Newspace parameters
Level: | \( N \) | \(=\) | \( 400 = 2^{4} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 400.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(64.1535279252\) |
Analytic rank: | \(1\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{11}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} - 11 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | no (minimal twist has level 5) |
Fricke sign: | \(1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{11}\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
0 | −19.8997 | 0 | 0 | 0 | 59.6992 | 0 | 153.000 | 0 | ||||||||||||||||||||||||
1.2 | 0 | 19.8997 | 0 | 0 | 0 | −59.6992 | 0 | 153.000 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(5\) | \(-1\) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 400.6.a.t | 2 | |
4.b | odd | 2 | 1 | 25.6.a.c | 2 | ||
5.b | even | 2 | 1 | inner | 400.6.a.t | 2 | |
5.c | odd | 4 | 2 | 80.6.c.a | 2 | ||
12.b | even | 2 | 1 | 225.6.a.n | 2 | ||
15.e | even | 4 | 2 | 720.6.f.f | 2 | ||
20.d | odd | 2 | 1 | 25.6.a.c | 2 | ||
20.e | even | 4 | 2 | 5.6.b.a | ✓ | 2 | |
40.i | odd | 4 | 2 | 320.6.c.g | 2 | ||
40.k | even | 4 | 2 | 320.6.c.f | 2 | ||
60.h | even | 2 | 1 | 225.6.a.n | 2 | ||
60.l | odd | 4 | 2 | 45.6.b.b | 2 | ||
140.j | odd | 4 | 2 | 245.6.b.a | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
5.6.b.a | ✓ | 2 | 20.e | even | 4 | 2 | |
25.6.a.c | 2 | 4.b | odd | 2 | 1 | ||
25.6.a.c | 2 | 20.d | odd | 2 | 1 | ||
45.6.b.b | 2 | 60.l | odd | 4 | 2 | ||
80.6.c.a | 2 | 5.c | odd | 4 | 2 | ||
225.6.a.n | 2 | 12.b | even | 2 | 1 | ||
225.6.a.n | 2 | 60.h | even | 2 | 1 | ||
245.6.b.a | 2 | 140.j | odd | 4 | 2 | ||
320.6.c.f | 2 | 40.k | even | 4 | 2 | ||
320.6.c.g | 2 | 40.i | odd | 4 | 2 | ||
400.6.a.t | 2 | 1.a | even | 1 | 1 | trivial | |
400.6.a.t | 2 | 5.b | even | 2 | 1 | inner | |
720.6.f.f | 2 | 15.e | even | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 396 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(400))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} - 396 \)
$5$
\( T^{2} \)
$7$
\( T^{2} - 3564 \)
$11$
\( (T + 252)^{2} \)
$13$
\( T^{2} - 14256 \)
$17$
\( T^{2} - 475904 \)
$19$
\( (T + 220)^{2} \)
$23$
\( T^{2} - 5926316 \)
$29$
\( (T - 6930)^{2} \)
$31$
\( (T + 6752)^{2} \)
$37$
\( T^{2} - 195150384 \)
$41$
\( (T + 198)^{2} \)
$43$
\( T^{2} - 174636 \)
$47$
\( T^{2} - 111096524 \)
$53$
\( T^{2} - 33918896 \)
$59$
\( (T + 24660)^{2} \)
$61$
\( (T + 5698)^{2} \)
$67$
\( T^{2} - 1904462604 \)
$71$
\( (T + 53352)^{2} \)
$73$
\( T^{2} - 5030030016 \)
$79$
\( (T - 51920)^{2} \)
$83$
\( T^{2} - 3824406476 \)
$89$
\( (T - 9990)^{2} \)
$97$
\( T^{2} - 10251546624 \)
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