Properties

Label 756.2.t
Level 756
Weight 2
Character orbit t
Rep. character \(\chi_{756}(269,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 22
Newform subspaces 5
Sturm bound 288
Trace bound 19

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Defining parameters

Level: \( N \) = \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 756.t (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 21 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 5 \)
Sturm bound: \(288\)
Trace bound: \(19\)
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

\( 22q - 7q^{7} + O(q^{10}) \) \( 22q - 7q^{7} - 9q^{19} - 11q^{25} + 21q^{31} - 8q^{37} - 26q^{43} + 37q^{49} + 15q^{61} - 4q^{67} + 27q^{73} + 32q^{79} + 60q^{85} + 18q^{91} + O(q^{100}) \)

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.t.a \(2\) \(6.037\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-5\) \(q+(-3+\zeta_{6})q^{7}+(4-8\zeta_{6})q^{13}+(10+\cdots)q^{19}+\cdots\)
756.2.t.b \(2\) \(6.037\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(4\) \(q+(1+2\zeta_{6})q^{7}+(-3+6\zeta_{6})q^{13}+(-4+\cdots)q^{19}+\cdots\)
756.2.t.c \(2\) \(6.037\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(4\) \(q+(3-2\zeta_{6})q^{7}+(1-2\zeta_{6})q^{13}+(4-2\zeta_{6})q^{19}+\cdots\)
756.2.t.d \(4\) \(6.037\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(-10\) \(q+\beta _{1}q^{5}+(-2+\beta _{2})q^{7}+(2+4\beta _{2}+\cdots)q^{13}+\cdots\)
756.2.t.e \(12\) \(6.037\) 12.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{2}q^{5}+\beta _{10}q^{7}+(-\beta _{3}-\beta _{5})q^{11}+\cdots\)

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))(\( \))(\( \))(\( \))
$3$ (\( \))(\( \))(\( \))(\( \))(\( \))
$5$ (\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 + 8 T^{2} + 39 T^{4} + 200 T^{6} + 625 T^{8} \))(\( 1 - 15 T^{2} + 96 T^{4} - 409 T^{6} + 1971 T^{8} - 11052 T^{10} + 56769 T^{12} - 276300 T^{14} + 1231875 T^{16} - 6390625 T^{18} + 37500000 T^{20} - 146484375 T^{22} + 244140625 T^{24} \))
$7$ (\( 1 + 5 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( ( 1 + 5 T + 7 T^{2} )^{2} \))(\( ( 1 - 7 T^{3} + 343 T^{6} )^{2} \))
$11$ (\( 1 + 11 T^{2} + 121 T^{4} \))(\( 1 + 11 T^{2} + 121 T^{4} \))(\( 1 + 11 T^{2} + 121 T^{4} \))(\( ( 1 + 11 T^{2} + 121 T^{4} )^{2} \))(\( 1 + 21 T^{2} + 120 T^{4} - 1085 T^{6} - 17973 T^{8} - 67788 T^{10} - 69951 T^{12} - 8202348 T^{14} - 263142693 T^{16} - 1922143685 T^{18} + 25723065720 T^{20} + 544685916621 T^{22} + 3138428376721 T^{24} \))
$13$ (\( ( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 36 T^{2} + 792 T^{4} - 11495 T^{6} + 133848 T^{8} - 1028196 T^{10} + 4826809 T^{12} )^{2} \))
$17$ (\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 16 T^{2} - 33 T^{4} - 4624 T^{6} + 83521 T^{8} \))(\( 1 + 12 T^{2} - 624 T^{4} - 4882 T^{6} + 283284 T^{8} + 1089036 T^{10} - 84522621 T^{12} + 314731404 T^{14} + 23660162964 T^{16} - 117839611858 T^{18} - 4352872643184 T^{20} + 24191926805388 T^{22} + 582622237229761 T^{24} \))
$19$ (\( ( 1 - 8 T + 19 T^{2} )( 1 - 7 T + 19 T^{2} ) \))(\( ( 1 - T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + T + 19 T^{2} ) \))(\( ( 1 + 3 T + 22 T^{2} + 57 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 + 9 T + 72 T^{2} + 405 T^{3} + 2223 T^{4} + 12546 T^{5} + 56077 T^{6} + 238374 T^{7} + 802503 T^{8} + 2777895 T^{9} + 9383112 T^{10} + 22284891 T^{11} + 47045881 T^{12} )^{2} \))
$23$ (\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 - 8 T^{2} - 465 T^{4} - 4232 T^{6} + 279841 T^{8} \))(\( 1 + 84 T^{2} + 3684 T^{4} + 103558 T^{6} + 2039616 T^{8} + 26787600 T^{10} + 378704571 T^{12} + 14170640400 T^{14} + 570768181056 T^{16} + 15330300593062 T^{18} + 288497669775204 T^{20} + 3479826941946516 T^{22} + 21914624432020321 T^{24} \))
$29$ (\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 4 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 21 T^{2} + 2481 T^{4} - 34153 T^{6} + 2086521 T^{8} - 14852901 T^{10} + 594823321 T^{12} )^{2} \))
$31$ (\( ( 1 - 11 T + 31 T^{2} )( 1 - 4 T + 31 T^{2} ) \))(\( ( 1 - 11 T + 31 T^{2} )( 1 - 4 T + 31 T^{2} ) \))(\( ( 1 - 4 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( ( 1 - 4 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \))(\( ( 1 + 30 T^{2} - 30 T^{4} + 1890 T^{5} - 30359 T^{6} + 58590 T^{7} - 28830 T^{8} + 27705630 T^{10} + 887503681 T^{12} )^{2} \))
$37$ (\( ( 1 - T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( ( 1 - 10 T + 37 T^{2} )( 1 - T + 37 T^{2} ) \))(\( ( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( ( 1 - 8 T + 27 T^{2} - 296 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 12 T + 48 T^{2} + 130 T^{3} - 468 T^{4} - 19764 T^{5} - 175701 T^{6} - 731268 T^{7} - 640692 T^{8} + 6584890 T^{9} + 89959728 T^{10} + 832127484 T^{11} + 2565726409 T^{12} )^{2} \))
$41$ (\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 64 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 60 T^{2} + 4836 T^{4} + 196567 T^{6} + 8129316 T^{8} + 169545660 T^{10} + 4750104241 T^{12} )^{2} \))
$43$ (\( ( 1 + 5 T + 43 T^{2} )^{2} \))(\( ( 1 + 5 T + 43 T^{2} )^{2} \))(\( ( 1 - 13 T + 43 T^{2} )^{2} \))(\( ( 1 + 5 T + 43 T^{2} )^{4} \))(\( ( 1 + 3 T + 111 T^{2} + 245 T^{3} + 4773 T^{4} + 5547 T^{5} + 79507 T^{6} )^{4} \))
$47$ (\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 76 T^{2} + 3567 T^{4} - 167884 T^{6} + 4879681 T^{8} \))(\( 1 - 267 T^{2} + 40920 T^{4} - 4343125 T^{6} + 352510407 T^{8} - 22655106144 T^{10} + 1180938336801 T^{12} - 50045129472096 T^{14} + 1720138335340167 T^{16} - 46815479575763125 T^{18} + 974357850199260120 T^{20} - 14043968306966623083 T^{22} + \)\(11\!\cdots\!41\)\( T^{24} \))
$53$ (\( 1 + 53 T^{2} + 2809 T^{4} \))(\( 1 + 53 T^{2} + 2809 T^{4} \))(\( 1 + 53 T^{2} + 2809 T^{4} \))(\( 1 + 52 T^{2} - 105 T^{4} + 146068 T^{6} + 7890481 T^{8} \))(\( 1 + 84 T^{2} + 2136 T^{4} - 35042 T^{6} - 8443404 T^{8} - 556134516 T^{10} - 29094135357 T^{12} - 1562181855444 T^{14} - 66622518837324 T^{16} - 776683542682418 T^{18} + 132986698718667096 T^{20} + 14690547510703096116 T^{22} + \)\(49\!\cdots\!41\)\( T^{24} \))
$59$ (\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 46 T^{2} - 1365 T^{4} - 160126 T^{6} + 12117361 T^{8} \))(\( 1 - 51 T^{2} - 1380 T^{4} + 158351 T^{6} - 7224741 T^{8} + 76185540 T^{10} + 30324904161 T^{12} + 265201864740 T^{14} - 87544794828501 T^{16} + 6679329682585991 T^{18} - 202626003893962980 T^{20} - 26066954418332711451 T^{22} + \)\(17\!\cdots\!81\)\( T^{24} \))
$61$ (\( ( 1 + 13 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \))(\( ( 1 - 14 T + 61 T^{2} )( 1 - T + 61 T^{2} ) \))(\( ( 1 - 14 T + 61 T^{2} )( 1 - 13 T + 61 T^{2} ) \))(\( ( 1 + 9 T + 88 T^{2} + 549 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 9 T + 135 T^{2} - 972 T^{3} + 9153 T^{4} - 113283 T^{5} + 772702 T^{6} - 6910263 T^{7} + 34058313 T^{8} - 220625532 T^{9} + 1869188535 T^{10} - 7601366709 T^{11} + 51520374361 T^{12} )^{2} \))
$67$ (\( ( 1 - 11 T + 67 T^{2} )( 1 - 5 T + 67 T^{2} ) \))(\( ( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 54 T^{2} - 994 T^{3} - 702 T^{4} + 26838 T^{5} + 653163 T^{6} + 1798146 T^{7} - 3151278 T^{8} - 298958422 T^{9} - 1088160534 T^{10} + 90458382169 T^{12} )^{2} \))
$71$ (\( ( 1 - 71 T^{2} )^{2} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( ( 1 + 74 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 237 T^{2} + 29877 T^{4} - 2533201 T^{6} + 150609957 T^{8} - 6022568397 T^{10} + 128100283921 T^{12} )^{2} \))
$73$ (\( ( 1 - 10 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))(\( ( 1 - 17 T + 73 T^{2} )( 1 - 7 T + 73 T^{2} ) \))(\( ( 1 + 7 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( ( 1 + 15 T + 148 T^{2} + 1095 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 - 27 T + 522 T^{2} - 7533 T^{3} + 94005 T^{4} - 988578 T^{5} + 9101239 T^{6} - 72166194 T^{7} + 500952645 T^{8} - 2930465061 T^{9} + 14823881802 T^{10} - 55972933011 T^{11} + 151334226289 T^{12} )^{2} \))
$79$ (\( ( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( ( 1 + 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \))(\( ( 1 - 17 T + 79 T^{2} )^{2}( 1 + 13 T + 79 T^{2} )^{2} \))(\( ( 1 - 12 T - 78 T^{2} + 374 T^{3} + 13518 T^{4} + 8802 T^{5} - 1498701 T^{6} + 695358 T^{7} + 84365838 T^{8} + 184396586 T^{9} - 3038106318 T^{10} - 36924676788 T^{11} + 243087455521 T^{12} )^{2} \))
$83$ (\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 - 122 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 123 T^{2} + 12585 T^{4} + 1116115 T^{6} + 86698065 T^{8} + 5837373483 T^{10} + 326940373369 T^{12} )^{2} \))
$89$ (\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 - 16 T^{2} - 7665 T^{4} - 126736 T^{6} + 62742241 T^{8} \))(\( ( 1 - 151 T^{2} + 14880 T^{4} - 1196071 T^{6} + 62742241 T^{8} )^{3} \))
$97$ (\( ( 1 - 19 T + 97 T^{2} )( 1 + 19 T + 97 T^{2} ) \))(\( ( 1 - 5 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} ) \))(\( ( 1 - 19 T + 97 T^{2} )( 1 + 19 T + 97 T^{2} ) \))(\( ( 1 + 49 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 378 T^{2} + 63423 T^{4} - 7014476 T^{6} + 596747007 T^{8} - 33464068218 T^{10} + 832972004929 T^{12} )^{2} \))
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