Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [484,3,Mod(243,484)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(484, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("484.243");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 484 = 2^{2} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 484.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: | \(13.1880447950\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 2x^{9} + 7x^{8} + 4x^{7} - 7x^{6} + 82x^{5} - 28x^{4} + 64x^{3} + 448x^{2} - 512x + 1024 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{9} \) |
| Twist minimal: | no (minimal twist has level 44) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 2x^{9} + 7x^{8} + 4x^{7} - 7x^{6} + 82x^{5} - 28x^{4} + 64x^{3} + 448x^{2} - 512x + 1024 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{9} - 2\nu^{8} + 7\nu^{7} + 4\nu^{6} - 7\nu^{5} + 82\nu^{4} - 28\nu^{3} + 64\nu^{2} + 448\nu - 512 ) / 128 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 33 \nu^{9} + 90 \nu^{8} - 103 \nu^{7} + 4 \nu^{6} + 1175 \nu^{5} - 954 \nu^{4} + 3196 \nu^{3} + \cdots + 27904 ) / 1792 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 149 \nu^{9} + 174 \nu^{8} + 275 \nu^{7} + 2076 \nu^{6} + 1693 \nu^{5} + 5458 \nu^{4} + 13332 \nu^{3} + \cdots + 35072 ) / 5376 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 53 \nu^{9} - 222 \nu^{8} - 403 \nu^{7} - 1772 \nu^{6} - 3197 \nu^{5} - 5442 \nu^{4} - 14708 \nu^{3} + \cdots - 48640 ) / 1792 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 41 \nu^{9} - 18 \nu^{8} - 167 \nu^{7} - 432 \nu^{6} - 193 \nu^{5} - 2038 \nu^{4} - 1692 \nu^{3} + \cdots + 5440 ) / 1344 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{9} + 2\nu^{8} - 3\nu^{7} - 4\nu^{6} + 19\nu^{5} - 42\nu^{4} + 32\nu^{3} + 112\nu^{2} - 224\nu + 480 ) / 32 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 29 \nu^{9} + 3 \nu^{8} + 113 \nu^{7} + 303 \nu^{6} + 73 \nu^{5} + 1231 \nu^{4} + 1374 \nu^{3} + \cdots - 2848 ) / 672 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 75 \nu^{9} - 174 \nu^{8} + 173 \nu^{7} + 164 \nu^{6} - 2141 \nu^{5} + 2158 \nu^{4} - 4596 \nu^{3} + \cdots - 49408 ) / 1792 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 85 \nu^{9} + 74 \nu^{8} - 323 \nu^{7} - 660 \nu^{6} + 515 \nu^{5} - 3674 \nu^{4} + 348 \nu^{3} + \cdots + 28160 ) / 1792 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{8} + \beta_{6} + \beta_{2} + \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} + 2\beta _1 - 2 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4\beta_{9} - \beta_{8} + 4\beta_{7} - 3\beta_{6} + 2\beta_{5} - \beta_{2} + \beta _1 - 17 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 4\beta_{9} - 7\beta_{8} - 8\beta_{7} - 15\beta_{6} - 11\beta_{5} - \beta_{4} - \beta_{3} - 5\beta_{2} - 9\beta _1 + 1 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -8\beta_{9} + 3\beta_{8} - 2\beta_{6} - 3\beta_{5} + 2\beta_{4} - 6\beta_{3} + 21\beta_{2} - 6\beta _1 - 7 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 28 \beta_{9} + 3 \beta_{8} - 24 \beta_{7} + 21 \beta_{6} + 27 \beta_{5} - 13 \beta_{4} + 9 \beta_{3} + \cdots + 7 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 4 \beta_{9} - 3 \beta_{8} + 76 \beta_{7} + 53 \beta_{6} + 38 \beta_{5} - 2 \beta_{4} - 42 \beta_{3} + \cdots + 201 ) / 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 48\beta_{9} - 9\beta_{8} - 32\beta_{7} + 22\beta_{6} - 105\beta_{5} + 7\beta_{3} - 80\beta_{2} - 19\beta _1 + 81 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 52 \beta_{9} + 113 \beta_{8} + 204 \beta_{7} + 175 \beta_{6} - 6 \beta_{5} + 176 \beta_{4} + \cdots + 89 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/484\mathbb{Z}\right)^\times\).
| \(n\) | \(243\) | \(365\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 243.1 |
|
−1.82909 | − | 0.808964i | 1.36518i | 2.69116 | + | 2.95934i | −4.78741 | 1.10438 | − | 2.49704i | 13.2453i | −2.52837 | − | 7.58995i | 7.13628 | 8.75661 | + | 3.87284i | ||||||||||||||||||||||||||||||||||||||
| 243.2 | −1.82909 | + | 0.808964i | − | 1.36518i | 2.69116 | − | 2.95934i | −4.78741 | 1.10438 | + | 2.49704i | − | 13.2453i | −2.52837 | + | 7.58995i | 7.13628 | 8.75661 | − | 3.87284i | |||||||||||||||||||||||||||||||||||||
| 243.3 | −1.32980 | − | 1.49386i | − | 5.74247i | −0.463244 | + | 3.97309i | 2.62387 | −8.57845 | + | 7.63636i | − | 5.65174i | 6.55126 | − | 4.59140i | −23.9759 | −3.48924 | − | 3.91971i | |||||||||||||||||||||||||||||||||||||
| 243.4 | −1.32980 | + | 1.49386i | 5.74247i | −0.463244 | − | 3.97309i | 2.62387 | −8.57845 | − | 7.63636i | 5.65174i | 6.55126 | + | 4.59140i | −23.9759 | −3.48924 | + | 3.91971i | |||||||||||||||||||||||||||||||||||||||
| 243.5 | −0.248646 | − | 1.98448i | 1.96580i | −3.87635 | + | 0.986869i | 1.68069 | 3.90109 | − | 0.488788i | − | 1.68671i | 2.92227 | + | 7.44717i | 5.13564 | −0.417899 | − | 3.33531i | ||||||||||||||||||||||||||||||||||||||
| 243.6 | −0.248646 | + | 1.98448i | − | 1.96580i | −3.87635 | − | 0.986869i | 1.68069 | 3.90109 | + | 0.488788i | 1.68671i | 2.92227 | − | 7.44717i | 5.13564 | −0.417899 | + | 3.33531i | ||||||||||||||||||||||||||||||||||||||
| 243.7 | 1.60747 | − | 1.18997i | − | 2.40388i | 1.16792 | − | 3.82570i | 5.80725 | −2.86056 | − | 3.86417i | 3.47504i | −2.67507 | − | 7.53949i | 3.22135 | 9.33499 | − | 6.91048i | ||||||||||||||||||||||||||||||||||||||
| 243.8 | 1.60747 | + | 1.18997i | 2.40388i | 1.16792 | + | 3.82570i | 5.80725 | −2.86056 | + | 3.86417i | − | 3.47504i | −2.67507 | + | 7.53949i | 3.22135 | 9.33499 | + | 6.91048i | ||||||||||||||||||||||||||||||||||||||
| 243.9 | 1.80007 | − | 0.871632i | 3.93920i | 2.48051 | − | 3.13800i | −7.32441 | 3.43354 | + | 7.09085i | − | 5.80518i | 1.72992 | − | 7.81072i | −6.51732 | −13.1845 | + | 6.38420i | ||||||||||||||||||||||||||||||||||||||
| 243.10 | 1.80007 | + | 0.871632i | − | 3.93920i | 2.48051 | + | 3.13800i | −7.32441 | 3.43354 | − | 7.09085i | 5.80518i | 1.72992 | + | 7.81072i | −6.51732 | −13.1845 | − | 6.38420i | ||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 484.3.b.h | 10 | |
| 4.b | odd | 2 | 1 | inner | 484.3.b.h | 10 | |
| 11.b | odd | 2 | 1 | 44.3.b.a | ✓ | 10 | |
| 33.d | even | 2 | 1 | 396.3.g.c | 10 | ||
| 44.c | even | 2 | 1 | 44.3.b.a | ✓ | 10 | |
| 88.b | odd | 2 | 1 | 704.3.d.d | 10 | ||
| 88.g | even | 2 | 1 | 704.3.d.d | 10 | ||
| 132.d | odd | 2 | 1 | 396.3.g.c | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 44.3.b.a | ✓ | 10 | 11.b | odd | 2 | 1 | |
| 44.3.b.a | ✓ | 10 | 44.c | even | 2 | 1 | |
| 396.3.g.c | 10 | 33.d | even | 2 | 1 | ||
| 396.3.g.c | 10 | 132.d | odd | 2 | 1 | ||
| 484.3.b.h | 10 | 1.a | even | 1 | 1 | trivial | |
| 484.3.b.h | 10 | 4.b | odd | 2 | 1 | inner | |
| 704.3.d.d | 10 | 88.b | odd | 2 | 1 | ||
| 704.3.d.d | 10 | 88.g | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(484, [\chi])\):
|
\( T_{3}^{10} + 60T_{3}^{8} + 1110T_{3}^{6} + 7884T_{3}^{4} + 22641T_{3}^{2} + 21296 \)
|
|
\( T_{5}^{5} + 2T_{5}^{4} - 58T_{5}^{3} - 24T_{5}^{2} + 721T_{5} - 898 \)
|
|
\( T_{13}^{5} - 2T_{13}^{4} - 472T_{13}^{3} - 80T_{13}^{2} + 56192T_{13} + 128384 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 2 T^{8} + \cdots + 1024 \)
|
| $3$ |
\( T^{10} + 60 T^{8} + \cdots + 21296 \)
|
| $5$ |
\( (T^{5} + 2 T^{4} + \cdots - 898)^{2} \)
|
| $7$ |
\( T^{10} + 256 T^{8} + \cdots + 6488064 \)
|
| $11$ |
\( T^{10} \)
|
| $13$ |
\( (T^{5} - 2 T^{4} + \cdots + 128384)^{2} \)
|
| $17$ |
\( (T^{5} + 10 T^{4} + \cdots + 12512)^{2} \)
|
| $19$ |
\( T^{10} + \cdots + 525533184 \)
|
| $23$ |
\( T^{10} + \cdots + 186356016 \)
|
| $29$ |
\( (T^{5} + 14 T^{4} + \cdots + 9476096)^{2} \)
|
| $31$ |
\( T^{10} + \cdots + 1383246375984 \)
|
| $37$ |
\( (T^{5} + 50 T^{4} + \cdots + 86437718)^{2} \)
|
| $41$ |
\( (T^{5} + 50 T^{4} + \cdots + 27499136)^{2} \)
|
| $43$ |
\( T^{10} + \cdots + 490601889017856 \)
|
| $47$ |
\( T^{10} + \cdots + 11\!\cdots\!56 \)
|
| $53$ |
\( (T^{5} - 50 T^{4} + \cdots - 94490272)^{2} \)
|
| $59$ |
\( T^{10} + \cdots + 37\!\cdots\!84 \)
|
| $61$ |
\( (T^{5} - 26 T^{4} + \cdots - 56868352)^{2} \)
|
| $67$ |
\( T^{10} + \cdots + 12\!\cdots\!56 \)
|
| $71$ |
\( T^{10} + \cdots + 21\!\cdots\!84 \)
|
| $73$ |
\( (T^{5} + 42 T^{4} + \cdots + 491204224)^{2} \)
|
| $79$ |
\( T^{10} + \cdots + 32\!\cdots\!36 \)
|
| $83$ |
\( T^{10} + \cdots + 13\!\cdots\!04 \)
|
| $89$ |
\( (T^{5} - 14 T^{4} + \cdots - 717570994)^{2} \)
|
| $97$ |
\( (T^{5} + 90 T^{4} + \cdots - 623378)^{2} \)
|
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