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Properties

Label 120.4.m.b
Level $120$
Weight $4$
Character orbit 120.m
Analytic conductor $7.080$
Analytic rank $0$
Dimension $64$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [120,4,Mod(59,120)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(120, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("120.59");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 120.m (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: \(7.08022920069\)
Analytic rank: \(0\)
Dimension: \(64\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 32 q^{4} - 72 q^{6} - 112 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 32 q^{4} - 72 q^{6} - 112 q^{9} - 72 q^{10} - 128 q^{16} + 672 q^{19} - 416 q^{24} + 496 q^{25} - 248 q^{30} + 240 q^{34} - 608 q^{36} - 1344 q^{40} + 336 q^{46} + 3520 q^{49} - 544 q^{51} - 952 q^{54} - 2064 q^{60} + 2176 q^{64} - 176 q^{66} + 672 q^{70} - 1600 q^{75} + 2304 q^{76} - 2304 q^{81} - 736 q^{84} - 1432 q^{90} - 2752 q^{91} + 4496 q^{94} + 640 q^{96} + 256 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1 −2.80496 − 0.363625i 3.73183 − 3.61572i 7.73555 + 2.03990i 2.66357 − 10.8584i −11.7824 + 8.78495i 26.3409 −20.9561 − 8.53468i 0.853113 − 26.9865i −11.4196 + 29.4889i
59.2 −2.80496 − 0.363625i 3.73183 + 3.61572i 7.73555 + 2.03990i −2.66357 − 10.8584i −9.15285 − 11.4989i −26.3409 −20.9561 − 8.53468i 0.853113 + 26.9865i 3.52280 + 31.4259i
59.3 −2.80496 + 0.363625i 3.73183 − 3.61572i 7.73555 − 2.03990i −2.66357 + 10.8584i −9.15285 + 11.4989i −26.3409 −20.9561 + 8.53468i 0.853113 − 26.9865i 3.52280 − 31.4259i
59.4 −2.80496 + 0.363625i 3.73183 + 3.61572i 7.73555 − 2.03990i 2.66357 + 10.8584i −11.7824 − 8.78495i 26.3409 −20.9561 + 8.53468i 0.853113 + 26.9865i −11.4196 − 29.4889i
59.5 −2.73183 − 0.732884i −2.64167 − 4.47455i 6.92576 + 4.00422i −10.4671 − 3.92924i 3.93726 + 14.1597i −4.10158 −15.9854 − 16.0146i −13.0432 + 23.6406i 25.7147 + 18.4052i
59.6 −2.73183 − 0.732884i −2.64167 + 4.47455i 6.92576 + 4.00422i 10.4671 − 3.92924i 10.4959 − 10.2877i 4.10158 −15.9854 − 16.0146i −13.0432 − 23.6406i −31.4741 + 3.06281i
59.7 −2.73183 + 0.732884i −2.64167 − 4.47455i 6.92576 − 4.00422i 10.4671 + 3.92924i 10.4959 + 10.2877i 4.10158 −15.9854 + 16.0146i −13.0432 + 23.6406i −31.4741 − 3.06281i
59.8 −2.73183 + 0.732884i −2.64167 + 4.47455i 6.92576 − 4.00422i −10.4671 + 3.92924i 3.93726 − 14.1597i −4.10158 −15.9854 + 16.0146i −13.0432 − 23.6406i 25.7147 − 18.4052i
59.9 −2.22835 − 1.74197i 4.99538 − 1.43044i 1.93108 + 7.76344i −10.2117 + 4.55196i −13.6232 − 5.51430i 3.90466 9.22055 − 20.6635i 22.9077 − 14.2912i 30.6847 + 7.64520i
59.10 −2.22835 − 1.74197i 4.99538 + 1.43044i 1.93108 + 7.76344i 10.2117 + 4.55196i −8.63968 − 11.8893i −3.90466 9.22055 − 20.6635i 22.9077 + 14.2912i −14.8260 − 27.9319i
59.11 −2.22835 + 1.74197i 4.99538 − 1.43044i 1.93108 − 7.76344i 10.2117 − 4.55196i −8.63968 + 11.8893i −3.90466 9.22055 + 20.6635i 22.9077 − 14.2912i −14.8260 + 27.9319i
59.12 −2.22835 + 1.74197i 4.99538 + 1.43044i 1.93108 − 7.76344i −10.2117 − 4.55196i −13.6232 + 5.51430i 3.90466 9.22055 + 20.6635i 22.9077 + 14.2912i 30.6847 − 7.64520i
59.13 −2.12434 − 1.86740i 0.659128 − 5.15418i 1.02565 + 7.93398i 10.6502 − 3.40199i −11.0251 + 9.71838i −28.2468 12.6371 − 18.7698i −26.1311 − 6.79452i −28.9775 − 12.6612i
59.14 −2.12434 − 1.86740i 0.659128 + 5.15418i 1.02565 + 7.93398i −10.6502 − 3.40199i 8.22469 − 12.1801i 28.2468 12.6371 − 18.7698i −26.1311 + 6.79452i 16.2718 + 27.1151i
59.15 −2.12434 + 1.86740i 0.659128 − 5.15418i 1.02565 − 7.93398i −10.6502 + 3.40199i 8.22469 + 12.1801i 28.2468 12.6371 + 18.7698i −26.1311 − 6.79452i 16.2718 − 27.1151i
59.16 −2.12434 + 1.86740i 0.659128 + 5.15418i 1.02565 − 7.93398i 10.6502 + 3.40199i −11.0251 − 9.71838i −28.2468 12.6371 + 18.7698i −26.1311 + 6.79452i −28.9775 + 12.6612i
59.17 −1.65669 − 2.29246i −4.82386 − 1.93142i −2.51074 + 7.59580i 7.20740 − 8.54713i 3.56395 + 14.2583i 30.0545 21.5726 − 6.82812i 19.5392 + 18.6338i −31.5344 − 2.36271i
59.18 −1.65669 − 2.29246i −4.82386 + 1.93142i −2.51074 + 7.59580i −7.20740 − 8.54713i 12.4193 + 7.85874i −30.0545 21.5726 − 6.82812i 19.5392 − 18.6338i −7.65352 + 30.6826i
59.19 −1.65669 + 2.29246i −4.82386 − 1.93142i −2.51074 − 7.59580i −7.20740 + 8.54713i 12.4193 − 7.85874i −30.0545 21.5726 + 6.82812i 19.5392 + 18.6338i −7.65352 − 30.6826i
59.20 −1.65669 + 2.29246i −4.82386 + 1.93142i −2.51074 − 7.59580i 7.20740 + 8.54713i 3.56395 − 14.2583i 30.0545 21.5726 + 6.82812i 19.5392 − 18.6338i −31.5344 + 2.36271i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
8.d odd 2 1 inner
15.d odd 2 1 inner
24.f even 2 1 inner
40.e odd 2 1 inner
120.m even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 120.4.m.b ✓ 64
3.b odd 2 1 inner 120.4.m.b ✓ 64
4.b odd 2 1 480.4.m.b 64
5.b even 2 1 inner 120.4.m.b ✓ 64
8.b even 2 1 480.4.m.b 64
8.d odd 2 1 inner 120.4.m.b ✓ 64
12.b even 2 1 480.4.m.b 64
15.d odd 2 1 inner 120.4.m.b ✓ 64
20.d odd 2 1 480.4.m.b 64
24.f even 2 1 inner 120.4.m.b ✓ 64
24.h odd 2 1 480.4.m.b 64
40.e odd 2 1 inner 120.4.m.b ✓ 64
40.f even 2 1 480.4.m.b 64
60.h even 2 1 480.4.m.b 64
120.i odd 2 1 480.4.m.b 64
120.m even 2 1 inner 120.4.m.b ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.m.b ✓ 64 1.a even 1 1 trivial
120.4.m.b ✓ 64 3.b odd 2 1 inner
120.4.m.b ✓ 64 5.b even 2 1 inner
120.4.m.b ✓ 64 8.d odd 2 1 inner
120.4.m.b ✓ 64 15.d odd 2 1 inner
120.4.m.b ✓ 64 24.f even 2 1 inner
120.4.m.b ✓ 64 40.e odd 2 1 inner
120.4.m.b ✓ 64 120.m even 2 1 inner
480.4.m.b 64 4.b odd 2 1
480.4.m.b 64 8.b even 2 1
480.4.m.b 64 12.b even 2 1
480.4.m.b 64 20.d odd 2 1
480.4.m.b 64 24.h odd 2 1
480.4.m.b 64 40.f even 2 1
480.4.m.b 64 60.h even 2 1
480.4.m.b 64 120.i odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} - 3184 T_{7}^{14} + 3946344 T_{7}^{12} - 2379623680 T_{7}^{10} + 707003826880 T_{7}^{8} + \cdots + 31\!\cdots\!00 \) acting on \(S_{4}^{\mathrm{new}}(120, [\chi])\). Copy content Toggle raw display