Newspace parameters
| Level: | \( N \) | \(=\) | \( 230 = 2 \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 230.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(1.83655924649\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{21}) \) |
| Defining polynomial: | \(x^{2} - x - 5\) |
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 |
|
−1.00000 | −2.79129 | 1.00000 | −1.00000 | 2.79129 | −1.79129 | −1.00000 | 4.79129 | 1.00000 | ||||||||||||||||||||||||
| 1.2 | −1.00000 | 1.79129 | 1.00000 | −1.00000 | −1.79129 | 2.79129 | −1.00000 | 0.208712 | 1.00000 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(1\) |
| \(23\) | \(-1\) |
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 | 230.2.a.a | ✓ | 2 |
| 3.b | odd | 2 | 1 | 2070.2.a.x | 2 | ||
| 4.b | odd | 2 | 1 | 1840.2.a.n | 2 | ||
| 5.b | even | 2 | 1 | 1150.2.a.o | 2 | ||
| 5.c | odd | 4 | 2 | 1150.2.b.g | 4 | ||
| 8.b | even | 2 | 1 | 7360.2.a.bq | 2 | ||
| 8.d | odd | 2 | 1 | 7360.2.a.bk | 2 | ||
| 20.d | odd | 2 | 1 | 9200.2.a.bs | 2 | ||
| 23.b | odd | 2 | 1 | 5290.2.a.e | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 230.2.a.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 1150.2.a.o | 2 | 5.b | even | 2 | 1 | ||
| 1150.2.b.g | 4 | 5.c | odd | 4 | 2 | ||
| 1840.2.a.n | 2 | 4.b | odd | 2 | 1 | ||
| 2070.2.a.x | 2 | 3.b | odd | 2 | 1 | ||
| 5290.2.a.e | 2 | 23.b | odd | 2 | 1 | ||
| 7360.2.a.bk | 2 | 8.d | odd | 2 | 1 | ||
| 7360.2.a.bq | 2 | 8.b | even | 2 | 1 | ||
| 9200.2.a.bs | 2 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} - 5 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(230))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ | \( ( 1 + T )^{2} \) |
| $3$ | \( -5 + T + T^{2} \) |
| $5$ | \( ( 1 + T )^{2} \) |
| $7$ | \( -5 - T + T^{2} \) |
| $11$ | \( -3 - 3 T + T^{2} \) |
| $13$ | \( 7 - 7 T + T^{2} \) |
| $17$ | \( -3 + 3 T + T^{2} \) |
| $19$ | \( 7 - 7 T + T^{2} \) |
| $23$ | \( ( -1 + T )^{2} \) |
| $29$ | \( -12 - 6 T + T^{2} \) |
| $31$ | \( -35 - 7 T + T^{2} \) |
| $37$ | \( ( 4 + T )^{2} \) |
| $41$ | \( 15 + 9 T + T^{2} \) |
| $43$ | \( -80 - 4 T + T^{2} \) |
| $47$ | \( 60 + 18 T + T^{2} \) |
| $53$ | \( ( -6 + T )^{2} \) |
| $59$ | \( 60 + 18 T + T^{2} \) |
| $61$ | \( -35 - 7 T + T^{2} \) |
| $67$ | \( -80 - 4 T + T^{2} \) |
| $71$ | \( -45 - 3 T + T^{2} \) |
| $73$ | \( -188 + 2 T + T^{2} \) |
| $79$ | \( ( -8 + T )^{2} \) |
| $83$ | \( ( 6 + T )^{2} \) |
| $89$ | \( -48 - 12 T + T^{2} \) |
| $97$ | \( -119 - 7 T + T^{2} \) |
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