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Properties

Label 720.4.w.c
Level $720$
Weight $4$
Character orbit 720.w
Analytic conductor $42.481$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [720,4,Mod(17,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.17");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 720.w (of order \(4\), degree \(2\), 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: \(42.4813752041\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.12745506816.8
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 71x^{4} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_{3} + 3 \beta_{2}) q^{5} + ( - \beta_{7} - 2 \beta_{6} + \beta_1 + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_{3} + 3 \beta_{2}) q^{5} + ( - \beta_{7} - 2 \beta_{6} + \beta_1 + 1) q^{7} + ( - 3 \beta_{5} - \beta_{4} + 23 \beta_{2}) q^{11} + (2 \beta_{7} - \beta_{6} - 36 \beta_1 + 36) q^{13} + (4 \beta_{5} - 2 \beta_{4} + \cdots + 19 \beta_{2}) q^{17}+ \cdots + ( - 20 \beta_{7} - 40 \beta_{6} + \cdots + 429) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{7} + 288 q^{13} + 544 q^{25} + 208 q^{31} - 1488 q^{37} - 1200 q^{43} - 1944 q^{55} - 3120 q^{61} + 1904 q^{67} + 2264 q^{73} + 2752 q^{85} + 8976 q^{91} + 3432 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 71x^{4} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 96\nu^{2} ) / 275 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 6\nu^{7} + 25\nu^{5} + 301\nu^{3} + 1025\nu ) / 1375 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6\nu^{7} - 25\nu^{5} + 301\nu^{3} - 1025\nu ) / 1375 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -16\nu^{7} + 50\nu^{5} - 1261\nu^{3} + 7550\nu ) / 1375 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 32\nu^{7} + 25\nu^{5} + 2522\nu^{3} + 3775\nu ) / 1375 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -11\nu^{6} - 150\nu^{4} - 506\nu^{2} - 5325 ) / 275 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -33\nu^{6} + 50\nu^{4} - 1518\nu^{2} + 1775 ) / 275 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{5} + 4\beta_{4} + 5\beta_{3} - 5\beta_{2} ) / 20 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{7} + \beta_{6} + 110\beta_1 ) / 20 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6\beta_{5} - 3\beta_{4} - 20\beta_{3} - 20\beta_{2} ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11\beta_{7} - 33\beta_{6} - 710 ) / 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -82\beta_{5} - 164\beta_{4} - 755\beta_{3} + 755\beta_{2} ) / 20 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -72\beta_{7} - 24\beta_{6} - 1265\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -1204\beta_{5} + 602\beta_{4} + 6305\beta_{3} + 6305\beta_{2} ) / 20 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{1}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
1.26663 + 1.26663i
1.97374 + 1.97374i
−1.97374 − 1.97374i
−1.26663 − 1.26663i
1.26663 − 1.26663i
1.97374 − 1.97374i
−1.97374 + 1.97374i
−1.26663 + 1.26663i
0 0 0 −11.1353 + 1.00227i 0 23.9129 + 23.9129i 0 0 0
17.2 0 0 0 −8.30690 − 7.48301i 0 −21.9129 − 21.9129i 0 0 0
17.3 0 0 0 8.30690 + 7.48301i 0 −21.9129 − 21.9129i 0 0 0
17.4 0 0 0 11.1353 − 1.00227i 0 23.9129 + 23.9129i 0 0 0
593.1 0 0 0 −11.1353 − 1.00227i 0 23.9129 − 23.9129i 0 0 0
593.2 0 0 0 −8.30690 + 7.48301i 0 −21.9129 + 21.9129i 0 0 0
593.3 0 0 0 8.30690 − 7.48301i 0 −21.9129 + 21.9129i 0 0 0
593.4 0 0 0 11.1353 + 1.00227i 0 23.9129 − 23.9129i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.4.w.c 8
3.b odd 2 1 inner 720.4.w.c 8
4.b odd 2 1 90.4.f.b ✓ 8
5.c odd 4 1 inner 720.4.w.c 8
12.b even 2 1 90.4.f.b ✓ 8
15.e even 4 1 inner 720.4.w.c 8
20.d odd 2 1 450.4.f.e 8
20.e even 4 1 90.4.f.b ✓ 8
20.e even 4 1 450.4.f.e 8
60.h even 2 1 450.4.f.e 8
60.l odd 4 1 90.4.f.b ✓ 8
60.l odd 4 1 450.4.f.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.4.f.b ✓ 8 4.b odd 2 1
90.4.f.b ✓ 8 12.b even 2 1
90.4.f.b ✓ 8 20.e even 4 1
90.4.f.b ✓ 8 60.l odd 4 1
450.4.f.e 8 20.d odd 2 1
450.4.f.e 8 20.e even 4 1
450.4.f.e 8 60.h even 2 1
450.4.f.e 8 60.l odd 4 1
720.4.w.c 8 1.a even 1 1 trivial
720.4.w.c 8 3.b odd 2 1 inner
720.4.w.c 8 5.c odd 4 1 inner
720.4.w.c 8 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 4T_{7}^{3} + 8T_{7}^{2} + 4192T_{7} + 1098304 \) acting on \(S_{4}^{\mathrm{new}}(720, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 272 T^{6} + \cdots + 244140625 \) Copy content Toggle raw display
$7$ \( (T^{4} - 4 T^{3} + \cdots + 1098304)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 4216 T^{2} + 64)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 144 T^{3} + \cdots + 2377764)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 185189072896 \) Copy content Toggle raw display
$19$ \( (T^{4} + 10928 T^{2} + 1597696)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 614656)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 53524 T^{2} + 662444644)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 52 T - 1424)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 744 T^{3} + \cdots + 3569106564)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 42436 T^{2} + 153313924)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 600 T^{3} + \cdots + 1664640000)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( (T^{4} - 578776 T^{2} + 33530004544)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 780 T - 102000)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 952 T^{3} + \cdots + 5698438144)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 295744 T^{2} + 13058089984)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 1132 T^{3} + \cdots + 2081366884)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 491568 T^{2} + 1598720256)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 15\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{4} - 757876 T^{2} + 92013942244)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 1716 T^{3} + \cdots + 2695478724)^{2} \) Copy content Toggle raw display
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