Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [441,3,Mod(55,441)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(441, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([0, 9]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("441.55");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 441 = 3^{2} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 441.v (of order \(14\), degree \(6\), 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: | \(12.0163796583\) |
Analytic rank: | \(0\) |
Dimension: | \(108\) |
Relative dimension: | \(18\) over \(\Q(\zeta_{14})\) |
Twist minimal: | no (minimal twist has level 147) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{14}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
55.1 | −2.34463 | + | 2.94007i | 0 | −2.25666 | − | 9.88706i | −1.32428 | + | 2.74990i | 0 | −6.90317 | + | 1.16027i | 20.8074 | + | 10.0203i | 0 | −4.97996 | − | 10.3410i | ||||||
55.2 | −2.15774 | + | 2.70572i | 0 | −1.77499 | − | 7.77675i | −1.41126 | + | 2.93051i | 0 | 1.51875 | + | 6.83326i | 12.3996 | + | 5.97132i | 0 | −4.88401 | − | 10.1418i | ||||||
55.3 | −1.90714 | + | 2.39148i | 0 | −1.19191 | − | 5.22209i | 0.609499 | − | 1.26564i | 0 | −1.84331 | − | 6.75294i | 3.73807 | + | 1.80016i | 0 | 1.86435 | + | 3.87136i | ||||||
55.4 | −1.68215 | + | 2.10935i | 0 | −0.729644 | − | 3.19678i | −1.89988 | + | 3.94513i | 0 | 5.36283 | − | 4.49889i | −1.75261 | − | 0.844013i | 0 | −5.12579 | − | 10.6438i | ||||||
55.5 | −1.45374 | + | 1.82294i | 0 | −0.319642 | − | 1.40044i | 2.35105 | − | 4.88201i | 0 | 1.00834 | + | 6.92699i | −5.38528 | − | 2.59341i | 0 | 5.48176 | + | 11.3830i | ||||||
55.6 | −0.988314 | + | 1.23931i | 0 | 0.330967 | + | 1.45006i | −1.75892 | + | 3.65243i | 0 | 5.32542 | + | 4.54311i | −7.83679 | − | 3.77400i | 0 | −2.78812 | − | 5.78959i | ||||||
55.7 | −0.622826 | + | 0.780999i | 0 | 0.668036 | + | 2.92686i | 3.23603 | − | 6.71968i | 0 | 6.96696 | − | 0.679280i | −6.30199 | − | 3.03488i | 0 | 3.23258 | + | 6.71253i | ||||||
55.8 | −0.310648 | + | 0.389540i | 0 | 0.834844 | + | 3.65769i | −3.33592 | + | 6.92711i | 0 | −5.25652 | − | 4.62266i | −3.47976 | − | 1.67576i | 0 | −1.66209 | − | 3.45137i | ||||||
55.9 | −0.192098 | + | 0.240883i | 0 | 0.868961 | + | 3.80717i | −3.14124 | + | 6.52285i | 0 | 2.01189 | + | 6.70465i | −2.19437 | − | 1.05675i | 0 | −0.967820 | − | 2.00970i | ||||||
55.10 | −0.141675 | + | 0.177654i | 0 | 0.878594 | + | 3.84937i | 1.28346 | − | 2.66513i | 0 | −6.17180 | − | 3.30286i | −1.62724 | − | 0.783636i | 0 | 0.291639 | + | 0.605594i | ||||||
55.11 | 0.573681 | − | 0.719373i | 0 | 0.701696 | + | 3.07433i | 1.16880 | − | 2.42703i | 0 | −4.29685 | + | 5.52604i | 5.93011 | + | 2.85579i | 0 | −1.07542 | − | 2.23314i | ||||||
55.12 | 0.903917 | − | 1.13348i | 0 | 0.422382 | + | 1.85058i | 3.47572 | − | 7.21742i | 0 | 6.98724 | + | 0.422439i | 7.70417 | + | 3.71013i | 0 | −5.03900 | − | 10.4636i | ||||||
55.13 | 1.18389 | − | 1.48455i | 0 | 0.0877927 | + | 0.384645i | −3.80801 | + | 7.90742i | 0 | 1.93330 | − | 6.72773i | 7.51802 | + | 3.62049i | 0 | 7.23068 | + | 15.0147i | ||||||
55.14 | 1.33639 | − | 1.67578i | 0 | −0.132215 | − | 0.579271i | −0.288815 | + | 0.599731i | 0 | −6.52547 | + | 2.53342i | 6.57713 | + | 3.16738i | 0 | 0.619047 | + | 1.28546i | ||||||
55.15 | 1.50073 | − | 1.88185i | 0 | −0.399101 | − | 1.74858i | −0.465335 | + | 0.966277i | 0 | 6.93265 | − | 0.968726i | 4.78495 | + | 2.30431i | 0 | 1.12005 | + | 2.32581i | ||||||
55.16 | 1.61994 | − | 2.03134i | 0 | −0.612049 | − | 2.68156i | −0.534134 | + | 1.10914i | 0 | −4.39676 | − | 5.44688i | 2.92487 | + | 1.40855i | 0 | 1.38777 | + | 2.88174i | ||||||
55.17 | 2.29233 | − | 2.87449i | 0 | −2.11783 | − | 9.27881i | 3.51947 | − | 7.30825i | 0 | 2.61430 | + | 6.49349i | −18.2765 | − | 8.80151i | 0 | −12.9397 | − | 26.8695i | ||||||
55.18 | 2.39011 | − | 2.99710i | 0 | −2.37991 | − | 10.4270i | 0.631746 | − | 1.31183i | 0 | −0.971914 | − | 6.93220i | −23.1239 | − | 11.1359i | 0 | −2.42176 | − | 5.02883i | ||||||
118.1 | −3.30424 | − | 1.59124i | 0 | 5.89202 | + | 7.38836i | 7.61062 | − | 1.73707i | 0 | 4.61863 | + | 5.26006i | −4.44770 | − | 19.4866i | 0 | −27.9114 | − | 6.37060i | ||||||
118.2 | −3.08824 | − | 1.48722i | 0 | 4.83146 | + | 6.05846i | 0.636277 | − | 0.145226i | 0 | −6.79347 | + | 1.68784i | −2.85954 | − | 12.5284i | 0 | −2.18096 | − | 0.497790i | ||||||
See next 80 embeddings (of 108 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
49.f | odd | 14 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 441.3.v.c | 108 | |
3.b | odd | 2 | 1 | 147.3.j.a | ✓ | 108 | |
49.f | odd | 14 | 1 | inner | 441.3.v.c | 108 | |
147.k | even | 14 | 1 | 147.3.j.a | ✓ | 108 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
147.3.j.a | ✓ | 108 | 3.b | odd | 2 | 1 | |
147.3.j.a | ✓ | 108 | 147.k | even | 14 | 1 | |
441.3.v.c | 108 | 1.a | even | 1 | 1 | trivial | |
441.3.v.c | 108 | 49.f | odd | 14 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{108} + 52 T_{2}^{106} + 4 T_{2}^{105} + 1616 T_{2}^{104} + 68 T_{2}^{103} + 38576 T_{2}^{102} - 776 T_{2}^{101} + 779621 T_{2}^{100} - 85794 T_{2}^{99} + 14214595 T_{2}^{98} - 2738404 T_{2}^{97} + \cdots + 32\!\cdots\!69 \)
acting on \(S_{3}^{\mathrm{new}}(441, [\chi])\).