Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [712,2,Mod(357,712)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(712, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("712.357");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 712 = 2^{3} \cdot 89 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 712.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: | \(5.68534862392\) |
Analytic rank: | \(0\) |
Dimension: | \(36\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
357.1 | −1.41365 | − | 0.0398817i | 1.27512i | 1.99682 | + | 0.112758i | − | 1.63611i | 0.0508541 | − | 1.80258i | 3.77754 | −2.81831 | − | 0.239037i | 1.37406 | −0.0652507 | + | 2.31288i | |||||||
357.2 | −1.41365 | + | 0.0398817i | − | 1.27512i | 1.99682 | − | 0.112758i | 1.63611i | 0.0508541 | + | 1.80258i | 3.77754 | −2.81831 | + | 0.239037i | 1.37406 | −0.0652507 | − | 2.31288i | |||||||
357.3 | −1.39812 | − | 0.212747i | 2.98776i | 1.90948 | + | 0.594892i | − | 2.44246i | 0.635637 | − | 4.17724i | −4.95507 | −2.54312 | − | 1.23797i | −5.92669 | −0.519627 | + | 3.41485i | |||||||
357.4 | −1.39812 | + | 0.212747i | − | 2.98776i | 1.90948 | − | 0.594892i | 2.44246i | 0.635637 | + | 4.17724i | −4.95507 | −2.54312 | + | 1.23797i | −5.92669 | −0.519627 | − | 3.41485i | |||||||
357.5 | −1.39187 | − | 0.250382i | 0.401361i | 1.87462 | + | 0.697000i | 2.41337i | 0.100494 | − | 0.558643i | 0.333867 | −2.43471 | − | 1.43951i | 2.83891 | 0.604264 | − | 3.35910i | ||||||||
357.6 | −1.39187 | + | 0.250382i | − | 0.401361i | 1.87462 | − | 0.697000i | − | 2.41337i | 0.100494 | + | 0.558643i | 0.333867 | −2.43471 | + | 1.43951i | 2.83891 | 0.604264 | + | 3.35910i | ||||||
357.7 | −1.08137 | − | 0.911395i | 1.16415i | 0.338718 | + | 1.97111i | − | 4.23108i | 1.06100 | − | 1.25888i | 0.524743 | 1.43018 | − | 2.44020i | 1.64475 | −3.85618 | + | 4.57536i | |||||||
357.8 | −1.08137 | + | 0.911395i | − | 1.16415i | 0.338718 | − | 1.97111i | 4.23108i | 1.06100 | + | 1.25888i | 0.524743 | 1.43018 | + | 2.44020i | 1.64475 | −3.85618 | − | 4.57536i | |||||||
357.9 | −1.02591 | − | 0.973398i | − | 1.45392i | 0.104992 | + | 1.99724i | 0.334048i | −1.41525 | + | 1.49160i | −0.482217 | 1.83640 | − | 2.15119i | 0.886105 | 0.325162 | − | 0.342704i | |||||||
357.10 | −1.02591 | + | 0.973398i | 1.45392i | 0.104992 | − | 1.99724i | − | 0.334048i | −1.41525 | − | 1.49160i | −0.482217 | 1.83640 | + | 2.15119i | 0.886105 | 0.325162 | + | 0.342704i | |||||||
357.11 | −0.697423 | − | 1.23028i | − | 1.45749i | −1.02720 | + | 1.71606i | 4.29817i | −1.79313 | + | 1.01649i | −4.23390 | 2.82764 | + | 0.0669315i | 0.875721 | 5.28797 | − | 2.99764i | |||||||
357.12 | −0.697423 | + | 1.23028i | 1.45749i | −1.02720 | − | 1.71606i | − | 4.29817i | −1.79313 | − | 1.01649i | −4.23390 | 2.82764 | − | 0.0669315i | 0.875721 | 5.28797 | + | 2.99764i | |||||||
357.13 | −0.599401 | − | 1.28091i | − | 2.00329i | −1.28144 | + | 1.53555i | − | 2.02712i | −2.56602 | + | 1.20077i | 2.51595 | 2.73499 | + | 0.720987i | −1.01317 | −2.59654 | + | 1.21506i | ||||||
357.14 | −0.599401 | + | 1.28091i | 2.00329i | −1.28144 | − | 1.53555i | 2.02712i | −2.56602 | − | 1.20077i | 2.51595 | 2.73499 | − | 0.720987i | −1.01317 | −2.59654 | − | 1.21506i | ||||||||
357.15 | −0.588837 | − | 1.28580i | 2.79991i | −1.30654 | + | 1.51425i | 3.56614i | 3.60011 | − | 1.64869i | −1.62232 | 2.71635 | + | 0.788304i | −4.83950 | 4.58533 | − | 2.09987i | ||||||||
357.16 | −0.588837 | + | 1.28580i | − | 2.79991i | −1.30654 | − | 1.51425i | − | 3.56614i | 3.60011 | + | 1.64869i | −1.62232 | 2.71635 | − | 0.788304i | −4.83950 | 4.58533 | + | 2.09987i | ||||||
357.17 | −0.456684 | − | 1.33845i | 3.12650i | −1.58288 | + | 1.22249i | − | 2.53893i | 4.18465 | − | 1.42782i | 0.0880153 | 2.35912 | + | 1.56031i | −6.77500 | −3.39822 | + | 1.15949i | |||||||
357.18 | −0.456684 | + | 1.33845i | − | 3.12650i | −1.58288 | − | 1.22249i | 2.53893i | 4.18465 | + | 1.42782i | 0.0880153 | 2.35912 | − | 1.56031i | −6.77500 | −3.39822 | − | 1.15949i | |||||||
357.19 | 0.103239 | − | 1.41044i | − | 3.25660i | −1.97868 | − | 0.291224i | − | 0.682759i | −4.59324 | − | 0.336207i | −1.89438 | −0.615031 | + | 2.76075i | −7.60545 | −0.962991 | − | 0.0704872i | ||||||
357.20 | 0.103239 | + | 1.41044i | 3.25660i | −1.97868 | + | 0.291224i | 0.682759i | −4.59324 | + | 0.336207i | −1.89438 | −0.615031 | − | 2.76075i | −7.60545 | −0.962991 | + | 0.0704872i | ||||||||
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 712.2.b.d | ✓ | 36 |
4.b | odd | 2 | 1 | 2848.2.b.d | 36 | ||
8.b | even | 2 | 1 | inner | 712.2.b.d | ✓ | 36 |
8.d | odd | 2 | 1 | 2848.2.b.d | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
712.2.b.d | ✓ | 36 | 1.a | even | 1 | 1 | trivial |
712.2.b.d | ✓ | 36 | 8.b | even | 2 | 1 | inner |
2848.2.b.d | 36 | 4.b | odd | 2 | 1 | ||
2848.2.b.d | 36 | 8.d | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(712, [\chi])\):
\( T_{3}^{36} + 79 T_{3}^{34} + 2827 T_{3}^{32} + 60679 T_{3}^{30} + 872187 T_{3}^{28} + 8879831 T_{3}^{26} + 66079399 T_{3}^{24} + 365769627 T_{3}^{22} + 1518590753 T_{3}^{20} + 4733929057 T_{3}^{18} + \cdots + 301088 \) |
\( T_{7}^{18} + 8 T_{7}^{17} - 46 T_{7}^{16} - 470 T_{7}^{15} + 624 T_{7}^{14} + 10606 T_{7}^{13} - 280 T_{7}^{12} - 117192 T_{7}^{11} - 54084 T_{7}^{10} + 667632 T_{7}^{9} + 402324 T_{7}^{8} - 1889052 T_{7}^{7} + \cdots - 10240 \) |