Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2015,4,Mod(1,2015)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2015, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2015.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 2015 = 5 \cdot 13 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 2015.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: | \(118.888848662\) |
Analytic rank: | \(0\) |
Dimension: | \(52\) |
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 | −5.51365 | −4.09405 | 22.4003 | 5.00000 | 22.5731 | 1.36887 | −79.3983 | −10.2388 | −27.5682 | ||||||||||||||||||
1.2 | −5.25783 | 1.11962 | 19.6447 | 5.00000 | −5.88679 | −8.36677 | −61.2259 | −25.7464 | −26.2891 | ||||||||||||||||||
1.3 | −5.24891 | 7.80149 | 19.5510 | 5.00000 | −40.9493 | 21.7487 | −60.6303 | 33.8632 | −26.2445 | ||||||||||||||||||
1.4 | −5.22415 | −3.79694 | 19.2918 | 5.00000 | 19.8358 | −29.2235 | −58.9898 | −12.5832 | −26.1208 | ||||||||||||||||||
1.5 | −4.92290 | 9.95497 | 16.2350 | 5.00000 | −49.0074 | −22.8992 | −40.5400 | 72.1015 | −24.6145 | ||||||||||||||||||
1.6 | −4.86800 | −10.2643 | 15.6974 | 5.00000 | 49.9665 | −2.22019 | −37.4712 | 78.3552 | −24.3400 | ||||||||||||||||||
1.7 | −4.77134 | 3.82224 | 14.7657 | 5.00000 | −18.2372 | −13.6823 | −32.2815 | −12.3905 | −23.8567 | ||||||||||||||||||
1.8 | −4.50504 | 7.24796 | 12.2954 | 5.00000 | −32.6524 | 28.1105 | −19.3510 | 25.5330 | −22.5252 | ||||||||||||||||||
1.9 | −4.02110 | −2.28061 | 8.16921 | 5.00000 | 9.17056 | 9.19591 | −0.680393 | −21.7988 | −20.1055 | ||||||||||||||||||
1.10 | −3.99913 | 2.07865 | 7.99306 | 5.00000 | −8.31281 | 10.0423 | 0.0277684 | −22.6792 | −19.9957 | ||||||||||||||||||
1.11 | −3.86222 | −7.81527 | 6.91678 | 5.00000 | 30.1843 | −18.3433 | 4.18364 | 34.0785 | −19.3111 | ||||||||||||||||||
1.12 | −3.72936 | 2.09223 | 5.90810 | 5.00000 | −7.80269 | −24.8983 | 7.80144 | −22.6226 | −18.6468 | ||||||||||||||||||
1.13 | −3.30845 | −6.11978 | 2.94581 | 5.00000 | 20.2469 | 26.7005 | 16.7215 | 10.4516 | −16.5422 | ||||||||||||||||||
1.14 | −3.08411 | −7.42027 | 1.51173 | 5.00000 | 22.8849 | 31.5045 | 20.0105 | 28.0605 | −15.4205 | ||||||||||||||||||
1.15 | −3.01441 | 5.87533 | 1.08664 | 5.00000 | −17.7106 | −32.1337 | 20.8397 | 7.51948 | −15.0720 | ||||||||||||||||||
1.16 | −2.79569 | −6.84932 | −0.184097 | 5.00000 | 19.1486 | 10.4445 | 22.8802 | 19.9132 | −13.9785 | ||||||||||||||||||
1.17 | −2.56347 | 5.79548 | −1.42863 | 5.00000 | −14.8565 | 11.0247 | 24.1700 | 6.58758 | −12.8173 | ||||||||||||||||||
1.18 | −2.02907 | 8.91392 | −3.88288 | 5.00000 | −18.0870 | 28.9989 | 24.1112 | 52.4580 | −10.1453 | ||||||||||||||||||
1.19 | −1.94767 | −0.204973 | −4.20660 | 5.00000 | 0.399220 | 7.04965 | 23.7744 | −26.9580 | −9.73833 | ||||||||||||||||||
1.20 | −1.68859 | 8.60461 | −5.14865 | 5.00000 | −14.5297 | −1.62336 | 22.2027 | 47.0394 | −8.44297 | ||||||||||||||||||
See all 52 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(5\) | \(-1\) |
\(13\) | \(1\) |
\(31\) | \(-1\) |
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 | 2015.4.a.g | ✓ | 52 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2015.4.a.g | ✓ | 52 | 1.a | even | 1 | 1 | trivial |