Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7581,2,Mod(1,7581)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7581, 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("7581.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7581 = 3 \cdot 7 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7581.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: | \(60.5345897723\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{12} - 3 x^{11} - 14 x^{10} + 45 x^{9} + 60 x^{8} - 229 x^{7} - 61 x^{6} + 453 x^{5} - 85 x^{4} + \cdots - 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{12} - 3 x^{11} - 14 x^{10} + 45 x^{9} + 60 x^{8} - 229 x^{7} - 61 x^{6} + 453 x^{5} - 85 x^{4} + \cdots - 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 211 \nu^{11} - 740 \nu^{10} - 3094 \nu^{9} + 11293 \nu^{8} + 14519 \nu^{7} - 57342 \nu^{6} + \cdots + 2013 ) / 6040 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 671 \nu^{11} + 1380 \nu^{10} + 11614 \nu^{9} - 20913 \nu^{8} - 74139 \nu^{7} + 110102 \nu^{6} + \cdots - 47193 ) / 18120 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 601 \nu^{11} - 1020 \nu^{10} - 9464 \nu^{9} + 13803 \nu^{8} + 50229 \nu^{7} - 59662 \nu^{6} + \cdots - 1437 ) / 4530 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 1101 \nu^{11} - 140 \nu^{10} - 21154 \nu^{9} - 557 \nu^{8} + 146809 \nu^{7} + 19678 \nu^{6} + \cdots - 517 ) / 6040 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 5371 \nu^{11} - 10020 \nu^{10} - 87374 \nu^{9} + 146253 \nu^{8} + 493959 \nu^{7} - 716662 \nu^{6} + \cdots + 81813 ) / 18120 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 659 \nu^{11} + 930 \nu^{10} + 11116 \nu^{9} - 13007 \nu^{8} - 66071 \nu^{7} + 60038 \nu^{6} + \cdots - 5507 ) / 1510 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 2339 \nu^{11} - 5040 \nu^{10} - 35536 \nu^{9} + 71907 \nu^{8} + 179361 \nu^{7} - 337418 \nu^{6} + \cdots + 14457 ) / 4530 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 1687 \nu^{11} - 3240 \nu^{10} - 26598 \nu^{9} + 46021 \nu^{8} + 143163 \nu^{7} - 214474 \nu^{6} + \cdots - 6119 ) / 3020 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 1333 \nu^{11} - 2196 \nu^{10} - 21722 \nu^{9} + 30687 \nu^{8} + 123201 \nu^{7} - 140074 \nu^{6} + \cdots + 8535 ) / 1812 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} - \beta_{9} - \beta_{6} + \beta_{4} + 6\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{11} - \beta_{10} + \beta_{8} + \beta_{7} + \beta_{4} + 7\beta_{2} + \beta _1 + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9 \beta_{11} - 9 \beta_{9} + 2 \beta_{8} + 2 \beta_{7} - 8 \beta_{6} + \beta_{5} + 9 \beta_{4} + \cdots + 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( 12 \beta_{11} - 9 \beta_{10} - 2 \beta_{9} + 13 \beta_{8} + 12 \beta_{7} - \beta_{5} + 14 \beta_{4} + \cdots + 99 \)
|
| \(\nu^{7}\) | \(=\) |
\( 70 \beta_{11} + \beta_{10} - 69 \beta_{9} + 27 \beta_{8} + 25 \beta_{7} - 56 \beta_{6} + 9 \beta_{5} + \cdots + 98 \)
|
| \(\nu^{8}\) | \(=\) |
\( 109 \beta_{11} - 65 \beta_{10} - 31 \beta_{9} + 121 \beta_{8} + 113 \beta_{7} - 4 \beta_{6} - 17 \beta_{5} + \cdots + 652 \)
|
| \(\nu^{9}\) | \(=\) |
\( 522 \beta_{11} + 13 \beta_{10} - 503 \beta_{9} + 265 \beta_{8} + 239 \beta_{7} - 382 \beta_{6} + \cdots + 810 \)
|
| \(\nu^{10}\) | \(=\) |
\( 902 \beta_{11} - 438 \beta_{10} - 337 \beta_{9} + 1007 \beta_{8} + 968 \beta_{7} - 72 \beta_{6} + \cdots + 4449 \)
|
| \(\nu^{11}\) | \(=\) |
\( 3835 \beta_{11} + 122 \beta_{10} - 3604 \beta_{9} + 2312 \beta_{8} + 2088 \beta_{7} - 2597 \beta_{6} + \cdots + 6462 \)
|
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 |
|
−2.52755 | 1.00000 | 4.38853 | −0.933729 | −2.52755 | 1.00000 | −6.03715 | 1.00000 | 2.36005 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.19502 | 1.00000 | 2.81812 | −3.27364 | −2.19502 | 1.00000 | −1.79580 | 1.00000 | 7.18570 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.73328 | 1.00000 | 1.00426 | 1.40955 | −1.73328 | 1.00000 | 1.72590 | 1.00000 | −2.44314 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.766288 | 1.00000 | −1.41280 | −3.61536 | −0.766288 | 1.00000 | 2.61519 | 1.00000 | 2.77041 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.496366 | 1.00000 | −1.75362 | 3.09192 | −0.496366 | 1.00000 | 1.86317 | 1.00000 | −1.53473 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.153964 | 1.00000 | −1.97629 | −3.86145 | 0.153964 | 1.00000 | −0.612208 | 1.00000 | −0.594527 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.703299 | 1.00000 | −1.50537 | 0.290190 | 0.703299 | 1.00000 | −2.46532 | 1.00000 | 0.204090 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.23150 | 1.00000 | −0.483402 | −1.93978 | 1.23150 | 1.00000 | −3.05832 | 1.00000 | −2.38884 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.31299 | 1.00000 | −0.276070 | 2.07532 | 1.31299 | 1.00000 | −2.98845 | 1.00000 | 2.72486 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.96094 | 1.00000 | 1.84529 | 0.320388 | 1.96094 | 1.00000 | −0.303379 | 1.00000 | 0.628261 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.61097 | 1.00000 | 4.81716 | 0.671563 | 2.61097 | 1.00000 | 7.35553 | 1.00000 | 1.75343 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.74485 | 1.00000 | 5.53419 | 3.76503 | 2.74485 | 1.00000 | 9.70083 | 1.00000 | 10.3344 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 7581.2.a.bo | yes | 12 |
| 19.b | odd | 2 | 1 | 7581.2.a.bj | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7581.2.a.bj | ✓ | 12 | 19.b | odd | 2 | 1 | |
| 7581.2.a.bo | yes | 12 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7581))\):
|
\( T_{2}^{12} - 3 T_{2}^{11} - 14 T_{2}^{10} + 45 T_{2}^{9} + 60 T_{2}^{8} - 229 T_{2}^{7} - 61 T_{2}^{6} + \cdots - 9 \)
|
|
\( T_{5}^{12} + 2 T_{5}^{11} - 35 T_{5}^{10} - 55 T_{5}^{9} + 437 T_{5}^{8} + 442 T_{5}^{7} - 2348 T_{5}^{6} + \cdots - 176 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} - 3 T^{11} + \cdots - 9 \)
|
| $3$ |
\( (T - 1)^{12} \)
|
| $5$ |
\( T^{12} + 2 T^{11} + \cdots - 176 \)
|
| $7$ |
\( (T - 1)^{12} \)
|
| $11$ |
\( T^{12} - 77 T^{10} + \cdots - 26224 \)
|
| $13$ |
\( T^{12} - 4 T^{11} + \cdots - 2862144 \)
|
| $17$ |
\( T^{12} + 8 T^{11} + \cdots - 106304 \)
|
| $19$ |
\( T^{12} \)
|
| $23$ |
\( T^{12} + 8 T^{11} + \cdots - 8452036 \)
|
| $29$ |
\( T^{12} - 267 T^{10} + \cdots - 64363100 \)
|
| $31$ |
\( T^{12} - 15 T^{11} + \cdots + 55510976 \)
|
| $37$ |
\( T^{12} - 5 T^{11} + \cdots - 1983344 \)
|
| $41$ |
\( T^{12} - 13 T^{11} + \cdots - 1647056 \)
|
| $43$ |
\( T^{12} + 2 T^{11} + \cdots + 21324196 \)
|
| $47$ |
\( T^{12} + \cdots - 4660818944 \)
|
| $53$ |
\( T^{12} + \cdots + 4118478121 \)
|
| $59$ |
\( T^{12} + \cdots - 815883520 \)
|
| $61$ |
\( T^{12} - 19 T^{11} + \cdots + 668176 \)
|
| $67$ |
\( T^{12} - 11 T^{11} + \cdots - 1161616 \)
|
| $71$ |
\( T^{12} + \cdots + 3757095936 \)
|
| $73$ |
\( T^{12} + \cdots + 332467344 \)
|
| $79$ |
\( T^{12} + \cdots - 6052014320 \)
|
| $83$ |
\( T^{12} + \cdots - 179804736 \)
|
| $89$ |
\( T^{12} - 15 T^{11} + \cdots + 57588880 \)
|
| $97$ |
\( T^{12} + \cdots + 6690018517264 \)
|
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