Newspace parameters
| Level: | \( N \) | \(=\) | \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 980.l (of order \(4\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(26.7030659073\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-1}) \) |
|
|
|
| Defining polynomial: |
\( x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 20) |
| 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 \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(197\) | \(491\) |
| \(\chi(n)\) | \(1\) | \(i\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 197.1 |
|
0 | −1.00000 | + | 1.00000i | 0 | 3.00000 | − | 4.00000i | 0 | 0 | 0 | 7.00000i | 0 | ||||||||||||||||||||
| 393.1 | 0 | −1.00000 | − | 1.00000i | 0 | 3.00000 | + | 4.00000i | 0 | 0 | 0 | − | 7.00000i | 0 | ||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 980.3.l.a | 2 | |
| 5.c | odd | 4 | 1 | inner | 980.3.l.a | 2 | |
| 7.b | odd | 2 | 1 | 20.3.f.a | ✓ | 2 | |
| 21.c | even | 2 | 1 | 180.3.l.a | 2 | ||
| 28.d | even | 2 | 1 | 80.3.p.a | 2 | ||
| 35.c | odd | 2 | 1 | 100.3.f.a | 2 | ||
| 35.f | even | 4 | 1 | 20.3.f.a | ✓ | 2 | |
| 35.f | even | 4 | 1 | 100.3.f.a | 2 | ||
| 56.e | even | 2 | 1 | 320.3.p.g | 2 | ||
| 56.h | odd | 2 | 1 | 320.3.p.c | 2 | ||
| 84.h | odd | 2 | 1 | 720.3.bh.e | 2 | ||
| 105.g | even | 2 | 1 | 900.3.l.a | 2 | ||
| 105.k | odd | 4 | 1 | 180.3.l.a | 2 | ||
| 105.k | odd | 4 | 1 | 900.3.l.a | 2 | ||
| 140.c | even | 2 | 1 | 400.3.p.d | 2 | ||
| 140.j | odd | 4 | 1 | 80.3.p.a | 2 | ||
| 140.j | odd | 4 | 1 | 400.3.p.d | 2 | ||
| 280.s | even | 4 | 1 | 320.3.p.c | 2 | ||
| 280.y | odd | 4 | 1 | 320.3.p.g | 2 | ||
| 420.w | even | 4 | 1 | 720.3.bh.e | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 20.3.f.a | ✓ | 2 | 7.b | odd | 2 | 1 | |
| 20.3.f.a | ✓ | 2 | 35.f | even | 4 | 1 | |
| 80.3.p.a | 2 | 28.d | even | 2 | 1 | ||
| 80.3.p.a | 2 | 140.j | odd | 4 | 1 | ||
| 100.3.f.a | 2 | 35.c | odd | 2 | 1 | ||
| 100.3.f.a | 2 | 35.f | even | 4 | 1 | ||
| 180.3.l.a | 2 | 21.c | even | 2 | 1 | ||
| 180.3.l.a | 2 | 105.k | odd | 4 | 1 | ||
| 320.3.p.c | 2 | 56.h | odd | 2 | 1 | ||
| 320.3.p.c | 2 | 280.s | even | 4 | 1 | ||
| 320.3.p.g | 2 | 56.e | even | 2 | 1 | ||
| 320.3.p.g | 2 | 280.y | odd | 4 | 1 | ||
| 400.3.p.d | 2 | 140.c | even | 2 | 1 | ||
| 400.3.p.d | 2 | 140.j | odd | 4 | 1 | ||
| 720.3.bh.e | 2 | 84.h | odd | 2 | 1 | ||
| 720.3.bh.e | 2 | 420.w | even | 4 | 1 | ||
| 900.3.l.a | 2 | 105.g | even | 2 | 1 | ||
| 900.3.l.a | 2 | 105.k | odd | 4 | 1 | ||
| 980.3.l.a | 2 | 1.a | even | 1 | 1 | trivial | |
| 980.3.l.a | 2 | 5.c | odd | 4 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 2T_{3} + 2 \)
acting on \(S_{3}^{\mathrm{new}}(980, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 2T + 2 \)
|
| $5$ |
\( T^{2} - 6T + 25 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( (T - 10)^{2} \)
|
| $13$ |
\( T^{2} + 18T + 162 \)
|
| $17$ |
\( T^{2} + 2T + 2 \)
|
| $19$ |
\( T^{2} + 64 \)
|
| $23$ |
\( T^{2} + 46T + 1058 \)
|
| $29$ |
\( T^{2} + 64 \)
|
| $31$ |
\( (T - 14)^{2} \)
|
| $37$ |
\( T^{2} - 66T + 2178 \)
|
| $41$ |
\( (T - 14)^{2} \)
|
| $43$ |
\( T^{2} + 30T + 450 \)
|
| $47$ |
\( T^{2} - 78T + 3042 \)
|
| $53$ |
\( T^{2} + 14T + 98 \)
|
| $59$ |
\( T^{2} + 3136 \)
|
| $61$ |
\( (T + 42)^{2} \)
|
| $67$ |
\( T^{2} + 14T + 98 \)
|
| $71$ |
\( (T - 98)^{2} \)
|
| $73$ |
\( T^{2} + 98T + 4802 \)
|
| $79$ |
\( T^{2} + 9216 \)
|
| $83$ |
\( T^{2} - 126T + 7938 \)
|
| $89$ |
\( T^{2} + 12544 \)
|
| $97$ |
\( T^{2} + 66T + 2178 \)
|
| show more | |
| show less |