Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7381,2,Mod(1,7381)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7381, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7381.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 7381 = 11^{2} \cdot 61 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 7381.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: | \(58.9375817319\) |
Analytic rank: | \(0\) |
Dimension: | \(21\) |
Twist minimal: | no (minimal twist has level 671) |
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 | −2.68161 | −2.98584 | 5.19104 | −3.56342 | 8.00685 | −3.18744 | −8.55713 | 5.91522 | 9.55570 | ||||||||||||||||||
1.2 | −2.67686 | 1.01551 | 5.16560 | 1.61394 | −2.71838 | 4.48909 | −8.47390 | −1.96874 | −4.32029 | ||||||||||||||||||
1.3 | −2.54333 | 0.588033 | 4.46852 | −0.268217 | −1.49556 | −2.27426 | −6.27827 | −2.65422 | 0.682163 | ||||||||||||||||||
1.4 | −2.16302 | −0.957308 | 2.67865 | 3.42546 | 2.07068 | −1.52570 | −1.46794 | −2.08356 | −7.40934 | ||||||||||||||||||
1.5 | −2.08860 | 3.36235 | 2.36225 | −0.0546248 | −7.02260 | 2.53368 | −0.756597 | 8.30537 | 0.114089 | ||||||||||||||||||
1.6 | −1.61525 | 2.06891 | 0.609041 | 2.50050 | −3.34180 | −4.06392 | 2.24675 | 1.28037 | −4.03894 | ||||||||||||||||||
1.7 | −1.29392 | 2.84985 | −0.325780 | −3.15298 | −3.68746 | −5.03078 | 3.00937 | 5.12162 | 4.07969 | ||||||||||||||||||
1.8 | −0.942064 | 2.26946 | −1.11252 | 4.16220 | −2.13797 | 0.914635 | 2.93219 | 2.15044 | −3.92106 | ||||||||||||||||||
1.9 | −0.731918 | −2.97561 | −1.46430 | 2.76447 | 2.17790 | 1.34613 | 2.53558 | 5.85424 | −2.02337 | ||||||||||||||||||
1.10 | −0.543169 | −1.77928 | −1.70497 | −0.382346 | 0.966447 | 4.84691 | 2.01242 | 0.165824 | 0.207678 | ||||||||||||||||||
1.11 | −0.472572 | −1.64615 | −1.77668 | −3.20957 | 0.777925 | −4.06194 | 1.78475 | −0.290187 | 1.51675 | ||||||||||||||||||
1.12 | 0.404310 | 1.66838 | −1.83653 | 0.0975766 | 0.674545 | −1.38056 | −1.55115 | −0.216497 | 0.0394512 | ||||||||||||||||||
1.13 | 0.634118 | −1.50740 | −1.59789 | −1.64555 | −0.955868 | 3.41198 | −2.28149 | −0.727754 | −1.04347 | ||||||||||||||||||
1.14 | 1.29453 | 3.40741 | −0.324186 | 2.94198 | 4.41100 | 0.962580 | −3.00873 | 8.61042 | 3.80849 | ||||||||||||||||||
1.15 | 1.45442 | −2.91272 | 0.115325 | 3.82594 | −4.23631 | −3.26042 | −2.74110 | 5.48395 | 5.56450 | ||||||||||||||||||
1.16 | 1.47389 | −0.831223 | 0.172365 | 0.593969 | −1.22513 | −3.31761 | −2.69374 | −2.30907 | 0.875448 | ||||||||||||||||||
1.17 | 2.19302 | 1.13601 | 2.80936 | −3.87665 | 2.49129 | −3.34814 | 1.77494 | −1.70949 | −8.50160 | ||||||||||||||||||
1.18 | 2.40042 | −0.672797 | 3.76203 | −3.90458 | −1.61500 | 3.05635 | 4.22962 | −2.54734 | −9.37264 | ||||||||||||||||||
1.19 | 2.55546 | 1.15142 | 4.53040 | 4.10969 | 2.94242 | 4.64385 | 6.46634 | −1.67423 | 10.5022 | ||||||||||||||||||
1.20 | 2.60374 | 2.97894 | 4.77947 | 0.851432 | 7.75638 | −3.04314 | 7.23702 | 5.87407 | 2.21691 | ||||||||||||||||||
See all 21 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(11\) | \(-1\) |
\(61\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 7381.2.a.j | 21 | |
11.b | odd | 2 | 1 | 671.2.a.d | ✓ | 21 | |
33.d | even | 2 | 1 | 6039.2.a.l | 21 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
671.2.a.d | ✓ | 21 | 11.b | odd | 2 | 1 | |
6039.2.a.l | 21 | 33.d | even | 2 | 1 | ||
7381.2.a.j | 21 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7381))\):
\( T_{2}^{21} - 37 T_{2}^{19} - 2 T_{2}^{18} + 582 T_{2}^{17} + 65 T_{2}^{16} - 5074 T_{2}^{15} - 876 T_{2}^{14} + 26808 T_{2}^{13} + 6340 T_{2}^{12} - 88228 T_{2}^{11} - 26710 T_{2}^{10} + 179236 T_{2}^{9} + 66498 T_{2}^{8} + \cdots + 2082 \) |
\( T_{7}^{21} + 5 T_{7}^{20} - 101 T_{7}^{19} - 540 T_{7}^{18} + 4187 T_{7}^{17} + 24602 T_{7}^{16} - 90995 T_{7}^{15} - 616649 T_{7}^{14} + 1079283 T_{7}^{13} + 9307761 T_{7}^{12} - 6077091 T_{7}^{11} + \cdots + 1454145536 \) |