Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [23,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("23.2");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 23 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
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: | \(1.35704393013\) |
Analytic rank: | \(0\) |
Dimension: | \(50\) |
Relative dimension: | \(5\) 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 | −3.24175 | − | 2.08335i | 1.21610 | + | 8.45817i | 2.84529 | + | 6.23031i | 12.7969 | + | 3.75752i | 13.6790 | − | 29.9528i | −11.3462 | + | 13.0942i | −0.631070 | + | 4.38919i | −44.1554 | + | 12.9652i | −33.6563 | − | 38.8414i |
2.2 | −2.68394 | − | 1.72487i | −0.220662 | − | 1.53474i | 0.905070 | + | 1.98183i | −18.5174 | − | 5.43720i | −2.05498 | + | 4.49977i | 13.0047 | − | 15.0082i | −2.64311 | + | 18.3833i | 23.5996 | − | 6.92946i | 40.3212 | + | 46.5331i |
2.3 | −0.442460 | − | 0.284352i | −0.784628 | − | 5.45721i | −3.20841 | − | 7.02543i | 12.7367 | + | 3.73985i | −1.20460 | + | 2.63770i | −5.26652 | + | 6.07789i | −1.17691 | + | 8.18558i | −3.25915 | + | 0.956973i | −4.57207 | − | 5.27645i |
2.4 | 1.91736 | + | 1.23221i | 0.776967 | + | 5.40392i | −1.16540 | − | 2.55188i | −0.907869 | − | 0.266574i | −5.16905 | + | 11.3186i | 8.82725 | − | 10.1872i | 3.50483 | − | 24.3766i | −2.69237 | + | 0.790552i | −1.41223 | − | 1.62980i |
2.5 | 3.73432 | + | 2.39990i | −0.655499 | − | 4.55909i | 4.86229 | + | 10.6469i | −9.99480 | − | 2.93474i | 8.49353 | − | 18.5982i | −17.7583 | + | 20.4942i | −2.34036 | + | 16.2775i | 5.55065 | − | 1.62982i | −30.2807 | − | 34.9458i |
3.1 | −0.589312 | − | 4.09876i | 1.03962 | − | 2.27645i | −8.77657 | + | 2.57703i | −3.66646 | + | 4.23132i | −9.94329 | − | 2.91961i | 16.9812 | − | 10.9132i | 1.97323 | + | 4.32077i | 13.5798 | + | 15.6719i | 19.5039 | + | 12.5344i |
3.2 | −0.132292 | − | 0.920110i | 0.209861 | − | 0.459531i | 6.84684 | − | 2.01041i | 6.87249 | − | 7.93128i | −0.450582 | − | 0.132303i | −25.4560 | + | 16.3596i | −5.84485 | − | 12.7984i | 17.5141 | + | 20.2124i | −8.20682 | − | 5.27420i |
3.3 | 0.0282746 | + | 0.196654i | −3.66027 | + | 8.01487i | 7.63807 | − | 2.24274i | −6.59261 | + | 7.60828i | −1.67965 | − | 0.493191i | 21.3237 | − | 13.7039i | 1.31727 | + | 2.88443i | −33.1594 | − | 38.2680i | −1.68261 | − | 1.08135i |
3.4 | 0.306014 | + | 2.12837i | 3.96829 | − | 8.68933i | 3.23961 | − | 0.951236i | −11.4648 | + | 13.2311i | 19.7085 | + | 5.78694i | −2.65189 | + | 1.70427i | 10.1620 | + | 22.2516i | −42.0760 | − | 48.5583i | −31.6691 | − | 20.3525i |
3.5 | 0.619909 | + | 4.31156i | −0.504759 | + | 1.10527i | −10.5293 | + | 3.09169i | 4.52322 | − | 5.22007i | −5.07833 | − | 1.49113i | 2.36872 | − | 1.52228i | −5.38116 | − | 11.7831i | 16.7144 | + | 19.2894i | 25.3106 | + | 16.2662i |
4.1 | −2.10729 | − | 4.61431i | 8.43369 | − | 2.47635i | −11.6123 | + | 13.4013i | −5.02341 | − | 3.22835i | −29.1989 | − | 33.6973i | 3.12135 | + | 21.7095i | 47.3705 | + | 13.9092i | 42.2809 | − | 27.1723i | −4.31085 | + | 29.9826i |
4.2 | −1.58245 | − | 3.46509i | −7.04544 | + | 2.06873i | −4.26380 | + | 4.92069i | −4.88780 | − | 3.14120i | 18.3174 | + | 21.1394i | −2.25200 | − | 15.6630i | −5.44232 | − | 1.59801i | 22.6448 | − | 14.5529i | −3.14982 | + | 21.9075i |
4.3 | −0.308069 | − | 0.674576i | 1.68484 | − | 0.494713i | 4.87874 | − | 5.63037i | 3.36936 | + | 2.16536i | −0.852766 | − | 0.984145i | 0.387311 | + | 2.69380i | −10.9935 | − | 3.22799i | −20.1199 | + | 12.9303i | 0.422704 | − | 2.93997i |
4.4 | 1.49850 | + | 3.28126i | −7.64281 | + | 2.24413i | −3.28227 | + | 3.78794i | 15.0482 | + | 9.67090i | −18.8163 | − | 21.7152i | −2.56161 | − | 17.8164i | 10.3412 | + | 3.03646i | 30.6625 | − | 19.7056i | −9.18297 | + | 63.8690i |
4.5 | 1.61009 | + | 3.52560i | 3.27467 | − | 0.961529i | −4.59861 | + | 5.30707i | −10.7624 | − | 6.91659i | 8.66248 | + | 9.99703i | −0.249078 | − | 1.73238i | 3.63606 | + | 1.06764i | −12.9149 | + | 8.29992i | 7.05669 | − | 49.0804i |
6.1 | −2.10729 | + | 4.61431i | 8.43369 | + | 2.47635i | −11.6123 | − | 13.4013i | −5.02341 | + | 3.22835i | −29.1989 | + | 33.6973i | 3.12135 | − | 21.7095i | 47.3705 | − | 13.9092i | 42.2809 | + | 27.1723i | −4.31085 | − | 29.9826i |
6.2 | −1.58245 | + | 3.46509i | −7.04544 | − | 2.06873i | −4.26380 | − | 4.92069i | −4.88780 | + | 3.14120i | 18.3174 | − | 21.1394i | −2.25200 | + | 15.6630i | −5.44232 | + | 1.59801i | 22.6448 | + | 14.5529i | −3.14982 | − | 21.9075i |
6.3 | −0.308069 | + | 0.674576i | 1.68484 | + | 0.494713i | 4.87874 | + | 5.63037i | 3.36936 | − | 2.16536i | −0.852766 | + | 0.984145i | 0.387311 | − | 2.69380i | −10.9935 | + | 3.22799i | −20.1199 | − | 12.9303i | 0.422704 | + | 2.93997i |
6.4 | 1.49850 | − | 3.28126i | −7.64281 | − | 2.24413i | −3.28227 | − | 3.78794i | 15.0482 | − | 9.67090i | −18.8163 | + | 21.7152i | −2.56161 | + | 17.8164i | 10.3412 | − | 3.03646i | 30.6625 | + | 19.7056i | −9.18297 | − | 63.8690i |
6.5 | 1.61009 | − | 3.52560i | 3.27467 | + | 0.961529i | −4.59861 | − | 5.30707i | −10.7624 | + | 6.91659i | 8.66248 | − | 9.99703i | −0.249078 | + | 1.73238i | 3.63606 | − | 1.06764i | −12.9149 | − | 8.29992i | 7.05669 | + | 49.0804i |
See all 50 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 | 23.4.c.a | ✓ | 50 |
3.b | odd | 2 | 1 | 207.4.i.a | 50 | ||
23.c | even | 11 | 1 | inner | 23.4.c.a | ✓ | 50 |
23.c | even | 11 | 1 | 529.4.a.n | 25 | ||
23.d | odd | 22 | 1 | 529.4.a.m | 25 | ||
69.h | odd | 22 | 1 | 207.4.i.a | 50 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
23.4.c.a | ✓ | 50 | 1.a | even | 1 | 1 | trivial |
23.4.c.a | ✓ | 50 | 23.c | even | 11 | 1 | inner |
207.4.i.a | 50 | 3.b | odd | 2 | 1 | ||
207.4.i.a | 50 | 69.h | odd | 22 | 1 | ||
529.4.a.m | 25 | 23.d | odd | 22 | 1 | ||
529.4.a.n | 25 | 23.c | even | 11 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(23, [\chi])\).