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Properties

Label 5292.2.j.g
Level $5292$
Weight $2$
Character orbit 5292.j
Analytic conductor $42.257$
Analytic rank $0$
Dimension $14$
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] = [5292,2,Mod(1765,5292)]
 
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, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5292.1765");
 
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.j (of order \(3\), 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: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 5x^{12} - 3x^{11} + 7x^{10} + 30x^{9} - 117x^{7} + 270x^{5} + 189x^{4} - 243x^{3} - 1215x^{2} + 2187 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 3^{8} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{5} + (\beta_{13} + \beta_{7}) q^{11} + (\beta_{11} - \beta_{8}) q^{13} + ( - \beta_{13} + \beta_{12} + \cdots - \beta_{5}) q^{17}+ \cdots + ( - \beta_{13} + \beta_{11} - \beta_{10} + \cdots + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 2 q^{5} - 2 q^{11} - 2 q^{13} - 4 q^{17} + 14 q^{19} - 11 q^{23} - 9 q^{25} - q^{29} + q^{31} - 20 q^{37} - 33 q^{41} + 7 q^{43} - 3 q^{47} - 30 q^{53} + 28 q^{55} - 14 q^{59} + 10 q^{61} - 15 q^{65} + 6 q^{67} - 2 q^{71} + 42 q^{73} - 10 q^{79} - 25 q^{83} + 8 q^{85} + 12 q^{89} + 28 q^{95} + 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 5x^{12} - 3x^{11} + 7x^{10} + 30x^{9} - 117x^{7} + 270x^{5} + 189x^{4} - 243x^{3} - 1215x^{2} + 2187 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 35 \nu^{13} + 72 \nu^{12} + 157 \nu^{11} + 312 \nu^{10} - 290 \nu^{9} - 1383 \nu^{8} + \cdots - 3645 ) / 43011 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 26 \nu^{13} + 7 \nu^{12} - 70 \nu^{11} - 266 \nu^{10} - 301 \nu^{9} + 955 \nu^{8} + 846 \nu^{7} + \cdots - 30618 ) / 14337 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 25 \nu^{13} - 294 \nu^{12} - 109 \nu^{11} + 924 \nu^{10} + 1715 \nu^{9} + 351 \nu^{8} + \cdots + 218700 ) / 43011 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 73 \nu^{13} - 360 \nu^{12} - 40 \nu^{11} + 804 \nu^{10} + 1703 \nu^{9} - 417 \nu^{8} + \cdots + 239841 ) / 43011 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 164 \nu^{13} + 138 \nu^{12} + 973 \nu^{11} + 963 \nu^{10} - 1517 \nu^{9} - 5925 \nu^{8} + \cdots - 24786 ) / 43011 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 88 \nu^{13} + 345 \nu^{12} + 476 \nu^{11} + 186 \nu^{10} - 1102 \nu^{9} - 4599 \nu^{8} + \cdots - 173502 ) / 43011 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 142 \nu^{13} - 351 \nu^{12} + 386 \nu^{11} + 1398 \nu^{10} + 869 \nu^{9} - 2478 \nu^{8} + \cdots + 187353 ) / 43011 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 251 \nu^{13} + 141 \nu^{12} - 796 \nu^{11} - 1404 \nu^{10} + 92 \nu^{9} + 6303 \nu^{8} + \cdots - 88209 ) / 43011 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 238 \nu^{13} + 138 \nu^{12} + 326 \nu^{11} + 24 \nu^{10} + 53 \nu^{9} - 1638 \nu^{8} + \cdots - 91854 ) / 43011 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 73 \nu^{13} - 69 \nu^{12} + 28 \nu^{11} - 36 \nu^{10} + 373 \nu^{9} + 1446 \nu^{8} - 2154 \nu^{7} + \cdots + 121257 ) / 14337 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 281 \nu^{13} + 6 \nu^{12} - 208 \nu^{11} - 657 \nu^{10} + 500 \nu^{9} + 6555 \nu^{8} - 4806 \nu^{7} + \cdots + 334611 ) / 43011 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 250 \nu^{13} + 639 \nu^{12} - 503 \nu^{11} - 2514 \nu^{10} - 2282 \nu^{9} + 4278 \nu^{8} + \cdots - 423549 ) / 43011 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 195 \nu^{13} + 299 \nu^{12} - 495 \nu^{11} - 1378 \nu^{10} - 987 \nu^{9} + 3587 \nu^{8} + \cdots - 200475 ) / 14337 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} - \beta_{3} - \beta_{2} + \beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + \beta_{5} + \cdots + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{12} + 3\beta_{7} - \beta_{6} + 3\beta_{4} - \beta_{3} - \beta_{2} + 4\beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2 \beta_{13} - \beta_{12} - \beta_{11} + \beta_{9} - \beta_{8} + \beta_{7} - 4 \beta_{6} + 3 \beta_{5} + \cdots + 8 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3 \beta_{13} + 6 \beta_{11} - 4 \beta_{10} - 9 \beta_{8} + 6 \beta_{7} + \beta_{6} - 4 \beta_{5} + \cdots - 11 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 8 \beta_{13} - 4 \beta_{12} - \beta_{11} - \beta_{10} + 4 \beta_{9} + 8 \beta_{8} + 13 \beta_{7} + \cdots + 19 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 21 \beta_{13} + 15 \beta_{12} + 3 \beta_{11} - 2 \beta_{10} - 9 \beta_{9} + 9 \beta_{8} + 18 \beta_{7} + \cdots + 33 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 17 \beta_{13} - 40 \beta_{12} + 11 \beta_{11} + \beta_{10} + 31 \beta_{9} + 11 \beta_{8} + 4 \beta_{7} + \cdots - 90 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 9 \beta_{13} + 57 \beta_{12} - 36 \beta_{11} + 69 \beta_{10} + 27 \beta_{9} + 18 \beta_{8} + \cdots - 58 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 16 \beta_{13} + 8 \beta_{12} - \beta_{11} - 9 \beta_{10} - 8 \beta_{9} + 80 \beta_{8} + 46 \beta_{7} + \cdots - 298 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 150 \beta_{13} + 90 \beta_{12} + 15 \beta_{11} + 122 \beta_{10} + 18 \beta_{9} - 27 \beta_{8} + \cdots - 560 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 134 \beta_{13} - 94 \beta_{12} + 368 \beta_{11} - 199 \beta_{10} + 148 \beta_{9} - 271 \beta_{8} + \cdots - 692 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 363 \beta_{13} + 267 \beta_{12} - 96 \beta_{11} + 196 \beta_{10} - 612 \beta_{9} + 612 \beta_{8} + \cdots - 822 ) / 3 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1765.1
−0.674693 − 1.59524i
1.68442 − 0.403398i
−1.73040 + 0.0755709i
1.64515 + 0.541745i
1.13119 − 1.31165i
−0.473632 + 1.66604i
−1.58203 − 0.705117i
−0.674693 + 1.59524i
1.68442 + 0.403398i
−1.73040 − 0.0755709i
1.64515 − 0.541745i
1.13119 + 1.31165i
−0.473632 − 1.66604i
−1.58203 + 0.705117i
0 0 0 −2.07260 + 3.58985i 0 0 0 0 0
1765.2 0 0 0 −1.80173 + 3.12069i 0 0 0 0 0
1765.3 0 0 0 −0.483929 + 0.838189i 0 0 0 0 0
1765.4 0 0 0 0.381918 − 0.661502i 0 0 0 0 0
1765.5 0 0 0 0.764702 − 1.32450i 0 0 0 0 0
1765.6 0 0 0 0.951504 − 1.64805i 0 0 0 0 0
1765.7 0 0 0 1.26013 − 2.18261i 0 0 0 0 0
3529.1 0 0 0 −2.07260 − 3.58985i 0 0 0 0 0
3529.2 0 0 0 −1.80173 − 3.12069i 0 0 0 0 0
3529.3 0 0 0 −0.483929 − 0.838189i 0 0 0 0 0
3529.4 0 0 0 0.381918 + 0.661502i 0 0 0 0 0
3529.5 0 0 0 0.764702 + 1.32450i 0 0 0 0 0
3529.6 0 0 0 0.951504 + 1.64805i 0 0 0 0 0
3529.7 0 0 0 1.26013 + 2.18261i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1765.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5292.2.j.g 14
3.b odd 2 1 1764.2.j.h 14
7.b odd 2 1 5292.2.j.h 14
7.c even 3 1 5292.2.i.i 14
7.c even 3 1 5292.2.l.i 14
7.d odd 6 1 756.2.i.b 14
7.d odd 6 1 756.2.l.b 14
9.c even 3 1 inner 5292.2.j.g 14
9.d odd 6 1 1764.2.j.h 14
21.c even 2 1 1764.2.j.g 14
21.g even 6 1 252.2.i.b ✓ 14
21.g even 6 1 252.2.l.b yes 14
21.h odd 6 1 1764.2.i.i 14
21.h odd 6 1 1764.2.l.i 14
28.f even 6 1 3024.2.q.j 14
28.f even 6 1 3024.2.t.j 14
63.g even 3 1 5292.2.i.i 14
63.h even 3 1 5292.2.l.i 14
63.i even 6 1 252.2.l.b yes 14
63.i even 6 1 2268.2.k.e 14
63.j odd 6 1 1764.2.l.i 14
63.k odd 6 1 756.2.i.b 14
63.k odd 6 1 2268.2.k.f 14
63.l odd 6 1 5292.2.j.h 14
63.n odd 6 1 1764.2.i.i 14
63.o even 6 1 1764.2.j.g 14
63.s even 6 1 252.2.i.b ✓ 14
63.s even 6 1 2268.2.k.e 14
63.t odd 6 1 756.2.l.b 14
63.t odd 6 1 2268.2.k.f 14
84.j odd 6 1 1008.2.q.j 14
84.j odd 6 1 1008.2.t.j 14
252.n even 6 1 3024.2.q.j 14
252.r odd 6 1 1008.2.t.j 14
252.bj even 6 1 3024.2.t.j 14
252.bn odd 6 1 1008.2.q.j 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.i.b ✓ 14 21.g even 6 1
252.2.i.b ✓ 14 63.s even 6 1
252.2.l.b yes 14 21.g even 6 1
252.2.l.b yes 14 63.i even 6 1
756.2.i.b 14 7.d odd 6 1
756.2.i.b 14 63.k odd 6 1
756.2.l.b 14 7.d odd 6 1
756.2.l.b 14 63.t odd 6 1
1008.2.q.j 14 84.j odd 6 1
1008.2.q.j 14 252.bn odd 6 1
1008.2.t.j 14 84.j odd 6 1
1008.2.t.j 14 252.r odd 6 1
1764.2.i.i 14 21.h odd 6 1
1764.2.i.i 14 63.n odd 6 1
1764.2.j.g 14 21.c even 2 1
1764.2.j.g 14 63.o even 6 1
1764.2.j.h 14 3.b odd 2 1
1764.2.j.h 14 9.d odd 6 1
1764.2.l.i 14 21.h odd 6 1
1764.2.l.i 14 63.j odd 6 1
2268.2.k.e 14 63.i even 6 1
2268.2.k.e 14 63.s even 6 1
2268.2.k.f 14 63.k odd 6 1
2268.2.k.f 14 63.t odd 6 1
3024.2.q.j 14 28.f even 6 1
3024.2.q.j 14 252.n even 6 1
3024.2.t.j 14 28.f even 6 1
3024.2.t.j 14 252.bj even 6 1
5292.2.i.i 14 7.c even 3 1
5292.2.i.i 14 63.g even 3 1
5292.2.j.g 14 1.a even 1 1 trivial
5292.2.j.g 14 9.c even 3 1 inner
5292.2.j.h 14 7.b odd 2 1
5292.2.j.h 14 63.l odd 6 1
5292.2.l.i 14 7.c even 3 1
5292.2.l.i 14 63.h even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{14} + 2 T_{5}^{13} + 24 T_{5}^{12} - 16 T_{5}^{11} + 295 T_{5}^{10} - 357 T_{5}^{9} + 2670 T_{5}^{8} + \cdots + 6561 \) acting on \(S_{2}^{\mathrm{new}}(5292, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + 2 T^{13} + \cdots + 6561 \) Copy content Toggle raw display
$7$ \( T^{14} \) Copy content Toggle raw display
$11$ \( T^{14} + 2 T^{13} + \cdots + 6561 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 150626529 \) Copy content Toggle raw display
$17$ \( (T^{7} + 2 T^{6} - 50 T^{5} + \cdots - 81)^{2} \) Copy content Toggle raw display
$19$ \( (T^{7} - 7 T^{6} + \cdots + 2021)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 105822369 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 145660761 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 13807190016 \) Copy content Toggle raw display
$37$ \( (T^{7} + 10 T^{6} + \cdots - 39584)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 1108290681 \) Copy content Toggle raw display
$43$ \( T^{14} - 7 T^{13} + \cdots + 4084441 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 136048896 \) Copy content Toggle raw display
$53$ \( (T^{7} + 15 T^{6} + \cdots + 30861)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 688747536 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 148644864 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 116985856 \) Copy content Toggle raw display
$71$ \( (T^{7} + T^{6} - 116 T^{5} + \cdots + 972)^{2} \) Copy content Toggle raw display
$73$ \( (T^{7} - 21 T^{6} + \cdots + 52427)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 54397165824 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 901054679121 \) Copy content Toggle raw display
$89$ \( (T^{7} - 6 T^{6} + \cdots + 128547)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 767677849 \) Copy content Toggle raw display
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