Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [68,5,Mod(15,68)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(68, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([4, 3]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("68.15");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 68 = 2^{2} \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 68.g (of order \(8\), 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: | \(7.02915748970\) |
Analytic rank: | \(0\) |
Dimension: | \(136\) |
Relative dimension: | \(34\) over \(\Q(\zeta_{8})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-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 | −3.97976 | + | 0.401912i | −5.14096 | + | 2.12946i | 15.6769 | − | 3.19903i | −30.8993 | + | 12.7989i | 19.6039 | − | 10.5409i | 8.44599 | − | 20.3904i | −61.1047 | + | 19.0321i | −35.3807 | + | 35.3807i | 117.828 | − | 63.3554i |
15.2 | −3.93941 | − | 0.693552i | 12.4503 | − | 5.15708i | 15.0380 | + | 5.46437i | −33.4720 | + | 13.8645i | −52.6236 | + | 11.6810i | −33.8486 | + | 81.7177i | −55.4510 | − | 31.9560i | 71.1386 | − | 71.1386i | 141.476 | − | 31.4036i |
15.3 | −3.92559 | + | 0.767965i | −14.3470 | + | 5.94274i | 14.8205 | − | 6.02942i | 24.9160 | − | 10.3205i | 51.7567 | − | 34.3468i | 6.13863 | − | 14.8200i | −53.5486 | + | 35.0506i | 113.246 | − | 113.246i | −89.8839 | + | 59.6487i |
15.4 | −3.88253 | − | 0.962278i | 6.09911 | − | 2.52633i | 14.1480 | + | 7.47214i | 44.1858 | − | 18.3024i | −26.1110 | + | 3.93952i | −18.8823 | + | 45.5859i | −47.7399 | − | 42.6251i | −26.4589 | + | 26.4589i | −189.164 | + | 28.5404i |
15.5 | −3.73480 | + | 1.43221i | 10.8531 | − | 4.49551i | 11.8975 | − | 10.6981i | 17.1956 | − | 7.12264i | −34.0957 | + | 32.3338i | 21.5256 | − | 51.9674i | −29.1130 | + | 56.9950i | 40.3050 | − | 40.3050i | −54.0210 | + | 51.2294i |
15.6 | −3.65611 | − | 1.62260i | −2.02334 | + | 0.838095i | 10.7344 | + | 11.8648i | 0.220874 | − | 0.0914891i | 8.75745 | + | 0.218896i | 15.3199 | − | 36.9854i | −19.9942 | − | 60.7966i | −53.8841 | + | 53.8841i | −0.955992 | − | 0.0238954i |
15.7 | −3.22362 | + | 2.36818i | −2.41525 | + | 1.00043i | 4.78349 | − | 15.2682i | 14.1064 | − | 5.84305i | 5.41666 | − | 8.94473i | −18.4314 | + | 44.4973i | 20.7376 | + | 60.5471i | −52.4431 | + | 52.4431i | −31.6362 | + | 52.2421i |
15.8 | −3.03366 | + | 2.60708i | 5.40922 | − | 2.24057i | 2.40622 | − | 15.8180i | −17.1875 | + | 7.11928i | −10.5684 | + | 20.8994i | −1.03674 | + | 2.50292i | 33.9393 | + | 54.2598i | −33.0362 | + | 33.0362i | 33.5804 | − | 66.4067i |
15.9 | −2.99412 | − | 2.65240i | −11.9983 | + | 4.96987i | 1.92950 | + | 15.8832i | −6.12124 | + | 2.53550i | 49.1065 | + | 16.9440i | −23.3729 | + | 56.4272i | 36.3516 | − | 52.6741i | 61.9846 | − | 61.9846i | 25.0529 | + | 8.64441i |
15.10 | −2.65240 | − | 2.99412i | 11.9983 | − | 4.96987i | −1.92950 | + | 15.8832i | −6.12124 | + | 2.53550i | −46.7048 | − | 22.7423i | 23.3729 | − | 56.4272i | 52.6741 | − | 36.3516i | 61.9846 | − | 61.9846i | 23.8276 | + | 11.6025i |
15.11 | −2.03932 | + | 3.44110i | −13.5696 | + | 5.62070i | −7.68235 | − | 14.0350i | −37.8294 | + | 15.6694i | 8.33129 | − | 58.1567i | −20.3620 | + | 49.1581i | 63.9627 | + | 2.18611i | 95.2653 | − | 95.2653i | 23.2260 | − | 162.130i |
15.12 | −1.62260 | − | 3.65611i | 2.02334 | − | 0.838095i | −10.7344 | + | 11.8648i | 0.220874 | − | 0.0914891i | −6.34724 | − | 6.03767i | −15.3199 | + | 36.9854i | 60.7966 | + | 19.9942i | −53.8841 | + | 53.8841i | −0.692885 | − | 0.659092i |
15.13 | −1.50860 | + | 3.70461i | −6.72667 | + | 2.78628i | −11.4482 | − | 11.1776i | 23.2295 | − | 9.62197i | −0.174207 | − | 29.1231i | 17.6038 | − | 42.4993i | 58.6793 | − | 25.5488i | −19.7909 | + | 19.7909i | 0.601596 | + | 100.572i |
15.14 | −1.34120 | + | 3.76845i | 15.7770 | − | 6.53505i | −12.4024 | − | 10.1085i | 7.80119 | − | 3.23136i | 3.46692 | + | 68.2196i | −15.2593 | + | 36.8392i | 54.7272 | − | 33.1803i | 148.931 | − | 148.931i | 1.71427 | + | 33.7323i |
15.15 | −0.962278 | − | 3.88253i | −6.09911 | + | 2.52633i | −14.1480 | + | 7.47214i | 44.1858 | − | 18.3024i | 15.6776 | + | 21.2489i | 18.8823 | − | 45.5859i | 42.6251 | + | 47.7399i | −26.4589 | + | 26.4589i | −113.578 | − | 153.941i |
15.16 | −0.693552 | − | 3.93941i | −12.4503 | + | 5.15708i | −15.0380 | + | 5.46437i | −33.4720 | + | 13.8645i | 28.9508 | + | 45.4702i | 33.8486 | − | 81.7177i | 31.9560 | + | 55.4510i | 71.1386 | − | 71.1386i | 77.8327 | + | 122.244i |
15.17 | −0.583213 | + | 3.95725i | 5.61605 | − | 2.32624i | −15.3197 | − | 4.61584i | −34.3663 | + | 14.2350i | 5.93019 | + | 23.5808i | 28.0507 | − | 67.7204i | 27.2007 | − | 57.9320i | −31.1470 | + | 31.1470i | −36.2886 | − | 144.298i |
15.18 | 0.211077 | + | 3.99443i | 1.44231 | − | 0.597422i | −15.9109 | + | 1.68627i | 11.7050 | − | 4.84835i | 2.69080 | + | 5.63508i | −27.8893 | + | 67.3306i | −10.0941 | − | 63.1990i | −55.5523 | + | 55.5523i | 21.8370 | + | 45.7312i |
15.19 | 0.401912 | − | 3.97976i | 5.14096 | − | 2.12946i | −15.6769 | − | 3.19903i | −30.8993 | + | 12.7989i | −6.40850 | − | 21.3156i | −8.44599 | + | 20.3904i | −19.0321 | + | 61.1047i | −35.3807 | + | 35.3807i | 38.5177 | + | 128.116i |
15.20 | 0.767965 | − | 3.92559i | 14.3470 | − | 5.94274i | −14.8205 | − | 6.02942i | 24.9160 | − | 10.3205i | −12.3107 | − | 60.8844i | −6.13863 | + | 14.8200i | −35.0506 | + | 53.5486i | 113.246 | − | 113.246i | −21.3795 | − | 105.736i |
See next 80 embeddings (of 136 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
17.d | even | 8 | 1 | inner |
68.g | odd | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 68.5.g.a | ✓ | 136 |
4.b | odd | 2 | 1 | inner | 68.5.g.a | ✓ | 136 |
17.d | even | 8 | 1 | inner | 68.5.g.a | ✓ | 136 |
68.g | odd | 8 | 1 | inner | 68.5.g.a | ✓ | 136 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
68.5.g.a | ✓ | 136 | 1.a | even | 1 | 1 | trivial |
68.5.g.a | ✓ | 136 | 4.b | odd | 2 | 1 | inner |
68.5.g.a | ✓ | 136 | 17.d | even | 8 | 1 | inner |
68.5.g.a | ✓ | 136 | 68.g | odd | 8 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(68, [\chi])\).