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Properties

Label 141.9.d.a
Level $141$
Weight $9$
Character orbit 141.d
Analytic conductor $57.440$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [141,9,Mod(46,141)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(141, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("141.46");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 141 = 3 \cdot 47 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 141.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(57.4403840186\)
Analytic rank: \(0\)
Dimension: \(64\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 8668 q^{4} - 2268 q^{6} + 4380 q^{7} + 17460 q^{8} + 139968 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 8668 q^{4} - 2268 q^{6} + 4380 q^{7} + 17460 q^{8} + 139968 q^{9} + 22680 q^{12} + 234828 q^{14} + 1146940 q^{16} + 23964 q^{17} - 137376 q^{24} - 4253372 q^{25} + 3659640 q^{28} + 6890220 q^{32} + 879072 q^{34} + 18956916 q^{36} + 2665092 q^{37} + 1679940 q^{42} + 4746228 q^{47} + 5806080 q^{48} + 42188644 q^{49} - 54716820 q^{50} - 17461332 q^{51} + 29865924 q^{53} - 4960116 q^{54} + 86943444 q^{55} + 82860360 q^{56} + 9007740 q^{59} + 25723776 q^{61} + 9579060 q^{63} + 128236396 q^{64} - 54261396 q^{65} - 39320088 q^{68} + 32343432 q^{71} + 38185020 q^{72} + 86323788 q^{74} - 64264104 q^{75} + 53614860 q^{79} + 306110016 q^{81} - 372504516 q^{83} + 4906656 q^{84} + 5752368 q^{89} - 392031636 q^{94} - 521065128 q^{95} - 233559288 q^{96} + 278551068 q^{97} - 191674284 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1 −30.3645 46.7654 666.003 704.918i −1420.01 1830.08 −12449.5 2187.00 − 21404.5i
46.2 −30.3645 46.7654 666.003 − 704.918i −1420.01 1830.08 −12449.5 2187.00 21404.5i
46.3 −30.3503 −46.7654 665.142 − 381.257i 1419.34 −1035.23 −12417.6 2187.00 11571.3i
46.4 −30.3503 −46.7654 665.142 381.257i 1419.34 −1035.23 −12417.6 2187.00 − 11571.3i
46.5 −27.7396 46.7654 513.487 194.593i −1297.25 −3761.43 −7142.61 2187.00 − 5397.94i
46.6 −27.7396 46.7654 513.487 − 194.593i −1297.25 −3761.43 −7142.61 2187.00 5397.94i
46.7 −25.2987 −46.7654 384.022 871.963i 1183.10 1170.30 −3238.80 2187.00 − 22059.5i
46.8 −25.2987 −46.7654 384.022 − 871.963i 1183.10 1170.30 −3238.80 2187.00 22059.5i
46.9 −22.7890 −46.7654 263.340 − 247.469i 1065.74 3685.69 −167.264 2187.00 5639.57i
46.10 −22.7890 −46.7654 263.340 247.469i 1065.74 3685.69 −167.264 2187.00 − 5639.57i
46.11 −22.7270 46.7654 260.518 341.222i −1062.84 644.296 −102.687 2187.00 − 7754.97i
46.12 −22.7270 46.7654 260.518 − 341.222i −1062.84 644.296 −102.687 2187.00 7754.97i
46.13 −22.4671 46.7654 248.772 1130.17i −1050.68 2719.09 162.395 2187.00 − 25391.6i
46.14 −22.4671 46.7654 248.772 − 1130.17i −1050.68 2719.09 162.395 2187.00 25391.6i
46.15 −20.4680 −46.7654 162.938 − 263.805i 957.192 −4274.50 1904.80 2187.00 5399.55i
46.16 −20.4680 −46.7654 162.938 263.805i 957.192 −4274.50 1904.80 2187.00 − 5399.55i
46.17 −17.9010 46.7654 64.4453 − 916.835i −837.146 −3333.31 3429.02 2187.00 16412.3i
46.18 −17.9010 46.7654 64.4453 916.835i −837.146 −3333.31 3429.02 2187.00 − 16412.3i
46.19 −13.9280 −46.7654 −62.0095 − 905.175i 651.350 −562.835 4429.25 2187.00 12607.3i
46.20 −13.9280 −46.7654 −62.0095 905.175i 651.350 −562.835 4429.25 2187.00 − 12607.3i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.64
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 141.9.d.a ✓ 64
47.b odd 2 1 inner 141.9.d.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
141.9.d.a ✓ 64 1.a even 1 1 trivial
141.9.d.a ✓ 64 47.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(141, [\chi])\).