Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [361,6,Mod(1,361)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(361, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("361.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 361 = 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 361.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: | \(57.8985589525\) |
| 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: |
\( x^{8} - 3x^{7} - 201x^{6} + 535x^{5} + 11664x^{4} - 29076x^{3} - 205704x^{2} + 365088x + 793440 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 19) |
| 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
\( x^{8} - 3x^{7} - 201x^{6} + 535x^{5} + 11664x^{4} - 29076x^{3} - 205704x^{2} + 365088x + 793440 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} - 5\nu^{6} - 343\nu^{5} + 917\nu^{4} + 30654\nu^{3} - 25632\nu^{2} - 595848\nu - 321936 ) / 58368 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 51 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 49\nu^{7} + 59\nu^{6} - 8903\nu^{5} - 11307\nu^{4} + 402174\nu^{3} + 283488\nu^{2} - 3988872\nu - 2160528 ) / 58368 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -6\nu^{7} + 11\nu^{6} + 1108\nu^{5} - 1075\nu^{4} - 54534\nu^{3} + 9240\nu^{2} + 658968\nu + 628368 ) / 7296 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 53 \nu^{7} - 343 \nu^{6} + 9667 \nu^{5} + 63879 \nu^{4} - 424470 \nu^{3} - 2654304 \nu^{2} + \cdots + 20944080 ) / 58368 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 59 \nu^{7} + 313 \nu^{6} - 11725 \nu^{5} - 58377 \nu^{4} + 666762 \nu^{3} + 2442144 \nu^{2} + \cdots - 19548720 ) / 58368 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 51 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} + \beta_{3} - 6\beta_{2} + 83\beta _1 - 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{7} + 6\beta_{6} + 8\beta_{5} + 12\beta_{4} + 119\beta_{3} - 4\beta_{2} + 56\beta _1 + 4319 \)
|
| \(\nu^{5}\) | \(=\) |
\( 137\beta_{7} + 145\beta_{6} + 16\beta_{4} + 265\beta_{3} - 1182\beta_{2} + 8211\beta _1 + 1442 \)
|
| \(\nu^{6}\) | \(=\) |
\( 426 \beta_{7} + 958 \beta_{6} + 1480 \beta_{5} + 1996 \beta_{4} + 13679 \beta_{3} - 1124 \beta_{2} + \cdots + 436767 \)
|
| \(\nu^{7}\) | \(=\) |
\( 16633 \beta_{7} + 18369 \beta_{6} + 64 \beta_{5} + 4464 \beta_{4} + 45145 \beta_{3} - 165086 \beta_{2} + \cdots + 531010 \)
|
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 |
|
−10.3204 | 28.8598 | 74.5105 | −37.2687 | −297.845 | −30.1607 | −438.725 | 589.889 | 384.627 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −8.08901 | −17.6867 | 33.4321 | −30.6969 | 143.068 | −20.6961 | −11.5845 | 69.8209 | 248.307 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.45593 | 4.03681 | −12.1447 | 69.0081 | −17.9877 | −107.943 | 196.706 | −226.704 | −307.495 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.39907 | 10.5598 | −30.0426 | −52.7075 | −14.7738 | 195.855 | 86.8018 | −131.491 | 73.7414 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 3.41907 | −21.2036 | −20.3100 | 31.4591 | −72.4966 | 24.4401 | −178.851 | 206.593 | 107.561 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 5.82008 | 7.06217 | 1.87332 | −72.6410 | 41.1024 | −226.435 | −175.340 | −193.126 | −422.776 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 6.86478 | 26.2552 | 15.1251 | 99.7688 | 180.236 | 92.8461 | −115.842 | 446.336 | 684.890 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 11.1605 | −9.88344 | 92.5562 | −16.9219 | −110.304 | 176.093 | 675.836 | −145.318 | −188.856 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(19\) | \( -1 \) |
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 | 361.6.a.h | 8 | |
| 19.b | odd | 2 | 1 | 361.6.a.g | 8 | ||
| 19.d | odd | 6 | 2 | 19.6.c.a | ✓ | 16 | |
| 57.f | even | 6 | 2 | 171.6.f.b | 16 | ||
| 76.f | even | 6 | 2 | 304.6.i.c | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 19.6.c.a | ✓ | 16 | 19.d | odd | 6 | 2 | |
| 171.6.f.b | 16 | 57.f | even | 6 | 2 | ||
| 304.6.i.c | 16 | 76.f | even | 6 | 2 | ||
| 361.6.a.g | 8 | 19.b | odd | 2 | 1 | ||
| 361.6.a.h | 8 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 3T_{2}^{7} - 201T_{2}^{6} + 535T_{2}^{5} + 11664T_{2}^{4} - 29076T_{2}^{3} - 205704T_{2}^{2} + 365088T_{2} + 793440 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(361))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 3 T^{7} + \cdots + 793440 \)
|
| $3$ |
\( T^{8} - 28 T^{7} + \cdots - 845485047 \)
|
| $5$ |
\( T^{8} + \cdots - 16053943025664 \)
|
| $7$ |
\( T^{8} + \cdots + 11\!\cdots\!92 \)
|
| $11$ |
\( T^{8} + \cdots + 20\!\cdots\!04 \)
|
| $13$ |
\( T^{8} + \cdots + 12\!\cdots\!00 \)
|
| $17$ |
\( T^{8} + \cdots + 40\!\cdots\!64 \)
|
| $19$ |
\( T^{8} \)
|
| $23$ |
\( T^{8} + \cdots - 70\!\cdots\!80 \)
|
| $29$ |
\( T^{8} + \cdots + 13\!\cdots\!12 \)
|
| $31$ |
\( T^{8} + \cdots - 13\!\cdots\!52 \)
|
| $37$ |
\( T^{8} + \cdots - 27\!\cdots\!88 \)
|
| $41$ |
\( T^{8} + \cdots - 28\!\cdots\!79 \)
|
| $43$ |
\( T^{8} + \cdots - 32\!\cdots\!76 \)
|
| $47$ |
\( T^{8} + \cdots + 12\!\cdots\!84 \)
|
| $53$ |
\( T^{8} + \cdots + 95\!\cdots\!76 \)
|
| $59$ |
\( T^{8} + \cdots + 22\!\cdots\!25 \)
|
| $61$ |
\( T^{8} + \cdots - 62\!\cdots\!00 \)
|
| $67$ |
\( T^{8} + \cdots + 81\!\cdots\!81 \)
|
| $71$ |
\( T^{8} + \cdots - 13\!\cdots\!28 \)
|
| $73$ |
\( T^{8} + \cdots - 74\!\cdots\!59 \)
|
| $79$ |
\( T^{8} + \cdots + 98\!\cdots\!00 \)
|
| $83$ |
\( T^{8} + \cdots - 22\!\cdots\!24 \)
|
| $89$ |
\( T^{8} + \cdots + 20\!\cdots\!40 \)
|
| $97$ |
\( T^{8} + \cdots + 11\!\cdots\!89 \)
|
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