Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9295,2,Mod(1,9295)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9295, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("9295.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9295 = 5 \cdot 11 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9295.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: | \(74.2209486788\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{10} - x^{9} - 13x^{8} + 10x^{7} + 56x^{6} - 31x^{5} - 94x^{4} + 29x^{3} + 57x^{2} - 8x - 9 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 715) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - x^{9} - 13x^{8} + 10x^{7} + 56x^{6} - 31x^{5} - 94x^{4} + 29x^{3} + 57x^{2} - 8x - 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 8\nu^{9} - 15\nu^{8} - 87\nu^{7} + 160\nu^{6} + 277\nu^{5} - 533\nu^{4} - 274\nu^{3} + 526\nu^{2} + 143\nu - 69 ) / 31 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{9} - 25\nu^{8} - 21\nu^{7} + 277\nu^{6} + 7\nu^{5} - 878\nu^{4} + 184\nu^{3} + 732\nu^{2} - 175\nu - 84 ) / 31 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 6 \nu^{9} + 19 \nu^{8} + 73 \nu^{7} - 213 \nu^{6} - 293 \nu^{5} + 702 \nu^{4} + 407 \nu^{3} + \cdots + 106 ) / 31 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 4\nu^{9} + 8\nu^{8} - 59\nu^{7} - 106\nu^{6} + 278\nu^{5} + 431\nu^{4} - 509\nu^{3} - 605\nu^{2} + 304\nu + 167 ) / 31 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -12\nu^{9} + 7\nu^{8} + 146\nu^{7} - 54\nu^{6} - 555\nu^{5} + 133\nu^{4} + 752\nu^{3} - 107\nu^{2} - 323\nu + 26 ) / 31 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 22 \nu^{9} - 18 \nu^{8} - 278 \nu^{7} + 161 \nu^{6} + 1126 \nu^{5} - 404 \nu^{4} - 1637 \nu^{3} + \cdots + 4 ) / 31 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 28 \nu^{9} - 37 \nu^{8} - 320 \nu^{7} + 374 \nu^{6} + 1078 \nu^{5} - 1168 \nu^{4} - 1052 \nu^{3} + \cdots - 195 ) / 31 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} + 2\beta_{7} + \beta_{5} + \beta_{3} + 6\beta_{2} + \beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{9} + 9\beta_{8} + 9\beta_{7} - 8\beta_{6} + 9\beta_{5} + 3\beta_{3} + 2\beta_{2} + 27\beta _1 + 10 \)
|
| \(\nu^{6}\) | \(=\) |
\( 12\beta_{8} + 23\beta_{7} - 3\beta_{6} + 11\beta_{5} - 2\beta_{4} + 12\beta_{3} + 36\beta_{2} + 15\beta _1 + 87 \)
|
| \(\nu^{7}\) | \(=\) |
\( -10\beta_{9} + 68\beta_{8} + 71\beta_{7} - 56\beta_{6} + 70\beta_{5} + 35\beta_{3} + 27\beta_{2} + 158\beta _1 + 90 \)
|
| \(\nu^{8}\) | \(=\) |
\( - \beta_{9} + 110 \beta_{8} + 200 \beta_{7} - 42 \beta_{6} + 100 \beta_{5} - 24 \beta_{4} + 106 \beta_{3} + \cdots + 548 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 76 \beta_{9} + 495 \beta_{8} + 543 \beta_{7} - 385 \beta_{6} + 518 \beta_{5} - 5 \beta_{4} + \cdots + 765 \)
|
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 |
|
−2.72755 | 1.45503 | 5.43954 | 1.00000 | −3.96867 | 4.18645 | −9.38152 | −0.882884 | −2.72755 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.87771 | −1.25504 | 1.52578 | 1.00000 | 2.35660 | −3.15071 | 0.890440 | −1.42487 | −1.87771 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.76174 | 2.61070 | 1.10373 | 1.00000 | −4.59938 | −1.68993 | 1.57899 | 3.81575 | −1.76174 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.901652 | −1.48451 | −1.18702 | 1.00000 | 1.33851 | −3.42196 | 2.87359 | −0.796232 | −0.901652 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.545526 | −1.92954 | −1.70240 | 1.00000 | 1.05261 | 3.49549 | 2.01976 | 0.723126 | −0.545526 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.419122 | 1.47716 | −1.82434 | 1.00000 | 0.619109 | 2.78382 | −1.60286 | −0.818002 | 0.419122 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.898362 | −3.10891 | −1.19295 | 1.00000 | −2.79292 | −0.237406 | −2.86842 | 6.66530 | 0.898362 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.14332 | −0.329530 | −0.692813 | 1.00000 | −0.376759 | 0.757607 | −3.07875 | −2.89141 | 1.14332 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.01155 | 1.84562 | 2.04633 | 1.00000 | 3.71256 | −2.81697 | 0.0931932 | 0.406326 | 2.01155 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.34182 | −2.28099 | 3.48414 | 1.00000 | −5.34167 | −2.90640 | 3.47559 | 2.20290 | 2.34182 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(11\) | \(-1\) |
| \(13\) | \(1\) |
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 | 9295.2.a.v | 10 | |
| 13.b | even | 2 | 1 | 9295.2.a.w | 10 | ||
| 13.e | even | 6 | 2 | 715.2.i.c | ✓ | 20 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 715.2.i.c | ✓ | 20 | 13.e | even | 6 | 2 | |
| 9295.2.a.v | 10 | 1.a | even | 1 | 1 | trivial | |
| 9295.2.a.w | 10 | 13.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9295))\):
|
\( T_{2}^{10} + T_{2}^{9} - 13T_{2}^{8} - 10T_{2}^{7} + 56T_{2}^{6} + 31T_{2}^{5} - 94T_{2}^{4} - 29T_{2}^{3} + 57T_{2}^{2} + 8T_{2} - 9 \)
|
|
\( T_{3}^{10} + 3T_{3}^{9} - 14T_{3}^{8} - 43T_{3}^{7} + 63T_{3}^{6} + 210T_{3}^{5} - 99T_{3}^{4} - 424T_{3}^{3} - T_{3}^{2} + 304T_{3} + 87 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + T^{9} + \cdots - 9 \)
|
| $3$ |
\( T^{10} + 3 T^{9} + \cdots + 87 \)
|
| $5$ |
\( (T - 1)^{10} \)
|
| $7$ |
\( T^{10} + 3 T^{9} + \cdots + 1093 \)
|
| $11$ |
\( (T - 1)^{10} \)
|
| $13$ |
\( T^{10} \)
|
| $17$ |
\( T^{10} + 17 T^{9} + \cdots - 4779 \)
|
| $19$ |
\( T^{10} - 7 T^{9} + \cdots - 15529 \)
|
| $23$ |
\( T^{10} + 16 T^{9} + \cdots + 82368 \)
|
| $29$ |
\( T^{10} + 17 T^{9} + \cdots - 286476 \)
|
| $31$ |
\( T^{10} + 9 T^{9} + \cdots + 492979 \)
|
| $37$ |
\( T^{10} + 20 T^{9} + \cdots - 48992 \)
|
| $41$ |
\( T^{10} + T^{9} + \cdots - 26973 \)
|
| $43$ |
\( T^{10} - 3 T^{9} + \cdots + 5709 \)
|
| $47$ |
\( T^{10} + 6 T^{9} + \cdots - 1771971 \)
|
| $53$ |
\( T^{10} + 22 T^{9} + \cdots + 301419 \)
|
| $59$ |
\( T^{10} - 319 T^{8} + \cdots - 48938151 \)
|
| $61$ |
\( T^{10} + \cdots + 107510464 \)
|
| $67$ |
\( T^{10} + 31 T^{9} + \cdots - 673049 \)
|
| $71$ |
\( T^{10} - 16 T^{9} + \cdots + 23560149 \)
|
| $73$ |
\( T^{10} + \cdots - 21074086811 \)
|
| $79$ |
\( T^{10} - T^{9} + \cdots - 3073601 \)
|
| $83$ |
\( T^{10} + \cdots - 1237170999 \)
|
| $89$ |
\( T^{10} + \cdots + 9637702773 \)
|
| $97$ |
\( T^{10} + \cdots - 616005687 \)
|
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