↑

Properties

Label 2280.2.bg.v
Level $2280$
Weight $2$
Character orbit 2280.bg
Analytic conductor $18.206$
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] = [2280,2,Mod(121,2280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2280, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2280.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2280.bg (of order \(3\), 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: \(18.2058916609\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 11x^{6} - 2x^{5} + 90x^{4} - 28x^{3} + 196x^{2} + 96x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{3} + 1) q^{3} + ( - \beta_{3} - 1) q^{5} + ( - \beta_{7} + \beta_{5} + \beta_{2}) q^{7} + \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 1) q^{3} + ( - \beta_{3} - 1) q^{5} + ( - \beta_{7} + \beta_{5} + \beta_{2}) q^{7} + \beta_{3} q^{9} + ( - \beta_{4} + 2) q^{11} + 2 \beta_1 q^{13} - \beta_{3} q^{15} + ( - \beta_{5} + \beta_1) q^{17} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \cdots - 1) q^{19}+ \cdots + (\beta_{6} + 2 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} - 4 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} - 4 q^{5} - 4 q^{9} + 14 q^{11} + 2 q^{13} + 4 q^{15} - q^{17} - 10 q^{19} - 4 q^{23} - 4 q^{25} - 8 q^{27} + 5 q^{29} - 36 q^{31} + 7 q^{33} - 36 q^{37} + 4 q^{39} - 6 q^{41} - 2 q^{43} + 8 q^{45} - 13 q^{47} + 28 q^{49} + q^{51} + 11 q^{53} - 7 q^{55} - 5 q^{57} - 11 q^{59} + 17 q^{61} - 4 q^{65} - 10 q^{67} - 8 q^{69} - 11 q^{71} - 14 q^{73} - 8 q^{75} - 32 q^{77} + 9 q^{79} - 4 q^{81} + 10 q^{83} - q^{85} + 10 q^{87} + q^{89} + 24 q^{91} - 18 q^{93} + 5 q^{95} + 20 q^{97} - 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 11x^{6} - 2x^{5} + 90x^{4} - 28x^{3} + 196x^{2} + 96x + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 165 \nu^{7} - 5159 \nu^{6} + 19229 \nu^{5} - 63066 \nu^{4} + 134134 \nu^{3} - 377076 \nu^{2} + \cdots - 600320 ) / 330896 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1771 \nu^{7} - 5291 \nu^{6} + 19721 \nu^{5} - 34518 \nu^{4} + 137566 \nu^{3} - 386724 \nu^{2} + \cdots - 615680 ) / 661792 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -205\nu^{7} + 484\nu^{6} - 1804\nu^{5} - 1271\nu^{4} - 12584\nu^{3} - 5986\nu^{2} - 7216\nu - 150490 ) / 41362 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -220\nu^{7} + 15\nu^{6} - 1936\nu^{5} - 1364\nu^{4} - 21071\nu^{3} - 6424\nu^{2} - 7744\nu - 28336 ) / 41362 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5575 \nu^{7} - 18711 \nu^{6} + 69741 \nu^{5} - 192926 \nu^{4} + 486486 \nu^{3} - 1367604 \nu^{2} + \cdots - 2177280 ) / 661792 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5935 \nu^{7} - 7455 \nu^{6} + 72909 \nu^{5} - 25246 \nu^{4} + 524726 \nu^{3} - 198956 \nu^{2} + \cdots + 516256 ) / 330896 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{4} - 5\beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{7} - 6\beta_{5} - \beta_{4} + 2\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -11\beta_{6} + 35\beta_{3} + 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 22\beta_{7} - 15\beta_{6} + 44\beta_{5} + 15\beta_{4} + 15\beta_{3} - 44\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 30\beta_{7} + 8\beta_{5} + 103\beta_{4} - 30\beta_{2} + 279 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 171\beta_{6} - 203\beta_{3} - 206\beta_{2} + 352\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2280\mathbb{Z}\right)^\times\).

\(n\) \(457\) \(761\) \(1141\) \(1711\) \(1921\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
−0.552497 + 0.956953i
1.53398 − 2.65693i
−1.35334 + 2.34405i
0.871855 − 1.51010i
−0.552497 − 0.956953i
1.53398 + 2.65693i
−1.35334 − 2.34405i
0.871855 + 1.51010i
0 0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0 −5.13486 0 −0.500000 + 0.866025i 0
121.2 0 0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0 0.539638 0 −0.500000 + 0.866025i 0
121.3 0 0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0 0.751020 0 −0.500000 + 0.866025i 0
121.4 0 0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0 3.84421 0 −0.500000 + 0.866025i 0
961.1 0 0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0 −5.13486 0 −0.500000 − 0.866025i 0
961.2 0 0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0 0.539638 0 −0.500000 − 0.866025i 0
961.3 0 0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0 0.751020 0 −0.500000 − 0.866025i 0
961.4 0 0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0 3.84421 0 −0.500000 − 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2280.2.bg.v ✓ 8
19.c even 3 1 inner 2280.2.bg.v ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2280.2.bg.v ✓ 8 1.a even 1 1 trivial
2280.2.bg.v ✓ 8 19.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2280, [\chi])\):

\( T_{7}^{4} - 21T_{7}^{2} + 26T_{7} - 8 \) Copy content Toggle raw display
\( T_{11}^{4} - 7T_{11}^{3} - 3T_{11}^{2} + 55T_{11} + 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} - 21 T^{2} + 26 T - 8)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 7 T^{3} - 3 T^{2} + \cdots + 18)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 2 T^{7} + \cdots + 65536 \) Copy content Toggle raw display
$17$ \( T^{8} + T^{7} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( T^{8} + 10 T^{7} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( T^{8} + 4 T^{7} + \cdots + 23104 \) Copy content Toggle raw display
$29$ \( T^{8} - 5 T^{7} + \cdots + 331776 \) Copy content Toggle raw display
$31$ \( (T^{4} + 18 T^{3} + 89 T^{2} + \cdots - 8)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 18 T^{3} + \cdots - 512)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 6 T^{7} + \cdots + 266256 \) Copy content Toggle raw display
$43$ \( T^{8} + 2 T^{7} + \cdots + 1401856 \) Copy content Toggle raw display
$47$ \( T^{8} + 13 T^{7} + \cdots + 2334784 \) Copy content Toggle raw display
$53$ \( T^{8} - 11 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$59$ \( T^{8} + 11 T^{7} + \cdots + 5184 \) Copy content Toggle raw display
$61$ \( T^{8} - 17 T^{7} + \cdots + 1948816 \) Copy content Toggle raw display
$67$ \( T^{8} + 10 T^{7} + \cdots + 65536 \) Copy content Toggle raw display
$71$ \( T^{8} + 11 T^{7} + \cdots + 26091664 \) Copy content Toggle raw display
$73$ \( T^{8} + 14 T^{7} + \cdots + 25563136 \) Copy content Toggle raw display
$79$ \( T^{8} - 9 T^{7} + \cdots + 26873856 \) Copy content Toggle raw display
$83$ \( (T^{4} - 5 T^{3} - 12 T^{2} + \cdots + 48)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - T^{7} + \cdots + 1267876 \) Copy content Toggle raw display
$97$ \( T^{8} - 20 T^{7} + \cdots + 64 \) Copy content Toggle raw display
show more
show less