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Properties

Label 5292.2.f.f
Level $5292$
Weight $2$
Character orbit 5292.f
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(2645,5292)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5292, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5292.2645");
 
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.f (of order \(2\), degree \(1\), 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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 12x^{14} + 106x^{12} - 384x^{10} + 1005x^{8} - 1200x^{6} + 1030x^{4} - 252x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 7^{2} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{10} q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{10} q^{5} + ( - \beta_{15} + \beta_{14}) q^{11} + (\beta_{8} - \beta_{7} + \beta_{2}) q^{13} + (\beta_{11} - \beta_{5}) q^{17} + (\beta_{8} - \beta_{7} + \beta_{6}) q^{19} + (\beta_{14} + \beta_{13} + \beta_{12}) q^{23} + (\beta_{3} - \beta_1) q^{25} + ( - \beta_{13} - \beta_{12}) q^{29} + ( - \beta_{8} - \beta_{7} + \cdots - \beta_{2}) q^{31}+ \cdots + (4 \beta_{8} - 2 \beta_{6} - 2 \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{37} - 32 q^{43} - 32 q^{67}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 12x^{14} + 106x^{12} - 384x^{10} + 1005x^{8} - 1200x^{6} + 1030x^{4} - 252x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 878446 \nu^{14} - 9796819 \nu^{12} + 83290551 \nu^{10} - 251050620 \nu^{8} + 538652244 \nu^{6} + \cdots - 272833414 ) / 198602691 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 12378 \nu^{14} - 136713 \nu^{12} + 1187651 \nu^{10} - 3705756 \nu^{8} + 9677932 \nu^{6} + \cdots - 1678362 ) / 2797221 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 5624 \nu^{14} + 62439 \nu^{12} - 533244 \nu^{10} + 1607280 \nu^{8} - 3592238 \nu^{6} + \cdots - 1259979 ) / 932407 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1711368 \nu^{14} - 18874018 \nu^{12} + 162264708 \nu^{10} - 489090960 \nu^{8} + 1147767525 \nu^{6} + \cdots + 618727571 ) / 198602691 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1097036 \nu^{15} - 16823687 \nu^{13} + 158768956 \nu^{11} - 790864651 \nu^{9} + \cdots - 1662217536 \nu ) / 198602691 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2228536 \nu^{14} + 26103500 \nu^{12} - 229219440 \nu^{10} + 795176982 \nu^{8} + \cdots + 312966731 ) / 198602691 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 38598 \nu^{14} + 462014 \nu^{12} - 4073324 \nu^{10} + 14662382 \nu^{8} - 38003215 \nu^{6} + \cdots + 5317207 ) / 2797221 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -28\nu^{14} + 343\nu^{12} - 3036\nu^{10} + 11298\nu^{8} - 29154\nu^{6} + 34818\nu^{4} - 25084\nu^{2} + 3955 ) / 1491 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2357007 \nu^{15} + 25590704 \nu^{13} - 218752844 \nu^{11} + 635778706 \nu^{9} + \cdots + 348259884 \nu ) / 198602691 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2507455 \nu^{15} - 29767959 \nu^{13} + 261887254 \nu^{11} - 930637220 \nu^{9} + \cdots - 1148510594 \nu ) / 198602691 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2811027 \nu^{15} + 35962699 \nu^{13} - 324517748 \nu^{11} + 1310072552 \nu^{9} + \cdots + 2072867706 \nu ) / 198602691 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3107570 \nu^{15} - 35765025 \nu^{13} + 314058996 \nu^{11} - 1064314686 \nu^{9} + \cdots - 156692886 \nu ) / 198602691 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 5201605 \nu^{15} - 62031725 \nu^{13} + 544711524 \nu^{11} - 1934539950 \nu^{9} + \cdots - 271771934 \nu ) / 198602691 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 6080639 \nu^{15} - 71693250 \nu^{13} + 629551080 \nu^{11} - 2203677654 \nu^{9} + \cdots - 314100780 \nu ) / 198602691 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 12122447 \nu^{15} - 143732450 \nu^{13} + 1262140008 \nu^{11} - 4448521683 \nu^{9} + \cdots - 629717228 \nu ) / 198602691 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} + \beta_{14} - 2\beta_{13} + 3\beta_{11} - 2\beta_{10} - \beta_{9} + 3\beta_{5} ) / 14 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} - 3\beta_{6} + \beta_{3} - 3\beta_{2} + \beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{15} + 9\beta_{14} - 11\beta_{13} - 7\beta_{12} ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9\beta_{8} - 6\beta_{7} - 17\beta_{6} - 2\beta_{4} - 9\beta_{3} - 24\beta_{2} - 8\beta _1 - 17 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4 \beta_{15} + 81 \beta_{14} - 71 \beta_{13} - 56 \beta_{12} - 75 \beta_{11} + 106 \beta_{10} + \cdots - 138 \beta_{5} ) / 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -21\beta_{4} - 74\beta_{3} - 60\beta _1 - 117 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 6 \beta_{15} - 659 \beta_{14} + 506 \beta_{13} + 420 \beta_{12} - 500 \beta_{11} + \cdots - 1018 \beta_{5} ) / 14 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -582\beta_{8} + 528\beta_{7} + 859\beta_{6} - 176\beta_{4} - 582\beta_{3} + 1356\beta_{2} - 452\beta _1 - 859 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 159\beta_{15} - 5147\beta_{14} + 3756\beta_{13} + 3164\beta_{12} ) / 7 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 4489 \beta_{8} + 4158 \beta_{7} + 6453 \beta_{6} + 1386 \beta_{4} + 4489 \beta_{3} + 10275 \beta_{2} + \cdots + 6453 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1550 \beta_{15} - 39575 \beta_{14} + 28323 \beta_{13} + 23975 \beta_{12} + 26773 \beta_{11} + \cdots + 58196 \beta_{5} ) / 14 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 10686\beta_{4} + 34353\beta_{3} + 26028\beta _1 + 48887 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 12802 \beta_{15} + 302471 \beta_{14} - 214867 \beta_{13} - 182196 \beta_{12} + \cdots + 442536 \beta_{5} ) / 14 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 262088 \beta_{8} - 245259 \beta_{7} - 371535 \beta_{6} + 81753 \beta_{4} + 262088 \beta_{3} + \cdots + 371535 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( -100406\beta_{15} + 2306481\beta_{14} - 1633804\beta_{13} - 1386266\beta_{12} ) / 7 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(785\) \(1081\) \(2647\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2645.1
−0.954123 − 0.550863i
−0.954123 + 0.550863i
−0.441700 + 0.255016i
−0.441700 − 0.255016i
−2.38976 − 1.37973i
−2.38976 + 1.37973i
1.47769 + 0.853147i
1.47769 − 0.853147i
−1.47769 + 0.853147i
−1.47769 − 0.853147i
2.38976 − 1.37973i
2.38976 + 1.37973i
0.441700 + 0.255016i
0.441700 − 0.255016i
0.954123 − 0.550863i
0.954123 + 0.550863i
0 0 0 −3.10075 0 0 0 0 0
2645.2 0 0 0 −3.10075 0 0 0 0 0
2645.3 0 0 0 −2.21618 0 0 0 0 0
2645.4 0 0 0 −2.21618 0 0 0 0 0
2645.5 0 0 0 −1.79271 0 0 0 0 0
2645.6 0 0 0 −1.79271 0 0 0 0 0
2645.7 0 0 0 −1.50337 0 0 0 0 0
2645.8 0 0 0 −1.50337 0 0 0 0 0
2645.9 0 0 0 1.50337 0 0 0 0 0
2645.10 0 0 0 1.50337 0 0 0 0 0
2645.11 0 0 0 1.79271 0 0 0 0 0
2645.12 0 0 0 1.79271 0 0 0 0 0
2645.13 0 0 0 2.21618 0 0 0 0 0
2645.14 0 0 0 2.21618 0 0 0 0 0
2645.15 0 0 0 3.10075 0 0 0 0 0
2645.16 0 0 0 3.10075 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2645.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5292.2.f.f ✓ 16
3.b odd 2 1 inner 5292.2.f.f ✓ 16
7.b odd 2 1 inner 5292.2.f.f ✓ 16
21.c even 2 1 inner 5292.2.f.f ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5292.2.f.f ✓ 16 1.a even 1 1 trivial
5292.2.f.f ✓ 16 3.b odd 2 1 inner
5292.2.f.f ✓ 16 7.b odd 2 1 inner
5292.2.f.f ✓ 16 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5292, [\chi])\):

\( T_{5}^{8} - 20T_{5}^{6} + 134T_{5}^{4} - 364T_{5}^{2} + 343 \) Copy content Toggle raw display
\( T_{13}^{8} + 40T_{13}^{6} + 392T_{13}^{4} + 608T_{13}^{2} + 16 \) 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
$5$ \( (T^{8} - 20 T^{6} + \cdots + 343)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + 76 T^{6} + \cdots + 16807)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 40 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 96 T^{6} + \cdots + 111132)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 44 T^{6} + 290 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 116 T^{6} + \cdots + 343)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 120 T^{6} + \cdots + 5488)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 52 T^{6} + \cdots + 10609)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 4 T^{3} + \cdots + 823)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 188 T^{6} + \cdots + 576583)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 8 T^{3} + \cdots + 478)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} - 320 T^{6} + \cdots + 14555548)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 376 T^{6} + \cdots + 268912)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 136 T^{6} + \cdots + 1372)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 40 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 8 T^{3} + \cdots + 388)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + 348 T^{6} + \cdots + 576583)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 296 T^{6} + \cdots + 148996)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 160 T^{2} + \cdots + 574)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} - 352 T^{6} + \cdots + 351232)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 628 T^{6} + \cdots + 81348967)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 464 T^{6} + \cdots + 37552384)^{2} \) Copy content Toggle raw display
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