Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2365,2,Mod(1,2365)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2365, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2365.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2365 = 5 \cdot 11 \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2365.a (trivial) |
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: | yes |
| Analytic conductor: | \(18.8846200780\) |
| Analytic rank: | \(1\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 7 x^{15} + 88 x^{13} - 91 x^{12} - 464 x^{11} + 590 x^{10} + 1345 x^{9} - 1609 x^{8} + \cdots + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 7 x^{15} + 88 x^{13} - 91 x^{12} - 464 x^{11} + 590 x^{10} + 1345 x^{9} - 1609 x^{8} + \cdots + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 117 \nu^{15} - 5475 \nu^{14} + 34352 \nu^{13} - 10888 \nu^{12} - 361711 \nu^{11} + 528612 \nu^{10} + \cdots - 33004 ) / 42604 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 117 \nu^{15} + 5475 \nu^{14} - 34352 \nu^{13} + 10888 \nu^{12} + 361711 \nu^{11} - 528612 \nu^{10} + \cdots + 160816 ) / 42604 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 328 \nu^{15} + 2786 \nu^{14} - 2993 \nu^{13} - 32381 \nu^{12} + 78287 \nu^{11} + 128565 \nu^{10} + \cdots - 67332 ) / 10651 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 1643 \nu^{15} + 15709 \nu^{14} - 24312 \nu^{13} - 170092 \nu^{12} + 470897 \nu^{11} + \cdots - 320520 ) / 42604 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 836 \nu^{15} + 5802 \nu^{14} - 2303 \nu^{13} - 55385 \nu^{12} + 67438 \nu^{11} + 203379 \nu^{10} + \cdots - 7693 ) / 10651 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 4539 \nu^{15} - 28877 \nu^{14} - 13168 \nu^{13} + 361952 \nu^{12} - 221189 \nu^{11} - 1856388 \nu^{10} + \cdots + 162688 ) / 42604 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 5969 \nu^{15} - 46089 \nu^{14} + 39822 \nu^{13} + 451364 \nu^{12} - 855079 \nu^{11} - 1693866 \nu^{10} + \cdots + 138008 ) / 42604 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 3091 \nu^{15} + 22293 \nu^{14} - 5572 \nu^{13} - 266022 \nu^{12} + 346043 \nu^{11} + \cdots - 156396 ) / 21302 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 3624 \nu^{15} - 29483 \nu^{14} + 33069 \nu^{13} + 282694 \nu^{12} - 642344 \nu^{11} - 981977 \nu^{10} + \cdots + 36814 ) / 21302 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 4007 \nu^{15} + 32931 \nu^{14} - 40558 \nu^{13} - 294754 \nu^{12} + 693395 \nu^{11} + \cdots - 106474 ) / 21302 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 5643 \nu^{15} - 44489 \nu^{14} + 39510 \nu^{13} + 456394 \nu^{12} - 897223 \nu^{11} - 1769558 \nu^{10} + \cdots + 81218 ) / 21302 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 22329 \nu^{15} - 182971 \nu^{14} + 221060 \nu^{13} + 1684328 \nu^{12} - 4043667 \nu^{11} + \cdots + 98468 ) / 42604 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 22575 \nu^{15} - 179735 \nu^{14} + 178038 \nu^{13} + 1780508 \nu^{12} - 3764213 \nu^{11} + \cdots + 564356 ) / 42604 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 2\beta_{2} + 6\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{10} + \beta_{8} - \beta_{6} + 3\beta_{4} + 3\beta_{3} + 10\beta_{2} + 13\beta _1 + 17 \)
|
| \(\nu^{5}\) | \(=\) |
\( 3\beta_{10} + 4\beta_{8} + \beta_{7} - 3\beta_{6} + \beta_{5} + 13\beta_{4} + 14\beta_{3} + 28\beta_{2} + 49\beta _1 + 37 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{15} - \beta_{14} + \beta_{11} + 15 \beta_{10} + 17 \beta_{8} + 3 \beta_{7} - 14 \beta_{6} + \cdots + 137 \)
|
| \(\nu^{7}\) | \(=\) |
\( 5 \beta_{15} - 5 \beta_{14} - \beta_{13} + 5 \beta_{11} + 48 \beta_{10} + 59 \beta_{8} + 14 \beta_{7} + \cdots + 385 \)
|
| \(\nu^{8}\) | \(=\) |
\( 26 \beta_{15} - 25 \beta_{14} - 5 \beta_{13} + \beta_{12} + 25 \beta_{11} + 177 \beta_{10} - \beta_{9} + \cdots + 1272 \)
|
| \(\nu^{9}\) | \(=\) |
\( 105 \beta_{15} - 100 \beta_{14} - 27 \beta_{13} + 7 \beta_{12} + 101 \beta_{11} + 572 \beta_{10} + \cdots + 3878 \)
|
| \(\nu^{10}\) | \(=\) |
\( 418 \beta_{15} - 389 \beta_{14} - 112 \beta_{13} + 42 \beta_{12} + 392 \beta_{11} + 1924 \beta_{10} + \cdots + 12428 \)
|
| \(\nu^{11}\) | \(=\) |
\( 1543 \beta_{15} - 1418 \beta_{14} - 460 \beta_{13} + 205 \beta_{12} + 1435 \beta_{11} + 6195 \beta_{10} + \cdots + 38914 \)
|
| \(\nu^{12}\) | \(=\) |
\( 5611 \beta_{15} - 5071 \beta_{14} - 1748 \beta_{13} + 927 \beta_{12} + 5118 \beta_{11} + 20156 \beta_{10} + \cdots + 123786 \)
|
| \(\nu^{13}\) | \(=\) |
\( 19814 \beta_{15} - 17679 \beta_{14} - 6538 \beta_{13} + 3897 \beta_{12} + 17843 \beta_{11} + \cdots + 391200 \)
|
| \(\nu^{14}\) | \(=\) |
\( 69234 \beta_{15} - 60934 \beta_{14} - 23711 \beta_{13} + 15705 \beta_{12} + 61309 \beta_{11} + \cdots + 1242738 \)
|
| \(\nu^{15}\) | \(=\) |
\( 238680 \beta_{15} - 207564 \beta_{14} - 84939 \beta_{13} + 61023 \beta_{12} + 208398 \beta_{11} + \cdots + 3941057 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.78586 | −3.14052 | 5.76099 | −1.00000 | 8.74904 | 0.873147 | −10.4776 | 6.86287 | 2.78586 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.74865 | 2.12364 | 5.55510 | −1.00000 | −5.83715 | −0.343906 | −9.77176 | 1.50984 | 2.74865 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.42839 | −0.652663 | 3.89709 | −1.00000 | 1.58492 | −4.62273 | −4.60688 | −2.57403 | 2.42839 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.28702 | −1.33416 | 3.23045 | −1.00000 | 3.05125 | 3.08714 | −2.81407 | −1.22002 | 2.28702 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −2.06770 | −2.89226 | 2.27538 | −1.00000 | 5.98033 | −4.36093 | −0.569407 | 5.36519 | 2.06770 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.91363 | 1.76460 | 1.66197 | −1.00000 | −3.37678 | 3.06888 | 0.646871 | 0.113798 | 1.91363 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.33874 | 0.167181 | −0.207774 | −1.00000 | −0.223812 | −1.42308 | 2.95564 | −2.97205 | 1.33874 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.10702 | −1.33218 | −0.774513 | −1.00000 | 1.47475 | 2.32719 | 3.07143 | −1.22529 | 1.10702 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −0.402629 | −1.30002 | −1.83789 | −1.00000 | 0.523424 | 0.0495835 | 1.54524 | −1.30996 | 0.402629 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.0410109 | −1.07697 | −1.99832 | −1.00000 | −0.0441677 | −3.68401 | −0.163975 | −1.84013 | −0.0410109 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0.348156 | −2.68165 | −1.87879 | −1.00000 | −0.933635 | −3.86416 | −1.35042 | 4.19127 | −0.348156 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0.951395 | 1.28017 | −1.09485 | −1.00000 | 1.21795 | 0.614407 | −2.94442 | −1.36116 | −0.951395 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 1.06483 | 2.53191 | −0.866142 | −1.00000 | 2.69605 | −1.97360 | −3.05195 | 3.41056 | −1.06483 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 1.28912 | −2.93657 | −0.338178 | −1.00000 | −3.78558 | 3.72295 | −3.01418 | 5.62345 | −1.28912 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 2.17270 | −1.88662 | 2.72061 | −1.00000 | −4.09906 | 0.654994 | 1.56568 | 0.559339 | −2.17270 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.21243 | 1.36613 | 2.89485 | −1.00000 | 3.02247 | −5.12586 | 1.97979 | −1.13368 | −2.21243 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(11\) | \(-1\) |
| \(43\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2365.2.a.j | ✓ | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2365.2.a.j | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2365))\):
|
\( T_{2}^{16} + 9 T_{2}^{15} + 15 T_{2}^{14} - 87 T_{2}^{13} - 312 T_{2}^{12} + 121 T_{2}^{11} + 1635 T_{2}^{10} + \cdots - 9 \)
|
|
\( T_{3}^{16} + 10 T_{3}^{15} + 19 T_{3}^{14} - 117 T_{3}^{13} - 488 T_{3}^{12} + 133 T_{3}^{11} + \cdots + 608 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + 9 T^{15} + \cdots - 9 \)
|
| $3$ |
\( T^{16} + 10 T^{15} + \cdots + 608 \)
|
| $5$ |
\( (T + 1)^{16} \)
|
| $7$ |
\( T^{16} + 11 T^{15} + \cdots + 2032 \)
|
| $11$ |
\( (T - 1)^{16} \)
|
| $13$ |
\( T^{16} + 6 T^{15} + \cdots + 27136 \)
|
| $17$ |
\( T^{16} + 17 T^{15} + \cdots + 384 \)
|
| $19$ |
\( T^{16} - 8 T^{15} + \cdots - 3760000 \)
|
| $23$ |
\( T^{16} + \cdots + 2327022048 \)
|
| $29$ |
\( T^{16} + 15 T^{15} + \cdots - 4684254 \)
|
| $31$ |
\( T^{16} + \cdots + 911845160000 \)
|
| $37$ |
\( T^{16} + \cdots - 57876258304 \)
|
| $41$ |
\( T^{16} + \cdots + 988617024 \)
|
| $43$ |
\( (T - 1)^{16} \)
|
| $47$ |
\( T^{16} + \cdots + 14028106056 \)
|
| $53$ |
\( T^{16} + \cdots - 1414987764 \)
|
| $59$ |
\( T^{16} + \cdots + 176358912 \)
|
| $61$ |
\( T^{16} + \cdots + 2109868378 \)
|
| $67$ |
\( T^{16} + \cdots - 100452249728 \)
|
| $71$ |
\( T^{16} + \cdots + 10619939547744 \)
|
| $73$ |
\( T^{16} + \cdots + 6924801251264 \)
|
| $79$ |
\( T^{16} + \cdots - 24147397307024 \)
|
| $83$ |
\( T^{16} + \cdots + 16524785568 \)
|
| $89$ |
\( T^{16} + \cdots + 141202409514816 \)
|
| $97$ |
\( T^{16} + 14 T^{15} + \cdots - 23107268 \)
|
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