Properties

Label 756.2.k
Level 756
Weight 2
Character orbit k
Rep. character \(\chi_{756}(109,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 22
Newform subspaces 6
Sturm bound 288
Trace bound 13

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 756.k (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 6 \)
Sturm bound: \(288\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(5\), \(13\)

Dimensions

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

Total New Old
Modular forms 324 22 302
Cusp forms 252 22 230
Eisenstein series 72 0 72

Trace form

\( 22q + 3q^{7} + O(q^{10}) \) \( 22q + 3q^{7} - 8q^{13} - 7q^{19} - 15q^{25} + q^{31} + 8q^{37} - 22q^{43} - 11q^{49} + 112q^{55} + 7q^{61} + 12q^{67} + 15q^{73} - 8q^{79} - 20q^{85} + 28q^{91} - 30q^{97} + O(q^{100}) \)

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.k.a \(2\) \(6.037\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-4\) \(q+(-3+2\zeta_{6})q^{7}-7q^{13}-8\zeta_{6}q^{19}+\cdots\)
756.2.k.b \(2\) \(6.037\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-4\) \(q+(-1-2\zeta_{6})q^{7}+5q^{13}-8\zeta_{6}q^{19}+\cdots\)
756.2.k.c \(2\) \(6.037\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(5\) \(q+(3-\zeta_{6})q^{7}+2q^{13}+\zeta_{6}q^{19}+(5+\cdots)q^{25}+\cdots\)
756.2.k.d \(4\) \(6.037\) \(\Q(\sqrt{-3}, \sqrt{10})\) None \(0\) \(0\) \(0\) \(2\) \(q+\beta _{1}q^{5}+(2+3\beta _{2})q^{7}+(-2\beta _{1}-2\beta _{3})q^{11}+\cdots\)
756.2.k.e \(6\) \(6.037\) 6.0.309123.1 None \(0\) \(0\) \(-1\) \(2\) \(q+(\beta _{1}+\beta _{2}+\beta _{4}-\beta _{5})q^{5}+(1+\beta _{2}+\cdots)q^{7}+\cdots\)
756.2.k.f \(6\) \(6.037\) 6.0.309123.1 None \(0\) \(0\) \(1\) \(2\) \(q+(-\beta _{1}-\beta _{2}-\beta _{4}+\beta _{5})q^{5}+(1+\beta _{2}+\cdots)q^{7}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(756, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(378, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 25 T^{4} + 625 T^{8} \))(\( 1 + T - 2 T^{2} - 35 T^{3} - 25 T^{4} + 52 T^{5} + 541 T^{6} + 260 T^{7} - 625 T^{8} - 4375 T^{9} - 1250 T^{10} + 3125 T^{11} + 15625 T^{12} \))(\( 1 - T - 2 T^{2} + 35 T^{3} - 25 T^{4} - 52 T^{5} + 541 T^{6} - 260 T^{7} - 625 T^{8} + 4375 T^{9} - 1250 T^{10} - 3125 T^{11} + 15625 T^{12} \))
$7$ (\( 1 + 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( ( 1 - T + 7 T^{2} )^{2} \))(\( 1 - 2 T + 2 T^{2} + 19 T^{3} + 14 T^{4} - 98 T^{5} + 343 T^{6} \))(\( 1 - 2 T + 2 T^{2} + 19 T^{3} + 14 T^{4} - 98 T^{5} + 343 T^{6} \))
$11$ (\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 + 18 T^{2} + 203 T^{4} + 2178 T^{6} + 14641 T^{8} \))(\( 1 + 5 T + 4 T^{2} + 11 T^{3} - 25 T^{4} - 718 T^{5} - 3005 T^{6} - 7898 T^{7} - 3025 T^{8} + 14641 T^{9} + 58564 T^{10} + 805255 T^{11} + 1771561 T^{12} \))(\( 1 - 5 T + 4 T^{2} - 11 T^{3} - 25 T^{4} + 718 T^{5} - 3005 T^{6} + 7898 T^{7} - 3025 T^{8} - 14641 T^{9} + 58564 T^{10} - 805255 T^{11} + 1771561 T^{12} \))
$13$ (\( ( 1 + 7 T + 13 T^{2} )^{2} \))(\( ( 1 - 5 T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 + 13 T^{2} )^{4} \))(\( ( 1 + 2 T + 28 T^{2} + 31 T^{3} + 364 T^{4} + 338 T^{5} + 2197 T^{6} )^{2} \))(\( ( 1 + 2 T + 28 T^{2} + 31 T^{3} + 364 T^{4} + 338 T^{5} + 2197 T^{6} )^{2} \))
$17$ (\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 24 T^{2} + 287 T^{4} - 6936 T^{6} + 83521 T^{8} \))(\( 1 - 4 T - 20 T^{2} + 146 T^{3} + 104 T^{4} - 1480 T^{5} + 4195 T^{6} - 25160 T^{7} + 30056 T^{8} + 717298 T^{9} - 1670420 T^{10} - 5679428 T^{11} + 24137569 T^{12} \))(\( 1 + 4 T - 20 T^{2} - 146 T^{3} + 104 T^{4} + 1480 T^{5} + 4195 T^{6} + 25160 T^{7} + 30056 T^{8} - 717298 T^{9} - 1670420 T^{10} + 5679428 T^{11} + 24137569 T^{12} \))
$19$ (\( ( 1 + T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( ( 1 + T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( ( 1 - 8 T + 19 T^{2} )^{2}( 1 + T + 19 T^{2} )^{2} \))(\( 1 + 3 T - 12 T^{2} - 67 T^{3} - 153 T^{4} + 54 T^{5} + 6315 T^{6} + 1026 T^{7} - 55233 T^{8} - 459553 T^{9} - 1563852 T^{10} + 7428297 T^{11} + 47045881 T^{12} \))(\( 1 + 3 T - 12 T^{2} - 67 T^{3} - 153 T^{4} + 54 T^{5} + 6315 T^{6} + 1026 T^{7} - 55233 T^{8} - 459553 T^{9} - 1563852 T^{10} + 7428297 T^{11} + 47045881 T^{12} \))
$23$ (\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 - 36 T^{2} + 767 T^{4} - 19044 T^{6} + 279841 T^{8} \))(\( 1 + 14 T + 88 T^{2} + 350 T^{3} + 1046 T^{4} + 602 T^{5} - 10061 T^{6} + 13846 T^{7} + 553334 T^{8} + 4258450 T^{9} + 24626008 T^{10} + 90108802 T^{11} + 148035889 T^{12} \))(\( 1 - 14 T + 88 T^{2} - 350 T^{3} + 1046 T^{4} - 602 T^{5} - 10061 T^{6} - 13846 T^{7} + 553334 T^{8} - 4258450 T^{9} + 24626008 T^{10} - 90108802 T^{11} + 148035889 T^{12} \))
$29$ (\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 48 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 5 T + 75 T^{2} + 227 T^{3} + 2175 T^{4} + 4205 T^{5} + 24389 T^{6} )^{2} \))(\( ( 1 - 5 T + 75 T^{2} - 227 T^{3} + 2175 T^{4} - 4205 T^{5} + 24389 T^{6} )^{2} \))
$31$ (\( ( 1 + 4 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( ( 1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4} )^{2} \))(\( 1 - 2 T - 78 T^{2} + 42 T^{3} + 3976 T^{4} - 200 T^{5} - 142097 T^{6} - 6200 T^{7} + 3820936 T^{8} + 1251222 T^{9} - 72034638 T^{10} - 57258302 T^{11} + 887503681 T^{12} \))(\( 1 - 2 T - 78 T^{2} + 42 T^{3} + 3976 T^{4} - 200 T^{5} - 142097 T^{6} - 6200 T^{7} + 3820936 T^{8} + 1251222 T^{9} - 72034638 T^{10} - 57258302 T^{11} + 887503681 T^{12} \))
$37$ (\( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( ( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + T + 37 T^{2} ) \))(\( ( 1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4} )^{2} \))(\( 1 - 54 T^{2} - 274 T^{3} + 918 T^{4} + 7398 T^{5} + 12183 T^{6} + 273726 T^{7} + 1256742 T^{8} - 13878922 T^{9} - 101204694 T^{10} + 2565726409 T^{12} \))(\( 1 - 54 T^{2} - 274 T^{3} + 918 T^{4} + 7398 T^{5} + 12183 T^{6} + 273726 T^{7} + 1256742 T^{8} - 13878922 T^{9} - 101204694 T^{10} + 2565726409 T^{12} \))
$41$ (\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 - 8 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 12 T + 132 T^{2} - 903 T^{3} + 5412 T^{4} - 20172 T^{5} + 68921 T^{6} )^{2} \))(\( ( 1 + 12 T + 132 T^{2} + 903 T^{3} + 5412 T^{4} + 20172 T^{5} + 68921 T^{6} )^{2} \))
$43$ (\( ( 1 + 13 T + 43 T^{2} )^{2} \))(\( ( 1 - 5 T + 43 T^{2} )^{2} \))(\( ( 1 - 5 T + 43 T^{2} )^{2} \))(\( ( 1 - 5 T + 43 T^{2} )^{4} \))(\( ( 1 + 9 T + 99 T^{2} + 493 T^{3} + 4257 T^{4} + 16641 T^{5} + 79507 T^{6} )^{2} \))(\( ( 1 + 9 T + 99 T^{2} + 493 T^{3} + 4257 T^{4} + 16641 T^{5} + 79507 T^{6} )^{2} \))
$47$ (\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 4 T^{2} - 2193 T^{4} - 8836 T^{6} + 4879681 T^{8} \))(\( 1 - 9 T - 6 T^{2} + 423 T^{3} - 1947 T^{4} + 1260 T^{5} + 36583 T^{6} + 59220 T^{7} - 4300923 T^{8} + 43917129 T^{9} - 29278086 T^{10} - 2064105063 T^{11} + 10779215329 T^{12} \))(\( 1 + 9 T - 6 T^{2} - 423 T^{3} - 1947 T^{4} - 1260 T^{5} + 36583 T^{6} - 59220 T^{7} - 4300923 T^{8} - 43917129 T^{9} - 29278086 T^{10} + 2064105063 T^{11} + 10779215329 T^{12} \))
$53$ (\( 1 - 53 T^{2} + 2809 T^{4} \))(\( 1 - 53 T^{2} + 2809 T^{4} \))(\( 1 - 53 T^{2} + 2809 T^{4} \))(\( 1 - 16 T^{2} - 2553 T^{4} - 44944 T^{6} + 7890481 T^{8} \))(\( 1 + 6 T - 24 T^{2} + 222 T^{3} + 6 T^{4} - 17166 T^{5} + 10591 T^{6} - 909798 T^{7} + 16854 T^{8} + 33050694 T^{9} - 189371544 T^{10} + 2509172958 T^{11} + 22164361129 T^{12} \))(\( 1 - 6 T - 24 T^{2} - 222 T^{3} + 6 T^{4} + 17166 T^{5} + 10591 T^{6} + 909798 T^{7} + 16854 T^{8} - 33050694 T^{9} - 189371544 T^{10} - 2509172958 T^{11} + 22164361129 T^{12} \))
$59$ (\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 + 42 T^{2} - 1717 T^{4} + 146202 T^{6} + 12117361 T^{8} \))(\( 1 - 5 T - 134 T^{2} + 223 T^{3} + 13355 T^{4} - 4526 T^{5} - 908765 T^{6} - 267034 T^{7} + 46488755 T^{8} + 45799517 T^{9} - 1623726374 T^{10} - 3574621495 T^{11} + 42180533641 T^{12} \))(\( 1 + 5 T - 134 T^{2} - 223 T^{3} + 13355 T^{4} + 4526 T^{5} - 908765 T^{6} + 267034 T^{7} + 46488755 T^{8} - 45799517 T^{9} - 1623726374 T^{10} + 3574621495 T^{11} + 42180533641 T^{12} \))
$61$ (\( ( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + T + 61 T^{2} ) \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( ( 1 - 3 T - 52 T^{2} - 183 T^{3} + 3721 T^{4} )^{2} \))(\( 1 + 7 T - 45 T^{2} + 84 T^{3} + 1093 T^{4} - 27251 T^{5} - 184994 T^{6} - 1662311 T^{7} + 4067053 T^{8} + 19066404 T^{9} - 623062845 T^{10} + 5912174107 T^{11} + 51520374361 T^{12} \))(\( 1 + 7 T - 45 T^{2} + 84 T^{3} + 1093 T^{4} - 27251 T^{5} - 184994 T^{6} - 1662311 T^{7} + 4067053 T^{8} + 19066404 T^{9} - 623062845 T^{10} + 5912174107 T^{11} + 51520374361 T^{12} \))
$67$ (\( ( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 11 T + 67 T^{2} )( 1 - 5 T + 67 T^{2} ) \))(\( ( 1 + 10 T + 33 T^{2} + 670 T^{3} + 4489 T^{4} )^{2} \))(\( 1 - 16 T - 4 T^{2} + 234 T^{3} + 15028 T^{4} - 85604 T^{5} - 250049 T^{6} - 5735468 T^{7} + 67460692 T^{8} + 70378542 T^{9} - 80604484 T^{10} - 21602001712 T^{11} + 90458382169 T^{12} \))(\( 1 - 16 T - 4 T^{2} + 234 T^{3} + 15028 T^{4} - 85604 T^{5} - 250049 T^{6} - 5735468 T^{7} + 67460692 T^{8} + 70378542 T^{9} - 80604484 T^{10} - 21602001712 T^{11} + 90458382169 T^{12} \))
$71$ (\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 18 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 11 T + 141 T^{2} - 1373 T^{3} + 10011 T^{4} - 55451 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 + 11 T + 141 T^{2} + 1373 T^{3} + 10011 T^{4} + 55451 T^{5} + 357911 T^{6} )^{2} \))
$73$ (\( ( 1 - 17 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))(\( ( 1 + 7 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \))(\( ( 1 - 5 T - 48 T^{2} - 365 T^{3} + 5329 T^{4} )^{2} \))(\( 1 - T - 136 T^{2} + 477 T^{3} + 8461 T^{4} - 27386 T^{5} - 503183 T^{6} - 1999178 T^{7} + 45088669 T^{8} + 185561109 T^{9} - 3862160776 T^{10} - 2073071593 T^{11} + 151334226289 T^{12} \))(\( 1 - T - 136 T^{2} + 477 T^{3} + 8461 T^{4} - 27386 T^{5} - 503183 T^{6} - 1999178 T^{7} + 45088669 T^{8} + 185561109 T^{9} - 3862160776 T^{10} - 2073071593 T^{11} + 151334226289 T^{12} \))
$79$ (\( ( 1 - 17 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \))(\( ( 1 + 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( ( 1 + 12 T + 65 T^{2} + 948 T^{3} + 6241 T^{4} )^{2} \))(\( 1 - 8 T - 18 T^{2} + 126 T^{3} - 2882 T^{4} + 21658 T^{5} + 208339 T^{6} + 1710982 T^{7} - 17986562 T^{8} + 62122914 T^{9} - 701101458 T^{10} - 24616451192 T^{11} + 243087455521 T^{12} \))(\( 1 - 8 T - 18 T^{2} + 126 T^{3} - 2882 T^{4} + 21658 T^{5} + 208339 T^{6} + 1710982 T^{7} - 17986562 T^{8} + 62122914 T^{9} - 701101458 T^{10} - 24616451192 T^{11} + 243087455521 T^{12} \))
$83$ (\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 126 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 17 T + 333 T^{2} - 2921 T^{3} + 27639 T^{4} - 117113 T^{5} + 571787 T^{6} )^{2} \))(\( ( 1 + 17 T + 333 T^{2} + 2921 T^{3} + 27639 T^{4} + 117113 T^{5} + 571787 T^{6} )^{2} \))
$89$ (\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 - 88 T^{2} - 177 T^{4} - 697048 T^{6} + 62742241 T^{8} \))(\( 1 - 3 T - 33 T^{2} - 1704 T^{3} + 393 T^{4} + 31803 T^{5} + 1804174 T^{6} + 2830467 T^{7} + 3112953 T^{8} - 1201267176 T^{9} - 2070493953 T^{10} - 16752178347 T^{11} + 496981290961 T^{12} \))(\( 1 + 3 T - 33 T^{2} + 1704 T^{3} + 393 T^{4} - 31803 T^{5} + 1804174 T^{6} - 2830467 T^{7} + 3112953 T^{8} + 1201267176 T^{9} - 2070493953 T^{10} + 16752178347 T^{11} + 496981290961 T^{12} \))
$97$ (\( ( 1 + 19 T + 97 T^{2} )^{2} \))(\( ( 1 - 5 T + 97 T^{2} )^{2} \))(\( ( 1 + 19 T + 97 T^{2} )^{2} \))(\( ( 1 + 5 T + 97 T^{2} )^{4} \))(\( ( 1 - 14 T + 251 T^{2} - 2660 T^{3} + 24347 T^{4} - 131726 T^{5} + 912673 T^{6} )^{2} \))(\( ( 1 - 14 T + 251 T^{2} - 2660 T^{3} + 24347 T^{4} - 131726 T^{5} + 912673 T^{6} )^{2} \))
show more
show less