Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [185,3,Mod(19,185)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(185, base_ring=CyclotomicField(36))
chi = DirichletCharacter(H, H._module([18, 35]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("185.19");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 185 = 5 \cdot 37 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 185.ba (of order \(36\), 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: | \(5.04088489067\) |
Analytic rank: | \(0\) |
Dimension: | \(432\) |
Relative dimension: | \(36\) over \(\Q(\zeta_{36})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{36}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 | −0.341572 | − | 3.90419i | −0.780895 | + | 0.655248i | −11.1868 | + | 1.97253i | −0.126406 | + | 4.99840i | 2.82494 | + | 2.82494i | 2.14262 | − | 5.88680i | 7.46486 | + | 27.8592i | −1.38239 | + | 7.83991i | 19.5579 | − | 1.21380i |
19.2 | −0.320071 | − | 3.65843i | 0.00857152 | − | 0.00719236i | −9.34241 | + | 1.64732i | 0.828602 | − | 4.93086i | −0.0290562 | − | 0.0290562i | −3.95176 | + | 10.8574i | 5.21487 | + | 19.4622i | −1.56281 | + | 8.86315i | −18.3044 | − | 1.45315i |
19.3 | −0.307827 | − | 3.51848i | 3.26477 | − | 2.73947i | −8.34573 | + | 1.47158i | −2.40766 | − | 4.38214i | −10.6438 | − | 10.6438i | 2.41283 | − | 6.62919i | 4.09024 | + | 15.2650i | 1.59121 | − | 9.02419i | −14.6773 | + | 9.82025i |
19.4 | −0.296650 | − | 3.39073i | −4.04044 | + | 3.39033i | −7.46981 | + | 1.31713i | −4.76055 | − | 1.52877i | 12.6943 | + | 12.6943i | −0.415189 | + | 1.14072i | 3.15820 | + | 11.7866i | 3.26798 | − | 18.5336i | −3.77141 | + | 16.5953i |
19.5 | −0.274090 | − | 3.13286i | 3.27219 | − | 2.74569i | −5.80047 | + | 1.02278i | 4.75259 | + | 1.55335i | −9.49874 | − | 9.49874i | −0.0635093 | + | 0.174490i | 1.53831 | + | 5.74106i | 1.60555 | − | 9.10555i | 3.56378 | − | 15.3150i |
19.6 | −0.263106 | − | 3.00731i | −2.52465 | + | 2.11843i | −5.03546 | + | 0.887888i | 4.15063 | − | 2.78788i | 7.03502 | + | 7.03502i | 2.92086 | − | 8.02500i | 0.869722 | + | 3.24585i | 0.323259 | − | 1.83329i | −9.47607 | − | 11.7487i |
19.7 | −0.234327 | − | 2.67837i | −0.0818783 | + | 0.0687040i | −3.17955 | + | 0.560640i | −4.91109 | + | 0.938739i | 0.203201 | + | 0.203201i | −3.08645 | + | 8.47996i | −0.536790 | − | 2.00333i | −1.56085 | + | 8.85202i | 3.66510 | + | 12.9338i |
19.8 | −0.230294 | − | 2.63227i | −0.796749 | + | 0.668552i | −2.93657 | + | 0.517797i | 3.20664 | + | 3.83634i | 1.94329 | + | 1.94329i | −2.03603 | + | 5.59395i | −0.696280 | − | 2.59855i | −1.37499 | + | 7.79793i | 9.35980 | − | 9.32421i |
19.9 | −0.223778 | − | 2.55780i | 1.45401 | − | 1.22006i | −2.55301 | + | 0.450165i | −4.75203 | + | 1.55507i | −3.44604 | − | 3.44604i | 2.93621 | − | 8.06716i | −0.935402 | − | 3.49097i | −0.937232 | + | 5.31531i | 5.04094 | + | 11.8067i |
19.10 | −0.179984 | − | 2.05723i | −3.68274 | + | 3.09019i | −0.260569 | + | 0.0459453i | 0.359327 | + | 4.98707i | 7.02006 | + | 7.02006i | −0.0988642 | + | 0.271627i | −1.99652 | − | 7.45111i | 2.45049 | − | 13.8974i | 10.1949 | − | 1.63681i |
19.11 | −0.160446 | − | 1.83391i | 1.95538 | − | 1.64076i | 0.601757 | − | 0.106106i | 4.02587 | − | 2.96519i | −3.32274 | − | 3.32274i | −0.170800 | + | 0.469270i | −2.19699 | − | 8.19928i | −0.431406 | + | 2.44662i | −6.08382 | − | 6.90732i |
19.12 | −0.122167 | − | 1.39637i | 4.30022 | − | 3.60832i | 2.00431 | − | 0.353413i | −3.99166 | − | 3.01108i | −5.56389 | − | 5.56389i | −4.03440 | + | 11.0844i | −2.18951 | − | 8.17135i | 3.90914 | − | 22.1699i | −3.71694 | + | 5.94169i |
19.13 | −0.110522 | − | 1.26327i | 0.575333 | − | 0.482762i | 2.35559 | − | 0.415353i | −1.04246 | − | 4.89012i | −0.673447 | − | 0.673447i | 1.57851 | − | 4.33691i | −2.09788 | − | 7.82940i | −1.46488 | + | 8.30777i | −6.06234 | + | 1.85738i |
19.14 | −0.102965 | − | 1.17690i | 3.28226 | − | 2.75414i | 2.56475 | − | 0.452234i | 0.167379 | + | 4.99720i | −3.57930 | − | 3.57930i | 0.859163 | − | 2.36053i | −2.01938 | − | 7.53643i | 1.62509 | − | 9.21633i | 5.86396 | − | 0.711526i |
19.15 | −0.0974015 | − | 1.11330i | −3.29672 | + | 2.76628i | 2.70927 | − | 0.477718i | 3.61437 | − | 3.45490i | 3.40082 | + | 3.40082i | −2.85674 | + | 7.84883i | −1.95271 | − | 7.28762i | 1.65325 | − | 9.37604i | −4.19840 | − | 3.68739i |
19.16 | −0.0841671 | − | 0.962034i | −2.51465 | + | 2.11004i | 3.02081 | − | 0.532650i | −3.73675 | − | 3.32216i | 2.24159 | + | 2.24159i | −1.33590 | + | 3.67037i | −1.76645 | − | 6.59250i | 0.308359 | − | 1.74879i | −2.88152 | + | 3.87450i |
19.17 | −0.0581376 | − | 0.664516i | −2.60569 | + | 2.18644i | 3.50103 | − | 0.617326i | −4.46481 | + | 2.25067i | 1.60441 | + | 1.60441i | 4.17366 | − | 11.4670i | −1.30435 | − | 4.86790i | 0.446303 | − | 2.53111i | 1.75518 | + | 2.83609i |
19.18 | −0.00858096 | − | 0.0980809i | −0.373760 | + | 0.313622i | 3.92968 | − | 0.692909i | 4.80438 | + | 1.38489i | 0.0339675 | + | 0.0339675i | 2.76903 | − | 7.60786i | −0.203610 | − | 0.759884i | −1.52150 | + | 8.62883i | 0.0946053 | − | 0.483101i |
19.19 | 0.00858096 | + | 0.0980809i | 0.373760 | − | 0.313622i | 3.92968 | − | 0.692909i | −2.02731 | + | 4.57056i | 0.0339675 | + | 0.0339675i | −2.76903 | + | 7.60786i | 0.203610 | + | 0.759884i | −1.52150 | + | 8.62883i | −0.465681 | − | 0.159620i |
19.20 | 0.0581376 | + | 0.664516i | 2.60569 | − | 2.18644i | 3.50103 | − | 0.617326i | 4.59403 | − | 1.97354i | 1.60441 | + | 1.60441i | −4.17366 | + | 11.4670i | 1.30435 | + | 4.86790i | 0.446303 | − | 2.53111i | 1.57853 | + | 2.93807i |
See next 80 embeddings (of 432 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
37.i | odd | 36 | 1 | inner |
185.ba | odd | 36 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 185.3.ba.a | ✓ | 432 |
5.b | even | 2 | 1 | inner | 185.3.ba.a | ✓ | 432 |
37.i | odd | 36 | 1 | inner | 185.3.ba.a | ✓ | 432 |
185.ba | odd | 36 | 1 | inner | 185.3.ba.a | ✓ | 432 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
185.3.ba.a | ✓ | 432 | 1.a | even | 1 | 1 | trivial |
185.3.ba.a | ✓ | 432 | 5.b | even | 2 | 1 | inner |
185.3.ba.a | ✓ | 432 | 37.i | odd | 36 | 1 | inner |
185.3.ba.a | ✓ | 432 | 185.ba | odd | 36 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(185, [\chi])\).