Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [950,6,Mod(1,950)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(950, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("950.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 950 = 2 \cdot 5^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 950.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: | \(152.364628822\) |
| 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: |
\( x^{12} - 4 x^{11} - 2039 x^{10} + 5340 x^{9} + 1506628 x^{8} - 1407112 x^{7} - 486931198 x^{6} + \cdots + 9535638376170 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{9}\cdot 5^{6} \) |
| Twist minimal: | no (minimal twist has level 190) |
| 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
\( x^{12} - 4 x^{11} - 2039 x^{10} + 5340 x^{9} + 1506628 x^{8} - 1407112 x^{7} - 486931198 x^{6} + \cdots + 9535638376170 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 341 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 26\!\cdots\!68 \nu^{11} + \cdots - 27\!\cdots\!70 ) / 98\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 64\!\cdots\!93 \nu^{11} + \cdots - 17\!\cdots\!40 ) / 79\!\cdots\!80 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 26\!\cdots\!39 \nu^{11} + \cdots + 51\!\cdots\!90 ) / 26\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 34\!\cdots\!21 \nu^{11} + \cdots - 63\!\cdots\!10 ) / 32\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 67\!\cdots\!27 \nu^{11} + \cdots - 91\!\cdots\!70 ) / 39\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 19\!\cdots\!91 \nu^{11} + \cdots - 34\!\cdots\!60 ) / 98\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 24\!\cdots\!41 \nu^{11} + \cdots - 18\!\cdots\!40 ) / 98\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 62\!\cdots\!03 \nu^{11} + \cdots - 36\!\cdots\!30 ) / 19\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 22\!\cdots\!73 \nu^{11} + \cdots - 49\!\cdots\!30 ) / 65\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 341 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{11} - \beta_{10} + \beta_{8} + 2\beta_{6} + 2\beta_{5} + 3\beta_{4} - 8\beta_{3} - 3\beta_{2} + 564\beta _1 + 517 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 9 \beta_{11} + 17 \beta_{10} + 4 \beta_{9} + 30 \beta_{8} - 59 \beta_{7} - 15 \beta_{6} + \cdots + 194363 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 1183 \beta_{11} - 591 \beta_{10} - 370 \beta_{9} + 1381 \beta_{8} - 1273 \beta_{7} + 1907 \beta_{6} + \cdots + 362783 \)
|
| \(\nu^{6}\) | \(=\) |
\( 4564 \beta_{11} + 13424 \beta_{10} + 3203 \beta_{9} + 27479 \beta_{8} - 80299 \beta_{7} + \cdots + 124603421 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 1093173 \beta_{11} - 361905 \beta_{10} - 625095 \beta_{9} + 1429014 \beta_{8} - 1675941 \beta_{7} + \cdots + 203951371 \)
|
| \(\nu^{8}\) | \(=\) |
\( 13743481 \beta_{11} + 7753297 \beta_{10} + 1708887 \beta_{9} + 19451255 \beta_{8} - 81427710 \beta_{7} + \cdots + 83888637603 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 909034253 \beta_{11} - 253272103 \beta_{10} - 693459235 \beta_{9} + 1312712528 \beta_{8} + \cdots + 113937550323 \)
|
| \(\nu^{10}\) | \(=\) |
\( 17540199442 \beta_{11} + 3368487538 \beta_{10} + 890992920 \beta_{9} + 12655832900 \beta_{8} + \cdots + 57912894028013 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 719224039671 \beta_{11} - 196380951131 \beta_{10} - 653117495572 \beta_{9} + 1130976368463 \beta_{8} + \cdots + 68077847563763 \)
|
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 |
|
−4.00000 | −30.2244 | 16.0000 | 0 | 120.897 | −2.25332 | −64.0000 | 670.512 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.00000 | −28.0392 | 16.0000 | 0 | 112.157 | −116.949 | −64.0000 | 543.198 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.00000 | −18.8308 | 16.0000 | 0 | 75.3231 | −36.9156 | −64.0000 | 111.598 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −4.00000 | −16.3199 | 16.0000 | 0 | 65.2795 | 183.824 | −64.0000 | 23.3387 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −4.00000 | −13.9208 | 16.0000 | 0 | 55.6830 | −193.662 | −64.0000 | −49.2126 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −4.00000 | −5.02941 | 16.0000 | 0 | 20.1176 | 76.4235 | −64.0000 | −217.705 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −4.00000 | −0.441586 | 16.0000 | 0 | 1.76634 | 138.874 | −64.0000 | −242.805 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −4.00000 | 1.68289 | 16.0000 | 0 | −6.73155 | −191.453 | −64.0000 | −240.168 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −4.00000 | 13.1963 | 16.0000 | 0 | −52.7850 | −170.932 | −64.0000 | −68.8590 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −4.00000 | 19.5162 | 16.0000 | 0 | −78.0649 | 85.1138 | −64.0000 | 137.883 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | −4.00000 | 21.9314 | 16.0000 | 0 | −87.7256 | −123.374 | −64.0000 | 237.987 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | −4.00000 | 24.4792 | 16.0000 | 0 | −97.9169 | 101.305 | −64.0000 | 356.233 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 950.6.a.x | 12 | |
| 5.b | even | 2 | 1 | 950.6.a.y | 12 | ||
| 5.c | odd | 4 | 2 | 190.6.b.b | ✓ | 24 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 190.6.b.b | ✓ | 24 | 5.c | odd | 4 | 2 | |
| 950.6.a.x | 12 | 1.a | even | 1 | 1 | trivial | |
| 950.6.a.y | 12 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + 32 T_{3}^{11} - 1577 T_{3}^{10} - 51870 T_{3}^{9} + 847288 T_{3}^{8} + 29961296 T_{3}^{7} + \cdots - 1873568071296 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(950))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{12} \)
|
| $3$ |
\( T^{12} + \cdots - 1873568071296 \)
|
| $5$ |
\( T^{12} \)
|
| $7$ |
\( T^{12} + \cdots - 12\!\cdots\!80 \)
|
| $11$ |
\( T^{12} + \cdots + 10\!\cdots\!00 \)
|
| $13$ |
\( T^{12} + \cdots + 63\!\cdots\!00 \)
|
| $17$ |
\( T^{12} + \cdots - 15\!\cdots\!52 \)
|
| $19$ |
\( (T - 361)^{12} \)
|
| $23$ |
\( T^{12} + \cdots + 26\!\cdots\!52 \)
|
| $29$ |
\( T^{12} + \cdots + 52\!\cdots\!00 \)
|
| $31$ |
\( T^{12} + \cdots - 33\!\cdots\!44 \)
|
| $37$ |
\( T^{12} + \cdots - 37\!\cdots\!40 \)
|
| $41$ |
\( T^{12} + \cdots - 66\!\cdots\!00 \)
|
| $43$ |
\( T^{12} + \cdots - 47\!\cdots\!76 \)
|
| $47$ |
\( T^{12} + \cdots + 14\!\cdots\!00 \)
|
| $53$ |
\( T^{12} + \cdots - 12\!\cdots\!24 \)
|
| $59$ |
\( T^{12} + \cdots - 49\!\cdots\!00 \)
|
| $61$ |
\( T^{12} + \cdots - 48\!\cdots\!00 \)
|
| $67$ |
\( T^{12} + \cdots - 12\!\cdots\!00 \)
|
| $71$ |
\( T^{12} + \cdots + 48\!\cdots\!00 \)
|
| $73$ |
\( T^{12} + \cdots - 40\!\cdots\!72 \)
|
| $79$ |
\( T^{12} + \cdots - 39\!\cdots\!00 \)
|
| $83$ |
\( T^{12} + \cdots + 29\!\cdots\!20 \)
|
| $89$ |
\( T^{12} + \cdots - 20\!\cdots\!00 \)
|
| $97$ |
\( T^{12} + \cdots + 23\!\cdots\!00 \)
|
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