Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [605,6,Mod(1,605)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(605, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("605.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 605 = 5 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 605.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: | \(97.0322109869\) |
| Analytic rank: | \(0\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - x^{8} - 207 x^{7} + 329 x^{6} + 13060 x^{5} - 23860 x^{4} - 271056 x^{3} + 558896 x^{2} + \cdots - 1723584 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - x^{8} - 207 x^{7} + 329 x^{6} + 13060 x^{5} - 23860 x^{4} - 271056 x^{3} + 558896 x^{2} + \cdots - 1723584 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 46 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2519 \nu^{8} + 28873 \nu^{7} - 614821 \nu^{6} - 5878965 \nu^{5} + 46232720 \nu^{4} + \cdots + 3731226432 ) / 275366976 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 6806 \nu^{8} - 5703 \nu^{7} + 1162979 \nu^{6} + 164877 \nu^{5} - 62076701 \nu^{4} + \cdots - 15258587712 ) / 68841744 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 28873 \nu^{8} + 122261 \nu^{7} + 6092895 \nu^{6} - 19213545 \nu^{5} - 351813604 \nu^{4} + \cdots - 18233968128 ) / 275366976 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 75293 \nu^{8} + 59087 \nu^{7} - 15819483 \nu^{6} - 140499 \nu^{5} + 1031533076 \nu^{4} + \cdots + 68052615744 ) / 275366976 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 21972 \nu^{8} - 50863 \nu^{7} + 4723397 \nu^{6} + 7383831 \nu^{5} - 315674169 \nu^{4} + \cdots - 24637381968 ) / 68841744 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 25644 \nu^{8} + 53097 \nu^{7} - 4915913 \nu^{6} - 6363523 \nu^{5} + 278485073 \nu^{4} + \cdots + 11608681776 ) / 22947248 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - \beta _1 + 46 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} + 5\beta_{3} - 2\beta_{2} + 82\beta _1 - 49 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{8} - 6\beta_{7} - 4\beta_{6} + \beta_{5} - 6\beta_{4} - 21\beta_{3} + 115\beta_{2} - 196\beta _1 + 3658 \)
|
| \(\nu^{5}\) | \(=\) |
\( 3 \beta_{8} + 147 \beta_{7} + 157 \beta_{6} + 19 \beta_{5} + 28 \beta_{4} + 590 \beta_{3} - 448 \beta_{2} + \cdots - 8847 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 116 \beta_{8} - 1064 \beta_{7} - 920 \beta_{6} + 122 \beta_{5} - 950 \beta_{4} - 4322 \beta_{3} + \cdots + 345120 \)
|
| \(\nu^{7}\) | \(=\) |
\( 722 \beta_{8} + 18223 \beta_{7} + 19651 \beta_{6} + 4102 \beta_{5} + 5708 \beta_{4} + 68937 \beta_{3} + \cdots - 1383327 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 11233 \beta_{8} - 149528 \beta_{7} - 144118 \beta_{6} + 8749 \beta_{5} - 121826 \beta_{4} + \cdots + 35584476 \)
|
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 |
|
−11.0401 | 15.0983 | 89.8838 | −25.0000 | −166.687 | −193.342 | −639.043 | −15.0416 | 276.003 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −7.41464 | −9.51592 | 22.9768 | −25.0000 | 70.5571 | 61.1284 | 66.9034 | −152.447 | 185.366 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −5.98120 | 14.8190 | 3.77470 | −25.0000 | −88.6352 | 51.3169 | 168.821 | −23.3980 | 149.530 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.03663 | −25.4690 | −27.8522 | −25.0000 | 51.8708 | −25.5642 | 121.896 | 405.669 | 50.9156 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.37054 | 14.0426 | −30.1216 | −25.0000 | 19.2461 | 231.699 | −85.1404 | −45.8041 | −34.2636 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.09682 | 30.8815 | −22.4097 | −25.0000 | 95.6343 | −178.755 | −168.497 | 710.667 | −77.4204 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 5.07389 | −7.47445 | −6.25564 | −25.0000 | −37.9245 | −7.57995 | −194.105 | −187.133 | −126.847 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 8.62083 | −27.0228 | 42.3186 | −25.0000 | −232.959 | 157.640 | 88.9552 | 487.232 | −215.521 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 9.31048 | 3.64075 | 54.6851 | −25.0000 | 33.8971 | −167.543 | 211.209 | −229.745 | −232.762 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(11\) | \(-1\) |
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 | 605.6.a.j | yes | 9 |
| 11.b | odd | 2 | 1 | 605.6.a.i | ✓ | 9 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 605.6.a.i | ✓ | 9 | 11.b | odd | 2 | 1 | |
| 605.6.a.j | yes | 9 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{9} - T_{2}^{8} - 207 T_{2}^{7} + 329 T_{2}^{6} + 13060 T_{2}^{5} - 23860 T_{2}^{4} + \cdots - 1723584 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(605))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} - T^{8} + \cdots - 1723584 \)
|
| $3$ |
\( T^{9} + \cdots - 17292403179 \)
|
| $5$ |
\( (T + 25)^{9} \)
|
| $7$ |
\( T^{9} + \cdots + 12\!\cdots\!67 \)
|
| $11$ |
\( T^{9} \)
|
| $13$ |
\( T^{9} + \cdots + 45\!\cdots\!84 \)
|
| $17$ |
\( T^{9} + \cdots - 13\!\cdots\!00 \)
|
| $19$ |
\( T^{9} + \cdots - 20\!\cdots\!16 \)
|
| $23$ |
\( T^{9} + \cdots - 34\!\cdots\!92 \)
|
| $29$ |
\( T^{9} + \cdots + 29\!\cdots\!32 \)
|
| $31$ |
\( T^{9} + \cdots + 76\!\cdots\!56 \)
|
| $37$ |
\( T^{9} + \cdots - 32\!\cdots\!16 \)
|
| $41$ |
\( T^{9} + \cdots - 22\!\cdots\!25 \)
|
| $43$ |
\( T^{9} + \cdots - 72\!\cdots\!25 \)
|
| $47$ |
\( T^{9} + \cdots - 95\!\cdots\!31 \)
|
| $53$ |
\( T^{9} + \cdots + 76\!\cdots\!48 \)
|
| $59$ |
\( T^{9} + \cdots + 11\!\cdots\!52 \)
|
| $61$ |
\( T^{9} + \cdots + 36\!\cdots\!49 \)
|
| $67$ |
\( T^{9} + \cdots + 55\!\cdots\!31 \)
|
| $71$ |
\( T^{9} + \cdots + 45\!\cdots\!52 \)
|
| $73$ |
\( T^{9} + \cdots - 27\!\cdots\!12 \)
|
| $79$ |
\( T^{9} + \cdots + 25\!\cdots\!00 \)
|
| $83$ |
\( T^{9} + \cdots + 42\!\cdots\!24 \)
|
| $89$ |
\( T^{9} + \cdots - 19\!\cdots\!49 \)
|
| $97$ |
\( T^{9} + \cdots - 41\!\cdots\!52 \)
|
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