Newspace parameters
Level: | \( N \) | \(=\) | \( 800 = 2^{5} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 800.f (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(47.2015280046\) |
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: | \( x^{12} - 4x^{11} + 7x^{10} - 12x^{9} + 21x^{8} - 68x^{6} + 336x^{4} - 768x^{3} + 1792x^{2} - 4096x + 4096 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
Coefficient ring index: | \( 2^{32} \) |
Twist minimal: | no (minimal twist has level 40) |
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 \( x^{12} - 4x^{11} + 7x^{10} - 12x^{9} + 21x^{8} - 68x^{6} + 336x^{4} - 768x^{3} + 1792x^{2} - 4096x + 4096 \) :
\(\beta_{1}\) | \(=\) | \( ( 47 \nu^{11} - 84 \nu^{10} + 217 \nu^{9} - 156 \nu^{8} + 715 \nu^{7} + 712 \nu^{6} - 3340 \nu^{5} - 2592 \nu^{4} + 8816 \nu^{3} - 11392 \nu^{2} + 51200 \nu - 130048 ) / 15360 \) |
\(\beta_{2}\) | \(=\) | \( ( - 23 \nu^{11} + 60 \nu^{10} - 81 \nu^{9} + 116 \nu^{8} - 51 \nu^{7} - 480 \nu^{6} + 684 \nu^{5} + 1024 \nu^{4} - 5616 \nu^{3} + 11520 \nu^{2} - 22528 \nu + 30720 ) / 7680 \) |
\(\beta_{3}\) | \(=\) | \( ( - 11 \nu^{11} + 28 \nu^{10} - 349 \nu^{9} + 852 \nu^{8} - 215 \nu^{7} - 400 \nu^{6} + 220 \nu^{5} - 960 \nu^{4} + 6992 \nu^{3} + 19456 \nu^{2} - 92672 \nu + 89088 ) / 3072 \) |
\(\beta_{4}\) | \(=\) | \( ( 97 \nu^{11} - 212 \nu^{10} + 215 \nu^{9} - 636 \nu^{8} + 709 \nu^{7} + 2096 \nu^{6} - 4244 \nu^{5} - 10176 \nu^{4} + 16400 \nu^{3} - 38912 \nu^{2} + 122368 \nu - 184320 ) / 15360 \) |
\(\beta_{5}\) | \(=\) | \( ( 69 \nu^{11} - 36 \nu^{10} - 205 \nu^{9} - 236 \nu^{8} + 857 \nu^{7} + 2288 \nu^{6} - 2948 \nu^{5} - 12736 \nu^{4} + 16336 \nu^{3} + 44032 \nu^{2} - 16384 \nu - 131072 ) / 7680 \) |
\(\beta_{6}\) | \(=\) | \( ( - 211 \nu^{11} + 468 \nu^{10} - 853 \nu^{9} + 476 \nu^{8} - 319 \nu^{7} - 2584 \nu^{6} - 164 \nu^{5} + 24160 \nu^{4} - 69296 \nu^{3} + 141184 \nu^{2} - 141312 \nu + 173056 ) / 15360 \) |
\(\beta_{7}\) | \(=\) | \( ( 19 \nu^{11} - 44 \nu^{10} + 85 \nu^{9} - 132 \nu^{8} + 63 \nu^{7} + 32 \nu^{6} - 1148 \nu^{5} - 192 \nu^{4} + 6320 \nu^{3} - 13824 \nu^{2} + 11776 \nu - 35840 ) / 1280 \) |
\(\beta_{8}\) | \(=\) | \( ( - 103 \nu^{11} + 428 \nu^{10} - 545 \nu^{9} + 1284 \nu^{8} - 2851 \nu^{7} - 1904 \nu^{6} + 5516 \nu^{5} + 9024 \nu^{4} - 18800 \nu^{3} + 48128 \nu^{2} - 143872 \nu + 376320 ) / 7680 \) |
\(\beta_{9}\) | \(=\) | \( ( 67 \nu^{11} + 60 \nu^{10} - 91 \nu^{9} - 684 \nu^{8} + 1199 \nu^{7} + 1640 \nu^{6} + 484 \nu^{5} - 5856 \nu^{4} - 7376 \nu^{3} + 8320 \nu^{2} + 59392 \nu - 158720 ) / 3840 \) |
\(\beta_{10}\) | \(=\) | \( ( - 271 \nu^{11} + 236 \nu^{10} + 1735 \nu^{9} - 3132 \nu^{8} - 1387 \nu^{7} + 4912 \nu^{6} + 812 \nu^{5} + 38208 \nu^{4} - 51440 \nu^{3} - 154624 \nu^{2} + 482816 \nu - 122880 ) / 15360 \) |
\(\beta_{11}\) | \(=\) | \( ( - 165 \nu^{11} + 492 \nu^{10} - 499 \nu^{9} + 1636 \nu^{8} - 2713 \nu^{7} - 2056 \nu^{6} + 8452 \nu^{5} + 11168 \nu^{4} - 53456 \nu^{3} + 48256 \nu^{2} - 292864 \nu + 523264 ) / 7680 \) |
\(\nu\) | \(=\) | \( ( -\beta_{11} + \beta_{8} - \beta_{7} - \beta_{5} + 2\beta_{4} + \beta_{2} + 2\beta _1 + 11 ) / 32 \) |
\(\nu^{2}\) | \(=\) | \( ( - \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} - 2 \beta_{7} + \beta_{6} + 2 \beta_{5} - 7 \beta_{4} - 2 \beta_{3} - 13 \beta_{2} + 5 \beta _1 + 5 ) / 32 \) |
\(\nu^{3}\) | \(=\) | \( ( - 2 \beta_{11} + \beta_{10} + \beta_{8} + \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 8 \beta_{4} + \beta_{3} + 4 \beta_{2} - 2 \beta _1 + 22 ) / 16 \) |
\(\nu^{4}\) | \(=\) | \( ( - 3 \beta_{11} + \beta_{10} + 5 \beta_{9} + \beta_{8} + 2 \beta_{7} + 11 \beta_{6} - 6 \beta_{5} - 47 \beta_{4} + 4 \beta_{3} - 115 \beta_{2} + 7 \beta _1 + 39 ) / 32 \) |
\(\nu^{5}\) | \(=\) | \( ( - 7 \beta_{11} - 2 \beta_{10} + 8 \beta_{9} - 3 \beta_{8} - 17 \beta_{7} - 20 \beta_{6} - 7 \beta_{5} - 82 \beta_{4} - 2 \beta_{3} - 109 \beta_{2} - 46 \beta _1 - 311 ) / 32 \) |
\(\nu^{6}\) | \(=\) | \( ( 11 \beta_{11} + 15 \beta_{10} + 5 \beta_{9} - 8 \beta_{8} - 8 \beta_{7} + 3 \beta_{6} + 18 \beta_{5} - 22 \beta_{4} + 15 \beta_{3} - 163 \beta_{2} + 3 \beta _1 - 3 ) / 16 \) |
\(\nu^{7}\) | \(=\) | \( ( - 11 \beta_{11} + 24 \beta_{10} + 44 \beta_{9} - 69 \beta_{8} + \beta_{7} - 36 \beta_{6} - 31 \beta_{5} - 174 \beta_{4} + 60 \beta_{3} + 315 \beta_{2} + 170 \beta _1 + 2397 ) / 32 \) |
\(\nu^{8}\) | \(=\) | \( ( \beta_{11} - 3 \beta_{10} - 3 \beta_{9} + 39 \beta_{8} - 138 \beta_{7} - 45 \beta_{6} - 134 \beta_{5} + 35 \beta_{4} + 74 \beta_{3} - 235 \beta_{2} + 823 \beta _1 - 1589 ) / 32 \) |
\(\nu^{9}\) | \(=\) | \( ( 54 \beta_{11} - 13 \beta_{10} - 56 \beta_{9} - 5 \beta_{8} - 57 \beta_{7} - 110 \beta_{6} + 82 \beta_{5} - 436 \beta_{4} - 113 \beta_{3} - 136 \beta_{2} + 778 \beta _1 + 366 ) / 16 \) |
\(\nu^{10}\) | \(=\) | \( ( - 37 \beta_{11} + 235 \beta_{10} + 311 \beta_{9} + 463 \beta_{8} + 386 \beta_{7} - 455 \beta_{6} + 154 \beta_{5} - 365 \beta_{4} + 124 \beta_{3} + 4147 \beta_{2} + 1941 \beta _1 + 5001 ) / 32 \) |
\(\nu^{11}\) | \(=\) | \( ( 207 \beta_{11} - 994 \beta_{10} + 844 \beta_{9} + 915 \beta_{8} + 117 \beta_{7} + 688 \beta_{6} - 1005 \beta_{5} - 2490 \beta_{4} - 638 \beta_{3} - 11795 \beta_{2} + 3698 \beta _1 + 13479 ) / 32 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(351\) | \(577\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
49.1 |
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0 | −7.99849 | 0 | 0 | 0 | − | 9.93501i | 0 | 36.9759 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
49.2 | 0 | −7.99849 | 0 | 0 | 0 | 9.93501i | 0 | 36.9759 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.3 | 0 | −4.24443 | 0 | 0 | 0 | − | 14.6308i | 0 | −8.98481 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.4 | 0 | −4.24443 | 0 | 0 | 0 | 14.6308i | 0 | −8.98481 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.5 | 0 | 0.888401 | 0 | 0 | 0 | − | 26.6173i | 0 | −26.2107 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.6 | 0 | 0.888401 | 0 | 0 | 0 | 26.6173i | 0 | −26.2107 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.7 | 0 | 1.51777 | 0 | 0 | 0 | − | 5.13620i | 0 | −24.6964 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.8 | 0 | 1.51777 | 0 | 0 | 0 | 5.13620i | 0 | −24.6964 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.9 | 0 | 6.25785 | 0 | 0 | 0 | − | 34.6280i | 0 | 12.1606 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.10 | 0 | 6.25785 | 0 | 0 | 0 | 34.6280i | 0 | 12.1606 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.11 | 0 | 9.57890 | 0 | 0 | 0 | − | 21.5703i | 0 | 64.7554 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.12 | 0 | 9.57890 | 0 | 0 | 0 | 21.5703i | 0 | 64.7554 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
40.f | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 800.4.f.c | 12 | |
4.b | odd | 2 | 1 | 200.4.f.b | 12 | ||
5.b | even | 2 | 1 | 800.4.f.b | 12 | ||
5.c | odd | 4 | 1 | 160.4.d.a | 12 | ||
5.c | odd | 4 | 1 | 800.4.d.d | 12 | ||
8.b | even | 2 | 1 | 800.4.f.b | 12 | ||
8.d | odd | 2 | 1 | 200.4.f.c | 12 | ||
15.e | even | 4 | 1 | 1440.4.k.c | 12 | ||
20.d | odd | 2 | 1 | 200.4.f.c | 12 | ||
20.e | even | 4 | 1 | 40.4.d.a | ✓ | 12 | |
20.e | even | 4 | 1 | 200.4.d.b | 12 | ||
40.e | odd | 2 | 1 | 200.4.f.b | 12 | ||
40.f | even | 2 | 1 | inner | 800.4.f.c | 12 | |
40.i | odd | 4 | 1 | 160.4.d.a | 12 | ||
40.i | odd | 4 | 1 | 800.4.d.d | 12 | ||
40.k | even | 4 | 1 | 40.4.d.a | ✓ | 12 | |
40.k | even | 4 | 1 | 200.4.d.b | 12 | ||
60.l | odd | 4 | 1 | 360.4.k.c | 12 | ||
80.i | odd | 4 | 1 | 1280.4.a.ba | 6 | ||
80.j | even | 4 | 1 | 1280.4.a.bb | 6 | ||
80.s | even | 4 | 1 | 1280.4.a.bc | 6 | ||
80.t | odd | 4 | 1 | 1280.4.a.bd | 6 | ||
120.q | odd | 4 | 1 | 360.4.k.c | 12 | ||
120.w | even | 4 | 1 | 1440.4.k.c | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
40.4.d.a | ✓ | 12 | 20.e | even | 4 | 1 | |
40.4.d.a | ✓ | 12 | 40.k | even | 4 | 1 | |
160.4.d.a | 12 | 5.c | odd | 4 | 1 | ||
160.4.d.a | 12 | 40.i | odd | 4 | 1 | ||
200.4.d.b | 12 | 20.e | even | 4 | 1 | ||
200.4.d.b | 12 | 40.k | even | 4 | 1 | ||
200.4.f.b | 12 | 4.b | odd | 2 | 1 | ||
200.4.f.b | 12 | 40.e | odd | 2 | 1 | ||
200.4.f.c | 12 | 8.d | odd | 2 | 1 | ||
200.4.f.c | 12 | 20.d | odd | 2 | 1 | ||
360.4.k.c | 12 | 60.l | odd | 4 | 1 | ||
360.4.k.c | 12 | 120.q | odd | 4 | 1 | ||
800.4.d.d | 12 | 5.c | odd | 4 | 1 | ||
800.4.d.d | 12 | 40.i | odd | 4 | 1 | ||
800.4.f.b | 12 | 5.b | even | 2 | 1 | ||
800.4.f.b | 12 | 8.b | even | 2 | 1 | ||
800.4.f.c | 12 | 1.a | even | 1 | 1 | trivial | |
800.4.f.c | 12 | 40.f | even | 2 | 1 | inner | |
1280.4.a.ba | 6 | 80.i | odd | 4 | 1 | ||
1280.4.a.bb | 6 | 80.j | even | 4 | 1 | ||
1280.4.a.bc | 6 | 80.s | even | 4 | 1 | ||
1280.4.a.bd | 6 | 80.t | odd | 4 | 1 | ||
1440.4.k.c | 12 | 15.e | even | 4 | 1 | ||
1440.4.k.c | 12 | 120.w | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 6T_{3}^{5} - 90T_{3}^{4} + 432T_{3}^{3} + 1428T_{3}^{2} - 4632T_{3} + 2744 \)
acting on \(S_{4}^{\mathrm{new}}(800, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} \)
$3$
\( (T^{6} - 6 T^{5} - 90 T^{4} + 432 T^{3} + \cdots + 2744)^{2} \)
$5$
\( T^{12} \)
$7$
\( T^{12} + \cdots + 220318448824384 \)
$11$
\( T^{12} + 10072 T^{10} + \cdots + 26\!\cdots\!00 \)
$13$
\( (T^{6} - 5572 T^{4} + 163840 T^{3} + \cdots - 157790400)^{2} \)
$17$
\( T^{12} + 29000 T^{10} + \cdots + 55\!\cdots\!64 \)
$19$
\( T^{12} + 46312 T^{10} + \cdots + 17\!\cdots\!24 \)
$23$
\( T^{12} + 65320 T^{10} + \cdots + 77\!\cdots\!56 \)
$29$
\( T^{12} + 92448 T^{10} + \cdots + 11\!\cdots\!00 \)
$31$
\( (T^{6} - 132 T^{5} + \cdots - 1437816300032)^{2} \)
$37$
\( (T^{6} + 68 T^{5} + \cdots + 45951464886848)^{2} \)
$41$
\( (T^{6} - 20 T^{5} + \cdots - 71667547865600)^{2} \)
$43$
\( (T^{6} + 602 T^{5} + \cdots - 508806074248)^{2} \)
$47$
\( T^{12} + 562088 T^{10} + \cdots + 51\!\cdots\!84 \)
$53$
\( (T^{6} + 528 T^{5} + \cdots + 929403110278976)^{2} \)
$59$
\( T^{12} + 1889384 T^{10} + \cdots + 10\!\cdots\!24 \)
$61$
\( T^{12} + 627448 T^{10} + \cdots + 61\!\cdots\!00 \)
$67$
\( (T^{6} + 1206 T^{5} + \cdots - 93747278347656)^{2} \)
$71$
\( (T^{6} - 796 T^{5} + \cdots - 36\!\cdots\!48)^{2} \)
$73$
\( T^{12} + 1958184 T^{10} + \cdots + 14\!\cdots\!76 \)
$79$
\( (T^{6} - 1008 T^{5} + \cdots - 44\!\cdots\!00)^{2} \)
$83$
\( (T^{6} - 1778 T^{5} + \cdots - 28\!\cdots\!68)^{2} \)
$89$
\( (T^{6} - 212 T^{5} + \cdots - 62\!\cdots\!00)^{2} \)
$97$
\( T^{12} + 3718376 T^{10} + \cdots + 11\!\cdots\!84 \)
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