↑

Properties

Label 241.4.a.a
Level $241$
Weight $4$
Character orbit 241.a
Self dual yes
Analytic conductor $14.219$
Analytic rank $1$
Dimension $27$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [241,4,Mod(1,241)]
 
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([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("241.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 241 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 241.a (trivial)

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: yes
Analytic conductor: \(14.2194603114\)
Analytic rank: \(1\)
Dimension: \(27\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 27 q - 11 q^{2} - 14 q^{3} + 89 q^{4} - 50 q^{5} - 63 q^{6} - 68 q^{7} - 129 q^{8} + 135 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 27 q - 11 q^{2} - 14 q^{3} + 89 q^{4} - 50 q^{5} - 63 q^{6} - 68 q^{7} - 129 q^{8} + 135 q^{9} - 79 q^{10} - 298 q^{11} - 125 q^{12} - 106 q^{13} - 284 q^{14} - 358 q^{15} + 117 q^{16} - 132 q^{17} - 405 q^{18} - 216 q^{19} - 570 q^{20} - 216 q^{21} - 38 q^{22} - 814 q^{23} - 873 q^{24} + 371 q^{25} - 608 q^{26} - 542 q^{27} - 575 q^{28} - 736 q^{29} - 155 q^{30} - 994 q^{31} - 1293 q^{32} - 248 q^{33} - 605 q^{34} - 1402 q^{35} + 44 q^{36} - 472 q^{37} - 200 q^{38} - 2086 q^{39} - 396 q^{40} - 1318 q^{41} - 939 q^{42} - 974 q^{43} - 2932 q^{44} - 1988 q^{45} - 553 q^{46} - 2012 q^{47} - 275 q^{48} + 443 q^{49} - 1713 q^{50} - 1564 q^{51} - 1617 q^{52} - 980 q^{53} - 2334 q^{54} - 1142 q^{55} - 3681 q^{56} - 2296 q^{57} - 83 q^{58} - 2590 q^{59} - 2423 q^{60} - 832 q^{61} + 1595 q^{62} - 420 q^{63} + 2783 q^{64} - 980 q^{65} + 4174 q^{66} - 132 q^{67} + 3331 q^{68} + 166 q^{69} + 6584 q^{70} - 5996 q^{71} + 2033 q^{72} + 932 q^{73} + 508 q^{74} + 3058 q^{75} + 3498 q^{76} - 732 q^{77} + 11548 q^{78} - 1666 q^{79} + 775 q^{80} + 2419 q^{81} + 3087 q^{82} + 878 q^{83} + 10226 q^{84} + 4330 q^{85} - 311 q^{86} + 2352 q^{87} + 7740 q^{88} - 2344 q^{89} + 14849 q^{90} + 144 q^{91} - 2496 q^{92} + 386 q^{93} + 10994 q^{94} - 4626 q^{95} + 3739 q^{96} + 1234 q^{97} + 6574 q^{98} - 3488 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −5.59631 −3.12024 23.3186 8.77168 17.4618 9.90189 −85.7278 −17.2641 −49.0890
1.2 −5.21700 9.42536 19.2171 −17.8519 −49.1722 19.3015 −58.5198 61.8375 93.1336
1.3 −4.87301 −4.30023 15.7463 −19.3755 20.9551 9.98357 −37.7476 −8.50805 94.4170
1.4 −4.64628 3.53512 13.5879 4.48471 −16.4252 6.79583 −25.9629 −14.5029 −20.8372
1.5 −4.28434 −6.33964 10.3556 10.6688 27.1611 −35.4898 −10.0920 13.1910 −45.7088
1.6 −4.14836 2.24806 9.20886 21.1319 −9.32575 −15.0304 −5.01477 −21.9462 −87.6627
1.7 −3.18179 4.58295 2.12378 −14.4646 −14.5820 12.2833 18.6969 −5.99653 46.0234
1.8 −2.84018 8.62520 0.0666386 −7.97639 −24.4972 −22.2917 22.5322 47.3942 22.6544
1.9 −2.63940 −6.71827 −1.03357 −8.36175 17.7322 −2.54579 23.8432 18.1352 22.0700
1.10 −2.58300 0.0893746 −1.32809 6.47115 −0.230855 −8.59546 24.0945 −26.9920 −16.7150
1.11 −2.40822 −10.0449 −2.20046 −12.6631 24.1903 −2.75837 24.5650 73.8996 30.4955
1.12 −1.15052 −0.498406 −6.67631 4.77789 0.573425 19.5308 16.8854 −26.7516 −5.49706
1.13 −1.10812 6.51375 −6.77208 8.81779 −7.21799 −21.1914 16.3692 15.4289 −9.77114
1.14 −0.811481 −5.90200 −7.34150 −7.73839 4.78936 32.9464 12.4493 7.83362 6.27956
1.15 −0.462625 −8.37696 −7.78598 11.3790 3.87539 −4.75271 7.30299 43.1735 −5.26423
1.16 0.152600 3.98082 −7.97671 −2.58440 0.607471 15.5864 −2.43804 −11.1531 −0.394379
1.17 1.23918 6.96372 −6.46444 −6.97348 8.62929 −12.0894 −17.9240 21.4934 −8.64138
1.18 1.69085 0.0833126 −5.14104 6.44616 0.140869 −0.0535401 −22.2195 −26.9931 10.8995
1.19 2.38301 −1.61433 −2.32128 18.2573 −3.84696 −23.9963 −24.5957 −24.3939 43.5074
1.20 2.85644 −9.32237 0.159225 8.64408 −26.6287 22.6615 −22.3967 59.9066 24.6913
See all 27 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.27
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(241\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 241.4.a.a ✓ 27
3.b odd 2 1 2169.4.a.b 27
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
241.4.a.a ✓ 27 1.a even 1 1 trivial
2169.4.a.b 27 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{27} + 11 T_{2}^{26} - 92 T_{2}^{25} - 1354 T_{2}^{24} + 2703 T_{2}^{23} + 72157 T_{2}^{22} + \cdots + 17638992000 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(241))\). Copy content Toggle raw display