Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [668,2,Mod(15,668)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(668, base_ring=CyclotomicField(166))
chi = DirichletCharacter(H, H._module([83, 95]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("668.15");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 668 = 2^{2} \cdot 167 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 668.h (of order \(166\), degree \(82\), 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: | \(5.33400685502\) |
Analytic rank: | \(0\) |
Dimension: | \(6724\) |
Relative dimension: | \(82\) over \(\Q(\zeta_{166})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{166}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
15.1 | −1.41330 | − | 0.0508355i | −1.89908 | − | 2.53233i | 1.99483 | + | 0.143692i | 0.0776745 | − | 0.336123i | 2.55524 | + | 3.67549i | 1.39922 | + | 0.132799i | −2.81199 | − | 0.304488i | −1.96595 | + | 6.73829i | −0.126864 | + | 0.471094i |
15.2 | −1.41324 | + | 0.0524176i | 0.519888 | + | 0.693245i | 1.99450 | − | 0.148157i | 0.730855 | − | 3.16265i | −0.771066 | − | 0.952472i | 3.90748 | + | 0.370857i | −2.81095 | + | 0.313930i | 0.629939 | − | 2.15911i | −0.867096 | + | 4.50789i |
15.3 | −1.40266 | − | 0.180378i | −1.32817 | − | 1.77105i | 1.93493 | + | 0.506019i | 0.679950 | − | 2.94236i | 1.54351 | + | 2.72375i | −1.65201 | − | 0.156792i | −2.62278 | − | 1.05879i | −0.532332 | + | 1.82456i | −1.48448 | + | 4.00449i |
15.4 | −1.40204 | − | 0.185136i | 0.267796 | + | 0.357093i | 1.93145 | + | 0.519139i | −0.750457 | + | 3.24747i | −0.309351 | − | 0.550239i | −1.50739 | − | 0.143066i | −2.61186 | − | 1.08544i | 0.784443 | − | 2.68867i | 1.65340 | − | 4.41416i |
15.5 | −1.39758 | + | 0.216291i | 1.40535 | + | 1.87397i | 1.90644 | − | 0.604567i | −0.352669 | + | 1.52611i | −2.36941 | − | 2.31505i | −3.78061 | − | 0.358817i | −2.53363 | + | 1.25727i | −0.696501 | + | 2.38725i | 0.162797 | − | 2.20914i |
15.6 | −1.37664 | − | 0.323827i | 1.92126 | + | 2.56191i | 1.79027 | + | 0.891587i | 0.303580 | − | 1.31369i | −1.81527 | − | 4.14898i | 1.06823 | + | 0.101385i | −2.17584 | − | 1.80713i | −2.03189 | + | 6.96427i | −0.843329 | + | 1.71017i |
15.7 | −1.36880 | − | 0.355518i | −0.199932 | − | 0.266599i | 1.74721 | + | 0.973265i | −0.435572 | + | 1.88486i | 0.178885 | + | 0.436000i | 2.28482 | + | 0.216852i | −2.04557 | − | 1.95337i | 0.809142 | − | 2.77333i | 1.26631 | − | 2.42514i |
15.8 | −1.35899 | + | 0.391340i | 1.57222 | + | 2.09648i | 1.69371 | − | 1.06365i | −0.776138 | + | 3.35860i | −2.95707 | − | 2.23382i | 4.43983 | + | 0.421383i | −1.88548 | + | 2.10831i | −1.08311 | + | 3.71235i | −0.259592 | − | 4.86804i |
15.9 | −1.35747 | − | 0.396595i | 1.06894 | + | 1.42538i | 1.68542 | + | 1.07673i | 0.441016 | − | 1.90842i | −0.885753 | − | 2.35885i | −3.35265 | − | 0.318199i | −1.86088 | − | 2.13005i | −0.0488343 | + | 0.167379i | −1.35553 | + | 2.41571i |
15.10 | −1.35512 | + | 0.404546i | −0.666088 | − | 0.888195i | 1.67269 | − | 1.09641i | 0.185473 | − | 0.802603i | 1.26194 | + | 0.934146i | 0.603158 | + | 0.0572456i | −1.82314 | + | 2.16245i | 0.495026 | − | 1.69670i | 0.0733515 | + | 1.16265i |
15.11 | −1.30737 | + | 0.539237i | 0.666088 | + | 0.888195i | 1.41845 | − | 1.40997i | 0.185473 | − | 0.802603i | −1.34977 | − | 0.802023i | −0.603158 | − | 0.0572456i | −1.09413 | + | 2.60823i | 0.495026 | − | 1.69670i | 0.190310 | + | 1.14932i |
15.12 | −1.30206 | + | 0.551933i | −1.57222 | − | 2.09648i | 1.39074 | − | 1.43730i | −0.776138 | + | 3.35860i | 3.20425 | + | 1.86199i | −4.43983 | − | 0.421383i | −1.01754 | + | 2.63906i | −1.08311 | + | 3.71235i | −0.843142 | − | 4.80149i |
15.13 | −1.22179 | + | 0.712204i | −1.40535 | − | 1.87397i | 0.985531 | − | 1.74032i | −0.352669 | + | 1.52611i | 3.05169 | + | 1.28870i | 3.78061 | + | 0.358817i | 0.0353553 | + | 2.82821i | −0.696501 | + | 2.38725i | −0.656017 | − | 2.11576i |
15.14 | −1.21190 | − | 0.728895i | −1.02052 | − | 1.36081i | 0.937425 | + | 1.76670i | −0.0807078 | + | 0.349249i | 0.244883 | + | 2.39303i | −3.47450 | − | 0.329764i | 0.151670 | − | 2.82436i | 0.0298911 | − | 0.102452i | 0.352376 | − | 0.364429i |
15.15 | −1.16508 | − | 0.801617i | −1.64821 | − | 2.19780i | 0.714821 | + | 1.86789i | −0.943687 | + | 4.08364i | 0.158496 | + | 3.88184i | 0.269428 | + | 0.0255713i | 0.664513 | − | 2.74926i | −1.27351 | + | 4.36494i | 4.37298 | − | 4.00129i |
15.16 | −1.13075 | + | 0.849359i | −0.519888 | − | 0.693245i | 0.557178 | − | 1.92082i | 0.730855 | − | 3.16265i | 1.17668 | + | 0.342313i | −3.90748 | − | 0.370857i | 1.00144 | + | 2.64521i | 0.629939 | − | 2.15911i | 1.85981 | + | 4.19691i |
15.17 | −1.12956 | − | 0.850944i | −0.939846 | − | 1.25324i | 0.551790 | + | 1.92238i | 0.224144 | − | 0.969945i | −0.00482749 | + | 2.21536i | 3.52889 | + | 0.334926i | 1.01256 | − | 2.64097i | 0.152947 | − | 0.524225i | −1.07855 | + | 0.904872i |
15.18 | −1.06576 | + | 0.929597i | 1.89908 | + | 2.53233i | 0.271697 | − | 1.98146i | 0.0776745 | − | 0.336123i | −4.37802 | − | 0.933483i | −1.39922 | − | 0.132799i | 1.55239 | + | 2.36433i | −1.96595 | + | 6.73829i | 0.229676 | + | 0.430433i |
15.19 | −1.03337 | − | 0.965479i | 0.777226 | + | 1.03639i | 0.135700 | + | 1.99539i | 0.786467 | − | 3.40330i | 0.197455 | − | 1.82137i | −2.18353 | − | 0.207238i | 1.78628 | − | 2.19299i | 0.370215 | − | 1.26891i | −4.09852 | + | 2.75754i |
15.20 | −1.02853 | − | 0.970630i | 0.994607 | + | 1.32626i | 0.115755 | + | 1.99665i | −0.166755 | + | 0.721602i | 0.264323 | − | 2.32950i | 4.08470 | + | 0.387677i | 1.81895 | − | 2.16597i | 0.0705222 | − | 0.241714i | 0.871922 | − | 0.580334i |
See next 80 embeddings (of 6724 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
167.d | odd | 166 | 1 | inner |
668.h | even | 166 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 668.2.h.a | ✓ | 6724 |
4.b | odd | 2 | 1 | inner | 668.2.h.a | ✓ | 6724 |
167.d | odd | 166 | 1 | inner | 668.2.h.a | ✓ | 6724 |
668.h | even | 166 | 1 | inner | 668.2.h.a | ✓ | 6724 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
668.2.h.a | ✓ | 6724 | 1.a | even | 1 | 1 | trivial |
668.2.h.a | ✓ | 6724 | 4.b | odd | 2 | 1 | inner |
668.2.h.a | ✓ | 6724 | 167.d | odd | 166 | 1 | inner |
668.2.h.a | ✓ | 6724 | 668.h | even | 166 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(668, [\chi])\).