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Properties

Label 9984.2.a.bu
Level $9984$
Weight $2$
Character orbit 9984.a
Self dual yes
Analytic conductor $79.723$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9984,2,Mod(1,9984)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9984, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("9984.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9984 = 2^{8} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9984.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: \(79.7226413780\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 14x^{6} + 24x^{5} + 65x^{4} - 82x^{3} - 126x^{2} + 84x + 92 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 312)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} - \beta_1 q^{5} + \beta_{3} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - \beta_1 q^{5} + \beta_{3} q^{7} + q^{9} - \beta_{4} q^{11} - q^{13} - \beta_1 q^{15} + ( - \beta_{2} + 1) q^{17} + (\beta_{7} + 1) q^{19} + \beta_{3} q^{21} + (\beta_{5} + \beta_{3} - \beta_1 + 1) q^{23} + ( - \beta_{6} + \beta_{4} - \beta_{2} + \cdots + 2) q^{25}+ \cdots - \beta_{4} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} - 2 q^{5} - 2 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{3} - 2 q^{5} - 2 q^{7} + 8 q^{9} - 8 q^{13} - 2 q^{15} + 8 q^{17} + 6 q^{19} - 2 q^{21} + 4 q^{23} + 16 q^{25} + 8 q^{27} - 4 q^{29} - 2 q^{31} - 4 q^{35} - 8 q^{37} - 8 q^{39} + 18 q^{41} + 16 q^{43} - 2 q^{45} + 12 q^{47} + 24 q^{49} + 8 q^{51} + 4 q^{53} - 12 q^{55} + 6 q^{57} - 4 q^{59} - 12 q^{61} - 2 q^{63} + 2 q^{65} + 22 q^{67} + 4 q^{69} + 16 q^{73} + 16 q^{75} + 20 q^{77} + 8 q^{81} + 16 q^{83} + 4 q^{85} - 4 q^{87} + 30 q^{89} + 2 q^{91} - 2 q^{93} - 12 q^{95} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 14x^{6} + 24x^{5} + 65x^{4} - 82x^{3} - 126x^{2} + 84x + 92 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - 16\nu^{5} - 4\nu^{4} + 79\nu^{3} + 40\nu^{2} - 114\nu - 80 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{7} + \nu^{6} + 14\nu^{5} - 8\nu^{4} - 59\nu^{3} + 3\nu^{2} + 72\nu + 29 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{7} - 15\nu^{5} - 4\nu^{4} + 69\nu^{3} + 38\nu^{2} - 93\nu - 72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{7} + 46\nu^{5} + 12\nu^{4} - 217\nu^{3} - 112\nu^{2} + 300\nu + 208 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{7} - 4\nu^{6} - 70\nu^{5} + 28\nu^{4} + 299\nu^{3} + 28\nu^{2} - 388\nu - 198 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 3\nu^{7} - 2\nu^{6} - 43\nu^{5} + 12\nu^{4} + 189\nu^{3} + 32\nu^{2} - 247\nu - 132 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -6\nu^{7} + 3\nu^{6} + 87\nu^{5} - 10\nu^{4} - 388\nu^{3} - 123\nu^{2} + 519\nu + 333 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + \beta_{3} + \beta _1 + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{6} - 5\beta_{5} + 5\beta_{4} + 3\beta_{3} + 4\beta_{2} + 9 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + 8\beta_{4} + 9\beta_{3} + \beta_{2} + 9\beta _1 + 45 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 49\beta_{6} - 29\beta_{5} + 33\beta_{4} + 17\beta_{3} + 40\beta_{2} - 4\beta _1 + 69 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 12\beta_{7} + 12\beta_{6} + 12\beta_{5} + 57\beta_{4} + 73\beta_{3} + 14\beta_{2} + 65\beta _1 + 298 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 8\beta_{7} + 353\beta_{6} - 175\beta_{5} + 231\beta_{4} + 141\beta_{3} + 332\beta_{2} - 64\beta _1 + 547 ) / 4 \) 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
−2.65121
2.54551
−1.30823
−0.829147
2.81444
1.60416
1.58297
−1.75849
0 1.00000 0 −4.33571 0 2.30442 0 1.00000 0
1.2 0 1.00000 0 −3.29521 0 −2.97802 0 1.00000 0
1.3 0 1.00000 0 −1.87654 0 −0.584696 0 1.00000 0
1.4 0 1.00000 0 −0.550135 0 −4.37841 0 1.00000 0
1.5 0 1.00000 0 0.218531 0 4.47783 0 1.00000 0
1.6 0 1.00000 0 1.47174 0 2.93973 0 1.00000 0
1.7 0 1.00000 0 3.05343 0 0.397397 0 1.00000 0
1.8 0 1.00000 0 3.31390 0 −4.17825 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(13\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9984.2.a.bu 8
4.b odd 2 1 9984.2.a.bs 8
8.b even 2 1 9984.2.a.bt 8
8.d odd 2 1 9984.2.a.bv 8
16.e even 4 2 312.2.g.b ✓ 16
16.f odd 4 2 1248.2.g.b 16
48.i odd 4 2 936.2.g.e 16
48.k even 4 2 3744.2.g.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.g.b ✓ 16 16.e even 4 2
936.2.g.e 16 48.i odd 4 2
1248.2.g.b 16 16.f odd 4 2
3744.2.g.e 16 48.k even 4 2
9984.2.a.bs 8 4.b odd 2 1
9984.2.a.bt 8 8.b even 2 1
9984.2.a.bu 8 1.a even 1 1 trivial
9984.2.a.bv 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9984))\):

\( T_{5}^{8} + 2T_{5}^{7} - 26T_{5}^{6} - 36T_{5}^{5} + 200T_{5}^{4} + 168T_{5}^{3} - 392T_{5}^{2} - 144T_{5} + 48 \) Copy content Toggle raw display
\( T_{7}^{8} + 2T_{7}^{7} - 38T_{7}^{6} - 60T_{7}^{5} + 444T_{7}^{4} + 416T_{7}^{3} - 1696T_{7}^{2} - 384T_{7} + 384 \) Copy content Toggle raw display
\( T_{11}^{8} - 64T_{11}^{6} + 8T_{11}^{5} + 1232T_{11}^{4} - 512T_{11}^{3} - 7008T_{11}^{2} + 6240T_{11} - 1296 \) Copy content Toggle raw display
\( T_{19}^{8} - 6T_{19}^{7} - 94T_{19}^{6} + 580T_{19}^{5} + 2204T_{19}^{4} - 15808T_{19}^{3} - 1632T_{19}^{2} + 75648T_{19} - 4736 \) Copy content Toggle raw display
\( T_{29}^{8} + 4T_{29}^{7} - 116T_{29}^{6} - 256T_{29}^{5} + 4224T_{29}^{4} - 1088T_{29}^{3} - 50752T_{29}^{2} + 114688T_{29} - 68352 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T - 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 2 T^{7} + \cdots + 48 \) Copy content Toggle raw display
$7$ \( T^{8} + 2 T^{7} + \cdots + 384 \) Copy content Toggle raw display
$11$ \( T^{8} - 64 T^{6} + \cdots - 1296 \) Copy content Toggle raw display
$13$ \( (T + 1)^{8} \) Copy content Toggle raw display
$17$ \( T^{8} - 8 T^{7} + \cdots - 64256 \) Copy content Toggle raw display
$19$ \( T^{8} - 6 T^{7} + \cdots - 4736 \) Copy content Toggle raw display
$23$ \( T^{8} - 4 T^{7} + \cdots - 166912 \) Copy content Toggle raw display
$29$ \( T^{8} + 4 T^{7} + \cdots - 68352 \) Copy content Toggle raw display
$31$ \( T^{8} + 2 T^{7} + \cdots + 110464 \) Copy content Toggle raw display
$37$ \( T^{8} + 8 T^{7} + \cdots + 12544 \) Copy content Toggle raw display
$41$ \( T^{8} - 18 T^{7} + \cdots - 139792 \) Copy content Toggle raw display
$43$ \( T^{8} - 16 T^{7} + \cdots - 749312 \) Copy content Toggle raw display
$47$ \( T^{8} - 12 T^{7} + \cdots + 48 \) Copy content Toggle raw display
$53$ \( T^{8} - 4 T^{7} + \cdots - 21504 \) Copy content Toggle raw display
$59$ \( T^{8} + 4 T^{7} + \cdots - 525968 \) Copy content Toggle raw display
$61$ \( T^{8} + 12 T^{7} + \cdots + 573696 \) Copy content Toggle raw display
$67$ \( T^{8} - 22 T^{7} + \cdots - 640128 \) Copy content Toggle raw display
$71$ \( T^{8} - 408 T^{6} + \cdots + 21431088 \) Copy content Toggle raw display
$73$ \( T^{8} - 16 T^{7} + \cdots + 3326208 \) Copy content Toggle raw display
$79$ \( T^{8} - 336 T^{6} + \cdots + 2166016 \) Copy content Toggle raw display
$83$ \( T^{8} - 16 T^{7} + \cdots - 2508752 \) Copy content Toggle raw display
$89$ \( T^{8} - 30 T^{7} + \cdots + 37581936 \) Copy content Toggle raw display
$97$ \( T^{8} - 20 T^{7} + \cdots - 637696 \) Copy content Toggle raw display
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