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Properties

Label 4732.2.g.e
Level $4732$
Weight $2$
Character orbit 4732.g
Analytic conductor $37.785$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4732,2,Mod(337,4732)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4732.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4732 = 2^{2} \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4732.g (of order \(2\), degree \(1\), 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: \(37.7852102365\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{21})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 11x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 364)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + ( - \beta_{2} + \beta_1) q^{5} - \beta_{2} q^{7} + (\beta_{3} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + ( - \beta_{2} + \beta_1) q^{5} - \beta_{2} q^{7} + (\beta_{3} + 2) q^{9} + (\beta_{2} - \beta_1) q^{11} + ( - 4 \beta_{2} + \beta_1) q^{15} - 3 q^{17} + ( - \beta_{2} - 3 \beta_1) q^{19} + (\beta_{2} + \beta_1) q^{21} - 6 q^{23} + (3 \beta_{3} - 4) q^{25} - 5 q^{27} + ( - \beta_{3} - 4) q^{29} - \beta_{2} q^{31} + (4 \beta_{2} - \beta_1) q^{33} + (\beta_{3} - 2) q^{35} + 4 \beta_{2} q^{37} + (4 \beta_{2} + 2 \beta_1) q^{41} + (3 \beta_{3} + 1) q^{43} + (2 \beta_{2} + \beta_1) q^{45} + (\beta_{2} - 4 \beta_1) q^{47} - q^{49} + 3 \beta_{3} q^{51} + (2 \beta_{3} - 7) q^{53} + ( - 3 \beta_{3} + 9) q^{55} + (16 \beta_{2} + \beta_1) q^{57} + 9 \beta_{2} q^{59} + 2 q^{61} + ( - 3 \beta_{2} - \beta_1) q^{63} - 7 \beta_{2} q^{67} + 6 \beta_{3} q^{69} + (2 \beta_{2} - 2 \beta_1) q^{71} - 14 \beta_{2} q^{73} + (\beta_{3} - 15) q^{75} + ( - \beta_{3} + 2) q^{77} - 4 q^{79} + (2 \beta_{3} - 6) q^{81} - 9 \beta_{2} q^{83} + (3 \beta_{2} - 3 \beta_1) q^{85} + (5 \beta_{3} + 5) q^{87} + ( - 3 \beta_{2} - 3 \beta_1) q^{89} + (\beta_{2} + \beta_1) q^{93} + ( - 5 \beta_{3} + 19) q^{95} + ( - 10 \beta_{2} + 3 \beta_1) q^{97} + ( - 2 \beta_{2} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} + 10 q^{9} - 12 q^{17} - 24 q^{23} - 10 q^{25} - 20 q^{27} - 18 q^{29} - 6 q^{35} + 10 q^{43} - 4 q^{49} + 6 q^{51} - 24 q^{53} + 30 q^{55} + 8 q^{61} + 12 q^{69} - 58 q^{75} + 6 q^{77} - 16 q^{79} - 20 q^{81} + 30 q^{87} + 66 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 11x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 6\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 6 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{2} - 6\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4732\mathbb{Z}\right)^\times\).

\(n\) \(2367\) \(2705\) \(4565\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
− 1.79129i
1.79129i
− 2.79129i
2.79129i
0 −2.79129 0 − 0.791288i 0 1.00000i 0 4.79129 0
337.2 0 −2.79129 0 0.791288i 0 − 1.00000i 0 4.79129 0
337.3 0 1.79129 0 − 3.79129i 0 − 1.00000i 0 0.208712 0
337.4 0 1.79129 0 3.79129i 0 1.00000i 0 0.208712 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4732.2.g.e 4
13.b even 2 1 inner 4732.2.g.e 4
13.d odd 4 1 4732.2.a.g 2
13.d odd 4 1 4732.2.a.h 2
13.f odd 12 2 364.2.k.d ✓ 4
39.k even 12 2 3276.2.z.e 4
52.l even 12 2 1456.2.s.k 4
91.w even 12 2 2548.2.l.l 4
91.x odd 12 2 2548.2.i.k 4
91.ba even 12 2 2548.2.i.i 4
91.bc even 12 2 2548.2.k.e 4
91.bd odd 12 2 2548.2.l.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
364.2.k.d ✓ 4 13.f odd 12 2
1456.2.s.k 4 52.l even 12 2
2548.2.i.i 4 91.ba even 12 2
2548.2.i.k 4 91.x odd 12 2
2548.2.k.e 4 91.bc even 12 2
2548.2.l.j 4 91.bd odd 12 2
2548.2.l.l 4 91.w even 12 2
3276.2.z.e 4 39.k even 12 2
4732.2.a.g 2 13.d odd 4 1
4732.2.a.h 2 13.d odd 4 1
4732.2.g.e 4 1.a even 1 1 trivial
4732.2.g.e 4 13.b 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 \(S_{2}^{\mathrm{new}}(4732, [\chi])\):

\( T_{3}^{2} + T_{3} - 5 \) Copy content Toggle raw display
\( T_{5}^{4} + 15T_{5}^{2} + 9 \) Copy content Toggle raw display
\( T_{17} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + T - 5)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 15T^{2} + 9 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 15T^{2} + 9 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T + 3)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 95T^{2} + 2209 \) Copy content Toggle raw display
$23$ \( (T + 6)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 9 T + 15)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 60T^{2} + 144 \) Copy content Toggle raw display
$43$ \( (T^{2} - 5 T - 41)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 186T^{2} + 5625 \) Copy content Toggle raw display
$53$ \( (T^{2} + 12 T + 15)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 60T^{2} + 144 \) Copy content Toggle raw display
$73$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$79$ \( (T + 4)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 99T^{2} + 2025 \) Copy content Toggle raw display
$97$ \( T^{4} + 359T^{2} + 7225 \) Copy content Toggle raw display
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