Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2312,1,Mod(115,2312)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2312, base_ring=CyclotomicField(68))
chi = DirichletCharacter(H, H._module([34, 34, 45]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2312.115");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2312 = 2^{3} \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2312.be (of order \(68\), degree \(32\), 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: | \(1.15383830921\) |
| Analytic rank: | \(0\) |
| Dimension: | \(32\) |
| Coefficient field: | \(\Q(\zeta_{68})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{32} - x^{30} + x^{28} - x^{26} + x^{24} - x^{22} + x^{20} - x^{18} + x^{16} - x^{14} + x^{12} + \cdots + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{68}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{68} - \cdots)\) |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2312\mathbb{Z}\right)^\times\).
| \(n\) | \(1157\) | \(1735\) | \(1737\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-\zeta_{68}^{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 115.1 |
|
0.673696 | − | 0.739009i | −0.0449462 | + | 0.0806938i | −0.0922684 | − | 0.995734i | 0 | 0.0293534 | + | 0.0875787i | 0 | −0.798017 | − | 0.602635i | 0.521941 | + | 0.842963i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 123.1 | 0.895163 | + | 0.445738i | 0.256725 | − | 0.581427i | 0.602635 | + | 0.798017i | 0 | 0.488975 | − | 0.406040i | 0 | 0.183750 | + | 0.982973i | 0.401546 | + | 0.440475i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 259.1 | 0.361242 | + | 0.932472i | 0.800095 | + | 0.111609i | −0.739009 | + | 0.673696i | 0 | 0.184956 | + | 0.786384i | 0 | −0.895163 | − | 0.445738i | −0.334130 | − | 0.0950681i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 387.1 | −0.673696 | − | 0.739009i | 1.74538 | − | 0.972171i | −0.0922684 | + | 0.995734i | 0 | −1.89430 | − | 0.634905i | 0 | 0.798017 | − | 0.602635i | 1.57481 | − | 2.54340i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 395.1 | −0.361242 | + | 0.932472i | −0.252769 | − | 1.81204i | −0.739009 | − | 0.673696i | 0 | 1.78099 | + | 0.418885i | 0 | 0.895163 | − | 0.445738i | −2.25778 | + | 0.642394i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 523.1 | −0.995734 | − | 0.0922684i | 0.352279 | + | 1.49780i | 0.982973 | + | 0.183750i | 0 | −0.212577 | − | 1.52391i | 0 | −0.961826 | − | 0.273663i | −1.22413 | + | 0.609547i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 531.1 | −0.895163 | + | 0.445738i | −1.73474 | + | 0.765964i | 0.602635 | − | 0.798017i | 0 | 1.21146 | − | 1.45890i | 0 | −0.183750 | + | 0.982973i | 1.74894 | − | 1.91849i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 659.1 | −0.798017 | + | 0.602635i | −0.276018 | − | 0.0127611i | 0.273663 | − | 0.961826i | 0 | 0.227957 | − | 0.156154i | 0 | 0.361242 | + | 0.932472i | −0.919711 | − | 0.0852238i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 667.1 | −0.961826 | − | 0.273663i | 0.621731 | + | 0.748723i | 0.850217 | + | 0.526432i | 0 | −0.393100 | − | 0.890286i | 0 | −0.673696 | − | 0.739009i | 0.00971379 | − | 0.0519642i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 795.1 | −0.183750 | + | 0.982973i | −0.359191 | + | 1.07168i | −0.932472 | − | 0.361242i | 0 | −0.987432 | − | 0.549996i | 0 | 0.526432 | − | 0.850217i | −0.221463 | − | 0.167241i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 803.1 | −0.526432 | − | 0.850217i | −0.258777 | + | 0.377767i | −0.445738 | + | 0.895163i | 0 | 0.457413 | + | 0.0211475i | 0 | 0.995734 | − | 0.0922684i | 0.285499 | + | 0.736958i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 931.1 | 0.526432 | + | 0.850217i | −1.60617 | − | 1.10025i | −0.445738 | + | 0.895163i | 0 | 0.0899135 | − | 1.94480i | 0 | −0.995734 | + | 0.0922684i | 1.00798 | + | 2.60190i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 939.1 | 0.183750 | − | 0.982973i | 1.56446 | + | 0.524354i | −0.932472 | − | 0.361242i | 0 | 0.802895 | − | 1.44147i | 0 | −0.526432 | + | 0.850217i | 1.37457 | + | 1.03803i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1067.1 | 0.961826 | + | 0.273663i | 1.34421 | − | 1.11622i | 0.850217 | + | 0.526432i | 0 | 1.59837 | − | 0.705749i | 0 | 0.673696 | + | 0.739009i | 0.377213 | − | 2.01791i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1075.1 | 0.798017 | − | 0.602635i | 0.0914812 | − | 1.97871i | 0.273663 | − | 0.961826i | 0 | −1.11943 | − | 1.63417i | 0 | −0.361242 | − | 0.932472i | −2.91118 | − | 0.269761i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1203.1 | 0.895163 | − | 0.445738i | 0.256725 | + | 0.581427i | 0.602635 | − | 0.798017i | 0 | 0.488975 | + | 0.406040i | 0 | 0.183750 | − | 0.982973i | 0.401546 | − | 0.440475i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1211.1 | 0.995734 | + | 0.0922684i | −1.24376 | + | 0.292529i | 0.982973 | + | 0.183750i | 0 | −1.26544 | + | 0.176521i | 0 | 0.961826 | + | 0.273663i | 0.566192 | − | 0.281930i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1339.1 | 0.361242 | − | 0.932472i | 0.800095 | − | 0.111609i | −0.739009 | − | 0.673696i | 0 | 0.184956 | − | 0.786384i | 0 | −0.895163 | + | 0.445738i | −0.334130 | + | 0.0950681i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1347.1 | 0.673696 | + | 0.739009i | −0.0449462 | − | 0.0806938i | −0.0922684 | + | 0.995734i | 0 | 0.0293534 | − | 0.0875787i | 0 | −0.798017 | + | 0.602635i | 0.521941 | − | 0.842963i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1475.1 | −0.361242 | − | 0.932472i | −0.252769 | + | 1.81204i | −0.739009 | + | 0.673696i | 0 | 1.78099 | − | 0.418885i | 0 | 0.895163 | + | 0.445738i | −2.25778 | − | 0.642394i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| See all 32 embeddings | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-2}) \) |
| 289.h | even | 68 | 1 | inner |
| 2312.be | odd | 68 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2312.1.be.a | ✓ | 32 |
| 8.d | odd | 2 | 1 | CM | 2312.1.be.a | ✓ | 32 |
| 289.h | even | 68 | 1 | inner | 2312.1.be.a | ✓ | 32 |
| 2312.be | odd | 68 | 1 | inner | 2312.1.be.a | ✓ | 32 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2312.1.be.a | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
| 2312.1.be.a | ✓ | 32 | 8.d | odd | 2 | 1 | CM |
| 2312.1.be.a | ✓ | 32 | 289.h | even | 68 | 1 | inner |
| 2312.1.be.a | ✓ | 32 | 2312.be | odd | 68 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2312, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{32} - T^{30} + \cdots + 1 \)
|
| $3$ |
\( T^{32} - 2 T^{31} + \cdots + 1 \)
|
| $5$ |
\( T^{32} \)
|
| $7$ |
\( T^{32} \)
|
| $11$ |
\( T^{32} + 2 T^{31} + \cdots + 65536 \)
|
| $13$ |
\( T^{32} \)
|
| $17$ |
\( T^{32} - T^{30} + \cdots + 1 \)
|
| $19$ |
\( T^{32} - 4 T^{30} + \cdots + 1 \)
|
| $23$ |
\( T^{32} \)
|
| $29$ |
\( T^{32} \)
|
| $31$ |
\( T^{32} \)
|
| $37$ |
\( T^{32} \)
|
| $41$ |
\( T^{32} - 2 T^{31} + \cdots + 1 \)
|
| $43$ |
\( (T^{16} + 51 T^{9} + \cdots + 17)^{2} \)
|
| $47$ |
\( T^{32} \)
|
| $53$ |
\( T^{32} \)
|
| $59$ |
\( (T^{16} + 17 T^{12} + \cdots + 17)^{2} \)
|
| $61$ |
\( T^{32} \)
|
| $67$ |
\( T^{32} + 17 T^{30} + \cdots + 289 \)
|
| $71$ |
\( T^{32} \)
|
| $73$ |
\( T^{32} - 2 T^{31} + \cdots + 1 \)
|
| $79$ |
\( T^{32} \)
|
| $83$ |
\( (T^{16} + 17 T^{12} + \cdots + 17)^{2} \)
|
| $89$ |
\( T^{32} - 34 T^{26} + \cdots + 289 \)
|
| $97$ |
\( T^{32} + 2 T^{31} + \cdots + 1 \)
|
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