Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [950,2,Mod(101,950)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(950, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([0, 14]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("950.101");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 950 = 2 \cdot 5^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 950.l (of order \(9\), degree \(6\), 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: | \(7.58578819202\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{9})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 24x^{10} + 264x^{8} - 1511x^{6} + 4812x^{4} - 7788x^{2} + 5329 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 190) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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
\( x^{12} - 24x^{10} + 264x^{8} - 1511x^{6} + 4812x^{4} - 7788x^{2} + 5329 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -25\nu^{10} + 510\nu^{8} - 4764\nu^{6} + 21478\nu^{4} - 53220\nu^{2} - 4267\nu + 54312 ) / 8534 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -25\nu^{10} + 510\nu^{8} - 4764\nu^{6} + 21478\nu^{4} - 53220\nu^{2} + 4267\nu + 54312 ) / 8534 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 8448 \nu^{11} - 93513 \nu^{10} - 223995 \nu^{9} + 2303442 \nu^{8} + 2546362 \nu^{7} + \cdots + 302962702 ) / 44231722 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 8990 \nu^{11} + 119136 \nu^{10} + 302119 \nu^{9} - 2818822 \nu^{8} - 4351295 \nu^{7} + \cdots - 386103862 ) / 44231722 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 8448 \nu^{11} - 93513 \nu^{10} + 223995 \nu^{9} + 2303442 \nu^{8} - 2546362 \nu^{7} + \cdots + 302962702 ) / 44231722 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 8990 \nu^{11} + 119136 \nu^{10} - 302119 \nu^{9} - 2818822 \nu^{8} + 4351295 \nu^{7} + \cdots - 386103862 ) / 44231722 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 18753 \nu^{11} - 79059 \nu^{10} - 398826 \nu^{9} + 1645785 \nu^{8} + 3920032 \nu^{7} + \cdots + 48113424 ) / 44231722 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 18753 \nu^{11} + 79059 \nu^{10} - 398826 \nu^{9} - 1645785 \nu^{8} + 3920032 \nu^{7} + \cdots - 48113424 ) / 44231722 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 43673 \nu^{11} - 25623 \nu^{10} - 1033698 \nu^{9} + 515380 \nu^{8} + 10872015 \nu^{7} + \cdots - 5322284 ) / 44231722 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 43673 \nu^{11} + 25623 \nu^{10} - 1033698 \nu^{9} - 515380 \nu^{8} + 10872015 \nu^{7} + \cdots + 5322284 ) / 44231722 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( -744\nu^{11} + 16031\nu^{9} - 159186\nu^{7} + 776412\nu^{5} - 2012234\nu^{3} + 1909212\nu + 311491 ) / 622982 \)
|
| \(\nu\) | \(=\) |
\( \beta_{2} - \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{10} + \beta_{9} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( 6\beta_{11} + 8\beta_{8} + 8\beta_{7} - \beta_{5} + \beta_{3} + 3\beta_{2} - 3\beta _1 - 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + 11 \beta_{6} + 11 \beta_{5} + 11 \beta_{4} + \cdots + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( 68 \beta_{11} + 5 \beta_{10} + 5 \beta_{9} + 85 \beta_{8} + 85 \beta_{7} + \beta_{6} + 3 \beta_{5} + \cdots - 34 \)
|
| \(\nu^{6}\) | \(=\) |
\( 116 \beta_{10} - 116 \beta_{9} - 16 \beta_{8} + 16 \beta_{7} + 82 \beta_{6} + 77 \beta_{5} + 82 \beta_{4} + \cdots - 106 \)
|
| \(\nu^{7}\) | \(=\) |
\( 392 \beta_{11} + 32 \beta_{10} + 32 \beta_{9} + 514 \beta_{8} + 514 \beta_{7} + 49 \beta_{6} + \cdots - 196 \)
|
| \(\nu^{8}\) | \(=\) |
\( 1976 \beta_{10} - 1976 \beta_{9} - 543 \beta_{8} + 543 \beta_{7} + 326 \beta_{6} + 294 \beta_{5} + \cdots - 1864 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 386 \beta_{11} - 300 \beta_{10} - 300 \beta_{9} + 536 \beta_{8} + 536 \beta_{7} + 690 \beta_{6} + \cdots + 193 \)
|
| \(\nu^{10}\) | \(=\) |
\( 18616 \beta_{10} - 18616 \beta_{9} - 6310 \beta_{8} + 6310 \beta_{7} - 1654 \beta_{6} - 1354 \beta_{5} + \cdots - 17296 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 38292 \beta_{11} - 8093 \beta_{10} - 8093 \beta_{9} - 31360 \beta_{8} - 31360 \beta_{7} + \cdots + 19146 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).
| \(n\) | \(77\) | \(401\) |
| \(\chi(n)\) | \(1\) | \(\beta_{10}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 101.1 |
|
−0.766044 | + | 0.642788i | −2.57374 | − | 0.936765i | 0.173648 | − | 0.984808i | 0 | 2.57374 | − | 0.936765i | 1.92448 | − | 3.33331i | 0.500000 | + | 0.866025i | 3.44848 | + | 2.89362i | 0 | ||||||||||||||||||||||||||||||||||||||||
| 101.2 | −0.766044 | + | 0.642788i | 1.13405 | + | 0.412760i | 0.173648 | − | 0.984808i | 0 | −1.13405 | + | 0.412760i | −1.09813 | + | 1.90202i | 0.500000 | + | 0.866025i | −1.18244 | − | 0.992183i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 251.1 | 0.939693 | + | 0.342020i | −0.397366 | − | 2.25357i | 0.766044 | + | 0.642788i | 0 | 0.397366 | − | 2.25357i | 1.38429 | − | 2.39766i | 0.500000 | + | 0.866025i | −2.10161 | + | 0.764925i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 251.2 | 0.939693 | + | 0.342020i | 0.0710139 | + | 0.402740i | 0.766044 | + | 0.642788i | 0 | −0.0710139 | + | 0.402740i | −1.15033 | + | 1.99244i | 0.500000 | + | 0.866025i | 2.66192 | − | 0.968860i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 301.1 | −0.766044 | − | 0.642788i | −2.57374 | + | 0.936765i | 0.173648 | + | 0.984808i | 0 | 2.57374 | + | 0.936765i | 1.92448 | + | 3.33331i | 0.500000 | − | 0.866025i | 3.44848 | − | 2.89362i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 301.2 | −0.766044 | − | 0.642788i | 1.13405 | − | 0.412760i | 0.173648 | + | 0.984808i | 0 | −1.13405 | − | 0.412760i | −1.09813 | − | 1.90202i | 0.500000 | − | 0.866025i | −1.18244 | + | 0.992183i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 351.1 | −0.173648 | − | 0.984808i | −2.00490 | + | 1.68231i | −0.939693 | + | 0.342020i | 0 | 2.00490 | + | 1.68231i | 0.485218 | − | 0.840422i | 0.500000 | + | 0.866025i | 0.668514 | − | 3.79133i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 351.2 | −0.173648 | − | 0.984808i | 2.27095 | − | 1.90555i | −0.939693 | + | 0.342020i | 0 | −2.27095 | − | 1.90555i | 1.45447 | − | 2.51922i | 0.500000 | + | 0.866025i | 1.00513 | − | 5.70040i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 651.1 | 0.939693 | − | 0.342020i | −0.397366 | + | 2.25357i | 0.766044 | − | 0.642788i | 0 | 0.397366 | + | 2.25357i | 1.38429 | + | 2.39766i | 0.500000 | − | 0.866025i | −2.10161 | − | 0.764925i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 651.2 | 0.939693 | − | 0.342020i | 0.0710139 | − | 0.402740i | 0.766044 | − | 0.642788i | 0 | −0.0710139 | − | 0.402740i | −1.15033 | − | 1.99244i | 0.500000 | − | 0.866025i | 2.66192 | + | 0.968860i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 701.1 | −0.173648 | + | 0.984808i | −2.00490 | − | 1.68231i | −0.939693 | − | 0.342020i | 0 | 2.00490 | − | 1.68231i | 0.485218 | + | 0.840422i | 0.500000 | − | 0.866025i | 0.668514 | + | 3.79133i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 701.2 | −0.173648 | + | 0.984808i | 2.27095 | + | 1.90555i | −0.939693 | − | 0.342020i | 0 | −2.27095 | + | 1.90555i | 1.45447 | + | 2.51922i | 0.500000 | − | 0.866025i | 1.00513 | + | 5.70040i | 0 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.e | even | 9 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 950.2.l.g | 12 | |
| 5.b | even | 2 | 1 | 190.2.k.c | ✓ | 12 | |
| 5.c | odd | 4 | 2 | 950.2.u.f | 24 | ||
| 19.e | even | 9 | 1 | inner | 950.2.l.g | 12 | |
| 95.o | odd | 18 | 1 | 3610.2.a.bd | 6 | ||
| 95.p | even | 18 | 1 | 190.2.k.c | ✓ | 12 | |
| 95.p | even | 18 | 1 | 3610.2.a.bf | 6 | ||
| 95.q | odd | 36 | 2 | 950.2.u.f | 24 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 190.2.k.c | ✓ | 12 | 5.b | even | 2 | 1 | |
| 190.2.k.c | ✓ | 12 | 95.p | even | 18 | 1 | |
| 950.2.l.g | 12 | 1.a | even | 1 | 1 | trivial | |
| 950.2.l.g | 12 | 19.e | even | 9 | 1 | inner | |
| 950.2.u.f | 24 | 5.c | odd | 4 | 2 | ||
| 950.2.u.f | 24 | 95.q | odd | 36 | 2 | ||
| 3610.2.a.bd | 6 | 95.o | odd | 18 | 1 | ||
| 3610.2.a.bf | 6 | 95.p | even | 18 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + 3 T_{3}^{11} - 4 T_{3}^{9} + 51 T_{3}^{8} + 189 T_{3}^{7} + 391 T_{3}^{6} + 210 T_{3}^{5} + \cdots + 576 \)
acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{6} - T^{3} + 1)^{2} \)
|
| $3$ |
\( T^{12} + 3 T^{11} + \cdots + 576 \)
|
| $5$ |
\( T^{12} \)
|
| $7$ |
\( T^{12} - 6 T^{11} + \cdots + 23104 \)
|
| $11$ |
\( T^{12} + 6 T^{11} + \cdots + 81 \)
|
| $13$ |
\( T^{12} - 18 T^{11} + \cdots + 64 \)
|
| $17$ |
\( T^{12} + 12 T^{11} + \cdots + 46656 \)
|
| $19$ |
\( T^{12} - 6 T^{11} + \cdots + 47045881 \)
|
| $23$ |
\( T^{12} + 3 T^{11} + \cdots + 5184 \)
|
| $29$ |
\( (T^{6} - 18 T^{5} + \cdots + 5184)^{2} \)
|
| $31$ |
\( T^{12} + \cdots + 1910738944 \)
|
| $37$ |
\( (T^{6} + 12 T^{5} + \cdots - 7624)^{2} \)
|
| $41$ |
\( T^{12} + \cdots + 1794623769 \)
|
| $43$ |
\( T^{12} - 12 T^{11} + \cdots + 27625536 \)
|
| $47$ |
\( T^{12} + \cdots + 39575532096 \)
|
| $53$ |
\( T^{12} + \cdots + 59962296384 \)
|
| $59$ |
\( T^{12} + 27 T^{11} + \cdots + 18429849 \)
|
| $61$ |
\( T^{12} + \cdots + 437981184 \)
|
| $67$ |
\( T^{12} + \cdots + 437981184 \)
|
| $71$ |
\( T^{12} - 24 T^{11} + \cdots + 2985984 \)
|
| $73$ |
\( T^{12} + \cdots + 345847495744 \)
|
| $79$ |
\( T^{12} + \cdots + 162205696 \)
|
| $83$ |
\( T^{12} + 219 T^{10} + \cdots + 5184 \)
|
| $89$ |
\( T^{12} + \cdots + 101062809 \)
|
| $97$ |
\( T^{12} + \cdots + 168169024 \)
|
| show more | |
| show less |