Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [354,6,Mod(353,354)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(354, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("354.353");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 354 = 2 \cdot 3 \cdot 59 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 354.c (of order \(2\), degree \(1\), 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: | \(56.7758722138\) |
Analytic rank: | \(0\) |
Dimension: | \(50\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
353.1 | −4.00000 | −15.5824 | − | 0.434789i | 16.0000 | − | 38.7472i | 62.3296 | + | 1.73916i | −41.9790 | −64.0000 | 242.622 | + | 13.5501i | 154.989i | |||||||||||
353.2 | −4.00000 | −15.5824 | + | 0.434789i | 16.0000 | 38.7472i | 62.3296 | − | 1.73916i | −41.9790 | −64.0000 | 242.622 | − | 13.5501i | − | 154.989i | |||||||||||
353.3 | −4.00000 | −15.5391 | − | 1.23947i | 16.0000 | 14.2214i | 62.1564 | + | 4.95787i | 129.028 | −64.0000 | 239.927 | + | 38.5204i | − | 56.8857i | |||||||||||
353.4 | −4.00000 | −15.5391 | + | 1.23947i | 16.0000 | − | 14.2214i | 62.1564 | − | 4.95787i | 129.028 | −64.0000 | 239.927 | − | 38.5204i | 56.8857i | |||||||||||
353.5 | −4.00000 | −14.5480 | − | 5.59965i | 16.0000 | 83.1853i | 58.1920 | + | 22.3986i | 242.629 | −64.0000 | 180.288 | + | 162.927i | − | 332.741i | |||||||||||
353.6 | −4.00000 | −14.5480 | + | 5.59965i | 16.0000 | − | 83.1853i | 58.1920 | − | 22.3986i | 242.629 | −64.0000 | 180.288 | − | 162.927i | 332.741i | |||||||||||
353.7 | −4.00000 | −13.3335 | − | 8.07581i | 16.0000 | 83.4718i | 53.3339 | + | 32.3032i | −123.882 | −64.0000 | 112.563 | + | 215.357i | − | 333.887i | |||||||||||
353.8 | −4.00000 | −13.3335 | + | 8.07581i | 16.0000 | − | 83.4718i | 53.3339 | − | 32.3032i | −123.882 | −64.0000 | 112.563 | − | 215.357i | 333.887i | |||||||||||
353.9 | −4.00000 | −12.9095 | − | 8.73750i | 16.0000 | − | 27.1818i | 51.6381 | + | 34.9500i | −249.363 | −64.0000 | 90.3121 | + | 225.594i | 108.727i | |||||||||||
353.10 | −4.00000 | −12.9095 | + | 8.73750i | 16.0000 | 27.1818i | 51.6381 | − | 34.9500i | −249.363 | −64.0000 | 90.3121 | − | 225.594i | − | 108.727i | |||||||||||
353.11 | −4.00000 | −11.6908 | − | 10.3115i | 16.0000 | − | 106.082i | 46.7630 | + | 41.2458i | 71.9340 | −64.0000 | 30.3476 | + | 241.098i | 424.326i | |||||||||||
353.12 | −4.00000 | −11.6908 | + | 10.3115i | 16.0000 | 106.082i | 46.7630 | − | 41.2458i | 71.9340 | −64.0000 | 30.3476 | − | 241.098i | − | 424.326i | |||||||||||
353.13 | −4.00000 | −11.5231 | − | 10.4984i | 16.0000 | − | 67.9871i | 46.0925 | + | 41.9938i | −61.6810 | −64.0000 | 22.5654 | + | 241.950i | 271.948i | |||||||||||
353.14 | −4.00000 | −11.5231 | + | 10.4984i | 16.0000 | 67.9871i | 46.0925 | − | 41.9938i | −61.6810 | −64.0000 | 22.5654 | − | 241.950i | − | 271.948i | |||||||||||
353.15 | −4.00000 | −9.75722 | − | 12.1572i | 16.0000 | 43.2791i | 39.0289 | + | 48.6287i | −5.13563 | −64.0000 | −52.5934 | + | 237.240i | − | 173.116i | |||||||||||
353.16 | −4.00000 | −9.75722 | + | 12.1572i | 16.0000 | − | 43.2791i | 39.0289 | − | 48.6287i | −5.13563 | −64.0000 | −52.5934 | − | 237.240i | 173.116i | |||||||||||
353.17 | −4.00000 | −9.68192 | − | 12.2172i | 16.0000 | − | 48.6180i | 38.7277 | + | 48.8689i | 149.149 | −64.0000 | −55.5209 | + | 236.572i | 194.472i | |||||||||||
353.18 | −4.00000 | −9.68192 | + | 12.2172i | 16.0000 | 48.6180i | 38.7277 | − | 48.8689i | 149.149 | −64.0000 | −55.5209 | − | 236.572i | − | 194.472i | |||||||||||
353.19 | −4.00000 | −4.24050 | − | 15.0006i | 16.0000 | 2.81364i | 16.9620 | + | 60.0024i | −187.063 | −64.0000 | −207.036 | + | 127.220i | − | 11.2546i | |||||||||||
353.20 | −4.00000 | −4.24050 | + | 15.0006i | 16.0000 | − | 2.81364i | 16.9620 | − | 60.0024i | −187.063 | −64.0000 | −207.036 | − | 127.220i | 11.2546i | |||||||||||
See all 50 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
177.d | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 354.6.c.a | ✓ | 50 |
3.b | odd | 2 | 1 | 354.6.c.b | yes | 50 | |
59.b | odd | 2 | 1 | 354.6.c.b | yes | 50 | |
177.d | even | 2 | 1 | inner | 354.6.c.a | ✓ | 50 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
354.6.c.a | ✓ | 50 | 1.a | even | 1 | 1 | trivial |
354.6.c.a | ✓ | 50 | 177.d | even | 2 | 1 | inner |
354.6.c.b | yes | 50 | 3.b | odd | 2 | 1 | |
354.6.c.b | yes | 50 | 59.b | odd | 2 | 1 |