Newspace parameters
Level: | \( N \) | \(=\) | \( 76 = 2^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 76.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(2.07085000914\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{57}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} - x - 14 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).
\(n\) | \(21\) | \(39\) |
\(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
37.1 |
|
0 | 0 | 0 | 0.725083 | 0 | 13.8248 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||
37.2 | 0 | 0 | 0 | 8.27492 | 0 | −8.82475 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-19}) \) |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 76.3.c.b | ✓ | 2 |
3.b | odd | 2 | 1 | 684.3.h.a | 2 | ||
4.b | odd | 2 | 1 | 304.3.e.e | 2 | ||
5.b | even | 2 | 1 | 1900.3.e.a | 2 | ||
5.c | odd | 4 | 2 | 1900.3.g.a | 4 | ||
8.b | even | 2 | 1 | 1216.3.e.f | 2 | ||
8.d | odd | 2 | 1 | 1216.3.e.e | 2 | ||
12.b | even | 2 | 1 | 2736.3.o.c | 2 | ||
19.b | odd | 2 | 1 | CM | 76.3.c.b | ✓ | 2 |
57.d | even | 2 | 1 | 684.3.h.a | 2 | ||
76.d | even | 2 | 1 | 304.3.e.e | 2 | ||
95.d | odd | 2 | 1 | 1900.3.e.a | 2 | ||
95.g | even | 4 | 2 | 1900.3.g.a | 4 | ||
152.b | even | 2 | 1 | 1216.3.e.e | 2 | ||
152.g | odd | 2 | 1 | 1216.3.e.f | 2 | ||
228.b | odd | 2 | 1 | 2736.3.o.c | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
76.3.c.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
76.3.c.b | ✓ | 2 | 19.b | odd | 2 | 1 | CM |
304.3.e.e | 2 | 4.b | odd | 2 | 1 | ||
304.3.e.e | 2 | 76.d | even | 2 | 1 | ||
684.3.h.a | 2 | 3.b | odd | 2 | 1 | ||
684.3.h.a | 2 | 57.d | even | 2 | 1 | ||
1216.3.e.e | 2 | 8.d | odd | 2 | 1 | ||
1216.3.e.e | 2 | 152.b | even | 2 | 1 | ||
1216.3.e.f | 2 | 8.b | even | 2 | 1 | ||
1216.3.e.f | 2 | 152.g | odd | 2 | 1 | ||
1900.3.e.a | 2 | 5.b | even | 2 | 1 | ||
1900.3.e.a | 2 | 95.d | odd | 2 | 1 | ||
1900.3.g.a | 4 | 5.c | odd | 4 | 2 | ||
1900.3.g.a | 4 | 95.g | even | 4 | 2 | ||
2736.3.o.c | 2 | 12.b | even | 2 | 1 | ||
2736.3.o.c | 2 | 228.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} \)
acting on \(S_{3}^{\mathrm{new}}(76, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} \)
$5$
\( T^{2} - 9T + 6 \)
$7$
\( T^{2} - 5T - 122 \)
$11$
\( T^{2} + 3T - 354 \)
$13$
\( T^{2} \)
$17$
\( T^{2} + 15T - 642 \)
$19$
\( (T + 19)^{2} \)
$23$
\( (T + 30)^{2} \)
$29$
\( T^{2} \)
$31$
\( T^{2} \)
$37$
\( T^{2} \)
$41$
\( T^{2} \)
$43$
\( T^{2} - 85T + 1678 \)
$47$
\( T^{2} + 75T - 1002 \)
$53$
\( T^{2} \)
$59$
\( T^{2} \)
$61$
\( T^{2} + 103T - 554 \)
$67$
\( T^{2} \)
$71$
\( T^{2} \)
$73$
\( T^{2} - 25T - 15362 \)
$79$
\( T^{2} \)
$83$
\( (T - 90)^{2} \)
$89$
\( T^{2} \)
$97$
\( T^{2} \)
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