Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1849,4,Mod(1,1849)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1849, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1849.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1849 = 43^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1849.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: | \(109.094531601\) |
| 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: |
\( x^{10} - 62x^{8} + 1289x^{6} - 11252x^{4} + 39376x^{2} - 35688 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 62x^{8} + 1289x^{6} - 11252x^{4} + 39376x^{2} - 35688 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -71\nu^{8} + 2915\nu^{6} - 22388\nu^{4} - 19240\nu^{2} + 6504 ) / 63744 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -35\nu^{9} + 2447\nu^{7} - 54692\nu^{5} + 407096\nu^{3} - 633336\nu ) / 31872 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 47\nu^{9} - 2603\nu^{7} + 43924\nu^{5} - 264984\nu^{3} + 441304\nu ) / 21248 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -157\nu^{9} + 8017\nu^{7} - 110332\nu^{5} + 336328\nu^{3} + 752376\nu ) / 63744 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -47\nu^{9} + 2603\nu^{7} - 43924\nu^{5} + 286232\nu^{3} - 866264\nu ) / 21248 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 149\nu^{8} - 7913\nu^{6} + 121052\nu^{4} - 565640\nu^{2} + 317640 ) / 31872 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 47\nu^{8} - 2603\nu^{6} + 43924\nu^{4} - 264984\nu^{2} + 420056 ) / 10624 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( -149\nu^{8} + 7913\nu^{6} - 121052\nu^{4} + 597512\nu^{2} - 731976 ) / 31872 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{9} + \beta_{7} + 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{4} + 20\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 38\beta_{9} + 8\beta_{8} + 29\beta_{7} - 6\beta_{2} + 268 \)
|
| \(\nu^{5}\) | \(=\) |
\( 38\beta_{6} + 9\beta_{5} + 51\beta_{4} + 6\beta_{3} + 503\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 1230\beta_{9} + 330\beta_{8} + 841\beta_{7} - 322\beta_{2} + 6852 \)
|
| \(\nu^{7}\) | \(=\) |
\( 1230\beta_{6} + 389\beta_{5} + 1823\beta_{4} + 322\beta_{3} + 14091\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 38246\beta_{9} + 11026\beta_{8} + 25113\beta_{7} - 12226\beta_{2} + 193380 \)
|
| \(\nu^{9}\) | \(=\) |
\( 38246\beta_{6} + 13133\beta_{5} + 59391\beta_{4} + 12226\beta_{3} + 413691\beta_1 \)
|
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 |
|
−5.51834 | −3.84074 | 22.4521 | −10.9281 | 21.1945 | −14.0538 | −79.7518 | −12.2488 | 60.3051 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −3.79990 | 3.62572 | 6.43924 | 1.22749 | −13.7774 | 33.4949 | 5.93074 | −13.8541 | −4.66434 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −3.05512 | −4.99167 | 1.33374 | 15.7813 | 15.2501 | −15.1156 | 20.3662 | −2.08325 | −48.2136 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.53388 | 6.96727 | −1.57947 | −11.2250 | −17.6542 | −27.5005 | 24.2732 | 21.5429 | 28.4428 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.16377 | −8.58157 | −6.64564 | −19.8751 | 9.98696 | 5.27515 | 17.0441 | 46.6433 | 23.1301 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.16377 | 8.58157 | −6.64564 | 19.8751 | 9.98696 | −5.27515 | −17.0441 | 46.6433 | 23.1301 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.53388 | −6.96727 | −1.57947 | 11.2250 | −17.6542 | 27.5005 | −24.2732 | 21.5429 | 28.4428 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 3.05512 | 4.99167 | 1.33374 | −15.7813 | 15.2501 | 15.1156 | −20.3662 | −2.08325 | −48.2136 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 3.79990 | −3.62572 | 6.43924 | −1.22749 | −13.7774 | −33.4949 | −5.93074 | −13.8541 | −4.66434 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 5.51834 | 3.84074 | 22.4521 | 10.9281 | 21.1945 | 14.0538 | 79.7518 | −12.2488 | 60.3051 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(43\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 43.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1849.4.a.e | ✓ | 10 |
| 43.b | odd | 2 | 1 | inner | 1849.4.a.e | ✓ | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1849.4.a.e | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 1849.4.a.e | ✓ | 10 | 43.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} - 62T_{2}^{8} + 1289T_{2}^{6} - 11252T_{2}^{4} + 39376T_{2}^{2} - 35688 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1849))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 62 T^{8} + \cdots - 35688 \)
|
| $3$ |
\( T^{10} - 175 T^{8} + \cdots - 17272992 \)
|
| $5$ |
\( T^{10} + \cdots - 2230500000 \)
|
| $7$ |
\( T^{10} + \cdots - 1065475803648 \)
|
| $11$ |
\( (T^{5} + 9 T^{4} + \cdots + 77102212)^{2} \)
|
| $13$ |
\( (T^{5} - 83 T^{4} + \cdots - 1633500)^{2} \)
|
| $17$ |
\( (T^{5} - 178 T^{4} + \cdots + 996296125)^{2} \)
|
| $19$ |
\( T^{10} + \cdots - 28\!\cdots\!00 \)
|
| $23$ |
\( (T^{5} - 218 T^{4} + \cdots - 400723625)^{2} \)
|
| $29$ |
\( T^{10} + \cdots - 57\!\cdots\!00 \)
|
| $31$ |
\( (T^{5} - 88 T^{4} + \cdots + 7413220473)^{2} \)
|
| $37$ |
\( T^{10} + \cdots - 85\!\cdots\!48 \)
|
| $41$ |
\( (T^{5} - 434 T^{4} + \cdots - 433937374063)^{2} \)
|
| $43$ |
\( T^{10} \)
|
| $47$ |
\( (T^{5} + \cdots + 1583082558000)^{2} \)
|
| $53$ |
\( (T^{5} + \cdots - 2628574415500)^{2} \)
|
| $59$ |
\( (T^{5} + \cdots - 4643121534208)^{2} \)
|
| $61$ |
\( T^{10} + \cdots - 21\!\cdots\!00 \)
|
| $67$ |
\( (T^{5} + \cdots + 41582685234000)^{2} \)
|
| $71$ |
\( T^{10} + \cdots - 11\!\cdots\!00 \)
|
| $73$ |
\( T^{10} + \cdots - 13\!\cdots\!92 \)
|
| $79$ |
\( (T^{5} + \cdots + 14740873920144)^{2} \)
|
| $83$ |
\( (T^{5} + \cdots + 9278139959500)^{2} \)
|
| $89$ |
\( T^{10} + \cdots - 10\!\cdots\!00 \)
|
| $97$ |
\( (T^{5} + 58 T^{4} + \cdots - 300508037125)^{2} \)
|
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