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Properties

Label 3584.2.b.h
Level $3584$
Weight $2$
Character orbit 3584.b
Analytic conductor $28.618$
Analytic rank $0$
Dimension $10$
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] = [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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4x^{9} + 12x^{8} - 12x^{7} + 3x^{6} - 48x^{5} + 158x^{4} - 80x^{3} - 66x^{2} + 72x + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{6} q^{5} + q^{7} + ( - \beta_{5} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + \beta_{6} q^{5} + q^{7} + ( - \beta_{5} - 2) q^{9} + \beta_{8} q^{11} + \beta_{9} q^{13} + ( - \beta_{5} - \beta_{4} + \beta_{2} - 1) q^{15} + (\beta_{3} + 2) q^{17} + ( - \beta_{9} + \beta_{7} + \cdots - \beta_1) q^{19}+ \cdots + ( - \beta_{8} + 4 \beta_{7} + \cdots - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{7} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{7} - 22 q^{9} - 16 q^{15} + 16 q^{17} + 8 q^{23} - 30 q^{25} + 8 q^{31} + 12 q^{33} - 20 q^{41} + 24 q^{47} + 10 q^{49} - 32 q^{55} - 28 q^{57} - 22 q^{63} + 32 q^{65} - 48 q^{73} - 40 q^{79} + 62 q^{81} - 8 q^{87} - 16 q^{89} + 32 q^{95} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4x^{9} + 12x^{8} - 12x^{7} + 3x^{6} - 48x^{5} + 158x^{4} - 80x^{3} - 66x^{2} + 72x + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1883 \nu^{9} + 39443 \nu^{8} - 140969 \nu^{7} + 463067 \nu^{6} - 313772 \nu^{5} + 12764 \nu^{4} + \cdots - 2273868 ) / 1451844 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 15829 \nu^{9} - 84722 \nu^{8} + 209765 \nu^{7} - 289322 \nu^{6} - 201334 \nu^{5} - 877304 \nu^{4} + \cdots - 9846 ) / 725922 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 19133 \nu^{9} + 66337 \nu^{8} - 198991 \nu^{7} + 125815 \nu^{6} - 18106 \nu^{5} + 934966 \nu^{4} + \cdots - 1858968 ) / 725922 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 22726 \nu^{9} - 88931 \nu^{8} + 257948 \nu^{7} - 199391 \nu^{6} - 159244 \nu^{5} - 620576 \nu^{4} + \cdots + 4740426 ) / 725922 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 31664 \nu^{9} + 124663 \nu^{8} - 353158 \nu^{7} + 285913 \nu^{6} + 124556 \nu^{5} + \cdots - 1480806 ) / 725922 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 77231 \nu^{9} - 432283 \nu^{8} + 1411729 \nu^{7} - 2393731 \nu^{6} + 1741774 \nu^{5} + \cdots + 9111492 ) / 1451844 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3820 \nu^{9} - 15476 \nu^{8} + 48281 \nu^{7} - 55454 \nu^{6} + 33773 \nu^{5} - 207770 \nu^{4} + \cdots + 165024 ) / 40329 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 76448 \nu^{9} + 287347 \nu^{8} - 878263 \nu^{7} + 850897 \nu^{6} - 413401 \nu^{5} + \cdots - 2399184 ) / 725922 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 209987 \nu^{9} + 975355 \nu^{8} - 3098881 \nu^{7} + 4281475 \nu^{6} - 2816962 \nu^{5} + \cdots - 15620868 ) / 1451844 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} + \beta_{8} + \beta_{7} - 2\beta_{6} + \beta_{3} - \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} + 3\beta_{8} + 3\beta_{7} - 2\beta_{6} + 2\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta _1 - 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{9} - 3 \beta_{8} - 6 \beta_{7} + 2 \beta_{6} + 8 \beta_{5} + 3 \beta_{4} - 7 \beta_{3} + \cdots - 22 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11\beta_{9} - 31\beta_{8} - 40\beta_{7} + 38\beta_{6} + 3\beta_{4} + \beta_{3} - \beta_{2} + 37\beta _1 - 18 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 19 \beta_{9} - 41 \beta_{8} - 52 \beta_{7} + 60 \beta_{6} - 71 \beta_{5} - 39 \beta_{4} + 48 \beta_{3} + \cdots + 230 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 99 \beta_{9} + 203 \beta_{8} + 220 \beta_{7} - 258 \beta_{6} - 186 \beta_{5} - 95 \beta_{4} + \cdots + 578 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 377 \beta_{9} + 915 \beta_{8} + 1048 \beta_{7} - 1064 \beta_{6} + 215 \beta_{5} + 131 \beta_{4} + \cdots - 870 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 103 \beta_{9} + 291 \beta_{8} + 372 \beta_{7} - 350 \beta_{6} + 2288 \beta_{5} + 1165 \beta_{4} + \cdots - 7654 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3395 \beta_{9} - 7965 \beta_{8} - 8946 \beta_{7} + 9296 \beta_{6} + 3431 \beta_{5} + 1827 \beta_{4} + \cdots - 11770 ) / 4 \) 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
0.875338 + 0.673961i
1.68909 + 0.861941i
−1.29337 − 1.34353i
−0.484731 − 0.120245i
1.21367 − 2.82548i
1.21367 + 2.82548i
−0.484731 + 0.120245i
−1.29337 + 1.34353i
1.68909 − 0.861941i
0.875338 − 0.673961i
0 − 3.17593i 0 − 4.23755i 0 1.00000 0 −7.08655 0
1793.2 0 − 2.99347i 0 0.369776i 0 1.00000 0 −5.96088 0
1793.3 0 − 2.47533i 0 2.59708i 0 1.00000 0 −3.12724 0
1793.4 0 − 0.783186i 0 − 3.56843i 0 1.00000 0 2.38662 0
1793.5 0 − 0.460386i 0 1.55819i 0 1.00000 0 2.78804 0
1793.6 0 0.460386i 0 − 1.55819i 0 1.00000 0 2.78804 0
1793.7 0 0.783186i 0 3.56843i 0 1.00000 0 2.38662 0
1793.8 0 2.47533i 0 − 2.59708i 0 1.00000 0 −3.12724 0
1793.9 0 2.99347i 0 − 0.369776i 0 1.00000 0 −5.96088 0
1793.10 0 3.17593i 0 4.23755i 0 1.00000 0 −7.08655 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1793.10
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.h 10
4.b odd 2 1 3584.2.b.g 10
8.b even 2 1 inner 3584.2.b.h 10
8.d odd 2 1 3584.2.b.g 10
16.e even 4 2 3584.2.a.o ✓ 10
16.f odd 4 2 3584.2.a.p yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3584.2.a.o ✓ 10 16.e even 4 2
3584.2.a.p yes 10 16.f odd 4 2
3584.2.b.g 10 4.b odd 2 1
3584.2.b.g 10 8.d odd 2 1
3584.2.b.h 10 1.a even 1 1 trivial
3584.2.b.h 10 8.b even 2 1 inner

Hecke kernels

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

\( T_{3}^{10} + 26T_{3}^{8} + 228T_{3}^{6} + 728T_{3}^{4} + 484T_{3}^{2} + 72 \) Copy content Toggle raw display
\( T_{23}^{5} - 4T_{23}^{4} - 60T_{23}^{3} + 112T_{23}^{2} + 644T_{23} - 1296 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 26 T^{8} + \cdots + 72 \) Copy content Toggle raw display
$5$ \( T^{10} + 40 T^{8} + \cdots + 512 \) Copy content Toggle raw display
$7$ \( (T - 1)^{10} \) Copy content Toggle raw display
$11$ \( T^{10} + 98 T^{8} + \cdots + 1492992 \) Copy content Toggle raw display
$13$ \( T^{10} + 104 T^{8} + \cdots + 4608 \) Copy content Toggle raw display
$17$ \( (T^{5} - 8 T^{4} + \cdots - 1152)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + 122 T^{8} + \cdots + 5832 \) Copy content Toggle raw display
$23$ \( (T^{5} - 4 T^{4} + \cdots - 1296)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + 184 T^{8} + \cdots + 18432 \) Copy content Toggle raw display
$31$ \( (T^{5} - 4 T^{4} + \cdots + 384)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 216 T^{8} + \cdots + 1492992 \) Copy content Toggle raw display
$41$ \( (T^{5} + 10 T^{4} + \cdots + 28128)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 274 T^{8} + \cdots + 18432 \) Copy content Toggle raw display
$47$ \( (T^{5} - 12 T^{4} + \cdots - 10368)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + 280 T^{8} + \cdots + 5752832 \) Copy content Toggle raw display
$59$ \( T^{10} + 202 T^{8} + \cdots + 52488 \) Copy content Toggle raw display
$61$ \( T^{10} + 200 T^{8} + \cdots + 7558272 \) Copy content Toggle raw display
$67$ \( T^{10} + 194 T^{8} + \cdots + 2654208 \) Copy content Toggle raw display
$71$ \( (T^{5} - 120 T^{3} + \cdots - 1152)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + 24 T^{4} + \cdots - 10496)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + 20 T^{4} + \cdots - 11584)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 170 T^{8} + \cdots + 7235208 \) Copy content Toggle raw display
$89$ \( (T^{5} + 8 T^{4} + \cdots - 2304)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} - 16 T^{4} + \cdots - 1536)^{2} \) Copy content Toggle raw display
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