Newspace parameters
Level: | \( N \) | \(=\) | \( 800 = 2^{5} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 800.ba (of order \(8\), degree \(4\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.38803216170\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\zeta_{8})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
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Defining polynomial: | \( x^{4} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 32) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(351\) | \(577\) |
\(\chi(n)\) | \(\zeta_{8}\) | \(1\) | \(-1\) |
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 |
|
1.41421 | −0.292893 | − | 0.707107i | 2.00000 | 0 | −0.414214 | − | 1.00000i | 1.00000 | − | 1.00000i | 2.82843 | 1.70711 | − | 1.70711i | 0 | ||||||||||||||||||||||
349.1 | 1.41421 | −0.292893 | + | 0.707107i | 2.00000 | 0 | −0.414214 | + | 1.00000i | 1.00000 | + | 1.00000i | 2.82843 | 1.70711 | + | 1.70711i | 0 | |||||||||||||||||||||||
549.1 | −1.41421 | −1.70711 | + | 0.707107i | 2.00000 | 0 | 2.41421 | − | 1.00000i | 1.00000 | − | 1.00000i | −2.82843 | 0.292893 | − | 0.292893i | 0 | |||||||||||||||||||||||
749.1 | −1.41421 | −1.70711 | − | 0.707107i | 2.00000 | 0 | 2.41421 | + | 1.00000i | 1.00000 | + | 1.00000i | −2.82843 | 0.292893 | + | 0.292893i | 0 | |||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
160.z | even | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 800.2.ba.a | 4 | |
5.b | even | 2 | 1 | 800.2.ba.b | 4 | ||
5.c | odd | 4 | 1 | 32.2.g.a | ✓ | 4 | |
5.c | odd | 4 | 1 | 800.2.y.a | 4 | ||
15.e | even | 4 | 1 | 288.2.v.a | 4 | ||
20.e | even | 4 | 1 | 128.2.g.a | 4 | ||
32.g | even | 8 | 1 | 800.2.ba.b | 4 | ||
40.i | odd | 4 | 1 | 256.2.g.b | 4 | ||
40.k | even | 4 | 1 | 256.2.g.a | 4 | ||
60.l | odd | 4 | 1 | 1152.2.v.a | 4 | ||
80.i | odd | 4 | 1 | 512.2.g.d | 4 | ||
80.j | even | 4 | 1 | 512.2.g.c | 4 | ||
80.s | even | 4 | 1 | 512.2.g.b | 4 | ||
80.t | odd | 4 | 1 | 512.2.g.a | 4 | ||
160.u | even | 8 | 1 | 128.2.g.a | 4 | ||
160.u | even | 8 | 1 | 256.2.g.a | 4 | ||
160.v | odd | 8 | 1 | 512.2.g.a | 4 | ||
160.v | odd | 8 | 1 | 512.2.g.d | 4 | ||
160.v | odd | 8 | 1 | 800.2.y.a | 4 | ||
160.z | even | 8 | 1 | inner | 800.2.ba.a | 4 | |
160.ba | even | 8 | 1 | 512.2.g.b | 4 | ||
160.ba | even | 8 | 1 | 512.2.g.c | 4 | ||
160.bb | odd | 8 | 1 | 32.2.g.a | ✓ | 4 | |
160.bb | odd | 8 | 1 | 256.2.g.b | 4 | ||
320.bc | odd | 16 | 2 | 4096.2.a.e | 4 | ||
320.bd | even | 16 | 2 | 4096.2.a.f | 4 | ||
480.bq | odd | 8 | 1 | 1152.2.v.a | 4 | ||
480.cb | even | 8 | 1 | 288.2.v.a | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
32.2.g.a | ✓ | 4 | 5.c | odd | 4 | 1 | |
32.2.g.a | ✓ | 4 | 160.bb | odd | 8 | 1 | |
128.2.g.a | 4 | 20.e | even | 4 | 1 | ||
128.2.g.a | 4 | 160.u | even | 8 | 1 | ||
256.2.g.a | 4 | 40.k | even | 4 | 1 | ||
256.2.g.a | 4 | 160.u | even | 8 | 1 | ||
256.2.g.b | 4 | 40.i | odd | 4 | 1 | ||
256.2.g.b | 4 | 160.bb | odd | 8 | 1 | ||
288.2.v.a | 4 | 15.e | even | 4 | 1 | ||
288.2.v.a | 4 | 480.cb | even | 8 | 1 | ||
512.2.g.a | 4 | 80.t | odd | 4 | 1 | ||
512.2.g.a | 4 | 160.v | odd | 8 | 1 | ||
512.2.g.b | 4 | 80.s | even | 4 | 1 | ||
512.2.g.b | 4 | 160.ba | even | 8 | 1 | ||
512.2.g.c | 4 | 80.j | even | 4 | 1 | ||
512.2.g.c | 4 | 160.ba | even | 8 | 1 | ||
512.2.g.d | 4 | 80.i | odd | 4 | 1 | ||
512.2.g.d | 4 | 160.v | odd | 8 | 1 | ||
800.2.y.a | 4 | 5.c | odd | 4 | 1 | ||
800.2.y.a | 4 | 160.v | odd | 8 | 1 | ||
800.2.ba.a | 4 | 1.a | even | 1 | 1 | trivial | |
800.2.ba.a | 4 | 160.z | even | 8 | 1 | inner | |
800.2.ba.b | 4 | 5.b | even | 2 | 1 | ||
800.2.ba.b | 4 | 32.g | even | 8 | 1 | ||
1152.2.v.a | 4 | 60.l | odd | 4 | 1 | ||
1152.2.v.a | 4 | 480.bq | odd | 8 | 1 | ||
4096.2.a.e | 4 | 320.bc | odd | 16 | 2 | ||
4096.2.a.f | 4 | 320.bd | even | 16 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 4T_{3}^{3} + 6T_{3}^{2} + 4T_{3} + 2 \)
acting on \(S_{2}^{\mathrm{new}}(800, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} - 2)^{2} \)
$3$
\( T^{4} + 4 T^{3} + 6 T^{2} + 4 T + 2 \)
$5$
\( T^{4} \)
$7$
\( (T^{2} - 2 T + 2)^{2} \)
$11$
\( T^{4} + 8 T^{3} + 18 T^{2} - 4 T + 2 \)
$13$
\( T^{4} + 2 T^{2} + 4 T + 2 \)
$17$
\( (T^{2} - 8)^{2} \)
$19$
\( T^{4} - 8 T^{3} + 18 T^{2} - 68 T + 578 \)
$23$
\( T^{4} - 12 T^{3} + 72 T^{2} - 24 T + 4 \)
$29$
\( T^{4} - 4 T^{3} + 6 T^{2} - 28 T + 98 \)
$31$
\( (T + 4)^{4} \)
$37$
\( T^{4} + 2 T^{2} + 4 T + 2 \)
$41$
\( T^{4} + 12 T^{3} + 72 T^{2} + 24 T + 4 \)
$43$
\( T^{4} - 12 T^{3} + 38 T^{2} + \cdots + 1922 \)
$47$
\( (T^{2} - 12 T + 4)^{2} \)
$53$
\( T^{4} - 16 T^{3} + 82 T^{2} - 84 T + 98 \)
$59$
\( T^{4} - 16 T^{3} + 114 T^{2} + \cdots + 1058 \)
$61$
\( T^{4} - 4 T^{3} + 6 T^{2} - 4 T + 2 \)
$67$
\( T^{4} - 12 T^{3} + 86 T^{2} + \cdots + 578 \)
$71$
\( T^{4} + 12 T^{3} + 72 T^{2} + 24 T + 4 \)
$73$
\( (T^{2} - 14 T + 98)^{2} \)
$79$
\( (T^{2} + 36)^{2} \)
$83$
\( T^{4} + 4 T^{3} + 22 T^{2} + \cdots + 1058 \)
$89$
\( T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 2116 \)
$97$
\( T^{4} + 344T^{2} + 784 \)
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