Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [23,6,Mod(2,23)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(23, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([2]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("23.2");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 23 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 23.c (of order \(11\), degree \(10\), 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: | \(3.68882785570\) |
Analytic rank: | \(0\) |
Dimension: | \(90\) |
Relative dimension: | \(9\) over \(\Q(\zeta_{11})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{11}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −8.49747 | − | 5.46100i | 1.17717 | + | 8.18741i | 29.0913 | + | 63.7011i | −4.12458 | − | 1.21109i | 34.7084 | − | 76.0009i | 66.5045 | − | 76.7503i | 54.6681 | − | 380.225i | 167.509 | − | 49.1850i | 28.4348 | + | 32.8155i |
2.2 | −5.98228 | − | 3.84458i | −4.25176 | − | 29.5716i | 7.71364 | + | 16.8905i | 10.2972 | + | 3.02354i | −88.2553 | + | 193.252i | 60.9133 | − | 70.2977i | −13.5930 | + | 94.5412i | −623.247 | + | 183.002i | −49.9767 | − | 57.6762i |
2.3 | −5.01311 | − | 3.22173i | −0.293619 | − | 2.04217i | 1.45843 | + | 3.19351i | −10.1657 | − | 2.98491i | −5.10736 | + | 11.1836i | −159.381 | + | 183.936i | −24.1608 | + | 168.042i | 229.073 | − | 67.2618i | 41.3450 | + | 47.7146i |
2.4 | −2.21730 | − | 1.42497i | 4.20469 | + | 29.2443i | −10.4074 | − | 22.7890i | −83.1134 | − | 24.4043i | 32.3492 | − | 70.8350i | 22.1618 | − | 25.5761i | −21.4006 | + | 148.845i | −604.392 | + | 177.466i | 149.512 | + | 172.546i |
2.5 | −1.85052 | − | 1.18926i | 1.12337 | + | 7.81321i | −11.2832 | − | 24.7067i | 82.9428 | + | 24.3542i | 7.21310 | − | 15.7945i | 98.7062 | − | 113.913i | −18.5206 | + | 128.813i | 173.372 | − | 50.9067i | −124.524 | − | 143.708i |
2.6 | 1.87482 | + | 1.20487i | −1.73793 | − | 12.0876i | −11.2301 | − | 24.5904i | −45.2299 | − | 13.2807i | 11.3057 | − | 24.7560i | 13.6774 | − | 15.7845i | 18.7232 | − | 130.223i | 90.0673 | − | 26.4461i | −68.7963 | − | 79.3951i |
2.7 | 5.39264 | + | 3.46564i | 2.36907 | + | 16.4773i | 3.77666 | + | 8.26972i | 27.3965 | + | 8.04434i | −44.3288 | + | 97.0664i | −81.9154 | + | 94.5354i | 20.8990 | − | 145.356i | −32.7312 | + | 9.61074i | 119.861 | + | 138.327i |
2.8 | 6.83356 | + | 4.39166i | −3.42127 | − | 23.7955i | 14.1176 | + | 30.9132i | 81.6604 | + | 23.9777i | 81.1223 | − | 177.633i | −26.6137 | + | 30.7138i | −2.29389 | + | 15.9544i | −321.363 | + | 94.3608i | 452.730 | + | 522.478i |
2.9 | 8.74318 | + | 5.61890i | 0.603375 | + | 4.19657i | 31.5779 | + | 69.1460i | −63.3504 | − | 18.6013i | −18.3047 | + | 40.0817i | 143.409 | − | 165.503i | −65.1024 | + | 452.797i | 215.910 | − | 63.3968i | −449.365 | − | 518.594i |
3.1 | −1.56617 | − | 10.8929i | 3.45588 | − | 7.56732i | −85.4995 | + | 25.1049i | 43.9262 | − | 50.6935i | −87.8428 | − | 25.7930i | −42.6312 | + | 27.3974i | 261.081 | + | 571.688i | 113.810 | + | 131.344i | −620.998 | − | 399.091i |
3.2 | −1.13164 | − | 7.87071i | −6.81486 | + | 14.9225i | −29.9637 | + | 8.79812i | −54.8415 | + | 63.2904i | 125.162 | + | 36.7510i | −3.45670 | + | 2.22148i | −2.54800 | − | 5.57934i | −17.1066 | − | 19.7421i | 560.201 | + | 360.019i |
3.3 | −0.696058 | − | 4.84119i | 8.29077 | − | 18.1543i | 7.75116 | − | 2.27595i | −11.8362 | + | 13.6597i | −93.6590 | − | 27.5008i | −29.6881 | + | 19.0794i | −81.4306 | − | 178.308i | −101.709 | − | 117.378i | 74.3679 | + | 47.7933i |
3.4 | −0.608938 | − | 4.23525i | −6.59663 | + | 14.4446i | 13.1372 | − | 3.85743i | 52.7594 | − | 60.8875i | 65.1935 | + | 19.1425i | 55.4232 | − | 35.6183i | −81.2163 | − | 177.839i | −5.99980 | − | 6.92413i | −290.001 | − | 186.373i |
3.5 | 0.330150 | + | 2.29625i | −5.46335 | + | 11.9631i | 25.5400 | − | 7.49923i | −32.3567 | + | 37.3416i | −29.2739 | − | 8.59559i | −167.445 | + | 107.610i | 56.4907 | + | 123.697i | 45.8643 | + | 52.9303i | −96.4282 | − | 61.9706i |
3.6 | 0.364275 | + | 2.53359i | 3.17904 | − | 6.96113i | 24.4174 | − | 7.16960i | −2.80660 | + | 3.23899i | 18.7947 | + | 5.51861i | 130.259 | − | 83.7125i | 61.0854 | + | 133.758i | 120.780 | + | 139.388i | −9.22864 | − | 5.93088i |
3.7 | 0.991600 | + | 6.89673i | 9.66731 | − | 21.1685i | −15.8778 | + | 4.66215i | 66.8592 | − | 77.1597i | 155.579 | + | 45.6822i | −163.317 | + | 104.958i | 44.7249 | + | 97.9339i | −195.515 | − | 225.637i | 598.447 | + | 384.598i |
3.8 | 1.13144 | + | 7.86930i | −9.75312 | + | 21.3564i | −29.9420 | + | 8.79177i | 30.9167 | − | 35.6798i | −179.095 | − | 52.5869i | 34.5697 | − | 22.2166i | 2.62201 | + | 5.74140i | −201.839 | − | 232.935i | 315.756 | + | 202.924i |
3.9 | 1.41793 | + | 9.86195i | 2.99089 | − | 6.54914i | −64.5437 | + | 18.9517i | −49.3449 | + | 56.9471i | 68.8281 | + | 20.2098i | 37.4214 | − | 24.0493i | −145.974 | − | 319.639i | 125.185 | + | 144.472i | −631.577 | − | 405.890i |
4.1 | −4.47298 | − | 9.79445i | −7.03817 | + | 2.06659i | −54.9683 | + | 63.4367i | 15.8872 | + | 10.2101i | 51.7227 | + | 59.6912i | −16.2855 | − | 113.268i | 536.597 | + | 157.559i | −159.160 | + | 102.286i | 28.9391 | − | 201.276i |
4.2 | −2.60966 | − | 5.71435i | −0.942040 | + | 0.276608i | −4.88796 | + | 5.64100i | −66.5354 | − | 42.7597i | 4.03903 | + | 4.66129i | 20.2510 | + | 140.849i | −147.892 | − | 43.4250i | −203.614 | + | 130.855i | −70.7094 | + | 491.795i |
See all 90 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
23.c | even | 11 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 23.6.c.a | ✓ | 90 |
23.c | even | 11 | 1 | inner | 23.6.c.a | ✓ | 90 |
23.c | even | 11 | 1 | 529.6.a.j | 45 | ||
23.d | odd | 22 | 1 | 529.6.a.k | 45 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
23.6.c.a | ✓ | 90 | 1.a | even | 1 | 1 | trivial |
23.6.c.a | ✓ | 90 | 23.c | even | 11 | 1 | inner |
529.6.a.j | 45 | 23.c | even | 11 | 1 | ||
529.6.a.k | 45 | 23.d | odd | 22 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(23, [\chi])\).