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Properties

Label 528.5.j.a
Level $528$
Weight $5$
Character orbit 528.j
Analytic conductor $54.579$
Analytic rank $0$
Dimension $8$
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] = [528,5,Mod(241,528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(528, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("528.241");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 528.j (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: \(54.5793405083\)
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: \( x^{8} + 102x^{6} + 2913x^{4} + 23292x^{2} + 41364 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 33)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + (\beta_{6} + \beta_{3} - 5) q^{5} + (\beta_{7} - \beta_{5} + \cdots - 3 \beta_1) q^{7}+ \cdots + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + (\beta_{6} + \beta_{3} - 5) q^{5} + (\beta_{7} - \beta_{5} + \cdots - 3 \beta_1) q^{7}+ \cdots + ( - 54 \beta_{7} - 27 \beta_{6} + \cdots - 108) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 36 q^{5} + 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 36 q^{5} + 216 q^{9} - 36 q^{11} - 108 q^{15} - 516 q^{23} - 2280 q^{25} - 2752 q^{31} + 1008 q^{33} + 5296 q^{37} - 972 q^{45} - 420 q^{47} - 6832 q^{49} + 3540 q^{53} - 3784 q^{55} + 16632 q^{59} + 3656 q^{67} + 9036 q^{69} + 13212 q^{71} + 9288 q^{75} + 23268 q^{77} + 5832 q^{81} + 15528 q^{89} + 19752 q^{91} - 21384 q^{93} + 7624 q^{97} - 972 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 102x^{6} + 2913x^{4} + 23292x^{2} + 41364 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{7} - 444\nu^{5} - 9669\nu^{3} - 41250\nu ) / 3216 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{6} - 444\nu^{4} - 9669\nu^{2} - 38034 ) / 3216 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{6} + 320\nu^{4} + 8535\nu^{2} + 1608\nu + 31182 ) / 1608 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} + 320\nu^{5} + 10143\nu^{3} + 93894\nu ) / 1608 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 196\nu^{4} + 10617\nu^{2} + 107946 ) / 3216 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{7} - 191\nu^{5} - 4551\nu^{3} - 16500\nu ) / 402 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{6} - 2\beta_{4} - 2\beta_{3} + \beta _1 - 52 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + 2\beta_{5} - 4\beta_{2} - 43\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -102\beta_{6} + 162\beta_{4} + 174\beta_{3} - 81\beta _1 + 2340 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -162\beta_{7} - 102\beta_{5} + 396\beta_{2} + 2193\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 5190\beta_{6} - 10518\beta_{4} - 12870\beta_{3} + 5259\beta _1 - 122448 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 10518\beta_{7} + 5190\beta_{5} - 28716\beta_{2} - 119835\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/528\mathbb{Z}\right)^\times\).

\(n\) \(133\) \(145\) \(353\) \(463\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
241.1
7.70102i
− 7.70102i
3.00247i
− 3.00247i
− 5.58567i
5.58567i
1.57474i
− 1.57474i
0 −5.19615 0 −12.5296 0 − 63.6540i 0 27.0000 0
241.2 0 −5.19615 0 −12.5296 0 63.6540i 0 27.0000 0
241.3 0 −5.19615 0 8.72578 0 − 1.45810i 0 27.0000 0
241.4 0 −5.19615 0 8.72578 0 1.45810i 0 27.0000 0
241.5 0 5.19615 0 −29.7487 0 − 12.9420i 0 27.0000 0
241.6 0 5.19615 0 −29.7487 0 12.9420i 0 27.0000 0
241.7 0 5.19615 0 15.5526 0 − 93.8006i 0 27.0000 0
241.8 0 5.19615 0 15.5526 0 93.8006i 0 27.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 241.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 528.5.j.a 8
4.b odd 2 1 33.5.c.a ✓ 8
11.b odd 2 1 inner 528.5.j.a 8
12.b even 2 1 99.5.c.c 8
44.c even 2 1 33.5.c.a ✓ 8
132.d odd 2 1 99.5.c.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.5.c.a ✓ 8 4.b odd 2 1
33.5.c.a ✓ 8 44.c even 2 1
99.5.c.c 8 12.b even 2 1
99.5.c.c 8 132.d odd 2 1
528.5.j.a 8 1.a even 1 1 trivial
528.5.j.a 8 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 18T_{5}^{3} - 518T_{5}^{2} - 3312T_{5} + 50584 \) acting on \(S_{5}^{\mathrm{new}}(528, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} - 27)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} + 18 T^{3} + \cdots + 50584)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 12695273424 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 45\!\cdots\!61 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 57\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( (T^{4} + 258 T^{3} + \cdots + 2066510872)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 32\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( (T^{4} + 1376 T^{3} + \cdots + 6236194624)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 2648 T^{3} + \cdots + 26585618752)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 25\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 12403328125408)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots - 15377555198312)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots - 39877809797408)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots - 192515994273728)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 855560872809632)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 28\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 76\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 12\!\cdots\!44)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 466619407059968)^{2} \) Copy content Toggle raw display
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