Newspace parameters
| Level: | \( N \) | \(=\) | \( 77 = 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 77.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(0.614848095564\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{5}) \) |
|
|
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| Defining polynomial: |
\( x^{2} - x - 1 \)
|
| 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 \(\beta = \sqrt{5}\). 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 |
|
−2.23607 | 3.23607 | 3.00000 | −2.00000 | −7.23607 | 1.00000 | −2.23607 | 7.47214 | 4.47214 | ||||||||||||||||||||||||
| 1.2 | 2.23607 | −1.23607 | 3.00000 | −2.00000 | −2.76393 | 1.00000 | 2.23607 | −1.47214 | −4.47214 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \( -1 \) |
| \(11\) | \( +1 \) |
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 | 77.2.a.d | ✓ | 2 |
| 3.b | odd | 2 | 1 | 693.2.a.h | 2 | ||
| 4.b | odd | 2 | 1 | 1232.2.a.m | 2 | ||
| 5.b | even | 2 | 1 | 1925.2.a.r | 2 | ||
| 5.c | odd | 4 | 2 | 1925.2.b.h | 4 | ||
| 7.b | odd | 2 | 1 | 539.2.a.f | 2 | ||
| 7.c | even | 3 | 2 | 539.2.e.i | 4 | ||
| 7.d | odd | 6 | 2 | 539.2.e.j | 4 | ||
| 8.b | even | 2 | 1 | 4928.2.a.bm | 2 | ||
| 8.d | odd | 2 | 1 | 4928.2.a.bv | 2 | ||
| 11.b | odd | 2 | 1 | 847.2.a.f | 2 | ||
| 11.c | even | 5 | 2 | 847.2.f.a | 4 | ||
| 11.c | even | 5 | 2 | 847.2.f.n | 4 | ||
| 11.d | odd | 10 | 2 | 847.2.f.b | 4 | ||
| 11.d | odd | 10 | 2 | 847.2.f.m | 4 | ||
| 21.c | even | 2 | 1 | 4851.2.a.y | 2 | ||
| 28.d | even | 2 | 1 | 8624.2.a.ce | 2 | ||
| 33.d | even | 2 | 1 | 7623.2.a.bl | 2 | ||
| 77.b | even | 2 | 1 | 5929.2.a.m | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 77.2.a.d | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 539.2.a.f | 2 | 7.b | odd | 2 | 1 | ||
| 539.2.e.i | 4 | 7.c | even | 3 | 2 | ||
| 539.2.e.j | 4 | 7.d | odd | 6 | 2 | ||
| 693.2.a.h | 2 | 3.b | odd | 2 | 1 | ||
| 847.2.a.f | 2 | 11.b | odd | 2 | 1 | ||
| 847.2.f.a | 4 | 11.c | even | 5 | 2 | ||
| 847.2.f.b | 4 | 11.d | odd | 10 | 2 | ||
| 847.2.f.m | 4 | 11.d | odd | 10 | 2 | ||
| 847.2.f.n | 4 | 11.c | even | 5 | 2 | ||
| 1232.2.a.m | 2 | 4.b | odd | 2 | 1 | ||
| 1925.2.a.r | 2 | 5.b | even | 2 | 1 | ||
| 1925.2.b.h | 4 | 5.c | odd | 4 | 2 | ||
| 4851.2.a.y | 2 | 21.c | even | 2 | 1 | ||
| 4928.2.a.bm | 2 | 8.b | even | 2 | 1 | ||
| 4928.2.a.bv | 2 | 8.d | odd | 2 | 1 | ||
| 5929.2.a.m | 2 | 77.b | even | 2 | 1 | ||
| 7623.2.a.bl | 2 | 33.d | even | 2 | 1 | ||
| 8624.2.a.ce | 2 | 28.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(77))\):
|
\( T_{2}^{2} - 5 \)
|
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\( T_{3}^{2} - 2T_{3} - 4 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 5 \)
|
| $3$ |
\( T^{2} - 2T - 4 \)
|
| $5$ |
\( (T + 2)^{2} \)
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| $7$ |
\( (T - 1)^{2} \)
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| $11$ |
\( (T + 1)^{2} \)
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| $13$ |
\( T^{2} - 2T - 4 \)
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| $17$ |
\( T^{2} + 2T - 4 \)
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| $19$ |
\( T^{2} - 4T - 16 \)
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| $23$ |
\( T^{2} + 4T - 16 \)
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| $29$ |
\( T^{2} - 8T - 4 \)
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| $31$ |
\( T^{2} + 10T + 20 \)
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| $37$ |
\( T^{2} + 8T - 4 \)
|
| $41$ |
\( T^{2} + 18T + 76 \)
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| $43$ |
\( (T - 8)^{2} \)
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| $47$ |
\( T^{2} - 10T + 20 \)
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| $53$ |
\( T^{2} - 8T - 4 \)
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| $59$ |
\( T^{2} - 2T - 4 \)
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| $61$ |
\( T^{2} + 10T + 20 \)
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| $67$ |
\( T^{2} - 20T + 80 \)
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| $71$ |
\( T^{2} + 12T + 16 \)
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| $73$ |
\( T^{2} + 6T + 4 \)
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| $79$ |
\( T^{2} - 80 \)
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| $83$ |
\( T^{2} - 4T - 176 \)
|
| $89$ |
\( (T - 2)^{2} \)
|
| $97$ |
\( T^{2} - 8T - 164 \)
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