Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [195,2,Mod(67,195)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(195, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 3, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("195.67");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 195 = 3 \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 195.bd (of order \(12\), degree \(4\), 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: | \(1.55708283941\) |
Analytic rank: | \(0\) |
Dimension: | \(56\) |
Relative dimension: | \(14\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
67.1 | −1.25215 | − | 2.16879i | 0.965926 | + | 0.258819i | −2.13577 | + | 3.69927i | −0.256829 | + | 2.22127i | −0.648162 | − | 2.41897i | 2.27192 | + | 1.31169i | 5.68865 | 0.866025 | + | 0.500000i | 5.13906 | − | 2.22436i | ||
67.2 | −1.00587 | − | 1.74222i | −0.965926 | − | 0.258819i | −1.02355 | + | 1.77285i | −1.62945 | + | 1.53131i | 0.520677 | + | 1.94319i | −0.165019 | − | 0.0952740i | 0.0947706 | 0.866025 | + | 0.500000i | 4.30689 | + | 1.29856i | ||
67.3 | −0.918199 | − | 1.59037i | −0.965926 | − | 0.258819i | −0.686179 | + | 1.18850i | 1.86952 | − | 1.22674i | 0.475295 | + | 1.77382i | −2.27688 | − | 1.31456i | −1.15260 | 0.866025 | + | 0.500000i | −3.66756 | − | 1.84683i | ||
67.4 | −0.853152 | − | 1.47770i | 0.965926 | + | 0.258819i | −0.455736 | + | 0.789358i | 1.35839 | − | 1.77617i | −0.441624 | − | 1.64816i | 4.26702 | + | 2.46357i | −1.85736 | 0.866025 | + | 0.500000i | −3.78356 | − | 0.491964i | ||
67.5 | −0.626089 | − | 1.08442i | 0.965926 | + | 0.258819i | 0.216026 | − | 0.374168i | −1.65898 | − | 1.49926i | −0.324087 | − | 1.20951i | −2.38272 | − | 1.37566i | −3.04536 | 0.866025 | + | 0.500000i | −0.587155 | + | 2.73770i | ||
67.6 | −0.314928 | − | 0.545471i | −0.965926 | − | 0.258819i | 0.801641 | − | 1.38848i | 1.08672 | + | 1.95424i | 0.163019 | + | 0.608393i | 3.19090 | + | 1.84227i | −2.26955 | 0.866025 | + | 0.500000i | 0.723739 | − | 1.20822i | ||
67.7 | 0.0801292 | + | 0.138788i | −0.965926 | − | 0.258819i | 0.987159 | − | 1.70981i | −2.23524 | + | 0.0607511i | −0.0414779 | − | 0.154798i | −3.84699 | − | 2.22106i | 0.636918 | 0.866025 | + | 0.500000i | −0.187540 | − | 0.305356i | ||
67.8 | 0.334760 | + | 0.579821i | 0.965926 | + | 0.258819i | 0.775871 | − | 1.34385i | −0.773477 | + | 2.09803i | 0.173285 | + | 0.646707i | 0.0515993 | + | 0.0297909i | 2.37796 | 0.866025 | + | 0.500000i | −1.47541 | + | 0.253859i | ||
67.9 | 0.482143 | + | 0.835096i | 0.965926 | + | 0.258819i | 0.535077 | − | 0.926780i | 0.137372 | − | 2.23184i | 0.249575 | + | 0.931428i | −1.17248 | − | 0.676930i | 2.96050 | 0.866025 | + | 0.500000i | 1.93004 | − | 0.961349i | ||
67.10 | 0.563303 | + | 0.975669i | −0.965926 | − | 0.258819i | 0.365380 | − | 0.632856i | −1.02941 | − | 1.98502i | −0.291587 | − | 1.08822i | 2.13490 | + | 1.23258i | 3.07649 | 0.866025 | + | 0.500000i | 1.35685 | − | 2.12254i | ||
67.11 | 0.861480 | + | 1.49213i | −0.965926 | − | 0.258819i | −0.484294 | + | 0.838822i | 2.23388 | + | 0.0988139i | −0.445935 | − | 1.66425i | 0.421075 | + | 0.243108i | 1.77708 | 0.866025 | + | 0.500000i | 1.77700 | + | 3.41836i | ||
67.12 | 1.13669 | + | 1.96880i | 0.965926 | + | 0.258819i | −1.58411 | + | 2.74376i | −2.22995 | − | 0.165292i | 0.588392 | + | 2.19591i | 1.84947 | + | 1.06779i | −2.65581 | 0.866025 | + | 0.500000i | −2.20933 | − | 4.57821i | ||
67.13 | 1.23409 | + | 2.13750i | −0.965926 | − | 0.258819i | −2.04594 | + | 3.54367i | −1.52807 | + | 1.63248i | −0.638810 | − | 2.38407i | −0.354557 | − | 0.204703i | −5.16311 | 0.866025 | + | 0.500000i | −5.37520 | − | 1.25162i | ||
67.14 | 1.27780 | + | 2.21322i | 0.965926 | + | 0.258819i | −2.26557 | + | 3.92408i | 2.19142 | − | 0.444615i | 0.661440 | + | 2.46853i | −3.98824 | − | 2.30261i | −6.46858 | 0.866025 | + | 0.500000i | 3.78424 | + | 4.28196i | ||
97.1 | −1.33341 | + | 2.30953i | −0.258819 | − | 0.965926i | −2.55595 | − | 4.42704i | 1.09012 | + | 1.95234i | 2.57595 | + | 0.690223i | 3.66094 | − | 2.11365i | 8.29887 | −0.866025 | + | 0.500000i | −5.96257 | − | 0.0856134i | ||
97.2 | −1.22286 | + | 2.11806i | 0.258819 | + | 0.965926i | −1.99078 | − | 3.44813i | 2.19831 | + | 0.409196i | −2.36239 | − | 0.633000i | −3.85373 | + | 2.22495i | 4.84635 | −0.866025 | + | 0.500000i | −3.55493 | + | 4.15576i | ||
97.3 | −0.856342 | + | 1.48323i | −0.258819 | − | 0.965926i | −0.466643 | − | 0.808249i | −2.21901 | + | 0.275658i | 1.65433 | + | 0.443275i | −0.592129 | + | 0.341866i | −1.82694 | −0.866025 | + | 0.500000i | 1.49137 | − | 3.52736i | ||
97.4 | −0.437238 | + | 0.757319i | 0.258819 | + | 0.965926i | 0.617645 | + | 1.06979i | 2.23600 | + | 0.0167872i | −0.844680 | − | 0.226331i | 1.89201 | − | 1.09235i | −2.82919 | −0.866025 | + | 0.500000i | −0.990380 | + | 1.68603i | ||
97.5 | −0.416908 | + | 0.722106i | 0.258819 | + | 0.965926i | 0.652375 | + | 1.12995i | −2.02394 | − | 0.950609i | −0.805405 | − | 0.215808i | −4.10424 | + | 2.36958i | −2.75555 | −0.866025 | + | 0.500000i | 1.53024 | − | 1.06518i | ||
97.6 | −0.278861 | + | 0.483002i | −0.258819 | − | 0.965926i | 0.844473 | + | 1.46267i | 1.99255 | + | 1.01476i | 0.538719 | + | 0.144349i | −1.80160 | + | 1.04015i | −2.05741 | −0.866025 | + | 0.500000i | −1.04578 | + | 0.679427i | ||
See all 56 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
65.o | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 195.2.bd.a | ✓ | 56 |
3.b | odd | 2 | 1 | 585.2.cf.b | 56 | ||
5.b | even | 2 | 1 | 975.2.bl.i | 56 | ||
5.c | odd | 4 | 1 | 195.2.bm.a | yes | 56 | |
5.c | odd | 4 | 1 | 975.2.bu.i | 56 | ||
13.f | odd | 12 | 1 | 195.2.bm.a | yes | 56 | |
15.e | even | 4 | 1 | 585.2.dp.c | 56 | ||
39.k | even | 12 | 1 | 585.2.dp.c | 56 | ||
65.o | even | 12 | 1 | inner | 195.2.bd.a | ✓ | 56 |
65.s | odd | 12 | 1 | 975.2.bu.i | 56 | ||
65.t | even | 12 | 1 | 975.2.bl.i | 56 | ||
195.bn | odd | 12 | 1 | 585.2.cf.b | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
195.2.bd.a | ✓ | 56 | 1.a | even | 1 | 1 | trivial |
195.2.bd.a | ✓ | 56 | 65.o | even | 12 | 1 | inner |
195.2.bm.a | yes | 56 | 5.c | odd | 4 | 1 | |
195.2.bm.a | yes | 56 | 13.f | odd | 12 | 1 | |
585.2.cf.b | 56 | 3.b | odd | 2 | 1 | ||
585.2.cf.b | 56 | 195.bn | odd | 12 | 1 | ||
585.2.dp.c | 56 | 15.e | even | 4 | 1 | ||
585.2.dp.c | 56 | 39.k | even | 12 | 1 | ||
975.2.bl.i | 56 | 5.b | even | 2 | 1 | ||
975.2.bl.i | 56 | 65.t | even | 12 | 1 | ||
975.2.bu.i | 56 | 5.c | odd | 4 | 1 | ||
975.2.bu.i | 56 | 65.s | odd | 12 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(195, [\chi])\).