Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [294,2,Mod(25,294)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(294, base_ring=CyclotomicField(42))
chi = DirichletCharacter(H, H._module([0, 16]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("294.25");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 294 = 2 \cdot 3 \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 294.m (of order \(21\), degree \(12\), 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: | \(36\) |
Relative dimension: | \(3\) over \(\Q(\zeta_{21})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{21}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
25.1 | −0.955573 | + | 0.294755i | 0.365341 | + | 0.930874i | 0.826239 | − | 0.563320i | −2.14348 | − | 0.323077i | −0.623490 | − | 0.781831i | 2.52421 | − | 0.792709i | −0.623490 | + | 0.781831i | −0.733052 | + | 0.680173i | 2.14348 | − | 0.323077i |
25.2 | −0.955573 | + | 0.294755i | 0.365341 | + | 0.930874i | 0.826239 | − | 0.563320i | −1.08148 | − | 0.163007i | −0.623490 | − | 0.781831i | −1.37377 | + | 2.26114i | −0.623490 | + | 0.781831i | −0.733052 | + | 0.680173i | 1.08148 | − | 0.163007i |
25.3 | −0.955573 | + | 0.294755i | 0.365341 | + | 0.930874i | 0.826239 | − | 0.563320i | 3.87015 | + | 0.583331i | −0.623490 | − | 0.781831i | 2.64466 | − | 0.0758986i | −0.623490 | + | 0.781831i | −0.733052 | + | 0.680173i | −3.87015 | + | 0.583331i |
37.1 | −0.826239 | + | 0.563320i | −0.733052 | + | 0.680173i | 0.365341 | − | 0.930874i | −2.59837 | − | 0.801491i | 0.222521 | − | 0.974928i | 2.59324 | − | 0.524514i | 0.222521 | + | 0.974928i | 0.0747301 | − | 0.997204i | 2.59837 | − | 0.801491i |
37.2 | −0.826239 | + | 0.563320i | −0.733052 | + | 0.680173i | 0.365341 | − | 0.930874i | −0.203744 | − | 0.0628466i | 0.222521 | − | 0.974928i | −1.93486 | − | 1.80452i | 0.222521 | + | 0.974928i | 0.0747301 | − | 0.997204i | 0.203744 | − | 0.0628466i |
37.3 | −0.826239 | + | 0.563320i | −0.733052 | + | 0.680173i | 0.365341 | − | 0.930874i | 3.05947 | + | 0.943720i | 0.222521 | − | 0.974928i | −1.76838 | + | 1.96795i | 0.222521 | + | 0.974928i | 0.0747301 | − | 0.997204i | −3.05947 | + | 0.943720i |
109.1 | 0.733052 | − | 0.680173i | −0.988831 | + | 0.149042i | 0.0747301 | − | 0.997204i | −1.05865 | + | 2.69740i | −0.623490 | + | 0.781831i | 0.494639 | + | 2.59910i | −0.623490 | − | 0.781831i | 0.955573 | − | 0.294755i | 1.05865 | + | 2.69740i |
109.2 | 0.733052 | − | 0.680173i | −0.988831 | + | 0.149042i | 0.0747301 | − | 0.997204i | 0.00944986 | − | 0.0240779i | −0.623490 | + | 0.781831i | 2.05870 | − | 1.66185i | −0.623490 | − | 0.781831i | 0.955573 | − | 0.294755i | −0.00944986 | − | 0.0240779i |
109.3 | 0.733052 | − | 0.680173i | −0.988831 | + | 0.149042i | 0.0747301 | − | 0.997204i | 1.35994 | − | 3.46506i | −0.623490 | + | 0.781831i | −2.55254 | + | 0.696071i | −0.623490 | − | 0.781831i | 0.955573 | − | 0.294755i | −1.35994 | − | 3.46506i |
121.1 | −0.0747301 | + | 0.997204i | 0.955573 | − | 0.294755i | −0.988831 | − | 0.149042i | −3.03776 | − | 2.81863i | 0.222521 | + | 0.974928i | −2.64177 | + | 0.145096i | 0.222521 | − | 0.974928i | 0.826239 | − | 0.563320i | 3.03776 | − | 2.81863i |
121.2 | −0.0747301 | + | 0.997204i | 0.955573 | − | 0.294755i | −0.988831 | − | 0.149042i | −0.0373053 | − | 0.0346142i | 0.222521 | + | 0.974928i | 0.835928 | − | 2.51022i | 0.222521 | − | 0.974928i | 0.826239 | − | 0.563320i | 0.0373053 | − | 0.0346142i |
121.3 | −0.0747301 | + | 0.997204i | 0.955573 | − | 0.294755i | −0.988831 | − | 0.149042i | 0.892287 | + | 0.827921i | 0.222521 | + | 0.974928i | 0.278782 | + | 2.63102i | 0.222521 | − | 0.974928i | 0.826239 | − | 0.563320i | −0.892287 | + | 0.827921i |
151.1 | −0.826239 | − | 0.563320i | −0.733052 | − | 0.680173i | 0.365341 | + | 0.930874i | −2.59837 | + | 0.801491i | 0.222521 | + | 0.974928i | 2.59324 | + | 0.524514i | 0.222521 | − | 0.974928i | 0.0747301 | + | 0.997204i | 2.59837 | + | 0.801491i |
151.2 | −0.826239 | − | 0.563320i | −0.733052 | − | 0.680173i | 0.365341 | + | 0.930874i | −0.203744 | + | 0.0628466i | 0.222521 | + | 0.974928i | −1.93486 | + | 1.80452i | 0.222521 | − | 0.974928i | 0.0747301 | + | 0.997204i | 0.203744 | + | 0.0628466i |
151.3 | −0.826239 | − | 0.563320i | −0.733052 | − | 0.680173i | 0.365341 | + | 0.930874i | 3.05947 | − | 0.943720i | 0.222521 | + | 0.974928i | −1.76838 | − | 1.96795i | 0.222521 | − | 0.974928i | 0.0747301 | + | 0.997204i | −3.05947 | − | 0.943720i |
163.1 | −0.365341 | − | 0.930874i | 0.0747301 | + | 0.997204i | −0.733052 | + | 0.680173i | −1.01252 | + | 0.690323i | 0.900969 | − | 0.433884i | −2.64531 | + | 0.0485591i | 0.900969 | + | 0.433884i | −0.988831 | + | 0.149042i | 1.01252 | + | 0.690323i |
163.2 | −0.365341 | − | 0.930874i | 0.0747301 | + | 0.997204i | −0.733052 | + | 0.680173i | −0.172293 | + | 0.117467i | 0.900969 | − | 0.433884i | −0.0217470 | − | 2.64566i | 0.900969 | + | 0.433884i | −0.988831 | + | 0.149042i | 0.172293 | + | 0.117467i |
163.3 | −0.365341 | − | 0.930874i | 0.0747301 | + | 0.997204i | −0.733052 | + | 0.680173i | 3.22240 | − | 2.19700i | 0.900969 | − | 0.433884i | 0.294655 | + | 2.62929i | 0.900969 | + | 0.433884i | −0.988831 | + | 0.149042i | −3.22240 | − | 2.19700i |
193.1 | −0.365341 | + | 0.930874i | 0.0747301 | − | 0.997204i | −0.733052 | − | 0.680173i | −1.01252 | − | 0.690323i | 0.900969 | + | 0.433884i | −2.64531 | − | 0.0485591i | 0.900969 | − | 0.433884i | −0.988831 | − | 0.149042i | 1.01252 | − | 0.690323i |
193.2 | −0.365341 | + | 0.930874i | 0.0747301 | − | 0.997204i | −0.733052 | − | 0.680173i | −0.172293 | − | 0.117467i | 0.900969 | + | 0.433884i | −0.0217470 | + | 2.64566i | 0.900969 | − | 0.433884i | −0.988831 | − | 0.149042i | 0.172293 | − | 0.117467i |
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
49.g | even | 21 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 294.2.m.c | ✓ | 36 |
3.b | odd | 2 | 1 | 882.2.z.f | 36 | ||
49.g | even | 21 | 1 | inner | 294.2.m.c | ✓ | 36 |
147.n | odd | 42 | 1 | 882.2.z.f | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
294.2.m.c | ✓ | 36 | 1.a | even | 1 | 1 | trivial |
294.2.m.c | ✓ | 36 | 49.g | even | 21 | 1 | inner |
882.2.z.f | 36 | 3.b | odd | 2 | 1 | ||
882.2.z.f | 36 | 147.n | odd | 42 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{36} - 2 T_{5}^{35} + 6 T_{5}^{34} - 44 T_{5}^{33} - 22 T_{5}^{32} - 613 T_{5}^{31} + 3398 T_{5}^{30} + \cdots + 729 \) acting on \(S_{2}^{\mathrm{new}}(294, [\chi])\).