Newspace parameters
Level: | \( N \) | \(=\) | \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1620.x (of order \(12\), degree \(4\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(12.9357651274\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{12})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{16} - 12x^{14} + 100x^{12} - 408x^{10} + 1191x^{8} - 2040x^{6} + 2500x^{4} - 1500x^{2} + 625 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 3^{4}\cdot 5^{2} \) |
Twist minimal: | no (minimal twist has level 540) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 \( x^{16} - 12x^{14} + 100x^{12} - 408x^{10} + 1191x^{8} - 2040x^{6} + 2500x^{4} - 1500x^{2} + 625 \) :
\(\beta_{1}\) | \(=\) | \( ( 162 \nu^{14} + 1386 \nu^{12} + 190 \nu^{10} + 31754 \nu^{8} + 610802 \nu^{6} - 1207925 \nu^{4} + 3359500 \nu^{2} + 1408250 ) / 3170625 \) |
\(\beta_{2}\) | \(=\) | \( ( 519 \nu^{15} - 11728 \nu^{13} + 111150 \nu^{11} - 674877 \nu^{9} + 2149629 \nu^{7} - 4634385 \nu^{5} + 4860000 \nu^{3} - 3584500 \nu ) / 634125 \) |
\(\beta_{3}\) | \(=\) | \( ( - 3896 \nu^{15} + 58387 \nu^{13} - 477945 \nu^{11} + 2122518 \nu^{9} - 4421091 \nu^{7} + 2915550 \nu^{5} + 7915750 \nu^{3} - 16439750 \nu ) / 3170625 \) |
\(\beta_{4}\) | \(=\) | \( ( - 4084 \nu^{14} + 46148 \nu^{12} - 378480 \nu^{10} + 1425447 \nu^{8} - 4067664 \nu^{6} + 6150300 \nu^{4} - 8357500 \nu^{2} + 4996625 ) / 3170625 \) |
\(\beta_{5}\) | \(=\) | \( ( 831 \nu^{14} - 5647 \nu^{12} + 31350 \nu^{10} + 79902 \nu^{8} - 618129 \nu^{6} + 2318760 \nu^{4} - 2891475 \nu^{2} + 2262875 ) / 634125 \) |
\(\beta_{6}\) | \(=\) | \( ( - 5812 \nu^{15} + 75864 \nu^{13} - 662340 \nu^{11} + 3059571 \nu^{9} - 10067427 \nu^{7} + 21289500 \nu^{5} - 32848375 \nu^{3} + 22664250 \nu ) / 3170625 \) |
\(\beta_{7}\) | \(=\) | \( ( -12\nu^{14} + 119\nu^{12} - 950\nu^{10} + 2996\nu^{8} - 8842\nu^{6} + 12730\nu^{4} - 19550\nu^{2} + 7750 ) / 7125 \) |
\(\beta_{8}\) | \(=\) | \( ( 8514 \nu^{15} - 71783 \nu^{13} + 526680 \nu^{11} - 891137 \nu^{9} + 1391094 \nu^{7} + 5307650 \nu^{5} - 6466875 \nu^{3} + 12366500 \nu ) / 3170625 \) |
\(\beta_{9}\) | \(=\) | \( ( 1903 \nu^{15} - 23126 \nu^{13} + 204630 \nu^{11} - 910374 \nu^{9} + 3154293 \nu^{7} - 6153435 \nu^{5} + 8646325 \nu^{3} - 4035500 \nu ) / 634125 \) |
\(\beta_{10}\) | \(=\) | \( ( - 10471 \nu^{14} + 144287 \nu^{12} - 1218945 \nu^{10} + 5567493 \nu^{8} - 15642291 \nu^{6} + 28273425 \nu^{4} - 27410125 \nu^{2} + 14057750 ) / 3170625 \) |
\(\beta_{11}\) | \(=\) | \( ( 12874 \nu^{14} - 123728 \nu^{12} + 939930 \nu^{10} - 2489517 \nu^{8} + 5326479 \nu^{6} - 1132875 \nu^{4} - 2093375 \nu^{2} + 7535875 ) / 3170625 \) |
\(\beta_{12}\) | \(=\) | \( ( 14343 \nu^{14} - 183596 \nu^{12} + 1544985 \nu^{10} - 6645669 \nu^{8} + 18837978 \nu^{6} - 30781425 \nu^{4} + 28939125 \nu^{2} - 11347625 ) / 3170625 \) |
\(\beta_{13}\) | \(=\) | \( ( 19846 \nu^{15} - 243562 \nu^{13} + 2011245 \nu^{11} - 8254868 \nu^{9} + 22837116 \nu^{7} - 37808575 \nu^{5} + 37381375 \nu^{3} - 19270875 \nu ) / 3170625 \) |
\(\beta_{14}\) | \(=\) | \( ( 20057 \nu^{15} - 242004 \nu^{13} + 2020365 \nu^{11} - 8294406 \nu^{9} + 24255447 \nu^{7} - 41296500 \nu^{5} + 47826875 \nu^{3} - 14753625 \nu ) / 3170625 \) |
\(\beta_{15}\) | \(=\) | \( ( - 26303 \nu^{15} + 315391 \nu^{13} - 2586660 \nu^{11} + 10264224 \nu^{9} - 27799788 \nu^{7} + 42033225 \nu^{5} - 40406750 \nu^{3} + 15650125 \nu ) / 3170625 \) |
\(\nu\) | \(=\) | \( ( 3\beta_{15} + \beta_{14} + 3\beta_{13} + 2\beta_{9} - 3\beta_{8} - \beta_{6} - 4\beta_{3} - 6\beta_{2} ) / 15 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{12} + \beta_{11} + \beta_{10} + 3\beta_{7} - 2\beta_{5} - 9\beta_{4} - 3\beta _1 + 9 ) / 3 \) |
\(\nu^{3}\) | \(=\) | \( ( -24\beta_{15} - 13\beta_{14} - 9\beta_{13} + 19\beta_{9} - 36\beta_{8} - 2\beta_{6} - 8\beta_{3} - 42\beta_{2} ) / 15 \) |
\(\nu^{4}\) | \(=\) | \( -6\beta_{12} + 6\beta_{11} - 4\beta_{10} + 14\beta_{7} - 4\beta_{5} - 14\beta_{4} + 7\beta_1 \) |
\(\nu^{5}\) | \(=\) | \( ( - 243 \beta_{15} + 14 \beta_{14} - 213 \beta_{13} + 43 \beta_{9} - 192 \beta_{8} + 106 \beta_{6} + 79 \beta_{3} - 114 \beta_{2} ) / 15 \) |
\(\nu^{6}\) | \(=\) | \( ( -95\beta_{12} + 38\beta_{11} - 133\beta_{10} + 135\beta_{7} + 38\beta_{5} + 270\beta _1 - 243 ) / 3 \) |
\(\nu^{7}\) | \(=\) | \( ( - 756 \beta_{15} + 778 \beta_{14} - 1296 \beta_{13} - 559 \beta_{9} + 36 \beta_{8} + 617 \beta_{6} + 818 \beta_{3} + 777 \beta_{2} ) / 15 \) |
\(\nu^{8}\) | \(=\) | \( 84\beta_{12} - 224\beta_{11} - 84\beta_{10} - 292\beta_{7} + 308\beta_{5} + 511\beta_{4} + 292\beta _1 - 511 \) |
\(\nu^{9}\) | \(=\) | \( ( 4938 \beta_{15} + 4111 \beta_{14} - 27 \beta_{13} - 5488 \beta_{9} + 8622 \beta_{8} - 1246 \beta_{6} + 1646 \beta_{3} + 9999 \beta_{2} ) / 15 \) |
\(\nu^{10}\) | \(=\) | \( ( 6266\beta_{12} - 6266\beta_{11} + 4579\beta_{10} - 11514\beta_{7} + 4579\beta_{5} + 9999\beta_{4} - 5757\beta_1 ) / 3 \) |
\(\nu^{11}\) | \(=\) | \( ( 66066 \beta_{15} - 8758 \beta_{14} + 56736 \beta_{13} - 10736 \beta_{9} + 57609 \beta_{8} - 36137 \beta_{6} - 25973 \beta_{3} + 33858 \beta_{2} ) / 15 \) |
\(\nu^{12}\) | \(=\) | \( 10260\beta_{12} - 3762\beta_{11} + 14022\beta_{10} - 12705\beta_{7} - 3762\beta_{5} - 25410\beta _1 + 22022 \) |
\(\nu^{13}\) | \(=\) | \( ( 226122 \beta_{15} - 241966 \beta_{14} + 383712 \beta_{13} + 174193 \beta_{9} + 7653 \beta_{8} - 182384 \beta_{6} - 245066 \beta_{3} - 212619 \beta_{2} ) / 15 \) |
\(\nu^{14}\) | \(=\) | \( ( - 75374 \beta_{12} + 205829 \beta_{11} + 75374 \beta_{10} + 253257 \beta_{7} - 281203 \beta_{5} - 438741 \beta_{4} - 253257 \beta _1 + 438741 ) / 3 \) |
\(\nu^{15}\) | \(=\) | \( ( - 1411131 \beta_{15} - 1215167 \beta_{14} + 55944 \beta_{13} + 1634426 \beta_{9} - 2500344 \beta_{8} + 400367 \beta_{6} - 470377 \beta_{3} - 2919603 \beta_{2} ) / 15 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).
\(n\) | \(811\) | \(1297\) | \(1541\) |
\(\chi(n)\) | \(1\) | \(-\beta_{7}\) | \(\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
53.1 |
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0 | 0 | 0 | −1.70289 | − | 1.44919i | 0 | −0.307007 | − | 0.0822623i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
53.2 | 0 | 0 | 0 | −0.0455319 | + | 2.23560i | 0 | 3.03906 | + | 0.814313i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
53.3 | 0 | 0 | 0 | 0.0455319 | − | 2.23560i | 0 | 3.03906 | + | 0.814313i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
53.4 | 0 | 0 | 0 | 1.70289 | + | 1.44919i | 0 | −0.307007 | − | 0.0822623i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
377.1 | 0 | 0 | 0 | −2.10648 | − | 0.750156i | 0 | 0.0822623 | − | 0.307007i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
377.2 | 0 | 0 | 0 | −1.91332 | + | 1.15723i | 0 | −0.814313 | + | 3.03906i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
377.3 | 0 | 0 | 0 | 1.91332 | − | 1.15723i | 0 | −0.814313 | + | 3.03906i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
377.4 | 0 | 0 | 0 | 2.10648 | + | 0.750156i | 0 | 0.0822623 | − | 0.307007i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
593.1 | 0 | 0 | 0 | −2.10648 | + | 0.750156i | 0 | 0.0822623 | + | 0.307007i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
593.2 | 0 | 0 | 0 | −1.91332 | − | 1.15723i | 0 | −0.814313 | − | 3.03906i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
593.3 | 0 | 0 | 0 | 1.91332 | + | 1.15723i | 0 | −0.814313 | − | 3.03906i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
593.4 | 0 | 0 | 0 | 2.10648 | − | 0.750156i | 0 | 0.0822623 | + | 0.307007i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
917.1 | 0 | 0 | 0 | −1.70289 | + | 1.44919i | 0 | −0.307007 | + | 0.0822623i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
917.2 | 0 | 0 | 0 | −0.0455319 | − | 2.23560i | 0 | 3.03906 | − | 0.814313i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
917.3 | 0 | 0 | 0 | 0.0455319 | + | 2.23560i | 0 | 3.03906 | − | 0.814313i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
917.4 | 0 | 0 | 0 | 1.70289 | − | 1.44919i | 0 | −0.307007 | + | 0.0822623i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
9.c | even | 3 | 1 | inner |
9.d | odd | 6 | 1 | inner |
15.e | even | 4 | 1 | inner |
45.k | odd | 12 | 1 | inner |
45.l | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1620.2.x.d | 16 | |
3.b | odd | 2 | 1 | inner | 1620.2.x.d | 16 | |
5.c | odd | 4 | 1 | inner | 1620.2.x.d | 16 | |
9.c | even | 3 | 1 | 540.2.j.a | ✓ | 8 | |
9.c | even | 3 | 1 | inner | 1620.2.x.d | 16 | |
9.d | odd | 6 | 1 | 540.2.j.a | ✓ | 8 | |
9.d | odd | 6 | 1 | inner | 1620.2.x.d | 16 | |
15.e | even | 4 | 1 | inner | 1620.2.x.d | 16 | |
36.f | odd | 6 | 1 | 2160.2.w.e | 8 | ||
36.h | even | 6 | 1 | 2160.2.w.e | 8 | ||
45.h | odd | 6 | 1 | 2700.2.j.j | 8 | ||
45.j | even | 6 | 1 | 2700.2.j.j | 8 | ||
45.k | odd | 12 | 1 | 540.2.j.a | ✓ | 8 | |
45.k | odd | 12 | 1 | inner | 1620.2.x.d | 16 | |
45.k | odd | 12 | 1 | 2700.2.j.j | 8 | ||
45.l | even | 12 | 1 | 540.2.j.a | ✓ | 8 | |
45.l | even | 12 | 1 | inner | 1620.2.x.d | 16 | |
45.l | even | 12 | 1 | 2700.2.j.j | 8 | ||
180.v | odd | 12 | 1 | 2160.2.w.e | 8 | ||
180.x | even | 12 | 1 | 2160.2.w.e | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
540.2.j.a | ✓ | 8 | 9.c | even | 3 | 1 | |
540.2.j.a | ✓ | 8 | 9.d | odd | 6 | 1 | |
540.2.j.a | ✓ | 8 | 45.k | odd | 12 | 1 | |
540.2.j.a | ✓ | 8 | 45.l | even | 12 | 1 | |
1620.2.x.d | 16 | 1.a | even | 1 | 1 | trivial | |
1620.2.x.d | 16 | 3.b | odd | 2 | 1 | inner | |
1620.2.x.d | 16 | 5.c | odd | 4 | 1 | inner | |
1620.2.x.d | 16 | 9.c | even | 3 | 1 | inner | |
1620.2.x.d | 16 | 9.d | odd | 6 | 1 | inner | |
1620.2.x.d | 16 | 15.e | even | 4 | 1 | inner | |
1620.2.x.d | 16 | 45.k | odd | 12 | 1 | inner | |
1620.2.x.d | 16 | 45.l | even | 12 | 1 | inner | |
2160.2.w.e | 8 | 36.f | odd | 6 | 1 | ||
2160.2.w.e | 8 | 36.h | even | 6 | 1 | ||
2160.2.w.e | 8 | 180.v | odd | 12 | 1 | ||
2160.2.w.e | 8 | 180.x | even | 12 | 1 | ||
2700.2.j.j | 8 | 45.h | odd | 6 | 1 | ||
2700.2.j.j | 8 | 45.j | even | 6 | 1 | ||
2700.2.j.j | 8 | 45.k | odd | 12 | 1 | ||
2700.2.j.j | 8 | 45.l | even | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} - 4T_{7}^{7} + 8T_{7}^{6} - 40T_{7}^{5} + 79T_{7}^{4} + 40T_{7}^{3} + 8T_{7}^{2} + 4T_{7} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(1620, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} \)
$3$
\( T^{16} \)
$5$
\( T^{16} - 4 T^{14} + 16 T^{12} + \cdots + 390625 \)
$7$
\( (T^{8} - 4 T^{7} + 8 T^{6} - 40 T^{5} + 79 T^{4} + \cdots + 1)^{2} \)
$11$
\( (T^{8} - 40 T^{6} + 1350 T^{4} + \cdots + 62500)^{2} \)
$13$
\( (T^{8} - 9 T^{4} + 81)^{2} \)
$17$
\( (T^{8} + 4400 T^{4} + 1000000)^{2} \)
$19$
\( (T^{4} + 50 T^{2} + 529)^{4} \)
$23$
\( T^{16} - 5900 T^{12} + \cdots + 3906250000 \)
$29$
\( (T^{8} + 40 T^{6} + 1350 T^{4} + \cdots + 62500)^{2} \)
$31$
\( (T^{4} - 4 T^{3} + 18 T^{2} + 8 T + 4)^{4} \)
$37$
\( (T^{4} + 729)^{4} \)
$41$
\( (T^{8} - 160 T^{6} + 19350 T^{4} + \cdots + 39062500)^{2} \)
$43$
\( (T^{8} + 12 T^{7} + 72 T^{6} + 720 T^{5} + \cdots + 1296)^{2} \)
$47$
\( T^{16} \)
$53$
\( (T^{8} + 5900 T^{4} + 62500)^{2} \)
$59$
\( (T^{8} + 160 T^{6} + 24600 T^{4} + \cdots + 1000000)^{2} \)
$61$
\( (T^{4} + 6 T^{3} + 123 T^{2} - 522 T + 7569)^{4} \)
$67$
\( (T^{8} + 20 T^{7} + 200 T^{6} + \cdots + 279841)^{2} \)
$71$
\( (T^{4} + 160 T^{2} + 6250)^{4} \)
$73$
\( (T^{4} + 20 T^{3} + 200 T^{2} + 940 T + 2209)^{4} \)
$79$
\( (T^{8} - 318 T^{6} + 81243 T^{4} + \cdots + 395254161)^{2} \)
$83$
\( T^{16} - 9900 T^{12} + \cdots + 25628906250000 \)
$89$
\( (T^{4} - 160 T^{2} + 1000)^{4} \)
$97$
\( (T^{8} - 24 T^{7} + 288 T^{6} + \cdots + 22667121)^{2} \)
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