Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [315,10,Mod(251,315)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(315, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("315.251");
S:= CuspForms(chi, 10);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
Character orbit: | \([\chi]\) | \(=\) | 315.b (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: | \(162.236288392\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
251.1 | − | 44.9460i | 0 | −1508.14 | 625.000 | 0 | 429.891 | − | 6337.89i | 44772.6i | 0 | − | 28091.2i | ||||||||||||||
251.2 | − | 43.4603i | 0 | −1376.80 | 625.000 | 0 | 5681.54 | + | 2841.43i | 37584.3i | 0 | − | 27162.7i | ||||||||||||||
251.3 | − | 41.7518i | 0 | −1231.22 | 625.000 | 0 | −6112.87 | − | 1728.13i | 30028.6i | 0 | − | 26094.9i | ||||||||||||||
251.4 | − | 38.7119i | 0 | −986.613 | 625.000 | 0 | −2221.76 | + | 5951.25i | 18373.2i | 0 | − | 24195.0i | ||||||||||||||
251.5 | − | 38.5224i | 0 | −971.978 | 625.000 | 0 | 5534.46 | + | 3118.23i | 17719.5i | 0 | − | 24076.5i | ||||||||||||||
251.6 | − | 38.0944i | 0 | −939.186 | 625.000 | 0 | −1841.41 | − | 6079.71i | 16273.4i | 0 | − | 23809.0i | ||||||||||||||
251.7 | − | 33.3124i | 0 | −597.718 | 625.000 | 0 | −1777.21 | + | 6098.78i | 2855.49i | 0 | − | 20820.3i | ||||||||||||||
251.8 | − | 33.1306i | 0 | −585.638 | 625.000 | 0 | 1836.41 | − | 6081.22i | 2439.66i | 0 | − | 20706.6i | ||||||||||||||
251.9 | − | 31.2804i | 0 | −466.463 | 625.000 | 0 | 4497.64 | − | 4486.07i | − | 1424.41i | 0 | − | 19550.2i | |||||||||||||
251.10 | − | 28.7659i | 0 | −315.478 | 625.000 | 0 | −6144.87 | − | 1610.66i | − | 5653.15i | 0 | − | 17978.7i | |||||||||||||
251.11 | − | 27.0678i | 0 | −220.664 | 625.000 | 0 | −2939.90 | + | 5631.22i | − | 7885.83i | 0 | − | 16917.3i | |||||||||||||
251.12 | − | 27.0056i | 0 | −217.303 | 625.000 | 0 | 6343.29 | − | 341.079i | − | 7958.48i | 0 | − | 16878.5i | |||||||||||||
251.13 | − | 25.1545i | 0 | −120.747 | 625.000 | 0 | −559.056 | − | 6327.80i | − | 9841.77i | 0 | − | 15721.5i | |||||||||||||
251.14 | − | 23.5339i | 0 | −41.8461 | 625.000 | 0 | −3605.99 | + | 5229.77i | − | 11064.6i | 0 | − | 14708.7i | |||||||||||||
251.15 | − | 22.4652i | 0 | 7.31397 | 625.000 | 0 | 6124.53 | + | 1686.34i | − | 11666.5i | 0 | − | 14040.8i | |||||||||||||
251.16 | − | 20.3785i | 0 | 96.7172 | 625.000 | 0 | −6209.18 | − | 1341.52i | − | 12404.7i | 0 | − | 12736.6i | |||||||||||||
251.17 | − | 16.2824i | 0 | 246.883 | 625.000 | 0 | −4032.57 | − | 4908.36i | − | 12356.4i | 0 | − | 10176.5i | |||||||||||||
251.18 | − | 13.2885i | 0 | 335.417 | 625.000 | 0 | 4209.35 | + | 4757.62i | − | 11260.9i | 0 | − | 8305.29i | |||||||||||||
251.19 | − | 10.2080i | 0 | 407.798 | 625.000 | 0 | −6351.60 | − | 103.891i | − | 9389.26i | 0 | − | 6379.98i | |||||||||||||
251.20 | − | 9.87231i | 0 | 414.537 | 625.000 | 0 | −1874.58 | − | 6069.56i | − | 9147.07i | 0 | − | 6170.19i | |||||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
21.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 315.10.b.b | yes | 48 |
3.b | odd | 2 | 1 | 315.10.b.a | ✓ | 48 | |
7.b | odd | 2 | 1 | 315.10.b.a | ✓ | 48 | |
21.c | even | 2 | 1 | inner | 315.10.b.b | yes | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
315.10.b.a | ✓ | 48 | 3.b | odd | 2 | 1 | |
315.10.b.a | ✓ | 48 | 7.b | odd | 2 | 1 | |
315.10.b.b | yes | 48 | 1.a | even | 1 | 1 | trivial |
315.10.b.b | yes | 48 | 21.c | even | 2 | 1 | inner |