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Properties

Label 1922.4.a.x
Level $1922$
Weight $4$
Character orbit 1922.a
Self dual yes
Analytic conductor $113.402$
Analytic rank $0$
Dimension $32$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1922,4,Mod(1,1922)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1922, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("1922.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1922 = 2 \cdot 31^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1922.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,-64,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(113.401671031\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 64 q^{2} + 128 q^{4} + 112 q^{7} - 256 q^{8} + 288 q^{9} - 224 q^{14} + 512 q^{16} - 576 q^{18} + 304 q^{19} + 1200 q^{25} + 448 q^{28} - 1024 q^{32} - 272 q^{33} + 1152 q^{36} - 608 q^{38} + 1616 q^{39}+ \cdots - 6176 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.00000 −10.0428 4.00000 1.39037 20.0856 −6.92491 −8.00000 73.8576 −2.78073
1.2 −2.00000 −9.30256 4.00000 18.9675 18.6051 21.8384 −8.00000 59.5377 −37.9350
1.3 −2.00000 −8.73949 4.00000 11.3839 17.4790 32.2218 −8.00000 49.3787 −22.7678
1.4 −2.00000 −7.83585 4.00000 −13.9902 15.6717 −28.2462 −8.00000 34.4006 27.9805
1.5 −2.00000 −7.26205 4.00000 2.99959 14.5241 27.7754 −8.00000 25.7374 −5.99919
1.6 −2.00000 −7.23390 4.00000 −20.3377 14.4678 31.7504 −8.00000 25.3293 40.6754
1.7 −2.00000 −6.24968 4.00000 −9.30955 12.4994 −9.12649 −8.00000 12.0585 18.6191
1.8 −2.00000 −5.36518 4.00000 14.2606 10.7304 −3.03268 −8.00000 1.78511 −28.5211
1.9 −2.00000 −5.13962 4.00000 6.34276 10.2792 33.0781 −8.00000 −0.584326 −12.6855
1.10 −2.00000 −5.10931 4.00000 −18.4069 10.2186 1.23685 −8.00000 −0.894964 36.8139
1.11 −2.00000 −3.78891 4.00000 −7.04778 7.57782 −19.1757 −8.00000 −12.6441 14.0956
1.12 −2.00000 −2.73675 4.00000 20.6657 5.47349 −27.1083 −8.00000 −19.5102 −41.3315
1.13 −2.00000 −1.21735 4.00000 9.21122 2.43469 −15.9940 −8.00000 −25.5181 −18.4224
1.14 −2.00000 −1.02445 4.00000 −8.88463 2.04891 2.25055 −8.00000 −25.9505 17.7693
1.15 −2.00000 −0.915774 4.00000 5.16716 1.83155 −2.76909 −8.00000 −26.1614 −10.3343
1.16 −2.00000 −0.422597 4.00000 −12.4119 0.845194 18.2258 −8.00000 −26.8214 24.8239
1.17 −2.00000 0.422597 4.00000 −12.4119 −0.845194 18.2258 −8.00000 −26.8214 24.8239
1.18 −2.00000 0.915774 4.00000 5.16716 −1.83155 −2.76909 −8.00000 −26.1614 −10.3343
1.19 −2.00000 1.02445 4.00000 −8.88463 −2.04891 2.25055 −8.00000 −25.9505 17.7693
1.20 −2.00000 1.21735 4.00000 9.21122 −2.43469 −15.9940 −8.00000 −25.5181 −18.4224
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.32
Significant digits:
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Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(31\) \( +1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1922.4.a.x ✓ 32
31.b odd 2 1 inner 1922.4.a.x ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1922.4.a.x ✓ 32 1.a even 1 1 trivial
1922.4.a.x ✓ 32 31.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} - 576 T_{3}^{30} + 147496 T_{3}^{28} - 22178064 T_{3}^{26} + 2178533848 T_{3}^{24} + \cdots + 21\!\cdots\!24 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1922))\). Copy content Toggle raw display