Newspace parameters
Level: | \( N \) | \(=\) | \( 400 = 2^{4} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 400.c (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(64.1535279252\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-1}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | no (minimal twist has level 10) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(177\) | \(351\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
49.1 |
|
0 | − | 6.00000i | 0 | 0 | 0 | − | 118.000i | 0 | 207.000 | 0 | ||||||||||||||||||||||
49.2 | 0 | 6.00000i | 0 | 0 | 0 | 118.000i | 0 | 207.000 | 0 | |||||||||||||||||||||||||
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.c.i | 2 | |
4.b | odd | 2 | 1 | 50.6.b.b | 2 | ||
5.b | even | 2 | 1 | inner | 400.6.c.i | 2 | |
5.c | odd | 4 | 1 | 80.6.a.c | 1 | ||
5.c | odd | 4 | 1 | 400.6.a.i | 1 | ||
12.b | even | 2 | 1 | 450.6.c.f | 2 | ||
15.e | even | 4 | 1 | 720.6.a.v | 1 | ||
20.d | odd | 2 | 1 | 50.6.b.b | 2 | ||
20.e | even | 4 | 1 | 10.6.a.c | ✓ | 1 | |
20.e | even | 4 | 1 | 50.6.a.b | 1 | ||
40.i | odd | 4 | 1 | 320.6.a.k | 1 | ||
40.k | even | 4 | 1 | 320.6.a.f | 1 | ||
60.h | even | 2 | 1 | 450.6.c.f | 2 | ||
60.l | odd | 4 | 1 | 90.6.a.b | 1 | ||
60.l | odd | 4 | 1 | 450.6.a.u | 1 | ||
140.j | odd | 4 | 1 | 490.6.a.k | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
10.6.a.c | ✓ | 1 | 20.e | even | 4 | 1 | |
50.6.a.b | 1 | 20.e | even | 4 | 1 | ||
50.6.b.b | 2 | 4.b | odd | 2 | 1 | ||
50.6.b.b | 2 | 20.d | odd | 2 | 1 | ||
80.6.a.c | 1 | 5.c | odd | 4 | 1 | ||
90.6.a.b | 1 | 60.l | odd | 4 | 1 | ||
320.6.a.f | 1 | 40.k | even | 4 | 1 | ||
320.6.a.k | 1 | 40.i | odd | 4 | 1 | ||
400.6.a.i | 1 | 5.c | odd | 4 | 1 | ||
400.6.c.i | 2 | 1.a | even | 1 | 1 | trivial | |
400.6.c.i | 2 | 5.b | even | 2 | 1 | inner | |
450.6.a.u | 1 | 60.l | odd | 4 | 1 | ||
450.6.c.f | 2 | 12.b | even | 2 | 1 | ||
450.6.c.f | 2 | 60.h | even | 2 | 1 | ||
490.6.a.k | 1 | 140.j | odd | 4 | 1 | ||
720.6.a.v | 1 | 15.e | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 36 \)
acting on \(S_{6}^{\mathrm{new}}(400, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} + 36 \)
$5$
\( T^{2} \)
$7$
\( T^{2} + 13924 \)
$11$
\( (T + 192)^{2} \)
$13$
\( T^{2} + 1223236 \)
$17$
\( T^{2} + 580644 \)
$19$
\( (T + 2740)^{2} \)
$23$
\( T^{2} + 2452356 \)
$29$
\( (T + 5910)^{2} \)
$31$
\( (T - 6868)^{2} \)
$37$
\( T^{2} + 30448324 \)
$41$
\( (T + 378)^{2} \)
$43$
\( T^{2} + 5924356 \)
$47$
\( T^{2} + 172186884 \)
$53$
\( T^{2} + 84162276 \)
$59$
\( (T + 34980)^{2} \)
$61$
\( (T + 9838)^{2} \)
$67$
\( T^{2} + 1137173284 \)
$71$
\( (T + 70212)^{2} \)
$73$
\( T^{2} + 483384196 \)
$79$
\( (T - 4520)^{2} \)
$83$
\( T^{2} + 11897137476 \)
$89$
\( (T + 38490)^{2} \)
$97$
\( T^{2} + 3678724 \)
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