Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6633,2,Mod(1,6633)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6633, 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("6633.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6633 = 3^{2} \cdot 11 \cdot 67 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6633.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: | \(52.9647716607\) |
| Analytic rank: | \(0\) |
| Dimension: | \(17\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{17} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{17} - x^{16} - 26 x^{15} + 25 x^{14} + 272 x^{13} - 244 x^{12} - 1472 x^{11} + 1186 x^{10} + 4406 x^{9} + \cdots - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 737) |
| 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 \(1,\beta_1,\ldots,\beta_{16}\) 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
\( x^{17} - x^{16} - 26 x^{15} + 25 x^{14} + 272 x^{13} - 244 x^{12} - 1472 x^{11} + 1186 x^{10} + 4406 x^{9} + \cdots - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 27575 \nu^{16} - 13959 \nu^{15} + 843560 \nu^{14} + 452672 \nu^{13} - 10291658 \nu^{12} + \cdots - 2955180 ) / 2721286 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 42559 \nu^{16} - 28638 \nu^{15} - 750284 \nu^{14} + 861143 \nu^{13} + 3455495 \nu^{12} + \cdots + 7475216 ) / 2721286 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 86001 \nu^{16} - 250136 \nu^{15} + 1897162 \nu^{14} + 5794521 \nu^{13} - 16740475 \nu^{12} + \cdots - 14046846 ) / 5442572 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 230329 \nu^{16} + 439848 \nu^{15} + 6997748 \nu^{14} - 10143163 \nu^{13} - 85202057 \nu^{12} + \cdots + 1304566 ) / 5442572 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 79595 \nu^{16} + 1558 \nu^{15} - 2027846 \nu^{14} + 122103 \nu^{13} + 21015197 \nu^{12} + \cdots + 4174317 ) / 1360643 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 214049 \nu^{16} + 210501 \nu^{15} + 5547392 \nu^{14} - 5156829 \nu^{13} - 57932009 \nu^{12} + \cdots - 5537703 ) / 1360643 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 436709 \nu^{16} - 320078 \nu^{15} - 11322680 \nu^{14} + 7612317 \nu^{13} + 118167911 \nu^{12} + \cdots + 11509244 ) / 2721286 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 508577 \nu^{16} - 17687 \nu^{15} - 12411743 \nu^{14} + 1463506 \nu^{13} + 121251307 \nu^{12} + \cdots + 3883726 ) / 2721286 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 1019239 \nu^{16} + 619284 \nu^{15} + 27458058 \nu^{14} - 14797609 \nu^{13} - 299668645 \nu^{12} + \cdots - 6108450 ) / 5442572 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 1221155 \nu^{16} - 143736 \nu^{15} + 31001512 \nu^{14} + 2650763 \nu^{13} - 317205775 \nu^{12} + \cdots - 28578490 ) / 5442572 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 1253765 \nu^{16} - 1040280 \nu^{15} - 32953894 \nu^{14} + 24351943 \nu^{13} + 347946495 \nu^{12} + \cdots - 2067402 ) / 5442572 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 335296 \nu^{16} - 87987 \nu^{15} - 8656255 \nu^{14} + 2325772 \nu^{13} + 90310300 \nu^{12} + \cdots - 4300557 ) / 1360643 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 676979 \nu^{16} - 359308 \nu^{15} - 17814320 \nu^{14} + 8700073 \nu^{13} + 189253867 \nu^{12} + \cdots + 4124632 ) / 2721286 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{15} - \beta_{13} - \beta_{7} + 2\beta_{4} + 7\beta_{2} + \beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{16} + \beta_{15} - \beta_{14} + \beta_{12} - \beta_{11} - \beta_{9} + \beta_{7} + \beta_{6} + \cdots - 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 10 \beta_{15} + \beta_{14} - 9 \beta_{13} + \beta_{11} + 2 \beta_{9} - 12 \beta_{7} - 2 \beta_{6} + \cdots + 87 \)
|
| \(\nu^{7}\) | \(=\) |
\( 11 \beta_{16} + 14 \beta_{15} - 13 \beta_{14} + \beta_{13} + 11 \beta_{12} - 13 \beta_{11} + \beta_{10} + \cdots - 30 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 83 \beta_{15} + 17 \beta_{14} - 65 \beta_{13} - \beta_{12} + 16 \beta_{11} - \beta_{10} + 30 \beta_{9} + \cdots + 543 \)
|
| \(\nu^{9}\) | \(=\) |
\( 92 \beta_{16} + 144 \beta_{15} - 126 \beta_{14} + 18 \beta_{13} + 90 \beta_{12} - 125 \beta_{11} + \cdots - 310 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 8 \beta_{16} - 656 \beta_{15} + 199 \beta_{14} - 446 \beta_{13} - 16 \beta_{12} + 179 \beta_{11} + \cdots + 3527 \)
|
| \(\nu^{11}\) | \(=\) |
\( 708 \beta_{16} + 1310 \beta_{15} - 1099 \beta_{14} + 219 \beta_{13} + 661 \beta_{12} - 1074 \beta_{11} + \cdots - 2795 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 174 \beta_{16} - 5094 \beta_{15} + 1985 \beta_{14} - 3043 \beta_{13} - 173 \beta_{12} + 1727 \beta_{11} + \cdots + 23492 \)
|
| \(\nu^{13}\) | \(=\) |
\( 5291 \beta_{16} + 11179 \beta_{15} - 9116 \beta_{14} + 2246 \beta_{13} + 4615 \beta_{12} - 8736 \beta_{11} + \cdots - 23633 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 2389 \beta_{16} - 39234 \beta_{15} + 18138 \beta_{14} - 20957 \beta_{13} - 1586 \beta_{12} + \cdots + 159389 \)
|
| \(\nu^{15}\) | \(=\) |
\( 39150 \beta_{16} + 91867 \beta_{15} - 73521 \beta_{14} + 20944 \beta_{13} + 31390 \beta_{12} + \cdots - 192950 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 26640 \beta_{16} - 300799 \beta_{15} + 157044 \beta_{14} - 146336 \beta_{13} - 13359 \beta_{12} + \cdots + 1097992 \)
|
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 |
|
−2.48380 | 0 | 4.16926 | 2.30083 | 0 | 0.224864 | −5.38802 | 0 | −5.71481 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.47482 | 0 | 4.12472 | −1.41072 | 0 | 1.07503 | −5.25828 | 0 | 3.49127 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.40304 | 0 | 3.77462 | −4.30776 | 0 | 2.07309 | −4.26450 | 0 | 10.3517 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.92669 | 0 | 1.71214 | −2.16600 | 0 | −2.75227 | 0.554608 | 0 | 4.17321 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.72368 | 0 | 0.971084 | −1.06718 | 0 | 4.32688 | 1.77352 | 0 | 1.83948 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.52171 | 0 | 0.315610 | 3.19472 | 0 | 1.20336 | 2.56316 | 0 | −4.86144 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.484318 | 0 | −1.76544 | −3.54277 | 0 | −0.730966 | 1.82367 | 0 | 1.71583 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.418937 | 0 | −1.82449 | −3.45455 | 0 | 4.84408 | 1.60222 | 0 | 1.44724 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −0.166502 | 0 | −1.97228 | 1.68184 | 0 | 1.74824 | 0.661391 | 0 | −0.280030 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.0213052 | 0 | −1.99955 | 1.68894 | 0 | −3.23910 | −0.0852109 | 0 | 0.0359832 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0.880523 | 0 | −1.22468 | −1.19978 | 0 | −2.13292 | −2.83940 | 0 | −1.05643 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.19652 | 0 | −0.568334 | 0.744253 | 0 | 4.07489 | −3.07307 | 0 | 0.890515 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 1.40401 | 0 | −0.0287695 | −2.32198 | 0 | −2.22330 | −2.84840 | 0 | −3.26008 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 1.69059 | 0 | 0.858090 | −2.30851 | 0 | 4.17978 | −1.93050 | 0 | −3.90275 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 2.20047 | 0 | 2.84207 | 3.56210 | 0 | 3.47229 | 1.85294 | 0 | 7.83830 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.45774 | 0 | 4.04047 | −3.82582 | 0 | −0.0899620 | 5.01493 | 0 | −9.40286 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 2.75236 | 0 | 5.57546 | 2.43239 | 0 | 3.94601 | 9.84095 | 0 | 6.69482 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(11\) | \(1\) |
| \(67\) | \(1\) |
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 | 6633.2.a.w | 17 | |
| 3.b | odd | 2 | 1 | 737.2.a.f | ✓ | 17 | |
| 33.d | even | 2 | 1 | 8107.2.a.o | 17 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 737.2.a.f | ✓ | 17 | 3.b | odd | 2 | 1 | |
| 6633.2.a.w | 17 | 1.a | even | 1 | 1 | trivial | |
| 8107.2.a.o | 17 | 33.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6633))\):
|
\( T_{2}^{17} + T_{2}^{16} - 26 T_{2}^{15} - 25 T_{2}^{14} + 272 T_{2}^{13} + 244 T_{2}^{12} - 1472 T_{2}^{11} + \cdots + 2 \)
|
|
\( T_{5}^{17} + 10 T_{5}^{16} - 9 T_{5}^{15} - 374 T_{5}^{14} - 622 T_{5}^{13} + 5228 T_{5}^{12} + \cdots - 569536 \)
|
|
\( T_{7}^{17} - 20 T_{7}^{16} + 129 T_{7}^{15} - 40 T_{7}^{14} - 3008 T_{7}^{13} + 10322 T_{7}^{12} + \cdots - 14336 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{17} + T^{16} + \cdots + 2 \)
|
| $3$ |
\( T^{17} \)
|
| $5$ |
\( T^{17} + 10 T^{16} + \cdots - 569536 \)
|
| $7$ |
\( T^{17} - 20 T^{16} + \cdots - 14336 \)
|
| $11$ |
\( (T + 1)^{17} \)
|
| $13$ |
\( T^{17} - T^{16} + \cdots + 1244 \)
|
| $17$ |
\( T^{17} + 2 T^{16} + \cdots - 10177376 \)
|
| $19$ |
\( T^{17} - 13 T^{16} + \cdots + 3615592 \)
|
| $23$ |
\( T^{17} + 16 T^{16} + \cdots - 28160 \)
|
| $29$ |
\( T^{17} - 5 T^{16} + \cdots - 19838176 \)
|
| $31$ |
\( T^{17} + \cdots - 553125251072 \)
|
| $37$ |
\( T^{17} + \cdots + 3657103090 \)
|
| $41$ |
\( T^{17} + \cdots - 13712702252 \)
|
| $43$ |
\( T^{17} + \cdots - 155446912 \)
|
| $47$ |
\( T^{17} + \cdots + 120416267195008 \)
|
| $53$ |
\( T^{17} + \cdots + 1306137045760 \)
|
| $59$ |
\( T^{17} + \cdots + 37887089920 \)
|
| $61$ |
\( T^{17} + \cdots - 16720492768 \)
|
| $67$ |
\( (T + 1)^{17} \)
|
| $71$ |
\( T^{17} + \cdots - 12416174080 \)
|
| $73$ |
\( T^{17} + \cdots - 158595558516640 \)
|
| $79$ |
\( T^{17} + \cdots - 4303519744 \)
|
| $83$ |
\( T^{17} + \cdots + 1453248512 \)
|
| $89$ |
\( T^{17} + \cdots + 10\!\cdots\!10 \)
|
| $97$ |
\( T^{17} + \cdots - 14277000901376 \)
|
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