↑

Properties

Label 1440.4.k.b
Level $1440$
Weight $4$
Character orbit 1440.k
Analytic conductor $84.963$
Analytic rank $0$
Dimension $8$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1440,4,Mod(721,1440)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1440.721"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1440.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(84.9627504083\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.55839580416.4
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 7x^{6} - 6x^{5} + 18x^{4} - 24x^{3} + 112x^{2} - 128x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q - 5 \beta_1 q^{5} + ( - \beta_{6} + 10) q^{7} + (3 \beta_{5} - \beta_{4} + \cdots + \beta_1) q^{11} + (2 \beta_{5} - 5 \beta_{4} + \cdots - 3 \beta_1) q^{13} + ( - 2 \beta_{7} - 4 \beta_{6} + \cdots + 27) q^{17}+ \cdots + (32 \beta_{7} + 48 \beta_{6} + \cdots - 532) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 80 q^{7} + 216 q^{17} - 32 q^{23} - 200 q^{25} + 136 q^{31} - 176 q^{41} - 848 q^{47} - 1320 q^{49} + 40 q^{55} - 120 q^{65} + 3088 q^{71} - 496 q^{73} + 2056 q^{79} - 3472 q^{89} + 320 q^{95} - 4256 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 7x^{6} - 6x^{5} + 18x^{4} - 24x^{3} + 112x^{2} - 128x + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 29\nu^{7} + 10\nu^{6} + 3\nu^{5} - 18\nu^{4} + 306\nu^{3} - 624\nu^{2} + 1040\nu + 1408 ) / 2880 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 3\nu^{5} + 8\nu^{4} + 6\nu^{3} - 20\nu^{2} + 64\nu + 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + 7\nu^{5} - 6\nu^{4} + 18\nu^{3} - 24\nu^{2} + 176\nu - 128 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 67\nu^{7} + 110\nu^{6} - 51\nu^{5} + 666\nu^{4} + 798\nu^{3} + 2208\nu^{2} + 880\nu + 13184 ) / 960 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} - 5\nu^{6} + 11\nu^{5} - 11\nu^{4} - 8\nu^{3} - 38\nu^{2} + 200\nu - 224 ) / 40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} - 5\nu^{5} + 5\nu^{4} - 12\nu^{3} + 34\nu^{2} - 80\nu + 128 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - 4\nu^{6} + 3\nu^{5} - 4\nu^{4} + 6\nu^{3} - 44\nu^{2} + 112\nu - 160 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + 2\beta_{6} + 2\beta_{3} + 4 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} + 4\beta_{4} + \beta_{3} - 4\beta_{2} - 24\beta _1 - 20 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} - 6\beta_{6} - 12\beta_{5} + 4\beta_{4} + 2\beta_{3} - 4\beta_{2} + 12\beta _1 - 32 ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{7} - 10\beta_{5} + 4\beta_{4} - 5\beta_{3} + 20\beta_{2} - 96\beta _1 - 44 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -27\beta_{7} - 26\beta_{6} + 12\beta_{5} - 12\beta_{4} + 2\beta_{3} + 28\beta_{2} - 180\beta _1 + 184 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -32\beta_{7} + 24\beta_{6} + 18\beta_{5} - 36\beta_{4} + 33\beta_{3} + 28\beta_{2} + 432\beta _1 - 404 ) / 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -9\beta_{7} - 14\beta_{6} + 156\beta_{5} + 60\beta_{4} - 86\beta_{3} - 44\beta_{2} + 756\beta _1 - 920 ) / 16 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(991\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
721.1
0.694547 + 1.87553i
1.61974 − 1.17321i
0.217599 − 1.98813i
−1.53189 + 1.28581i
0.694547 − 1.87553i
1.61974 + 1.17321i
0.217599 + 1.98813i
−1.53189 − 1.28581i
0 0 0 − 5.00000i 0 −1.05849 0 0 0
721.2 0 0 0 − 5.00000i 0 3.73490 0 0 0
721.3 0 0 0 − 5.00000i 0 18.2117 0 0 0
721.4 0 0 0 − 5.00000i 0 19.1119 0 0 0
721.5 0 0 0 5.00000i 0 −1.05849 0 0 0
721.6 0 0 0 5.00000i 0 3.73490 0 0 0
721.7 0 0 0 5.00000i 0 18.2117 0 0 0
721.8 0 0 0 5.00000i 0 19.1119 0 0 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 721.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.4.k.b 8
3.b odd 2 1 480.4.k.b 8
4.b odd 2 1 360.4.k.b 8
8.b even 2 1 inner 1440.4.k.b 8
8.d odd 2 1 360.4.k.b 8
12.b even 2 1 120.4.k.b ✓ 8
24.f even 2 1 120.4.k.b ✓ 8
24.h odd 2 1 480.4.k.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.k.b ✓ 8 12.b even 2 1
120.4.k.b ✓ 8 24.f even 2 1
360.4.k.b 8 4.b odd 2 1
360.4.k.b 8 8.d odd 2 1
480.4.k.b 8 3.b odd 2 1
480.4.k.b 8 24.h odd 2 1
1440.4.k.b 8 1.a even 1 1 trivial
1440.4.k.b 8 8.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 40T_{7}^{3} + 444T_{7}^{2} - 784T_{7} - 1376 \) acting on \(S_{4}^{\mathrm{new}}(1440, [\chi])\). 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
$5$ \( (T^{2} + 25)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} - 40 T^{3} + \cdots - 1376)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 62236278784 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 38353305600 \) Copy content Toggle raw display
$17$ \( (T^{4} - 108 T^{3} + \cdots - 61632)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 31548801716224 \) Copy content Toggle raw display
$23$ \( (T^{4} + 16 T^{3} + \cdots + 7277888)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 29\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( (T^{4} - 68 T^{3} + \cdots - 140627840)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 82\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( (T^{4} + 88 T^{3} + \cdots - 3143107120)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 70\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( (T^{4} + 424 T^{3} + \cdots - 10676481472)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 33\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 90\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( (T^{4} - 1544 T^{3} + \cdots - 11209493248)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 248 T^{3} + \cdots + 112584267280)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 1028 T^{3} + \cdots - 443106379136)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{4} + 1736 T^{3} + \cdots - 68323863856)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 1040188142960)^{2} \) Copy content Toggle raw display
show more
show less