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Properties

Label 961.6.a.l
Level $961$
Weight $6$
Character orbit 961.a
Self dual yes
Analytic conductor $154.129$
Analytic rank $0$
Dimension $52$
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] = [961,6,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 = [52,32,0] 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: \(0\)
Dimension: \(52\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 52 q + 32 q^{2} + 896 q^{4} + 200 q^{5} + 392 q^{7} + 2028 q^{8} + 4860 q^{9} - 268 q^{10} + 388 q^{14} + 8024 q^{16} + 10368 q^{18} + 5776 q^{19} + 11236 q^{20} + 28612 q^{25} + 52204 q^{28} + 73088 q^{32}+ \cdots + 386748 q^{98}+O(q^{100}) \) 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 −10.1854 −0.443736 71.7426 51.7990 4.51963 −206.138 −404.794 −242.803 −527.594
1.2 −10.1854 0.443736 71.7426 51.7990 −4.51963 −206.138 −404.794 −242.803 −527.594
1.3 −9.55715 −9.91144 59.3390 −39.0320 94.7250 −11.6595 −261.283 −144.763 373.034
1.4 −9.55715 9.91144 59.3390 −39.0320 −94.7250 −11.6595 −261.283 −144.763 373.034
1.5 −9.15350 −28.9143 51.7865 −19.7901 264.667 174.599 −181.116 593.037 181.149
1.6 −9.15350 28.9143 51.7865 −19.7901 −264.667 174.599 −181.116 593.037 181.149
1.7 −8.48205 −16.4016 39.9451 −23.2376 139.119 251.807 −67.3906 26.0119 197.102
1.8 −8.48205 16.4016 39.9451 −23.2376 −139.119 251.807 −67.3906 26.0119 197.102
1.9 −7.26685 −12.8731 20.8072 −68.5105 93.5467 −58.7482 81.3367 −77.2840 497.855
1.10 −7.26685 12.8731 20.8072 −68.5105 −93.5467 −58.7482 81.3367 −77.2840 497.855
1.11 −6.55672 −8.63603 10.9906 51.3900 56.6240 −148.160 137.753 −168.419 −336.950
1.12 −6.55672 8.63603 10.9906 51.3900 −56.6240 −148.160 137.753 −168.419 −336.950
1.13 −6.15581 −26.5983 5.89406 78.3608 163.734 125.916 160.703 464.471 −482.375
1.14 −6.15581 26.5983 5.89406 78.3608 −163.734 125.916 160.703 464.471 −482.375
1.15 −6.14476 −25.4665 5.75811 93.4668 156.486 154.574 161.250 405.542 −574.331
1.16 −6.14476 25.4665 5.75811 93.4668 −156.486 154.574 161.250 405.542 −574.331
1.17 −2.96630 −13.4742 −23.2011 82.2390 39.9685 −22.1290 163.743 −61.4459 −243.946
1.18 −2.96630 13.4742 −23.2011 82.2390 −39.9685 −22.1290 163.743 −61.4459 −243.946
1.19 −2.62846 −22.2058 −25.0912 −60.5934 58.3669 −159.767 150.062 250.097 159.267
1.20 −2.62846 22.2058 −25.0912 −60.5934 −58.3669 −159.767 150.062 250.097 159.267
See all 52 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.52
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 961.6.a.l ✓ 52
31.b odd 2 1 inner 961.6.a.l ✓ 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
961.6.a.l ✓ 52 1.a even 1 1 trivial
961.6.a.l ✓ 52 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(961))\):

\( T_{2}^{26} - 16 T_{2}^{25} - 512 T_{2}^{24} + 8878 T_{2}^{23} + 111413 T_{2}^{22} + \cdots - 17\!\cdots\!88 \) Copy content Toggle raw display
\( T_{3}^{52} - 8748 T_{3}^{50} + 35543006 T_{3}^{48} - 89097940744 T_{3}^{46} + 154408423260132 T_{3}^{44} + \cdots + 25\!\cdots\!08 \) Copy content Toggle raw display