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Properties

Label 4732.2.a.u
Level $4732$
Weight $2$
Character orbit 4732.a
Self dual yes
Analytic conductor $37.785$
Analytic rank $1$
Dimension $9$
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] = [4732,2,Mod(1,4732)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4732, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4732.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4732 = 2^{2} \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4732.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: \(37.7852102365\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 15x^{7} - 2x^{6} + 68x^{5} + 13x^{4} - 112x^{3} - 36x^{2} + 61x + 29 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_1 - 1) q^{3} + \beta_{7} q^{5} + q^{7} + ( - \beta_{6} + \beta_{5} + 2 \beta_{4} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_1 - 1) q^{3} + \beta_{7} q^{5} + q^{7} + ( - \beta_{6} + \beta_{5} + 2 \beta_{4} + \cdots + 1) q^{9}+ \cdots + (\beta_{8} + \beta_{7} - 4 \beta_{6} + \cdots - 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 6 q^{3} - 4 q^{5} + 9 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 6 q^{3} - 4 q^{5} + 9 q^{7} + 7 q^{9} - 5 q^{11} + 5 q^{15} + q^{17} - q^{19} - 6 q^{21} - 27 q^{23} + 7 q^{25} - 30 q^{27} - 14 q^{29} - 21 q^{31} + 19 q^{33} - 4 q^{35} - 14 q^{37} + 14 q^{41} - 19 q^{43} - 13 q^{45} + 2 q^{47} + 9 q^{49} - 10 q^{51} - 30 q^{53} - 2 q^{55} - 28 q^{57} + 28 q^{59} - 12 q^{61} + 7 q^{63} + 31 q^{67} + 19 q^{69} - 18 q^{71} - 3 q^{73} - 27 q^{75} - 5 q^{77} + q^{79} + 53 q^{81} - 5 q^{83} + 21 q^{85} - 31 q^{87} + 4 q^{89} + 3 q^{93} - 41 q^{95} - 3 q^{97} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 15x^{7} - 2x^{6} + 68x^{5} + 13x^{4} - 112x^{3} - 36x^{2} + 61x + 29 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -15\nu^{8} + 163\nu^{7} + 296\nu^{6} - 2198\nu^{5} - 2118\nu^{4} + 7723\nu^{3} + 4860\nu^{2} - 6440\nu - 3838 ) / 337 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 43\nu^{8} + 27\nu^{7} - 534\nu^{6} - 484\nu^{5} + 1421\nu^{4} + 1945\nu^{3} + 222\nu^{2} - 2208\nu - 1287 ) / 337 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -56\nu^{8} - 43\nu^{7} + 813\nu^{6} + 646\nu^{5} - 3324\nu^{4} - 2149\nu^{3} + 4327\nu^{2} + 1794\nu - 1208 ) / 337 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 114\nu^{8} - 93\nu^{7} - 1643\nu^{6} + 1068\nu^{5} + 6863\nu^{4} - 3629\nu^{3} - 8628\nu^{2} + 2438\nu + 2411 ) / 337 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 118 \nu^{8} + 114 \nu^{7} + 1677 \nu^{6} - 1407 \nu^{5} - 6956 \nu^{4} + 5329 \nu^{3} + 9587 \nu^{2} + \cdots - 4760 ) / 337 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 199 \nu^{8} - 118 \nu^{7} - 2871 \nu^{6} + 1279 \nu^{5} + 12125 \nu^{4} - 4369 \nu^{3} - 16959 \nu^{2} + \cdots + 7085 ) / 337 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 303 \nu^{8} + 327 \nu^{7} + 4429 \nu^{6} - 4027 \nu^{5} - 19598 \nu^{4} + 14869 \nu^{3} + \cdots - 15924 ) / 337 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + 2\beta_{7} - \beta_{6} + 2\beta_{4} - 2\beta_{3} + \beta_{2} + 5\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{7} + 8\beta_{5} + 12\beta_{4} - 11\beta_{3} + 11\beta_{2} + 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11\beta_{8} + 19\beta_{7} - 15\beta_{6} + 2\beta_{5} + 21\beta_{4} - 25\beta_{3} + 13\beta_{2} + 33\beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{8} + 27\beta_{7} - 5\beta_{6} + 63\beta_{5} + 118\beta_{4} - 102\beta_{3} + 103\beta_{2} + 5\beta _1 + 212 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 101 \beta_{8} + 165 \beta_{7} - 154 \beta_{6} + 34 \beta_{5} + 200 \beta_{4} - 255 \beta_{3} + \cdots + 234 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 40 \beta_{8} + 289 \beta_{7} - 89 \beta_{6} + 514 \beta_{5} + 1084 \beta_{4} - 921 \beta_{3} + \cdots + 1760 \) Copy content Toggle raw display

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

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
−1.11607
−2.72084
−0.794150
−2.13429
1.14100
−0.634469
3.02399
1.82297
1.41186
0 −3.36305 0 2.18646 0 1.00000 0 8.31012 0
1.2 0 −3.27580 0 −3.07655 0 1.00000 0 7.73088 0
1.3 0 −1.34911 0 0.413269 0 1.00000 0 −1.17991 0
1.4 0 −1.33235 0 −1.52015 0 1.00000 0 −1.22484 0
1.5 0 −1.10598 0 −4.06755 0 1.00000 0 −1.77680 0
1.6 0 0.167469 0 3.79534 0 1.00000 0 −2.97195 0
1.7 0 0.777013 0 0.436047 0 1.00000 0 −2.39625 0
1.8 0 1.26801 0 −0.138655 0 1.00000 0 −1.39214 0
1.9 0 2.21380 0 −2.02821 0 1.00000 0 1.90090 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( -1 \)
\(13\) \( +1 \)

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 4732.2.a.u ✓ 9
13.b even 2 1 4732.2.a.v yes 9
13.d odd 4 2 4732.2.g.l 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4732.2.a.u ✓ 9 1.a even 1 1 trivial
4732.2.a.v yes 9 13.b even 2 1
4732.2.g.l 18 13.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4732))\):

\( T_{3}^{9} + 6T_{3}^{8} + T_{3}^{7} - 44T_{3}^{6} - 45T_{3}^{5} + 77T_{3}^{4} + 93T_{3}^{3} - 40T_{3}^{2} - 44T_{3} + 8 \) Copy content Toggle raw display
\( T_{5}^{9} + 4T_{5}^{8} - 18T_{5}^{7} - 79T_{5}^{6} + 44T_{5}^{5} + 328T_{5}^{4} + 89T_{5}^{3} - 208T_{5}^{2} + 28T_{5} + 8 \) Copy content Toggle raw display
\( T_{11}^{9} + 5 T_{11}^{8} - 32 T_{11}^{7} - 206 T_{11}^{6} - 39 T_{11}^{5} + 1678 T_{11}^{4} + \cdots - 181 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( T^{9} + 6 T^{8} + \cdots + 8 \) Copy content Toggle raw display
$5$ \( T^{9} + 4 T^{8} + \cdots + 8 \) Copy content Toggle raw display
$7$ \( (T - 1)^{9} \) Copy content Toggle raw display
$11$ \( T^{9} + 5 T^{8} + \cdots - 181 \) Copy content Toggle raw display
$13$ \( T^{9} \) Copy content Toggle raw display
$17$ \( T^{9} - T^{8} + \cdots + 21064 \) Copy content Toggle raw display
$19$ \( T^{9} + T^{8} + \cdots + 1352 \) Copy content Toggle raw display
$23$ \( T^{9} + 27 T^{8} + \cdots + 897001 \) Copy content Toggle raw display
$29$ \( T^{9} + 14 T^{8} + \cdots + 710137 \) Copy content Toggle raw display
$31$ \( T^{9} + 21 T^{8} + \cdots + 3138008 \) Copy content Toggle raw display
$37$ \( T^{9} + 14 T^{8} + \cdots + 12147239 \) Copy content Toggle raw display
$41$ \( T^{9} - 14 T^{8} + \cdots + 248968 \) Copy content Toggle raw display
$43$ \( T^{9} + 19 T^{8} + \cdots - 29107 \) Copy content Toggle raw display
$47$ \( T^{9} - 2 T^{8} + \cdots - 1490488 \) Copy content Toggle raw display
$53$ \( T^{9} + 30 T^{8} + \cdots - 2644993 \) Copy content Toggle raw display
$59$ \( T^{9} - 28 T^{8} + \cdots + 7224952 \) Copy content Toggle raw display
$61$ \( T^{9} + 12 T^{8} + \cdots + 40768 \) Copy content Toggle raw display
$67$ \( T^{9} - 31 T^{8} + \cdots + 425852167 \) Copy content Toggle raw display
$71$ \( T^{9} + 18 T^{8} + \cdots - 244223 \) Copy content Toggle raw display
$73$ \( T^{9} + 3 T^{8} + \cdots + 428731192 \) Copy content Toggle raw display
$79$ \( T^{9} - T^{8} + \cdots - 22399 \) Copy content Toggle raw display
$83$ \( T^{9} + 5 T^{8} + \cdots - 2484616 \) Copy content Toggle raw display
$89$ \( T^{9} - 4 T^{8} + \cdots - 719368 \) Copy content Toggle raw display
$97$ \( T^{9} + 3 T^{8} + \cdots - 271496 \) Copy content Toggle raw display
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