Properties

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

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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: \( x^{12} + 14x^{10} + 67x^{8} + 141x^{6} + 129x^{4} + 39x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_{8} q^{2} - \beta_{5} q^{3} + ( - \beta_{7} + 1) q^{4} + ( - \beta_{2} + \beta_1) q^{5} - \beta_{3} q^{6} + ( - \beta_{11} - \beta_{4}) q^{7} - q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{2} - \beta_{5} q^{3} + ( - \beta_{7} + 1) q^{4} + ( - \beta_{2} + \beta_1) q^{5} - \beta_{3} q^{6} + ( - \beta_{11} - \beta_{4}) q^{7} - q^{8} - q^{9} + ( - \beta_{11} - \beta_{9} + \cdots + 2 \beta_1) q^{10}+ \cdots + \beta_{4} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{4} - 12 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 8 q^{4} - 12 q^{8} - 12 q^{9} - 16 q^{16} - 16 q^{23} + 24 q^{25} + 16 q^{29} + 16 q^{32} - 12 q^{35} - 8 q^{36} + 28 q^{39} - 76 q^{46} - 16 q^{49} + 92 q^{50} + 44 q^{58} - 84 q^{64} + 64 q^{70} + 76 q^{71} + 12 q^{72} - 36 q^{77} - 52 q^{78} + 12 q^{81} - 36 q^{85} + 56 q^{92} - 80 q^{93} - 140 q^{95} - 108 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 14x^{10} + 67x^{8} + 141x^{6} + 129x^{4} + 39x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3\nu^{10} + 31\nu^{8} + 78\nu^{6} + 39\nu^{4} - 8\nu^{2} + 25 ) / 14 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{10} + 23\nu^{8} + 80\nu^{6} + 124\nu^{4} + 95\nu^{2} + 12 ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{11} - 15\nu^{9} - 82\nu^{7} - 209\nu^{5} - 212\nu^{3} - 27\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{11} + 23\nu^{9} + 73\nu^{7} + 54\nu^{5} - 66\nu^{3} - 58\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{11} - 45\nu^{9} - 232\nu^{7} - 501\nu^{5} - 426\nu^{3} - 95\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\nu^{9} - 12\nu^{7} - 43\nu^{5} - 55\nu^{3} - 19\nu \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -6\nu^{10} - 69\nu^{8} - 226\nu^{6} - 239\nu^{4} - 26\nu^{2} + 20 ) / 7 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -5\nu^{10} - 61\nu^{8} - 228\nu^{6} - 324\nu^{4} - 143\nu^{2} - 2 ) / 7 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 21 \nu^{11} + 13 \nu^{10} + 259 \nu^{9} + 167 \nu^{8} + 994 \nu^{7} + 674 \nu^{6} + 1519 \nu^{5} + \cdots + 43 ) / 28 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -9\nu^{11} - 121\nu^{9} - 542\nu^{7} - 1041\nu^{5} - 830\nu^{3} - 173\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 21 \nu^{11} + 13 \nu^{10} - 259 \nu^{9} + 167 \nu^{8} - 994 \nu^{7} + 674 \nu^{6} - 1519 \nu^{5} + \cdots + 43 ) / 28 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{8} + \beta_{7} - \beta_{2} + 2\beta _1 - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{11} + 3\beta_{10} + \beta_{9} + 4\beta_{6} - 7\beta_{5} - 5\beta_{4} - 5\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{11} + 2\beta_{9} + 11\beta_{8} - 7\beta_{7} + 9\beta_{2} - 12\beta _1 + 23 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{11} - 13\beta_{10} - 4\beta_{9} - 11\beta_{6} + 24\beta_{5} + 14\beta_{4} + 17\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -19\beta_{11} - 19\beta_{9} - 86\beta_{8} + 49\beta_{7} - 64\beta_{2} + 77\beta _1 - 134 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -57\beta_{11} + 189\beta_{10} + 57\beta_{9} + 137\beta_{6} - 324\beta_{5} - 176\beta_{4} - 236\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 144\beta_{11} + 144\beta_{9} + 613\beta_{8} - 337\beta_{7} + 439\beta_{2} - 514\beta _1 + 861 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 395\beta_{11} - 1315\beta_{10} - 395\beta_{9} - 901\beta_{6} + 2190\beta_{5} + 1164\beta_{4} + 1626\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -510\beta_{11} - 510\beta_{9} - 2122\beta_{8} + 1151\beta_{7} - 1496\beta_{2} + 1740\beta _1 - 2871 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 2711 \beta_{11} + 9025 \beta_{10} + 2711 \beta_{9} + 6058 \beta_{6} - 14857 \beta_{5} + \cdots - 11139 \beta_{3} ) / 2 \) 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
1.64421i
1.28628i
1.64421i
1.28628i
1.51343i
0.167898i
1.51343i
0.167898i
2.61062i
0.712783i
2.61062i
0.712783i
−2.11491 1.00000i 2.47283 −1.98969 2.11491i 2.62846 0.302029i −1.00000 −1.00000 4.20801
160.2 −2.11491 1.00000i 2.47283 1.98969 2.11491i −2.62846 + 0.302029i −1.00000 −1.00000 −4.20801
160.3 −2.11491 1.00000i 2.47283 −1.98969 2.11491i 2.62846 + 0.302029i −1.00000 −1.00000 4.20801
160.4 −2.11491 1.00000i 2.47283 1.98969 2.11491i −2.62846 0.302029i −1.00000 −1.00000 −4.20801
160.5 0.254102 1.00000i −1.93543 −0.458377 0.254102i 1.07291 + 2.41844i −1.00000 −1.00000 −0.116474
160.6 0.254102 1.00000i −1.93543 0.458377 0.254102i −1.07291 2.41844i −1.00000 −1.00000 0.116474
160.7 0.254102 1.00000i −1.93543 −0.458377 0.254102i 1.07291 2.41844i −1.00000 −1.00000 −0.116474
160.8 0.254102 1.00000i −1.93543 0.458377 0.254102i −1.07291 + 2.41844i −1.00000 −1.00000 0.116474
160.9 1.86081 1.00000i 1.46260 −4.10256 1.86081i −0.663393 2.56123i −1.00000 −1.00000 −7.63407
160.10 1.86081 1.00000i 1.46260 4.10256 1.86081i 0.663393 + 2.56123i −1.00000 −1.00000 7.63407
160.11 1.86081 1.00000i 1.46260 −4.10256 1.86081i −0.663393 + 2.56123i −1.00000 −1.00000 −7.63407
160.12 1.86081 1.00000i 1.46260 4.10256 1.86081i 0.663393 2.56123i −1.00000 −1.00000 7.63407
\(n\): 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.b 12
3.b odd 2 1 1449.2.h.h 12
7.b odd 2 1 inner 483.2.h.b 12
21.c even 2 1 1449.2.h.h 12
23.b odd 2 1 inner 483.2.h.b 12
69.c even 2 1 1449.2.h.h 12
161.c even 2 1 inner 483.2.h.b 12
483.c odd 2 1 1449.2.h.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.h.b 12 1.a even 1 1 trivial
483.2.h.b 12 7.b odd 2 1 inner
483.2.h.b 12 23.b odd 2 1 inner
483.2.h.b 12 161.c even 2 1 inner
1449.2.h.h 12 3.b odd 2 1
1449.2.h.h 12 21.c even 2 1
1449.2.h.h 12 69.c even 2 1
1449.2.h.h 12 483.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 4T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(483, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{3} - 4 T + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$5$ \( (T^{6} - 21 T^{4} + \cdots - 14)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + 8 T^{10} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( (T^{6} + 14 T^{4} + \cdots + 56)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 47 T^{4} + \cdots + 2116)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 52 T^{4} + \cdots - 14)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 84 T^{4} + \cdots - 19166)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 8 T^{5} + \cdots + 12167)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 4 T^{2} - 7 T + 26)^{4} \) Copy content Toggle raw display
$31$ \( (T^{6} + 146 T^{4} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 166 T^{4} + \cdots + 157304)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 158 T^{4} + \cdots + 51076)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 241 T^{4} + \cdots + 240254)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 176 T^{4} + \cdots + 153664)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 217 T^{4} + \cdots + 270494)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 119 T^{4} + \cdots + 33124)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 119 T^{4} + \cdots - 2744)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 141 T^{4} + \cdots + 39326)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} - 19 T^{2} + \cdots - 134)^{4} \) Copy content Toggle raw display
$73$ \( (T^{6} + 114 T^{4} + \cdots + 2704)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 324 T^{4} + \cdots + 734174)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 54 T^{4} + \cdots - 3584)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 277 T^{4} + \cdots - 14)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 374 T^{4} + \cdots - 251384)^{2} \) Copy content Toggle raw display
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