Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [49,26,Mod(1,49)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(49, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 26, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("49.1");
S:= CuspForms(chi, 26);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 49 = 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 26 \) |
| Character orbit: | \([\chi]\) | \(=\) | 49.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: | \(194.038422177\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 7 x^{15} - 102767646 x^{14} - 8353831787 x^{13} + \cdots - 19\!\cdots\!25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | multiple of \( 2^{65}\cdot 3^{19}\cdot 5^{9}\cdot 7^{31} \) |
| Twist minimal: | no (minimal twist has level 7) |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 7 x^{15} - 102767646 x^{14} - 8353831787 x^{13} + \cdots - 19\!\cdots\!25 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 21\!\cdots\!43 \nu^{15} + \cdots + 21\!\cdots\!09 ) / 19\!\cdots\!48 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 21\!\cdots\!43 \nu^{15} + \cdots - 32\!\cdots\!71 ) / 66\!\cdots\!16 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 63\!\cdots\!71 \nu^{15} + \cdots - 19\!\cdots\!25 ) / 19\!\cdots\!48 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 15\!\cdots\!05 \nu^{15} + \cdots - 10\!\cdots\!77 ) / 66\!\cdots\!16 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 43\!\cdots\!09 \nu^{15} + \cdots - 57\!\cdots\!15 ) / 19\!\cdots\!12 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 12\!\cdots\!21 \nu^{15} + \cdots - 46\!\cdots\!89 ) / 19\!\cdots\!48 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 17\!\cdots\!97 \nu^{15} + \cdots - 20\!\cdots\!37 ) / 19\!\cdots\!48 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 29\!\cdots\!69 \nu^{15} + \cdots - 19\!\cdots\!13 ) / 49\!\cdots\!12 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 75\!\cdots\!65 \nu^{15} + \cdots + 44\!\cdots\!41 ) / 12\!\cdots\!28 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 17\!\cdots\!43 \nu^{15} + \cdots + 11\!\cdots\!03 ) / 28\!\cdots\!64 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 10\!\cdots\!77 \nu^{15} + \cdots - 18\!\cdots\!03 ) / 11\!\cdots\!36 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 12\!\cdots\!53 \nu^{15} + \cdots - 27\!\cdots\!19 ) / 49\!\cdots\!12 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 99\!\cdots\!17 \nu^{15} + \cdots - 49\!\cdots\!21 ) / 19\!\cdots\!48 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 47\!\cdots\!93 \nu^{15} + \cdots + 31\!\cdots\!17 ) / 19\!\cdots\!48 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 3\beta_{2} + 263\beta _1 + 51383868 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{6} - 3\beta_{5} - 7\beta_{4} + 941\beta_{3} + 21965\beta_{2} + 85243557\beta _1 + 13620463462 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{15} + \beta_{14} - \beta_{13} - 19 \beta_{12} - 4 \beta_{11} + 7 \beta_{10} + \cdots + 43\!\cdots\!11 ) / 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 93 \beta_{15} - 5741 \beta_{14} - 1011 \beta_{13} - 8163 \beta_{12} + 69214 \beta_{11} + \cdots + 85\!\cdots\!39 ) / 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 39195651 \beta_{15} + 40815523 \beta_{14} - 46487883 \beta_{13} - 851762469 \beta_{12} + \cdots + 10\!\cdots\!09 ) / 16 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 18189430247 \beta_{15} - 592389286503 \beta_{14} - 126130822553 \beta_{13} - 1258441117827 \beta_{12} + \cdots + 78\!\cdots\!19 ) / 16 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 12\!\cdots\!83 \beta_{15} + \cdots + 29\!\cdots\!01 ) / 16 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 85\!\cdots\!47 \beta_{15} + \cdots + 32\!\cdots\!39 ) / 16 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 17\!\cdots\!42 \beta_{15} + \cdots + 42\!\cdots\!28 ) / 8 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 29\!\cdots\!17 \beta_{15} + \cdots + 12\!\cdots\!77 ) / 16 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 50\!\cdots\!95 \beta_{15} + \cdots + 12\!\cdots\!85 ) / 8 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 42\!\cdots\!47 \beta_{15} + \cdots + 23\!\cdots\!67 ) / 8 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 14\!\cdots\!31 \beta_{15} + \cdots + 37\!\cdots\!65 ) / 8 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 10\!\cdots\!75 \beta_{15} + \cdots + 87\!\cdots\!25 ) / 8 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−11302.7 | 854307. | 9.41976e7 | −4.59088e8 | −9.65602e9 | 0 | −6.85434e11 | −1.17448e11 | 5.18896e12 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −9617.18 | −1.62258e6 | 5.89356e7 | 7.42240e8 | 1.56046e10 | 0 | −2.44095e11 | 1.78548e12 | −7.13825e12 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −8277.22 | −79373.8 | 3.49580e7 | 7.14800e7 | 6.56995e8 | 0 | −1.16173e10 | −8.40988e11 | −5.91656e11 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −6675.56 | 1.38614e6 | 1.10086e7 | 6.74635e8 | −9.25329e9 | 0 | 1.50506e11 | 1.07411e12 | −4.50356e12 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −6148.57 | −1.04706e6 | 4.25045e6 | −8.38110e8 | 6.43792e9 | 0 | 1.80178e11 | 2.49046e11 | 5.15317e12 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −3305.11 | 1.44999e6 | −2.26307e7 | −7.72886e8 | −4.79237e9 | 0 | 1.85698e11 | 1.25518e12 | 2.55447e12 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −2961.62 | 212163. | −2.47832e7 | 3.66500e8 | −6.28345e8 | 0 | 1.72774e11 | −8.02276e11 | −1.08543e12 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 299.698 | −1.04757e6 | −3.34646e7 | 8.64767e8 | −3.13954e8 | 0 | −2.00855e10 | 2.50105e11 | 2.59169e11 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 367.198 | −1.08049e6 | −3.34196e7 | −3.08176e8 | −3.96756e8 | 0 | −2.45928e10 | 3.20179e11 | −1.13162e11 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2791.04 | 617931. | −2.57645e7 | −4.34224e8 | 1.72467e9 | 0 | −1.65562e11 | −4.65450e11 | −1.21194e12 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 4923.84 | 1.40850e6 | −9.31023e6 | 7.35037e8 | 6.93521e9 | 0 | −2.11059e11 | 1.13657e12 | 3.61920e12 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 6955.51 | 218993. | 1.48247e7 | 2.94492e8 | 1.52321e9 | 0 | −1.30275e11 | −7.99331e11 | 2.04834e12 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 7087.00 | −1.56615e6 | 1.66712e7 | −7.06414e7 | −1.10993e10 | 0 | −1.19652e11 | 1.60553e12 | −5.00636e11 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 8511.53 | −353422. | 3.88917e7 | −7.70711e8 | −3.00816e9 | 0 | 4.54286e10 | −7.22382e11 | −6.55993e12 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 10620.7 | −462798. | 7.92447e7 | 7.26756e8 | −4.91524e9 | 0 | 4.85262e11 | −6.33106e11 | 7.71865e12 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 10781.5 | 1.64286e6 | 8.26859e7 | −5.33897e8 | 1.77124e10 | 0 | 5.29710e11 | 1.85169e12 | −5.75620e12 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(-1\) |
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 | 49.26.a.g | 16 | |
| 7.b | odd | 2 | 1 | 49.26.a.f | 16 | ||
| 7.d | odd | 6 | 2 | 7.26.c.a | ✓ | 32 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.26.c.a | ✓ | 32 | 7.d | odd | 6 | 2 | |
| 49.26.a.f | 16 | 7.b | odd | 2 | 1 | ||
| 49.26.a.g | 16 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{26}^{\mathrm{new}}(\Gamma_0(49))\):
|
\( T_{2}^{16} - 4050 T_{2}^{15} - 403382004 T_{2}^{14} + 1519515733680 T_{2}^{13} + \cdots - 26\!\cdots\!24 \)
|
|
\( T_{3}^{16} - 531440 T_{3}^{15} - 9210542930352 T_{3}^{14} + \cdots + 44\!\cdots\!61 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + \cdots - 26\!\cdots\!24 \)
|
| $3$ |
\( T^{16} + \cdots + 44\!\cdots\!61 \)
|
| $5$ |
\( T^{16} + \cdots + 20\!\cdots\!25 \)
|
| $7$ |
\( T^{16} \)
|
| $11$ |
\( T^{16} + \cdots + 23\!\cdots\!25 \)
|
| $13$ |
\( T^{16} + \cdots + 92\!\cdots\!76 \)
|
| $17$ |
\( T^{16} + \cdots - 24\!\cdots\!59 \)
|
| $19$ |
\( T^{16} + \cdots - 73\!\cdots\!31 \)
|
| $23$ |
\( T^{16} + \cdots + 22\!\cdots\!01 \)
|
| $29$ |
\( T^{16} + \cdots + 14\!\cdots\!00 \)
|
| $31$ |
\( T^{16} + \cdots - 54\!\cdots\!79 \)
|
| $37$ |
\( T^{16} + \cdots + 93\!\cdots\!25 \)
|
| $41$ |
\( T^{16} + \cdots + 88\!\cdots\!00 \)
|
| $43$ |
\( T^{16} + \cdots - 42\!\cdots\!96 \)
|
| $47$ |
\( T^{16} + \cdots + 13\!\cdots\!25 \)
|
| $53$ |
\( T^{16} + \cdots + 13\!\cdots\!29 \)
|
| $59$ |
\( T^{16} + \cdots - 30\!\cdots\!75 \)
|
| $61$ |
\( T^{16} + \cdots - 59\!\cdots\!71 \)
|
| $67$ |
\( T^{16} + \cdots + 15\!\cdots\!25 \)
|
| $71$ |
\( T^{16} + \cdots - 70\!\cdots\!44 \)
|
| $73$ |
\( T^{16} + \cdots - 16\!\cdots\!79 \)
|
| $79$ |
\( T^{16} + \cdots + 75\!\cdots\!61 \)
|
| $83$ |
\( T^{16} + \cdots + 15\!\cdots\!96 \)
|
| $89$ |
\( T^{16} + \cdots + 83\!\cdots\!29 \)
|
| $97$ |
\( T^{16} + \cdots + 15\!\cdots\!16 \)
|
| show more | |
| show less |