Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [294,2,Mod(41,294)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(294, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([7, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("294.41");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 294 = 2 \cdot 3 \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 294.j (of order \(14\), degree \(6\), 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: | \(2.34760181943\) |
Analytic rank: | \(0\) |
Dimension: | \(120\) |
Relative dimension: | \(20\) over \(\Q(\zeta_{14})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{14}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
41.1 | −0.781831 | + | 0.623490i | −1.59440 | + | 0.676684i | 0.222521 | − | 0.974928i | −2.59923 | + | 1.25172i | 0.824644 | − | 1.52314i | −2.01156 | + | 1.71861i | 0.433884 | + | 0.900969i | 2.08420 | − | 2.15780i | 1.25172 | − | 2.59923i |
41.2 | −0.781831 | + | 0.623490i | −1.54527 | − | 0.782396i | 0.222521 | − | 0.974928i | 0.415766 | − | 0.200223i | 1.69596 | − | 0.351758i | −2.57018 | − | 0.627843i | 0.433884 | + | 0.900969i | 1.77571 | + | 2.41802i | −0.200223 | + | 0.415766i |
41.3 | −0.781831 | + | 0.623490i | −0.995627 | + | 1.41730i | 0.222521 | − | 0.974928i | 1.32694 | − | 0.639022i | −0.105257 | − | 1.72885i | 1.06200 | − | 2.42325i | 0.433884 | + | 0.900969i | −1.01745 | − | 2.82220i | −0.639022 | + | 1.32694i |
41.4 | −0.781831 | + | 0.623490i | −0.580532 | − | 1.63186i | 0.222521 | − | 0.974928i | −3.14996 | + | 1.51694i | 1.47133 | + | 0.913887i | 2.47808 | + | 0.926885i | 0.433884 | + | 0.900969i | −2.32596 | + | 1.89470i | 1.51694 | − | 3.14996i |
41.5 | −0.781831 | + | 0.623490i | −0.0435112 | − | 1.73150i | 0.222521 | − | 0.974928i | 3.37524 | − | 1.62543i | 1.11359 | + | 1.32662i | −0.0659865 | − | 2.64493i | 0.433884 | + | 0.900969i | −2.99621 | + | 0.150680i | −1.62543 | + | 3.37524i |
41.6 | −0.781831 | + | 0.623490i | 0.169937 | + | 1.72369i | 0.222521 | − | 0.974928i | −0.210425 | + | 0.101335i | −1.20757 | − | 1.24168i | 0.981077 | + | 2.45713i | 0.433884 | + | 0.900969i | −2.94224 | + | 0.585838i | 0.101335 | − | 0.210425i |
41.7 | −0.781831 | + | 0.623490i | 1.06385 | + | 1.36683i | 0.222521 | − | 0.974928i | −2.67532 | + | 1.28836i | −1.68396 | − | 0.405333i | −1.08169 | − | 2.41453i | 0.433884 | + | 0.900969i | −0.736454 | + | 2.90820i | 1.28836 | − | 2.67532i |
41.8 | −0.781831 | + | 0.623490i | 1.34258 | − | 1.09430i | 0.222521 | − | 0.974928i | −2.13620 | + | 1.02874i | −0.367384 | + | 1.69264i | −1.53826 | − | 2.15262i | 0.433884 | + | 0.900969i | 0.605021 | − | 2.93836i | 1.02874 | − | 2.13620i |
41.9 | −0.781831 | + | 0.623490i | 1.38755 | − | 1.03668i | 0.222521 | − | 0.974928i | 1.24567 | − | 0.599884i | −0.438472 | + | 1.67563i | 2.35011 | + | 1.21532i | 0.433884 | + | 0.900969i | 0.850594 | − | 2.87689i | −0.599884 | + | 1.24567i |
41.10 | −0.781831 | + | 0.623490i | 1.57726 | + | 0.715728i | 0.222521 | − | 0.974928i | 2.99870 | − | 1.44410i | −1.67940 | + | 0.423824i | −2.37504 | + | 1.16585i | 0.433884 | + | 0.900969i | 1.97547 | + | 2.25777i | −1.44410 | + | 2.99870i |
41.11 | 0.781831 | − | 0.623490i | −1.69264 | + | 0.367384i | 0.222521 | − | 0.974928i | 2.13620 | − | 1.02874i | −1.09430 | + | 1.34258i | −1.53826 | − | 2.15262i | −0.433884 | − | 0.900969i | 2.73006 | − | 1.24370i | 1.02874 | − | 2.13620i |
41.12 | 0.781831 | − | 0.623490i | −1.67563 | + | 0.438472i | 0.222521 | − | 0.974928i | −1.24567 | + | 0.599884i | −1.03668 | + | 1.38755i | 2.35011 | + | 1.21532i | −0.433884 | − | 0.900969i | 2.61548 | − | 1.46944i | −0.599884 | + | 1.24567i |
41.13 | 0.781831 | − | 0.623490i | −1.32662 | − | 1.11359i | 0.222521 | − | 0.974928i | −3.37524 | + | 1.62543i | −1.73150 | − | 0.0435112i | −0.0659865 | − | 2.64493i | −0.433884 | − | 0.900969i | 0.519818 | + | 2.95462i | −1.62543 | + | 3.37524i |
41.14 | 0.781831 | − | 0.623490i | −0.913887 | − | 1.47133i | 0.222521 | − | 0.974928i | 3.14996 | − | 1.51694i | −1.63186 | − | 0.580532i | 2.47808 | + | 0.926885i | −0.433884 | − | 0.900969i | −1.32962 | + | 2.68926i | 1.51694 | − | 3.14996i |
41.15 | 0.781831 | − | 0.623490i | −0.423824 | + | 1.67940i | 0.222521 | − | 0.974928i | −2.99870 | + | 1.44410i | 0.715728 | + | 1.57726i | −2.37504 | + | 1.16585i | −0.433884 | − | 0.900969i | −2.64075 | − | 1.42354i | −1.44410 | + | 2.99870i |
41.16 | 0.781831 | − | 0.623490i | 0.351758 | − | 1.69596i | 0.222521 | − | 0.974928i | −0.415766 | + | 0.200223i | −0.782396 | − | 1.54527i | −2.57018 | − | 0.627843i | −0.433884 | − | 0.900969i | −2.75253 | − | 1.19313i | −0.200223 | + | 0.415766i |
41.17 | 0.781831 | − | 0.623490i | 0.405333 | + | 1.68396i | 0.222521 | − | 0.974928i | 2.67532 | − | 1.28836i | 1.36683 | + | 1.06385i | −1.08169 | − | 2.41453i | −0.433884 | − | 0.900969i | −2.67141 | + | 1.36512i | 1.28836 | − | 2.67532i |
41.18 | 0.781831 | − | 0.623490i | 1.24168 | + | 1.20757i | 0.222521 | − | 0.974928i | 0.210425 | − | 0.101335i | 1.72369 | + | 0.169937i | 0.981077 | + | 2.45713i | −0.433884 | − | 0.900969i | 0.0835604 | + | 2.99884i | 0.101335 | − | 0.210425i |
41.19 | 0.781831 | − | 0.623490i | 1.52314 | − | 0.824644i | 0.222521 | − | 0.974928i | 2.59923 | − | 1.25172i | 0.676684 | − | 1.59440i | −2.01156 | + | 1.71861i | −0.433884 | − | 0.900969i | 1.63993 | − | 2.51210i | 1.25172 | − | 2.59923i |
41.20 | 0.781831 | − | 0.623490i | 1.72885 | + | 0.105257i | 0.222521 | − | 0.974928i | −1.32694 | + | 0.639022i | 1.41730 | − | 0.995627i | 1.06200 | − | 2.42325i | −0.433884 | − | 0.900969i | 2.97784 | + | 0.363947i | −0.639022 | + | 1.32694i |
See next 80 embeddings (of 120 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
49.f | odd | 14 | 1 | inner |
147.k | even | 14 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 294.2.j.a | ✓ | 120 |
3.b | odd | 2 | 1 | inner | 294.2.j.a | ✓ | 120 |
49.f | odd | 14 | 1 | inner | 294.2.j.a | ✓ | 120 |
147.k | even | 14 | 1 | inner | 294.2.j.a | ✓ | 120 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
294.2.j.a | ✓ | 120 | 1.a | even | 1 | 1 | trivial |
294.2.j.a | ✓ | 120 | 3.b | odd | 2 | 1 | inner |
294.2.j.a | ✓ | 120 | 49.f | odd | 14 | 1 | inner |
294.2.j.a | ✓ | 120 | 147.k | even | 14 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(294, [\chi])\).