Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [289,10,Mod(1,289)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(289, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("289.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 289 = 17^{2} \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
Character orbit: | \([\chi]\) | \(=\) | 289.a (trivial) |
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: | yes |
Analytic conductor: | \(148.845356651\) |
Analytic rank: | \(1\) |
Dimension: | \(24\) |
Twist minimal: | no (minimal twist has level 17) |
Fricke sign: | \(1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 | −38.1697 | −135.302 | 944.927 | 33.6875 | 5164.44 | −4816.06 | −16524.7 | −1376.34 | −1285.84 | ||||||||||||||||||
1.2 | −38.1697 | 135.302 | 944.927 | −33.6875 | −5164.44 | 4816.06 | −16524.7 | −1376.34 | 1285.84 | ||||||||||||||||||
1.3 | −36.7615 | −213.730 | 839.411 | −2187.52 | 7857.06 | −1444.09 | −12036.1 | 25997.7 | 80416.5 | ||||||||||||||||||
1.4 | −36.7615 | 213.730 | 839.411 | 2187.52 | −7857.06 | 1444.09 | −12036.1 | 25997.7 | −80416.5 | ||||||||||||||||||
1.5 | −31.4159 | −60.4230 | 474.957 | 1582.21 | 1898.24 | −10488.5 | 1163.73 | −16032.1 | −49706.4 | ||||||||||||||||||
1.6 | −31.4159 | 60.4230 | 474.957 | −1582.21 | −1898.24 | 10488.5 | 1163.73 | −16032.1 | 49706.4 | ||||||||||||||||||
1.7 | −17.1503 | −137.714 | −217.866 | 1247.89 | 2361.84 | 10551.1 | 12517.4 | −717.820 | −21401.7 | ||||||||||||||||||
1.8 | −17.1503 | 137.714 | −217.866 | −1247.89 | −2361.84 | −10551.1 | 12517.4 | −717.820 | 21401.7 | ||||||||||||||||||
1.9 | −12.8587 | −204.729 | −346.654 | 120.661 | 2632.54 | 4525.78 | 11041.2 | 22230.9 | −1551.54 | ||||||||||||||||||
1.10 | −12.8587 | 204.729 | −346.654 | −120.661 | −2632.54 | −4525.78 | 11041.2 | 22230.9 | 1551.54 | ||||||||||||||||||
1.11 | −9.57683 | −3.76279 | −420.284 | 1873.25 | 36.0355 | −1553.61 | 8928.33 | −19668.8 | −17939.8 | ||||||||||||||||||
1.12 | −9.57683 | 3.76279 | −420.284 | −1873.25 | −36.0355 | 1553.61 | 8928.33 | −19668.8 | 17939.8 | ||||||||||||||||||
1.13 | 7.42963 | −213.375 | −456.801 | 298.376 | −1585.29 | −7723.69 | −7197.83 | 25845.7 | 2216.82 | ||||||||||||||||||
1.14 | 7.42963 | 213.375 | −456.801 | −298.376 | 1585.29 | 7723.69 | −7197.83 | 25845.7 | −2216.82 | ||||||||||||||||||
1.15 | 12.9147 | −101.422 | −345.210 | −2078.59 | −1309.84 | −2161.61 | −11070.6 | −9396.49 | −26844.4 | ||||||||||||||||||
1.16 | 12.9147 | 101.422 | −345.210 | 2078.59 | 1309.84 | 2161.61 | −11070.6 | −9396.49 | 26844.4 | ||||||||||||||||||
1.17 | 21.4252 | −32.0790 | −52.9591 | −1493.58 | −687.300 | −1220.42 | −12104.4 | −18653.9 | −32000.3 | ||||||||||||||||||
1.18 | 21.4252 | 32.0790 | −52.9591 | 1493.58 | 687.300 | 1220.42 | −12104.4 | −18653.9 | 32000.3 | ||||||||||||||||||
1.19 | 28.8329 | −253.694 | 319.336 | 1602.54 | −7314.73 | −6961.05 | −5555.06 | 44677.5 | 46205.9 | ||||||||||||||||||
1.20 | 28.8329 | 253.694 | 319.336 | −1602.54 | 7314.73 | 6961.05 | −5555.06 | 44677.5 | −46205.9 | ||||||||||||||||||
See all 24 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(17\) | \(1\) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
17.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 289.10.a.f | 24 | |
17.b | even | 2 | 1 | inner | 289.10.a.f | 24 | |
17.d | even | 8 | 2 | 17.10.c.a | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
17.10.c.a | ✓ | 24 | 17.d | even | 8 | 2 | |
289.10.a.f | 24 | 1.a | even | 1 | 1 | trivial | |
289.10.a.f | 24 | 17.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(289))\):
\( T_{2}^{12} - 4353 T_{2}^{10} + 6952624 T_{2}^{8} - 201024 T_{2}^{7} - 4976376048 T_{2}^{6} + \cdots + 78\!\cdots\!12 \) |
\( T_{3}^{24} - 290380 T_{3}^{22} + 35998259964 T_{3}^{20} + \cdots + 10\!\cdots\!00 \) |