Newspace parameters
Level: | \( N \) | \(=\) | \( 164 = 2^{2} \cdot 41 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 164.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(0.0818466620718\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{2}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} - 2 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{4}\) |
Projective field: | Galois closure of 4.2.6724.1 |
Artin image: | $D_8$ |
Artin field: | Galois closure of 8.0.17643776.1 |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/164\mathbb{Z}\right)^\times\).
\(n\) | \(83\) | \(129\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
163.1 |
|
−1.00000 | −1.41421 | 1.00000 | 0 | 1.41421 | 1.41421 | −1.00000 | 1.00000 | 0 | ||||||||||||||||||||||||
163.2 | −1.00000 | 1.41421 | 1.00000 | 0 | −1.41421 | −1.41421 | −1.00000 | 1.00000 | 0 | |||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
164.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-41}) \) |
4.b | odd | 2 | 1 | inner |
41.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 164.1.d.b | ✓ | 2 |
3.b | odd | 2 | 1 | 1476.1.h.b | 2 | ||
4.b | odd | 2 | 1 | inner | 164.1.d.b | ✓ | 2 |
8.b | even | 2 | 1 | 2624.1.h.b | 2 | ||
8.d | odd | 2 | 1 | 2624.1.h.b | 2 | ||
12.b | even | 2 | 1 | 1476.1.h.b | 2 | ||
41.b | even | 2 | 1 | inner | 164.1.d.b | ✓ | 2 |
123.b | odd | 2 | 1 | 1476.1.h.b | 2 | ||
164.d | odd | 2 | 1 | CM | 164.1.d.b | ✓ | 2 |
328.c | odd | 2 | 1 | 2624.1.h.b | 2 | ||
328.g | even | 2 | 1 | 2624.1.h.b | 2 | ||
492.d | even | 2 | 1 | 1476.1.h.b | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
164.1.d.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
164.1.d.b | ✓ | 2 | 4.b | odd | 2 | 1 | inner |
164.1.d.b | ✓ | 2 | 41.b | even | 2 | 1 | inner |
164.1.d.b | ✓ | 2 | 164.d | odd | 2 | 1 | CM |
1476.1.h.b | 2 | 3.b | odd | 2 | 1 | ||
1476.1.h.b | 2 | 12.b | even | 2 | 1 | ||
1476.1.h.b | 2 | 123.b | odd | 2 | 1 | ||
1476.1.h.b | 2 | 492.d | even | 2 | 1 | ||
2624.1.h.b | 2 | 8.b | even | 2 | 1 | ||
2624.1.h.b | 2 | 8.d | odd | 2 | 1 | ||
2624.1.h.b | 2 | 328.c | odd | 2 | 1 | ||
2624.1.h.b | 2 | 328.g | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 2 \)
acting on \(S_{1}^{\mathrm{new}}(164, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T + 1)^{2} \)
$3$
\( T^{2} - 2 \)
$5$
\( T^{2} \)
$7$
\( T^{2} - 2 \)
$11$
\( T^{2} - 2 \)
$13$
\( T^{2} \)
$17$
\( T^{2} \)
$19$
\( T^{2} - 2 \)
$23$
\( T^{2} \)
$29$
\( T^{2} \)
$31$
\( T^{2} \)
$37$
\( T^{2} \)
$41$
\( (T - 1)^{2} \)
$43$
\( T^{2} \)
$47$
\( T^{2} - 2 \)
$53$
\( T^{2} \)
$59$
\( T^{2} \)
$61$
\( (T + 2)^{2} \)
$67$
\( T^{2} - 2 \)
$71$
\( T^{2} - 2 \)
$73$
\( T^{2} \)
$79$
\( T^{2} - 2 \)
$83$
\( T^{2} \)
$89$
\( T^{2} \)
$97$
\( T^{2} \)
show more
show less