Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [77,4,Mod(15,77)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(77, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("77.15");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 77 = 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 77.f (of order \(5\), degree \(4\), 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: | \(4.54314707044\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(10\) over \(\Q(\zeta_{5})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
15.1 | −4.23611 | − | 3.07771i | −2.01219 | + | 6.19288i | 6.00015 | + | 18.4666i | −14.2597 | + | 10.3603i | 27.5837 | − | 20.0407i | −2.16312 | − | 6.65740i | 18.4731 | − | 56.8544i | −12.4593 | − | 9.05225i | 92.2919 | ||
15.2 | −3.13241 | − | 2.27583i | 2.57135 | − | 7.91381i | 2.16046 | + | 6.64921i | −14.0082 | + | 10.1776i | −26.0650 | + | 18.9373i | −2.16312 | − | 6.65740i | −1.20677 | + | 3.71406i | −34.1731 | − | 24.8282i | 67.0418 | ||
15.3 | −2.08555 | − | 1.51524i | −1.27325 | + | 3.91867i | −0.418581 | − | 1.28826i | 0.733295 | − | 0.532770i | 8.59314 | − | 6.24328i | −2.16312 | − | 6.65740i | −7.45191 | + | 22.9346i | 8.10867 | + | 5.89129i | −2.33659 | ||
15.4 | −1.91048 | − | 1.38804i | 1.19654 | − | 3.68258i | −0.748877 | − | 2.30480i | 12.0553 | − | 8.75871i | −7.39754 | + | 5.37463i | −2.16312 | − | 6.65740i | −7.60635 | + | 23.4100i | 9.71380 | + | 7.05749i | −35.1889 | ||
15.5 | −0.281932 | − | 0.204836i | −2.51790 | + | 7.74930i | −2.43461 | − | 7.49295i | −4.21014 | + | 3.05884i | 2.29721 | − | 1.66902i | −2.16312 | − | 6.65740i | −1.70994 | + | 5.26265i | −31.8684 | − | 23.1538i | 1.81354 | ||
15.6 | 1.06715 | + | 0.775331i | 2.12611 | − | 6.54350i | −1.93446 | − | 5.95366i | −8.06046 | + | 5.85627i | 7.34226 | − | 5.33447i | −2.16312 | − | 6.65740i | 5.81262 | − | 17.8894i | −16.4536 | − | 11.9543i | −13.1423 | ||
15.7 | 2.07504 | + | 1.50761i | 0.186723 | − | 0.574674i | −0.439213 | − | 1.35176i | 11.4299 | − | 8.30432i | 1.25384 | − | 0.910969i | −2.16312 | − | 6.65740i | 7.46730 | − | 22.9820i | 21.5481 | + | 15.6556i | 36.2372 | ||
15.8 | 3.62073 | + | 2.63062i | −0.489110 | + | 1.50533i | 3.71743 | + | 11.4411i | −16.0185 | + | 11.6381i | −5.73088 | + | 4.16373i | −2.16312 | − | 6.65740i | −5.57330 | + | 17.1529i | 19.8167 | + | 14.3977i | −88.6140 | ||
15.9 | 3.84003 | + | 2.78994i | −2.63098 | + | 8.09734i | 4.48990 | + | 13.8185i | 11.7384 | − | 8.52841i | −32.6942 | + | 23.7537i | −2.16312 | − | 6.65740i | −9.57738 | + | 29.4761i | −36.8013 | − | 26.7377i | 68.8694 | ||
15.10 | 4.16155 | + | 3.02354i | 2.81484 | − | 8.66318i | 5.70454 | + | 17.5568i | 1.18370 | − | 0.860011i | 37.9076 | − | 27.5415i | −2.16312 | − | 6.65740i | −16.6274 | + | 51.1738i | −45.2840 | − | 32.9007i | 7.52632 | ||
36.1 | −4.23611 | + | 3.07771i | −2.01219 | − | 6.19288i | 6.00015 | − | 18.4666i | −14.2597 | − | 10.3603i | 27.5837 | + | 20.0407i | −2.16312 | + | 6.65740i | 18.4731 | + | 56.8544i | −12.4593 | + | 9.05225i | 92.2919 | ||
36.2 | −3.13241 | + | 2.27583i | 2.57135 | + | 7.91381i | 2.16046 | − | 6.64921i | −14.0082 | − | 10.1776i | −26.0650 | − | 18.9373i | −2.16312 | + | 6.65740i | −1.20677 | − | 3.71406i | −34.1731 | + | 24.8282i | 67.0418 | ||
36.3 | −2.08555 | + | 1.51524i | −1.27325 | − | 3.91867i | −0.418581 | + | 1.28826i | 0.733295 | + | 0.532770i | 8.59314 | + | 6.24328i | −2.16312 | + | 6.65740i | −7.45191 | − | 22.9346i | 8.10867 | − | 5.89129i | −2.33659 | ||
36.4 | −1.91048 | + | 1.38804i | 1.19654 | + | 3.68258i | −0.748877 | + | 2.30480i | 12.0553 | + | 8.75871i | −7.39754 | − | 5.37463i | −2.16312 | + | 6.65740i | −7.60635 | − | 23.4100i | 9.71380 | − | 7.05749i | −35.1889 | ||
36.5 | −0.281932 | + | 0.204836i | −2.51790 | − | 7.74930i | −2.43461 | + | 7.49295i | −4.21014 | − | 3.05884i | 2.29721 | + | 1.66902i | −2.16312 | + | 6.65740i | −1.70994 | − | 5.26265i | −31.8684 | + | 23.1538i | 1.81354 | ||
36.6 | 1.06715 | − | 0.775331i | 2.12611 | + | 6.54350i | −1.93446 | + | 5.95366i | −8.06046 | − | 5.85627i | 7.34226 | + | 5.33447i | −2.16312 | + | 6.65740i | 5.81262 | + | 17.8894i | −16.4536 | + | 11.9543i | −13.1423 | ||
36.7 | 2.07504 | − | 1.50761i | 0.186723 | + | 0.574674i | −0.439213 | + | 1.35176i | 11.4299 | + | 8.30432i | 1.25384 | + | 0.910969i | −2.16312 | + | 6.65740i | 7.46730 | + | 22.9820i | 21.5481 | − | 15.6556i | 36.2372 | ||
36.8 | 3.62073 | − | 2.63062i | −0.489110 | − | 1.50533i | 3.71743 | − | 11.4411i | −16.0185 | − | 11.6381i | −5.73088 | − | 4.16373i | −2.16312 | + | 6.65740i | −5.57330 | − | 17.1529i | 19.8167 | − | 14.3977i | −88.6140 | ||
36.9 | 3.84003 | − | 2.78994i | −2.63098 | − | 8.09734i | 4.48990 | − | 13.8185i | 11.7384 | + | 8.52841i | −32.6942 | − | 23.7537i | −2.16312 | + | 6.65740i | −9.57738 | − | 29.4761i | −36.8013 | + | 26.7377i | 68.8694 | ||
36.10 | 4.16155 | − | 3.02354i | 2.81484 | + | 8.66318i | 5.70454 | − | 17.5568i | 1.18370 | + | 0.860011i | 37.9076 | + | 27.5415i | −2.16312 | + | 6.65740i | −16.6274 | − | 51.1738i | −45.2840 | + | 32.9007i | 7.52632 | ||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.c | even | 5 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 77.4.f.b | ✓ | 40 |
11.c | even | 5 | 1 | inner | 77.4.f.b | ✓ | 40 |
11.c | even | 5 | 1 | 847.4.a.q | 20 | ||
11.d | odd | 10 | 1 | 847.4.a.r | 20 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
77.4.f.b | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
77.4.f.b | ✓ | 40 | 11.c | even | 5 | 1 | inner |
847.4.a.q | 20 | 11.c | even | 5 | 1 | ||
847.4.a.r | 20 | 11.d | odd | 10 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} - 8 T_{2}^{39} + 89 T_{2}^{38} - 480 T_{2}^{37} + 3827 T_{2}^{36} - 17852 T_{2}^{35} + \cdots + 11\!\cdots\!76 \) acting on \(S_{4}^{\mathrm{new}}(77, [\chi])\).