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Properties

Label 1250.2.b.e
Level $1250$
Weight $2$
Character orbit 1250.b
Analytic conductor $9.981$
Analytic rank $0$
Dimension $8$
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] = [1250,2,Mod(1249,1250)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1250, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1250.1249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1250 = 2 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1250.b (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: \(9.98130025266\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.14884000000.15
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 19x^{6} + 121x^{4} + 304x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 50)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + \beta_1 q^{3} - q^{4} + ( - \beta_{7} + \beta_{4} + 1) q^{6} - \beta_{5} q^{7} + \beta_{3} q^{8} + (\beta_{7} + \beta_{6} - \beta_{4} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + \beta_1 q^{3} - q^{4} + ( - \beta_{7} + \beta_{4} + 1) q^{6} - \beta_{5} q^{7} + \beta_{3} q^{8} + (\beta_{7} + \beta_{6} - \beta_{4} - 2) q^{9} + \beta_{6} q^{11} - \beta_1 q^{12} + ( - \beta_{5} + 4 \beta_{3} + \cdots - \beta_1) q^{13}+ \cdots + ( - \beta_{7} - 2 \beta_{6} + \beta_{4} + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} + 2 q^{6} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} + 2 q^{6} - 14 q^{9} - 4 q^{11} - 4 q^{14} + 8 q^{16} + 10 q^{19} - 14 q^{21} - 2 q^{24} + 22 q^{26} - 30 q^{29} - 24 q^{31} - 24 q^{34} + 14 q^{36} + 22 q^{39} + 26 q^{41} + 4 q^{44} - 8 q^{46} + 4 q^{49} + 26 q^{51} - 20 q^{54} + 4 q^{56} + 16 q^{61} - 8 q^{64} + 14 q^{66} + 62 q^{69} - 44 q^{71} - 24 q^{74} - 10 q^{76} + 20 q^{79} - 32 q^{81} + 14 q^{84} + 12 q^{86} + 10 q^{89} - 24 q^{91} - 4 q^{94} + 2 q^{96} + 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 19x^{6} + 121x^{4} + 304x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 19\nu^{5} + 121\nu^{3} + 240\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{7} - 79\nu^{5} - 365\nu^{3} - 480\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 19\nu^{4} + 105\nu^{2} + 144 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 19\nu^{5} + 105\nu^{3} + 144\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} - 15\nu^{4} - 61\nu^{2} - 64 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{6} + 79\nu^{4} + 365\nu^{2} + 480 ) / 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} - \beta_{4} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + 4\beta_{2} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -11\beta_{7} - 10\beta_{6} + 15\beta_{4} + 35 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 15\beta_{5} + 4\beta_{3} - 40\beta_{2} + 45\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 104\beta_{7} + 85\beta_{6} - 164\beta_{4} - 284 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -164\beta_{5} - 76\beta_{3} + 340\beta_{2} - 369\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1250\mathbb{Z}\right)^\times\).

\(n\) \(627\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
− 2.33275i
− 1.34841i
1.71472i
2.96645i
− 2.96645i
− 1.71472i
1.34841i
2.33275i
− 1.00000i − 2.33275i −1.00000 0 −2.33275 − 3.77447i 1.00000i −2.44172 0
1249.2 − 1.00000i − 1.34841i −1.00000 0 −1.34841 0.833366i 1.00000i 1.18178 0
1249.3 − 1.00000i 1.71472i −1.00000 0 1.71472 2.77447i 1.00000i 0.0597522 0
1249.4 − 1.00000i 2.96645i −1.00000 0 2.96645 − 1.83337i 1.00000i −5.79981 0
1249.5 1.00000i − 2.96645i −1.00000 0 2.96645 1.83337i − 1.00000i −5.79981 0
1249.6 1.00000i − 1.71472i −1.00000 0 1.71472 − 2.77447i − 1.00000i 0.0597522 0
1249.7 1.00000i 1.34841i −1.00000 0 −1.34841 − 0.833366i − 1.00000i 1.18178 0
1249.8 1.00000i 2.33275i −1.00000 0 −2.33275 3.77447i − 1.00000i −2.44172 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1249.8
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 1250.2.b.e 8
5.b even 2 1 inner 1250.2.b.e 8
5.c odd 4 1 1250.2.a.f 4
5.c odd 4 1 1250.2.a.l 4
20.e even 4 1 10000.2.a.t 4
20.e even 4 1 10000.2.a.x 4
25.d even 5 2 250.2.e.c 16
25.e even 10 2 250.2.e.c 16
25.f odd 20 2 50.2.d.b ✓ 8
25.f odd 20 2 250.2.d.d 8
75.l even 20 2 450.2.h.e 8
100.l even 20 2 400.2.u.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.d.b ✓ 8 25.f odd 20 2
250.2.d.d 8 25.f odd 20 2
250.2.e.c 16 25.d even 5 2
250.2.e.c 16 25.e even 10 2
400.2.u.d 8 100.l even 20 2
450.2.h.e 8 75.l even 20 2
1250.2.a.f 4 5.c odd 4 1
1250.2.a.l 4 5.c odd 4 1
1250.2.b.e 8 1.a even 1 1 trivial
1250.2.b.e 8 5.b even 2 1 inner
10000.2.a.t 4 20.e even 4 1
10000.2.a.x 4 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 19T_{3}^{6} + 121T_{3}^{4} + 304T_{3}^{2} + 256 \) acting on \(S_{2}^{\mathrm{new}}(1250, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + 19 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 26 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( (T^{4} + 2 T^{3} - 11 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 69 T^{6} + \cdots + 39601 \) Copy content Toggle raw display
$17$ \( T^{8} + 66 T^{6} + \cdots + 11881 \) Copy content Toggle raw display
$19$ \( (T^{4} - 5 T^{3} - 35 T^{2} + \cdots + 80)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 74 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( (T^{4} + 15 T^{3} + 70 T^{2} + \cdots + 5)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 12 T^{3} + \cdots - 1264)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 106 T^{6} + \cdots + 5041 \) Copy content Toggle raw display
$41$ \( (T^{4} - 13 T^{3} + \cdots - 89)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 114 T^{6} + \cdots + 30976 \) Copy content Toggle raw display
$47$ \( T^{8} + 26 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$53$ \( T^{8} + 219 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$59$ \( (T^{4} - 140 T^{2} + \cdots - 320)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 8 T^{3} + \cdots - 1709)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 346 T^{6} + \cdots + 891136 \) Copy content Toggle raw display
$71$ \( (T^{4} + 22 T^{3} + \cdots - 64)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 279 T^{6} + \cdots + 1175056 \) Copy content Toggle raw display
$79$ \( (T^{4} - 10 T^{3} + \cdots - 320)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 314 T^{6} + \cdots + 13424896 \) Copy content Toggle raw display
$89$ \( (T^{4} - 5 T^{3} + \cdots - 3100)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 123 T^{2} + 1)^{2} \) Copy content Toggle raw display
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