Newspace parameters
Level: | \( N \) | \(=\) | \( 40 = 2^{3} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 40.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(12.4954010194\) |
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: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | yes |
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 \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).
\(n\) | \(17\) | \(21\) | \(31\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
9.1 |
|
0 | − | 34.0000i | 0 | 275.000 | − | 50.0000i | 0 | 106.000i | 0 | 1031.00 | 0 | |||||||||||||||||||||
9.2 | 0 | 34.0000i | 0 | 275.000 | + | 50.0000i | 0 | − | 106.000i | 0 | 1031.00 | 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 | 40.8.c.a | ✓ | 2 |
3.b | odd | 2 | 1 | 360.8.f.a | 2 | ||
4.b | odd | 2 | 1 | 80.8.c.b | 2 | ||
5.b | even | 2 | 1 | inner | 40.8.c.a | ✓ | 2 |
5.c | odd | 4 | 1 | 200.8.a.c | 1 | ||
5.c | odd | 4 | 1 | 200.8.a.f | 1 | ||
8.b | even | 2 | 1 | 320.8.c.b | 2 | ||
8.d | odd | 2 | 1 | 320.8.c.a | 2 | ||
15.d | odd | 2 | 1 | 360.8.f.a | 2 | ||
20.d | odd | 2 | 1 | 80.8.c.b | 2 | ||
20.e | even | 4 | 1 | 400.8.a.h | 1 | ||
20.e | even | 4 | 1 | 400.8.a.n | 1 | ||
40.e | odd | 2 | 1 | 320.8.c.a | 2 | ||
40.f | even | 2 | 1 | 320.8.c.b | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
40.8.c.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
40.8.c.a | ✓ | 2 | 5.b | even | 2 | 1 | inner |
80.8.c.b | 2 | 4.b | odd | 2 | 1 | ||
80.8.c.b | 2 | 20.d | odd | 2 | 1 | ||
200.8.a.c | 1 | 5.c | odd | 4 | 1 | ||
200.8.a.f | 1 | 5.c | odd | 4 | 1 | ||
320.8.c.a | 2 | 8.d | odd | 2 | 1 | ||
320.8.c.a | 2 | 40.e | odd | 2 | 1 | ||
320.8.c.b | 2 | 8.b | even | 2 | 1 | ||
320.8.c.b | 2 | 40.f | even | 2 | 1 | ||
360.8.f.a | 2 | 3.b | odd | 2 | 1 | ||
360.8.f.a | 2 | 15.d | odd | 2 | 1 | ||
400.8.a.h | 1 | 20.e | even | 4 | 1 | ||
400.8.a.n | 1 | 20.e | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 1156 \)
acting on \(S_{8}^{\mathrm{new}}(40, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} + 1156 \)
$5$
\( T^{2} - 550T + 78125 \)
$7$
\( T^{2} + 11236 \)
$11$
\( (T + 1324)^{2} \)
$13$
\( T^{2} + 77933584 \)
$17$
\( T^{2} + 576000000 \)
$19$
\( (T - 4876)^{2} \)
$23$
\( T^{2} + 2175849316 \)
$29$
\( (T - 110902)^{2} \)
$31$
\( (T + 247680)^{2} \)
$37$
\( T^{2} + 129666248464 \)
$41$
\( (T - 104402)^{2} \)
$43$
\( T^{2} + 509256358884 \)
$47$
\( T^{2} + 24611961924 \)
$53$
\( T^{2} + 1136927447824 \)
$59$
\( (T + 832572)^{2} \)
$61$
\( (T - 529070)^{2} \)
$67$
\( T^{2} + 17425765638724 \)
$71$
\( (T - 5176568)^{2} \)
$73$
\( T^{2} + 56632576576 \)
$79$
\( (T - 3742736)^{2} \)
$83$
\( T^{2} + 61809251476996 \)
$89$
\( (T + 4300854)^{2} \)
$97$
\( T^{2} + 1317426475264 \)
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