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Properties

Label 5120.2.a.t
Level $5120$
Weight $2$
Character orbit 5120.a
Self dual yes
Analytic conductor $40.883$
Analytic rank $1$
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] = [5120,2,Mod(1,5120)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5120, 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("5120.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5120 = 2^{10} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5120.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: \(40.8834058349\)
Analytic rank: \(1\)
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: \( x^{8} - 12x^{6} - 8x^{5} + 21x^{4} + 12x^{3} - 10x^{2} - 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 80)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} - 1) q^{3} + q^{5} + \beta_{4} q^{7} + ( - \beta_{7} - \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 1) q^{3} + q^{5} + \beta_{4} q^{7} + ( - \beta_{7} - \beta_{2} + 1) q^{9} + (\beta_{7} + \beta_{5} - 1) q^{11} + ( - \beta_{5} - \beta_{4} + \cdots + \beta_1) q^{13}+ \cdots + (\beta_{7} - \beta_{6} - 2 \beta_{4} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} + 8 q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} + 8 q^{5} - 4 q^{7} + 8 q^{9} - 8 q^{11} - 4 q^{15} - 16 q^{19} - 12 q^{23} + 8 q^{25} - 16 q^{27} - 4 q^{35} - 28 q^{43} + 8 q^{45} - 20 q^{47} + 8 q^{49} - 24 q^{51} - 8 q^{55} - 16 q^{59} - 20 q^{63} - 36 q^{67} - 16 q^{69} + 8 q^{71} - 4 q^{75} + 8 q^{79} + 8 q^{81} - 20 q^{83} - 40 q^{91} - 16 q^{95} - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 12x^{6} - 8x^{5} + 21x^{4} + 12x^{3} - 10x^{2} - 4x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{7} + 9\nu^{6} + 9\nu^{5} - 84\nu^{4} + 47\nu^{3} + 198\nu^{2} - 55\nu - 59 ) / 17 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -8\nu^{7} + 19\nu^{6} + 87\nu^{5} - 149\nu^{4} - 220\nu^{3} + 214\nu^{2} + 86\nu - 49 ) / 17 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14\nu^{7} - 12\nu^{6} - 165\nu^{5} + 27\nu^{4} + 351\nu^{3} - 43\nu^{2} - 159\nu + 5 ) / 17 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10\nu^{7} + 6\nu^{6} - 113\nu^{5} - 158\nu^{4} + 88\nu^{3} + 251\nu^{2} + 54\nu - 96 ) / 17 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -44\nu^{7} + 28\nu^{6} + 504\nu^{5} + 39\nu^{4} - 887\nu^{3} + 4\nu^{2} + 371\nu + 11 ) / 17 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 38\nu^{7} - \nu^{6} - 443\nu^{5} - 291\nu^{4} + 654\nu^{3} + 318\nu^{2} - 162\nu - 35 ) / 17 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 54\nu^{7} - 22\nu^{6} - 617\nu^{5} - 180\nu^{4} + 958\nu^{3} + 77\nu^{2} - 300\nu + 29 ) / 17 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + \beta_{4} + 4\beta_{3} + 2\beta_{2} - 2\beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -10\beta_{7} + 8\beta_{6} - 3\beta_{5} - \beta_{4} + 9\beta_{3} + 4\beta_{2} - 10\beta _1 + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -17\beta_{7} + 7\beta_{6} - \beta_{5} + 7\beta_{4} + 48\beta_{3} + 24\beta_{2} - 29\beta _1 + 38 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -107\beta_{7} + 78\beta_{6} - 32\beta_{5} - 3\beta_{4} + 120\beta_{3} + 60\beta_{2} - 118\beta _1 + 75 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -252\beta_{7} + 137\beta_{6} - 35\beta_{5} + 58\beta_{4} + 554\beta_{3} + 278\beta_{2} - 375\beta _1 + 399 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -1208\beta_{7} + 834\beta_{6} - 330\beta_{5} + 29\beta_{4} + 1597\beta_{3} + 805\beta_{2} - 1423\beta _1 + 1067 ) / 2 \) 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
1.06941
0.187687
−1.50013
−0.602887
0.731397
−2.56993
3.51762
−0.833171
0 −3.28980 0 1.00000 0 0.982011 0 7.82281 0
1.2 0 −2.58464 0 1.00000 0 −4.50961 0 3.68037 0
1.3 0 −1.93785 0 1.00000 0 2.73482 0 0.755274 0
1.4 0 −1.01919 0 1.00000 0 −4.02840 0 −1.96126 0
1.5 0 0.169718 0 1.00000 0 2.66881 0 −2.97120 0
1.6 0 0.296378 0 1.00000 0 1.73696 0 −2.91216 0
1.7 0 2.01261 0 1.00000 0 −0.690576 0 1.05061 0
1.8 0 2.35278 0 1.00000 0 −2.89402 0 2.53555 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
\(2\) \(-1\)
\(5\) \(-1\)

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 5120.2.a.t 8
4.b odd 2 1 5120.2.a.v 8
8.b even 2 1 5120.2.a.u 8
8.d odd 2 1 5120.2.a.s 8
32.g even 8 2 320.2.l.a 16
32.g even 8 2 640.2.l.a 16
32.h odd 8 2 80.2.l.a ✓ 16
32.h odd 8 2 640.2.l.b 16
96.o even 8 2 720.2.t.c 16
96.p odd 8 2 2880.2.t.c 16
160.u even 8 2 400.2.q.h 16
160.v odd 8 2 1600.2.q.h 16
160.y odd 8 2 400.2.l.h 16
160.z even 8 2 1600.2.l.i 16
160.ba even 8 2 400.2.q.g 16
160.bb odd 8 2 1600.2.q.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.l.a ✓ 16 32.h odd 8 2
320.2.l.a 16 32.g even 8 2
400.2.l.h 16 160.y odd 8 2
400.2.q.g 16 160.ba even 8 2
400.2.q.h 16 160.u even 8 2
640.2.l.a 16 32.g even 8 2
640.2.l.b 16 32.h odd 8 2
720.2.t.c 16 96.o even 8 2
1600.2.l.i 16 160.z even 8 2
1600.2.q.g 16 160.bb odd 8 2
1600.2.q.h 16 160.v odd 8 2
2880.2.t.c 16 96.p odd 8 2
5120.2.a.s 8 8.d odd 2 1
5120.2.a.t 8 1.a even 1 1 trivial
5120.2.a.u 8 8.b even 2 1
5120.2.a.v 8 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5120))\):

\( T_{3}^{8} + 4T_{3}^{7} - 8T_{3}^{6} - 40T_{3}^{5} + 8T_{3}^{4} + 104T_{3}^{3} + 32T_{3}^{2} - 32T_{3} + 4 \) Copy content Toggle raw display
\( T_{7}^{8} + 4T_{7}^{7} - 24T_{7}^{6} - 72T_{7}^{5} + 232T_{7}^{4} + 328T_{7}^{3} - 896T_{7}^{2} - 32T_{7} + 452 \) Copy content Toggle raw display
\( T_{13}^{8} - 56T_{13}^{6} - 96T_{13}^{5} + 768T_{13}^{4} + 2176T_{13}^{3} - 960T_{13}^{2} - 6912T_{13} - 4544 \) Copy content Toggle raw display
\( T_{29}^{8} - 104T_{29}^{6} + 3064T_{29}^{4} + 1024T_{29}^{3} - 27424T_{29}^{2} - 14336T_{29} + 54928 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 4 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T - 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 4 T^{7} + \cdots + 452 \) Copy content Toggle raw display
$11$ \( T^{8} + 8 T^{7} + \cdots - 1136 \) Copy content Toggle raw display
$13$ \( T^{8} - 56 T^{6} + \cdots - 4544 \) Copy content Toggle raw display
$17$ \( T^{8} - 72 T^{6} + \cdots + 13888 \) Copy content Toggle raw display
$19$ \( T^{8} + 16 T^{7} + \cdots + 784 \) Copy content Toggle raw display
$23$ \( T^{8} + 12 T^{7} + \cdots + 1316 \) Copy content Toggle raw display
$29$ \( T^{8} - 104 T^{6} + \cdots + 54928 \) Copy content Toggle raw display
$31$ \( T^{8} - 96 T^{6} + \cdots - 20224 \) Copy content Toggle raw display
$37$ \( T^{8} - 160 T^{6} + \cdots + 4352 \) Copy content Toggle raw display
$41$ \( T^{8} - 192 T^{6} + \cdots - 332656 \) Copy content Toggle raw display
$43$ \( T^{8} + 28 T^{7} + \cdots - 7324 \) Copy content Toggle raw display
$47$ \( T^{8} + 20 T^{7} + \cdots + 575044 \) Copy content Toggle raw display
$53$ \( T^{8} - 200 T^{6} + \cdots - 619456 \) Copy content Toggle raw display
$59$ \( T^{8} + 16 T^{7} + \cdots - 110576 \) Copy content Toggle raw display
$61$ \( T^{8} - 240 T^{6} + \cdots + 1180672 \) Copy content Toggle raw display
$67$ \( T^{8} + 36 T^{7} + \cdots + 6791204 \) Copy content Toggle raw display
$71$ \( T^{8} - 8 T^{7} + \cdots - 1825792 \) Copy content Toggle raw display
$73$ \( T^{8} - 280 T^{6} + \cdots - 125888 \) Copy content Toggle raw display
$79$ \( T^{8} - 8 T^{7} + \cdots + 4352 \) Copy content Toggle raw display
$83$ \( T^{8} + 20 T^{7} + \cdots + 45284 \) Copy content Toggle raw display
$89$ \( T^{8} - 208 T^{6} + \cdots - 827136 \) Copy content Toggle raw display
$97$ \( T^{8} - 440 T^{6} + \cdots - 8549312 \) Copy content Toggle raw display
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