Newspace parameters
| Level: | \( N \) | \(=\) | \( 105 = 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 105.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(6.19520055060\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{2}) \) |
| Defining polynomial: | \(x^{2} - 2\) |
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{2}\). 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 |
|
−3.82843 | −3.00000 | 6.65685 | 5.00000 | 11.4853 | −7.00000 | 5.14214 | 9.00000 | −19.1421 | ||||||||||||||||||||||||
| 1.2 | 1.82843 | −3.00000 | −4.65685 | 5.00000 | −5.48528 | −7.00000 | −23.1421 | 9.00000 | 9.14214 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
| \(7\) | \(1\) |
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 | 105.4.a.e | ✓ | 2 |
| 3.b | odd | 2 | 1 | 315.4.a.k | 2 | ||
| 4.b | odd | 2 | 1 | 1680.4.a.bo | 2 | ||
| 5.b | even | 2 | 1 | 525.4.a.l | 2 | ||
| 5.c | odd | 4 | 2 | 525.4.d.l | 4 | ||
| 7.b | odd | 2 | 1 | 735.4.a.o | 2 | ||
| 15.d | odd | 2 | 1 | 1575.4.a.q | 2 | ||
| 21.c | even | 2 | 1 | 2205.4.a.bb | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 105.4.a.e | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 315.4.a.k | 2 | 3.b | odd | 2 | 1 | ||
| 525.4.a.l | 2 | 5.b | even | 2 | 1 | ||
| 525.4.d.l | 4 | 5.c | odd | 4 | 2 | ||
| 735.4.a.o | 2 | 7.b | odd | 2 | 1 | ||
| 1575.4.a.q | 2 | 15.d | odd | 2 | 1 | ||
| 1680.4.a.bo | 2 | 4.b | odd | 2 | 1 | ||
| 2205.4.a.bb | 2 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2 T_{2} - 7 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(105))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ | \( -7 + 2 T + T^{2} \) |
| $3$ | \( ( 3 + T )^{2} \) |
| $5$ | \( ( -5 + T )^{2} \) |
| $7$ | \( ( 7 + T )^{2} \) |
| $11$ | \( -3136 + 16 T + T^{2} \) |
| $13$ | \( 1412 + 76 T + T^{2} \) |
| $17$ | \( 3812 + 124 T + T^{2} \) |
| $19$ | \( 256 + 96 T + T^{2} \) |
| $23$ | \( -9184 + 16 T + T^{2} \) |
| $29$ | \( -14492 - 188 T + T^{2} \) |
| $31$ | \( 1008 + 120 T + T^{2} \) |
| $37$ | \( -30492 + 132 T + T^{2} \) |
| $41$ | \( -48700 - 100 T + T^{2} \) |
| $43$ | \( 41072 + 536 T + T^{2} \) |
| $47$ | \( 201184 + 928 T + T^{2} \) |
| $53$ | \( 162596 - 884 T + T^{2} \) |
| $59$ | \( -330224 - 104 T + T^{2} \) |
| $61$ | \( 40644 + 468 T + T^{2} \) |
| $67$ | \( 700784 + 1688 T + T^{2} \) |
| $71$ | \( -175376 + 136 T + T^{2} \) |
| $73$ | \( -764956 - 508 T + T^{2} \) |
| $79$ | \( -383936 + 432 T + T^{2} \) |
| $83$ | \( 28816 + 584 T + T^{2} \) |
| $89$ | \( 392452 + 1404 T + T^{2} \) |
| $97$ | \( 335908 + 1188 T + T^{2} \) |
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