Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [105,3,Mod(62,105)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(105, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1, 2]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("105.62");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 105 = 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 105.k (of order \(4\), degree \(2\), 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: | \(2.86104277578\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} + 433x^{8} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{16} + 433x^{8} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{10} + 479\nu^{2} ) / 210 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{10} + \nu^{8} + 374\nu^{2} + 164 ) / 105 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{12} - 437\nu^{4} ) / 84 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{12} + \nu^{8} - 853\nu^{4} + 164 ) / 105 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 4\nu^{12} - 3\nu^{10} + 1706\nu^{4} - 1227\nu^{2} ) / 210 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -5\nu^{13} + 2\nu^{11} - 2185\nu^{5} + 958\nu^{3} ) / 420 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -13\nu^{13} + 4\nu^{9} - 5597\nu^{5} + 1916\nu ) / 840 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 13\nu^{13} + 4\nu^{9} + 5597\nu^{5} + 1916\nu ) / 840 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 23\nu^{14} + 9967\nu^{6} ) / 840 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( -23\nu^{13} - 4\nu^{9} - 9967\nu^{5} - 1076\nu ) / 840 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 23\nu^{13} - 4\nu^{9} + 9967\nu^{5} - 1076\nu ) / 840 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 3\nu^{14} - \nu^{10} + 1297\nu^{6} - 409\nu^{2} ) / 70 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 23\nu^{15} - 10\nu^{13} + 9967\nu^{7} - 4370\nu^{5} ) / 840 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 21\nu^{15} - 4\nu^{13} + 4\nu^{11} + 9093\nu^{7} - 1748\nu^{5} + 1748\nu^{3} ) / 336 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 21\nu^{15} - 4\nu^{13} - 4\nu^{11} + 9093\nu^{7} - 1748\nu^{5} - 1748\nu^{3} ) / 336 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{11} + \beta_{10} + \beta_{8} + \beta_{7} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} - \beta_{2} + 5\beta_1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{15} - 2\beta_{14} + 5\beta_{11} - 5\beta_{10} - 5\beta_{8} + 5\beta_{7} + 10\beta_{6} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -5\beta_{5} + 5\beta_{4} - 16\beta_{3} - 5\beta_{2} - 5\beta_1 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 13\beta_{11} - 13\beta_{10} - 23\beta_{8} + 23\beta_{7} ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -46\beta_{12} + 72\beta_{9} + 23\beta_{5} + 23\beta_{4} - 23\beta_{2} - 23\beta_1 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -46\beta_{15} - 46\beta_{14} + 210\beta_{13} + 59\beta_{11} - 59\beta_{10} - 59\beta_{8} + 59\beta_{7} ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 105\beta_{5} + 105\beta_{4} + 105\beta_{2} + 105\beta _1 - 328 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( -479\beta_{11} - 479\beta_{10} - 269\beta_{8} - 269\beta_{7} ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( -479\beta_{5} - 479\beta_{4} + 479\beta_{2} - 1975\beta_1 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 958 \beta_{15} + 958 \beta_{14} - 2185 \beta_{11} + 2185 \beta_{10} + 2185 \beta_{8} + \cdots - 4370 \beta_{6} ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 2185\beta_{5} - 2185\beta_{4} + 6824\beta_{3} + 2185\beta_{2} + 2185\beta_1 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( -5597\beta_{11} + 5597\beta_{10} + 9967\beta_{8} - 9967\beta_{7} ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 19934\beta_{12} - 31128\beta_{9} - 9967\beta_{5} - 9967\beta_{4} + 9967\beta_{2} + 9967\beta_1 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 19934 \beta_{15} + 19934 \beta_{14} - 90930 \beta_{13} - 25531 \beta_{11} + 25531 \beta_{10} + \cdots - 25531 \beta_{7} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).
| \(n\) | \(22\) | \(31\) | \(71\) |
| \(\chi(n)\) | \(\beta_{3}\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 62.1 |
|
−0.707107 | + | 0.707107i | −2.87568 | + | 0.854662i | 3.00000i | −4.57796 | − | 2.01054i | 1.42908 | − | 2.63775i | 3.94887 | − | 5.77983i | −4.94975 | − | 4.94975i | 7.53910 | − | 4.91548i | 4.65877 | − | 1.81544i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 62.2 | −0.707107 | + | 0.707107i | −0.152778 | − | 2.99611i | 3.00000i | −4.24762 | + | 2.63775i | 2.22660 | + | 2.01054i | 4.33402 | + | 5.49694i | −4.94975 | − | 4.94975i | −8.95332 | + | 0.915476i | 1.13835 | − | 4.86869i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 62.3 | −0.707107 | + | 0.707107i | 0.152778 | + | 2.99611i | 3.00000i | 4.24762 | − | 2.63775i | −2.22660 | − | 2.01054i | 5.49694 | + | 4.33402i | −4.94975 | − | 4.94975i | −8.95332 | + | 0.915476i | −1.13835 | + | 4.86869i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 62.4 | −0.707107 | + | 0.707107i | 2.87568 | − | 0.854662i | 3.00000i | 4.57796 | + | 2.01054i | −1.42908 | + | 2.63775i | −5.77983 | + | 3.94887i | −4.94975 | − | 4.94975i | 7.53910 | − | 4.91548i | −4.65877 | + | 1.81544i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 62.5 | 0.707107 | − | 0.707107i | −2.99611 | − | 0.152778i | 3.00000i | 4.24762 | − | 2.63775i | −2.22660 | + | 2.01054i | 4.33402 | + | 5.49694i | 4.94975 | + | 4.94975i | 8.95332 | + | 0.915476i | 1.13835 | − | 4.86869i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 62.6 | 0.707107 | − | 0.707107i | −0.854662 | + | 2.87568i | 3.00000i | −4.57796 | − | 2.01054i | 1.42908 | + | 2.63775i | −5.77983 | + | 3.94887i | 4.94975 | + | 4.94975i | −7.53910 | − | 4.91548i | −4.65877 | + | 1.81544i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 62.7 | 0.707107 | − | 0.707107i | 0.854662 | − | 2.87568i | 3.00000i | 4.57796 | + | 2.01054i | −1.42908 | − | 2.63775i | 3.94887 | − | 5.77983i | 4.94975 | + | 4.94975i | −7.53910 | − | 4.91548i | 4.65877 | − | 1.81544i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 62.8 | 0.707107 | − | 0.707107i | 2.99611 | + | 0.152778i | 3.00000i | −4.24762 | + | 2.63775i | 2.22660 | − | 2.01054i | 5.49694 | + | 4.33402i | 4.94975 | + | 4.94975i | 8.95332 | + | 0.915476i | −1.13835 | + | 4.86869i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.1 | −0.707107 | − | 0.707107i | −2.87568 | − | 0.854662i | − | 3.00000i | −4.57796 | + | 2.01054i | 1.42908 | + | 2.63775i | 3.94887 | + | 5.77983i | −4.94975 | + | 4.94975i | 7.53910 | + | 4.91548i | 4.65877 | + | 1.81544i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.2 | −0.707107 | − | 0.707107i | −0.152778 | + | 2.99611i | − | 3.00000i | −4.24762 | − | 2.63775i | 2.22660 | − | 2.01054i | 4.33402 | − | 5.49694i | −4.94975 | + | 4.94975i | −8.95332 | − | 0.915476i | 1.13835 | + | 4.86869i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.3 | −0.707107 | − | 0.707107i | 0.152778 | − | 2.99611i | − | 3.00000i | 4.24762 | + | 2.63775i | −2.22660 | + | 2.01054i | 5.49694 | − | 4.33402i | −4.94975 | + | 4.94975i | −8.95332 | − | 0.915476i | −1.13835 | − | 4.86869i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.4 | −0.707107 | − | 0.707107i | 2.87568 | + | 0.854662i | − | 3.00000i | 4.57796 | − | 2.01054i | −1.42908 | − | 2.63775i | −5.77983 | − | 3.94887i | −4.94975 | + | 4.94975i | 7.53910 | + | 4.91548i | −4.65877 | − | 1.81544i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.5 | 0.707107 | + | 0.707107i | −2.99611 | + | 0.152778i | − | 3.00000i | 4.24762 | + | 2.63775i | −2.22660 | − | 2.01054i | 4.33402 | − | 5.49694i | 4.94975 | − | 4.94975i | 8.95332 | − | 0.915476i | 1.13835 | + | 4.86869i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.6 | 0.707107 | + | 0.707107i | −0.854662 | − | 2.87568i | − | 3.00000i | −4.57796 | + | 2.01054i | 1.42908 | − | 2.63775i | −5.77983 | − | 3.94887i | 4.94975 | − | 4.94975i | −7.53910 | + | 4.91548i | −4.65877 | − | 1.81544i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.7 | 0.707107 | + | 0.707107i | 0.854662 | + | 2.87568i | − | 3.00000i | 4.57796 | − | 2.01054i | −1.42908 | + | 2.63775i | 3.94887 | + | 5.77983i | 4.94975 | − | 4.94975i | −7.53910 | + | 4.91548i | 4.65877 | + | 1.81544i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.8 | 0.707107 | + | 0.707107i | 2.99611 | − | 0.152778i | − | 3.00000i | −4.24762 | − | 2.63775i | 2.22660 | + | 2.01054i | 5.49694 | − | 4.33402i | 4.94975 | − | 4.94975i | 8.95332 | − | 0.915476i | −1.13835 | − | 4.86869i | ||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 5.c | odd | 4 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 15.e | even | 4 | 1 | inner |
| 21.c | even | 2 | 1 | inner |
| 35.f | even | 4 | 1 | inner |
| 105.k | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 105.3.k.c | ✓ | 16 |
| 3.b | odd | 2 | 1 | inner | 105.3.k.c | ✓ | 16 |
| 5.c | odd | 4 | 1 | inner | 105.3.k.c | ✓ | 16 |
| 7.b | odd | 2 | 1 | inner | 105.3.k.c | ✓ | 16 |
| 15.e | even | 4 | 1 | inner | 105.3.k.c | ✓ | 16 |
| 21.c | even | 2 | 1 | inner | 105.3.k.c | ✓ | 16 |
| 35.f | even | 4 | 1 | inner | 105.3.k.c | ✓ | 16 |
| 105.k | odd | 4 | 1 | inner | 105.3.k.c | ✓ | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 105.3.k.c | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 105.3.k.c | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
| 105.3.k.c | ✓ | 16 | 5.c | odd | 4 | 1 | inner |
| 105.3.k.c | ✓ | 16 | 7.b | odd | 2 | 1 | inner |
| 105.3.k.c | ✓ | 16 | 15.e | even | 4 | 1 | inner |
| 105.3.k.c | ✓ | 16 | 21.c | even | 2 | 1 | inner |
| 105.3.k.c | ✓ | 16 | 35.f | even | 4 | 1 | inner |
| 105.3.k.c | ✓ | 16 | 105.k | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(105, [\chi])\):
|
\( T_{2}^{4} + 1 \)
|
|
\( T_{11}^{4} + 200T_{11}^{2} + 1296 \)
|
|
\( T_{13}^{8} + 208844T_{13}^{4} + 4685402500 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 1)^{4} \)
|
| $3$ |
\( T^{16} - 224 T^{12} + \cdots + 43046721 \)
|
| $5$ |
\( (T^{8} - 56 T^{6} + \cdots + 390625)^{2} \)
|
| $7$ |
\( (T^{8} - 16 T^{7} + \cdots + 5764801)^{2} \)
|
| $11$ |
\( (T^{4} + 200 T^{2} + 1296)^{4} \)
|
| $13$ |
\( (T^{8} + 208844 T^{4} + 4685402500)^{2} \)
|
| $17$ |
\( (T^{8} + 2240 T^{4} + 1024)^{2} \)
|
| $19$ |
\( (T^{4} - 748 T^{2} + 130050)^{4} \)
|
| $23$ |
\( (T^{8} + 1976072 T^{4} + 78074896)^{2} \)
|
| $29$ |
\( (T^{4} - 920 T^{2} + 104976)^{4} \)
|
| $31$ |
\( (T^{4} + 1248 T^{2} + 282752)^{4} \)
|
| $37$ |
\( (T^{4} + 96 T^{3} + \cdots + 291600)^{4} \)
|
| $41$ |
\( (T^{4} - 1008 T^{2} + 209952)^{4} \)
|
| $43$ |
\( (T^{4} - 16 T^{3} + \cdots + 2782224)^{4} \)
|
| $47$ |
\( (T^{8} + \cdots + 7967567545344)^{2} \)
|
| $53$ |
\( (T^{8} + \cdots + 118592100000000)^{2} \)
|
| $59$ |
\( (T^{4} + 3148 T^{2} + 1996002)^{4} \)
|
| $61$ |
\( (T^{4} + 6684 T^{2} + 2770658)^{4} \)
|
| $67$ |
\( (T^{4} - 80 T^{3} + \cdots + 22165264)^{4} \)
|
| $71$ |
\( (T^{4} + 2660 T^{2} + 617796)^{4} \)
|
| $73$ |
\( (T^{8} + \cdots + 79585811103744)^{2} \)
|
| $79$ |
\( (T^{2} + 5746)^{8} \)
|
| $83$ |
\( (T^{8} + \cdots + 968830988602500)^{2} \)
|
| $89$ |
\( (T^{4} + 30576 T^{2} + 5999648)^{4} \)
|
| $97$ |
\( (T^{8} + 2127040 T^{4} + 561132831744)^{2} \)
|
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