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: | \(1\) |
| 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: |
\( x^{8} - 3x^{7} - 42x^{6} + 89x^{5} + 543x^{4} - 536x^{3} - 2344x^{2} - 448x + 640 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 3x^{7} - 42x^{6} + 89x^{5} + 543x^{4} - 536x^{3} - 2344x^{2} - 448x + 640 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5\nu^{7} - 19\nu^{6} - 118\nu^{5} + 437\nu^{4} - 41\nu^{3} - 1188\nu^{2} + 3464\nu - 2144 ) / 1024 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{7} - 27\nu^{6} + 250\nu^{5} + 941\nu^{4} - 5057\nu^{3} - 7556\nu^{2} + 25672\nu + 9888 ) / 2048 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 13\nu^{7} - 43\nu^{6} - 486\nu^{5} + 1277\nu^{4} + 4623\nu^{3} - 8132\nu^{2} - 7736\nu + 4256 ) / 1024 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -7\nu^{7} + 33\nu^{6} + 242\nu^{5} - 983\nu^{4} - 2381\nu^{3} + 6092\nu^{2} + 8616\nu - 992 ) / 512 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -39\nu^{7} + 161\nu^{6} + 1458\nu^{5} - 5175\nu^{4} - 15565\nu^{3} + 39756\nu^{2} + 53160\nu - 35296 ) / 2048 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} - 2\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 21\beta _1 + 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( -7\beta_{7} + 7\beta_{6} - 4\beta_{5} + \beta_{4} + 3\beta_{3} + 28\beta_{2} + 39\beta _1 + 219 \)
|
| \(\nu^{5}\) | \(=\) |
\( -42\beta_{7} + 16\beta_{6} - 68\beta_{5} - 26\beta_{4} + 50\beta_{3} + 47\beta_{2} + 511\beta _1 + 337 \)
|
| \(\nu^{6}\) | \(=\) |
\( -283\beta_{7} + 294\beta_{6} - 178\beta_{5} - 11\beta_{4} + 179\beta_{3} + 749\beta_{2} + 1335\beta _1 + 5099 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 1463 \beta_{7} + 883 \beta_{6} - 1948 \beta_{5} - 751 \beta_{4} + 1811 \beta_{3} + 1754 \beta_{2} + \cdots + 11305 \)
|
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 |
|
−5.62840 | −3.00000 | 23.6788 | 17.9040 | 16.8852 | −27.4918 | −88.2467 | 9.00000 | −100.771 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.04157 | −3.00000 | 8.33432 | −1.86981 | 12.1247 | −14.7771 | −1.35117 | 9.00000 | 7.55696 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.87848 | −3.00000 | 0.285638 | −11.9859 | 8.63544 | 8.98033 | 22.2056 | 9.00000 | 34.5013 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.76899 | −3.00000 | −4.87068 | 6.53865 | 5.30696 | 0.312554 | 22.7681 | 9.00000 | −11.5668 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.573528 | −3.00000 | −7.67107 | 1.44806 | 1.72058 | 16.5512 | 8.98779 | 9.00000 | −0.830503 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.19503 | −3.00000 | −3.18183 | 15.4172 | −6.58510 | −19.9526 | −24.5445 | 9.00000 | 33.8411 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 3.21113 | −3.00000 | 2.31138 | −5.94257 | −9.63340 | 20.8450 | −18.2669 | 9.00000 | −19.0824 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 4.48480 | −3.00000 | 12.1134 | −9.50960 | −13.4544 | −26.4676 | 18.4478 | 9.00000 | −42.6486 | |||||||||||||||||||||||||||||||||||||||||||
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.c | ✓ | 8 |
| 3.b | odd | 2 | 1 | 639.4.a.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 213.4.a.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 639.4.a.d | 8 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 5T_{2}^{7} - 35T_{2}^{6} - 170T_{2}^{5} + 323T_{2}^{4} + 1637T_{2}^{3} - 469T_{2}^{2} - 4392T_{2} - 2100 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(213))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 5 T^{7} + \cdots - 2100 \)
|
| $3$ |
\( (T + 3)^{8} \)
|
| $5$ |
\( T^{8} - 12 T^{7} + \cdots + 3310060 \)
|
| $7$ |
\( T^{8} + 42 T^{7} + \cdots + 207755820 \)
|
| $11$ |
\( T^{8} + \cdots + 322421504232 \)
|
| $13$ |
\( T^{8} + \cdots + 2410653776 \)
|
| $17$ |
\( T^{8} + \cdots - 151908497859552 \)
|
| $19$ |
\( T^{8} + \cdots - 186087722816000 \)
|
| $23$ |
\( T^{8} + \cdots + 70372464469632 \)
|
| $29$ |
\( T^{8} + \cdots - 12\!\cdots\!20 \)
|
| $31$ |
\( T^{8} + \cdots - 11\!\cdots\!20 \)
|
| $37$ |
\( T^{8} + \cdots + 24\!\cdots\!80 \)
|
| $41$ |
\( T^{8} + \cdots - 11\!\cdots\!80 \)
|
| $43$ |
\( T^{8} + \cdots - 10\!\cdots\!00 \)
|
| $47$ |
\( T^{8} + \cdots + 11\!\cdots\!80 \)
|
| $53$ |
\( T^{8} + \cdots - 74\!\cdots\!28 \)
|
| $59$ |
\( T^{8} + \cdots + 31\!\cdots\!20 \)
|
| $61$ |
\( T^{8} + \cdots - 68\!\cdots\!60 \)
|
| $67$ |
\( T^{8} + \cdots + 74\!\cdots\!00 \)
|
| $71$ |
\( (T - 71)^{8} \)
|
| $73$ |
\( T^{8} + \cdots - 16\!\cdots\!84 \)
|
| $79$ |
\( T^{8} + \cdots + 90\!\cdots\!00 \)
|
| $83$ |
\( T^{8} + \cdots + 68\!\cdots\!28 \)
|
| $89$ |
\( T^{8} + \cdots + 31\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots + 75\!\cdots\!12 \)
|
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