Newspace parameters
Level: | \( N \) | \(=\) | \( 400 = 2^{4} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 400.q (of order \(4\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.19401608085\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(i)\) |
Coefficient field: | \(\Q(i, \sqrt{11})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - 5x^{2} + 9 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \( x^{4} - 5x^{2} + 9 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{3} - 2\nu ) / 3 \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} + \nu - 3 \) |
\(\beta_{3}\) | \(=\) | \( -\nu^{2} + \nu + 3 \) |
\(\nu\) | \(=\) | \( ( \beta_{3} + \beta_{2} ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{3} + \beta_{2} + 6 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{3} + \beta_{2} + 3\beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(177\) | \(351\) |
\(\chi(n)\) | \(\beta_{1}\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
149.1 |
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−1.00000 | − | 1.00000i | −1.15831 | − | 1.15831i | 2.00000i | 0 | 2.31662i | −4.31662 | 2.00000 | − | 2.00000i | − | 0.316625i | 0 | |||||||||||||||||||||||
149.2 | −1.00000 | − | 1.00000i | 2.15831 | + | 2.15831i | 2.00000i | 0 | − | 4.31662i | 2.31662 | 2.00000 | − | 2.00000i | 6.31662i | 0 | ||||||||||||||||||||||||
349.1 | −1.00000 | + | 1.00000i | −1.15831 | + | 1.15831i | − | 2.00000i | 0 | − | 2.31662i | −4.31662 | 2.00000 | + | 2.00000i | 0.316625i | 0 | |||||||||||||||||||||||
349.2 | −1.00000 | + | 1.00000i | 2.15831 | − | 2.15831i | − | 2.00000i | 0 | 4.31662i | 2.31662 | 2.00000 | + | 2.00000i | − | 6.31662i | 0 | |||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
80.q | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 400.2.q.c | 4 | |
4.b | odd | 2 | 1 | 1600.2.q.c | 4 | ||
5.b | even | 2 | 1 | 400.2.q.d | 4 | ||
5.c | odd | 4 | 1 | 400.2.l.d | ✓ | 4 | |
5.c | odd | 4 | 1 | 400.2.l.e | yes | 4 | |
16.e | even | 4 | 1 | 400.2.q.d | 4 | ||
16.f | odd | 4 | 1 | 1600.2.q.d | 4 | ||
20.d | odd | 2 | 1 | 1600.2.q.d | 4 | ||
20.e | even | 4 | 1 | 1600.2.l.d | 4 | ||
20.e | even | 4 | 1 | 1600.2.l.e | 4 | ||
80.i | odd | 4 | 1 | 400.2.l.d | ✓ | 4 | |
80.j | even | 4 | 1 | 1600.2.l.d | 4 | ||
80.k | odd | 4 | 1 | 1600.2.q.c | 4 | ||
80.q | even | 4 | 1 | inner | 400.2.q.c | 4 | |
80.s | even | 4 | 1 | 1600.2.l.e | 4 | ||
80.t | odd | 4 | 1 | 400.2.l.e | yes | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
400.2.l.d | ✓ | 4 | 5.c | odd | 4 | 1 | |
400.2.l.d | ✓ | 4 | 80.i | odd | 4 | 1 | |
400.2.l.e | yes | 4 | 5.c | odd | 4 | 1 | |
400.2.l.e | yes | 4 | 80.t | odd | 4 | 1 | |
400.2.q.c | 4 | 1.a | even | 1 | 1 | trivial | |
400.2.q.c | 4 | 80.q | even | 4 | 1 | inner | |
400.2.q.d | 4 | 5.b | even | 2 | 1 | ||
400.2.q.d | 4 | 16.e | even | 4 | 1 | ||
1600.2.l.d | 4 | 20.e | even | 4 | 1 | ||
1600.2.l.d | 4 | 80.j | even | 4 | 1 | ||
1600.2.l.e | 4 | 20.e | even | 4 | 1 | ||
1600.2.l.e | 4 | 80.s | even | 4 | 1 | ||
1600.2.q.c | 4 | 4.b | odd | 2 | 1 | ||
1600.2.q.c | 4 | 80.k | odd | 4 | 1 | ||
1600.2.q.d | 4 | 16.f | odd | 4 | 1 | ||
1600.2.q.d | 4 | 20.d | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 2T_{3}^{3} + 2T_{3}^{2} + 10T_{3} + 25 \)
acting on \(S_{2}^{\mathrm{new}}(400, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 2 T + 2)^{2} \)
$3$
\( T^{4} - 2 T^{3} + 2 T^{2} + 10 T + 25 \)
$5$
\( T^{4} \)
$7$
\( (T^{2} + 2 T - 10)^{2} \)
$11$
\( T^{4} - 6 T^{3} + 18 T^{2} + 6 T + 1 \)
$13$
\( T^{4} + 4 T^{3} + 8 T^{2} - 80 T + 400 \)
$17$
\( T^{4} + 30T^{2} + 49 \)
$19$
\( T^{4} + 6 T^{3} + 18 T^{2} - 6 T + 1 \)
$23$
\( (T^{2} - 6 T - 2)^{2} \)
$29$
\( (T^{2} + 4 T + 8)^{2} \)
$31$
\( (T^{2} - 2 T - 10)^{2} \)
$37$
\( T^{4} - 16 T^{3} + 128 T^{2} + \cdots + 100 \)
$41$
\( (T^{2} + 25)^{2} \)
$43$
\( T^{4} - 4 T^{3} + 8 T^{2} + 344 T + 7396 \)
$47$
\( (T^{2} + 64)^{2} \)
$53$
\( T^{4} + 484 \)
$59$
\( T^{4} + 8 T^{3} + 32 T^{2} - 112 T + 196 \)
$61$
\( T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 4900 \)
$67$
\( T^{4} + 30 T^{3} + 450 T^{2} + \cdots + 11449 \)
$71$
\( T^{4} + 96T^{2} + 1600 \)
$73$
\( (T^{2} + 20 T + 89)^{2} \)
$79$
\( (T^{2} + 2 T - 10)^{2} \)
$83$
\( T^{4} + 22 T^{3} + 242 T^{2} + \cdots + 3025 \)
$89$
\( T^{4} + 270T^{2} + 3969 \)
$97$
\( (T^{2} + 44)^{2} \)
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