Newspace parameters
Level: | \( N \) | \(=\) | \( 176 = 2^{4} \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 176.h (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.79565265274\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-2}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{2} + 2 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2\cdot 3 \) |
Twist minimal: | no (minimal twist has level 22) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 6\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/176\mathbb{Z}\right)^\times\).
\(n\) | \(111\) | \(133\) | \(145\) |
\(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
65.1 |
|
0 | −1.00000 | 0 | −1.00000 | 0 | − | 8.48528i | 0 | −8.00000 | 0 | |||||||||||||||||||||||
65.2 | 0 | −1.00000 | 0 | −1.00000 | 0 | 8.48528i | 0 | −8.00000 | 0 | |||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 176.3.h.c | 2 | |
3.b | odd | 2 | 1 | 1584.3.j.d | 2 | ||
4.b | odd | 2 | 1 | 22.3.b.a | ✓ | 2 | |
8.b | even | 2 | 1 | 704.3.h.e | 2 | ||
8.d | odd | 2 | 1 | 704.3.h.d | 2 | ||
11.b | odd | 2 | 1 | inner | 176.3.h.c | 2 | |
12.b | even | 2 | 1 | 198.3.d.b | 2 | ||
20.d | odd | 2 | 1 | 550.3.d.a | 2 | ||
20.e | even | 4 | 2 | 550.3.c.a | 4 | ||
28.d | even | 2 | 1 | 1078.3.d.a | 2 | ||
33.d | even | 2 | 1 | 1584.3.j.d | 2 | ||
44.c | even | 2 | 1 | 22.3.b.a | ✓ | 2 | |
44.g | even | 10 | 4 | 242.3.d.b | 8 | ||
44.h | odd | 10 | 4 | 242.3.d.b | 8 | ||
88.b | odd | 2 | 1 | 704.3.h.e | 2 | ||
88.g | even | 2 | 1 | 704.3.h.d | 2 | ||
132.d | odd | 2 | 1 | 198.3.d.b | 2 | ||
220.g | even | 2 | 1 | 550.3.d.a | 2 | ||
220.i | odd | 4 | 2 | 550.3.c.a | 4 | ||
308.g | odd | 2 | 1 | 1078.3.d.a | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
22.3.b.a | ✓ | 2 | 4.b | odd | 2 | 1 | |
22.3.b.a | ✓ | 2 | 44.c | even | 2 | 1 | |
176.3.h.c | 2 | 1.a | even | 1 | 1 | trivial | |
176.3.h.c | 2 | 11.b | odd | 2 | 1 | inner | |
198.3.d.b | 2 | 12.b | even | 2 | 1 | ||
198.3.d.b | 2 | 132.d | odd | 2 | 1 | ||
242.3.d.b | 8 | 44.g | even | 10 | 4 | ||
242.3.d.b | 8 | 44.h | odd | 10 | 4 | ||
550.3.c.a | 4 | 20.e | even | 4 | 2 | ||
550.3.c.a | 4 | 220.i | odd | 4 | 2 | ||
550.3.d.a | 2 | 20.d | odd | 2 | 1 | ||
550.3.d.a | 2 | 220.g | even | 2 | 1 | ||
704.3.h.d | 2 | 8.d | odd | 2 | 1 | ||
704.3.h.d | 2 | 88.g | even | 2 | 1 | ||
704.3.h.e | 2 | 8.b | even | 2 | 1 | ||
704.3.h.e | 2 | 88.b | odd | 2 | 1 | ||
1078.3.d.a | 2 | 28.d | even | 2 | 1 | ||
1078.3.d.a | 2 | 308.g | odd | 2 | 1 | ||
1584.3.j.d | 2 | 3.b | odd | 2 | 1 | ||
1584.3.j.d | 2 | 33.d | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} + 1 \)
acting on \(S_{3}^{\mathrm{new}}(176, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( (T + 1)^{2} \)
$5$
\( (T + 1)^{2} \)
$7$
\( T^{2} + 72 \)
$11$
\( T^{2} + 14T + 121 \)
$13$
\( T^{2} + 72 \)
$17$
\( T^{2} + 648 \)
$19$
\( T^{2} + 648 \)
$23$
\( (T + 17)^{2} \)
$29$
\( T^{2} + 1152 \)
$31$
\( (T + 17)^{2} \)
$37$
\( (T - 47)^{2} \)
$41$
\( T^{2} + 72 \)
$43$
\( T^{2} + 288 \)
$47$
\( (T - 58)^{2} \)
$53$
\( (T - 2)^{2} \)
$59$
\( (T - 55)^{2} \)
$61$
\( T^{2} + 7200 \)
$67$
\( (T + 89)^{2} \)
$71$
\( (T - 7)^{2} \)
$73$
\( T^{2} + 16200 \)
$79$
\( T^{2} + 1152 \)
$83$
\( T^{2} + 1152 \)
$89$
\( (T + 97)^{2} \)
$97$
\( (T + 121)^{2} \)
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