Newspace parameters
Level: | \( N \) | \(=\) | \( 77 = 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 77.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.614848095564\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-7}) \) |
Defining polynomial: |
\( x^{2} - x + 2 \)
|
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-7}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/77\mathbb{Z}\right)^\times\).
\(n\) | \(45\) | \(57\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
76.1 |
|
− | 2.64575i | 0 | −5.00000 | 0 | 0 | − | 2.64575i | 7.93725i | 3.00000 | 0 | ||||||||||||||||||||||
76.2 | 2.64575i | 0 | −5.00000 | 0 | 0 | 2.64575i | − | 7.93725i | 3.00000 | 0 | ||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-7}) \) |
11.b | odd | 2 | 1 | inner |
77.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 77.2.b.a | ✓ | 2 |
3.b | odd | 2 | 1 | 693.2.c.a | 2 | ||
4.b | odd | 2 | 1 | 1232.2.e.a | 2 | ||
7.b | odd | 2 | 1 | CM | 77.2.b.a | ✓ | 2 |
7.c | even | 3 | 2 | 539.2.i.a | 4 | ||
7.d | odd | 6 | 2 | 539.2.i.a | 4 | ||
11.b | odd | 2 | 1 | inner | 77.2.b.a | ✓ | 2 |
11.c | even | 5 | 4 | 847.2.l.b | 8 | ||
11.d | odd | 10 | 4 | 847.2.l.b | 8 | ||
21.c | even | 2 | 1 | 693.2.c.a | 2 | ||
28.d | even | 2 | 1 | 1232.2.e.a | 2 | ||
33.d | even | 2 | 1 | 693.2.c.a | 2 | ||
44.c | even | 2 | 1 | 1232.2.e.a | 2 | ||
77.b | even | 2 | 1 | inner | 77.2.b.a | ✓ | 2 |
77.h | odd | 6 | 2 | 539.2.i.a | 4 | ||
77.i | even | 6 | 2 | 539.2.i.a | 4 | ||
77.j | odd | 10 | 4 | 847.2.l.b | 8 | ||
77.l | even | 10 | 4 | 847.2.l.b | 8 | ||
231.h | odd | 2 | 1 | 693.2.c.a | 2 | ||
308.g | odd | 2 | 1 | 1232.2.e.a | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
77.2.b.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
77.2.b.a | ✓ | 2 | 7.b | odd | 2 | 1 | CM |
77.2.b.a | ✓ | 2 | 11.b | odd | 2 | 1 | inner |
77.2.b.a | ✓ | 2 | 77.b | even | 2 | 1 | inner |
539.2.i.a | 4 | 7.c | even | 3 | 2 | ||
539.2.i.a | 4 | 7.d | odd | 6 | 2 | ||
539.2.i.a | 4 | 77.h | odd | 6 | 2 | ||
539.2.i.a | 4 | 77.i | even | 6 | 2 | ||
693.2.c.a | 2 | 3.b | odd | 2 | 1 | ||
693.2.c.a | 2 | 21.c | even | 2 | 1 | ||
693.2.c.a | 2 | 33.d | even | 2 | 1 | ||
693.2.c.a | 2 | 231.h | odd | 2 | 1 | ||
847.2.l.b | 8 | 11.c | even | 5 | 4 | ||
847.2.l.b | 8 | 11.d | odd | 10 | 4 | ||
847.2.l.b | 8 | 77.j | odd | 10 | 4 | ||
847.2.l.b | 8 | 77.l | even | 10 | 4 | ||
1232.2.e.a | 2 | 4.b | odd | 2 | 1 | ||
1232.2.e.a | 2 | 28.d | even | 2 | 1 | ||
1232.2.e.a | 2 | 44.c | even | 2 | 1 | ||
1232.2.e.a | 2 | 308.g | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 7 \)
acting on \(S_{2}^{\mathrm{new}}(77, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 7 \)
$3$
\( T^{2} \)
$5$
\( T^{2} \)
$7$
\( T^{2} + 7 \)
$11$
\( T^{2} - 4T + 11 \)
$13$
\( T^{2} \)
$17$
\( T^{2} \)
$19$
\( T^{2} \)
$23$
\( (T + 8)^{2} \)
$29$
\( T^{2} + 112 \)
$31$
\( T^{2} \)
$37$
\( (T + 6)^{2} \)
$41$
\( T^{2} \)
$43$
\( T^{2} + 28 \)
$47$
\( T^{2} \)
$53$
\( (T - 10)^{2} \)
$59$
\( T^{2} \)
$61$
\( T^{2} \)
$67$
\( (T + 4)^{2} \)
$71$
\( (T + 16)^{2} \)
$73$
\( T^{2} \)
$79$
\( T^{2} + 252 \)
$83$
\( T^{2} \)
$89$
\( T^{2} \)
$97$
\( T^{2} \)
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