Properties

Label 483.2.h.c
Level $483$
Weight $2$
Character orbit 483.h
Analytic conductor $3.857$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [483,2,Mod(160,483)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(483, 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("483.160");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.h (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: \(3.85677441763\)
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: \( x^{12} + 17x^{10} + 92x^{8} + 180x^{6} + 92x^{4} + 17x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + \beta_{9} q^{3} + (\beta_{7} + 3) q^{4} - \beta_{5} q^{5} + \beta_{4} q^{6} + (\beta_{3} + \beta_1) q^{7} + (\beta_{7} - 3 \beta_{2} - 3) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + \beta_{9} q^{3} + (\beta_{7} + 3) q^{4} - \beta_{5} q^{5} + \beta_{4} q^{6} + (\beta_{3} + \beta_1) q^{7} + (\beta_{7} - 3 \beta_{2} - 3) q^{8} - q^{9} + ( - \beta_{6} + \beta_{5} - \beta_1) q^{10} + ( - \beta_{10} + \beta_{3}) q^{11} + (3 \beta_{9} + \beta_{8}) q^{12} + (2 \beta_{9} - \beta_{4}) q^{13} + (\beta_{11} + \beta_{5} + \cdots + 2 \beta_1) q^{14}+ \cdots + (\beta_{10} - \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{2} + 36 q^{4} - 24 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{2} + 36 q^{4} - 24 q^{8} - 12 q^{9} + 68 q^{16} - 4 q^{18} + 12 q^{23} - 16 q^{25} - 16 q^{29} - 44 q^{32} + 8 q^{35} - 36 q^{36} - 20 q^{39} + 32 q^{46} - 4 q^{49} - 64 q^{50} - 92 q^{58} + 112 q^{64} - 28 q^{70} + 20 q^{71} + 24 q^{72} - 52 q^{77} + 52 q^{78} + 12 q^{81} - 44 q^{85} - 44 q^{92} + 40 q^{93} + 52 q^{95} + 116 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 17x^{10} + 92x^{8} + 180x^{6} + 92x^{4} + 17x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -19\nu^{10} - 322\nu^{8} - 1730\nu^{6} - 3310\nu^{4} - 1478\nu^{2} - 141 ) / 20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{10} + 17\nu^{8} + 92\nu^{6} + 180\nu^{4} + 91\nu^{2} + 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{11} - 17\nu^{9} - 92\nu^{7} - 180\nu^{5} - 92\nu^{3} - 16\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6\nu^{11} + 98\nu^{9} + 485\nu^{7} + 730\nu^{5} - 63\nu^{3} - 56\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 33\nu^{10} + 554\nu^{8} + 2920\nu^{6} + 5340\nu^{4} + 1966\nu^{2} + 187 ) / 10 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 93\nu^{10} + 1554\nu^{8} + 8110\nu^{6} + 14450\nu^{4} + 4526\nu^{2} + 267 ) / 20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 5\nu^{10} + 84\nu^{8} + 443\nu^{6} + 809\nu^{4} + 292\nu^{2} + 26 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -63\nu^{11} - 1054\nu^{9} - 5510\nu^{7} - 9830\nu^{5} - 3046\nu^{3} - 97\nu ) / 20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 83\nu^{11} + 1394\nu^{9} + 7350\nu^{7} + 13430\nu^{5} + 4886\nu^{3} + 457\nu ) / 20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -11\nu^{11} - 184\nu^{9} - 962\nu^{7} - 1720\nu^{5} - 550\nu^{3} - 39\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( 9\nu^{11} + 152\nu^{9} + 811\nu^{7} + 1528\nu^{5} + 649\nu^{3} + 69\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{9} + \beta_{8} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + 2\beta_{5} - \beta_{2} + \beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} - 8\beta_{9} - 4\beta_{8} + 3\beta_{4} - 8\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{7} + 8\beta_{6} - 17\beta_{5} + 12\beta_{2} - 2\beta _1 + 39 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{11} + \beta_{10} + 37\beta_{9} + 11\beta_{8} - 17\beta_{4} + 30\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -17\beta_{7} - 64\beta_{6} + 147\beta_{5} - 112\beta_{2} - 10\beta _1 - 291 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 143\beta_{11} - 36\beta_{10} - 672\beta_{9} - 140\beta_{8} + 319\beta_{4} - 480\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 196\beta_{7} + 523\beta_{6} - 1278\beta_{5} + 985\beta_{2} + 189\beta _1 + 2318 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -1324\beta_{11} + 428\beta_{10} + 5973\beta_{9} + 997\beta_{8} - 2832\beta_{4} + 3979\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -974\beta_{7} - 2176\beta_{6} + 5540\beta_{5} - 4254\beta_{2} - 1012\beta _1 - 9565 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 11780\beta_{11} - 4324\beta_{10} - 52317\beta_{9} - 7677\beta_{8} + 24640\beta_{4} - 33565\beta_{3} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\) \(323\) \(346\) \(442\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
160.1
0.341398i
2.92914i
0.341398i
2.92914i
2.14584i
0.466018i
2.14584i
0.466018i
0.562016i
1.77931i
0.562016i
1.77931i
−2.69639 1.00000i 5.27053 −2.58774 2.69639i −0.551006 + 2.58774i −8.81864 −1.00000 6.97756
160.2 −2.69639 1.00000i 5.27053 2.58774 2.69639i 0.551006 2.58774i −8.81864 −1.00000 −6.97756
160.3 −2.69639 1.00000i 5.27053 −2.58774 2.69639i −0.551006 2.58774i −8.81864 −1.00000 6.97756
160.4 −2.69639 1.00000i 5.27053 2.58774 2.69639i 0.551006 + 2.58774i −8.81864 −1.00000 −6.97756
160.5 1.17819 1.00000i −0.611859 −1.67982 1.17819i −2.04406 + 1.67982i −3.07728 −1.00000 −1.97916
160.6 1.17819 1.00000i −0.611859 1.67982 1.17819i 2.04406 1.67982i −3.07728 −1.00000 1.97916
160.7 1.17819 1.00000i −0.611859 −1.67982 1.17819i −2.04406 1.67982i −3.07728 −1.00000 −1.97916
160.8 1.17819 1.00000i −0.611859 1.67982 1.17819i 2.04406 + 1.67982i −3.07728 −1.00000 1.97916
160.9 2.51820 1.00000i 4.34132 −1.21729 2.51820i 2.34908 + 1.21729i 5.89592 −1.00000 −3.06538
160.10 2.51820 1.00000i 4.34132 1.21729 2.51820i −2.34908 1.21729i 5.89592 −1.00000 3.06538
160.11 2.51820 1.00000i 4.34132 −1.21729 2.51820i 2.34908 1.21729i 5.89592 −1.00000 −3.06538
160.12 2.51820 1.00000i 4.34132 1.21729 2.51820i −2.34908 + 1.21729i 5.89592 −1.00000 3.06538
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 160.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
23.b odd 2 1 inner
161.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 483.2.h.c 12
3.b odd 2 1 1449.2.h.e 12
7.b odd 2 1 inner 483.2.h.c 12
21.c even 2 1 1449.2.h.e 12
23.b odd 2 1 inner 483.2.h.c 12
69.c even 2 1 1449.2.h.e 12
161.c even 2 1 inner 483.2.h.c 12
483.c odd 2 1 1449.2.h.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.h.c 12 1.a even 1 1 trivial
483.2.h.c 12 7.b odd 2 1 inner
483.2.h.c 12 23.b odd 2 1 inner
483.2.h.c 12 161.c even 2 1 inner
1449.2.h.e 12 3.b odd 2 1
1449.2.h.e 12 21.c even 2 1
1449.2.h.e 12 69.c even 2 1
1449.2.h.e 12 483.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - T_{2}^{2} - 7T_{2} + 8 \) acting on \(S_{2}^{\mathrm{new}}(483, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{3} - T^{2} - 7 T + 8)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$5$ \( (T^{6} - 11 T^{4} + \cdots - 28)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + 2 T^{10} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( (T^{6} + 25 T^{4} + \cdots + 175)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 23 T^{4} + 21 T^{2} + 4)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 62 T^{4} + \cdots - 1792)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 25 T^{4} + \cdots - 175)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 6 T^{5} + \cdots + 12167)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 4 T^{2} - 19 T - 50)^{4} \) Copy content Toggle raw display
$31$ \( (T^{6} + 54 T^{4} + \cdots + 100)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 114 T^{4} + \cdots + 1372)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 25)^{6} \) Copy content Toggle raw display
$43$ \( (T^{6} + 95 T^{4} + \cdots + 1372)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 197 T^{4} + \cdots + 71824)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 86 T^{4} + 1662 T^{2} + 7)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 132 T^{4} + \cdots + 49)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 194 T^{4} + \cdots - 229327)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 99 T^{4} + \cdots + 20412)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} - 5 T^{2} + \cdots + 128)^{4} \) Copy content Toggle raw display
$73$ \( (T^{6} + 450 T^{4} + \cdots + 2458624)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 386 T^{4} + \cdots + 51772)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 470 T^{4} + \cdots - 2903152)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 171 T^{4} + \cdots - 17500)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 614 T^{4} + \cdots - 4435312)^{2} \) Copy content Toggle raw display
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