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Properties

Label 250.2.d.c
Level $250$
Weight $2$
Character orbit 250.d
Analytic conductor $1.996$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [250,2,Mod(51,250)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(250, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("250.51");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 250 = 2 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 250.d (of order \(5\), degree \(4\), 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: \(1.99626005053\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5 \)
Twist minimal: no (minimal twist has level 50)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + (\beta_{2} - \beta_1) q^{3} - \beta_1 q^{4} + ( - \beta_{7} + \beta_{5} - \beta_{4} + \cdots + 1) q^{6} + (\beta_{7} - \beta_{5} + \beta_{2} + 2) q^{7} + ( - \beta_{4} + \beta_{3} - \beta_1 + 1) q^{8}+ \cdots + (3 \beta_{7} - \beta_{5} + \cdots + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{6} + 16 q^{7} + 2 q^{8} + 4 q^{9} - 4 q^{11} - 2 q^{12} - 2 q^{13} + 4 q^{14} - 2 q^{16} - 4 q^{17} - 4 q^{18} - 10 q^{19} + 16 q^{21} - 6 q^{22} - 12 q^{23} - 8 q^{24}+ \cdots + 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{20}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{20}^{3} + \zeta_{20} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{20}^{4} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{20}^{6} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{20}^{7} + \zeta_{20} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{20}^{5} + \zeta_{20} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} + 2\zeta_{20} \) Copy content Toggle raw display

Display \(\zeta_{20}^j\) in terms of \(\beta_i\)

\(\zeta_{20}\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{2} ) / 5 \) Copy content Toggle raw display
\(\zeta_{20}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{20}^{3}\)\(=\) \( ( -\beta_{7} - \beta_{6} - \beta_{5} + 4\beta_{2} ) / 5 \) Copy content Toggle raw display
\(\zeta_{20}^{4}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{20}^{5}\)\(=\) \( ( \beta_{7} - 4\beta_{6} + \beta_{5} + \beta_{2} ) / 5 \) Copy content Toggle raw display
\(\zeta_{20}^{6}\)\(=\) \( \beta_{4} \) Copy content Toggle raw display
\(\zeta_{20}^{7}\)\(=\) \( ( -\beta_{7} - \beta_{6} + 4\beta_{5} - \beta_{2} ) / 5 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{20}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/250\mathbb{Z}\right)^\times\).

\(n\) \(127\)
\(\chi(n)\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
0.587785 + 0.809017i
−0.587785 − 0.809017i
−0.951057 − 0.309017i
0.951057 + 0.309017i
−0.951057 + 0.309017i
0.951057 − 0.309017i
0.587785 − 0.809017i
−0.587785 + 0.809017i
0.809017 − 0.587785i −0.0542543 + 0.166977i 0.309017 − 0.951057i 0 0.0542543 + 0.166977i 1.27346 −0.309017 − 0.951057i 2.40211 + 1.74524i 0
51.2 0.809017 − 0.587785i 0.672288 − 2.06909i 0.309017 − 0.951057i 0 −0.672288 − 2.06909i 2.72654 −0.309017 − 0.951057i −1.40211 − 1.01869i 0
101.1 −0.309017 + 0.951057i −2.34786 − 1.70582i −0.809017 − 0.587785i 0 2.34786 − 1.70582i −1.07768 0.809017 − 0.587785i 1.67557 + 5.15688i 0
101.2 −0.309017 + 0.951057i 0.729825 + 0.530249i −0.809017 − 0.587785i 0 −0.729825 + 0.530249i 5.07768 0.809017 − 0.587785i −0.675571 − 2.07919i 0
151.1 −0.309017 − 0.951057i −2.34786 + 1.70582i −0.809017 + 0.587785i 0 2.34786 + 1.70582i −1.07768 0.809017 + 0.587785i 1.67557 − 5.15688i 0
151.2 −0.309017 − 0.951057i 0.729825 − 0.530249i −0.809017 + 0.587785i 0 −0.729825 − 0.530249i 5.07768 0.809017 + 0.587785i −0.675571 + 2.07919i 0
201.1 0.809017 + 0.587785i −0.0542543 − 0.166977i 0.309017 + 0.951057i 0 0.0542543 − 0.166977i 1.27346 −0.309017 + 0.951057i 2.40211 − 1.74524i 0
201.2 0.809017 + 0.587785i 0.672288 + 2.06909i 0.309017 + 0.951057i 0 −0.672288 + 2.06909i 2.72654 −0.309017 + 0.951057i −1.40211 + 1.01869i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 250.2.d.c 8
5.b even 2 1 250.2.d.b 8
5.c odd 4 1 50.2.e.a ✓ 8
5.c odd 4 1 250.2.e.a 8
15.e even 4 1 450.2.l.b 8
20.e even 4 1 400.2.y.a 8
25.d even 5 1 inner 250.2.d.c 8
25.d even 5 1 1250.2.a.h 4
25.e even 10 1 250.2.d.b 8
25.e even 10 1 1250.2.a.i 4
25.f odd 20 1 50.2.e.a ✓ 8
25.f odd 20 1 250.2.e.a 8
25.f odd 20 2 1250.2.b.c 8
75.l even 20 1 450.2.l.b 8
100.h odd 10 1 10000.2.a.bb 4
100.j odd 10 1 10000.2.a.o 4
100.l even 20 1 400.2.y.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.e.a ✓ 8 5.c odd 4 1
50.2.e.a ✓ 8 25.f odd 20 1
250.2.d.b 8 5.b even 2 1
250.2.d.b 8 25.e even 10 1
250.2.d.c 8 1.a even 1 1 trivial
250.2.d.c 8 25.d even 5 1 inner
250.2.e.a 8 5.c odd 4 1
250.2.e.a 8 25.f odd 20 1
400.2.y.a 8 20.e even 4 1
400.2.y.a 8 100.l even 20 1
450.2.l.b 8 15.e even 4 1
450.2.l.b 8 75.l even 20 1
1250.2.a.h 4 25.d even 5 1
1250.2.a.i 4 25.e even 10 1
1250.2.b.c 8 25.f odd 20 2
10000.2.a.o 4 100.j odd 10 1
10000.2.a.bb 4 100.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 2T_{3}^{7} + 3T_{3}^{6} + 4T_{3}^{5} + 30T_{3}^{4} - 46T_{3}^{3} + 28T_{3}^{2} + 2T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(250, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 8 T^{3} + 14 T^{2} + \cdots - 19)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 4 T^{7} + \cdots + 3481 \) Copy content Toggle raw display
$13$ \( T^{8} + 2 T^{7} + \cdots + 1681 \) Copy content Toggle raw display
$17$ \( T^{8} + 4 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{8} + 10 T^{7} + \cdots + 25 \) Copy content Toggle raw display
$23$ \( T^{8} + 12 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{8} + 10 T^{7} + \cdots + 9025 \) Copy content Toggle raw display
$31$ \( T^{8} - 6 T^{7} + \cdots + 361 \) Copy content Toggle raw display
$37$ \( T^{8} + 4 T^{7} + \cdots + 58081 \) Copy content Toggle raw display
$41$ \( T^{8} + 14 T^{7} + \cdots + 28504921 \) Copy content Toggle raw display
$43$ \( (T^{4} - 14 T^{3} + \cdots - 1919)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 24 T^{7} + \cdots + 8637721 \) Copy content Toggle raw display
$53$ \( T^{8} - 8 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$59$ \( T^{8} + 2560 T^{4} + \cdots + 1638400 \) Copy content Toggle raw display
$61$ \( T^{8} + 14 T^{7} + \cdots + 32761 \) Copy content Toggle raw display
$67$ \( T^{8} + 4 T^{7} + \cdots + 3481 \) Copy content Toggle raw display
$71$ \( T^{8} + 34 T^{7} + \cdots + 1042441 \) Copy content Toggle raw display
$73$ \( T^{8} + 12 T^{7} + \cdots + 92416 \) Copy content Toggle raw display
$79$ \( T^{8} + 20 T^{6} + \cdots + 22278400 \) Copy content Toggle raw display
$83$ \( T^{8} + 32 T^{7} + \cdots + 17131321 \) Copy content Toggle raw display
$89$ \( (T^{4} + 10 T^{3} + \cdots + 400)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 4 T^{7} + \cdots + 130321 \) Copy content Toggle raw display
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