Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [74,2,Mod(7,74)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(74, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([16]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("74.7");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 74 = 2 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 74.f (of order \(9\), degree \(6\), 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: | \(0.590892974957\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{9})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 24x^{10} + 264x^{8} - 1687x^{6} + 6600x^{4} - 15000x^{2} + 15625 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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
\( x^{12} - 24x^{10} + 264x^{8} - 1687x^{6} + 6600x^{4} - 15000x^{2} + 15625 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -12\nu^{11} + 263\nu^{9} - 2568\nu^{7} + 13644\nu^{5} - 40150\nu^{3} + 52500\nu + 3125 ) / 6250 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{10} - 24\nu^{8} + 264\nu^{6} - 1562\nu^{4} + 5100\nu^{2} + 125\nu - 7500 ) / 250 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{10} - 24\nu^{8} + 264\nu^{6} - 1562\nu^{4} + 5100\nu^{2} - 125\nu - 7500 ) / 250 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{11} - 45 \nu^{10} - 24 \nu^{9} + 955 \nu^{8} + 264 \nu^{7} - 8880 \nu^{6} - 1687 \nu^{5} + \cdots + 175000 ) / 6250 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{11} + 45 \nu^{10} - 24 \nu^{9} - 955 \nu^{8} + 264 \nu^{7} + 8880 \nu^{6} - 1687 \nu^{5} + \cdots - 175000 ) / 6250 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 12 \nu^{11} + 40 \nu^{10} - 188 \nu^{9} - 835 \nu^{8} + 1393 \nu^{7} + 7560 \nu^{6} - 5719 \nu^{5} + \cdots - 125000 ) / 6250 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 12 \nu^{11} - 40 \nu^{10} - 188 \nu^{9} + 835 \nu^{8} + 1393 \nu^{7} - 7560 \nu^{6} - 5719 \nu^{5} + \cdots + 125000 ) / 6250 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( \nu^{11} - 20 \nu^{10} - 24 \nu^{9} + 355 \nu^{8} + 264 \nu^{7} - 2905 \nu^{6} - 1687 \nu^{5} + \cdots + 34375 ) / 1250 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - \nu^{11} - 20 \nu^{10} + 24 \nu^{9} + 355 \nu^{8} - 264 \nu^{7} - 2905 \nu^{6} + 1687 \nu^{5} + \cdots + 34375 ) / 1250 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 28 \nu^{11} + 100 \nu^{10} - 572 \nu^{9} - 1775 \nu^{8} + 4992 \nu^{7} + 14525 \nu^{6} + \cdots - 184375 ) / 6250 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 28 \nu^{11} - 100 \nu^{10} - 572 \nu^{9} + 1775 \nu^{8} + 4992 \nu^{7} - 14525 \nu^{6} + \cdots + 184375 ) / 6250 \)
|
| \(\nu\) | \(=\) |
\( -\beta_{3} + \beta_{2} \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} - \beta_{8} + 5\beta_{5} + 5\beta_{4} - 3\beta_{3} + 3\beta_{2} \)
|
| \(\nu^{4}\) | \(=\) |
\( - 7 \beta_{11} + 7 \beta_{10} + 7 \beta_{9} + 7 \beta_{8} - 5 \beta_{7} + 5 \beta_{6} - 5 \beta_{5} + \cdots + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{11} - \beta_{10} + 11\beta_{9} - 11\beta_{8} + 35\beta_{5} + 35\beta_{4} - \beta_{3} + \beta_{2} - 8\beta _1 + 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 26 \beta_{11} + 26 \beta_{10} + 25 \beta_{9} + 25 \beta_{8} - 50 \beta_{7} + 50 \beta_{6} - 55 \beta_{5} + \cdots - 21 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 19 \beta_{11} - 19 \beta_{10} + 65 \beta_{9} - 65 \beta_{8} + 5 \beta_{7} + 5 \beta_{6} + 125 \beta_{5} + \cdots + 56 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 38 \beta_{11} + 38 \beta_{10} + 14 \beta_{9} + 14 \beta_{8} - 230 \beta_{7} + 230 \beta_{6} + \cdots - 268 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 192 \beta_{11} - 192 \beta_{10} + 218 \beta_{9} - 218 \beta_{8} + 120 \beta_{7} + 120 \beta_{6} + \cdots + 413 \)
|
| \(\nu^{10}\) | \(=\) |
\( 118 \beta_{11} - 118 \beta_{10} - 430 \beta_{9} - 430 \beta_{8} - 130 \beta_{7} + 130 \beta_{6} + \cdots - 1292 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 1279 \beta_{11} - 1279 \beta_{10} + 29 \beta_{9} - 29 \beta_{8} + 1560 \beta_{7} + 1560 \beta_{6} + \cdots + 1876 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).
| \(n\) | \(39\) |
| \(\chi(n)\) | \(-\beta_{6}\) |
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 |
|
−0.939693 | + | 0.342020i | −2.04963 | − | 0.746005i | 0.766044 | − | 0.642788i | −0.326352 | − | 1.85083i | 2.18117 | −0.711830 | − | 4.03699i | −0.500000 | + | 0.866025i | 1.34633 | + | 1.12971i | 0.939693 | + | 1.62760i | ||||||||||||||||||||||||||||||||||||||
| 7.2 | −0.939693 | + | 0.342020i | 1.72328 | + | 0.627223i | 0.766044 | − | 0.642788i | −0.326352 | − | 1.85083i | −1.83388 | 0.598489 | + | 3.39420i | −0.500000 | + | 0.866025i | 0.278152 | + | 0.233397i | 0.939693 | + | 1.62760i | |||||||||||||||||||||||||||||||||||||||
| 9.1 | 0.173648 | + | 0.984808i | −0.238878 | + | 1.35474i | −0.939693 | + | 0.342020i | 0.266044 | + | 0.223238i | −1.37564 | −0.365982 | − | 0.307095i | −0.500000 | − | 0.866025i | 1.04081 | + | 0.378824i | −0.173648 | + | 0.300767i | |||||||||||||||||||||||||||||||||||||||
| 9.2 | 0.173648 | + | 0.984808i | 0.504922 | − | 2.86356i | −0.939693 | + | 0.342020i | 0.266044 | + | 0.223238i | 2.90773 | 0.773586 | + | 0.649116i | −0.500000 | − | 0.866025i | −5.12593 | − | 1.86569i | −0.173648 | + | 0.300767i | |||||||||||||||||||||||||||||||||||||||
| 33.1 | 0.173648 | − | 0.984808i | −0.238878 | − | 1.35474i | −0.939693 | − | 0.342020i | 0.266044 | − | 0.223238i | −1.37564 | −0.365982 | + | 0.307095i | −0.500000 | + | 0.866025i | 1.04081 | − | 0.378824i | −0.173648 | − | 0.300767i | |||||||||||||||||||||||||||||||||||||||
| 33.2 | 0.173648 | − | 0.984808i | 0.504922 | + | 2.86356i | −0.939693 | − | 0.342020i | 0.266044 | − | 0.223238i | 2.90773 | 0.773586 | − | 0.649116i | −0.500000 | + | 0.866025i | −5.12593 | + | 1.86569i | −0.173648 | − | 0.300767i | |||||||||||||||||||||||||||||||||||||||
| 49.1 | 0.766044 | − | 0.642788i | −2.41262 | − | 2.02443i | 0.173648 | − | 0.984808i | −1.43969 | + | 0.524005i | −3.14945 | 4.53424 | − | 1.65033i | −0.500000 | − | 0.866025i | 1.20148 | + | 6.81391i | −0.766044 | + | 1.32683i | |||||||||||||||||||||||||||||||||||||||
| 49.2 | 0.766044 | − | 0.642788i | 0.972925 | + | 0.816381i | 0.173648 | − | 0.984808i | −1.43969 | + | 0.524005i | 1.27006 | −1.82850 | + | 0.665520i | −0.500000 | − | 0.866025i | −0.240839 | − | 1.36587i | −0.766044 | + | 1.32683i | |||||||||||||||||||||||||||||||||||||||
| 53.1 | −0.939693 | − | 0.342020i | −2.04963 | + | 0.746005i | 0.766044 | + | 0.642788i | −0.326352 | + | 1.85083i | 2.18117 | −0.711830 | + | 4.03699i | −0.500000 | − | 0.866025i | 1.34633 | − | 1.12971i | 0.939693 | − | 1.62760i | |||||||||||||||||||||||||||||||||||||||
| 53.2 | −0.939693 | − | 0.342020i | 1.72328 | − | 0.627223i | 0.766044 | + | 0.642788i | −0.326352 | + | 1.85083i | −1.83388 | 0.598489 | − | 3.39420i | −0.500000 | − | 0.866025i | 0.278152 | − | 0.233397i | 0.939693 | − | 1.62760i | |||||||||||||||||||||||||||||||||||||||
| 71.1 | 0.766044 | + | 0.642788i | −2.41262 | + | 2.02443i | 0.173648 | + | 0.984808i | −1.43969 | − | 0.524005i | −3.14945 | 4.53424 | + | 1.65033i | −0.500000 | + | 0.866025i | 1.20148 | − | 6.81391i | −0.766044 | − | 1.32683i | |||||||||||||||||||||||||||||||||||||||
| 71.2 | 0.766044 | + | 0.642788i | 0.972925 | − | 0.816381i | 0.173648 | + | 0.984808i | −1.43969 | − | 0.524005i | 1.27006 | −1.82850 | − | 0.665520i | −0.500000 | + | 0.866025i | −0.240839 | + | 1.36587i | −0.766044 | − | 1.32683i | |||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 37.f | even | 9 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 74.2.f.b | ✓ | 12 |
| 3.b | odd | 2 | 1 | 666.2.x.g | 12 | ||
| 4.b | odd | 2 | 1 | 592.2.bc.d | 12 | ||
| 37.f | even | 9 | 1 | inner | 74.2.f.b | ✓ | 12 |
| 37.f | even | 9 | 1 | 2738.2.a.t | 6 | ||
| 37.h | even | 18 | 1 | 2738.2.a.q | 6 | ||
| 111.p | odd | 18 | 1 | 666.2.x.g | 12 | ||
| 148.p | odd | 18 | 1 | 592.2.bc.d | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 74.2.f.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 74.2.f.b | ✓ | 12 | 37.f | even | 9 | 1 | inner |
| 592.2.bc.d | 12 | 4.b | odd | 2 | 1 | ||
| 592.2.bc.d | 12 | 148.p | odd | 18 | 1 | ||
| 666.2.x.g | 12 | 3.b | odd | 2 | 1 | ||
| 666.2.x.g | 12 | 111.p | odd | 18 | 1 | ||
| 2738.2.a.q | 6 | 37.h | even | 18 | 1 | ||
| 2738.2.a.t | 6 | 37.f | even | 9 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + 3 T_{3}^{11} + 6 T_{3}^{10} + 8 T_{3}^{9} + 24 T_{3}^{8} - 126 T_{3}^{7} - 151 T_{3}^{6} + \cdots + 4096 \)
acting on \(S_{2}^{\mathrm{new}}(74, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{6} + T^{3} + 1)^{2} \)
|
| $3$ |
\( T^{12} + 3 T^{11} + \cdots + 4096 \)
|
| $5$ |
\( (T^{6} + 3 T^{5} + 6 T^{4} + \cdots + 1)^{2} \)
|
| $7$ |
\( T^{12} - 6 T^{11} + \cdots + 4096 \)
|
| $11$ |
\( T^{12} - 3 T^{11} + \cdots + 4096 \)
|
| $13$ |
\( T^{12} + 6 T^{11} + \cdots + 516961 \)
|
| $17$ |
\( T^{12} + 3 T^{11} + \cdots + 26569 \)
|
| $19$ |
\( T^{12} + 3 T^{11} + \cdots + 4096 \)
|
| $23$ |
\( T^{12} + 21 T^{11} + \cdots + 4096 \)
|
| $29$ |
\( T^{12} - 6 T^{11} + \cdots + 1369 \)
|
| $31$ |
\( (T^{6} - 21 T^{5} + \cdots + 130112)^{2} \)
|
| $37$ |
\( T^{12} + \cdots + 2565726409 \)
|
| $41$ |
\( T^{12} + \cdots + 466905664 \)
|
| $43$ |
\( (T^{6} - 18 T^{5} + \cdots + 8704)^{2} \)
|
| $47$ |
\( T^{12} + \cdots + 6890328064 \)
|
| $53$ |
\( T^{12} + 6 T^{11} + \cdots + 289 \)
|
| $59$ |
\( T^{12} + 6 T^{11} + \cdots + 1183744 \)
|
| $61$ |
\( T^{12} + \cdots + 2801373184 \)
|
| $67$ |
\( T^{12} + 27 T^{11} + \cdots + 262144 \)
|
| $71$ |
\( T^{12} + 18 T^{11} + \cdots + 262144 \)
|
| $73$ |
\( (T^{6} - 27 T^{5} + \cdots + 216289)^{2} \)
|
| $79$ |
\( T^{12} + 12 T^{11} + \cdots + 95883264 \)
|
| $83$ |
\( T^{12} + 6 T^{11} + \cdots + 1183744 \)
|
| $89$ |
\( T^{12} + \cdots + 48144697561 \)
|
| $97$ |
\( T^{12} + 42 T^{11} + \cdots + 7529536 \)
|
| show more | |
| show less |