Newspace parameters
Level: | \( N \) | \(=\) | \( 261 = 3^{2} \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 261.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.08409549276\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-5}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{2} + 5 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 29) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/261\mathbb{Z}\right)^\times\).
\(n\) | \(118\) | \(146\) |
\(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
28.1 |
|
− | 2.23607i | 0 | −3.00000 | 3.00000 | 0 | 2.00000 | 2.23607i | 0 | − | 6.70820i | ||||||||||||||||||||||
28.2 | 2.23607i | 0 | −3.00000 | 3.00000 | 0 | 2.00000 | − | 2.23607i | 0 | 6.70820i | ||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
29.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 261.2.c.a | 2 | |
3.b | odd | 2 | 1 | 29.2.b.a | ✓ | 2 | |
4.b | odd | 2 | 1 | 4176.2.o.k | 2 | ||
12.b | even | 2 | 1 | 464.2.e.a | 2 | ||
15.d | odd | 2 | 1 | 725.2.c.c | 2 | ||
15.e | even | 4 | 2 | 725.2.d.a | 4 | ||
21.c | even | 2 | 1 | 1421.2.b.b | 2 | ||
24.f | even | 2 | 1 | 1856.2.e.f | 2 | ||
24.h | odd | 2 | 1 | 1856.2.e.g | 2 | ||
29.b | even | 2 | 1 | inner | 261.2.c.a | 2 | |
29.c | odd | 4 | 2 | 7569.2.a.i | 2 | ||
87.d | odd | 2 | 1 | 29.2.b.a | ✓ | 2 | |
87.f | even | 4 | 2 | 841.2.a.b | 2 | ||
87.h | odd | 14 | 6 | 841.2.e.g | 12 | ||
87.j | odd | 14 | 6 | 841.2.e.g | 12 | ||
87.k | even | 28 | 12 | 841.2.d.h | 12 | ||
116.d | odd | 2 | 1 | 4176.2.o.k | 2 | ||
348.b | even | 2 | 1 | 464.2.e.a | 2 | ||
435.b | odd | 2 | 1 | 725.2.c.c | 2 | ||
435.p | even | 4 | 2 | 725.2.d.a | 4 | ||
609.h | even | 2 | 1 | 1421.2.b.b | 2 | ||
696.l | even | 2 | 1 | 1856.2.e.f | 2 | ||
696.n | odd | 2 | 1 | 1856.2.e.g | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
29.2.b.a | ✓ | 2 | 3.b | odd | 2 | 1 | |
29.2.b.a | ✓ | 2 | 87.d | odd | 2 | 1 | |
261.2.c.a | 2 | 1.a | even | 1 | 1 | trivial | |
261.2.c.a | 2 | 29.b | even | 2 | 1 | inner | |
464.2.e.a | 2 | 12.b | even | 2 | 1 | ||
464.2.e.a | 2 | 348.b | even | 2 | 1 | ||
725.2.c.c | 2 | 15.d | odd | 2 | 1 | ||
725.2.c.c | 2 | 435.b | odd | 2 | 1 | ||
725.2.d.a | 4 | 15.e | even | 4 | 2 | ||
725.2.d.a | 4 | 435.p | even | 4 | 2 | ||
841.2.a.b | 2 | 87.f | even | 4 | 2 | ||
841.2.d.h | 12 | 87.k | even | 28 | 12 | ||
841.2.e.g | 12 | 87.h | odd | 14 | 6 | ||
841.2.e.g | 12 | 87.j | odd | 14 | 6 | ||
1421.2.b.b | 2 | 21.c | even | 2 | 1 | ||
1421.2.b.b | 2 | 609.h | even | 2 | 1 | ||
1856.2.e.f | 2 | 24.f | even | 2 | 1 | ||
1856.2.e.f | 2 | 696.l | even | 2 | 1 | ||
1856.2.e.g | 2 | 24.h | odd | 2 | 1 | ||
1856.2.e.g | 2 | 696.n | odd | 2 | 1 | ||
4176.2.o.k | 2 | 4.b | odd | 2 | 1 | ||
4176.2.o.k | 2 | 116.d | odd | 2 | 1 | ||
7569.2.a.i | 2 | 29.c | odd | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 5 \)
acting on \(S_{2}^{\mathrm{new}}(261, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 5 \)
$3$
\( T^{2} \)
$5$
\( (T - 3)^{2} \)
$7$
\( (T - 2)^{2} \)
$11$
\( T^{2} + 5 \)
$13$
\( (T + 1)^{2} \)
$17$
\( T^{2} + 20 \)
$19$
\( T^{2} \)
$23$
\( (T + 6)^{2} \)
$29$
\( T^{2} - 6T + 29 \)
$31$
\( T^{2} + 45 \)
$37$
\( T^{2} \)
$41$
\( T^{2} + 20 \)
$43$
\( T^{2} + 45 \)
$47$
\( T^{2} + 5 \)
$53$
\( (T - 9)^{2} \)
$59$
\( (T + 6)^{2} \)
$61$
\( T^{2} + 180 \)
$67$
\( (T - 8)^{2} \)
$71$
\( T^{2} \)
$73$
\( T^{2} \)
$79$
\( T^{2} + 45 \)
$83$
\( (T - 6)^{2} \)
$89$
\( T^{2} + 20 \)
$97$
\( T^{2} + 180 \)
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