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Properties

Label 961.4.a.n
Level $961$
Weight $4$
Character orbit 961.a
Self dual yes
Analytic conductor $56.701$
Analytic rank $0$
Dimension $64$
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] = [961,4,Mod(1,961)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(961, 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("961.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 961 = 31^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 961.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [64,16,0] 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: \(56.7008355155\)
Analytic rank: \(0\)
Dimension: \(64\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 16 q^{2} + 288 q^{4} + 80 q^{5} + 112 q^{7} + 24 q^{8} + 720 q^{9} + 136 q^{10} + 712 q^{14} + 1408 q^{16} + 432 q^{18} + 608 q^{19} + 1656 q^{20} + 2000 q^{25} + 2168 q^{28} + 448 q^{32} + 1056 q^{33}+ \cdots + 16312 q^{98}+O(q^{100}) \) 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 −5.39615 −5.75704 21.1185 −7.08279 31.0659 −23.7309 −70.7893 6.14347 38.2198
1.2 −5.39615 5.75704 21.1185 −7.08279 −31.0659 −23.7309 −70.7893 6.14347 38.2198
1.3 −5.27970 −8.77892 19.8752 8.89688 46.3500 −5.59079 −62.6975 50.0694 −46.9729
1.4 −5.27970 8.77892 19.8752 8.89688 −46.3500 −5.59079 −62.6975 50.0694 −46.9729
1.5 −5.09719 −7.55189 17.9813 15.0229 38.4934 22.9621 −50.8767 30.0311 −76.5743
1.6 −5.09719 7.55189 17.9813 15.0229 −38.4934 22.9621 −50.8767 30.0311 −76.5743
1.7 −5.00848 −0.719557 17.0848 −1.61888 3.60388 13.7664 −45.5012 −26.4822 8.10812
1.8 −5.00848 0.719557 17.0848 −1.61888 −3.60388 13.7664 −45.5012 −26.4822 8.10812
1.9 −4.14958 −4.87079 9.21898 −9.62932 20.2117 −3.07553 −5.05823 −3.27538 39.9576
1.10 −4.14958 4.87079 9.21898 −9.62932 −20.2117 −3.07553 −5.05823 −3.27538 39.9576
1.11 −3.72239 −6.81554 5.85620 13.4288 25.3701 −17.7878 7.98006 19.4515 −49.9871
1.12 −3.72239 6.81554 5.85620 13.4288 −25.3701 −17.7878 7.98006 19.4515 −49.9871
1.13 −3.71688 −3.30202 5.81519 20.0470 12.2732 24.1013 8.12069 −16.0967 −74.5124
1.14 −3.71688 3.30202 5.81519 20.0470 −12.2732 24.1013 8.12069 −16.0967 −74.5124
1.15 −2.98296 −5.85952 0.898072 −19.1029 17.4787 0.598005 21.1848 7.33402 56.9832
1.16 −2.98296 5.85952 0.898072 −19.1029 −17.4787 0.598005 21.1848 7.33402 56.9832
1.17 −2.81607 −8.78771 −0.0697240 −5.38333 24.7468 −34.4668 22.7249 50.2239 15.1599
1.18 −2.81607 8.78771 −0.0697240 −5.38333 −24.7468 −34.4668 22.7249 50.2239 15.1599
1.19 −2.27500 −1.47099 −2.82438 −7.84575 3.34649 4.99479 24.6255 −24.8362 17.8491
1.20 −2.27500 1.47099 −2.82438 −7.84575 −3.34649 4.99479 24.6255 −24.8362 17.8491
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.64
Significant digits:
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Atkin-Lehner signs

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

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 961.4.a.n ✓ 64
31.b odd 2 1 inner 961.4.a.n ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
961.4.a.n ✓ 64 1.a even 1 1 trivial
961.4.a.n ✓ 64 31.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(961))\):

\( T_{2}^{32} - 8 T_{2}^{31} - 168 T_{2}^{30} + 1468 T_{2}^{29} + 12080 T_{2}^{28} - 119488 T_{2}^{27} + \cdots + 5845588901888 \) Copy content Toggle raw display
\( T_{3}^{64} - 1224 T_{3}^{62} + 707884 T_{3}^{60} - 257341744 T_{3}^{58} + 65999300318 T_{3}^{56} + \cdots + 28\!\cdots\!96 \) Copy content Toggle raw display