Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [92,4,Mod(7,92)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(92, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([11, 19]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("92.7");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 92 = 2^{2} \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 92.h (of order \(22\), degree \(10\), 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: | \(5.42817572053\) |
Analytic rank: | \(0\) |
Dimension: | \(340\) |
Relative dimension: | \(34\) over \(\Q(\zeta_{22})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{22}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7.1 | −2.82761 | + | 0.0677825i | 1.81046 | + | 2.81713i | 7.99081 | − | 0.383326i | 12.5112 | − | 5.71369i | −5.31024 | − | 7.84305i | −5.24129 | + | 1.53898i | −22.5690 | + | 1.62553i | 6.55773 | − | 14.3594i | −34.9897 | + | 17.0042i |
7.2 | −2.81852 | − | 0.236485i | 1.10868 | + | 1.72513i | 7.88815 | + | 1.33308i | −11.1319 | + | 5.08378i | −2.71686 | − | 5.12452i | −3.75061 | + | 1.10128i | −21.9177 | − | 5.62274i | 9.46928 | − | 20.7348i | 32.5778 | − | 11.6962i |
7.3 | −2.76439 | + | 0.598463i | −3.90833 | − | 6.08148i | 7.28368 | − | 3.30877i | 2.35661 | − | 1.07623i | 14.4437 | + | 14.4726i | 30.6621 | − | 9.00321i | −18.1547 | + | 13.5057i | −10.4932 | + | 22.9768i | −5.87051 | + | 4.38546i |
7.4 | −2.62792 | − | 1.04596i | −3.88100 | − | 6.03895i | 5.81193 | + | 5.49740i | −5.21193 | + | 2.38021i | 3.88245 | + | 19.9293i | −18.1422 | + | 5.32703i | −9.52324 | − | 20.5258i | −10.1906 | + | 22.3143i | 16.1861 | − | 0.803526i |
7.5 | −2.44815 | − | 1.41653i | 5.07187 | + | 7.89198i | 3.98688 | + | 6.93576i | 3.38966 | − | 1.54801i | −1.23745 | − | 26.5052i | 22.4823 | − | 6.60139i | 0.0642622 | − | 22.6273i | −25.3433 | + | 55.4942i | −10.4912 | − | 1.01181i |
7.6 | −2.42335 | + | 1.45856i | 4.96798 | + | 7.73032i | 3.74522 | − | 7.06918i | −11.1313 | + | 5.08348i | −23.3143 | − | 11.4872i | −5.60740 | + | 1.64648i | 1.23482 | + | 22.5937i | −23.8609 | + | 52.2481i | 19.5604 | − | 28.5546i |
7.7 | −2.33549 | + | 1.59546i | −0.208207 | − | 0.323976i | 2.90901 | − | 7.45236i | −6.50566 | + | 2.97103i | 1.00315 | + | 0.424456i | 6.33191 | − | 1.85922i | 5.09597 | + | 22.0461i | 11.1546 | − | 24.4252i | 10.4537 | − | 17.3183i |
7.8 | −2.18875 | + | 1.79147i | −3.00503 | − | 4.67592i | 1.58124 | − | 7.84217i | 13.7050 | − | 6.25888i | 14.9540 | + | 4.85098i | −30.1106 | + | 8.84128i | 10.5881 | + | 19.9973i | −1.61781 | + | 3.54250i | −18.7843 | + | 38.2513i |
7.9 | −1.95779 | − | 2.04134i | −2.24967 | − | 3.50055i | −0.334121 | + | 7.99302i | 15.6896 | − | 7.16519i | −2.74143 | + | 11.4457i | 8.54748 | − | 2.50977i | 16.9706 | − | 14.9666i | 4.02336 | − | 8.80993i | −45.3434 | − | 17.9998i |
7.10 | −1.87947 | − | 2.11367i | 0.164413 | + | 0.255832i | −0.935191 | + | 7.94515i | −17.0252 | + | 7.77513i | 0.231734 | − | 0.828343i | 28.9597 | − | 8.50335i | 18.5511 | − | 12.9560i | 11.1778 | − | 24.4759i | 48.4323 | + | 21.3724i |
7.11 | −1.83494 | − | 2.15244i | 2.38392 | + | 3.70945i | −1.26598 | + | 7.89920i | −0.833192 | + | 0.380506i | 3.61001 | − | 11.9379i | −32.5869 | + | 9.56837i | 19.3255 | − | 11.7696i | 3.13926 | − | 6.87402i | 2.34787 | + | 1.09519i |
7.12 | −1.46175 | + | 2.42142i | 3.00503 | + | 4.67592i | −3.72659 | − | 7.07902i | 13.7050 | − | 6.25888i | −15.7150 | + | 0.441437i | 30.1106 | − | 8.84128i | 22.5886 | + | 1.32411i | −1.61781 | + | 3.54250i | −4.87791 | + | 42.3346i |
7.13 | −1.24685 | + | 2.53877i | 0.208207 | + | 0.323976i | −4.89075 | − | 6.33092i | −6.50566 | + | 2.97103i | −1.08210 | + | 0.124642i | −6.33191 | + | 1.85922i | 22.1708 | − | 4.52283i | 11.1546 | − | 24.4252i | 0.568767 | − | 20.2208i |
7.14 | −1.09883 | + | 2.60626i | −4.96798 | − | 7.73032i | −5.58513 | − | 5.72768i | −11.1313 | + | 5.08348i | 25.6062 | − | 4.45349i | 5.60740 | − | 1.64648i | 21.0649 | − | 8.26253i | −23.8609 | + | 52.2481i | −1.01745 | − | 34.5968i |
7.15 | −0.722671 | − | 2.73455i | 2.75621 | + | 4.28874i | −6.95549 | + | 3.95235i | 5.20290 | − | 2.37608i | 9.73593 | − | 10.6363i | −4.87145 | + | 1.43039i | 15.8344 | + | 16.1639i | 0.419588 | − | 0.918770i | −10.2575 | − | 12.5104i |
7.16 | −0.437400 | − | 2.79440i | −3.57074 | − | 5.55619i | −7.61736 | + | 2.44454i | −5.68745 | + | 2.59737i | −13.9644 | + | 12.4084i | −0.744179 | + | 0.218511i | 10.1629 | + | 20.2167i | −6.90479 | + | 15.1194i | 9.74579 | + | 14.7569i |
7.17 | −0.198958 | + | 2.82142i | 3.90833 | + | 6.08148i | −7.92083 | − | 1.12269i | 2.35661 | − | 1.07623i | −17.9360 | + | 9.81708i | −30.6621 | + | 9.00321i | 4.74350 | − | 22.1246i | −10.4932 | + | 22.9768i | 2.56763 | + | 6.86313i |
7.18 | 0.335319 | + | 2.80848i | −1.81046 | − | 2.81713i | −7.77512 | + | 1.88347i | 12.5112 | − | 5.71369i | 7.30478 | − | 6.02929i | 5.24129 | − | 1.53898i | −7.89684 | − | 21.2047i | 6.55773 | − | 14.3594i | 20.2420 | + | 33.2217i |
7.19 | 0.426542 | − | 2.79608i | 0.814062 | + | 1.26671i | −7.63612 | − | 2.38529i | 9.73779 | − | 4.44710i | 3.88904 | − | 1.73588i | 15.5856 | − | 4.57635i | −9.92658 | + | 20.3338i | 10.2744 | − | 22.4977i | −8.28087 | − | 29.1245i |
7.20 | 0.635196 | + | 2.75618i | −1.10868 | − | 1.72513i | −7.19305 | + | 3.50143i | −11.1319 | + | 5.08378i | 4.05055 | − | 4.15151i | 3.75061 | − | 1.10128i | −14.2196 | − | 17.6013i | 9.46928 | − | 20.7348i | −21.0827 | − | 27.4524i |
See next 80 embeddings (of 340 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
23.d | odd | 22 | 1 | inner |
92.h | even | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 92.4.h.a | ✓ | 340 |
4.b | odd | 2 | 1 | inner | 92.4.h.a | ✓ | 340 |
23.d | odd | 22 | 1 | inner | 92.4.h.a | ✓ | 340 |
92.h | even | 22 | 1 | inner | 92.4.h.a | ✓ | 340 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
92.4.h.a | ✓ | 340 | 1.a | even | 1 | 1 | trivial |
92.4.h.a | ✓ | 340 | 4.b | odd | 2 | 1 | inner |
92.4.h.a | ✓ | 340 | 23.d | odd | 22 | 1 | inner |
92.4.h.a | ✓ | 340 | 92.h | even | 22 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(92, [\chi])\).