Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4522,2,Mod(1,4522)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4522, 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("4522.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4522 = 2 \cdot 7 \cdot 17 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4522.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: | \(36.1083517940\) |
| Analytic rank: | \(0\) |
| Dimension: | \(11\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{11} - 25x^{9} - 7x^{8} + 217x^{7} + 127x^{6} - 745x^{5} - 650x^{4} + 785x^{3} + 776x^{2} - 240x - 256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - 25x^{9} - 7x^{8} + 217x^{7} + 127x^{6} - 745x^{5} - 650x^{4} + 785x^{3} + 776x^{2} - 240x - 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 79 \nu^{10} + 68 \nu^{9} + 1919 \nu^{8} - 1095 \nu^{7} - 16253 \nu^{6} + 3893 \nu^{5} + 55849 \nu^{4} + \cdots + 23872 ) / 50 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 373 \nu^{10} - 316 \nu^{9} - 9053 \nu^{8} + 5065 \nu^{7} + 76561 \nu^{6} - 17641 \nu^{5} + \cdots - 110064 ) / 200 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 679 \nu^{10} + 568 \nu^{9} + 16519 \nu^{8} - 9095 \nu^{7} - 140103 \nu^{6} + 31543 \nu^{5} + \cdots + 202872 ) / 200 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 679 \nu^{10} + 568 \nu^{9} + 16519 \nu^{8} - 9095 \nu^{7} - 140103 \nu^{6} + 31543 \nu^{5} + \cdots + 203872 ) / 200 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 1773 \nu^{10} - 1616 \nu^{9} - 42853 \nu^{8} + 26565 \nu^{7} + 360661 \nu^{6} - 101741 \nu^{5} + \cdots - 518064 ) / 400 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 1917 \nu^{10} + 1664 \nu^{9} + 46437 \nu^{8} - 26885 \nu^{7} - 391669 \nu^{6} + 96589 \nu^{5} + \cdots + 557056 ) / 400 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 1487 \nu^{10} + 1304 \nu^{9} + 36007 \nu^{8} - 21135 \nu^{7} - 303559 \nu^{6} + 76879 \nu^{5} + \cdots + 429216 ) / 200 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 3131 \nu^{10} - 2752 \nu^{9} - 75891 \nu^{8} + 44755 \nu^{7} + 640867 \nu^{6} - 164827 \nu^{5} + \cdots - 923408 ) / 400 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 5769 \nu^{10} - 5048 \nu^{9} - 139809 \nu^{8} + 81945 \nu^{7} + 1180233 \nu^{6} - 299873 \nu^{5} + \cdots - 1693792 ) / 400 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + \beta _1 + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{9} + \beta_{6} - \beta_{4} + 7\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{8} + 2\beta_{7} - 11\beta_{5} + 11\beta_{4} + 2\beta_{3} + \beta_{2} + 11\beta _1 + 39 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 2 \beta_{10} - 10 \beta_{9} - \beta_{7} + 13 \beta_{6} - \beta_{5} - 12 \beta_{4} + 2 \beta_{3} + \cdots + 13 \)
|
| \(\nu^{6}\) | \(=\) |
\( - \beta_{9} - 15 \beta_{8} + 33 \beta_{7} + 2 \beta_{6} - 110 \beta_{5} + 110 \beta_{4} + 38 \beta_{3} + \cdots + 339 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 31 \beta_{10} - 94 \beta_{9} - 12 \beta_{7} + 143 \beta_{6} - 20 \beta_{5} - 124 \beta_{4} + \cdots + 127 \)
|
| \(\nu^{8}\) | \(=\) |
\( 4 \beta_{10} - 24 \beta_{9} - 175 \beta_{8} + 413 \beta_{7} + 39 \beta_{6} - 1084 \beta_{5} + \cdots + 3057 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 357 \beta_{10} - 902 \beta_{9} + 8 \beta_{8} - 108 \beta_{7} + 1493 \beta_{6} - 276 \beta_{5} + \cdots + 1153 \)
|
| \(\nu^{10}\) | \(=\) |
\( 121 \beta_{10} - 389 \beta_{9} - 1865 \beta_{8} + 4681 \beta_{7} + 525 \beta_{6} - 10653 \beta_{5} + \cdots + 28136 \)
|
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 |
|
1.00000 | −3.13332 | 1.00000 | −0.710133 | −3.13332 | −1.00000 | 1.00000 | 6.81768 | −0.710133 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.60324 | 1.00000 | 4.38904 | −2.60324 | −1.00000 | 1.00000 | 3.77686 | 4.38904 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.57659 | 1.00000 | −1.85635 | −1.57659 | −1.00000 | 1.00000 | −0.514372 | −1.85635 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −1.44318 | 1.00000 | −3.11494 | −1.44318 | −1.00000 | 1.00000 | −0.917218 | −3.11494 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | −0.873941 | 1.00000 | 0.875943 | −0.873941 | −1.00000 | 1.00000 | −2.23623 | 0.875943 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | −0.869101 | 1.00000 | 3.27237 | −0.869101 | −1.00000 | 1.00000 | −2.24466 | 3.27237 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 0.754497 | 1.00000 | −3.68402 | 0.754497 | −1.00000 | 1.00000 | −2.43073 | −3.68402 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 0.969860 | 1.00000 | −0.828379 | 0.969860 | −1.00000 | 1.00000 | −2.05937 | −0.828379 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.00000 | 2.65124 | 1.00000 | 1.57496 | 2.65124 | −1.00000 | 1.00000 | 4.02906 | 1.57496 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.00000 | 2.94214 | 1.00000 | 2.69528 | 2.94214 | −1.00000 | 1.00000 | 5.65622 | 2.69528 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.00000 | 3.18163 | 1.00000 | −1.61378 | 3.18163 | −1.00000 | 1.00000 | 7.12278 | −1.61378 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
| \(17\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 4522.2.a.be | ✓ | 11 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4522.2.a.be | ✓ | 11 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4522))\):
|
\( T_{3}^{11} - 25 T_{3}^{9} - 7 T_{3}^{8} + 217 T_{3}^{7} + 127 T_{3}^{6} - 745 T_{3}^{5} - 650 T_{3}^{4} + \cdots - 256 \)
|
|
\( T_{5}^{11} - T_{5}^{10} - 35 T_{5}^{9} + 18 T_{5}^{8} + 419 T_{5}^{7} - 37 T_{5}^{6} - 2040 T_{5}^{5} + \cdots - 1080 \)
|
|
\( T_{11}^{11} - 3 T_{11}^{10} - 61 T_{11}^{9} + 187 T_{11}^{8} + 1039 T_{11}^{7} - 2852 T_{11}^{6} + \cdots + 480 \)
|
|
\( T_{13}^{11} - 8 T_{13}^{10} - 36 T_{13}^{9} + 343 T_{13}^{8} + 422 T_{13}^{7} - 4769 T_{13}^{6} + \cdots - 128 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{11} \)
|
| $3$ |
\( T^{11} - 25 T^{9} + \cdots - 256 \)
|
| $5$ |
\( T^{11} - T^{10} + \cdots - 1080 \)
|
| $7$ |
\( (T + 1)^{11} \)
|
| $11$ |
\( T^{11} - 3 T^{10} + \cdots + 480 \)
|
| $13$ |
\( T^{11} - 8 T^{10} + \cdots - 128 \)
|
| $17$ |
\( (T - 1)^{11} \)
|
| $19$ |
\( (T - 1)^{11} \)
|
| $23$ |
\( T^{11} + 6 T^{10} + \cdots - 1792 \)
|
| $29$ |
\( T^{11} - 187 T^{9} + \cdots + 297696 \)
|
| $31$ |
\( T^{11} + 7 T^{10} + \cdots - 4438528 \)
|
| $37$ |
\( T^{11} - 3 T^{10} + \cdots - 5473152 \)
|
| $41$ |
\( T^{11} - 16 T^{10} + \cdots - 5350080 \)
|
| $43$ |
\( T^{11} + 10 T^{10} + \cdots - 74473472 \)
|
| $47$ |
\( T^{11} - 22 T^{10} + \cdots + 14798464 \)
|
| $53$ |
\( T^{11} + \cdots - 995605792 \)
|
| $59$ |
\( T^{11} + \cdots - 541334400 \)
|
| $61$ |
\( T^{11} + \cdots + 119876656 \)
|
| $67$ |
\( T^{11} + T^{10} + \cdots + 132736 \)
|
| $71$ |
\( T^{11} + \cdots + 207271936 \)
|
| $73$ |
\( T^{11} - 9 T^{10} + \cdots + 41801280 \)
|
| $79$ |
\( T^{11} - 11 T^{10} + \cdots - 18486272 \)
|
| $83$ |
\( T^{11} + \cdots + 28397983616 \)
|
| $89$ |
\( T^{11} - 32 T^{10} + \cdots + 10980004 \)
|
| $97$ |
\( T^{11} + \cdots - 302198784 \)
|
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