Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [245,6,Mod(1,245)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(245, 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("245.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 245 = 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 245.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: | \(39.2940358542\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} - 225x^{6} + 537x^{5} + 15166x^{4} - 25016x^{3} - 317696x^{2} + 463952x + 1012704 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{5}\cdot 7^{3} \) |
| Twist minimal: | no (minimal twist has level 35) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 3x^{7} - 225x^{6} + 537x^{5} + 15166x^{4} - 25016x^{3} - 317696x^{2} + 463952x + 1012704 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 57 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5\nu^{7} - 52\nu^{6} - 762\nu^{5} + 8738\nu^{4} + 16161\nu^{3} - 304774\nu^{2} + 386020\nu + 915528 ) / 10464 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 67 \nu^{7} - 457 \nu^{6} - 12151 \nu^{5} + 75255 \nu^{4} + 524526 \nu^{3} - 2509532 \nu^{2} + \cdots + 9588768 ) / 20928 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 11\nu^{7} + 60\nu^{6} - 2374\nu^{5} - 12866\nu^{4} + 142287\nu^{3} + 736382\nu^{2} - 1822564\nu - 6403080 ) / 10464 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 42 \nu^{7} - 415 \nu^{6} - 6815 \nu^{5} + 68407 \nu^{4} + 206319 \nu^{3} - 2289738 \nu^{2} + \cdots + 6429000 ) / 10464 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 42 \nu^{7} + 415 \nu^{6} + 6815 \nu^{5} - 68407 \nu^{4} - 195855 \nu^{3} + 2279274 \nu^{2} + \cdots - 6041832 ) / 10464 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 57 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} + \beta_{2} + 93\beta _1 + 20 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{7} - 4\beta_{6} - 2\beta_{5} + 4\beta_{4} + 28\beta_{3} + 125\beta_{2} + 141\beta _1 + 5231 \)
|
| \(\nu^{5}\) | \(=\) |
\( 169\beta_{7} + 155\beta_{6} + 10\beta_{5} - 12\beta_{4} + 176\beta_{3} + 89\beta_{2} + 9841\beta _1 + 3640 \)
|
| \(\nu^{6}\) | \(=\) |
\( 432\beta_{7} - 728\beta_{6} - 268\beta_{5} + 688\beta_{4} + 5724\beta_{3} + 14677\beta_{2} + 15645\beta _1 + 553269 \)
|
| \(\nu^{7}\) | \(=\) |
\( 23521 \beta_{7} + 19809 \beta_{6} + 2232 \beta_{5} - 1664 \beta_{4} + 39512 \beta_{3} + 5477 \beta_{2} + \cdots + 393712 \)
|
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 |
|
−10.9703 | −4.62268 | 88.3466 | 25.0000 | 50.7120 | 0 | −618.137 | −221.631 | −274.256 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −6.77005 | −23.4913 | 13.8335 | 25.0000 | 159.037 | 0 | 122.988 | 308.841 | −169.251 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −6.63433 | 30.5785 | 12.0144 | 25.0000 | −202.868 | 0 | 132.591 | 692.045 | −165.858 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.26911 | 7.00999 | −30.3894 | 25.0000 | −8.89643 | 0 | 79.1787 | −193.860 | −31.7277 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 3.11373 | −20.6063 | −22.3047 | 25.0000 | −64.1623 | 0 | −169.090 | 181.619 | 77.8431 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 4.91557 | 9.91717 | −7.83720 | 25.0000 | 48.7485 | 0 | −195.822 | −144.650 | 122.889 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 9.65190 | 22.8595 | 61.1592 | 25.0000 | 220.637 | 0 | 281.441 | 279.555 | 241.297 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 10.9626 | −23.6449 | 88.1776 | 25.0000 | −259.208 | 0 | 615.850 | 316.080 | 274.064 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(7\) | \(1\) |
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 | 245.6.a.j | 8 | |
| 7.b | odd | 2 | 1 | 245.6.a.k | 8 | ||
| 7.c | even | 3 | 2 | 35.6.e.b | ✓ | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.6.e.b | ✓ | 16 | 7.c | even | 3 | 2 | |
| 245.6.a.j | 8 | 1.a | even | 1 | 1 | trivial | |
| 245.6.a.k | 8 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(245))\):
|
\( T_{2}^{8} - 3T_{2}^{7} - 225T_{2}^{6} + 537T_{2}^{5} + 15166T_{2}^{4} - 25016T_{2}^{3} - 317696T_{2}^{2} + 463952T_{2} + 1012704 \)
|
|
\( T_{3}^{8} + 2 T_{3}^{7} - 1579 T_{3}^{6} - 5436 T_{3}^{5} + 752739 T_{3}^{4} + 2145922 T_{3}^{3} + \cdots + 2571136884 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 3 T^{7} + \cdots + 1012704 \)
|
| $3$ |
\( T^{8} + \cdots + 2571136884 \)
|
| $5$ |
\( (T - 25)^{8} \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( T^{8} + \cdots + 12\!\cdots\!00 \)
|
| $13$ |
\( T^{8} + \cdots - 52\!\cdots\!44 \)
|
| $17$ |
\( T^{8} + \cdots - 17\!\cdots\!56 \)
|
| $19$ |
\( T^{8} + \cdots - 14\!\cdots\!32 \)
|
| $23$ |
\( T^{8} + \cdots - 73\!\cdots\!19 \)
|
| $29$ |
\( T^{8} + \cdots + 54\!\cdots\!00 \)
|
| $31$ |
\( T^{8} + \cdots - 13\!\cdots\!16 \)
|
| $37$ |
\( T^{8} + \cdots - 35\!\cdots\!00 \)
|
| $41$ |
\( T^{8} + \cdots - 44\!\cdots\!75 \)
|
| $43$ |
\( T^{8} + \cdots - 19\!\cdots\!12 \)
|
| $47$ |
\( T^{8} + \cdots + 45\!\cdots\!00 \)
|
| $53$ |
\( T^{8} + \cdots + 18\!\cdots\!48 \)
|
| $59$ |
\( T^{8} + \cdots - 35\!\cdots\!00 \)
|
| $61$ |
\( T^{8} + \cdots - 26\!\cdots\!32 \)
|
| $67$ |
\( T^{8} + \cdots + 29\!\cdots\!00 \)
|
| $71$ |
\( T^{8} + \cdots - 11\!\cdots\!24 \)
|
| $73$ |
\( T^{8} + \cdots - 21\!\cdots\!24 \)
|
| $79$ |
\( T^{8} + \cdots - 56\!\cdots\!64 \)
|
| $83$ |
\( T^{8} + \cdots + 20\!\cdots\!56 \)
|
| $89$ |
\( T^{8} + \cdots - 10\!\cdots\!12 \)
|
| $97$ |
\( T^{8} + \cdots + 10\!\cdots\!16 \)
|
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