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Properties

Label 97.6.b.a
Level $97$
Weight $6$
Character orbit 97.b
Analytic conductor $15.557$
Analytic rank $0$
Dimension $40$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [97,6,Mod(96,97)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(97, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 6, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("97.96"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Level: \( N \) \(=\) \( 97 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 97.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.5572305219\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 2 q^{2} + 40 q^{3} + 638 q^{4} - 130 q^{6} + 180 q^{8} + 3300 q^{9} + 382 q^{11} + 2586 q^{12} + 10174 q^{16} + 4738 q^{18} + 1996 q^{22} - 3102 q^{24} - 25178 q^{25} + 3046 q^{27} + 14796 q^{31}+ \cdots - 562238 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
96.1 −10.9603 12.4768 88.1276 − 77.3203i −136.749 − 129.228i −615.173 −87.3297 847.452i
96.2 −10.9603 12.4768 88.1276 77.3203i −136.749 129.228i −615.173 −87.3297 − 847.452i
96.3 −9.84944 −20.3089 65.0114 − 53.6529i 200.031 − 56.8568i −325.144 169.450 528.451i
96.4 −9.84944 −20.3089 65.0114 53.6529i 200.031 56.8568i −325.144 169.450 − 528.451i
96.5 −8.53310 4.72888 40.8138 − 19.3202i −40.3520 95.2642i −75.2089 −220.638 164.861i
96.6 −8.53310 4.72888 40.8138 19.3202i −40.3520 − 95.2642i −75.2089 −220.638 − 164.861i
96.7 −8.40356 27.7780 38.6199 − 56.8413i −233.434 175.534i −55.6305 528.618 477.669i
96.8 −8.40356 27.7780 38.6199 56.8413i −233.434 − 175.534i −55.6305 528.618 − 477.669i
96.9 −6.78566 −6.83487 14.0452 − 60.7818i 46.3791 183.728i 121.835 −196.285 412.445i
96.10 −6.78566 −6.83487 14.0452 60.7818i 46.3791 − 183.728i 121.835 −196.285 − 412.445i
96.11 −5.09653 −8.23001 −6.02536 − 103.917i 41.9445 − 183.907i 193.797 −175.267 529.617i
96.12 −5.09653 −8.23001 −6.02536 103.917i 41.9445 183.907i 193.797 −175.267 − 529.617i
96.13 −4.70798 18.5644 −9.83488 − 69.5512i −87.4008 − 119.658i 196.958 101.636 327.446i
96.14 −4.70798 18.5644 −9.83488 69.5512i −87.4008 119.658i 196.958 101.636 − 327.446i
96.15 −4.61222 −26.0544 −10.7275 − 41.0639i 120.169 132.637i 197.068 435.832 189.396i
96.16 −4.61222 −26.0544 −10.7275 41.0639i 120.169 − 132.637i 197.068 435.832 − 189.396i
96.17 −2.57129 15.1707 −25.3885 − 24.3473i −39.0083 87.8859i 147.562 −12.8494 62.6040i
96.18 −2.57129 15.1707 −25.3885 24.3473i −39.0083 − 87.8859i 147.562 −12.8494 − 62.6040i
96.19 −0.466290 −9.91982 −31.7826 − 23.6226i 4.62551 − 97.7284i 29.7412 −144.597 11.0150i
96.20 −0.466290 −9.91982 −31.7826 23.6226i 4.62551 97.7284i 29.7412 −144.597 − 11.0150i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 96.40
Significant digits:
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Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 97.6.b.a ✓ 40
97.b even 2 1 inner 97.6.b.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
97.6.b.a ✓ 40 1.a even 1 1 trivial
97.6.b.a ✓ 40 97.b even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(97, [\chi])\).