Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [252,11,Mod(181,252)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(252, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 11, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("252.181");
S:= CuspForms(chi, 11);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 11 \) |
| Character orbit: | \([\chi]\) | \(=\) | 252.d (of order \(2\), degree \(1\), 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: | \(160.110027674\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 2 x^{11} + 8700283 x^{10} + 1743363790 x^{9} + 25853580960505 x^{8} + \cdots + 27\!\cdots\!64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{32}\cdot 3^{17}\cdot 7^{10} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 2 x^{11} + 8700283 x^{10} + 1743363790 x^{9} + 25853580960505 x^{8} + \cdots + 27\!\cdots\!64 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 19\!\cdots\!20 \nu^{11} + \cdots + 24\!\cdots\!00 ) / 79\!\cdots\!96 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 39\!\cdots\!79 \nu^{11} + \cdots - 12\!\cdots\!44 ) / 10\!\cdots\!72 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 52\!\cdots\!08 \nu^{11} + \cdots + 90\!\cdots\!36 ) / 11\!\cdots\!28 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 95\!\cdots\!61 \nu^{11} + \cdots + 27\!\cdots\!48 ) / 13\!\cdots\!84 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 97\!\cdots\!17 \nu^{11} + \cdots + 23\!\cdots\!08 ) / 13\!\cdots\!84 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 13\!\cdots\!37 \nu^{11} + \cdots + 12\!\cdots\!76 ) / 13\!\cdots\!84 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 20\!\cdots\!95 \nu^{11} + \cdots - 60\!\cdots\!28 ) / 81\!\cdots\!24 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 11\!\cdots\!40 \nu^{11} + \cdots + 41\!\cdots\!32 ) / 26\!\cdots\!32 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 93\!\cdots\!35 \nu^{11} + \cdots + 48\!\cdots\!44 ) / 12\!\cdots\!84 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 35\!\cdots\!17 \nu^{11} + \cdots + 10\!\cdots\!20 ) / 24\!\cdots\!72 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 20\!\cdots\!35 \nu^{11} + \cdots - 61\!\cdots\!12 ) / 34\!\cdots\!24 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{11} - 336 \beta_{10} + 113 \beta_{9} - 840 \beta_{7} - 56 \beta_{6} - 952 \beta_{5} + \cdots + 28224 ) / 169344 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 971 \beta_{11} - 45360 \beta_{10} + 859 \beta_{9} - 5472 \beta_{8} - 279216 \beta_{7} + \cdots - 245556730944 ) / 169344 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 624759 \beta_{11} + 132175323 \beta_{10} - 29918919 \beta_{9} - 175230 \beta_{8} + \cdots - 9317964957840 ) / 21168 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 5510291267 \beta_{11} + 1327048680384 \beta_{10} - 63492811219 \beta_{9} + 24785644032 \beta_{8} + \cdots + 67\!\cdots\!12 ) / 169344 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 19577515022339 \beta_{11} + \cdots + 53\!\cdots\!44 ) / 169344 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 10\!\cdots\!85 \beta_{11} + \cdots - 79\!\cdots\!16 ) / 6048 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 74\!\cdots\!71 \beta_{11} + \cdots - 31\!\cdots\!28 ) / 169344 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 15\!\cdots\!89 \beta_{11} + \cdots + 76\!\cdots\!28 ) / 169344 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 65\!\cdots\!19 \beta_{11} + \cdots + 42\!\cdots\!00 ) / 42336 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 78\!\cdots\!13 \beta_{11} + \cdots - 25\!\cdots\!04 ) / 169344 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 83\!\cdots\!97 \beta_{11} + \cdots - 85\!\cdots\!28 ) / 169344 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(73\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 181.1 |
|
0 | 0 | 0 | − | 4686.55i | 0 | −15193.6 | − | 7185.43i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.2 | 0 | 0 | 0 | − | 4686.55i | 0 | −15193.6 | + | 7185.43i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.3 | 0 | 0 | 0 | − | 4316.86i | 0 | 16495.1 | − | 3223.01i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.4 | 0 | 0 | 0 | − | 4316.86i | 0 | 16495.1 | + | 3223.01i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.5 | 0 | 0 | 0 | − | 1732.73i | 0 | 5145.51 | − | 16000.0i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.6 | 0 | 0 | 0 | − | 1732.73i | 0 | 5145.51 | + | 16000.0i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.7 | 0 | 0 | 0 | 1732.73i | 0 | 5145.51 | − | 16000.0i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.8 | 0 | 0 | 0 | 1732.73i | 0 | 5145.51 | + | 16000.0i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.9 | 0 | 0 | 0 | 4316.86i | 0 | 16495.1 | − | 3223.01i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.10 | 0 | 0 | 0 | 4316.86i | 0 | 16495.1 | + | 3223.01i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.11 | 0 | 0 | 0 | 4686.55i | 0 | −15193.6 | − | 7185.43i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.12 | 0 | 0 | 0 | 4686.55i | 0 | −15193.6 | + | 7185.43i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 21.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 252.11.d.c | ✓ | 12 |
| 3.b | odd | 2 | 1 | inner | 252.11.d.c | ✓ | 12 |
| 7.b | odd | 2 | 1 | inner | 252.11.d.c | ✓ | 12 |
| 21.c | even | 2 | 1 | inner | 252.11.d.c | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 252.11.d.c | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 252.11.d.c | ✓ | 12 | 3.b | odd | 2 | 1 | inner |
| 252.11.d.c | ✓ | 12 | 7.b | odd | 2 | 1 | inner |
| 252.11.d.c | ✓ | 12 | 21.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + 43601460T_{5}^{4} + 531195301721700T_{5}^{2} + 1228875093816588288000 \)
acting on \(S_{11}^{\mathrm{new}}(252, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( (T^{6} + \cdots + 12\!\cdots\!00)^{2} \)
|
| $7$ |
\( (T^{6} + \cdots + 22\!\cdots\!49)^{2} \)
|
| $11$ |
\( (T^{6} + \cdots - 12\!\cdots\!40)^{2} \)
|
| $13$ |
\( (T^{6} + \cdots + 47\!\cdots\!08)^{2} \)
|
| $17$ |
\( (T^{6} + \cdots + 63\!\cdots\!80)^{2} \)
|
| $19$ |
\( (T^{6} + \cdots + 23\!\cdots\!32)^{2} \)
|
| $23$ |
\( (T^{6} + \cdots - 12\!\cdots\!60)^{2} \)
|
| $29$ |
\( (T^{6} + \cdots - 12\!\cdots\!60)^{2} \)
|
| $31$ |
\( (T^{6} + \cdots + 28\!\cdots\!72)^{2} \)
|
| $37$ |
\( (T^{3} + \cdots + 48\!\cdots\!76)^{4} \)
|
| $41$ |
\( (T^{6} + \cdots + 14\!\cdots\!00)^{2} \)
|
| $43$ |
\( (T^{3} + \cdots + 39\!\cdots\!32)^{4} \)
|
| $47$ |
\( (T^{6} + \cdots + 45\!\cdots\!80)^{2} \)
|
| $53$ |
\( (T^{6} + \cdots - 43\!\cdots\!40)^{2} \)
|
| $59$ |
\( (T^{6} + \cdots + 11\!\cdots\!80)^{2} \)
|
| $61$ |
\( (T^{6} + \cdots + 11\!\cdots\!52)^{2} \)
|
| $67$ |
\( (T^{3} + \cdots + 16\!\cdots\!72)^{4} \)
|
| $71$ |
\( (T^{6} + \cdots - 12\!\cdots\!60)^{2} \)
|
| $73$ |
\( (T^{6} + \cdots + 74\!\cdots\!00)^{2} \)
|
| $79$ |
\( (T^{3} + \cdots - 22\!\cdots\!32)^{4} \)
|
| $83$ |
\( (T^{6} + \cdots + 32\!\cdots\!00)^{2} \)
|
| $89$ |
\( (T^{6} + \cdots + 20\!\cdots\!80)^{2} \)
|
| $97$ |
\( (T^{6} + \cdots + 15\!\cdots\!88)^{2} \)
|
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