Properties

Label 1512.2.j.b
Level $1512$
Weight $2$
Character orbit 1512.j
Analytic conductor $12.073$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1512,2,Mod(323,1512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1512, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1512.323");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1512.j (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: \(12.0733807856\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.56070144.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) 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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_{3} + 1) q^{2} + (\beta_{5} - \beta_{3} - \beta_1) q^{4} + (\beta_{4} + 1) q^{5} + \beta_{3} q^{7} + ( - 2 \beta_{3} + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_{3} + 1) q^{2} + (\beta_{5} - \beta_{3} - \beta_1) q^{4} + (\beta_{4} + 1) q^{5} + \beta_{3} q^{7} + ( - 2 \beta_{3} + 2) q^{8} + (\beta_{7} + \beta_{5} + \beta_{3} + \beta_{2} + 1) q^{10} + (\beta_{5} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{11} + (\beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{3} + \beta_1 + 1) q^{13} + ( - \beta_1 - 1) q^{14} + (2 \beta_{5} + 2 \beta_{3} + 2 \beta_1 + 4) q^{16} - 2 \beta_{3} q^{17} + (\beta_{5} - 2 \beta_{4} - \beta_1 + 2) q^{19} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{20} + ( - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - 1) q^{22} + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_1 - 2) q^{23} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_1 + 4) q^{25} + (\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{26} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{28} + (\beta_{5} - \beta_1 + 3) q^{29} + ( - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - \beta_1 - 1) q^{31} + 4 \beta_{5} q^{32} + (2 \beta_1 + 2) q^{34} + (\beta_{3} + \beta_{2}) q^{35} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} + 3 \beta_{3} + 2 \beta_1 + 2) q^{37} + ( - 2 \beta_{7} + 2 \beta_{5} - 2 \beta_{2} - \beta_1 + 3) q^{38} + (2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 2) q^{40} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} + 5 \beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{41} + (\beta_{7} - \beta_{6} - 3 \beta_{5} + 3 \beta_1 + 1) q^{43} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - 3 \beta_{3} - \beta_{2} + \beta_1) q^{44} + ( - 2 \beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{4} - 4 \beta_{3} - \beta_1 - 1) q^{46} + (3 \beta_{5} + 2 \beta_{4} - 3 \beta_1 + 1) q^{47} - q^{49} + (2 \beta_{7} + 4 \beta_{5} + 8 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{50} + (2 \beta_{6} - 2 \beta_{5} - 4 \beta_{3} - 2 \beta_{2}) q^{52} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_1) q^{53} + (\beta_{7} + \beta_{6} + 3 \beta_{5} + 11 \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 3) q^{55} + (2 \beta_{3} + 2) q^{56} + (3 \beta_{5} + \beta_{3} - \beta_1 + 4) q^{58} + ( - \beta_{5} + 4 \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{59} + ( - 2 \beta_{5} + 6 \beta_{3} - 2 \beta_1 - 2) q^{61} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{62} - 8 \beta_{3} q^{64} + (2 \beta_{7} + 2 \beta_{6} + 7 \beta_{5} + 6 \beta_{3} + 2 \beta_{2} + 7 \beta_1 + 7) q^{65} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \beta_1 + 1) q^{67} + (2 \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{68} + ( - \beta_{6} - \beta_{4} - \beta_1 - 1) q^{70} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_1 - 4) q^{71} + ( - \beta_{7} + \beta_{6} - 3 \beta_{5} + 2 \beta_{4} + 3 \beta_1 + 1) q^{73} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 - 3) q^{74} + ( - 2 \beta_{7} + 2 \beta_{6} + 3 \beta_{5} + 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + \cdots + 4) q^{76}+ \cdots + ( - \beta_{5} - \beta_{3} - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} + 8 q^{5} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} + 8 q^{5} + 16 q^{8} + 4 q^{10} - 4 q^{14} + 16 q^{16} + 16 q^{19} - 12 q^{22} - 16 q^{23} + 32 q^{25} - 16 q^{26} + 8 q^{28} + 24 q^{29} - 16 q^{32} + 8 q^{34} + 20 q^{38} + 16 q^{40} + 8 q^{43} + 4 q^{46} + 8 q^{47} - 8 q^{49} - 8 q^{50} + 8 q^{52} + 16 q^{56} + 24 q^{58} + 12 q^{62} + 8 q^{67} - 16 q^{68} - 4 q^{70} - 32 q^{71} + 8 q^{73} - 28 q^{74} + 24 q^{76} + 16 q^{80} - 36 q^{82} - 32 q^{86} - 8 q^{91} + 24 q^{92} + 40 q^{94} - 112 q^{95} - 32 q^{97} - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -5\nu^{7} - \nu^{6} - 25\nu^{5} - 46\nu^{4} - 5\nu^{3} - 132\nu^{2} + 28\nu - 92 ) / 37 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -6\nu^{7} + 21\nu^{6} - 67\nu^{5} + 115\nu^{4} - 117\nu^{3} + 71\nu^{2} + 115\nu - 66 ) / 37 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -6\nu^{7} + 21\nu^{6} - 67\nu^{5} + 115\nu^{4} - 117\nu^{3} + 71\nu^{2} + 41\nu - 29 ) / 37 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{6} + 3\nu^{5} - 11\nu^{4} + 17\nu^{3} - 23\nu^{2} + 15\nu - 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{7} + 36\nu^{6} - 136\nu^{5} + 361\nu^{4} - 634\nu^{3} + 793\nu^{2} - 601\nu + 241 ) / 37 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -42\nu^{7} + 147\nu^{6} - 543\nu^{5} + 953\nu^{4} - 1485\nu^{3} + 1163\nu^{2} - 749\nu + 56 ) / 37 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -42\nu^{7} + 147\nu^{6} - 543\nu^{5} + 1027\nu^{4} - 1633\nu^{3} + 1681\nu^{2} - 1193\nu + 500 ) / 37 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} - \beta_{6} + 6\beta_{5} + 3\beta_{4} + 14\beta_{3} - 5\beta_{2} - 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{6} - \beta_{5} - 4\beta_{4} + 15\beta_{3} - 6\beta_{2} + 7\beta _1 + 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 9\beta_{7} - \beta_{6} - 40\beta_{5} - 25\beta_{4} - 42\beta_{3} + 11\beta_{2} + 10\beta _1 + 52 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 5\beta_{7} + 12\beta_{6} - 21\beta_{5} + 7\beta_{4} - 101\beta_{3} + 32\beta_{2} - 39\beta _1 - 40 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -23\beta_{7} + 19\beta_{6} + 84\beta_{5} + 70\beta_{4} + \beta_{3} + 3\beta_{2} - 70\beta _1 - 153 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1081\) \(1135\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
323.1
0.500000 + 2.19293i
0.500000 1.19293i
0.500000 2.19293i
0.500000 + 1.19293i
0.500000 1.56488i
0.500000 + 0.564882i
0.500000 + 1.56488i
0.500000 0.564882i
−0.366025 1.36603i 0 −1.73205 + 1.00000i −2.38587 0 1.00000i 2.00000 + 2.00000i 0 0.873288 + 3.25916i
323.2 −0.366025 1.36603i 0 −1.73205 + 1.00000i 4.38587 0 1.00000i 2.00000 + 2.00000i 0 −1.60534 5.99121i
323.3 −0.366025 + 1.36603i 0 −1.73205 1.00000i −2.38587 0 1.00000i 2.00000 2.00000i 0 0.873288 3.25916i
323.4 −0.366025 + 1.36603i 0 −1.73205 1.00000i 4.38587 0 1.00000i 2.00000 2.00000i 0 −1.60534 + 5.99121i
323.5 1.36603 0.366025i 0 1.73205 1.00000i −1.12976 0 1.00000i 2.00000 2.00000i 0 −1.54329 + 0.413523i
323.6 1.36603 0.366025i 0 1.73205 1.00000i 3.12976 0 1.00000i 2.00000 2.00000i 0 4.27534 1.14557i
323.7 1.36603 + 0.366025i 0 1.73205 + 1.00000i −1.12976 0 1.00000i 2.00000 + 2.00000i 0 −1.54329 0.413523i
323.8 1.36603 + 0.366025i 0 1.73205 + 1.00000i 3.12976 0 1.00000i 2.00000 + 2.00000i 0 4.27534 + 1.14557i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 323.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1512.2.j.b yes 8
3.b odd 2 1 1512.2.j.a 8
4.b odd 2 1 6048.2.j.b 8
8.b even 2 1 6048.2.j.a 8
8.d odd 2 1 1512.2.j.a 8
12.b even 2 1 6048.2.j.a 8
24.f even 2 1 inner 1512.2.j.b yes 8
24.h odd 2 1 6048.2.j.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.j.a 8 3.b odd 2 1
1512.2.j.a 8 8.d odd 2 1
1512.2.j.b yes 8 1.a even 1 1 trivial
1512.2.j.b yes 8 24.f even 2 1 inner
6048.2.j.a 8 8.b even 2 1
6048.2.j.a 8 12.b even 2 1
6048.2.j.b 8 4.b odd 2 1
6048.2.j.b 8 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 4T_{5}^{3} - 10T_{5}^{2} + 28T_{5} + 37 \) acting on \(S_{2}^{\mathrm{new}}(1512, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 4 T^{3} - 10 T^{2} + 28 T + 37)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} + 44 T^{6} + 510 T^{4} + \cdots + 169 \) Copy content Toggle raw display
$13$ \( T^{8} + 96 T^{6} + 2984 T^{4} + \cdots + 1936 \) Copy content Toggle raw display
$17$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 8 T^{3} - 46 T^{2} + 344 T + 193)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 8 T^{3} - 46 T^{2} - 464 T - 827)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 6 T + 6)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} + 92 T^{6} + 2310 T^{4} + \cdots + 9409 \) Copy content Toggle raw display
$37$ \( T^{8} + 132 T^{6} + 5126 T^{4} + \cdots + 146689 \) Copy content Toggle raw display
$41$ \( T^{8} + 212 T^{6} + 16446 T^{4} + \cdots + 6651241 \) Copy content Toggle raw display
$43$ \( (T^{4} - 4 T^{3} - 88 T^{2} + 472 T - 524)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 4 T^{3} - 112 T^{2} + 520 T - 572)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 88 T^{2} + 192 T + 208)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 240 T^{6} + 16328 T^{4} + \cdots + 913936 \) Copy content Toggle raw display
$61$ \( (T^{4} + 152 T^{2} + 2704)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 4 T^{3} - 40 T^{2} + 184 T - 44)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 16 T^{3} + 26 T^{2} - 88 T - 131)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 4 T^{3} - 136 T^{2} - 584 T - 716)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 384 T^{6} + \cdots + 18215824 \) Copy content Toggle raw display
$83$ \( T^{8} + 432 T^{6} + 56840 T^{4} + \cdots + 6697744 \) Copy content Toggle raw display
$89$ \( T^{8} + 420 T^{6} + 50382 T^{4} + \cdots + 9865881 \) Copy content Toggle raw display
$97$ \( (T^{4} + 16 T^{3} - 88 T^{2} - 448 T + 1936)^{2} \) Copy content Toggle raw display
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