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Properties

Label 2240.2.g.n
Level $2240$
Weight $2$
Character orbit 2240.g
Analytic conductor $17.886$
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] = [2240,2,Mod(449,2240)]
 
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, 0, 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("2240.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.g (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: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 13x^{8} + 56x^{6} + 97x^{4} + 61x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1120)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{4} q^{5} - \beta_{5} q^{7} + (\beta_{9} - \beta_{8} - \beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{4} q^{5} - \beta_{5} q^{7} + (\beta_{9} - \beta_{8} - \beta_{3} - 1) q^{9} + (\beta_{9} - \beta_{8} + \beta_{2} - 1) q^{11} + ( - \beta_{7} - \beta_{5} + 2 \beta_1) q^{13} + (\beta_{9} - \beta_{7} - \beta_{6} + \cdots + 1) q^{15}+ \cdots + (4 \beta_{3} + 2 \beta_{2} + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{5} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 2 q^{5} - 14 q^{9} - 8 q^{11} + 4 q^{15} + 24 q^{19} + 4 q^{21} + 6 q^{25} - 24 q^{29} + 24 q^{31} - 64 q^{39} - 4 q^{41} - 10 q^{45} - 10 q^{49} - 24 q^{51} + 16 q^{55} + 32 q^{59} + 20 q^{61} - 8 q^{65} + 8 q^{69} + 8 q^{71} - 64 q^{75} - 64 q^{79} + 2 q^{81} + 12 q^{85} - 4 q^{89} + 60 q^{95} + 80 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 13x^{8} + 56x^{6} + 97x^{4} + 61x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{9} + 12\nu^{7} + 44\nu^{5} + 55\nu^{3} + 20\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{8} - 34\nu^{6} - 112\nu^{4} - 107\nu^{2} - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{8} - 34\nu^{6} - 112\nu^{4} - 111\nu^{2} - 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{9} - 3\nu^{8} - 34\nu^{7} - 34\nu^{6} - 110\nu^{5} - 110\nu^{4} - 93\nu^{3} - 97\nu^{2} + 16\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{9} - 23\nu^{7} - 78\nu^{5} - 82\nu^{3} - 13\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{9} + 3\nu^{8} - 34\nu^{7} + 34\nu^{6} - 110\nu^{5} + 110\nu^{4} - 93\nu^{3} + 97\nu^{2} + 16\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{9} + 35\nu^{7} + 122\nu^{5} + 137\nu^{3} + 29\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5\nu^{9} + 5\nu^{8} + 56\nu^{7} + 58\nu^{6} + 178\nu^{5} + 198\nu^{4} + 151\nu^{3} + 203\nu^{2} - 14\nu + 20 ) / 4 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5\nu^{9} - 5\nu^{8} + 56\nu^{7} - 58\nu^{6} + 178\nu^{5} - 198\nu^{4} + 151\nu^{3} - 203\nu^{2} - 14\nu - 20 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} - \beta_{5} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + \beta_{2} - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{9} + \beta_{8} + 4\beta_{7} + \beta_{6} + 6\beta_{5} + \beta_{4} - 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{6} + 2\beta_{4} + 5\beta_{3} - 7\beta_{2} + 23 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -8\beta_{9} - 8\beta_{8} - 21\beta_{7} - 6\beta_{6} - 41\beta_{5} - 6\beta_{4} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -3\beta_{9} + 3\beta_{8} + 17\beta_{6} - 17\beta_{4} - 24\beta_{3} + 46\beta_{2} - 128 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 52\beta_{9} + 52\beta_{8} + 125\beta_{7} + 32\beta_{6} + 273\beta_{5} + 32\beta_{4} + 7\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 34\beta_{9} - 34\beta_{8} - 118\beta_{6} + 118\beta_{4} + 121\beta_{3} - 297\beta_{2} + 769 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -327\beta_{9} - 327\beta_{8} - 776\beta_{7} - 175\beta_{6} - 1782\beta_{5} - 175\beta_{4} - 122\beta_1 ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
− 1.84576i
− 0.271831i
2.52064i
− 1.28447i
− 1.23118i
1.23118i
1.28447i
− 2.52064i
0.271831i
1.84576i
0 − 3.25260i 0 1.49436 − 1.66340i 0 1.00000i 0 −7.57939 0
449.2 0 − 2.19794i 0 −1.64514 + 1.51444i 0 − 1.00000i 0 −1.83094 0
449.3 0 − 1.83297i 0 −1.86302 + 1.23660i 0 1.00000i 0 −0.359777 0
449.4 0 − 1.63460i 0 2.23081 + 0.153266i 0 − 1.00000i 0 0.328072 0
449.5 0 − 0.746976i 0 0.782984 + 2.09450i 0 1.00000i 0 2.44203 0
449.6 0 0.746976i 0 0.782984 − 2.09450i 0 − 1.00000i 0 2.44203 0
449.7 0 1.63460i 0 2.23081 − 0.153266i 0 1.00000i 0 0.328072 0
449.8 0 1.83297i 0 −1.86302 − 1.23660i 0 − 1.00000i 0 −0.359777 0
449.9 0 2.19794i 0 −1.64514 − 1.51444i 0 1.00000i 0 −1.83094 0
449.10 0 3.25260i 0 1.49436 + 1.66340i 0 − 1.00000i 0 −7.57939 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.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.g.n 10
4.b odd 2 1 2240.2.g.o 10
5.b even 2 1 inner 2240.2.g.n 10
8.b even 2 1 1120.2.g.c yes 10
8.d odd 2 1 1120.2.g.b ✓ 10
20.d odd 2 1 2240.2.g.o 10
40.e odd 2 1 1120.2.g.b ✓ 10
40.f even 2 1 1120.2.g.c yes 10
40.i odd 4 1 5600.2.a.bv 5
40.i odd 4 1 5600.2.a.bx 5
40.k even 4 1 5600.2.a.bu 5
40.k even 4 1 5600.2.a.bw 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1120.2.g.b ✓ 10 8.d odd 2 1
1120.2.g.b ✓ 10 40.e odd 2 1
1120.2.g.c yes 10 8.b even 2 1
1120.2.g.c yes 10 40.f even 2 1
2240.2.g.n 10 1.a even 1 1 trivial
2240.2.g.n 10 5.b even 2 1 inner
2240.2.g.o 10 4.b odd 2 1
2240.2.g.o 10 20.d odd 2 1
5600.2.a.bu 5 40.k even 4 1
5600.2.a.bv 5 40.i odd 4 1
5600.2.a.bw 5 40.k even 4 1
5600.2.a.bx 5 40.i odd 4 1

Hecke kernels

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

\( T_{3}^{10} + 22T_{3}^{8} + 165T_{3}^{6} + 532T_{3}^{4} + 708T_{3}^{2} + 256 \) Copy content Toggle raw display
\( T_{11}^{5} + 4T_{11}^{4} - 25T_{11}^{3} - 152T_{11}^{2} - 248T_{11} - 128 \) Copy content Toggle raw display
\( T_{19}^{5} - 12T_{19}^{4} + 30T_{19}^{3} + 72T_{19}^{2} - 184T_{19} - 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 22 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{10} - 2 T^{9} + \cdots + 3125 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$11$ \( (T^{5} + 4 T^{4} + \cdots - 128)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 94 T^{8} + \cdots + 678976 \) Copy content Toggle raw display
$17$ \( T^{10} + 66 T^{8} + \cdots + 1024 \) Copy content Toggle raw display
$19$ \( (T^{5} - 12 T^{4} + \cdots - 256)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 104 T^{8} + \cdots + 262144 \) Copy content Toggle raw display
$29$ \( (T^{5} + 12 T^{4} + \cdots + 4744)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 12 T^{4} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 136 T^{8} + \cdots + 65536 \) Copy content Toggle raw display
$41$ \( (T^{5} + 2 T^{4} + \cdots - 128)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 228130816 \) Copy content Toggle raw display
$47$ \( T^{10} + 122 T^{8} + \cdots + 4096 \) Copy content Toggle raw display
$53$ \( T^{10} + 208 T^{8} + \cdots + 3444736 \) Copy content Toggle raw display
$59$ \( (T^{5} - 16 T^{4} + \cdots - 58112)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 10 T^{4} + \cdots - 1648)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 232 T^{8} + \cdots + 262144 \) Copy content Toggle raw display
$71$ \( (T^{5} - 4 T^{4} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 224 T^{8} + \cdots + 4194304 \) Copy content Toggle raw display
$79$ \( (T^{5} + 32 T^{4} + \cdots + 3904)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 750321664 \) Copy content Toggle raw display
$89$ \( (T^{5} + 2 T^{4} + \cdots + 3104)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 274 T^{8} + \cdots + 11723776 \) Copy content Toggle raw display
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