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Properties

Label 1440.2.q.p
Level $1440$
Weight $2$
Character orbit 1440.q
Analytic conductor $11.498$
Analytic rank $0$
Dimension $10$
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(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: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.8528759163648.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + x^{8} + 9x^{6} - 36x^{5} + 27x^{4} + 27x^{2} - 162x + 243 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} - \beta_{2} q^{5} + (\beta_{9} - \beta_{7} + \cdots - \beta_{4}) q^{7}+ \cdots + ( - \beta_{9} + \beta_{8} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} - \beta_{2} q^{5} + (\beta_{9} - \beta_{7} + \cdots - \beta_{4}) q^{7}+ \cdots + (3 \beta_{9} - 3 \beta_{7} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{3} - 5 q^{5} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + q^{3} - 5 q^{5} - q^{9} - 9 q^{11} - 4 q^{13} - 2 q^{15} + 6 q^{17} + 14 q^{19} + 12 q^{21} - 10 q^{23} - 5 q^{25} - 2 q^{27} - 2 q^{29} - 8 q^{31} - 27 q^{33} + 16 q^{37} + 26 q^{39} - q^{41} - q^{43} - q^{45} - 10 q^{47} - q^{49} - 3 q^{51} - 20 q^{53} + 18 q^{55} - 19 q^{57} - 13 q^{59} - 14 q^{61} - 30 q^{63} - 4 q^{65} - 15 q^{67} + 8 q^{69} + 32 q^{71} - 14 q^{73} + q^{75} + 12 q^{77} - 18 q^{79} + 35 q^{81} + 8 q^{83} - 3 q^{85} - 32 q^{87} + 28 q^{89} - 36 q^{91} + 16 q^{93} - 7 q^{95} + 17 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} + x^{8} + 9x^{6} - 36x^{5} + 27x^{4} + 27x^{2} - 162x + 243 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} + \nu^{8} + 4\nu^{7} + 12\nu^{6} + 45\nu^{5} - 63\nu^{4} + 27\nu - 567 ) / 486 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{9} - \nu^{8} - 4\nu^{7} + 15\nu^{6} - 18\nu^{5} + 9\nu^{4} + 81\nu^{3} + 162\nu^{2} - 270\nu + 81 ) / 243 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{9} - \nu^{8} - 4\nu^{7} - 12\nu^{6} + 9\nu^{5} + 9\nu^{4} + 297\nu + 81 ) / 162 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{9} + 2\nu^{8} - \nu^{7} - 9\nu^{5} + 36\nu^{4} - 27\nu^{3} - 27\nu + 162 ) / 81 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{9} + 7\nu^{8} + 28\nu^{7} - 24\nu^{6} + 45\nu^{5} - 63\nu^{4} + 162\nu^{3} - 648\nu^{2} - 297\nu - 567 ) / 486 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{9} + 11\nu^{8} - 10\nu^{7} - 12\nu^{6} - 99\nu^{5} + 117\nu^{4} - 189\nu + 1053 ) / 486 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -4\nu^{9} - 4\nu^{8} + 11\nu^{7} + 6\nu^{6} - 45\nu^{5} + 36\nu^{4} + 81\nu^{3} - 81\nu^{2} - 108\nu + 324 ) / 243 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 17 \nu^{9} + 19 \nu^{8} + 22 \nu^{7} - 60 \nu^{6} - 171 \nu^{5} + 369 \nu^{4} + 162 \nu^{3} + \cdots + 2349 ) / 486 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{4} + 2\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{7} + \beta_{6} + 3\beta_{5} + \beta_{4} + 2\beta_{3} - 3\beta_{2} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{7} + \beta_{6} + 3\beta_{5} + 4\beta_{4} + 2\beta_{3} + 6\beta_{2} - 5\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -6\beta_{9} + 6\beta_{8} + 3\beta_{7} + 4\beta_{6} + 9\beta_{5} + \beta_{4} + 5\beta_{3} + 6\beta_{2} + 7\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 15 \beta_{9} - 6 \beta_{8} - 18 \beta_{7} - 11 \beta_{6} - 12 \beta_{5} - 20 \beta_{4} - 10 \beta_{3} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 9 \beta_{9} - 18 \beta_{8} + 15 \beta_{7} + \beta_{6} - 6 \beta_{5} - 14 \beta_{4} + 2 \beta_{3} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3\beta_{9} - 30\beta_{8} - 33\beta_{7} + 13\beta_{6} - 35\beta_{4} + 14\beta_{3} - 102\beta_{2} + 70\beta _1 + 6 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(577\) \(641\) \(901\) \(991\)
\(\chi(n)\) \(1\) \(-\beta_{2}\) \(1\) \(1\)

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
−1.13593 + 1.30754i
1.06839 + 1.36328i
−1.41743 − 0.995434i
1.72806 − 0.117480i
0.756905 − 1.55791i
−1.13593 − 1.30754i
1.06839 − 1.36328i
−1.41743 + 0.995434i
1.72806 + 0.117480i
0.756905 + 1.55791i
0 −1.70033 − 0.329969i 0 −0.500000 + 0.866025i 0 −0.876687 − 1.51847i 0 2.78224 + 1.12211i 0
481.2 0 −0.646443 + 1.60689i 0 −0.500000 + 0.866025i 0 −0.445362 − 0.771389i 0 −2.16422 − 2.07753i 0
481.3 0 0.153356 − 1.72525i 0 −0.500000 + 0.866025i 0 2.31980 + 4.01801i 0 −2.95296 − 0.529154i 0
481.4 0 0.965772 + 1.43781i 0 −0.500000 + 0.866025i 0 0.538872 + 0.933353i 0 −1.13457 + 2.77718i 0
481.5 0 1.72765 − 0.123458i 0 −0.500000 + 0.866025i 0 −1.53662 − 2.66151i 0 2.96952 − 0.426584i 0
961.1 0 −1.70033 + 0.329969i 0 −0.500000 − 0.866025i 0 −0.876687 + 1.51847i 0 2.78224 − 1.12211i 0
961.2 0 −0.646443 − 1.60689i 0 −0.500000 − 0.866025i 0 −0.445362 + 0.771389i 0 −2.16422 + 2.07753i 0
961.3 0 0.153356 + 1.72525i 0 −0.500000 − 0.866025i 0 2.31980 − 4.01801i 0 −2.95296 + 0.529154i 0
961.4 0 0.965772 − 1.43781i 0 −0.500000 − 0.866025i 0 0.538872 − 0.933353i 0 −1.13457 − 2.77718i 0
961.5 0 1.72765 + 0.123458i 0 −0.500000 − 0.866025i 0 −1.53662 + 2.66151i 0 2.96952 + 0.426584i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 481.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.2.q.p yes 10
3.b odd 2 1 4320.2.q.p 10
4.b odd 2 1 1440.2.q.o ✓ 10
9.c even 3 1 inner 1440.2.q.p yes 10
9.d odd 6 1 4320.2.q.p 10
12.b even 2 1 4320.2.q.o 10
36.f odd 6 1 1440.2.q.o ✓ 10
36.h even 6 1 4320.2.q.o 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.2.q.o ✓ 10 4.b odd 2 1
1440.2.q.o ✓ 10 36.f odd 6 1
1440.2.q.p yes 10 1.a even 1 1 trivial
1440.2.q.p yes 10 9.c even 3 1 inner
4320.2.q.o 10 12.b even 2 1
4320.2.q.o 10 36.h even 6 1
4320.2.q.p 10 3.b odd 2 1
4320.2.q.p 10 9.d odd 6 1

Hecke kernels

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

\( T_{7}^{10} + 18T_{7}^{8} + 44T_{7}^{7} + 303T_{7}^{6} + 420T_{7}^{5} + 862T_{7}^{4} + 402T_{7}^{3} + 969T_{7}^{2} + 504T_{7} + 576 \) Copy content Toggle raw display
\( T_{11}^{10} + 9 T_{11}^{9} + 63 T_{11}^{8} + 194 T_{11}^{7} + 504 T_{11}^{6} + 348 T_{11}^{5} + \cdots + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - T^{9} + \cdots + 243 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{10} + 18 T^{8} + \cdots + 576 \) Copy content Toggle raw display
$11$ \( T^{10} + 9 T^{9} + \cdots + 144 \) Copy content Toggle raw display
$13$ \( T^{10} + 4 T^{9} + \cdots + 2304 \) Copy content Toggle raw display
$17$ \( (T^{5} - 3 T^{4} + \cdots - 108)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} - 7 T^{4} + \cdots + 284)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 10 T^{9} + \cdots + 36 \) Copy content Toggle raw display
$29$ \( T^{10} + 2 T^{9} + \cdots + 293764 \) Copy content Toggle raw display
$31$ \( T^{10} + 8 T^{9} + \cdots + 678976 \) Copy content Toggle raw display
$37$ \( (T^{5} - 8 T^{4} + \cdots - 2528)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + T^{9} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{10} + T^{9} + \cdots + 9265936 \) Copy content Toggle raw display
$47$ \( T^{10} + 10 T^{9} + \cdots + 19642624 \) Copy content Toggle raw display
$53$ \( (T^{5} + 10 T^{4} + \cdots + 16832)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + 13 T^{9} + \cdots + 2214144 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 9115284676 \) Copy content Toggle raw display
$67$ \( T^{10} + 15 T^{9} + \cdots + 81 \) Copy content Toggle raw display
$71$ \( (T^{5} - 16 T^{4} + \cdots - 6792)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + 7 T^{4} + \cdots + 8768)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + 18 T^{9} + \cdots + 2985984 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 3175998736 \) Copy content Toggle raw display
$89$ \( (T^{5} - 14 T^{4} + \cdots + 18)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 114211554304 \) Copy content Toggle raw display
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