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Properties

Label 50.12.b.a
Level $50$
Weight $12$
Character orbit 50.b
Analytic conductor $38.417$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [50,12,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, 12); 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, 12, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
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,-2048,0,-47232] 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: \(38.4171590280\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 10)
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.

\(f(q)\) \(=\) \( q + 16 \beta q^{2} + 369 \beta q^{3} - 1024 q^{4} - 23616 q^{6} - 12787 \beta q^{7} - 16384 \beta q^{8} - 367497 q^{9} + 769152 q^{11} - 377856 \beta q^{12} - 459491 \beta q^{13} + 818368 q^{14} + 1048576 q^{16} + \cdots - 282661052544 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2048 q^{4} - 47232 q^{6} - 734994 q^{9} + 1538304 q^{11} + 1636736 q^{14} + 2097152 q^{16} + 11043320 q^{19} + 37747224 q^{21} + 48365568 q^{24} + 58814848 q^{26} + 30538020 q^{29} - 483167576 q^{31}+ \cdots - 565322105088 q^{99}+O(q^{100}) \) 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
− 32.0000i − 738.000i −1024.00 0 −23616.0 25574.0i 32768.0i −367497. 0
49.2 32.0000i 738.000i −1024.00 0 −23616.0 − 25574.0i − 32768.0i −367497. 0
\(n\): e.g. 2-40 or 990-1000
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 50.12.b.a 2
5.b even 2 1 inner 50.12.b.a 2
5.c odd 4 1 10.12.a.b ✓ 1
5.c odd 4 1 50.12.a.c 1
15.e even 4 1 90.12.a.k 1
20.e even 4 1 80.12.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.12.a.b ✓ 1 5.c odd 4 1
50.12.a.c 1 5.c odd 4 1
50.12.b.a 2 1.a even 1 1 trivial
50.12.b.a 2 5.b even 2 1 inner
80.12.a.a 1 20.e even 4 1
90.12.a.k 1 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 544644 \) acting on \(S_{12}^{\mathrm{new}}(50, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1024 \) Copy content Toggle raw display
$3$ \( T^{2} + 544644 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 654029476 \) Copy content Toggle raw display
$11$ \( (T - 769152)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 844527916324 \) Copy content Toggle raw display
$17$ \( T^{2} + 106353720086436 \) Copy content Toggle raw display
$19$ \( (T - 5521660)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 15\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( (T - 15269010)^{2} \) Copy content Toggle raw display
$31$ \( (T + 241583788)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 663136971090916 \) Copy content Toggle raw display
$41$ \( (T + 1217700138)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 46\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{2} + 23\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{2} + 12\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( (T - 1069039020)^{2} \) Copy content Toggle raw display
$61$ \( (T + 2091535078)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 21\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T - 9660178332)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 31\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( (T + 5026936280)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 14\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T + 35558583210)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 11\!\cdots\!96 \) Copy content Toggle raw display
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