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Properties

Label 2240.2.e.d
Level $2240$
Weight $2$
Character orbit 2240.e
Analytic conductor $17.886$
Analytic rank $0$
Dimension $8$
CM discriminant -35
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2240,2,Mod(2239,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2240.2239");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.e (of order \(2\), degree \(1\), 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: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.121550625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 4x^{6} - 9x^{5} + 23x^{4} + 18x^{3} - 16x^{2} + 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 560)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} + \beta_{2} q^{5} + ( - \beta_{6} + \beta_1) q^{7} + (\beta_{5} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{3} + \beta_{2} q^{5} + ( - \beta_{6} + \beta_1) q^{7} + (\beta_{5} - 3) q^{9} + (\beta_{7} + \beta_{4}) q^{11} + (\beta_{3} - \beta_{2}) q^{13} + ( - \beta_{7} + 3 \beta_{4}) q^{15} + (3 \beta_{3} + \beta_{2}) q^{17} + ( - \beta_{5} + 4) q^{21} + 5 q^{25} + ( - 5 \beta_{6} + 2 \beta_1) q^{27} + ( - \beta_{5} - 4) q^{29} + ( - \beta_{3} - \beta_{2}) q^{33} + (2 \beta_{7} - \beta_{4}) q^{35} + (3 \beta_{7} - 7 \beta_{4}) q^{39} + 5 \beta_{3} q^{45} + ( - \beta_{6} - 4 \beta_1) q^{47} - 7 q^{49} + (5 \beta_{7} - 9 \beta_{4}) q^{51} + ( - \beta_{6} + 4 \beta_1) q^{55} + (6 \beta_{6} + \beta_1) q^{63} + (\beta_{5} - 8) q^{65} + (4 \beta_{7} - 2 \beta_{4}) q^{71} - 6 \beta_{2} q^{73} + 5 \beta_{6} q^{75} + ( - 3 \beta_{3} - 5 \beta_{2}) q^{77} + (3 \beta_{7} - \beta_{4}) q^{79} + ( - 2 \beta_{5} + 17) q^{81} + (6 \beta_{6} - 6 \beta_1) q^{83} + (3 \beta_{5} - 4) q^{85} + (\beta_{6} - 2 \beta_1) q^{87} + ( - 3 \beta_{7} + 5 \beta_{4}) q^{91} + (7 \beta_{3} + 5 \beta_{2}) q^{97} + (2 \beta_{7} + 4 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 20 q^{9} + 28 q^{21} + 40 q^{25} - 36 q^{29} - 56 q^{49} - 60 q^{65} + 128 q^{81} - 20 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 4x^{6} - 9x^{5} + 23x^{4} + 18x^{3} - 16x^{2} + 8x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -7\nu^{7} - 93\nu^{6} + 244\nu^{5} + 547\nu^{4} + 659\nu^{3} - 3622\nu^{2} - 1884\nu + 1240 ) / 1224 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 25\nu^{7} - 3\nu^{6} - 70\nu^{5} - 205\nu^{4} - 95\nu^{3} + 40\nu^{2} - 120\nu + 2624 ) / 1224 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -89\nu^{7} + 129\nu^{6} + 392\nu^{5} + 485\nu^{4} - 2375\nu^{3} - 1550\nu^{2} + 3324\nu - 1720 ) / 1224 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 41\nu^{7} - 111\nu^{6} - 74\nu^{5} - 173\nu^{4} + 1517\nu^{3} - 1036\nu^{2} - 768\nu + 1072 ) / 408 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -65\nu^{7} + 69\nu^{6} + 284\nu^{5} + 533\nu^{4} - 1691\nu^{3} - 1226\nu^{2} + 2556\nu + 32 ) / 408 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 313\nu^{7} - 621\nu^{6} - 652\nu^{5} - 2077\nu^{4} + 9235\nu^{3} - 3110\nu^{2} - 3420\nu + 5696 ) / 1224 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -107\nu^{7} + 123\nu^{6} + 320\nu^{5} + 959\nu^{4} - 2429\nu^{3} - 722\nu^{2} + 228\nu - 1096 ) / 408 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 3\beta_{6} + \beta_{5} - 5\beta_{4} - 3\beta_{3} - 3\beta_{2} + 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} - 2\beta_{6} + \beta_{4} - 5\beta_{2} + 2\beta _1 + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -9\beta_{7} - 9\beta_{6} + 9\beta_{5} + \beta_{4} - 21\beta_{3} - 9\beta_{2} + 12\beta _1 - 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{7} + 33\beta_{6} + 5\beta_{5} - 63\beta_{4} - 11\beta_{3} - 33\beta_{2} + 22\beta _1 + 58 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -40\beta_{7} - 45\beta_{6} + 20\beta_{4} - 36\beta_{2} + 45\beta _1 + 81 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -91\beta_{7} - 17\beta_{6} + 91\beta_{5} - 143\beta_{4} - 203\beta_{3} - 17\beta_{2} + 186\beta _1 - 234 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2239.1
−1.44918 + 1.77086i
−0.862555 − 0.141174i
2.25820 − 0.369600i
0.553538 + 0.676408i
2.25820 + 0.369600i
0.553538 − 0.676408i
−1.44918 − 1.77086i
−0.862555 + 0.141174i
0 − 3.25937i 0 −2.23607 0 2.64575i 0 −7.62348 0
2239.2 0 − 3.25937i 0 2.23607 0 2.64575i 0 −7.62348 0
2239.3 0 − 0.613616i 0 −2.23607 0 − 2.64575i 0 2.62348 0
2239.4 0 − 0.613616i 0 2.23607 0 − 2.64575i 0 2.62348 0
2239.5 0 0.613616i 0 −2.23607 0 2.64575i 0 2.62348 0
2239.6 0 0.613616i 0 2.23607 0 2.64575i 0 2.62348 0
2239.7 0 3.25937i 0 −2.23607 0 − 2.64575i 0 −7.62348 0
2239.8 0 3.25937i 0 2.23607 0 − 2.64575i 0 −7.62348 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2239.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
140.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.2.e.d 8
4.b odd 2 1 inner 2240.2.e.d 8
5.b even 2 1 inner 2240.2.e.d 8
7.b odd 2 1 inner 2240.2.e.d 8
8.b even 2 1 560.2.e.c ✓ 8
8.d odd 2 1 560.2.e.c ✓ 8
20.d odd 2 1 inner 2240.2.e.d 8
28.d even 2 1 inner 2240.2.e.d 8
35.c odd 2 1 CM 2240.2.e.d 8
40.e odd 2 1 560.2.e.c ✓ 8
40.f even 2 1 560.2.e.c ✓ 8
40.i odd 4 2 2800.2.k.p 8
40.k even 4 2 2800.2.k.p 8
56.e even 2 1 560.2.e.c ✓ 8
56.h odd 2 1 560.2.e.c ✓ 8
140.c even 2 1 inner 2240.2.e.d 8
280.c odd 2 1 560.2.e.c ✓ 8
280.n even 2 1 560.2.e.c ✓ 8
280.s even 4 2 2800.2.k.p 8
280.y odd 4 2 2800.2.k.p 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.e.c ✓ 8 8.b even 2 1
560.2.e.c ✓ 8 8.d odd 2 1
560.2.e.c ✓ 8 40.e odd 2 1
560.2.e.c ✓ 8 40.f even 2 1
560.2.e.c ✓ 8 56.e even 2 1
560.2.e.c ✓ 8 56.h odd 2 1
560.2.e.c ✓ 8 280.c odd 2 1
560.2.e.c ✓ 8 280.n even 2 1
2240.2.e.d 8 1.a even 1 1 trivial
2240.2.e.d 8 4.b odd 2 1 inner
2240.2.e.d 8 5.b even 2 1 inner
2240.2.e.d 8 7.b odd 2 1 inner
2240.2.e.d 8 20.d odd 2 1 inner
2240.2.e.d 8 28.d even 2 1 inner
2240.2.e.d 8 35.c odd 2 1 CM
2240.2.e.d 8 140.c even 2 1 inner
2800.2.k.p 8 40.i odd 4 2
2800.2.k.p 8 40.k even 4 2
2800.2.k.p 8 280.s even 4 2
2800.2.k.p 8 280.y odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2240, [\chi])\):

\( T_{3}^{4} + 11T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} + 31T_{11}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 11 T^{2} + 4)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 7)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 31 T^{2} + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 33 T^{2} + 36)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 97 T^{2} + 2116)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( (T^{2} + 9 T - 6)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} + 219 T^{2} + 6084)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( (T^{2} + 140)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 180)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 159 T^{2} + 6084)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 252)^{4} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} - 537 T^{2} + 60516)^{2} \) Copy content Toggle raw display
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