Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [585,2,Mod(163,585)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(585, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 9, 11]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("585.163");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 585 = 3^{2} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 585.cf (of order \(12\), degree \(4\), 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: | \(4.67124851824\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(5\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} + 26 x^{18} + 279 x^{16} + 1604 x^{14} + 5353 x^{12} + 10466 x^{10} + 11441 x^{8} + 6176 x^{6} + \cdots + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 65) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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
\( x^{20} + 26 x^{18} + 279 x^{16} + 1604 x^{14} + 5353 x^{12} + 10466 x^{10} + 11441 x^{8} + 6176 x^{6} + \cdots + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 20 \nu^{18} - 389 \nu^{16} - 2695 \nu^{14} - 7125 \nu^{12} + 1214 \nu^{10} + 39860 \nu^{8} + \cdots - 409 ) / 5992 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 20 \nu^{18} + 389 \nu^{16} + 2695 \nu^{14} + 7125 \nu^{12} - 1214 \nu^{10} - 39860 \nu^{8} + \cdots + 409 ) / 5992 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 69 \nu^{18} - 1647 \nu^{16} - 15798 \nu^{14} - 78322 \nu^{12} - 214723 \nu^{10} - 324081 \nu^{8} + \cdots - 539 ) / 1712 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 321 \nu^{19} - 483 \nu^{18} + 9951 \nu^{17} - 11529 \nu^{16} + 127330 \nu^{15} - 110586 \nu^{14} + \cdots - 3773 ) / 23968 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 700 \nu^{19} + 617 \nu^{18} + 17360 \nu^{17} + 16607 \nu^{16} + 174468 \nu^{15} + 185192 \nu^{14} + \cdots + 26961 ) / 11984 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 409 \nu^{19} - 10614 \nu^{17} - 113722 \nu^{15} - 653341 \nu^{13} - 2182252 \nu^{11} - 4281808 \nu^{9} + \cdots - 2996 ) / 5992 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 1635 \nu^{19} - 309 \nu^{18} + 41725 \nu^{17} - 7171 \nu^{16} + 436590 \nu^{15} - 66542 \nu^{14} + \cdots - 30699 ) / 23968 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 700 \nu^{19} - 617 \nu^{18} + 17360 \nu^{17} - 16607 \nu^{16} + 174468 \nu^{15} - 185192 \nu^{14} + \cdots - 26961 ) / 11984 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 419 \nu^{19} + 4593 \nu^{18} - 12681 \nu^{17} + 118807 \nu^{16} - 158886 \nu^{15} + 1266230 \nu^{14} + \cdots + 83703 ) / 23968 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 3773 \nu^{19} - 565 \nu^{18} - 97615 \nu^{17} - 11551 \nu^{16} - 1041138 \nu^{15} - 83062 \nu^{14} + \cdots + 81509 ) / 23968 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 3773 \nu^{19} + 321 \nu^{18} + 97615 \nu^{17} + 9951 \nu^{16} + 1041138 \nu^{15} + 127330 \nu^{14} + \cdots + 39483 ) / 23968 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 3773 \nu^{19} - 321 \nu^{18} + 97615 \nu^{17} - 9951 \nu^{16} + 1041138 \nu^{15} - 127330 \nu^{14} + \cdots - 39483 ) / 23968 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 2753 \nu^{19} - 867 \nu^{18} - 71035 \nu^{17} - 22593 \nu^{16} - 754642 \nu^{15} - 243222 \nu^{14} + \cdots - 15221 ) / 11984 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 2753 \nu^{19} - 867 \nu^{18} + 71035 \nu^{17} - 22593 \nu^{16} + 754642 \nu^{15} - 243222 \nu^{14} + \cdots - 15221 ) / 11984 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 6965 \nu^{19} + 1863 \nu^{18} - 180971 \nu^{17} + 49605 \nu^{16} - 1940134 \nu^{15} + 545958 \nu^{14} + \cdots + 1285 ) / 23968 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 8297 \nu^{19} - 175 \nu^{18} + 216915 \nu^{17} - 2093 \nu^{16} + 2342074 \nu^{15} + 9562 \nu^{14} + \cdots + 23947 ) / 23968 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 4979 \nu^{19} - 129835 \nu^{17} - 1398012 \nu^{15} - 8068552 \nu^{13} - 27041293 \nu^{11} + \cdots + 11984 ) / 11984 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 10417 \nu^{19} - 483 \nu^{18} + 268635 \nu^{17} - 11529 \nu^{16} + 2853942 \nu^{15} - 110586 \nu^{14} + \cdots + 20195 ) / 23968 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 16083 \nu^{19} - 483 \nu^{18} - 419809 \nu^{17} - 11529 \nu^{16} - 4525598 \nu^{15} + \cdots - 15757 ) / 23968 \)
|
| \(\nu\) | \(=\) |
\( \beta_{2} + \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{12} + \beta_{10} - \beta_{3} + \beta _1 - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{19} + 2 \beta_{18} - \beta_{13} - 3 \beta_{12} - 2 \beta_{11} - 2 \beta_{10} - \beta_{9} + \cdots - 7 \beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{19} - \beta_{18} + \beta_{17} + 2 \beta_{16} - 2 \beta_{15} + 3 \beta_{14} - 6 \beta_{12} + \cdots + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 9 \beta_{19} - 17 \beta_{18} - \beta_{17} - 3 \beta_{14} + 11 \beta_{13} + 26 \beta_{12} + 17 \beta_{11} + \cdots - 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 13 \beta_{19} + 13 \beta_{18} - 13 \beta_{17} - 26 \beta_{16} + 26 \beta_{15} - 32 \beta_{14} + \cdots - 39 \)
|
| \(\nu^{7}\) | \(=\) |
\( 66 \beta_{19} + 122 \beta_{18} + 14 \beta_{17} + 37 \beta_{14} - 89 \beta_{13} - 189 \beta_{12} + \cdots + 21 \)
|
| \(\nu^{8}\) | \(=\) |
\( 120 \beta_{19} - 116 \beta_{18} + 116 \beta_{17} + 232 \beta_{16} - 240 \beta_{15} + 260 \beta_{14} + \cdots + 213 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 462 \beta_{19} - 840 \beta_{18} - 136 \beta_{17} - 332 \beta_{14} + 657 \beta_{13} + 1306 \beta_{12} + \cdots - 170 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 974 \beta_{19} + 914 \beta_{18} - 914 \beta_{17} - 1828 \beta_{16} + 1948 \beta_{15} - 1941 \beta_{14} + \cdots - 1255 \)
|
| \(\nu^{11}\) | \(=\) |
\( 3210 \beta_{19} + 5733 \beta_{18} + 1135 \beta_{17} + 2648 \beta_{14} - 4690 \beta_{13} - 8890 \beta_{12} + \cdots + 1267 \)
|
| \(\nu^{12}\) | \(=\) |
\( 7438 \beta_{19} - 6832 \beta_{18} + 6832 \beta_{17} + 13664 \beta_{16} - 14876 \beta_{15} + 14032 \beta_{14} + \cdots + 7797 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 22356 \beta_{19} - 39188 \beta_{18} - 8780 \beta_{17} - 19984 \beta_{14} + 33058 \beta_{13} + \cdots - 9130 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 54982 \beta_{19} + 49778 \beta_{18} - 49778 \beta_{17} - 99556 \beta_{16} + 109964 \beta_{15} + \cdots - 50290 \)
|
| \(\nu^{15}\) | \(=\) |
\( 156271 \beta_{19} + 269080 \beta_{18} + 65190 \beta_{17} + 146494 \beta_{14} - 231975 \beta_{13} + \cdots + 64782 \)
|
| \(\nu^{16}\) | \(=\) |
\( 398897 \beta_{19} - 357747 \beta_{18} + 357747 \beta_{17} + 715494 \beta_{16} - 797794 \beta_{15} + \cdots + 333034 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 1095413 \beta_{19} - 1856511 \beta_{18} - 472963 \beta_{17} - 1056441 \beta_{14} + 1625819 \beta_{13} + \cdots - 456258 \)
|
| \(\nu^{18}\) | \(=\) |
\( - 2861437 \beta_{19} + 2550751 \beta_{18} - 2550751 \beta_{17} - 5101502 \beta_{16} + 5722874 \beta_{15} + \cdots - 2246355 \)
|
| \(\nu^{19}\) | \(=\) |
\( 7691530 \beta_{19} + 12862874 \beta_{18} + 3385870 \beta_{17} + 7545155 \beta_{14} - 11394317 \beta_{13} + \cdots + 3202109 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).
| \(n\) | \(326\) | \(352\) | \(496\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{11} - \beta_{12}\) | \(-\beta_{11}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 163.1 |
|
−0.759023 | + | 1.31467i | 0 | −0.152233 | − | 0.263675i | −0.600231 | + | 2.15400i | 0 | −2.24723 | + | 1.29744i | −2.57390 | 0 | −2.37621 | − | 2.42404i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.2 | 0.137404 | − | 0.237991i | 0 | 0.962240 | + | 1.66665i | −1.69883 | − | 1.45395i | 0 | −0.334376 | + | 0.193052i | 1.07848 | 0 | −0.579454 | + | 0.204528i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.3 | 0.246951 | − | 0.427732i | 0 | 0.878030 | + | 1.52079i | 2.21791 | + | 0.284413i | 0 | 3.18307 | − | 1.83775i | 1.85513 | 0 | 0.669366 | − | 0.878433i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.4 | 0.915816 | − | 1.58624i | 0 | −0.677439 | − | 1.17336i | −1.45480 | + | 1.69810i | 0 | −3.06478 | + | 1.76945i | 1.18163 | 0 | 1.36126 | + | 3.86282i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.5 | 1.32488 | − | 2.29475i | 0 | −2.51060 | − | 4.34849i | 1.30391 | − | 1.81654i | 0 | 0.0972962 | − | 0.0561740i | −8.00544 | 0 | −2.44100 | − | 5.39885i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 262.1 | −0.759023 | − | 1.31467i | 0 | −0.152233 | + | 0.263675i | −0.600231 | − | 2.15400i | 0 | −2.24723 | − | 1.29744i | −2.57390 | 0 | −2.37621 | + | 2.42404i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 262.2 | 0.137404 | + | 0.237991i | 0 | 0.962240 | − | 1.66665i | −1.69883 | + | 1.45395i | 0 | −0.334376 | − | 0.193052i | 1.07848 | 0 | −0.579454 | − | 0.204528i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 262.3 | 0.246951 | + | 0.427732i | 0 | 0.878030 | − | 1.52079i | 2.21791 | − | 0.284413i | 0 | 3.18307 | + | 1.83775i | 1.85513 | 0 | 0.669366 | + | 0.878433i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 262.4 | 0.915816 | + | 1.58624i | 0 | −0.677439 | + | 1.17336i | −1.45480 | − | 1.69810i | 0 | −3.06478 | − | 1.76945i | 1.18163 | 0 | 1.36126 | − | 3.86282i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 262.5 | 1.32488 | + | 2.29475i | 0 | −2.51060 | + | 4.34849i | 1.30391 | + | 1.81654i | 0 | 0.0972962 | + | 0.0561740i | −8.00544 | 0 | −2.44100 | + | 5.39885i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 388.1 | −1.12540 | − | 1.94926i | 0 | −1.53307 | + | 2.65535i | 2.22228 | − | 0.247944i | 0 | −1.10607 | − | 0.638592i | 2.39966 | 0 | −2.98427 | − | 4.05275i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 388.2 | −0.511309 | − | 0.885613i | 0 | 0.477126 | − | 0.826407i | 1.69584 | + | 1.45744i | 0 | 0.834479 | + | 0.481787i | −3.02107 | 0 | 0.423625 | − | 2.24706i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 388.3 | −0.0656513 | − | 0.113711i | 0 | 0.991380 | − | 1.71712i | −0.813169 | − | 2.08297i | 0 | −2.40874 | − | 1.39069i | −0.522947 | 0 | −0.183472 | + | 0.229216i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 388.4 | 0.792369 | + | 1.37242i | 0 | −0.255697 | + | 0.442881i | −0.0672627 | + | 2.23506i | 0 | −0.474866 | − | 0.274164i | 2.35905 | 0 | −3.12074 | + | 1.67868i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 388.5 | 1.04397 | + | 1.80821i | 0 | −1.17974 | + | 2.04338i | 0.194361 | − | 2.22760i | 0 | 2.52122 | + | 1.45563i | −0.750585 | 0 | 4.23088 | − | 1.97411i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 487.1 | −1.12540 | + | 1.94926i | 0 | −1.53307 | − | 2.65535i | 2.22228 | + | 0.247944i | 0 | −1.10607 | + | 0.638592i | 2.39966 | 0 | −2.98427 | + | 4.05275i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 487.2 | −0.511309 | + | 0.885613i | 0 | 0.477126 | + | 0.826407i | 1.69584 | − | 1.45744i | 0 | 0.834479 | − | 0.481787i | −3.02107 | 0 | 0.423625 | + | 2.24706i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 487.3 | −0.0656513 | + | 0.113711i | 0 | 0.991380 | + | 1.71712i | −0.813169 | + | 2.08297i | 0 | −2.40874 | + | 1.39069i | −0.522947 | 0 | −0.183472 | − | 0.229216i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 487.4 | 0.792369 | − | 1.37242i | 0 | −0.255697 | − | 0.442881i | −0.0672627 | − | 2.23506i | 0 | −0.474866 | + | 0.274164i | 2.35905 | 0 | −3.12074 | − | 1.67868i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 487.5 | 1.04397 | − | 1.80821i | 0 | −1.17974 | − | 2.04338i | 0.194361 | + | 2.22760i | 0 | 2.52122 | − | 1.45563i | −0.750585 | 0 | 4.23088 | + | 1.97411i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 65.o | even | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 585.2.cf.a | 20 | |
| 3.b | odd | 2 | 1 | 65.2.o.a | ✓ | 20 | |
| 5.c | odd | 4 | 1 | 585.2.dp.a | 20 | ||
| 13.f | odd | 12 | 1 | 585.2.dp.a | 20 | ||
| 15.d | odd | 2 | 1 | 325.2.s.b | 20 | ||
| 15.e | even | 4 | 1 | 65.2.t.a | yes | 20 | |
| 15.e | even | 4 | 1 | 325.2.x.b | 20 | ||
| 39.d | odd | 2 | 1 | 845.2.o.g | 20 | ||
| 39.f | even | 4 | 1 | 845.2.t.e | 20 | ||
| 39.f | even | 4 | 1 | 845.2.t.f | 20 | ||
| 39.h | odd | 6 | 1 | 845.2.k.d | 20 | ||
| 39.h | odd | 6 | 1 | 845.2.o.f | 20 | ||
| 39.i | odd | 6 | 1 | 845.2.k.e | 20 | ||
| 39.i | odd | 6 | 1 | 845.2.o.e | 20 | ||
| 39.k | even | 12 | 1 | 65.2.t.a | yes | 20 | |
| 39.k | even | 12 | 1 | 845.2.f.d | 20 | ||
| 39.k | even | 12 | 1 | 845.2.f.e | 20 | ||
| 39.k | even | 12 | 1 | 845.2.t.g | 20 | ||
| 65.o | even | 12 | 1 | inner | 585.2.cf.a | 20 | |
| 195.j | odd | 4 | 1 | 845.2.o.e | 20 | ||
| 195.s | even | 4 | 1 | 845.2.t.g | 20 | ||
| 195.u | odd | 4 | 1 | 845.2.o.f | 20 | ||
| 195.bc | odd | 12 | 1 | 325.2.s.b | 20 | ||
| 195.bc | odd | 12 | 1 | 845.2.k.d | 20 | ||
| 195.bc | odd | 12 | 1 | 845.2.o.g | 20 | ||
| 195.bf | even | 12 | 1 | 845.2.f.d | 20 | ||
| 195.bf | even | 12 | 1 | 845.2.t.e | 20 | ||
| 195.bh | even | 12 | 1 | 325.2.x.b | 20 | ||
| 195.bl | even | 12 | 1 | 845.2.f.e | 20 | ||
| 195.bl | even | 12 | 1 | 845.2.t.f | 20 | ||
| 195.bn | odd | 12 | 1 | 65.2.o.a | ✓ | 20 | |
| 195.bn | odd | 12 | 1 | 845.2.k.e | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 65.2.o.a | ✓ | 20 | 3.b | odd | 2 | 1 | |
| 65.2.o.a | ✓ | 20 | 195.bn | odd | 12 | 1 | |
| 65.2.t.a | yes | 20 | 15.e | even | 4 | 1 | |
| 65.2.t.a | yes | 20 | 39.k | even | 12 | 1 | |
| 325.2.s.b | 20 | 15.d | odd | 2 | 1 | ||
| 325.2.s.b | 20 | 195.bc | odd | 12 | 1 | ||
| 325.2.x.b | 20 | 15.e | even | 4 | 1 | ||
| 325.2.x.b | 20 | 195.bh | even | 12 | 1 | ||
| 585.2.cf.a | 20 | 1.a | even | 1 | 1 | trivial | |
| 585.2.cf.a | 20 | 65.o | even | 12 | 1 | inner | |
| 585.2.dp.a | 20 | 5.c | odd | 4 | 1 | ||
| 585.2.dp.a | 20 | 13.f | odd | 12 | 1 | ||
| 845.2.f.d | 20 | 39.k | even | 12 | 1 | ||
| 845.2.f.d | 20 | 195.bf | even | 12 | 1 | ||
| 845.2.f.e | 20 | 39.k | even | 12 | 1 | ||
| 845.2.f.e | 20 | 195.bl | even | 12 | 1 | ||
| 845.2.k.d | 20 | 39.h | odd | 6 | 1 | ||
| 845.2.k.d | 20 | 195.bc | odd | 12 | 1 | ||
| 845.2.k.e | 20 | 39.i | odd | 6 | 1 | ||
| 845.2.k.e | 20 | 195.bn | odd | 12 | 1 | ||
| 845.2.o.e | 20 | 39.i | odd | 6 | 1 | ||
| 845.2.o.e | 20 | 195.j | odd | 4 | 1 | ||
| 845.2.o.f | 20 | 39.h | odd | 6 | 1 | ||
| 845.2.o.f | 20 | 195.u | odd | 4 | 1 | ||
| 845.2.o.g | 20 | 39.d | odd | 2 | 1 | ||
| 845.2.o.g | 20 | 195.bc | odd | 12 | 1 | ||
| 845.2.t.e | 20 | 39.f | even | 4 | 1 | ||
| 845.2.t.e | 20 | 195.bf | even | 12 | 1 | ||
| 845.2.t.f | 20 | 39.f | even | 4 | 1 | ||
| 845.2.t.f | 20 | 195.bl | even | 12 | 1 | ||
| 845.2.t.g | 20 | 39.k | even | 12 | 1 | ||
| 845.2.t.g | 20 | 195.s | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{20} - 4 T_{2}^{19} + 21 T_{2}^{18} - 48 T_{2}^{17} + 170 T_{2}^{16} - 322 T_{2}^{15} + 893 T_{2}^{14} + \cdots + 1 \)
acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} - 4 T^{19} + \cdots + 1 \)
|
| $3$ |
\( T^{20} \)
|
| $5$ |
\( T^{20} - 6 T^{19} + \cdots + 9765625 \)
|
| $7$ |
\( T^{20} + 6 T^{19} + \cdots + 64 \)
|
| $11$ |
\( T^{20} - 16 T^{19} + \cdots + 256 \)
|
| $13$ |
\( T^{20} + \cdots + 137858491849 \)
|
| $17$ |
\( T^{20} - 10 T^{19} + \cdots + 1168561 \)
|
| $19$ |
\( T^{20} + \cdots + 1583721616 \)
|
| $23$ |
\( T^{20} - 2 T^{19} + \cdots + 144 \)
|
| $29$ |
\( T^{20} + \cdots + 206213167449 \)
|
| $31$ |
\( T^{20} - 104 T^{17} + \cdots + 2166784 \)
|
| $37$ |
\( T^{20} + \cdots + 4508182449 \)
|
| $41$ |
\( T^{20} + \cdots + 3748255729 \)
|
| $43$ |
\( T^{20} + \cdots + 1370772640000 \)
|
| $47$ |
\( T^{20} + \cdots + 807469056 \)
|
| $53$ |
\( T^{20} + \cdots + 2978634160384 \)
|
| $59$ |
\( T^{20} + 16 T^{19} + \cdots + 33856 \)
|
| $61$ |
\( T^{20} + \cdots + 826457355409 \)
|
| $67$ |
\( T^{20} + \cdots + 15478905336976 \)
|
| $71$ |
\( T^{20} + \cdots + 11\!\cdots\!44 \)
|
| $73$ |
\( (T^{10} - 36 T^{9} + \cdots + 253113232)^{2} \)
|
| $79$ |
\( T^{20} + \cdots + 75\!\cdots\!36 \)
|
| $83$ |
\( T^{20} + \cdots + 11512611864576 \)
|
| $89$ |
\( T^{20} + \cdots + 329648222500 \)
|
| $97$ |
\( T^{20} + \cdots + 28\!\cdots\!36 \)
|
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