Newspace parameters
Level: | \( N \) | \(=\) | \( 845 = 5 \cdot 13^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 845.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(6.74735897080\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | 4.4.4752.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - 2x^{3} - 3x^{2} + 4x + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 65) |
Fricke sign: | \(-1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 3x^{2} + 4x + 1 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} - \nu - 2 \) |
\(\beta_{3}\) | \(=\) | \( \nu^{3} - \nu^{2} - 3\nu + 1 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} + \beta _1 + 2 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 |
|
−1.49551 | 0.0947876 | 0.236543 | −1.00000 | −0.141756 | 4.82684 | 2.63726 | −2.99102 | 1.49551 | ||||||||||||||||||||||||||||||
1.2 | −0.219687 | −1.60020 | −1.95174 | −1.00000 | 0.351542 | −0.332247 | 0.868145 | −0.439374 | 0.219687 | |||||||||||||||||||||||||||||||
1.3 | 1.21969 | 2.33225 | −0.512364 | −1.00000 | 2.84461 | 3.60020 | −3.06430 | 2.43937 | −1.21969 | |||||||||||||||||||||||||||||||
1.4 | 2.49551 | −2.82684 | 4.22756 | −1.00000 | −7.05440 | 1.90521 | 5.55889 | 4.99102 | −2.49551 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(5\) | \(1\) |
\(13\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 845.2.a.m | 4 | |
3.b | odd | 2 | 1 | 7605.2.a.cf | 4 | ||
5.b | even | 2 | 1 | 4225.2.a.bi | 4 | ||
13.b | even | 2 | 1 | 845.2.a.l | 4 | ||
13.c | even | 3 | 2 | 845.2.e.m | 8 | ||
13.d | odd | 4 | 2 | 845.2.c.g | 8 | ||
13.e | even | 6 | 2 | 845.2.e.n | 8 | ||
13.f | odd | 12 | 2 | 65.2.m.a | ✓ | 8 | |
13.f | odd | 12 | 2 | 845.2.m.g | 8 | ||
39.d | odd | 2 | 1 | 7605.2.a.cj | 4 | ||
39.k | even | 12 | 2 | 585.2.bu.c | 8 | ||
52.l | even | 12 | 2 | 1040.2.da.b | 8 | ||
65.d | even | 2 | 1 | 4225.2.a.bl | 4 | ||
65.o | even | 12 | 2 | 325.2.m.b | 8 | ||
65.s | odd | 12 | 2 | 325.2.n.d | 8 | ||
65.t | even | 12 | 2 | 325.2.m.c | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
65.2.m.a | ✓ | 8 | 13.f | odd | 12 | 2 | |
325.2.m.b | 8 | 65.o | even | 12 | 2 | ||
325.2.m.c | 8 | 65.t | even | 12 | 2 | ||
325.2.n.d | 8 | 65.s | odd | 12 | 2 | ||
585.2.bu.c | 8 | 39.k | even | 12 | 2 | ||
845.2.a.l | 4 | 13.b | even | 2 | 1 | ||
845.2.a.m | 4 | 1.a | even | 1 | 1 | trivial | |
845.2.c.g | 8 | 13.d | odd | 4 | 2 | ||
845.2.e.m | 8 | 13.c | even | 3 | 2 | ||
845.2.e.n | 8 | 13.e | even | 6 | 2 | ||
845.2.m.g | 8 | 13.f | odd | 12 | 2 | ||
1040.2.da.b | 8 | 52.l | even | 12 | 2 | ||
4225.2.a.bi | 4 | 5.b | even | 2 | 1 | ||
4225.2.a.bl | 4 | 65.d | even | 2 | 1 | ||
7605.2.a.cf | 4 | 3.b | odd | 2 | 1 | ||
7605.2.a.cj | 4 | 39.d | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 2T_{2}^{3} - 3T_{2}^{2} + 4T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(845))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 2 T^{3} - 3 T^{2} + 4 T + 1 \)
$3$
\( T^{4} + 2 T^{3} - 6 T^{2} - 10 T + 1 \)
$5$
\( (T + 1)^{4} \)
$7$
\( T^{4} - 10 T^{3} + 30 T^{2} - 22 T - 11 \)
$11$
\( T^{4} - 30T^{2} + 33 \)
$13$
\( T^{4} \)
$17$
\( T^{4} + 2 T^{3} - 18 T^{2} - 10 T + 13 \)
$19$
\( (T^{2} - 8 T + 13)^{2} \)
$23$
\( T^{4} + 10 T^{3} + 6 T^{2} - 146 T - 299 \)
$29$
\( T^{4} - 8 T^{3} - 18 T^{2} + 40 T + 1 \)
$31$
\( (T^{2} - 4 T - 8)^{2} \)
$37$
\( T^{4} + 2 T^{3} - 54 T^{2} + 38 T + 1 \)
$41$
\( (T^{2} - 4 T + 1)^{2} \)
$43$
\( T^{4} + 2 T^{3} - 18 T^{2} - 10 T + 13 \)
$47$
\( T^{4} - 8 T^{3} - 72 T^{2} + \cdots - 1328 \)
$53$
\( T^{4} + 12 T^{3} + 36 T^{2} - 48 \)
$59$
\( T^{4} - 12 T^{3} + 30 T^{2} - 12 T - 3 \)
$61$
\( T^{4} - 28 T^{3} + 258 T^{2} + \cdots + 1261 \)
$67$
\( T^{4} - 30 T^{3} + 330 T^{2} + \cdots + 2769 \)
$71$
\( T^{4} - 4 T^{3} - 210 T^{2} + \cdots + 10477 \)
$73$
\( T^{4} + 8 T^{3} - 84 T^{2} + \cdots - 1712 \)
$79$
\( T^{4} + 8 T^{3} - 132 T^{2} + \cdots + 4432 \)
$83$
\( T^{4} + 12 T^{3} - 24 T^{2} + \cdots - 192 \)
$89$
\( T^{4} + 12 T^{3} - 234 T^{2} + \cdots + 8853 \)
$97$
\( T^{4} - 2 T^{3} - 90 T^{2} - 374 T - 443 \)
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