Newspace parameters
Level: | \( N \) | \(=\) | \( 170 = 2 \cdot 5 \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 170.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.35745683436\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-1}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
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Defining polynomial: | \( x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/170\mathbb{Z}\right)^\times\).
\(n\) | \(71\) | \(137\) |
\(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
101.1 |
|
1.00000 | − | 3.00000i | 1.00000 | − | 1.00000i | − | 3.00000i | 4.00000i | 1.00000 | −6.00000 | − | 1.00000i | ||||||||||||||||||||
101.2 | 1.00000 | 3.00000i | 1.00000 | 1.00000i | 3.00000i | − | 4.00000i | 1.00000 | −6.00000 | 1.00000i | ||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
17.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 170.2.b.b | ✓ | 2 |
3.b | odd | 2 | 1 | 1530.2.c.b | 2 | ||
4.b | odd | 2 | 1 | 1360.2.c.a | 2 | ||
5.b | even | 2 | 1 | 850.2.b.a | 2 | ||
5.c | odd | 4 | 1 | 850.2.d.a | 2 | ||
5.c | odd | 4 | 1 | 850.2.d.h | 2 | ||
17.b | even | 2 | 1 | inner | 170.2.b.b | ✓ | 2 |
17.c | even | 4 | 1 | 2890.2.a.a | 1 | ||
17.c | even | 4 | 1 | 2890.2.a.l | 1 | ||
51.c | odd | 2 | 1 | 1530.2.c.b | 2 | ||
68.d | odd | 2 | 1 | 1360.2.c.a | 2 | ||
85.c | even | 2 | 1 | 850.2.b.a | 2 | ||
85.g | odd | 4 | 1 | 850.2.d.a | 2 | ||
85.g | odd | 4 | 1 | 850.2.d.h | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
170.2.b.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
170.2.b.b | ✓ | 2 | 17.b | even | 2 | 1 | inner |
850.2.b.a | 2 | 5.b | even | 2 | 1 | ||
850.2.b.a | 2 | 85.c | even | 2 | 1 | ||
850.2.d.a | 2 | 5.c | odd | 4 | 1 | ||
850.2.d.a | 2 | 85.g | odd | 4 | 1 | ||
850.2.d.h | 2 | 5.c | odd | 4 | 1 | ||
850.2.d.h | 2 | 85.g | odd | 4 | 1 | ||
1360.2.c.a | 2 | 4.b | odd | 2 | 1 | ||
1360.2.c.a | 2 | 68.d | odd | 2 | 1 | ||
1530.2.c.b | 2 | 3.b | odd | 2 | 1 | ||
1530.2.c.b | 2 | 51.c | odd | 2 | 1 | ||
2890.2.a.a | 1 | 17.c | even | 4 | 1 | ||
2890.2.a.l | 1 | 17.c | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(170, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T - 1)^{2} \)
$3$
\( T^{2} + 9 \)
$5$
\( T^{2} + 1 \)
$7$
\( T^{2} + 16 \)
$11$
\( T^{2} + 4 \)
$13$
\( (T - 1)^{2} \)
$17$
\( T^{2} + 8T + 17 \)
$19$
\( (T - 7)^{2} \)
$23$
\( T^{2} + 36 \)
$29$
\( T^{2} + 9 \)
$31$
\( T^{2} + 49 \)
$37$
\( T^{2} + 4 \)
$41$
\( T^{2} + 64 \)
$43$
\( (T + 8)^{2} \)
$47$
\( (T - 9)^{2} \)
$53$
\( (T + 11)^{2} \)
$59$
\( (T + 5)^{2} \)
$61$
\( T^{2} + 1 \)
$67$
\( (T + 10)^{2} \)
$71$
\( T^{2} + 1 \)
$73$
\( T^{2} + 81 \)
$79$
\( T^{2} \)
$83$
\( (T - 6)^{2} \)
$89$
\( (T + 1)^{2} \)
$97$
\( T^{2} + 1 \)
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