Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2299,4,Mod(1,2299)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2299, 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("2299.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2299 = 11^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2299.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: | \(135.645391103\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 43x^{6} - 12x^{5} + 545x^{4} + 264x^{3} - 2017x^{2} - 660x + 2178 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 209) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 43x^{6} - 12x^{5} + 545x^{4} + 264x^{3} - 2017x^{2} - 660x + 2178 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{7} - 165\nu^{6} - 334\nu^{5} + 6318\nu^{4} + 6389\nu^{3} - 64911\nu^{2} - 45634\nu + 129690 ) / 6864 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 17\nu^{7} - 33\nu^{6} - 566\nu^{5} + 390\nu^{4} + 4975\nu^{3} + 6237\nu^{2} - 12278\nu - 22110 ) / 3432 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} - 43\nu^{5} - 12\nu^{4} + 512\nu^{3} + 264\nu^{2} - 1258\nu - 462 ) / 66 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 43\nu^{7} - 33\nu^{6} - 1684\nu^{5} + 936\nu^{4} + 18287\nu^{3} - 6633\nu^{2} - 46702\nu + 17358 ) / 1716 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 5\nu^{7} - 12\nu^{6} - 194\nu^{5} + 390\nu^{4} + 2023\nu^{3} - 3114\nu^{2} - 4070\nu + 5688 ) / 156 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} + \beta_{6} + \beta_{4} + 2\beta_{3} + \beta_{2} + 18\beta _1 + 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{6} - 2\beta_{5} - 4\beta_{4} + 23\beta_{2} + 2\beta _1 + 193 \)
|
| \(\nu^{5}\) | \(=\) |
\( -25\beta_{7} + 33\beta_{6} - 14\beta_{5} + 29\beta_{4} + 42\beta_{3} + 29\beta_{2} + 362\beta _1 + 192 \)
|
| \(\nu^{6}\) | \(=\) |
\( -12\beta_{7} + 88\beta_{6} - 72\beta_{5} - 144\beta_{4} - 16\beta_{3} + 499\beta_{2} + 88\beta _1 + 3907 \)
|
| \(\nu^{7}\) | \(=\) |
\( -563\beta_{7} + 931\beta_{6} - 560\beta_{5} + 687\beta_{4} + 782\beta_{3} + 747\beta_{2} + 7632\beta _1 + 5058 \)
|
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 |
|
−4.65458 | −3.27619 | 13.6651 | −13.6377 | 15.2493 | 11.6190 | −26.3687 | −16.2666 | 63.4778 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −3.50438 | −0.605984 | 4.28065 | 0.861140 | 2.12360 | 30.2736 | 13.0340 | −26.6328 | −3.01776 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.09608 | 0.225665 | −3.60644 | −3.33943 | −0.473012 | −21.2626 | 24.3281 | −26.9491 | 6.99972 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.60842 | 7.48922 | −5.41298 | −10.7091 | −12.0458 | 24.0540 | 21.5737 | 29.0884 | 17.2248 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.14081 | 1.79882 | −6.69855 | 12.6906 | 2.05212 | 10.6895 | −16.7683 | −23.7642 | 14.4776 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.73547 | −9.28551 | −4.98814 | −1.59392 | −16.1147 | −8.61614 | −22.5405 | 59.2207 | −2.76620 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.06018 | 5.61545 | 8.48503 | −6.03806 | 22.7997 | −4.30385 | 1.96930 | 4.53328 | −24.5156 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 4.92700 | −2.96147 | 16.2753 | 9.76654 | −14.5912 | 16.5465 | 40.7725 | −18.2297 | 48.1197 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \( -1 \) |
| \(19\) | \( +1 \) |
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 | 2299.4.a.i | 8 | |
| 11.b | odd | 2 | 1 | 209.4.a.a | ✓ | 8 | |
| 33.d | even | 2 | 1 | 1881.4.a.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 209.4.a.a | ✓ | 8 | 11.b | odd | 2 | 1 | |
| 1881.4.a.a | 8 | 33.d | even | 2 | 1 | ||
| 2299.4.a.i | 8 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2299))\):
|
\( T_{2}^{8} - 43T_{2}^{6} - 12T_{2}^{5} + 545T_{2}^{4} + 264T_{2}^{3} - 2017T_{2}^{2} - 660T_{2} + 2178 \)
|
|
\( T_{5}^{8} + 12T_{5}^{7} - 232T_{5}^{6} - 3010T_{5}^{5} + 8324T_{5}^{4} + 175704T_{5}^{3} + 471697T_{5}^{2} + 41724T_{5} - 500988 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 43 T^{6} + \cdots + 2178 \)
|
| $3$ |
\( T^{8} + T^{7} + \cdots + 932 \)
|
| $5$ |
\( T^{8} + 12 T^{7} + \cdots - 500988 \)
|
| $7$ |
\( T^{8} + \cdots - 1179970928 \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( T^{8} + \cdots - 928182024856 \)
|
| $17$ |
\( T^{8} + \cdots - 5700281587968 \)
|
| $19$ |
\( (T + 19)^{8} \)
|
| $23$ |
\( T^{8} + \cdots + 25\!\cdots\!76 \)
|
| $29$ |
\( T^{8} + \cdots + 39\!\cdots\!76 \)
|
| $31$ |
\( T^{8} + \cdots - 43\!\cdots\!44 \)
|
| $37$ |
\( T^{8} + \cdots - 80\!\cdots\!72 \)
|
| $41$ |
\( T^{8} + \cdots - 50\!\cdots\!52 \)
|
| $43$ |
\( T^{8} + \cdots + 60\!\cdots\!36 \)
|
| $47$ |
\( T^{8} + \cdots + 11\!\cdots\!36 \)
|
| $53$ |
\( T^{8} + \cdots + 64\!\cdots\!96 \)
|
| $59$ |
\( T^{8} + \cdots - 11\!\cdots\!48 \)
|
| $61$ |
\( T^{8} + \cdots + 19\!\cdots\!08 \)
|
| $67$ |
\( T^{8} + \cdots - 14\!\cdots\!68 \)
|
| $71$ |
\( T^{8} + \cdots - 15\!\cdots\!12 \)
|
| $73$ |
\( T^{8} + \cdots - 29\!\cdots\!12 \)
|
| $79$ |
\( T^{8} + \cdots + 69\!\cdots\!52 \)
|
| $83$ |
\( T^{8} + \cdots - 31\!\cdots\!64 \)
|
| $89$ |
\( T^{8} + \cdots - 20\!\cdots\!44 \)
|
| $97$ |
\( T^{8} + \cdots - 11\!\cdots\!12 \)
|
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