Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [637,2,Mod(491,637)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(637, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("637.491");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 637 = 7^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 637.q (of order \(6\), degree \(2\), 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: | \(5.08647060876\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
491.1 | −2.08022 | + | 1.20101i | −0.888571 | − | 1.53905i | 1.88487 | − | 3.26470i | − | 0.706642i | 3.69684 | + | 2.13437i | 0 | 4.25098i | −0.0791164 | + | 0.137034i | 0.848687 | + | 1.46997i | |||||
491.2 | −2.08022 | + | 1.20101i | 0.888571 | + | 1.53905i | 1.88487 | − | 3.26470i | 0.706642i | −3.69684 | − | 2.13437i | 0 | 4.25098i | −0.0791164 | + | 0.137034i | −0.848687 | − | 1.46997i | ||||||
491.3 | −2.04719 | + | 1.18194i | −1.49347 | − | 2.58676i | 1.79399 | − | 3.10727i | − | 3.20263i | 6.11481 | + | 3.53039i | 0 | 3.75379i | −2.96088 | + | 5.12839i | 3.78533 | + | 6.55639i | |||||
491.4 | −2.04719 | + | 1.18194i | 1.49347 | + | 2.58676i | 1.79399 | − | 3.10727i | 3.20263i | −6.11481 | − | 3.53039i | 0 | 3.75379i | −2.96088 | + | 5.12839i | −3.78533 | − | 6.55639i | ||||||
491.5 | −1.12902 | + | 0.651838i | −0.134969 | − | 0.233774i | −0.150215 | + | 0.260179i | 1.56818i | 0.304765 | + | 0.175956i | 0 | − | 2.99901i | 1.46357 | − | 2.53497i | −1.02220 | − | 1.77050i | |||||
491.6 | −1.12902 | + | 0.651838i | 0.134969 | + | 0.233774i | −0.150215 | + | 0.260179i | − | 1.56818i | −0.304765 | − | 0.175956i | 0 | − | 2.99901i | 1.46357 | − | 2.53497i | 1.02220 | + | 1.77050i | ||||
491.7 | −0.250157 | + | 0.144428i | −0.969921 | − | 1.67995i | −0.958281 | + | 1.65979i | 4.29339i | 0.485264 | + | 0.280167i | 0 | − | 1.13132i | −0.381493 | + | 0.660765i | −0.620085 | − | 1.07402i | |||||
491.8 | −0.250157 | + | 0.144428i | 0.969921 | + | 1.67995i | −0.958281 | + | 1.65979i | − | 4.29339i | −0.485264 | − | 0.280167i | 0 | − | 1.13132i | −0.381493 | + | 0.660765i | 0.620085 | + | 1.07402i | ||||
491.9 | 0.489742 | − | 0.282753i | −1.54556 | − | 2.67698i | −0.840102 | + | 1.45510i | 1.80514i | −1.51385 | − | 0.874021i | 0 | 2.08118i | −3.27749 | + | 5.67679i | 0.510408 | + | 0.884053i | ||||||
491.10 | 0.489742 | − | 0.282753i | 1.54556 | + | 2.67698i | −0.840102 | + | 1.45510i | − | 1.80514i | 1.51385 | + | 0.874021i | 0 | 2.08118i | −3.27749 | + | 5.67679i | −0.510408 | − | 0.884053i | |||||
491.11 | 0.900699 | − | 0.520019i | −0.384681 | − | 0.666288i | −0.459161 | + | 0.795291i | − | 1.67669i | −0.692964 | − | 0.400083i | 0 | 3.03516i | 1.20404 | − | 2.08546i | −0.871910 | − | 1.51019i | |||||
491.12 | 0.900699 | − | 0.520019i | 0.384681 | + | 0.666288i | −0.459161 | + | 0.795291i | 1.67669i | 0.692964 | + | 0.400083i | 0 | 3.03516i | 1.20404 | − | 2.08546i | 0.871910 | + | 1.51019i | ||||||
491.13 | 1.81104 | − | 1.04560i | −0.663994 | − | 1.15007i | 1.18657 | − | 2.05519i | − | 3.48900i | −2.40503 | − | 1.38855i | 0 | − | 0.780297i | 0.618224 | − | 1.07080i | −3.64811 | − | 6.31871i | ||||
491.14 | 1.81104 | − | 1.04560i | 0.663994 | + | 1.15007i | 1.18657 | − | 2.05519i | 3.48900i | 2.40503 | + | 1.38855i | 0 | − | 0.780297i | 0.618224 | − | 1.07080i | 3.64811 | + | 6.31871i | |||||
491.15 | 2.30510 | − | 1.33085i | −1.59481 | − | 2.76230i | 2.54233 | − | 4.40345i | − | 0.329389i | −7.35241 | − | 4.24492i | 0 | − | 8.21047i | −3.58685 | + | 6.21261i | −0.438368 | − | 0.759275i | ||||
491.16 | 2.30510 | − | 1.33085i | 1.59481 | + | 2.76230i | 2.54233 | − | 4.40345i | 0.329389i | 7.35241 | + | 4.24492i | 0 | − | 8.21047i | −3.58685 | + | 6.21261i | 0.438368 | + | 0.759275i | |||||
589.1 | −2.08022 | − | 1.20101i | −0.888571 | + | 1.53905i | 1.88487 | + | 3.26470i | 0.706642i | 3.69684 | − | 2.13437i | 0 | − | 4.25098i | −0.0791164 | − | 0.137034i | 0.848687 | − | 1.46997i | |||||
589.2 | −2.08022 | − | 1.20101i | 0.888571 | − | 1.53905i | 1.88487 | + | 3.26470i | − | 0.706642i | −3.69684 | + | 2.13437i | 0 | − | 4.25098i | −0.0791164 | − | 0.137034i | −0.848687 | + | 1.46997i | ||||
589.3 | −2.04719 | − | 1.18194i | −1.49347 | + | 2.58676i | 1.79399 | + | 3.10727i | 3.20263i | 6.11481 | − | 3.53039i | 0 | − | 3.75379i | −2.96088 | − | 5.12839i | 3.78533 | − | 6.55639i | |||||
589.4 | −2.04719 | − | 1.18194i | 1.49347 | − | 2.58676i | 1.79399 | + | 3.10727i | − | 3.20263i | −6.11481 | + | 3.53039i | 0 | − | 3.75379i | −2.96088 | − | 5.12839i | −3.78533 | + | 6.55639i | ||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
13.e | even | 6 | 1 | inner |
91.t | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 637.2.q.j | ✓ | 32 |
7.b | odd | 2 | 1 | inner | 637.2.q.j | ✓ | 32 |
7.c | even | 3 | 1 | 637.2.k.j | 32 | ||
7.c | even | 3 | 1 | 637.2.u.j | 32 | ||
7.d | odd | 6 | 1 | 637.2.k.j | 32 | ||
7.d | odd | 6 | 1 | 637.2.u.j | 32 | ||
13.e | even | 6 | 1 | inner | 637.2.q.j | ✓ | 32 |
13.f | odd | 12 | 2 | 8281.2.a.cx | 32 | ||
91.k | even | 6 | 1 | 637.2.u.j | 32 | ||
91.l | odd | 6 | 1 | 637.2.u.j | 32 | ||
91.p | odd | 6 | 1 | 637.2.k.j | 32 | ||
91.t | odd | 6 | 1 | inner | 637.2.q.j | ✓ | 32 |
91.u | even | 6 | 1 | 637.2.k.j | 32 | ||
91.bc | even | 12 | 2 | 8281.2.a.cx | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
637.2.k.j | 32 | 7.c | even | 3 | 1 | ||
637.2.k.j | 32 | 7.d | odd | 6 | 1 | ||
637.2.k.j | 32 | 91.p | odd | 6 | 1 | ||
637.2.k.j | 32 | 91.u | even | 6 | 1 | ||
637.2.q.j | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
637.2.q.j | ✓ | 32 | 7.b | odd | 2 | 1 | inner |
637.2.q.j | ✓ | 32 | 13.e | even | 6 | 1 | inner |
637.2.q.j | ✓ | 32 | 91.t | odd | 6 | 1 | inner |
637.2.u.j | 32 | 7.c | even | 3 | 1 | ||
637.2.u.j | 32 | 7.d | odd | 6 | 1 | ||
637.2.u.j | 32 | 91.k | even | 6 | 1 | ||
637.2.u.j | 32 | 91.l | odd | 6 | 1 | ||
8281.2.a.cx | 32 | 13.f | odd | 12 | 2 | ||
8281.2.a.cx | 32 | 91.bc | even | 12 | 2 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\):
\( T_{2}^{16} - 13 T_{2}^{14} + 119 T_{2}^{12} + 6 T_{2}^{11} - 544 T_{2}^{10} - 18 T_{2}^{9} + 1804 T_{2}^{8} + \cdots + 49 \) |
\( T_{3}^{32} + 38 T_{3}^{30} + 873 T_{3}^{28} + 13066 T_{3}^{26} + 144668 T_{3}^{24} + 1181268 T_{3}^{22} + \cdots + 614656 \) |