Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [693,2,Mod(32,693)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(693, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([5, 4, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("693.32");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 693 = 3^{2} \cdot 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 693.r (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: | \(5.53363286007\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
32.1 | −1.38013 | + | 2.39046i | 1.47497 | − | 0.907994i | −2.80952 | − | 4.86624i | 2.33301i | 0.134863 | + | 4.77901i | −0.854297 | + | 2.50403i | 9.98953 | 1.35110 | − | 2.67853i | −5.57696 | − | 3.21986i | ||||
32.2 | −1.37952 | + | 2.38940i | 0.310009 | + | 1.70408i | −2.80615 | − | 4.86039i | − | 1.17845i | −4.49939 | − | 1.61008i | 1.90327 | + | 1.83781i | 9.96648 | −2.80779 | + | 1.05656i | 2.81578 | + | 1.62569i | |||
32.3 | −1.31516 | + | 2.27793i | −1.72198 | − | 0.186503i | −2.45930 | − | 4.25964i | − | 3.09923i | 2.68953 | − | 3.67727i | 2.47997 | + | 0.921806i | 7.67689 | 2.93043 | + | 0.642311i | 7.05983 | + | 4.07599i | |||
32.4 | −1.28452 | + | 2.22485i | 1.30058 | − | 1.14389i | −2.29998 | − | 3.98368i | − | 3.14155i | 0.874377 | + | 4.36295i | −0.353471 | − | 2.62203i | 6.67940 | 0.383013 | − | 2.97545i | 6.98949 | + | 4.03539i | |||
32.5 | −1.28452 | + | 2.22485i | −0.522595 | + | 1.65133i | −2.29997 | − | 3.98366i | 1.91712i | −3.00268 | − | 3.28386i | 1.33384 | − | 2.28493i | 6.67931 | −2.45379 | − | 1.72596i | −4.26530 | − | 2.46257i | ||||
32.6 | −1.27299 | + | 2.20489i | 1.43960 | + | 0.963096i | −2.24103 | − | 3.88157i | − | 1.47270i | −3.95612 | + | 1.94814i | −2.60051 | − | 0.487186i | 6.31928 | 1.14489 | + | 2.77294i | 3.24714 | + | 1.87474i | |||
32.7 | −1.24873 | + | 2.16286i | −1.46670 | + | 0.921304i | −2.11865 | − | 3.66961i | 2.56481i | −0.161146 | − | 4.32272i | −1.99852 | + | 1.73375i | 5.58756 | 1.30240 | − | 2.70255i | −5.54734 | − | 3.20276i | ||||
32.8 | −1.23056 | + | 2.13139i | −0.546512 | − | 1.64357i | −2.02854 | − | 3.51353i | − | 1.18855i | 4.17560 | + | 0.857679i | −1.88916 | + | 1.85232i | 5.06268 | −2.40265 | + | 1.79646i | 2.53326 | + | 1.46258i | |||
32.9 | −1.20107 | + | 2.08032i | −0.261612 | − | 1.71218i | −1.88515 | − | 3.26517i | − | 0.488480i | 3.87609 | + | 1.51221i | 2.64507 | − | 0.0598399i | 4.25250 | −2.86312 | + | 0.895855i | 1.01619 | + | 0.586700i | |||
32.10 | −1.15050 | + | 1.99273i | 1.59728 | + | 0.669852i | −1.64732 | − | 2.85324i | 2.83395i | −3.17251 | + | 2.41228i | −0.128725 | − | 2.64262i | 2.97898 | 2.10260 | + | 2.13988i | −5.64730 | − | 3.26047i | ||||
32.11 | −1.14599 | + | 1.98492i | −0.858225 | + | 1.50448i | −1.62660 | − | 2.81735i | − | 4.06236i | −2.00274 | − | 3.42762i | −2.60212 | + | 0.478508i | 2.87230 | −1.52690 | − | 2.58236i | 8.06344 | + | 4.65543i | |||
32.12 | −1.12947 | + | 1.95630i | −1.53540 | + | 0.801594i | −1.55142 | − | 2.68714i | 0.256197i | 0.166029 | − | 3.90909i | −0.237628 | − | 2.63506i | 2.49125 | 1.71489 | − | 2.46153i | −0.501199 | − | 0.289367i | ||||
32.13 | −1.06743 | + | 1.84885i | 1.34388 | + | 1.09269i | −1.27883 | − | 2.21501i | 2.69562i | −3.45473 | + | 1.31826i | 2.01333 | + | 1.71654i | 1.19055 | 0.612049 | + | 2.93690i | −4.98380 | − | 2.87740i | ||||
32.14 | −1.06376 | + | 1.84248i | −1.72517 | − | 0.154244i | −1.26315 | − | 2.18784i | 1.71991i | 2.11935 | − | 3.01451i | 0.874708 | + | 2.49698i | 1.11971 | 2.95242 | + | 0.532195i | −3.16889 | − | 1.82956i | ||||
32.15 | −1.04326 | + | 1.80699i | 1.32970 | − | 1.10991i | −1.17680 | − | 2.03828i | 0.781280i | 0.618370 | + | 3.56068i | 2.63996 | − | 0.174906i | 0.737804 | 0.536192 | − | 2.95169i | −1.41176 | − | 0.815082i | ||||
32.16 | −1.00475 | + | 1.74028i | 1.72811 | + | 0.116771i | −1.01906 | − | 1.76506i | − | 2.82365i | −1.93954 | + | 2.89007i | 1.45176 | + | 2.21187i | 0.0765903 | 2.97273 | + | 0.403587i | 4.91395 | + | 2.83707i | |||
32.17 | −0.997772 | + | 1.72819i | 0.870661 | − | 1.49731i | −0.991098 | − | 1.71663i | 3.70831i | 1.71892 | + | 2.99865i | −1.88891 | − | 1.85257i | −0.0355293 | −1.48390 | − | 2.60731i | −6.40867 | − | 3.70005i | ||||
32.18 | −0.989463 | + | 1.71380i | −1.24040 | − | 1.20889i | −0.958072 | − | 1.65943i | 2.85900i | 3.29912 | − | 0.929647i | −2.61998 | − | 0.368369i | −0.165943 | 0.0771826 | + | 2.99901i | −4.89974 | − | 2.82887i | ||||
32.19 | −0.888640 | + | 1.53917i | 0.650782 | + | 1.60514i | −0.579364 | − | 1.00349i | − | 3.18894i | −3.04890 | − | 0.424730i | 1.56859 | − | 2.13062i | −1.49518 | −2.15297 | + | 2.08920i | 4.90832 | + | 2.83382i | |||
32.20 | −0.886651 | + | 1.53572i | −1.43789 | − | 0.965645i | −0.572300 | − | 0.991253i | 0.473605i | 2.75787 | − | 1.35201i | 1.06688 | − | 2.42111i | −1.51688 | 1.13506 | + | 2.77698i | −0.727327 | − | 0.419923i | ||||
See next 80 embeddings (of 184 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.b | odd | 2 | 1 | inner |
63.n | odd | 6 | 1 | inner |
693.r | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 693.2.r.a | ✓ | 184 |
7.c | even | 3 | 1 | 693.2.bn.a | yes | 184 | |
9.d | odd | 6 | 1 | 693.2.bn.a | yes | 184 | |
11.b | odd | 2 | 1 | inner | 693.2.r.a | ✓ | 184 |
63.n | odd | 6 | 1 | inner | 693.2.r.a | ✓ | 184 |
77.h | odd | 6 | 1 | 693.2.bn.a | yes | 184 | |
99.g | even | 6 | 1 | 693.2.bn.a | yes | 184 | |
693.r | even | 6 | 1 | inner | 693.2.r.a | ✓ | 184 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
693.2.r.a | ✓ | 184 | 1.a | even | 1 | 1 | trivial |
693.2.r.a | ✓ | 184 | 11.b | odd | 2 | 1 | inner |
693.2.r.a | ✓ | 184 | 63.n | odd | 6 | 1 | inner |
693.2.r.a | ✓ | 184 | 693.r | even | 6 | 1 | inner |
693.2.bn.a | yes | 184 | 7.c | even | 3 | 1 | |
693.2.bn.a | yes | 184 | 9.d | odd | 6 | 1 | |
693.2.bn.a | yes | 184 | 77.h | odd | 6 | 1 | |
693.2.bn.a | yes | 184 | 99.g | even | 6 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(693, [\chi])\).