Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2057,4,Mod(1,2057)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2057, 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("2057.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2057 = 11^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2057.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: | \(121.366928882\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{12} - 2 x^{11} - 72 x^{10} + 90 x^{9} + 1924 x^{8} - 1096 x^{7} - 22484 x^{6} + 1486 x^{5} + \cdots + 75328 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 187) |
| 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
\( x^{12} - 2 x^{11} - 72 x^{10} + 90 x^{9} + 1924 x^{8} - 1096 x^{7} - 22484 x^{6} + 1486 x^{5} + \cdots + 75328 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 20\nu + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 133541 \nu^{11} - 1719716 \nu^{10} - 5711968 \nu^{9} + 101156802 \nu^{8} + 104507376 \nu^{7} + \cdots + 42607780512 ) / 1259512320 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 25463 \nu^{11} - 23258 \nu^{10} - 1979564 \nu^{9} + 941646 \nu^{8} + 55417988 \nu^{7} + \cdots + 185225216 ) / 157439040 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 251857 \nu^{11} + 1418932 \nu^{10} + 20300256 \nu^{9} - 109146234 \nu^{8} + \cdots - 82523776544 ) / 1259512320 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 53131 \nu^{11} - 169486 \nu^{10} - 3329588 \nu^{9} + 7368822 \nu^{8} + 78510636 \nu^{7} + \cdots - 1103095488 ) / 157439040 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 21861 \nu^{11} + 81816 \nu^{10} + 1400488 \nu^{9} - 4137522 \nu^{8} - 33269576 \nu^{7} + \cdots - 2364204512 ) / 28625280 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 964573 \nu^{11} + 4189428 \nu^{10} + 61953024 \nu^{9} - 240675826 \nu^{8} + \cdots - 79083552416 ) / 1259512320 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 1211857 \nu^{11} + 4083732 \nu^{10} + 82870176 \nu^{9} - 216144634 \nu^{8} + \cdots - 46705439264 ) / 1259512320 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 724677 \nu^{11} + 2105472 \nu^{10} + 49336216 \nu^{9} - 108458034 \nu^{8} + \cdots - 38713874144 ) / 314878080 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 21\beta _1 + 11 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{11} + \beta_{10} - \beta_{9} - 2\beta_{8} + 6\beta_{5} + 27\beta_{2} + 45\beta _1 + 249 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} - 6 \beta_{8} + 8 \beta_{7} - 4 \beta_{6} + 12 \beta_{5} + \cdots + 484 \)
|
| \(\nu^{6}\) | \(=\) |
\( 54 \beta_{11} + 40 \beta_{10} - 52 \beta_{9} - 88 \beta_{8} + 30 \beta_{7} - 8 \beta_{6} + 280 \beta_{5} + \cdots + 5930 \)
|
| \(\nu^{7}\) | \(=\) |
\( 236 \beta_{11} + 146 \beta_{10} - 146 \beta_{9} - 344 \beta_{8} + 434 \beta_{7} - 196 \beta_{6} + \cdots + 16767 \)
|
| \(\nu^{8}\) | \(=\) |
\( 2169 \beta_{11} + 1451 \beta_{10} - 2051 \beta_{9} - 3222 \beta_{8} + 1902 \beta_{7} - 596 \beta_{6} + \cdots + 151391 \)
|
| \(\nu^{9}\) | \(=\) |
\( 10202 \beta_{11} + 6978 \beta_{10} - 7326 \beta_{9} - 14638 \beta_{8} + 17690 \beta_{7} - 7364 \beta_{6} + \cdots + 538208 \)
|
| \(\nu^{10}\) | \(=\) |
\( 78132 \beta_{11} + 52232 \beta_{10} - 73376 \beta_{9} - 111396 \beta_{8} + 85256 \beta_{7} + \cdots + 4066652 \)
|
| \(\nu^{11}\) | \(=\) |
\( 388528 \beta_{11} + 282324 \beta_{10} - 308772 \beta_{9} - 552192 \beta_{8} + 650660 \beta_{7} + \cdots + 16805831 \)
|
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 |
|
−5.54055 | −9.16274 | 22.6977 | 14.8257 | 50.7667 | 20.0444 | −81.4336 | 56.9559 | −82.1427 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −5.24119 | 5.22307 | 19.4700 | −14.7961 | −27.3751 | −15.7778 | −60.1166 | 0.280412 | 77.5491 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.72673 | 0.151093 | 14.3419 | 13.7842 | −0.714177 | −15.3969 | −29.9766 | −26.9772 | −65.1542 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −3.76170 | 7.49421 | 6.15038 | 8.81271 | −28.1910 | 10.8225 | 6.95770 | 29.1632 | −33.1508 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −2.28193 | −3.55867 | −2.79279 | 0.141464 | 8.12064 | −31.0601 | 24.6284 | −14.3359 | −0.322812 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −2.00243 | −9.34368 | −3.99028 | −12.7089 | 18.7100 | 10.0991 | 24.0097 | 60.3043 | 25.4486 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.255402 | 8.17844 | −7.93477 | 4.15484 | −2.08879 | −20.4156 | 4.06976 | 39.8869 | −1.06115 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.470624 | −2.60876 | −7.77851 | 21.7227 | −1.22775 | 11.5058 | −7.42575 | −20.1944 | 10.2232 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.06516 | −1.29693 | −6.86543 | −15.6676 | −1.38144 | 10.5402 | −15.8341 | −25.3180 | −16.6886 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 3.71127 | 6.19465 | 5.77353 | 15.0518 | 22.9900 | −31.3904 | −8.26304 | 11.3736 | 55.8611 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 3.91619 | 6.35856 | 7.33657 | −7.95223 | 24.9013 | 25.4745 | −2.59810 | 13.4312 | −31.1425 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 4.64667 | −4.62923 | 13.5916 | 1.63140 | −21.5105 | −13.4459 | 25.9823 | −5.57022 | 7.58059 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \(-1\) |
| \(17\) | \(1\) |
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 | 2057.4.a.j | 12 | |
| 11.b | odd | 2 | 1 | 187.4.a.f | ✓ | 12 | |
| 33.d | even | 2 | 1 | 1683.4.a.n | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 187.4.a.f | ✓ | 12 | 11.b | odd | 2 | 1 | |
| 1683.4.a.n | 12 | 33.d | even | 2 | 1 | ||
| 2057.4.a.j | 12 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2057))\):
|
\( T_{2}^{12} + 10 T_{2}^{11} - 28 T_{2}^{10} - 520 T_{2}^{9} - 341 T_{2}^{8} + 9028 T_{2}^{7} + \cdots - 20400 \)
|
|
\( T_{3}^{12} - 3 T_{3}^{11} - 217 T_{3}^{10} + 708 T_{3}^{9} + 16689 T_{3}^{8} - 51902 T_{3}^{7} + \cdots + 9091456 \)
|
|
\( T_{5}^{12} - 29 T_{5}^{11} - 527 T_{5}^{10} + 19860 T_{5}^{9} + 57197 T_{5}^{8} - 4790070 T_{5}^{7} + \cdots + 13228722912 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + 10 T^{11} + \cdots - 20400 \)
|
| $3$ |
\( T^{12} - 3 T^{11} + \cdots + 9091456 \)
|
| $5$ |
\( T^{12} + \cdots + 13228722912 \)
|
| $7$ |
\( T^{12} + \cdots + 440051897483264 \)
|
| $11$ |
\( T^{12} \)
|
| $13$ |
\( T^{12} + \cdots + 12\!\cdots\!84 \)
|
| $17$ |
\( (T + 17)^{12} \)
|
| $19$ |
\( T^{12} + \cdots + 61\!\cdots\!64 \)
|
| $23$ |
\( T^{12} + \cdots + 14\!\cdots\!44 \)
|
| $29$ |
\( T^{12} + \cdots + 18\!\cdots\!12 \)
|
| $31$ |
\( T^{12} + \cdots + 17\!\cdots\!92 \)
|
| $37$ |
\( T^{12} + \cdots + 10\!\cdots\!00 \)
|
| $41$ |
\( T^{12} + \cdots - 26\!\cdots\!76 \)
|
| $43$ |
\( T^{12} + \cdots + 29\!\cdots\!32 \)
|
| $47$ |
\( T^{12} + \cdots + 42\!\cdots\!36 \)
|
| $53$ |
\( T^{12} + \cdots + 79\!\cdots\!88 \)
|
| $59$ |
\( T^{12} + \cdots - 83\!\cdots\!92 \)
|
| $61$ |
\( T^{12} + \cdots + 31\!\cdots\!12 \)
|
| $67$ |
\( T^{12} + \cdots + 25\!\cdots\!88 \)
|
| $71$ |
\( T^{12} + \cdots + 36\!\cdots\!00 \)
|
| $73$ |
\( T^{12} + \cdots - 10\!\cdots\!00 \)
|
| $79$ |
\( T^{12} + \cdots + 30\!\cdots\!32 \)
|
| $83$ |
\( T^{12} + \cdots - 13\!\cdots\!64 \)
|
| $89$ |
\( T^{12} + \cdots - 11\!\cdots\!06 \)
|
| $97$ |
\( T^{12} + \cdots - 53\!\cdots\!48 \)
|
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