Properties

Label 440.2.b.d
Level $440$
Weight $2$
Character orbit 440.b
Analytic conductor $3.513$
Analytic rank $0$
Dimension $8$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [440,2,Mod(89,440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(440, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("440.89");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 440.b (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: \(3.51341768894\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.47985531136.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 19x^{6} + 91x^{4} + 45x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{3} q^{5} + ( - \beta_{4} - \beta_{3} + \beta_1) q^{7} + (\beta_{6} + \beta_{5} - \beta_{3} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{3} q^{5} + ( - \beta_{4} - \beta_{3} + \beta_1) q^{7} + (\beta_{6} + \beta_{5} - \beta_{3} - 2) q^{9} + q^{11} + (\beta_{7} - \beta_{6} + \cdots + \beta_{3}) q^{13}+ \cdots + (\beta_{6} + \beta_{5} - \beta_{3} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 14 q^{9} + 8 q^{11} - 12 q^{15} + 18 q^{19} - 14 q^{21} - 2 q^{25} - 6 q^{29} - 30 q^{31} + 30 q^{35} - 8 q^{39} + 20 q^{41} - 22 q^{45} - 18 q^{49} - 46 q^{51} - 12 q^{59} + 58 q^{61} - 8 q^{65} + 60 q^{69} - 2 q^{71} + 26 q^{75} + 40 q^{79} + 88 q^{81} - 26 q^{85} - 42 q^{89} + 8 q^{91} + 28 q^{95} - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 19x^{6} + 91x^{4} + 45x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 18\nu^{5} - 73\nu^{3} + 32\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{7} + \nu^{6} + 38\nu^{5} + 18\nu^{4} + 180\nu^{3} + 77\nu^{2} + 72\nu + 12 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{7} - \nu^{6} + 38\nu^{5} - 18\nu^{4} + 180\nu^{3} - 77\nu^{2} + 72\nu - 12 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} - \nu^{6} - 56\nu^{5} - 20\nu^{4} - 257\nu^{3} - 95\nu^{2} - 76\nu - 4 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{7} + 2\nu^{6} + 94\nu^{5} + 38\nu^{4} + 437\nu^{3} + 180\nu^{2} + 148\nu + 56 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{7} + 2\nu^{6} - 94\nu^{5} + 38\nu^{4} - 437\nu^{3} + 180\nu^{2} - 148\nu + 56 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} - \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} - 2\beta_{4} - 2\beta_{3} + \beta_{2} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{7} - 11\beta_{6} - 13\beta_{5} + 4\beta_{4} + 9\beta_{3} + 49 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 17\beta_{7} - 17\beta_{6} + 36\beta_{4} + 36\beta_{3} - 13\beta_{2} + 85\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -36\beta_{7} + 121\beta_{6} + 157\beta_{5} - 76\beta_{4} - 81\beta_{3} - 509 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -233\beta_{7} + 233\beta_{6} - 502\beta_{4} - 502\beta_{3} + 157\beta_{2} - 841\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/440\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(177\) \(221\) \(321\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1
3.36007i
2.67673i
0.655762i
0.339102i
0.339102i
0.655762i
2.67673i
3.36007i
0 3.36007i 0 −0.256321 2.22133i 0 1.08258i 0 −8.29009 0
89.2 0 2.67673i 0 2.06639 + 0.854430i 0 4.38559i 0 −4.16490 0
89.3 0 0.655762i 0 −2.23285 0.119978i 0 0.415806i 0 2.56998 0
89.4 0 0.339102i 0 0.422782 2.19574i 0 4.05237i 0 2.88501 0
89.5 0 0.339102i 0 0.422782 + 2.19574i 0 4.05237i 0 2.88501 0
89.6 0 0.655762i 0 −2.23285 + 0.119978i 0 0.415806i 0 2.56998 0
89.7 0 2.67673i 0 2.06639 0.854430i 0 4.38559i 0 −4.16490 0
89.8 0 3.36007i 0 −0.256321 + 2.22133i 0 1.08258i 0 −8.29009 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 89.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 440.2.b.d 8
3.b odd 2 1 3960.2.d.f 8
4.b odd 2 1 880.2.b.j 8
5.b even 2 1 inner 440.2.b.d 8
5.c odd 4 1 2200.2.a.x 4
5.c odd 4 1 2200.2.a.y 4
15.d odd 2 1 3960.2.d.f 8
20.d odd 2 1 880.2.b.j 8
20.e even 4 1 4400.2.a.cb 4
20.e even 4 1 4400.2.a.ce 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.b.d 8 1.a even 1 1 trivial
440.2.b.d 8 5.b even 2 1 inner
880.2.b.j 8 4.b odd 2 1
880.2.b.j 8 20.d odd 2 1
2200.2.a.x 4 5.c odd 4 1
2200.2.a.y 4 5.c odd 4 1
3960.2.d.f 8 3.b odd 2 1
3960.2.d.f 8 15.d odd 2 1
4400.2.a.cb 4 20.e even 4 1
4400.2.a.ce 4 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(440, [\chi])\):

\( T_{3}^{8} + 19T_{3}^{6} + 91T_{3}^{4} + 45T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{8} + 37T_{7}^{6} + 364T_{7}^{4} + 432T_{7}^{2} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 19 T^{6} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{8} + T^{6} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} + 37 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( (T - 1)^{8} \) Copy content Toggle raw display
$13$ \( (T^{2} + 16)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} + 137 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$19$ \( (T^{4} - 9 T^{3} + \cdots - 128)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 154 T^{6} + \cdots + 732736 \) Copy content Toggle raw display
$29$ \( (T^{4} + 3 T^{3} - 38 T^{2} + 32)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 15 T^{3} + \cdots + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 135 T^{6} + \cdots + 21904 \) Copy content Toggle raw display
$41$ \( (T^{4} - 10 T^{3} + \cdots - 1024)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 192 T^{6} + \cdots + 1290496 \) Copy content Toggle raw display
$47$ \( T^{8} + 124 T^{6} + \cdots + 409600 \) Copy content Toggle raw display
$53$ \( T^{8} + 177 T^{6} + \cdots + 1638400 \) Copy content Toggle raw display
$59$ \( (T^{4} + 6 T^{3} - 47 T^{2} + \cdots - 32)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 29 T^{3} + \cdots + 160)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 202 T^{6} + \cdots + 322624 \) Copy content Toggle raw display
$71$ \( (T^{4} + T^{3} - 137 T^{2} + \cdots - 1336)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 228 T^{6} + \cdots + 1048576 \) Copy content Toggle raw display
$79$ \( (T^{4} - 20 T^{3} + \cdots - 19456)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 524 T^{6} + \cdots + 38142976 \) Copy content Toggle raw display
$89$ \( (T^{4} + 21 T^{3} + \cdots - 346)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 310 T^{6} + \cdots + 262144 \) Copy content Toggle raw display
show more
show less