↑

Properties

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

Related objects

Downloads

Learn more

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: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 102x^{12} + 1769x^{8} + 8100x^{4} + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{10} \)
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{5} - \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{5} - \beta_{2} q^{7} + ( - \beta_{12} - \beta_{7}) q^{11} + ( - \beta_{10} + \beta_{8} - \beta_{2} + \cdots - 1) q^{13}+ \cdots + ( - \beta_{14} + \beta_{13} - \beta_{10} + \cdots + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{7} - 6 q^{19} - 24 q^{25} + 24 q^{31} - 26 q^{37} - 16 q^{43} + 2 q^{49} + 60 q^{61} + 2 q^{67} + 30 q^{73} + 10 q^{79} - 32 q^{85} - 84 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 102x^{12} + 1769x^{8} + 8100x^{4} + 4096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 327 \nu^{15} + 5280 \nu^{13} + 41600 \nu^{12} - 56522 \nu^{11} + 452032 \nu^{9} + \cdots + 118536192 ) / 33471488 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 327 \nu^{15} - 5280 \nu^{13} + 41600 \nu^{12} + 56522 \nu^{11} - 452032 \nu^{9} + \cdots + 118536192 ) / 33471488 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 549\nu^{14} + 56510\nu^{10} + 1027693\nu^{6} + 6087412\nu^{2} ) / 2091968 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 549 \nu^{15} - 7072 \nu^{12} + 56510 \nu^{11} - 649824 \nu^{8} + 1027693 \nu^{7} - 5531584 \nu^{4} + \cdots + 14324480 ) / 8367872 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 549 \nu^{15} - 1064 \nu^{14} - 56510 \nu^{11} - 84752 \nu^{10} - 1027693 \nu^{7} + \cdots + 4183936 \nu ) / 8367872 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 549 \nu^{15} - 1064 \nu^{14} + 56510 \nu^{11} - 84752 \nu^{10} + 1027693 \nu^{7} + \cdots - 4183936 \nu ) / 8367872 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2179\nu^{14} - 211906\nu^{10} - 2908171\nu^{6} - 8080812\nu^{2} ) / 2091968 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5817 \nu^{15} - 4256 \nu^{13} + 621622 \nu^{11} - 339008 \nu^{9} + 12889569 \nu^{7} + \cdots + 33471488 ) / 66942976 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5817 \nu^{15} + 4392 \nu^{14} + 4256 \nu^{13} + 621622 \nu^{11} + 452080 \nu^{10} + \cdots - 47979904 \nu ) / 33471488 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 14601 \nu^{15} + 4256 \nu^{13} - 1525782 \nu^{11} + 339008 \nu^{9} - 29332657 \nu^{7} + \cdots - 33471488 ) / 66942976 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 14465 \nu^{15} + 41736 \nu^{14} - 26080 \nu^{13} + 1412710 \nu^{11} + 4181680 \nu^{10} + \cdots - 48630144 \nu ) / 33471488 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 15119 \nu^{15} + 17432 \nu^{14} - 15520 \nu^{13} + 1525754 \nu^{11} + 1695248 \nu^{10} + \cdots - 69906048 \nu ) / 33471488 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 34747 \nu^{15} + 47904 \nu^{13} - 64768 \nu^{12} + 3447042 \nu^{11} + 4633664 \nu^{9} + \cdots - 150668288 ) / 66942976 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 35401 \nu^{15} - 37344 \nu^{13} + 18432 \nu^{12} - 3560086 \nu^{11} - 3729600 \nu^{9} + \cdots + 86404096 ) / 66942976 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 10141 \nu^{15} - 18672 \nu^{14} - 10912 \nu^{13} + 1017166 \nu^{11} - 1864800 \nu^{10} + \cdots - 48305024 \nu ) / 16735744 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{10} - \beta_{8} - \beta_{6} + \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} - \beta_{5} + 3\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{15} + 2 \beta_{12} - \beta_{11} - 5 \beta_{10} - 3 \beta_{9} - 13 \beta_{8} + \beta_{7} + \cdots + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -13\beta_{14} - 13\beta_{13} + 9\beta_{10} + 9\beta_{8} + 18\beta_{4} - 4\beta_{2} + 9\beta _1 - 61 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 11 \beta_{15} + 2 \beta_{14} - 2 \beta_{13} + 26 \beta_{12} - 11 \beta_{11} + 35 \beta_{10} + \cdots - 42 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 26\beta_{15} - 26\beta_{11} - 26\beta_{9} - 107\beta_{7} + 77\beta_{6} + 77\beta_{5} - 103\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 107 \beta_{15} + 24 \beta_{14} - 24 \beta_{13} - 262 \beta_{12} + 107 \beta_{11} + 285 \beta_{10} + \cdots - 388 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1261 \beta_{14} + 1261 \beta_{13} - 673 \beta_{10} - 673 \beta_{8} - 1346 \beta_{4} + 524 \beta_{2} + \cdots + 4269 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 999 \beta_{15} - 230 \beta_{14} + 230 \beta_{13} - 2458 \beta_{12} + 999 \beta_{11} - 2471 \beta_{10} + \cdots + 3510 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2458 \beta_{15} + 2458 \beta_{11} + 2458 \beta_{9} + 9155 \beta_{7} - 5981 \beta_{6} + \cdots + 7183 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 9155 \beta_{15} - 2100 \beta_{14} + 2100 \beta_{13} + 22510 \beta_{12} - 9155 \beta_{11} + \cdots + 31656 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 105701 \beta_{14} - 105701 \beta_{13} + 53617 \beta_{10} + 53617 \beta_{8} + 107234 \beta_{4} + \cdots - 340501 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 83191 \beta_{15} + 18978 \beta_{14} - 18978 \beta_{13} + 204338 \beta_{12} - 83191 \beta_{11} + \cdots - 285530 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 204338 \beta_{15} - 204338 \beta_{11} - 204338 \beta_{9} - 753139 \beta_{7} + 482589 \beta_{6} + \cdots - 572199 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 753139 \beta_{15} + 171232 \beta_{14} - 171232 \beta_{13} - 1848742 \beta_{12} + 753139 \beta_{11} + \cdots - 2576476 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(757\) \(785\) \(1081\) \(1135\)
\(\chi(n)\) \(1\) \(-1\) \(\beta_{8}\) \(1\)

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
2.12464 + 2.12464i
1.35296 + 1.35296i
1.12962 − 1.12962i
0.615924 − 0.615924i
−0.615924 − 0.615924i
−1.12962 − 1.12962i
−1.35296 + 1.35296i
−2.12464 + 2.12464i
2.12464 − 2.12464i
1.35296 − 1.35296i
1.12962 + 1.12962i
0.615924 + 0.615924i
−0.615924 + 0.615924i
−1.12962 + 1.12962i
−1.35296 − 1.35296i
−2.12464 − 2.12464i
0 0 0 −2.12464 + 3.67999i 0 −0.715432 − 2.54719i 0 0 0
593.2 0 0 0 −1.35296 + 2.34339i 0 0.204106 + 2.63787i 0 0 0
593.3 0 0 0 −1.12962 + 1.95656i 0 2.62257 + 0.349446i 0 0 0
593.4 0 0 0 −0.615924 + 1.06681i 0 −2.61125 + 0.425899i 0 0 0
593.5 0 0 0 0.615924 − 1.06681i 0 −2.61125 + 0.425899i 0 0 0
593.6 0 0 0 1.12962 − 1.95656i 0 2.62257 + 0.349446i 0 0 0
593.7 0 0 0 1.35296 − 2.34339i 0 0.204106 + 2.63787i 0 0 0
593.8 0 0 0 2.12464 − 3.67999i 0 −0.715432 − 2.54719i 0 0 0
1025.1 0 0 0 −2.12464 − 3.67999i 0 −0.715432 + 2.54719i 0 0 0
1025.2 0 0 0 −1.35296 − 2.34339i 0 0.204106 − 2.63787i 0 0 0
1025.3 0 0 0 −1.12962 − 1.95656i 0 2.62257 − 0.349446i 0 0 0
1025.4 0 0 0 −0.615924 − 1.06681i 0 −2.61125 − 0.425899i 0 0 0
1025.5 0 0 0 0.615924 + 1.06681i 0 −2.61125 − 0.425899i 0 0 0
1025.6 0 0 0 1.12962 + 1.95656i 0 2.62257 − 0.349446i 0 0 0
1025.7 0 0 0 1.35296 + 2.34339i 0 0.204106 − 2.63787i 0 0 0
1025.8 0 0 0 2.12464 + 3.67999i 0 −0.715432 + 2.54719i 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.a ✓ 16
3.b odd 2 1 inner 1512.2.bl.a ✓ 16
7.d odd 6 1 inner 1512.2.bl.a ✓ 16
21.g even 6 1 inner 1512.2.bl.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.bl.a ✓ 16 1.a even 1 1 trivial
1512.2.bl.a ✓ 16 3.b odd 2 1 inner
1512.2.bl.a ✓ 16 7.d odd 6 1 inner
1512.2.bl.a ✓ 16 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 32 T_{5}^{14} + 716 T_{5}^{12} + 7712 T_{5}^{10} + 59536 T_{5}^{8} + 264640 T_{5}^{6} + \cdots + 1048576 \) 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
$5$ \( T^{16} + 32 T^{14} + \cdots + 1048576 \) Copy content Toggle raw display
$7$ \( (T^{8} + T^{7} - 7 T^{5} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 157351936 \) Copy content Toggle raw display
$13$ \( (T^{8} + 61 T^{6} + \cdots + 12544)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 533794816 \) Copy content Toggle raw display
$19$ \( (T^{8} + 3 T^{7} + \cdots + 256)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 4 T^{2} + 16)^{4} \) Copy content Toggle raw display
$29$ \( (T^{8} + 252 T^{6} + \cdots + 1032256)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 12 T^{7} + \cdots + 9409)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 13 T^{7} + \cdots + 196)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 172 T^{6} + \cdots + 440896)^{2} \) Copy content Toggle raw display
$43$ \( (T + 1)^{16} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 578183827456 \) Copy content Toggle raw display
$53$ \( T^{16} - 288 T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 1032386052096 \) Copy content Toggle raw display
$61$ \( (T^{8} - 30 T^{7} + \cdots + 1996569)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - T^{7} + 50 T^{6} + \cdots + 784)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 524 T^{6} + \cdots + 205520896)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 15 T^{7} + \cdots + 21381376)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 5 T^{7} + \cdots + 839056)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 244 T^{6} + \cdots + 3211264)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + 376 T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$97$ \( (T^{8} + 140 T^{6} + \cdots + 167281)^{2} \) Copy content Toggle raw display
show more
show less