Newspace parameters
| Level: | \( N \) | \(=\) | \( 71 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 71.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(1.93460987696\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.2836736.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 108x^{2} - 40x + 2825 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 108x^{2} - 40x + 2825 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 10\nu^{2} - 53\nu + 570 ) / 155 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 10\nu^{2} + 363\nu - 570 ) / 155 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{3} + 11\nu^{2} + 106\nu + 534 ) / 31 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 10\beta _1 - 54 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 20\beta_{3} - 53\beta_{2} - 163\beta _1 + 60 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/71\mathbb{Z}\right)^\times\).
| \(n\) | \(7\) |
| \(\chi(n)\) | \(-1\) |
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.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 70.1 |
|
−2.41421 | 0.414214 | 1.82843 | −0.585786 | −1.00000 | − | 8.98419i | 5.24264 | −8.82843 | 1.41421 | |||||||||||||||||||||||||||||
| 70.2 | −2.41421 | 0.414214 | 1.82843 | −0.585786 | −1.00000 | 8.98419i | 5.24264 | −8.82843 | 1.41421 | |||||||||||||||||||||||||||||||
| 70.3 | 0.414214 | −2.41421 | −3.82843 | −3.41421 | −1.00000 | − | 11.7168i | −3.24264 | −3.17157 | −1.41421 | ||||||||||||||||||||||||||||||
| 70.4 | 0.414214 | −2.41421 | −3.82843 | −3.41421 | −1.00000 | 11.7168i | −3.24264 | −3.17157 | −1.41421 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 71.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 71.3.b.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 639.3.d.a | 4 | ||
| 4.b | odd | 2 | 1 | 1136.3.h.a | 4 | ||
| 71.b | odd | 2 | 1 | inner | 71.3.b.a | ✓ | 4 |
| 213.b | even | 2 | 1 | 639.3.d.a | 4 | ||
| 284.c | even | 2 | 1 | 1136.3.h.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 71.3.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 71.3.b.a | ✓ | 4 | 71.b | odd | 2 | 1 | inner |
| 639.3.d.a | 4 | 3.b | odd | 2 | 1 | ||
| 639.3.d.a | 4 | 213.b | even | 2 | 1 | ||
| 1136.3.h.a | 4 | 4.b | odd | 2 | 1 | ||
| 1136.3.h.a | 4 | 284.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 2T_{2} - 1 \)
acting on \(S_{3}^{\mathrm{new}}(71, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 2 T - 1)^{2} \)
$3$
\( (T^{2} + 2 T - 1)^{2} \)
$5$
\( (T^{2} + 4 T + 2)^{2} \)
$7$
\( T^{4} + 218 T^{2} + 11081 \)
$11$
\( T^{4} + 436 T^{2} + 44324 \)
$13$
\( T^{4} + 494 T^{2} + 11081 \)
$17$
\( T^{4} + 436 T^{2} + 44324 \)
$19$
\( (T^{2} - 6 T - 233)^{2} \)
$23$
\( T^{4} + 1642 T^{2} + 542969 \)
$29$
\( (T^{2} - 98)^{2} \)
$31$
\( T^{4} + 1962 T^{2} + 897561 \)
$37$
\( (T^{2} + 12 T - 2702)^{2} \)
$41$
\( T^{4} + 3836 T^{2} + \cdots + 2171876 \)
$43$
\( (T^{2} + 30 T - 497)^{2} \)
$47$
\( T^{4} + 5450 T^{2} + \cdots + 6925625 \)
$53$
\( T^{4} + 6512 T^{2} + 709184 \)
$59$
\( T^{4} + 3488 T^{2} + \cdots + 2836736 \)
$61$
\( T^{4} + 11512 T^{2} + \cdots + 8687504 \)
$67$
\( T^{4} + 7324 T^{2} + \cdots + 12809636 \)
$71$
\( T^{4} - 152 T^{3} + \cdots + 25411681 \)
$73$
\( (T^{2} - 34 T - 6439)^{2} \)
$79$
\( (T^{2} - 96 T + 126)^{2} \)
$83$
\( (T^{2} - 164 T + 5572)^{2} \)
$89$
\( (T^{2} + 94 T + 1241)^{2} \)
$97$
\( T^{4} + 21364 T^{2} + \cdots + 106421924 \)
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