Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [61,4,Mod(1,61)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(61, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("61.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 61 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 61.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: | \(3.59911651035\) |
| Analytic rank: | \(0\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 3x^{8} - 58x^{7} + 154x^{6} + 1145x^{5} - 2511x^{4} - 9096x^{3} + 13216x^{2} + 24608x - 3712 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 3x^{8} - 58x^{7} + 154x^{6} + 1145x^{5} - 2511x^{4} - 9096x^{3} + 13216x^{2} + 24608x - 3712 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 14 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 39 \nu^{8} + 6 \nu^{7} - 2176 \nu^{6} - 1058 \nu^{5} + 38501 \nu^{4} + 31144 \nu^{3} - 223720 \nu^{2} + \cdots + 50496 ) / 6976 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 293 \nu^{8} - 196 \nu^{7} + 16292 \nu^{6} + 15014 \nu^{5} - 283203 \nu^{4} - 315774 \nu^{3} + \cdots - 296192 ) / 6976 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 343 \nu^{8} - 254 \nu^{7} + 19160 \nu^{6} + 18338 \nu^{5} - 334229 \nu^{4} - 371728 \nu^{3} + \cdots - 323904 ) / 6976 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 51 \nu^{8} - 33 \nu^{7} + 2812 \nu^{6} + 2440 \nu^{5} - 48201 \nu^{4} - 49103 \nu^{3} + \cdots - 48392 ) / 872 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 213 \nu^{8} - 125 \nu^{7} + 11834 \nu^{6} + 9870 \nu^{5} - 205093 \nu^{4} - 210617 \nu^{3} + \cdots - 180000 ) / 3488 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 234 \nu^{8} + 145 \nu^{7} - 13056 \nu^{6} - 11144 \nu^{5} + 227736 \nu^{4} + 233843 \nu^{3} + \cdots + 248912 ) / 1744 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 14 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{8} - \beta_{7} - 2\beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} + 19\beta _1 + 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} + 7\beta_{7} - 2\beta_{6} - 3\beta_{4} + 9\beta_{3} + 26\beta_{2} + 2\beta _1 + 279 \)
|
| \(\nu^{5}\) | \(=\) |
\( -36\beta_{8} - 30\beta_{7} - 6\beta_{6} - 82\beta_{5} + 50\beta_{4} + 128\beta_{3} - 32\beta_{2} + 409\beta _1 + 198 \)
|
| \(\nu^{6}\) | \(=\) |
\( 10 \beta_{8} + 262 \beta_{7} - 100 \beta_{6} - 24 \beta_{5} - 126 \beta_{4} + 418 \beta_{3} + \cdots + 6336 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 1107 \beta_{8} - 679 \beta_{7} - 324 \beta_{6} - 2746 \beta_{5} + 1679 \beta_{4} + 4225 \beta_{3} + \cdots + 7219 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 437 \beta_{8} + 7797 \beta_{7} - 3718 \beta_{6} - 1544 \beta_{5} - 3769 \beta_{4} + 15043 \beta_{3} + \cdots + 157369 \)
|
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 |
|
−5.02597 | 10.0504 | 17.2604 | 9.94427 | −50.5132 | −19.9021 | −46.5426 | 74.0111 | −49.9797 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.04088 | −7.89658 | 8.32871 | −13.7088 | 31.9091 | −30.5005 | −1.32826 | 35.3560 | 55.3958 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −3.19644 | 2.57244 | 2.21724 | 0.285247 | −8.22266 | 10.2737 | 18.4843 | −20.3825 | −0.911776 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.45778 | −7.48742 | −5.87487 | 5.30991 | 10.9150 | 22.2683 | 20.2265 | 29.0615 | −7.74068 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.141214 | 7.19407 | −7.98006 | 11.4411 | 1.01591 | 13.6814 | −2.25661 | 24.7547 | 1.61565 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.59165 | 6.84896 | 4.89994 | −9.86624 | 24.5990 | 15.7105 | −11.1343 | 19.9082 | −35.4361 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 3.64840 | −3.53813 | 5.31083 | 18.6915 | −12.9085 | 22.3984 | −9.81118 | −14.4816 | 68.1942 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 3.88832 | 4.82743 | 7.11906 | 8.00408 | 18.7706 | −35.4823 | −3.42538 | −3.69593 | 31.1225 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 5.45149 | −1.57119 | 21.7187 | −6.10105 | −8.56533 | 2.55248 | 74.7876 | −24.5314 | −33.2598 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(61\) | \(1\) |
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 | 61.4.a.b | ✓ | 9 |
| 3.b | odd | 2 | 1 | 549.4.a.f | 9 | ||
| 4.b | odd | 2 | 1 | 976.4.a.h | 9 | ||
| 5.b | even | 2 | 1 | 1525.4.a.b | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 61.4.a.b | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 549.4.a.f | 9 | 3.b | odd | 2 | 1 | ||
| 976.4.a.h | 9 | 4.b | odd | 2 | 1 | ||
| 1525.4.a.b | 9 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{9} - 3 T_{2}^{8} - 58 T_{2}^{7} + 154 T_{2}^{6} + 1145 T_{2}^{5} - 2511 T_{2}^{4} - 9096 T_{2}^{3} + \cdots - 3712 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(61))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} - 3 T^{8} + \cdots - 3712 \)
|
| $3$ |
\( T^{9} - 11 T^{8} + \cdots - 2021248 \)
|
| $5$ |
\( T^{9} - 24 T^{8} + \cdots + 21274656 \)
|
| $7$ |
\( T^{9} + \cdots + 60552697842 \)
|
| $11$ |
\( T^{9} + \cdots + 590470684120 \)
|
| $13$ |
\( T^{9} + \cdots + 33553151014536 \)
|
| $17$ |
\( T^{9} + \cdots - 22\!\cdots\!72 \)
|
| $19$ |
\( T^{9} + \cdots + 59\!\cdots\!68 \)
|
| $23$ |
\( T^{9} + \cdots - 25\!\cdots\!94 \)
|
| $29$ |
\( T^{9} + \cdots - 26\!\cdots\!76 \)
|
| $31$ |
\( T^{9} + \cdots - 82\!\cdots\!60 \)
|
| $37$ |
\( T^{9} + \cdots - 26\!\cdots\!28 \)
|
| $41$ |
\( T^{9} + \cdots - 73\!\cdots\!30 \)
|
| $43$ |
\( T^{9} + \cdots + 22\!\cdots\!16 \)
|
| $47$ |
\( T^{9} + \cdots - 11\!\cdots\!68 \)
|
| $53$ |
\( T^{9} + \cdots + 15\!\cdots\!16 \)
|
| $59$ |
\( T^{9} + \cdots + 89\!\cdots\!08 \)
|
| $61$ |
\( (T + 61)^{9} \)
|
| $67$ |
\( T^{9} + \cdots + 12\!\cdots\!68 \)
|
| $71$ |
\( T^{9} + \cdots - 27\!\cdots\!96 \)
|
| $73$ |
\( T^{9} + \cdots - 32\!\cdots\!18 \)
|
| $79$ |
\( T^{9} + \cdots + 57\!\cdots\!64 \)
|
| $83$ |
\( T^{9} + \cdots - 54\!\cdots\!44 \)
|
| $89$ |
\( T^{9} + \cdots - 52\!\cdots\!92 \)
|
| $97$ |
\( T^{9} + \cdots + 92\!\cdots\!92 \)
|
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