Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [77,4,Mod(17,77)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(77, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([5, 27]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("77.17");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 77 = 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 77.n (of order \(30\), degree \(8\), 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: | \(4.54314707044\) |
Analytic rank: | \(0\) |
Dimension: | \(176\) |
Relative dimension: | \(22\) over \(\Q(\zeta_{30})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{30}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | −3.87348 | − | 3.48769i | 0.846259 | + | 1.90073i | 2.00358 | + | 19.0628i | 0.657021 | + | 3.09104i | 3.35119 | − | 10.3139i | −3.03387 | − | 18.2701i | 34.2149 | − | 47.0927i | 15.1699 | − | 16.8479i | 8.23565 | − | 14.2646i |
17.2 | −3.63700 | − | 3.27477i | −3.23523 | − | 7.26645i | 1.66743 | + | 15.8645i | 1.97468 | + | 9.29016i | −12.0294 | + | 37.0227i | 11.9536 | + | 14.1461i | 22.8750 | − | 31.4847i | −24.2680 | + | 26.9523i | 23.2412 | − | 40.2550i |
17.3 | −3.33798 | − | 3.00553i | −1.94365 | − | 4.36551i | 1.27267 | + | 12.1086i | −3.69623 | − | 17.3894i | −6.63281 | + | 20.4137i | −17.7850 | − | 5.16648i | 11.0235 | − | 15.1726i | 2.78664 | − | 3.09487i | −39.9264 | + | 69.1546i |
17.4 | −3.25272 | − | 2.92876i | 3.30357 | + | 7.41994i | 1.16631 | + | 11.0967i | −3.29794 | − | 15.5156i | 10.9857 | − | 33.8104i | 3.14028 | + | 18.2521i | 8.12421 | − | 11.1820i | −26.0755 | + | 28.9597i | −34.7141 | + | 60.1267i |
17.5 | −2.74310 | − | 2.46990i | 0.806254 | + | 1.81088i | 0.587969 | + | 5.59415i | 0.434945 | + | 2.04626i | 2.26104 | − | 6.95877i | 18.4556 | + | 1.54614i | −5.15296 | + | 7.09244i | 15.4373 | − | 17.1449i | 3.86094 | − | 6.68734i |
17.6 | −2.29364 | − | 2.06520i | 4.04421 | + | 9.08345i | 0.159492 | + | 1.51746i | 3.09625 | + | 14.5667i | 9.48319 | − | 29.1863i | −8.23970 | − | 16.5864i | −11.7450 | + | 16.1657i | −48.0869 | + | 53.4059i | 22.9815 | − | 39.8051i |
17.7 | −2.16383 | − | 1.94832i | −0.570022 | − | 1.28029i | 0.0499798 | + | 0.475526i | 3.23118 | + | 15.2015i | −1.26099 | + | 3.88092i | −15.1390 | + | 10.6683i | −12.8734 | + | 17.7187i | 16.7523 | − | 18.6053i | 22.6257 | − | 39.1889i |
17.8 | −1.30063 | − | 1.17109i | −2.30577 | − | 5.17884i | −0.516047 | − | 4.90986i | 0.329952 | + | 1.55230i | −3.06595 | + | 9.43603i | 1.71585 | − | 18.4406i | −13.3085 | + | 18.3176i | −3.43732 | + | 3.81753i | 1.38875 | − | 2.40538i |
17.9 | −1.27371 | − | 1.14685i | −2.99266 | − | 6.72163i | −0.529166 | − | 5.03467i | −3.10847 | − | 14.6242i | −3.89693 | + | 11.9935i | 16.0251 | + | 9.28422i | −13.1594 | + | 18.1124i | −18.1577 | + | 20.1662i | −12.8125 | + | 22.1919i |
17.10 | −1.04448 | − | 0.940451i | 1.04120 | + | 2.33857i | −0.629744 | − | 5.99162i | −1.55589 | − | 7.31990i | 1.11180 | − | 3.42177i | −15.9430 | + | 9.42442i | −11.5860 | + | 15.9468i | 13.6817 | − | 15.1951i | −5.25891 | + | 9.10871i |
17.11 | −0.101647 | − | 0.0915238i | 2.43582 | + | 5.47095i | −0.834272 | − | 7.93757i | −2.91889 | − | 13.7323i | 0.253127 | − | 0.779044i | 0.0206170 | − | 18.5202i | −1.28485 | + | 1.76845i | −5.93152 | + | 6.58762i | −0.960133 | + | 1.66300i |
17.12 | 0.105220 | + | 0.0947403i | 2.36435 | + | 5.31043i | −0.834132 | − | 7.93624i | 1.95794 | + | 9.21137i | −0.254335 | + | 0.782762i | 12.6485 | + | 13.5283i | 1.32990 | − | 1.83045i | −4.54395 | + | 5.04656i | −0.666675 | + | 1.15471i |
17.13 | 0.631608 | + | 0.568703i | −3.77378 | − | 8.47605i | −0.760721 | − | 7.23778i | 1.53376 | + | 7.21577i | 2.43680 | − | 7.49970i | −14.3667 | + | 11.6875i | 7.63220 | − | 10.5048i | −39.5355 | + | 43.9086i | −3.13489 | + | 5.42980i |
17.14 | 0.677422 | + | 0.609954i | −0.922770 | − | 2.07258i | −0.749370 | − | 7.12978i | 3.21414 | + | 15.1213i | 0.639070 | − | 1.96686i | 12.8687 | − | 13.3190i | 8.12762 | − | 11.1867i | 14.6225 | − | 16.2399i | −7.04599 | + | 12.2040i |
17.15 | 1.76800 | + | 1.59191i | −0.459138 | − | 1.03124i | −0.244598 | − | 2.32720i | −2.74908 | − | 12.9334i | 0.829889 | − | 2.55413i | 13.9884 | + | 12.1377i | 14.4593 | − | 19.9015i | 17.2139 | − | 19.1179i | 15.7284 | − | 27.2425i |
17.16 | 1.95261 | + | 1.75814i | −1.42219 | − | 3.19430i | −0.114589 | − | 1.09024i | −1.58425 | − | 7.45329i | 2.83903 | − | 8.73764i | −16.4147 | − | 8.57654i | 14.0483 | − | 19.3358i | 9.88562 | − | 10.9791i | 10.0105 | − | 17.3387i |
17.17 | 2.24857 | + | 2.02462i | 2.64431 | + | 5.93922i | 0.120749 | + | 1.14885i | 1.63909 | + | 7.71131i | −6.07876 | + | 18.7085i | −16.6915 | + | 8.02459i | 12.1735 | − | 16.7553i | −10.2154 | + | 11.3453i | −11.9269 | + | 20.6580i |
17.18 | 2.86713 | + | 2.58158i | 3.38904 | + | 7.61192i | 0.719679 | + | 6.84729i | −0.771490 | − | 3.62958i | −9.93392 | + | 30.5735i | 14.6454 | − | 11.3363i | 2.52852 | − | 3.48021i | −28.3892 | + | 31.5293i | 7.15807 | − | 12.3981i |
17.19 | 3.11976 | + | 2.80905i | −1.91488 | − | 4.30089i | 1.00595 | + | 9.57096i | 3.24497 | + | 15.2664i | 6.10744 | − | 18.7968i | 8.74619 | + | 16.3250i | −4.00652 | + | 5.51451i | 3.23561 | − | 3.59351i | −32.7605 | + | 56.7428i |
17.20 | 3.42618 | + | 3.08494i | −3.94032 | − | 8.85010i | 1.38558 | + | 13.1829i | −2.35545 | − | 11.0815i | 13.8018 | − | 42.4777i | 11.0019 | − | 14.8983i | −14.2421 | + | 19.6026i | −44.7316 | + | 49.6795i | 26.1156 | − | 45.2336i |
See next 80 embeddings (of 176 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
11.d | odd | 10 | 1 | inner |
77.n | even | 30 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 77.4.n.a | ✓ | 176 |
7.d | odd | 6 | 1 | inner | 77.4.n.a | ✓ | 176 |
11.d | odd | 10 | 1 | inner | 77.4.n.a | ✓ | 176 |
77.n | even | 30 | 1 | inner | 77.4.n.a | ✓ | 176 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
77.4.n.a | ✓ | 176 | 1.a | even | 1 | 1 | trivial |
77.4.n.a | ✓ | 176 | 7.d | odd | 6 | 1 | inner |
77.4.n.a | ✓ | 176 | 11.d | odd | 10 | 1 | inner |
77.4.n.a | ✓ | 176 | 77.n | even | 30 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(77, [\chi])\).