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Properties

Label 3584.2.b.k
Level $3584$
Weight $2$
Character orbit 3584.b
Analytic conductor $28.618$
Analytic rank $0$
Dimension $12$
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 4 x^{10} + 8 x^{9} - 24 x^{8} + 24 x^{7} + 176 x^{6} + 160 x^{5} + 145 x^{4} + \cdots + 8 \) 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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} - \beta_{7} q^{5} + q^{7} + (\beta_{5} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{3} - \beta_{7} q^{5} + q^{7} + (\beta_{5} - 2) q^{9} + (\beta_{7} - \beta_{3}) q^{11} - \beta_{2} q^{13} + (\beta_{10} - \beta_1 + 1) q^{15} - \beta_{10} q^{17} + (\beta_{7} - \beta_{4} - \beta_{2}) q^{19} - \beta_{6} q^{21} + ( - \beta_{9} - \beta_{8}) q^{23} + (\beta_{8} + \beta_1) q^{25} + ( - \beta_{11} - \beta_{7} + \cdots + \beta_{2}) q^{27}+ \cdots + (\beta_{11} - \beta_{7} + \cdots - 2 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{7} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{7} - 20 q^{9} + 16 q^{15} - 4 q^{25} + 16 q^{31} + 8 q^{33} + 8 q^{41} - 16 q^{47} + 12 q^{49} + 32 q^{55} + 8 q^{57} - 20 q^{63} - 16 q^{65} - 16 q^{71} - 32 q^{73} + 48 q^{79} + 20 q^{81} - 16 q^{87} - 32 q^{89} + 32 q^{95} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 4 x^{10} + 8 x^{9} - 24 x^{8} + 24 x^{7} + 176 x^{6} + 160 x^{5} + 145 x^{4} + \cdots + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 419637583 \nu^{11} - 1440544428 \nu^{10} + 635340644 \nu^{9} + 4866667572 \nu^{8} + \cdots + 11421355507 ) / 9640197931 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 638917040 \nu^{11} - 2376776737 \nu^{10} + 1975208857 \nu^{9} + 5237100284 \nu^{8} + \cdots + 36460193404 ) / 9640197931 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8615178688 \nu^{11} - 52103353005 \nu^{10} + 115591508302 \nu^{9} - 43547709868 \nu^{8} + \cdots - 39713864336 ) / 96401979310 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 894988184 \nu^{11} + 5710664665 \nu^{10} - 13387713576 \nu^{9} + 6452256724 \nu^{8} + \cdots + 8956064308 ) / 7415536870 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 495332004 \nu^{11} - 2247729805 \nu^{10} + 2867331746 \nu^{9} + 3802764346 \nu^{8} + \cdots + 6862964757 ) / 3707768435 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1410256356 \nu^{11} + 5921534715 \nu^{10} - 6862437384 \nu^{9} - 9554539644 \nu^{8} + \cdots - 25484705788 ) / 7415536870 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13383971527 \nu^{11} + 62676653525 \nu^{10} - 94344403828 \nu^{9} - 51680040438 \nu^{8} + \cdots - 86421877996 ) / 48200989655 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6278889 \nu^{11} + 25797695 \nu^{10} - 26888646 \nu^{9} - 52239876 \nu^{8} + 164047612 \nu^{7} + \cdots - 96252252 ) / 22141015 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2182781366 \nu^{11} - 9596353535 \nu^{10} + 11698649024 \nu^{9} + 16394832844 \nu^{8} + \cdots - 8063859872 ) / 6885855665 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3571797907 \nu^{11} + 15225607627 \nu^{10} - 17186575048 \nu^{9} - 29155049066 \nu^{8} + \cdots - 63122961648 ) / 9640197931 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3742088406 \nu^{11} + 17147484255 \nu^{10} - 24666155409 \nu^{9} - 16843522124 \nu^{8} + \cdots - 34789559518 ) / 6885855665 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - 3\beta_{6} + \beta_{5} + \beta_{3} + 2\beta _1 + 3 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{11} + 4\beta_{10} - 4\beta_{8} - 4\beta_{7} - \beta_{6} + 4\beta_{5} + 4\beta_{4} + 7\beta_{3} + 4 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{11} + 7 \beta_{10} - 3 \beta_{9} - 7 \beta_{8} - 4 \beta_{7} + 15 \beta_{6} + 11 \beta_{5} + \cdots - 9 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{11} - 14\beta_{7} + 42\beta_{6} + 43\beta_{4} + 51\beta_{3} + 14\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 11 \beta_{11} - 65 \beta_{10} + 9 \beta_{9} + 53 \beta_{8} - 68 \beta_{7} + 143 \beta_{6} - 85 \beta_{5} + \cdots - 13 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 111 \beta_{11} - 340 \beta_{10} - 72 \beta_{9} + 188 \beta_{8} - 236 \beta_{7} + 67 \beta_{6} + \cdots - 700 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 315 \beta_{11} - 1259 \beta_{10} - 615 \beta_{9} + 393 \beta_{8} - 420 \beta_{7} - 209 \beta_{6} + \cdots - 4397 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -1064\beta_{10} - 594\beta_{9} + 279\beta_{8} - 625\beta_{5} - 995\beta _1 - 4114 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 2147 \beta_{11} - 12873 \beta_{10} - 5921 \beta_{9} + 4367 \beta_{8} + 3980 \beta_{7} - 383 \beta_{6} + \cdots - 43091 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 5767 \beta_{11} - 32820 \beta_{10} - 10864 \beta_{9} + 14660 \beta_{8} + 21420 \beta_{7} + \cdots - 87764 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 5365 \beta_{11} - 57879 \beta_{10} - 15021 \beta_{9} + 29199 \beta_{8} + 82804 \beta_{7} + \cdots - 133047 ) / 8 \) 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
0.960396 − 2.31860i
−1.45042 − 0.600784i
−0.232297 − 0.560814i
−0.484138 − 0.200537i
2.93456 − 1.21553i
0.271901 + 0.656426i
0.271901 − 0.656426i
2.93456 + 1.21553i
−0.484138 + 0.200537i
−0.232297 + 0.560814i
−1.45042 + 0.600784i
0.960396 + 2.31860i
0 − 3.22299i 0 − 0.725063i 0 1.00000 0 −7.38766 0
1793.2 0 − 2.61578i 0 3.96111i 0 1.00000 0 −3.84231 0
1793.3 0 − 2.53584i 0 1.47134i 0 1.00000 0 −3.43049 0
1793.4 0 − 1.81529i 0 − 2.66965i 0 1.00000 0 −0.295267 0
1793.5 0 − 1.01685i 0 1.29145i 0 1.00000 0 1.96601 0
1793.6 0 − 0.101362i 0 − 2.19640i 0 1.00000 0 2.98973 0
1793.7 0 0.101362i 0 2.19640i 0 1.00000 0 2.98973 0
1793.8 0 1.01685i 0 − 1.29145i 0 1.00000 0 1.96601 0
1793.9 0 1.81529i 0 2.66965i 0 1.00000 0 −0.295267 0
1793.10 0 2.53584i 0 − 1.47134i 0 1.00000 0 −3.43049 0
1793.11 0 2.61578i 0 − 3.96111i 0 1.00000 0 −3.84231 0
1793.12 0 3.22299i 0 0.725063i 0 1.00000 0 −7.38766 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.k 12
4.b odd 2 1 3584.2.b.i 12
8.b even 2 1 inner 3584.2.b.k 12
8.d odd 2 1 3584.2.b.i 12
16.e even 4 1 3584.2.a.e ✓ 6
16.e even 4 1 3584.2.a.k yes 6
16.f odd 4 1 3584.2.a.f yes 6
16.f odd 4 1 3584.2.a.l yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3584.2.a.e ✓ 6 16.e even 4 1
3584.2.a.f yes 6 16.f odd 4 1
3584.2.a.k yes 6 16.e even 4 1
3584.2.a.l yes 6 16.f odd 4 1
3584.2.b.i 12 4.b odd 2 1
3584.2.b.i 12 8.d odd 2 1
3584.2.b.k 12 1.a even 1 1 trivial
3584.2.b.k 12 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 \(S_{2}^{\mathrm{new}}(3584, [\chi])\):

\( T_{3}^{12} + 28T_{3}^{10} + 288T_{3}^{8} + 1328T_{3}^{6} + 2612T_{3}^{4} + 1584T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{23}^{6} - 76T_{23}^{4} + 96T_{23}^{3} + 1060T_{23}^{2} - 832T_{23} - 2848 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 28 T^{10} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{12} + 32 T^{10} + \cdots + 1024 \) Copy content Toggle raw display
$7$ \( (T - 1)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 68 T^{10} + \cdots + 4096 \) Copy content Toggle raw display
$13$ \( T^{12} + 48 T^{10} + \cdots + 1024 \) Copy content Toggle raw display
$17$ \( (T^{6} - 48 T^{4} + \cdots - 896)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 140 T^{10} + \cdots + 300304 \) Copy content Toggle raw display
$23$ \( (T^{6} - 76 T^{4} + \cdots - 2848)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 168 T^{10} + \cdots + 16384 \) Copy content Toggle raw display
$31$ \( (T^{6} - 8 T^{5} + \cdots + 11776)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 168 T^{10} + \cdots + 16384 \) Copy content Toggle raw display
$41$ \( (T^{6} - 4 T^{5} + \cdots + 5696)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 2117472256 \) Copy content Toggle raw display
$47$ \( (T^{6} + 8 T^{5} + \cdots - 179200)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 2390818816 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 241740304 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 2466512896 \) Copy content Toggle raw display
$67$ \( T^{12} + 356 T^{10} + \cdots + 18939904 \) Copy content Toggle raw display
$71$ \( (T^{6} + 8 T^{5} + \cdots - 51200)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 16 T^{5} + \cdots + 899200)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 24 T^{5} + \cdots - 21632)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 72755351824 \) Copy content Toggle raw display
$89$ \( (T^{6} + 16 T^{5} + \cdots + 6272)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 16 T^{5} + \cdots + 232576)^{2} \) Copy content Toggle raw display
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