Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2624,2,Mod(1313,2624)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2624, 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("2624.1313");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2624 = 2^{6} \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2624.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: | \(20.9527454904\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{15} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 227 \nu^{11} - 3780 \nu^{10} + 5634 \nu^{9} - 4752 \nu^{8} + 8816 \nu^{7} + 54696 \nu^{6} + \cdots - 130392 ) / 103944 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 473 \nu^{11} + 516 \nu^{10} - 1118 \nu^{9} + 3014 \nu^{8} + 4564 \nu^{7} + 5676 \nu^{6} + \cdots + 194196 ) / 51972 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 748 \nu^{11} - 816 \nu^{10} + 1768 \nu^{9} - 4162 \nu^{8} - 6311 \nu^{7} - 8976 \nu^{6} + \cdots - 98004 ) / 51972 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 6423 \nu^{11} + 3088 \nu^{10} - 6170 \nu^{9} + 4872 \nu^{8} - 106508 \nu^{7} - 198344 \nu^{6} + \cdots + 579384 ) / 103944 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 3941 \nu^{11} + 3060 \nu^{10} - 2086 \nu^{9} + 1372 \nu^{8} + 61953 \nu^{7} + 47292 \nu^{6} + \cdots - 146880 ) / 51972 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 34 \nu^{11} + 4 \nu^{10} - 17 \nu^{9} + 12 \nu^{8} - 546 \nu^{7} - 896 \nu^{6} + 196 \nu^{5} + \cdots + 2736 ) / 366 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 10271 \nu^{11} + 6480 \nu^{10} - 5378 \nu^{9} + 4008 \nu^{8} + 162468 \nu^{7} + 140576 \nu^{6} + \cdots - 643320 ) / 103944 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 1688 \nu^{11} - 1054 \nu^{10} + 840 \nu^{9} - 684 \nu^{8} - 26378 \nu^{7} - 24587 \nu^{6} + \cdots + 105120 ) / 12993 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 21257 \nu^{11} - 13740 \nu^{10} + 12446 \nu^{9} - 7536 \nu^{8} - 345936 \nu^{7} - 272408 \nu^{6} + \cdots + 1349640 ) / 103944 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 118 \nu^{11} - 90 \nu^{10} + 53 \nu^{9} - 32 \nu^{8} - 1866 \nu^{7} - 1416 \nu^{6} + 1364 \nu^{5} + \cdots + 4320 ) / 426 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 29059 \nu^{11} + 21864 \nu^{10} - 11304 \nu^{9} + 4470 \nu^{8} + 462430 \nu^{7} + 348708 \nu^{6} + \cdots - 1049472 ) / 51972 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{9} + \beta_{8} - 3\beta_{7} + 2\beta_{5} - \beta_{4} - 2\beta_{3} - 2\beta_{2} - \beta_1 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{8} - \beta_{7} + \beta_{6} - \beta_{4} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{10} + \beta_{6} - 4\beta_{5} - 2\beta_{4} - 2\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -2\beta_{11} - 6\beta_{10} - 7\beta_{5} - \beta_{3} - 3\beta_{2} + 10 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - \beta_{11} - 7 \beta_{10} - 8 \beta_{9} - 10 \beta_{8} - 30 \beta_{7} - 6 \beta_{6} - 18 \beta_{5} + \cdots + 39 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( -4\beta_{9} - 15\beta_{8} - 28\beta_{7} \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 8 \beta_{11} - 40 \beta_{10} - 35 \beta_{9} - 61 \beta_{8} - 153 \beta_{7} + 32 \beta_{6} + \cdots - 216 ) / 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( -35\beta_{11} - 129\beta_{10} - 214\beta_{5} + 48\beta_{3} + 83\beta_{2} - 223 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( -24\beta_{11} - 107\beta_{10} - 83\beta_{6} - 212\beta_{5} + 130\beta_{4} + 82\beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( -131\beta_{9} - 341\beta_{8} - 721\beta_{7} - 472\beta_{6} + 721\beta_{4} + 393\beta_1 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 131 \beta_{11} + 557 \beta_{10} - 400 \beta_{9} - 878 \beta_{8} - 1986 \beta_{7} - 426 \beta_{6} + \cdots - 2949 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2624\mathbb{Z}\right)^\times\).
| \(n\) | \(129\) | \(575\) | \(1477\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1313.1 |
|
0 | − | 3.08613i | 0 | − | 3.18940i | 0 | −4.49756 | 0 | −6.52420 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1313.2 | 0 | − | 3.08613i | 0 | 3.18940i | 0 | 4.49756 | 0 | −6.52420 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1313.3 | 0 | − | 1.51414i | 0 | − | 0.985762i | 0 | −4.11514 | 0 | 0.707389 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1313.4 | 0 | − | 1.51414i | 0 | 0.985762i | 0 | 4.11514 | 0 | 0.707389 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1313.5 | 0 | − | 0.428007i | 0 | − | 2.20364i | 0 | −1.68450 | 0 | 2.81681 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1313.6 | 0 | − | 0.428007i | 0 | 2.20364i | 0 | 1.68450 | 0 | 2.81681 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1313.7 | 0 | 0.428007i | 0 | − | 2.20364i | 0 | 1.68450 | 0 | 2.81681 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1313.8 | 0 | 0.428007i | 0 | 2.20364i | 0 | −1.68450 | 0 | 2.81681 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1313.9 | 0 | 1.51414i | 0 | − | 0.985762i | 0 | 4.11514 | 0 | 0.707389 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1313.10 | 0 | 1.51414i | 0 | 0.985762i | 0 | −4.11514 | 0 | 0.707389 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1313.11 | 0 | 3.08613i | 0 | − | 3.18940i | 0 | 4.49756 | 0 | −6.52420 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1313.12 | 0 | 3.08613i | 0 | 3.18940i | 0 | −4.49756 | 0 | −6.52420 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 8.b | even | 2 | 1 | inner |
| 8.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2624.2.b.e | ✓ | 12 |
| 4.b | odd | 2 | 1 | inner | 2624.2.b.e | ✓ | 12 |
| 8.b | even | 2 | 1 | inner | 2624.2.b.e | ✓ | 12 |
| 8.d | odd | 2 | 1 | inner | 2624.2.b.e | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2624.2.b.e | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 2624.2.b.e | ✓ | 12 | 4.b | odd | 2 | 1 | inner |
| 2624.2.b.e | ✓ | 12 | 8.b | even | 2 | 1 | inner |
| 2624.2.b.e | ✓ | 12 | 8.d | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2624, [\chi])\):
|
\( T_{3}^{6} + 12T_{3}^{4} + 24T_{3}^{2} + 4 \)
|
|
\( T_{7}^{6} - 40T_{7}^{4} + 448T_{7}^{2} - 972 \)
|
|
\( T_{23} \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( (T^{6} + 12 T^{4} + 24 T^{2} + 4)^{2} \)
|
| $5$ |
\( (T^{6} + 16 T^{4} + \cdots + 48)^{2} \)
|
| $7$ |
\( (T^{6} - 40 T^{4} + \cdots - 972)^{2} \)
|
| $11$ |
\( (T^{6} + 32 T^{4} + \cdots + 36)^{2} \)
|
| $13$ |
\( (T^{2} + 12)^{6} \)
|
| $17$ |
\( (T^{3} - 2 T^{2} - 20 T + 24)^{4} \)
|
| $19$ |
\( (T^{6} + 84 T^{4} + \cdots + 21316)^{2} \)
|
| $23$ |
\( T^{12} \)
|
| $29$ |
\( (T^{6} + 52 T^{4} + \cdots + 192)^{2} \)
|
| $31$ |
\( (T^{2} - 48)^{6} \)
|
| $37$ |
\( (T^{6} + 288 T^{4} + \cdots + 726192)^{2} \)
|
| $41$ |
\( (T + 1)^{12} \)
|
| $43$ |
\( (T^{6} + 156 T^{4} + \cdots + 118336)^{2} \)
|
| $47$ |
\( (T^{6} - 28 T^{4} + \cdots - 12)^{2} \)
|
| $53$ |
\( (T^{6} + 52 T^{4} + \cdots + 192)^{2} \)
|
| $59$ |
\( (T^{2} + 36)^{6} \)
|
| $61$ |
\( (T^{6} + 64 T^{4} + \cdots + 6912)^{2} \)
|
| $67$ |
\( (T^{6} + 16 T^{4} + \cdots + 36)^{2} \)
|
| $71$ |
\( (T^{6} - 340 T^{4} + \cdots - 4332)^{2} \)
|
| $73$ |
\( (T^{3} - 10 T^{2} - 4 T + 4)^{4} \)
|
| $79$ |
\( (T^{6} - 172 T^{4} + \cdots - 38988)^{2} \)
|
| $83$ |
\( (T^{6} + 300 T^{4} + \cdots + 5184)^{2} \)
|
| $89$ |
\( (T^{3} - 6 T^{2} - 60 T - 72)^{4} \)
|
| $97$ |
\( (T^{3} - 2 T^{2} + \cdots + 536)^{4} \)
|
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