Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [671,2,Mod(14,671)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(671, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([24, 25]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("671.14");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 671 = 11 \cdot 61 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 671.ck (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: | \(5.35796197563\) |
Analytic rank: | \(0\) |
Dimension: | \(480\) |
Relative dimension: | \(60\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
14.1 | −1.09795 | + | 2.46603i | −0.791281 | − | 2.43531i | −3.53757 | − | 3.92887i | −0.357447 | + | 3.40089i | 6.87435 | + | 0.722523i | −0.902579 | − | 4.24630i | 8.43822 | − | 2.74174i | −2.87757 | + | 2.09068i | −7.99424 | − | 4.61548i |
14.2 | −1.09088 | + | 2.45015i | 0.874736 | + | 2.69216i | −3.47496 | − | 3.85933i | 0.162454 | − | 1.54565i | −7.55043 | − | 0.793582i | 0.643881 | + | 3.02922i | 8.14518 | − | 2.64653i | −4.05552 | + | 2.94651i | 3.60985 | + | 2.08415i |
14.3 | −1.07778 | + | 2.42074i | 0.0630027 | + | 0.193902i | −3.36011 | − | 3.73178i | −0.187024 | + | 1.77942i | −0.537291 | − | 0.0564716i | 0.849246 | + | 3.99539i | 7.61488 | − | 2.47422i | 2.39342 | − | 1.73892i | −4.10594 | − | 2.37056i |
14.4 | −1.04401 | + | 2.34488i | −0.583396 | − | 1.79551i | −3.07026 | − | 3.40986i | 0.320382 | − | 3.04823i | 4.81932 | + | 0.506531i | −0.266419 | − | 1.25340i | 6.31877 | − | 2.05309i | −0.456447 | + | 0.331628i | 6.81326 | + | 3.93364i |
14.5 | −1.01763 | + | 2.28564i | 0.541493 | + | 1.66655i | −2.85032 | − | 3.16560i | 0.0850514 | − | 0.809210i | −4.36017 | − | 0.458272i | −0.423656 | − | 1.99315i | 5.37704 | − | 1.74711i | −0.0571068 | + | 0.0414905i | 1.76301 | + | 1.01788i |
14.6 | −1.01486 | + | 2.27942i | −0.379729 | − | 1.16869i | −2.82754 | − | 3.14030i | −0.118877 | + | 1.13104i | 3.04929 | + | 0.320494i | 0.262125 | + | 1.23320i | 5.28160 | − | 1.71609i | 1.20542 | − | 0.875789i | −2.45746 | − | 1.41881i |
14.7 | −0.944063 | + | 2.12040i | 0.949585 | + | 2.92252i | −2.26658 | − | 2.51729i | −0.349328 | + | 3.32363i | −7.09338 | − | 0.745545i | −0.530768 | − | 2.49707i | 3.06252 | − | 0.995074i | −5.21238 | + | 3.78701i | −6.71764 | − | 3.87843i |
14.8 | −0.914585 | + | 2.05419i | −1.05536 | − | 3.24806i | −2.04497 | − | 2.27117i | 0.186406 | − | 1.77354i | 7.63736 | + | 0.802719i | 0.492311 | + | 2.31614i | 2.25865 | − | 0.733881i | −7.00909 | + | 5.09240i | 3.47270 | + | 2.00497i |
14.9 | −0.841616 | + | 1.89030i | 0.392484 | + | 1.20794i | −1.52666 | − | 1.69552i | 0.322713 | − | 3.07041i | −2.61369 | − | 0.274710i | 0.450217 | + | 2.11810i | 0.554068 | − | 0.180027i | 1.12197 | − | 0.815161i | 5.53239 | + | 3.19413i |
14.10 | −0.840089 | + | 1.88687i | 0.422744 | + | 1.30107i | −1.51627 | − | 1.68399i | −0.407298 | + | 3.87518i | −2.81010 | − | 0.295353i | −0.114564 | − | 0.538981i | 0.522572 | − | 0.169794i | 0.912972 | − | 0.663313i | −6.96980 | − | 4.02402i |
14.11 | −0.831389 | + | 1.86733i | −0.191751 | − | 0.590149i | −1.45745 | − | 1.61867i | −0.146061 | + | 1.38968i | 1.26142 | + | 0.132581i | −0.394481 | − | 1.85589i | 0.346285 | − | 0.112515i | 2.11554 | − | 1.53703i | −2.47356 | − | 1.42811i |
14.12 | −0.791444 | + | 1.77761i | −0.629136 | − | 1.93628i | −1.19526 | − | 1.32747i | −0.0926998 | + | 0.881979i | 3.93989 | + | 0.414099i | 0.135633 | + | 0.638103i | −0.395488 | + | 0.128502i | −0.926325 | + | 0.673015i | −1.49445 | − | 0.862822i |
14.13 | −0.785203 | + | 1.76360i | 0.203004 | + | 0.624781i | −1.15546 | − | 1.28327i | 0.119183 | − | 1.13395i | −1.26126 | − | 0.132564i | −0.590630 | − | 2.77870i | −0.501574 | + | 0.162971i | 2.07791 | − | 1.50969i | 1.90624 | + | 1.10057i |
14.14 | −0.759127 | + | 1.70503i | −0.0220173 | − | 0.0677622i | −0.992581 | − | 1.10237i | 0.430383 | − | 4.09482i | 0.132250 | + | 0.0139001i | 0.867496 | + | 4.08125i | −0.917003 | + | 0.297952i | 2.42294 | − | 1.76037i | 6.65506 | + | 3.84230i |
14.15 | −0.641762 | + | 1.44142i | 0.766175 | + | 2.35804i | −0.327577 | − | 0.363811i | 0.308577 | − | 2.93592i | −3.89064 | − | 0.408923i | −1.02482 | − | 4.82142i | −2.26658 | + | 0.736458i | −2.54630 | + | 1.85000i | 4.03386 | + | 2.32895i |
14.16 | −0.616002 | + | 1.38356i | −0.492021 | − | 1.51429i | −0.196528 | − | 0.218267i | −0.0864920 | + | 0.822917i | 2.39820 | + | 0.252061i | 0.972024 | + | 4.57301i | −2.45770 | + | 0.798555i | 0.376076 | − | 0.273235i | −1.08528 | − | 0.626586i |
14.17 | −0.615500 | + | 1.38244i | 0.695450 | + | 2.14037i | −0.194028 | − | 0.215490i | −0.177745 | + | 1.69113i | −3.38698 | − | 0.355986i | 0.875518 | + | 4.11899i | −2.46108 | + | 0.799652i | −1.67050 | + | 1.21369i | −2.22847 | − | 1.28661i |
14.18 | −0.584331 | + | 1.31243i | −0.155559 | − | 0.478760i | −0.0427671 | − | 0.0474976i | −0.236278 | + | 2.24803i | 0.719236 | + | 0.0755948i | −0.890049 | − | 4.18735i | −2.64531 | + | 0.859514i | 2.22204 | − | 1.61441i | −2.81232 | − | 1.62369i |
14.19 | −0.534745 | + | 1.20106i | 0.955432 | + | 2.94052i | 0.181677 | + | 0.201773i | 0.132663 | − | 1.26221i | −4.04264 | − | 0.424899i | 0.245631 | + | 1.15560i | −2.84024 | + | 0.922849i | −5.30675 | + | 3.85558i | 1.44504 | + | 0.834296i |
14.20 | −0.480268 | + | 1.07870i | −0.530935 | − | 1.63405i | 0.405327 | + | 0.450161i | 0.369451 | − | 3.51509i | 2.01764 | + | 0.212062i | −0.518425 | − | 2.43900i | −2.92624 | + | 0.950792i | 0.0388275 | − | 0.0282099i | 3.61429 | + | 2.08671i |
See next 80 embeddings (of 480 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.c | even | 5 | 1 | inner |
61.f | even | 6 | 1 | inner |
671.ck | even | 30 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 671.2.ck.a | ✓ | 480 |
11.c | even | 5 | 1 | inner | 671.2.ck.a | ✓ | 480 |
61.f | even | 6 | 1 | inner | 671.2.ck.a | ✓ | 480 |
671.ck | even | 30 | 1 | inner | 671.2.ck.a | ✓ | 480 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
671.2.ck.a | ✓ | 480 | 1.a | even | 1 | 1 | trivial |
671.2.ck.a | ✓ | 480 | 11.c | even | 5 | 1 | inner |
671.2.ck.a | ✓ | 480 | 61.f | even | 6 | 1 | inner |
671.2.ck.a | ✓ | 480 | 671.ck | even | 30 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(671, [\chi])\).