Newspace parameters
Level: | \( N \) | \(=\) | \( 24 = 2^{3} \cdot 3 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 24.h (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.48087911401\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{12} + \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{12} + 10x^{10} + 216x^{8} + 6848x^{6} + 55296x^{4} + 655360x^{2} + 16777216 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{17}\cdot 3^{4} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) 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 \( x^{12} + 10x^{10} + 216x^{8} + 6848x^{6} + 55296x^{4} + 655360x^{2} + 16777216 \) :
\(\beta_{1}\) | \(=\) | \( 4\nu \) |
\(\beta_{2}\) | \(=\) | \( ( - \nu^{11} - 38 \nu^{10} + 6 \nu^{9} + 4 \nu^{8} - 2104 \nu^{7} + 3824 \nu^{6} + 8896 \nu^{5} + 2944 \nu^{4} - 125952 \nu^{3} + 790528 \nu^{2} - 6324224 \nu + 24379392 ) / 5505024 \) |
\(\beta_{3}\) | \(=\) | \( ( -\nu^{11} - 10\nu^{9} - 216\nu^{7} - 6848\nu^{5} - 55296\nu^{3} - 655360\nu ) / 1048576 \) |
\(\beta_{4}\) | \(=\) | \( ( - 13 \nu^{11} - 152 \nu^{10} + 1086 \nu^{9} + 16 \nu^{8} + 25736 \nu^{7} + 15296 \nu^{6} + 75328 \nu^{5} + 11776 \nu^{4} + 3545088 \nu^{3} + 3162112 \nu^{2} + \cdots + 97517568 ) / 22020096 \) |
\(\beta_{5}\) | \(=\) | \( ( -\nu^{10} - 10\nu^{8} - 216\nu^{6} - 6848\nu^{4} - 55296\nu^{2} - 524288 ) / 65536 \) |
\(\beta_{6}\) | \(=\) | \( ( -\nu^{10} - 74\nu^{8} - 856\nu^{6} - 4288\nu^{4} - 264192\nu^{2} - 1638400 ) / 65536 \) |
\(\beta_{7}\) | \(=\) | \( ( - 25 \nu^{11} + 76 \nu^{10} + 486 \nu^{9} - 8 \nu^{8} - 6232 \nu^{7} - 7648 \nu^{6} + 36928 \nu^{5} - 5888 \nu^{4} + 184320 \nu^{3} - 1581056 \nu^{2} - 7405568 \nu - 48758784 ) / 11010048 \) |
\(\beta_{8}\) | \(=\) | \( ( - 55 \nu^{11} + 32 \nu^{10} - 1126 \nu^{9} - 3776 \nu^{8} - 1256 \nu^{7} + 31488 \nu^{6} - 140608 \nu^{5} - 141312 \nu^{4} - 956416 \nu^{3} - 22347776 \nu^{2} + \cdots + 66584576 ) / 11010048 \) |
\(\beta_{9}\) | \(=\) | \( ( -3\nu^{11} + 34\nu^{9} - 8\nu^{7} - 6720\nu^{5} + 534528\nu^{3} + 2097152\nu ) / 524288 \) |
\(\beta_{10}\) | \(=\) | \( ( - 135 \nu^{11} - 64 \nu^{10} - 1094 \nu^{9} + 384 \nu^{8} - 2024 \nu^{7} - 134656 \nu^{6} - 99136 \nu^{5} + 569344 \nu^{4} + 120832 \nu^{3} - 8060928 \nu^{2} + \cdots - 294649856 ) / 22020096 \) |
\(\beta_{11}\) | \(=\) | \( ( - 57 \nu^{11} - 212 \nu^{10} - 1114 \nu^{9} - 5448 \nu^{8} - 5464 \nu^{7} + 2848 \nu^{6} - 122816 \nu^{5} - 1285888 \nu^{4} - 1208320 \nu^{3} + 13983744 \nu^{2} + \cdots + 93323264 ) / 11010048 \) |
\(\nu\) | \(=\) | \( ( \beta_1 ) / 4 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{11} - \beta_{8} - \beta_{5} - \beta_{2} - 6 ) / 4 \) |
\(\nu^{3}\) | \(=\) | \( ( 2\beta_{9} - 2\beta_{7} - 2\beta_{4} - 6\beta_{3} - \beta_1 ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( ( - 3 \beta_{11} + 4 \beta_{10} - \beta_{8} - 8 \beta_{7} + 4 \beta_{6} - 17 \beta_{5} + 4 \beta_{4} + 8 \beta_{3} + 23 \beta_{2} - 106 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( 4 \beta_{11} + 12 \beta_{10} - 6 \beta_{9} + 8 \beta_{8} + 22 \beta_{7} - 4 \beta_{6} - 6 \beta_{4} - 158 \beta_{3} + 24 \beta_{2} - 21 \beta _1 - 4 \) |
\(\nu^{6}\) | \(=\) | \( \beta_{11} - 68 \beta_{10} + 67 \beta_{8} + 24 \beta_{7} - 20 \beta_{6} - 13 \beta_{5} + 44 \beta_{4} - 24 \beta_{3} + 107 \beta_{2} - 2490 \) |
\(\nu^{7}\) | \(=\) | \( 24 \beta_{11} + 72 \beta_{10} - 92 \beta_{9} + 48 \beta_{8} - 196 \beta_{7} - 24 \beta_{6} + 532 \beta_{4} - 12 \beta_{3} - 752 \beta_{2} - 882 \beta _1 - 24 \) |
\(\nu^{8}\) | \(=\) | \( - 886 \beta_{11} + 760 \beta_{10} + 126 \beta_{8} - 400 \beta_{7} - 744 \beta_{6} + 1630 \beta_{5} - 360 \beta_{4} + 400 \beta_{3} + 206 \beta_{2} + 10268 \) |
\(\nu^{9}\) | \(=\) | \( - 1104 \beta_{11} - 3312 \beta_{10} - 536 \beta_{9} - 2208 \beta_{8} + 8152 \beta_{7} + 1104 \beta_{6} + 6920 \beta_{4} + 17928 \beta_{3} + 2336 \beta_{2} + 2956 \beta _1 + 1104 \) |
\(\nu^{10}\) | \(=\) | \( 5092 \beta_{11} - 6608 \beta_{10} + 1516 \beta_{8} + 26208 \beta_{7} - 1936 \beta_{6} - 6996 \beta_{5} - 19600 \beta_{4} - 26208 \beta_{3} - 90100 \beta_{2} + 356760 \) |
\(\nu^{11}\) | \(=\) | \( - 21536 \beta_{11} - 64608 \beta_{10} + 11024 \beta_{9} - 43072 \beta_{8} - 134544 \beta_{7} + 21536 \beta_{6} - 87728 \beta_{4} + 22608 \beta_{3} - 25280 \beta_{2} + 168568 \beta _1 + 21536 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/24\mathbb{Z}\right)^\times\).
\(n\) | \(7\) | \(13\) | \(17\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 |
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−3.49180 | − | 1.95124i | 0.370354 | − | 8.99238i | 8.38535 | + | 13.6267i | −31.6607 | −18.8395 | + | 30.6769i | −41.0115 | −2.69111 | − | 63.9434i | −80.7257 | − | 6.66073i | 110.553 | + | 61.7775i | ||||||||||||||||||||||||||||||||||||||||
5.2 | −3.49180 | + | 1.95124i | 0.370354 | + | 8.99238i | 8.38535 | − | 13.6267i | −31.6607 | −18.8395 | − | 30.6769i | −41.0115 | −2.69111 | + | 63.9434i | −80.7257 | + | 6.66073i | 110.553 | − | 61.7775i | |||||||||||||||||||||||||||||||||||||||||
5.3 | −2.97547 | − | 2.67331i | 6.08044 | + | 6.63538i | 1.70680 | + | 15.9087i | 10.1873 | −0.353701 | − | 35.9983i | 73.1393 | 37.4504 | − | 51.8986i | −7.05649 | + | 80.6920i | −30.3121 | − | 27.2339i | |||||||||||||||||||||||||||||||||||||||||
5.4 | −2.97547 | + | 2.67331i | 6.08044 | − | 6.63538i | 1.70680 | − | 15.9087i | 10.1873 | −0.353701 | + | 35.9983i | 73.1393 | 37.4504 | + | 51.8986i | −7.05649 | − | 80.6920i | −30.3121 | + | 27.2339i | |||||||||||||||||||||||||||||||||||||||||
5.5 | −0.673742 | − | 3.94285i | −7.99319 | + | 4.13629i | −15.0921 | + | 5.31293i | −14.4851 | 21.6941 | + | 28.7292i | −34.1278 | 31.1163 | + | 55.9265i | 46.7822 | − | 66.1243i | 9.75922 | + | 57.1126i | |||||||||||||||||||||||||||||||||||||||||
5.6 | −0.673742 | + | 3.94285i | −7.99319 | − | 4.13629i | −15.0921 | − | 5.31293i | −14.4851 | 21.6941 | − | 28.7292i | −34.1278 | 31.1163 | − | 55.9265i | 46.7822 | + | 66.1243i | 9.75922 | − | 57.1126i | |||||||||||||||||||||||||||||||||||||||||
5.7 | 0.673742 | − | 3.94285i | 7.99319 | − | 4.13629i | −15.0921 | − | 5.31293i | 14.4851 | −10.9234 | − | 34.3027i | −34.1278 | −31.1163 | + | 55.9265i | 46.7822 | − | 66.1243i | 9.75922 | − | 57.1126i | |||||||||||||||||||||||||||||||||||||||||
5.8 | 0.673742 | + | 3.94285i | 7.99319 | + | 4.13629i | −15.0921 | + | 5.31293i | 14.4851 | −10.9234 | + | 34.3027i | −34.1278 | −31.1163 | − | 55.9265i | 46.7822 | + | 66.1243i | 9.75922 | + | 57.1126i | |||||||||||||||||||||||||||||||||||||||||
5.9 | 2.97547 | − | 2.67331i | −6.08044 | − | 6.63538i | 1.70680 | − | 15.9087i | −10.1873 | −35.8306 | − | 3.48842i | 73.1393 | −37.4504 | − | 51.8986i | −7.05649 | + | 80.6920i | −30.3121 | + | 27.2339i | |||||||||||||||||||||||||||||||||||||||||
5.10 | 2.97547 | + | 2.67331i | −6.08044 | + | 6.63538i | 1.70680 | + | 15.9087i | −10.1873 | −35.8306 | + | 3.48842i | 73.1393 | −37.4504 | + | 51.8986i | −7.05649 | − | 80.6920i | −30.3121 | − | 27.2339i | |||||||||||||||||||||||||||||||||||||||||
5.11 | 3.49180 | − | 1.95124i | −0.370354 | + | 8.99238i | 8.38535 | − | 13.6267i | 31.6607 | 16.2531 | + | 32.1222i | −41.0115 | 2.69111 | − | 63.9434i | −80.7257 | − | 6.66073i | 110.553 | − | 61.7775i | |||||||||||||||||||||||||||||||||||||||||
5.12 | 3.49180 | + | 1.95124i | −0.370354 | − | 8.99238i | 8.38535 | + | 13.6267i | 31.6607 | 16.2531 | − | 32.1222i | −41.0115 | 2.69111 | + | 63.9434i | −80.7257 | + | 6.66073i | 110.553 | + | 61.7775i | |||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
8.b | even | 2 | 1 | inner |
24.h | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 24.5.h.c | ✓ | 12 |
3.b | odd | 2 | 1 | inner | 24.5.h.c | ✓ | 12 |
4.b | odd | 2 | 1 | 96.5.h.c | 12 | ||
8.b | even | 2 | 1 | inner | 24.5.h.c | ✓ | 12 |
8.d | odd | 2 | 1 | 96.5.h.c | 12 | ||
12.b | even | 2 | 1 | 96.5.h.c | 12 | ||
24.f | even | 2 | 1 | 96.5.h.c | 12 | ||
24.h | odd | 2 | 1 | inner | 24.5.h.c | ✓ | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
24.5.h.c | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
24.5.h.c | ✓ | 12 | 3.b | odd | 2 | 1 | inner |
24.5.h.c | ✓ | 12 | 8.b | even | 2 | 1 | inner |
24.5.h.c | ✓ | 12 | 24.h | odd | 2 | 1 | inner |
96.5.h.c | 12 | 4.b | odd | 2 | 1 | ||
96.5.h.c | 12 | 8.d | odd | 2 | 1 | ||
96.5.h.c | 12 | 12.b | even | 2 | 1 | ||
96.5.h.c | 12 | 24.f | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} - 1316T_{5}^{4} + 336128T_{5}^{2} - 21827584 \)
acting on \(S_{5}^{\mathrm{new}}(24, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} + 10 T^{10} + 216 T^{8} + \cdots + 16777216 \)
$3$
\( T^{12} + 82 T^{10} + \cdots + 282429536481 \)
$5$
\( (T^{6} - 1316 T^{4} + 336128 T^{2} + \cdots - 21827584)^{2} \)
$7$
\( (T^{3} + 2 T^{2} - 4096 T - 102368)^{4} \)
$11$
\( (T^{6} - 42004 T^{4} + \cdots - 2501306728704)^{2} \)
$13$
\( (T^{6} + 97568 T^{4} + \cdots + 1287403978752)^{2} \)
$17$
\( (T^{6} + 306112 T^{4} + \cdots + 469646435155968)^{2} \)
$19$
\( (T^{6} + 492344 T^{4} + \cdots + 21\!\cdots\!68)^{2} \)
$23$
\( (T^{6} + 875008 T^{4} + \cdots + 922297011535872)^{2} \)
$29$
\( (T^{6} - 1237924 T^{4} + \cdots - 20\!\cdots\!64)^{2} \)
$31$
\( (T^{3} - 270 T^{2} - 1276896 T + 248591392)^{4} \)
$37$
\( (T^{6} + 2024480 T^{4} + \cdots + 11\!\cdots\!52)^{2} \)
$41$
\( (T^{6} + 7845376 T^{4} + \cdots + 70\!\cdots\!08)^{2} \)
$43$
\( (T^{6} + 5093048 T^{4} + \cdots + 71\!\cdots\!88)^{2} \)
$47$
\( (T^{6} + 23731200 T^{4} + \cdots + 26\!\cdots\!28)^{2} \)
$53$
\( (T^{6} - 25745188 T^{4} + \cdots - 32\!\cdots\!24)^{2} \)
$59$
\( (T^{6} - 29687060 T^{4} + \cdots - 14\!\cdots\!16)^{2} \)
$61$
\( (T^{6} + 23252256 T^{4} + \cdots + 74\!\cdots\!92)^{2} \)
$67$
\( (T^{6} + 25233336 T^{4} + \cdots + 19\!\cdots\!48)^{2} \)
$71$
\( (T^{6} + 92256768 T^{4} + \cdots + 30\!\cdots\!72)^{2} \)
$73$
\( (T^{3} - 3158 T^{2} + \cdots + 55695205944)^{4} \)
$79$
\( (T^{3} - 1678 T^{2} + \cdots + 2204573728)^{4} \)
$83$
\( (T^{6} - 117411860 T^{4} + \cdots - 18\!\cdots\!16)^{2} \)
$89$
\( (T^{6} + 136507072 T^{4} + \cdots + 14\!\cdots\!88)^{2} \)
$97$
\( (T^{3} + 7826 T^{2} - 7078852 T + 286711096)^{4} \)
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