Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2768,2,Mod(1,2768)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2768, 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("2768.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2768 = 2^{4} \cdot 173 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2768.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: | \(22.1025912795\) |
| Analytic rank: | \(0\) |
| Dimension: | \(13\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{13} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{13} - 2 x^{12} - 25 x^{11} + 47 x^{10} + 219 x^{9} - 389 x^{8} - 808 x^{7} + 1392 x^{6} + 1192 x^{5} + \cdots - 128 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1384) |
| 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 \(1,\beta_1,\ldots,\beta_{12}\) 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
\( x^{13} - 2 x^{12} - 25 x^{11} + 47 x^{10} + 219 x^{9} - 389 x^{8} - 808 x^{7} + 1392 x^{6} + 1192 x^{5} + \cdots - 128 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 1869349 \nu^{12} + 5551626 \nu^{11} + 53149549 \nu^{10} - 143676231 \nu^{9} + \cdots - 1825756048 ) / 243811482 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5838067 \nu^{12} - 17195027 \nu^{11} - 145420007 \nu^{10} + 408073567 \nu^{9} + \cdots + 1119194516 ) / 203176235 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 35152996 \nu^{12} - 26640891 \nu^{11} - 951046306 \nu^{10} + 547443921 \nu^{9} + \cdots + 4830075178 ) / 1219057410 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3711800 \nu^{12} - 4723800 \nu^{11} - 94976291 \nu^{10} + 102239826 \nu^{9} + 858636342 \nu^{8} + \cdots + 864855083 ) / 121905741 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 43665101 \nu^{12} - 72221406 \nu^{11} - 1104746891 \nu^{10} + 1603187331 \nu^{9} + \cdots + 5718640898 ) / 1219057410 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 15352682 \nu^{12} + 30562717 \nu^{11} + 383440262 \nu^{10} - 705956957 \nu^{9} + \cdots - 3518772266 ) / 406352470 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 807513 \nu^{12} - 693868 \nu^{11} - 21384253 \nu^{10} + 14497773 \nu^{9} + 203234885 \nu^{8} + \cdots + 232628564 ) / 13774660 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 136719679 \nu^{12} + 146046294 \nu^{11} + 3528817609 \nu^{10} - 3136741839 \nu^{9} + \cdots - 16875696922 ) / 1219057410 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 145252532 \nu^{12} + 206826567 \nu^{11} + 3749662472 \nu^{10} - 4656312747 \nu^{9} + \cdots - 32855905406 ) / 1219057410 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 87107768 \nu^{12} - 124618293 \nu^{11} - 2235263408 \nu^{10} + 2794394253 \nu^{9} + \cdots + 12788869364 ) / 609528705 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{11} + 2\beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + 8\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{11} + \beta_{10} + 2\beta_{8} + \beta_{7} - 2\beta_{5} - \beta_{4} + 11\beta_{2} - \beta _1 + 30 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2 \beta_{12} - 13 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} + 26 \beta_{8} - 14 \beta_{7} + 3 \beta_{6} + \cdots + 19 \)
|
| \(\nu^{6}\) | \(=\) |
\( 8 \beta_{12} - 36 \beta_{11} + 23 \beta_{10} - 4 \beta_{9} + 39 \beta_{8} + 16 \beta_{7} - 6 \beta_{6} + \cdots + 273 \)
|
| \(\nu^{7}\) | \(=\) |
\( 37 \beta_{12} - 160 \beta_{11} + 43 \beta_{10} - 46 \beta_{9} + 297 \beta_{8} - 166 \beta_{7} + \cdots + 275 \)
|
| \(\nu^{8}\) | \(=\) |
\( 175 \beta_{12} - 506 \beta_{11} + 367 \beta_{10} - 86 \beta_{9} + 557 \beta_{8} + 185 \beta_{7} + \cdots + 2699 \)
|
| \(\nu^{9}\) | \(=\) |
\( 545 \beta_{12} - 1949 \beta_{11} + 685 \beta_{10} - 734 \beta_{9} + 3327 \beta_{8} - 1887 \beta_{7} + \cdots + 3556 \)
|
| \(\nu^{10}\) | \(=\) |
\( 2731 \beta_{12} - 6503 \beta_{11} + 5053 \beta_{10} - 1370 \beta_{9} + 7158 \beta_{8} + 1874 \beta_{7} + \cdots + 27870 \)
|
| \(\nu^{11}\) | \(=\) |
\( 7425 \beta_{12} - 23487 \beta_{11} + 9742 \beta_{10} - 10106 \beta_{9} + 37272 \beta_{8} - 21050 \beta_{7} + \cdots + 43543 \)
|
| \(\nu^{12}\) | \(=\) |
\( 37290 \beta_{12} - 80035 \beta_{11} + 64576 \beta_{10} - 19484 \beta_{9} + 87757 \beta_{8} + \cdots + 295269 \)
|
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 | −3.20689 | 0 | 0.820135 | 0 | −1.01692 | 0 | 7.28414 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.59222 | 0 | 3.32984 | 0 | 4.68611 | 0 | 3.71959 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.15923 | 0 | 2.59078 | 0 | −2.44699 | 0 | 1.66228 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.29709 | 0 | 3.63024 | 0 | −1.45019 | 0 | −1.31755 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.986894 | 0 | −2.22725 | 0 | 4.72949 | 0 | −2.02604 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −0.328140 | 0 | −0.343976 | 0 | −3.44582 | 0 | −2.89232 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 0.544729 | 0 | −2.23978 | 0 | −3.11683 | 0 | −2.70327 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 0.749062 | 0 | −4.45333 | 0 | 3.45608 | 0 | −2.43891 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 1.25133 | 0 | 3.46724 | 0 | −0.247813 | 0 | −1.43416 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 1.60702 | 0 | 3.88676 | 0 | 4.54904 | 0 | −0.417473 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 2.04281 | 0 | −0.636677 | 0 | 2.31109 | 0 | 1.17307 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0 | 3.00457 | 0 | 2.21040 | 0 | −2.53439 | 0 | 6.02744 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 0 | 3.37094 | 0 | −0.0343863 | 0 | 4.52715 | 0 | 8.36321 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(173\) | \( -1 \) |
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 | 2768.2.a.o | 13 | |
| 4.b | odd | 2 | 1 | 1384.2.a.e | ✓ | 13 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1384.2.a.e | ✓ | 13 | 4.b | odd | 2 | 1 | |
| 2768.2.a.o | 13 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{13} - 2 T_{3}^{12} - 25 T_{3}^{11} + 47 T_{3}^{10} + 219 T_{3}^{9} - 389 T_{3}^{8} - 808 T_{3}^{7} + \cdots - 128 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2768))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{13} \)
|
| $3$ |
\( T^{13} - 2 T^{12} + \cdots - 128 \)
|
| $5$ |
\( T^{13} - 10 T^{12} + \cdots - 128 \)
|
| $7$ |
\( T^{13} - 10 T^{12} + \cdots + 88740 \)
|
| $11$ |
\( T^{13} + 9 T^{12} + \cdots - 87424 \)
|
| $13$ |
\( T^{13} - 9 T^{12} + \cdots - 140984 \)
|
| $17$ |
\( T^{13} - 14 T^{12} + \cdots - 8704 \)
|
| $19$ |
\( T^{13} - T^{12} + \cdots + 4376992 \)
|
| $23$ |
\( T^{13} - 11 T^{12} + \cdots - 3571712 \)
|
| $29$ |
\( T^{13} + \cdots + 697536792 \)
|
| $31$ |
\( T^{13} - 5 T^{12} + \cdots - 67284992 \)
|
| $37$ |
\( T^{13} + \cdots + 199177448 \)
|
| $41$ |
\( T^{13} - 36 T^{12} + \cdots + 79402 \)
|
| $43$ |
\( T^{13} + \cdots + 227190528 \)
|
| $47$ |
\( T^{13} + \cdots + 7150131200 \)
|
| $53$ |
\( T^{13} + \cdots + 3963367552 \)
|
| $59$ |
\( T^{13} + \cdots - 638828032 \)
|
| $61$ |
\( T^{13} + \cdots + 71352460160 \)
|
| $67$ |
\( T^{13} + 21 T^{12} + \cdots - 7038208 \)
|
| $71$ |
\( T^{13} + 5 T^{12} + \cdots + 30273300 \)
|
| $73$ |
\( T^{13} + \cdots - 29130341470 \)
|
| $79$ |
\( T^{13} + \cdots + 9024805060 \)
|
| $83$ |
\( T^{13} + 19 T^{12} + \cdots + 10412288 \)
|
| $89$ |
\( T^{13} + \cdots - 6964860310 \)
|
| $97$ |
\( T^{13} + \cdots + 6762354176 \)
|
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