Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [840,2,Mod(811,840)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(840, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("840.811");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 840.z (of order \(2\), degree \(1\), 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: | \(6.70743376979\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
811.1 | −1.41307 | − | 0.0568977i | − | 1.00000i | 1.99353 | + | 0.160801i | 1.00000 | −0.0568977 | + | 1.41307i | −2.57870 | − | 0.591873i | −2.80784 | − | 0.340650i | −1.00000 | −1.41307 | − | 0.0568977i | |||||
811.2 | −1.41307 | + | 0.0568977i | 1.00000i | 1.99353 | − | 0.160801i | 1.00000 | −0.0568977 | − | 1.41307i | −2.57870 | + | 0.591873i | −2.80784 | + | 0.340650i | −1.00000 | −1.41307 | + | 0.0568977i | ||||||
811.3 | −1.35569 | − | 0.402636i | 1.00000i | 1.67577 | + | 1.09170i | 1.00000 | 0.402636 | − | 1.35569i | 2.48628 | + | 0.904667i | −1.83226 | − | 2.15472i | −1.00000 | −1.35569 | − | 0.402636i | ||||||
811.4 | −1.35569 | + | 0.402636i | − | 1.00000i | 1.67577 | − | 1.09170i | 1.00000 | 0.402636 | + | 1.35569i | 2.48628 | − | 0.904667i | −1.83226 | + | 2.15472i | −1.00000 | −1.35569 | + | 0.402636i | |||||
811.5 | −1.22597 | − | 0.704981i | − | 1.00000i | 1.00600 | + | 1.72857i | 1.00000 | −0.704981 | + | 1.22597i | 2.37591 | − | 1.16407i | −0.0147234 | − | 2.82839i | −1.00000 | −1.22597 | − | 0.704981i | |||||
811.6 | −1.22597 | + | 0.704981i | 1.00000i | 1.00600 | − | 1.72857i | 1.00000 | −0.704981 | − | 1.22597i | 2.37591 | + | 1.16407i | −0.0147234 | + | 2.82839i | −1.00000 | −1.22597 | + | 0.704981i | ||||||
811.7 | −0.767567 | − | 1.18779i | 1.00000i | −0.821681 | + | 1.82341i | 1.00000 | 1.18779 | − | 0.767567i | −1.33527 | + | 2.28409i | 2.79653 | − | 0.423611i | −1.00000 | −0.767567 | − | 1.18779i | ||||||
811.8 | −0.767567 | + | 1.18779i | − | 1.00000i | −0.821681 | − | 1.82341i | 1.00000 | 1.18779 | + | 0.767567i | −1.33527 | − | 2.28409i | 2.79653 | + | 0.423611i | −1.00000 | −0.767567 | + | 1.18779i | |||||
811.9 | −0.718490 | − | 1.21810i | 1.00000i | −0.967544 | + | 1.75039i | 1.00000 | 1.21810 | − | 0.718490i | 1.92209 | − | 1.81812i | 2.82732 | − | 0.0790702i | −1.00000 | −0.718490 | − | 1.21810i | ||||||
811.10 | −0.718490 | + | 1.21810i | − | 1.00000i | −0.967544 | − | 1.75039i | 1.00000 | 1.21810 | + | 0.718490i | 1.92209 | + | 1.81812i | 2.82732 | + | 0.0790702i | −1.00000 | −0.718490 | + | 1.21810i | |||||
811.11 | −0.135580 | − | 1.40770i | 1.00000i | −1.96324 | + | 0.381712i | 1.00000 | 1.40770 | − | 0.135580i | −2.43116 | − | 1.04376i | 0.803512 | + | 2.71189i | −1.00000 | −0.135580 | − | 1.40770i | ||||||
811.12 | −0.135580 | + | 1.40770i | − | 1.00000i | −1.96324 | − | 0.381712i | 1.00000 | 1.40770 | + | 0.135580i | −2.43116 | + | 1.04376i | 0.803512 | − | 2.71189i | −1.00000 | −0.135580 | + | 1.40770i | |||||
811.13 | 0.0653764 | − | 1.41270i | − | 1.00000i | −1.99145 | − | 0.184715i | 1.00000 | −1.41270 | − | 0.0653764i | 1.57556 | + | 2.12547i | −0.391141 | + | 2.80125i | −1.00000 | 0.0653764 | − | 1.41270i | |||||
811.14 | 0.0653764 | + | 1.41270i | 1.00000i | −1.99145 | + | 0.184715i | 1.00000 | −1.41270 | + | 0.0653764i | 1.57556 | − | 2.12547i | −0.391141 | − | 2.80125i | −1.00000 | 0.0653764 | + | 1.41270i | ||||||
811.15 | 0.403065 | − | 1.35556i | 1.00000i | −1.67508 | − | 1.09276i | 1.00000 | 1.35556 | + | 0.403065i | −0.527819 | + | 2.59257i | −2.15646 | + | 1.83021i | −1.00000 | 0.403065 | − | 1.35556i | ||||||
811.16 | 0.403065 | + | 1.35556i | − | 1.00000i | −1.67508 | + | 1.09276i | 1.00000 | 1.35556 | − | 0.403065i | −0.527819 | − | 2.59257i | −2.15646 | − | 1.83021i | −1.00000 | 0.403065 | + | 1.35556i | |||||
811.17 | 0.531410 | − | 1.31057i | − | 1.00000i | −1.43521 | − | 1.39290i | 1.00000 | −1.31057 | − | 0.531410i | −2.64060 | − | 0.165018i | −2.58819 | + | 1.14074i | −1.00000 | 0.531410 | − | 1.31057i | |||||
811.18 | 0.531410 | + | 1.31057i | 1.00000i | −1.43521 | + | 1.39290i | 1.00000 | −1.31057 | + | 0.531410i | −2.64060 | + | 0.165018i | −2.58819 | − | 1.14074i | −1.00000 | 0.531410 | + | 1.31057i | ||||||
811.19 | 0.818985 | − | 1.15294i | 1.00000i | −0.658528 | − | 1.88848i | 1.00000 | 1.15294 | + | 0.818985i | −0.864047 | − | 2.50068i | −2.71662 | − | 0.787391i | −1.00000 | 0.818985 | − | 1.15294i | ||||||
811.20 | 0.818985 | + | 1.15294i | − | 1.00000i | −0.658528 | + | 1.88848i | 1.00000 | 1.15294 | − | 0.818985i | −0.864047 | + | 2.50068i | −2.71662 | + | 0.787391i | −1.00000 | 0.818985 | + | 1.15294i | |||||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
56.e | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 840.2.z.d | yes | 28 |
4.b | odd | 2 | 1 | 3360.2.z.d | 28 | ||
7.b | odd | 2 | 1 | 840.2.z.c | ✓ | 28 | |
8.b | even | 2 | 1 | 3360.2.z.c | 28 | ||
8.d | odd | 2 | 1 | 840.2.z.c | ✓ | 28 | |
28.d | even | 2 | 1 | 3360.2.z.c | 28 | ||
56.e | even | 2 | 1 | inner | 840.2.z.d | yes | 28 |
56.h | odd | 2 | 1 | 3360.2.z.d | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
840.2.z.c | ✓ | 28 | 7.b | odd | 2 | 1 | |
840.2.z.c | ✓ | 28 | 8.d | odd | 2 | 1 | |
840.2.z.d | yes | 28 | 1.a | even | 1 | 1 | trivial |
840.2.z.d | yes | 28 | 56.e | even | 2 | 1 | inner |
3360.2.z.c | 28 | 8.b | even | 2 | 1 | ||
3360.2.z.c | 28 | 28.d | even | 2 | 1 | ||
3360.2.z.d | 28 | 4.b | odd | 2 | 1 | ||
3360.2.z.d | 28 | 56.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\):
\( T_{11}^{14} - 84 T_{11}^{12} - 40 T_{11}^{11} + 2516 T_{11}^{10} + 2512 T_{11}^{9} - 33200 T_{11}^{8} + \cdots + 16384 \) |
\( T_{13}^{14} + 4 T_{13}^{13} - 68 T_{13}^{12} - 176 T_{13}^{11} + 1812 T_{13}^{10} + 2160 T_{13}^{9} + \cdots - 32768 \) |