Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [241,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("241.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 241 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
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: | \(38.6525005749\) |
Analytic rank: | \(1\) |
Dimension: | \(47\) |
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.
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 | −10.7206 | 11.5622 | 82.9310 | −65.4773 | −123.953 | 98.2220 | −546.010 | −109.317 | 701.956 | ||||||||||||||||||
1.2 | −10.2895 | 26.3385 | 73.8735 | −43.0235 | −271.010 | −187.960 | −430.856 | 450.718 | 442.690 | ||||||||||||||||||
1.3 | −10.2411 | −23.2816 | 72.8806 | −9.01991 | 238.430 | 168.636 | −418.664 | 299.033 | 92.3741 | ||||||||||||||||||
1.4 | −9.93843 | −19.3294 | 66.7723 | −104.407 | 192.103 | −127.795 | −345.582 | 130.624 | 1037.64 | ||||||||||||||||||
1.5 | −9.92645 | −12.6407 | 66.5344 | 62.3940 | 125.477 | 188.856 | −342.804 | −83.2134 | −619.351 | ||||||||||||||||||
1.6 | −9.63170 | 21.5246 | 60.7697 | 21.9387 | −207.319 | −64.7970 | −277.101 | 220.310 | −211.307 | ||||||||||||||||||
1.7 | −9.03131 | −24.7487 | 49.5645 | 88.0875 | 223.513 | −61.1400 | −158.630 | 369.498 | −795.546 | ||||||||||||||||||
1.8 | −8.56012 | −10.4236 | 41.2757 | −41.0951 | 89.2271 | −137.755 | −79.4012 | −134.349 | 351.779 | ||||||||||||||||||
1.9 | −8.38800 | 3.30188 | 38.3585 | −47.3447 | −27.6961 | 224.163 | −53.3350 | −232.098 | 397.127 | ||||||||||||||||||
1.10 | −7.94385 | 22.5005 | 31.1048 | 46.4850 | −178.741 | 51.0662 | 7.11130 | 263.273 | −369.270 | ||||||||||||||||||
1.11 | −7.40336 | 13.2418 | 22.8098 | 34.5473 | −98.0338 | −151.594 | 68.0385 | −67.6549 | −255.767 | ||||||||||||||||||
1.12 | −7.15052 | −11.7753 | 19.1300 | 36.8177 | 84.1996 | −10.3585 | 92.0272 | −104.342 | −263.266 | ||||||||||||||||||
1.13 | −6.39218 | 3.09349 | 8.85997 | −97.8228 | −19.7741 | −190.960 | 147.915 | −233.430 | 625.301 | ||||||||||||||||||
1.14 | −6.09017 | 5.61699 | 5.09012 | 70.8471 | −34.2084 | 179.955 | 163.886 | −211.449 | −431.471 | ||||||||||||||||||
1.15 | −5.45931 | −11.8068 | −2.19591 | 83.2099 | 64.4571 | −5.76248 | 186.686 | −103.599 | −454.269 | ||||||||||||||||||
1.16 | −3.90261 | 16.1434 | −16.7696 | −47.7344 | −63.0012 | 130.876 | 190.329 | 17.6081 | 186.289 | ||||||||||||||||||
1.17 | −3.85750 | −23.7750 | −17.1197 | −53.9863 | 91.7120 | −208.479 | 189.479 | 322.250 | 208.252 | ||||||||||||||||||
1.18 | −3.57364 | −12.1250 | −19.2291 | −91.1682 | 43.3305 | 66.0046 | 183.074 | −95.9836 | 325.802 | ||||||||||||||||||
1.19 | −3.36464 | 21.4437 | −20.6792 | −7.37461 | −72.1504 | −58.5773 | 177.247 | 216.832 | 24.8130 | ||||||||||||||||||
1.20 | −2.85211 | 16.8003 | −23.8655 | −2.03317 | −47.9163 | −89.3811 | 159.334 | 39.2509 | 5.79881 | ||||||||||||||||||
See all 47 embeddings |
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 | 241.6.a.a | ✓ | 47 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
241.6.a.a | ✓ | 47 | 1.a | even | 1 | 1 | trivial |