Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [380,2,Mod(31,380)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(380, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("380.31");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 380 = 2^{2} \cdot 5 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 380.n (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: | \(3.03431527681\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(20\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
31.1 | −1.41089 | − | 0.0968610i | 0.717455 | + | 1.24267i | 1.98124 | + | 0.273321i | −0.500000 | − | 0.866025i | −0.891886 | − | 1.82277i | − | 2.27676i | −2.76884 | − | 0.577531i | 0.470516 | − | 0.814958i | 0.621562 | + | 1.27030i | |
31.2 | −1.33235 | − | 0.474182i | −0.435478 | − | 0.754271i | 1.55030 | + | 1.26355i | −0.500000 | − | 0.866025i | 0.222547 | + | 1.21145i | 3.15791i | −1.46639 | − | 2.41862i | 1.12072 | − | 1.94114i | 0.255520 | + | 1.39094i | ||
31.3 | −1.31582 | + | 0.518293i | −0.736860 | − | 1.27628i | 1.46274 | − | 1.36396i | −0.500000 | − | 0.866025i | 1.63106 | + | 1.29744i | 0.456236i | −1.21777 | + | 2.55285i | 0.414074 | − | 0.717197i | 1.10676 | + | 0.880384i | ||
31.4 | −1.18980 | − | 0.764449i | −1.38246 | − | 2.39450i | 0.831236 | + | 1.81908i | −0.500000 | − | 0.866025i | −0.185619 | + | 3.90579i | − | 2.18272i | 0.401590 | − | 2.79977i | −2.32241 | + | 4.02254i | −0.0671334 | + | 1.41262i | |
31.5 | −1.10676 | + | 0.880384i | 0.736860 | + | 1.27628i | 0.449849 | − | 1.94875i | −0.500000 | − | 0.866025i | −1.93915 | − | 0.763819i | − | 0.456236i | 1.21777 | + | 2.55285i | 0.414074 | − | 0.717197i | 1.31582 | + | 0.518293i | |
31.6 | −0.785164 | − | 1.17623i | 1.08269 | + | 1.87527i | −0.767036 | + | 1.84707i | −0.500000 | − | 0.866025i | 1.35566 | − | 2.74588i | − | 3.94804i | 2.77483 | − | 0.548038i | −0.844418 | + | 1.46258i | −0.626064 | + | 1.26809i | |
31.7 | −0.621562 | + | 1.27030i | −0.717455 | − | 1.24267i | −1.22732 | − | 1.57914i | −0.500000 | − | 0.866025i | 2.02450 | − | 0.138987i | 2.27676i | 2.76884 | − | 0.577531i | 0.470516 | − | 0.814958i | 1.41089 | − | 0.0968610i | ||
31.8 | −0.255520 | + | 1.39094i | 0.435478 | + | 0.754271i | −1.86942 | − | 0.710826i | −0.500000 | − | 0.866025i | −1.16042 | + | 0.412992i | − | 3.15791i | 1.46639 | − | 2.41862i | 1.12072 | − | 1.94114i | 1.33235 | − | 0.474182i | |
31.9 | −0.0867802 | − | 1.41155i | 0.980171 | + | 1.69771i | −1.98494 | + | 0.244989i | −0.500000 | − | 0.866025i | 2.31133 | − | 1.53089i | 1.97181i | 0.518067 | + | 2.78058i | −0.421469 | + | 0.730006i | −1.17905 | + | 0.780928i | ||
31.10 | 0.0669309 | − | 1.41263i | −0.665205 | − | 1.15217i | −1.99104 | − | 0.189097i | −0.500000 | − | 0.866025i | −1.67211 | + | 0.862572i | − | 4.68878i | −0.400386 | + | 2.79994i | 0.615006 | − | 1.06522i | −1.25684 | + | 0.648351i | |
31.11 | 0.0671334 | + | 1.41262i | 1.38246 | + | 2.39450i | −1.99099 | + | 0.189668i | −0.500000 | − | 0.866025i | −3.28970 | + | 2.11365i | 2.18272i | −0.401590 | − | 2.79977i | −2.32241 | + | 4.02254i | 1.18980 | − | 0.764449i | ||
31.12 | 0.337532 | − | 1.37334i | −1.50046 | − | 2.59888i | −1.77214 | − | 0.927096i | −0.500000 | − | 0.866025i | −4.07560 | + | 1.18344i | 4.06445i | −1.87138 | + | 2.12084i | −3.00277 | + | 5.20095i | −1.35812 | + | 0.394360i | ||
31.13 | 0.626064 | + | 1.26809i | −1.08269 | − | 1.87527i | −1.21609 | + | 1.58781i | −0.500000 | − | 0.866025i | 1.70017 | − | 2.54698i | 3.94804i | −2.77483 | − | 0.548038i | −0.844418 | + | 1.46258i | 0.785164 | − | 1.17623i | ||
31.14 | 0.864116 | − | 1.11951i | 0.0154003 | + | 0.0266742i | −0.506608 | − | 1.93477i | −0.500000 | − | 0.866025i | 0.0431697 | + | 0.00580873i | 0.370928i | −2.60377 | − | 1.10472i | 1.49953 | − | 2.59725i | −1.40158 | − | 0.188591i | ||
31.15 | 1.17905 | + | 0.780928i | −0.980171 | − | 1.69771i | 0.780303 | + | 1.84150i | −0.500000 | − | 0.866025i | 0.170119 | − | 2.76712i | − | 1.97181i | −0.518067 | + | 2.78058i | −0.421469 | + | 0.730006i | 0.0867802 | − | 1.41155i | |
31.16 | 1.19364 | − | 0.758433i | 1.50479 | + | 2.60637i | 0.849559 | − | 1.81059i | −0.500000 | − | 0.866025i | 3.77293 | + | 1.96979i | 2.59896i | −0.359145 | − | 2.80553i | −3.02877 | + | 5.24598i | −1.25364 | − | 0.654507i | ||
31.17 | 1.25364 | − | 0.654507i | −1.50479 | − | 2.60637i | 1.14324 | − | 1.64104i | −0.500000 | − | 0.866025i | −3.59235 | − | 2.28256i | − | 2.59896i | 0.359145 | − | 2.80553i | −3.02877 | + | 5.24598i | −1.19364 | − | 0.758433i | |
31.18 | 1.25684 | + | 0.648351i | 0.665205 | + | 1.15217i | 1.15928 | + | 1.62974i | −0.500000 | − | 0.866025i | 0.0890455 | + | 1.87937i | 4.68878i | 0.400386 | + | 2.79994i | 0.615006 | − | 1.06522i | −0.0669309 | − | 1.41263i | ||
31.19 | 1.35812 | + | 0.394360i | 1.50046 | + | 2.59888i | 1.68896 | + | 1.07117i | −0.500000 | − | 0.866025i | 1.01291 | + | 4.12130i | − | 4.06445i | 1.87138 | + | 2.12084i | −3.00277 | + | 5.20095i | −0.337532 | − | 1.37334i | |
31.20 | 1.40158 | − | 0.188591i | −0.0154003 | − | 0.0266742i | 1.92887 | − | 0.528652i | −0.500000 | − | 0.866025i | −0.0266153 | − | 0.0344817i | − | 0.370928i | 2.60377 | − | 1.10472i | 1.49953 | − | 2.59725i | −0.864116 | − | 1.11951i | |
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
19.d | odd | 6 | 1 | inner |
76.f | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 380.2.n.b | ✓ | 40 |
4.b | odd | 2 | 1 | inner | 380.2.n.b | ✓ | 40 |
19.d | odd | 6 | 1 | inner | 380.2.n.b | ✓ | 40 |
76.f | even | 6 | 1 | inner | 380.2.n.b | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
380.2.n.b | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
380.2.n.b | ✓ | 40 | 4.b | odd | 2 | 1 | inner |
380.2.n.b | ✓ | 40 | 19.d | odd | 6 | 1 | inner |
380.2.n.b | ✓ | 40 | 76.f | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} + 41 T_{3}^{38} + 976 T_{3}^{36} + 15639 T_{3}^{34} + 187640 T_{3}^{32} + 1737687 T_{3}^{30} + \cdots + 4096 \) acting on \(S_{2}^{\mathrm{new}}(380, [\chi])\).