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Properties

Label 1512.2.bl.d
Level $1512$
Weight $2$
Character orbit 1512.bl
Analytic conductor $12.073$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1512,2,Mod(593,1512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1512, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1512.593");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1512.bl (of order \(6\), degree \(2\), 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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: 16.0.81094542259068665856.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{14} + 9x^{12} - 8x^{10} + 44x^{8} - 32x^{6} + 144x^{4} - 64x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{10} + \beta_{4}) q^{5} - \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{10} + \beta_{4}) q^{5} - \beta_{2} q^{7} + (\beta_{8} - \beta_{6} - \beta_1) q^{11} + \beta_{5} q^{13} + (\beta_{12} + \beta_{6}) q^{17} + (\beta_{14} - 3 \beta_{13} + 3 \beta_{5} + \cdots + 1) q^{19}+ \cdots + ( - 3 \beta_{14} - 3 \beta_{11} + \cdots - 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{7} + 12 q^{19} - 12 q^{31} + 16 q^{37} - 16 q^{43} - 28 q^{49} - 60 q^{61} - 4 q^{67} + 12 q^{73} - 32 q^{79} - 32 q^{85} + 24 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{14} + 9x^{12} - 8x^{10} + 44x^{8} - 32x^{6} + 144x^{4} - 64x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{15} + 107\nu^{13} - 3\nu^{11} + 688\nu^{9} - 276\nu^{7} + 2384\nu^{5} + 176\nu^{3} + 4544\nu ) / 2240 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{14} - 11\nu^{12} - \nu^{10} - 4\nu^{8} - 92\nu^{6} - 372\nu^{4} - 128\nu^{2} - 912 ) / 560 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{14} + 46\nu^{12} - 34\nu^{10} + 319\nu^{8} - 188\nu^{6} + 1352\nu^{4} - 432\nu^{2} + 3152 ) / 560 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{15} - 11\nu^{13} + 27\nu^{11} - 60\nu^{9} + 76\nu^{7} - 176\nu^{5} + 208\nu^{3} - 576\nu ) / 448 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{14} + 13\nu^{12} - 37\nu^{10} + 27\nu^{8} - 184\nu^{6} - 44\nu^{4} - 816\nu^{2} - 144 ) / 560 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9\nu^{15} - 41\nu^{13} + 9\nu^{11} - 384\nu^{9} + 548\nu^{7} - 992\nu^{5} + 1712\nu^{3} - 2432\nu ) / 2240 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{15} + 9\nu^{13} - \nu^{11} + 16\nu^{9} - 12\nu^{7} + 48\nu^{5} - 48\nu^{3} + 128\nu ) / 160 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 17\nu^{15} + 47\nu^{13} + 17\nu^{11} + 208\nu^{9} - 396\nu^{7} + 1424\nu^{5} - 624\nu^{3} + 4864\nu ) / 2240 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -23\nu^{14} + 27\nu^{12} - 163\nu^{10} + 188\nu^{8} - 436\nu^{6} + 544\nu^{4} - 1264\nu^{2} + 1984 ) / 2240 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -29\nu^{15} + 101\nu^{13} - 309\nu^{11} + 584\nu^{9} - 1268\nu^{7} + 2512\nu^{5} - 3152\nu^{3} + 4352\nu ) / 2240 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -33\nu^{14} + 57\nu^{12} - 33\nu^{10} + 288\nu^{8} - 236\nu^{6} - 96\nu^{4} + 816\nu^{2} + 704 ) / 2240 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -17\nu^{15} + 23\nu^{13} - 87\nu^{11} + 142\nu^{9} - 444\nu^{7} + 816\nu^{5} - 1056\nu^{3} + 2976\nu ) / 1120 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -39\nu^{14} - 9\nu^{12} - 319\nu^{10} - 296\nu^{8} - 1348\nu^{6} - 928\nu^{4} - 3312\nu^{2} - 3648 ) / 2240 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 37\nu^{14} - 153\nu^{12} + 177\nu^{10} - 972\nu^{8} + 1164\nu^{6} - 4016\nu^{4} + 4176\nu^{2} - 10496 ) / 2240 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -29\nu^{15} - 39\nu^{13} - 169\nu^{11} - 116\nu^{9} - 708\nu^{7} - 848\nu^{5} - 912\nu^{3} - 2368\nu ) / 1120 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{12} - \beta_{10} - \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{14} + \beta_{13} + \beta_{11} - \beta_{9} - \beta_{5} + \beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} - \beta_{10} + \beta_{8} + 2\beta_{6} + 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{14} + \beta_{13} - \beta_{11} - \beta_{9} - 2\beta_{5} - \beta_{2} - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{15} - \beta_{12} + 4\beta_{10} + \beta_{8} - \beta_{7} + 2\beta_{6} + 5\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -6\beta_{13} - 2\beta_{11} + 12\beta_{9} + 3\beta_{5} - 3\beta_{3} - 3\beta_{2} - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -4\beta_{15} + 3\beta_{12} + \beta_{10} - 8\beta_{8} - \beta_{7} - 3\beta_{4} - 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( \beta_{14} - 7\beta_{13} - \beta_{11} + 13\beta_{9} + \beta_{5} + \beta_{3} + 10\beta_{2} - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( \beta_{15} - 7\beta_{10} - 9\beta_{8} - 4\beta_{7} - 10\beta_{6} - 12\beta_{4} + 6\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -19\beta_{14} + 3\beta_{13} + 5\beta_{11} - 43\beta_{9} - 6\beta_{5} - 8\beta_{3} + 5\beta_{2} + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( \beta_{15} - 3\beta_{12} - 4\beta_{10} + 11\beta_{8} + 5\beta_{7} - 18\beta_{6} + 23\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 16\beta_{14} + 6\beta_{13} + 42\beta_{11} - 52\beta_{9} + 49\beta_{5} + 7\beta_{3} - 49\beta_{2} + 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 4\beta_{15} - 15\beta_{12} + 3\beta_{10} + 16\beta_{8} + 45\beta_{7} + 16\beta_{6} - \beta_{4} - 6\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 83\beta_{14} + 11\beta_{13} - 35\beta_{11} - 41\beta_{9} + 59\beta_{5} + 75\beta_{3} + 22\beta_{2} + 83 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 3\beta_{15} - 88\beta_{12} + 19\beta_{10} + 85\beta_{8} - 20\beta_{7} + 2\beta_{6} - 76\beta_{4} - 30\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(757\) \(785\) \(1081\) \(1135\)
\(\chi(n)\) \(1\) \(-1\) \(1 - \beta_{9}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
593.1
0.929502 − 1.06584i
−0.638735 − 1.26175i
1.30439 + 0.546424i
−1.12988 + 0.850516i
1.12988 − 0.850516i
−1.30439 − 0.546424i
0.638735 + 1.26175i
−0.929502 + 1.06584i
0.929502 + 1.06584i
−0.638735 + 1.26175i
1.30439 − 0.546424i
−1.12988 − 0.850516i
1.12988 + 0.850516i
−1.30439 + 0.546424i
0.638735 − 1.26175i
−0.929502 − 1.06584i
0 0 0 −1.68461 + 2.91782i 0 −1.07993 + 2.41532i 0 0 0
593.2 0 0 0 −1.19260 + 2.06564i 0 2.48792 − 0.900140i 0 0 0
593.3 0 0 0 −0.802235 + 1.38951i 0 1.03309 + 2.43572i 0 0 0
593.4 0 0 0 −0.310225 + 0.537326i 0 −1.44109 − 2.21884i 0 0 0
593.5 0 0 0 0.310225 − 0.537326i 0 −1.44109 − 2.21884i 0 0 0
593.6 0 0 0 0.802235 − 1.38951i 0 1.03309 + 2.43572i 0 0 0
593.7 0 0 0 1.19260 − 2.06564i 0 2.48792 − 0.900140i 0 0 0
593.8 0 0 0 1.68461 − 2.91782i 0 −1.07993 + 2.41532i 0 0 0
1025.1 0 0 0 −1.68461 − 2.91782i 0 −1.07993 − 2.41532i 0 0 0
1025.2 0 0 0 −1.19260 − 2.06564i 0 2.48792 + 0.900140i 0 0 0
1025.3 0 0 0 −0.802235 − 1.38951i 0 1.03309 − 2.43572i 0 0 0
1025.4 0 0 0 −0.310225 − 0.537326i 0 −1.44109 + 2.21884i 0 0 0
1025.5 0 0 0 0.310225 + 0.537326i 0 −1.44109 + 2.21884i 0 0 0
1025.6 0 0 0 0.802235 + 1.38951i 0 1.03309 − 2.43572i 0 0 0
1025.7 0 0 0 1.19260 + 2.06564i 0 2.48792 + 0.900140i 0 0 0
1025.8 0 0 0 1.68461 + 2.91782i 0 −1.07993 − 2.41532i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 593.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1512.2.bl.d ✓ 16
3.b odd 2 1 inner 1512.2.bl.d ✓ 16
7.d odd 6 1 inner 1512.2.bl.d ✓ 16
21.g even 6 1 inner 1512.2.bl.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.bl.d ✓ 16 1.a even 1 1 trivial
1512.2.bl.d ✓ 16 3.b odd 2 1 inner
1512.2.bl.d ✓ 16 7.d odd 6 1 inner
1512.2.bl.d ✓ 16 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 20T_{5}^{14} + 284T_{5}^{12} + 1904T_{5}^{10} + 9232T_{5}^{8} + 21568T_{5}^{6} + 35840T_{5}^{4} + 13312T_{5}^{2} + 4096 \) acting on \(S_{2}^{\mathrm{new}}(1512, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 20 T^{14} + \cdots + 4096 \) Copy content Toggle raw display
$7$ \( (T^{8} - 2 T^{7} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} - 60 T^{14} + \cdots + 16777216 \) Copy content Toggle raw display
$13$ \( (T^{8} + 22 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 52 T^{14} + \cdots + 2560000 \) Copy content Toggle raw display
$19$ \( (T^{8} - 6 T^{7} + \cdots + 250000)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 280883040256 \) Copy content Toggle raw display
$29$ \( (T^{8} + 60 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 6 T^{7} + \cdots + 169)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 8 T^{7} + 73 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 340 T^{6} + \cdots + 24522304)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 4 T^{3} + \cdots + 349)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 7676563456 \) Copy content Toggle raw display
$53$ \( T^{16} - 120 T^{14} + \cdots + 2560000 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{8} + 30 T^{7} + \cdots + 77841)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 2 T^{7} + \cdots + 8880400)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 140 T^{6} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 6 T^{7} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 16 T^{7} + \cdots + 1081600)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 280 T^{6} + \cdots + 409600)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 79285774913536 \) Copy content Toggle raw display
$97$ \( (T^{8} + 404 T^{6} + \cdots + 80227849)^{2} \) Copy content Toggle raw display
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