Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1710,2,Mod(647,1710)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1710, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1710.647");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1710.n (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: | \(13.6544187456\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 4 x^{19} - 10 x^{18} + 56 x^{17} + 50 x^{16} - 336 x^{15} + 672 x^{14} - 776 x^{13} + 626 x^{12} + \cdots + 32 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{10} \) |
| 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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 4 x^{19} - 10 x^{18} + 56 x^{17} + 50 x^{16} - 336 x^{15} + 672 x^{14} - 776 x^{13} + 626 x^{12} + \cdots + 32 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 72\!\cdots\!24 \nu^{19} + \cdots - 12\!\cdots\!06 ) / 27\!\cdots\!18 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 17\!\cdots\!69 \nu^{19} + \cdots - 23\!\cdots\!40 ) / 55\!\cdots\!36 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 15\!\cdots\!44 \nu^{19} + \cdots - 11\!\cdots\!40 ) / 27\!\cdots\!18 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 19\!\cdots\!80 \nu^{19} + \cdots - 14\!\cdots\!86 ) / 27\!\cdots\!18 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 29\!\cdots\!52 \nu^{19} + \cdots - 22\!\cdots\!30 ) / 27\!\cdots\!18 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 65\!\cdots\!87 \nu^{19} + \cdots - 26\!\cdots\!04 ) / 55\!\cdots\!36 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 14\!\cdots\!11 \nu^{19} + \cdots + 58\!\cdots\!44 ) / 11\!\cdots\!72 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 37\!\cdots\!76 \nu^{19} + \cdots + 78\!\cdots\!96 ) / 27\!\cdots\!18 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 81\!\cdots\!29 \nu^{19} + \cdots - 44\!\cdots\!84 ) / 55\!\cdots\!36 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 22\!\cdots\!16 \nu^{19} + \cdots + 33\!\cdots\!04 ) / 12\!\cdots\!36 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 39\!\cdots\!85 \nu^{19} + \cdots - 26\!\cdots\!92 ) / 11\!\cdots\!72 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 21\!\cdots\!23 \nu^{19} + \cdots - 28\!\cdots\!60 ) / 55\!\cdots\!36 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 21\!\cdots\!69 \nu^{19} + \cdots + 25\!\cdots\!96 ) / 55\!\cdots\!36 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 44\!\cdots\!85 \nu^{19} + \cdots + 12\!\cdots\!60 ) / 11\!\cdots\!72 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 30\!\cdots\!00 \nu^{19} + \cdots - 90\!\cdots\!64 ) / 55\!\cdots\!36 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 32\!\cdots\!00 \nu^{19} + \cdots + 16\!\cdots\!80 ) / 55\!\cdots\!36 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 19\!\cdots\!96 \nu^{19} + \cdots + 12\!\cdots\!12 ) / 27\!\cdots\!18 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 21\!\cdots\!80 \nu^{19} + \cdots + 66\!\cdots\!68 ) / 27\!\cdots\!18 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 57\!\cdots\!19 \nu^{19} + \cdots + 71\!\cdots\!96 ) / 55\!\cdots\!36 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{16} + \beta_{15} + \beta_{14} + \beta_{11} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{19} - \beta_{17} + 4\beta_{10} - \beta_{9} - \beta_{6} + 8\beta_{5} + \beta_{3} - \beta_{2} + 4 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2 \beta_{16} + 5 \beta_{15} + 10 \beta_{14} - 2 \beta_{13} - 5 \beta_{12} + 11 \beta_{11} + 3 \beta_{10} + \cdots + 1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 16 \beta_{19} + 12 \beta_{18} - 12 \beta_{17} + 4 \beta_{14} - 4 \beta_{13} - 4 \beta_{12} + \cdots + 6 \beta_1 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 7 \beta_{19} + 7 \beta_{18} - 3 \beta_{17} - 74 \beta_{16} - 34 \beta_{15} + 34 \beta_{14} - 127 \beta_{13} + \cdots - 32 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 155 \beta_{19} + 217 \beta_{18} - 119 \beta_{16} - 105 \beta_{15} - 10 \beta_{14} - 160 \beta_{13} + \cdots - 770 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 85 \beta_{19} + 204 \beta_{18} + 85 \beta_{17} - 1936 \beta_{16} - 1719 \beta_{15} - 1016 \beta_{14} + \cdots - 1615 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 2091 \beta_{18} + 2091 \beta_{17} - 3213 \beta_{16} - 3213 \beta_{15} - 2472 \beta_{14} - 2472 \beta_{13} + \cdots - 15198 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 1751 \beta_{19} + 1751 \beta_{18} + 4223 \beta_{17} - 23928 \beta_{16} - 26880 \beta_{15} + \cdots - 29058 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 28971 \beta_{19} + 40959 \beta_{17} - 36490 \beta_{16} - 44936 \beta_{15} - 57594 \beta_{14} + \cdots - 152110 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 76720 \beta_{19} - 31784 \beta_{18} + 76720 \beta_{17} - 99769 \beta_{16} - 198670 \beta_{15} + \cdots - 203811 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 579460 \beta_{19} - 409761 \beta_{18} + 409761 \beta_{17} + 108504 \beta_{16} - 108504 \beta_{15} + \cdots - 973170 \beta_1 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 1305657 \beta_{19} - 1305657 \beta_{18} + 540837 \beta_{17} + 2851151 \beta_{16} + 1452169 \beta_{15} + \cdots + 3331860 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 5894213 \beta_{19} - 8335603 \beta_{18} + 12537223 \beta_{16} + 9925909 \beta_{15} + 1846494 \beta_{14} + \cdots + 31996842 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 8865233 \beta_{19} - 21402456 \beta_{18} - 8865233 \beta_{17} + 80056861 \beta_{16} + \cdots + 128586181 ) / 2 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( - 85951074 \beta_{18} - 85951074 \beta_{17} + 253671703 \beta_{16} + 253671703 \beta_{15} + \cdots + 667647854 ) / 2 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( 141969696 \beta_{19} - 141969696 \beta_{18} - 342744864 \beta_{17} + 1059472869 \beta_{16} + \cdots + 2025632800 ) / 2 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( 1266976935 \beta_{19} - 1791775503 \beta_{17} + 2481169810 \beta_{16} + 3166659538 \beta_{15} + \cdots + 7022575476 ) / 2 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( 5405826548 \beta_{19} + 2239167010 \beta_{18} - 5405826548 \beta_{17} + 4742172932 \beta_{16} + \cdots + 13087434199 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1710\mathbb{Z}\right)^\times\).
| \(n\) | \(191\) | \(1027\) | \(1351\) |
| \(\chi(n)\) | \(-1\) | \(\beta_{10}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 647.1 |
|
−0.707107 | + | 0.707107i | 0 | − | 1.00000i | −2.03893 | + | 0.918023i | 0 | −1.07805 | − | 1.07805i | 0.707107 | + | 0.707107i | 0 | 0.792601 | − | 2.09088i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 647.2 | −0.707107 | + | 0.707107i | 0 | − | 1.00000i | −1.06972 | + | 1.96359i | 0 | 3.58826 | + | 3.58826i | 0.707107 | + | 0.707107i | 0 | −0.632067 | − | 2.14488i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 647.3 | −0.707107 | + | 0.707107i | 0 | − | 1.00000i | −0.975056 | − | 2.01228i | 0 | −3.02518 | − | 3.02518i | 0.707107 | + | 0.707107i | 0 | 2.11236 | + | 0.733428i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 647.4 | −0.707107 | + | 0.707107i | 0 | − | 1.00000i | 1.16531 | − | 1.90842i | 0 | 2.34049 | + | 2.34049i | 0.707107 | + | 0.707107i | 0 | 0.525458 | + | 2.17345i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 647.5 | −0.707107 | + | 0.707107i | 0 | − | 1.00000i | 2.21129 | + | 0.331972i | 0 | −1.82552 | − | 1.82552i | 0.707107 | + | 0.707107i | 0 | −1.79836 | + | 1.32888i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 647.6 | 0.707107 | − | 0.707107i | 0 | − | 1.00000i | −2.21129 | − | 0.331972i | 0 | −1.82552 | − | 1.82552i | −0.707107 | − | 0.707107i | 0 | −1.79836 | + | 1.32888i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 647.7 | 0.707107 | − | 0.707107i | 0 | − | 1.00000i | −1.16531 | + | 1.90842i | 0 | 2.34049 | + | 2.34049i | −0.707107 | − | 0.707107i | 0 | 0.525458 | + | 2.17345i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 647.8 | 0.707107 | − | 0.707107i | 0 | − | 1.00000i | 0.975056 | + | 2.01228i | 0 | −3.02518 | − | 3.02518i | −0.707107 | − | 0.707107i | 0 | 2.11236 | + | 0.733428i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 647.9 | 0.707107 | − | 0.707107i | 0 | − | 1.00000i | 1.06972 | − | 1.96359i | 0 | 3.58826 | + | 3.58826i | −0.707107 | − | 0.707107i | 0 | −0.632067 | − | 2.14488i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 647.10 | 0.707107 | − | 0.707107i | 0 | − | 1.00000i | 2.03893 | − | 0.918023i | 0 | −1.07805 | − | 1.07805i | −0.707107 | − | 0.707107i | 0 | 0.792601 | − | 2.09088i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1673.1 | −0.707107 | − | 0.707107i | 0 | 1.00000i | −2.03893 | − | 0.918023i | 0 | −1.07805 | + | 1.07805i | 0.707107 | − | 0.707107i | 0 | 0.792601 | + | 2.09088i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1673.2 | −0.707107 | − | 0.707107i | 0 | 1.00000i | −1.06972 | − | 1.96359i | 0 | 3.58826 | − | 3.58826i | 0.707107 | − | 0.707107i | 0 | −0.632067 | + | 2.14488i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1673.3 | −0.707107 | − | 0.707107i | 0 | 1.00000i | −0.975056 | + | 2.01228i | 0 | −3.02518 | + | 3.02518i | 0.707107 | − | 0.707107i | 0 | 2.11236 | − | 0.733428i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1673.4 | −0.707107 | − | 0.707107i | 0 | 1.00000i | 1.16531 | + | 1.90842i | 0 | 2.34049 | − | 2.34049i | 0.707107 | − | 0.707107i | 0 | 0.525458 | − | 2.17345i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1673.5 | −0.707107 | − | 0.707107i | 0 | 1.00000i | 2.21129 | − | 0.331972i | 0 | −1.82552 | + | 1.82552i | 0.707107 | − | 0.707107i | 0 | −1.79836 | − | 1.32888i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1673.6 | 0.707107 | + | 0.707107i | 0 | 1.00000i | −2.21129 | + | 0.331972i | 0 | −1.82552 | + | 1.82552i | −0.707107 | + | 0.707107i | 0 | −1.79836 | − | 1.32888i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1673.7 | 0.707107 | + | 0.707107i | 0 | 1.00000i | −1.16531 | − | 1.90842i | 0 | 2.34049 | − | 2.34049i | −0.707107 | + | 0.707107i | 0 | 0.525458 | − | 2.17345i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1673.8 | 0.707107 | + | 0.707107i | 0 | 1.00000i | 0.975056 | − | 2.01228i | 0 | −3.02518 | + | 3.02518i | −0.707107 | + | 0.707107i | 0 | 2.11236 | − | 0.733428i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1673.9 | 0.707107 | + | 0.707107i | 0 | 1.00000i | 1.06972 | + | 1.96359i | 0 | 3.58826 | − | 3.58826i | −0.707107 | + | 0.707107i | 0 | −0.632067 | + | 2.14488i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1673.10 | 0.707107 | + | 0.707107i | 0 | 1.00000i | 2.03893 | + | 0.918023i | 0 | −1.07805 | + | 1.07805i | −0.707107 | + | 0.707107i | 0 | 0.792601 | + | 2.09088i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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 |
| 15.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1710.2.n.i | ✓ | 20 |
| 3.b | odd | 2 | 1 | inner | 1710.2.n.i | ✓ | 20 |
| 5.c | odd | 4 | 1 | inner | 1710.2.n.i | ✓ | 20 |
| 15.e | even | 4 | 1 | inner | 1710.2.n.i | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1710.2.n.i | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 1710.2.n.i | ✓ | 20 | 3.b | odd | 2 | 1 | inner |
| 1710.2.n.i | ✓ | 20 | 5.c | odd | 4 | 1 | inner |
| 1710.2.n.i | ✓ | 20 | 15.e | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1710, [\chi])\):
|
\( T_{7}^{10} + 32T_{7}^{7} + 584T_{7}^{6} + 624T_{7}^{5} + 512T_{7}^{4} + 5760T_{7}^{3} + 48400T_{7}^{2} + 88000T_{7} + 80000 \)
|
|
\( T_{17}^{20} + 2288T_{17}^{16} + 1179008T_{17}^{12} + 165189632T_{17}^{8} + 6196826112T_{17}^{4} + 68719476736 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 1)^{5} \)
|
| $3$ |
\( T^{20} \)
|
| $5$ |
\( T^{20} + 13 T^{16} + \cdots + 9765625 \)
|
| $7$ |
\( (T^{10} + 32 T^{7} + \cdots + 80000)^{2} \)
|
| $11$ |
\( (T^{10} + 64 T^{8} + \cdots + 128)^{2} \)
|
| $13$ |
\( (T^{10} + 6 T^{9} + \cdots + 128)^{2} \)
|
| $17$ |
\( T^{20} + \cdots + 68719476736 \)
|
| $19$ |
\( (T^{2} + 1)^{10} \)
|
| $23$ |
\( T^{20} + \cdots + 4294967296 \)
|
| $29$ |
\( (T^{10} - 218 T^{8} + \cdots - 21727232)^{2} \)
|
| $31$ |
\( (T^{5} - 4 T^{4} - 56 T^{3} + \cdots - 64)^{4} \)
|
| $37$ |
\( (T^{10} + 6 T^{9} + \cdots + 128)^{2} \)
|
| $41$ |
\( (T^{10} + 226 T^{8} + \cdots + 123008)^{2} \)
|
| $43$ |
\( (T^{10} + 248 T^{7} + \cdots + 76880000)^{2} \)
|
| $47$ |
\( T^{20} + \cdots + 10\!\cdots\!56 \)
|
| $53$ |
\( T^{20} + \cdots + 68719476736 \)
|
| $59$ |
\( (T^{10} - 104 T^{8} + \cdots - 8192)^{2} \)
|
| $61$ |
\( (T^{5} + 4 T^{4} + \cdots + 8896)^{4} \)
|
| $67$ |
\( T^{20} \)
|
| $71$ |
\( (T^{10} + 512 T^{8} + \cdots + 22151168)^{2} \)
|
| $73$ |
\( (T^{10} - 10 T^{9} + \cdots + 4921113632)^{2} \)
|
| $79$ |
\( (T^{10} + 640 T^{8} + \cdots + 904806400)^{2} \)
|
| $83$ |
\( T^{20} + 47248 T^{16} + \cdots + 16777216 \)
|
| $89$ |
\( (T^{10} - 578 T^{8} + \cdots - 4739484800)^{2} \)
|
| $97$ |
\( (T^{10} + 10 T^{9} + \cdots + 1073512448)^{2} \)
|
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