Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2070,2,Mod(1241,2070)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2070, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2070.1241");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2070.e (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: | \(16.5290332184\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} + 32x^{14} + 392x^{12} + 2324x^{10} + 6930x^{8} + 9856x^{6} + 5740x^{4} + 1108x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
| Coefficient ring index: | \( 2^{11} \) |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} + 32x^{14} + 392x^{12} + 2324x^{10} + 6930x^{8} + 9856x^{6} + 5740x^{4} + 1108x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{14} + 31\nu^{12} + 353\nu^{10} + 1747\nu^{8} + 3031\nu^{6} - 1479\nu^{4} - 4661\nu^{2} - 71 ) / 1568 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -13\nu^{14} - 411\nu^{12} - 4925\nu^{10} - 27999\nu^{8} - 76603\nu^{6} - 88749\nu^{4} - 24879\nu^{2} + 2195 ) / 1568 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5\nu^{14} + 153\nu^{12} + 1751\nu^{10} + 9303\nu^{8} + 22823\nu^{6} + 21583\nu^{4} + 3137\nu^{2} - 275 ) / 392 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{15} - 91\nu^{13} - 1017\nu^{11} - 5039\nu^{9} - 9481\nu^{7} + 2899\nu^{5} + 22933\nu^{3} + 9247\nu ) / 392 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 13 \nu^{15} + 411 \nu^{13} + 4925 \nu^{11} + 27999 \nu^{9} + 76603 \nu^{7} + 88749 \nu^{5} + \cdots - 2195 \nu ) / 1568 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 19\nu^{14} + 593\nu^{12} + 7015\nu^{10} + 39505\nu^{8} + 108221\nu^{6} + 129879\nu^{4} + 45877\nu^{2} - 557 ) / 784 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 55 \nu^{14} + 1729 \nu^{12} + 20591 \nu^{10} + 116373 \nu^{8} + 317281 \nu^{6} + 371943 \nu^{4} + \cdots - 1057 ) / 1568 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 8\nu^{15} + 255\nu^{13} + 3104\nu^{11} + 18204\nu^{9} + 53214\nu^{7} + 72735\nu^{5} + 38816\nu^{3} + 6544\nu ) / 196 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 69 \nu^{15} + 2195 \nu^{13} + 26653 \nu^{11} + 155823 \nu^{9} + 453411 \nu^{7} + 613797 \nu^{5} + \cdots + 39117 \nu ) / 1568 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 71 \nu^{15} - 2273 \nu^{13} - 27863 \nu^{11} - 165357 \nu^{9} - 493777 \nu^{7} - 702807 \nu^{5} + \cdots - 74007 \nu ) / 1568 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 41 \nu^{15} + 1303 \nu^{13} + 15789 \nu^{11} + 91911 \nu^{9} + 265007 \nu^{7} + 351273 \nu^{5} + \cdots + 20029 \nu ) / 784 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 85 \nu^{15} + 12 \nu^{14} - 2747 \nu^{13} + 388 \nu^{12} - 34149 \nu^{11} + 4796 \nu^{10} + \cdots + 7804 ) / 1568 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 85 \nu^{15} + 25 \nu^{14} + 2747 \nu^{13} + 799 \nu^{12} + 34149 \nu^{11} + 9721 \nu^{10} + \cdots + 5609 ) / 1568 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 169 \nu^{15} + 18 \nu^{14} - 5399 \nu^{13} + 574 \nu^{12} - 65985 \nu^{11} + 6970 \nu^{10} + \cdots - 3766 ) / 1568 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 169 \nu^{15} - 18 \nu^{14} - 5399 \nu^{13} - 574 \nu^{12} - 65985 \nu^{11} - 6970 \nu^{10} + \cdots + 3766 ) / 1568 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{11} - \beta_{9} - \beta_{5} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{13} + \beta_{12} - 2\beta_{7} + \beta_{6} + \beta_{3} - \beta_{2} + 2\beta _1 - 7 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{15} - \beta_{14} - 8\beta_{11} + 7\beta_{9} - 5\beta_{8} + 11\beta_{5} - \beta_{4} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - \beta_{15} + \beta_{14} - 9 \beta_{13} - 9 \beta_{12} + 22 \beta_{7} - 12 \beta_{6} - 8 \beta_{3} + \cdots + 52 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 14 \beta_{15} + 14 \beta_{14} - 2 \beta_{13} + 2 \beta_{12} + 69 \beta_{11} - 14 \beta_{10} + \cdots - 2 \beta_{2} ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 19 \beta_{15} - 19 \beta_{14} + 86 \beta_{13} + 86 \beta_{12} - 228 \beta_{7} + 133 \beta_{6} + \cdots - 461 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 173 \beta_{15} - 173 \beta_{14} + 42 \beta_{13} - 42 \beta_{12} - 642 \beta_{11} + \cdots + 42 \beta_{2} ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 126 \beta_{15} + 126 \beta_{14} - 429 \beta_{13} - 429 \beta_{12} + 1192 \beta_{7} - 729 \beta_{6} + \cdots + 2212 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 2038 \beta_{15} + 2038 \beta_{14} - 628 \beta_{13} + 628 \beta_{12} + 6291 \beta_{11} + \cdots - 628 \beta_{2} ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 2998 \beta_{15} - 2998 \beta_{14} + 8833 \beta_{13} + 8833 \beta_{12} - 25326 \beta_{7} + 16039 \beta_{6} + \cdots - 44493 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 23423 \beta_{15} - 23423 \beta_{14} + 8184 \beta_{13} - 8184 \beta_{12} - 63958 \beta_{11} + \cdots + 8184 \beta_{2} ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 34197 \beta_{15} + 34197 \beta_{14} - 92959 \beta_{13} - 92959 \beta_{12} + 272502 \beta_{7} + \cdots + 461354 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 265372 \beta_{15} + 265372 \beta_{14} - 99474 \beta_{13} + 99474 \beta_{12} + 667523 \beta_{11} + \cdots - 99474 \beta_{2} ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 382989 \beta_{15} - 382989 \beta_{14} + 993290 \beta_{13} + 993290 \beta_{12} - 2958048 \beta_{7} + \cdots - 4882959 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 2979759 \beta_{15} - 2979759 \beta_{14} + 1163166 \beta_{13} - 1163166 \beta_{12} + \cdots + 1163166 \beta_{2} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2070\mathbb{Z}\right)^\times\).
| \(n\) | \(461\) | \(1657\) | \(1891\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1241.1 |
|
− | 1.00000i | 0 | −1.00000 | 1.00000 | 0 | − | 4.69659i | 1.00000i | 0 | − | 1.00000i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1241.2 | − | 1.00000i | 0 | −1.00000 | 1.00000 | 0 | − | 2.68734i | 1.00000i | 0 | − | 1.00000i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1241.3 | − | 1.00000i | 0 | −1.00000 | 1.00000 | 0 | − | 2.10311i | 1.00000i | 0 | − | 1.00000i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1241.4 | − | 1.00000i | 0 | −1.00000 | 1.00000 | 0 | − | 0.0425859i | 1.00000i | 0 | − | 1.00000i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1241.5 | − | 1.00000i | 0 | −1.00000 | 1.00000 | 0 | 0.891877i | 1.00000i | 0 | − | 1.00000i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1241.6 | − | 1.00000i | 0 | −1.00000 | 1.00000 | 0 | 1.12555i | 1.00000i | 0 | − | 1.00000i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1241.7 | − | 1.00000i | 0 | −1.00000 | 1.00000 | 0 | 3.66489i | 1.00000i | 0 | − | 1.00000i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1241.8 | − | 1.00000i | 0 | −1.00000 | 1.00000 | 0 | 3.84729i | 1.00000i | 0 | − | 1.00000i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1241.9 | 1.00000i | 0 | −1.00000 | 1.00000 | 0 | − | 3.84729i | − | 1.00000i | 0 | 1.00000i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1241.10 | 1.00000i | 0 | −1.00000 | 1.00000 | 0 | − | 3.66489i | − | 1.00000i | 0 | 1.00000i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1241.11 | 1.00000i | 0 | −1.00000 | 1.00000 | 0 | − | 1.12555i | − | 1.00000i | 0 | 1.00000i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1241.12 | 1.00000i | 0 | −1.00000 | 1.00000 | 0 | − | 0.891877i | − | 1.00000i | 0 | 1.00000i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1241.13 | 1.00000i | 0 | −1.00000 | 1.00000 | 0 | 0.0425859i | − | 1.00000i | 0 | 1.00000i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1241.14 | 1.00000i | 0 | −1.00000 | 1.00000 | 0 | 2.10311i | − | 1.00000i | 0 | 1.00000i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1241.15 | 1.00000i | 0 | −1.00000 | 1.00000 | 0 | 2.68734i | − | 1.00000i | 0 | 1.00000i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1241.16 | 1.00000i | 0 | −1.00000 | 1.00000 | 0 | 4.69659i | − | 1.00000i | 0 | 1.00000i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 69.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2070.2.e.b | yes | 16 |
| 3.b | odd | 2 | 1 | 2070.2.e.a | ✓ | 16 | |
| 23.b | odd | 2 | 1 | 2070.2.e.a | ✓ | 16 | |
| 69.c | even | 2 | 1 | inner | 2070.2.e.b | yes | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2070.2.e.a | ✓ | 16 | 3.b | odd | 2 | 1 | |
| 2070.2.e.a | ✓ | 16 | 23.b | odd | 2 | 1 | |
| 2070.2.e.b | yes | 16 | 1.a | even | 1 | 1 | trivial |
| 2070.2.e.b | yes | 16 | 69.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{8} + 12T_{11}^{7} + 22T_{11}^{6} - 200T_{11}^{5} - 732T_{11}^{4} + 192T_{11}^{3} + 2568T_{11}^{2} + 896T_{11} - 1792 \)
acting on \(S_{2}^{\mathrm{new}}(2070, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{8} \)
|
| $3$ |
\( T^{16} \)
|
| $5$ |
\( (T - 1)^{16} \)
|
| $7$ |
\( T^{16} + 64 T^{14} + \cdots + 256 \)
|
| $11$ |
\( (T^{8} + 12 T^{7} + \cdots - 1792)^{2} \)
|
| $13$ |
\( (T^{8} - 46 T^{6} + \cdots - 1568)^{2} \)
|
| $17$ |
\( (T^{8} - 40 T^{6} + \cdots - 128)^{2} \)
|
| $19$ |
\( T^{16} + \cdots + 4228120576 \)
|
| $23$ |
\( T^{16} + \cdots + 78310985281 \)
|
| $29$ |
\( T^{16} + \cdots + 45474709504 \)
|
| $31$ |
\( (T^{8} + 4 T^{7} + \cdots + 28672)^{2} \)
|
| $37$ |
\( T^{16} + \cdots + 286015744 \)
|
| $41$ |
\( T^{16} + \cdots + 4787974164736 \)
|
| $43$ |
\( T^{16} + \cdots + 1584676864 \)
|
| $47$ |
\( T^{16} + \cdots + 1670008274944 \)
|
| $53$ |
\( (T^{8} - 4 T^{7} + \cdots + 1035776)^{2} \)
|
| $59$ |
\( T^{16} + \cdots + 716888201453824 \)
|
| $61$ |
\( T^{16} + \cdots + 8469889024 \)
|
| $67$ |
\( T^{16} + \cdots + 203119673344 \)
|
| $71$ |
\( T^{16} + \cdots + 20025696600064 \)
|
| $73$ |
\( (T^{8} - 16 T^{7} + \cdots + 4600064)^{2} \)
|
| $79$ |
\( T^{16} + \cdots + 825757696 \)
|
| $83$ |
\( (T^{8} + 28 T^{7} + \cdots - 17706752)^{2} \)
|
| $89$ |
\( (T^{8} - 20 T^{7} + \cdots - 180512)^{2} \)
|
| $97$ |
\( T^{16} + \cdots + 71572141441024 \)
|
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