Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [213,4,Mod(1,213)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(213, 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("213.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 213 = 3 \cdot 71 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 213.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: | \(12.5674068312\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 3 x^{9} - 59 x^{8} + 124 x^{7} + 1247 x^{6} - 1563 x^{5} - 10389 x^{4} + 7234 x^{3} + \cdots + 2816 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 3 x^{9} - 59 x^{8} + 124 x^{7} + 1247 x^{6} - 1563 x^{5} - 10389 x^{4} + 7234 x^{3} + \cdots + 2816 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 19\nu + 12 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 9 \nu^{9} - 47 \nu^{8} - 279 \nu^{7} + 1736 \nu^{6} + 2791 \nu^{5} - 26319 \nu^{4} - 20849 \nu^{3} + \cdots - 203200 ) / 10624 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 17 \nu^{9} - 15 \nu^{8} - 1191 \nu^{7} + 992 \nu^{6} + 25487 \nu^{5} - 12751 \nu^{4} - 186273 \nu^{3} + \cdots - 69824 ) / 5312 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 21 \nu^{9} - 331 \nu^{8} + 677 \nu^{7} + 10912 \nu^{6} - 41517 \nu^{5} - 105235 \nu^{4} + \cdots + 698048 ) / 10624 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 43 \nu^{9} + 77 \nu^{8} + 2661 \nu^{7} - 2392 \nu^{6} - 57749 \nu^{5} + 6669 \nu^{4} + \cdots + 13504 ) / 10624 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 31 \nu^{9} - 125 \nu^{8} - 1625 \nu^{7} + 4836 \nu^{6} + 31341 \nu^{5} - 54577 \nu^{4} + \cdots - 187008 ) / 5312 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 89 \nu^{9} + 391 \nu^{8} + 4087 \nu^{7} - 13552 \nu^{6} - 69727 \nu^{5} + 132335 \nu^{4} + \cdots + 314304 ) / 10624 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 21\beta _1 + 12 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{9} + \beta_{8} - 2\beta_{7} + \beta_{6} + \beta_{4} + 2\beta_{3} + 28\beta_{2} + 48\beta _1 + 244 \)
|
| \(\nu^{5}\) | \(=\) |
\( 6 \beta_{9} + 4 \beta_{8} - 6 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 6 \beta_{4} + 34 \beta_{3} + \cdots + 564 \)
|
| \(\nu^{6}\) | \(=\) |
\( 86 \beta_{9} + 46 \beta_{8} - 78 \beta_{7} + 40 \beta_{6} + 2 \beta_{5} + 60 \beta_{4} + 105 \beta_{3} + \cdots + 5844 \)
|
| \(\nu^{7}\) | \(=\) |
\( 316 \beta_{9} + 177 \beta_{8} - 316 \beta_{7} + 123 \beta_{6} - 62 \beta_{5} + 343 \beta_{4} + \cdots + 19860 \)
|
| \(\nu^{8}\) | \(=\) |
\( 2864 \beta_{9} + 1422 \beta_{8} - 2616 \beta_{7} + 1298 \beta_{6} + 124 \beta_{5} + 2530 \beta_{4} + \cdots + 153436 \)
|
| \(\nu^{9}\) | \(=\) |
\( 12152 \beta_{9} + 5724 \beta_{8} - 12400 \beta_{7} + 5180 \beta_{6} - 1040 \beta_{5} + 14516 \beta_{4} + \cdots + 640212 \)
|
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 |
|
−4.53447 | 3.00000 | 12.5615 | 12.1281 | −13.6034 | 6.94994 | −20.6838 | 9.00000 | −54.9947 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.35866 | 3.00000 | 10.9979 | −5.47891 | −13.0760 | −22.7930 | −13.0670 | 9.00000 | 23.8807 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.66137 | 3.00000 | −0.917123 | 9.97702 | −7.98410 | 27.8943 | 23.7317 | 9.00000 | −26.5525 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.718207 | 3.00000 | −7.48418 | −10.9984 | −2.15462 | −18.9488 | 11.1208 | 9.00000 | 7.89913 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.477838 | 3.00000 | −7.77167 | −17.5267 | 1.43351 | 26.1595 | −7.53630 | 9.00000 | −8.37493 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.790458 | 3.00000 | −7.37518 | 15.6028 | 2.37137 | 2.17537 | −12.1534 | 9.00000 | 12.3334 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 3.51180 | 3.00000 | 4.33270 | 1.14142 | 10.5354 | 16.1595 | −12.8788 | 9.00000 | 4.00844 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 4.17792 | 3.00000 | 9.45505 | 21.5483 | 12.5338 | 5.41669 | 6.07910 | 9.00000 | 90.0271 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 5.10978 | 3.00000 | 18.1098 | −4.33627 | 15.3293 | 17.4412 | 51.6591 | 9.00000 | −22.1574 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 5.20492 | 3.00000 | 19.0912 | 5.94261 | 15.6147 | −32.4547 | 57.7285 | 9.00000 | 30.9308 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(71\) | \( -1 \) |
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 | 213.4.a.e | ✓ | 10 |
| 3.b | odd | 2 | 1 | 639.4.a.g | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 213.4.a.e | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 639.4.a.g | 10 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} - 7 T_{2}^{9} - 41 T_{2}^{8} + 336 T_{2}^{7} + 421 T_{2}^{6} - 5093 T_{2}^{5} + 543 T_{2}^{4} + \cdots + 5568 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(213))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 7 T^{9} + \cdots + 5568 \)
|
| $3$ |
\( (T - 3)^{10} \)
|
| $5$ |
\( T^{10} + \cdots + 1263793040 \)
|
| $7$ |
\( T^{10} + \cdots - 236079176544 \)
|
| $11$ |
\( T^{10} + \cdots + 96698187852096 \)
|
| $13$ |
\( T^{10} + \cdots - 18\!\cdots\!44 \)
|
| $17$ |
\( T^{10} + \cdots - 11\!\cdots\!32 \)
|
| $19$ |
\( T^{10} + \cdots - 229236937889408 \)
|
| $23$ |
\( T^{10} + \cdots + 63\!\cdots\!96 \)
|
| $29$ |
\( T^{10} + \cdots - 36\!\cdots\!56 \)
|
| $31$ |
\( T^{10} + \cdots + 26\!\cdots\!68 \)
|
| $37$ |
\( T^{10} + \cdots + 30\!\cdots\!28 \)
|
| $41$ |
\( T^{10} + \cdots + 16\!\cdots\!76 \)
|
| $43$ |
\( T^{10} + \cdots + 31\!\cdots\!00 \)
|
| $47$ |
\( T^{10} + \cdots - 55\!\cdots\!80 \)
|
| $53$ |
\( T^{10} + \cdots + 46\!\cdots\!96 \)
|
| $59$ |
\( T^{10} + \cdots + 25\!\cdots\!08 \)
|
| $61$ |
\( T^{10} + \cdots + 55\!\cdots\!96 \)
|
| $67$ |
\( T^{10} + \cdots - 22\!\cdots\!60 \)
|
| $71$ |
\( (T - 71)^{10} \)
|
| $73$ |
\( T^{10} + \cdots + 15\!\cdots\!24 \)
|
| $79$ |
\( T^{10} + \cdots + 14\!\cdots\!60 \)
|
| $83$ |
\( T^{10} + \cdots + 26\!\cdots\!92 \)
|
| $89$ |
\( T^{10} + \cdots - 48\!\cdots\!40 \)
|
| $97$ |
\( T^{10} + \cdots + 59\!\cdots\!56 \)
|
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