Newspace parameters
Level: | \( N \) | \(=\) | \( 80 = 2^{4} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 80.q (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.638803216170\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-1}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
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Defining polynomial: | \( x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).
\(n\) | \(17\) | \(21\) | \(31\) |
\(\chi(n)\) | \(-1\) | \(i\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
29.1 |
|
−1.00000 | + | 1.00000i | 1.00000 | − | 1.00000i | − | 2.00000i | 1.00000 | − | 2.00000i | 2.00000i | 0 | 2.00000 | + | 2.00000i | 1.00000i | 1.00000 | + | 3.00000i | |||||||||||||
69.1 | −1.00000 | − | 1.00000i | 1.00000 | + | 1.00000i | 2.00000i | 1.00000 | + | 2.00000i | − | 2.00000i | 0 | 2.00000 | − | 2.00000i | − | 1.00000i | 1.00000 | − | 3.00000i | |||||||||||||
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 | 80.2.q.a | ✓ | 2 |
3.b | odd | 2 | 1 | 720.2.bm.b | 2 | ||
4.b | odd | 2 | 1 | 320.2.q.a | 2 | ||
5.b | even | 2 | 1 | 80.2.q.b | yes | 2 | |
5.c | odd | 4 | 1 | 400.2.l.a | 2 | ||
5.c | odd | 4 | 1 | 400.2.l.b | 2 | ||
8.b | even | 2 | 1 | 640.2.q.b | 2 | ||
8.d | odd | 2 | 1 | 640.2.q.d | 2 | ||
15.d | odd | 2 | 1 | 720.2.bm.a | 2 | ||
16.e | even | 4 | 1 | 80.2.q.b | yes | 2 | |
16.e | even | 4 | 1 | 640.2.q.c | 2 | ||
16.f | odd | 4 | 1 | 320.2.q.b | 2 | ||
16.f | odd | 4 | 1 | 640.2.q.a | 2 | ||
20.d | odd | 2 | 1 | 320.2.q.b | 2 | ||
20.e | even | 4 | 1 | 1600.2.l.b | 2 | ||
20.e | even | 4 | 1 | 1600.2.l.c | 2 | ||
40.e | odd | 2 | 1 | 640.2.q.a | 2 | ||
40.f | even | 2 | 1 | 640.2.q.c | 2 | ||
48.i | odd | 4 | 1 | 720.2.bm.a | 2 | ||
80.i | odd | 4 | 1 | 400.2.l.a | 2 | ||
80.j | even | 4 | 1 | 1600.2.l.b | 2 | ||
80.k | odd | 4 | 1 | 320.2.q.a | 2 | ||
80.k | odd | 4 | 1 | 640.2.q.d | 2 | ||
80.q | even | 4 | 1 | inner | 80.2.q.a | ✓ | 2 |
80.q | even | 4 | 1 | 640.2.q.b | 2 | ||
80.s | even | 4 | 1 | 1600.2.l.c | 2 | ||
80.t | odd | 4 | 1 | 400.2.l.b | 2 | ||
240.bm | odd | 4 | 1 | 720.2.bm.b | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
80.2.q.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
80.2.q.a | ✓ | 2 | 80.q | even | 4 | 1 | inner |
80.2.q.b | yes | 2 | 5.b | even | 2 | 1 | |
80.2.q.b | yes | 2 | 16.e | even | 4 | 1 | |
320.2.q.a | 2 | 4.b | odd | 2 | 1 | ||
320.2.q.a | 2 | 80.k | odd | 4 | 1 | ||
320.2.q.b | 2 | 16.f | odd | 4 | 1 | ||
320.2.q.b | 2 | 20.d | odd | 2 | 1 | ||
400.2.l.a | 2 | 5.c | odd | 4 | 1 | ||
400.2.l.a | 2 | 80.i | odd | 4 | 1 | ||
400.2.l.b | 2 | 5.c | odd | 4 | 1 | ||
400.2.l.b | 2 | 80.t | odd | 4 | 1 | ||
640.2.q.a | 2 | 16.f | odd | 4 | 1 | ||
640.2.q.a | 2 | 40.e | odd | 2 | 1 | ||
640.2.q.b | 2 | 8.b | even | 2 | 1 | ||
640.2.q.b | 2 | 80.q | even | 4 | 1 | ||
640.2.q.c | 2 | 16.e | even | 4 | 1 | ||
640.2.q.c | 2 | 40.f | even | 2 | 1 | ||
640.2.q.d | 2 | 8.d | odd | 2 | 1 | ||
640.2.q.d | 2 | 80.k | odd | 4 | 1 | ||
720.2.bm.a | 2 | 15.d | odd | 2 | 1 | ||
720.2.bm.a | 2 | 48.i | odd | 4 | 1 | ||
720.2.bm.b | 2 | 3.b | odd | 2 | 1 | ||
720.2.bm.b | 2 | 240.bm | odd | 4 | 1 | ||
1600.2.l.b | 2 | 20.e | even | 4 | 1 | ||
1600.2.l.b | 2 | 80.j | even | 4 | 1 | ||
1600.2.l.c | 2 | 20.e | even | 4 | 1 | ||
1600.2.l.c | 2 | 80.s | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 2T_{3} + 2 \)
acting on \(S_{2}^{\mathrm{new}}(80, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 2T + 2 \)
$3$
\( T^{2} - 2T + 2 \)
$5$
\( T^{2} - 2T + 5 \)
$7$
\( T^{2} \)
$11$
\( T^{2} + 6T + 18 \)
$13$
\( T^{2} - 6T + 18 \)
$17$
\( T^{2} + 16 \)
$19$
\( T^{2} + 2T + 2 \)
$23$
\( (T + 8)^{2} \)
$29$
\( T^{2} - 6T + 18 \)
$31$
\( T^{2} \)
$37$
\( T^{2} - 6T + 18 \)
$41$
\( T^{2} \)
$43$
\( T^{2} - 6T + 18 \)
$47$
\( T^{2} + 4 \)
$53$
\( T^{2} + 18T + 162 \)
$59$
\( T^{2} - 18T + 162 \)
$61$
\( T^{2} + 10T + 50 \)
$67$
\( T^{2} + 6T + 18 \)
$71$
\( T^{2} + 36 \)
$73$
\( (T - 6)^{2} \)
$79$
\( (T - 8)^{2} \)
$83$
\( T^{2} - 18T + 162 \)
$89$
\( T^{2} + 144 \)
$97$
\( T^{2} + 144 \)
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