Newspace parameters
Level: | \( N \) | \(=\) | \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 390.bn (of order \(12\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.11416567883\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{12})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
Defining polynomial: |
\( x^{16} - 4 x^{15} + 12 x^{14} - 48 x^{13} + 67 x^{12} - 24 x^{11} + 118 x^{10} - 176 x^{9} + 351 x^{8} - 180 x^{7} + 358 x^{6} - 336 x^{5} + 390 x^{4} - 344 x^{3} + 164 x^{2} - 40 x + 4 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
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
\( x^{16} - 4 x^{15} + 12 x^{14} - 48 x^{13} + 67 x^{12} - 24 x^{11} + 118 x^{10} - 176 x^{9} + 351 x^{8} - 180 x^{7} + 358 x^{6} - 336 x^{5} + 390 x^{4} - 344 x^{3} + 164 x^{2} - 40 x + 4 \)
:
\(\beta_{1}\) | \(=\) |
\( ( 1434138866353 \nu^{15} - 74224360591111 \nu^{14} + 217326371958939 \nu^{13} - 648847305484636 \nu^{12} + \cdots + 46\!\cdots\!94 ) / 51\!\cdots\!90 \)
|
\(\beta_{2}\) | \(=\) |
\( ( - 123434729989149 \nu^{15} + 492304781090243 \nu^{14} + \cdots + 195044141171618 ) / 51\!\cdots\!90 \)
|
\(\beta_{3}\) | \(=\) |
\( ( - 137809854283897 \nu^{15} + 666808683532056 \nu^{14} + \cdots + 69\!\cdots\!44 ) / 10\!\cdots\!78 \)
|
\(\beta_{4}\) | \(=\) |
\( ( 3909951128714 \nu^{15} - 13126607646508 \nu^{14} + 38145193554947 \nu^{13} - 162143292202288 \nu^{12} + \cdots - 16575048540328 ) / 28585593772190 \)
|
\(\beta_{5}\) | \(=\) |
\( ( - 4143762135082 \nu^{15} + 12665097411614 \nu^{14} - 36598537974476 \nu^{13} + 160755388928989 \nu^{12} + \cdots + 3631882693804 ) / 28585593772190 \)
|
\(\beta_{6}\) | \(=\) |
\( ( - 14\!\cdots\!27 \nu^{15} + \cdots + 35\!\cdots\!44 ) / 51\!\cdots\!90 \)
|
\(\beta_{7}\) | \(=\) |
\( ( 6100180981 \nu^{15} - 22259671750 \nu^{14} + 65510725326 \nu^{13} - 270219726184 \nu^{12} + 314950092835 \nu^{11} - 40514369326 \nu^{10} + \cdots - 77929949288 ) / 18265555126 \)
|
\(\beta_{8}\) | \(=\) |
\( ( - 6072104849199 \nu^{15} + 21780610045393 \nu^{14} - 63731636791632 \nu^{13} + 264606089742658 \nu^{12} + \cdots + 65633529574178 ) / 16530327389030 \)
|
\(\beta_{9}\) | \(=\) |
\( ( 20\!\cdots\!87 \nu^{15} + \cdots - 28\!\cdots\!14 ) / 51\!\cdots\!90 \)
|
\(\beta_{10}\) | \(=\) |
\( ( 30320067 \nu^{15} - 111020724 \nu^{14} + 326080511 \nu^{13} - 1344589864 \nu^{12} + 1575363211 \nu^{11} - 188871116 \nu^{10} + 3515753069 \nu^{9} + \cdots - 330526984 ) / 63923990 \)
|
\(\beta_{11}\) | \(=\) |
\( ( 567889621787826 \nu^{15} + \cdots - 77\!\cdots\!96 ) / 10\!\cdots\!78 \)
|
\(\beta_{12}\) | \(=\) |
\( ( 52\!\cdots\!58 \nu^{15} + \cdots - 60\!\cdots\!06 ) / 51\!\cdots\!90 \)
|
\(\beta_{13}\) | \(=\) |
\( ( - 19482487322 \nu^{15} + 71829768307 \nu^{14} - 211530176114 \nu^{13} + 869648666130 \nu^{12} - 1035106924390 \nu^{11} + \cdots + 258618081490 ) / 18265555126 \)
|
\(\beta_{14}\) | \(=\) |
\( ( - 82631746 \nu^{15} + 300206917 \nu^{14} - 880560228 \nu^{13} + 3640243297 \nu^{12} - 4191737118 \nu^{11} + 407798693 \nu^{10} + \cdots + 791533522 ) / 63923990 \)
|
\(\beta_{15}\) | \(=\) |
\( ( 69\!\cdots\!34 \nu^{15} + \cdots - 83\!\cdots\!68 ) / 51\!\cdots\!90 \)
|
\(\nu\) | \(=\) |
\( \beta_{6} - \beta_{5} - \beta_{3} \)
|
\(\nu^{2}\) | \(=\) |
\( \beta_{14} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + \beta_{6} + \beta_{5} + \beta_{4} + 3\beta_{2} \)
|
\(\nu^{3}\) | \(=\) |
\( 2\beta_{15} + 4\beta_{14} + 2\beta_{13} + 3\beta_{12} + 2\beta_{11} - 2\beta_{8} + \beta_{6} - 5\beta _1 + 10 \)
|
\(\nu^{4}\) | \(=\) |
\( - 2 \beta_{15} - 6 \beta_{14} - 11 \beta_{12} - 4 \beta_{11} - 4 \beta_{10} + 9 \beta_{9} - 10 \beta_{8} - 9 \beta_{7} - 3 \beta_{6} - 15 \beta_{5} + 9 \beta_{4} - 10 \beta_{3} - 20 \beta_{2} - 9 \beta _1 + 17 \)
|
\(\nu^{5}\) | \(=\) |
\( - 23 \beta_{15} - 16 \beta_{14} - 22 \beta_{13} - 21 \beta_{12} - 2 \beta_{11} + 21 \beta_{10} - 2 \beta_{9} + 5 \beta_{8} + 22 \beta_{6} - 2 \beta_{5} + 39 \beta_{4} - 21 \beta_{3} + 14 \beta_{2} + 7 \beta _1 - 37 \)
|
\(\nu^{6}\) | \(=\) |
\( 28 \beta_{15} + 154 \beta_{14} + 23 \beta_{13} + 140 \beta_{12} + 89 \beta_{11} + 95 \beta_{10} - 56 \beta_{9} + 12 \beta_{8} + 89 \beta_{7} + 84 \beta_{6} + 121 \beta_{5} - 12 \beta_{4} + 56 \beta_{3} + 173 \beta_{2} - 39 \beta _1 + 93 \)
|
\(\nu^{7}\) | \(=\) |
\( 163 \beta_{15} + 184 \beta_{14} + 240 \beta_{13} + 50 \beta_{12} - 94 \beta_{10} + 200 \beta_{9} - 351 \beta_{8} - 35 \beta_{7} - 219 \beta_{6} - 128 \beta_{5} - 94 \beta_{4} + 35 \beta_{3} - 440 \beta_{2} - 400 \beta _1 + 985 \)
|
\(\nu^{8}\) | \(=\) |
\( - 645 \beta_{15} - 1488 \beta_{14} - 329 \beta_{13} - 1794 \beta_{12} - 914 \beta_{11} - 242 \beta_{10} + 914 \beta_{9} - 672 \beta_{8} - 663 \beta_{7} - 645 \beta_{6} - 1243 \beta_{5} + 914 \beta_{4} - 914 \beta_{3} + \cdots + 9 \)
|
\(\nu^{9}\) | \(=\) |
\( - 1828 \beta_{15} - 186 \beta_{14} - 1819 \beta_{13} - 195 \beta_{12} + 477 \beta_{11} + 3325 \beta_{10} - 1299 \beta_{9} + 1776 \beta_{8} + 1915 \beta_{7} + 2296 \beta_{6} + 2491 \beta_{5} + 1915 \beta_{4} + \cdots - 5566 \)
|
\(\nu^{10}\) | \(=\) |
\( 6400 \beta_{15} + 16412 \beta_{14} + 6539 \beta_{13} + 14863 \beta_{12} + 7481 \beta_{11} + 6042 \beta_{10} - 3465 \beta_{9} - 1493 \beta_{8} + 8974 \beta_{7} + 1665 \beta_{6} + 11899 \beta_{5} - 7481 \beta_{4} + \cdots + 17408 \)
|
\(\nu^{11}\) | \(=\) |
\( 8014 \beta_{15} - 12665 \beta_{14} + 18481 \beta_{13} - 24749 \beta_{12} - 18707 \beta_{11} - 21849 \beta_{10} + 32416 \beta_{9} - 37414 \beta_{8} - 15992 \beta_{7} - 38812 \beta_{6} - 32352 \beta_{5} + \cdots + 77085 \)
|
\(\nu^{12}\) | \(=\) |
\( - 91679 \beta_{15} - 170862 \beta_{14} - 70257 \beta_{13} - 181429 \beta_{12} - 89750 \beta_{11} + 7240 \beta_{10} + 55995 \beta_{9} - 7240 \beta_{8} - 43472 \beta_{7} - 31804 \beta_{6} + \cdots - 166538 \)
|
\(\nu^{13}\) | \(=\) |
\( - 55995 \beta_{15} + 241852 \beta_{14} - 92227 \beta_{13} + 278084 \beta_{12} + 185857 \beta_{11} + 372602 \beta_{10} - 253893 \beta_{9} + 253893 \beta_{8} + 322709 \beta_{7} + 281000 \beta_{6} + \cdots - 469625 \)
|
\(\nu^{14}\) | \(=\) |
\( 880388 \beta_{15} + 1459906 \beta_{14} + 949204 \beta_{13} + 1273161 \beta_{12} + 510702 \beta_{11} + 13265 \beta_{10} + 13265 \beta_{9} - 533645 \beta_{8} + 533645 \beta_{7} - 400341 \beta_{6} + \cdots + 2650257 \)
|
\(\nu^{15}\) | \(=\) |
\( - 533645 \beta_{15} - 4295841 \beta_{14} + 533645 \beta_{13} - 5220149 \beta_{12} - 3264759 \beta_{11} - 2855121 \beta_{10} + 3769764 \beta_{9} - 3264759 \beta_{8} - 2855121 \beta_{7} + \cdots + 3840272 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/390\mathbb{Z}\right)^\times\).
\(n\) | \(131\) | \(157\) | \(301\) |
\(\chi(n)\) | \(1\) | \(-\beta_{13}\) | \(\beta_{5} - \beta_{13}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
67.1 |
|
0.500000 | + | 0.866025i | −0.965926 | − | 0.258819i | −0.500000 | + | 0.866025i | −1.90502 | − | 1.17085i | −0.258819 | − | 0.965926i | 3.41596 | + | 1.97220i | −1.00000 | 0.866025 | + | 0.500000i | 0.0614757 | − | 2.23522i | ||||||||||||||||||||||||||||||||||||||||||||||||||
67.2 | 0.500000 | + | 0.866025i | −0.965926 | − | 0.258819i | −0.500000 | + | 0.866025i | 2.06394 | + | 0.860320i | −0.258819 | − | 0.965926i | 0.450069 | + | 0.259847i | −1.00000 | 0.866025 | + | 0.500000i | 0.286912 | + | 2.21758i | |||||||||||||||||||||||||||||||||||||||||||||||||||
67.3 | 0.500000 | + | 0.866025i | 0.965926 | + | 0.258819i | −0.500000 | + | 0.866025i | −0.0421887 | − | 2.23567i | 0.258819 | + | 0.965926i | 2.21194 | + | 1.27707i | −1.00000 | 0.866025 | + | 0.500000i | 1.91505 | − | 1.15437i | |||||||||||||||||||||||||||||||||||||||||||||||||||
67.4 | 0.500000 | + | 0.866025i | 0.965926 | + | 0.258819i | −0.500000 | + | 0.866025i | 1.61532 | + | 1.54620i | 0.258819 | + | 0.965926i | 1.65408 | + | 0.954985i | −1.00000 | 0.866025 | + | 0.500000i | −0.531389 | + | 2.17201i | |||||||||||||||||||||||||||||||||||||||||||||||||||
97.1 | 0.500000 | − | 0.866025i | −0.258819 | − | 0.965926i | −0.500000 | − | 0.866025i | −1.66815 | − | 1.48905i | −0.965926 | − | 0.258819i | −0.568824 | + | 0.328411i | −1.00000 | −0.866025 | + | 0.500000i | −2.12363 | + | 0.700141i | |||||||||||||||||||||||||||||||||||||||||||||||||||
97.2 | 0.500000 | − | 0.866025i | −0.258819 | − | 0.965926i | −0.500000 | − | 0.866025i | 1.50924 | − | 1.64991i | −0.965926 | − | 0.258819i | 2.70280 | − | 1.56046i | −1.00000 | −0.866025 | + | 0.500000i | −0.674247 | − | 2.13199i | |||||||||||||||||||||||||||||||||||||||||||||||||||
97.3 | 0.500000 | − | 0.866025i | 0.258819 | + | 0.965926i | −0.500000 | − | 0.866025i | −2.23607 | + | 0.00342112i | 0.965926 | + | 0.258819i | 3.25724 | − | 1.88057i | −1.00000 | −0.866025 | + | 0.500000i | −1.11507 | + | 1.93820i | |||||||||||||||||||||||||||||||||||||||||||||||||||
97.4 | 0.500000 | − | 0.866025i | 0.258819 | + | 0.965926i | −0.500000 | − | 0.866025i | 0.662933 | + | 2.13554i | 0.965926 | + | 0.258819i | −1.12326 | + | 0.648516i | −1.00000 | −0.866025 | + | 0.500000i | 2.18090 | + | 0.493652i | |||||||||||||||||||||||||||||||||||||||||||||||||||
163.1 | 0.500000 | − | 0.866025i | −0.965926 | + | 0.258819i | −0.500000 | − | 0.866025i | −1.90502 | + | 1.17085i | −0.258819 | + | 0.965926i | 3.41596 | − | 1.97220i | −1.00000 | 0.866025 | − | 0.500000i | 0.0614757 | + | 2.23522i | |||||||||||||||||||||||||||||||||||||||||||||||||||
163.2 | 0.500000 | − | 0.866025i | −0.965926 | + | 0.258819i | −0.500000 | − | 0.866025i | 2.06394 | − | 0.860320i | −0.258819 | + | 0.965926i | 0.450069 | − | 0.259847i | −1.00000 | 0.866025 | − | 0.500000i | 0.286912 | − | 2.21758i | |||||||||||||||||||||||||||||||||||||||||||||||||||
163.3 | 0.500000 | − | 0.866025i | 0.965926 | − | 0.258819i | −0.500000 | − | 0.866025i | −0.0421887 | + | 2.23567i | 0.258819 | − | 0.965926i | 2.21194 | − | 1.27707i | −1.00000 | 0.866025 | − | 0.500000i | 1.91505 | + | 1.15437i | |||||||||||||||||||||||||||||||||||||||||||||||||||
163.4 | 0.500000 | − | 0.866025i | 0.965926 | − | 0.258819i | −0.500000 | − | 0.866025i | 1.61532 | − | 1.54620i | 0.258819 | − | 0.965926i | 1.65408 | − | 0.954985i | −1.00000 | 0.866025 | − | 0.500000i | −0.531389 | − | 2.17201i | |||||||||||||||||||||||||||||||||||||||||||||||||||
193.1 | 0.500000 | + | 0.866025i | −0.258819 | + | 0.965926i | −0.500000 | + | 0.866025i | −1.66815 | + | 1.48905i | −0.965926 | + | 0.258819i | −0.568824 | − | 0.328411i | −1.00000 | −0.866025 | − | 0.500000i | −2.12363 | − | 0.700141i | |||||||||||||||||||||||||||||||||||||||||||||||||||
193.2 | 0.500000 | + | 0.866025i | −0.258819 | + | 0.965926i | −0.500000 | + | 0.866025i | 1.50924 | + | 1.64991i | −0.965926 | + | 0.258819i | 2.70280 | + | 1.56046i | −1.00000 | −0.866025 | − | 0.500000i | −0.674247 | + | 2.13199i | |||||||||||||||||||||||||||||||||||||||||||||||||||
193.3 | 0.500000 | + | 0.866025i | 0.258819 | − | 0.965926i | −0.500000 | + | 0.866025i | −2.23607 | − | 0.00342112i | 0.965926 | − | 0.258819i | 3.25724 | + | 1.88057i | −1.00000 | −0.866025 | − | 0.500000i | −1.11507 | − | 1.93820i | |||||||||||||||||||||||||||||||||||||||||||||||||||
193.4 | 0.500000 | + | 0.866025i | 0.258819 | − | 0.965926i | −0.500000 | + | 0.866025i | 0.662933 | − | 2.13554i | 0.965926 | − | 0.258819i | −1.12326 | − | 0.648516i | −1.00000 | −0.866025 | − | 0.500000i | 2.18090 | − | 0.493652i | |||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
65.o | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 390.2.bn.b | yes | 16 |
5.c | odd | 4 | 1 | 390.2.bd.b | ✓ | 16 | |
13.f | odd | 12 | 1 | 390.2.bd.b | ✓ | 16 | |
65.o | even | 12 | 1 | inner | 390.2.bn.b | yes | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
390.2.bd.b | ✓ | 16 | 5.c | odd | 4 | 1 | |
390.2.bd.b | ✓ | 16 | 13.f | odd | 12 | 1 | |
390.2.bn.b | yes | 16 | 1.a | even | 1 | 1 | trivial |
390.2.bn.b | yes | 16 | 65.o | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{16} - 24 T_{7}^{15} + 262 T_{7}^{14} - 1680 T_{7}^{13} + 6823 T_{7}^{12} - 17400 T_{7}^{11} + 24326 T_{7}^{10} - 5472 T_{7}^{9} - 33171 T_{7}^{8} + 17400 T_{7}^{7} + 87604 T_{7}^{6} - 136800 T_{7}^{5} + \cdots + 10000 \)
acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} - T + 1)^{8} \)
$3$
\( (T^{8} - T^{4} + 1)^{2} \)
$5$
\( T^{16} - 2 T^{14} + 16 T^{13} + \cdots + 390625 \)
$7$
\( T^{16} - 24 T^{15} + 262 T^{14} + \cdots + 10000 \)
$11$
\( T^{16} - 12 T^{15} + 78 T^{14} + \cdots + 10000 \)
$13$
\( T^{16} - 8 T^{15} + 20 T^{14} + \cdots + 815730721 \)
$17$
\( T^{16} + 4 T^{15} - 58 T^{14} + \cdots + 160000 \)
$19$
\( T^{16} + 4 T^{15} + 26 T^{14} + 68 T^{13} + \cdots + 16 \)
$23$
\( T^{16} - 4 T^{15} - 28 T^{14} + 64 T^{13} + \cdots + 256 \)
$29$
\( T^{16} + 48 T^{15} + 1030 T^{14} + \cdots + 24010000 \)
$31$
\( T^{16} - 16 T^{15} + 128 T^{14} + \cdots + 21827584 \)
$37$
\( T^{16} - 12 T^{15} + 576 T^{13} + \cdots + 418609 \)
$41$
\( T^{16} + 12 T^{15} + 90 T^{14} + \cdots + 85264 \)
$43$
\( T^{16} - 20 T^{15} + \cdots + 151388416 \)
$47$
\( T^{16} + \cdots + 610001280825616 \)
$53$
\( T^{16} - 32 T^{15} + \cdots + 14952153841 \)
$59$
\( T^{16} - 4 T^{15} + \cdots + 2922403926016 \)
$61$
\( T^{16} - 4 T^{15} + \cdots + 1536975103504 \)
$67$
\( T^{16} + 28 T^{15} + \cdots + 2412728463616 \)
$71$
\( T^{16} + 36 T^{15} + \cdots + 43\!\cdots\!16 \)
$73$
\( (T^{8} - 16 T^{7} - 54 T^{6} + 864 T^{5} + \cdots - 86528)^{2} \)
$79$
\( T^{16} + \cdots + 243571706822656 \)
$83$
\( T^{16} + 232 T^{14} + \cdots + 64770304 \)
$89$
\( T^{16} + 24 T^{15} + \cdots + 30073246952464 \)
$97$
\( T^{16} + \cdots + 344958409179136 \)
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