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Properties

Label 520.6.a.f
Level $520$
Weight $6$
Character orbit 520.a
Self dual yes
Analytic conductor $83.400$
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] = [520,6,Mod(1,520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(520, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("520.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 520.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: \(83.3995863027\)
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: \( x^{8} - 1344x^{6} - 288x^{5} + 542568x^{4} - 84480x^{3} - 68942848x^{2} + 12558720x + 2674054800 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{15}\cdot 3\cdot 5 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - 25 q^{5} + ( - \beta_{3} - 2 \beta_1 - 23) q^{7} + (\beta_{3} + \beta_{2} + \beta_1 + 93) q^{9} + (\beta_{5} + \beta_{3} - \beta_{2} + \cdots - 90) q^{11} + 169 q^{13} + 25 \beta_1 q^{15}+ \cdots + ( - 42 \beta_{7} - 122 \beta_{6} + \cdots - 49476) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 200 q^{5} - 184 q^{7} + 744 q^{9} - 720 q^{11} + 1352 q^{13} - 248 q^{17} - 2544 q^{19} + 4520 q^{21} - 88 q^{23} + 5000 q^{25} - 864 q^{27} + 2224 q^{29} + 5368 q^{31} + 9720 q^{33} + 4600 q^{35}+ \cdots - 395808 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 1344x^{6} - 288x^{5} + 542568x^{4} - 84480x^{3} - 68942848x^{2} + 12558720x + 2674054800 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 473 \nu^{7} - 1460 \nu^{6} - 602542 \nu^{5} + 1351006 \nu^{4} + 213834404 \nu^{3} + \cdots - 31533938280 ) / 185935680 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 473 \nu^{7} + 1460 \nu^{6} + 602542 \nu^{5} - 1351006 \nu^{4} - 213834404 \nu^{3} + \cdots - 30940450200 ) / 185935680 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 151 \nu^{7} - 2104 \nu^{6} - 171062 \nu^{5} + 2306702 \nu^{4} + 45687292 \nu^{3} + \cdots + 43954000344 ) / 37187136 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 290 \nu^{7} + 5793 \nu^{6} - 391808 \nu^{5} - 6765044 \nu^{4} + 141171240 \nu^{3} + \cdots - 139545379920 ) / 61978560 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1091 \nu^{7} + 32257 \nu^{6} - 1271866 \nu^{5} - 42903404 \nu^{4} + 384627068 \nu^{3} + \cdots - 1092276422640 ) / 185935680 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1459 \nu^{7} + 5324 \nu^{6} - 1792655 \nu^{5} - 6891499 \nu^{4} + 595167412 \nu^{3} + \cdots - 109674850140 ) / 92967840 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 + 336 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} - 4\beta_{5} - 5\beta_{4} + \beta_{3} + 4\beta_{2} + 538\beta _1 + 108 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 48\beta_{7} - 32\beta_{6} - 24\beta_{4} + 936\beta_{3} + 752\beta_{2} + 1408\beta _1 + 180300 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2496 \beta_{7} + 32 \beta_{6} - 4128 \beta_{5} - 3984 \beta_{4} + 5968 \beta_{3} + 4448 \beta_{2} + \cdots + 294720 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 59840 \beta_{7} - 36224 \beta_{6} + 1472 \beta_{5} - 33056 \beta_{4} + 748588 \beta_{3} + \cdots + 111758592 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2323032 \beta_{7} + 20352 \beta_{6} - 3445680 \beta_{5} - 2848188 \beta_{4} + 7333932 \beta_{3} + \cdots + 406829136 \) 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
27.4182
20.1115
10.3870
9.12057
−8.52201
−10.3988
−22.9775
−25.1390
0 −27.4182 0 −25.0000 0 −211.374 0 508.759 0
1.2 0 −20.1115 0 −25.0000 0 125.327 0 161.473 0
1.3 0 −10.3870 0 −25.0000 0 −38.6052 0 −135.111 0
1.4 0 −9.12057 0 −25.0000 0 −144.052 0 −159.815 0
1.5 0 8.52201 0 −25.0000 0 192.988 0 −170.375 0
1.6 0 10.3988 0 −25.0000 0 −42.4705 0 −134.866 0
1.7 0 22.9775 0 −25.0000 0 8.46221 0 284.964 0
1.8 0 25.1390 0 −25.0000 0 −74.2765 0 388.971 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
\(2\) \( -1 \)
\(5\) \( +1 \)
\(13\) \( -1 \)

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 520.6.a.f ✓ 8
4.b odd 2 1 1040.6.a.y 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.6.a.f ✓ 8 1.a even 1 1 trivial
1040.6.a.y 8 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 1344T_{3}^{6} + 288T_{3}^{5} + 542568T_{3}^{4} + 84480T_{3}^{3} - 68942848T_{3}^{2} - 12558720T_{3} + 2674054800 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(520))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 2674054800 \) Copy content Toggle raw display
$5$ \( (T + 25)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots - 758955843705600 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 49\!\cdots\!72 \) Copy content Toggle raw display
$13$ \( (T - 169)^{8} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 76\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 87\!\cdots\!12 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots - 22\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 47\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 20\!\cdots\!08 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 17\!\cdots\!80 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots - 72\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 51\!\cdots\!80 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 18\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 29\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots - 36\!\cdots\!08 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots - 29\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots - 95\!\cdots\!60 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots - 52\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 70\!\cdots\!40 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 37\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots - 60\!\cdots\!12 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots - 90\!\cdots\!00 \) Copy content Toggle raw display
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