Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1444,2,Mod(429,1444)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1444, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1444.429");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1444 = 2^{2} \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1444.e (of order \(3\), degree \(2\), 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: | \(11.5303980519\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 6 x^{11} - 3 x^{10} + 70 x^{9} - 15 x^{8} - 426 x^{7} + 64 x^{6} + 1659 x^{5} + 267 x^{4} + \cdots + 4161 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | no (minimal twist has level 76) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{12} - 6 x^{11} - 3 x^{10} + 70 x^{9} - 15 x^{8} - 426 x^{7} + 64 x^{6} + 1659 x^{5} + 267 x^{4} + \cdots + 4161 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - \nu^{10} + 5 \nu^{9} + 9 \nu^{8} - 66 \nu^{7} + 49 \nu^{6} + 105 \nu^{5} - 553 \nu^{4} + \cdots - 5578 ) / 1853 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 22 \nu^{10} - 110 \nu^{9} - 89 \nu^{8} + 1016 \nu^{7} + 230 \nu^{6} - 4708 \nu^{5} - 1786 \nu^{4} + \cdots - 13970 ) / 1853 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 995 \nu^{11} + 7831 \nu^{10} + 46995 \nu^{9} - 326814 \nu^{8} + 54349 \nu^{7} + 1703408 \nu^{6} + \cdots + 10936061 ) / 8740601 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3014 \nu^{11} + 25876 \nu^{10} - 250822 \nu^{9} - 84243 \nu^{8} + 2839567 \nu^{7} + \cdots - 91337490 ) / 8740601 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 3014 \nu^{11} + 59030 \nu^{10} - 173708 \nu^{9} - 679911 \nu^{8} + 2764229 \nu^{7} + \cdots - 108083119 ) / 8740601 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 5549 \nu^{11} + 28161 \nu^{10} + 42520 \nu^{9} - 88867 \nu^{8} - 1402242 \nu^{7} + \cdots + 13272986 ) / 8740601 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 5549 \nu^{11} + 32878 \nu^{10} + 18935 \nu^{9} - 645473 \nu^{8} + 965692 \nu^{7} + \cdots - 41137609 ) / 8740601 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 24220 \nu^{11} - 81323 \nu^{10} - 427002 \nu^{9} + 1543220 \nu^{8} + 2521526 \nu^{7} + \cdots - 9775054 ) / 8740601 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 24441 \nu^{11} + 84897 \nu^{10} + 198319 \nu^{9} - 340479 \nu^{8} - 2357880 \nu^{7} + \cdots + 137176039 ) / 8740601 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 24441 \nu^{11} + 183954 \nu^{10} - 296966 \nu^{9} - 1231992 \nu^{8} + 4179882 \nu^{7} + \cdots - 114419307 ) / 8740601 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 1506 \nu^{11} + 8283 \nu^{10} + 6301 \nu^{9} - 90477 \nu^{8} - 1422 \nu^{7} + 485184 \nu^{6} + \cdots - 1259576 ) / 514153 \)
|
| \(\nu\) | \(=\) |
\( ( 3\beta_{11} + 4\beta_{8} + 3\beta_{5} - 3\beta_{4} + 2\beta_{3} - 2\beta_{2} + \beta _1 + 3 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3 \beta_{11} + \beta_{10} - \beta_{9} + 4 \beta_{8} - \beta_{7} + \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + \cdots + 12 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4 \beta_{11} + \beta_{10} + 7 \beta_{8} + \beta_{7} + 2 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + 8 \beta_{3} + \cdots + 7 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 21 \beta_{11} + 8 \beta_{10} - 2 \beta_{9} + 38 \beta_{8} + \beta_{7} + 17 \beta_{6} + 18 \beta_{5} + \cdots + 39 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 21 \beta_{11} + 25 \beta_{10} + 5 \beta_{9} + 85 \beta_{8} + 29 \beta_{7} + 64 \beta_{6} + 27 \beta_{5} + \cdots + 57 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( 4 \beta_{11} + 15 \beta_{10} + 10 \beta_{9} + 54 \beta_{8} + 21 \beta_{7} + 57 \beta_{6} + 24 \beta_{5} + \cdots + 9 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 270 \beta_{11} + 119 \beta_{10} + 133 \beta_{9} + 155 \beta_{8} + 211 \beta_{7} + 470 \beta_{6} + \cdots - 207 ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 1089 \beta_{11} + 166 \beta_{10} + 506 \beta_{9} - 50 \beta_{8} + 470 \beta_{7} + 1204 \beta_{6} + \cdots - 1071 ) / 3 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 1477 \beta_{11} + 86 \beta_{10} + 542 \beta_{9} - 627 \beta_{8} + 306 \beta_{7} + 936 \beta_{6} + \cdots - 1329 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 14004 \beta_{11} - 130 \beta_{10} + 5005 \beta_{9} - 8080 \beta_{8} + 1003 \beta_{7} + 6620 \beta_{6} + \cdots - 12339 ) / 3 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 43776 \beta_{11} - 2189 \beta_{10} + 13871 \beta_{9} - 30497 \beta_{8} - 2023 \beta_{7} + \cdots - 36225 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1444\mathbb{Z}\right)^\times\).
| \(n\) | \(723\) | \(1085\) |
| \(\chi(n)\) | \(1\) | \(\beta_{11}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 429.1 |
|
0 | −1.50920 | + | 2.61402i | 0 | 1.89668 | − | 3.28515i | 0 | 0.139023 | 0 | −3.05539 | − | 5.29210i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 429.2 | 0 | −1.21809 | + | 2.10979i | 0 | −1.10017 | + | 1.90556i | 0 | 2.96827 | 0 | −1.46748 | − | 2.54175i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 429.3 | 0 | −0.906288 | + | 1.56974i | 0 | −0.141102 | + | 0.244396i | 0 | 1.15987 | 0 | −0.142716 | − | 0.247192i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 429.4 | 0 | 0.243160 | − | 0.421165i | 0 | −1.39668 | + | 2.41913i | 0 | −3.36570 | 0 | 1.38175 | + | 2.39326i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 429.5 | 0 | 0.544440 | − | 0.942998i | 0 | 1.60017 | − | 2.77158i | 0 | −0.556791 | 0 | 0.907170 | + | 1.57126i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 429.6 | 0 | 1.34598 | − | 2.33131i | 0 | 0.641102 | − | 1.11042i | 0 | −3.34467 | 0 | −2.12333 | − | 3.67771i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 653.1 | 0 | −1.50920 | − | 2.61402i | 0 | 1.89668 | + | 3.28515i | 0 | 0.139023 | 0 | −3.05539 | + | 5.29210i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 653.2 | 0 | −1.21809 | − | 2.10979i | 0 | −1.10017 | − | 1.90556i | 0 | 2.96827 | 0 | −1.46748 | + | 2.54175i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 653.3 | 0 | −0.906288 | − | 1.56974i | 0 | −0.141102 | − | 0.244396i | 0 | 1.15987 | 0 | −0.142716 | + | 0.247192i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 653.4 | 0 | 0.243160 | + | 0.421165i | 0 | −1.39668 | − | 2.41913i | 0 | −3.36570 | 0 | 1.38175 | − | 2.39326i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 653.5 | 0 | 0.544440 | + | 0.942998i | 0 | 1.60017 | + | 2.77158i | 0 | −0.556791 | 0 | 0.907170 | − | 1.57126i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 653.6 | 0 | 1.34598 | + | 2.33131i | 0 | 0.641102 | + | 1.11042i | 0 | −3.34467 | 0 | −2.12333 | + | 3.67771i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1444.2.e.g | 12 | |
| 19.b | odd | 2 | 1 | 1444.2.e.h | 12 | ||
| 19.c | even | 3 | 1 | 1444.2.a.h | 6 | ||
| 19.c | even | 3 | 1 | inner | 1444.2.e.g | 12 | |
| 19.d | odd | 6 | 1 | 1444.2.a.g | 6 | ||
| 19.d | odd | 6 | 1 | 1444.2.e.h | 12 | ||
| 19.e | even | 9 | 2 | 76.2.i.a | ✓ | 12 | |
| 57.l | odd | 18 | 2 | 684.2.bo.c | 12 | ||
| 76.f | even | 6 | 1 | 5776.2.a.by | 6 | ||
| 76.g | odd | 6 | 1 | 5776.2.a.bw | 6 | ||
| 76.l | odd | 18 | 2 | 304.2.u.e | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 76.2.i.a | ✓ | 12 | 19.e | even | 9 | 2 | |
| 304.2.u.e | 12 | 76.l | odd | 18 | 2 | ||
| 684.2.bo.c | 12 | 57.l | odd | 18 | 2 | ||
| 1444.2.a.g | 6 | 19.d | odd | 6 | 1 | ||
| 1444.2.a.h | 6 | 19.c | even | 3 | 1 | ||
| 1444.2.e.g | 12 | 1.a | even | 1 | 1 | trivial | |
| 1444.2.e.g | 12 | 19.c | even | 3 | 1 | inner | |
| 1444.2.e.h | 12 | 19.b | odd | 2 | 1 | ||
| 1444.2.e.h | 12 | 19.d | odd | 6 | 1 | ||
| 5776.2.a.bw | 6 | 76.g | odd | 6 | 1 | ||
| 5776.2.a.by | 6 | 76.f | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + 3 T_{3}^{11} + 18 T_{3}^{10} + 27 T_{3}^{9} + 147 T_{3}^{8} + 192 T_{3}^{7} + 709 T_{3}^{6} + \cdots + 361 \)
acting on \(S_{2}^{\mathrm{new}}(1444, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} + 3 T^{11} + \cdots + 361 \)
|
| $5$ |
\( T^{12} - 3 T^{11} + \cdots + 729 \)
|
| $7$ |
\( (T^{6} + 3 T^{5} - 12 T^{4} + \cdots - 3)^{2} \)
|
| $11$ |
\( (T^{6} + 3 T^{5} - 24 T^{4} + \cdots - 27)^{2} \)
|
| $13$ |
\( T^{12} + 12 T^{11} + \cdots + 130321 \)
|
| $17$ |
\( T^{12} - 6 T^{11} + \cdots + 8982009 \)
|
| $19$ |
\( T^{12} \)
|
| $23$ |
\( T^{12} + \cdots + 151585344 \)
|
| $29$ |
\( T^{12} + 21 T^{11} + \cdots + 263169 \)
|
| $31$ |
\( (T^{6} + 6 T^{5} + \cdots - 361)^{2} \)
|
| $37$ |
\( (T^{6} + 6 T^{5} + \cdots + 11096)^{2} \)
|
| $41$ |
\( T^{12} + \cdots + 360278361 \)
|
| $43$ |
\( T^{12} - 18 T^{11} + \cdots + 1203409 \)
|
| $47$ |
\( T^{12} + 30 T^{11} + \cdots + 729 \)
|
| $53$ |
\( T^{12} + 18 T^{11} + \cdots + 95004009 \)
|
| $59$ |
\( T^{12} + \cdots + 12621848409 \)
|
| $61$ |
\( T^{12} + \cdots + 1418049649 \)
|
| $67$ |
\( T^{12} + \cdots + 6080256576 \)
|
| $71$ |
\( T^{12} - 12 T^{11} + \cdots + 16842816 \)
|
| $73$ |
\( T^{12} - 24 T^{11} + \cdots + 36864 \)
|
| $79$ |
\( T^{12} + \cdots + 1548343801 \)
|
| $83$ |
\( (T^{6} + 3 T^{5} + \cdots - 109269)^{2} \)
|
| $89$ |
\( T^{12} + \cdots + 7695324729 \)
|
| $97$ |
\( T^{12} + \cdots + 16983563041 \)
|
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