Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [832,4,Mod(417,832)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(832, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("832.417");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 832 = 2^{6} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 832.b (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(49.0895891248\) |
| 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: |
\( x^{12} + 182x^{10} + 12293x^{8} + 383396x^{6} + 5552404x^{4} + 33084576x^{2} + 52881984 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{9} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{12} + 182x^{10} + 12293x^{8} + 383396x^{6} + 5552404x^{4} + 33084576x^{2} + 52881984 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 34625 \nu^{10} - 1643858 \nu^{8} - 631611203 \nu^{6} - 30194788916 \nu^{4} + \cdots - 1515029864112 ) / 45508698144 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -220\nu^{10} - 36947\nu^{8} - 2161340\nu^{6} - 52066343\nu^{4} - 465157978\nu^{2} - 986730408 ) / 83349264 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 154745 \nu^{10} + 18529204 \nu^{8} + 548480437 \nu^{6} - 1766565638 \nu^{4} + \cdots + 434494581120 ) / 45508698144 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 731099 \nu^{10} - 117643138 \nu^{8} - 6729777919 \nu^{6} - 167696709916 \nu^{4} + \cdots - 4173411796560 ) / 45508698144 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 261689 \nu^{10} + 42677528 \nu^{8} + 2441460173 \nu^{6} + 58069021622 \nu^{4} + \cdots + 701564085408 ) / 15169566048 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 135689 \nu^{11} + 23095558 \nu^{9} + 1399346293 \nu^{7} + 36305355364 \nu^{5} + \cdots + 1106584216848 \nu ) / 606115847808 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 11394004 \nu^{11} + 1771636211 \nu^{9} + 96919780820 \nu^{7} + 2323147925711 \nu^{5} + \cdots + 112963760751096 \nu ) / 6894567768816 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 116648657 \nu^{11} + 20396793454 \nu^{9} + 1308085489405 \nu^{7} + \cdots + 13\!\cdots\!40 \nu ) / 55156542150528 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 48680399 \nu^{11} - 8843242762 \nu^{9} - 580642192771 \nu^{7} + \cdots - 540049903638384 \nu ) / 18385514050176 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 215247049 \nu^{11} - 34448010206 \nu^{9} - 1919280721205 \nu^{7} + \cdots + 131994259892784 \nu ) / 55156542150528 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} - \beta_{2} - 31 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} - 4\beta_{10} - 4\beta_{9} + 5\beta_{8} - 29\beta_{7} - 50\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -18\beta_{6} - 9\beta_{5} - 68\beta_{4} - 130\beta_{3} + 71\beta_{2} + 1481 \)
|
| \(\nu^{5}\) | \(=\) |
\( -35\beta_{11} + 382\beta_{10} + 332\beta_{9} - 399\beta_{8} + 3717\beta_{7} + 2850\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 1754\beta_{6} + 667\beta_{5} + 4326\beta_{4} + 11604\beta_{3} - 4763\beta_{2} - 82445 \)
|
| \(\nu^{7}\) | \(=\) |
\( 835\beta_{11} - 29418\beta_{10} - 22600\beta_{9} + 28891\beta_{8} - 333157\beta_{7} - 172646\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( -135878\beta_{6} - 40579\beta_{5} - 279722\beta_{4} - 910596\beta_{3} + 315115\beta_{2} + 4943821 \)
|
| \(\nu^{9}\) | \(=\) |
\( 11361\beta_{11} + 2108370\beta_{10} + 1487868\beta_{9} - 2048811\beta_{8} + 26203245\beta_{7} + 10858378\beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( 9847674\beta_{6} + 2392071\beta_{5} + 18455842\beta_{4} + 67198720\beta_{3} - 20816563\beta_{2} - 309748933 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 3942317 \beta_{11} - 146640946 \beta_{10} - 97960768 \beta_{9} + 143724899 \beta_{8} + \cdots - 700328846 \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/832\mathbb{Z}\right)^\times\).
| \(n\) | \(261\) | \(703\) | \(769\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 417.1 |
|
0 | − | 9.23760i | 0 | 16.4741i | 0 | 2.41511 | 0 | −58.3333 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 417.2 | 0 | − | 5.70026i | 0 | − | 16.8511i | 0 | 17.8439 | 0 | −5.49297 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 417.3 | 0 | − | 5.27597i | 0 | − | 4.21449i | 0 | 26.6367 | 0 | −0.835875 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 417.4 | 0 | − | 5.19764i | 0 | 10.6342i | 0 | 3.22506 | 0 | −0.0155092 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 417.5 | 0 | − | 2.56371i | 0 | − | 11.5030i | 0 | −10.9222 | 0 | 20.4274 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 417.6 | 0 | − | 2.17938i | 0 | − | 9.02645i | 0 | −29.1985 | 0 | 22.2503 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 417.7 | 0 | 2.17938i | 0 | 9.02645i | 0 | −29.1985 | 0 | 22.2503 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 417.8 | 0 | 2.56371i | 0 | 11.5030i | 0 | −10.9222 | 0 | 20.4274 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 417.9 | 0 | 5.19764i | 0 | − | 10.6342i | 0 | 3.22506 | 0 | −0.0155092 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 417.10 | 0 | 5.27597i | 0 | 4.21449i | 0 | 26.6367 | 0 | −0.835875 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 417.11 | 0 | 5.70026i | 0 | 16.8511i | 0 | 17.8439 | 0 | −5.49297 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 417.12 | 0 | 9.23760i | 0 | − | 16.4741i | 0 | 2.41511 | 0 | −58.3333 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 832.4.b.b | yes | 12 |
| 4.b | odd | 2 | 1 | 832.4.b.a | ✓ | 12 | |
| 8.b | even | 2 | 1 | inner | 832.4.b.b | yes | 12 |
| 8.d | odd | 2 | 1 | 832.4.b.a | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 832.4.b.a | ✓ | 12 | 4.b | odd | 2 | 1 | |
| 832.4.b.a | ✓ | 12 | 8.d | odd | 2 | 1 | |
| 832.4.b.b | yes | 12 | 1.a | even | 1 | 1 | trivial |
| 832.4.b.b | yes | 12 | 8.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(832, [\chi])\):
|
\( T_{3}^{12} + 184T_{3}^{10} + 11974T_{3}^{8} + 359168T_{3}^{6} + 5122105T_{3}^{4} + 31122136T_{3}^{2} + 65092624 \)
|
|
\( T_{7}^{6} - 10T_{7}^{5} - 958T_{7}^{4} + 10436T_{7}^{3} + 116319T_{7}^{2} - 816894T_{7} + 1180634 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} + 184 T^{10} + \cdots + 65092624 \)
|
| $5$ |
\( T^{12} + \cdots + 1668840250896 \)
|
| $7$ |
\( (T^{6} - 10 T^{5} + \cdots + 1180634)^{2} \)
|
| $11$ |
\( T^{12} + \cdots + 59\!\cdots\!76 \)
|
| $13$ |
\( (T^{2} + 169)^{6} \)
|
| $17$ |
\( (T^{6} + 80 T^{5} + \cdots - 10608098728)^{2} \)
|
| $19$ |
\( T^{12} + \cdots + 26\!\cdots\!84 \)
|
| $23$ |
\( (T^{6} + 224 T^{5} + \cdots + 92489173504)^{2} \)
|
| $29$ |
\( T^{12} + \cdots + 22\!\cdots\!56 \)
|
| $31$ |
\( (T^{6} + \cdots - 34340210008832)^{2} \)
|
| $37$ |
\( T^{12} + \cdots + 54\!\cdots\!96 \)
|
| $41$ |
\( (T^{6} + \cdots - 486720257517824)^{2} \)
|
| $43$ |
\( T^{12} + \cdots + 93\!\cdots\!84 \)
|
| $47$ |
\( (T^{6} + \cdots - 322057737480934)^{2} \)
|
| $53$ |
\( T^{12} + \cdots + 10\!\cdots\!64 \)
|
| $59$ |
\( T^{12} + \cdots + 37\!\cdots\!36 \)
|
| $61$ |
\( T^{12} + \cdots + 21\!\cdots\!64 \)
|
| $67$ |
\( T^{12} + \cdots + 52\!\cdots\!76 \)
|
| $71$ |
\( (T^{6} + \cdots + 36\!\cdots\!82)^{2} \)
|
| $73$ |
\( (T^{6} + \cdots - 52\!\cdots\!32)^{2} \)
|
| $79$ |
\( (T^{6} + \cdots - 11\!\cdots\!08)^{2} \)
|
| $83$ |
\( T^{12} + \cdots + 75\!\cdots\!84 \)
|
| $89$ |
\( (T^{6} + \cdots - 50\!\cdots\!16)^{2} \)
|
| $97$ |
\( (T^{6} + \cdots - 77\!\cdots\!16)^{2} \)
|
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