Newspace parameters
Level: | \( N \) | \(=\) | \( 289 = 17^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 289.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.30767661842\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-1}) \) |
Defining polynomial: |
\( x^{2} + 1 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | no (minimal twist has level 17) |
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 \(\left(\mathbb{Z}/289\mathbb{Z}\right)^\times\).
\(n\) | \(3\) |
\(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
288.1 |
|
1.00000 | 0 | −1.00000 | − | 2.00000i | 0 | − | 4.00000i | −3.00000 | 3.00000 | − | 2.00000i | |||||||||||||||||||||
288.2 | 1.00000 | 0 | −1.00000 | 2.00000i | 0 | 4.00000i | −3.00000 | 3.00000 | 2.00000i | |||||||||||||||||||||||||
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 | 289.2.b.a | 2 | |
17.b | even | 2 | 1 | inner | 289.2.b.a | 2 | |
17.c | even | 4 | 1 | 17.2.a.a | ✓ | 1 | |
17.c | even | 4 | 1 | 289.2.a.a | 1 | ||
17.d | even | 8 | 4 | 289.2.c.a | 4 | ||
17.e | odd | 16 | 8 | 289.2.d.d | 8 | ||
51.f | odd | 4 | 1 | 153.2.a.c | 1 | ||
51.f | odd | 4 | 1 | 2601.2.a.g | 1 | ||
68.f | odd | 4 | 1 | 272.2.a.b | 1 | ||
68.f | odd | 4 | 1 | 4624.2.a.d | 1 | ||
85.f | odd | 4 | 1 | 425.2.b.b | 2 | ||
85.i | odd | 4 | 1 | 425.2.b.b | 2 | ||
85.j | even | 4 | 1 | 425.2.a.d | 1 | ||
85.j | even | 4 | 1 | 7225.2.a.g | 1 | ||
119.f | odd | 4 | 1 | 833.2.a.a | 1 | ||
119.m | odd | 12 | 2 | 833.2.e.a | 2 | ||
119.n | even | 12 | 2 | 833.2.e.b | 2 | ||
136.i | even | 4 | 1 | 1088.2.a.i | 1 | ||
136.j | odd | 4 | 1 | 1088.2.a.h | 1 | ||
187.f | odd | 4 | 1 | 2057.2.a.e | 1 | ||
204.l | even | 4 | 1 | 2448.2.a.o | 1 | ||
221.k | even | 4 | 1 | 2873.2.a.c | 1 | ||
255.i | odd | 4 | 1 | 3825.2.a.d | 1 | ||
323.g | odd | 4 | 1 | 6137.2.a.b | 1 | ||
340.n | odd | 4 | 1 | 6800.2.a.n | 1 | ||
357.l | even | 4 | 1 | 7497.2.a.l | 1 | ||
391.f | odd | 4 | 1 | 8993.2.a.a | 1 | ||
408.q | even | 4 | 1 | 9792.2.a.i | 1 | ||
408.t | odd | 4 | 1 | 9792.2.a.n | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
17.2.a.a | ✓ | 1 | 17.c | even | 4 | 1 | |
153.2.a.c | 1 | 51.f | odd | 4 | 1 | ||
272.2.a.b | 1 | 68.f | odd | 4 | 1 | ||
289.2.a.a | 1 | 17.c | even | 4 | 1 | ||
289.2.b.a | 2 | 1.a | even | 1 | 1 | trivial | |
289.2.b.a | 2 | 17.b | even | 2 | 1 | inner | |
289.2.c.a | 4 | 17.d | even | 8 | 4 | ||
289.2.d.d | 8 | 17.e | odd | 16 | 8 | ||
425.2.a.d | 1 | 85.j | even | 4 | 1 | ||
425.2.b.b | 2 | 85.f | odd | 4 | 1 | ||
425.2.b.b | 2 | 85.i | odd | 4 | 1 | ||
833.2.a.a | 1 | 119.f | odd | 4 | 1 | ||
833.2.e.a | 2 | 119.m | odd | 12 | 2 | ||
833.2.e.b | 2 | 119.n | even | 12 | 2 | ||
1088.2.a.h | 1 | 136.j | odd | 4 | 1 | ||
1088.2.a.i | 1 | 136.i | even | 4 | 1 | ||
2057.2.a.e | 1 | 187.f | odd | 4 | 1 | ||
2448.2.a.o | 1 | 204.l | even | 4 | 1 | ||
2601.2.a.g | 1 | 51.f | odd | 4 | 1 | ||
2873.2.a.c | 1 | 221.k | even | 4 | 1 | ||
3825.2.a.d | 1 | 255.i | odd | 4 | 1 | ||
4624.2.a.d | 1 | 68.f | odd | 4 | 1 | ||
6137.2.a.b | 1 | 323.g | odd | 4 | 1 | ||
6800.2.a.n | 1 | 340.n | odd | 4 | 1 | ||
7225.2.a.g | 1 | 85.j | even | 4 | 1 | ||
7497.2.a.l | 1 | 357.l | even | 4 | 1 | ||
8993.2.a.a | 1 | 391.f | odd | 4 | 1 | ||
9792.2.a.i | 1 | 408.q | even | 4 | 1 | ||
9792.2.a.n | 1 | 408.t | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(289, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T - 1)^{2} \)
$3$
\( T^{2} \)
$5$
\( T^{2} + 4 \)
$7$
\( T^{2} + 16 \)
$11$
\( T^{2} \)
$13$
\( (T + 2)^{2} \)
$17$
\( T^{2} \)
$19$
\( (T - 4)^{2} \)
$23$
\( T^{2} + 16 \)
$29$
\( T^{2} + 36 \)
$31$
\( T^{2} + 16 \)
$37$
\( T^{2} + 4 \)
$41$
\( T^{2} + 36 \)
$43$
\( (T + 4)^{2} \)
$47$
\( T^{2} \)
$53$
\( (T + 6)^{2} \)
$59$
\( (T - 12)^{2} \)
$61$
\( T^{2} + 100 \)
$67$
\( (T - 4)^{2} \)
$71$
\( T^{2} + 16 \)
$73$
\( T^{2} + 36 \)
$79$
\( T^{2} + 144 \)
$83$
\( (T - 4)^{2} \)
$89$
\( (T - 10)^{2} \)
$97$
\( T^{2} + 4 \)
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