↑

Properties

Label 720.2.t.b
Level $720$
Weight $2$
Character orbit 720.t
Analytic conductor $5.749$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [720,2,Mod(181,720)]
 
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, 1, 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("720.181");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.t (of order \(4\), degree \(2\), 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: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.18939904.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 240)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{4}) q^{2} + ( - \beta_{7} - \beta_{6} - \beta_{3} + \cdots - 1) q^{4}+ \cdots + (\beta_{6} - \beta_{5} - \beta_{4} + \cdots + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_{4}) q^{2} + ( - \beta_{7} - \beta_{6} - \beta_{3} + \cdots - 1) q^{4}+ \cdots + ( - 5 \beta_{5} + 5 \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{4} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{4} + 12 q^{8} - 8 q^{11} + 8 q^{13} + 4 q^{14} - 8 q^{17} + 8 q^{19} + 8 q^{20} + 16 q^{22} + 20 q^{26} - 8 q^{28} - 24 q^{29} + 8 q^{31} + 16 q^{34} + 8 q^{35} - 8 q^{37} + 16 q^{38} - 4 q^{40} + 16 q^{44} + 24 q^{46} + 40 q^{49} - 4 q^{50} + 16 q^{52} + 16 q^{56} - 8 q^{59} - 16 q^{61} - 28 q^{62} + 8 q^{64} + 8 q^{65} + 8 q^{68} + 4 q^{70} + 36 q^{74} - 40 q^{76} + 8 q^{77} - 40 q^{79} - 16 q^{80} + 64 q^{82} - 32 q^{83} + 8 q^{85} - 16 q^{86} - 16 q^{88} - 8 q^{91} - 16 q^{92} + 32 q^{94} + 16 q^{95} - 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu^{7} - 7\nu^{6} + 24\nu^{5} - 42\nu^{4} + 59\nu^{3} - 48\nu^{2} + 24\nu - 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\nu^{7} + 7\nu^{6} - 24\nu^{5} + 43\nu^{4} - 61\nu^{3} + 54\nu^{2} - 29\nu + 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -3\nu^{7} + 10\nu^{6} - 35\nu^{5} + 60\nu^{4} - 87\nu^{3} + 73\nu^{2} - 42\nu + 11 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -3\nu^{7} + 11\nu^{6} - 38\nu^{5} + 70\nu^{4} - 102\nu^{3} + 91\nu^{2} - 53\nu + 13 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 5\nu^{7} - 17\nu^{6} + 60\nu^{5} - 105\nu^{4} + 155\nu^{3} - 133\nu^{2} + 77\nu - 19 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -5\nu^{7} + 18\nu^{6} - 63\nu^{5} + 115\nu^{4} - 170\nu^{3} + 152\nu^{2} - 89\nu + 23 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -8\nu^{7} + 28\nu^{6} - 98\nu^{5} + 175\nu^{4} - 256\nu^{3} + 223\nu^{2} - 126\nu + 31 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + 3\beta_{6} + \beta_{5} - 3\beta_{4} + \beta_{3} + \beta_{2} - \beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{7} - \beta_{6} + 7\beta_{5} - \beta_{4} + 5\beta_{3} - 3\beta_{2} + 3\beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11\beta_{7} - 15\beta_{6} + 3\beta_{5} + 11\beta_{4} - \beta_{3} - 5\beta_{2} + 9\beta _1 + 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -13\beta_{7} - 11\beta_{6} - 29\beta_{5} + 17\beta_{4} - 23\beta_{3} + 13\beta_{2} - 3\beta _1 + 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -67\beta_{7} + 59\beta_{6} - 41\beta_{5} - 29\beta_{4} - 15\beta_{3} + 37\beta_{2} - 47\beta _1 - 48 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -7\beta_{7} + 113\beta_{6} + 97\beta_{5} - 105\beta_{4} + 91\beta_{3} - 31\beta_{2} - 39\beta _1 - 122 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(-\beta_{7}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
0.500000 + 0.0297061i
0.500000 − 0.691860i
0.500000 − 1.44392i
0.500000 + 2.10607i
0.500000 − 0.0297061i
0.500000 + 0.691860i
0.500000 + 1.44392i
0.500000 − 2.10607i
−0.874559 − 1.11137i 0 −0.470294 + 1.94392i −0.707107 − 0.707107i 0 1.41421i 2.57172 − 1.17740i 0 −0.167452 + 1.40426i
181.2 −0.635665 + 1.26330i 0 −1.19186 − 1.60607i 0.707107 + 0.707107i 0 − 1.41421i 2.78658 − 0.484753i 0 −1.34277 + 0.443806i
181.3 0.167452 + 1.40426i 0 −1.94392 + 0.470294i −0.707107 − 0.707107i 0 1.41421i −0.985930 − 2.65103i 0 0.874559 − 1.11137i
181.4 1.34277 + 0.443806i 0 1.60607 + 1.19186i 0.707107 + 0.707107i 0 − 1.41421i 1.62764 + 2.31318i 0 0.635665 + 1.26330i
541.1 −0.874559 + 1.11137i 0 −0.470294 − 1.94392i −0.707107 + 0.707107i 0 − 1.41421i 2.57172 + 1.17740i 0 −0.167452 − 1.40426i
541.2 −0.635665 − 1.26330i 0 −1.19186 + 1.60607i 0.707107 − 0.707107i 0 1.41421i 2.78658 + 0.484753i 0 −1.34277 − 0.443806i
541.3 0.167452 − 1.40426i 0 −1.94392 − 0.470294i −0.707107 + 0.707107i 0 − 1.41421i −0.985930 + 2.65103i 0 0.874559 + 1.11137i
541.4 1.34277 − 0.443806i 0 1.60607 − 1.19186i 0.707107 − 0.707107i 0 1.41421i 1.62764 − 2.31318i 0 0.635665 − 1.26330i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.t.b 8
3.b odd 2 1 240.2.s.b ✓ 8
4.b odd 2 1 2880.2.t.b 8
12.b even 2 1 960.2.s.b 8
16.e even 4 1 inner 720.2.t.b 8
16.f odd 4 1 2880.2.t.b 8
24.f even 2 1 1920.2.s.d 8
24.h odd 2 1 1920.2.s.c 8
48.i odd 4 1 240.2.s.b ✓ 8
48.i odd 4 1 1920.2.s.c 8
48.k even 4 1 960.2.s.b 8
48.k even 4 1 1920.2.s.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.s.b ✓ 8 3.b odd 2 1
240.2.s.b ✓ 8 48.i odd 4 1
720.2.t.b 8 1.a even 1 1 trivial
720.2.t.b 8 16.e even 4 1 inner
960.2.s.b 8 12.b even 2 1
960.2.s.b 8 48.k even 4 1
1920.2.s.c 8 24.h odd 2 1
1920.2.s.c 8 48.i odd 4 1
1920.2.s.d 8 24.f even 2 1
1920.2.s.d 8 48.k even 4 1
2880.2.t.b 8 4.b odd 2 1
2880.2.t.b 8 16.f odd 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 2 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} + 8 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( T^{8} - 8 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( (T^{4} + 4 T^{3} - 12 T^{2} + 8)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} - 8 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( T^{8} + 128 T^{6} + \cdots + 200704 \) Copy content Toggle raw display
$29$ \( T^{8} + 24 T^{7} + \cdots + 135424 \) Copy content Toggle raw display
$31$ \( (T^{4} - 4 T^{3} - 40 T^{2} + \cdots - 28)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 8 T^{7} + \cdots + 414736 \) Copy content Toggle raw display
$41$ \( T^{8} + 304 T^{6} + \cdots + 17707264 \) Copy content Toggle raw display
$43$ \( T^{8} + 256 T^{5} + \cdots + 12544 \) Copy content Toggle raw display
$47$ \( (T^{4} - 152 T^{2} + \cdots - 224)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 33312 T^{4} + 73984 \) Copy content Toggle raw display
$59$ \( T^{8} + 8 T^{7} + \cdots + 7529536 \) Copy content Toggle raw display
$61$ \( T^{8} + 16 T^{7} + \cdots + 4343056 \) Copy content Toggle raw display
$67$ \( T^{8} + 4224 T^{4} + 1183744 \) Copy content Toggle raw display
$71$ \( T^{8} + 272 T^{6} + \cdots + 5161984 \) Copy content Toggle raw display
$73$ \( T^{8} + 176 T^{6} + \cdots + 50176 \) Copy content Toggle raw display
$79$ \( (T^{4} + 20 T^{3} + \cdots + 164)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 32 T^{7} + \cdots + 2166784 \) Copy content Toggle raw display
$89$ \( T^{8} + 592 T^{6} + \cdots + 118026496 \) Copy content Toggle raw display
$97$ \( (T^{4} + 24 T^{3} + \cdots - 736)^{2} \) Copy content Toggle raw display
show more
show less