Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [405,3,Mod(28,405)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(405, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([4, 9]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("405.28");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 405 = 3^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 405.l (of order \(12\), degree \(4\), not minimal) |
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: | no |
| Analytic conductor: | \(11.0354507066\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 2 x^{15} + 2 x^{14} - 36 x^{13} - 109 x^{12} + 482 x^{11} - 98 x^{10} + 3204 x^{9} + \cdots + 2560000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 2 x^{15} + 2 x^{14} - 36 x^{13} - 109 x^{12} + 482 x^{11} - 98 x^{10} + 3204 x^{9} + \cdots + 2560000 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 29\!\cdots\!19 \nu^{15} - 555400748554380 \nu^{14} + \cdots - 17\!\cdots\!24 ) / 42\!\cdots\!23 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 13\!\cdots\!35 \nu^{15} + \cdots - 17\!\cdots\!00 ) / 13\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 16\!\cdots\!27 \nu^{15} + \cdots - 42\!\cdots\!00 ) / 13\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 16\!\cdots\!95 \nu^{15} + \cdots - 36\!\cdots\!00 ) / 13\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 6774588553737 \nu^{15} + 29057612144714 \nu^{14} - 216072965574754 \nu^{13} + \cdots - 44\!\cdots\!00 ) / 51\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 33\!\cdots\!93 \nu^{15} + \cdots + 41\!\cdots\!00 ) / 24\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 30\!\cdots\!85 \nu^{15} + \cdots + 41\!\cdots\!00 ) / 16\!\cdots\!20 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 347520844456759 \nu^{15} + 830533459988258 \nu^{14} + \cdots - 16\!\cdots\!00 ) / 10\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 14\!\cdots\!43 \nu^{15} + \cdots - 95\!\cdots\!00 ) / 27\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 90\!\cdots\!27 \nu^{15} + \cdots - 81\!\cdots\!00 ) / 16\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 14\!\cdots\!27 \nu^{15} + \cdots - 89\!\cdots\!00 ) / 13\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 79\!\cdots\!39 \nu^{15} + \cdots - 58\!\cdots\!00 ) / 67\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 10\!\cdots\!39 \nu^{15} + \cdots - 27\!\cdots\!00 ) / 67\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 34\!\cdots\!39 \nu^{15} + \cdots + 11\!\cdots\!00 ) / 33\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{12} - \beta_{6} + 7\beta_{5} + \beta_{3} \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{14} - \beta_{11} - 4\beta_{7} + 12\beta_{4} + \beta_{2} + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{15} - 20\beta_{12} - 17\beta_{10} + 71\beta_{9} - \beta_{8} + 20\beta_{4} + 17\beta_{2} + 20\beta _1 + 17 \)
|
| \(\nu^{5}\) | \(=\) |
\( 17\beta_{13} - 25\beta_{10} + 67\beta_{9} - 17\beta_{8} - 166\beta_{6} + 67\beta_{5} + 25\beta_{3} - 67 \)
|
| \(\nu^{6}\) | \(=\) |
\( -25\beta_{14} + 25\beta_{13} - 251\beta_{11} - 1128\beta_{7} - 342\beta_{6} + 342\beta_{4} - 342\beta_1 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 251 \beta_{15} - 2414 \beta_{12} - 467 \beta_{11} - 467 \beta_{10} + 1265 \beta_{9} - 1732 \beta_{7} + \cdots + 467 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 467 \beta_{15} + 467 \beta_{14} + 467 \beta_{13} - 5582 \beta_{12} - 3669 \beta_{10} + 11907 \beta_{9} + \cdots - 11907 \)
|
| \(\nu^{9}\) | \(=\) |
\( 3669\beta_{13} - 7917\beta_{11} - 29980\beta_{7} - 35946\beta_{6} - 7917\beta_{2} - 35946\beta _1 - 29980 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 7917 \beta_{15} - 89098 \beta_{12} - 54291 \beta_{11} + 7917 \beta_{8} - 224232 \beta_{7} + \cdots - 89098 \beta_1 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 54291 \beta_{15} + 54291 \beta_{14} - 542678 \beta_{12} - 128683 \beta_{10} + 366937 \beta_{9} + \cdots - 366937 \)
|
| \(\nu^{12}\) | \(=\) |
\( 128683 \beta_{14} + 128683 \beta_{13} - 1404246 \beta_{6} - 1404246 \beta_{4} - 814133 \beta_{2} + \cdots - 3311960 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 2047661 \beta_{11} + 2047661 \beta_{10} - 5929775 \beta_{9} + 814133 \beta_{8} - 7977436 \beta_{7} + \cdots - 2047661 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 2047661 \beta_{15} + 2047661 \beta_{14} - 2047661 \beta_{13} - 21964474 \beta_{12} + \cdots - 12334115 \beta_{3} \)
|
| \(\nu^{15}\) | \(=\) |
\( 12334115 \beta_{14} + 32202779 \beta_{11} + 126379332 \beta_{7} - 126514134 \beta_{4} - 32202779 \beta_{2} - 126379332 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).
| \(n\) | \(82\) | \(326\) |
| \(\chi(n)\) | \(\beta_{7}\) | \(-1 + \beta_{9}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 28.1 |
|
−3.00035 | − | 0.803942i | 0 | 4.89170 | + | 2.82422i | −4.69060 | + | 1.73154i | 0 | −1.09772 | − | 0.294134i | −3.62067 | − | 3.62067i | 0 | 15.4655 | − | 1.42426i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 28.2 | −1.48661 | − | 0.398337i | 0 | −1.41275 | − | 0.815652i | 0.463012 | − | 4.97852i | 0 | −2.61146 | − | 0.699739i | 6.12842 | + | 6.12842i | 0 | −2.67145 | + | 7.21670i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 28.3 | 2.05601 | + | 0.550907i | 0 | 0.459586 | + | 0.265342i | 2.92464 | − | 4.05542i | 0 | −6.15409 | − | 1.64898i | −5.22169 | − | 5.22169i | 0 | 8.24726 | − | 6.72679i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 28.4 | 3.79698 | + | 1.01740i | 0 | 9.91787 | + | 5.72609i | −0.795129 | + | 4.93637i | 0 | −7.89506 | − | 2.11547i | 20.7139 | + | 20.7139i | 0 | −8.04135 | + | 17.9344i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 217.1 | −3.00035 | + | 0.803942i | 0 | 4.89170 | − | 2.82422i | −4.69060 | − | 1.73154i | 0 | −1.09772 | + | 0.294134i | −3.62067 | + | 3.62067i | 0 | 15.4655 | + | 1.42426i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 217.2 | −1.48661 | + | 0.398337i | 0 | −1.41275 | + | 0.815652i | 0.463012 | + | 4.97852i | 0 | −2.61146 | + | 0.699739i | 6.12842 | − | 6.12842i | 0 | −2.67145 | − | 7.21670i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 217.3 | 2.05601 | − | 0.550907i | 0 | 0.459586 | − | 0.265342i | 2.92464 | + | 4.05542i | 0 | −6.15409 | + | 1.64898i | −5.22169 | + | 5.22169i | 0 | 8.24726 | + | 6.72679i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 217.4 | 3.79698 | − | 1.01740i | 0 | 9.91787 | − | 5.72609i | −0.795129 | − | 4.93637i | 0 | −7.89506 | + | 2.11547i | 20.7139 | − | 20.7139i | 0 | −8.04135 | − | 17.9344i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 298.1 | −1.01740 | − | 3.79698i | 0 | −9.91787 | + | 5.72609i | −3.87746 | − | 3.15679i | 0 | 2.11547 | + | 7.89506i | 20.7139 | + | 20.7139i | 0 | −8.04135 | + | 17.9344i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 298.2 | −0.550907 | − | 2.05601i | 0 | −0.459586 | + | 0.265342i | 2.04978 | + | 4.56053i | 0 | 1.64898 | + | 6.15409i | −5.22169 | − | 5.22169i | 0 | 8.24726 | − | 6.72679i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 298.3 | 0.398337 | + | 1.48661i | 0 | 1.41275 | − | 0.815652i | 4.08001 | + | 2.89024i | 0 | 0.699739 | + | 2.61146i | 6.12842 | + | 6.12842i | 0 | −2.67145 | + | 7.21670i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 298.4 | 0.803942 | + | 3.00035i | 0 | −4.89170 | + | 2.82422i | 0.845743 | − | 4.92795i | 0 | 0.294134 | + | 1.09772i | −3.62067 | − | 3.62067i | 0 | 15.4655 | − | 1.42426i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 352.1 | −1.01740 | + | 3.79698i | 0 | −9.91787 | − | 5.72609i | −3.87746 | + | 3.15679i | 0 | 2.11547 | − | 7.89506i | 20.7139 | − | 20.7139i | 0 | −8.04135 | − | 17.9344i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 352.2 | −0.550907 | + | 2.05601i | 0 | −0.459586 | − | 0.265342i | 2.04978 | − | 4.56053i | 0 | 1.64898 | − | 6.15409i | −5.22169 | + | 5.22169i | 0 | 8.24726 | + | 6.72679i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 352.3 | 0.398337 | − | 1.48661i | 0 | 1.41275 | + | 0.815652i | 4.08001 | − | 2.89024i | 0 | 0.699739 | − | 2.61146i | 6.12842 | − | 6.12842i | 0 | −2.67145 | − | 7.21670i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 352.4 | 0.803942 | − | 3.00035i | 0 | −4.89170 | − | 2.82422i | 0.845743 | + | 4.92795i | 0 | 0.294134 | − | 1.09772i | −3.62067 | + | 3.62067i | 0 | 15.4655 | + | 1.42426i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
| 9.c | even | 3 | 1 | inner |
| 45.k | odd | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 405.3.l.l | 16 | |
| 3.b | odd | 2 | 1 | 405.3.l.k | 16 | ||
| 5.c | odd | 4 | 1 | inner | 405.3.l.l | 16 | |
| 9.c | even | 3 | 1 | 405.3.g.c | ✓ | 8 | |
| 9.c | even | 3 | 1 | inner | 405.3.l.l | 16 | |
| 9.d | odd | 6 | 1 | 405.3.g.e | yes | 8 | |
| 9.d | odd | 6 | 1 | 405.3.l.k | 16 | ||
| 15.e | even | 4 | 1 | 405.3.l.k | 16 | ||
| 45.k | odd | 12 | 1 | 405.3.g.c | ✓ | 8 | |
| 45.k | odd | 12 | 1 | inner | 405.3.l.l | 16 | |
| 45.l | even | 12 | 1 | 405.3.g.e | yes | 8 | |
| 45.l | even | 12 | 1 | 405.3.l.k | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 405.3.g.c | ✓ | 8 | 9.c | even | 3 | 1 | |
| 405.3.g.c | ✓ | 8 | 45.k | odd | 12 | 1 | |
| 405.3.g.e | yes | 8 | 9.d | odd | 6 | 1 | |
| 405.3.g.e | yes | 8 | 45.l | even | 12 | 1 | |
| 405.3.l.k | 16 | 3.b | odd | 2 | 1 | ||
| 405.3.l.k | 16 | 9.d | odd | 6 | 1 | ||
| 405.3.l.k | 16 | 15.e | even | 4 | 1 | ||
| 405.3.l.k | 16 | 45.l | even | 12 | 1 | ||
| 405.3.l.l | 16 | 1.a | even | 1 | 1 | trivial | |
| 405.3.l.l | 16 | 5.c | odd | 4 | 1 | inner | |
| 405.3.l.l | 16 | 9.c | even | 3 | 1 | inner | |
| 405.3.l.l | 16 | 45.k | odd | 12 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} - 2 T_{2}^{15} + 2 T_{2}^{14} - 36 T_{2}^{13} - 109 T_{2}^{12} + 482 T_{2}^{11} + \cdots + 2560000 \)
acting on \(S_{3}^{\mathrm{new}}(405, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} - 2 T^{15} + \cdots + 2560000 \)
|
| $3$ |
\( T^{16} \)
|
| $5$ |
\( T^{16} + \cdots + 152587890625 \)
|
| $7$ |
\( T^{16} + \cdots + 655360000 \)
|
| $11$ |
\( (T^{8} + 10 T^{7} + \cdots + 181710400)^{2} \)
|
| $13$ |
\( T^{16} + \cdots + 203295582846976 \)
|
| $17$ |
\( (T^{8} + 38 T^{7} + \cdots + 109160704)^{2} \)
|
| $19$ |
\( (T^{8} + 1020 T^{6} + \cdots + 1071225)^{2} \)
|
| $23$ |
\( T^{16} + \cdots + 64\!\cdots\!76 \)
|
| $29$ |
\( T^{16} + \cdots + 3626739360000 \)
|
| $31$ |
\( (T^{8} - 58 T^{7} + \cdots + 175304665636)^{2} \)
|
| $37$ |
\( (T^{8} - 50 T^{7} + \cdots + 349328281600)^{2} \)
|
| $41$ |
\( (T^{8} - 158 T^{7} + \cdots + 6380654641)^{2} \)
|
| $43$ |
\( T^{16} + \cdots + 16\!\cdots\!96 \)
|
| $47$ |
\( T^{16} + \cdots + 97\!\cdots\!76 \)
|
| $53$ |
\( (T^{8} - 118 T^{7} + \cdots + 376151609344)^{2} \)
|
| $59$ |
\( T^{16} + \cdots + 81\!\cdots\!81 \)
|
| $61$ |
\( (T^{8} + \cdots + 16609863270400)^{2} \)
|
| $67$ |
\( T^{16} + \cdots + 88\!\cdots\!56 \)
|
| $71$ |
\( (T^{4} - 124 T^{3} + \cdots - 1371818)^{4} \)
|
| $73$ |
\( (T^{8} + \cdots + 92297676980224)^{2} \)
|
| $79$ |
\( T^{16} + \cdots + 83\!\cdots\!16 \)
|
| $83$ |
\( T^{16} + \cdots + 48\!\cdots\!36 \)
|
| $89$ |
\( (T^{8} + \cdots + 21\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{16} + \cdots + 14\!\cdots\!16 \)
|
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