5292.2.bm.a 16
63.t odd 6 1 1764.2.x.a 16
84.h odd 2 1 1008.2.df.d 16
84.n even 6 1 1008.2.ca.d 16
252.o even 6 1 3024.2.df.d 16
252.s odd 6 1 3024.2.ca.d 16
252.bi even 6 1 1008.2.ca.d 16
252.bl odd 6 1 1008.2.df.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.w.a ✓ 16 21.h odd 6 1
252.2.w.a ✓ 16 63.l odd 6 1
252.2.bm.a yes 16 21.c even 2 1
252.2.bm.a yes 16 63.g even 3 1
756.2.w.a 16 7.c even 3 1
756.2.w.a 16 63.o even 6 1
756.2.bm.a 16 7.b odd 2 1
756.2.bm.a 16 63.n odd 6 1
1008.2.ca.d 16 84.n even 6 1
1008.2.ca.d 16 252.bi even 6 1
1008.2.df.d 16 84.h odd 2 1
1008.2.df.d 16 252.bl odd 6 1
1764.2.w.b 16 9.c even 3 1
1764.2.w.b 16 21.g even 6 1
1764.2.x.a 16 21.h odd 6 1
1764.2.x.a 16 63.t odd 6 1
1764.2.x.b 16 21.g even 6 1
1764.2.x.b 16 63.h even 3 1
1764.2.bm.a 16 3.b odd 2 1
1764.2.bm.a 16 63.k odd 6 1
2268.2.t.a 16 63.h even 3 1
2268.2.t.a 16 63.o even 6 1
2268.2.t.b 16 63.j odd 6 1
2268.2.t.b 16 63.l odd 6 1
3024.2.ca.d 16 28.g odd 6 1
3024.2.ca.d 16 252.s odd 6 1
3024.2.df.d 16 28.d even 2 1
3024.2.df.d 16 252.o even 6 1
5292.2.w.b 16 7.d odd 6 1
5292.2.w.b 16 9.d odd 6 1
5292.2.x.a 16 7.c even 3 1
5292.2.x.a 16 63.i even 6 1
5292.2.x.b 16 7.d odd 6 1
5292.2.x.b 16 63.j odd 6 1
5292.2.bm.a 16 1.a even 1 1 trivial
5292.2.bm.a 16 63.s even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 24T_{5}^{6} + 12T_{5}^{5} + 171T_{5}^{4} - 135T_{5}^{3} - 324T_{5}^{2} + 261T_{5} - 18 \) 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
$5$ \( (T^{8} - 24 T^{6} + \cdots - 18)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} + 90 T^{14} + \cdots + 26244 \) Copy content Toggle raw display
$13$ \( T^{16} + 3 T^{15} + \cdots + 3337929 \) Copy content Toggle raw display
$17$ \( T^{16} + 9 T^{15} + \cdots + 13549761 \) Copy content Toggle raw display
$19$ \( T^{16} - 93 T^{14} + \cdots + 2099601 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 15198451524 \) Copy content Toggle raw display
$29$ \( T^{16} + 6 T^{15} + \cdots + 15752961 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 3910251024 \) Copy content Toggle raw display
$37$ \( T^{16} - T^{15} + \cdots + 52765696 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 91647269289 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 28009034881 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 1971620372736 \) Copy content Toggle raw display
$53$ \( T^{16} - 153 T^{14} + \cdots + 531441 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 165574120464 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 1475481744 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 2114953586944 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 780959242139904 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 7523023152969 \) Copy content Toggle raw display
$79$ \( T^{16} + T^{15} + \cdots + 10549504 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 669184533369 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 7161826993281 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 22864161681 \) Copy content Toggle raw display
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