Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6128,2,Mod(1,6128)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6128, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6128.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6128 = 2^{4} \cdot 383 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6128.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: | \(48.9323263586\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 5 x^{15} - 19 x^{14} + 120 x^{13} + 105 x^{12} - 1092 x^{11} - 28 x^{10} + 4827 x^{9} + \cdots + 293 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1532) |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 5 x^{15} - 19 x^{14} + 120 x^{13} + 105 x^{12} - 1092 x^{11} - 28 x^{10} + 4827 x^{9} + \cdots + 293 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 1578298941 \nu^{15} - 13868464344 \nu^{14} - 1306890308 \nu^{13} + 290599475517 \nu^{12} + \cdots + 601881107165 ) / 51974813039 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 3692998010 \nu^{15} + 91032341297 \nu^{14} - 223966515582 \nu^{13} - 1902438030482 \nu^{12} + \cdots - 8358151592940 ) / 51974813039 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 4300050470 \nu^{15} - 97356261976 \nu^{14} + 226199488848 \nu^{13} + 2043642817780 \nu^{12} + \cdots + 9565970733240 ) / 51974813039 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 5948961979 \nu^{15} - 53174099843 \nu^{14} - 3280506541 \nu^{13} + 1125768272476 \nu^{12} + \cdots + 3146319753010 ) / 51974813039 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 7492853149 \nu^{15} - 23891464237 \nu^{14} - 176180510614 \nu^{13} + 542859478604 \nu^{12} + \cdots - 492938067810 ) / 51974813039 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 7851749601 \nu^{15} - 6553989699 \nu^{14} + 308351466690 \nu^{13} + 83510415430 \nu^{12} + \cdots + 4245751259920 ) / 51974813039 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 10190983653 \nu^{15} + 26264443746 \nu^{14} + 266340241337 \nu^{13} - 620134876645 \nu^{12} + \cdots + 1379370256161 ) / 51974813039 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 11055257580 \nu^{15} + 20237461469 \nu^{14} + 320170280148 \nu^{13} - 498156060467 \nu^{12} + \cdots + 2503737441477 ) / 51974813039 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 13283761879 \nu^{15} + 41760801771 \nu^{14} + 316042874288 \nu^{13} - 954760454350 \nu^{12} + \cdots + 771124918111 ) / 51974813039 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 18548110729 \nu^{15} - 44128925706 \nu^{14} - 496350790762 \nu^{13} + 1041015539071 \nu^{12} + \cdots - 3152599948404 ) / 51974813039 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 31240840918 \nu^{15} + 172070028821 \nu^{14} + 449667044142 \nu^{13} - 3756440129461 \nu^{12} + \cdots - 5816752903371 ) / 51974813039 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 34413934960 \nu^{15} - 188037859006 \nu^{14} - 503890010063 \nu^{13} + 4117386055286 \nu^{12} + \cdots + 6052941253609 ) / 51974813039 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 36033693937 \nu^{15} - 198779747916 \nu^{14} - 514348602828 \nu^{13} + 4323044184824 \nu^{12} + \cdots + 5901443684273 ) / 51974813039 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 42683228554 \nu^{15} - 250482559542 \nu^{14} - 553877354465 \nu^{13} + 5445406579337 \nu^{12} + \cdots + 9103386308163 ) / 51974813039 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{11} + \beta_{9} - \beta_{6} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{15} - \beta_{14} + \beta_{12} + 2\beta_{11} + \beta_{9} + \beta_{8} + 2\beta_{5} - 2\beta_{2} + 8\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{13} + \beta_{12} + 10 \beta_{11} - \beta_{10} + 11 \beta_{9} + 2 \beta_{8} - \beta_{7} + \cdots + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( 13 \beta_{15} - 11 \beta_{14} - 2 \beta_{13} + 11 \beta_{12} + 26 \beta_{11} + 15 \beta_{9} + 12 \beta_{8} + \cdots - 15 \)
|
| \(\nu^{6}\) | \(=\) |
\( 4 \beta_{15} - 4 \beta_{14} + 12 \beta_{13} + 13 \beta_{12} + 95 \beta_{11} - 11 \beta_{10} + 104 \beta_{9} + \cdots + 102 \)
|
| \(\nu^{7}\) | \(=\) |
\( 144 \beta_{15} - 112 \beta_{14} - 32 \beta_{13} + 105 \beta_{12} + 281 \beta_{11} + \beta_{10} + \cdots - 166 \)
|
| \(\nu^{8}\) | \(=\) |
\( 97 \beta_{15} - 92 \beta_{14} + 122 \beta_{13} + 149 \beta_{12} + 904 \beta_{11} - 90 \beta_{10} + \cdots + 692 \)
|
| \(\nu^{9}\) | \(=\) |
\( 1539 \beta_{15} - 1151 \beta_{14} - 371 \beta_{13} + 979 \beta_{12} + 2885 \beta_{11} + 45 \beta_{10} + \cdots - 1668 \)
|
| \(\nu^{10}\) | \(=\) |
\( 1610 \beta_{15} - 1457 \beta_{14} + 1184 \beta_{13} + 1657 \beta_{12} + 8701 \beta_{11} - 589 \beta_{10} + \cdots + 4743 \)
|
| \(\nu^{11}\) | \(=\) |
\( 16281 \beta_{15} - 11964 \beta_{14} - 3774 \beta_{13} + 9210 \beta_{12} + 29047 \beta_{11} + \cdots - 16187 \)
|
| \(\nu^{12}\) | \(=\) |
\( 22716 \beta_{15} - 19769 \beta_{14} + 11226 \beta_{13} + 18101 \beta_{12} + 84723 \beta_{11} + \cdots + 31510 \)
|
| \(\nu^{13}\) | \(=\) |
\( 171706 \beta_{15} - 125225 \beta_{14} - 35811 \beta_{13} + 88125 \beta_{12} + 289979 \beta_{11} + \cdots - 155722 \)
|
| \(\nu^{14}\) | \(=\) |
\( 293197 \beta_{15} - 247050 \beta_{14} + 104763 \beta_{13} + 194976 \beta_{12} + 832869 \beta_{11} + \cdots + 190725 \)
|
| \(\nu^{15}\) | \(=\) |
\( 1809618 \beta_{15} - 1315884 \beta_{14} - 325361 \beta_{13} + 857356 \beta_{12} + 2884140 \beta_{11} + \cdots - 1503870 \)
|
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 | −2.89444 | 0 | −3.18445 | 0 | 1.04122 | 0 | 5.37776 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.66599 | 0 | −1.45842 | 0 | 5.07062 | 0 | 4.10753 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.86965 | 0 | −2.57673 | 0 | 2.60832 | 0 | 0.495609 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.18145 | 0 | −1.57341 | 0 | −1.81551 | 0 | −1.60418 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −1.16448 | 0 | 0.505324 | 0 | 0.507216 | 0 | −1.64398 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −1.11256 | 0 | 3.88639 | 0 | 5.27642 | 0 | −1.76221 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | −0.429998 | 0 | 1.29780 | 0 | 2.55748 | 0 | −2.81510 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | −0.217333 | 0 | 0.711589 | 0 | 0.553517 | 0 | −2.95277 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 0.718088 | 0 | −3.05261 | 0 | −0.100553 | 0 | −2.48435 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 1.01660 | 0 | −1.13666 | 0 | −4.18222 | 0 | −1.96652 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 1.64894 | 0 | 2.14574 | 0 | −2.65197 | 0 | −0.280993 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0 | 2.23106 | 0 | 0.977854 | 0 | 4.56692 | 0 | 1.97764 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 0 | 2.23552 | 0 | −3.10243 | 0 | −1.55126 | 0 | 1.99756 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 0 | 2.38050 | 0 | 4.04378 | 0 | 2.78993 | 0 | 2.66676 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 0 | 3.08362 | 0 | −2.45571 | 0 | 3.25166 | 0 | 6.50872 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 0 | 3.22157 | 0 | 1.97192 | 0 | 0.0782131 | 0 | 7.37852 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(383\) | \( +1 \) |
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 | 6128.2.a.m | 16 | |
| 4.b | odd | 2 | 1 | 1532.2.a.c | ✓ | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1532.2.a.c | ✓ | 16 | 4.b | odd | 2 | 1 | |
| 6128.2.a.m | 16 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6128))\):
|
\( T_{3}^{16} - 5 T_{3}^{15} - 19 T_{3}^{14} + 120 T_{3}^{13} + 105 T_{3}^{12} - 1092 T_{3}^{11} + \cdots + 293 \)
|
|
\( T_{5}^{16} + 3 T_{5}^{15} - 41 T_{5}^{14} - 137 T_{5}^{13} + 591 T_{5}^{12} + 2235 T_{5}^{11} + \cdots + 15104 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} \)
|
| $3$ |
\( T^{16} - 5 T^{15} + \cdots + 293 \)
|
| $5$ |
\( T^{16} + 3 T^{15} + \cdots + 15104 \)
|
| $7$ |
\( T^{16} - 18 T^{15} + \cdots - 531 \)
|
| $11$ |
\( T^{16} - 6 T^{15} + \cdots - 5465856 \)
|
| $13$ |
\( T^{16} + 14 T^{15} + \cdots + 8241152 \)
|
| $17$ |
\( T^{16} + T^{15} + \cdots + 996388 \)
|
| $19$ |
\( T^{16} - 18 T^{15} + \cdots + 55259851 \)
|
| $23$ |
\( T^{16} + \cdots + 2095229091 \)
|
| $29$ |
\( T^{16} + \cdots + 132663516 \)
|
| $31$ |
\( T^{16} + \cdots - 3759736147 \)
|
| $37$ |
\( T^{16} + \cdots - 1149947648 \)
|
| $41$ |
\( T^{16} + \cdots + 14656850688 \)
|
| $43$ |
\( T^{16} + \cdots + 630594267549 \)
|
| $47$ |
\( T^{16} + \cdots + 30852972544 \)
|
| $53$ |
\( T^{16} + \cdots - 4066764544 \)
|
| $59$ |
\( T^{16} + \cdots - 89484260096 \)
|
| $61$ |
\( T^{16} + \cdots - 2227622719232 \)
|
| $67$ |
\( T^{16} + \cdots - 554978869443 \)
|
| $71$ |
\( T^{16} + \cdots - 135528796933 \)
|
| $73$ |
\( T^{16} + \cdots + 112894640916 \)
|
| $79$ |
\( T^{16} + \cdots + 27768943788032 \)
|
| $83$ |
\( T^{16} + \cdots - 412962523392 \)
|
| $89$ |
\( T^{16} + \cdots - 84811145472 \)
|
| $97$ |
\( T^{16} + \cdots + 193121395968 \)
|
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