Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [555,2,Mod(334,555)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(555, 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("555.334");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 555 = 3 \cdot 5 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 555.c (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: | \(4.43169731218\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.309760000.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} + 8x^{6} - 2x^{5} - x^{4} - 2x^{3} + 18x^{2} + 6x + 1 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 4x^{7} + 8x^{6} - 2x^{5} - x^{4} - 2x^{3} + 18x^{2} + 6x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -8\nu^{7} + 44\nu^{6} - 130\nu^{5} + 211\nu^{4} - 186\nu^{3} + 50\nu^{2} + 26\nu + 403 ) / 245 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 11\nu^{7} + 13\nu^{6} - 152\nu^{5} + 451\nu^{4} - 222\nu^{3} - 32\nu^{2} + \nu + 530 ) / 245 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 13\nu^{7} - 47\nu^{6} + 101\nu^{5} - 55\nu^{4} + 143\nu^{3} - 69\nu^{2} - 30\nu + 319 ) / 245 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -61\nu^{7} + 262\nu^{6} - 538\nu^{5} + 194\nu^{4} + 162\nu^{3} - 23\nu^{2} - 941\nu - 155 ) / 245 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 74\nu^{7} - 309\nu^{6} + 639\nu^{5} - 249\nu^{4} - 19\nu^{3} - 291\nu^{2} + 1401\nu + 229 ) / 245 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 87\nu^{7} - 356\nu^{6} + 740\nu^{5} - 304\nu^{4} + 124\nu^{3} - 360\nu^{2} + 1861\nu + 303 ) / 245 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 178\nu^{7} - 734\nu^{6} + 1545\nu^{5} - 591\nu^{4} - 149\nu^{3} - 10\nu^{2} + 3219\nu + 527 ) / 245 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} - \beta_{5} - \beta_{3} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - 2\beta_{5} - \beta_{4} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{7} + 4\beta_{6} - 5\beta_{5} - 3\beta_{4} + 4\beta_{3} - \beta_{2} + 3\beta _1 - 8 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 6\beta_{3} - 2\beta_{2} + 7\beta _1 - 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 9\beta_{7} - 21\beta_{6} + 24\beta_{5} + 25\beta_{4} + 21\beta_{3} - 9\beta_{2} + 25\beta _1 - 49 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 17\beta_{7} - 35\beta_{6} + 39\beta_{5} + 47\beta_{4} \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 64\beta_{7} - 121\beta_{6} + 127\beta_{5} + 168\beta_{4} - 121\beta_{3} + 64\beta_{2} - 168\beta _1 + 295 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/555\mathbb{Z}\right)^\times\).
| \(n\) | \(76\) | \(112\) | \(371\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 334.1 |
|
− | 1.61803i | 1.00000i | −0.618034 | −2.14896 | + | 0.618034i | 1.61803 | 0.671869i | − | 2.23607i | −1.00000 | 1.00000 | + | 3.47709i | ||||||||||||||||||||||||||||||||||||
| 334.2 | − | 1.61803i | 1.00000i | −0.618034 | 2.14896 | + | 0.618034i | 1.61803 | 3.32813i | − | 2.23607i | −1.00000 | 1.00000 | − | 3.47709i | |||||||||||||||||||||||||||||||||||||
| 334.3 | − | 0.618034i | − | 1.00000i | 1.61803 | −1.54336 | + | 1.61803i | −0.618034 | − | 4.49721i | − | 2.23607i | −1.00000 | 1.00000 | + | 0.953850i | |||||||||||||||||||||||||||||||||||
| 334.4 | − | 0.618034i | − | 1.00000i | 1.61803 | 1.54336 | + | 1.61803i | −0.618034 | 0.497212i | − | 2.23607i | −1.00000 | 1.00000 | − | 0.953850i | ||||||||||||||||||||||||||||||||||||
| 334.5 | 0.618034i | 1.00000i | 1.61803 | −1.54336 | − | 1.61803i | −0.618034 | 4.49721i | 2.23607i | −1.00000 | 1.00000 | − | 0.953850i | |||||||||||||||||||||||||||||||||||||||
| 334.6 | 0.618034i | 1.00000i | 1.61803 | 1.54336 | − | 1.61803i | −0.618034 | − | 0.497212i | 2.23607i | −1.00000 | 1.00000 | + | 0.953850i | ||||||||||||||||||||||||||||||||||||||
| 334.7 | 1.61803i | − | 1.00000i | −0.618034 | −2.14896 | − | 0.618034i | 1.61803 | − | 0.671869i | 2.23607i | −1.00000 | 1.00000 | − | 3.47709i | |||||||||||||||||||||||||||||||||||||
| 334.8 | 1.61803i | − | 1.00000i | −0.618034 | 2.14896 | − | 0.618034i | 1.61803 | − | 3.32813i | 2.23607i | −1.00000 | 1.00000 | + | 3.47709i | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 555.2.c.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1665.2.c.d | 8 | ||
| 5.b | even | 2 | 1 | inner | 555.2.c.b | ✓ | 8 |
| 5.c | odd | 4 | 1 | 2775.2.a.w | 4 | ||
| 5.c | odd | 4 | 1 | 2775.2.a.y | 4 | ||
| 15.d | odd | 2 | 1 | 1665.2.c.d | 8 | ||
| 15.e | even | 4 | 1 | 8325.2.a.bt | 4 | ||
| 15.e | even | 4 | 1 | 8325.2.a.bw | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 555.2.c.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 555.2.c.b | ✓ | 8 | 5.b | even | 2 | 1 | inner |
| 1665.2.c.d | 8 | 3.b | odd | 2 | 1 | ||
| 1665.2.c.d | 8 | 15.d | odd | 2 | 1 | ||
| 2775.2.a.w | 4 | 5.c | odd | 4 | 1 | ||
| 2775.2.a.y | 4 | 5.c | odd | 4 | 1 | ||
| 8325.2.a.bt | 4 | 15.e | even | 4 | 1 | ||
| 8325.2.a.bw | 4 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 3T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(555, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 3 T^{2} + 1)^{2} \)
|
| $3$ |
\( (T^{2} + 1)^{4} \)
|
| $5$ |
\( T^{8} - 8 T^{6} + \cdots + 625 \)
|
| $7$ |
\( T^{8} + 32 T^{6} + \cdots + 25 \)
|
| $11$ |
\( (T^{4} + 4 T^{3} - T^{2} + \cdots + 5)^{2} \)
|
| $13$ |
\( T^{8} + 32 T^{6} + \cdots + 25 \)
|
| $17$ |
\( T^{8} + 62 T^{6} + \cdots + 21025 \)
|
| $19$ |
\( (T^{4} - 4 T^{3} - 11 T^{2} + \cdots - 5)^{2} \)
|
| $23$ |
\( T^{8} + 84 T^{6} + \cdots + 1681 \)
|
| $29$ |
\( (T^{4} - 12 T^{3} + \cdots - 49)^{2} \)
|
| $31$ |
\( (T^{4} + 2 T^{3} - 28 T^{2} + \cdots + 59)^{2} \)
|
| $37$ |
\( (T^{2} + 1)^{4} \)
|
| $41$ |
\( (T^{4} + 26 T^{3} + \cdots + 1289)^{2} \)
|
| $43$ |
\( T^{8} + 78 T^{6} + \cdots + 25 \)
|
| $47$ |
\( T^{8} + 180 T^{6} + \cdots + 1 \)
|
| $53$ |
\( (T^{4} + 68 T^{2} + 176)^{2} \)
|
| $59$ |
\( (T^{4} - 12 T^{3} + \cdots - 1949)^{2} \)
|
| $61$ |
\( (T^{4} + 6 T^{3} - 16 T^{2} + \cdots - 25)^{2} \)
|
| $67$ |
\( T^{8} + 142 T^{6} + \cdots + 429025 \)
|
| $71$ |
\( (T^{4} + 34 T^{3} + \cdots + 2801)^{2} \)
|
| $73$ |
\( T^{8} + 290 T^{6} + \cdots + 13140625 \)
|
| $79$ |
\( (T^{4} + 6 T^{3} + \cdots - 6245)^{2} \)
|
| $83$ |
\( T^{8} + 126 T^{6} + \cdots + 121 \)
|
| $89$ |
\( (T^{4} - 24 T^{3} + \cdots - 725)^{2} \)
|
| $97$ |
\( T^{8} + 200 T^{6} + \cdots + 398161 \)
|
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