Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1840,4,Mod(1,1840)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1840, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1840.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1840 = 2^{4} \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1840.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: | \(108.563514411\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 5 x^{9} - 192 x^{8} + 762 x^{7} + 12246 x^{6} - 33828 x^{5} - 298243 x^{4} + 383603 x^{3} + \cdots + 57408 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{7} \) |
| Twist minimal: | no (minimal twist has level 920) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 5 x^{9} - 192 x^{8} + 762 x^{7} + 12246 x^{6} - 33828 x^{5} - 298243 x^{4} + 383603 x^{3} + \cdots + 57408 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 40 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 13492675 \nu^{9} - 90064369 \nu^{8} - 2424760468 \nu^{7} + 14223759494 \nu^{6} + \cdots - 13787240781840 ) / 189638109072 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 133928635 \nu^{9} - 932238831 \nu^{8} - 23988456900 \nu^{7} + 148152752502 \nu^{6} + \cdots + 46471969427424 ) / 1706742981648 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 74568380 \nu^{9} + 381349725 \nu^{8} + 14353347492 \nu^{7} - 60353430060 \nu^{6} + \cdots + 7611310191408 ) / 853371490824 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 73632521 \nu^{9} - 380531223 \nu^{8} - 13939762884 \nu^{7} + 57963763818 \nu^{6} + \cdots + 30379739386224 ) / 568914327216 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 157136465 \nu^{9} - 873082230 \nu^{8} - 30030328044 \nu^{7} + 136872598350 \nu^{6} + \cdots + 44170193069544 ) / 853371490824 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 88362863 \nu^{9} + 431383173 \nu^{8} + 17149709286 \nu^{7} - 66106410600 \nu^{6} + \cdots - 61611158517060 ) / 426685745412 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 230763412 \nu^{9} + 1153582707 \nu^{8} + 44674875612 \nu^{7} - 177340372980 \nu^{6} + \cdots - 98477372031504 ) / 853371490824 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 40 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{9} - \beta_{8} - 2\beta_{7} - 2\beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + 72\beta _1 + 36 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{9} - 10\beta_{8} - 8\beta_{7} - 7\beta_{6} + 11\beta_{4} - 4\beta_{3} + 99\beta_{2} + 145\beta _1 + 2829 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 157 \beta_{9} - 91 \beta_{8} - 266 \beta_{7} - 245 \beta_{6} - 9 \beta_{5} + 160 \beta_{4} + \cdots + 5097 \)
|
| \(\nu^{6}\) | \(=\) |
\( 34 \beta_{9} - 1390 \beta_{8} - 1430 \beta_{7} - 1243 \beta_{6} - 555 \beta_{5} + 1757 \beta_{4} + \cdots + 232185 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 17788 \beta_{9} - 7819 \beta_{8} - 30512 \beta_{7} - 24080 \beta_{6} - 3465 \beta_{5} + 20305 \beta_{4} + \cdots + 653064 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 7706 \beta_{9} - 156058 \beta_{8} - 199196 \beta_{7} - 151990 \beta_{6} - 110946 \beta_{5} + \cdots + 20541678 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 1822591 \beta_{9} - 746035 \beta_{8} - 3374474 \beta_{7} - 2265212 \beta_{6} - 686178 \beta_{5} + \cdots + 79069830 \)
|
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 |
|
0 | −10.3184 | 0 | 5.00000 | 0 | −13.9200 | 0 | 79.4692 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −8.50847 | 0 | 5.00000 | 0 | −14.9751 | 0 | 45.3940 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −6.17442 | 0 | 5.00000 | 0 | 21.9366 | 0 | 11.1235 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −4.57895 | 0 | 5.00000 | 0 | −22.7390 | 0 | −6.03324 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.0875767 | 0 | 5.00000 | 0 | 28.3995 | 0 | −26.9923 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0.292216 | 0 | 5.00000 | 0 | −23.7015 | 0 | −26.9146 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 2.78233 | 0 | 5.00000 | 0 | 11.8146 | 0 | −19.2586 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 4.67252 | 0 | 5.00000 | 0 | −26.1910 | 0 | −5.16760 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 7.60722 | 0 | 5.00000 | 0 | 20.6846 | 0 | 30.8698 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 9.13838 | 0 | 5.00000 | 0 | 4.69126 | 0 | 56.5099 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(-1\) |
| \(23\) | \(-1\) |
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 | 1840.4.a.ba | 10 | |
| 4.b | odd | 2 | 1 | 920.4.a.h | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 920.4.a.h | ✓ | 10 | 4.b | odd | 2 | 1 | |
| 1840.4.a.ba | 10 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1840))\):
|
\( T_{3}^{10} + 5 T_{3}^{9} - 192 T_{3}^{8} - 762 T_{3}^{7} + 12246 T_{3}^{6} + 33828 T_{3}^{5} + \cdots + 57408 \)
|
|
\( T_{7}^{10} + 14 T_{7}^{9} - 1932 T_{7}^{8} - 26286 T_{7}^{7} + 1329984 T_{7}^{6} + 16807960 T_{7}^{5} + \cdots - 2101566628800 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} \)
|
| $3$ |
\( T^{10} + 5 T^{9} + \cdots + 57408 \)
|
| $5$ |
\( (T - 5)^{10} \)
|
| $7$ |
\( T^{10} + \cdots - 2101566628800 \)
|
| $11$ |
\( T^{10} + \cdots + 80040845736576 \)
|
| $13$ |
\( T^{10} + \cdots - 923318227263000 \)
|
| $17$ |
\( T^{10} + \cdots - 83\!\cdots\!48 \)
|
| $19$ |
\( T^{10} + \cdots + 65\!\cdots\!48 \)
|
| $23$ |
\( (T - 23)^{10} \)
|
| $29$ |
\( T^{10} + \cdots + 25\!\cdots\!64 \)
|
| $31$ |
\( T^{10} + \cdots - 90\!\cdots\!16 \)
|
| $37$ |
\( T^{10} + \cdots - 71\!\cdots\!28 \)
|
| $41$ |
\( T^{10} + \cdots - 66\!\cdots\!14 \)
|
| $43$ |
\( T^{10} + \cdots + 42\!\cdots\!00 \)
|
| $47$ |
\( T^{10} + \cdots - 34\!\cdots\!68 \)
|
| $53$ |
\( T^{10} + \cdots + 10\!\cdots\!32 \)
|
| $59$ |
\( T^{10} + \cdots - 43\!\cdots\!92 \)
|
| $61$ |
\( T^{10} + \cdots - 21\!\cdots\!68 \)
|
| $67$ |
\( T^{10} + \cdots + 16\!\cdots\!24 \)
|
| $71$ |
\( T^{10} + \cdots + 79\!\cdots\!88 \)
|
| $73$ |
\( T^{10} + \cdots + 41\!\cdots\!12 \)
|
| $79$ |
\( T^{10} + \cdots - 30\!\cdots\!68 \)
|
| $83$ |
\( T^{10} + \cdots - 83\!\cdots\!12 \)
|
| $89$ |
\( T^{10} + \cdots - 56\!\cdots\!08 \)
|
| $97$ |
\( T^{10} + \cdots + 10\!\cdots\!96 \)
|
| show more | |
| show less |