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Properties

Label 2240.2.b.h
Level $2240$
Weight $2$
Character orbit 2240.b
Analytic conductor $17.886$
Analytic rank $0$
Dimension $12$
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] = [2240,2,Mod(1121,2240)]
 
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, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2240.1121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.b (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: \( x^{12} - 6 x^{11} + 35 x^{10} - 120 x^{9} + 362 x^{8} - 794 x^{7} + 1463 x^{6} - 2048 x^{5} + 2258 x^{4} + \cdots + 61 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{12} \)
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.

\(f(q)\) \(=\) \( q - \beta_{11} q^{3} - \beta_{9} q^{5} + q^{7} + ( - \beta_{5} - \beta_{4} - \beta_{2} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{11} q^{3} - \beta_{9} q^{5} + q^{7} + ( - \beta_{5} - \beta_{4} - \beta_{2} + \cdots - 1) q^{9}+ \cdots + ( - 8 \beta_{11} + 2 \beta_{9} + \cdots + 2 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{7} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{7} - 20 q^{9} + 16 q^{17} + 8 q^{23} - 12 q^{25} - 8 q^{31} + 72 q^{33} - 32 q^{39} - 32 q^{47} + 12 q^{49} + 8 q^{55} + 8 q^{57} - 20 q^{63} + 8 q^{65} + 48 q^{71} + 8 q^{73} - 16 q^{79} + 92 q^{81} - 32 q^{87} + 16 q^{89} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 35 x^{10} - 120 x^{9} + 362 x^{8} - 794 x^{7} + 1463 x^{6} - 2048 x^{5} + 2258 x^{4} + \cdots + 61 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{4} + 2\nu^{3} - 9\nu^{2} + 8\nu - 9 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{4} + 2\nu^{3} - 7\nu^{2} + 6\nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10 \nu^{11} - 55 \nu^{10} + 251 \nu^{9} - 717 \nu^{8} + 1474 \nu^{7} - 2198 \nu^{6} + 1376 \nu^{5} + \cdots + 782 ) / 286 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{8} + 4\nu^{7} - 18\nu^{6} + 40\nu^{5} - 77\nu^{4} + 92\nu^{3} - 68\nu^{2} + 28\nu - 3 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{8} - 4\nu^{7} + 20\nu^{6} - 46\nu^{5} + 105\nu^{4} - 138\nu^{3} + 152\nu^{2} - 90\nu + 29 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 34 \nu^{11} + 187 \nu^{10} - 1025 \nu^{9} + 3210 \nu^{8} - 8844 \nu^{7} + 17283 \nu^{6} + \cdots - 2716 ) / 572 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8 \nu^{11} - 44 \nu^{10} + 258 \nu^{9} - 831 \nu^{8} + 2552 \nu^{7} - 5362 \nu^{6} + 10310 \nu^{5} + \cdots - 1977 ) / 143 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 40 \nu^{11} - 220 \nu^{10} + 1290 \nu^{9} - 4155 \nu^{8} + 12188 \nu^{7} - 24808 \nu^{6} + 42112 \nu^{5} + \cdots - 1019 ) / 572 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 56 \nu^{11} - 308 \nu^{10} + 1806 \nu^{9} - 5817 \nu^{8} + 17292 \nu^{7} - 35532 \nu^{6} + 62732 \nu^{5} + \cdots - 4401 ) / 572 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{10} - 5 \nu^{9} + 28 \nu^{8} - 82 \nu^{7} + 229 \nu^{6} - 421 \nu^{5} + 682 \nu^{4} - 748 \nu^{3} + \cdots + 82 ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 162 \nu^{11} - 891 \nu^{10} + 5153 \nu^{9} - 16506 \nu^{8} + 48532 \nu^{7} - 99071 \nu^{6} + \cdots - 13186 ) / 572 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{9} + 2\beta_{8} + \beta_{7} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{9} + 2\beta_{8} + \beta_{7} + 2\beta_{2} - 4\beta _1 - 12 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{11} + 15\beta_{9} - 9\beta_{8} - 4\beta_{7} + 2\beta_{6} + 2\beta_{3} + 3\beta_{2} - 6\beta _1 - 19 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{11} + 32\beta_{9} - 20\beta_{8} - 9\beta_{7} + 4\beta_{6} + 4\beta_{3} - 12\beta_{2} + 16\beta _1 + 50 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 14\beta_{11} - 73\beta_{9} + 35\beta_{8} + 13\beta_{7} - 6\beta_{6} - 10\beta_{3} - 35\beta_{2} + 50\beta _1 + 157 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 26 \beta_{11} - 150 \beta_{9} + 78 \beta_{8} + 31 \beta_{7} - 14 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} + \cdots - 76 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 48 \beta_{11} + 180 \beta_{9} - 64 \beta_{8} - 7 \beta_{7} + 14 \beta_{5} + 28 \beta_{4} + \cdots - 1104 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 444 \beta_{11} + 2196 \beta_{9} - 1032 \beta_{8} - 339 \beta_{7} + 140 \beta_{6} - 16 \beta_{5} + \cdots - 138 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 142 \beta_{11} + 1063 \beta_{9} - 609 \beta_{8} - 302 \beta_{7} + 150 \beta_{6} - 156 \beta_{5} + \cdots + 6701 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 3004 \beta_{11} + 16 \beta_{10} - 13418 \beta_{9} + 5890 \beta_{8} + 1513 \beta_{7} - 516 \beta_{6} + \cdots + 7420 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 4226 \beta_{11} + 88 \beta_{10} - 20663 \beta_{9} + 9613 \beta_{8} + 3127 \beta_{7} - 1290 \beta_{6} + \cdots - 34593 ) / 4 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1121.1
0.500000 + 0.234551i
0.500000 − 2.14588i
0.500000 − 0.631151i
0.500000 + 2.40029i
0.500000 − 1.16542i
0.500000 + 2.51441i
0.500000 − 2.51441i
0.500000 + 1.16542i
0.500000 − 2.40029i
0.500000 + 0.631151i
0.500000 + 2.14588i
0.500000 − 0.234551i
0 − 3.32646i 0 − 1.00000i 0 1.00000 0 −8.06534 0
1121.2 0 − 3.13466i 0 1.00000i 0 1.00000 0 −6.82607 0
1121.3 0 − 2.11677i 0 − 1.00000i 0 1.00000 0 −1.48073 0
1121.4 0 − 1.47713i 0 1.00000i 0 1.00000 0 0.818075 0
1121.5 0 − 0.639640i 0 1.00000i 0 1.00000 0 2.59086 0
1121.6 0 − 0.191804i 0 1.00000i 0 1.00000 0 2.96321 0
1121.7 0 0.191804i 0 − 1.00000i 0 1.00000 0 2.96321 0
1121.8 0 0.639640i 0 − 1.00000i 0 1.00000 0 2.59086 0
1121.9 0 1.47713i 0 − 1.00000i 0 1.00000 0 0.818075 0
1121.10 0 2.11677i 0 1.00000i 0 1.00000 0 −1.48073 0
1121.11 0 3.13466i 0 − 1.00000i 0 1.00000 0 −6.82607 0
1121.12 0 3.32646i 0 1.00000i 0 1.00000 0 −8.06534 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1121.12
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 2240.2.b.h yes 12
4.b odd 2 1 2240.2.b.g ✓ 12
8.b even 2 1 inner 2240.2.b.h yes 12
8.d odd 2 1 2240.2.b.g ✓ 12
16.e even 4 1 8960.2.a.cb 6
16.e even 4 1 8960.2.a.ce 6
16.f odd 4 1 8960.2.a.cc 6
16.f odd 4 1 8960.2.a.ch 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2240.2.b.g ✓ 12 4.b odd 2 1
2240.2.b.g ✓ 12 8.d odd 2 1
2240.2.b.h yes 12 1.a even 1 1 trivial
2240.2.b.h yes 12 8.b even 2 1 inner
8960.2.a.cb 6 16.e even 4 1
8960.2.a.cc 6 16.f odd 4 1
8960.2.a.ce 6 16.e even 4 1
8960.2.a.ch 6 16.f odd 4 1

Hecke kernels

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

\( T_{3}^{12} + 28T_{3}^{10} + 270T_{3}^{8} + 1044T_{3}^{6} + 1481T_{3}^{4} + 488T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{23}^{6} - 4T_{23}^{5} - 80T_{23}^{4} + 160T_{23}^{3} + 1504T_{23}^{2} + 2304T_{23} + 1024 \) Copy content Toggle raw display
\( T_{31}^{6} + 4T_{31}^{5} - 80T_{31}^{4} - 400T_{31}^{3} + 784T_{31}^{2} + 4608T_{31} + 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 28 T^{10} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$7$ \( (T - 1)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 100 T^{10} + \cdots + 7311616 \) Copy content Toggle raw display
$13$ \( T^{12} + 116 T^{10} + \cdots + 595984 \) Copy content Toggle raw display
$17$ \( (T^{6} - 8 T^{5} - 34 T^{4} + \cdots + 88)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 208 T^{10} + \cdots + 34668544 \) Copy content Toggle raw display
$23$ \( (T^{6} - 4 T^{5} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 252 T^{10} + \cdots + 891136 \) Copy content Toggle raw display
$31$ \( (T^{6} + 4 T^{5} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 3421782016 \) Copy content Toggle raw display
$41$ \( (T^{6} - 140 T^{4} + \cdots - 32192)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 432 T^{10} + \cdots + 31719424 \) Copy content Toggle raw display
$47$ \( (T^{6} + 16 T^{5} + \cdots + 50272)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 5858983936 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 6461587456 \) Copy content Toggle raw display
$61$ \( T^{12} + 344 T^{10} + \cdots + 28217344 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 15352201216 \) Copy content Toggle raw display
$71$ \( (T^{6} - 24 T^{5} + \cdots + 29248)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 4 T^{5} + \cdots + 22528)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 8 T^{5} + \cdots - 13628)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 184913760256 \) Copy content Toggle raw display
$89$ \( (T^{6} - 8 T^{5} + \cdots + 11584)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 16 T^{5} + \cdots + 2776)^{2} \) Copy content Toggle raw display
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