Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [29,12,Mod(1,29)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(29, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("29.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 29 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 29.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: | \(22.2819522362\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{11} - x^{10} - 15790 x^{9} + 14666 x^{8} + 87206462 x^{7} - 14008334 x^{6} - 203974096304 x^{5} + \cdots - 75\!\cdots\!58 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{13}\cdot 3^{2} \) |
| 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
\( x^{11} - x^{10} - 15790 x^{9} + 14666 x^{8} + 87206462 x^{7} - 14008334 x^{6} - 203974096304 x^{5} + \cdots - 75\!\cdots\!58 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 56\!\cdots\!01 \nu^{10} + \cdots - 20\!\cdots\!66 ) / 44\!\cdots\!80 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 56\!\cdots\!01 \nu^{10} + \cdots - 20\!\cdots\!86 ) / 44\!\cdots\!80 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 86\!\cdots\!11 \nu^{10} + \cdots + 31\!\cdots\!78 ) / 22\!\cdots\!40 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 18\!\cdots\!63 \nu^{10} + \cdots + 67\!\cdots\!06 ) / 44\!\cdots\!80 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 84\!\cdots\!85 \nu^{10} + \cdots - 30\!\cdots\!82 ) / 14\!\cdots\!40 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 92\!\cdots\!01 \nu^{10} + \cdots + 33\!\cdots\!10 ) / 14\!\cdots\!96 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 27\!\cdots\!59 \nu^{10} + \cdots + 10\!\cdots\!26 ) / 44\!\cdots\!80 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 33\!\cdots\!19 \nu^{10} + \cdots - 12\!\cdots\!98 ) / 44\!\cdots\!80 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 53\!\cdots\!49 \nu^{10} + \cdots - 19\!\cdots\!26 ) / 56\!\cdots\!60 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 2871 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2 \beta_{9} + 6 \beta_{8} + 5 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} - 4 \beta_{4} + 91 \beta_{3} + \cdots - 142 \)
|
| \(\nu^{4}\) | \(=\) |
\( 290 \beta_{10} - 215 \beta_{9} + 96 \beta_{8} + 133 \beta_{7} + 238 \beta_{6} + 283 \beta_{5} + \cdots + 13621517 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 15382 \beta_{10} + 9579 \beta_{9} + 40210 \beta_{8} + 35952 \beta_{7} + 33935 \beta_{6} + \cdots - 27734024 \)
|
| \(\nu^{6}\) | \(=\) |
\( 3055256 \beta_{10} - 2314236 \beta_{9} + 541720 \beta_{8} + 1625600 \beta_{7} + 2571828 \beta_{6} + \cdots + 75938887363 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 179336896 \beta_{10} + 49362454 \beta_{9} + 251488850 \beta_{8} + 235468575 \beta_{7} + \cdots - 258006933466 \)
|
| \(\nu^{8}\) | \(=\) |
\( 24799739046 \beta_{10} - 18301020869 \beta_{9} + 1812340208 \beta_{8} + 14744018455 \beta_{7} + \cdots + 456567119229865 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 1548427764434 \beta_{10} + 309797115173 \beta_{9} + 1598854878186 \beta_{8} + 1528245562510 \beta_{7} + \cdots - 20\!\cdots\!84 \)
|
| \(\nu^{10}\) | \(=\) |
\( 184293327453748 \beta_{10} - 131088051301278 \beta_{9} + 413168836200 \beta_{8} + 117756171358734 \beta_{7} + \cdots + 28\!\cdots\!91 \)
|
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 |
|
−85.7218 | 234.308 | 5300.23 | −1506.26 | −20085.3 | −7069.49 | −278787. | −122247. | 129120. | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −68.3340 | 537.644 | 2621.54 | 8265.40 | −36739.4 | −68659.4 | −39192.0 | 111914. | −564808. | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −59.7551 | −392.705 | 1522.68 | −6379.08 | 23466.1 | −23005.8 | 31390.8 | −22929.7 | 381183. | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −23.7638 | −595.785 | −1483.28 | −5297.67 | 14158.1 | 80499.9 | 83916.8 | 177813. | 125893. | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −23.7036 | −804.018 | −1486.14 | 6643.74 | 19058.1 | −61145.1 | 83771.9 | 469298. | −157481. | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −21.9987 | 310.394 | −1564.06 | −886.321 | −6828.28 | 44263.6 | 79460.6 | −80802.4 | 19497.9 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 27.4581 | −457.582 | −1294.05 | 9990.24 | −12564.3 | 28164.0 | −91766.4 | 32234.4 | 274313. | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 30.9131 | 543.939 | −1092.38 | −6165.86 | 16814.8 | 8452.88 | −97078.9 | 118722. | −190606. | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 51.9918 | 135.432 | 655.142 | 3682.79 | 7041.33 | −83752.5 | −72417.1 | −158805. | 191474. | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 62.2743 | −384.675 | 1830.09 | 1377.47 | −23955.4 | 33050.3 | −13570.0 | −29172.4 | 85780.8 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 78.6399 | −108.951 | 4136.23 | −12464.4 | −8567.92 | −230.276 | 164218. | −165277. | −980202. | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(29\) | \(-1\) |
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 | 29.12.a.a | ✓ | 11 |
| 3.b | odd | 2 | 1 | 261.12.a.a | 11 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 29.12.a.a | ✓ | 11 | 1.a | even | 1 | 1 | trivial |
| 261.12.a.a | 11 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{11} + 32 T_{2}^{10} - 15325 T_{2}^{9} - 407614 T_{2}^{8} + 82465976 T_{2}^{7} + \cdots - 93\!\cdots\!40 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(29))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} + \cdots - 93\!\cdots\!40 \)
|
| $3$ |
\( T^{11} + \cdots - 10\!\cdots\!92 \)
|
| $5$ |
\( T^{11} + \cdots - 96\!\cdots\!50 \)
|
| $7$ |
\( T^{11} + \cdots - 36\!\cdots\!16 \)
|
| $11$ |
\( T^{11} + \cdots - 73\!\cdots\!76 \)
|
| $13$ |
\( T^{11} + \cdots + 12\!\cdots\!62 \)
|
| $17$ |
\( T^{11} + \cdots + 18\!\cdots\!04 \)
|
| $19$ |
\( T^{11} + \cdots + 81\!\cdots\!72 \)
|
| $23$ |
\( T^{11} + \cdots - 23\!\cdots\!08 \)
|
| $29$ |
\( (T - 20511149)^{11} \)
|
| $31$ |
\( T^{11} + \cdots + 15\!\cdots\!48 \)
|
| $37$ |
\( T^{11} + \cdots - 31\!\cdots\!88 \)
|
| $41$ |
\( T^{11} + \cdots - 36\!\cdots\!28 \)
|
| $43$ |
\( T^{11} + \cdots - 14\!\cdots\!60 \)
|
| $47$ |
\( T^{11} + \cdots + 21\!\cdots\!68 \)
|
| $53$ |
\( T^{11} + \cdots + 17\!\cdots\!82 \)
|
| $59$ |
\( T^{11} + \cdots + 21\!\cdots\!00 \)
|
| $61$ |
\( T^{11} + \cdots + 21\!\cdots\!20 \)
|
| $67$ |
\( T^{11} + \cdots - 21\!\cdots\!80 \)
|
| $71$ |
\( T^{11} + \cdots + 26\!\cdots\!92 \)
|
| $73$ |
\( T^{11} + \cdots - 11\!\cdots\!48 \)
|
| $79$ |
\( T^{11} + \cdots - 12\!\cdots\!80 \)
|
| $83$ |
\( T^{11} + \cdots - 45\!\cdots\!64 \)
|
| $89$ |
\( T^{11} + \cdots - 45\!\cdots\!40 \)
|
| $97$ |
\( T^{11} + \cdots - 18\!\cdots\!04 \)
|
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