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Properties

Label 4732.2.g.j
Level $4732$
Weight $2$
Character orbit 4732.g
Analytic conductor $37.785$
Analytic rank $0$
Dimension $12$
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 21x^{10} + 152x^{8} + 490x^{6} + 776x^{4} + 588x^{2} + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{8} - \beta_{5} + 1) q^{3} + ( - \beta_{6} - \beta_{4} + \cdots - \beta_1) q^{5}+ \cdots + ( - \beta_{11} + \beta_{9} + \beta_{8} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{8} - \beta_{5} + 1) q^{3} + ( - \beta_{6} - \beta_{4} + \cdots - \beta_1) q^{5}+ \cdots + (\beta_{10} + 2 \beta_{6} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{3} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{3} + 40 q^{9} + 6 q^{17} - 36 q^{23} - 4 q^{25} + 48 q^{27} + 34 q^{29} - 28 q^{43} - 12 q^{49} - 4 q^{51} + 6 q^{53} + 36 q^{55} - 8 q^{61} - 78 q^{69} - 74 q^{75} - 8 q^{77} + 44 q^{79} + 12 q^{81} + 102 q^{87} - 38 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 21x^{10} + 152x^{8} + 490x^{6} + 776x^{4} + 588x^{2} + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 12\nu^{11} + 229\nu^{9} + 1383\nu^{7} + 3191\nu^{5} + 2984\nu^{3} + 937\nu ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\nu^{11} + 294\nu^{9} + 1870\nu^{7} + 4762\nu^{5} + 5145\nu^{3} + 1954\nu ) / 13 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -22\nu^{11} - 424\nu^{9} - 2612\nu^{7} - 6274\nu^{5} - 6257\nu^{3} - 2143\nu ) / 13 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -36\nu^{10} - 688\nu^{8} - 4171\nu^{6} - 9734\nu^{4} - 9395\nu^{2} - 3146 ) / 13 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -42\nu^{11} - 804\nu^{9} - 4889\nu^{7} - 11467\nu^{5} - 11116\nu^{3} - 3727\nu ) / 13 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 57\nu^{10} + 1098\nu^{8} + 6759\nu^{6} + 16229\nu^{4} + 16261\nu^{2} + 5668 ) / 13 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -70\nu^{10} - 1345\nu^{8} - 8241\nu^{6} - 19635\nu^{4} - 19537\nu^{2} - 6786 ) / 13 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( 6\nu^{10} + 115\nu^{8} + 701\nu^{6} + 1652\nu^{4} + 1613\nu^{2} + 546 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -93\nu^{11} - 1786\nu^{9} - 10930\nu^{7} - 25963\nu^{5} - 25656\nu^{3} - 8827\nu ) / 13 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 116\nu^{10} + 2227\nu^{8} + 13619\nu^{6} + 32291\nu^{4} + 31762\nu^{2} + 10803 ) / 13 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} + \beta_{8} + 2\beta_{7} - 2\beta_{5} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{10} + 4\beta_{6} + 3\beta_{4} + 4\beta_{2} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{11} + 3\beta_{9} - 11\beta_{8} - 20\beta_{7} + 22\beta_{5} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 25\beta_{10} - 52\beta_{6} - 27\beta_{4} + 5\beta_{3} - 44\beta_{2} + 36\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -63\beta_{11} - 50\beta_{9} + 105\beta_{8} + 186\beta_{7} - 221\beta_{5} - 115 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -271\beta_{10} + 571\beta_{6} + 231\beta_{4} - 85\beta_{3} + 428\beta_{2} - 301\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 532\beta_{11} + 611\beta_{9} - 1000\beta_{8} - 1758\beta_{7} + 2199\beta_{5} + 948 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2810\beta_{10} - 5949\beta_{6} - 2064\beta_{4} + 1052\beta_{3} - 4143\beta_{2} + 2706\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -4770\beta_{11} - 6695\beta_{9} + 9659\beta_{8} + 16933\beta_{7} - 21847\beta_{5} - 8424 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -28542\beta_{10} + 60552\beta_{6} + 19199\beta_{4} - 11609\beta_{3} + 40442\beta_{2} - 25357\beta_1 \) 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
3.15013i
− 3.15013i
0.894429i
− 0.894429i
2.33921i
− 2.33921i
− 1.33947i
1.33947i
1.34819i
− 1.34819i
1.09223i
− 1.09223i
0 −2.48311 0 − 1.84698i 0 − 1.00000i 0 3.16581 0
337.2 0 −2.48311 0 1.84698i 0 1.00000i 0 3.16581 0
337.3 0 −1.85869 0 − 0.131645i 0 1.00000i 0 0.454712 0
337.4 0 −1.85869 0 0.131645i 0 − 1.00000i 0 0.454712 0
337.5 0 1.76089 0 − 2.41317i 0 1.00000i 0 0.100737 0
337.6 0 1.76089 0 2.41317i 0 − 1.00000i 0 0.100737 0
337.7 0 2.16666 0 − 2.91727i 0 1.00000i 0 1.69443 0
337.8 0 2.16666 0 2.91727i 0 − 1.00000i 0 1.69443 0
337.9 0 3.12621 0 − 0.154960i 0 1.00000i 0 6.77319 0
337.10 0 3.12621 0 0.154960i 0 − 1.00000i 0 6.77319 0
337.11 0 3.28803 0 − 3.77007i 0 − 1.00000i 0 7.81112 0
337.12 0 3.28803 0 3.77007i 0 1.00000i 0 7.81112 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.12
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.j 12
13.b even 2 1 inner 4732.2.g.j 12
13.d odd 4 1 4732.2.a.q ✓ 6
13.d odd 4 1 4732.2.a.r yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4732.2.a.q ✓ 6 13.d odd 4 1
4732.2.a.r yes 6 13.d odd 4 1
4732.2.g.j 12 1.a even 1 1 trivial
4732.2.g.j 12 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 \(S_{2}^{\mathrm{new}}(4732, [\chi])\):

\( T_{3}^{6} - 6T_{3}^{5} - T_{3}^{4} + 58T_{3}^{3} - 61T_{3}^{2} - 129T_{3} + 181 \) Copy content Toggle raw display
\( T_{5}^{12} + 32T_{5}^{10} + 352T_{5}^{8} + 1583T_{5}^{6} + 2468T_{5}^{4} + 100T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{17}^{6} - 3T_{17}^{5} - 44T_{17}^{4} + 9T_{17}^{3} + 603T_{17}^{2} + 1268T_{17} + 727 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} - 6 T^{5} + \cdots + 181)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + 32 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + 44 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( (T^{6} - 3 T^{5} + \cdots + 727)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 39 T^{10} + \cdots + 169 \) Copy content Toggle raw display
$23$ \( (T^{6} + 18 T^{5} + \cdots - 97)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 17 T^{5} + \cdots + 29)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 111 T^{10} + \cdots + 877969 \) Copy content Toggle raw display
$37$ \( T^{12} + 119 T^{10} + \cdots + 78961 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 1335244681 \) Copy content Toggle raw display
$43$ \( (T^{6} + 14 T^{5} + \cdots + 4067)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 248 T^{10} + \cdots + 83594449 \) Copy content Toggle raw display
$53$ \( (T^{6} - 3 T^{5} + \cdots + 3653)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 10660356001 \) Copy content Toggle raw display
$61$ \( (T^{6} + 4 T^{5} + \cdots + 3263)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 380 T^{10} + \cdots + 52809289 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 467813641 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 5564115649 \) Copy content Toggle raw display
$79$ \( (T^{6} - 22 T^{5} + \cdots - 22919)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 369312659521 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 14321148241 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 2029721650489 \) Copy content Toggle raw display
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