Newspace parameters
| Level: | \( N \) | \(=\) | \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 600.y (of order \(5\), degree \(4\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(4.79102412128\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{16} - 8 x^{15} + 29 x^{14} - 32 x^{13} - 144 x^{12} + 670 x^{11} - 790 x^{10} - 2180 x^{9} + \cdots + 390625 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 5^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 8 x^{15} + 29 x^{14} - 32 x^{13} - 144 x^{12} + 670 x^{11} - 790 x^{10} - 2180 x^{9} + \cdots + 390625 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 26549 \nu^{15} - 155928 \nu^{14} + 3297564 \nu^{13} - 12751987 \nu^{12} + 3701621 \nu^{11} + \cdots - 73592734375 ) / 2229375000 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - \nu^{15} + 8 \nu^{14} - 29 \nu^{13} + 32 \nu^{12} + 144 \nu^{11} - 670 \nu^{10} + 790 \nu^{9} + \cdots + 546875 ) / 78125 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 30757 \nu^{15} - 440601 \nu^{14} + 2418963 \nu^{13} - 4488604 \nu^{12} - 10105093 \nu^{11} + \cdots - 22639375000 ) / 2229375000 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 77371 \nu^{15} + 1204447 \nu^{14} - 9417611 \nu^{13} + 23423688 \nu^{12} + 23668721 \nu^{11} + \cdots + 128883750000 ) / 2229375000 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 94244 \nu^{15} - 2475003 \nu^{14} + 17327739 \nu^{13} - 40542037 \nu^{12} + \cdots - 241055078125 ) / 2229375000 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 92189 \nu^{15} - 317947 \nu^{14} - 799339 \nu^{13} + 7212612 \nu^{12} - 11731121 \nu^{11} + \cdots + 31866562500 ) / 2229375000 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 144197 \nu^{15} - 690899 \nu^{14} + 7047787 \nu^{13} - 18403446 \nu^{12} + \cdots - 89950781250 ) / 2229375000 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 53064 \nu^{15} + 358877 \nu^{14} - 1083201 \nu^{13} + 299533 \nu^{12} + 9445336 \nu^{11} + \cdots + 2848515625 ) / 743125000 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 55051 \nu^{15} - 750343 \nu^{14} + 3240659 \nu^{13} - 4190222 \nu^{12} - 17911999 \nu^{11} + \cdots - 32345843750 ) / 445875000 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 289784 \nu^{15} - 2164487 \nu^{14} + 6200731 \nu^{13} + 2821727 \nu^{12} - 64171916 \nu^{11} + \cdots - 1683203125 ) / 2229375000 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 296788 \nu^{15} + 2324069 \nu^{14} - 6800947 \nu^{13} - 47799 \nu^{12} + 59556592 \nu^{11} + \cdots + 45834140625 ) / 2229375000 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 75427 \nu^{15} - 656696 \nu^{14} + 2199928 \nu^{13} - 913449 \nu^{12} - 16938883 \nu^{11} + \cdots - 16438828125 ) / 445875000 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 370426 \nu^{15} - 5040313 \nu^{14} + 22444169 \nu^{13} - 30256727 \nu^{12} + \cdots - 220732109375 ) / 2229375000 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 664121 \nu^{15} - 5356873 \nu^{14} + 15287099 \nu^{13} + 3444708 \nu^{12} - 142556189 \nu^{11} + \cdots - 73313437500 ) / 2229375000 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 1376678 \nu^{15} + 10045149 \nu^{14} - 27207537 \nu^{13} - 11672929 \nu^{12} + \cdots + 95085546875 ) / 2229375000 \)
|
| \(\nu\) | \(=\) |
\( ( - \beta_{15} - \beta_{13} - 4 \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} + 3 \beta_{5} - \beta_{4} + \cdots + 2 ) / 5 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 2 \beta_{15} - 5 \beta_{14} - 2 \beta_{13} + 5 \beta_{12} - 3 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} + \cdots + 4 ) / 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 6 \beta_{15} - 10 \beta_{14} - 4 \beta_{13} + 10 \beta_{12} - 35 \beta_{11} - \beta_{9} + \beta_{8} + \cdots - 37 ) / 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 7 \beta_{15} - 10 \beta_{14} - 13 \beta_{13} - 30 \beta_{11} + 15 \beta_{10} + 28 \beta_{9} + 7 \beta_{8} + \cdots - 39 ) / 5 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 19 \beta_{15} + 5 \beta_{14} - 41 \beta_{13} - 75 \beta_{12} - 105 \beta_{11} + 70 \beta_{10} + \cdots - 78 ) / 5 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 77 \beta_{15} + 85 \beta_{14} - 127 \beta_{13} - 75 \beta_{12} + 345 \beta_{11} + 55 \beta_{10} + \cdots - 41 ) / 5 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 109 \beta_{15} + 105 \beta_{14} - 159 \beta_{13} - 145 \beta_{12} + 430 \beta_{11} + 185 \beta_{10} + \cdots - 2 ) / 5 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 537 \beta_{15} + 665 \beta_{14} + 617 \beta_{13} + 725 \beta_{12} + 700 \beta_{11} - 440 \beta_{10} + \cdots - 1224 ) / 5 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 999 \beta_{15} + 1315 \beta_{14} + 1799 \beta_{13} - 1495 \beta_{12} - 875 \beta_{11} + 580 \beta_{10} + \cdots - 1518 ) / 5 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 4358 \beta_{15} + 3185 \beta_{14} + 6488 \beta_{13} - 1525 \beta_{12} - 3385 \beta_{11} - 185 \beta_{10} + \cdots - 6096 ) / 5 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 8664 \beta_{15} + 870 \beta_{14} - 3664 \beta_{13} - 11200 \beta_{12} + 16540 \beta_{11} + \cdots + 4438 ) / 5 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 9813 \beta_{15} - 9940 \beta_{14} - 8663 \beta_{13} + 15210 \beta_{12} + 30985 \beta_{11} + \cdots + 16811 ) / 5 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 54166 \beta_{15} - 20795 \beta_{14} - 73156 \beta_{13} + 50225 \beta_{12} + 110600 \beta_{11} + \cdots + 79882 ) / 5 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 28968 \beta_{15} - 20720 \beta_{14} + 2368 \beta_{13} + 85760 \beta_{12} - 49250 \beta_{11} + \cdots + 100799 ) / 5 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 81606 \beta_{15} + 42820 \beta_{14} + 85866 \beta_{13} + 33150 \beta_{12} - 249320 \beta_{11} + \cdots + 80003 ) / 5 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).
| \(n\) | \(151\) | \(301\) | \(401\) | \(577\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 121.1 |
|
0 | 0.809017 | + | 0.587785i | 0 | −1.51589 | − | 1.64380i | 0 | −1.85988 | 0 | 0.309017 | + | 0.951057i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.2 | 0 | 0.809017 | + | 0.587785i | 0 | −0.557573 | + | 2.16544i | 0 | −5.24701 | 0 | 0.309017 | + | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.3 | 0 | 0.809017 | + | 0.587785i | 0 | 2.16283 | − | 0.567585i | 0 | −1.62470 | 0 | 0.309017 | + | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.4 | 0 | 0.809017 | + | 0.587785i | 0 | 2.21965 | + | 0.270458i | 0 | 2.87749 | 0 | 0.309017 | + | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 241.1 | 0 | −0.309017 | − | 0.951057i | 0 | −2.19110 | − | 0.446198i | 0 | −2.82731 | 0 | −0.809017 | + | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 241.2 | 0 | −0.309017 | − | 0.951057i | 0 | −0.000769711 | − | 2.23607i | 0 | 3.15968 | 0 | −0.809017 | + | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 241.3 | 0 | −0.309017 | − | 0.951057i | 0 | 1.60857 | + | 1.55322i | 0 | 4.44278 | 0 | −0.809017 | + | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 241.4 | 0 | −0.309017 | − | 0.951057i | 0 | 1.77428 | − | 1.36086i | 0 | −3.92105 | 0 | −0.809017 | + | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 361.1 | 0 | −0.309017 | + | 0.951057i | 0 | −2.19110 | + | 0.446198i | 0 | −2.82731 | 0 | −0.809017 | − | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 361.2 | 0 | −0.309017 | + | 0.951057i | 0 | −0.000769711 | 2.23607i | 0 | 3.15968 | 0 | −0.809017 | − | 0.587785i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 361.3 | 0 | −0.309017 | + | 0.951057i | 0 | 1.60857 | − | 1.55322i | 0 | 4.44278 | 0 | −0.809017 | − | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 361.4 | 0 | −0.309017 | + | 0.951057i | 0 | 1.77428 | + | 1.36086i | 0 | −3.92105 | 0 | −0.809017 | − | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 481.1 | 0 | 0.809017 | − | 0.587785i | 0 | −1.51589 | + | 1.64380i | 0 | −1.85988 | 0 | 0.309017 | − | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 481.2 | 0 | 0.809017 | − | 0.587785i | 0 | −0.557573 | − | 2.16544i | 0 | −5.24701 | 0 | 0.309017 | − | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 481.3 | 0 | 0.809017 | − | 0.587785i | 0 | 2.16283 | + | 0.567585i | 0 | −1.62470 | 0 | 0.309017 | − | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 481.4 | 0 | 0.809017 | − | 0.587785i | 0 | 2.21965 | − | 0.270458i | 0 | 2.87749 | 0 | 0.309017 | − | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 25.d | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 600.2.y.f | ✓ | 16 |
| 25.d | even | 5 | 1 | inner | 600.2.y.f | ✓ | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 600.2.y.f | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 600.2.y.f | ✓ | 16 | 25.d | even | 5 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} + 5T_{7}^{7} - 35T_{7}^{6} - 185T_{7}^{5} + 310T_{7}^{4} + 2100T_{7}^{3} + 125T_{7}^{2} - 7550T_{7} - 7100 \)
acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} \)
|
| $3$ |
\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \)
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| $5$ |
\( T^{16} - 7 T^{15} + \cdots + 390625 \)
|
| $7$ |
\( (T^{8} + 5 T^{7} + \cdots - 7100)^{2} \)
|
| $11$ |
\( T^{16} - 9 T^{15} + \cdots + 4477456 \)
|
| $13$ |
\( T^{16} + \cdots + 148254976 \)
|
| $17$ |
\( T^{16} - 7 T^{15} + \cdots + 6533136 \)
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| $19$ |
\( T^{16} + \cdots + 1306388736 \)
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| $23$ |
\( T^{16} + \cdots + 1306388736 \)
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| $29$ |
\( T^{16} + 29 T^{15} + \cdots + 31315216 \)
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| $31$ |
\( T^{16} + \cdots + 189778724496 \)
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| $37$ |
\( T^{16} + \cdots + 24007643136 \)
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| $41$ |
\( T^{16} - 4 T^{15} + \cdots + 156816 \)
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| $43$ |
\( (T^{8} + 8 T^{7} + \cdots + 405136)^{2} \)
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| $47$ |
\( T^{16} + \cdots + 500697760000 \)
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| $53$ |
\( T^{16} - 8 T^{15} + \cdots + 58079641 \)
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| $59$ |
\( T^{16} - 19 T^{15} + \cdots + 20736 \)
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| $61$ |
\( T^{16} + \cdots + 3186667695376 \)
|
| $67$ |
\( T^{16} + \cdots + 99857970466816 \)
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| $71$ |
\( T^{16} + \cdots + 533794816 \)
|
| $73$ |
\( T^{16} + \cdots + 24014900250000 \)
|
| $79$ |
\( T^{16} + \cdots + 11764439296 \)
|
| $83$ |
\( T^{16} + \cdots + 40\!\cdots\!16 \)
|
| $89$ |
\( T^{16} + \cdots + 14295550096 \)
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| $97$ |
\( T^{16} + \cdots + 2314776630721 \)
|
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