Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [378,2,Mod(143,378)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(378, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("378.143");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 378 = 2 \cdot 3^{3} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 378.l (of order \(6\), degree \(2\), not 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: | \(3.01834519640\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 8 x^{15} + 23 x^{14} - 8 x^{13} - 131 x^{12} + 380 x^{11} - 289 x^{10} - 880 x^{9} + \cdots + 6561 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 3^{4} \) |
| Twist minimal: | no (minimal twist has level 126) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 8 x^{15} + 23 x^{14} - 8 x^{13} - 131 x^{12} + 380 x^{11} - 289 x^{10} - 880 x^{9} + \cdots + 6561 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 50 \nu^{15} + 1352 \nu^{14} - 6827 \nu^{13} + 7676 \nu^{12} + 27422 \nu^{11} - 107246 \nu^{10} + \cdots + 2825604 ) / 142155 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 169 \nu^{15} - 866 \nu^{14} + 1319 \nu^{13} + 2308 \nu^{12} - 13199 \nu^{11} + 19055 \nu^{10} + \cdots - 244944 ) / 47385 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 154 \nu^{15} - 1325 \nu^{14} + 3608 \nu^{13} - 224 \nu^{12} - 22478 \nu^{11} + 55022 \nu^{10} + \cdots - 1285227 ) / 47385 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 1445 \nu^{15} - 9836 \nu^{14} + 21081 \nu^{13} + 15627 \nu^{12} - 172766 \nu^{11} + 334353 \nu^{10} + \cdots - 7117227 ) / 47385 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2858 \nu^{15} - 19265 \nu^{14} + 40866 \nu^{13} + 31392 \nu^{12} - 338066 \nu^{11} + 649239 \nu^{10} + \cdots - 14041269 ) / 47385 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 11192 \nu^{15} + 70123 \nu^{14} - 136087 \nu^{13} - 145859 \nu^{12} + 1215277 \nu^{11} + \cdots + 42998607 ) / 142155 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 16898 \nu^{15} - 108472 \nu^{14} + 217033 \nu^{13} + 208526 \nu^{12} - 1883968 \nu^{11} + \cdots - 69445998 ) / 142155 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 4120 \nu^{15} + 25571 \nu^{14} - 48788 \nu^{13} - 55006 \nu^{12} + 441224 \nu^{11} + \cdots + 14963454 ) / 28431 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 8586 \nu^{15} - 53254 \nu^{14} + 101261 \nu^{13} + 115567 \nu^{12} - 919066 \nu^{11} + \cdots - 30865131 ) / 47385 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 26068 \nu^{15} + 165707 \nu^{14} - 325403 \nu^{13} - 334861 \nu^{12} + 2873363 \nu^{11} + \cdots + 102032298 ) / 142155 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 31130 \nu^{15} + 194263 \nu^{14} - 374983 \nu^{13} - 405716 \nu^{12} + 3354958 \nu^{11} + \cdots + 117273501 ) / 142155 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 33311 \nu^{15} - 207064 \nu^{14} + 396466 \nu^{13} + 441842 \nu^{12} - 3574291 \nu^{11} + \cdots - 121811526 ) / 142155 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 5368 \nu^{15} - 32959 \nu^{14} + 62025 \nu^{13} + 72930 \nu^{12} - 567820 \nu^{11} + 981351 \nu^{10} + \cdots - 18826182 ) / 15795 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 62668 \nu^{15} + 397070 \nu^{14} - 778856 \nu^{13} - 802747 \nu^{12} + 6878696 \nu^{11} + \cdots + 244346949 ) / 142155 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 5105 \nu^{15} - 31804 \nu^{14} + 61039 \nu^{13} + 67583 \nu^{12} - 549514 \nu^{11} + 965497 \nu^{10} + \cdots - 18816948 ) / 10935 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{15} + 2\beta_{9} + \beta_{8} + \beta_{7} - 2\beta_{6} - \beta_{4} - \beta_{3} + 2\beta_{2} - \beta _1 + 2 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{14} - 2 \beta_{13} - \beta_{11} + \beta_{10} + 3 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - \beta_{6} + \cdots + 3 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 2 \beta_{13} + \beta_{12} + 2 \beta_{11} + 3 \beta_{10} + 6 \beta_{9} + 2 \beta_{8} + 5 \beta_{7} + \cdots - 4 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 5 \beta_{15} - 3 \beta_{13} - \beta_{12} + 3 \beta_{11} + 6 \beta_{10} + 2 \beta_{9} - 3 \beta_{8} + \cdots + 8 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 10 \beta_{14} + 3 \beta_{13} + 3 \beta_{12} + 9 \beta_{11} + 5 \beta_{10} + 6 \beta_{9} + 30 \beta_{8} + \cdots - 1 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 7 \beta_{15} - 2 \beta_{14} + 7 \beta_{13} - 2 \beta_{12} + 11 \beta_{11} - 2 \beta_{10} + 8 \beta_{9} + \cdots + 20 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 36 \beta_{15} + 3 \beta_{14} + 6 \beta_{13} + 37 \beta_{12} + 24 \beta_{11} - 21 \beta_{10} + \cdots - 3 ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 42 \beta_{15} + 68 \beta_{14} + 10 \beta_{13} + 66 \beta_{12} + 50 \beta_{11} - 25 \beta_{10} + \cdots + 94 ) / 3 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 44 \beta_{15} + 32 \beta_{14} + 216 \beta_{12} + 72 \beta_{11} + 14 \beta_{10} + 40 \beta_{9} + \cdots + 168 ) / 3 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 58 \beta_{15} - 18 \beta_{14} + 90 \beta_{13} + 172 \beta_{12} + 126 \beta_{11} + 162 \beta_{10} + \cdots + 188 ) / 3 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 126 \beta_{15} - 351 \beta_{14} - 100 \beta_{13} + 231 \beta_{12} + 13 \beta_{11} + 333 \beta_{10} + \cdots + 217 ) / 3 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 209 \beta_{15} - 418 \beta_{14} - 251 \beta_{13} - 170 \beta_{12} + 68 \beta_{11} + 416 \beta_{10} + \cdots - 97 ) / 3 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 385 \beta_{15} - 609 \beta_{14} - 1440 \beta_{13} + 402 \beta_{12} - 186 \beta_{11} + 414 \beta_{10} + \cdots + 1243 ) / 3 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 1281 \beta_{15} - 511 \beta_{14} - 835 \beta_{13} + 960 \beta_{12} + 1042 \beta_{11} + 812 \beta_{10} + \cdots - 1641 ) / 3 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 2826 \beta_{15} - 1179 \beta_{14} - 1012 \beta_{13} + 2828 \beta_{12} + 1522 \beta_{11} + 2838 \beta_{10} + \cdots + 208 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(325\) |
| \(\chi(n)\) | \(\beta_{8}\) | \(\beta_{8}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 143.1 |
|
− | 1.00000i | 0 | −1.00000 | −1.14095 | + | 1.97618i | 0 | 1.42337 | − | 2.23025i | 1.00000i | 0 | 1.97618 | + | 1.14095i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 143.2 | − | 1.00000i | 0 | −1.00000 | −0.714925 | + | 1.23829i | 0 | 0.327442 | + | 2.62541i | 1.00000i | 0 | 1.23829 | + | 0.714925i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 143.3 | − | 1.00000i | 0 | −1.00000 | 0.0338034 | − | 0.0585493i | 0 | 1.19767 | + | 2.35915i | 1.00000i | 0 | −0.0585493 | − | 0.0338034i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 143.4 | − | 1.00000i | 0 | −1.00000 | 1.82207 | − | 3.15592i | 0 | −1.58246 | − | 2.12034i | 1.00000i | 0 | −3.15592 | − | 1.82207i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 143.5 | 1.00000i | 0 | −1.00000 | −1.77612 | + | 3.07634i | 0 | 2.63804 | + | 0.201867i | − | 1.00000i | 0 | −3.07634 | − | 1.77612i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 143.6 | 1.00000i | 0 | −1.00000 | −0.483662 | + | 0.837727i | 0 | −2.16249 | − | 1.52435i | − | 1.00000i | 0 | −0.837727 | − | 0.483662i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 143.7 | 1.00000i | 0 | −1.00000 | 0.450129 | − | 0.779646i | 0 | 1.57151 | − | 2.12847i | − | 1.00000i | 0 | 0.779646 | + | 0.450129i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 143.8 | 1.00000i | 0 | −1.00000 | 1.80966 | − | 3.13442i | 0 | −2.41308 | + | 1.08492i | − | 1.00000i | 0 | 3.13442 | + | 1.80966i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 341.1 | − | 1.00000i | 0 | −1.00000 | −1.77612 | − | 3.07634i | 0 | 2.63804 | − | 0.201867i | 1.00000i | 0 | −3.07634 | + | 1.77612i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 341.2 | − | 1.00000i | 0 | −1.00000 | −0.483662 | − | 0.837727i | 0 | −2.16249 | + | 1.52435i | 1.00000i | 0 | −0.837727 | + | 0.483662i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 341.3 | − | 1.00000i | 0 | −1.00000 | 0.450129 | + | 0.779646i | 0 | 1.57151 | + | 2.12847i | 1.00000i | 0 | 0.779646 | − | 0.450129i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 341.4 | − | 1.00000i | 0 | −1.00000 | 1.80966 | + | 3.13442i | 0 | −2.41308 | − | 1.08492i | 1.00000i | 0 | 3.13442 | − | 1.80966i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 341.5 | 1.00000i | 0 | −1.00000 | −1.14095 | − | 1.97618i | 0 | 1.42337 | + | 2.23025i | − | 1.00000i | 0 | 1.97618 | − | 1.14095i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 341.6 | 1.00000i | 0 | −1.00000 | −0.714925 | − | 1.23829i | 0 | 0.327442 | − | 2.62541i | − | 1.00000i | 0 | 1.23829 | − | 0.714925i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 341.7 | 1.00000i | 0 | −1.00000 | 0.0338034 | + | 0.0585493i | 0 | 1.19767 | − | 2.35915i | − | 1.00000i | 0 | −0.0585493 | + | 0.0338034i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 341.8 | 1.00000i | 0 | −1.00000 | 1.82207 | + | 3.15592i | 0 | −1.58246 | + | 2.12034i | − | 1.00000i | 0 | −3.15592 | + | 1.82207i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 63.i | even | 6 | 1 | inner |
Twists
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(378, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{8} \)
|
| $3$ |
\( T^{16} \)
|
| $5$ |
\( T^{16} + 24 T^{14} + \cdots + 81 \)
|
| $7$ |
\( T^{16} - 2 T^{15} + \cdots + 5764801 \)
|
| $11$ |
\( T^{16} + 12 T^{15} + \cdots + 61732449 \)
|
| $13$ |
\( T^{16} + \cdots + 390971529 \)
|
| $17$ |
\( T^{16} - 18 T^{15} + \cdots + 56070144 \)
|
| $19$ |
\( T^{16} - 72 T^{14} + \cdots + 9199089 \)
|
| $23$ |
\( T^{16} + \cdots + 187388721 \)
|
| $29$ |
\( T^{16} + 6 T^{15} + \cdots + 1108809 \)
|
| $31$ |
\( T^{16} + 204 T^{14} + \cdots + 65610000 \)
|
| $37$ |
\( T^{16} + \cdots + 32746159681 \)
|
| $41$ |
\( T^{16} - 6 T^{15} + \cdots + 81 \)
|
| $43$ |
\( T^{16} + \cdots + 2999643361 \)
|
| $47$ |
\( (T^{8} - 18 T^{7} + \cdots + 766944)^{2} \)
|
| $53$ |
\( T^{16} + \cdots + 36759242529 \)
|
| $59$ |
\( (T^{8} + 30 T^{7} + \cdots + 465300)^{2} \)
|
| $61$ |
\( T^{16} + \cdots + 547560000 \)
|
| $67$ |
\( (T^{8} + 14 T^{7} + \cdots + 51028)^{2} \)
|
| $71$ |
\( T^{16} + 486 T^{14} + \cdots + 65610000 \)
|
| $73$ |
\( T^{16} - 150 T^{14} + \cdots + 71115489 \)
|
| $79$ |
\( (T^{8} - 16 T^{7} + \cdots - 985100)^{2} \)
|
| $83$ |
\( T^{16} + \cdots + 953512641 \)
|
| $89$ |
\( T^{16} + \cdots + 131145120363321 \)
|
| $97$ |
\( T^{16} + \cdots + 9120206721024 \)
|
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