Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1156,4,Mod(1,1156)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1156, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1156.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1156 = 2^{2} \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1156.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: | \(68.2062079666\) |
| 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: |
\( x^{12} - 216 x^{10} - 74 x^{9} + 17391 x^{8} + 9408 x^{7} - 659646 x^{6} - 424698 x^{5} + \cdots + 168035561 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3\cdot 17^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 216 x^{10} - 74 x^{9} + 17391 x^{8} + 9408 x^{7} - 659646 x^{6} - 424698 x^{5} + \cdots + 168035561 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 1322028016 \nu^{11} + 26931127768 \nu^{10} - 196627363553 \nu^{9} - 5281807462045 \nu^{8} + \cdots - 19\!\cdots\!91 ) / 27\!\cdots\!08 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 1522749596 \nu^{11} - 1084843796 \nu^{10} + 287896091251 \nu^{9} + \cdots + 40\!\cdots\!85 ) / 906184282652136 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 2979478513 \nu^{11} - 19809322756 \nu^{10} + 556064951855 \nu^{9} + \cdots + 14\!\cdots\!39 ) / 302061427550712 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 33893595556 \nu^{11} - 187843301125 \nu^{10} + 6470939919098 \nu^{9} + \cdots + 12\!\cdots\!21 ) / 27\!\cdots\!08 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 646256360 \nu^{11} - 3446150744 \nu^{10} + 122285279671 \nu^{9} + 700065395921 \nu^{8} + \cdots + 20\!\cdots\!75 ) / 50343571258452 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 58337119987 \nu^{11} - 288118657627 \nu^{10} + 11239969034648 \nu^{9} + \cdots + 20\!\cdots\!27 ) / 27\!\cdots\!08 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 72355439900 \nu^{11} + 378941133638 \nu^{10} - 13805568111949 \nu^{9} + \cdots - 26\!\cdots\!85 ) / 27\!\cdots\!08 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 3008211239 \nu^{11} - 17048030149 \nu^{10} + 570023238330 \nu^{9} + \cdots + 11\!\cdots\!39 ) / 100687142516904 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 104623377893 \nu^{11} - 479796495254 \nu^{10} + 20083045265209 \nu^{9} + \cdots + 28\!\cdots\!83 ) / 27\!\cdots\!08 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 60651008107 \nu^{11} - 273297020614 \nu^{10} + 11596906294202 \nu^{9} + \cdots + 18\!\cdots\!90 ) / 906184282652136 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{8} + 2\beta_{5} + \beta_{3} + 36 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{10} - \beta_{9} + \beta_{8} - 2\beta_{6} + 3\beta_{5} + 2\beta_{3} - 6\beta_{2} + 52\beta _1 + 17 \)
|
| \(\nu^{4}\) | \(=\) |
\( - \beta_{11} - 5 \beta_{10} - 26 \beta_{9} + 85 \beta_{8} - 6 \beta_{7} + 2 \beta_{6} + 250 \beta_{5} + \cdots + 1986 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 6 \beta_{11} + 133 \beta_{10} - 163 \beta_{9} + 188 \beta_{8} + 39 \beta_{7} - 209 \beta_{6} + \cdots + 2608 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 433 \beta_{11} - 355 \beta_{10} - 3504 \beta_{9} + 6863 \beta_{8} - 270 \beta_{7} + 750 \beta_{6} + \cdots + 133271 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 2302 \beta_{11} + 14093 \beta_{10} - 18705 \beta_{9} + 22838 \beta_{8} + 5985 \beta_{7} + \cdots + 316852 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 62833 \beta_{11} - 13249 \beta_{10} - 365202 \beta_{9} + 563748 \beta_{8} + 9450 \beta_{7} + \cdots + 9971269 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 367432 \beta_{11} + 1352322 \beta_{10} - 1985080 \beta_{9} + 2426661 \beta_{8} + 690831 \beta_{7} + \cdots + 34417299 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 6990746 \beta_{11} + 497688 \beta_{10} - 34876136 \beta_{9} + 47354745 \beta_{8} + 3268002 \beta_{7} + \cdots + 794656045 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 44903158 \beta_{11} + 123299253 \beta_{10} - 202486159 \beta_{9} + 243729817 \beta_{8} + \cdots + 3517403753 \)
|
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 |
|
0 | −8.58863 | 0 | 18.4886 | 0 | −18.4337 | 0 | 46.7646 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −7.00964 | 0 | −11.7257 | 0 | 7.23337 | 0 | 22.1350 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −6.53488 | 0 | −6.25385 | 0 | 18.5396 | 0 | 15.7046 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −4.67156 | 0 | 16.9558 | 0 | 16.0325 | 0 | −5.17655 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −4.10622 | 0 | 11.3149 | 0 | −32.1558 | 0 | −10.1390 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −2.20468 | 0 | −8.02973 | 0 | 0.312143 | 0 | −22.1394 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 1.14445 | 0 | −4.44584 | 0 | −23.8731 | 0 | −25.6902 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 4.83992 | 0 | 21.6443 | 0 | 19.7255 | 0 | −3.57522 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 5.00468 | 0 | 13.5445 | 0 | 20.8587 | 0 | −1.95322 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 5.10659 | 0 | −10.5769 | 0 | −4.18296 | 0 | −0.922756 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 7.47257 | 0 | 1.69815 | 0 | 12.2977 | 0 | 28.8393 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0 | 9.54740 | 0 | −12.6142 | 0 | −34.3539 | 0 | 64.1529 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(17\) | \(-1\) |
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 | 1156.4.a.k | yes | 12 |
| 17.b | even | 2 | 1 | 1156.4.a.j | ✓ | 12 | |
| 17.c | even | 4 | 2 | 1156.4.b.h | 24 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1156.4.a.j | ✓ | 12 | 17.b | even | 2 | 1 | |
| 1156.4.a.k | yes | 12 | 1.a | even | 1 | 1 | trivial |
| 1156.4.b.h | 24 | 17.c | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} - 216 T_{3}^{10} - 74 T_{3}^{9} + 17391 T_{3}^{8} + 9408 T_{3}^{7} - 659646 T_{3}^{6} + \cdots + 168035561 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1156))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} + \cdots + 168035561 \)
|
| $5$ |
\( T^{12} + \cdots + 616758225399 \)
|
| $7$ |
\( T^{12} + \cdots - 6905150730069 \)
|
| $11$ |
\( T^{12} + \cdots - 13\!\cdots\!68 \)
|
| $13$ |
\( T^{12} + \cdots + 11\!\cdots\!83 \)
|
| $17$ |
\( T^{12} \)
|
| $19$ |
\( T^{12} + \cdots - 72\!\cdots\!19 \)
|
| $23$ |
\( T^{12} + \cdots - 39\!\cdots\!37 \)
|
| $29$ |
\( T^{12} + \cdots - 30\!\cdots\!93 \)
|
| $31$ |
\( T^{12} + \cdots - 50\!\cdots\!17 \)
|
| $37$ |
\( T^{12} + \cdots - 29\!\cdots\!77 \)
|
| $41$ |
\( T^{12} + \cdots - 18\!\cdots\!07 \)
|
| $43$ |
\( T^{12} + \cdots + 43\!\cdots\!97 \)
|
| $47$ |
\( T^{12} + \cdots + 65\!\cdots\!19 \)
|
| $53$ |
\( T^{12} + \cdots + 38\!\cdots\!16 \)
|
| $59$ |
\( T^{12} + \cdots + 56\!\cdots\!21 \)
|
| $61$ |
\( T^{12} + \cdots - 13\!\cdots\!41 \)
|
| $67$ |
\( T^{12} + \cdots - 22\!\cdots\!31 \)
|
| $71$ |
\( T^{12} + \cdots + 13\!\cdots\!71 \)
|
| $73$ |
\( T^{12} + \cdots + 34\!\cdots\!89 \)
|
| $79$ |
\( T^{12} + \cdots + 32\!\cdots\!56 \)
|
| $83$ |
\( T^{12} + \cdots + 66\!\cdots\!21 \)
|
| $89$ |
\( T^{12} + \cdots + 12\!\cdots\!93 \)
|
| $97$ |
\( T^{12} + \cdots - 33\!\cdots\!32 \)
|
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