Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [304,5,Mod(65,304)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(304, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("304.65");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 304 = 2^{4} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 304.r (of order \(6\), degree \(2\), not 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: | \(31.4244687775\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(20\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 152) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
65.1 | 0 | −14.4186 | − | 8.32460i | 0 | 10.1950 | − | 17.6583i | 0 | −80.1603 | 0 | 98.0979 | + | 169.911i | 0 | ||||||||||||
65.2 | 0 | −13.5253 | − | 7.80881i | 0 | 14.4872 | − | 25.0925i | 0 | 26.1842 | 0 | 81.4552 | + | 141.085i | 0 | ||||||||||||
65.3 | 0 | −12.4606 | − | 7.19410i | 0 | −20.6420 | + | 35.7530i | 0 | −1.47481 | 0 | 63.0102 | + | 109.137i | 0 | ||||||||||||
65.4 | 0 | −10.6119 | − | 6.12677i | 0 | 3.27153 | − | 5.66646i | 0 | 87.4966 | 0 | 34.5747 | + | 59.8851i | 0 | ||||||||||||
65.5 | 0 | −9.48048 | − | 5.47356i | 0 | −15.3284 | + | 26.5496i | 0 | −33.8970 | 0 | 19.4197 | + | 33.6359i | 0 | ||||||||||||
65.6 | 0 | −7.37289 | − | 4.25674i | 0 | 7.14367 | − | 12.3732i | 0 | 35.7108 | 0 | −4.26032 | − | 7.37909i | 0 | ||||||||||||
65.7 | 0 | −5.77348 | − | 3.33332i | 0 | −1.87999 | + | 3.25623i | 0 | 40.7434 | 0 | −18.2780 | − | 31.6584i | 0 | ||||||||||||
65.8 | 0 | −5.09915 | − | 2.94399i | 0 | 0.570982 | − | 0.988970i | 0 | −80.7369 | 0 | −23.1658 | − | 40.1243i | 0 | ||||||||||||
65.9 | 0 | −2.52583 | − | 1.45829i | 0 | 22.1213 | − | 38.3151i | 0 | 11.1904 | 0 | −36.2468 | − | 62.7813i | 0 | ||||||||||||
65.10 | 0 | −1.93870 | − | 1.11931i | 0 | −15.6195 | + | 27.0538i | 0 | 6.74912 | 0 | −37.9943 | − | 65.8080i | 0 | ||||||||||||
65.11 | 0 | −0.620551 | − | 0.358275i | 0 | 16.9339 | − | 29.3305i | 0 | −38.8265 | 0 | −40.2433 | − | 69.7034i | 0 | ||||||||||||
65.12 | 0 | 0.962404 | + | 0.555644i | 0 | −13.4131 | + | 23.2322i | 0 | 62.8585 | 0 | −39.8825 | − | 69.0785i | 0 | ||||||||||||
65.13 | 0 | 5.28375 | + | 3.05057i | 0 | 1.28449 | − | 2.22480i | 0 | 6.16432 | 0 | −21.8880 | − | 37.9111i | 0 | ||||||||||||
65.14 | 0 | 5.59346 | + | 3.22938i | 0 | −4.68830 | + | 8.12037i | 0 | −33.5286 | 0 | −19.6422 | − | 34.0212i | 0 | ||||||||||||
65.15 | 0 | 7.81020 | + | 4.50922i | 0 | 15.3964 | − | 26.6673i | 0 | 70.8202 | 0 | 0.166103 | + | 0.287698i | 0 | ||||||||||||
65.16 | 0 | 9.62676 | + | 5.55801i | 0 | −23.6855 | + | 41.0244i | 0 | −82.2157 | 0 | 21.2830 | + | 36.8632i | 0 | ||||||||||||
65.17 | 0 | 10.7629 | + | 6.21398i | 0 | 20.5828 | − | 35.6505i | 0 | −44.5677 | 0 | 36.7272 | + | 63.6134i | 0 | ||||||||||||
65.18 | 0 | 11.3222 | + | 6.53690i | 0 | −19.8361 | + | 34.3572i | 0 | 76.8642 | 0 | 44.9621 | + | 77.8766i | 0 | ||||||||||||
65.19 | 0 | 12.1979 | + | 7.04247i | 0 | 0.573898 | − | 0.994020i | 0 | 40.4604 | 0 | 58.6927 | + | 101.659i | 0 | ||||||||||||
65.20 | 0 | 14.2678 | + | 8.23749i | 0 | 2.53183 | − | 4.38526i | 0 | −53.8347 | 0 | 95.2125 | + | 164.913i | 0 | ||||||||||||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.d | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 304.5.r.d | 40 | |
4.b | odd | 2 | 1 | 152.5.n.a | ✓ | 40 | |
19.d | odd | 6 | 1 | inner | 304.5.r.d | 40 | |
76.f | even | 6 | 1 | 152.5.n.a | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
152.5.n.a | ✓ | 40 | 4.b | odd | 2 | 1 | |
152.5.n.a | ✓ | 40 | 76.f | even | 6 | 1 | |
304.5.r.d | 40 | 1.a | even | 1 | 1 | trivial | |
304.5.r.d | 40 | 19.d | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{40} + 12 T_{3}^{39} - 1050 T_{3}^{38} - 13176 T_{3}^{37} + 653455 T_{3}^{36} + 8261832 T_{3}^{35} - 273020834 T_{3}^{34} - 3478133916 T_{3}^{33} + 85744483199 T_{3}^{32} + \cdots + 16\!\cdots\!21 \)
acting on \(S_{5}^{\mathrm{new}}(304, [\chi])\).