Newspace parameters
| Level: | \( N \) | \(=\) | \( 1875 = 3 \cdot 5^{4} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1875.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(110.628581261\) |
| Analytic rank: | \(1\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{10} - x^{9} - 61 x^{8} + 54 x^{7} + 1219 x^{6} - 1069 x^{5} - 8923 x^{4} + 9432 x^{3} + 18852 x^{2} + \cdots + 10944 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 5 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - x^{9} - 61 x^{8} + 54 x^{7} + 1219 x^{6} - 1069 x^{5} - 8923 x^{4} + 9432 x^{3} + 18852 x^{2} + \cdots + 10944 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 79 \nu^{9} - 39 \nu^{8} + 6221 \nu^{7} + 4296 \nu^{6} - 175309 \nu^{5} - 85407 \nu^{4} + \cdots + 4591008 ) / 692160 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 839 \nu^{9} - 681 \nu^{8} - 51101 \nu^{7} + 32864 \nu^{6} + 1014149 \nu^{5} - 546273 \nu^{4} + \cdots - 12423648 ) / 1384320 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 997 \nu^{9} - 603 \nu^{8} - 63543 \nu^{7} + 24272 \nu^{6} + 1364767 \nu^{5} - 375459 \nu^{4} + \cdots - 38217504 ) / 1384320 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 10671 \nu^{9} + 6049 \nu^{8} + 647189 \nu^{7} - 269216 \nu^{6} - 12844221 \nu^{5} + \cdots + 161162592 ) / 1384320 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 10421 \nu^{9} + 879 \nu^{8} + 622939 \nu^{7} + 48484 \nu^{6} - 11974031 \nu^{5} + \cdots + 81838272 ) / 692160 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 26193 \nu^{9} + 2767 \nu^{8} + 1578907 \nu^{7} + 30192 \nu^{6} - 30747363 \nu^{5} + \cdots + 292159776 ) / 1384320 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 13863 \nu^{9} - 2463 \nu^{8} + 858757 \nu^{7} + 209192 \nu^{6} - 17509413 \nu^{5} + \cdots + 204857376 ) / 692160 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 34619 \nu^{9} - 9059 \nu^{8} + 2116481 \nu^{7} + 749416 \nu^{6} - 42055169 \nu^{5} + \cdots + 348159648 ) / 1384320 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} + \beta_{2} + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{9} + \beta_{8} + \beta_{6} - 2\beta_{5} - 9\beta_{3} - \beta_{2} + 20\beta _1 - 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{9} + 3\beta_{7} - \beta_{6} - 2\beta_{5} + 26\beta_{4} - 22\beta_{3} + 36\beta_{2} + 11\beta _1 + 251 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 33 \beta_{9} + 32 \beta_{8} + 5 \beta_{7} + 31 \beta_{6} - 80 \beta_{5} - 7 \beta_{4} - 393 \beta_{3} + \cdots - 93 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 48 \beta_{9} + 3 \beta_{8} + 117 \beta_{7} - 18 \beta_{6} - 114 \beta_{5} + 650 \beta_{4} + \cdots + 5967 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 920 \beta_{9} + 884 \beta_{8} + 234 \beta_{7} + 824 \beta_{6} - 2614 \beta_{5} - 264 \beta_{4} + \cdots - 2276 \)
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| \(\nu^{8}\) | \(=\) |
\( - 1820 \beta_{9} + 204 \beta_{8} + 3732 \beta_{7} - 20 \beta_{6} - 4672 \beta_{5} + 16465 \beta_{4} + \cdots + 148792 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 25113 \beta_{9} + 23929 \beta_{8} + 8608 \beta_{7} + 21473 \beta_{6} - 80578 \beta_{5} - 7460 \beta_{4} + \cdots - 51900 \)
|
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 |
|
−5.46982 | −3.00000 | 21.9189 | 0 | 16.4094 | 31.1300 | −76.1337 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.54955 | −3.00000 | 12.6984 | 0 | 13.6487 | 8.49133 | −21.3758 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.87444 | −3.00000 | 0.262378 | 0 | 8.62331 | 8.99984 | 22.2413 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.35653 | −3.00000 | −6.15983 | 0 | 4.06959 | 19.2974 | 19.2082 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.04884 | −3.00000 | −6.89993 | 0 | 3.14653 | −32.0322 | 15.6277 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.658275 | −3.00000 | −7.56667 | 0 | 1.97482 | 0.266664 | 10.2471 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.44553 | −3.00000 | −2.01939 | 0 | −7.33659 | −8.35285 | −24.5027 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.91436 | −3.00000 | 0.493465 | 0 | −8.74307 | −19.3569 | −21.8767 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 4.47014 | −3.00000 | 11.9821 | 0 | −13.4104 | 30.1752 | 17.8006 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 5.12743 | −3.00000 | 18.2905 | 0 | −15.3823 | 12.3816 | 52.7640 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1875.4.a.a | ✓ | 10 |
| 5.b | even | 2 | 1 | 1875.4.a.d | yes | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1875.4.a.a | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 1875.4.a.d | yes | 10 | 5.b | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + T_{2}^{9} - 61 T_{2}^{8} - 54 T_{2}^{7} + 1219 T_{2}^{6} + 1069 T_{2}^{5} - 8923 T_{2}^{4} + \cdots + 10944 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1875))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + T^{9} + \cdots + 10944 \)
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| $3$ |
\( (T + 3)^{10} \)
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| $5$ |
\( T^{10} \)
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| $7$ |
\( T^{10} + \cdots - 23688338000 \)
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| $11$ |
\( T^{10} + \cdots + 43932717994176 \)
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| $13$ |
\( T^{10} + \cdots - 13\!\cdots\!89 \)
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| $17$ |
\( T^{10} + \cdots - 30\!\cdots\!64 \)
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| $19$ |
\( T^{10} + \cdots - 128742355320975 \)
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| $23$ |
\( T^{10} + \cdots + 25\!\cdots\!00 \)
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| $29$ |
\( T^{10} + \cdots - 65\!\cdots\!20 \)
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| $31$ |
\( T^{10} + \cdots - 85\!\cdots\!25 \)
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| $37$ |
\( T^{10} + \cdots - 71\!\cdots\!00 \)
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| $41$ |
\( T^{10} + \cdots + 34\!\cdots\!00 \)
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| $43$ |
\( T^{10} + \cdots + 11\!\cdots\!69 \)
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| $47$ |
\( T^{10} + \cdots + 22\!\cdots\!96 \)
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| $53$ |
\( T^{10} + \cdots - 47\!\cdots\!00 \)
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| $59$ |
\( T^{10} + \cdots - 94\!\cdots\!80 \)
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| $61$ |
\( T^{10} + \cdots - 20\!\cdots\!61 \)
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| $67$ |
\( T^{10} + \cdots - 28\!\cdots\!09 \)
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| $71$ |
\( T^{10} + \cdots - 29\!\cdots\!96 \)
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| $73$ |
\( T^{10} + \cdots - 82\!\cdots\!00 \)
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| $79$ |
\( T^{10} + \cdots - 47\!\cdots\!25 \)
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| $83$ |
\( T^{10} + \cdots - 19\!\cdots\!04 \)
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| $89$ |
\( T^{10} + \cdots + 45\!\cdots\!20 \)
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| $97$ |
\( T^{10} + \cdots + 49\!\cdots\!11 \)
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