Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [740,2,Mod(3,740)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(740, base_ring=CyclotomicField(36))
chi = DirichletCharacter(H, H._module([18, 27, 26]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("740.3");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 740 = 2^{2} \cdot 5 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 740.ce (of order \(36\), degree \(12\), 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: | \(5.90892974957\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\Q(\zeta_{36})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - x^{6} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{36}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{36}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/740\mathbb{Z}\right)^\times\).
| \(n\) | \(261\) | \(297\) | \(371\) |
| \(\chi(n)\) | \(\zeta_{36}^{2}\) | \(\zeta_{36}^{9}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
−0.123257 | − | 1.40883i | 0 | −1.96962 | + | 0.347296i | 1.53737 | − | 1.62373i | 0 | 0 | 0.732051 | + | 2.73205i | 2.95442 | + | 0.520945i | −2.47706 | − | 1.96575i | ||||||||||||||||||||||||||||||||||||||||||
| 67.1 | 0.597672 | − | 1.28171i | 0 | −1.28558 | − | 1.53209i | −1.33210 | + | 1.79597i | 0 | 0 | −2.73205 | + | 0.732051i | 1.92836 | − | 2.29813i | 1.50575 | + | 2.78078i | |||||||||||||||||||||||||||||||||||||||||||
| 243.1 | 0.597672 | + | 1.28171i | 0 | −1.28558 | + | 1.53209i | −1.33210 | − | 1.79597i | 0 | 0 | −2.73205 | − | 0.732051i | 1.92836 | + | 2.29813i | 1.50575 | − | 2.78078i | |||||||||||||||||||||||||||||||||||||||||||
| 247.1 | −0.123257 | + | 1.40883i | 0 | −1.96962 | − | 0.347296i | 1.53737 | + | 1.62373i | 0 | 0 | 0.732051 | − | 2.73205i | 2.95442 | − | 0.520945i | −2.47706 | + | 1.96575i | |||||||||||||||||||||||||||||||||||||||||||
| 263.1 | −1.15846 | + | 0.811160i | 0 | 0.684040 | − | 1.87939i | −2.17488 | − | 0.519531i | 0 | 0 | 0.732051 | + | 2.73205i | −1.02606 | − | 2.81908i | 2.94092 | − | 1.16232i | |||||||||||||||||||||||||||||||||||||||||||
| 287.1 | −1.15846 | − | 0.811160i | 0 | 0.684040 | + | 1.87939i | −2.17488 | + | 0.519531i | 0 | 0 | 0.732051 | − | 2.73205i | −1.02606 | + | 2.81908i | 2.94092 | + | 1.16232i | |||||||||||||||||||||||||||||||||||||||||||
| 363.1 | 1.28171 | + | 0.597672i | 0 | 1.28558 | + | 1.53209i | 0.637511 | + | 2.14326i | 0 | 0 | 0.732051 | + | 2.73205i | −1.92836 | + | 2.29813i | −0.463863 | + | 3.12807i | |||||||||||||||||||||||||||||||||||||||||||
| 447.1 | −1.40883 | + | 0.123257i | 0 | 1.96962 | − | 0.347296i | 2.22141 | + | 0.255652i | 0 | 0 | −2.73205 | + | 0.732051i | −2.95442 | − | 0.520945i | −3.16110 | − | 0.0863678i | |||||||||||||||||||||||||||||||||||||||||||
| 543.1 | −1.40883 | − | 0.123257i | 0 | 1.96962 | + | 0.347296i | 2.22141 | − | 0.255652i | 0 | 0 | −2.73205 | − | 0.732051i | −2.95442 | + | 0.520945i | −3.16110 | + | 0.0863678i | |||||||||||||||||||||||||||||||||||||||||||
| 583.1 | 0.811160 | − | 1.15846i | 0 | −0.684040 | − | 1.87939i | −0.889301 | + | 2.05162i | 0 | 0 | −2.73205 | − | 0.732051i | 1.02606 | − | 2.81908i | 1.65535 | + | 2.69441i | |||||||||||||||||||||||||||||||||||||||||||
| 687.1 | 1.28171 | − | 0.597672i | 0 | 1.28558 | − | 1.53209i | 0.637511 | − | 2.14326i | 0 | 0 | 0.732051 | − | 2.73205i | −1.92836 | − | 2.29813i | −0.463863 | − | 3.12807i | |||||||||||||||||||||||||||||||||||||||||||
| 707.1 | 0.811160 | + | 1.15846i | 0 | −0.684040 | + | 1.87939i | −0.889301 | − | 2.05162i | 0 | 0 | −2.73205 | + | 0.732051i | 1.02606 | + | 2.81908i | 1.65535 | − | 2.69441i | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-1}) \) |
| 185.y | odd | 36 | 1 | inner |
| 740.ce | even | 36 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 740.2.ce.b | yes | 12 |
| 4.b | odd | 2 | 1 | CM | 740.2.ce.b | yes | 12 |
| 5.c | odd | 4 | 1 | 740.2.ce.a | ✓ | 12 | |
| 20.e | even | 4 | 1 | 740.2.ce.a | ✓ | 12 | |
| 37.h | even | 18 | 1 | 740.2.ce.a | ✓ | 12 | |
| 148.o | odd | 18 | 1 | 740.2.ce.a | ✓ | 12 | |
| 185.y | odd | 36 | 1 | inner | 740.2.ce.b | yes | 12 |
| 740.ce | even | 36 | 1 | inner | 740.2.ce.b | yes | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 740.2.ce.a | ✓ | 12 | 5.c | odd | 4 | 1 | |
| 740.2.ce.a | ✓ | 12 | 20.e | even | 4 | 1 | |
| 740.2.ce.a | ✓ | 12 | 37.h | even | 18 | 1 | |
| 740.2.ce.a | ✓ | 12 | 148.o | odd | 18 | 1 | |
| 740.2.ce.b | yes | 12 | 1.a | even | 1 | 1 | trivial |
| 740.2.ce.b | yes | 12 | 4.b | odd | 2 | 1 | CM |
| 740.2.ce.b | yes | 12 | 185.y | odd | 36 | 1 | inner |
| 740.2.ce.b | yes | 12 | 740.ce | even | 36 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(740, [\chi])\):
|
\( T_{3} \)
|
|
\( T_{13}^{12} + 698T_{13}^{9} + 177497T_{13}^{6} + 23908216T_{13}^{3} + 2565726409 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + 4 T^{9} + \cdots + 64 \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( T^{12} - 4 T^{9} + \cdots + 15625 \)
|
| $7$ |
\( T^{12} \)
|
| $11$ |
\( T^{12} \)
|
| $13$ |
\( T^{12} + \cdots + 2565726409 \)
|
| $17$ |
\( T^{12} + 6 T^{11} + \cdots + 23648769 \)
|
| $19$ |
\( T^{12} \)
|
| $23$ |
\( T^{12} \)
|
| $29$ |
\( T^{12} + \cdots + 3165975289 \)
|
| $31$ |
\( T^{12} \)
|
| $37$ |
\( T^{12} + \cdots + 2565726409 \)
|
| $41$ |
\( T^{12} + \cdots + 256672441 \)
|
| $43$ |
\( T^{12} \)
|
| $47$ |
\( T^{12} \)
|
| $53$ |
\( T^{12} + \cdots + 1677100110841 \)
|
| $59$ |
\( T^{12} \)
|
| $61$ |
\( T^{12} - 72 T^{11} + \cdots + 63313849 \)
|
| $67$ |
\( T^{12} \)
|
| $71$ |
\( T^{12} \)
|
| $73$ |
\( (T^{4} - 10 T^{3} + \cdots + 28561)^{3} \)
|
| $79$ |
\( T^{12} \)
|
| $83$ |
\( T^{12} \)
|
| $89$ |
\( T^{12} + \cdots + 454266564049 \)
|
| $97$ |
\( T^{12} + \cdots + 3965582356129 \)
|
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