Newform invariants
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\).
We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).
| \(n\) |
\(29\) |
\(325\) |
| \(\chi(n)\) |
\(\zeta_{18}^{2}\) |
\(1\) |
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.
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 3 T_{5}^{5} + 3 T_{5}^{3} + 9 T_{5}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\).
Hecke Characteristic Polynomials
| $p$ |
$F_p(T)$ |
| $2$
| \( 1 + T^{3} + T^{6} \)
|
| $3$
| \( 1 + 9 T^{3} + 27 T^{6} \)
|
| $5$
| \( 1 - 3 T + 18 T^{3} - 36 T^{4} - 75 T^{5} + 379 T^{6} - 375 T^{7} - 900 T^{8} + 2250 T^{9} - 9375 T^{11} + 15625 T^{12} \)
|
| $7$
| \( 1 + T^{3} + T^{6} \)
|
| $11$
| \( 1 - 6 T + 18 T^{2} - 36 T^{3} - 108 T^{4} + 912 T^{5} - 3203 T^{6} + 10032 T^{7} - 13068 T^{8} - 47916 T^{9} + 263538 T^{10} - 966306 T^{11} + 1771561 T^{12} \)
|
| $13$
| \( 1 + 3 T + 6 T^{2} + 8 T^{3} - 27 T^{4} - 891 T^{5} - 3483 T^{6} - 11583 T^{7} - 4563 T^{8} + 17576 T^{9} + 171366 T^{10} + 1113879 T^{11} + 4826809 T^{12} \)
|
| $17$
| \( 1 - 42 T^{2} + 18 T^{3} + 1050 T^{4} - 378 T^{5} - 20081 T^{6} - 6426 T^{7} + 303450 T^{8} + 88434 T^{9} - 3507882 T^{10} + 24137569 T^{12} \)
|
| $19$
| \( 1 - 3 T - 24 T^{2} + 23 T^{3} + 279 T^{4} + 666 T^{5} - 6501 T^{6} + 12654 T^{7} + 100719 T^{8} + 157757 T^{9} - 3127704 T^{10} - 7428297 T^{11} + 47045881 T^{12} \)
|
| $23$
| \( 1 - 6 T - 36 T^{2} + 396 T^{3} - 810 T^{4} - 5874 T^{5} + 51733 T^{6} - 135102 T^{7} - 428490 T^{8} + 4818132 T^{9} - 10074276 T^{10} - 38618058 T^{11} + 148035889 T^{12} \)
|
| $29$
| \( 1 + 3 T + 108 T^{2} + 90 T^{3} + 4851 T^{4} - 3507 T^{5} + 149293 T^{6} - 101703 T^{7} + 4079691 T^{8} + 2195010 T^{9} + 76386348 T^{10} + 61533447 T^{11} + 594823321 T^{12} \)
|
| $31$
| \( 1 + 3 T + 24 T^{2} + 296 T^{3} + 1530 T^{4} + 7119 T^{5} + 55665 T^{6} + 220689 T^{7} + 1470330 T^{8} + 8818136 T^{9} + 22164504 T^{10} + 85887453 T^{11} + 887503681 T^{12} \)
|
| $37$
| \( 1 - 12 T + 21 T^{2} + 284 T^{3} + 18 T^{4} - 15444 T^{5} + 118797 T^{6} - 571428 T^{7} + 24642 T^{8} + 14385452 T^{9} + 39357381 T^{10} - 832127484 T^{11} + 2565726409 T^{12} \)
|
| $41$
| \( 1 + 12 T + 144 T^{2} + 1044 T^{3} + 9216 T^{4} + 58152 T^{5} + 440263 T^{6} + 2384232 T^{7} + 15492096 T^{8} + 71953524 T^{9} + 406909584 T^{10} + 1390274412 T^{11} + 4750104241 T^{12} \)
|
| $43$
| \( 1 + 6 T - 6 T^{2} + 284 T^{3} - 684 T^{4} - 7920 T^{5} + 123837 T^{6} - 340560 T^{7} - 1264716 T^{8} + 22579988 T^{9} - 20512806 T^{10} + 882050658 T^{11} + 6321363049 T^{12} \)
|
| $47$
| \( 1 - 24 T + 306 T^{2} - 2844 T^{3} + 25254 T^{4} - 215106 T^{5} + 1621333 T^{6} - 10109982 T^{7} + 55786086 T^{8} - 295272612 T^{9} + 1493182386 T^{10} - 5504280168 T^{11} + 10779215329 T^{12} \)
|
| $53$
| \( ( 1 + 42 T^{2} + 153 T^{3} + 2226 T^{4} + 148877 T^{6} )^{2} \)
|
| $59$
| \( 1 + 24 T + 252 T^{2} + 1395 T^{3} - 2475 T^{4} - 150699 T^{5} - 1633823 T^{6} - 8891241 T^{7} - 8615475 T^{8} + 286503705 T^{9} + 3053574972 T^{10} + 17158183176 T^{11} + 42180533641 T^{12} \)
|
| $61$
| \( 1 + 3 T + 114 T^{2} + 926 T^{3} + 12897 T^{4} + 70029 T^{5} + 1006281 T^{6} + 4271769 T^{7} + 47989737 T^{8} + 210184406 T^{9} + 1578425874 T^{10} + 2533788903 T^{11} + 51520374361 T^{12} \)
|
| $67$
| \( 1 - 3 T - 96 T^{2} + 1094 T^{3} - 3735 T^{4} - 48879 T^{5} + 940191 T^{6} - 3274893 T^{7} - 16766415 T^{8} + 329034722 T^{9} - 1934507616 T^{10} - 4050375321 T^{11} + 90458382169 T^{12} \)
|
| $71$
| \( 1 + 12 T - 24 T^{2} - 738 T^{3} - 228 T^{4} + 5556 T^{5} - 117857 T^{6} + 394476 T^{7} - 1149348 T^{8} - 264138318 T^{9} - 609880344 T^{10} + 21650752212 T^{11} + 128100283921 T^{12} \)
|
| $73$
| \( 1 - 3 T - 96 T^{2} + 635 T^{3} + 1935 T^{4} - 19872 T^{5} + 150873 T^{6} - 1450656 T^{7} + 10311615 T^{8} + 247025795 T^{9} - 2726231136 T^{10} - 6219214779 T^{11} + 151334226289 T^{12} \)
|
| $79$
| \( 1 - 33 T + 492 T^{2} - 4402 T^{3} + 17532 T^{4} + 134901 T^{5} - 2452347 T^{6} + 10657179 T^{7} + 109417212 T^{8} - 2170357678 T^{9} + 19163439852 T^{10} - 101542861167 T^{11} + 243087455521 T^{12} \)
|
| $83$
| \( 1 + 144 T^{2} + 774 T^{3} + 20916 T^{4} + 72810 T^{5} + 2086741 T^{6} + 6043230 T^{7} + 144090324 T^{8} + 442563138 T^{9} + 6833998224 T^{10} + 326940373369 T^{12} \)
|
| $89$
| \( 1 - 21 T + 138 T^{2} + 891 T^{3} - 11325 T^{4} - 132708 T^{5} + 2900905 T^{6} - 11811012 T^{7} - 89705325 T^{8} + 628127379 T^{9} + 8658429258 T^{10} - 117265248429 T^{11} + 496981290961 T^{12} \)
|
| $97$
| \( 1 + 15 T + 120 T^{2} + 2138 T^{3} + 14679 T^{4} + 127917 T^{5} + 2485869 T^{6} + 12407949 T^{7} + 138114711 T^{8} + 1951294874 T^{9} + 10623513720 T^{10} + 128810103855 T^{11} + 832972004929 T^{12} \)
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