Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [38,8,Mod(7,38)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(38, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("38.7");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 38 = 2 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 38.c (of order \(3\), degree \(2\), minimal) |
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: | no |
| Analytic conductor: | \(11.8706309684\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} + 10165 x^{10} - 228496 x^{9} + 79937698 x^{8} - 1362528496 x^{7} + 222057453949 x^{6} + \cdots + 20\!\cdots\!25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{3}\cdot 7^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{12} + 10165 x^{10} - 228496 x^{9} + 79937698 x^{8} - 1362528496 x^{7} + 222057453949 x^{6} + \cdots + 20\!\cdots\!25 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 14\!\cdots\!36 \nu^{11} + \cdots - 21\!\cdots\!25 ) / 18\!\cdots\!95 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 19\!\cdots\!06 \nu^{11} + \cdots - 90\!\cdots\!00 ) / 58\!\cdots\!05 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 22\!\cdots\!62 \nu^{11} + \cdots + 17\!\cdots\!40 ) / 12\!\cdots\!60 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 91\!\cdots\!95 \nu^{11} + \cdots - 13\!\cdots\!00 ) / 41\!\cdots\!20 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 70\!\cdots\!57 \nu^{11} + \cdots - 36\!\cdots\!00 ) / 12\!\cdots\!60 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 97\!\cdots\!03 \nu^{11} + \cdots - 10\!\cdots\!50 ) / 13\!\cdots\!40 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 22\!\cdots\!68 \nu^{11} + \cdots + 68\!\cdots\!35 ) / 12\!\cdots\!60 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 25\!\cdots\!90 \nu^{11} + \cdots + 58\!\cdots\!70 ) / 13\!\cdots\!40 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 44\!\cdots\!61 \nu^{11} + \cdots + 44\!\cdots\!80 ) / 12\!\cdots\!80 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 54\!\cdots\!16 \nu^{11} + \cdots - 12\!\cdots\!82 ) / 77\!\cdots\!08 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{10} + \beta_{5} - 3\beta_{4} + 17\beta_{3} - 3389\beta_{2} - 3389 \)
|
| \(\nu^{3}\) | \(=\) |
\( - 12 \beta_{11} - 33 \beta_{10} - 90 \beta_{9} + 90 \beta_{7} - 321 \beta_{6} + 321 \beta_{4} + \cdots + 57246 \)
|
| \(\nu^{4}\) | \(=\) |
\( 600\beta_{11} - 600\beta_{8} + 288\beta_{7} + 30984\beta_{6} - 8248\beta_{5} + 18856673\beta_{2} + 238424\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 396768 \beta_{10} + 699552 \beta_{9} + 108960 \beta_{8} + 396768 \beta_{5} - 3373896 \beta_{4} + \cdots - 801300360 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 7469976 \beta_{11} - 64221577 \beta_{10} - 7354800 \beta_{9} + 7354800 \beta_{7} + \cdots + 133326061805 \)
|
| \(\nu^{7}\) | \(=\) |
\( 895452876 \beta_{11} - 895452876 \beta_{8} - 4972985226 \beta_{7} + 29723881977 \beta_{6} + \cdots + 314109844408 \beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 512960788720 \beta_{10} + 143312360832 \beta_{9} + 71932681008 \beta_{8} + 512960788720 \beta_{5} + \cdots - 10\!\cdots\!33 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 7458965056128 \beta_{11} - 37556383808832 \beta_{10} - 36380156730624 \beta_{9} + \cdots + 77\!\cdots\!72 \)
|
| \(\nu^{10}\) | \(=\) |
\( 645482617454256 \beta_{11} - 645482617454256 \beta_{8} + \cdots + 20\!\cdots\!61 \beta_1 \)
|
| \(\nu^{11}\) | \(=\) |
\( 33\!\cdots\!89 \beta_{10} + \cdots - 69\!\cdots\!38 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).
| \(n\) | \(21\) |
| \(\chi(n)\) | \(-1 - \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
−4.00000 | − | 6.92820i | −45.4369 | − | 78.6990i | −32.0000 | + | 55.4256i | 149.770 | + | 259.410i | −363.495 | + | 629.592i | −708.303 | 512.000 | −3035.52 | + | 5257.68i | 1198.16 | − | 2075.28i | ||||||||||||||||||||||||||||||||||||||||
| 7.2 | −4.00000 | − | 6.92820i | −20.5062 | − | 35.5178i | −32.0000 | + | 55.4256i | −273.130 | − | 473.076i | −164.050 | + | 284.143i | −154.247 | 512.000 | 252.489 | − | 437.324i | −2185.04 | + | 3784.60i | |||||||||||||||||||||||||||||||||||||||||
| 7.3 | −4.00000 | − | 6.92820i | −11.7080 | − | 20.2789i | −32.0000 | + | 55.4256i | 68.7885 | + | 119.145i | −93.6643 | + | 162.231i | 1024.63 | 512.000 | 819.344 | − | 1419.14i | 550.308 | − | 953.162i | |||||||||||||||||||||||||||||||||||||||||
| 7.4 | −4.00000 | − | 6.92820i | 20.2647 | + | 35.0995i | −32.0000 | + | 55.4256i | −75.0460 | − | 129.983i | 162.117 | − | 280.796i | −892.194 | 512.000 | 272.185 | − | 471.439i | −600.368 | + | 1039.87i | |||||||||||||||||||||||||||||||||||||||||
| 7.5 | −4.00000 | − | 6.92820i | 26.8545 | + | 46.5134i | −32.0000 | + | 55.4256i | 253.607 | + | 439.261i | 214.836 | − | 372.107i | −1033.15 | 512.000 | −348.830 | + | 604.192i | 2028.86 | − | 3514.09i | |||||||||||||||||||||||||||||||||||||||||
| 7.6 | −4.00000 | − | 6.92820i | 36.5320 | + | 63.2752i | −32.0000 | + | 55.4256i | −61.9899 | − | 107.370i | 292.256 | − | 506.202i | 1245.26 | 512.000 | −1575.67 | + | 2729.14i | −495.919 | + | 858.957i | |||||||||||||||||||||||||||||||||||||||||
| 11.1 | −4.00000 | + | 6.92820i | −45.4369 | + | 78.6990i | −32.0000 | − | 55.4256i | 149.770 | − | 259.410i | −363.495 | − | 629.592i | −708.303 | 512.000 | −3035.52 | − | 5257.68i | 1198.16 | + | 2075.28i | |||||||||||||||||||||||||||||||||||||||||
| 11.2 | −4.00000 | + | 6.92820i | −20.5062 | + | 35.5178i | −32.0000 | − | 55.4256i | −273.130 | + | 473.076i | −164.050 | − | 284.143i | −154.247 | 512.000 | 252.489 | + | 437.324i | −2185.04 | − | 3784.60i | |||||||||||||||||||||||||||||||||||||||||
| 11.3 | −4.00000 | + | 6.92820i | −11.7080 | + | 20.2789i | −32.0000 | − | 55.4256i | 68.7885 | − | 119.145i | −93.6643 | − | 162.231i | 1024.63 | 512.000 | 819.344 | + | 1419.14i | 550.308 | + | 953.162i | |||||||||||||||||||||||||||||||||||||||||
| 11.4 | −4.00000 | + | 6.92820i | 20.2647 | − | 35.0995i | −32.0000 | − | 55.4256i | −75.0460 | + | 129.983i | 162.117 | + | 280.796i | −892.194 | 512.000 | 272.185 | + | 471.439i | −600.368 | − | 1039.87i | |||||||||||||||||||||||||||||||||||||||||
| 11.5 | −4.00000 | + | 6.92820i | 26.8545 | − | 46.5134i | −32.0000 | − | 55.4256i | 253.607 | − | 439.261i | 214.836 | + | 372.107i | −1033.15 | 512.000 | −348.830 | − | 604.192i | 2028.86 | + | 3514.09i | |||||||||||||||||||||||||||||||||||||||||
| 11.6 | −4.00000 | + | 6.92820i | 36.5320 | − | 63.2752i | −32.0000 | − | 55.4256i | −61.9899 | + | 107.370i | 292.256 | + | 506.202i | 1245.26 | 512.000 | −1575.67 | − | 2729.14i | −495.919 | − | 858.957i | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 38.8.c.a | ✓ | 12 |
| 19.c | even | 3 | 1 | inner | 38.8.c.a | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.8.c.a | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 38.8.c.a | ✓ | 12 | 19.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} - 12 T_{3}^{11} + 10249 T_{3}^{10} - 269556 T_{3}^{9} + 81995602 T_{3}^{8} + \cdots + 19\!\cdots\!25 \)
acting on \(S_{8}^{\mathrm{new}}(38, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 8 T + 64)^{6} \)
|
| $3$ |
\( T^{12} + \cdots + 19\!\cdots\!25 \)
|
| $5$ |
\( T^{12} + \cdots + 45\!\cdots\!00 \)
|
| $7$ |
\( (T^{6} + \cdots + 12\!\cdots\!88)^{2} \)
|
| $11$ |
\( (T^{6} + \cdots + 28\!\cdots\!60)^{2} \)
|
| $13$ |
\( T^{12} + \cdots + 51\!\cdots\!24 \)
|
| $17$ |
\( T^{12} + \cdots + 16\!\cdots\!36 \)
|
| $19$ |
\( T^{12} + \cdots + 51\!\cdots\!61 \)
|
| $23$ |
\( T^{12} + \cdots + 52\!\cdots\!24 \)
|
| $29$ |
\( T^{12} + \cdots + 23\!\cdots\!00 \)
|
| $31$ |
\( (T^{6} + \cdots + 75\!\cdots\!16)^{2} \)
|
| $37$ |
\( (T^{6} + \cdots + 42\!\cdots\!20)^{2} \)
|
| $41$ |
\( T^{12} + \cdots + 27\!\cdots\!21 \)
|
| $43$ |
\( T^{12} + \cdots + 49\!\cdots\!44 \)
|
| $47$ |
\( T^{12} + \cdots + 40\!\cdots\!00 \)
|
| $53$ |
\( T^{12} + \cdots + 11\!\cdots\!00 \)
|
| $59$ |
\( T^{12} + \cdots + 20\!\cdots\!21 \)
|
| $61$ |
\( T^{12} + \cdots + 26\!\cdots\!24 \)
|
| $67$ |
\( T^{12} + \cdots + 27\!\cdots\!25 \)
|
| $71$ |
\( T^{12} + \cdots + 37\!\cdots\!76 \)
|
| $73$ |
\( T^{12} + \cdots + 23\!\cdots\!25 \)
|
| $79$ |
\( T^{12} + \cdots + 98\!\cdots\!00 \)
|
| $83$ |
\( (T^{6} + \cdots - 59\!\cdots\!16)^{2} \)
|
| $89$ |
\( T^{12} + \cdots + 28\!\cdots\!24 \)
|
| $97$ |
\( T^{12} + \cdots + 78\!\cdots\!25 \)
|
| show more | |
| show less |