Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [418,2,Mod(107,418)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(418, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([9, 25]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("418.107");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 418 = 2 \cdot 11 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 418.s (of order \(30\), 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: | \(3.33774680449\) |
Analytic rank: | \(0\) |
Dimension: | \(80\) |
Relative dimension: | \(10\) over \(\Q(\zeta_{30})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{30}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
107.1 | −0.913545 | − | 0.406737i | −1.94271 | − | 1.74922i | 0.669131 | + | 0.743145i | 0.100715 | − | 0.958243i | 1.06328 | + | 2.38816i | −0.0448322 | − | 0.0145669i | −0.309017 | − | 0.951057i | 0.400750 | + | 3.81288i | −0.481761 | + | 0.834434i |
107.2 | −0.913545 | − | 0.406737i | −1.85282 | − | 1.66828i | 0.669131 | + | 0.743145i | −0.211376 | + | 2.01110i | 1.01408 | + | 2.27766i | 0.508834 | + | 0.165330i | −0.309017 | − | 0.951057i | 0.336173 | + | 3.19847i | 1.01109 | − | 1.75126i |
107.3 | −0.913545 | − | 0.406737i | −1.47425 | − | 1.32742i | 0.669131 | + | 0.743145i | 0.214664 | − | 2.04239i | 0.806884 | + | 1.81229i | −4.92396 | − | 1.59989i | −0.309017 | − | 0.951057i | 0.0977827 | + | 0.930340i | −1.02682 | + | 1.77850i |
107.4 | −0.913545 | − | 0.406737i | −0.808456 | − | 0.727937i | 0.669131 | + | 0.743145i | 0.425216 | − | 4.04566i | 0.442482 | + | 0.993832i | 4.38637 | + | 1.42522i | −0.309017 | − | 0.951057i | −0.189877 | − | 1.80656i | −2.03397 | + | 3.52294i |
107.5 | −0.913545 | − | 0.406737i | −0.370496 | − | 0.333596i | 0.669131 | + | 0.743145i | −0.348800 | + | 3.31861i | 0.202779 | + | 0.455450i | 2.27900 | + | 0.740493i | −0.309017 | − | 0.951057i | −0.287604 | − | 2.73637i | 1.66844 | − | 2.88983i |
107.6 | −0.913545 | − | 0.406737i | 0.0608391 | + | 0.0547798i | 0.669131 | + | 0.743145i | −0.122891 | + | 1.16923i | −0.0332984 | − | 0.0747894i | −0.576821 | − | 0.187421i | −0.309017 | − | 0.951057i | −0.312885 | − | 2.97690i | 0.587836 | − | 1.01816i |
107.7 | −0.913545 | − | 0.406737i | 0.116360 | + | 0.104771i | 0.669131 | + | 0.743145i | 0.258207 | − | 2.45667i | −0.0636859 | − | 0.143041i | −0.618608 | − | 0.200998i | −0.309017 | − | 0.951057i | −0.311023 | − | 2.95918i | −1.23510 | + | 2.13926i |
107.8 | −0.913545 | − | 0.406737i | 1.34081 | + | 1.20727i | 0.669131 | + | 0.743145i | 0.264977 | − | 2.52108i | −0.733850 | − | 1.64825i | −0.512875 | − | 0.166643i | −0.309017 | − | 0.951057i | 0.0266838 | + | 0.253879i | −1.26749 | + | 2.19535i |
107.9 | −0.913545 | − | 0.406737i | 2.10303 | + | 1.89358i | 0.669131 | + | 0.743145i | −0.360171 | + | 3.42680i | −1.15103 | − | 2.58525i | 3.62558 | + | 1.17802i | −0.309017 | − | 0.951057i | 0.523516 | + | 4.98092i | 1.72284 | − | 2.98404i |
107.10 | −0.913545 | − | 0.406737i | 2.22316 | + | 2.00174i | 0.669131 | + | 0.743145i | −0.155938 | + | 1.48365i | −1.21677 | − | 2.73292i | −4.91364 | − | 1.59654i | −0.309017 | − | 0.951057i | 0.621881 | + | 5.91681i | 0.745913 | − | 1.29196i |
145.1 | 0.978148 | − | 0.207912i | −1.24186 | + | 2.78926i | 0.913545 | − | 0.406737i | −1.24858 | + | 1.38669i | −0.634802 | + | 2.98651i | −0.788641 | + | 1.08547i | 0.809017 | − | 0.587785i | −4.23038 | − | 4.69832i | −0.932989 | + | 1.61598i |
145.2 | 0.978148 | − | 0.207912i | −0.879298 | + | 1.97493i | 0.913545 | − | 0.406737i | 0.155783 | − | 0.173014i | −0.449471 | + | 2.11459i | 2.76336 | − | 3.80344i | 0.809017 | − | 0.587785i | −1.11981 | − | 1.24368i | 0.116407 | − | 0.201622i |
145.3 | 0.978148 | − | 0.207912i | −0.766319 | + | 1.72118i | 0.913545 | − | 0.406737i | 1.42537 | − | 1.58303i | −0.391719 | + | 1.84289i | 0.188148 | − | 0.258963i | 0.809017 | − | 0.587785i | −0.367824 | − | 0.408510i | 1.06509 | − | 1.84479i |
145.4 | 0.978148 | − | 0.207912i | −0.222075 | + | 0.498789i | 0.913545 | − | 0.406737i | −0.572934 | + | 0.636307i | −0.113518 | + | 0.534061i | −2.17900 | + | 2.99913i | 0.809017 | − | 0.587785i | 1.80792 | + | 2.00790i | −0.428118 | + | 0.741522i |
145.5 | 0.978148 | − | 0.207912i | −0.167101 | + | 0.375316i | 0.913545 | − | 0.406737i | 1.82285 | − | 2.02448i | −0.0854173 | + | 0.401857i | 0.891945 | − | 1.22766i | 0.809017 | − | 0.587785i | 1.89445 | + | 2.10400i | 1.36210 | − | 2.35923i |
145.6 | 0.978148 | − | 0.207912i | 0.100523 | − | 0.225779i | 0.913545 | − | 0.406737i | −2.22917 | + | 2.47574i | 0.0513844 | − | 0.241745i | −1.25716 | + | 1.73034i | 0.809017 | − | 0.587785i | 1.96652 | + | 2.18404i | −1.66572 | + | 2.88511i |
145.7 | 0.978148 | − | 0.207912i | 0.411849 | − | 0.925027i | 0.913545 | − | 0.406737i | 2.71323 | − | 3.01335i | 0.210525 | − | 0.990442i | −2.18824 | + | 3.01185i | 0.809017 | − | 0.587785i | 1.32134 | + | 1.46749i | 2.02743 | − | 3.51161i |
145.8 | 0.978148 | − | 0.207912i | 0.588176 | − | 1.32107i | 0.913545 | − | 0.406737i | −1.28088 | + | 1.42257i | 0.300658 | − | 1.41449i | 1.62246 | − | 2.23312i | 0.809017 | − | 0.587785i | 0.608128 | + | 0.675395i | −0.957126 | + | 1.65779i |
145.9 | 0.978148 | − | 0.207912i | 1.00011 | − | 2.24629i | 0.913545 | − | 0.406737i | 0.0731199 | − | 0.0812078i | 0.511227 | − | 2.40514i | 1.06798 | − | 1.46995i | 0.809017 | − | 0.587785i | −2.03819 | − | 2.26364i | 0.0546380 | − | 0.0946357i |
145.10 | 0.978148 | − | 0.207912i | 1.34512 | − | 3.02120i | 0.913545 | − | 0.406737i | 0.223897 | − | 0.248663i | 0.687587 | − | 3.23484i | −2.45911 | + | 3.38468i | 0.809017 | − | 0.587785i | −5.31088 | − | 5.89833i | 0.167304 | − | 0.289780i |
See all 80 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
209.t | even | 30 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 418.2.s.a | ✓ | 80 |
11.d | odd | 10 | 1 | 418.2.s.b | yes | 80 | |
19.d | odd | 6 | 1 | 418.2.s.b | yes | 80 | |
209.t | even | 30 | 1 | inner | 418.2.s.a | ✓ | 80 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
418.2.s.a | ✓ | 80 | 1.a | even | 1 | 1 | trivial |
418.2.s.a | ✓ | 80 | 209.t | even | 30 | 1 | inner |
418.2.s.b | yes | 80 | 11.d | odd | 10 | 1 | |
418.2.s.b | yes | 80 | 19.d | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{80} + 3 T_{3}^{79} + 25 T_{3}^{78} + 91 T_{3}^{77} + 289 T_{3}^{76} + 940 T_{3}^{75} + \cdots + 429981696 \) acting on \(S_{2}^{\mathrm{new}}(418, [\chi])\).