Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [510,2,Mod(121,510)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(510, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([0, 0, 7]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("510.121");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 510.u (of order \(8\), degree \(4\), 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: | \(4.07237050309\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{8})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} + 24x^{14} + 220x^{12} + 1016x^{10} + 2568x^{8} + 3552x^{6} + 2472x^{4} + 656x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-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
\( x^{16} + 24x^{14} + 220x^{12} + 1016x^{10} + 2568x^{8} + 3552x^{6} + 2472x^{4} + 656x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{15} + 26\nu^{13} + 272\nu^{11} + 1526\nu^{9} + 4974\nu^{7} + 9352\nu^{5} + 9276\nu^{3} + 3500\nu ) / 272 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{14} + 26\nu^{12} + 255\nu^{10} + 1203\nu^{8} + 2900\nu^{6} + 3402\nu^{4} + 1422\nu^{2} + 136\nu - 2 ) / 136 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{14} + 26\nu^{12} + 255\nu^{10} + 1203\nu^{8} + 2900\nu^{6} + 3402\nu^{4} + 1422\nu^{2} - 136\nu - 2 ) / 136 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 33 \nu^{15} - 10 \nu^{14} + 739 \nu^{13} - 226 \nu^{12} + 6086 \nu^{11} - 1887 \nu^{10} + 24025 \nu^{9} + \cdots + 224 ) / 272 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 17 \nu^{15} + 30 \nu^{14} - 374 \nu^{13} + 678 \nu^{12} - 2975 \nu^{11} + 5644 \nu^{10} + \cdots + 280 ) / 272 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 12 \nu^{15} + 35 \nu^{14} - 278 \nu^{13} + 774 \nu^{12} - 2414 \nu^{11} + 6239 \nu^{10} - 10305 \nu^{9} + \cdots - 36 ) / 272 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 33 \nu^{15} + 10 \nu^{14} + 739 \nu^{13} + 226 \nu^{12} + 6086 \nu^{11} + 1887 \nu^{10} + 24025 \nu^{9} + \cdots - 224 ) / 272 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 12 \nu^{15} - 35 \nu^{14} - 278 \nu^{13} - 774 \nu^{12} - 2414 \nu^{11} - 6239 \nu^{10} - 10305 \nu^{9} + \cdots + 36 ) / 272 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 17 \nu^{15} + 30 \nu^{14} + 374 \nu^{13} + 678 \nu^{12} + 2975 \nu^{11} + 5644 \nu^{10} + 10948 \nu^{9} + \cdots + 280 ) / 272 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 62 \nu^{15} - 13 \nu^{14} + 1374 \nu^{13} - 270 \nu^{12} + 11101 \nu^{11} - 1972 \nu^{10} + 42286 \nu^{9} + \cdots + 196 ) / 272 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( -15\nu^{15} - 339\nu^{13} - 2822\nu^{11} - 11228\nu^{9} - 22454\nu^{7} - 21110\nu^{5} - 7152\nu^{3} - 140\nu ) / 68 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 2\nu^{14} + 45\nu^{12} + 372\nu^{10} + 1464\nu^{8} + 2872\nu^{6} + 2598\nu^{4} + 800\nu^{2} + 4 ) / 8 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 8 \nu^{15} + 93 \nu^{14} - 174 \nu^{13} + 2078 \nu^{12} - 1377 \nu^{11} + 17034 \nu^{10} + \cdots + 732 ) / 272 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 8 \nu^{15} + 93 \nu^{14} + 174 \nu^{13} + 2078 \nu^{12} + 1377 \nu^{11} + 17034 \nu^{10} + 5238 \nu^{9} + \cdots + 460 ) / 272 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 62 \nu^{15} - 73 \nu^{14} - 1374 \nu^{13} - 1626 \nu^{12} - 11101 \nu^{11} - 13260 \nu^{10} + \cdots - 92 ) / 272 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{15} + \beta_{14} + \beta_{13} + \beta_{10} - \beta_{9} - 2\beta_{7} - \beta_{5} + 2\beta_{4} - 6 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{15} + \beta_{14} - \beta_{13} + 2 \beta_{11} - \beta_{10} + 5 \beta_{9} - 4 \beta_{8} + \cdots - 4 \beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 3 \beta_{15} - 4 \beta_{14} - 4 \beta_{13} - 2 \beta_{12} - 3 \beta_{10} + 7 \beta_{9} - \beta_{8} + \cdots + 17 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 8 \beta_{15} - 5 \beta_{14} + 5 \beta_{13} - 11 \beta_{11} + 8 \beta_{10} - 33 \beta_{9} + 20 \beta_{8} + \cdots + 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( 20 \beta_{15} + 35 \beta_{14} + 35 \beta_{13} + 28 \beta_{12} + 20 \beta_{10} - 73 \beta_{9} + \cdots - 129 \)
|
| \(\nu^{7}\) | \(=\) |
\( 91 \beta_{15} + 46 \beta_{14} - 46 \beta_{13} + 108 \beta_{11} - 91 \beta_{10} + 354 \beta_{9} + \cdots - 45 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 156 \beta_{15} - 322 \beta_{14} - 322 \beta_{13} - 308 \beta_{12} - 156 \beta_{10} + 720 \beta_{9} + \cdots + 1120 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 930 \beta_{15} - 424 \beta_{14} + 424 \beta_{13} - 1042 \beta_{11} + 930 \beta_{10} - 3544 \beta_{9} + \cdots + 506 \)
|
| \(\nu^{10}\) | \(=\) |
\( 1351 \beta_{15} + 3033 \beta_{14} + 3033 \beta_{13} + 3120 \beta_{12} + 1351 \beta_{10} - 6977 \beta_{9} + \cdots - 10288 \)
|
| \(\nu^{11}\) | \(=\) |
\( 9133 \beta_{15} + 3959 \beta_{14} - 3959 \beta_{13} + 10010 \beta_{11} - 9133 \beta_{10} + 34531 \beta_{9} + \cdots - 5174 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 12350 \beta_{15} - 28844 \beta_{14} - 28844 \beta_{13} - 30572 \beta_{12} - 12350 \beta_{10} + \cdots + 96722 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 88284 \beta_{15} - 37354 \beta_{14} + 37354 \beta_{13} - 95974 \beta_{11} + 88284 \beta_{10} + \cdots + 50930 \)
|
| \(\nu^{14}\) | \(=\) |
\( 115758 \beta_{15} + 275292 \beta_{14} + 275292 \beta_{13} + 295400 \beta_{12} + 115758 \beta_{10} + \cdots - 918158 \)
|
| \(\nu^{15}\) | \(=\) |
\( 847932 \beta_{15} + 354698 \beta_{14} - 354698 \beta_{13} + 919100 \beta_{11} - 847932 \beta_{10} + \cdots - 493234 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/510\mathbb{Z}\right)^\times\).
| \(n\) | \(241\) | \(307\) | \(341\) |
| \(\chi(n)\) | \(\beta_{9}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 121.1 |
|
−0.707107 | − | 0.707107i | −0.923880 | + | 0.382683i | 1.00000i | 0.382683 | + | 0.923880i | 0.923880 | + | 0.382683i | −1.65331 | + | 3.99145i | 0.707107 | − | 0.707107i | 0.707107 | − | 0.707107i | 0.382683 | − | 0.923880i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.2 | −0.707107 | − | 0.707107i | −0.923880 | + | 0.382683i | 1.00000i | 0.382683 | + | 0.923880i | 0.923880 | + | 0.382683i | 0.336289 | − | 0.811874i | 0.707107 | − | 0.707107i | 0.707107 | − | 0.707107i | 0.382683 | − | 0.923880i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.3 | −0.707107 | − | 0.707107i | 0.923880 | − | 0.382683i | 1.00000i | −0.382683 | − | 0.923880i | −0.923880 | − | 0.382683i | −1.88714 | + | 4.55596i | 0.707107 | − | 0.707107i | 0.707107 | − | 0.707107i | −0.382683 | + | 0.923880i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.4 | −0.707107 | − | 0.707107i | 0.923880 | − | 0.382683i | 1.00000i | −0.382683 | − | 0.923880i | −0.923880 | − | 0.382683i | 1.20417 | − | 2.90712i | 0.707107 | − | 0.707107i | 0.707107 | − | 0.707107i | −0.382683 | + | 0.923880i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.1 | 0.707107 | + | 0.707107i | −0.382683 | − | 0.923880i | 1.00000i | −0.923880 | + | 0.382683i | 0.382683 | − | 0.923880i | −3.25356 | − | 1.34767i | −0.707107 | + | 0.707107i | −0.707107 | + | 0.707107i | −0.923880 | − | 0.382683i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.2 | 0.707107 | + | 0.707107i | −0.382683 | − | 0.923880i | 1.00000i | −0.923880 | + | 0.382683i | 0.382683 | − | 0.923880i | −2.20732 | − | 0.914303i | −0.707107 | + | 0.707107i | −0.707107 | + | 0.707107i | −0.923880 | − | 0.382683i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.3 | 0.707107 | + | 0.707107i | 0.382683 | + | 0.923880i | 1.00000i | 0.923880 | − | 0.382683i | −0.382683 | + | 0.923880i | −0.980215 | − | 0.406018i | −0.707107 | + | 0.707107i | −0.707107 | + | 0.707107i | 0.923880 | + | 0.382683i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.4 | 0.707107 | + | 0.707107i | 0.382683 | + | 0.923880i | 1.00000i | 0.923880 | − | 0.382683i | −0.382683 | + | 0.923880i | 4.44110 | + | 1.83956i | −0.707107 | + | 0.707107i | −0.707107 | + | 0.707107i | 0.923880 | + | 0.382683i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 331.1 | 0.707107 | − | 0.707107i | −0.382683 | + | 0.923880i | − | 1.00000i | −0.923880 | − | 0.382683i | 0.382683 | + | 0.923880i | −3.25356 | + | 1.34767i | −0.707107 | − | 0.707107i | −0.707107 | − | 0.707107i | −0.923880 | + | 0.382683i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 331.2 | 0.707107 | − | 0.707107i | −0.382683 | + | 0.923880i | − | 1.00000i | −0.923880 | − | 0.382683i | 0.382683 | + | 0.923880i | −2.20732 | + | 0.914303i | −0.707107 | − | 0.707107i | −0.707107 | − | 0.707107i | −0.923880 | + | 0.382683i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 331.3 | 0.707107 | − | 0.707107i | 0.382683 | − | 0.923880i | − | 1.00000i | 0.923880 | + | 0.382683i | −0.382683 | − | 0.923880i | −0.980215 | + | 0.406018i | −0.707107 | − | 0.707107i | −0.707107 | − | 0.707107i | 0.923880 | − | 0.382683i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 331.4 | 0.707107 | − | 0.707107i | 0.382683 | − | 0.923880i | − | 1.00000i | 0.923880 | + | 0.382683i | −0.382683 | − | 0.923880i | 4.44110 | − | 1.83956i | −0.707107 | − | 0.707107i | −0.707107 | − | 0.707107i | 0.923880 | − | 0.382683i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 451.1 | −0.707107 | + | 0.707107i | −0.923880 | − | 0.382683i | − | 1.00000i | 0.382683 | − | 0.923880i | 0.923880 | − | 0.382683i | −1.65331 | − | 3.99145i | 0.707107 | + | 0.707107i | 0.707107 | + | 0.707107i | 0.382683 | + | 0.923880i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 451.2 | −0.707107 | + | 0.707107i | −0.923880 | − | 0.382683i | − | 1.00000i | 0.382683 | − | 0.923880i | 0.923880 | − | 0.382683i | 0.336289 | + | 0.811874i | 0.707107 | + | 0.707107i | 0.707107 | + | 0.707107i | 0.382683 | + | 0.923880i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 451.3 | −0.707107 | + | 0.707107i | 0.923880 | + | 0.382683i | − | 1.00000i | −0.382683 | + | 0.923880i | −0.923880 | + | 0.382683i | −1.88714 | − | 4.55596i | 0.707107 | + | 0.707107i | 0.707107 | + | 0.707107i | −0.382683 | − | 0.923880i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 451.4 | −0.707107 | + | 0.707107i | 0.923880 | + | 0.382683i | − | 1.00000i | −0.382683 | + | 0.923880i | −0.923880 | + | 0.382683i | 1.20417 | + | 2.90712i | 0.707107 | + | 0.707107i | 0.707107 | + | 0.707107i | −0.382683 | − | 0.923880i | ||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.d | even | 8 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 510.2.u.c | ✓ | 16 |
| 17.d | even | 8 | 1 | inner | 510.2.u.c | ✓ | 16 |
| 17.e | odd | 16 | 1 | 8670.2.a.cj | 8 | ||
| 17.e | odd | 16 | 1 | 8670.2.a.ck | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 510.2.u.c | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 510.2.u.c | ✓ | 16 | 17.d | even | 8 | 1 | inner |
| 8670.2.a.cj | 8 | 17.e | odd | 16 | 1 | ||
| 8670.2.a.ck | 8 | 17.e | odd | 16 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{16} + 8 T_{7}^{15} + 40 T_{7}^{14} + 32 T_{7}^{13} - 480 T_{7}^{12} - 1664 T_{7}^{11} + \cdots + 6390784 \)
acting on \(S_{2}^{\mathrm{new}}(510, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 1)^{4} \)
|
| $3$ |
\( (T^{8} + 1)^{2} \)
|
| $5$ |
\( (T^{8} + 1)^{2} \)
|
| $7$ |
\( T^{16} + 8 T^{15} + \cdots + 6390784 \)
|
| $11$ |
\( T^{16} + 16 T^{14} + \cdots + 73984 \)
|
| $13$ |
\( T^{16} + 112 T^{14} + \cdots + 4624 \)
|
| $17$ |
\( T^{16} + \cdots + 6975757441 \)
|
| $19$ |
\( T^{16} + \cdots + 9563275264 \)
|
| $23$ |
\( T^{16} + \cdots + 155068988944 \)
|
| $29$ |
\( T^{16} + \cdots + 720651583744 \)
|
| $31$ |
\( T^{16} + \cdots + 18634434064 \)
|
| $37$ |
\( (T^{8} - 16 T^{7} + \cdots + 1024)^{2} \)
|
| $41$ |
\( T^{16} + \cdots + 83701433344 \)
|
| $43$ |
\( T^{16} + \cdots + 626400784 \)
|
| $47$ |
\( T^{16} + \cdots + 21818834944 \)
|
| $53$ |
\( T^{16} + \cdots + 3743481856 \)
|
| $59$ |
\( T^{16} + \cdots + 4332009497104 \)
|
| $61$ |
\( T^{16} + \cdots + 3808805011456 \)
|
| $67$ |
\( (T^{8} - 8 T^{7} + \cdots - 2043388)^{2} \)
|
| $71$ |
\( T^{16} + \cdots + 208912103179264 \)
|
| $73$ |
\( T^{16} + \cdots + 6237526230016 \)
|
| $79$ |
\( T^{16} + \cdots + 11376238545424 \)
|
| $83$ |
\( T^{16} + \cdots + 282989953024 \)
|
| $89$ |
\( T^{16} + \cdots + 88978319540224 \)
|
| $97$ |
\( T^{16} + \cdots + 1877294580736 \)
|
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