↑

Properties

Label 1440.2.q.q
Level $1440$
Weight $2$
Character orbit 1440.q
Analytic conductor $11.498$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1440,2,Mod(481,1440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1440, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1440.481");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1440.q (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: \(11.4984578911\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + x^{10} + 4x^{8} - 21x^{6} + 36x^{4} + 81x^{2} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{11} q^{3} + \beta_{3} q^{5} - \beta_{9} q^{7} - \beta_{10} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{11} q^{3} + \beta_{3} q^{5} - \beta_{9} q^{7} - \beta_{10} q^{9} + (\beta_{5} - \beta_1) q^{11} + ( - \beta_{6} + \beta_{4} - \beta_{3}) q^{13} - \beta_{5} q^{15} + (\beta_{10} - \beta_{6} - \beta_{4} - 3) q^{17} + (\beta_{11} - \beta_{7} + \beta_{5} - \beta_1) q^{19} + ( - 2 \beta_{6} - \beta_{3} + \beta_{2} - 1) q^{21} + ( - \beta_{9} + \beta_{8} + \cdots - 2 \beta_1) q^{23}+ \cdots + (2 \beta_{11} - \beta_{9} + \cdots - \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{5} - 2 q^{9} - 4 q^{13} - 28 q^{17} - 12 q^{21} - 6 q^{25} + 8 q^{29} + 34 q^{33} + 40 q^{37} + 6 q^{41} + 2 q^{45} - 14 q^{49} - 8 q^{53} + 14 q^{57} - 12 q^{61} + 4 q^{65} + 28 q^{69} - 28 q^{73} + 12 q^{77} - 14 q^{81} - 14 q^{85} + 16 q^{89} - 12 q^{93} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + x^{10} + 4x^{8} - 21x^{6} + 36x^{4} + 81x^{2} + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{10} - 4\nu^{8} + 29\nu^{6} + 18\nu^{4} + 63\nu^{2} - 162 ) / 810 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{10} + 7\nu^{8} + 28\nu^{6} - 219\nu^{4} - 234\nu^{2} - 729 ) / 810 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{11} - 7\nu^{9} - 28\nu^{7} + 219\nu^{5} + 234\nu^{3} + 729\nu ) / 2430 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{10} + 11\nu^{8} - \nu^{6} + 33\nu^{4} - 27\nu^{2} + 243 ) / 270 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{11} - 4\nu^{9} + 29\nu^{7} + 18\nu^{5} + 63\nu^{3} - 972\nu ) / 810 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{11} - 5\nu^{9} - 11\nu^{7} - 27\nu^{5} + 315\nu^{3} + 81\nu ) / 486 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 7\nu^{11} - 62\nu^{9} - 23\nu^{7} - 216\nu^{5} + 909\nu^{3} - 2106\nu ) / 2430 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{10} + \nu^{8} + 4\nu^{6} - 21\nu^{4} + 36\nu^{2} + 81 ) / 81 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{11} + \nu^{9} + 4\nu^{7} - 21\nu^{5} + 36\nu^{3} + 81\nu ) / 243 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} + 2\beta_{8} + \beta_{7} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} - 3\beta_{4} + 3\beta_{3} - \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{9} - \beta_{8} + 2\beta_{7} + 9\beta_{5} - \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{10} + 2\beta_{6} + 24\beta_{3} - 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 9\beta_{11} - \beta_{9} - 4\beta_{8} + 21\beta_{7} + 22\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 6\beta_{10} + 21\beta_{6} + 3\beta_{4} - 9\beta_{3} + \beta_{2} - 24 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 18\beta_{11} - 43\beta_{9} + 23\beta_{8} - 8\beta_{7} - 9\beta_{5} - 54\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 63\beta_{10} - 8\beta_{6} - 66\beta_{4} - 24\beta_{3} - 46\beta_{2} - 120 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 189\beta_{11} + 62\beta_{9} - 100\beta_{8} - 70\beta_{7} + 198\beta_{5} - 136\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(577\) \(641\) \(901\) \(991\)
\(\chi(n)\) \(1\) \(-\beta_{3}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
481.1
1.66309 − 0.483872i
1.02330 + 1.39744i
0.661048 − 1.60094i
−0.661048 + 1.60094i
−1.02330 − 1.39744i
−1.66309 + 0.483872i
1.66309 + 0.483872i
1.02330 − 1.39744i
0.661048 + 1.60094i
−0.661048 − 1.60094i
−1.02330 + 1.39744i
−1.66309 − 0.483872i
0 −1.66309 − 0.483872i 0 0.500000 − 0.866025i 0 −1.56078 − 2.70334i 0 2.53174 + 1.60944i 0
481.2 0 −1.02330 + 1.39744i 0 0.500000 − 0.866025i 0 1.87403 + 3.24592i 0 −0.905704 − 2.86002i 0
481.3 0 −0.661048 − 1.60094i 0 0.500000 − 0.866025i 0 1.02566 + 1.77649i 0 −2.12603 + 2.11660i 0
481.4 0 0.661048 + 1.60094i 0 0.500000 − 0.866025i 0 −1.02566 − 1.77649i 0 −2.12603 + 2.11660i 0
481.5 0 1.02330 − 1.39744i 0 0.500000 − 0.866025i 0 −1.87403 − 3.24592i 0 −0.905704 − 2.86002i 0
481.6 0 1.66309 + 0.483872i 0 0.500000 − 0.866025i 0 1.56078 + 2.70334i 0 2.53174 + 1.60944i 0
961.1 0 −1.66309 + 0.483872i 0 0.500000 + 0.866025i 0 −1.56078 + 2.70334i 0 2.53174 − 1.60944i 0
961.2 0 −1.02330 − 1.39744i 0 0.500000 + 0.866025i 0 1.87403 − 3.24592i 0 −0.905704 + 2.86002i 0
961.3 0 −0.661048 + 1.60094i 0 0.500000 + 0.866025i 0 1.02566 − 1.77649i 0 −2.12603 − 2.11660i 0
961.4 0 0.661048 − 1.60094i 0 0.500000 + 0.866025i 0 −1.02566 + 1.77649i 0 −2.12603 − 2.11660i 0
961.5 0 1.02330 + 1.39744i 0 0.500000 + 0.866025i 0 −1.87403 + 3.24592i 0 −0.905704 + 2.86002i 0
961.6 0 1.66309 − 0.483872i 0 0.500000 + 0.866025i 0 1.56078 − 2.70334i 0 2.53174 − 1.60944i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 481.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.2.q.q ✓ 12
3.b odd 2 1 4320.2.q.q 12
4.b odd 2 1 inner 1440.2.q.q ✓ 12
9.c even 3 1 inner 1440.2.q.q ✓ 12
9.d odd 6 1 4320.2.q.q 12
12.b even 2 1 4320.2.q.q 12
36.f odd 6 1 inner 1440.2.q.q ✓ 12
36.h even 6 1 4320.2.q.q 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.2.q.q ✓ 12 1.a even 1 1 trivial
1440.2.q.q ✓ 12 4.b odd 2 1 inner
1440.2.q.q ✓ 12 9.c even 3 1 inner
1440.2.q.q ✓ 12 36.f odd 6 1 inner
4320.2.q.q 12 3.b odd 2 1
4320.2.q.q 12 9.d odd 6 1
4320.2.q.q 12 12.b even 2 1
4320.2.q.q 12 36.h even 6 1

Hecke kernels

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

\( T_{7}^{12} + 28T_{7}^{10} + 547T_{7}^{8} + 5484T_{7}^{6} + 40041T_{7}^{4} + 136512T_{7}^{2} + 331776 \) Copy content Toggle raw display
\( T_{11}^{12} + 17T_{11}^{10} + 225T_{11}^{8} + 1016T_{11}^{6} + 3484T_{11}^{4} + 2304T_{11}^{2} + 1296 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + T^{10} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{12} + 28 T^{10} + \cdots + 331776 \) Copy content Toggle raw display
$11$ \( T^{12} + 17 T^{10} + \cdots + 1296 \) Copy content Toggle raw display
$13$ \( (T^{6} + 2 T^{5} + \cdots + 256)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} + 7 T^{2} - 8 T - 54)^{4} \) Copy content Toggle raw display
$19$ \( (T^{6} - 17 T^{4} + \cdots - 36)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 285610000 \) Copy content Toggle raw display
$29$ \( (T^{6} - 4 T^{5} + \cdots + 6084)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 64 T^{10} + \cdots + 1679616 \) Copy content Toggle raw display
$37$ \( (T^{3} - 10 T^{2} + \cdots + 72)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} - 3 T^{5} + \cdots + 164025)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 113 T^{10} + \cdots + 810000 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 11019960576 \) Copy content Toggle raw display
$53$ \( (T^{3} + 2 T^{2} + \cdots - 144)^{4} \) Copy content Toggle raw display
$59$ \( T^{12} + 61 T^{10} + \cdots + 20736 \) Copy content Toggle raw display
$61$ \( (T^{6} + 6 T^{5} + \cdots + 145924)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 332150625 \) Copy content Toggle raw display
$71$ \( (T^{6} - 424 T^{4} + \cdots - 767376)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 7 T^{2} + \cdots - 1384)^{4} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 4111875506176 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 53084160000 \) Copy content Toggle raw display
$89$ \( (T^{3} - 4 T^{2} + \cdots - 486)^{4} \) Copy content Toggle raw display
$97$ \( (T^{6} - T^{5} + 33 T^{4} + \cdots + 2304)^{2} \) Copy content Toggle raw display
show more
show less