Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [79,5,Mod(12,79)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(79, base_ring=CyclotomicField(26))
chi = DirichletCharacter(H, H._module([3]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("79.12");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 79 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 79.f (of order \(26\), degree \(12\), 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: | \(8.16622708362\) |
Analytic rank: | \(0\) |
Dimension: | \(300\) |
Relative dimension: | \(25\) over \(\Q(\zeta_{26})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{26}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
12.1 | −0.880563 | − | 7.25209i | −9.65770 | − | 3.66268i | −36.2824 | + | 8.94280i | 10.7508 | + | 5.64246i | −18.0579 | + | 73.2637i | −10.6557 | − | 4.04117i | 55.3547 | + | 145.958i | 19.2265 | + | 17.0332i | 31.4529 | − | 82.9344i |
12.2 | −0.857180 | − | 7.05952i | 6.71608 | + | 2.54707i | −33.5669 | + | 8.27351i | 33.3680 | + | 17.5129i | 12.2242 | − | 49.5956i | 53.6484 | + | 20.3461i | 46.8323 | + | 123.487i | −22.0112 | − | 19.5002i | 95.0300 | − | 250.573i |
12.3 | −0.837468 | − | 6.89717i | 8.79602 | + | 3.33589i | −31.3345 | + | 7.72327i | −21.8137 | − | 11.4487i | 15.6418 | − | 63.4613i | −13.1124 | − | 4.97287i | 40.0906 | + | 105.710i | 5.61243 | + | 4.97218i | −60.6953 | + | 160.040i |
12.4 | −0.710891 | − | 5.85472i | −9.91873 | − | 3.76168i | −18.2373 | + | 4.49508i | −37.2186 | − | 19.5338i | −14.9724 | + | 60.7455i | 17.7079 | + | 6.71570i | 5.82039 | + | 15.3471i | 23.6017 | + | 20.9092i | −87.9066 | + | 231.791i |
12.5 | −0.570567 | − | 4.69905i | −1.80843 | − | 0.685848i | −6.22041 | + | 1.53319i | 25.7388 | + | 13.5087i | −2.19100 | + | 8.88923i | −74.0027 | − | 28.0655i | −16.1030 | − | 42.4601i | −57.8293 | − | 51.2323i | 48.7925 | − | 128.655i |
12.6 | −0.562063 | − | 4.62900i | 16.5307 | + | 6.26928i | −5.57667 | + | 1.37453i | 16.3245 | + | 8.56774i | 19.7292 | − | 80.0446i | −54.0880 | − | 20.5129i | −16.9592 | − | 44.7178i | 173.332 | + | 153.559i | 30.4847 | − | 80.3816i |
12.7 | −0.516103 | − | 4.25049i | −2.41955 | − | 0.917613i | −2.26526 | + | 0.558337i | −2.95535 | − | 1.55109i | −2.65157 | + | 10.7578i | −5.92784 | − | 2.24813i | −20.7507 | − | 54.7152i | −55.6172 | − | 49.2725i | −5.06762 | + | 13.3622i |
12.8 | −0.502346 | − | 4.13719i | 5.96938 | + | 2.26389i | −1.32894 | + | 0.327553i | −8.58865 | − | 4.50767i | 6.36744 | − | 25.8337i | 68.4200 | + | 25.9483i | −21.6228 | − | 57.0145i | −30.1211 | − | 26.6850i | −14.3346 | + | 37.7973i |
12.9 | −0.419088 | − | 3.45150i | −12.6530 | − | 4.79867i | 3.79786 | − | 0.936088i | 29.4454 | + | 15.4541i | −11.2599 | + | 45.6830i | 78.2376 | + | 29.6716i | −24.5491 | − | 64.7306i | 76.4429 | + | 67.7225i | 40.9997 | − | 108.107i |
12.10 | −0.233315 | − | 1.92152i | −15.7837 | − | 5.98596i | 11.8973 | − | 2.93241i | −12.0970 | − | 6.34898i | −7.81957 | + | 31.7252i | −70.4513 | − | 26.7186i | −19.3926 | − | 51.1342i | 152.664 | + | 135.248i | −9.37728 | + | 24.7259i |
12.11 | −0.173468 | − | 1.42863i | 6.53699 | + | 2.47915i | 13.5242 | − | 3.33341i | −34.1964 | − | 17.9477i | 2.40785 | − | 9.76902i | −59.0770 | − | 22.4049i | −15.2734 | − | 40.2725i | −24.0433 | − | 21.3005i | −19.7087 | + | 51.9675i |
12.12 | −0.131802 | − | 1.08548i | 10.9130 | + | 4.13876i | 14.3742 | − | 3.54291i | 8.25526 | + | 4.33270i | 3.05420 | − | 12.3914i | 31.8441 | + | 12.0769i | −11.9442 | − | 31.4943i | 41.3350 | + | 36.6196i | 3.61501 | − | 9.53201i |
12.13 | 0.0188772 | + | 0.155468i | −7.84131 | − | 2.97382i | 15.5113 | − | 3.82318i | −24.6914 | − | 12.9591i | 0.314310 | − | 1.27521i | 42.2786 | + | 16.0341i | 1.77574 | + | 4.68224i | −7.98675 | − | 7.07564i | 1.54861 | − | 4.08335i |
12.14 | 0.0344746 | + | 0.283924i | 3.49904 | + | 1.32701i | 15.4556 | − | 3.80947i | 39.0664 | + | 20.5036i | −0.256143 | + | 1.03921i | −9.85170 | − | 3.73626i | 3.23715 | + | 8.53568i | −50.1470 | − | 44.4264i | −4.47468 | + | 11.7988i |
12.15 | 0.0505850 | + | 0.416605i | −8.05722 | − | 3.05570i | 15.3641 | − | 3.78690i | 10.7923 | + | 5.66421i | 0.865446 | − | 3.51125i | −22.4759 | − | 8.52398i | 4.73588 | + | 12.4875i | −5.04789 | − | 4.47204i | −1.81381 | + | 4.78263i |
12.16 | 0.264349 | + | 2.17711i | −2.01838 | − | 0.765469i | 10.8652 | − | 2.67802i | −12.7192 | − | 6.67557i | 1.13295 | − | 4.59657i | 44.8959 | + | 17.0268i | 21.1455 | + | 55.7560i | −57.1415 | − | 50.6229i | 11.1711 | − | 29.4558i |
12.17 | 0.347113 | + | 2.85873i | 14.7159 | + | 5.58100i | 7.48322 | − | 1.84445i | −30.9987 | − | 16.2694i | −10.8465 | + | 44.0060i | 42.9932 | + | 16.3052i | 24.2089 | + | 63.8337i | 124.780 | + | 110.546i | 35.7497 | − | 94.2643i |
12.18 | 0.486643 | + | 4.00786i | 11.0169 | + | 4.17816i | −0.291090 | + | 0.0717472i | 13.9031 | + | 7.29689i | −11.3842 | + | 46.1875i | −26.1724 | − | 9.92589i | 22.4771 | + | 59.2673i | 43.2855 | + | 38.3476i | −22.4791 | + | 59.2726i |
12.19 | 0.495058 | + | 4.07717i | −12.1243 | − | 4.59813i | −0.843140 | + | 0.207815i | 27.6718 | + | 14.5233i | 12.7451 | − | 51.7090i | −7.53215 | − | 2.85657i | 22.0377 | + | 58.1087i | 65.2256 | + | 57.7849i | −45.5148 | + | 120.013i |
12.20 | 0.498028 | + | 4.10163i | −0.324100 | − | 0.122915i | −1.04024 | + | 0.256397i | −12.1103 | − | 6.35598i | 0.342740 | − | 1.39055i | −66.6642 | − | 25.2824i | 21.8725 | + | 57.6731i | −60.5394 | − | 53.6333i | 20.0386 | − | 52.8374i |
See next 80 embeddings (of 300 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
79.f | odd | 26 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 79.5.f.a | ✓ | 300 |
79.f | odd | 26 | 1 | inner | 79.5.f.a | ✓ | 300 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
79.5.f.a | ✓ | 300 | 1.a | even | 1 | 1 | trivial |
79.5.f.a | ✓ | 300 | 79.f | odd | 26 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(79, [\chi])\).