Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [547,2,Mod(46,547)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(547, base_ring=CyclotomicField(26))
chi = DirichletCharacter(H, H._module([14]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("547.46");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 547 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 547.f (of order \(13\), degree \(12\), 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: | \(4.36781699056\) |
Analytic rank: | \(0\) |
Dimension: | \(528\) |
Relative dimension: | \(44\) over \(\Q(\zeta_{13})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{13}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
46.1 | −0.335888 | + | 2.76629i | 1.13158 | −5.59764 | − | 1.37970i | −0.153172 | − | 0.403882i | −0.380085 | + | 3.13028i | −0.393363 | − | 3.23964i | 3.72053 | − | 9.81022i | −1.71952 | 1.16870 | − | 0.288060i | ||||
46.2 | −0.311246 | + | 2.56334i | −2.48877 | −4.53197 | − | 1.11703i | 0.771833 | + | 2.03516i | 0.774620 | − | 6.37957i | −0.224653 | − | 1.85018i | 2.44259 | − | 6.44059i | 3.19397 | −5.45703 | + | 1.34504i | ||||
46.3 | −0.305966 | + | 2.51985i | 0.544519 | −4.31417 | − | 1.06335i | 0.299331 | + | 0.789270i | −0.166604 | + | 1.37211i | 0.466827 | + | 3.84467i | 2.19924 | − | 5.79891i | −2.70350 | −2.08043 | + | 0.512780i | ||||
46.4 | −0.304286 | + | 2.50602i | −1.95070 | −4.24565 | − | 1.04646i | −1.15766 | − | 3.05250i | 0.593571 | − | 4.88850i | 0.109988 | + | 0.905836i | 2.12398 | − | 5.60049i | 0.805247 | 8.00187 | − | 1.97228i | ||||
46.5 | −0.268734 | + | 2.21322i | 2.12681 | −2.88424 | − | 0.710902i | −0.947553 | − | 2.49849i | −0.571546 | + | 4.70710i | −0.292661 | − | 2.41028i | 0.767309 | − | 2.02323i | 1.52333 | 5.78435 | − | 1.42572i | ||||
46.6 | −0.255736 | + | 2.10617i | 1.99966 | −2.42868 | − | 0.598615i | 0.984637 | + | 2.59628i | −0.511385 | + | 4.21164i | 0.216641 | + | 1.78420i | 0.377195 | − | 0.994582i | 0.998658 | −5.72001 | + | 1.40986i | ||||
46.7 | −0.245329 | + | 2.02046i | 0.136449 | −2.08020 | − | 0.512724i | 1.38996 | + | 3.66502i | −0.0334750 | + | 0.275691i | −0.418952 | − | 3.45038i | 0.102818 | − | 0.271108i | −2.98138 | −7.74605 | + | 1.90923i | ||||
46.8 | −0.236626 | + | 1.94879i | −2.82258 | −1.79991 | − | 0.443638i | 0.468139 | + | 1.23438i | 0.667896 | − | 5.50062i | 0.189324 | + | 1.55922i | −0.101791 | + | 0.268400i | 4.96697 | −2.51632 | + | 0.620218i | ||||
46.9 | −0.234924 | + | 1.93478i | 0.630029 | −1.74629 | − | 0.430421i | −1.10146 | − | 2.90432i | −0.148009 | + | 1.21897i | −0.214681 | − | 1.76806i | −0.139228 | + | 0.367113i | −2.60306 | 5.87797 | − | 1.44879i | ||||
46.10 | −0.192434 | + | 1.58483i | 2.35170 | −0.532783 | − | 0.131319i | −0.750458 | − | 1.97880i | −0.452547 | + | 3.72706i | 0.413115 | + | 3.40231i | −0.821590 | + | 2.16636i | 2.53051 | 3.28047 | − | 0.808564i | ||||
46.11 | −0.188009 | + | 1.54840i | −0.908881 | −0.420298 | − | 0.103594i | −0.0244243 | − | 0.0644017i | 0.170878 | − | 1.40731i | −0.367947 | − | 3.03032i | −0.866778 | + | 2.28551i | −2.17394 | 0.104311 | − | 0.0257104i | ||||
46.12 | −0.167985 | + | 1.38348i | 2.55202 | 0.0560731 | + | 0.0138208i | 0.714436 | + | 1.88381i | −0.428702 | + | 3.53068i | −0.0490013 | − | 0.403562i | −1.01693 | + | 2.68142i | 3.51279 | −2.72624 | + | 0.671958i | ||||
46.13 | −0.166856 | + | 1.37418i | −1.48276 | 0.0813480 | + | 0.0200505i | −0.394072 | − | 1.03908i | 0.247408 | − | 2.03759i | −0.324019 | − | 2.66853i | −1.02287 | + | 2.69708i | −0.801408 | 1.49364 | − | 0.368149i | ||||
46.14 | −0.149053 | + | 1.22756i | 0.296875 | 0.457198 | + | 0.112689i | −0.175522 | − | 0.462814i | −0.0442500 | + | 0.364432i | 0.391638 | + | 3.22543i | −1.08347 | + | 2.85688i | −2.91187 | 0.594294 | − | 0.146480i | ||||
46.15 | −0.142855 | + | 1.17652i | −1.73902 | 0.578096 | + | 0.142488i | 1.43759 | + | 3.79062i | 0.248428 | − | 2.04599i | 0.544189 | + | 4.48180i | −1.09075 | + | 2.87607i | 0.0242007 | −4.66510 | + | 1.14984i | ||||
46.16 | −0.137944 | + | 1.13607i | −2.73564 | 0.670252 | + | 0.165202i | −1.11149 | − | 2.93075i | 0.377365 | − | 3.10788i | 0.373963 | + | 3.07987i | −1.09177 | + | 2.87876i | 4.48371 | 3.48287 | − | 0.858450i | ||||
46.17 | −0.0945775 | + | 0.778916i | 3.19456 | 1.34412 | + | 0.331295i | −0.702506 | − | 1.85236i | −0.302134 | + | 2.48830i | −0.187976 | − | 1.54812i | −0.941647 | + | 2.48292i | 7.20523 | 1.50927 | − | 0.372002i | ||||
46.18 | −0.0547827 | + | 0.451176i | 1.64947 | 1.74132 | + | 0.429198i | 0.558494 | + | 1.47263i | −0.0903624 | + | 0.744201i | −0.347117 | − | 2.85877i | −0.611367 | + | 1.61204i | −0.279256 | −0.695011 | + | 0.171305i | ||||
46.19 | −0.0443771 | + | 0.365478i | −0.533473 | 1.81028 | + | 0.446194i | 0.569048 | + | 1.50046i | 0.0236740 | − | 0.194973i | 0.225842 | + | 1.85997i | −0.504513 | + | 1.33029i | −2.71541 | −0.573637 | + | 0.141389i | ||||
46.20 | −0.0420575 | + | 0.346375i | −3.30487 | 1.82368 | + | 0.449496i | 1.10374 | + | 2.91033i | 0.138995 | − | 1.14473i | −0.459333 | − | 3.78294i | −0.479850 | + | 1.26526i | 7.92218 | −1.05449 | + | 0.259908i | ||||
See next 80 embeddings (of 528 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
547.f | even | 13 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 547.2.f.a | ✓ | 528 |
547.f | even | 13 | 1 | inner | 547.2.f.a | ✓ | 528 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
547.2.f.a | ✓ | 528 | 1.a | even | 1 | 1 | trivial |
547.2.f.a | ✓ | 528 | 547.f | even | 13 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(547, [\chi])\).