Newspace parameters
| Level: | \( N \) | \(=\) | \( 2025 = 3^{4} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2025.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(119.478867762\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{8} - 44x^{6} + 567x^{4} - 2024x^{2} + 1900 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 405) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 44x^{6} + 567x^{4} - 2024x^{2} + 1900 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 11 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - 39\nu^{5} + 402\nu^{3} - 704\nu ) / 30 \)
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| \(\beta_{4}\) | \(=\) |
\( ( -4\nu^{7} + 171\nu^{5} - 1953\nu^{3} + 3506\nu ) / 90 \)
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| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} - 45\nu^{4} + 540\nu^{2} - 1016 ) / 18 \)
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| \(\beta_{6}\) | \(=\) |
\( ( 4\nu^{7} - 171\nu^{5} + 2043\nu^{3} - 5126\nu ) / 90 \)
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| \(\beta_{7}\) | \(=\) |
\( ( \nu^{6} - 36\nu^{4} + 333\nu^{2} - 548 ) / 9 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 11 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{4} + 18\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{7} - 2\beta_{5} + 23\beta_{2} + 201 \)
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| \(\nu^{5}\) | \(=\) |
\( 23\beta_{6} + 29\beta_{4} + 8\beta_{3} + 368\beta_1 \)
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| \(\nu^{6}\) | \(=\) |
\( 45\beta_{7} - 72\beta_{5} + 495\beta_{2} + 4121 \)
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| \(\nu^{7}\) | \(=\) |
\( 495\beta_{6} + 729\beta_{4} + 342\beta_{3} + 7820\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−4.80346 | 0 | 15.0732 | 0 | 0 | 5.92463 | −33.9759 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −3.99806 | 0 | 7.98447 | 0 | 0 | 7.79840 | 0.0620886 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.85698 | 0 | −4.55164 | 0 | 0 | 1.02402 | 23.3081 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.22227 | 0 | −6.50606 | 0 | 0 | −33.1668 | 17.7303 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.22227 | 0 | −6.50606 | 0 | 0 | 33.1668 | −17.7303 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.85698 | 0 | −4.55164 | 0 | 0 | −1.02402 | −23.3081 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 3.99806 | 0 | 7.98447 | 0 | 0 | −7.79840 | −0.0620886 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 4.80346 | 0 | 15.0732 | 0 | 0 | −5.92463 | 33.9759 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
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 | 2025.4.a.bc | 8 | |
| 3.b | odd | 2 | 1 | 2025.4.a.bd | 8 | ||
| 5.b | even | 2 | 1 | inner | 2025.4.a.bc | 8 | |
| 5.c | odd | 4 | 2 | 405.4.b.b | ✓ | 8 | |
| 15.d | odd | 2 | 1 | 2025.4.a.bd | 8 | ||
| 15.e | even | 4 | 2 | 405.4.b.c | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 405.4.b.b | ✓ | 8 | 5.c | odd | 4 | 2 | |
| 405.4.b.c | yes | 8 | 15.e | even | 4 | 2 | |
| 2025.4.a.bc | 8 | 1.a | even | 1 | 1 | trivial | |
| 2025.4.a.bc | 8 | 5.b | even | 2 | 1 | inner | |
| 2025.4.a.bd | 8 | 3.b | odd | 2 | 1 | ||
| 2025.4.a.bd | 8 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2025))\):
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\( T_{2}^{8} - 44T_{2}^{6} + 567T_{2}^{4} - 2024T_{2}^{2} + 1900 \)
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\( T_{7}^{8} - 1197T_{7}^{6} + 108900T_{7}^{4} - 2461104T_{7}^{2} + 2462400 \)
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\( T_{11}^{4} - 1755T_{11}^{2} - 7038T_{11} + 502920 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 44 T^{6} + \cdots + 1900 \)
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| $3$ |
\( T^{8} \)
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| $5$ |
\( T^{8} \)
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| $7$ |
\( T^{8} - 1197 T^{6} + \cdots + 2462400 \)
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| $11$ |
\( (T^{4} - 1755 T^{2} + \cdots + 502920)^{2} \)
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| $13$ |
\( T^{8} + \cdots + 68838460416 \)
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| $17$ |
\( T^{8} + \cdots + 24363576945664 \)
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| $19$ |
\( (T^{4} + 59 T^{3} + \cdots + 2964730)^{2} \)
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| $23$ |
\( T^{8} + \cdots + 533400694426624 \)
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| $29$ |
\( (T^{4} - 159 T^{3} + \cdots - 16354800)^{2} \)
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| $31$ |
\( (T^{4} + 208 T^{3} + \cdots - 27072584)^{2} \)
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| $37$ |
\( T^{8} + \cdots + 53\!\cdots\!00 \)
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| $41$ |
\( (T^{4} + 243 T^{3} + \cdots + 54333414)^{2} \)
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| $43$ |
\( T^{8} + \cdots + 57\!\cdots\!96 \)
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| $47$ |
\( T^{8} + \cdots + 18\!\cdots\!16 \)
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| $53$ |
\( T^{8} + \cdots + 21\!\cdots\!24 \)
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| $59$ |
\( (T^{4} + 573 T^{3} + \cdots + 77712038244)^{2} \)
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| $61$ |
\( (T^{4} + 199 T^{3} + \cdots + 37684911880)^{2} \)
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| $67$ |
\( T^{8} + \cdots + 68\!\cdots\!56 \)
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| $71$ |
\( (T^{4} - 864 T^{3} + \cdots + 17785914552)^{2} \)
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| $73$ |
\( T^{8} + \cdots + 37\!\cdots\!04 \)
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| $79$ |
\( (T^{4} + 1298 T^{3} + \cdots - 256210144256)^{2} \)
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| $83$ |
\( T^{8} + \cdots + 50\!\cdots\!44 \)
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| $89$ |
\( (T^{4} - 543 T^{3} + \cdots + 438625777050)^{2} \)
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| $97$ |
\( T^{8} + \cdots + 31\!\cdots\!36 \)
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