Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [935,2,Mod(6,935)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(935, base_ring=CyclotomicField(80))
chi = DirichletCharacter(H, H._module([0, 72, 75]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("935.6");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 935 = 5 \cdot 11 \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 935.cq (of order \(80\), degree \(32\), 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: | \(7.46601258899\) |
Analytic rank: | \(0\) |
Dimension: | \(2304\) |
Relative dimension: | \(72\) over \(\Q(\zeta_{80})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{80}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
6.1 | −2.75247 | − | 0.216624i | 2.06942 | + | 0.954014i | 5.55378 | + | 0.879632i | 0.619094 | + | 0.785317i | −5.48934 | − | 3.07418i | 1.32001 | − | 3.57803i | −9.72667 | − | 2.33517i | 1.42400 | + | 1.66729i | −1.53392 | − | 2.29567i |
6.2 | −2.74161 | − | 0.215769i | −2.38115 | − | 1.09773i | 5.49447 | + | 0.870239i | 0.619094 | + | 0.785317i | 6.29133 | + | 3.52331i | −0.698491 | + | 1.89334i | −9.52771 | − | 2.28740i | 2.51654 | + | 2.94649i | −1.52786 | − | 2.28661i |
6.3 | −2.71024 | − | 0.213300i | −0.0605571 | − | 0.0279172i | 5.32452 | + | 0.843321i | −0.619094 | − | 0.785317i | 0.158169 | + | 0.0885791i | −1.76341 | + | 4.77993i | −8.96383 | − | 2.15202i | −1.94546 | − | 2.27784i | 1.51038 | + | 2.26045i |
6.4 | −2.58467 | − | 0.203418i | 1.02211 | + | 0.471199i | 4.66379 | + | 0.738671i | −0.619094 | − | 0.785317i | −2.54597 | − | 1.42581i | 1.33454 | − | 3.61742i | −6.86205 | − | 1.64743i | −1.12566 | − | 1.31798i | 1.44041 | + | 2.15572i |
6.5 | −2.44267 | − | 0.192243i | 2.80452 | + | 1.29290i | 3.95432 | + | 0.626303i | 0.619094 | + | 0.785317i | −6.60198 | − | 3.69729i | −1.30609 | + | 3.54031i | −4.77366 | − | 1.14605i | 4.24540 | + | 4.97073i | −1.36127 | − | 2.03729i |
6.6 | −2.38548 | − | 0.187741i | 1.96792 | + | 0.907224i | 3.67989 | + | 0.582837i | −0.619094 | − | 0.785317i | −4.52411 | − | 2.53362i | −0.170398 | + | 0.461884i | −4.01540 | − | 0.964011i | 1.10131 | + | 1.28947i | 1.32940 | + | 1.98959i |
6.7 | −2.34874 | − | 0.184849i | −0.633787 | − | 0.292180i | 3.50701 | + | 0.555456i | −0.619094 | − | 0.785317i | 1.43459 | + | 0.803408i | 0.260599 | − | 0.706386i | −3.55256 | − | 0.852895i | −1.63203 | − | 1.91086i | 1.30892 | + | 1.95894i |
6.8 | −2.34721 | − | 0.184730i | 2.55658 | + | 1.17860i | 3.49991 | + | 0.554332i | −0.619094 | − | 0.785317i | −5.78311 | − | 3.23870i | 0.286675 | − | 0.777066i | −3.53381 | − | 0.848392i | 3.19865 | + | 3.74514i | 1.30807 | + | 1.95767i |
6.9 | −2.28388 | − | 0.179745i | −1.59650 | − | 0.735999i | 3.20841 | + | 0.508162i | 0.619094 | + | 0.785317i | 3.51393 | + | 1.96789i | −0.172703 | + | 0.468131i | −2.78099 | − | 0.667657i | 0.0587886 | + | 0.0688327i | −1.27278 | − | 1.90485i |
6.10 | −2.27895 | − | 0.179357i | −2.01672 | − | 0.929721i | 3.18605 | + | 0.504620i | −0.619094 | − | 0.785317i | 4.42925 | + | 2.48050i | 0.761722 | − | 2.06474i | −2.72466 | − | 0.654133i | 1.25444 | + | 1.46876i | 1.27003 | + | 1.90073i |
6.11 | −2.23081 | − | 0.175569i | 1.40132 | + | 0.646019i | 2.97032 | + | 0.470452i | 0.619094 | + | 0.785317i | −3.01267 | − | 1.68717i | −1.08832 | + | 2.95002i | −2.19186 | − | 0.526218i | −0.401978 | − | 0.470656i | −1.24320 | − | 1.86059i |
6.12 | −2.12071 | − | 0.166904i | −0.539372 | − | 0.248654i | 2.49418 | + | 0.395040i | 0.619094 | + | 0.785317i | 1.10235 | + | 0.617346i | 1.13191 | − | 3.06817i | −1.08652 | − | 0.260851i | −1.71925 | − | 2.01298i | −1.18185 | − | 1.76876i |
6.13 | −1.87118 | − | 0.147265i | 1.38650 | + | 0.639187i | 1.50425 | + | 0.238249i | 0.619094 | + | 0.785317i | −2.50026 | − | 1.40022i | 0.00615221 | − | 0.0166763i | 0.870581 | + | 0.209008i | −0.434513 | − | 0.508750i | −1.04279 | − | 1.56064i |
6.14 | −1.83460 | − | 0.144386i | −1.63655 | − | 0.754461i | 1.36952 | + | 0.216911i | 0.619094 | + | 0.785317i | 2.89348 | + | 1.62043i | −1.71992 | + | 4.66204i | 1.09765 | + | 0.263522i | 0.160748 | + | 0.188212i | −1.02240 | − | 1.53013i |
6.15 | −1.80742 | − | 0.142247i | −0.928595 | − | 0.428088i | 1.27116 | + | 0.201333i | 0.619094 | + | 0.785317i | 1.61747 | + | 0.905826i | 1.66255 | − | 4.50654i | 1.25694 | + | 0.301766i | −1.26932 | − | 1.48618i | −1.00725 | − | 1.50746i |
6.16 | −1.79039 | − | 0.140907i | −3.04749 | − | 1.40491i | 1.21028 | + | 0.191689i | 0.619094 | + | 0.785317i | 5.25824 | + | 2.94476i | 0.451335 | − | 1.22340i | 1.35276 | + | 0.324768i | 5.36507 | + | 6.28168i | −0.997765 | − | 1.49326i |
6.17 | −1.74796 | − | 0.137567i | −0.786637 | − | 0.362645i | 1.06106 | + | 0.168056i | −0.619094 | − | 0.785317i | 1.32512 | + | 0.742104i | −0.688996 | + | 1.86760i | 1.57826 | + | 0.378907i | −1.46106 | − | 1.71068i | 0.974118 | + | 1.45787i |
6.18 | −1.70423 | − | 0.134126i | −3.09649 | − | 1.42750i | 0.911048 | + | 0.144296i | −0.619094 | − | 0.785317i | 5.08568 | + | 2.84812i | −1.01619 | + | 2.75451i | 1.79126 | + | 0.430043i | 5.60215 | + | 6.55927i | 0.949750 | + | 1.42140i |
6.19 | −1.66536 | − | 0.131067i | 2.87013 | + | 1.32315i | 0.780884 | + | 0.123680i | −0.619094 | − | 0.785317i | −4.60640 | − | 2.57971i | −0.441550 | + | 1.19687i | 1.96447 | + | 0.471627i | 4.53859 | + | 5.31401i | 0.928088 | + | 1.38898i |
6.20 | −1.52246 | − | 0.119820i | 0.407330 | + | 0.187782i | 0.328144 | + | 0.0519728i | −0.619094 | − | 0.785317i | −0.597642 | − | 0.334696i | 1.14264 | − | 3.09726i | 2.47658 | + | 0.594575i | −1.81769 | − | 2.12824i | 0.848448 | + | 1.26979i |
See next 80 embeddings (of 2304 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.d | odd | 10 | 1 | inner |
17.e | odd | 16 | 1 | inner |
187.t | even | 80 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 935.2.cq.a | ✓ | 2304 |
11.d | odd | 10 | 1 | inner | 935.2.cq.a | ✓ | 2304 |
17.e | odd | 16 | 1 | inner | 935.2.cq.a | ✓ | 2304 |
187.t | even | 80 | 1 | inner | 935.2.cq.a | ✓ | 2304 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
935.2.cq.a | ✓ | 2304 | 1.a | even | 1 | 1 | trivial |
935.2.cq.a | ✓ | 2304 | 11.d | odd | 10 | 1 | inner |
935.2.cq.a | ✓ | 2304 | 17.e | odd | 16 | 1 | inner |
935.2.cq.a | ✓ | 2304 | 187.t | even | 80 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(935, [\chi])\).