↑

Properties

Label 3584.2.b.j
Level $3584$
Weight $2$
Character orbit 3584.b
Analytic conductor $28.618$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3584,2,Mod(1793,3584)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3584, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3584.1793");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3584 = 2^{9} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3584.b (of order \(2\), degree \(1\), not 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: \(28.6183840844\)
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: \( x^{12} - 4 x^{11} + 2 x^{10} + 8 x^{9} + 2 x^{8} + 12 x^{7} + 40 x^{6} + 4 x^{5} + 49 x^{4} - 56 x^{3} + \cdots + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{4} q^{5} - q^{7} - \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{4} q^{5} - q^{7} - \beta_{2} q^{9} + \beta_{7} q^{11} + \beta_{8} q^{13} + (\beta_{3} - \beta_{2} - 1) q^{15} + (\beta_{6} - \beta_{5} + \beta_{2}) q^{17} + (\beta_{9} + \beta_{7} - \beta_1) q^{19} - \beta_1 q^{21} + ( - \beta_{11} + \beta_{2}) q^{23} + (\beta_{11} + \beta_{5} - 2) q^{25} + (\beta_{9} + \beta_{7}) q^{27} + (\beta_{10} + \beta_{9} + \beta_{7}) q^{29} + (\beta_{5} + \beta_{2} + 1) q^{31} + ( - \beta_{6} - 1) q^{33} - \beta_{4} q^{35} + ( - \beta_{9} + 2 \beta_{7} + \cdots - \beta_1) q^{37}+ \cdots + ( - \beta_{10} - \beta_{9} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{7} - 4 q^{9} - 16 q^{15} - 20 q^{25} + 16 q^{31} - 8 q^{33} - 8 q^{41} + 16 q^{47} + 12 q^{49} + 32 q^{55} + 24 q^{57} + 4 q^{63} + 16 q^{71} + 16 q^{73} - 48 q^{79} - 28 q^{81} - 16 q^{87} - 16 q^{89} + 32 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 2 x^{10} + 8 x^{9} + 2 x^{8} + 12 x^{7} + 40 x^{6} + 4 x^{5} + 49 x^{4} - 56 x^{3} + \cdots + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 57365257 \nu^{11} + 234079984 \nu^{10} - 135078180 \nu^{9} - 426165507 \nu^{8} + \cdots + 956458910 ) / 1092011357 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 80846544 \nu^{11} - 323793039 \nu^{10} + 200978248 \nu^{9} + 485124412 \nu^{8} + \cdots - 3531534701 ) / 1092011357 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 573246170 \nu^{11} - 2294903120 \nu^{10} + 1096597106 \nu^{9} + 4751301658 \nu^{8} + \cdots - 4444512520 ) / 1092011357 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 658197437 \nu^{11} + 2194250807 \nu^{10} + 325448325 \nu^{9} - 5678219629 \nu^{8} + \cdots - 40306606 ) / 1092011357 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 770254572 \nu^{11} - 3049019197 \nu^{10} + 1242928968 \nu^{9} + 6873092542 \nu^{8} + \cdots - 1891631756 ) / 1092011357 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 813376454 \nu^{11} - 3234081458 \nu^{10} + 1377620178 \nu^{9} + 7165126334 \nu^{8} + \cdots - 4411599275 ) / 1092011357 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1259935989 \nu^{11} - 4247324243 \nu^{10} - 436257400 \nu^{9} + 10784206962 \nu^{8} + \cdots - 983567068 ) / 1092011357 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1406385910 \nu^{11} - 5322839349 \nu^{10} + 1762791222 \nu^{9} + 11315529253 \nu^{8} + \cdots - 9850874438 ) / 1092011357 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1638376128 \nu^{11} + 5841864389 \nu^{10} - 666519353 \nu^{9} - 13630410092 \nu^{8} + \cdots + 6383899508 ) / 1092011357 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2068797341 \nu^{11} + 7447421152 \nu^{10} - 1119554207 \nu^{9} - 17154929956 \nu^{8} + \cdots + 8082402408 ) / 1092011357 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2171793544 \nu^{11} + 8621122746 \nu^{10} - 3709150224 \nu^{9} - 18834920612 \nu^{8} + \cdots + 12455232595 ) / 1092011357 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{6} + 2\beta_{5} - \beta_{3} + 2\beta_{2} - 4\beta _1 + 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{11} - 3\beta_{10} + 3\beta_{9} - \beta_{8} + 6\beta_{6} - 2\beta_{3} + 4\beta_{2} - 7\beta _1 + 8 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4 \beta_{11} - 9 \beta_{10} - \beta_{9} - 9 \beta_{8} - 10 \beta_{7} + 8 \beta_{6} + 2 \beta_{5} + \cdots + 10 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -17\beta_{10} + 3\beta_{9} - 15\beta_{8} - 8\beta_{7} + 2\beta_{4} - 39\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 16 \beta_{11} - 45 \beta_{10} + 5 \beta_{9} - 41 \beta_{8} - 34 \beta_{7} - 36 \beta_{6} - 10 \beta_{5} + \cdots - 60 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 86 \beta_{11} - 110 \beta_{10} + 27 \beta_{9} - 93 \beta_{8} - 63 \beta_{7} - 206 \beta_{6} + \cdots - 376 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 349 \beta_{11} - 179 \beta_{10} + 35 \beta_{9} - 157 \beta_{8} - 116 \beta_{7} - 807 \beta_{6} + \cdots - 1482 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -578\beta_{11} - 1359\beta_{6} - 361\beta_{5} + 431\beta_{3} - 1471\beta_{2} - 2526 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3312 \beta_{11} + 1720 \beta_{10} - 326 \beta_{9} + 1506 \beta_{8} + 1100 \beta_{7} - 7722 \beta_{6} + \cdots - 14217 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 7808 \beta_{11} + 9843 \beta_{10} - 1888 \beta_{9} + 8596 \beta_{8} + 6207 \beta_{7} - 18292 \beta_{6} + \cdots - 33708 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 13050 \beta_{11} + 39614 \beta_{10} - 7398 \beta_{9} + 34696 \beta_{8} + 25252 \beta_{7} + \cdots - 56117 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3584\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1793.1
0.568642 + 1.37282i
2.84980 + 1.18043i
−0.445186 + 1.07477i
−1.48663 + 0.615783i
0.169437 + 0.409058i
0.343932 + 0.142462i
0.343932 − 0.142462i
0.169437 − 0.409058i
−1.48663 − 0.615783i
−0.445186 − 1.07477i
2.84980 − 1.18043i
0.568642 − 1.37282i
0 − 2.74564i 0 − 2.98707i 0 −1.00000 0 −4.53856 0
1793.2 0 − 2.36086i 0 − 0.711757i 0 −1.00000 0 −2.57364 0
1793.3 0 − 2.14955i 0 2.09671i 0 −1.00000 0 −1.62055 0
1793.4 0 − 1.23157i 0 − 3.90423i 0 −1.00000 0 1.48325 0
1793.5 0 − 0.818115i 0 3.08378i 0 −1.00000 0 2.33069 0
1793.6 0 − 0.284923i 0 − 1.19247i 0 −1.00000 0 2.91882 0
1793.7 0 0.284923i 0 1.19247i 0 −1.00000 0 2.91882 0
1793.8 0 0.818115i 0 − 3.08378i 0 −1.00000 0 2.33069 0
1793.9 0 1.23157i 0 3.90423i 0 −1.00000 0 1.48325 0
1793.10 0 2.14955i 0 − 2.09671i 0 −1.00000 0 −1.62055 0
1793.11 0 2.36086i 0 0.711757i 0 −1.00000 0 −2.57364 0
1793.12 0 2.74564i 0 2.98707i 0 −1.00000 0 −4.53856 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1793.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3584.2.b.j 12
4.b odd 2 1 3584.2.b.l 12
8.b even 2 1 inner 3584.2.b.j 12
8.d odd 2 1 3584.2.b.l 12
16.e even 4 1 3584.2.a.h yes 6
16.e even 4 1 3584.2.a.j yes 6
16.f odd 4 1 3584.2.a.g ✓ 6
16.f odd 4 1 3584.2.a.i yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3584.2.a.g ✓ 6 16.f odd 4 1
3584.2.a.h yes 6 16.e even 4 1
3584.2.a.i yes 6 16.f odd 4 1
3584.2.a.j yes 6 16.e even 4 1
3584.2.b.j 12 1.a even 1 1 trivial
3584.2.b.j 12 8.b even 2 1 inner
3584.2.b.l 12 4.b odd 2 1
3584.2.b.l 12 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3584, [\chi])\):

\( T_{3}^{12} + 20T_{3}^{10} + 144T_{3}^{8} + 448T_{3}^{6} + 564T_{3}^{4} + 240T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{23}^{6} - 68T_{23}^{4} + 64T_{23}^{3} + 772T_{23}^{2} + 128T_{23} - 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 20 T^{10} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{12} + 40 T^{10} + \cdots + 4096 \) Copy content Toggle raw display
$7$ \( (T + 1)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 44 T^{10} + \cdots + 4096 \) Copy content Toggle raw display
$13$ \( T^{12} + 104 T^{10} + \cdots + 4096 \) Copy content Toggle raw display
$17$ \( (T^{6} - 56 T^{4} + \cdots - 512)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 84 T^{10} + \cdots + 26896 \) Copy content Toggle raw display
$23$ \( (T^{6} - 68 T^{4} + \cdots - 1024)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 208 T^{10} + \cdots + 16777216 \) Copy content Toggle raw display
$31$ \( (T^{6} - 8 T^{5} + \cdots + 2048)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 240 T^{10} + \cdots + 4194304 \) Copy content Toggle raw display
$41$ \( (T^{6} + 4 T^{5} + \cdots - 3136)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 268 T^{10} + \cdots + 43454464 \) Copy content Toggle raw display
$47$ \( (T^{6} - 8 T^{5} + \cdots + 14336)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 240 T^{10} + \cdots + 4194304 \) Copy content Toggle raw display
$59$ \( T^{12} + 180 T^{10} + \cdots + 5895184 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 554298118144 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 5473632256 \) Copy content Toggle raw display
$71$ \( (T^{6} - 8 T^{5} + \cdots - 203776)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 8 T^{5} + \cdots + 512)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 24 T^{5} + \cdots - 171008)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 724 T^{10} + \cdots + 17875984 \) Copy content Toggle raw display
$89$ \( (T^{6} + 8 T^{5} + \cdots - 52736)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 520 T^{4} + \cdots - 2576896)^{2} \) Copy content Toggle raw display
show more
show less