Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [855,2,Mod(23,855)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(855, base_ring=CyclotomicField(36))
chi = DirichletCharacter(H, H._module([30, 27, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("855.23");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 855 = 3^{2} \cdot 5 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 855.dm (of order \(36\), degree \(12\), 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: | \(6.82720937282\) |
Analytic rank: | \(0\) |
Dimension: | \(1392\) |
Relative dimension: | \(116\) over \(\Q(\zeta_{36})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{36}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
23.1 | −1.59218 | + | 2.27387i | 1.72050 | − | 0.199721i | −1.95139 | − | 5.36141i | 0.349170 | + | 2.20864i | −2.28520 | + | 4.23017i | −0.906543 | + | 0.906543i | 9.93551 | + | 2.66221i | 2.92022 | − | 0.687238i | −5.57808 | − | 2.72258i |
23.2 | −1.56831 | + | 2.23979i | 1.06544 | + | 1.36559i | −1.87299 | − | 5.14599i | 0.136322 | − | 2.23191i | −4.72958 | + | 0.244681i | −0.497242 | + | 0.497242i | 9.18115 | + | 2.46008i | −0.729676 | + | 2.90991i | 4.78520 | + | 3.80567i |
23.3 | −1.56629 | + | 2.23690i | −0.759229 | + | 1.55678i | −1.86640 | − | 5.12789i | −2.19457 | + | 0.428784i | −2.29319 | − | 4.13670i | 1.04965 | − | 1.04965i | 9.11850 | + | 2.44330i | −1.84714 | − | 2.36391i | 2.47820 | − | 5.58064i |
23.4 | −1.55563 | + | 2.22167i | 0.173604 | − | 1.72333i | −1.83179 | − | 5.03281i | −2.23603 | − | 0.0130878i | 3.55860 | + | 3.06655i | 2.79066 | − | 2.79066i | 8.79135 | + | 2.35564i | −2.93972 | − | 0.598353i | 3.50751 | − | 4.94736i |
23.5 | −1.52991 | + | 2.18494i | −1.70742 | − | 0.291075i | −1.74929 | − | 4.80612i | 1.54920 | + | 1.61245i | 3.24817 | − | 3.28528i | 1.73999 | − | 1.73999i | 8.02446 | + | 2.15015i | 2.83055 | + | 0.993975i | −5.89323 | + | 0.917997i |
23.6 | −1.52131 | + | 2.17265i | −0.567477 | − | 1.63645i | −1.72199 | − | 4.73114i | 2.18223 | − | 0.487740i | 4.41874 | + | 1.25661i | −2.45590 | + | 2.45590i | 7.77491 | + | 2.08328i | −2.35594 | + | 1.85730i | −2.26014 | + | 5.48321i |
23.7 | −1.50312 | + | 2.14667i | −1.71871 | − | 0.214558i | −1.66481 | − | 4.57402i | −2.11765 | − | 0.718012i | 3.04401 | − | 3.36700i | −3.08170 | + | 3.08170i | 7.25871 | + | 1.94497i | 2.90793 | + | 0.737527i | 4.72442 | − | 3.46666i |
23.8 | −1.43964 | + | 2.05602i | 1.41154 | − | 1.00377i | −1.47062 | − | 4.04050i | 1.69445 | − | 1.45905i | 0.0316545 | + | 4.34723i | 3.00329 | − | 3.00329i | 5.57569 | + | 1.49400i | 0.984899 | − | 2.83372i | 0.560430 | + | 5.58435i |
23.9 | −1.39379 | + | 1.99054i | 1.30945 | − | 1.13373i | −1.33556 | − | 3.66943i | −0.411456 | − | 2.19789i | 0.431635 | + | 4.18670i | −1.73617 | + | 1.73617i | 4.47124 | + | 1.19807i | 0.429315 | − | 2.96912i | 4.94847 | + | 2.24438i |
23.10 | −1.38197 | + | 1.97366i | 0.494643 | + | 1.65992i | −1.30144 | − | 3.57568i | 1.86774 | + | 1.22945i | −3.95969 | − | 1.31770i | 3.08858 | − | 3.08858i | 4.20114 | + | 1.12569i | −2.51066 | + | 1.64214i | −5.00767 | + | 1.98722i |
23.11 | −1.35498 | + | 1.93512i | −1.21999 | + | 1.22948i | −1.22466 | − | 3.36472i | 0.0895790 | + | 2.23427i | −0.726112 | − | 4.02676i | −2.22451 | + | 2.22451i | 3.60684 | + | 0.966449i | −0.0232303 | − | 2.99991i | −4.44496 | − | 2.85406i |
23.12 | −1.33479 | + | 1.90628i | −1.37861 | + | 1.04854i | −1.16820 | − | 3.20959i | 1.60441 | − | 1.55752i | −0.158641 | − | 4.02760i | −1.89917 | + | 1.89917i | 3.18200 | + | 0.852615i | 0.801148 | − | 2.89105i | 0.827521 | + | 5.13742i |
23.13 | −1.32656 | + | 1.89453i | 1.36695 | + | 1.06369i | −1.14543 | − | 3.14704i | −1.71105 | + | 1.43955i | −3.82854 | + | 1.17868i | 1.25759 | − | 1.25759i | 3.01366 | + | 0.807508i | 0.737120 | + | 2.90803i | −0.457458 | − | 5.15129i |
23.14 | −1.30398 | + | 1.86227i | −0.278008 | + | 1.70959i | −1.08366 | − | 2.97734i | −0.802200 | − | 2.08722i | −2.82121 | − | 2.74700i | 0.0328215 | − | 0.0328215i | 2.56579 | + | 0.687500i | −2.84542 | − | 0.950560i | 4.93302 | + | 1.22777i |
23.15 | −1.27465 | + | 1.82039i | 0.514103 | − | 1.65399i | −1.00504 | − | 2.76132i | −1.41493 | + | 1.73147i | 2.35561 | + | 3.04413i | −2.97934 | + | 2.97934i | 2.01462 | + | 0.539816i | −2.47140 | − | 1.70065i | −1.34840 | − | 4.78273i |
23.16 | −1.27321 | + | 1.81834i | −0.671321 | − | 1.59666i | −1.00124 | − | 2.75089i | −1.34077 | − | 1.78951i | 3.75801 | + | 0.812204i | −0.0538499 | + | 0.0538499i | 1.98855 | + | 0.532830i | −2.09866 | + | 2.14374i | 4.96102 | − | 0.159555i |
23.17 | −1.26950 | + | 1.81303i | −1.51913 | − | 0.832010i | −0.991417 | − | 2.72390i | −1.47644 | + | 1.67932i | 3.43700 | − | 1.69800i | 0.703811 | − | 0.703811i | 1.92134 | + | 0.514822i | 1.61552 | + | 2.52787i | −1.17033 | − | 4.80873i |
23.18 | −1.25732 | + | 1.79565i | 1.55144 | + | 0.770089i | −0.959438 | − | 2.63604i | 2.08904 | − | 0.797428i | −3.33347 | + | 1.81758i | −0.518266 | + | 0.518266i | 1.70494 | + | 0.456838i | 1.81393 | + | 2.38949i | −1.19471 | + | 4.75381i |
23.19 | −1.22192 | + | 1.74508i | −0.146112 | + | 1.72588i | −0.868180 | − | 2.38531i | 0.636498 | + | 2.14356i | −2.83326 | − | 2.36386i | −1.26059 | + | 1.26059i | 1.10787 | + | 0.296854i | −2.95730 | − | 0.504343i | −4.51844 | − | 1.50852i |
23.20 | −1.20798 | + | 1.72518i | 1.17981 | + | 1.26809i | −0.832974 | − | 2.28858i | −2.23348 | + | 0.107608i | −3.61286 | + | 0.503553i | −3.54890 | + | 3.54890i | 0.885837 | + | 0.237359i | −0.216097 | + | 2.99221i | 2.51236 | − | 3.98313i |
See next 80 embeddings (of 1392 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
171.bf | odd | 18 | 1 | inner |
855.dm | even | 36 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 855.2.dm.a | yes | 1392 |
5.c | odd | 4 | 1 | inner | 855.2.dm.a | yes | 1392 |
9.d | odd | 6 | 1 | 855.2.dh.a | ✓ | 1392 | |
19.e | even | 9 | 1 | 855.2.dh.a | ✓ | 1392 | |
45.l | even | 12 | 1 | 855.2.dh.a | ✓ | 1392 | |
95.q | odd | 36 | 1 | 855.2.dh.a | ✓ | 1392 | |
171.bf | odd | 18 | 1 | inner | 855.2.dm.a | yes | 1392 |
855.dm | even | 36 | 1 | inner | 855.2.dm.a | yes | 1392 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
855.2.dh.a | ✓ | 1392 | 9.d | odd | 6 | 1 | |
855.2.dh.a | ✓ | 1392 | 19.e | even | 9 | 1 | |
855.2.dh.a | ✓ | 1392 | 45.l | even | 12 | 1 | |
855.2.dh.a | ✓ | 1392 | 95.q | odd | 36 | 1 | |
855.2.dm.a | yes | 1392 | 1.a | even | 1 | 1 | trivial |
855.2.dm.a | yes | 1392 | 5.c | odd | 4 | 1 | inner |
855.2.dm.a | yes | 1392 | 171.bf | odd | 18 | 1 | inner |
855.2.dm.a | yes | 1392 | 855.dm | even | 36 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(855, [\chi])\).