Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4000,2,Mod(3249,4000)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4000, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4000.3249");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4000 = 2^{5} \cdot 5^{3} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4000.f (of order \(2\), degree \(1\), not 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: | \(31.9401608085\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 3 x^{19} + 6 x^{18} - 5 x^{17} - 3 x^{16} + 20 x^{15} - 28 x^{14} + 24 x^{13} + 16 x^{12} + \cdots + 1024 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{14} \) |
| Twist minimal: | no (minimal twist has level 1000) |
| 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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 3 x^{19} + 6 x^{18} - 5 x^{17} - 3 x^{16} + 20 x^{15} - 28 x^{14} + 24 x^{13} + 16 x^{12} + \cdots + 1024 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{19} + 11 \nu^{18} - 32 \nu^{17} + 43 \nu^{16} + 23 \nu^{15} - 90 \nu^{14} + 200 \nu^{13} + \cdots + 10752 ) / 3072 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{19} - 11 \nu^{18} + 6 \nu^{17} - 5 \nu^{16} - 19 \nu^{15} + 20 \nu^{14} - 44 \nu^{13} + \cdots - 2048 ) / 1536 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 5 \nu^{19} + 7 \nu^{18} - 30 \nu^{17} + 25 \nu^{16} - \nu^{15} - 100 \nu^{14} + 124 \nu^{13} + \cdots + 7168 ) / 1536 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 17 \nu^{19} - 61 \nu^{18} + 80 \nu^{17} + 3 \nu^{16} - 145 \nu^{15} + 278 \nu^{14} - 256 \nu^{13} + \cdots - 9728 ) / 3072 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 25 \nu^{19} + 53 \nu^{18} - 40 \nu^{17} - 3 \nu^{16} + 185 \nu^{15} - 238 \nu^{14} + 152 \nu^{13} + \cdots + 8704 ) / 3072 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 11 \nu^{19} + 11 \nu^{18} - 12 \nu^{17} - 17 \nu^{16} + 47 \nu^{15} - 62 \nu^{14} - 24 \nu^{13} + \cdots + 2560 ) / 1024 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 25 \nu^{19} + 41 \nu^{18} - 132 \nu^{17} + 69 \nu^{16} + 101 \nu^{15} - 394 \nu^{14} + \cdots + 32256 ) / 3072 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 9 \nu^{19} - 18 \nu^{18} + 25 \nu^{17} + 11 \nu^{16} - 52 \nu^{15} + 119 \nu^{14} - 30 \nu^{13} + \cdots + 1024 ) / 768 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 21 \nu^{19} + 23 \nu^{18} - 46 \nu^{17} - 31 \nu^{16} + 95 \nu^{15} - 188 \nu^{14} + 52 \nu^{13} + \cdots + 3072 ) / 1536 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 17 \nu^{19} + 39 \nu^{18} - 70 \nu^{17} + 17 \nu^{16} + 111 \nu^{15} - 268 \nu^{14} + 256 \nu^{13} + \cdots + 10752 ) / 1536 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 19 \nu^{19} - 33 \nu^{18} + 86 \nu^{17} - 43 \nu^{16} - 57 \nu^{15} + 272 \nu^{14} - 368 \nu^{13} + \cdots - 19968 ) / 1536 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 41 \nu^{19} - 61 \nu^{18} + 168 \nu^{17} - 21 \nu^{16} - 193 \nu^{15} + 590 \nu^{14} - 528 \nu^{13} + \cdots - 26112 ) / 3072 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 7 \nu^{19} - 25 \nu^{18} + 58 \nu^{17} - 35 \nu^{16} - 49 \nu^{15} + 192 \nu^{14} - 220 \nu^{13} + \cdots - 8448 ) / 768 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 15 \nu^{19} - 39 \nu^{18} + 60 \nu^{17} + 5 \nu^{16} - 107 \nu^{15} + 262 \nu^{14} - 176 \nu^{13} + \cdots - 4608 ) / 1024 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 51 \nu^{19} - 123 \nu^{18} + 212 \nu^{17} - 23 \nu^{16} - 335 \nu^{15} + 886 \nu^{14} - 696 \nu^{13} + \cdots - 24064 ) / 3072 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 25 \nu^{19} - 73 \nu^{18} + 100 \nu^{17} - 29 \nu^{16} - 181 \nu^{15} + 450 \nu^{14} - 400 \nu^{13} + \cdots - 13824 ) / 1536 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 65 \nu^{19} - 121 \nu^{18} + 148 \nu^{17} + 187 \nu^{16} - 405 \nu^{15} + 762 \nu^{14} + 32 \nu^{13} + \cdots + 13824 ) / 3072 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 51 \nu^{19} - 91 \nu^{18} + 164 \nu^{17} - 7 \nu^{16} - 271 \nu^{15} + 646 \nu^{14} - 488 \nu^{13} + \cdots - 21504 ) / 1536 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 103 \nu^{19} - 311 \nu^{18} + 444 \nu^{17} - 203 \nu^{16} - 811 \nu^{15} + 1814 \nu^{14} + \cdots - 77312 ) / 3072 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{11} - \beta_{7} - \beta_{3} + \beta_{2} + \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{18} - \beta_{16} + \beta_{15} - \beta_{12} + \beta_{11} - 3\beta_{10} - \beta_{9} + \beta_{7} - \beta_{4} + \beta_{2} ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - \beta_{18} + \beta_{16} + \beta_{15} - 2 \beta_{14} - \beta_{12} + \beta_{10} - \beta_{9} + 2 \beta_{8} + \cdots - 2 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2 \beta_{18} + 2 \beta_{17} - 2 \beta_{14} + \beta_{11} + 4 \beta_{9} - 2 \beta_{8} + \beta_{7} + \cdots + 2 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 2 \beta_{19} + 3 \beta_{18} - 2 \beta_{17} + \beta_{16} + 3 \beta_{15} - 2 \beta_{14} - 2 \beta_{13} + \cdots + 2 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 3 \beta_{18} - 3 \beta_{16} + \beta_{15} + 2 \beta_{14} - 5 \beta_{12} + 2 \beta_{11} + \beta_{10} + \cdots - 6 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 4 \beta_{19} + 2 \beta_{17} - 2 \beta_{16} - 6 \beta_{15} - 2 \beta_{14} - 4 \beta_{13} + 6 \beta_{12} + \cdots - 10 ) / 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 2 \beta_{19} + 3 \beta_{18} + 2 \beta_{17} + \beta_{16} - \beta_{15} - 2 \beta_{14} + 6 \beta_{13} + \cdots - 26 ) / 4 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 11 \beta_{18} + 19 \beta_{16} - 9 \beta_{15} - 2 \beta_{14} + 8 \beta_{13} + 9 \beta_{12} - 4 \beta_{11} + \cdots + 18 ) / 4 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 6 \beta_{18} - 10 \beta_{17} + 4 \beta_{16} - 4 \beta_{15} + 10 \beta_{14} + 8 \beta_{13} + 4 \beta_{12} + \cdots - 26 ) / 4 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 18 \beta_{19} - 5 \beta_{18} - 14 \beta_{17} - 19 \beta_{16} + 7 \beta_{15} + 18 \beta_{14} - 6 \beta_{13} + \cdots - 18 ) / 4 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 21 \beta_{18} - 3 \beta_{16} + 17 \beta_{15} + 2 \beta_{14} + 8 \beta_{13} + 19 \beta_{12} + \cdots + 42 ) / 4 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 12 \beta_{19} - 16 \beta_{18} + 42 \beta_{17} + 6 \beta_{16} - 46 \beta_{15} - 10 \beta_{14} + 52 \beta_{13} + \cdots - 34 ) / 4 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 50 \beta_{19} - 41 \beta_{18} - 30 \beta_{17} + 69 \beta_{16} + 43 \beta_{15} + 46 \beta_{14} + \cdots + 6 ) / 4 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 16 \beta_{19} + 49 \beta_{18} - 48 \beta_{17} + 7 \beta_{16} + 43 \beta_{15} + 70 \beta_{14} + \cdots - 86 ) / 4 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( - 16 \beta_{19} + 90 \beta_{18} + 94 \beta_{17} - 68 \beta_{16} + 4 \beta_{15} + 66 \beta_{14} + \cdots + 126 ) / 4 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 118 \beta_{19} + 223 \beta_{18} + 26 \beta_{17} - 311 \beta_{16} + 43 \beta_{15} - 54 \beta_{14} + \cdots + 102 ) / 4 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 64 \beta_{19} + 23 \beta_{18} + 128 \beta_{17} + 113 \beta_{16} - 235 \beta_{15} + 106 \beta_{14} + \cdots + 770 ) / 4 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( 44 \beta_{19} + 216 \beta_{18} - 110 \beta_{17} - 90 \beta_{16} - 430 \beta_{15} - 18 \beta_{14} + \cdots + 678 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times\).
| \(n\) | \(1377\) | \(2501\) | \(2751\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3249.1 |
|
0 | −2.62662 | 0 | 0 | 0 | − | 0.269237i | 0 | 3.89913 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3249.2 | 0 | −2.62662 | 0 | 0 | 0 | 0.269237i | 0 | 3.89913 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3249.3 | 0 | −2.52102 | 0 | 0 | 0 | − | 0.987050i | 0 | 3.35556 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3249.4 | 0 | −2.52102 | 0 | 0 | 0 | 0.987050i | 0 | 3.35556 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3249.5 | 0 | −1.69676 | 0 | 0 | 0 | − | 4.40040i | 0 | −0.120998 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3249.6 | 0 | −1.69676 | 0 | 0 | 0 | 4.40040i | 0 | −0.120998 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3249.7 | 0 | −0.513027 | 0 | 0 | 0 | − | 1.12889i | 0 | −2.73680 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3249.8 | 0 | −0.513027 | 0 | 0 | 0 | 1.12889i | 0 | −2.73680 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3249.9 | 0 | 0.207209 | 0 | 0 | 0 | − | 2.73181i | 0 | −2.95706 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3249.10 | 0 | 0.207209 | 0 | 0 | 0 | 2.73181i | 0 | −2.95706 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3249.11 | 0 | 0.939252 | 0 | 0 | 0 | − | 1.90117i | 0 | −2.11781 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3249.12 | 0 | 0.939252 | 0 | 0 | 0 | 1.90117i | 0 | −2.11781 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3249.13 | 0 | 1.21236 | 0 | 0 | 0 | − | 4.63239i | 0 | −1.53019 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3249.14 | 0 | 1.21236 | 0 | 0 | 0 | 4.63239i | 0 | −1.53019 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3249.15 | 0 | 1.83526 | 0 | 0 | 0 | − | 1.31564i | 0 | 0.368171 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3249.16 | 0 | 1.83526 | 0 | 0 | 0 | 1.31564i | 0 | 0.368171 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3249.17 | 0 | 2.07677 | 0 | 0 | 0 | − | 2.55636i | 0 | 1.31298 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3249.18 | 0 | 2.07677 | 0 | 0 | 0 | 2.55636i | 0 | 1.31298 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3249.19 | 0 | 3.08659 | 0 | 0 | 0 | − | 3.24242i | 0 | 6.52702 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3249.20 | 0 | 3.08659 | 0 | 0 | 0 | 3.24242i | 0 | 6.52702 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 40.f | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 4000.2.f.d | 20 | |
| 4.b | odd | 2 | 1 | 1000.2.f.d | 20 | ||
| 5.b | even | 2 | 1 | 4000.2.f.c | 20 | ||
| 5.c | odd | 4 | 2 | 4000.2.d.c | 40 | ||
| 8.b | even | 2 | 1 | 4000.2.f.c | 20 | ||
| 8.d | odd | 2 | 1 | 1000.2.f.c | 20 | ||
| 20.d | odd | 2 | 1 | 1000.2.f.c | 20 | ||
| 20.e | even | 4 | 2 | 1000.2.d.c | ✓ | 40 | |
| 40.e | odd | 2 | 1 | 1000.2.f.d | 20 | ||
| 40.f | even | 2 | 1 | inner | 4000.2.f.d | 20 | |
| 40.i | odd | 4 | 2 | 4000.2.d.c | 40 | ||
| 40.k | even | 4 | 2 | 1000.2.d.c | ✓ | 40 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1000.2.d.c | ✓ | 40 | 20.e | even | 4 | 2 | |
| 1000.2.d.c | ✓ | 40 | 40.k | even | 4 | 2 | |
| 1000.2.f.c | 20 | 8.d | odd | 2 | 1 | ||
| 1000.2.f.c | 20 | 20.d | odd | 2 | 1 | ||
| 1000.2.f.d | 20 | 4.b | odd | 2 | 1 | ||
| 1000.2.f.d | 20 | 40.e | odd | 2 | 1 | ||
| 4000.2.d.c | 40 | 5.c | odd | 4 | 2 | ||
| 4000.2.d.c | 40 | 40.i | odd | 4 | 2 | ||
| 4000.2.f.c | 20 | 5.b | even | 2 | 1 | ||
| 4000.2.f.c | 20 | 8.b | even | 2 | 1 | ||
| 4000.2.f.d | 20 | 1.a | even | 1 | 1 | trivial | |
| 4000.2.f.d | 20 | 40.f | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} - 2T_{3}^{9} - 16T_{3}^{8} + 32T_{3}^{7} + 77T_{3}^{6} - 168T_{3}^{5} - 94T_{3}^{4} + 292T_{3}^{3} - 61T_{3}^{2} - 76T_{3} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(4000, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( (T^{10} - 2 T^{9} + \cdots + 16)^{2} \)
|
| $5$ |
\( T^{20} \)
|
| $7$ |
\( T^{20} + 73 T^{18} + \cdots + 119961 \)
|
| $11$ |
\( T^{20} + \cdots + 242921025 \)
|
| $13$ |
\( (T^{10} + 3 T^{9} + \cdots - 74475)^{2} \)
|
| $17$ |
\( T^{20} + \cdots + 93273522176 \)
|
| $19$ |
\( T^{20} + \cdots + 14131585001 \)
|
| $23$ |
\( T^{20} + \cdots + 198733991936 \)
|
| $29$ |
\( T^{20} + \cdots + 188118041760000 \)
|
| $31$ |
\( (T^{10} + 12 T^{9} + \cdots - 13424)^{2} \)
|
| $37$ |
\( (T^{10} + 9 T^{9} + \cdots - 97344)^{2} \)
|
| $41$ |
\( (T^{10} - 11 T^{9} + \cdots + 4900275)^{2} \)
|
| $43$ |
\( (T^{10} + 30 T^{9} + \cdots - 22343616)^{2} \)
|
| $47$ |
\( T^{20} + \cdots + 155998693721081 \)
|
| $53$ |
\( (T^{10} + 5 T^{9} + \cdots - 21379)^{2} \)
|
| $59$ |
\( T^{20} + \cdots + 58948955361 \)
|
| $61$ |
\( T^{20} + \cdots + 29661335302400 \)
|
| $67$ |
\( (T^{10} + 20 T^{9} + \cdots + 292107024)^{2} \)
|
| $71$ |
\( (T^{10} + 24 T^{9} + \cdots - 2766384)^{2} \)
|
| $73$ |
\( T^{20} + \cdots + 12\!\cdots\!96 \)
|
| $79$ |
\( (T^{10} - 24 T^{9} + \cdots - 2000)^{2} \)
|
| $83$ |
\( (T^{10} - 20 T^{9} + \cdots - 44548096)^{2} \)
|
| $89$ |
\( (T^{10} - 11 T^{9} + \cdots + 24825600)^{2} \)
|
| $97$ |
\( T^{20} + \cdots + 20198622433536 \)
|
| show more | |
| show less |