↑

Properties

Label 50.22.b.c
Level $50$
Weight $22$
Character orbit 50.b
Analytic conductor $139.739$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [50,22,Mod(49,50)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("50.49"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(50, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 50.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-2097152,0,121479168] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(139.738672144\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 512 \beta q^{2} + 29658 \beta q^{3} - 1048576 q^{4} + 60739584 q^{6} - 713712916 \beta q^{7} + 536870912 \beta q^{8} + 6941965347 q^{9} - 106767894948 q^{11} - 31098667008 \beta q^{12} - 75075282737 \beta q^{13} + \cdots - 74\!\cdots\!56 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2097152 q^{4} + 121479168 q^{6} + 13883930694 q^{9} - 213535789896 q^{11} - 2923368103936 q^{14} + 2199023255552 q^{16} - 22048111910920 q^{19} + 169338381301824 q^{21} - 127380140064768 q^{24} - 307508358090752 q^{26}+ \cdots - 14\!\cdots\!12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.00000i
− 1.00000i
− 1024.00i 59316.0i −1.04858e6 0 6.07396e7 − 1.42743e9i 1.07374e9i 6.94197e9 0
49.2 1024.00i − 59316.0i −1.04858e6 0 6.07396e7 1.42743e9i − 1.07374e9i 6.94197e9 0
\(n\): e.g. 2-40 or 80-90
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 50.22.b.c 2
5.b even 2 1 inner 50.22.b.c 2
5.c odd 4 1 2.22.a.b ✓ 1
5.c odd 4 1 50.22.a.a 1
15.e even 4 1 18.22.a.b 1
20.e even 4 1 16.22.a.b 1
40.i odd 4 1 64.22.a.c 1
40.k even 4 1 64.22.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.22.a.b ✓ 1 5.c odd 4 1
16.22.a.b 1 20.e even 4 1
18.22.a.b 1 15.e even 4 1
50.22.a.a 1 5.c odd 4 1
50.22.b.c 2 1.a even 1 1 trivial
50.22.b.c 2 5.b even 2 1 inner
64.22.a.c 1 40.i odd 4 1
64.22.a.e 1 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 3518387856 \) acting on \(S_{22}^{\mathrm{new}}(50, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1048576 \) Copy content Toggle raw display
$3$ \( T^{2} + 3518387856 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 20\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( (T + 106767894948)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 22\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{2} + 12\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( (T + 11024055955460)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 16\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T + 23\!\cdots\!10)^{2} \) Copy content Toggle raw display
$31$ \( (T + 878552957377888)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 96\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( (T + 24\!\cdots\!38)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 17\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{2} + 37\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{2} + 35\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( (T - 29\!\cdots\!80)^{2} \) Copy content Toggle raw display
$61$ \( (T - 79\!\cdots\!22)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 23\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( (T - 88\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 13\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T + 33\!\cdots\!20)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 41\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T - 41\!\cdots\!10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 52\!\cdots\!04 \) Copy content Toggle raw display
show more
show less