Properties

Label 1440.2.b.d
Level $1440$
Weight $2$
Character orbit 1440.b
Analytic conductor $11.498$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1440,2,Mod(431,1440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1440, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 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("1440.431");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1440.b (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: \(11.4984578911\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.2580992.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + x^{4} + 2x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 360)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} - \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} - \beta_1 q^{7} + ( - \beta_{3} - \beta_1) q^{11} + ( - \beta_{3} + \beta_1) q^{13} + (\beta_{5} + \beta_1) q^{17} + (\beta_{2} - 3) q^{19} + (\beta_{4} - 1) q^{23} + q^{25} + (\beta_{4} - \beta_{2} - 2) q^{29} + (\beta_{5} + \beta_1) q^{31} - \beta_1 q^{35} - \beta_{5} q^{37} + (\beta_{5} - \beta_{3} - 2 \beta_1) q^{41} + (\beta_{4} + \beta_{2} + 2) q^{43} + (\beta_{4} - 2 \beta_{2} - 1) q^{47} + 5 q^{49} + ( - \beta_{4} - \beta_{2} + 2) q^{53} + ( - \beta_{3} - \beta_1) q^{55} + (\beta_{5} - 2 \beta_1) q^{59} + ( - 2 \beta_{3} - 2 \beta_1) q^{61} + ( - \beta_{3} + \beta_1) q^{65} + ( - \beta_{4} + \beta_{2}) q^{67} + (\beta_{4} - \beta_{2} + 8) q^{71} + ( - 2 \beta_{4} + 2 \beta_{2} - 2) q^{73} + (\beta_{4} - \beta_{2} - 2) q^{77} + ( - \beta_{5} - 2 \beta_{3} + 3 \beta_1) q^{79} + 2 \beta_{5} q^{83} + (\beta_{5} + \beta_1) q^{85} + (\beta_{5} - \beta_{3} + 2 \beta_1) q^{89} + (\beta_{4} - \beta_{2} + 2) q^{91} + (\beta_{2} - 3) q^{95} + ( - \beta_{4} + 3 \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{5} - 16 q^{19} - 4 q^{23} + 6 q^{25} - 12 q^{29} + 16 q^{43} - 8 q^{47} + 30 q^{49} + 8 q^{53} + 48 q^{71} - 12 q^{73} - 12 q^{77} + 12 q^{91} - 16 q^{95} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( -\nu^{5} - \nu^{3} + 2\nu^{2} - 2\nu + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 2\nu^{4} - \nu^{3} + 2\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 2\nu^{4} + \nu^{3} + 6\nu - 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{5} + \nu^{4} + \nu^{3} + \nu^{2} - 2\nu + 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7\nu^{5} - 4\nu^{4} - \nu^{3} + 6\nu^{2} + 14\nu - 36 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{4} + 3\beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{4} + \beta_{3} - \beta_{2} - 4\beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{5} + \beta_{4} - 2\beta_{3} + 2\beta_{2} - 5\beta _1 - 5 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{5} - 3\beta_{3} - \beta_{2} - 6\beta _1 + 15 ) / 4 \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
431.1
0.681664 + 1.23909i
1.38078 0.305697i
−1.06244 0.933389i
−1.06244 + 0.933389i
1.38078 + 0.305697i
0.681664 1.23909i
0 0 0 1.00000 0 1.41421i 0 0 0
431.2 0 0 0 1.00000 0 1.41421i 0 0 0
431.3 0 0 0 1.00000 0 1.41421i 0 0 0
431.4 0 0 0 1.00000 0 1.41421i 0 0 0
431.5 0 0 0 1.00000 0 1.41421i 0 0 0
431.6 0 0 0 1.00000 0 1.41421i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 431.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.2.b.d 6
3.b odd 2 1 1440.2.b.c 6
4.b odd 2 1 360.2.b.d yes 6
5.b even 2 1 7200.2.b.e 6
5.c odd 4 2 7200.2.m.e 12
8.b even 2 1 360.2.b.c 6
8.d odd 2 1 1440.2.b.c 6
12.b even 2 1 360.2.b.c 6
15.d odd 2 1 7200.2.b.d 6
15.e even 4 2 7200.2.m.d 12
20.d odd 2 1 1800.2.b.d 6
20.e even 4 2 1800.2.m.e 12
24.f even 2 1 inner 1440.2.b.d 6
24.h odd 2 1 360.2.b.d yes 6
40.e odd 2 1 7200.2.b.d 6
40.f even 2 1 1800.2.b.e 6
40.i odd 4 2 1800.2.m.d 12
40.k even 4 2 7200.2.m.d 12
60.h even 2 1 1800.2.b.e 6
60.l odd 4 2 1800.2.m.d 12
120.i odd 2 1 1800.2.b.d 6
120.m even 2 1 7200.2.b.e 6
120.q odd 4 2 7200.2.m.e 12
120.w even 4 2 1800.2.m.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.b.c 6 8.b even 2 1
360.2.b.c 6 12.b even 2 1
360.2.b.d yes 6 4.b odd 2 1
360.2.b.d yes 6 24.h odd 2 1
1440.2.b.c 6 3.b odd 2 1
1440.2.b.c 6 8.d odd 2 1
1440.2.b.d 6 1.a even 1 1 trivial
1440.2.b.d 6 24.f even 2 1 inner
1800.2.b.d 6 20.d odd 2 1
1800.2.b.d 6 120.i odd 2 1
1800.2.b.e 6 40.f even 2 1
1800.2.b.e 6 60.h even 2 1
1800.2.m.d 12 40.i odd 4 2
1800.2.m.d 12 60.l odd 4 2
1800.2.m.e 12 20.e even 4 2
1800.2.m.e 12 120.w even 4 2
7200.2.b.d 6 15.d odd 2 1
7200.2.b.d 6 40.e odd 2 1
7200.2.b.e 6 5.b even 2 1
7200.2.b.e 6 120.m even 2 1
7200.2.m.d 12 15.e even 4 2
7200.2.m.d 12 40.k even 4 2
7200.2.m.e 12 5.c odd 4 2
7200.2.m.e 12 120.q odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1440, [\chi])\):

\( T_{7}^{2} + 2 \) Copy content Toggle raw display
\( T_{23}^{3} + 2T_{23}^{2} - 32T_{23} - 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T - 1)^{6} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2)^{3} \) Copy content Toggle raw display
$11$ \( T^{6} + 46 T^{4} + 220 T^{2} + 8 \) Copy content Toggle raw display
$13$ \( T^{6} + 46 T^{4} + 604 T^{2} + \cdots + 2312 \) Copy content Toggle raw display
$17$ \( T^{6} + 80 T^{4} + 2000 T^{2} + \cdots + 15488 \) Copy content Toggle raw display
$19$ \( (T^{3} + 8 T^{2} - 4 T - 16)^{2} \) Copy content Toggle raw display
$23$ \( (T^{3} + 2 T^{2} - 32 T - 32)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 6 T^{2} - 28 T - 8)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 80 T^{4} + 2000 T^{2} + \cdots + 15488 \) Copy content Toggle raw display
$37$ \( T^{6} + 78 T^{4} + 1244 T^{2} + \cdots + 5000 \) Copy content Toggle raw display
$41$ \( T^{6} + 134 T^{4} + 4748 T^{2} + \cdots + 49928 \) Copy content Toggle raw display
$43$ \( (T^{3} - 8 T^{2} - 56 T - 64)^{2} \) Copy content Toggle raw display
$47$ \( (T^{3} + 4 T^{2} - 92 T - 352)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 4 T^{2} - 72 T + 352)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 110 T^{4} + 92 T^{2} + 8 \) Copy content Toggle raw display
$61$ \( T^{6} + 184 T^{4} + 3520 T^{2} + \cdots + 512 \) Copy content Toggle raw display
$67$ \( (T^{3} - 40 T - 64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} - 24 T^{2} + 152 T - 128)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 6 T^{2} - 148 T - 824)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 336 T^{4} + 31184 T^{2} + \cdots + 881792 \) Copy content Toggle raw display
$83$ \( T^{6} + 312 T^{4} + 19904 T^{2} + \cdots + 320000 \) Copy content Toggle raw display
$89$ \( T^{6} + 118 T^{4} + 1356 T^{2} + \cdots + 200 \) Copy content Toggle raw display
$97$ \( (T^{3} - 2 T^{2} - 204 T + 1208)^{2} \) Copy content Toggle raw display
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