Properties

Label 756.2.j
Level 756
Weight 2
Character orbit j
Rep. character \(\chi_{756}(253,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 12
Newform subspaces 2
Sturm bound 288
Trace bound 5

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Defining parameters

Level: \( N \) = \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 756.j (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(288\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(756, [\chi])\).

Total New Old
Modular forms 324 12 312
Cusp forms 252 12 240
Eisenstein series 72 0 72

Trace form

\( 12q - 2q^{5} + O(q^{10}) \) \( 12q - 2q^{5} - 4q^{11} - 4q^{17} + 12q^{19} + 8q^{23} - 14q^{29} + 6q^{31} + 8q^{35} - 12q^{37} - 6q^{41} - 6q^{43} + 6q^{47} - 6q^{49} + 24q^{53} - 12q^{55} + 28q^{59} + 18q^{65} - 4q^{71} + 12q^{73} - 8q^{77} + 6q^{79} + 2q^{83} + 30q^{85} - 36q^{89} - 12q^{91} - 4q^{95} + 24q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(756, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
756.2.j.a \(6\) \(6.037\) 6.0.309123.1 None \(0\) \(0\) \(-3\) \(-3\) \(q+(-1+\beta _{1}+\beta _{3})q^{5}-\beta _{3}q^{7}+(-\beta _{2}+\cdots)q^{11}+\cdots\)
756.2.j.b \(6\) \(6.037\) 6.0.309123.1 None \(0\) \(0\) \(1\) \(3\) \(q+(-\beta _{1}+\beta _{5})q^{5}+(1-\beta _{4})q^{7}+(1-2\beta _{2}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(756, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(756, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(378, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))
$3$ (\( \))(\( \))
$5$ (\( 1 + 3 T - 15 T^{3} - 27 T^{4} - 6 T^{5} + 61 T^{6} - 30 T^{7} - 675 T^{8} - 1875 T^{9} + 9375 T^{11} + 15625 T^{12} \))(\( 1 - T - 10 T^{2} + 7 T^{3} + 57 T^{4} - 14 T^{5} - 299 T^{6} - 70 T^{7} + 1425 T^{8} + 875 T^{9} - 6250 T^{10} - 3125 T^{11} + 15625 T^{12} \))
$7$ (\( ( 1 + T + T^{2} )^{3} \))(\( ( 1 - T + T^{2} )^{3} \))
$11$ (\( 1 + 6 T - 30 T^{3} + 162 T^{4} + 402 T^{5} - 821 T^{6} + 4422 T^{7} + 19602 T^{8} - 39930 T^{9} + 966306 T^{11} + 1771561 T^{12} \))(\( 1 - 2 T - 4 T^{2} - 46 T^{3} + 6 T^{4} + 230 T^{5} + 1699 T^{6} + 2530 T^{7} + 726 T^{8} - 61226 T^{9} - 58564 T^{10} - 322102 T^{11} + 1771561 T^{12} \))
$13$ (\( ( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} )^{3} \))(\( 1 + 3 T + 3 T^{2} - 84 T^{3} - 195 T^{4} + 345 T^{5} + 5006 T^{6} + 4485 T^{7} - 32955 T^{8} - 184548 T^{9} + 85683 T^{10} + 1113879 T^{11} + 4826809 T^{12} \))
$17$ (\( ( 1 + 18 T^{2} - 9 T^{3} + 306 T^{4} + 4913 T^{6} )^{2} \))(\( ( 1 + 2 T + 32 T^{2} + 21 T^{3} + 544 T^{4} + 578 T^{5} + 4913 T^{6} )^{2} \))
$19$ (\( ( 1 - 3 T + 21 T^{2} - 65 T^{3} + 399 T^{4} - 1083 T^{5} + 6859 T^{6} )^{2} \))(\( ( 1 - 3 T + 33 T^{2} - 35 T^{3} + 627 T^{4} - 1083 T^{5} + 6859 T^{6} )^{2} \))
$23$ (\( 1 + 6 T - 18 T^{2} - 30 T^{3} + 612 T^{4} - 2172 T^{5} - 30449 T^{6} - 49956 T^{7} + 323748 T^{8} - 365010 T^{9} - 5037138 T^{10} + 38618058 T^{11} + 148035889 T^{12} \))(\( 1 - 14 T + 74 T^{2} - 358 T^{3} + 2628 T^{4} - 11188 T^{5} + 33943 T^{6} - 257324 T^{7} + 1390212 T^{8} - 4355786 T^{9} + 20708234 T^{10} - 90108802 T^{11} + 148035889 T^{12} \))
$29$ (\( 1 + 15 T + 90 T^{2} + 411 T^{3} + 2205 T^{4} + 4110 T^{5} - 17723 T^{6} + 119190 T^{7} + 1854405 T^{8} + 10023879 T^{9} + 63655290 T^{10} + 307667235 T^{11} + 594823321 T^{12} \))(\( 1 - T - 46 T^{2} - 149 T^{3} + 897 T^{4} + 4282 T^{5} - 13523 T^{6} + 124178 T^{7} + 754377 T^{8} - 3633961 T^{9} - 32534926 T^{10} - 20511149 T^{11} + 594823321 T^{12} \))
$31$ (\( 1 - 3 T - 6 T^{2} + 221 T^{3} - 639 T^{4} - 2088 T^{5} + 61647 T^{6} - 64728 T^{7} - 614079 T^{8} + 6583811 T^{9} - 5541126 T^{10} - 85887453 T^{11} + 887503681 T^{12} \))(\( 1 - 3 T - 48 T^{2} + 147 T^{3} + 1005 T^{4} - 1344 T^{5} - 24505 T^{6} - 41664 T^{7} + 965805 T^{8} + 4379277 T^{9} - 44329008 T^{10} - 85887453 T^{11} + 887503681 T^{12} \))
$37$ (\( ( 1 + 3 T + 33 T^{2} + 115 T^{3} + 1221 T^{4} + 4107 T^{5} + 50653 T^{6} )^{2} \))(\( ( 1 + 3 T + 81 T^{2} + 199 T^{3} + 2997 T^{4} + 4107 T^{5} + 50653 T^{6} )^{2} \))
$41$ (\( 1 + 6 T - 90 T^{2} - 210 T^{3} + 7812 T^{4} + 8952 T^{5} - 340301 T^{6} + 367032 T^{7} + 13131972 T^{8} - 14473410 T^{9} - 254318490 T^{10} + 695137206 T^{11} + 4750104241 T^{12} \))(\( 1 - 90 T^{2} - 18 T^{3} + 4410 T^{4} + 810 T^{5} - 194177 T^{6} + 33210 T^{7} + 7413210 T^{8} - 1240578 T^{9} - 254318490 T^{10} + 4750104241 T^{12} \))
$43$ (\( 1 + 3 T - 12 T^{2} + 473 T^{3} + 153 T^{4} - 4176 T^{5} + 165435 T^{6} - 179568 T^{7} + 282897 T^{8} + 37606811 T^{9} - 41025612 T^{10} + 441025329 T^{11} + 6321363049 T^{12} \))(\( 1 + 3 T - 24 T^{2} - 979 T^{3} - 1947 T^{4} + 14820 T^{5} + 386067 T^{6} + 637260 T^{7} - 3600003 T^{8} - 77837353 T^{9} - 82051224 T^{10} + 441025329 T^{11} + 6321363049 T^{12} \))
$47$ (\( 1 + 15 T + 48 T^{2} - 3 T^{3} + 3075 T^{4} + 18798 T^{5} + 4399 T^{6} + 883506 T^{7} + 6792675 T^{8} - 311469 T^{9} + 234224688 T^{10} + 3440175105 T^{11} + 10779215329 T^{12} \))(\( 1 - 21 T + 180 T^{2} - 1119 T^{3} + 10053 T^{4} - 100416 T^{5} + 788551 T^{6} - 4719552 T^{7} + 22207077 T^{8} - 116177937 T^{9} + 878342580 T^{10} - 4816245147 T^{11} + 10779215329 T^{12} \))
$53$ (\( ( 1 - 18 T + 198 T^{2} - 1521 T^{3} + 10494 T^{4} - 50562 T^{5} + 148877 T^{6} )^{2} \))(\( ( 1 + 6 T + 162 T^{2} + 627 T^{3} + 8586 T^{4} + 16854 T^{5} + 148877 T^{6} )^{2} \))
$59$ (\( 1 + 3 T - 114 T^{2} - 501 T^{3} + 6567 T^{4} + 20406 T^{5} - 323957 T^{6} + 1203954 T^{7} + 22859727 T^{8} - 102894879 T^{9} - 1381379154 T^{10} + 2144772897 T^{11} + 42180533641 T^{12} \))(\( 1 - 31 T + 476 T^{2} - 5741 T^{3} + 62553 T^{4} - 587576 T^{5} + 4781851 T^{6} - 34666984 T^{7} + 217746993 T^{8} - 1179080839 T^{9} + 5767863836 T^{10} - 22162653269 T^{11} + 42180533641 T^{12} \))
$61$ (\( 1 - 6 T - 102 T^{2} + 698 T^{3} + 6048 T^{4} - 26604 T^{5} - 259509 T^{6} - 1622844 T^{7} + 22504608 T^{8} + 158432738 T^{9} - 1412275782 T^{10} - 5067577806 T^{11} + 51520374361 T^{12} \))(\( 1 + 6 T + 48 T^{2} + 642 T^{3} + 3018 T^{4} + 35394 T^{5} + 438671 T^{6} + 2159034 T^{7} + 11229978 T^{8} + 145721802 T^{9} + 664600368 T^{10} + 5067577806 T^{11} + 51520374361 T^{12} \))
$67$ (\( 1 - 6 T - 138 T^{2} + 446 T^{3} + 14148 T^{4} - 16668 T^{5} - 1033545 T^{6} - 1116756 T^{7} + 63510372 T^{8} + 134140298 T^{9} - 2780854698 T^{10} - 8100750642 T^{11} + 90458382169 T^{12} \))(\( 1 + 6 T - 150 T^{2} - 506 T^{3} + 17268 T^{4} + 28236 T^{5} - 1220289 T^{6} + 1891812 T^{7} + 77516052 T^{8} - 152186078 T^{9} - 3022668150 T^{10} + 8100750642 T^{11} + 90458382169 T^{12} \))
$71$ (\( ( 1 - 15 T + 231 T^{2} - 1833 T^{3} + 16401 T^{4} - 75615 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 + 17 T + 119 T^{2} + 507 T^{3} + 8449 T^{4} + 85697 T^{5} + 357911 T^{6} )^{2} \))
$73$ (\( ( 1 - 9 T + 207 T^{2} - 1235 T^{3} + 15111 T^{4} - 47961 T^{5} + 389017 T^{6} )^{2} \))(\( ( 1 + 3 T + 195 T^{2} + 359 T^{3} + 14235 T^{4} + 15987 T^{5} + 389017 T^{6} )^{2} \))
$79$ (\( 1 + 3 T + 6 T^{2} + 1787 T^{3} + 2439 T^{4} + 14634 T^{5} + 1719519 T^{6} + 1156086 T^{7} + 15221799 T^{8} + 881060693 T^{9} + 233700486 T^{10} + 9231169197 T^{11} + 243087455521 T^{12} \))(\( 1 - 9 T - 114 T^{2} + 351 T^{3} + 13143 T^{4} + 15786 T^{5} - 1414609 T^{6} + 1247094 T^{7} + 82025463 T^{8} + 173056689 T^{9} - 4440309234 T^{10} - 27693507591 T^{11} + 243087455521 T^{12} \))
$83$ (\( 1 + 18 T - 198 T^{3} + 23814 T^{4} + 132750 T^{5} - 711245 T^{6} + 11018250 T^{7} + 164054646 T^{8} - 113213826 T^{9} + 70902731574 T^{11} + 326940373369 T^{12} \))(\( 1 - 20 T + 38 T^{2} - 346 T^{3} + 32058 T^{4} - 183754 T^{5} - 606869 T^{6} - 15251582 T^{7} + 220847562 T^{8} - 197838302 T^{9} + 1803416198 T^{10} - 78780812860 T^{11} + 326940373369 T^{12} \))
$89$ (\( ( 1 + 6 T + 72 T^{2} - 21 T^{3} + 6408 T^{4} + 47526 T^{5} + 704969 T^{6} )^{2} \))(\( ( 1 + 12 T + 216 T^{2} + 1425 T^{3} + 19224 T^{4} + 95052 T^{5} + 704969 T^{6} )^{2} \))
$97$ (\( 1 - 15 T - 84 T^{2} + 1139 T^{3} + 22203 T^{4} - 134028 T^{5} - 1218567 T^{6} - 13000716 T^{7} + 208908027 T^{8} + 1039534547 T^{9} - 7436459604 T^{10} - 128810103855 T^{11} + 832972004929 T^{12} \))(\( 1 - 9 T - 66 T^{2} + 2023 T^{3} - 7707 T^{4} - 73950 T^{5} + 1766073 T^{6} - 7173150 T^{7} - 72515163 T^{8} + 1846337479 T^{9} - 5842932546 T^{10} - 77286062313 T^{11} + 832972004929 T^{12} \))
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