Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [77,5,Mod(12,77)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(77, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([5, 0]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("77.12");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 77 = 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 77.g (of order \(6\), degree \(2\), 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: | \(7.95948715746\) |
Analytic rank: | \(0\) |
Dimension: | \(52\) |
Relative dimension: | \(26\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
12.1 | −3.91011 | + | 6.77251i | −3.50234 | + | 2.02208i | −22.5780 | − | 39.1062i | −35.9654 | − | 20.7647i | − | 31.6262i | −26.0446 | + | 41.5052i | 228.006 | −32.3224 | + | 55.9840i | 281.258 | − | 162.384i | |||
12.2 | −3.81165 | + | 6.60197i | −11.4172 | + | 6.59174i | −21.0573 | − | 36.4724i | 31.8098 | + | 18.3654i | − | 100.502i | 14.2292 | − | 46.8885i | 199.080 | 46.4021 | − | 80.3707i | −242.495 | + | 140.005i | |||
12.3 | −3.10533 | + | 5.37860i | 2.83706 | − | 1.63798i | −11.2862 | − | 19.5483i | 12.3670 | + | 7.14007i | 20.3459i | 48.2167 | + | 8.72619i | 40.8189 | −35.1341 | + | 60.8540i | −76.8071 | + | 44.3446i | ||||
12.4 | −2.89521 | + | 5.01465i | 13.3474 | − | 7.70612i | −8.76446 | − | 15.1805i | −30.7835 | − | 17.7729i | 89.2433i | 47.2459 | + | 12.9933i | 8.85302 | 78.2687 | − | 135.565i | 178.249 | − | 102.912i | ||||
12.5 | −2.61130 | + | 4.52290i | −3.04027 | + | 1.75530i | −5.63773 | − | 9.76484i | 17.2419 | + | 9.95462i | − | 18.3344i | −18.9050 | + | 45.2062i | −24.6743 | −34.3378 | + | 59.4749i | −90.0474 | + | 51.9889i | |||
12.6 | −2.56846 | + | 4.44871i | −9.34644 | + | 5.39617i | −5.19399 | − | 8.99625i | −26.7270 | − | 15.4308i | − | 55.4394i | 46.0085 | − | 16.8589i | −28.8285 | 17.7373 | − | 30.7219i | 137.294 | − | 79.2669i | |||
12.7 | −2.30271 | + | 3.98841i | −12.0294 | + | 6.94516i | −2.60495 | − | 4.51190i | −3.86397 | − | 2.23086i | − | 63.9708i | −47.7715 | − | 10.9032i | −49.6930 | 55.9705 | − | 96.9438i | 17.7952 | − | 10.2741i | |||
12.8 | −1.49777 | + | 2.59422i | 13.6319 | − | 7.87037i | 3.51335 | + | 6.08530i | 29.8673 | + | 17.2439i | 47.1521i | −23.1068 | + | 43.2097i | −68.9776 | 83.3855 | − | 144.428i | −89.4689 | + | 51.6549i | ||||
12.9 | −1.47029 | + | 2.54662i | −1.32001 | + | 0.762107i | 3.67649 | + | 6.36786i | −14.7545 | − | 8.51849i | − | 4.48208i | −3.88861 | − | 48.8455i | −68.6713 | −39.3384 | + | 68.1361i | 43.3867 | − | 25.0493i | |||
12.10 | −1.38186 | + | 2.39345i | 7.25851 | − | 4.19071i | 4.18094 | + | 7.24159i | −27.2738 | − | 15.7465i | 23.1638i | −48.3883 | + | 7.71829i | −67.3293 | −5.37598 | + | 9.31148i | 75.3770 | − | 43.5189i | ||||
12.11 | −0.836255 | + | 1.44844i | 6.34113 | − | 3.66105i | 6.60135 | + | 11.4339i | 26.0396 | + | 15.0340i | 12.2463i | 27.6858 | − | 40.4289i | −48.8418 | −13.6934 | + | 23.7176i | −43.5516 | + | 25.1445i | ||||
12.12 | −0.741768 | + | 1.28478i | −10.7216 | + | 6.19010i | 6.89956 | + | 11.9504i | 30.8633 | + | 17.8190i | − | 18.3665i | 46.3081 | + | 16.0176i | −44.2080 | 36.1346 | − | 62.5870i | −45.7869 | + | 26.4351i | |||
12.13 | −0.495228 | + | 0.857760i | 2.90703 | − | 1.67837i | 7.50950 | + | 13.0068i | −23.3408 | − | 13.4758i | 3.32471i | 25.0856 | + | 42.0917i | −30.7229 | −34.8661 | + | 60.3899i | 23.1180 | − | 13.3472i | ||||
12.14 | 0.111598 | − | 0.193293i | −12.8748 | + | 7.43329i | 7.97509 | + | 13.8133i | −24.4063 | − | 14.0910i | 3.31816i | −2.15536 | + | 48.9526i | 7.13114 | 70.0077 | − | 121.257i | −5.44738 | + | 3.14505i | ||||
12.15 | 0.175467 | − | 0.303917i | −6.65697 | + | 3.84340i | 7.93842 | + | 13.7498i | 18.2997 | + | 10.5653i | 2.69756i | −43.7356 | − | 22.0953i | 11.1867 | −10.9565 | + | 18.9773i | 6.42197 | − | 3.70773i | ||||
12.16 | 0.924861 | − | 1.60191i | 13.9207 | − | 8.03710i | 6.28926 | + | 10.8933i | −18.9656 | − | 10.9498i | − | 29.7328i | −16.2859 | − | 46.2144i | 52.8623 | 88.6900 | − | 153.616i | −35.0811 | + | 20.2541i | |||
12.17 | 1.21125 | − | 2.09794i | 0.957269 | − | 0.552680i | 5.06576 | + | 8.77415i | 15.8155 | + | 9.13109i | − | 2.67773i | −36.5703 | + | 32.6131i | 63.3035 | −39.8891 | + | 69.0899i | 38.3130 | − | 22.1200i | |||
12.18 | 1.30121 | − | 2.25377i | 9.51544 | − | 5.49374i | 4.61369 | + | 7.99115i | 1.46113 | + | 0.843586i | − | 28.5941i | 41.5151 | + | 26.0288i | 65.6524 | 19.8624 | − | 34.4028i | 3.80249 | − | 2.19537i | |||
12.19 | 2.28068 | − | 3.95025i | −14.8447 | + | 8.57057i | −2.40298 | − | 4.16209i | 5.82284 | + | 3.36182i | 78.1868i | −10.4517 | − | 47.8723i | 51.0600 | 106.409 | − | 184.306i | 26.5600 | − | 15.3345i | ||||
12.20 | 2.53920 | − | 4.39802i | 2.02982 | − | 1.17192i | −4.89508 | − | 8.47852i | −33.5620 | − | 19.3771i | − | 11.9029i | −48.4739 | − | 7.16097i | 31.5361 | −37.7532 | + | 65.3905i | −170.441 | + | 98.4044i | |||
See all 52 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 77.5.g.a | ✓ | 52 |
7.d | odd | 6 | 1 | inner | 77.5.g.a | ✓ | 52 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
77.5.g.a | ✓ | 52 | 1.a | even | 1 | 1 | trivial |
77.5.g.a | ✓ | 52 | 7.d | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(77, [\chi])\).