Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [425,2,Mod(49,425)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(425, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([4, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("425.49");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 425 = 5^{2} \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 425.n (of order \(8\), degree \(4\), not 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: | \(3.39364208590\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{8})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
49.1 | −1.33351 | + | 1.33351i | −1.74560 | + | 0.723051i | − | 1.55648i | 0 | 1.36358 | − | 3.29196i | 0.128325 | − | 0.309803i | −0.591431 | − | 0.591431i | 0.402996 | − | 0.402996i | 0 | |||||
49.2 | −0.917051 | + | 0.917051i | 1.69803 | − | 0.703348i | 0.318036i | 0 | −0.912176 | + | 2.20219i | −1.34200 | + | 3.23987i | −2.12576 | − | 2.12576i | 0.267297 | − | 0.267297i | 0 | ||||||
49.3 | 0.187572 | − | 0.187572i | 1.67148 | − | 0.692349i | 1.92963i | 0 | 0.183657 | − | 0.443388i | 1.88562 | − | 4.55230i | 0.737090 | + | 0.737090i | 0.193173 | − | 0.193173i | 0 | ||||||
49.4 | 0.639117 | − | 0.639117i | −2.66226 | + | 1.10274i | 1.18306i | 0 | −0.996713 | + | 2.40628i | 0.278207 | − | 0.671652i | 2.03435 | + | 2.03435i | 3.75027 | − | 3.75027i | 0 | ||||||
49.5 | 0.921271 | − | 0.921271i | 0.309218 | − | 0.128082i | 0.302521i | 0 | 0.166875 | − | 0.402872i | −1.49424 | + | 3.60742i | 2.12124 | + | 2.12124i | −2.04211 | + | 2.04211i | 0 | ||||||
49.6 | 1.91681 | − | 1.91681i | −0.977976 | + | 0.405091i | − | 5.34834i | 0 | −1.09811 | + | 2.65108i | 0.544082 | − | 1.31353i | −6.41814 | − | 6.41814i | −1.32898 | + | 1.32898i | 0 | |||||
274.1 | −1.82721 | + | 1.82721i | 1.17935 | + | 2.84719i | − | 4.67741i | 0 | −7.35734 | − | 3.04751i | −2.60148 | − | 1.07757i | 4.89219 | + | 4.89219i | −4.59433 | + | 4.59433i | 0 | |||||
274.2 | −1.52941 | + | 1.52941i | −1.06191 | − | 2.56367i | − | 2.67820i | 0 | 5.54499 | + | 2.29681i | −2.90308 | − | 1.20249i | 1.03724 | + | 1.03724i | −3.32343 | + | 3.32343i | 0 | |||||
274.3 | −0.982785 | + | 0.982785i | 0.0424107 | + | 0.102388i | 0.0682683i | 0 | −0.142306 | − | 0.0589452i | 1.58527 | + | 0.656642i | −2.03266 | − | 2.03266i | 2.11264 | − | 2.11264i | 0 | ||||||
274.4 | 0.392996 | − | 0.392996i | −0.958007 | − | 2.31283i | 1.69111i | 0 | −1.28543 | − | 0.532441i | 3.72866 | + | 1.54446i | 1.45059 | + | 1.45059i | −2.31010 | + | 2.31010i | 0 | ||||||
274.5 | 0.813283 | − | 0.813283i | 0.786634 | + | 1.89910i | 0.677141i | 0 | 2.18426 | + | 0.904752i | −0.346882 | − | 0.143683i | 2.17727 | + | 2.17727i | −0.866476 | + | 0.866476i | 0 | ||||||
274.6 | 1.71892 | − | 1.71892i | −0.281370 | − | 0.679288i | − | 3.90934i | 0 | −1.65129 | − | 0.683987i | 0.537508 | + | 0.222643i | −3.28199 | − | 3.28199i | 1.73906 | − | 1.73906i | 0 | |||||
349.1 | −1.82721 | − | 1.82721i | 1.17935 | − | 2.84719i | 4.67741i | 0 | −7.35734 | + | 3.04751i | −2.60148 | + | 1.07757i | 4.89219 | − | 4.89219i | −4.59433 | − | 4.59433i | 0 | ||||||
349.2 | −1.52941 | − | 1.52941i | −1.06191 | + | 2.56367i | 2.67820i | 0 | 5.54499 | − | 2.29681i | −2.90308 | + | 1.20249i | 1.03724 | − | 1.03724i | −3.32343 | − | 3.32343i | 0 | ||||||
349.3 | −0.982785 | − | 0.982785i | 0.0424107 | − | 0.102388i | − | 0.0682683i | 0 | −0.142306 | + | 0.0589452i | 1.58527 | − | 0.656642i | −2.03266 | + | 2.03266i | 2.11264 | + | 2.11264i | 0 | |||||
349.4 | 0.392996 | + | 0.392996i | −0.958007 | + | 2.31283i | − | 1.69111i | 0 | −1.28543 | + | 0.532441i | 3.72866 | − | 1.54446i | 1.45059 | − | 1.45059i | −2.31010 | − | 2.31010i | 0 | |||||
349.5 | 0.813283 | + | 0.813283i | 0.786634 | − | 1.89910i | − | 0.677141i | 0 | 2.18426 | − | 0.904752i | −0.346882 | + | 0.143683i | 2.17727 | − | 2.17727i | −0.866476 | − | 0.866476i | 0 | |||||
349.6 | 1.71892 | + | 1.71892i | −0.281370 | + | 0.679288i | 3.90934i | 0 | −1.65129 | + | 0.683987i | 0.537508 | − | 0.222643i | −3.28199 | + | 3.28199i | 1.73906 | + | 1.73906i | 0 | ||||||
399.1 | −1.33351 | − | 1.33351i | −1.74560 | − | 0.723051i | 1.55648i | 0 | 1.36358 | + | 3.29196i | 0.128325 | + | 0.309803i | −0.591431 | + | 0.591431i | 0.402996 | + | 0.402996i | 0 | ||||||
399.2 | −0.917051 | − | 0.917051i | 1.69803 | + | 0.703348i | − | 0.318036i | 0 | −0.912176 | − | 2.20219i | −1.34200 | − | 3.23987i | −2.12576 | + | 2.12576i | 0.267297 | + | 0.267297i | 0 | |||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
85.m | even | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 425.2.n.d | 24 | |
5.b | even | 2 | 1 | 425.2.n.e | 24 | ||
5.c | odd | 4 | 1 | 425.2.m.c | ✓ | 24 | |
5.c | odd | 4 | 1 | 425.2.m.d | yes | 24 | |
17.d | even | 8 | 1 | 425.2.n.e | 24 | ||
85.k | odd | 8 | 1 | 425.2.m.d | yes | 24 | |
85.m | even | 8 | 1 | inner | 425.2.n.d | 24 | |
85.n | odd | 8 | 1 | 425.2.m.c | ✓ | 24 | |
85.o | even | 16 | 2 | 7225.2.a.cb | 24 | ||
85.r | even | 16 | 2 | 7225.2.a.bx | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
425.2.m.c | ✓ | 24 | 5.c | odd | 4 | 1 | |
425.2.m.c | ✓ | 24 | 85.n | odd | 8 | 1 | |
425.2.m.d | yes | 24 | 5.c | odd | 4 | 1 | |
425.2.m.d | yes | 24 | 85.k | odd | 8 | 1 | |
425.2.n.d | 24 | 1.a | even | 1 | 1 | trivial | |
425.2.n.d | 24 | 85.m | even | 8 | 1 | inner | |
425.2.n.e | 24 | 5.b | even | 2 | 1 | ||
425.2.n.e | 24 | 17.d | even | 8 | 1 | ||
7225.2.a.bx | 24 | 85.r | even | 16 | 2 | ||
7225.2.a.cb | 24 | 85.o | even | 16 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 90 T_{2}^{20} + 12 T_{2}^{19} - 100 T_{2}^{17} + 2351 T_{2}^{16} + 392 T_{2}^{15} + \cdots + 625 \) acting on \(S_{2}^{\mathrm{new}}(425, [\chi])\).