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Properties

Label 5292.2.f.g
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: \( x^{16} - 20 x^{14} + 214 x^{12} - 120 x^{11} - 1516 x^{10} - 432 x^{9} + 12117 x^{8} - 9744 x^{7} + \cdots + 25921 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 7^{4} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{5} + \beta_{11} q^{11} + \beta_{15} q^{13} - \beta_{2} q^{17} - \beta_{10} q^{19} - \beta_{7} q^{23} + (\beta_{4} + 3) q^{25} + ( - \beta_{12} - \beta_{7}) q^{29} + (\beta_{10} - \beta_{3}) q^{31} + (\beta_{8} - 2) q^{37} + (\beta_{6} - \beta_{2} - \beta_1) q^{41} + ( - \beta_{8} + \beta_{4} + 1) q^{43} + (\beta_{6} + 2 \beta_{2} + 2 \beta_1) q^{47} + (\beta_{12} - \beta_{11}) q^{53} + (\beta_{15} - \beta_{13} - \beta_{3}) q^{55} + ( - \beta_{6} - 2 \beta_{2}) q^{59} + (\beta_{15} + \beta_{3}) q^{61} + ( - 2 \beta_{14} + 3 \beta_{11} + \beta_{7}) q^{65} + (\beta_{8} + 1) q^{67} + ( - \beta_{14} - \beta_{12} + \cdots + \beta_{7}) q^{71}+ \cdots + (\beta_{13} - \beta_{10} + 2 \beta_{3}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 48 q^{25} - 32 q^{37} + 16 q^{43} + 16 q^{67}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 20 x^{14} + 214 x^{12} - 120 x^{11} - 1516 x^{10} - 432 x^{9} + 12117 x^{8} - 9744 x^{7} + \cdots + 25921 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 32\!\cdots\!94 \nu^{15} + \cdots + 18\!\cdots\!11 ) / 54\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 86\!\cdots\!08 \nu^{15} + \cdots - 24\!\cdots\!14 ) / 13\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 21\!\cdots\!78 \nu^{15} + \cdots + 63\!\cdots\!42 ) / 32\!\cdots\!07 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 21\!\cdots\!48 \nu^{15} + \cdots + 13\!\cdots\!35 ) / 32\!\cdots\!33 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 33\!\cdots\!98 \nu^{15} + \cdots - 12\!\cdots\!36 ) / 32\!\cdots\!33 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 69\!\cdots\!40 \nu^{15} + \cdots - 44\!\cdots\!00 ) / 54\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 14\!\cdots\!53 \nu^{15} + \cdots - 17\!\cdots\!51 ) / 75\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 70\!\cdots\!58 \nu^{15} + \cdots - 19\!\cdots\!98 ) / 32\!\cdots\!33 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12\!\cdots\!52 \nu^{15} + \cdots - 32\!\cdots\!68 ) / 54\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 56\!\cdots\!21 \nu^{15} + \cdots + 94\!\cdots\!49 ) / 22\!\cdots\!49 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 28\!\cdots\!59 \nu^{15} + \cdots - 26\!\cdots\!85 ) / 75\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 28\!\cdots\!03 \nu^{15} + \cdots + 83\!\cdots\!33 ) / 75\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 19\!\cdots\!46 \nu^{15} + \cdots - 18\!\cdots\!22 ) / 32\!\cdots\!07 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 45\!\cdots\!81 \nu^{15} + \cdots + 58\!\cdots\!19 ) / 75\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 15\!\cdots\!66 \nu^{15} + \cdots + 16\!\cdots\!36 ) / 22\!\cdots\!49 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} + 2 \beta_{13} - 2 \beta_{12} - 4 \beta_{11} + 2 \beta_{9} - \beta_{8} + \beta_{7} + \cdots + 4 \beta_1 ) / 14 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{14} - 2 \beta_{13} + 4 \beta_{12} - 6 \beta_{11} - 14 \beta_{10} + 3 \beta_{9} + \beta_{8} + \cdots + 35 ) / 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 9 \beta_{14} + 17 \beta_{13} - 24 \beta_{12} - 69 \beta_{11} + 21 \beta_{10} + 3 \beta_{9} + \cdots - 8 \beta_1 ) / 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 56 \beta_{15} + 32 \beta_{14} + 4 \beta_{13} + 48 \beta_{12} + 180 \beta_{11} - 84 \beta_{10} + \cdots - 49 ) / 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 70 \beta_{15} - 242 \beta_{14} + 109 \beta_{13} - 188 \beta_{12} - 649 \beta_{11} + 385 \beta_{10} + \cdots + 525 ) / 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 756 \beta_{15} + 736 \beta_{14} + 298 \beta_{13} + 754 \beta_{12} + 3286 \beta_{11} - 588 \beta_{10} + \cdots - 511 ) / 14 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 882 \beta_{15} - 2248 \beta_{14} + 1605 \beta_{13} - 2217 \beta_{12} - 6835 \beta_{11} + \cdots + 17346 ) / 14 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 7504 \beta_{15} + 3136 \beta_{14} + 3760 \beta_{13} + 6440 \beta_{12} + 19936 \beta_{11} - 7224 \beta_{10} + \cdots - 31493 ) / 14 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 7686 \beta_{15} - 11430 \beta_{14} + 18154 \beta_{13} - 17985 \beta_{12} - 38784 \beta_{11} + \cdots + 320544 ) / 14 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 36330 \beta_{15} - 57056 \beta_{14} + 55756 \beta_{13} - 45334 \beta_{12} - 150196 \beta_{11} + \cdots - 843885 ) / 14 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 375067 \beta_{15} + 140094 \beta_{14} + 124994 \beta_{13} + 230623 \beta_{12} + 823384 \beta_{11} + \cdots + 3711939 ) / 14 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 607572 \beta_{15} - 1643288 \beta_{14} + 991288 \beta_{13} - 2171744 \beta_{12} - 6296432 \beta_{11} + \cdots - 9644131 ) / 14 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 7145593 \beta_{15} + 5608094 \beta_{14} + 334530 \beta_{13} + 8823370 \beta_{12} + 27613158 \beta_{11} + \cdots + 23542974 ) / 14 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 17100622 \beta_{15} - 28423022 \beta_{14} + 14722328 \beta_{13} - 38563186 \beta_{12} + \cdots - 15003667 ) / 14 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 98762685 \beta_{15} + 91458303 \beta_{14} - 3927396 \beta_{13} + 135168413 \beta_{12} + \cdots - 152672940 ) / 14 \) 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
0.416116 + 1.21752i
0.416116 − 1.21752i
−0.0215130 − 0.261052i
−0.0215130 + 0.261052i
2.14817 + 1.98289i
2.14817 − 1.98289i
−1.75356 − 1.58671i
−1.75356 + 1.58671i
3.16778 + 0.261052i
3.16778 − 0.261052i
−3.56238 + 1.21752i
−3.56238 − 1.21752i
1.43573 + 1.58671i
1.43573 − 1.58671i
−1.83033 + 1.98289i
−1.83033 − 1.98289i
0 0 0 −4.03797 0 0 0 0 0
2645.2 0 0 0 −4.03797 0 0 0 0 0
2645.3 0 0 0 −3.47991 0 0 0 0 0
2645.4 0 0 0 −3.47991 0 0 0 0 0
2645.5 0 0 0 −1.58848 0 0 0 0 0
2645.6 0 0 0 −1.58848 0 0 0 0 0
2645.7 0 0 0 −1.03042 0 0 0 0 0
2645.8 0 0 0 −1.03042 0 0 0 0 0
2645.9 0 0 0 1.03042 0 0 0 0 0
2645.10 0 0 0 1.03042 0 0 0 0 0
2645.11 0 0 0 1.58848 0 0 0 0 0
2645.12 0 0 0 1.58848 0 0 0 0 0
2645.13 0 0 0 3.47991 0 0 0 0 0
2645.14 0 0 0 3.47991 0 0 0 0 0
2645.15 0 0 0 4.03797 0 0 0 0 0
2645.16 0 0 0 4.03797 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.g ✓ 16
3.b odd 2 1 inner 5292.2.f.g ✓ 16
7.b odd 2 1 inner 5292.2.f.g ✓ 16
21.c even 2 1 inner 5292.2.f.g ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5292.2.f.g ✓ 16 1.a even 1 1 trivial
5292.2.f.g ✓ 16 3.b odd 2 1 inner
5292.2.f.g ✓ 16 7.b odd 2 1 inner
5292.2.f.g ✓ 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 \(S_{2}^{\mathrm{new}}(5292, [\chi])\):

\( T_{5}^{8} - 32T_{5}^{6} + 302T_{5}^{4} - 784T_{5}^{2} + 529 \) Copy content Toggle raw display
\( T_{13}^{8} + 76T_{13}^{6} + 1694T_{13}^{4} + 12152T_{13}^{2} + 9604 \) 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
$5$ \( (T^{8} - 32 T^{6} + \cdots + 529)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + 28 T^{6} + 246 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 76 T^{6} + \cdots + 9604)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 3)^{8} \) Copy content Toggle raw display
$19$ \( (T^{8} + 92 T^{6} + \cdots + 21316)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 56 T^{6} + \cdots + 2116)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 108 T^{6} + \cdots + 20164)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 124 T^{6} + \cdots + 9604)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 8 T^{3} - 24 T^{2} + \cdots - 41)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 224 T^{6} + \cdots + 390625)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 4 T^{3} + \cdots + 343)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} - 260 T^{6} + \cdots + 9186961)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 136 T^{6} + \cdots + 122500)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 196 T^{6} + \cdots + 677329)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 148 T^{6} + \cdots + 498436)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 4 T^{3} + \cdots + 196)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + 240 T^{6} + \cdots + 153664)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 296 T^{6} + \cdots + 31684)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 82 T^{2} + \cdots + 1297)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} - 592 T^{6} + \cdots + 330985249)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 364 T^{6} + \cdots + 3794704)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 308 T^{6} + \cdots + 948676)^{2} \) Copy content Toggle raw display
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