Newspace parameters
| Level: | \( N \) | \(=\) | \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 180.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(73.3281498110\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
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|
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| Defining polynomial: |
\( x^{16} + 15630052 x^{14} + 100431843210026 x^{12} + \cdots + 41\!\cdots\!25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{58}\cdot 3^{4}\cdot 5^{16} \) |
| Twist minimal: | no (minimal twist has level 20) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{16} + 15630052 x^{14} + 100431843210026 x^{12} + \cdots + 41\!\cdots\!25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 77\!\cdots\!71 \nu^{15} + \cdots + 41\!\cdots\!00 ) / 35\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 77\!\cdots\!71 \nu^{15} + \cdots + 12\!\cdots\!00 ) / 17\!\cdots\!00 \)
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| \(\beta_{3}\) | \(=\) |
\( ( - 32\!\cdots\!14 \nu^{15} + \cdots + 16\!\cdots\!75 ) / 59\!\cdots\!00 \)
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| \(\beta_{4}\) | \(=\) |
\( ( 32\!\cdots\!33 \nu^{15} + \cdots - 54\!\cdots\!50 ) / 51\!\cdots\!00 \)
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| \(\beta_{5}\) | \(=\) |
\( ( - 88\!\cdots\!31 \nu^{15} + \cdots + 24\!\cdots\!00 ) / 71\!\cdots\!00 \)
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| \(\beta_{6}\) | \(=\) |
\( ( - 25\!\cdots\!16 \nu^{15} + \cdots + 91\!\cdots\!25 ) / 17\!\cdots\!00 \)
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| \(\beta_{7}\) | \(=\) |
\( ( 20\!\cdots\!97 \nu^{15} + \cdots + 20\!\cdots\!50 ) / 35\!\cdots\!00 \)
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| \(\beta_{8}\) | \(=\) |
\( ( - 51\!\cdots\!57 \nu^{15} + \cdots + 46\!\cdots\!50 ) / 71\!\cdots\!00 \)
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| \(\beta_{9}\) | \(=\) |
\( ( 36\!\cdots\!47 \nu^{15} + \cdots + 53\!\cdots\!00 ) / 35\!\cdots\!00 \)
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| \(\beta_{10}\) | \(=\) |
\( ( - 39\!\cdots\!01 \nu^{15} + \cdots - 78\!\cdots\!00 ) / 35\!\cdots\!00 \)
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| \(\beta_{11}\) | \(=\) |
\( ( 66\!\cdots\!43 \nu^{15} + \cdots - 48\!\cdots\!50 ) / 51\!\cdots\!00 \)
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| \(\beta_{12}\) | \(=\) |
\( ( 50\!\cdots\!80 \nu^{15} + \cdots - 31\!\cdots\!75 ) / 35\!\cdots\!00 \)
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| \(\beta_{13}\) | \(=\) |
\( ( 29\!\cdots\!27 \nu^{15} + \cdots + 52\!\cdots\!25 ) / 17\!\cdots\!00 \)
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| \(\beta_{14}\) | \(=\) |
\( ( 66\!\cdots\!09 \nu^{15} + \cdots + 73\!\cdots\!50 ) / 35\!\cdots\!00 \)
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| \(\beta_{15}\) | \(=\) |
\( ( - 79\!\cdots\!17 \nu^{15} + \cdots - 18\!\cdots\!75 ) / 17\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} - \beta_{5} - \beta_{3} - 17\beta _1 + 6 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( ( 149 \beta_{15} + 287 \beta_{14} - 98 \beta_{13} - 323 \beta_{12} + 51 \beta_{11} - 744 \beta_{10} + \cdots - 7790132 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 76669 \beta_{15} - 65047 \beta_{14} - 40665 \beta_{13} + 59240 \beta_{12} - 95867 \beta_{11} + \cdots - 14557334 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 613416900 \beta_{15} - 795040988 \beta_{14} + 598331122 \beta_{13} + 1222179726 \beta_{12} + \cdots + 21619464685439 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 291150171010 \beta_{15} + 354906269520 \beta_{14} + 411738118373 \beta_{13} - 175156164389 \beta_{12} + \cdots + 27932728834607 ) / 4 \)
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| \(\nu^{6}\) | \(=\) |
\( ( 45\!\cdots\!42 \beta_{15} + \cdots - 13\!\cdots\!69 ) / 8 \)
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| \(\nu^{7}\) | \(=\) |
\( ( - 41\!\cdots\!63 \beta_{15} + \cdots - 72\!\cdots\!15 ) / 2 \)
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| \(\nu^{8}\) | \(=\) |
\( ( - 39\!\cdots\!46 \beta_{15} + \cdots + 10\!\cdots\!40 ) / 2 \)
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| \(\nu^{9}\) | \(=\) |
\( ( 17\!\cdots\!92 \beta_{15} + \cdots + 88\!\cdots\!03 ) / 4 \)
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| \(\nu^{10}\) | \(=\) |
\( ( 54\!\cdots\!76 \beta_{15} + \cdots - 14\!\cdots\!01 ) / 8 \)
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| \(\nu^{11}\) | \(=\) |
\( ( - 16\!\cdots\!33 \beta_{15} + \cdots - 48\!\cdots\!76 ) / 4 \)
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| \(\nu^{12}\) | \(=\) |
\( ( - 92\!\cdots\!34 \beta_{15} + \cdots + 24\!\cdots\!71 ) / 4 \)
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| \(\nu^{13}\) | \(=\) |
\( ( - 58\!\cdots\!38 \beta_{15} + \cdots + 11\!\cdots\!12 ) / 2 \)
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| \(\nu^{14}\) | \(=\) |
\( ( 31\!\cdots\!55 \beta_{15} + \cdots - 88\!\cdots\!04 ) / 4 \)
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| \(\nu^{15}\) | \(=\) |
\( ( 10\!\cdots\!85 \beta_{15} + \cdots - 10\!\cdots\!72 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/180\mathbb{Z}\right)^\times\).
| \(n\) | \(37\) | \(91\) | \(101\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 91.1 |
|
−14.5274 | − | 6.70489i | 0 | 166.089 | + | 194.809i | −279.508 | 0 | − | 2626.96i | −1106.66 | − | 3943.67i | 0 | 4060.52 | + | 1874.07i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.2 | −14.5274 | + | 6.70489i | 0 | 166.089 | − | 194.809i | −279.508 | 0 | 2626.96i | −1106.66 | + | 3943.67i | 0 | 4060.52 | − | 1874.07i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.3 | −11.5676 | − | 11.0540i | 0 | 11.6196 | + | 255.736i | −279.508 | 0 | 3325.58i | 2692.49 | − | 3086.70i | 0 | 3233.25 | + | 3089.68i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.4 | −11.5676 | + | 11.0540i | 0 | 11.6196 | − | 255.736i | −279.508 | 0 | − | 3325.58i | 2692.49 | + | 3086.70i | 0 | 3233.25 | − | 3089.68i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.5 | −11.3498 | − | 11.2775i | 0 | 1.63618 | + | 255.995i | 279.508 | 0 | − | 3940.57i | 2868.41 | − | 2923.94i | 0 | −3172.37 | − | 3152.16i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.6 | −11.3498 | + | 11.2775i | 0 | 1.63618 | − | 255.995i | 279.508 | 0 | 3940.57i | 2868.41 | + | 2923.94i | 0 | −3172.37 | + | 3152.16i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.7 | −4.64016 | − | 15.3124i | 0 | −212.938 | + | 142.104i | 279.508 | 0 | 210.345i | 3164.01 | + | 2601.20i | 0 | −1296.96 | − | 4279.94i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.8 | −4.64016 | + | 15.3124i | 0 | −212.938 | − | 142.104i | 279.508 | 0 | − | 210.345i | 3164.01 | − | 2601.20i | 0 | −1296.96 | + | 4279.94i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.9 | 1.41320 | − | 15.9375i | 0 | −252.006 | − | 45.0455i | −279.508 | 0 | − | 2633.20i | −1074.04 | + | 3952.68i | 0 | −395.000 | + | 4454.66i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.10 | 1.41320 | + | 15.9375i | 0 | −252.006 | + | 45.0455i | −279.508 | 0 | 2633.20i | −1074.04 | − | 3952.68i | 0 | −395.000 | − | 4454.66i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.11 | 7.35022 | − | 14.2118i | 0 | −147.949 | − | 208.919i | 279.508 | 0 | 3540.70i | −4056.56 | + | 567.011i | 0 | 2054.45 | − | 3972.31i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.12 | 7.35022 | + | 14.2118i | 0 | −147.949 | + | 208.919i | 279.508 | 0 | − | 3540.70i | −4056.56 | − | 567.011i | 0 | 2054.45 | + | 3972.31i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.13 | 14.9660 | − | 5.65855i | 0 | 191.962 | − | 169.372i | 279.508 | 0 | − | 2973.76i | 1914.50 | − | 3621.04i | 0 | 4183.12 | − | 1581.61i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.14 | 14.9660 | + | 5.65855i | 0 | 191.962 | + | 169.372i | 279.508 | 0 | 2973.76i | 1914.50 | + | 3621.04i | 0 | 4183.12 | + | 1581.61i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.15 | 15.3556 | − | 4.49522i | 0 | 215.586 | − | 138.053i | −279.508 | 0 | − | 820.952i | 2689.86 | − | 3088.99i | 0 | −4292.01 | + | 1256.45i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.16 | 15.3556 | + | 4.49522i | 0 | 215.586 | + | 138.053i | −279.508 | 0 | 820.952i | 2689.86 | + | 3088.99i | 0 | −4292.01 | − | 1256.45i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 180.9.c.a | 16 | |
| 3.b | odd | 2 | 1 | 20.9.b.a | ✓ | 16 | |
| 4.b | odd | 2 | 1 | inner | 180.9.c.a | 16 | |
| 12.b | even | 2 | 1 | 20.9.b.a | ✓ | 16 | |
| 15.d | odd | 2 | 1 | 100.9.b.d | 16 | ||
| 15.e | even | 4 | 2 | 100.9.d.c | 32 | ||
| 24.f | even | 2 | 1 | 320.9.b.d | 16 | ||
| 24.h | odd | 2 | 1 | 320.9.b.d | 16 | ||
| 60.h | even | 2 | 1 | 100.9.b.d | 16 | ||
| 60.l | odd | 4 | 2 | 100.9.d.c | 32 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 20.9.b.a | ✓ | 16 | 3.b | odd | 2 | 1 | |
| 20.9.b.a | ✓ | 16 | 12.b | even | 2 | 1 | |
| 100.9.b.d | 16 | 15.d | odd | 2 | 1 | ||
| 100.9.b.d | 16 | 60.h | even | 2 | 1 | ||
| 100.9.d.c | 32 | 15.e | even | 4 | 2 | ||
| 100.9.d.c | 32 | 60.l | odd | 4 | 2 | ||
| 180.9.c.a | 16 | 1.a | even | 1 | 1 | trivial | |
| 180.9.c.a | 16 | 4.b | odd | 2 | 1 | inner | |
| 320.9.b.d | 16 | 24.f | even | 2 | 1 | ||
| 320.9.b.d | 16 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{16} + 62520208 T_{7}^{14} + \cdots + 27\!\cdots\!00 \)
acting on \(S_{9}^{\mathrm{new}}(180, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + \cdots + 18\!\cdots\!16 \)
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| $3$ |
\( T^{16} \)
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| $5$ |
\( (T^{2} - 78125)^{8} \)
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| $7$ |
\( T^{16} + \cdots + 27\!\cdots\!00 \)
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| $11$ |
\( T^{16} + \cdots + 69\!\cdots\!00 \)
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| $13$ |
\( (T^{8} + \cdots + 36\!\cdots\!00)^{2} \)
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| $17$ |
\( (T^{8} + \cdots + 92\!\cdots\!00)^{2} \)
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| $19$ |
\( T^{16} + \cdots + 22\!\cdots\!00 \)
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| $23$ |
\( T^{16} + \cdots + 34\!\cdots\!00 \)
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| $29$ |
\( (T^{8} + \cdots - 61\!\cdots\!04)^{2} \)
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| $31$ |
\( T^{16} + \cdots + 81\!\cdots\!00 \)
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| $37$ |
\( (T^{8} + \cdots - 37\!\cdots\!00)^{2} \)
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| $41$ |
\( (T^{8} + \cdots - 56\!\cdots\!64)^{2} \)
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| $43$ |
\( T^{16} + \cdots + 26\!\cdots\!00 \)
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| $47$ |
\( T^{16} + \cdots + 11\!\cdots\!00 \)
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| $53$ |
\( (T^{8} + \cdots - 18\!\cdots\!00)^{2} \)
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| $59$ |
\( T^{16} + \cdots + 60\!\cdots\!00 \)
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| $61$ |
\( (T^{8} + \cdots + 83\!\cdots\!76)^{2} \)
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| $67$ |
\( T^{16} + \cdots + 75\!\cdots\!00 \)
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| $71$ |
\( T^{16} + \cdots + 29\!\cdots\!00 \)
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| $73$ |
\( (T^{8} + \cdots - 26\!\cdots\!00)^{2} \)
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| $79$ |
\( T^{16} + \cdots + 10\!\cdots\!00 \)
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| $83$ |
\( T^{16} + \cdots + 24\!\cdots\!00 \)
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| $89$ |
\( (T^{8} + \cdots - 36\!\cdots\!24)^{2} \)
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| $97$ |
\( (T^{8} + \cdots + 40\!\cdots\!00)^{2} \)
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