Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [717,4,Mod(1,717)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(717, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("717.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 717 = 3 \cdot 239 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 717.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: | \(42.3043694741\) |
Analytic rank: | \(1\) |
Dimension: | \(28\) |
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 | −5.49015 | 3.00000 | 22.1417 | −8.06682 | −16.4704 | 32.6199 | −77.6401 | 9.00000 | 44.2880 | ||||||||||||||||||
1.2 | −5.29084 | 3.00000 | 19.9930 | −20.9477 | −15.8725 | −16.3120 | −63.4528 | 9.00000 | 110.831 | ||||||||||||||||||
1.3 | −5.18205 | 3.00000 | 18.8537 | 5.12080 | −15.5462 | 5.19788 | −56.2444 | 9.00000 | −26.5363 | ||||||||||||||||||
1.4 | −4.77227 | 3.00000 | 14.7746 | 17.3570 | −14.3168 | −36.1394 | −32.3302 | 9.00000 | −82.8322 | ||||||||||||||||||
1.5 | −4.22853 | 3.00000 | 9.88049 | −18.4854 | −12.6856 | 24.6174 | −7.95172 | 9.00000 | 78.1662 | ||||||||||||||||||
1.6 | −3.94640 | 3.00000 | 7.57406 | 3.12472 | −11.8392 | 13.3014 | 1.68092 | 9.00000 | −12.3314 | ||||||||||||||||||
1.7 | −3.72975 | 3.00000 | 5.91105 | 8.83024 | −11.1893 | −20.7717 | 7.79125 | 9.00000 | −32.9346 | ||||||||||||||||||
1.8 | −3.71471 | 3.00000 | 5.79908 | −14.6178 | −11.1441 | −2.58079 | 8.17580 | 9.00000 | 54.3008 | ||||||||||||||||||
1.9 | −3.06927 | 3.00000 | 1.42044 | 10.0975 | −9.20782 | 1.48241 | 20.1945 | 9.00000 | −30.9919 | ||||||||||||||||||
1.10 | −2.60026 | 3.00000 | −1.23865 | −11.0636 | −7.80078 | −16.8612 | 24.0229 | 9.00000 | 28.7682 | ||||||||||||||||||
1.11 | −2.03138 | 3.00000 | −3.87348 | 11.5727 | −6.09415 | −13.0852 | 24.1196 | 9.00000 | −23.5086 | ||||||||||||||||||
1.12 | −1.64745 | 3.00000 | −5.28592 | 5.57976 | −4.94234 | 35.6078 | 21.8878 | 9.00000 | −9.19235 | ||||||||||||||||||
1.13 | −1.27488 | 3.00000 | −6.37468 | −17.5425 | −3.82464 | −26.4099 | 18.3260 | 9.00000 | 22.3646 | ||||||||||||||||||
1.14 | −0.797577 | 3.00000 | −7.36387 | −7.16277 | −2.39273 | 0.569206 | 12.2539 | 9.00000 | 5.71286 | ||||||||||||||||||
1.15 | −0.608170 | 3.00000 | −7.63013 | −1.98658 | −1.82451 | −5.67520 | 9.50578 | 9.00000 | 1.20818 | ||||||||||||||||||
1.16 | 0.404541 | 3.00000 | −7.83635 | −0.184451 | 1.21362 | 12.9030 | −6.40644 | 9.00000 | −0.0746179 | ||||||||||||||||||
1.17 | 0.456628 | 3.00000 | −7.79149 | 14.2736 | 1.36988 | −13.0550 | −7.21084 | 9.00000 | 6.51773 | ||||||||||||||||||
1.18 | 0.686651 | 3.00000 | −7.52851 | −17.8875 | 2.05995 | 15.8088 | −10.6627 | 9.00000 | −12.2825 | ||||||||||||||||||
1.19 | 1.55579 | 3.00000 | −5.57952 | 17.3600 | 4.66737 | −1.16704 | −21.1269 | 9.00000 | 27.0085 | ||||||||||||||||||
1.20 | 2.01135 | 3.00000 | −3.95446 | −6.74089 | 6.03406 | 7.15060 | −24.0446 | 9.00000 | −13.5583 | ||||||||||||||||||
See all 28 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(239\) | \(1\) |
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 | 717.4.a.a | ✓ | 28 |
3.b | odd | 2 | 1 | 2151.4.a.c | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
717.4.a.a | ✓ | 28 | 1.a | even | 1 | 1 | trivial |
2151.4.a.c | 28 | 3.b | odd | 2 | 1 |