Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4225,2,Mod(1,4225)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4225, 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("4225.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4225 = 5^{2} \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4225.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: | \(33.7367948540\) |
| Analytic rank: | \(0\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 17x^{7} - 9x^{6} + 59x^{5} + 32x^{4} - 44x^{3} - 23x^{2} + x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 845) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 17x^{7} - 9x^{6} + 59x^{5} + 32x^{4} - 44x^{3} - 23x^{2} + x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -27\nu^{8} - 10\nu^{7} + 554\nu^{6} + 310\nu^{5} - 2939\nu^{4} - 926\nu^{3} + 4655\nu^{2} + 904\nu + 446 ) / 1066 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 297 \nu^{8} - 110 \nu^{7} + 5028 \nu^{6} + 4476 \nu^{5} - 16339 \nu^{4} - 14450 \nu^{3} + \cdots - 424 ) / 1066 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 446 \nu^{8} - 27 \nu^{7} + 7572 \nu^{6} + 4568 \nu^{5} - 26004 \nu^{4} - 17211 \nu^{3} + \cdots + 458 ) / 1066 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 480 \nu^{8} + 296 \nu^{7} + 8013 \nu^{6} - 648 \nu^{5} - 28560 \nu^{4} + 2252 \nu^{3} + \cdots - 2435 ) / 1066 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 620 \nu^{8} - 27 \nu^{7} - 10550 \nu^{6} - 5026 \nu^{5} + 36890 \nu^{4} + 16901 \nu^{3} - 28206 \nu^{2} + \cdots + 1524 ) / 1066 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 1369 \nu^{8} - 480 \nu^{7} - 22977 \nu^{6} - 4308 \nu^{5} + 80123 \nu^{4} + 15248 \nu^{3} + \cdots + 621 ) / 1066 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 1793 \nu^{8} - 777 \nu^{7} - 30295 \nu^{6} - 3096 \nu^{5} + 109615 \nu^{4} + 12477 \nu^{3} + \cdots + 6725 ) / 1066 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} - 2\beta_{6} + \beta_{5} - \beta_{3} - 2\beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{8} - \beta_{6} - 3\beta_{5} - 2\beta_{4} - 3\beta_{2} + 9\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 9\beta_{7} - 27\beta_{6} + 5\beta_{5} - 3\beta_{4} - 16\beta_{3} - 27\beta_{2} + 6\beta _1 + 51 \)
|
| \(\nu^{5}\) | \(=\) |
\( -16\beta_{8} + 6\beta_{7} - 29\beta_{6} - 44\beta_{5} - 27\beta_{4} - 12\beta_{3} - 64\beta_{2} + 103\beta _1 + 72 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 12 \beta_{8} + 103 \beta_{7} - 354 \beta_{6} + 5 \beta_{5} - 64 \beta_{4} - 215 \beta_{3} + \cdots + 630 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 215 \beta_{8} + 154 \beta_{7} - 613 \beta_{6} - 555 \beta_{5} - 370 \beta_{4} - 316 \beta_{3} + \cdots + 1365 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 316 \beta_{8} + 1318 \beta_{7} - 4741 \beta_{6} - 466 \beta_{5} - 1091 \beta_{4} - 2863 \beta_{3} + \cdots + 8472 \)
|
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 |
|
−2.58648 | −0.884825 | 4.68989 | 0 | 2.28858 | 0.858383 | −6.95735 | −2.21708 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.28079 | −3.21428 | 3.20199 | 0 | 7.33108 | 2.30369 | −2.74149 | 7.33158 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.04721 | 2.75868 | −0.903360 | 0 | −2.88890 | 3.42366 | 3.04042 | 4.61031 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.271374 | 0.319618 | −1.92636 | 0 | −0.0867358 | −3.38151 | 1.06551 | −2.89784 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.0240266 | −2.93017 | −1.99942 | 0 | −0.0704021 | −1.66541 | −0.0960927 | 5.58589 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.53088 | −2.88726 | 0.343581 | 0 | −4.42003 | 3.86493 | −2.53577 | 5.33625 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.20556 | 0.0130567 | 2.86449 | 0 | 0.0287972 | 4.60897 | 1.90669 | −2.99983 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.63597 | −1.98944 | 4.94835 | 0 | −5.24412 | −3.28231 | 7.77176 | 0.957886 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.78942 | 1.81462 | 5.78084 | 0 | 5.06173 | 0.269601 | 10.5463 | 0.292842 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(13\) | \(-1\) |
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 | 4225.2.a.bt | 9 | |
| 5.b | even | 2 | 1 | 845.2.a.n | ✓ | 9 | |
| 13.b | even | 2 | 1 | 4225.2.a.bs | 9 | ||
| 15.d | odd | 2 | 1 | 7605.2.a.cs | 9 | ||
| 65.d | even | 2 | 1 | 845.2.a.o | yes | 9 | |
| 65.g | odd | 4 | 2 | 845.2.c.h | 18 | ||
| 65.l | even | 6 | 2 | 845.2.e.o | 18 | ||
| 65.n | even | 6 | 2 | 845.2.e.p | 18 | ||
| 65.s | odd | 12 | 4 | 845.2.m.j | 36 | ||
| 195.e | odd | 2 | 1 | 7605.2.a.cp | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 845.2.a.n | ✓ | 9 | 5.b | even | 2 | 1 | |
| 845.2.a.o | yes | 9 | 65.d | even | 2 | 1 | |
| 845.2.c.h | 18 | 65.g | odd | 4 | 2 | ||
| 845.2.e.o | 18 | 65.l | even | 6 | 2 | ||
| 845.2.e.p | 18 | 65.n | even | 6 | 2 | ||
| 845.2.m.j | 36 | 65.s | odd | 12 | 4 | ||
| 4225.2.a.bs | 9 | 13.b | even | 2 | 1 | ||
| 4225.2.a.bt | 9 | 1.a | even | 1 | 1 | trivial | |
| 7605.2.a.cp | 9 | 195.e | odd | 2 | 1 | ||
| 7605.2.a.cs | 9 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4225))\):
|
\( T_{2}^{9} - 3T_{2}^{8} - 13T_{2}^{7} + 40T_{2}^{6} + 49T_{2}^{5} - 156T_{2}^{4} - 51T_{2}^{3} + 152T_{2}^{2} + 38T_{2} - 1 \)
|
|
\( T_{3}^{9} + 7T_{3}^{8} + 3T_{3}^{7} - 72T_{3}^{6} - 130T_{3}^{5} + 129T_{3}^{4} + 385T_{3}^{3} + 103T_{3}^{2} - 78T_{3} + 1 \)
|
|
\( T_{7}^{9} - 7T_{7}^{8} - 15T_{7}^{7} + 176T_{7}^{6} - 58T_{7}^{5} - 1305T_{7}^{4} + 1385T_{7}^{3} + 2321T_{7}^{2} - 2930T_{7} + 601 \)
|
|
\( T_{11}^{9} - 9 T_{11}^{8} - 12 T_{11}^{7} + 241 T_{11}^{6} - 76 T_{11}^{5} - 2176 T_{11}^{4} + \cdots - 8896 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} - 3 T^{8} + \cdots - 1 \)
|
| $3$ |
\( T^{9} + 7 T^{8} + \cdots + 1 \)
|
| $5$ |
\( T^{9} \)
|
| $7$ |
\( T^{9} - 7 T^{8} + \cdots + 601 \)
|
| $11$ |
\( T^{9} - 9 T^{8} + \cdots - 8896 \)
|
| $13$ |
\( T^{9} \)
|
| $17$ |
\( T^{9} - T^{8} + \cdots - 6656 \)
|
| $19$ |
\( T^{9} - 4 T^{8} + \cdots - 10816 \)
|
| $23$ |
\( T^{9} + 14 T^{8} + \cdots + 43693 \)
|
| $29$ |
\( T^{9} - 12 T^{8} + \cdots + 1 \)
|
| $31$ |
\( T^{9} - 7 T^{8} + \cdots + 10816 \)
|
| $37$ |
\( T^{9} + 5 T^{8} + \cdots + 10018112 \)
|
| $41$ |
\( T^{9} - 10 T^{8} + \cdots - 993287 \)
|
| $43$ |
\( T^{9} + 39 T^{8} + \cdots + 71513 \)
|
| $47$ |
\( T^{9} - 36 T^{8} + \cdots + 11677 \)
|
| $53$ |
\( T^{9} - 8 T^{8} + \cdots + 4469312 \)
|
| $59$ |
\( T^{9} - 21 T^{8} + \cdots + 272896 \)
|
| $61$ |
\( T^{9} + 3 T^{8} + \cdots - 183247 \)
|
| $67$ |
\( T^{9} - T^{8} + \cdots + 115123 \)
|
| $71$ |
\( T^{9} - T^{8} + \cdots + 5586496 \)
|
| $73$ |
\( T^{9} - 256 T^{7} + \cdots - 866816 \)
|
| $79$ |
\( T^{9} - 39 T^{8} + \cdots - 10816 \)
|
| $83$ |
\( T^{9} - 7 T^{8} + \cdots - 49784561 \)
|
| $89$ |
\( T^{9} - 19 T^{8} + \cdots + 1049 \)
|
| $97$ |
\( T^{9} + 34 T^{8} + \cdots - 106251776 \)
|
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