Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1805,4,Mod(1,1805)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1805, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1805.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1805 = 5 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1805.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: | \(106.498447560\) |
| 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: |
\( x^{10} - 3 x^{9} - 57 x^{8} + 148 x^{7} + 1138 x^{6} - 2387 x^{5} - 9690 x^{4} + 14326 x^{3} + \cdots - 48336 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 95) |
| 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
\( x^{10} - 3 x^{9} - 57 x^{8} + 148 x^{7} + 1138 x^{6} - 2387 x^{5} - 9690 x^{4} + 14326 x^{3} + \cdots - 48336 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 213 \nu^{9} + 917 \nu^{8} + 10179 \nu^{7} - 42118 \nu^{6} - 147186 \nu^{5} + 588687 \nu^{4} + \cdots + 2462888 ) / 56200 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 213 \nu^{9} - 917 \nu^{8} - 10179 \nu^{7} + 42118 \nu^{6} + 147186 \nu^{5} - 588687 \nu^{4} + \cdots - 2631488 ) / 56200 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 139 \nu^{9} - 981 \nu^{8} - 5297 \nu^{7} + 47604 \nu^{6} + 40128 \nu^{5} - 728451 \nu^{4} + \cdots - 5175884 ) / 28100 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 59 \nu^{9} + 473 \nu^{8} + 2495 \nu^{7} - 23764 \nu^{6} - 27266 \nu^{5} + 376861 \nu^{4} + \cdots + 2747160 ) / 11240 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 21 \nu^{9} - 35 \nu^{8} - 1150 \nu^{7} + 1481 \nu^{6} + 20721 \nu^{5} - 19127 \nu^{4} - 135326 \nu^{3} + \cdots - 90700 ) / 2810 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 517 \nu^{9} - 2173 \nu^{8} - 24311 \nu^{7} + 102362 \nu^{6} + 331054 \nu^{5} - 1490303 \nu^{4} + \cdots - 8906992 ) / 56200 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 633 \nu^{9} - 1617 \nu^{8} - 33179 \nu^{7} + 71738 \nu^{6} + 561606 \nu^{5} - 915027 \nu^{4} + \cdots + 1511712 ) / 56200 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 19\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{9} - \beta_{7} + \beta_{3} + 27\beta_{2} + 4\beta _1 + 221 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{9} - 5\beta_{8} + \beta_{7} + 4\beta_{5} + 31\beta_{4} + 32\beta_{3} + 12\beta_{2} + 412\beta _1 + 116 \)
|
| \(\nu^{6}\) | \(=\) |
\( 41 \beta_{9} + 2 \beta_{8} - 33 \beta_{7} - 4 \beta_{6} + 6 \beta_{5} - 31 \beta_{4} + 44 \beta_{3} + \cdots + 4796 \)
|
| \(\nu^{7}\) | \(=\) |
\( 99 \beta_{9} - 205 \beta_{8} + 36 \beta_{7} + 20 \beta_{6} + 210 \beta_{5} + 754 \beta_{4} + 868 \beta_{3} + \cdots + 4088 \)
|
| \(\nu^{8}\) | \(=\) |
\( 1312 \beta_{9} + 115 \beta_{8} - 855 \beta_{7} - 140 \beta_{6} + 396 \beta_{5} - 1714 \beta_{4} + \cdots + 110871 \)
|
| \(\nu^{9}\) | \(=\) |
\( 3654 \beta_{9} - 6242 \beta_{8} + 1110 \beta_{7} + 1144 \beta_{6} + 7790 \beta_{5} + 16476 \beta_{4} + \cdots + 132594 \)
|
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 |
|
−4.80335 | −3.28547 | 15.0722 | −5.00000 | 15.7813 | −11.3286 | −33.9701 | −16.2057 | 24.0167 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.19346 | 7.45528 | 9.58510 | −5.00000 | −31.2634 | 33.0991 | −6.64705 | 28.5812 | 20.9673 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.37046 | −7.40326 | −2.38094 | −5.00000 | 17.5491 | 9.37372 | 24.6076 | 27.8082 | 11.8523 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.19968 | 8.75512 | −3.16141 | −5.00000 | −19.2584 | −33.5680 | 24.5515 | 49.6521 | 10.9984 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.15104 | 0.368561 | −6.67511 | −5.00000 | −0.424227 | −2.58808 | 16.8916 | −26.8642 | 5.75519 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.87520 | 1.27242 | −4.48363 | −5.00000 | 2.38604 | 29.8625 | −23.4093 | −25.3809 | −9.37599 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.71312 | −8.60185 | −0.638955 | −5.00000 | −23.3379 | 10.4999 | −23.4386 | 46.9918 | −13.5656 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 3.18057 | 0.0718266 | 2.11603 | −5.00000 | 0.228450 | −23.3899 | −18.7144 | −26.9948 | −15.9029 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 4.78143 | 9.23102 | 14.8621 | −5.00000 | 44.1375 | 5.17574 | 32.8105 | 58.2118 | −23.9071 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 5.16766 | −2.86365 | 18.7047 | −5.00000 | −14.7984 | −14.1365 | 55.3182 | −18.7995 | −25.8383 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(19\) | \( +1 \) |
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 | 1805.4.a.r | 10 | |
| 19.b | odd | 2 | 1 | 1805.4.a.p | 10 | ||
| 19.c | even | 3 | 2 | 95.4.e.c | ✓ | 20 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 95.4.e.c | ✓ | 20 | 19.c | even | 3 | 2 | |
| 1805.4.a.p | 10 | 19.b | odd | 2 | 1 | ||
| 1805.4.a.r | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1805))\):
|
\( T_{2}^{10} - 3 T_{2}^{9} - 57 T_{2}^{8} + 148 T_{2}^{7} + 1138 T_{2}^{6} - 2387 T_{2}^{5} - 9690 T_{2}^{4} + \cdots - 48336 \)
|
|
\( T_{3}^{10} - 5 T_{3}^{9} - 171 T_{3}^{8} + 639 T_{3}^{7} + 9634 T_{3}^{6} - 19528 T_{3}^{5} + \cdots + 12160 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 3 T^{9} + \cdots - 48336 \)
|
| $3$ |
\( T^{10} - 5 T^{9} + \cdots + 12160 \)
|
| $5$ |
\( (T + 5)^{10} \)
|
| $7$ |
\( T^{10} + \cdots - 163855792416 \)
|
| $11$ |
\( T^{10} + \cdots - 20\!\cdots\!39 \)
|
| $13$ |
\( T^{10} + \cdots + 6382352340208 \)
|
| $17$ |
\( T^{10} + \cdots - 14\!\cdots\!92 \)
|
| $19$ |
\( T^{10} \)
|
| $23$ |
\( T^{10} + \cdots + 36\!\cdots\!52 \)
|
| $29$ |
\( T^{10} + \cdots + 17\!\cdots\!16 \)
|
| $31$ |
\( T^{10} + \cdots + 61\!\cdots\!92 \)
|
| $37$ |
\( T^{10} + \cdots + 38\!\cdots\!48 \)
|
| $41$ |
\( T^{10} + \cdots - 11\!\cdots\!12 \)
|
| $43$ |
\( T^{10} + \cdots - 11\!\cdots\!56 \)
|
| $47$ |
\( T^{10} + \cdots - 61\!\cdots\!96 \)
|
| $53$ |
\( T^{10} + \cdots - 16\!\cdots\!76 \)
|
| $59$ |
\( T^{10} + \cdots - 22\!\cdots\!00 \)
|
| $61$ |
\( T^{10} + \cdots - 41\!\cdots\!24 \)
|
| $67$ |
\( T^{10} + \cdots + 25\!\cdots\!20 \)
|
| $71$ |
\( T^{10} + \cdots + 94\!\cdots\!76 \)
|
| $73$ |
\( T^{10} + \cdots - 14\!\cdots\!80 \)
|
| $79$ |
\( T^{10} + \cdots - 23\!\cdots\!04 \)
|
| $83$ |
\( T^{10} + \cdots - 24\!\cdots\!28 \)
|
| $89$ |
\( T^{10} + \cdots - 28\!\cdots\!36 \)
|
| $97$ |
\( T^{10} + \cdots - 20\!\cdots\!68 \)
|
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