Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [406,2,Mod(233,406)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(406, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("406.233");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 406 = 2 \cdot 7 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 406.e (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: | \(3.24192632206\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Relative dimension: | \(5\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 10.0.3118758597603.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 4x^{8} - 16x^{6} - 34x^{5} + 43x^{4} + 155x^{3} + 199x^{2} + 124x + 43 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3 \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 4x^{8} - 16x^{6} - 34x^{5} + 43x^{4} + 155x^{3} + 199x^{2} + 124x + 43 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 244169 \nu^{9} - 750258 \nu^{8} + 198422 \nu^{7} + 3041052 \nu^{6} - 8027063 \nu^{5} + \cdots - 17035541 ) / 27779803 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 313192 \nu^{9} + 3118568 \nu^{8} - 2157819 \nu^{7} - 10818671 \nu^{6} + 20747939 \nu^{5} + \cdots + 139659043 ) / 27779803 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 893358 \nu^{9} + 475480 \nu^{8} + 5106016 \nu^{7} - 5222834 \nu^{6} + 14656748 \nu^{5} + \cdots + 29238847 ) / 27779803 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 5726 \nu^{9} + 8728 \nu^{8} + 15695 \nu^{7} - 30701 \nu^{6} + 113887 \nu^{5} + 47565 \nu^{4} + \cdots + 163662 ) / 139597 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1616901 \nu^{9} - 715422 \nu^{8} - 6561850 \nu^{7} + 4318132 \nu^{6} - 26461030 \nu^{5} + \cdots + 75110912 ) / 27779803 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 1768918 \nu^{9} + 516157 \nu^{8} + 5805510 \nu^{7} + 1407113 \nu^{6} + 25732292 \nu^{5} + \cdots - 138763851 ) / 27779803 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 1850159 \nu^{9} - 1056286 \nu^{8} + 10198817 \nu^{7} + 514287 \nu^{6} + 22231484 \nu^{5} + \cdots - 199826103 ) / 27779803 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 2396972 \nu^{9} - 3681779 \nu^{8} - 7578548 \nu^{7} + 14487182 \nu^{6} - 48162975 \nu^{5} + \cdots - 77176853 ) / 27779803 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 2611328 \nu^{9} - 2640778 \nu^{8} - 9404480 \nu^{7} + 11479543 \nu^{6} - 50294265 \nu^{5} + \cdots + 13015032 ) / 27779803 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + 2\beta_{5} + 2\beta_{4} + 2\beta_{3} + \beta_{2} - \beta _1 - 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{9} - \beta_{8} + 2\beta_{7} - \beta_{6} + 4\beta_{5} + \beta_{4} - 2\beta_{3} - \beta_{2} - 2\beta _1 + 1 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5 \beta_{9} - 7 \beta_{8} + 5 \beta_{7} + 5 \beta_{6} + 10 \beta_{5} - 11 \beta_{4} + 4 \beta_{3} + \cdots + 4 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 8\beta_{9} - 6\beta_{8} + 7\beta_{7} + 2\beta_{6} + 2\beta_{5} - 7\beta_{4} - \beta_{3} + 6\beta_{2} - \beta _1 + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 32 \beta_{9} + 2 \beta_{8} + 41 \beta_{7} + 26 \beta_{6} + 31 \beta_{5} - 20 \beta_{4} + 25 \beta_{3} + \cdots + 61 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( \beta_{9} - 2 \beta_{8} + 64 \beta_{7} + 16 \beta_{6} + 113 \beta_{5} + 29 \beta_{4} - 7 \beta_{3} + \cdots + 53 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 2 \beta_{9} - 143 \beta_{8} + 121 \beta_{7} + 55 \beta_{6} + 302 \beta_{5} - 202 \beta_{4} + \cdots + 23 ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( 124 \beta_{9} - 213 \beta_{8} + 134 \beta_{7} + 56 \beta_{6} + 131 \beta_{5} - 275 \beta_{4} + \cdots + 184 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 1048 \beta_{9} - 890 \beta_{8} + 1075 \beta_{7} + 544 \beta_{6} + 455 \beta_{5} - 1864 \beta_{4} + \cdots + 1898 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/406\mathbb{Z}\right)^\times\).
| \(n\) | \(59\) | \(379\) |
| \(\chi(n)\) | \(-1 + \beta_{4}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 233.1 |
|
−0.500000 | + | 0.866025i | −1.61748 | − | 2.80156i | −0.500000 | − | 0.866025i | −0.572197 | + | 0.991074i | 3.23497 | 0.469216 | + | 2.60381i | 1.00000 | −3.73251 | + | 6.46489i | −0.572197 | − | 0.991074i | ||||||||||||||||||||||||||||||||||
| 233.2 | −0.500000 | + | 0.866025i | −1.20348 | − | 2.08450i | −0.500000 | − | 0.866025i | 1.10394 | − | 1.91209i | 2.40697 | −1.36646 | − | 2.26557i | 1.00000 | −1.39675 | + | 2.41924i | 1.10394 | + | 1.91209i | |||||||||||||||||||||||||||||||||||
| 233.3 | −0.500000 | + | 0.866025i | −0.257545 | − | 0.446080i | −0.500000 | − | 0.866025i | −1.84343 | + | 3.19291i | 0.515089 | −2.63485 | + | 0.239979i | 1.00000 | 1.36734 | − | 2.36831i | −1.84343 | − | 3.19291i | |||||||||||||||||||||||||||||||||||
| 233.4 | −0.500000 | + | 0.866025i | 0.357289 | + | 0.618843i | −0.500000 | − | 0.866025i | −0.116584 | + | 0.201930i | −0.714579 | 2.36799 | + | 1.18009i | 1.00000 | 1.24469 | − | 2.15586i | −0.116584 | − | 0.201930i | |||||||||||||||||||||||||||||||||||
| 233.5 | −0.500000 | + | 0.866025i | 1.22122 | + | 2.11522i | −0.500000 | − | 0.866025i | −2.07173 | + | 3.58835i | −2.44245 | −0.335903 | − | 2.62434i | 1.00000 | −1.48277 | + | 2.56824i | −2.07173 | − | 3.58835i | |||||||||||||||||||||||||||||||||||
| 291.1 | −0.500000 | − | 0.866025i | −1.61748 | + | 2.80156i | −0.500000 | + | 0.866025i | −0.572197 | − | 0.991074i | 3.23497 | 0.469216 | − | 2.60381i | 1.00000 | −3.73251 | − | 6.46489i | −0.572197 | + | 0.991074i | |||||||||||||||||||||||||||||||||||
| 291.2 | −0.500000 | − | 0.866025i | −1.20348 | + | 2.08450i | −0.500000 | + | 0.866025i | 1.10394 | + | 1.91209i | 2.40697 | −1.36646 | + | 2.26557i | 1.00000 | −1.39675 | − | 2.41924i | 1.10394 | − | 1.91209i | |||||||||||||||||||||||||||||||||||
| 291.3 | −0.500000 | − | 0.866025i | −0.257545 | + | 0.446080i | −0.500000 | + | 0.866025i | −1.84343 | − | 3.19291i | 0.515089 | −2.63485 | − | 0.239979i | 1.00000 | 1.36734 | + | 2.36831i | −1.84343 | + | 3.19291i | |||||||||||||||||||||||||||||||||||
| 291.4 | −0.500000 | − | 0.866025i | 0.357289 | − | 0.618843i | −0.500000 | + | 0.866025i | −0.116584 | − | 0.201930i | −0.714579 | 2.36799 | − | 1.18009i | 1.00000 | 1.24469 | + | 2.15586i | −0.116584 | + | 0.201930i | |||||||||||||||||||||||||||||||||||
| 291.5 | −0.500000 | − | 0.866025i | 1.22122 | − | 2.11522i | −0.500000 | + | 0.866025i | −2.07173 | − | 3.58835i | −2.44245 | −0.335903 | + | 2.62434i | 1.00000 | −1.48277 | − | 2.56824i | −2.07173 | + | 3.58835i | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 406.2.e.a | ✓ | 10 |
| 7.c | even | 3 | 1 | inner | 406.2.e.a | ✓ | 10 |
| 7.c | even | 3 | 1 | 2842.2.a.z | 5 | ||
| 7.d | odd | 6 | 1 | 2842.2.a.x | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 406.2.e.a | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 406.2.e.a | ✓ | 10 | 7.c | even | 3 | 1 | inner |
| 2842.2.a.x | 5 | 7.d | odd | 6 | 1 | ||
| 2842.2.a.z | 5 | 7.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} + 3 T_{3}^{9} + 16 T_{3}^{8} + 17 T_{3}^{7} + 100 T_{3}^{6} + 104 T_{3}^{5} + 382 T_{3}^{4} + \cdots + 49 \)
acting on \(S_{2}^{\mathrm{new}}(406, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T + 1)^{5} \)
|
| $3$ |
\( T^{10} + 3 T^{9} + \cdots + 49 \)
|
| $5$ |
\( T^{10} + 7 T^{9} + \cdots + 81 \)
|
| $7$ |
\( T^{10} + 3 T^{9} + \cdots + 16807 \)
|
| $11$ |
\( T^{10} + 33 T^{8} + \cdots + 81 \)
|
| $13$ |
\( (T^{5} - 10 T^{4} + \cdots + 103)^{2} \)
|
| $17$ |
\( T^{10} + 8 T^{9} + \cdots + 9 \)
|
| $19$ |
\( T^{10} + 2 T^{9} + \cdots + 2634129 \)
|
| $23$ |
\( T^{10} + T^{9} + \cdots + 1089 \)
|
| $29$ |
\( (T + 1)^{10} \)
|
| $31$ |
\( T^{10} + 11 T^{9} + \cdots + 149769 \)
|
| $37$ |
\( T^{10} - 8 T^{9} + \cdots + 121801 \)
|
| $41$ |
\( (T^{5} - 23 T^{4} + \cdots + 7113)^{2} \)
|
| $43$ |
\( (T^{5} + 3 T^{4} + \cdots + 16843)^{2} \)
|
| $47$ |
\( T^{10} + 16 T^{9} + \cdots + 3969 \)
|
| $53$ |
\( T^{10} + 7 T^{9} + \cdots + 33489 \)
|
| $59$ |
\( T^{10} - 9 T^{9} + \cdots + 431649 \)
|
| $61$ |
\( T^{10} + 15 T^{9} + \cdots + 28376929 \)
|
| $67$ |
\( T^{10} + \cdots + 131813361 \)
|
| $71$ |
\( (T^{5} + 22 T^{4} + \cdots + 18189)^{2} \)
|
| $73$ |
\( T^{10} + 247 T^{8} + \cdots + 116281 \)
|
| $79$ |
\( T^{10} - 13 T^{9} + \cdots + 2442969 \)
|
| $83$ |
\( (T^{5} - 28 T^{4} + \cdots + 25851)^{2} \)
|
| $89$ |
\( T^{10} + \cdots + 26494398441 \)
|
| $97$ |
\( (T^{5} - 42 T^{4} + \cdots + 15007)^{2} \)
|
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