Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [546,2,Mod(209,546)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(546, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 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("546.209");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 546.g (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: | \(4.35983195036\) |
| 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: |
\( x^{12} - 2x^{11} + 4x^{10} - 8x^{8} + 26x^{7} - 50x^{6} + 78x^{5} - 72x^{4} + 324x^{2} - 486x + 729 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{12} - 2x^{11} + 4x^{10} - 8x^{8} + 26x^{7} - 50x^{6} + 78x^{5} - 72x^{4} + 324x^{2} - 486x + 729 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{10} - 5 \nu^{9} + 10 \nu^{8} + 15 \nu^{7} + 19 \nu^{6} - 4 \nu^{5} - 47 \nu^{4} - 69 \nu^{3} + \cdots - 405 ) / 648 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 7 \nu^{10} + 29 \nu^{9} - 58 \nu^{8} + 87 \nu^{7} - 79 \nu^{6} - 32 \nu^{5} + 173 \nu^{4} + \cdots + 1701 ) / 648 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - \nu^{11} + 17 \nu^{10} - 16 \nu^{9} - 3 \nu^{8} + 53 \nu^{7} - 92 \nu^{6} + 215 \nu^{5} + \cdots + 2916 ) / 972 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 7 \nu^{11} - 29 \nu^{10} + 58 \nu^{9} - 87 \nu^{8} + 79 \nu^{7} + 32 \nu^{6} - 173 \nu^{5} + \cdots - 1701 \nu ) / 1944 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{11} - 2\nu^{10} + 4\nu^{9} - 8\nu^{7} + 26\nu^{6} - 50\nu^{5} + 78\nu^{4} - 72\nu^{3} + 324\nu - 486 ) / 243 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - \nu^{11} + \nu^{10} - 2 \nu^{9} - 7 \nu^{8} + 5 \nu^{7} - 12 \nu^{6} + 15 \nu^{5} - 31 \nu^{4} + \cdots + 216 ) / 216 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 7 \nu^{11} - 59 \nu^{10} + 208 \nu^{9} - 387 \nu^{8} + 601 \nu^{7} - 538 \nu^{6} - 53 \nu^{5} + \cdots + 12150 ) / 1944 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 7 \nu^{11} - 17 \nu^{10} + 43 \nu^{9} - 84 \nu^{8} + 88 \nu^{7} - 10 \nu^{6} - 257 \nu^{5} + \cdots + 972 ) / 972 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 5 \nu^{11} - 10 \nu^{10} + 29 \nu^{9} - 45 \nu^{8} + 50 \nu^{7} - 59 \nu^{6} - 79 \nu^{5} + \cdots + 1701 ) / 648 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 5 \nu^{11} + 10 \nu^{10} + \nu^{9} - 15 \nu^{8} + 70 \nu^{7} - 103 \nu^{6} + 163 \nu^{5} + \cdots + 1701 ) / 648 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{10} - \beta_{9} + \beta_{7} + \beta_{6} + \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{6} - 2\beta_{5} + \beta_{4} + \beta_{3} - \beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{11} + \beta_{9} + 2\beta_{8} + \beta_{7} + \beta_{6} - 2\beta_{5} - \beta_{4} - 5\beta_{3} - 2\beta _1 + 1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{11} - 2\beta_{10} - 2\beta_{7} - 3\beta_{6} + 10\beta_{5} + 2\beta_{4} - 4\beta_{3} - 4\beta_{2} - 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -8\beta_{10} + 6\beta_{6} + 12\beta_{5} + 6\beta_{4} - 6\beta _1 + 15 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 12 \beta_{11} - 12 \beta_{10} + 8 \beta_{8} - 6 \beta_{7} - 14 \beta_{5} + 6 \beta_{4} + 6 \beta_{3} + \cdots - 6 \)
|
| \(\nu^{8}\) | \(=\) |
\( 21 \beta_{10} - 25 \beta_{9} + 6 \beta_{8} + \beta_{7} + 17 \beta_{6} - 22 \beta_{5} - 10 \beta_{4} + \cdots + 16 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 8 \beta_{11} - 27 \beta_{10} - 21 \beta_{9} + 5 \beta_{8} - 40 \beta_{7} + 37 \beta_{6} - 10 \beta_{5} + \cdots - 9 \)
|
| \(\nu^{10}\) | \(=\) |
\( 57 \beta_{11} + 33 \beta_{9} + 8 \beta_{8} + 9 \beta_{7} + 95 \beta_{6} - 40 \beta_{5} + 31 \beta_{4} + \cdots - 25 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 72 \beta_{11} + 48 \beta_{10} - 24 \beta_{8} - 48 \beta_{7} - 27 \beta_{6} + 48 \beta_{5} + \cdots - 216 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).
| \(n\) | \(157\) | \(365\) | \(379\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 209.1 |
|
− | 1.00000i | −1.52865 | − | 0.814390i | −1.00000 | −1.76081 | −0.814390 | + | 1.52865i | 2.58450 | + | 0.566014i | 1.00000i | 1.67354 | + | 2.48983i | 1.76081i | |||||||||||||||||||||||||||||||||||||||||||||
| 209.2 | − | 1.00000i | −1.35967 | + | 1.07299i | −1.00000 | 0.745238 | 1.07299 | + | 1.35967i | −2.35778 | + | 1.20037i | 1.00000i | 0.697395 | − | 2.91781i | − | 0.745238i | |||||||||||||||||||||||||||||||||||||||||||||
| 209.3 | − | 1.00000i | 0.0198536 | − | 1.73194i | −1.00000 | −1.44804 | −1.73194 | − | 0.0198536i | −2.59654 | − | 0.507916i | 1.00000i | −2.99921 | − | 0.0687703i | 1.44804i | ||||||||||||||||||||||||||||||||||||||||||||||
| 209.4 | − | 1.00000i | 0.551597 | + | 1.64187i | −1.00000 | −0.124244 | 1.64187 | − | 0.551597i | 1.46370 | + | 2.20399i | 1.00000i | −2.39148 | + | 1.81130i | 0.124244i | ||||||||||||||||||||||||||||||||||||||||||||||
| 209.5 | − | 1.00000i | 1.59106 | + | 0.684481i | −1.00000 | 4.42062 | 0.684481 | − | 1.59106i | −2.43879 | − | 1.02583i | 1.00000i | 2.06297 | + | 2.17811i | − | 4.42062i | |||||||||||||||||||||||||||||||||||||||||||||
| 209.6 | − | 1.00000i | 1.72580 | + | 0.146987i | −1.00000 | −3.83276 | 0.146987 | − | 1.72580i | −0.655092 | + | 2.56337i | 1.00000i | 2.95679 | + | 0.507343i | 3.83276i | ||||||||||||||||||||||||||||||||||||||||||||||
| 209.7 | 1.00000i | −1.52865 | + | 0.814390i | −1.00000 | −1.76081 | −0.814390 | − | 1.52865i | 2.58450 | − | 0.566014i | − | 1.00000i | 1.67354 | − | 2.48983i | − | 1.76081i | |||||||||||||||||||||||||||||||||||||||||||||
| 209.8 | 1.00000i | −1.35967 | − | 1.07299i | −1.00000 | 0.745238 | 1.07299 | − | 1.35967i | −2.35778 | − | 1.20037i | − | 1.00000i | 0.697395 | + | 2.91781i | 0.745238i | ||||||||||||||||||||||||||||||||||||||||||||||
| 209.9 | 1.00000i | 0.0198536 | + | 1.73194i | −1.00000 | −1.44804 | −1.73194 | + | 0.0198536i | −2.59654 | + | 0.507916i | − | 1.00000i | −2.99921 | + | 0.0687703i | − | 1.44804i | |||||||||||||||||||||||||||||||||||||||||||||
| 209.10 | 1.00000i | 0.551597 | − | 1.64187i | −1.00000 | −0.124244 | 1.64187 | + | 0.551597i | 1.46370 | − | 2.20399i | − | 1.00000i | −2.39148 | − | 1.81130i | − | 0.124244i | |||||||||||||||||||||||||||||||||||||||||||||
| 209.11 | 1.00000i | 1.59106 | − | 0.684481i | −1.00000 | 4.42062 | 0.684481 | + | 1.59106i | −2.43879 | + | 1.02583i | − | 1.00000i | 2.06297 | − | 2.17811i | 4.42062i | ||||||||||||||||||||||||||||||||||||||||||||||
| 209.12 | 1.00000i | 1.72580 | − | 0.146987i | −1.00000 | −3.83276 | 0.146987 | + | 1.72580i | −0.655092 | − | 2.56337i | − | 1.00000i | 2.95679 | − | 0.507343i | − | 3.83276i | |||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 21.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 546.2.g.d | yes | 12 |
| 3.b | odd | 2 | 1 | 546.2.g.c | ✓ | 12 | |
| 7.b | odd | 2 | 1 | 546.2.g.c | ✓ | 12 | |
| 21.c | even | 2 | 1 | inner | 546.2.g.d | yes | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 546.2.g.c | ✓ | 12 | 3.b | odd | 2 | 1 | |
| 546.2.g.c | ✓ | 12 | 7.b | odd | 2 | 1 | |
| 546.2.g.d | yes | 12 | 1.a | even | 1 | 1 | trivial |
| 546.2.g.d | yes | 12 | 21.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + 2T_{5}^{5} - 18T_{5}^{4} - 46T_{5}^{3} - 7T_{5}^{2} + 32T_{5} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{6} \)
|
| $3$ |
\( T^{12} - 2 T^{11} + \cdots + 729 \)
|
| $5$ |
\( (T^{6} + 2 T^{5} - 18 T^{4} + \cdots + 4)^{2} \)
|
| $7$ |
\( T^{12} + 8 T^{11} + \cdots + 117649 \)
|
| $11$ |
\( T^{12} + 100 T^{10} + \cdots + 9709456 \)
|
| $13$ |
\( (T^{2} + 1)^{6} \)
|
| $17$ |
\( (T^{6} - 6 T^{5} + \cdots + 3032)^{2} \)
|
| $19$ |
\( T^{12} + 110 T^{10} + \cdots + 1893376 \)
|
| $23$ |
\( T^{12} + 180 T^{10} + \cdots + 96983104 \)
|
| $29$ |
\( T^{12} + 126 T^{10} + \cdots + 565504 \)
|
| $31$ |
\( T^{12} + \cdots + 378535936 \)
|
| $37$ |
\( (T^{6} - 8 T^{5} + \cdots - 12743)^{2} \)
|
| $41$ |
\( (T^{6} - 14 T^{5} + \cdots + 1024)^{2} \)
|
| $43$ |
\( (T^{6} + 4 T^{5} - 54 T^{4} + \cdots + 68)^{2} \)
|
| $47$ |
\( (T^{6} + 34 T^{5} + \cdots + 16000)^{2} \)
|
| $53$ |
\( T^{12} + \cdots + 18969001984 \)
|
| $59$ |
\( (T^{6} - 4 T^{5} + \cdots + 512)^{2} \)
|
| $61$ |
\( T^{12} + \cdots + 2567651584 \)
|
| $67$ |
\( (T^{6} - 4 T^{5} + \cdots + 15376)^{2} \)
|
| $71$ |
\( T^{12} + \cdots + 195552256 \)
|
| $73$ |
\( T^{12} + \cdots + 117948286096 \)
|
| $79$ |
\( (T^{6} - 18 T^{5} + \cdots - 64)^{2} \)
|
| $83$ |
\( (T^{6} + 16 T^{5} + \cdots + 22912)^{2} \)
|
| $89$ |
\( (T^{6} + 24 T^{5} + \cdots - 41344)^{2} \)
|
| $97$ |
\( T^{12} + \cdots + 1435500544 \)
|
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