Newspace parameters
| Level: | \( N \) | \(=\) | \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 360.q (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(21.2406876021\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{20} - 6 x^{19} + 9 x^{18} + 228 x^{17} - 1491 x^{16} + 5274 x^{15} + 540 x^{14} + \cdots + 205891132094649 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{22}\cdot 3^{15} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 6 x^{19} + 9 x^{18} + 228 x^{17} - 1491 x^{16} + 5274 x^{15} + 540 x^{14} + \cdots + 205891132094649 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 69942469 \nu^{19} - 831856812 \nu^{18} - 3102014088 \nu^{17} + 7604625702 \nu^{16} + \cdots - 89\!\cdots\!50 ) / 20\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 45800222 \nu^{19} - 414610701 \nu^{18} - 926917470 \nu^{17} - 13264257012 \nu^{16} + \cdots - 16\!\cdots\!09 ) / 22\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 183704849 \nu^{19} - 364556514 \nu^{18} - 4327571322 \nu^{17} + 47644060641 \nu^{16} + \cdots - 64\!\cdots\!69 ) / 32\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3966795971 \nu^{19} - 80626820583 \nu^{18} - 217206929142 \nu^{17} - 313327140582 \nu^{16} + \cdots - 66\!\cdots\!25 ) / 70\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 9652855402 \nu^{19} - 165675490107 \nu^{18} - 382742725626 \nu^{17} + 1257257421588 \nu^{16} + \cdots - 16\!\cdots\!07 ) / 14\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 398397850 \nu^{19} - 1760904879 \nu^{18} - 3901130658 \nu^{17} + 62916583008 \nu^{16} + \cdots - 21\!\cdots\!23 ) / 22\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 13216742975 \nu^{19} + 104595404433 \nu^{18} + 398726313066 \nu^{17} + \cdots + 89\!\cdots\!71 ) / 47\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 946792921 \nu^{19} - 746375406 \nu^{18} + 3407287662 \nu^{17} + 156945116964 \nu^{16} + \cdots - 26\!\cdots\!26 ) / 32\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 3137325383 \nu^{19} + 1900291218 \nu^{18} + 11147121738 \nu^{17} + 583495728810 \nu^{16} + \cdots - 46\!\cdots\!76 ) / 10\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 6438691679 \nu^{19} + 3401425515 \nu^{18} + 95481515754 \nu^{17} - 955904256078 \nu^{16} + \cdots + 73\!\cdots\!01 ) / 20\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 45381525235 \nu^{19} - 6987927657 \nu^{18} + 270375342246 \nu^{17} + \cdots + 12\!\cdots\!81 ) / 14\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 308765071 \nu^{19} + 1300086609 \nu^{18} + 3007378494 \nu^{17} - 48314719470 \nu^{16} + \cdots + 16\!\cdots\!87 ) / 91\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 8509128289 \nu^{19} + 40329537483 \nu^{18} + 124028552250 \nu^{17} + \cdots + 42\!\cdots\!57 ) / 23\!\cdots\!00 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 8688149081 \nu^{19} - 10884822996 \nu^{18} + 52232107572 \nu^{17} + 2034588732414 \nu^{16} + \cdots - 16\!\cdots\!46 ) / 20\!\cdots\!00 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 67720850041 \nu^{19} - 8284472709 \nu^{18} - 470602071414 \nu^{17} + 10551607581810 \nu^{16} + \cdots - 18\!\cdots\!67 ) / 14\!\cdots\!00 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 71203592759 \nu^{19} - 46149291291 \nu^{18} + 223632315414 \nu^{17} + 14537087258190 \nu^{16} + \cdots - 21\!\cdots\!33 ) / 14\!\cdots\!00 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 44525631335 \nu^{19} + 62064657354 \nu^{18} + 87294900618 \nu^{17} + \cdots + 12\!\cdots\!88 ) / 70\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{19} + 2 \beta_{18} - \beta_{17} - \beta_{14} - \beta_{13} - \beta_{11} - \beta_{10} - \beta_{9} + \cdots - 27 \)
|
| \(\nu^{4}\) | \(=\) |
\( - \beta_{19} + \beta_{18} - 6 \beta_{17} - 3 \beta_{16} - 4 \beta_{14} + \beta_{13} - 5 \beta_{12} + \cdots + 148 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 5 \beta_{19} + 25 \beta_{18} - 9 \beta_{17} - 3 \beta_{16} + 19 \beta_{15} + 11 \beta_{14} + \cdots - 64 \)
|
| \(\nu^{6}\) | \(=\) |
\( 27 \beta_{19} - 3 \beta_{18} + 168 \beta_{17} + 84 \beta_{16} + 186 \beta_{15} + 54 \beta_{14} + \cdots + 699 \)
|
| \(\nu^{7}\) | \(=\) |
\( 54 \beta_{19} - 225 \beta_{18} + 750 \beta_{17} + 531 \beta_{16} - 936 \beta_{15} + 150 \beta_{14} + \cdots - 65001 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 819 \beta_{19} - 4245 \beta_{18} - 846 \beta_{17} - 927 \beta_{16} - 1992 \beta_{15} - 12 \beta_{14} + \cdots - 147825 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 8364 \beta_{19} - 4314 \beta_{18} - 11682 \beta_{17} - 8829 \beta_{16} + 5154 \beta_{15} + \cdots - 143025 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 76248 \beta_{19} - 108153 \beta_{18} + 37908 \beta_{17} + 10521 \beta_{16} + 106713 \beta_{15} + \cdots + 6896304 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 451251 \beta_{19} + 127647 \beta_{18} + 269343 \beta_{17} - 220590 \beta_{16} - 135882 \beta_{15} + \cdots + 8859789 \)
|
| \(\nu^{12}\) | \(=\) |
\( 75690 \beta_{19} + 732159 \beta_{18} + 1611414 \beta_{17} - 99279 \beta_{16} + 2132127 \beta_{15} + \cdots - 103394565 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 574119 \beta_{19} + 11916054 \beta_{18} - 12264075 \beta_{17} - 8587512 \beta_{16} - 21226203 \beta_{15} + \cdots - 143519913 \)
|
| \(\nu^{14}\) | \(=\) |
\( 20538360 \beta_{19} + 7933626 \beta_{18} - 151691022 \beta_{17} - 70470783 \beta_{16} + \cdots + 1690224813 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 100287666 \beta_{19} + 206294067 \beta_{18} - 112085424 \beta_{17} - 137254662 \beta_{16} + \cdots + 5015926458 \)
|
| \(\nu^{16}\) | \(=\) |
\( 2203596117 \beta_{19} + 1699529580 \beta_{18} + 2674311066 \beta_{17} + 3024693738 \beta_{16} + \cdots + 43715415582 \)
|
| \(\nu^{17}\) | \(=\) |
\( 4824317772 \beta_{19} + 2145650121 \beta_{18} + 7236830142 \beta_{17} + 860462109 \beta_{16} + \cdots - 412441091865 \)
|
| \(\nu^{18}\) | \(=\) |
\( 50812534035 \beta_{19} - 56779419933 \beta_{18} + 8929800288 \beta_{17} + 32867884917 \beta_{16} + \cdots - 3951307894545 \)
|
| \(\nu^{19}\) | \(=\) |
\( 77574126627 \beta_{19} - 299498687298 \beta_{18} - 46785680919 \beta_{17} + 70233632685 \beta_{16} + \cdots + 14354716981608 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).
| \(n\) | \(181\) | \(217\) | \(271\) | \(281\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1 - \beta_{3}\) |
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 | −5.16520 | − | 0.566318i | 0 | 2.50000 | − | 4.33013i | 0 | 1.57018 | + | 2.71963i | 0 | 26.3586 | + | 5.85030i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.2 | 0 | −3.71666 | − | 3.63131i | 0 | 2.50000 | − | 4.33013i | 0 | −0.754378 | − | 1.30662i | 0 | 0.627193 | + | 26.9927i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.3 | 0 | −3.19865 | + | 4.09495i | 0 | 2.50000 | − | 4.33013i | 0 | −6.25621 | − | 10.8361i | 0 | −6.53729 | − | 26.1966i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.4 | 0 | −2.25193 | + | 4.68282i | 0 | 2.50000 | − | 4.33013i | 0 | 17.4237 | + | 30.1787i | 0 | −16.8576 | − | 21.0908i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.5 | 0 | −0.163780 | − | 5.19357i | 0 | 2.50000 | − | 4.33013i | 0 | 12.4893 | + | 21.6321i | 0 | −26.9464 | + | 1.70121i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.6 | 0 | 0.797147 | − | 5.13464i | 0 | 2.50000 | − | 4.33013i | 0 | −15.7660 | − | 27.3074i | 0 | −25.7291 | − | 8.18613i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.7 | 0 | 1.86398 | + | 4.85032i | 0 | 2.50000 | − | 4.33013i | 0 | −5.21366 | − | 9.03032i | 0 | −20.0512 | + | 18.0818i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.8 | 0 | 4.57310 | − | 2.46713i | 0 | 2.50000 | − | 4.33013i | 0 | 14.1295 | + | 24.4730i | 0 | 14.8265 | − | 22.5649i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.9 | 0 | 5.11851 | + | 0.894893i | 0 | 2.50000 | − | 4.33013i | 0 | −12.4200 | − | 21.5121i | 0 | 25.3983 | + | 9.16104i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 121.10 | 0 | 5.14349 | + | 0.737938i | 0 | 2.50000 | − | 4.33013i | 0 | 5.79756 | + | 10.0417i | 0 | 25.9109 | + | 7.59115i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 241.1 | 0 | −5.16520 | + | 0.566318i | 0 | 2.50000 | + | 4.33013i | 0 | 1.57018 | − | 2.71963i | 0 | 26.3586 | − | 5.85030i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 241.2 | 0 | −3.71666 | + | 3.63131i | 0 | 2.50000 | + | 4.33013i | 0 | −0.754378 | + | 1.30662i | 0 | 0.627193 | − | 26.9927i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 241.3 | 0 | −3.19865 | − | 4.09495i | 0 | 2.50000 | + | 4.33013i | 0 | −6.25621 | + | 10.8361i | 0 | −6.53729 | + | 26.1966i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 241.4 | 0 | −2.25193 | − | 4.68282i | 0 | 2.50000 | + | 4.33013i | 0 | 17.4237 | − | 30.1787i | 0 | −16.8576 | + | 21.0908i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 241.5 | 0 | −0.163780 | + | 5.19357i | 0 | 2.50000 | + | 4.33013i | 0 | 12.4893 | − | 21.6321i | 0 | −26.9464 | − | 1.70121i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 241.6 | 0 | 0.797147 | + | 5.13464i | 0 | 2.50000 | + | 4.33013i | 0 | −15.7660 | + | 27.3074i | 0 | −25.7291 | + | 8.18613i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 241.7 | 0 | 1.86398 | − | 4.85032i | 0 | 2.50000 | + | 4.33013i | 0 | −5.21366 | + | 9.03032i | 0 | −20.0512 | − | 18.0818i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 241.8 | 0 | 4.57310 | + | 2.46713i | 0 | 2.50000 | + | 4.33013i | 0 | 14.1295 | − | 24.4730i | 0 | 14.8265 | + | 22.5649i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 241.9 | 0 | 5.11851 | − | 0.894893i | 0 | 2.50000 | + | 4.33013i | 0 | −12.4200 | + | 21.5121i | 0 | 25.3983 | − | 9.16104i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 241.10 | 0 | 5.14349 | − | 0.737938i | 0 | 2.50000 | + | 4.33013i | 0 | 5.79756 | − | 10.0417i | 0 | 25.9109 | − | 7.59115i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 360.4.q.e | ✓ | 20 |
| 3.b | odd | 2 | 1 | 1080.4.q.e | 20 | ||
| 9.c | even | 3 | 1 | inner | 360.4.q.e | ✓ | 20 |
| 9.d | odd | 6 | 1 | 1080.4.q.e | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 360.4.q.e | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 360.4.q.e | ✓ | 20 | 9.c | even | 3 | 1 | inner |
| 1080.4.q.e | 20 | 3.b | odd | 2 | 1 | ||
| 1080.4.q.e | 20 | 9.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{20} - 22 T_{7}^{19} + 2572 T_{7}^{18} - 31512 T_{7}^{17} + 3795924 T_{7}^{16} + \cdots + 19\!\cdots\!64 \)
acting on \(S_{4}^{\mathrm{new}}(360, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( T^{20} + \cdots + 205891132094649 \)
|
| $5$ |
\( (T^{2} - 5 T + 25)^{10} \)
|
| $7$ |
\( T^{20} + \cdots + 19\!\cdots\!64 \)
|
| $11$ |
\( T^{20} + \cdots + 78\!\cdots\!76 \)
|
| $13$ |
\( T^{20} + \cdots + 11\!\cdots\!24 \)
|
| $17$ |
\( (T^{10} + \cdots + 19\!\cdots\!32)^{2} \)
|
| $19$ |
\( (T^{10} + \cdots - 69\!\cdots\!28)^{2} \)
|
| $23$ |
\( T^{20} + \cdots + 13\!\cdots\!56 \)
|
| $29$ |
\( T^{20} + \cdots + 27\!\cdots\!84 \)
|
| $31$ |
\( T^{20} + \cdots + 16\!\cdots\!84 \)
|
| $37$ |
\( (T^{10} + \cdots + 34\!\cdots\!48)^{2} \)
|
| $41$ |
\( T^{20} + \cdots + 36\!\cdots\!25 \)
|
| $43$ |
\( T^{20} + \cdots + 20\!\cdots\!76 \)
|
| $47$ |
\( T^{20} + \cdots + 39\!\cdots\!84 \)
|
| $53$ |
\( (T^{10} + \cdots + 68\!\cdots\!92)^{2} \)
|
| $59$ |
\( T^{20} + \cdots + 53\!\cdots\!24 \)
|
| $61$ |
\( T^{20} + \cdots + 12\!\cdots\!56 \)
|
| $67$ |
\( T^{20} + \cdots + 43\!\cdots\!25 \)
|
| $71$ |
\( (T^{10} + \cdots - 48\!\cdots\!80)^{2} \)
|
| $73$ |
\( (T^{10} + \cdots + 10\!\cdots\!44)^{2} \)
|
| $79$ |
\( T^{20} + \cdots + 63\!\cdots\!96 \)
|
| $83$ |
\( T^{20} + \cdots + 18\!\cdots\!00 \)
|
| $89$ |
\( (T^{10} + \cdots - 16\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{20} + \cdots + 29\!\cdots\!04 \)
|
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