Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [445,2,Mod(266,445)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(445, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("445.266");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 445 = 5 \cdot 89 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 445.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: | \(3.55334288995\) |
| 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: |
\( x^{12} + 22x^{10} + 154x^{8} + 421x^{6} + 477x^{4} + 224x^{2} + 36 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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
\( x^{12} + 22x^{10} + 154x^{8} + 421x^{6} + 477x^{4} + 224x^{2} + 36 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -11\nu^{10} - 260\nu^{8} - 2042\nu^{6} - 6475\nu^{4} - 7813\nu^{2} - 2598 ) / 284 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -11\nu^{10} - 260\nu^{8} - 2042\nu^{6} - 6475\nu^{4} - 7529\nu^{2} - 1462 ) / 284 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 11\nu^{11} + 260\nu^{9} + 2042\nu^{7} + 6475\nu^{5} + 7813\nu^{3} + 2882\nu ) / 284 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 10\nu^{10} + 217\nu^{8} + 1482\nu^{6} + 3879\nu^{4} + 4011\nu^{2} + 1271 ) / 71 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -43\nu^{10} - 926\nu^{8} - 6188\nu^{6} - 15139\nu^{4} - 12895\nu^{2} - 3030 ) / 142 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -107\nu^{11} - 2258\nu^{9} - 14480\nu^{7} - 32609\nu^{5} - 25047\nu^{3} - 7870\nu ) / 426 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 161\nu^{10} + 3444\nu^{8} + 22710\nu^{6} + 54223\nu^{4} + 45471\nu^{2} + 11226 ) / 142 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 100\nu^{11} + 2170\nu^{9} + 14749\nu^{7} + 37654\nu^{5} + 36063\nu^{3} + 10367\nu ) / 213 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( -505\nu^{11} - 10852\nu^{9} - 72214\nu^{7} - 175477\nu^{5} - 150051\nu^{3} - 35750\nu ) / 852 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 188\nu^{11} + 4037\nu^{9} + 26825\nu^{7} + 65030\nu^{5} + 55683\nu^{3} + 13330\nu ) / 213 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta_{2} - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{11} - \beta_{10} - \beta_{7} + \beta_{4} - 8\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} + 4\beta_{6} - 12\beta_{3} + 10\beta_{2} + 36 \)
|
| \(\nu^{5}\) | \(=\) |
\( 17\beta_{11} + 20\beta_{10} + 11\beta_{7} - 10\beta_{4} + 80\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -17\beta_{8} - 65\beta_{6} + 7\beta_{5} + 136\beta_{3} - 100\beta_{2} - 383 \)
|
| \(\nu^{7}\) | \(=\) |
\( -225\beta_{11} - 273\beta_{10} + 7\beta_{9} - 119\beta_{7} + 93\beta_{4} - 857\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 225\beta_{8} + 842\beta_{6} - 139\beta_{5} - 1545\beta_{3} + 1043\beta_{2} + 4255 \)
|
| \(\nu^{9}\) | \(=\) |
\( 2751\beta_{11} + 3368\beta_{10} - 139\beta_{9} + 1320\beta_{7} - 904\beta_{4} + 9483\beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( -2751\beta_{8} - 10190\beta_{6} + 1986\beta_{5} + 17625\beta_{3} - 11291\beta_{2} - 48060 \)
|
| \(\nu^{11}\) | \(=\) |
\( -32552\beta_{11} - 39991\beta_{10} + 1986\beta_{9} - 14874\beta_{7} + 9305\beta_{4} - 106724\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/445\mathbb{Z}\right)^\times\).
| \(n\) | \(181\) | \(357\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 266.1 |
|
−2.55847 | − | 1.59667i | 4.54579 | 1.00000 | 4.08504i | − | 2.47694i | −6.51334 | 0.450646 | −2.55847 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 266.2 | −2.55847 | 1.59667i | 4.54579 | 1.00000 | − | 4.08504i | 2.47694i | −6.51334 | 0.450646 | −2.55847 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 266.3 | −1.88426 | − | 0.599654i | 1.55042 | 1.00000 | 1.12990i | 4.55493i | 0.847122 | 2.64042 | −1.88426 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 266.4 | −1.88426 | 0.599654i | 1.55042 | 1.00000 | − | 1.12990i | − | 4.55493i | 0.847122 | 2.64042 | −1.88426 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 266.5 | 0.154676 | − | 0.737942i | −1.97608 | 1.00000 | − | 0.114142i | − | 1.04301i | −0.615004 | 2.45544 | 0.154676 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 266.6 | 0.154676 | 0.737942i | −1.97608 | 1.00000 | 0.114142i | 1.04301i | −0.615004 | 2.45544 | 0.154676 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 266.7 | 0.832572 | − | 3.38691i | −1.30682 | 1.00000 | − | 2.81985i | − | 1.05870i | −2.75317 | −8.47119 | 0.832572 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 266.8 | 0.832572 | 3.38691i | −1.30682 | 1.00000 | 2.81985i | 1.05870i | −2.75317 | −8.47119 | 0.832572 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 266.9 | 1.86647 | − | 1.02078i | 1.48372 | 1.00000 | − | 1.90525i | − | 2.77249i | −0.963620 | 1.95801 | 1.86647 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 266.10 | 1.86647 | 1.02078i | 1.48372 | 1.00000 | 1.90525i | 2.77249i | −0.963620 | 1.95801 | 1.86647 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 266.11 | 2.58901 | − | 2.45628i | 4.70297 | 1.00000 | − | 6.35934i | 1.79499i | 6.99801 | −3.03333 | 2.58901 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 266.12 | 2.58901 | 2.45628i | 4.70297 | 1.00000 | 6.35934i | − | 1.79499i | 6.99801 | −3.03333 | 2.58901 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 89.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 445.2.d.b | ✓ | 12 |
| 89.b | even | 2 | 1 | inner | 445.2.d.b | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 445.2.d.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 445.2.d.b | ✓ | 12 | 89.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - T_{2}^{5} - 10T_{2}^{4} + 10T_{2}^{3} + 22T_{2}^{2} - 23T_{2} + 3 \)
acting on \(S_{2}^{\mathrm{new}}(445, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{6} - T^{5} - 10 T^{4} + \cdots + 3)^{2} \)
|
| $3$ |
\( T^{12} + 22 T^{10} + \cdots + 36 \)
|
| $5$ |
\( (T - 1)^{12} \)
|
| $7$ |
\( T^{12} + 40 T^{10} + \cdots + 3844 \)
|
| $11$ |
\( (T^{6} + 8 T^{5} - 4 T^{4} + \cdots - 96)^{2} \)
|
| $13$ |
\( T^{12} + 67 T^{10} + \cdots + 14400 \)
|
| $17$ |
\( (T^{6} + 5 T^{5} + \cdots + 248)^{2} \)
|
| $19$ |
\( T^{12} + 41 T^{10} + \cdots + 4 \)
|
| $23$ |
\( T^{12} + 205 T^{10} + \cdots + 35307364 \)
|
| $29$ |
\( T^{12} + 122 T^{10} + \cdots + 2849344 \)
|
| $31$ |
\( T^{12} + 171 T^{10} + \cdots + 605284 \)
|
| $37$ |
\( T^{12} + \cdots + 620607744 \)
|
| $41$ |
\( T^{12} + \cdots + 3060302400 \)
|
| $43$ |
\( T^{12} + \cdots + 45848230884 \)
|
| $47$ |
\( (T^{6} + 10 T^{5} + \cdots + 104140)^{2} \)
|
| $53$ |
\( (T^{6} - 5 T^{5} + \cdots - 175128)^{2} \)
|
| $59$ |
\( T^{12} + \cdots + 4488732004 \)
|
| $61$ |
\( T^{12} + 459 T^{10} + \cdots + 15116544 \)
|
| $67$ |
\( (T^{6} - 5 T^{5} + \cdots - 18916)^{2} \)
|
| $71$ |
\( (T^{6} - 2 T^{5} + \cdots - 1008)^{2} \)
|
| $73$ |
\( (T^{6} + 13 T^{5} + \cdots + 20100)^{2} \)
|
| $79$ |
\( (T^{6} - T^{5} - 109 T^{4} + \cdots + 3264)^{2} \)
|
| $83$ |
\( T^{12} + 196 T^{10} + \cdots + 7584516 \)
|
| $89$ |
\( T^{12} + \cdots + 496981290961 \)
|
| $97$ |
\( (T^{6} - 25 T^{5} + \cdots + 783912)^{2} \)
|
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