Newspace parameters
| Level: | \( N \) | \(=\) | \( 2900 = 2^{2} \cdot 5^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2900.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(23.1566165862\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-7}) \) |
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|
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| Defining polynomial: |
\( x^{2} - x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 116) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-7}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2900\mathbb{Z}\right)^\times\).
| \(n\) | \(901\) | \(1277\) | \(1451\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1101.1 |
|
0 | − | 2.64575i | 0 | 0 | 0 | 2.00000 | 0 | −4.00000 | 0 | |||||||||||||||||||||||
| 1101.2 | 0 | 2.64575i | 0 | 0 | 0 | 2.00000 | 0 | −4.00000 | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 29.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2900.2.d.a | 2 | |
| 5.b | even | 2 | 1 | 116.2.c.a | ✓ | 2 | |
| 5.c | odd | 4 | 2 | 2900.2.f.b | 4 | ||
| 15.d | odd | 2 | 1 | 1044.2.h.c | 2 | ||
| 20.d | odd | 2 | 1 | 464.2.e.b | 2 | ||
| 29.b | even | 2 | 1 | inner | 2900.2.d.a | 2 | |
| 40.e | odd | 2 | 1 | 1856.2.e.c | 2 | ||
| 40.f | even | 2 | 1 | 1856.2.e.a | 2 | ||
| 60.h | even | 2 | 1 | 4176.2.o.e | 2 | ||
| 145.d | even | 2 | 1 | 116.2.c.a | ✓ | 2 | |
| 145.f | odd | 4 | 2 | 3364.2.a.f | 2 | ||
| 145.h | odd | 4 | 2 | 2900.2.f.b | 4 | ||
| 435.b | odd | 2 | 1 | 1044.2.h.c | 2 | ||
| 580.e | odd | 2 | 1 | 464.2.e.b | 2 | ||
| 1160.e | even | 2 | 1 | 1856.2.e.a | 2 | ||
| 1160.o | odd | 2 | 1 | 1856.2.e.c | 2 | ||
| 1740.k | even | 2 | 1 | 4176.2.o.e | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 116.2.c.a | ✓ | 2 | 5.b | even | 2 | 1 | |
| 116.2.c.a | ✓ | 2 | 145.d | even | 2 | 1 | |
| 464.2.e.b | 2 | 20.d | odd | 2 | 1 | ||
| 464.2.e.b | 2 | 580.e | odd | 2 | 1 | ||
| 1044.2.h.c | 2 | 15.d | odd | 2 | 1 | ||
| 1044.2.h.c | 2 | 435.b | odd | 2 | 1 | ||
| 1856.2.e.a | 2 | 40.f | even | 2 | 1 | ||
| 1856.2.e.a | 2 | 1160.e | even | 2 | 1 | ||
| 1856.2.e.c | 2 | 40.e | odd | 2 | 1 | ||
| 1856.2.e.c | 2 | 1160.o | odd | 2 | 1 | ||
| 2900.2.d.a | 2 | 1.a | even | 1 | 1 | trivial | |
| 2900.2.d.a | 2 | 29.b | even | 2 | 1 | inner | |
| 2900.2.f.b | 4 | 5.c | odd | 4 | 2 | ||
| 2900.2.f.b | 4 | 145.h | odd | 4 | 2 | ||
| 3364.2.a.f | 2 | 145.f | odd | 4 | 2 | ||
| 4176.2.o.e | 2 | 60.h | even | 2 | 1 | ||
| 4176.2.o.e | 2 | 1740.k | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2900, [\chi])\):
|
\( T_{3}^{2} + 7 \)
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\( T_{7} - 2 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 7 \)
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| $5$ |
\( T^{2} \)
|
| $7$ |
\( (T - 2)^{2} \)
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| $11$ |
\( T^{2} + 7 \)
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| $13$ |
\( (T + 5)^{2} \)
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| $17$ |
\( T^{2} + 28 \)
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| $19$ |
\( T^{2} + 28 \)
|
| $23$ |
\( (T + 6)^{2} \)
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| $29$ |
\( T^{2} + 2T + 29 \)
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| $31$ |
\( T^{2} + 7 \)
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| $37$ |
\( T^{2} + 112 \)
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| $41$ |
\( T^{2} + 28 \)
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| $43$ |
\( T^{2} + 63 \)
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| $47$ |
\( T^{2} + 63 \)
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| $53$ |
\( (T + 5)^{2} \)
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| $59$ |
\( (T + 14)^{2} \)
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| $61$ |
\( T^{2} + 28 \)
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| $67$ |
\( (T - 4)^{2} \)
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| $71$ |
\( (T + 8)^{2} \)
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| $73$ |
\( T^{2} + 112 \)
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| $79$ |
\( T^{2} + 7 \)
|
| $83$ |
\( (T + 2)^{2} \)
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| $89$ |
\( T^{2} + 28 \)
|
| $97$ |
\( T^{2} + 28 \)
|
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