Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [98,10,Mod(1,98)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(98, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("98.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 98 = 2 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 98.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: | \(50.4735119441\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 7165x^{4} + 16168x^{3} + 12807408x^{2} - 32180328x - 5372164 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{2}\cdot 7^{6} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 7165x^{4} + 16168x^{3} + 12807408x^{2} - 32180328x - 5372164 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 16557 \nu^{5} + 1210854 \nu^{4} - 100542547 \nu^{3} - 8636476956 \nu^{2} + \cdots + 10243942375796 ) / 3101451260 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -10717\nu^{5} + 2982\nu^{4} + 64030099\nu^{3} - 46012148\nu^{2} - 91498140874\nu + 114577250368 ) / 1860870756 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 142991 \nu^{5} - 3910461 \nu^{4} + 1169730299 \nu^{3} + 13263914945 \nu^{2} + \cdots + 2902579423454 ) / 18608707560 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 26311 \nu^{5} - 1222782 \nu^{4} - 155577849 \nu^{3} + 8820525548 \nu^{2} + \cdots - 10581294778128 ) / 1240580504 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 374123 \nu^{5} + 15601227 \nu^{4} + 3145018667 \nu^{3} - 58197203515 \nu^{2} + \cdots + 8276645371622 ) / 4652176890 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{4} - 12\beta_{2} - 5\beta _1 + 195 ) / 588 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 28\beta_{5} + 58\beta_{4} - 336\beta_{3} + 16\beta_{2} - 275\beta _1 + 1404621 ) / 588 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 1302\beta_{5} - 7114\beta_{4} + 19656\beta_{3} - 87480\beta_{2} - 18205\beta _1 - 543285 ) / 588 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 202328\beta_{5} + 210234\beta_{4} - 2380896\beta_{3} + 214016\beta_{2} - 983475\beta _1 + 5039754837 ) / 588 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 7715050 \beta_{5} - 25618702 \beta_{4} + 118217400 \beta_{3} - 522316448 \beta_{2} - 65172775 \beta _1 - 3252625455 ) / 588 \)
|
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 | −245.181 | 256.000 | 2662.60 | −3922.90 | 0 | 4096.00 | 40431.0 | 42601.6 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 16.0000 | −143.106 | 256.000 | −474.347 | −2289.70 | 0 | 4096.00 | 796.313 | −7589.55 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 16.0000 | −121.370 | 256.000 | −2666.02 | −1941.92 | 0 | 4096.00 | −4952.27 | −42656.3 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 16.0000 | 121.370 | 256.000 | 2666.02 | 1941.92 | 0 | 4096.00 | −4952.27 | 42656.3 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 16.0000 | 143.106 | 256.000 | 474.347 | 2289.70 | 0 | 4096.00 | 796.313 | 7589.55 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 16.0000 | 245.181 | 256.000 | −2662.60 | 3922.90 | 0 | 4096.00 | 40431.0 | −42601.6 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 98.10.a.l | ✓ | 6 |
| 7.b | odd | 2 | 1 | inner | 98.10.a.l | ✓ | 6 |
| 7.c | even | 3 | 2 | 98.10.c.n | 12 | ||
| 7.d | odd | 6 | 2 | 98.10.c.n | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 98.10.a.l | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 98.10.a.l | ✓ | 6 | 7.b | odd | 2 | 1 | inner |
| 98.10.c.n | 12 | 7.c | even | 3 | 2 | ||
| 98.10.c.n | 12 | 7.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 95324T_{3}^{4} + 2418290244T_{3}^{2} - 18134891776800 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(98))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 16)^{6} \)
|
| $3$ |
\( T^{6} + \cdots - 18134891776800 \)
|
| $5$ |
\( T^{6} + \cdots - 11\!\cdots\!00 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( (T^{3} + \cdots + 13071858183056)^{2} \)
|
| $13$ |
\( T^{6} + \cdots - 85\!\cdots\!00 \)
|
| $17$ |
\( T^{6} + \cdots - 83\!\cdots\!72 \)
|
| $19$ |
\( T^{6} + \cdots - 81\!\cdots\!00 \)
|
| $23$ |
\( (T^{3} + \cdots - 95\!\cdots\!60)^{2} \)
|
| $29$ |
\( (T^{3} + \cdots + 15\!\cdots\!76)^{2} \)
|
| $31$ |
\( T^{6} + \cdots - 11\!\cdots\!08 \)
|
| $37$ |
\( (T^{3} + \cdots - 24\!\cdots\!00)^{2} \)
|
| $41$ |
\( T^{6} + \cdots - 53\!\cdots\!68 \)
|
| $43$ |
\( (T^{3} + \cdots + 20\!\cdots\!48)^{2} \)
|
| $47$ |
\( T^{6} + \cdots - 28\!\cdots\!00 \)
|
| $53$ |
\( (T^{3} + \cdots + 73\!\cdots\!68)^{2} \)
|
| $59$ |
\( T^{6} + \cdots - 50\!\cdots\!92 \)
|
| $61$ |
\( T^{6} + \cdots - 50\!\cdots\!00 \)
|
| $67$ |
\( (T^{3} + \cdots - 72\!\cdots\!12)^{2} \)
|
| $71$ |
\( (T^{3} + \cdots + 12\!\cdots\!48)^{2} \)
|
| $73$ |
\( T^{6} + \cdots - 11\!\cdots\!32 \)
|
| $79$ |
\( (T^{3} + \cdots - 37\!\cdots\!20)^{2} \)
|
| $83$ |
\( T^{6} + \cdots - 61\!\cdots\!52 \)
|
| $89$ |
\( T^{6} + \cdots - 13\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots - 13\!\cdots\!72 \)
|
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