Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [261,12,Mod(1,261)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(261, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("261.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 261 = 3^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 261.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: | \(200.537570126\) |
| 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: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{13}\cdot 3^{4} \) |
| Twist minimal: | no (minimal twist has level 29) |
| 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
\( x^{11} - x^{10} - 15790 x^{9} + 14666 x^{8} + 87206462 x^{7} - 14008334 x^{6} - 203974096304 x^{5} + \cdots - 75\!\cdots\!58 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2871 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 56\!\cdots\!01 \nu^{10} + \cdots - 20\!\cdots\!46 ) / 14\!\cdots\!60 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 80\!\cdots\!33 \nu^{10} + \cdots - 29\!\cdots\!10 ) / 37\!\cdots\!40 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 91\!\cdots\!07 \nu^{10} + \cdots + 33\!\cdots\!70 ) / 14\!\cdots\!60 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 92\!\cdots\!01 \nu^{10} + \cdots - 33\!\cdots\!10 ) / 14\!\cdots\!96 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 17\!\cdots\!75 \nu^{10} + \cdots - 65\!\cdots\!62 ) / 14\!\cdots\!60 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 10\!\cdots\!81 \nu^{10} + \cdots + 36\!\cdots\!54 ) / 74\!\cdots\!80 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 21\!\cdots\!05 \nu^{10} + \cdots + 77\!\cdots\!54 ) / 14\!\cdots\!60 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 25\!\cdots\!87 \nu^{10} + \cdots - 92\!\cdots\!14 ) / 14\!\cdots\!96 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2871 \)
|
| \(\nu^{3}\) | \(=\) |
\( - 2 \beta_{10} - 2 \beta_{9} - 6 \beta_{8} - 4 \beta_{7} - 9 \beta_{6} - 3 \beta_{5} + 27 \beta_{3} + \cdots - 111 \)
|
| \(\nu^{4}\) | \(=\) |
\( 10 \beta_{10} + 324 \beta_{9} - 55 \beta_{8} + 225 \beta_{7} - 89 \beta_{6} - 238 \beta_{5} + \cdots + 13621663 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 28740 \beta_{10} - 32530 \beta_{9} - 51677 \beta_{8} - 38319 \beta_{7} - 81840 \beta_{6} + \cdots - 27519793 \)
|
| \(\nu^{6}\) | \(=\) |
\( 154704 \beta_{10} + 3391688 \beta_{9} - 431612 \beta_{8} + 2468940 \beta_{7} - 1134464 \beta_{6} + \cdots + 75939786039 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 256494054 \beta_{10} - 298313766 \beta_{9} - 383013666 \beta_{8} - 305856508 \beta_{7} + \cdots - 256706722429 \)
|
| \(\nu^{8}\) | \(=\) |
\( 1821509134 \beta_{10} + 27269535996 \beta_{9} - 2855699909 \beta_{8} + 20122530003 \beta_{7} + \cdots + 456572329667143 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 1975942874424 \beta_{10} - 2327288256370 \beta_{9} - 2713255099587 \beta_{8} - 2285739989597 \beta_{7} + \cdots - 20\!\cdots\!69 \)
|
| \(\nu^{10}\) | \(=\) |
\( 17582847741900 \beta_{10} + 201013362531376 \beta_{9} - 18039580668670 \beta_{8} + 148670899043178 \beta_{7} + \cdots + 28\!\cdots\!11 \)
|
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 |
|
−78.6399 | 0 | 4136.23 | 12464.4 | 0 | −230.276 | −164218. | 0 | −980202. | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −62.2743 | 0 | 1830.09 | −1377.47 | 0 | 33050.3 | 13570.0 | 0 | 85780.8 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −51.9918 | 0 | 655.142 | −3682.79 | 0 | −83752.5 | 72417.1 | 0 | 191474. | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −30.9131 | 0 | −1092.38 | 6165.86 | 0 | 8452.88 | 97078.9 | 0 | −190606. | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −27.4581 | 0 | −1294.05 | −9990.24 | 0 | 28164.0 | 91766.4 | 0 | 274313. | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 21.9987 | 0 | −1564.06 | 886.321 | 0 | 44263.6 | −79460.6 | 0 | 19497.9 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 23.7036 | 0 | −1486.14 | −6643.74 | 0 | −61145.1 | −83771.9 | 0 | −157481. | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 23.7638 | 0 | −1483.28 | 5297.67 | 0 | 80499.9 | −83916.8 | 0 | 125893. | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 59.7551 | 0 | 1522.68 | 6379.08 | 0 | −23005.8 | −31390.8 | 0 | 381183. | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 68.3340 | 0 | 2621.54 | −8265.40 | 0 | −68659.4 | 39192.0 | 0 | −564808. | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 85.7218 | 0 | 5300.23 | 1506.26 | 0 | −7069.49 | 278787. | 0 | 129120. | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(29\) | \(1\) |
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 | 261.12.a.a | 11 | |
| 3.b | odd | 2 | 1 | 29.12.a.a | ✓ | 11 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 29.12.a.a | ✓ | 11 | 3.b | odd | 2 | 1 | |
| 261.12.a.a | 11 | 1.a | even | 1 | 1 | trivial | |
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 \(S_{12}^{\mathrm{new}}(\Gamma_0(261))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} + \cdots + 93\!\cdots\!40 \)
|
| $3$ |
\( T^{11} \)
|
| $5$ |
\( T^{11} + \cdots + 96\!\cdots\!50 \)
|
| $7$ |
\( T^{11} + \cdots - 36\!\cdots\!16 \)
|
| $11$ |
\( T^{11} + \cdots + 73\!\cdots\!76 \)
|
| $13$ |
\( T^{11} + \cdots + 12\!\cdots\!62 \)
|
| $17$ |
\( T^{11} + \cdots - 18\!\cdots\!04 \)
|
| $19$ |
\( T^{11} + \cdots + 81\!\cdots\!72 \)
|
| $23$ |
\( T^{11} + \cdots + 23\!\cdots\!08 \)
|
| $29$ |
\( (T + 20511149)^{11} \)
|
| $31$ |
\( T^{11} + \cdots + 15\!\cdots\!48 \)
|
| $37$ |
\( T^{11} + \cdots - 31\!\cdots\!88 \)
|
| $41$ |
\( T^{11} + \cdots + 36\!\cdots\!28 \)
|
| $43$ |
\( T^{11} + \cdots - 14\!\cdots\!60 \)
|
| $47$ |
\( T^{11} + \cdots - 21\!\cdots\!68 \)
|
| $53$ |
\( T^{11} + \cdots - 17\!\cdots\!82 \)
|
| $59$ |
\( T^{11} + \cdots - 21\!\cdots\!00 \)
|
| $61$ |
\( T^{11} + \cdots + 21\!\cdots\!20 \)
|
| $67$ |
\( T^{11} + \cdots - 21\!\cdots\!80 \)
|
| $71$ |
\( T^{11} + \cdots - 26\!\cdots\!92 \)
|
| $73$ |
\( T^{11} + \cdots - 11\!\cdots\!48 \)
|
| $79$ |
\( T^{11} + \cdots - 12\!\cdots\!80 \)
|
| $83$ |
\( T^{11} + \cdots + 45\!\cdots\!64 \)
|
| $89$ |
\( T^{11} + \cdots + 45\!\cdots\!40 \)
|
| $97$ |
\( T^{11} + \cdots - 18\!\cdots\!04 \)
|
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