Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [201,4,Mod(1,201)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(201, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("201.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 201 = 3 \cdot 67 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 201.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: | \(11.8593839112\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - x^{6} - 38x^{5} + 18x^{4} + 373x^{3} - 151x^{2} - 956x + 498 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} - 38x^{5} + 18x^{4} + 373x^{3} - 151x^{2} - 956x + 498 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 47\nu^{5} + 27\nu^{4} + 1339\nu^{3} - 408\nu^{2} - 6314\nu + 1966 ) / 466 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -25\nu^{6} + 10\nu^{5} + 956\nu^{4} + 310\nu^{3} - 9139\nu^{2} - 5250\nu + 17954 ) / 932 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -35\nu^{6} + 14\nu^{5} + 1152\nu^{4} + 434\nu^{3} - 7389\nu^{2} - 4554\nu + 4818 ) / 932 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -87\nu^{6} + 128\nu^{5} + 3010\nu^{4} - 2556\nu^{3} - 23919\nu^{2} + 13418\nu + 32134 ) / 1864 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{6} + \beta_{5} + 2\beta_{4} - \beta_{3} + \beta_{2} + 18\beta _1 + 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( -5\beta_{5} + 7\beta_{4} + 29\beta_{2} + 44\beta _1 + 210 \)
|
| \(\nu^{5}\) | \(=\) |
\( -58\beta_{6} + 22\beta_{5} + 67\beta_{4} - 39\beta_{3} + 53\beta_{2} + 427\beta _1 + 343 \)
|
| \(\nu^{6}\) | \(=\) |
\( -48\beta_{6} - 170\beta_{5} + 282\beta_{4} - 28\beta_{3} + 777\beta_{2} + 1501\beta _1 + 4939 \)
|
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 |
|
−5.41567 | −3.00000 | 21.3295 | −5.98460 | 16.2470 | 0.240383 | −72.1883 | 9.00000 | 32.4106 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −3.01477 | −3.00000 | 1.08886 | 14.7190 | 9.04432 | −32.2394 | 20.8355 | 9.00000 | −44.3744 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.85609 | −3.00000 | −4.55494 | −4.35854 | 5.56826 | 17.6123 | 23.3031 | 9.00000 | 8.08983 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.535294 | −3.00000 | −7.71346 | 12.7037 | 1.60588 | −8.67573 | 8.41132 | 9.00000 | −6.80020 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.11972 | −3.00000 | −3.50678 | 1.76139 | −6.35917 | 22.2326 | −24.3912 | 9.00000 | 3.73365 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.26131 | −3.00000 | 2.63612 | 7.83356 | −9.78392 | −27.8825 | −17.4933 | 9.00000 | 25.5476 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.44080 | −3.00000 | 11.7207 | −15.6745 | −13.3224 | −4.28763 | 16.5228 | 9.00000 | −69.6071 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(67\) | \(-1\) |
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 | 201.4.a.c | ✓ | 7 |
| 3.b | odd | 2 | 1 | 603.4.a.c | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 201.4.a.c | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 603.4.a.c | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} + T_{2}^{6} - 38T_{2}^{5} - 18T_{2}^{4} + 373T_{2}^{3} + 151T_{2}^{2} - 956T_{2} - 498 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(201))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + T^{6} + \cdots - 498 \)
|
| $3$ |
\( (T + 3)^{7} \)
|
| $5$ |
\( T^{7} - 11 T^{6} + \cdots + 1054848 \)
|
| $7$ |
\( T^{7} + 33 T^{6} + \cdots - 3147392 \)
|
| $11$ |
\( T^{7} + \cdots + 3855752448 \)
|
| $13$ |
\( T^{7} + \cdots + 36060533248 \)
|
| $17$ |
\( T^{7} + \cdots + 47177769168 \)
|
| $19$ |
\( T^{7} + \cdots + 2352293777728 \)
|
| $23$ |
\( T^{7} + \cdots + 176992942356 \)
|
| $29$ |
\( T^{7} + \cdots - 154337966321664 \)
|
| $31$ |
\( T^{7} + \cdots + 6536744991936 \)
|
| $37$ |
\( T^{7} + \cdots + 18\!\cdots\!52 \)
|
| $41$ |
\( T^{7} + \cdots + 17\!\cdots\!28 \)
|
| $43$ |
\( T^{7} + \cdots + 15\!\cdots\!28 \)
|
| $47$ |
\( T^{7} + \cdots - 50\!\cdots\!08 \)
|
| $53$ |
\( T^{7} + \cdots + 42\!\cdots\!76 \)
|
| $59$ |
\( T^{7} + \cdots + 18\!\cdots\!88 \)
|
| $61$ |
\( T^{7} + \cdots + 34\!\cdots\!16 \)
|
| $67$ |
\( (T - 67)^{7} \)
|
| $71$ |
\( T^{7} + \cdots - 10\!\cdots\!92 \)
|
| $73$ |
\( T^{7} + \cdots + 22\!\cdots\!52 \)
|
| $79$ |
\( T^{7} + \cdots + 18\!\cdots\!84 \)
|
| $83$ |
\( T^{7} + \cdots + 50\!\cdots\!32 \)
|
| $89$ |
\( T^{7} + \cdots - 19\!\cdots\!92 \)
|
| $97$ |
\( T^{7} + \cdots - 88\!\cdots\!16 \)
|
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