Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [350,10,Mod(1,350)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(350, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("350.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 350 = 2 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 350.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: | \(180.262542657\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3 x^{7} - 104167 x^{6} + 2076739 x^{5} + 2783854950 x^{4} - 135630349144 x^{3} + \cdots - 24\!\cdots\!56 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3^{4}\cdot 5^{9} \) |
| Twist minimal: | no (minimal twist has level 70) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 3 x^{7} - 104167 x^{6} + 2076739 x^{5} + 2783854950 x^{4} - 135630349144 x^{3} + \cdots - 24\!\cdots\!56 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 26\nu - 26053 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3854241458 \nu^{7} + 361157628713 \nu^{6} - 349728214792329 \nu^{5} + \cdots + 13\!\cdots\!32 ) / 91\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 589001672 \nu^{7} - 63230675867 \nu^{6} + 54934221802986 \nu^{5} + \cdots - 33\!\cdots\!88 ) / 95\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1932288647 \nu^{7} + 249028664567 \nu^{6} - 173357115118761 \nu^{5} + \cdots + 21\!\cdots\!88 ) / 24\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 57618020549 \nu^{7} + 5354004131189 \nu^{6} + \cdots + 13\!\cdots\!96 ) / 36\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 58960299999 \nu^{7} - 5838681607489 \nu^{6} + \cdots - 16\!\cdots\!96 ) / 20\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 26\beta _1 + 26053 \)
|
| \(\nu^{3}\) | \(=\) |
\( 10\beta_{7} + 38\beta_{6} + 24\beta_{5} + 56\beta_{4} - 37\beta_{3} + 14\beta_{2} + 50603\beta _1 - 680592 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 2385 \beta_{7} - 4634 \beta_{6} - 1257 \beta_{5} + 899 \beta_{4} + 4569 \beta_{3} + 77214 \beta_{2} + \cdots + 1318669055 \)
|
| \(\nu^{5}\) | \(=\) |
\( 1085830 \beta_{7} + 3487598 \beta_{6} + 2128740 \beta_{5} + 4582028 \beta_{4} - 2909509 \beta_{3} + \cdots - 36500944194 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 243390305 \beta_{7} - 490440102 \beta_{6} - 115072461 \beta_{5} + 19113127 \beta_{4} + \cdots + 81858455930717 \)
|
| \(\nu^{7}\) | \(=\) |
\( 87976366360 \beta_{7} + 265298044110 \beta_{6} + 153777768138 \beta_{5} + 322570281586 \beta_{4} + \cdots - 23\!\cdots\!16 \)
|
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 |
|
−16.0000 | −273.592 | 256.000 | 0 | 4377.46 | −2401.00 | −4096.00 | 55169.3 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −16.0000 | −177.275 | 256.000 | 0 | 2836.40 | −2401.00 | −4096.00 | 11743.4 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −16.0000 | −138.348 | 256.000 | 0 | 2213.57 | −2401.00 | −4096.00 | −542.877 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −16.0000 | 10.3031 | 256.000 | 0 | −164.849 | −2401.00 | −4096.00 | −19576.8 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −16.0000 | 62.1763 | 256.000 | 0 | −994.820 | −2401.00 | −4096.00 | −15817.1 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −16.0000 | 91.7804 | 256.000 | 0 | −1468.49 | −2401.00 | −4096.00 | −11259.4 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −16.0000 | 100.724 | 256.000 | 0 | −1611.58 | −2401.00 | −4096.00 | −9537.74 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −16.0000 | 247.231 | 256.000 | 0 | −3955.70 | −2401.00 | −4096.00 | 41440.2 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(-1\) |
| \(7\) | \(1\) |
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 | 350.10.a.w | 8 | |
| 5.b | even | 2 | 1 | 350.10.a.x | 8 | ||
| 5.c | odd | 4 | 2 | 70.10.c.b | ✓ | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 70.10.c.b | ✓ | 16 | 5.c | odd | 4 | 2 | |
| 350.10.a.w | 8 | 1.a | even | 1 | 1 | trivial | |
| 350.10.a.x | 8 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 77 T_{3}^{7} - 101577 T_{3}^{6} - 4123581 T_{3}^{5} + 2732036400 T_{3}^{4} + \cdots - 98\!\cdots\!76 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(350))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 16)^{8} \)
|
| $3$ |
\( T^{8} + \cdots - 98\!\cdots\!76 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( (T + 2401)^{8} \)
|
| $11$ |
\( T^{8} + \cdots - 58\!\cdots\!16 \)
|
| $13$ |
\( T^{8} + \cdots - 13\!\cdots\!52 \)
|
| $17$ |
\( T^{8} + \cdots + 14\!\cdots\!36 \)
|
| $19$ |
\( T^{8} + \cdots - 73\!\cdots\!00 \)
|
| $23$ |
\( T^{8} + \cdots - 17\!\cdots\!00 \)
|
| $29$ |
\( T^{8} + \cdots - 99\!\cdots\!00 \)
|
| $31$ |
\( T^{8} + \cdots - 91\!\cdots\!52 \)
|
| $37$ |
\( T^{8} + \cdots + 48\!\cdots\!76 \)
|
| $41$ |
\( T^{8} + \cdots + 69\!\cdots\!28 \)
|
| $43$ |
\( T^{8} + \cdots - 17\!\cdots\!88 \)
|
| $47$ |
\( T^{8} + \cdots + 16\!\cdots\!28 \)
|
| $53$ |
\( T^{8} + \cdots + 54\!\cdots\!32 \)
|
| $59$ |
\( T^{8} + \cdots + 24\!\cdots\!00 \)
|
| $61$ |
\( T^{8} + \cdots + 12\!\cdots\!00 \)
|
| $67$ |
\( T^{8} + \cdots - 24\!\cdots\!64 \)
|
| $71$ |
\( T^{8} + \cdots + 18\!\cdots\!68 \)
|
| $73$ |
\( T^{8} + \cdots - 46\!\cdots\!00 \)
|
| $79$ |
\( T^{8} + \cdots - 21\!\cdots\!00 \)
|
| $83$ |
\( T^{8} + \cdots - 15\!\cdots\!56 \)
|
| $89$ |
\( T^{8} + \cdots - 31\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots + 38\!\cdots\!00 \)
|
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