Properties

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

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [720,4,Mod(719,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.719");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 720.o (of order \(2\), degree \(1\), 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: \(8\)
Coefficient field: 8.0.928445276160000.218
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 130x^{6} - 376x^{5} + 5673x^{4} - 10724x^{3} + 96952x^{2} - 91652x + 530329 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + (5 \beta_{5} + 5 \beta_1) q^{5} - \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (5 \beta_{5} + 5 \beta_1) q^{5} - \beta_{2} q^{7} + \beta_{3} q^{11} - 13 \beta_{4} q^{13} + 28 \beta_1 q^{17} + 2 \beta_{6} q^{19} - \beta_{7} q^{23} + ( - 50 \beta_{4} + 25) q^{25} + 38 \beta_{5} q^{29} - 3 \beta_{6} q^{31} + ( - 5 \beta_{7} - 5 \beta_{3}) q^{35} - 59 \beta_{4} q^{37} + 109 \beta_{5} q^{41} + 6 \beta_{2} q^{43} + 16 \beta_{7} q^{47} + 347 q^{49} - 384 \beta_1 q^{53} + ( - 5 \beta_{6} + 15 \beta_{2}) q^{55} + 11 \beta_{3} q^{59} + 550 q^{61} + (195 \beta_{5} - 130 \beta_1) q^{65} - 32 \beta_{2} q^{67} - 12 \beta_{3} q^{71} - 242 \beta_{4} q^{73} - 690 \beta_1 q^{77} + 7 \beta_{6} q^{79} - 22 \beta_{7} q^{83} + ( - 140 \beta_{4} + 420) q^{85} + 227 \beta_{5} q^{89} + 13 \beta_{6} q^{91} + ( - 30 \beta_{7} + 20 \beta_{3}) q^{95} - 774 \beta_{4} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 200 q^{25} + 2776 q^{49} + 4400 q^{61} + 3360 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 130x^{6} - 376x^{5} + 5673x^{4} - 10724x^{3} + 96952x^{2} - 91652x + 530329 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{6} - 6\nu^{5} + 307\nu^{4} - 604\nu^{3} + 12719\nu^{2} - 12418\nu + 130554 ) / 12317 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} + 3\nu^{5} - 97\nu^{4} + 189\nu^{3} - 2461\nu^{2} + 2367\nu - 13636 ) / 113 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{6} - 9\nu^{5} + 297\nu^{4} - 579\nu^{3} + 8451\nu^{2} - 8163\nu + 66666 ) / 109 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 174 \nu^{7} + 609 \nu^{6} - 16615 \nu^{5} + 40015 \nu^{4} - 402745 \nu^{3} + 564407 \nu^{2} - 1771845 \nu + 793174 ) / 1242709 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 222 \nu^{7} + 777 \nu^{6} - 22771 \nu^{5} + 54985 \nu^{4} - 678965 \nu^{3} + 963851 \nu^{2} - 5724175 \nu + 2703260 ) / 1288313 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 27216 \nu^{7} - 95256 \nu^{6} + 3627264 \nu^{5} - 8830020 \nu^{4} + 170982000 \nu^{3} - 247690608 \nu^{2} + 4227412944 \nu - 2072716770 ) / 140426117 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -16\nu^{7} + 56\nu^{6} - 2052\nu^{5} + 4990\nu^{4} - 76872\nu^{3} + 110346\nu^{2} - 777796\nu + 370672 ) / 11401 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} + 6\beta_{5} - 6\beta_{4} + 6 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 6\beta_{5} - 6\beta_{4} + 2\beta_{3} + 6\beta_{2} - 12\beta _1 - 366 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9\beta_{7} - 34\beta_{6} - 543\beta_{5} + 531\beta_{4} + 3\beta_{3} + 9\beta_{2} - 18\beta _1 - 552 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6\beta_{7} - 23\beta_{6} - 364\beta_{5} + 356\beta_{4} - 44\beta_{3} - 124\beta_{2} + 700\beta _1 + 4530 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 900 \beta_{7} + 1350 \beta_{6} + 32169 \beta_{5} - 28701 \beta_{4} - 335 \beta_{3} - 945 \beta_{2} + 5280 \beta _1 + 34896 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 2745 \beta_{7} + 4223 \beta_{6} + 99240 \beta_{5} - 88776 \beta_{4} + 7444 \beta_{3} + 18828 \beta_{2} - 161730 \beta _1 - 566574 ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 59640 \beta_{7} - 58239 \beta_{6} - 1760352 \beta_{5} + 1402374 \beta_{4} + 27230 \beta_{3} + 69216 \beta_{2} - 584556 \beta _1 - 2105790 ) / 12 \) Copy content Toggle raw display

Character values

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

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
719.1
0.500000 4.84426i
0.500000 + 5.87954i
0.500000 + 4.84426i
0.500000 5.87954i
0.500000 + 3.43005i
0.500000 7.29375i
0.500000 3.43005i
0.500000 + 7.29375i
0 0 0 −8.66025 7.07107i 0 −26.2679 0 0 0
719.2 0 0 0 −8.66025 7.07107i 0 26.2679 0 0 0
719.3 0 0 0 −8.66025 + 7.07107i 0 −26.2679 0 0 0
719.4 0 0 0 −8.66025 + 7.07107i 0 26.2679 0 0 0
719.5 0 0 0 8.66025 7.07107i 0 −26.2679 0 0 0
719.6 0 0 0 8.66025 7.07107i 0 26.2679 0 0 0
719.7 0 0 0 8.66025 + 7.07107i 0 −26.2679 0 0 0
719.8 0 0 0 8.66025 + 7.07107i 0 26.2679 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 719.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.4.o.b 8
3.b odd 2 1 inner 720.4.o.b 8
4.b odd 2 1 inner 720.4.o.b 8
5.b even 2 1 inner 720.4.o.b 8
12.b even 2 1 inner 720.4.o.b 8
15.d odd 2 1 inner 720.4.o.b 8
20.d odd 2 1 inner 720.4.o.b 8
60.h even 2 1 inner 720.4.o.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.4.o.b 8 1.a even 1 1 trivial
720.4.o.b 8 3.b odd 2 1 inner
720.4.o.b 8 4.b odd 2 1 inner
720.4.o.b 8 5.b even 2 1 inner
720.4.o.b 8 12.b even 2 1 inner
720.4.o.b 8 15.d odd 2 1 inner
720.4.o.b 8 20.d odd 2 1 inner
720.4.o.b 8 60.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 690 \) 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
$5$ \( (T^{4} - 50 T^{2} + 15625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 690)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 2070)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1014)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 2352)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 16560)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1380)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2888)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 37260)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 20886)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 23762)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 24840)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 353280)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 442368)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 250470)^{4} \) Copy content Toggle raw display
$61$ \( (T - 550)^{8} \) Copy content Toggle raw display
$67$ \( (T^{2} - 706560)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 298080)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 351384)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 202860)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 667920)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 103058)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 3594456)^{4} \) Copy content Toggle raw display
show more
show less