↑

Properties

Label 1250.2.b.f
Level $1250$
Weight $2$
Character orbit 1250.b
Analytic conductor $9.981$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: 8.0.324000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 9x^{6} + 26x^{4} + 24x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + ( - \beta_{7} - \beta_{3}) q^{3} - q^{4} - \beta_{2} q^{6} + (2 \beta_{7} + 2 \beta_{3}) q^{7} + \beta_{5} q^{8} + ( - \beta_{4} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + ( - \beta_{7} - \beta_{3}) q^{3} - q^{4} - \beta_{2} q^{6} + (2 \beta_{7} + 2 \beta_{3}) q^{7} + \beta_{5} q^{8} + ( - \beta_{4} + 1) q^{9} + ( - 2 \beta_{6} + 2 \beta_{2} + 1) q^{11} + (\beta_{7} + \beta_{3}) q^{12} + ( - 2 \beta_{7} - 4 \beta_{3}) q^{13} + 2 \beta_{2} q^{14} + q^{16} + ( - 4 \beta_{5} - 4 \beta_{3} - \beta_1) q^{17} + (\beta_{3} + \beta_1) q^{18} + ( - 3 \beta_{6} + 2 \beta_{4} + \cdots - 3) q^{19}+ \cdots + ( - 4 \beta_{6} - \beta_{4} + 2 \beta_{2} + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} + 2 q^{6} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} + 2 q^{6} + 6 q^{9} - 4 q^{11} - 4 q^{14} + 8 q^{16} - 30 q^{19} + 36 q^{21} - 2 q^{24} + 12 q^{26} - 20 q^{29} + 16 q^{31} - 14 q^{34} - 6 q^{36} - 48 q^{39} + 6 q^{41} + 4 q^{44} + 12 q^{46} - 16 q^{49} - 14 q^{51} + 10 q^{54} + 4 q^{56} + 40 q^{59} + 36 q^{61} - 8 q^{64} - 26 q^{66} + 32 q^{69} + 16 q^{71} - 24 q^{74} + 30 q^{76} - 20 q^{79} - 32 q^{81} - 36 q^{84} - 28 q^{86} - 10 q^{89} + 96 q^{91} - 4 q^{94} + 2 q^{96} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 9x^{6} + 26x^{4} + 24x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + 4\nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} + 5\nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} + 6\nu^{4} + 9\nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} + 7\nu^{5} + 14\nu^{3} + 7\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - 4\beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 5\beta_{3} + 10\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{6} - 6\beta_{4} + 15\beta_{2} - 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{7} - 7\beta_{5} + 21\beta_{3} - 35\beta_1 \) 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
0.209057i
− 1.33826i
− 1.82709i
1.95630i
− 1.95630i
1.82709i
1.33826i
− 0.209057i
− 1.00000i − 1.95630i −1.00000 0 −1.95630 3.91259i 1.00000i −0.827091 0
1249.2 − 1.00000i − 0.209057i −1.00000 0 −0.209057 0.418114i 1.00000i 2.95630 0
1249.3 − 1.00000i 1.33826i −1.00000 0 1.33826 − 2.67652i 1.00000i 1.20906 0
1249.4 − 1.00000i 1.82709i −1.00000 0 1.82709 − 3.65418i 1.00000i −0.338261 0
1249.5 1.00000i − 1.82709i −1.00000 0 1.82709 3.65418i − 1.00000i −0.338261 0
1249.6 1.00000i − 1.33826i −1.00000 0 1.33826 2.67652i − 1.00000i 1.20906 0
1249.7 1.00000i 0.209057i −1.00000 0 −0.209057 − 0.418114i − 1.00000i 2.95630 0
1249.8 1.00000i 1.95630i −1.00000 0 −1.95630 − 3.91259i − 1.00000i −0.827091 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.f 8
5.b even 2 1 inner 1250.2.b.f 8
5.c odd 4 1 1250.2.a.e ✓ 4
5.c odd 4 1 1250.2.a.k yes 4
20.e even 4 1 10000.2.a.s 4
20.e even 4 1 10000.2.a.w 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1250.2.a.e ✓ 4 5.c odd 4 1
1250.2.a.k yes 4 5.c odd 4 1
1250.2.b.f 8 1.a even 1 1 trivial
1250.2.b.f 8 5.b even 2 1 inner
10000.2.a.s 4 20.e even 4 1
10000.2.a.w 4 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 9T_{3}^{6} + 26T_{3}^{4} + 24T_{3}^{2} + 1 \) 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
$3$ \( T^{8} + 9 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 36 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( (T^{4} + 2 T^{3} - 16 T^{2} + \cdots + 31)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 84 T^{6} + \cdots + 20736 \) Copy content Toggle raw display
$17$ \( T^{8} + 81 T^{6} + \cdots + 3721 \) Copy content Toggle raw display
$19$ \( (T^{4} + 15 T^{3} + \cdots - 755)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 164 T^{6} + \cdots + 215296 \) Copy content Toggle raw display
$29$ \( (T^{4} + 10 T^{3} + \cdots - 2480)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 8 T^{3} + \cdots - 464)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 176 T^{6} + \cdots + 2027776 \) Copy content Toggle raw display
$41$ \( (T^{4} - 3 T^{3} + \cdots - 359)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 154 T^{6} + \cdots + 1661521 \) Copy content Toggle raw display
$47$ \( T^{8} + 156 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$53$ \( T^{8} + 144 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$59$ \( (T^{4} - 20 T^{3} + \cdots - 905)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 18 T^{3} + \cdots - 9584)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 261 T^{6} + \cdots + 4157521 \) Copy content Toggle raw display
$71$ \( (T^{4} - 8 T^{3} - 76 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 249 T^{6} + \cdots + 3481 \) Copy content Toggle raw display
$79$ \( (T^{4} + 10 T^{3} + \cdots - 2480)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 324 T^{6} + \cdots + 9541921 \) Copy content Toggle raw display
$89$ \( (T^{4} + 5 T^{3} - 160 T^{2} + \cdots + 25)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 203 T^{2} + 10201)^{2} \) Copy content Toggle raw display
show more
show less