↑

Properties

Label 961.6.a.m
Level $961$
Weight $6$
Character orbit 961.a
Self dual yes
Analytic conductor $154.129$
Analytic rank $1$
Dimension $96$
Inner twists $2$

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] = [961,6,Mod(1,961)] 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("961.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
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, 6, names="a")
 
Level: \( N \) \(=\) \( 961 = 31^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 961.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [96,-32,0] 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: \(154.128850840\)
Analytic rank: \(1\)
Dimension: \(96\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 32 q^{2} + 1408 q^{4} - 400 q^{5} - 784 q^{7} - 552 q^{8} + 6480 q^{9} - 3736 q^{10} - 11768 q^{14} + 18432 q^{16} - 7776 q^{18} - 11552 q^{19} - 22536 q^{20} + 50000 q^{25} - 12104 q^{28} - 25856 q^{32}+ \cdots - 648248 q^{98}+O(q^{100}) \) 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 −10.9911 −20.3152 88.8040 −31.4181 223.286 186.306 −624.338 169.708 345.320
1.2 −10.9911 20.3152 88.8040 −31.4181 −223.286 186.306 −624.338 169.708 345.320
1.3 −10.7348 −25.8744 83.2358 73.0830 277.756 −54.5118 −550.006 426.483 −784.531
1.4 −10.7348 25.8744 83.2358 73.0830 −277.756 −54.5118 −550.006 426.483 −784.531
1.5 −10.5373 −17.4666 79.0352 −18.8321 184.052 −108.058 −495.625 62.0837 198.440
1.6 −10.5373 17.4666 79.0352 −18.8321 −184.052 −108.058 −495.625 62.0837 198.440
1.7 −9.89035 −7.43672 65.8190 75.2162 73.5518 169.795 −334.482 −187.695 −743.914
1.8 −9.89035 7.43672 65.8190 75.2162 −73.5518 169.795 −334.482 −187.695 −743.914
1.9 −9.42447 −4.47208 56.8206 −81.4003 42.1470 216.537 −233.921 −223.000 767.154
1.10 −9.42447 4.47208 56.8206 −81.4003 −42.1470 216.537 −233.921 −223.000 767.154
1.11 −9.28516 −3.09095 54.2143 −52.7948 28.7000 −200.879 −206.263 −233.446 490.209
1.12 −9.28516 3.09095 54.2143 −52.7948 −28.7000 −200.879 −206.263 −233.446 490.209
1.13 −9.06981 −23.1422 50.2614 19.6200 209.896 −2.92686 −165.627 292.563 −177.950
1.14 −9.06981 23.1422 50.2614 19.6200 −209.896 −2.92686 −165.627 292.563 −177.950
1.15 −8.72589 −11.7855 44.1412 53.6863 102.839 −26.7279 −105.943 −104.103 −468.461
1.16 −8.72589 11.7855 44.1412 53.6863 −102.839 −26.7279 −105.943 −104.103 −468.461
1.17 −8.20693 −12.4216 35.3537 14.9587 101.944 −175.447 −27.5236 −88.7029 −122.765
1.18 −8.20693 12.4216 35.3537 14.9587 −101.944 −175.447 −27.5236 −88.7029 −122.765
1.19 −6.69058 −4.35345 12.7639 −51.8354 29.1271 75.5149 128.701 −224.047 346.809
1.20 −6.69058 4.35345 12.7639 −51.8354 −29.1271 75.5149 128.701 −224.047 346.809
See all 96 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.96
Significant digits:
Format:

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 961.6.a.m ✓ 96
31.b odd 2 1 inner 961.6.a.m ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
961.6.a.m ✓ 96 1.a even 1 1 trivial
961.6.a.m ✓ 96 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(961))\):

\( T_{2}^{48} + 16 T_{2}^{47} - 992 T_{2}^{46} - 16804 T_{2}^{45} + 449728 T_{2}^{44} + \cdots - 20\!\cdots\!04 \) Copy content Toggle raw display
\( T_{3}^{96} - 14904 T_{3}^{94} + 107044348 T_{3}^{92} - 493587076864 T_{3}^{90} + \cdots + 84\!\cdots\!76 \) Copy content Toggle raw display