Properties

Label 4592.2.a.bg
Level $4592$
Weight $2$
Character orbit 4592.a
Self dual yes
Analytic conductor $36.667$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4592,2,Mod(1,4592)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4592, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4592.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4592 = 2^{4} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4592.a (trivial)

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: yes
Analytic conductor: \(36.6673046082\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.185257757.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 10x^{4} + 10x^{3} + 23x^{2} - 24x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 287)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{3} + ( - \beta_{4} + \beta_{3}) q^{5} + q^{7} + (\beta_{4} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 1) q^{3} + ( - \beta_{4} + \beta_{3}) q^{5} + q^{7} + (\beta_{4} + 2) q^{9} + ( - \beta_{5} - \beta_{2} + \beta_1 - 1) q^{11} + (\beta_{5} + 2 \beta_1 + 1) q^{13} + (\beta_{5} - \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + \beta_1 + 1) q^{15} + (\beta_{3} + 1) q^{17} + (\beta_{5} + \beta_{4} - \beta_{2} - \beta_1) q^{19} + ( - \beta_{2} + 1) q^{21} + ( - \beta_{2} - 3) q^{23} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1 + 4) q^{25} + ( - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1) q^{27} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{29} + (\beta_{4} + \beta_{3} + 4) q^{31} + ( - 2 \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_1 + 2) q^{33} + ( - \beta_{4} + \beta_{3}) q^{35} + ( - 2 \beta_{4} - \beta_{2} + \beta_1 + 4) q^{37} + ( - \beta_{5} - 3 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{39} + q^{41} + (2 \beta_{5} + \beta_{3} - 3) q^{43} + (\beta_{5} - 3 \beta_{4} + 3 \beta_{3} + \beta_{2} - 3 \beta_1 - 5) q^{45} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 + 4) q^{47} + q^{49} + (2 \beta_{3} - \beta_1 + 3) q^{51} + (\beta_{4} - 3 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{53} + ( - 3 \beta_{4} + 3 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 2) q^{55} + (\beta_{5} + 2 \beta_{4} + \beta_{2} - 4 \beta_1 + 6) q^{57} + ( - \beta_{4} - 3 \beta_{3} + 2) q^{59} + ( - \beta_{5} - 2 \beta_{4} - 3 \beta_{2} + 3 \beta_1 - 1) q^{61} + (\beta_{4} + 2) q^{63} + ( - 2 \beta_{5} - 3 \beta_{4} - \beta_{3} - 4 \beta_{2}) q^{65} + (\beta_{4} - \beta_{3} - 2 \beta_{2} - 4) q^{67} + (\beta_{4} + 4 \beta_{2} + 1) q^{69} + (\beta_{5} + 2 \beta_{4} - \beta_{2} - \beta_1 + 1) q^{71} + (\beta_{5} + \beta_{2} - 3 \beta_1 + 9) q^{73} + ( - 5 \beta_{5} - 2 \beta_{3} - 8 \beta_{2} - \beta_1 + 10) q^{75} + ( - \beta_{5} - \beta_{2} + \beta_1 - 1) q^{77} + ( - 2 \beta_{5} - 2 \beta_1) q^{79} + (2 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 4) q^{81} + ( - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{83} + ( - \beta_{4} + 3 \beta_{3} + 4) q^{85} + (3 \beta_{5} - 2 \beta_{4} + 3 \beta_{2} + \beta_1 - 7) q^{87} + (\beta_{4} + \beta_{3} - \beta_{2} - 3 \beta_1 - 6) q^{89} + (\beta_{5} + 2 \beta_1 + 1) q^{91} + ( - \beta_{5} + \beta_{4} + \beta_{3} - 5 \beta_{2} - 3 \beta_1 + 7) q^{93} + (3 \beta_{5} + 3 \beta_{4} + \beta_{3} + 3 \beta_{2} - 5 \beta_1 - 1) q^{95} + (2 \beta_{4} - 4 \beta_{3} - 3 \beta_{2} - 4 \beta_1 + 3) q^{97} + ( - 2 \beta_{5} + \beta_{4} - 5 \beta_{3} - 6 \beta_{2} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{3} - q^{5} + 6 q^{7} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{3} - q^{5} + 6 q^{7} + 14 q^{9} - 6 q^{11} + 7 q^{13} + 13 q^{15} + 7 q^{17} - 2 q^{19} + 4 q^{21} - 20 q^{23} + 29 q^{25} - 2 q^{27} - 9 q^{29} + 27 q^{31} + 17 q^{33} - q^{35} + 19 q^{37} - q^{39} + 6 q^{41} - 19 q^{43} - 35 q^{45} + 19 q^{47} + 6 q^{49} + 19 q^{51} + 5 q^{53} - 3 q^{55} + 37 q^{57} + 7 q^{59} - 12 q^{61} + 14 q^{63} - 13 q^{65} - 27 q^{67} + 16 q^{69} + 6 q^{71} + 52 q^{73} + 46 q^{75} - 6 q^{77} - 22 q^{81} - 12 q^{83} + 25 q^{85} - 42 q^{87} - 38 q^{89} + 7 q^{91} + 33 q^{93} - q^{95} + 8 q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 10x^{4} + 10x^{3} + 23x^{2} - 24x + 5 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 5\nu + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + \nu^{2} + 5\nu - 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{5} - 10\nu^{3} + 2\nu^{2} + 24\nu - 10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{5} + \nu^{4} + 10\nu^{3} - 8\nu^{2} - 24\nu + 13 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} + 5\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + \beta_{4} + 6\beta_{3} + 6\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{4} - 2\beta_{3} + 8\beta_{2} + 26\beta _1 - 6 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.47904
−2.01956
0.306800
2.05073
−2.46179
0.644787
0 −2.84004 0 −3.76023 0 1.00000 0 5.06582 0
1.2 0 −1.86075 0 −0.244521 0 1.00000 0 0.462403 0
1.3 0 1.50512 0 0.333855 0 1.00000 0 −0.734606 0
1.4 0 1.62935 0 4.18004 0 1.00000 0 −0.345215 0
1.5 0 2.61045 0 2.85638 0 1.00000 0 3.81447 0
1.6 0 2.95586 0 −4.36552 0 1.00000 0 5.73713 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(41\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4592.2.a.bg 6
4.b odd 2 1 287.2.a.f 6
12.b even 2 1 2583.2.a.t 6
20.d odd 2 1 7175.2.a.p 6
28.d even 2 1 2009.2.a.o 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.2.a.f 6 4.b odd 2 1
2009.2.a.o 6 28.d even 2 1
2583.2.a.t 6 12.b even 2 1
4592.2.a.bg 6 1.a even 1 1 trivial
7175.2.a.p 6 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4592))\):

\( T_{3}^{6} - 4T_{3}^{5} - 8T_{3}^{4} + 46T_{3}^{3} - 13T_{3}^{2} - 111T_{3} + 100 \) Copy content Toggle raw display
\( T_{5}^{6} + T_{5}^{5} - 29T_{5}^{4} - 16T_{5}^{3} + 200T_{5}^{2} - 16T_{5} - 16 \) Copy content Toggle raw display
\( T_{11}^{6} + 6T_{11}^{5} - 29T_{11}^{4} - 218T_{11}^{3} + 28T_{11}^{2} + 1928T_{11} + 2720 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 4 T^{5} - 8 T^{4} + 46 T^{3} + \cdots + 100 \) Copy content Toggle raw display
$5$ \( T^{6} + T^{5} - 29 T^{4} - 16 T^{3} + \cdots - 16 \) Copy content Toggle raw display
$7$ \( (T - 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 6 T^{5} - 29 T^{4} + \cdots + 2720 \) Copy content Toggle raw display
$13$ \( T^{6} - 7 T^{5} - 49 T^{4} + \cdots - 1546 \) Copy content Toggle raw display
$17$ \( T^{6} - 7 T^{5} + 3 T^{4} + 26 T^{3} + \cdots + 2 \) Copy content Toggle raw display
$19$ \( T^{6} + 2 T^{5} - 68 T^{4} + \cdots - 3212 \) Copy content Toggle raw display
$23$ \( T^{6} + 20 T^{5} + 152 T^{4} + \cdots + 344 \) Copy content Toggle raw display
$29$ \( T^{6} + 9 T^{5} - 79 T^{4} + \cdots + 10448 \) Copy content Toggle raw display
$31$ \( T^{6} - 27 T^{5} + 257 T^{4} + \cdots - 1280 \) Copy content Toggle raw display
$37$ \( T^{6} - 19 T^{5} + 54 T^{4} + \cdots - 376 \) Copy content Toggle raw display
$41$ \( (T - 1)^{6} \) Copy content Toggle raw display
$43$ \( T^{6} + 19 T^{5} + 27 T^{4} + \cdots + 29756 \) Copy content Toggle raw display
$47$ \( T^{6} - 19 T^{5} + 94 T^{4} + \cdots - 512 \) Copy content Toggle raw display
$53$ \( T^{6} - 5 T^{5} - 127 T^{4} + \cdots - 28432 \) Copy content Toggle raw display
$59$ \( T^{6} - 7 T^{5} - 183 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$61$ \( T^{6} + 12 T^{5} - 183 T^{4} + \cdots - 55952 \) Copy content Toggle raw display
$67$ \( T^{6} + 27 T^{5} + 243 T^{4} + \cdots - 26848 \) Copy content Toggle raw display
$71$ \( T^{6} - 6 T^{5} - 105 T^{4} + \cdots + 4672 \) Copy content Toggle raw display
$73$ \( T^{6} - 52 T^{5} + 1009 T^{4} + \cdots + 656 \) Copy content Toggle raw display
$79$ \( T^{6} - 152 T^{4} + 96 T^{3} + \cdots + 2048 \) Copy content Toggle raw display
$83$ \( T^{6} + 12 T^{5} - 127 T^{4} + \cdots - 19744 \) Copy content Toggle raw display
$89$ \( T^{6} + 38 T^{5} + 404 T^{4} + \cdots - 345082 \) Copy content Toggle raw display
$97$ \( T^{6} - 8 T^{5} - 316 T^{4} + \cdots + 303494 \) Copy content Toggle raw display
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