Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [91,8,Mod(22,91)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(91, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("91.22");
S:= CuspForms(chi, 8);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 91 = 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 91.f (of order \(3\), degree \(2\), 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: | \(28.4270373191\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(24\) over \(\Q(\zeta_{3})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
22.1 | −11.0832 | + | 19.1967i | −25.9723 | + | 44.9854i | −181.677 | − | 314.673i | −486.993 | −575.715 | − | 997.168i | 171.500 | + | 297.047i | 5216.95 | −255.623 | − | 442.751i | 5397.46 | − | 9348.68i | ||||
22.2 | −10.1427 | + | 17.5677i | −14.5450 | + | 25.1926i | −141.749 | − | 245.517i | 326.304 | −295.051 | − | 511.043i | 171.500 | + | 297.047i | 3154.36 | 670.388 | + | 1161.15i | −3309.61 | + | 5732.41i | ||||
22.3 | −9.35205 | + | 16.1982i | 31.2473 | − | 54.1220i | −110.922 | − | 192.122i | −263.837 | 584.454 | + | 1012.30i | 171.500 | + | 297.047i | 1755.26 | −859.293 | − | 1488.34i | 2467.42 | − | 4273.70i | ||||
22.4 | −8.31541 | + | 14.4027i | 18.6917 | − | 32.3751i | −74.2921 | − | 128.678i | 517.036 | 310.859 | + | 538.424i | 171.500 | + | 297.047i | 342.331 | 394.737 | + | 683.705i | −4299.36 | + | 7446.72i | ||||
22.5 | −8.31356 | + | 14.3995i | 3.55199 | − | 6.15223i | −74.2305 | − | 128.571i | −126.313 | 59.0594 | + | 102.294i | 171.500 | + | 297.047i | 340.208 | 1068.27 | + | 1850.29i | 1050.11 | − | 1818.85i | ||||
22.6 | −6.42253 | + | 11.1242i | −13.9142 | + | 24.1002i | −18.4978 | − | 32.0392i | −93.0462 | −178.729 | − | 309.568i | 171.500 | + | 297.047i | −1168.96 | 706.288 | + | 1223.33i | 597.592 | − | 1035.06i | ||||
22.7 | −6.02827 | + | 10.4413i | −33.3343 | + | 57.7367i | −8.68004 | − | 15.0343i | −353.391 | −401.896 | − | 696.105i | 171.500 | + | 297.047i | −1333.93 | −1128.85 | − | 1955.23i | 2130.33 | − | 3689.85i | ||||
22.8 | −5.90267 | + | 10.2237i | 36.7895 | − | 63.7213i | −5.68308 | − | 9.84339i | −76.4660 | 434.313 | + | 752.252i | 171.500 | + | 297.047i | −1376.90 | −1613.43 | − | 2794.55i | 451.354 | − | 781.768i | ||||
22.9 | −3.64211 | + | 6.30832i | −23.2226 | + | 40.2227i | 37.4701 | + | 64.9001i | 324.801 | −169.158 | − | 292.991i | 171.500 | + | 297.047i | −1478.26 | 14.9215 | + | 25.8448i | −1182.96 | + | 2048.95i | ||||
22.10 | −1.96989 | + | 3.41195i | 18.4244 | − | 31.9119i | 56.2391 | + | 97.4089i | 280.776 | 72.5879 | + | 125.726i | 171.500 | + | 297.047i | −947.431 | 414.587 | + | 718.085i | −553.099 | + | 957.995i | ||||
22.11 | −0.951312 | + | 1.64772i | −41.0916 | + | 71.1728i | 62.1900 | + | 107.716i | 94.6646 | −78.1819 | − | 135.415i | 171.500 | + | 297.047i | −480.185 | −2283.54 | − | 3955.21i | −90.0557 | + | 155.981i | ||||
22.12 | −0.463694 | + | 0.803142i | −3.61457 | + | 6.26061i | 63.5700 | + | 110.106i | −525.610 | −3.35211 | − | 5.80602i | 171.500 | + | 297.047i | −236.614 | 1067.37 | + | 1848.74i | 243.723 | − | 422.140i | ||||
22.13 | −0.408038 | + | 0.706743i | 34.4693 | − | 59.7025i | 63.6670 | + | 110.274i | −447.050 | 28.1296 | + | 48.7218i | 171.500 | + | 297.047i | −208.372 | −1282.76 | − | 2221.80i | 182.414 | − | 315.950i | ||||
22.14 | 0.786346 | − | 1.36199i | 46.3303 | − | 80.2465i | 62.7633 | + | 108.709i | 412.529 | −72.8633 | − | 126.203i | 171.500 | + | 297.047i | 398.719 | −3199.50 | − | 5541.69i | 324.391 | − | 561.861i | ||||
22.15 | 0.904301 | − | 1.56630i | −10.7648 | + | 18.6452i | 62.3645 | + | 108.018i | −102.650 | 19.4692 | + | 33.7217i | 171.500 | + | 297.047i | 457.086 | 861.739 | + | 1492.58i | −92.8265 | + | 160.780i | ||||
22.16 | 3.76119 | − | 6.51457i | −16.5227 | + | 28.6182i | 35.7070 | + | 61.8463i | 415.555 | 124.290 | + | 215.276i | 171.500 | + | 297.047i | 1500.07 | 547.501 | + | 948.299i | 1562.98 | − | 2707.16i | ||||
22.17 | 4.98688 | − | 8.63753i | 13.2496 | − | 22.9490i | 14.2620 | + | 24.7026i | 86.0031 | −132.148 | − | 228.888i | 171.500 | + | 297.047i | 1561.13 | 742.397 | + | 1285.87i | 428.887 | − | 742.854i | ||||
22.18 | 5.17651 | − | 8.96598i | 22.2003 | − | 38.4521i | 10.4074 | + | 18.0262i | −162.299 | −229.841 | − | 398.096i | 171.500 | + | 297.047i | 1540.68 | 107.790 | + | 186.698i | −840.144 | + | 1455.17i | ||||
22.19 | 6.11392 | − | 10.5896i | −32.6404 | + | 56.5348i | −10.7599 | − | 18.6367i | −72.0010 | 399.121 | + | 691.298i | 171.500 | + | 297.047i | 1302.02 | −1037.29 | − | 1796.63i | −440.208 | + | 762.462i | ||||
22.20 | 8.46862 | − | 14.6681i | 17.8769 | − | 30.9637i | −79.4351 | − | 137.586i | 457.543 | −302.785 | − | 524.440i | 171.500 | + | 297.047i | −522.854 | 454.332 | + | 786.927i | 3874.76 | − | 6711.28i | ||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 91.8.f.a | ✓ | 48 |
13.c | even | 3 | 1 | inner | 91.8.f.a | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
91.8.f.a | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
91.8.f.a | ✓ | 48 | 13.c | even | 3 | 1 | inner |