Newspace parameters
| Level: | \( N \) | \(=\) | \( 1104 = 2^{4} \cdot 3 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1104.o (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(30.0818211854\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{3}, \sqrt{23})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 13x^{2} + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} - 13x^{2} + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{3} - 6\nu ) / 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -6\nu^{3} + 58\nu ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( 4\nu^{2} - 26 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 3\beta_1 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 26 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{2} + 29\beta_1 ) / 8 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1104\mathbb{Z}\right)^\times\).
| \(n\) | \(97\) | \(277\) | \(415\) | \(737\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1103.1 |
|
0 | 3.00000 | 0 | −9.99198 | 0 | −3.86442 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||
| 1103.2 | 0 | 3.00000 | 0 | −0.400321 | 0 | −13.4561 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1103.3 | 0 | 3.00000 | 0 | 0.400321 | 0 | 13.4561 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1103.4 | 0 | 3.00000 | 0 | 9.99198 | 0 | 3.86442 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 276.h | odd | 2 | 1 | CM by \(\Q(\sqrt{-69}) \) |
| 12.b | even | 2 | 1 | inner |
| 23.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1104.3.o.b | yes | 4 |
| 3.b | odd | 2 | 1 | 1104.3.o.a | ✓ | 4 | |
| 4.b | odd | 2 | 1 | 1104.3.o.a | ✓ | 4 | |
| 12.b | even | 2 | 1 | inner | 1104.3.o.b | yes | 4 |
| 23.b | odd | 2 | 1 | inner | 1104.3.o.b | yes | 4 |
| 69.c | even | 2 | 1 | 1104.3.o.a | ✓ | 4 | |
| 92.b | even | 2 | 1 | 1104.3.o.a | ✓ | 4 | |
| 276.h | odd | 2 | 1 | CM | 1104.3.o.b | yes | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1104.3.o.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 1104.3.o.a | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 1104.3.o.a | ✓ | 4 | 69.c | even | 2 | 1 | |
| 1104.3.o.a | ✓ | 4 | 92.b | even | 2 | 1 | |
| 1104.3.o.b | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 1104.3.o.b | yes | 4 | 12.b | even | 2 | 1 | inner |
| 1104.3.o.b | yes | 4 | 23.b | odd | 2 | 1 | inner |
| 1104.3.o.b | yes | 4 | 276.h | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1104, [\chi])\):
|
\( T_{5}^{4} - 100T_{5}^{2} + 16 \)
|
|
\( T_{71} + 50 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T - 3)^{4} \)
|
| $5$ |
\( T^{4} - 100T^{2} + 16 \)
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| $7$ |
\( T^{4} - 196T^{2} + 2704 \)
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| $11$ |
\( T^{4} \)
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| $13$ |
\( (T^{2} - 276)^{2} \)
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| $17$ |
\( T^{4} - 1156 T^{2} + 327184 \)
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| $19$ |
\( T^{4} - 1444 T^{2} + 183184 \)
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| $23$ |
\( (T - 23)^{4} \)
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| $29$ |
\( T^{4} \)
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| $31$ |
\( T^{4} \)
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| $37$ |
\( T^{4} \)
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| $41$ |
\( T^{4} \)
|
| $43$ |
\( T^{4} - 7396 T^{2} + 13337104 \)
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| $47$ |
\( (T^{2} - 6900)^{2} \)
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| $53$ |
\( T^{4} - 11236 T^{2} + 19963024 \)
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| $59$ |
\( (T^{2} - 13524)^{2} \)
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| $61$ |
\( T^{4} \)
|
| $67$ |
\( T^{4} - 17956 T^{2} + 58186384 \)
|
| $71$ |
\( (T + 50)^{4} \)
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| $73$ |
\( (T - 130)^{4} \)
|
| $79$ |
\( T^{4} - 24964 T^{2} + 128414224 \)
|
| $83$ |
\( T^{4} \)
|
| $89$ |
\( T^{4} - 31684 T^{2} + 72114064 \)
|
| $97$ |
\( T^{4} \)
|
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