Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1445,2,Mod(1,1445)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1445, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1445.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1445 = 5 \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1445.a (trivial) |
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: | yes |
| Analytic conductor: | \(11.5383830921\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{12} - 4 x^{11} - 10 x^{10} + 52 x^{9} + 21 x^{8} - 232 x^{7} + 44 x^{6} + 424 x^{5} - 137 x^{4} + \cdots + 17 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 85) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{12} - 4 x^{11} - 10 x^{10} + 52 x^{9} + 21 x^{8} - 232 x^{7} + 44 x^{6} + 424 x^{5} - 137 x^{4} + \cdots + 17 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 54 \nu^{11} - 195 \nu^{10} - 568 \nu^{9} + 2455 \nu^{8} + 1476 \nu^{7} - 10314 \nu^{6} + 1071 \nu^{5} + \cdots + 1470 ) / 41 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 141 \nu^{11} + 352 \nu^{10} + 1925 \nu^{9} - 4399 \nu^{8} - 9389 \nu^{7} + 18116 \nu^{6} + \cdots + 1191 ) / 41 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 212 \nu^{11} + 515 \nu^{10} + 2933 \nu^{9} - 6428 \nu^{8} - 14596 \nu^{7} + 26388 \nu^{6} + \cdots + 2356 ) / 41 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 232 \nu^{11} + 569 \nu^{10} + 3195 \nu^{9} - 7111 \nu^{8} - 15785 \nu^{7} + 29265 \nu^{6} + \cdots + 2217 ) / 41 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 332 \nu^{11} + 839 \nu^{10} + 4505 \nu^{9} - 10485 \nu^{8} - 21771 \nu^{7} + 43158 \nu^{6} + \cdots + 2793 ) / 41 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 332 \nu^{11} - 839 \nu^{10} - 4505 \nu^{9} + 10485 \nu^{8} + 21771 \nu^{7} - 43158 \nu^{6} + \cdots - 2916 ) / 41 \)
|
| \(\beta_{8}\) | \(=\) |
\( - 8 \nu^{11} + 19 \nu^{10} + 112 \nu^{9} - 237 \nu^{8} - 568 \nu^{7} + 971 \nu^{6} + 1294 \nu^{5} + \cdots + 108 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 362 \nu^{11} - 879 \nu^{10} - 5021 \nu^{9} + 10997 \nu^{8} + 25092 \nu^{7} - 45321 \nu^{6} + \cdots - 3999 ) / 41 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 391 \nu^{11} + 986 \nu^{10} + 5323 \nu^{9} - 12342 \nu^{8} - 25871 \nu^{7} + 50942 \nu^{6} + \cdots + 3328 ) / 41 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 489 \nu^{11} + 1185 \nu^{10} + 6779 \nu^{9} - 14799 \nu^{8} - 33866 \nu^{7} + 60804 \nu^{6} + \cdots + 5644 ) / 41 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + \beta_{6} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{10} - \beta_{9} + \beta_{7} + 2\beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + 7\beta_{7} + 8\beta_{6} - 2\beta_{5} + \beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{11} - 8 \beta_{10} - 8 \beta_{9} - 2 \beta_{8} + 10 \beta_{7} + 19 \beta_{6} - 10 \beta_{5} + \cdots + 10 \)
|
| \(\nu^{6}\) | \(=\) |
\( 10 \beta_{11} - 11 \beta_{10} - 11 \beta_{9} - 10 \beta_{8} + 48 \beta_{7} + 58 \beta_{6} - 22 \beta_{5} + \cdots + 88 \)
|
| \(\nu^{7}\) | \(=\) |
\( 9 \beta_{11} - 58 \beta_{10} - 59 \beta_{9} - 21 \beta_{8} + 83 \beta_{7} + 151 \beta_{6} - 80 \beta_{5} + \cdots + 91 \)
|
| \(\nu^{8}\) | \(=\) |
\( 71 \beta_{11} - 98 \beta_{10} - 99 \beta_{9} - 72 \beta_{8} + 335 \beta_{7} + 421 \beta_{6} - 189 \beta_{5} + \cdots + 560 \)
|
| \(\nu^{9}\) | \(=\) |
\( 55 \beta_{11} - 417 \beta_{10} - 436 \beta_{9} - 158 \beta_{8} + 657 \beta_{7} + 1146 \beta_{6} + \cdots + 772 \)
|
| \(\nu^{10}\) | \(=\) |
\( 432 \beta_{11} - 810 \beta_{10} - 841 \beta_{9} - 458 \beta_{8} + 2378 \beta_{7} + 3087 \beta_{6} + \cdots + 3733 \)
|
| \(\nu^{11}\) | \(=\) |
\( 248 \beta_{11} - 3015 \beta_{10} - 3254 \beta_{9} - 1032 \beta_{8} + 5096 \beta_{7} + 8565 \beta_{6} + \cdots + 6241 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.73061 | 1.54691 | 5.45623 | 1.00000 | −4.22400 | −1.26445 | −9.43761 | −0.607075 | −2.73061 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.63994 | −2.12055 | 4.96928 | 1.00000 | 5.59814 | −4.06194 | −7.83873 | 1.49675 | −2.63994 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.80583 | 0.687917 | 1.26102 | 1.00000 | −1.24226 | −4.34193 | 1.33447 | −2.52677 | −1.80583 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.55555 | −3.00797 | 0.419729 | 1.00000 | 4.67904 | −3.45467 | 2.45819 | 6.04787 | −1.55555 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.55041 | −1.14040 | 0.403772 | 1.00000 | 1.76809 | 3.74146 | 2.47481 | −1.69949 | −1.55041 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.962871 | 2.64897 | −1.07288 | 1.00000 | −2.55062 | −3.09463 | 2.95879 | 4.01706 | −0.962871 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.301687 | 1.06101 | −1.90899 | 1.00000 | 0.320094 | 2.50984 | −1.17929 | −1.87425 | 0.301687 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.360254 | 0.0542373 | −1.87022 | 1.00000 | 0.0195392 | 0.298718 | −1.39426 | −2.99706 | 0.360254 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.747914 | −3.07503 | −1.44062 | 1.00000 | −2.29986 | −3.23262 | −2.57329 | 6.45581 | 0.747914 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.43840 | 0.109907 | 0.0689897 | 1.00000 | 0.158090 | −0.695085 | −2.77756 | −2.98792 | 1.43840 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.04505 | −3.19566 | 2.18224 | 1.00000 | −6.53528 | 1.17743 | 0.372688 | 7.21221 | 2.04505 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.35190 | −1.56935 | 3.53144 | 1.00000 | −3.69096 | −3.58212 | 3.60181 | −0.537139 | 2.35190 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(17\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1445.2.a.p | 12 | |
| 5.b | even | 2 | 1 | 7225.2.a.bs | 12 | ||
| 17.b | even | 2 | 1 | 1445.2.a.q | 12 | ||
| 17.c | even | 4 | 2 | 1445.2.d.j | 24 | ||
| 17.e | odd | 16 | 2 | 85.2.l.a | ✓ | 24 | |
| 51.i | even | 16 | 2 | 765.2.be.b | 24 | ||
| 85.c | even | 2 | 1 | 7225.2.a.bq | 12 | ||
| 85.o | even | 16 | 2 | 425.2.n.c | 24 | ||
| 85.p | odd | 16 | 2 | 425.2.m.b | 24 | ||
| 85.r | even | 16 | 2 | 425.2.n.f | 24 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 85.2.l.a | ✓ | 24 | 17.e | odd | 16 | 2 | |
| 425.2.m.b | 24 | 85.p | odd | 16 | 2 | ||
| 425.2.n.c | 24 | 85.o | even | 16 | 2 | ||
| 425.2.n.f | 24 | 85.r | even | 16 | 2 | ||
| 765.2.be.b | 24 | 51.i | even | 16 | 2 | ||
| 1445.2.a.p | 12 | 1.a | even | 1 | 1 | trivial | |
| 1445.2.a.q | 12 | 17.b | even | 2 | 1 | ||
| 1445.2.d.j | 24 | 17.c | even | 4 | 2 | ||
| 7225.2.a.bq | 12 | 85.c | even | 2 | 1 | ||
| 7225.2.a.bs | 12 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1445))\):
|
\( T_{2}^{12} + 4 T_{2}^{11} - 10 T_{2}^{10} - 52 T_{2}^{9} + 21 T_{2}^{8} + 232 T_{2}^{7} + 44 T_{2}^{6} + \cdots + 17 \)
|
|
\( T_{3}^{12} + 8 T_{3}^{11} + 8 T_{3}^{10} - 80 T_{3}^{9} - 186 T_{3}^{8} + 176 T_{3}^{7} + 680 T_{3}^{6} + \cdots + 2 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + 4 T^{11} + \cdots + 17 \)
|
| $3$ |
\( T^{12} + 8 T^{11} + \cdots + 2 \)
|
| $5$ |
\( (T - 1)^{12} \)
|
| $7$ |
\( T^{12} + 16 T^{11} + \cdots + 6338 \)
|
| $11$ |
\( T^{12} + 16 T^{11} + \cdots - 1598 \)
|
| $13$ |
\( T^{12} + 8 T^{11} + \cdots - 23932 \)
|
| $17$ |
\( T^{12} \)
|
| $19$ |
\( T^{12} - 92 T^{10} + \cdots - 3008 \)
|
| $23$ |
\( T^{12} + 16 T^{11} + \cdots - 2206 \)
|
| $29$ |
\( T^{12} + 16 T^{11} + \cdots - 6008 \)
|
| $31$ |
\( T^{12} + 24 T^{11} + \cdots + 352546 \)
|
| $37$ |
\( T^{12} + 24 T^{11} + \cdots + 26248 \)
|
| $41$ |
\( T^{12} + 8 T^{11} + \cdots - 427904 \)
|
| $43$ |
\( T^{12} + 16 T^{11} + \cdots + 63172 \)
|
| $47$ |
\( T^{12} + 32 T^{11} + \cdots + 45712388 \)
|
| $53$ |
\( T^{12} - 340 T^{10} + \cdots - 8342512 \)
|
| $59$ |
\( T^{12} + \cdots + 296882176 \)
|
| $61$ |
\( T^{12} + 24 T^{11} + \cdots - 12501472 \)
|
| $67$ |
\( T^{12} + \cdots - 961394684 \)
|
| $71$ |
\( T^{12} - 16 T^{11} + \cdots + 1907842 \)
|
| $73$ |
\( T^{12} + 16 T^{11} + \cdots + 15430176 \)
|
| $79$ |
\( T^{12} + \cdots + 23846133218 \)
|
| $83$ |
\( T^{12} + \cdots - 3068795644 \)
|
| $89$ |
\( T^{12} - 8 T^{11} + \cdots + 186436 \)
|
| $97$ |
\( T^{12} + \cdots - 3976256888 \)
|
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