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Properties

Label 2240.2.l.b
Level $2240$
Weight $2$
Character orbit 2240.l
Analytic conductor $17.886$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2240,2,Mod(1569,2240)]
 
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([0, 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("2240.1569");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.l (of order \(2\), degree \(1\), 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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 3x^{10} + 11x^{8} + 22x^{6} + 99x^{4} + 243x^{2} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{9} q^{3} + \beta_{4} q^{5} - \beta_{3} q^{7} + ( - \beta_{10} - \beta_{5} + \beta_{4} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{3} + \beta_{4} q^{5} - \beta_{3} q^{7} + ( - \beta_{10} - \beta_{5} + \beta_{4} + 1) q^{9} + ( - \beta_{8} - \beta_{7} - \beta_{2}) q^{11} + ( - \beta_{10} - 2) q^{13} + (\beta_{11} + \beta_{9} + \cdots + \beta_{3}) q^{15}+ \cdots + (2 \beta_{8} - 4 \beta_{7} + \cdots - 4 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{5} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{5} + 24 q^{9} - 20 q^{13} - 12 q^{25} - 24 q^{37} + 24 q^{41} + 48 q^{45} - 12 q^{49} + 8 q^{53} - 12 q^{65} + 4 q^{77} + 20 q^{81} + 56 q^{85} + 64 q^{89} - 120 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 3x^{10} + 11x^{8} + 22x^{6} + 99x^{4} + 243x^{2} + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -5\nu^{10} - 51\nu^{8} - 82\nu^{6} - 587\nu^{4} - 1773\nu^{2} - 3888 ) / 2268 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -10\nu^{11} - 39\nu^{9} + 25\nu^{7} - 1048\nu^{5} - 1593\nu^{3} - 2673\nu ) / 6804 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -23\nu^{11} + 30\nu^{9} - 37\nu^{7} + 97\nu^{5} + 954\nu^{3} + 2187\nu ) / 13608 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -23\nu^{10} - 96\nu^{8} - 415\nu^{6} - 155\nu^{4} - 2952\nu^{2} - 3483 ) / 4536 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -23\nu^{10} + 30\nu^{8} - 37\nu^{6} + 97\nu^{4} - 3582\nu^{2} - 2349 ) / 4536 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10\nu^{10} + 39\nu^{8} - 25\nu^{6} - 86\nu^{4} + 459\nu^{2} + 1539 ) / 1134 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 25\nu^{11} + 129\nu^{9} + 32\nu^{7} + 415\nu^{5} + 423\nu^{3} + 4698\nu ) / 6804 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{11} + 3\nu^{9} + 11\nu^{7} + 22\nu^{5} + 99\nu^{3} + 486\nu ) / 243 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{11} + 3\nu^{9} + 11\nu^{7} + 22\nu^{5} + 99\nu^{3} ) / 243 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4\nu^{10} + 3\nu^{8} + 17\nu^{6} + 70\nu^{4} + 117\nu^{2} + 243 ) / 324 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{11} + 7\nu^{7} + 11\nu^{5} - 96\nu^{3} - 45\nu ) / 216 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} + \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{10} - 2\beta_{5} - \beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + \beta_{9} - \beta_{7} + 3\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{10} - 2\beta_{6} + 2\beta_{5} + 4\beta_{4} - 5\beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4\beta_{11} - \beta_{9} - \beta_{8} - 2\beta_{7} - 8\beta_{3} - 14\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{10} - 5\beta_{6} + 3\beta_{5} - 11\beta_{4} + 4\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 16\beta_{11} + 21\beta_{9} + 11\beta_{8} - 14\beta_{7} - 12\beta_{3} + 18\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -17\beta_{10} + 34\beta_{6} + 40\beta_{5} - 14\beta_{4} - 19\beta _1 - 56 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3\beta_{11} + 15\beta_{9} - 20\beta_{8} + 54\beta_{7} + 77\beta_{3} + 23\beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 143\beta_{10} + 52\beta_{6} - 32\beta_{5} + 34\beta_{4} + 97\beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -84\beta_{11} - 11\beta_{9} + 21\beta_{8} + 72\beta_{7} - 748\beta_{3} - 28\beta_{2} ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1569.1
1.55635 + 0.760115i
1.55635 − 0.760115i
1.02957 + 1.39283i
1.02957 − 1.39283i
0.517456 + 1.65295i
0.517456 − 1.65295i
−0.517456 − 1.65295i
−0.517456 + 1.65295i
−1.02957 − 1.39283i
−1.02957 + 1.39283i
−1.55635 − 0.760115i
−1.55635 + 0.760115i
0 −3.11270 0 2.08433 − 0.809661i 0 − 1.00000i 0 6.68890 0
1569.2 0 −3.11270 0 2.08433 + 0.809661i 0 1.00000i 0 6.68890 0
1569.3 0 −2.05914 0 −1.27280 − 1.83847i 0 − 1.00000i 0 1.24006 0
1569.4 0 −2.05914 0 −1.27280 + 1.83847i 0 1.00000i 0 1.24006 0
1569.5 0 −1.03491 0 0.188470 − 2.22811i 0 1.00000i 0 −1.92896 0
1569.6 0 −1.03491 0 0.188470 + 2.22811i 0 − 1.00000i 0 −1.92896 0
1569.7 0 1.03491 0 0.188470 − 2.22811i 0 − 1.00000i 0 −1.92896 0
1569.8 0 1.03491 0 0.188470 + 2.22811i 0 1.00000i 0 −1.92896 0
1569.9 0 2.05914 0 −1.27280 − 1.83847i 0 1.00000i 0 1.24006 0
1569.10 0 2.05914 0 −1.27280 + 1.83847i 0 − 1.00000i 0 1.24006 0
1569.11 0 3.11270 0 2.08433 − 0.809661i 0 1.00000i 0 6.68890 0
1569.12 0 3.11270 0 2.08433 + 0.809661i 0 − 1.00000i 0 6.68890 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1569.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
40.e odd 2 1 inner
40.f 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.l.b yes 12
4.b odd 2 1 inner 2240.2.l.b yes 12
5.b even 2 1 2240.2.l.a ✓ 12
8.b even 2 1 2240.2.l.a ✓ 12
8.d odd 2 1 2240.2.l.a ✓ 12
20.d odd 2 1 2240.2.l.a ✓ 12
40.e odd 2 1 inner 2240.2.l.b yes 12
40.f even 2 1 inner 2240.2.l.b yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2240.2.l.a ✓ 12 5.b even 2 1
2240.2.l.a ✓ 12 8.b even 2 1
2240.2.l.a ✓ 12 8.d odd 2 1
2240.2.l.a ✓ 12 20.d odd 2 1
2240.2.l.b yes 12 1.a even 1 1 trivial
2240.2.l.b yes 12 4.b odd 2 1 inner
2240.2.l.b yes 12 40.e odd 2 1 inner
2240.2.l.b yes 12 40.f even 2 1 inner

Hecke kernels

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

\( T_{3}^{6} - 15T_{3}^{4} + 56T_{3}^{2} - 44 \) Copy content Toggle raw display
\( T_{13}^{3} + 5T_{13}^{2} - 2T_{13} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} - 15 T^{4} + \cdots - 44)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} - 2 T^{5} + \cdots + 125)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$11$ \( (T^{6} + 49 T^{4} + \cdots + 2704)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} + 5 T^{2} - 2 T - 2)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} + 35 T^{4} + \cdots + 176)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 28 T^{4} + \cdots + 256)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 92 T^{4} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 139 T^{4} + \cdots + 85184)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 60 T^{4} + \cdots - 2816)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 6 T^{2} - 64 T - 32)^{4} \) Copy content Toggle raw display
$41$ \( (T - 2)^{12} \) Copy content Toggle raw display
$43$ \( (T^{6} - 216 T^{4} + \cdots - 45056)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 65 T^{4} + \cdots + 256)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 2 T^{2} + \cdots + 112)^{4} \) Copy content Toggle raw display
$59$ \( (T^{6} + 128 T^{4} + \cdots + 38416)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 212 T^{4} + \cdots + 180224)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 340 T^{4} + \cdots - 1301696)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 196 T^{4} + \cdots - 704)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 216 T^{4} + \cdots + 45056)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 331 T^{4} + \cdots - 790064)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 232 T^{4} + \cdots - 240944)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} - 16 T^{2} + \cdots - 16)^{4} \) Copy content Toggle raw display
$97$ \( (T^{6} + 387 T^{4} + \cdots + 1655984)^{2} \) Copy content Toggle raw display
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