Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [43,8,Mod(1,43)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(43, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("43.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 43 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 43.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: | \(13.4325560958\) |
| Analytic rank: | \(1\) |
| Dimension: | \(11\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{11} - 2 x^{10} - 977 x^{9} + 2592 x^{8} + 344686 x^{7} - 1160956 x^{6} - 53409536 x^{5} + \cdots + 238240894976 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 7 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - 2 x^{10} - 977 x^{9} + 2592 x^{8} + 344686 x^{7} - 1160956 x^{6} - 53409536 x^{5} + \cdots + 238240894976 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 178 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 152109332450999 \nu^{10} + \cdots + 15\!\cdots\!68 ) / 10\!\cdots\!80 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 26\!\cdots\!79 \nu^{10} + \cdots + 37\!\cdots\!08 ) / 83\!\cdots\!40 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 15\!\cdots\!37 \nu^{10} + \cdots + 36\!\cdots\!96 ) / 83\!\cdots\!40 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 20\!\cdots\!37 \nu^{10} + \cdots - 46\!\cdots\!36 ) / 10\!\cdots\!80 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 995986480737037 \nu^{10} + \cdots + 45\!\cdots\!28 ) / 41\!\cdots\!72 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 690593231497871 \nu^{10} + \cdots + 10\!\cdots\!68 ) / 26\!\cdots\!20 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 31\!\cdots\!91 \nu^{10} + \cdots + 10\!\cdots\!68 ) / 83\!\cdots\!40 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 76\!\cdots\!15 \nu^{10} + \cdots - 10\!\cdots\!44 ) / 16\!\cdots\!88 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - \beta _1 + 178 \)
|
| \(\nu^{3}\) | \(=\) |
\( - 2 \beta_{10} + 4 \beta_{9} + \beta_{8} - 3 \beta_{7} - 18 \beta_{5} + 4 \beta_{4} - 3 \beta_{3} + \cdots - 216 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 10 \beta_{10} + 44 \beta_{9} - 30 \beta_{8} - 5 \beta_{7} - 29 \beta_{6} - 112 \beta_{5} + \cdots + 47769 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 762 \beta_{10} + 2120 \beta_{9} + 306 \beta_{8} - 1711 \beta_{7} + 89 \beta_{6} - 8560 \beta_{5} + \cdots - 82473 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 5154 \beta_{10} + 26856 \beta_{9} - 14456 \beta_{8} - 3327 \beta_{7} - 17529 \beta_{6} + \cdots + 14953881 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 283250 \beta_{10} + 917648 \beta_{9} + 85284 \beta_{8} - 735667 \beta_{7} + 92087 \beta_{6} + \cdots - 23795839 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 1867314 \beta_{10} + 12553584 \beta_{9} - 5561184 \beta_{8} - 1698147 \beta_{7} - 8240085 \beta_{6} + \cdots + 5074980349 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 110006946 \beta_{10} + 372850128 \beta_{9} + 22506528 \beta_{8} - 292040643 \beta_{7} + \cdots - 5320106523 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 587476050 \beta_{10} + 5358368064 \beta_{9} - 2031998808 \beta_{8} - 802180395 \beta_{7} + \cdots + 1804293884701 \)
|
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 |
|
−21.7860 | 54.2998 | 346.631 | −409.924 | −1182.98 | 476.571 | −4763.10 | 761.467 | 8930.61 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −16.1572 | −48.3227 | 133.057 | −272.935 | 780.762 | 1101.65 | −81.7025 | 148.083 | 4409.88 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −15.1261 | 62.8737 | 100.799 | 12.0619 | −951.035 | −1248.65 | 411.441 | 1766.10 | −182.450 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −13.4671 | −87.0645 | 53.3630 | 131.138 | 1172.51 | −712.280 | 1005.14 | 5393.23 | −1766.05 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −8.31419 | 11.6940 | −58.8742 | 148.086 | −97.2261 | 122.189 | 1553.71 | −2050.25 | −1231.21 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −5.52766 | −57.7031 | −97.4450 | 304.619 | 318.963 | 1421.71 | 1246.18 | 1142.65 | −1683.83 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 3.14919 | 34.1076 | −118.083 | −39.2891 | 107.411 | 434.472 | −774.961 | −1023.67 | −123.729 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 8.30971 | 47.1073 | −58.9487 | −187.284 | 391.449 | −1501.21 | −1553.49 | 32.1014 | −1556.27 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 12.3182 | −46.7197 | 23.7379 | 330.186 | −575.503 | −467.397 | −1284.32 | −4.26512 | 4067.29 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 15.2185 | −9.20709 | 103.603 | −537.911 | −140.118 | 1471.62 | −371.288 | −2102.23 | −8186.21 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 17.3827 | −29.0652 | 174.160 | −230.747 | −505.234 | −1110.68 | 802.384 | −1342.21 | −4011.02 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(43\) | \(-1\) |
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 | 43.8.a.a | ✓ | 11 |
| 3.b | odd | 2 | 1 | 387.8.a.b | 11 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 43.8.a.a | ✓ | 11 | 1.a | even | 1 | 1 | trivial |
| 387.8.a.b | 11 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{11} + 24 T_{2}^{10} - 717 T_{2}^{9} - 18498 T_{2}^{8} + 169726 T_{2}^{7} + 5061216 T_{2}^{6} + \cdots - 281015823360 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(43))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} + \cdots - 281015823360 \)
|
| $3$ |
\( T^{11} + \cdots - 19\!\cdots\!12 \)
|
| $5$ |
\( T^{11} + \cdots - 24\!\cdots\!00 \)
|
| $7$ |
\( T^{11} + \cdots + 40\!\cdots\!60 \)
|
| $11$ |
\( T^{11} + \cdots - 21\!\cdots\!12 \)
|
| $13$ |
\( T^{11} + \cdots + 43\!\cdots\!00 \)
|
| $17$ |
\( T^{11} + \cdots + 77\!\cdots\!91 \)
|
| $19$ |
\( T^{11} + \cdots + 19\!\cdots\!00 \)
|
| $23$ |
\( T^{11} + \cdots - 90\!\cdots\!15 \)
|
| $29$ |
\( T^{11} + \cdots - 19\!\cdots\!00 \)
|
| $31$ |
\( T^{11} + \cdots + 21\!\cdots\!45 \)
|
| $37$ |
\( T^{11} + \cdots + 13\!\cdots\!00 \)
|
| $41$ |
\( T^{11} + \cdots + 14\!\cdots\!75 \)
|
| $43$ |
\( (T - 79507)^{11} \)
|
| $47$ |
\( T^{11} + \cdots - 61\!\cdots\!00 \)
|
| $53$ |
\( T^{11} + \cdots - 15\!\cdots\!00 \)
|
| $59$ |
\( T^{11} + \cdots - 47\!\cdots\!00 \)
|
| $61$ |
\( T^{11} + \cdots + 46\!\cdots\!80 \)
|
| $67$ |
\( T^{11} + \cdots - 63\!\cdots\!64 \)
|
| $71$ |
\( T^{11} + \cdots + 11\!\cdots\!56 \)
|
| $73$ |
\( T^{11} + \cdots - 53\!\cdots\!28 \)
|
| $79$ |
\( T^{11} + \cdots + 65\!\cdots\!60 \)
|
| $83$ |
\( T^{11} + \cdots - 62\!\cdots\!80 \)
|
| $89$ |
\( T^{11} + \cdots + 11\!\cdots\!00 \)
|
| $97$ |
\( T^{11} + \cdots - 28\!\cdots\!07 \)
|
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