Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [845,2,Mod(268,845)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(845, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("845.268");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 845 = 5 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 845.k (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: | \(6.74735897080\) |
| 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: |
\( x^{20} + 26 x^{18} + 279 x^{16} + 1604 x^{14} + 5353 x^{12} + 10466 x^{10} + 11441 x^{8} + 6176 x^{6} + \cdots + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 65) |
| 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
\( x^{20} + 26 x^{18} + 279 x^{16} + 1604 x^{14} + 5353 x^{12} + 10466 x^{10} + 11441 x^{8} + 6176 x^{6} + \cdots + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 69 \nu^{18} + 1647 \nu^{16} + 15798 \nu^{14} + 78322 \nu^{12} + 214723 \nu^{10} + 324081 \nu^{8} + \cdots + 539 ) / 1712 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 2051 \nu^{19} - 1625 \nu^{18} - 52213 \nu^{17} - 40407 \nu^{16} - 546574 \nu^{15} - 409402 \nu^{14} + \cdots - 11323 ) / 23968 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1625 \nu^{18} - 40407 \nu^{16} - 409402 \nu^{14} - 2186822 \nu^{12} - 6647231 \nu^{10} + \cdots - 35291 ) / 11984 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2137 \nu^{19} + 321 \nu^{18} + 55159 \nu^{17} + 9951 \nu^{16} + 586250 \nu^{15} + 127330 \nu^{14} + \cdots + 27499 ) / 23968 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 908 \nu^{19} - 41 \nu^{18} - 22604 \nu^{17} - 11 \nu^{16} - 229460 \nu^{15} + 13762 \nu^{14} + \cdots + 12681 ) / 11984 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 2137 \nu^{19} + 321 \nu^{18} - 55159 \nu^{17} + 9951 \nu^{16} - 586250 \nu^{15} + 127330 \nu^{14} + \cdots + 51467 ) / 23968 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 2051 \nu^{19} + 1625 \nu^{18} - 52213 \nu^{17} + 40407 \nu^{16} - 546574 \nu^{15} + 409402 \nu^{14} + \cdots + 11323 ) / 23968 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 2753 \nu^{19} - 867 \nu^{18} - 71035 \nu^{17} - 22593 \nu^{16} - 754642 \nu^{15} - 243222 \nu^{14} + \cdots - 15221 ) / 11984 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 2753 \nu^{19} + 867 \nu^{18} - 71035 \nu^{17} + 22593 \nu^{16} - 754642 \nu^{15} + 243222 \nu^{14} + \cdots + 15221 ) / 11984 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 6161 \nu^{19} + 1625 \nu^{18} - 162487 \nu^{17} + 40407 \nu^{16} - 1774122 \nu^{15} + \cdots + 11323 ) / 23968 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 5315 \nu^{19} + 3731 \nu^{18} - 137269 \nu^{17} + 93877 \nu^{16} - 1459766 \nu^{15} + \cdots + 89369 ) / 23968 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 539 \nu^{19} + 13945 \nu^{17} + 148734 \nu^{15} + 848758 \nu^{13} + 2806945 \nu^{11} + \cdots + 22081 \nu ) / 1712 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 10681 \nu^{19} - 2701 \nu^{18} - 278863 \nu^{17} - 65979 \nu^{16} - 3007858 \nu^{15} + \cdots - 76919 ) / 23968 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 11655 \nu^{19} - 191 \nu^{18} + 301777 \nu^{17} - 4801 \nu^{16} + 3219622 \nu^{15} - 49518 \nu^{14} + \cdots + 54179 ) / 23968 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 5223 \nu^{19} + 1330 \nu^{18} - 135929 \nu^{17} + 34482 \nu^{16} - 1460102 \nu^{15} + \cdots + 34314 ) / 11984 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 1767 \nu^{19} + 203 \nu^{18} + 46090 \nu^{17} + 5334 \nu^{16} + 496321 \nu^{15} + 57876 \nu^{14} + \cdots - 455 ) / 2996 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 14121 \nu^{19} + 1997 \nu^{18} - 366743 \nu^{17} + 48691 \nu^{16} - 3929786 \nu^{15} + \cdots + 34959 ) / 23968 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 17385 \nu^{19} - 3359 \nu^{18} - 451799 \nu^{17} - 85593 \nu^{16} - 4842978 \nu^{15} + \cdots - 41765 ) / 23968 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{8} - \beta_{4} + \beta_{3} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{19} + \beta_{18} + \beta_{17} - \beta_{14} + 2 \beta_{13} - \beta_{12} + \beta_{11} + \cdots - 3 \beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3 \beta_{19} - \beta_{18} + 2 \beta_{15} + 2 \beta_{13} - \beta_{12} + 2 \beta_{10} - 2 \beta_{9} + \cdots + 9 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 8 \beta_{19} - 9 \beta_{18} - 8 \beta_{17} + \beta_{15} + 8 \beta_{14} - 15 \beta_{13} + 9 \beta_{12} + \cdots - 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 32 \beta_{19} + 13 \beta_{18} + 2 \beta_{17} + 2 \beta_{16} - 19 \beta_{15} - 19 \beta_{13} + \cdots - 50 \)
|
| \(\nu^{7}\) | \(=\) |
\( 51 \beta_{19} + 68 \beta_{18} + 54 \beta_{17} - 15 \beta_{15} - 54 \beta_{14} + 99 \beta_{13} + \cdots + 36 \)
|
| \(\nu^{8}\) | \(=\) |
\( 267 \beta_{19} - 120 \beta_{18} - 28 \beta_{17} - 24 \beta_{16} + 147 \beta_{15} - 4 \beta_{14} + \cdots + 304 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 308 \beta_{19} - 488 \beta_{18} - 352 \beta_{17} + 154 \beta_{15} + 352 \beta_{14} - 648 \beta_{13} + \cdots - 323 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 2044 \beta_{19} + 974 \beta_{18} + 274 \beta_{17} + 214 \beta_{16} - 1070 \beta_{15} + 60 \beta_{14} + \cdots - 1940 \)
|
| \(\nu^{11}\) | \(=\) |
\( 1854 \beta_{19} + 3434 \beta_{18} + 2299 \beta_{17} - 1356 \beta_{15} - 2299 \beta_{14} + 4298 \beta_{13} + \cdots + 2590 \)
|
| \(\nu^{12}\) | \(=\) |
\( 15044 \beta_{19} - 7438 \beta_{18} - 2330 \beta_{17} - 1724 \beta_{16} + 7606 \beta_{15} - 606 \beta_{14} + \cdots + 12755 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 11342 \beta_{19} - 23984 \beta_{18} - 15204 \beta_{17} + 11014 \beta_{15} + 15204 \beta_{14} + \cdots - 19642 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 108504 \beta_{19} + 54982 \beta_{18} + 18452 \beta_{17} + 13248 \beta_{16} - 53522 \beta_{15} + \cdots - 85534 \)
|
| \(\nu^{15}\) | \(=\) |
\( 71003 \beta_{19} + 167135 \beta_{18} + 101945 \beta_{17} - 85268 \beta_{15} - 101945 \beta_{14} + \cdots + 144450 \)
|
| \(\nu^{16}\) | \(=\) |
\( 773909 \beta_{19} - 398897 \beta_{18} - 140250 \beta_{17} - 99100 \beta_{16} + 375012 \beta_{15} + \cdots + 581415 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 455154 \beta_{19} - 1164737 \beta_{18} - 691774 \beta_{17} + 640259 \beta_{15} + 691774 \beta_{14} + \cdots - 1043445 \)
|
| \(\nu^{18}\) | \(=\) |
\( - 5484892 \beta_{19} + 2861437 \beta_{18} + 1039156 \beta_{17} + 728470 \beta_{16} - 2623455 \beta_{15} + \cdots - 3989746 \)
|
| \(\nu^{19}\) | \(=\) |
\( 2980455 \beta_{19} + 8124372 \beta_{18} + 4738502 \beta_{17} - 4711075 \beta_{15} - 4738502 \beta_{14} + \cdots + 7456686 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).
| \(n\) | \(171\) | \(677\) |
| \(\chi(n)\) | \(-\beta_{13}\) | \(\beta_{13}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 268.1 |
|
−2.25081 | 1.40490 | + | 1.40490i | 3.06613 | −2.22228 | + | 0.247944i | −3.16216 | − | 3.16216i | 1.27718i | −2.39966 | 0.947480i | 5.00192 | − | 0.558075i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 268.2 | −1.51805 | 0.478298 | + | 0.478298i | 0.304465 | 0.600231 | − | 2.15400i | −0.726078 | − | 0.726078i | − | 2.59488i | 2.57390 | − | 2.54246i | −0.911178 | + | 3.26987i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 268.3 | −1.02262 | −1.97063 | − | 1.97063i | −0.954253 | −1.69584 | − | 1.45744i | 2.01520 | + | 2.01520i | − | 0.963574i | 3.02107 | 4.76674i | 1.73420 | + | 1.49040i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 268.4 | −0.131303 | −0.243172 | − | 0.243172i | −1.98276 | 0.813169 | + | 2.08297i | 0.0319291 | + | 0.0319291i | 2.78137i | 0.522947 | − | 2.88174i | −0.106771 | − | 0.273499i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 268.5 | 0.274809 | 1.67095 | + | 1.67095i | −1.92448 | 1.69883 | + | 1.45395i | 0.459191 | + | 0.459191i | − | 0.386104i | −1.07848 | 2.58414i | 0.466854 | + | 0.399558i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 268.6 | 0.493902 | −0.664960 | − | 0.664960i | −1.75606 | −2.21791 | − | 0.284413i | −0.328425 | − | 0.328425i | 3.67549i | −1.85513 | − | 2.11566i | −1.09543 | − | 0.140472i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 268.7 | 1.58474 | −0.139520 | − | 0.139520i | 0.511395 | 0.0672627 | − | 2.23506i | −0.221103 | − | 0.221103i | 0.548328i | −2.35905 | − | 2.96107i | 0.106594 | − | 3.54198i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 268.8 | 1.83163 | −1.40138 | − | 1.40138i | 1.35488 | 1.45480 | − | 1.69810i | −2.56682 | − | 2.56682i | − | 3.53890i | −1.18163 | 0.927746i | 2.66466 | − | 3.11030i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 268.9 | 2.08794 | 1.94842 | + | 1.94842i | 2.35949 | −0.194361 | + | 2.22760i | 4.06818 | + | 4.06818i | − | 2.91126i | 0.750585 | 4.59268i | −0.405815 | + | 4.65110i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 268.10 | 2.64975 | 0.917096 | + | 0.917096i | 5.02120 | −1.30391 | + | 1.81654i | 2.43008 | + | 2.43008i | 0.112348i | 8.00544 | − | 1.31787i | −3.45504 | + | 4.81339i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 577.1 | −2.25081 | 1.40490 | − | 1.40490i | 3.06613 | −2.22228 | − | 0.247944i | −3.16216 | + | 3.16216i | − | 1.27718i | −2.39966 | − | 0.947480i | 5.00192 | + | 0.558075i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 577.2 | −1.51805 | 0.478298 | − | 0.478298i | 0.304465 | 0.600231 | + | 2.15400i | −0.726078 | + | 0.726078i | 2.59488i | 2.57390 | 2.54246i | −0.911178 | − | 3.26987i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 577.3 | −1.02262 | −1.97063 | + | 1.97063i | −0.954253 | −1.69584 | + | 1.45744i | 2.01520 | − | 2.01520i | 0.963574i | 3.02107 | − | 4.76674i | 1.73420 | − | 1.49040i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 577.4 | −0.131303 | −0.243172 | + | 0.243172i | −1.98276 | 0.813169 | − | 2.08297i | 0.0319291 | − | 0.0319291i | − | 2.78137i | 0.522947 | 2.88174i | −0.106771 | + | 0.273499i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 577.5 | 0.274809 | 1.67095 | − | 1.67095i | −1.92448 | 1.69883 | − | 1.45395i | 0.459191 | − | 0.459191i | 0.386104i | −1.07848 | − | 2.58414i | 0.466854 | − | 0.399558i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 577.6 | 0.493902 | −0.664960 | + | 0.664960i | −1.75606 | −2.21791 | + | 0.284413i | −0.328425 | + | 0.328425i | − | 3.67549i | −1.85513 | 2.11566i | −1.09543 | + | 0.140472i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 577.7 | 1.58474 | −0.139520 | + | 0.139520i | 0.511395 | 0.0672627 | + | 2.23506i | −0.221103 | + | 0.221103i | − | 0.548328i | −2.35905 | 2.96107i | 0.106594 | + | 3.54198i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 577.8 | 1.83163 | −1.40138 | + | 1.40138i | 1.35488 | 1.45480 | + | 1.69810i | −2.56682 | + | 2.56682i | 3.53890i | −1.18163 | − | 0.927746i | 2.66466 | + | 3.11030i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 577.9 | 2.08794 | 1.94842 | − | 1.94842i | 2.35949 | −0.194361 | − | 2.22760i | 4.06818 | − | 4.06818i | 2.91126i | 0.750585 | − | 4.59268i | −0.405815 | − | 4.65110i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 577.10 | 2.64975 | 0.917096 | − | 0.917096i | 5.02120 | −1.30391 | − | 1.81654i | 2.43008 | − | 2.43008i | − | 0.112348i | 8.00544 | 1.31787i | −3.45504 | − | 4.81339i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 65.k | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 845.2.k.e | 20 | |
| 5.c | odd | 4 | 1 | 845.2.f.e | 20 | ||
| 13.b | even | 2 | 1 | 845.2.k.d | 20 | ||
| 13.c | even | 3 | 1 | 65.2.o.a | ✓ | 20 | |
| 13.c | even | 3 | 1 | 845.2.o.e | 20 | ||
| 13.d | odd | 4 | 1 | 845.2.f.d | 20 | ||
| 13.d | odd | 4 | 1 | 845.2.f.e | 20 | ||
| 13.e | even | 6 | 1 | 845.2.o.f | 20 | ||
| 13.e | even | 6 | 1 | 845.2.o.g | 20 | ||
| 13.f | odd | 12 | 1 | 65.2.t.a | yes | 20 | |
| 13.f | odd | 12 | 1 | 845.2.t.e | 20 | ||
| 13.f | odd | 12 | 1 | 845.2.t.f | 20 | ||
| 13.f | odd | 12 | 1 | 845.2.t.g | 20 | ||
| 39.i | odd | 6 | 1 | 585.2.cf.a | 20 | ||
| 39.k | even | 12 | 1 | 585.2.dp.a | 20 | ||
| 65.f | even | 4 | 1 | 845.2.k.d | 20 | ||
| 65.h | odd | 4 | 1 | 845.2.f.d | 20 | ||
| 65.k | even | 4 | 1 | inner | 845.2.k.e | 20 | |
| 65.n | even | 6 | 1 | 325.2.s.b | 20 | ||
| 65.o | even | 12 | 1 | 65.2.o.a | ✓ | 20 | |
| 65.o | even | 12 | 1 | 845.2.o.e | 20 | ||
| 65.q | odd | 12 | 1 | 65.2.t.a | yes | 20 | |
| 65.q | odd | 12 | 1 | 325.2.x.b | 20 | ||
| 65.q | odd | 12 | 1 | 845.2.t.f | 20 | ||
| 65.r | odd | 12 | 1 | 845.2.t.e | 20 | ||
| 65.r | odd | 12 | 1 | 845.2.t.g | 20 | ||
| 65.s | odd | 12 | 1 | 325.2.x.b | 20 | ||
| 65.t | even | 12 | 1 | 325.2.s.b | 20 | ||
| 65.t | even | 12 | 1 | 845.2.o.f | 20 | ||
| 65.t | even | 12 | 1 | 845.2.o.g | 20 | ||
| 195.bl | even | 12 | 1 | 585.2.dp.a | 20 | ||
| 195.bn | odd | 12 | 1 | 585.2.cf.a | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 65.2.o.a | ✓ | 20 | 13.c | even | 3 | 1 | |
| 65.2.o.a | ✓ | 20 | 65.o | even | 12 | 1 | |
| 65.2.t.a | yes | 20 | 13.f | odd | 12 | 1 | |
| 65.2.t.a | yes | 20 | 65.q | odd | 12 | 1 | |
| 325.2.s.b | 20 | 65.n | even | 6 | 1 | ||
| 325.2.s.b | 20 | 65.t | even | 12 | 1 | ||
| 325.2.x.b | 20 | 65.q | odd | 12 | 1 | ||
| 325.2.x.b | 20 | 65.s | odd | 12 | 1 | ||
| 585.2.cf.a | 20 | 39.i | odd | 6 | 1 | ||
| 585.2.cf.a | 20 | 195.bn | odd | 12 | 1 | ||
| 585.2.dp.a | 20 | 39.k | even | 12 | 1 | ||
| 585.2.dp.a | 20 | 195.bl | even | 12 | 1 | ||
| 845.2.f.d | 20 | 13.d | odd | 4 | 1 | ||
| 845.2.f.d | 20 | 65.h | odd | 4 | 1 | ||
| 845.2.f.e | 20 | 5.c | odd | 4 | 1 | ||
| 845.2.f.e | 20 | 13.d | odd | 4 | 1 | ||
| 845.2.k.d | 20 | 13.b | even | 2 | 1 | ||
| 845.2.k.d | 20 | 65.f | even | 4 | 1 | ||
| 845.2.k.e | 20 | 1.a | even | 1 | 1 | trivial | |
| 845.2.k.e | 20 | 65.k | even | 4 | 1 | inner | |
| 845.2.o.e | 20 | 13.c | even | 3 | 1 | ||
| 845.2.o.e | 20 | 65.o | even | 12 | 1 | ||
| 845.2.o.f | 20 | 13.e | even | 6 | 1 | ||
| 845.2.o.f | 20 | 65.t | even | 12 | 1 | ||
| 845.2.o.g | 20 | 13.e | even | 6 | 1 | ||
| 845.2.o.g | 20 | 65.t | even | 12 | 1 | ||
| 845.2.t.e | 20 | 13.f | odd | 12 | 1 | ||
| 845.2.t.e | 20 | 65.r | odd | 12 | 1 | ||
| 845.2.t.f | 20 | 13.f | odd | 12 | 1 | ||
| 845.2.t.f | 20 | 65.q | odd | 12 | 1 | ||
| 845.2.t.g | 20 | 13.f | odd | 12 | 1 | ||
| 845.2.t.g | 20 | 65.r | odd | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} - 4T_{2}^{9} - 5T_{2}^{8} + 34T_{2}^{7} - 9T_{2}^{6} - 80T_{2}^{5} + 51T_{2}^{4} + 52T_{2}^{3} - 37T_{2}^{2} + 2T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{10} - 4 T^{9} - 5 T^{8} + \cdots + 1)^{2} \)
|
| $3$ |
\( T^{20} - 4 T^{19} + \cdots + 16 \)
|
| $5$ |
\( T^{20} + 6 T^{19} + \cdots + 9765625 \)
|
| $7$ |
\( T^{20} + 52 T^{18} + \cdots + 64 \)
|
| $11$ |
\( T^{20} - 8 T^{19} + \cdots + 256 \)
|
| $13$ |
\( T^{20} \)
|
| $17$ |
\( T^{20} - 14 T^{19} + \cdots + 1168561 \)
|
| $19$ |
\( T^{20} + \cdots + 1583721616 \)
|
| $23$ |
\( T^{20} + 8 T^{19} + \cdots + 144 \)
|
| $29$ |
\( T^{20} + \cdots + 206213167449 \)
|
| $31$ |
\( T^{20} - 104 T^{17} + \cdots + 2166784 \)
|
| $37$ |
\( T^{20} + \cdots + 4508182449 \)
|
| $41$ |
\( T^{20} + \cdots + 3748255729 \)
|
| $43$ |
\( T^{20} + \cdots + 1370772640000 \)
|
| $47$ |
\( T^{20} + \cdots + 807469056 \)
|
| $53$ |
\( T^{20} + \cdots + 2978634160384 \)
|
| $59$ |
\( T^{20} + 8 T^{19} + \cdots + 33856 \)
|
| $61$ |
\( (T^{10} - 16 T^{9} + \cdots + 909097)^{2} \)
|
| $67$ |
\( (T^{10} - 58 T^{9} + \cdots + 3934324)^{2} \)
|
| $71$ |
\( T^{20} + \cdots + 11\!\cdots\!44 \)
|
| $73$ |
\( (T^{10} - 36 T^{9} + \cdots + 253113232)^{2} \)
|
| $79$ |
\( T^{20} + \cdots + 75\!\cdots\!36 \)
|
| $83$ |
\( T^{20} + \cdots + 11512611864576 \)
|
| $89$ |
\( T^{20} + \cdots + 329648222500 \)
|
| $97$ |
\( (T^{10} - 22 T^{9} + \cdots + 169592356)^{2} \)
|
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