Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [700,2,Mod(81,700)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(700, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([0, 12, 20]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("700.81");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 700.bg (of order \(15\), degree \(8\), 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: | \(5.58952814149\) |
Analytic rank: | \(0\) |
Dimension: | \(160\) |
Relative dimension: | \(20\) over \(\Q(\zeta_{15})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{15}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
81.1 | 0 | −0.297443 | − | 2.82998i | 0 | 0.228995 | + | 2.22431i | 0 | −1.76280 | − | 1.97295i | 0 | −4.98590 | + | 1.05978i | 0 | ||||||||||
81.2 | 0 | −0.296036 | − | 2.81659i | 0 | −1.05498 | − | 1.97155i | 0 | −1.22751 | − | 2.34376i | 0 | −4.91111 | + | 1.04389i | 0 | ||||||||||
81.3 | 0 | −0.272347 | − | 2.59120i | 0 | 1.41505 | + | 1.73137i | 0 | −0.381973 | + | 2.61803i | 0 | −3.70573 | + | 0.787676i | 0 | ||||||||||
81.4 | 0 | −0.260227 | − | 2.47589i | 0 | −1.74304 | + | 1.40065i | 0 | 2.53743 | − | 0.749316i | 0 | −3.12789 | + | 0.664854i | 0 | ||||||||||
81.5 | 0 | −0.221467 | − | 2.10712i | 0 | −2.22616 | − | 0.210296i | 0 | 1.31775 | + | 2.29424i | 0 | −1.45644 | + | 0.309577i | 0 | ||||||||||
81.6 | 0 | −0.164810 | − | 1.56806i | 0 | 1.06004 | − | 1.96884i | 0 | 2.59162 | − | 0.532438i | 0 | 0.502795 | − | 0.106872i | 0 | ||||||||||
81.7 | 0 | −0.115689 | − | 1.10070i | 0 | −0.0767094 | − | 2.23475i | 0 | −0.987418 | + | 2.45459i | 0 | 1.73628 | − | 0.369057i | 0 | ||||||||||
81.8 | 0 | −0.0806619 | − | 0.767447i | 0 | 2.11029 | − | 0.739383i | 0 | 0.187493 | − | 2.63910i | 0 | 2.35197 | − | 0.499928i | 0 | ||||||||||
81.9 | 0 | −0.0610054 | − | 0.580427i | 0 | 2.23452 | + | 0.0832050i | 0 | −2.51126 | − | 0.832804i | 0 | 2.60127 | − | 0.552917i | 0 | ||||||||||
81.10 | 0 | −0.0323219 | − | 0.307522i | 0 | −2.23362 | − | 0.104508i | 0 | −2.61097 | + | 0.427582i | 0 | 2.84092 | − | 0.603856i | 0 | ||||||||||
81.11 | 0 | −0.0229598 | − | 0.218448i | 0 | 1.54824 | + | 1.61337i | 0 | 2.37030 | + | 1.17545i | 0 | 2.88725 | − | 0.613704i | 0 | ||||||||||
81.12 | 0 | 0.0118737 | + | 0.112971i | 0 | −1.67957 | + | 1.47616i | 0 | 1.14088 | − | 2.38713i | 0 | 2.92182 | − | 0.621052i | 0 | ||||||||||
81.13 | 0 | 0.160475 | + | 1.52682i | 0 | −2.09374 | − | 0.785010i | 0 | 1.96337 | + | 1.77347i | 0 | 0.629027 | − | 0.133704i | 0 | ||||||||||
81.14 | 0 | 0.162396 | + | 1.54509i | 0 | 0.272993 | + | 2.21934i | 0 | −0.0456598 | − | 2.64536i | 0 | 0.573508 | − | 0.121903i | 0 | ||||||||||
81.15 | 0 | 0.173464 | + | 1.65040i | 0 | 0.0779163 | − | 2.23471i | 0 | 2.53599 | − | 0.754166i | 0 | 0.240704 | − | 0.0511632i | 0 | ||||||||||
81.16 | 0 | 0.180708 | + | 1.71932i | 0 | −0.0662493 | − | 2.23509i | 0 | −2.63897 | − | 0.189349i | 0 | 0.0110297 | − | 0.00234443i | 0 | ||||||||||
81.17 | 0 | 0.212873 | + | 2.02535i | 0 | 2.17143 | − | 0.533747i | 0 | −0.472995 | + | 2.60313i | 0 | −1.12230 | + | 0.238553i | 0 | ||||||||||
81.18 | 0 | 0.225525 | + | 2.14572i | 0 | −0.830377 | + | 2.07617i | 0 | −0.864256 | + | 2.50061i | 0 | −1.61882 | + | 0.344092i | 0 | ||||||||||
81.19 | 0 | 0.340003 | + | 3.23492i | 0 | −2.21544 | − | 0.302996i | 0 | −1.66018 | − | 2.06005i | 0 | −7.41464 | + | 1.57603i | 0 | ||||||||||
81.20 | 0 | 0.357649 | + | 3.40280i | 0 | 2.12228 | + | 0.704219i | 0 | 2.44622 | − | 1.00797i | 0 | −8.51670 | + | 1.81028i | 0 | ||||||||||
See next 80 embeddings (of 160 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
25.d | even | 5 | 1 | inner |
175.q | even | 15 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 700.2.bg.a | ✓ | 160 |
7.c | even | 3 | 1 | inner | 700.2.bg.a | ✓ | 160 |
25.d | even | 5 | 1 | inner | 700.2.bg.a | ✓ | 160 |
175.q | even | 15 | 1 | inner | 700.2.bg.a | ✓ | 160 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
700.2.bg.a | ✓ | 160 | 1.a | even | 1 | 1 | trivial |
700.2.bg.a | ✓ | 160 | 7.c | even | 3 | 1 | inner |
700.2.bg.a | ✓ | 160 | 25.d | even | 5 | 1 | inner |
700.2.bg.a | ✓ | 160 | 175.q | even | 15 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(700, [\chi])\).