Properties

Label 950.4.a.h
Level $950$
Weight $4$
Character orbit 950.a
Self dual yes
Analytic conductor $56.052$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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] = [950,4,Mod(1,950)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(950, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("950.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 950.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,4,-1,8,0,-2,-57,16,35] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(56.0518145055\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{177}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{177})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} - \beta q^{3} + 4 q^{4} - 2 \beta q^{6} + ( - \beta - 28) q^{7} + 8 q^{8} + (\beta + 17) q^{9} + (2 \beta + 4) q^{11} - 4 \beta q^{12} + (7 \beta - 10) q^{13} + ( - 2 \beta - 56) q^{14} + 16 q^{16} + \cdots + (40 \beta + 156) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} - q^{3} + 8 q^{4} - 2 q^{6} - 57 q^{7} + 16 q^{8} + 35 q^{9} + 10 q^{11} - 4 q^{12} - 13 q^{13} - 114 q^{14} + 32 q^{16} + 51 q^{17} + 70 q^{18} - 38 q^{19} + 117 q^{21} + 20 q^{22} + 155 q^{23}+ \cdots + 352 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
7.15207
−6.15207
2.00000 −7.15207 4.00000 0 −14.3041 −35.1521 8.00000 24.1521 0
1.2 2.00000 6.15207 4.00000 0 12.3041 −21.8479 8.00000 10.8479 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( +1 \)
\(19\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.4.a.h 2
5.b even 2 1 38.4.a.b 2
5.c odd 4 2 950.4.b.g 4
15.d odd 2 1 342.4.a.k 2
20.d odd 2 1 304.4.a.d 2
35.c odd 2 1 1862.4.a.b 2
40.e odd 2 1 1216.4.a.l 2
40.f even 2 1 1216.4.a.j 2
95.d odd 2 1 722.4.a.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.a.b 2 5.b even 2 1
304.4.a.d 2 20.d odd 2 1
342.4.a.k 2 15.d odd 2 1
722.4.a.i 2 95.d odd 2 1
950.4.a.h 2 1.a even 1 1 trivial
950.4.b.g 4 5.c odd 4 2
1216.4.a.j 2 40.f even 2 1
1216.4.a.l 2 40.e odd 2 1
1862.4.a.b 2 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(950))\):

\( T_{3}^{2} + T_{3} - 44 \) Copy content Toggle raw display
\( T_{7}^{2} + 57T_{7} + 768 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T - 44 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 57T + 768 \) Copy content Toggle raw display
$11$ \( T^{2} - 10T - 152 \) Copy content Toggle raw display
$13$ \( T^{2} + 13T - 2126 \) Copy content Toggle raw display
$17$ \( T^{2} - 51T - 9306 \) Copy content Toggle raw display
$19$ \( (T + 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 155T - 1472 \) Copy content Toggle raw display
$29$ \( T^{2} + 79T - 35654 \) Copy content Toggle raw display
$31$ \( T^{2} + 16T - 11264 \) Copy content Toggle raw display
$37$ \( T^{2} + 380T + 24772 \) Copy content Toggle raw display
$41$ \( T^{2} + 790T + 154432 \) Copy content Toggle raw display
$43$ \( T^{2} + 296T - 80048 \) Copy content Toggle raw display
$47$ \( T^{2} - 200T - 60800 \) Copy content Toggle raw display
$53$ \( T^{2} + 397T + 35818 \) Copy content Toggle raw display
$59$ \( T^{2} - 201T - 212964 \) Copy content Toggle raw display
$61$ \( T^{2} + 680T + 29932 \) Copy content Toggle raw display
$67$ \( T^{2} - 939T + 138612 \) Copy content Toggle raw display
$71$ \( T^{2} - 406T + 19792 \) Copy content Toggle raw display
$73$ \( T^{2} + 123T + 3738 \) Copy content Toggle raw display
$79$ \( T^{2} - 106T - 146048 \) Copy content Toggle raw display
$83$ \( T^{2} + 2226 T + 1237176 \) Copy content Toggle raw display
$89$ \( T^{2} + 870T + 184800 \) Copy content Toggle raw display
$97$ \( T^{2} - 1864 T + 613036 \) Copy content Toggle raw display
show more
show less