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Properties

Label 3584.2.a.o
Level $3584$
Weight $2$
Character orbit 3584.a
Self dual yes
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(1,3584)]
 
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, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3584.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3584 = 2^{9} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3584.a (trivial)

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: yes
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: \( x^{10} - 2x^{9} - 19x^{8} + 44x^{7} + 86x^{6} - 236x^{5} - 58x^{4} + 368x^{3} - 194x^{2} - 12x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} - \beta_{3} q^{5} - q^{7} + (\beta_{6} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{3} - \beta_{3} q^{5} - q^{7} + (\beta_{6} + 2) q^{9} - \beta_{7} q^{11} - \beta_{8} q^{13} + ( - \beta_{9} - \beta_{6} - \beta_{2} - 1) q^{15} + (\beta_{4} + 2) q^{17} + (\beta_{8} - \beta_{5} - \beta_{3} + \beta_1) q^{19} + \beta_{5} q^{21} + (\beta_{2} - 1) q^{23} + (\beta_{9} + \beta_{6} - \beta_{4} + \cdots + 2) q^{25}+ \cdots + ( - \beta_{7} - 2 \beta_{5} + \cdots + 4 \beta_1) q^{99}+O(q^{100}) \) 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 \( x^{10} - 2x^{9} - 19x^{8} + 44x^{7} + 86x^{6} - 236x^{5} - 58x^{4} + 368x^{3} - 194x^{2} - 12x + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 365 \nu^{9} + 475 \nu^{8} + 7148 \nu^{7} - 11086 \nu^{6} - 37174 \nu^{5} + 59872 \nu^{4} + \cdots + 12126 ) / 723 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 270 \nu^{9} + 424 \nu^{8} + 5380 \nu^{7} - 9399 \nu^{6} - 28522 \nu^{5} + 49165 \nu^{4} + \cdots + 7963 ) / 241 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 821 \nu^{9} + 1009 \nu^{8} + 16502 \nu^{7} - 23377 \nu^{6} - 90850 \nu^{5} + 124135 \nu^{4} + \cdots + 18516 ) / 723 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 910 \nu^{9} - 1072 \nu^{8} - 18204 \nu^{7} + 25064 \nu^{6} + 99468 \nu^{5} - 133080 \nu^{4} + \cdots - 20724 ) / 241 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5947 \nu^{9} - 7145 \nu^{8} - 118678 \nu^{7} + 166772 \nu^{6} + 644474 \nu^{5} - 886628 \nu^{4} + \cdots - 129066 ) / 1446 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1938 \nu^{9} + 2390 \nu^{8} + 38670 \nu^{7} - 55671 \nu^{6} - 209566 \nu^{5} + 297519 \nu^{4} + \cdots + 46317 ) / 241 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4723 \nu^{9} - 5737 \nu^{8} - 94128 \nu^{7} + 133996 \nu^{6} + 509326 \nu^{5} - 715642 \nu^{4} + \cdots - 109098 ) / 482 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3738 \nu^{9} + 4574 \nu^{8} + 74617 \nu^{7} - 106522 \nu^{6} - 404854 \nu^{5} + 567767 \nu^{4} + \cdots + 85506 ) / 241 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4952 \nu^{9} - 6002 \nu^{8} - 98816 \nu^{7} + 140021 \nu^{6} + 536250 \nu^{5} - 746099 \nu^{4} + \cdots - 114105 ) / 241 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + \beta_{6} - 2\beta_{5} - \beta_{4} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} - 2\beta_{6} + 2\beta_{4} + 4\beta_{3} - \beta _1 + 18 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8 \beta_{9} + 4 \beta_{8} + 6 \beta_{7} + 8 \beta_{6} - 16 \beta_{5} - 8 \beta_{4} - 4 \beta_{3} + \cdots - 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2 \beta_{9} - 3 \beta_{8} - 17 \beta_{7} - 14 \beta_{6} + 9 \beta_{5} + 10 \beta_{4} + 26 \beta_{3} + \cdots + 76 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 82 \beta_{9} + 66 \beta_{8} + 104 \beta_{7} + 91 \beta_{6} - 160 \beta_{5} - 91 \beta_{4} - 98 \beta_{3} + \cdots - 200 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 48 \beta_{9} - 59 \beta_{8} - 235 \beta_{7} - 189 \beta_{6} + 171 \beta_{5} + 123 \beta_{4} + \cdots + 798 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 465 \beta_{9} + 441 \beta_{8} + 757 \beta_{7} + 593 \beta_{6} - 919 \beta_{5} - 569 \beta_{4} + \cdots - 1646 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 841 \beta_{9} - 946 \beta_{8} - 3130 \beta_{7} - 2508 \beta_{6} + 2598 \beta_{5} + 1657 \beta_{4} + \cdots + 9392 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 5590 \beta_{9} + 5650 \beta_{8} + 10524 \beta_{7} + 8044 \beta_{6} - 11434 \beta_{5} - 7333 \beta_{4} + \cdots - 24644 ) / 2 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0.508920
−2.00182
−0.269488
0.776589
−3.66546
−1.79086
2.97399
0.791624
2.40586
2.27064
0 −3.17593 0 4.23755 0 −1.00000 0 7.08655 0
1.2 0 −2.99347 0 −0.369776 0 −1.00000 0 5.96088 0
1.3 0 −2.47533 0 −2.59708 0 −1.00000 0 3.12724 0
1.4 0 −0.783186 0 3.56843 0 −1.00000 0 −2.38662 0
1.5 0 −0.460386 0 −1.55819 0 −1.00000 0 −2.78804 0
1.6 0 0.460386 0 1.55819 0 −1.00000 0 −2.78804 0
1.7 0 0.783186 0 −3.56843 0 −1.00000 0 −2.38662 0
1.8 0 2.47533 0 2.59708 0 −1.00000 0 3.12724 0
1.9 0 2.99347 0 0.369776 0 −1.00000 0 5.96088 0
1.10 0 3.17593 0 −4.23755 0 −1.00000 0 7.08655 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)

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.a.o ✓ 10
4.b odd 2 1 3584.2.a.p yes 10
8.b even 2 1 inner 3584.2.a.o ✓ 10
8.d odd 2 1 3584.2.a.p yes 10
16.e even 4 2 3584.2.b.h 10
16.f odd 4 2 3584.2.b.g 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3584.2.a.o ✓ 10 1.a even 1 1 trivial
3584.2.a.o ✓ 10 8.b even 2 1 inner
3584.2.a.p yes 10 4.b odd 2 1
3584.2.a.p yes 10 8.d odd 2 1
3584.2.b.g 10 16.f odd 4 2
3584.2.b.h 10 16.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3584))\):

\( T_{3}^{10} - 26T_{3}^{8} + 228T_{3}^{6} - 728T_{3}^{4} + 484T_{3}^{2} - 72 \) Copy content Toggle raw display
\( T_{5}^{10} - 40T_{5}^{8} + 532T_{5}^{6} - 2672T_{5}^{4} + 4100T_{5}^{2} - 512 \) 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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