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Properties

Label 241.6.b.a
Level $241$
Weight $6$
Character orbit 241.b
Analytic conductor $38.653$
Analytic rank $0$
Dimension $100$
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] = [241,6,Mod(240,241)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(241, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("241.240");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 241 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 241.b (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: \(38.6525005749\)
Analytic rank: \(0\)
Dimension: \(100\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 100 q - 2 q^{2} - 6 q^{3} + 1578 q^{4} + 32 q^{5} - 406 q^{6} - 66 q^{8} + 8382 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 100 q - 2 q^{2} - 6 q^{3} + 1578 q^{4} + 32 q^{5} - 406 q^{6} - 66 q^{8} + 8382 q^{9} - 1158 q^{10} - 1834 q^{12} - 698 q^{15} + 23674 q^{16} - 3282 q^{18} + 3536 q^{20} - 16222 q^{24} + 63128 q^{25} + 354 q^{27} - 13472 q^{29} - 20350 q^{30} + 13586 q^{32} + 146832 q^{36} - 65964 q^{40} - 6746 q^{41} + 11410 q^{45} - 2516 q^{47} + 60430 q^{48} - 217658 q^{49} + 58686 q^{50} - 11434 q^{53} - 127996 q^{54} - 32314 q^{58} + 81476 q^{59} - 95334 q^{60} + 28116 q^{61} + 444870 q^{64} + 71780 q^{67} - 5418 q^{72} - 294384 q^{75} + 88486 q^{77} + 386602 q^{79} - 98726 q^{80} + 706868 q^{81} + 1094 q^{82} + 164244 q^{83} - 91756 q^{87} - 335490 q^{90} + 412900 q^{91} - 448088 q^{94} - 310990 q^{96} + 183998 q^{97} + 397288 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
240.1 −10.9592 −0.358981 88.1050 48.3353 3.93416 − 184.095i −614.868 −242.871 −529.718
240.2 −10.9592 −0.358981 88.1050 48.3353 3.93416 184.095i −614.868 −242.871 −529.718
240.3 −10.4365 28.4483 76.9204 1.57259 −296.900 231.955i −468.811 566.305 −16.4124
240.4 −10.4365 28.4483 76.9204 1.57259 −296.900 − 231.955i −468.811 566.305 −16.4124
240.5 −10.4191 −21.3579 76.5573 −37.6384 222.530 45.4470i −464.247 213.161 392.157
240.6 −10.4191 −21.3579 76.5573 −37.6384 222.530 − 45.4470i −464.247 213.161 392.157
240.7 −10.3085 11.9800 74.2659 −75.5387 −123.496 38.7253i −435.700 −99.4793 778.693
240.8 −10.3085 11.9800 74.2659 −75.5387 −123.496 − 38.7253i −435.700 −99.4793 778.693
240.9 −9.65754 −24.8947 61.2681 87.4411 240.422 − 156.878i −282.658 376.746 −844.466
240.10 −9.65754 −24.8947 61.2681 87.4411 240.422 156.878i −282.658 376.746 −844.466
240.11 −9.45040 17.4801 57.3100 91.1260 −165.194 34.8903i −239.190 62.5547 −861.177
240.12 −9.45040 17.4801 57.3100 91.1260 −165.194 − 34.8903i −239.190 62.5547 −861.177
240.13 −8.97550 −4.19240 48.5597 4.20038 37.6289 − 91.8949i −148.632 −225.424 −37.7005
240.14 −8.97550 −4.19240 48.5597 4.20038 37.6289 91.8949i −148.632 −225.424 −37.7005
240.15 −8.19134 −14.1407 35.0980 −99.2355 115.831 167.657i −25.3768 −43.0413 812.871
240.16 −8.19134 −14.1407 35.0980 −99.2355 115.831 − 167.657i −25.3768 −43.0413 812.871
240.17 −7.87460 18.5063 30.0093 16.2350 −145.730 28.8753i 15.6760 99.4824 −127.844
240.18 −7.87460 18.5063 30.0093 16.2350 −145.730 − 28.8753i 15.6760 99.4824 −127.844
240.19 −7.67714 4.25657 26.9385 −41.5977 −32.6783 − 228.026i 38.8577 −224.882 319.351
240.20 −7.67714 4.25657 26.9385 −41.5977 −32.6783 228.026i 38.8577 −224.882 319.351
See all 100 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 240.100
Significant digits:
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Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 241.6.b.a ✓ 100
241.b even 2 1 inner 241.6.b.a ✓ 100
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
241.6.b.a ✓ 100 1.a even 1 1 trivial
241.6.b.a ✓ 100 241.b even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(241, [\chi])\).