Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [168,4,Mod(11,168)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(168, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 3, 3, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("168.11");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 168 = 2^{3} \cdot 3 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 168.v (of order \(6\), 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: | \(9.91232088096\) |
Analytic rank: | \(0\) |
Dimension: | \(184\) |
Relative dimension: | \(92\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
11.1 | −2.82802 | − | 0.0477234i | 0.799088 | + | 5.13434i | 7.99544 | + | 0.269926i | 0.139084 | + | 0.240900i | −2.01481 | − | 14.5582i | 4.18856 | − | 18.0404i | −22.5984 | − | 1.14493i | −25.7229 | + | 8.20558i | −0.381836 | − | 0.687910i |
11.2 | −2.82363 | − | 0.164748i | −0.360706 | − | 5.18362i | 7.94572 | + | 0.930371i | 10.4965 | + | 18.1805i | 0.164511 | + | 14.6960i | −18.5109 | − | 0.588189i | −22.2824 | − | 3.93606i | −26.7398 | + | 3.73953i | −26.6430 | − | 53.0642i |
11.3 | −2.82183 | − | 0.193129i | −5.18144 | − | 0.390781i | 7.92540 | + | 1.08995i | 7.95416 | + | 13.7770i | 14.5456 | + | 2.10340i | 18.4515 | − | 1.59414i | −22.1536 | − | 4.60628i | 26.6946 | + | 4.04961i | −19.7845 | − | 40.4125i |
11.4 | −2.81518 | − | 0.273405i | 4.60128 | + | 2.41418i | 7.85050 | + | 1.53937i | −7.18350 | − | 12.4422i | −12.2934 | − | 8.05436i | −15.0501 | + | 10.7932i | −21.6797 | − | 6.47998i | 15.3435 | + | 22.2166i | 16.8211 | + | 36.9910i |
11.5 | −2.79847 | − | 0.410550i | −5.18237 | + | 0.378250i | 7.66290 | + | 2.29783i | −9.90213 | − | 17.1510i | 14.6580 | + | 1.06910i | 1.86933 | − | 18.4257i | −20.5010 | − | 9.57641i | 26.7139 | − | 3.92046i | 20.6695 | + | 52.0619i |
11.6 | −2.78006 | + | 0.520843i | 1.08553 | + | 5.08150i | 7.45745 | − | 2.89595i | 6.68614 | + | 11.5807i | −5.66451 | − | 13.5615i | 4.08452 | + | 18.0642i | −19.2238 | + | 11.9351i | −24.6432 | + | 11.0323i | −24.6196 | − | 28.7127i |
11.7 | −2.74932 | + | 0.664275i | 4.62157 | − | 2.37509i | 7.11748 | − | 3.65260i | 3.04263 | + | 5.26998i | −11.1284 | + | 9.59988i | 17.0330 | − | 7.27154i | −17.1419 | + | 14.7701i | 15.7179 | − | 21.9533i | −11.8659 | − | 12.4677i |
11.8 | −2.72868 | + | 0.744517i | 1.45164 | − | 4.98926i | 6.89139 | − | 4.06310i | −6.36074 | − | 11.0171i | −0.246461 | + | 14.6949i | −10.4252 | − | 15.3073i | −15.7793 | + | 16.2177i | −22.7855 | − | 14.4852i | 25.5589 | + | 25.3265i |
11.9 | −2.72824 | − | 0.746131i | −3.42298 | + | 3.90938i | 6.88658 | + | 4.07125i | −0.646968 | − | 1.12058i | 12.2556 | − | 8.11173i | −16.4591 | + | 8.49107i | −15.7505 | − | 16.2456i | −3.56646 | − | 26.7634i | 0.928982 | + | 3.53993i |
11.10 | −2.70695 | + | 0.820003i | −3.91011 | − | 3.42214i | 6.65519 | − | 4.43942i | −2.88032 | − | 4.98886i | 13.3907 | + | 6.05726i | −5.96675 | + | 17.5328i | −14.3749 | + | 17.4746i | 3.57796 | + | 26.7619i | 11.8878 | + | 11.1427i |
11.11 | −2.68148 | − | 0.899817i | 3.33150 | − | 3.98762i | 6.38066 | + | 4.82568i | −3.71534 | − | 6.43516i | −12.5215 | + | 7.69498i | 3.81353 | + | 18.1234i | −12.7674 | − | 18.6814i | −4.80222 | − | 26.5695i | 4.17215 | + | 20.5989i |
11.12 | −2.62607 | − | 1.05059i | 5.17720 | − | 0.443396i | 5.79251 | + | 5.51787i | 4.08846 | + | 7.08141i | −14.0615 | − | 4.27474i | −7.44249 | − | 16.9591i | −9.41451 | − | 20.5759i | 26.6068 | − | 4.59110i | −3.29690 | − | 22.8916i |
11.13 | −2.61525 | − | 1.07724i | −1.39745 | − | 5.00471i | 5.67911 | + | 5.63451i | −1.03843 | − | 1.79862i | −1.73659 | + | 14.5940i | 18.4984 | + | 0.899114i | −8.78260 | − | 20.8534i | −23.0943 | + | 13.9877i | 0.778223 | + | 5.82248i |
11.14 | −2.50772 | + | 1.30819i | 4.70191 | + | 2.21179i | 4.57730 | − | 6.56112i | −6.88079 | − | 11.9179i | −14.6845 | + | 0.604435i | 18.3466 | + | 2.52991i | −2.89542 | + | 22.4414i | 17.2160 | + | 20.7993i | 32.8459 | + | 20.8853i |
11.15 | −2.49348 | + | 1.33513i | −4.07521 | + | 3.22376i | 4.43487 | − | 6.65822i | −4.58233 | − | 7.93683i | 5.85732 | − | 13.4793i | 15.9950 | + | 9.33591i | −2.16868 | + | 22.5233i | 6.21469 | − | 26.2750i | 22.0226 | + | 13.6723i |
11.16 | −2.40299 | + | 1.49185i | −4.07521 | + | 3.22376i | 3.54875 | − | 7.16982i | 4.58233 | + | 7.93683i | 4.98333 | − | 13.8263i | −15.9950 | − | 9.33591i | 2.16868 | + | 22.5233i | 6.21469 | − | 26.2750i | −22.8519 | − | 12.2360i |
11.17 | −2.38678 | + | 1.51765i | 4.70191 | + | 2.21179i | 3.39345 | − | 7.24462i | 6.88079 | + | 11.9179i | −14.5792 | + | 1.85683i | −18.3466 | − | 2.52991i | 2.89542 | + | 22.4414i | 17.2160 | + | 20.7993i | −34.5102 | − | 18.0027i |
11.18 | −2.28518 | − | 1.66672i | 4.42614 | + | 2.72200i | 2.44409 | + | 7.61751i | 7.35113 | + | 12.7325i | −5.57772 | − | 13.5974i | 12.0391 | + | 14.0734i | 7.11106 | − | 21.4810i | 12.1815 | + | 24.0959i | 4.42291 | − | 41.3484i |
11.19 | −2.21188 | − | 1.76283i | −0.903574 | + | 5.11699i | 1.78486 | + | 7.79835i | −7.29019 | − | 12.6270i | 11.0190 | − | 9.72533i | 15.6852 | + | 9.84762i | 9.79926 | − | 20.3955i | −25.3671 | − | 9.24716i | −6.13417 | + | 40.7808i |
11.20 | −2.21019 | − | 1.76495i | −5.16921 | − | 0.528471i | 1.76987 | + | 7.80177i | 2.23983 | + | 3.87950i | 10.4922 | + | 10.2914i | −11.5640 | + | 14.4663i | 9.85801 | − | 20.3671i | 26.4414 | + | 5.46355i | 1.89669 | − | 12.5276i |
See next 80 embeddings (of 184 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.c | even | 3 | 1 | inner |
8.d | odd | 2 | 1 | inner |
21.h | odd | 6 | 1 | inner |
24.f | even | 2 | 1 | inner |
56.k | odd | 6 | 1 | inner |
168.v | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 168.4.v.a | ✓ | 184 |
3.b | odd | 2 | 1 | inner | 168.4.v.a | ✓ | 184 |
7.c | even | 3 | 1 | inner | 168.4.v.a | ✓ | 184 |
8.d | odd | 2 | 1 | inner | 168.4.v.a | ✓ | 184 |
21.h | odd | 6 | 1 | inner | 168.4.v.a | ✓ | 184 |
24.f | even | 2 | 1 | inner | 168.4.v.a | ✓ | 184 |
56.k | odd | 6 | 1 | inner | 168.4.v.a | ✓ | 184 |
168.v | even | 6 | 1 | inner | 168.4.v.a | ✓ | 184 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
168.4.v.a | ✓ | 184 | 1.a | even | 1 | 1 | trivial |
168.4.v.a | ✓ | 184 | 3.b | odd | 2 | 1 | inner |
168.4.v.a | ✓ | 184 | 7.c | even | 3 | 1 | inner |
168.4.v.a | ✓ | 184 | 8.d | odd | 2 | 1 | inner |
168.4.v.a | ✓ | 184 | 21.h | odd | 6 | 1 | inner |
168.4.v.a | ✓ | 184 | 24.f | even | 2 | 1 | inner |
168.4.v.a | ✓ | 184 | 56.k | odd | 6 | 1 | inner |
168.4.v.a | ✓ | 184 | 168.v | even | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(168, [\chi])\).