Properties

Label 950.4.b.g
Level $950$
Weight $4$
Character orbit 950.b
Analytic conductor $56.052$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [950,4,Mod(799,950)] 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([1, 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("950.799"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 950.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-16,0,-4,0,0,-70,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(56.0518145055\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{177})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 89x^{2} + 1936 \) 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 38)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + \beta_1 q^{3} - 4 q^{4} + (2 \beta_{3} - 2) q^{6} + (14 \beta_{2} - \beta_1) q^{7} + 4 \beta_{2} q^{8} + (\beta_{3} - 18) q^{9} + ( - 2 \beta_{3} + 6) q^{11} - 4 \beta_1 q^{12} + ( - 5 \beta_{2} - 7 \beta_1) q^{13}+ \cdots + (40 \beta_{3} - 196) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{4} - 4 q^{6} - 70 q^{9} + 20 q^{11} + 228 q^{14} + 64 q^{16} + 76 q^{19} + 234 q^{21} + 16 q^{24} - 52 q^{26} + 158 q^{29} - 32 q^{31} - 204 q^{34} + 280 q^{36} + 1226 q^{39} - 1580 q^{41} - 80 q^{44}+ \cdots - 704 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 89x^{2} + 1936 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 45\nu ) / 22 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 45 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 45 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 22\beta_{2} - 45\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
799.1
7.15207i
6.15207i
6.15207i
7.15207i
2.00000i 7.15207i −4.00000 0 −14.3041 35.1521i 8.00000i −24.1521 0
799.2 2.00000i 6.15207i −4.00000 0 12.3041 21.8479i 8.00000i −10.8479 0
799.3 2.00000i 6.15207i −4.00000 0 12.3041 21.8479i 8.00000i −10.8479 0
799.4 2.00000i 7.15207i −4.00000 0 −14.3041 35.1521i 8.00000i −24.1521 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 950.4.b.g 4
5.b even 2 1 inner 950.4.b.g 4
5.c odd 4 1 38.4.a.b 2
5.c odd 4 1 950.4.a.h 2
15.e even 4 1 342.4.a.k 2
20.e even 4 1 304.4.a.d 2
35.f even 4 1 1862.4.a.b 2
40.i odd 4 1 1216.4.a.j 2
40.k even 4 1 1216.4.a.l 2
95.g even 4 1 722.4.a.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.a.b 2 5.c odd 4 1
304.4.a.d 2 20.e even 4 1
342.4.a.k 2 15.e even 4 1
722.4.a.i 2 95.g even 4 1
950.4.a.h 2 5.c odd 4 1
950.4.b.g 4 1.a even 1 1 trivial
950.4.b.g 4 5.b even 2 1 inner
1216.4.a.j 2 40.i odd 4 1
1216.4.a.l 2 40.k even 4 1
1862.4.a.b 2 35.f even 4 1

Hecke kernels

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

\( T_{3}^{4} + 89T_{3}^{2} + 1936 \) Copy content Toggle raw display
\( T_{7}^{4} + 1713T_{7}^{2} + 589824 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 89T^{2} + 1936 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 1713 T^{2} + 589824 \) Copy content Toggle raw display
$11$ \( (T^{2} - 10 T - 152)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 4421 T^{2} + 4519876 \) Copy content Toggle raw display
$17$ \( T^{4} + 21213 T^{2} + 86601636 \) Copy content Toggle raw display
$19$ \( (T - 19)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 26969 T^{2} + 2166784 \) Copy content Toggle raw display
$29$ \( (T^{2} - 79 T - 35654)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 16 T - 11264)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 94856 T^{2} + 613651984 \) Copy content Toggle raw display
$41$ \( (T^{2} + 790 T + 154432)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 6407682304 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 3696640000 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 1282929124 \) Copy content Toggle raw display
$59$ \( (T^{2} + 201 T - 212964)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 680 T + 29932)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 19213286544 \) Copy content Toggle raw display
$71$ \( (T^{2} - 406 T + 19792)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 7653 T^{2} + 13972644 \) Copy content Toggle raw display
$79$ \( (T^{2} + 106 T - 146048)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 1530604454976 \) Copy content Toggle raw display
$89$ \( (T^{2} - 870 T + 184800)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 375813137296 \) Copy content Toggle raw display
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