Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [45,26,Mod(19,45)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(45, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 26, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("45.19");
S:= CuspForms(chi, 26);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 45 = 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 26 \) |
| Character orbit: | \([\chi]\) | \(=\) | 45.b (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: | \(178.198550979\) |
| 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: |
\( x^{12} + 71168091 x^{10} + \cdots + 10\!\cdots\!36 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{44}\cdot 3^{28}\cdot 5^{29} \) |
| Twist minimal: | no (minimal twist has level 5) |
| 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
\( x^{12} + 71168091 x^{10} + \cdots + 10\!\cdots\!36 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( 4\nu^{2} + 47445394 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 17\!\cdots\!09 \nu^{11} + \cdots + 26\!\cdots\!44 \nu ) / 44\!\cdots\!40 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 35\!\cdots\!59 \nu^{11} + \cdots - 17\!\cdots\!28 ) / 12\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 51\!\cdots\!03 \nu^{11} + \cdots + 78\!\cdots\!76 ) / 63\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 52\!\cdots\!33 \nu^{11} + \cdots + 87\!\cdots\!64 ) / 18\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 10\!\cdots\!13 \nu^{11} + \cdots + 24\!\cdots\!96 ) / 31\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 10\!\cdots\!93 \nu^{11} + \cdots + 45\!\cdots\!44 ) / 12\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 13\!\cdots\!29 \nu^{11} + \cdots - 40\!\cdots\!32 ) / 63\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 22\!\cdots\!13 \nu^{11} + \cdots + 29\!\cdots\!96 ) / 39\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 19\!\cdots\!31 \nu^{11} + \cdots - 26\!\cdots\!48 ) / 31\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} - 47445394 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{6} + 34\beta_{4} + 986\beta_{3} - 80807892\beta_1 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 23 \beta_{11} + 23 \beta_{10} + 124 \beta_{9} - 98 \beta_{8} - 790 \beta_{7} - 222 \beta_{5} + \cdots + 19\!\cdots\!68 ) / 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 133434 \beta_{11} + 14826 \beta_{10} + 1528688 \beta_{9} + 59304 \beta_{7} - 32252507 \beta_{6} + \cdots + 19\!\cdots\!88 \beta_1 ) / 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 822527133 \beta_{11} - 822527133 \beta_{10} + 1588508268 \beta_{9} + 5604157590 \beta_{8} + \cdots - 45\!\cdots\!48 ) / 8 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 10015603660062 \beta_{11} - 1112844851118 \beta_{10} - 64099974788304 \beta_{9} + \cdots - 48\!\cdots\!56 \beta_1 ) / 8 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 23\!\cdots\!11 \beta_{11} + \cdots + 11\!\cdots\!28 ) / 8 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 45\!\cdots\!90 \beta_{11} + \cdots + 12\!\cdots\!56 \beta_1 ) / 8 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 63\!\cdots\!05 \beta_{11} + \cdots - 30\!\cdots\!08 ) / 8 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 16\!\cdots\!70 \beta_{11} + \cdots - 33\!\cdots\!44 \beta_1 ) / 8 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).
| \(n\) | \(11\) | \(37\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 19.1 |
|
− | 10595.3i | 0 | −7.87068e7 | −5.09516e8 | + | 1.96001e8i | 0 | 3.45094e10i | 4.78405e11i | 0 | 2.07670e12 | + | 5.39850e12i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 19.2 | − | 9361.55i | 0 | −5.40841e7 | 5.44720e8 | + | 3.61012e7i | 0 | − | 1.37587e10i | 1.92189e11i | 0 | 3.37963e11 | − | 5.09942e12i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 19.3 | − | 6313.90i | 0 | −6.31093e6 | −3.25528e8 | − | 4.38241e8i | 0 | − | 3.34630e10i | − | 1.72013e11i | 0 | −2.76701e12 | + | 2.05535e12i | ||||||||||||||||||||||||||||||||||||||||||||||||
| 19.4 | − | 5681.18i | 0 | 1.27857e6 | −4.20624e8 | + | 3.47992e8i | 0 | 6.01315e10i | − | 1.97893e11i | 0 | 1.97701e12 | + | 2.38965e12i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 19.5 | − | 2921.94i | 0 | 2.50167e7 | 5.18558e8 | − | 1.70647e8i | 0 | 4.60230e10i | − | 1.71141e11i | 0 | −4.98620e11 | − | 1.51520e12i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 19.6 | − | 2023.27i | 0 | 2.94608e7 | −8.23814e7 | + | 5.39663e8i | 0 | − | 2.78390e10i | − | 1.27497e11i | 0 | 1.09188e12 | + | 1.66680e11i | ||||||||||||||||||||||||||||||||||||||||||||||||
| 19.7 | 2023.27i | 0 | 2.94608e7 | −8.23814e7 | − | 5.39663e8i | 0 | 2.78390e10i | 1.27497e11i | 0 | 1.09188e12 | − | 1.66680e11i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.8 | 2921.94i | 0 | 2.50167e7 | 5.18558e8 | + | 1.70647e8i | 0 | − | 4.60230e10i | 1.71141e11i | 0 | −4.98620e11 | + | 1.51520e12i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.9 | 5681.18i | 0 | 1.27857e6 | −4.20624e8 | − | 3.47992e8i | 0 | − | 6.01315e10i | 1.97893e11i | 0 | 1.97701e12 | − | 2.38965e12i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.10 | 6313.90i | 0 | −6.31093e6 | −3.25528e8 | + | 4.38241e8i | 0 | 3.34630e10i | 1.72013e11i | 0 | −2.76701e12 | − | 2.05535e12i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.11 | 9361.55i | 0 | −5.40841e7 | 5.44720e8 | − | 3.61012e7i | 0 | 1.37587e10i | − | 1.92189e11i | 0 | 3.37963e11 | + | 5.09942e12i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.12 | 10595.3i | 0 | −7.87068e7 | −5.09516e8 | − | 1.96001e8i | 0 | − | 3.45094e10i | − | 4.78405e11i | 0 | 2.07670e12 | − | 5.39850e12i | |||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 45.26.b.b | 12 | |
| 3.b | odd | 2 | 1 | 5.26.b.a | ✓ | 12 | |
| 5.b | even | 2 | 1 | inner | 45.26.b.b | 12 | |
| 15.d | odd | 2 | 1 | 5.26.b.a | ✓ | 12 | |
| 15.e | even | 4 | 2 | 25.26.a.f | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.26.b.a | ✓ | 12 | 3.b | odd | 2 | 1 | |
| 5.26.b.a | ✓ | 12 | 15.d | odd | 2 | 1 | |
| 25.26.a.f | 12 | 15.e | even | 4 | 2 | ||
| 45.26.b.b | 12 | 1.a | even | 1 | 1 | trivial | |
| 45.26.b.b | 12 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + 284672364 T_{2}^{10} + \cdots + 44\!\cdots\!56 \)
acting on \(S_{26}^{\mathrm{new}}(45, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + \cdots + 44\!\cdots\!56 \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( T^{12} + \cdots + 70\!\cdots\!25 \)
|
| $7$ |
\( T^{12} + \cdots + 14\!\cdots\!36 \)
|
| $11$ |
\( (T^{6} + \cdots - 68\!\cdots\!36)^{2} \)
|
| $13$ |
\( T^{12} + \cdots + 63\!\cdots\!56 \)
|
| $17$ |
\( T^{12} + \cdots + 22\!\cdots\!96 \)
|
| $19$ |
\( (T^{6} + \cdots - 21\!\cdots\!00)^{2} \)
|
| $23$ |
\( T^{12} + \cdots + 15\!\cdots\!16 \)
|
| $29$ |
\( (T^{6} + \cdots - 11\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{6} + \cdots - 29\!\cdots\!36)^{2} \)
|
| $37$ |
\( T^{12} + \cdots + 23\!\cdots\!16 \)
|
| $41$ |
\( (T^{6} + \cdots + 13\!\cdots\!64)^{2} \)
|
| $43$ |
\( T^{12} + \cdots + 18\!\cdots\!36 \)
|
| $47$ |
\( T^{12} + \cdots + 43\!\cdots\!76 \)
|
| $53$ |
\( T^{12} + \cdots + 41\!\cdots\!96 \)
|
| $59$ |
\( (T^{6} + \cdots - 44\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{6} + \cdots + 66\!\cdots\!64)^{2} \)
|
| $67$ |
\( T^{12} + \cdots + 75\!\cdots\!96 \)
|
| $71$ |
\( (T^{6} + \cdots + 14\!\cdots\!64)^{2} \)
|
| $73$ |
\( T^{12} + \cdots + 98\!\cdots\!16 \)
|
| $79$ |
\( (T^{6} + \cdots + 10\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{12} + \cdots + 27\!\cdots\!76 \)
|
| $89$ |
\( (T^{6} + \cdots + 32\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{12} + \cdots + 10\!\cdots\!76 \)
|
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