Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [123,6,Mod(1,123)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(123, 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("123.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 123 = 3 \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 123.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: | \(19.7272098370\) |
| 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: |
\( x^{9} - 4x^{8} - 209x^{7} + 394x^{6} + 13998x^{5} - 1172x^{4} - 279108x^{3} - 81704x^{2} + 1331104x - 904704 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{9} - 4x^{8} - 209x^{7} + 394x^{6} + 13998x^{5} - 1172x^{4} - 279108x^{3} - 81704x^{2} + 1331104x - 904704 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3\nu - 47 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 12097 \nu^{8} + 304912 \nu^{7} + 684341 \nu^{6} - 51353658 \nu^{5} + 59797818 \nu^{4} + \cdots + 30302279552 ) / 261481120 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 23627 \nu^{8} - 178626 \nu^{7} - 4302505 \nu^{6} + 24058042 \nu^{5} + 245217574 \nu^{4} + \cdots - 1470534304 ) / 130740560 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 56927 \nu^{8} + 548108 \nu^{7} + 8342727 \nu^{6} - 64836254 \nu^{5} - 351508690 \nu^{4} + \cdots + 16430506720 ) / 261481120 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 84309 \nu^{8} + 423388 \nu^{7} + 16112661 \nu^{6} - 46625210 \nu^{5} - 935360166 \nu^{4} + \cdots - 2346245344 ) / 261481120 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 29392 \nu^{8} - 115483 \nu^{7} - 6111587 \nu^{6} + 10410234 \nu^{5} + 397725270 \nu^{4} + \cdots + 17717368760 ) / 65370280 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 29677 \nu^{8} - 64170 \nu^{7} - 6400874 \nu^{6} + 906581 \nu^{5} + 431752966 \nu^{4} + \cdots + 26869933944 ) / 65370280 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3\beta _1 + 47 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - 3\beta_{4} - 2\beta_{3} + 4\beta_{2} + 88\beta _1 + 115 \)
|
| \(\nu^{4}\) | \(=\) |
\( 8\beta_{8} - 3\beta_{7} + 6\beta_{6} - 5\beta_{4} - 12\beta_{3} + 110\beta_{2} + 500\beta _1 + 4083 \)
|
| \(\nu^{5}\) | \(=\) |
\( 32 \beta_{8} + 95 \beta_{7} - 4 \beta_{6} + 18 \beta_{5} - 379 \beta_{4} - 300 \beta_{3} + 646 \beta_{2} + \cdots + 20797 \)
|
| \(\nu^{6}\) | \(=\) |
\( 1320 \beta_{8} - 555 \beta_{7} + 644 \beta_{6} + 2 \beta_{5} - 1349 \beta_{4} - 2208 \beta_{3} + \cdots + 418863 \)
|
| \(\nu^{7}\) | \(=\) |
\( 8296 \beta_{8} + 5771 \beta_{7} - 1360 \beta_{6} + 3306 \beta_{5} - 43475 \beta_{4} - 38408 \beta_{3} + \cdots + 2933917 \)
|
| \(\nu^{8}\) | \(=\) |
\( 187480 \beta_{8} - 87495 \beta_{7} + 48792 \beta_{6} + 7030 \beta_{5} - 237609 \beta_{4} + \cdots + 45989663 \)
|
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 |
|
−9.19566 | 9.00000 | 52.5602 | −41.6733 | −82.7610 | −192.549 | −189.065 | 81.0000 | 383.214 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −8.52897 | 9.00000 | 40.7433 | 42.6367 | −76.7607 | 149.348 | −74.5709 | 81.0000 | −363.647 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.71245 | 9.00000 | −24.6426 | 97.6999 | −24.4120 | 208.104 | 153.640 | 81.0000 | −265.006 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.380557 | 9.00000 | −31.8552 | −52.0181 | 3.42501 | 73.6737 | −24.3005 | 81.0000 | −19.7958 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.15317 | 9.00000 | −30.6702 | 27.4046 | 10.3785 | −155.646 | −72.2693 | 81.0000 | 31.6021 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 5.71341 | 9.00000 | 0.643094 | 102.498 | 51.4207 | 8.52219 | −179.155 | 81.0000 | 585.616 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 6.80983 | 9.00000 | 14.3738 | −63.9944 | 61.2885 | 198.434 | −120.032 | 81.0000 | −435.791 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 9.42415 | 9.00000 | 56.8146 | 34.8936 | 84.8174 | 113.937 | 233.857 | 81.0000 | 328.842 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 10.9560 | 9.00000 | 88.0330 | 2.55248 | 98.6036 | −167.824 | 613.896 | 81.0000 | 27.9649 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(41\) | \(1\) |
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 | 123.6.a.c | ✓ | 9 |
| 3.b | odd | 2 | 1 | 369.6.a.d | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 123.6.a.c | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 369.6.a.d | 9 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{9} - 14 T_{2}^{8} - 129 T_{2}^{7} + 2308 T_{2}^{6} + 1394 T_{2}^{5} - 103480 T_{2}^{4} + \cdots + 375040 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(123))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} - 14 T^{8} + \cdots + 375040 \)
|
| $3$ |
\( (T - 9)^{9} \)
|
| $5$ |
\( T^{9} + \cdots + 144570686583008 \)
|
| $7$ |
\( T^{9} + \cdots + 22\!\cdots\!00 \)
|
| $11$ |
\( T^{9} + \cdots - 50\!\cdots\!80 \)
|
| $13$ |
\( T^{9} + \cdots - 13\!\cdots\!00 \)
|
| $17$ |
\( T^{9} + \cdots + 12\!\cdots\!00 \)
|
| $19$ |
\( T^{9} + \cdots - 98\!\cdots\!36 \)
|
| $23$ |
\( T^{9} + \cdots + 15\!\cdots\!00 \)
|
| $29$ |
\( T^{9} + \cdots + 40\!\cdots\!20 \)
|
| $31$ |
\( T^{9} + \cdots + 17\!\cdots\!40 \)
|
| $37$ |
\( T^{9} + \cdots - 27\!\cdots\!24 \)
|
| $41$ |
\( (T + 1681)^{9} \)
|
| $43$ |
\( T^{9} + \cdots + 33\!\cdots\!12 \)
|
| $47$ |
\( T^{9} + \cdots + 12\!\cdots\!44 \)
|
| $53$ |
\( T^{9} + \cdots + 23\!\cdots\!60 \)
|
| $59$ |
\( T^{9} + \cdots - 73\!\cdots\!40 \)
|
| $61$ |
\( T^{9} + \cdots - 66\!\cdots\!48 \)
|
| $67$ |
\( T^{9} + \cdots + 13\!\cdots\!96 \)
|
| $71$ |
\( T^{9} + \cdots - 35\!\cdots\!68 \)
|
| $73$ |
\( T^{9} + \cdots + 50\!\cdots\!32 \)
|
| $79$ |
\( T^{9} + \cdots + 38\!\cdots\!00 \)
|
| $83$ |
\( T^{9} + \cdots + 46\!\cdots\!88 \)
|
| $89$ |
\( T^{9} + \cdots - 40\!\cdots\!04 \)
|
| $97$ |
\( T^{9} + \cdots + 43\!\cdots\!48 \)
|
| show more | |
| show less |