Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [660,2,Mod(571,660)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(660, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("660.571");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 660 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 660.k (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: | \(5.27012653340\) |
| 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: |
\( x^{20} + 6x^{16} - 12x^{14} + 13x^{12} - 84x^{10} + 52x^{8} - 192x^{6} + 384x^{4} + 1024 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{9} \) |
| 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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} + 6x^{16} - 12x^{14} + 13x^{12} - 84x^{10} + 52x^{8} - 192x^{6} + 384x^{4} + 1024 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{19} + 40 \nu^{17} - 74 \nu^{15} + 164 \nu^{13} - 819 \nu^{11} + 756 \nu^{9} - 2300 \nu^{7} + \cdots + 11264 \nu ) / 3584 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 3 \nu^{19} - 26 \nu^{18} + 20 \nu^{17} + 24 \nu^{16} - 2 \nu^{15} + 132 \nu^{14} + 348 \nu^{13} + \cdots - 9728 ) / 14336 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3 \nu^{19} + 34 \nu^{18} - 20 \nu^{17} + 184 \nu^{16} + 2 \nu^{15} - 276 \nu^{14} - 348 \nu^{13} + \cdots + 35328 ) / 14336 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 5 \nu^{18} - 10 \nu^{16} + 22 \nu^{14} - 104 \nu^{12} + 105 \nu^{10} - 294 \nu^{8} + 484 \nu^{6} + \cdots - 128 ) / 1792 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 3 \nu^{19} + 34 \nu^{18} + 20 \nu^{17} + 184 \nu^{16} - 2 \nu^{15} - 276 \nu^{14} + 348 \nu^{13} + \cdots + 35328 ) / 14336 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 7 \nu^{19} - 46 \nu^{18} + 84 \nu^{17} + 232 \nu^{16} - 154 \nu^{15} - 180 \nu^{14} + \cdots + 15872 ) / 14336 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 9 \nu^{18} + 32 \nu^{16} - 118 \nu^{14} + 204 \nu^{12} - 693 \nu^{10} + 980 \nu^{8} - 1812 \nu^{6} + \cdots + 6144 ) / 1792 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 5 \nu^{18} - 4 \nu^{16} - 22 \nu^{14} + 20 \nu^{12} + 63 \nu^{10} + 112 \nu^{8} + 244 \nu^{6} + \cdots - 2560 ) / 896 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 7 \nu^{19} - 78 \nu^{18} + 84 \nu^{17} - 40 \nu^{16} - 154 \nu^{15} - 52 \nu^{14} + 140 \nu^{13} + \cdots - 7680 ) / 14336 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 3 \nu^{19} + 98 \nu^{18} - 20 \nu^{17} - 168 \nu^{16} + 2 \nu^{15} + 364 \nu^{14} - 348 \nu^{13} + \cdots - 17920 ) / 14336 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( -\nu^{18} - 6\nu^{14} + 12\nu^{12} - 13\nu^{10} + 84\nu^{8} - 52\nu^{6} + 192\nu^{4} - 256\nu^{2} ) / 128 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( \nu^{19} + 4 \nu^{17} + 6 \nu^{15} + 12 \nu^{13} - 35 \nu^{11} - 32 \nu^{9} - 284 \nu^{7} + \cdots + 1024 \nu ) / 512 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( \nu^{19} - 4 \nu^{17} + 6 \nu^{15} - 36 \nu^{13} + 61 \nu^{11} - 136 \nu^{9} + 388 \nu^{7} + \cdots - 1024 \nu ) / 512 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 5 \nu^{19} + 10 \nu^{17} - 22 \nu^{15} + 104 \nu^{13} - 105 \nu^{11} + 294 \nu^{9} + \cdots + 128 \nu ) / 1792 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 23 \nu^{19} + 4 \nu^{17} - 90 \nu^{15} - 20 \nu^{13} - 315 \nu^{11} + 372 \nu^{7} + \cdots - 2816 \nu ) / 7168 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 73 \nu^{19} - 78 \nu^{18} - 76 \nu^{17} - 40 \nu^{16} + 198 \nu^{15} - 52 \nu^{14} - 628 \nu^{13} + \cdots - 7680 ) / 14336 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{14} + \beta_{13} - \beta_{11} - \beta_{7} + \beta_{5} - 1 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{19} + \beta_{17} - \beta_{16} + \beta_{13} + \beta_{9} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{14} + 2\beta_{13} - 2\beta_{12} + 2\beta_{9} - 2\beta_{7} - 2\beta_{6} + 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( -2\beta_{18} + 4\beta_{17} + 2\beta_{16} - 2\beta_{8} + 2\beta_{6} - 3\beta_{3} + 2\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( \beta_{14} - 3 \beta_{13} - 2 \beta_{12} + \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + 5 \beta_{7} + \cdots + 1 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 5 \beta_{19} - 6 \beta_{18} - 5 \beta_{17} - 3 \beta_{16} - 2 \beta_{15} - 3 \beta_{13} + \cdots + \beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( 3 \beta_{14} + 2 \beta_{12} + 2 \beta_{11} - 6 \beta_{10} - 2 \beta_{9} + 4 \beta_{8} - 4 \beta_{7} + \cdots + 6 \)
|
| \(\nu^{11}\) | \(=\) |
\( 2 \beta_{19} - 6 \beta_{17} - 8 \beta_{16} - 2 \beta_{15} + 6 \beta_{13} + 4 \beta_{12} + 6 \beta_{9} + \cdots + 8 \beta_1 \)
|
| \(\nu^{12}\) | \(=\) |
\( -\beta_{14} + 17\beta_{13} - \beta_{11} + 8\beta_{8} - 45\beta_{7} - 12\beta_{6} - 3\beta_{5} + 10\beta_{2} + 23 \)
|
| \(\nu^{13}\) | \(=\) |
\( 17 \beta_{19} - 4 \beta_{18} + 49 \beta_{17} + 19 \beta_{16} + 5 \beta_{13} - 12 \beta_{12} + \cdots + 3 \beta_1 \)
|
| \(\nu^{14}\) | \(=\) |
\( 9 \beta_{14} - 10 \beta_{13} - 10 \beta_{12} - 36 \beta_{11} - 4 \beta_{10} + 10 \beta_{9} - 24 \beta_{8} + \cdots - 32 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 20 \beta_{19} - 46 \beta_{18} + 40 \beta_{17} - 18 \beta_{16} + 36 \beta_{15} - 20 \beta_{13} + \cdots - 18 \beta_1 \)
|
| \(\nu^{16}\) | \(=\) |
\( 29 \beta_{14} - 11 \beta_{13} - 54 \beta_{12} + 5 \beta_{11} - 46 \beta_{10} + 54 \beta_{9} - 23 \beta_{7} + \cdots - 11 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 65 \beta_{19} - 66 \beta_{18} - 17 \beta_{17} - 15 \beta_{16} + 66 \beta_{15} + 29 \beta_{13} + \cdots + 29 \beta_1 \)
|
| \(\nu^{18}\) | \(=\) |
\( - 9 \beta_{14} + 100 \beta_{13} - 30 \beta_{12} + 70 \beta_{11} - 66 \beta_{10} + 30 \beta_{9} + \cdots + 74 \)
|
| \(\nu^{19}\) | \(=\) |
\( 70 \beta_{19} - 172 \beta_{18} - 10 \beta_{17} + 148 \beta_{16} - 102 \beta_{15} + 42 \beta_{13} + \cdots - 172 \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/660\mathbb{Z}\right)^\times\).
| \(n\) | \(221\) | \(331\) | \(397\) | \(541\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 571.1 |
|
−1.40859 | − | 0.125938i | 1.00000i | 1.96828 | + | 0.354792i | −1.00000 | 0.125938 | − | 1.40859i | 1.24340 | −2.72783 | − | 0.747640i | −1.00000 | 1.40859 | + | 0.125938i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 571.2 | −1.40859 | + | 0.125938i | − | 1.00000i | 1.96828 | − | 0.354792i | −1.00000 | 0.125938 | + | 1.40859i | 1.24340 | −2.72783 | + | 0.747640i | −1.00000 | 1.40859 | − | 0.125938i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 571.3 | −1.13671 | − | 0.841364i | 1.00000i | 0.584212 | + | 1.91277i | −1.00000 | 0.841364 | − | 1.13671i | 3.64887 | 0.945260 | − | 2.66580i | −1.00000 | 1.13671 | + | 0.841364i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 571.4 | −1.13671 | + | 0.841364i | − | 1.00000i | 0.584212 | − | 1.91277i | −1.00000 | 0.841364 | + | 1.13671i | 3.64887 | 0.945260 | + | 2.66580i | −1.00000 | 1.13671 | − | 0.841364i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 571.5 | −0.934128 | − | 1.06179i | − | 1.00000i | −0.254810 | + | 1.98370i | −1.00000 | −1.06179 | + | 0.934128i | −1.04136 | 2.34431 | − | 1.58247i | −1.00000 | 0.934128 | + | 1.06179i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 571.6 | −0.934128 | + | 1.06179i | 1.00000i | −0.254810 | − | 1.98370i | −1.00000 | −1.06179 | − | 0.934128i | −1.04136 | 2.34431 | + | 1.58247i | −1.00000 | 0.934128 | − | 1.06179i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 571.7 | −0.724960 | − | 1.21426i | − | 1.00000i | −0.948867 | + | 1.76058i | −1.00000 | −1.21426 | + | 0.724960i | 2.18017 | 2.82570 | − | 0.124178i | −1.00000 | 0.724960 | + | 1.21426i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 571.8 | −0.724960 | + | 1.21426i | 1.00000i | −0.948867 | − | 1.76058i | −1.00000 | −1.21426 | − | 0.724960i | 2.18017 | 2.82570 | + | 0.124178i | −1.00000 | 0.724960 | − | 1.21426i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 571.9 | −0.570608 | − | 1.29399i | 1.00000i | −1.34881 | + | 1.47672i | −1.00000 | 1.29399 | − | 0.570608i | −4.39342 | 2.68050 | + | 0.902721i | −1.00000 | 0.570608 | + | 1.29399i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 571.10 | −0.570608 | + | 1.29399i | − | 1.00000i | −1.34881 | − | 1.47672i | −1.00000 | 1.29399 | + | 0.570608i | −4.39342 | 2.68050 | − | 0.902721i | −1.00000 | 0.570608 | − | 1.29399i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 571.11 | 0.570608 | − | 1.29399i | − | 1.00000i | −1.34881 | − | 1.47672i | −1.00000 | −1.29399 | − | 0.570608i | 4.39342 | −2.68050 | + | 0.902721i | −1.00000 | −0.570608 | + | 1.29399i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 571.12 | 0.570608 | + | 1.29399i | 1.00000i | −1.34881 | + | 1.47672i | −1.00000 | −1.29399 | + | 0.570608i | 4.39342 | −2.68050 | − | 0.902721i | −1.00000 | −0.570608 | − | 1.29399i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 571.13 | 0.724960 | − | 1.21426i | 1.00000i | −0.948867 | − | 1.76058i | −1.00000 | 1.21426 | + | 0.724960i | −2.18017 | −2.82570 | − | 0.124178i | −1.00000 | −0.724960 | + | 1.21426i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 571.14 | 0.724960 | + | 1.21426i | − | 1.00000i | −0.948867 | + | 1.76058i | −1.00000 | 1.21426 | − | 0.724960i | −2.18017 | −2.82570 | + | 0.124178i | −1.00000 | −0.724960 | − | 1.21426i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 571.15 | 0.934128 | − | 1.06179i | 1.00000i | −0.254810 | − | 1.98370i | −1.00000 | 1.06179 | + | 0.934128i | 1.04136 | −2.34431 | − | 1.58247i | −1.00000 | −0.934128 | + | 1.06179i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 571.16 | 0.934128 | + | 1.06179i | − | 1.00000i | −0.254810 | + | 1.98370i | −1.00000 | 1.06179 | − | 0.934128i | 1.04136 | −2.34431 | + | 1.58247i | −1.00000 | −0.934128 | − | 1.06179i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 571.17 | 1.13671 | − | 0.841364i | − | 1.00000i | 0.584212 | − | 1.91277i | −1.00000 | −0.841364 | − | 1.13671i | −3.64887 | −0.945260 | − | 2.66580i | −1.00000 | −1.13671 | + | 0.841364i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 571.18 | 1.13671 | + | 0.841364i | 1.00000i | 0.584212 | + | 1.91277i | −1.00000 | −0.841364 | + | 1.13671i | −3.64887 | −0.945260 | + | 2.66580i | −1.00000 | −1.13671 | − | 0.841364i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 571.19 | 1.40859 | − | 0.125938i | − | 1.00000i | 1.96828 | − | 0.354792i | −1.00000 | −0.125938 | − | 1.40859i | −1.24340 | 2.72783 | − | 0.747640i | −1.00000 | −1.40859 | + | 0.125938i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 571.20 | 1.40859 | + | 0.125938i | 1.00000i | 1.96828 | + | 0.354792i | −1.00000 | −0.125938 | + | 1.40859i | −1.24340 | 2.72783 | + | 0.747640i | −1.00000 | −1.40859 | − | 0.125938i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 11.b | odd | 2 | 1 | inner |
| 44.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 660.2.k.d | ✓ | 20 |
| 4.b | odd | 2 | 1 | inner | 660.2.k.d | ✓ | 20 |
| 11.b | odd | 2 | 1 | inner | 660.2.k.d | ✓ | 20 |
| 44.c | even | 2 | 1 | inner | 660.2.k.d | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 660.2.k.d | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 660.2.k.d | ✓ | 20 | 4.b | odd | 2 | 1 | inner |
| 660.2.k.d | ✓ | 20 | 11.b | odd | 2 | 1 | inner |
| 660.2.k.d | ✓ | 20 | 44.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{10} - 40T_{7}^{8} + 512T_{7}^{6} - 2368T_{7}^{4} + 3904T_{7}^{2} - 2048 \)
acting on \(S_{2}^{\mathrm{new}}(660, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} + 6 T^{16} + \cdots + 1024 \)
|
| $3$ |
\( (T^{2} + 1)^{10} \)
|
| $5$ |
\( (T + 1)^{20} \)
|
| $7$ |
\( (T^{10} - 40 T^{8} + \cdots - 2048)^{2} \)
|
| $11$ |
\( T^{20} + \cdots + 25937424601 \)
|
| $13$ |
\( (T^{10} + 82 T^{8} + \cdots + 36992)^{2} \)
|
| $17$ |
\( (T^{10} + 88 T^{8} + \cdots + 8192)^{2} \)
|
| $19$ |
\( (T^{10} - 106 T^{8} + \cdots - 700928)^{2} \)
|
| $23$ |
\( (T^{10} + 96 T^{8} + \cdots + 16384)^{2} \)
|
| $29$ |
\( (T^{10} + 106 T^{8} + \cdots + 86528)^{2} \)
|
| $31$ |
\( (T^{10} + 288 T^{8} + \cdots + 328044544)^{2} \)
|
| $37$ |
\( (T^{5} - 14 T^{4} + \cdots - 6752)^{4} \)
|
| $41$ |
\( (T^{10} + 170 T^{8} + \cdots + 147968)^{2} \)
|
| $43$ |
\( (T^{10} - 200 T^{8} + \cdots - 8192)^{2} \)
|
| $47$ |
\( (T^{10} + 304 T^{8} + \cdots + 67108864)^{2} \)
|
| $53$ |
\( (T^{5} - 56 T^{3} + \cdots + 384)^{4} \)
|
| $59$ |
\( (T^{10} + 240 T^{8} + \cdots + 36864)^{2} \)
|
| $61$ |
\( (T^{10} + 216 T^{8} + \cdots + 663552)^{2} \)
|
| $67$ |
\( (T^{10} + 292 T^{8} + \cdots + 589824)^{2} \)
|
| $71$ |
\( (T^{10} + 452 T^{8} + \cdots + 49112064)^{2} \)
|
| $73$ |
\( (T^{10} + 226 T^{8} + \cdots + 70662272)^{2} \)
|
| $79$ |
\( (T^{10} - 410 T^{8} + \cdots - 700928)^{2} \)
|
| $83$ |
\( (T^{10} - 418 T^{8} + \cdots - 11751552)^{2} \)
|
| $89$ |
\( (T^{5} - 26 T^{4} + \cdots + 116704)^{4} \)
|
| $97$ |
\( (T^{5} - 14 T^{4} + \cdots - 27616)^{4} \)
|
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