↑

Properties

Label 241.4.b.a
Level $241$
Weight $4$
Character orbit 241.b
Analytic conductor $14.219$
Analytic rank $0$
Dimension $60$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [241,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("241.240");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 241 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(14.2194603114\)
Analytic rank: \(0\)
Dimension: \(60\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 2 q^{2} + 250 q^{4} - 4 q^{5} - 22 q^{6} - 18 q^{8} + 540 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 2 q^{2} + 250 q^{4} - 4 q^{5} - 22 q^{6} - 18 q^{8} + 540 q^{9} + 90 q^{10} + 86 q^{12} + 4 q^{15} + 906 q^{16} + 126 q^{18} - 352 q^{20} + 2 q^{24} + 1364 q^{25} + 444 q^{27} - 176 q^{29} - 454 q^{30} - 286 q^{32} + 1488 q^{36} + 484 q^{40} - 1544 q^{41} - 704 q^{45} + 64 q^{47} - 1106 q^{48} - 3060 q^{49} - 690 q^{50} - 568 q^{53} + 2252 q^{54} + 1478 q^{58} + 92 q^{59} - 3030 q^{60} + 2360 q^{61} - 362 q^{64} - 1048 q^{67} - 810 q^{72} + 1128 q^{75} + 1216 q^{77} - 1868 q^{79} - 230 q^{80} + 2828 q^{81} - 66 q^{82} - 516 q^{83} - 88 q^{87} - 3690 q^{90} - 740 q^{91} + 4032 q^{94} - 574 q^{96} - 628 q^{97} - 4328 q^{98}+O(q^{100}) \) 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 −5.36366 −3.59088 20.7688 −13.5101 19.2602 − 31.6111i −68.4875 −14.1056 72.4637
240.2 −5.36366 −3.59088 20.7688 −13.5101 19.2602 31.6111i −68.4875 −14.1056 72.4637
240.3 −5.23552 6.75835 19.4106 3.14536 −35.3835 − 14.6001i −59.7406 18.6753 −16.4676
240.4 −5.23552 6.75835 19.4106 3.14536 −35.3835 14.6001i −59.7406 18.6753 −16.4676
240.5 −4.94173 −0.922070 16.4207 14.9069 4.55662 − 14.3104i −41.6126 −26.1498 −73.6658
240.6 −4.94173 −0.922070 16.4207 14.9069 4.55662 14.3104i −41.6126 −26.1498 −73.6658
240.7 −4.72573 −8.64489 14.3325 4.14724 40.8534 − 15.2635i −29.9258 47.7341 −19.5988
240.8 −4.72573 −8.64489 14.3325 4.14724 40.8534 15.2635i −29.9258 47.7341 −19.5988
240.9 −4.37268 7.53609 11.1203 −19.0827 −32.9529 − 26.4165i −13.6440 29.7926 83.4425
240.10 −4.37268 7.53609 11.1203 −19.0827 −32.9529 26.4165i −13.6440 29.7926 83.4425
240.11 −3.99746 0.395539 7.97969 −7.68030 −1.58115 − 2.04074i 0.0811834 −26.8435 30.7017
240.12 −3.99746 0.395539 7.97969 −7.68030 −1.58115 2.04074i 0.0811834 −26.8435 30.7017
240.13 −3.55806 8.17862 4.65976 14.1923 −29.1000 − 18.9980i 11.8848 39.8898 −50.4971
240.14 −3.55806 8.17862 4.65976 14.1923 −29.1000 18.9980i 11.8848 39.8898 −50.4971
240.15 −3.09145 −7.42815 1.55708 −16.1808 22.9638 − 5.06123i 19.9180 28.1775 50.0223
240.16 −3.09145 −7.42815 1.55708 −16.1808 22.9638 5.06123i 19.9180 28.1775 50.0223
240.17 −2.99074 3.33857 0.944545 1.09828 −9.98481 − 24.4755i 21.1011 −15.8539 −3.28467
240.18 −2.99074 3.33857 0.944545 1.09828 −9.98481 24.4755i 21.1011 −15.8539 −3.28467
240.19 −2.94735 −5.22531 0.686849 14.2133 15.4008 − 22.2246i 21.5544 0.303818 −41.8915
240.20 −2.94735 −5.22531 0.686849 14.2133 15.4008 22.2246i 21.5544 0.303818 −41.8915
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 240.60
Significant digits:
Format:

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 241.4.b.a ✓ 60
241.b even 2 1 inner 241.4.b.a ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
241.4.b.a ✓ 60 1.a even 1 1 trivial
241.4.b.a ✓ 60 241.b even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(241, [\chi])\).