Newspace parameters
| Level: | \( N \) | \(=\) | \( 231 = 3 \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 231.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(1.84454428669\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{21}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 5 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{21})\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.79129 | −1.00000 | 5.79129 | 3.00000 | 2.79129 | 1.00000 | −10.5826 | 1.00000 | −8.37386 | ||||||||||||||||||||||||
| 1.2 | 1.79129 | −1.00000 | 1.20871 | 3.00000 | −1.79129 | 1.00000 | −1.41742 | 1.00000 | 5.37386 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(7\) | \(-1\) |
| \(11\) | \(1\) |
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 | 231.2.a.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | 693.2.a.j | 2 | ||
| 4.b | odd | 2 | 1 | 3696.2.a.bl | 2 | ||
| 5.b | even | 2 | 1 | 5775.2.a.bn | 2 | ||
| 7.b | odd | 2 | 1 | 1617.2.a.o | 2 | ||
| 11.b | odd | 2 | 1 | 2541.2.a.z | 2 | ||
| 21.c | even | 2 | 1 | 4851.2.a.ba | 2 | ||
| 33.d | even | 2 | 1 | 7623.2.a.bf | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 231.2.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 693.2.a.j | 2 | 3.b | odd | 2 | 1 | ||
| 1617.2.a.o | 2 | 7.b | odd | 2 | 1 | ||
| 2541.2.a.z | 2 | 11.b | odd | 2 | 1 | ||
| 3696.2.a.bl | 2 | 4.b | odd | 2 | 1 | ||
| 4851.2.a.ba | 2 | 21.c | even | 2 | 1 | ||
| 5775.2.a.bn | 2 | 5.b | even | 2 | 1 | ||
| 7623.2.a.bf | 2 | 33.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + T_{2} - 5 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(231))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + T - 5 \)
$3$
\( (T + 1)^{2} \)
$5$
\( (T - 3)^{2} \)
$7$
\( (T - 1)^{2} \)
$11$
\( (T + 1)^{2} \)
$13$
\( (T - 1)^{2} \)
$17$
\( T^{2} - 6T - 12 \)
$19$
\( T^{2} + 4T - 17 \)
$23$
\( T^{2} + 2T - 20 \)
$29$
\( T^{2} - 2T - 83 \)
$31$
\( T^{2} + 2T - 20 \)
$37$
\( (T - 1)^{2} \)
$41$
\( T^{2} + 4T - 80 \)
$43$
\( T^{2} + 6T - 12 \)
$47$
\( T^{2} - 12T + 15 \)
$53$
\( T^{2} + 10T + 4 \)
$59$
\( T^{2} - 21 \)
$61$
\( (T - 10)^{2} \)
$67$
\( T^{2} - 8T - 5 \)
$71$
\( T^{2} - 4T - 80 \)
$73$
\( (T - 7)^{2} \)
$79$
\( T^{2} + 4T - 80 \)
$83$
\( T^{2} + 14T + 28 \)
$89$
\( T^{2} - 84 \)
$97$
\( T^{2} + 14T + 28 \)
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