Properties

Label 804.2.a
Level 804
Weight 2
Character orbit a
Rep. character \(\chi_{804}(1,\cdot)\)
Character field \(\Q\)
Dimension 12
Newform subspaces 6
Sturm bound 272
Trace bound 5

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Defining parameters

Level: \( N \) = \( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 804.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(272\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(804))\).

Total New Old
Modular forms 142 12 130
Cusp forms 131 12 119
Eisenstein series 11 0 11

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(67\)FrickeDim.
\(-\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(5\)
Plus space\(+\)\(3\)
Minus space\(-\)\(9\)

Trace form

\( 12q + 12q^{9} + O(q^{10}) \) \( 12q + 12q^{9} + 4q^{11} + 4q^{13} + 4q^{15} - 2q^{17} + 10q^{19} + 4q^{21} + 6q^{23} + 20q^{25} + 2q^{29} + 8q^{31} + 4q^{33} + 4q^{35} + 18q^{37} + 24q^{41} - 20q^{43} - 2q^{47} + 36q^{49} - 4q^{51} + 4q^{53} + 8q^{57} - 6q^{59} + 12q^{65} + 2q^{67} + 8q^{69} - 4q^{71} - 18q^{73} - 8q^{75} - 4q^{77} - 24q^{79} + 12q^{81} - 12q^{83} + 24q^{85} - 12q^{87} + 30q^{89} + 16q^{91} + 4q^{93} - 8q^{95} + 4q^{97} + 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(804))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 67
804.2.a.a \(1\) \(6.420\) \(\Q\) None \(0\) \(-1\) \(-3\) \(3\) \(-\) \(+\) \(-\) \(q-q^{3}-3q^{5}+3q^{7}+q^{9}-2q^{11}+\cdots\)
804.2.a.b \(1\) \(6.420\) \(\Q\) None \(0\) \(-1\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q-q^{3}+q^{9}-2q^{11}-4q^{13}-3q^{17}+\cdots\)
804.2.a.c \(1\) \(6.420\) \(\Q\) None \(0\) \(-1\) \(4\) \(0\) \(-\) \(+\) \(+\) \(q-q^{3}+4q^{5}+q^{9}+2q^{13}-4q^{15}+\cdots\)
804.2.a.d \(1\) \(6.420\) \(\Q\) None \(0\) \(1\) \(-1\) \(-3\) \(-\) \(-\) \(+\) \(q+q^{3}-q^{5}-3q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
804.2.a.e \(3\) \(6.420\) 3.3.1076.1 None \(0\) \(-3\) \(-3\) \(-5\) \(-\) \(+\) \(+\) \(q-q^{3}+(-1-\beta _{1})q^{5}+(-2-\beta _{1}+\beta _{2})q^{7}+\cdots\)
804.2.a.f \(5\) \(6.420\) 5.5.24571284.1 None \(0\) \(5\) \(3\) \(5\) \(-\) \(-\) \(-\) \(q+q^{3}+(1-\beta _{1})q^{5}+(1-\beta _{2})q^{7}+q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(804))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(804)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(67))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(134))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(201))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(268))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(402))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$3$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{5} \))
$5$ (\( 1 + 3 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 + 3 T + 10 T^{2} + 29 T^{3} + 50 T^{4} + 75 T^{5} + 125 T^{6} \))(\( 1 - 3 T + 12 T^{2} - 23 T^{3} + 91 T^{4} - 176 T^{5} + 455 T^{6} - 575 T^{7} + 1500 T^{8} - 1875 T^{9} + 3125 T^{10} \))
$7$ (\( 1 - 3 T + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 + 3 T + 7 T^{2} \))(\( 1 + 5 T + 12 T^{2} + 21 T^{3} + 84 T^{4} + 245 T^{5} + 343 T^{6} \))(\( 1 - 5 T + 18 T^{2} - 51 T^{3} + 133 T^{4} - 288 T^{5} + 931 T^{6} - 2499 T^{7} + 6174 T^{8} - 12005 T^{9} + 16807 T^{10} \))
$11$ (\( 1 + 2 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 - 4 T + 17 T^{2} - 60 T^{3} + 187 T^{4} - 484 T^{5} + 1331 T^{6} \))(\( 1 - 6 T + 39 T^{2} - 164 T^{3} + 754 T^{4} - 2540 T^{5} + 8294 T^{6} - 19844 T^{7} + 51909 T^{8} - 87846 T^{9} + 161051 T^{10} \))
$13$ (\( 1 - 2 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 2 T + 19 T^{2} - 40 T^{3} + 247 T^{4} - 338 T^{5} + 2197 T^{6} \))(\( 1 - 4 T + 41 T^{2} - 148 T^{3} + 922 T^{4} - 2464 T^{5} + 11986 T^{6} - 25012 T^{7} + 90077 T^{8} - 114244 T^{9} + 371293 T^{10} \))
$17$ (\( 1 + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 - 2 T + 31 T^{2} - 56 T^{3} + 527 T^{4} - 578 T^{5} + 4913 T^{6} \))(\( 1 - T + 21 T^{2} - 24 T^{3} + 478 T^{4} - 1366 T^{5} + 8126 T^{6} - 6936 T^{7} + 103173 T^{8} - 83521 T^{9} + 1419857 T^{10} \))
$19$ (\( 1 + 4 T + 19 T^{2} \))(\( 1 - 5 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 4 T + 45 T^{2} + 156 T^{3} + 855 T^{4} + 1444 T^{5} + 6859 T^{6} \))(\( 1 - 13 T + 99 T^{2} - 572 T^{3} + 3194 T^{4} - 15246 T^{5} + 60686 T^{6} - 206492 T^{7} + 679041 T^{8} - 1694173 T^{9} + 2476099 T^{10} \))
$23$ (\( 1 + 5 T + 23 T^{2} \))(\( 1 - T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 - 7 T + 23 T^{2} \))(\( 1 - 9 T + 88 T^{2} - 411 T^{3} + 2024 T^{4} - 4761 T^{5} + 12167 T^{6} \))(\( 1 + 55 T^{2} + 72 T^{3} + 1447 T^{4} + 3474 T^{5} + 33281 T^{6} + 38088 T^{7} + 669185 T^{8} + 6436343 T^{10} \))
$29$ (\( 1 + 4 T + 29 T^{2} \))(\( 1 + 7 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( ( 1 - 8 T + 29 T^{2} )^{3} \))(\( 1 - 3 T + 87 T^{2} - 212 T^{3} + 4036 T^{4} - 8402 T^{5} + 117044 T^{6} - 178292 T^{7} + 2121843 T^{8} - 2121843 T^{9} + 20511149 T^{10} \))
$31$ (\( 1 - 5 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 - 3 T + 31 T^{2} \))(\( 1 - T + 48 T^{2} - 153 T^{3} + 1488 T^{4} - 961 T^{5} + 29791 T^{6} \))(\( 1 - 3 T + 18 T^{2} + 11 T^{3} + 705 T^{4} - 4656 T^{5} + 21855 T^{6} + 10571 T^{7} + 536238 T^{8} - 2770563 T^{9} + 28629151 T^{10} \))
$37$ (\( 1 + 11 T + 37 T^{2} \))(\( 1 - 7 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 + 3 T + 37 T^{2} \))(\( 1 - 19 T + 214 T^{2} - 1575 T^{3} + 7918 T^{4} - 26011 T^{5} + 50653 T^{6} \))(\( 1 - 12 T + 193 T^{2} - 1530 T^{3} + 13997 T^{4} - 80018 T^{5} + 517889 T^{6} - 2094570 T^{7} + 9776029 T^{8} - 22489932 T^{9} + 69343957 T^{10} \))
$41$ (\( 1 + 5 T + 41 T^{2} \))(\( 1 + 8 T + 41 T^{2} \))(\( 1 - 12 T + 41 T^{2} \))(\( 1 - T + 41 T^{2} \))(\( 1 - 13 T + 132 T^{2} - 1037 T^{3} + 5412 T^{4} - 21853 T^{5} + 68921 T^{6} \))(\( 1 - 11 T + 94 T^{2} - 341 T^{3} + 2905 T^{4} - 13348 T^{5} + 119105 T^{6} - 573221 T^{7} + 6478574 T^{8} - 31083371 T^{9} + 115856201 T^{10} \))
$43$ (\( 1 + 5 T + 43 T^{2} \))(\( 1 + 2 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 11 T + 43 T^{2} \))(\( 1 + T + 60 T^{2} - 135 T^{3} + 2580 T^{4} + 1849 T^{5} + 79507 T^{6} \))(\( 1 - 3 T + 108 T^{2} - 133 T^{3} + 5901 T^{4} - 1440 T^{5} + 253743 T^{6} - 245917 T^{7} + 8586756 T^{8} - 10256403 T^{9} + 147008443 T^{10} \))
$47$ (\( 1 + 8 T + 47 T^{2} \))(\( 1 - 7 T + 47 T^{2} \))(\( 1 - 10 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 - 2 T + 65 T^{2} - 32 T^{3} + 3055 T^{4} - 4418 T^{5} + 103823 T^{6} \))(\( 1 + 13 T + 123 T^{2} + 600 T^{3} + 3814 T^{4} + 18094 T^{5} + 179258 T^{6} + 1325400 T^{7} + 12770229 T^{8} + 63435853 T^{9} + 229345007 T^{10} \))
$53$ (\( 1 - T + 53 T^{2} \))(\( 1 + 14 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 - 11 T + 53 T^{2} \))(\( 1 - 3 T + 60 T^{2} + 141 T^{3} + 3180 T^{4} - 8427 T^{5} + 148877 T^{6} \))(\( 1 + 9 T + 240 T^{2} + 1675 T^{3} + 24397 T^{4} + 126916 T^{5} + 1293041 T^{6} + 4705075 T^{7} + 35730480 T^{8} + 71014329 T^{9} + 418195493 T^{10} \))
$59$ (\( 1 - 9 T + 59 T^{2} \))(\( 1 + 15 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 + 3 T + 59 T^{2} \))(\( 1 - 9 T + 102 T^{2} - 1141 T^{3} + 6018 T^{4} - 31329 T^{5} + 205379 T^{6} \))(\( 1 + 12 T + 253 T^{2} + 2696 T^{3} + 27307 T^{4} + 234602 T^{5} + 1611113 T^{6} + 9384776 T^{7} + 51960887 T^{8} + 145408332 T^{9} + 714924299 T^{10} \))
$61$ (\( 1 + 12 T + 61 T^{2} \))(\( 1 + 6 T + 61 T^{2} \))(\( 1 - 6 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 67 T^{2} + 452 T^{3} + 4087 T^{4} + 226981 T^{6} \))(\( 1 - 4 T + 137 T^{2} - 628 T^{3} + 12754 T^{4} - 55984 T^{5} + 777994 T^{6} - 2336788 T^{7} + 31096397 T^{8} - 55383364 T^{9} + 844596301 T^{10} \))
$67$ (\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{5} \))
$71$ (\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 2 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 - 14 T + 57 T^{2} + 200 T^{3} + 4047 T^{4} - 70574 T^{5} + 357911 T^{6} \))(\( 1 + 24 T + 555 T^{2} + 7492 T^{3} + 93850 T^{4} + 822088 T^{5} + 6663350 T^{6} + 37767172 T^{7} + 198640605 T^{8} + 609880344 T^{9} + 1804229351 T^{10} \))
$73$ (\( 1 + 9 T + 73 T^{2} \))(\( 1 - 9 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 9 T + 73 T^{2} \))(\( 1 + 23 T + 310 T^{2} + 3139 T^{3} + 22630 T^{4} + 122567 T^{5} + 389017 T^{6} \))(\( 1 - 12 T + 357 T^{2} - 3118 T^{3} + 50553 T^{4} - 326970 T^{5} + 3690369 T^{6} - 16615822 T^{7} + 138879069 T^{8} - 340778892 T^{9} + 2073071593 T^{10} \))
$79$ (\( 1 + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 20 T + 325 T^{2} + 3048 T^{3} + 25675 T^{4} + 124820 T^{5} + 493039 T^{6} \))(\( 1 + 4 T + 179 T^{2} - 96 T^{3} + 13138 T^{4} - 48712 T^{5} + 1037902 T^{6} - 599136 T^{7} + 88253981 T^{8} + 155800324 T^{9} + 3077056399 T^{10} \))
$83$ (\( 1 + T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 - 11 T + 83 T^{2} \))(\( 1 - 15 T + 222 T^{2} - 2463 T^{3} + 18426 T^{4} - 103335 T^{5} + 571787 T^{6} \))(\( 1 + 27 T + 558 T^{2} + 8237 T^{3} + 99931 T^{4} + 998672 T^{5} + 8294273 T^{6} + 56744693 T^{7} + 319057146 T^{8} + 1281374667 T^{9} + 3939040643 T^{10} \))
$89$ (\( 1 - 14 T + 89 T^{2} \))(\( 1 - 5 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 12 T + 199 T^{2} - 1284 T^{3} + 17711 T^{4} - 95052 T^{5} + 704969 T^{6} \))(\( 1 + 5 T + 315 T^{2} + 612 T^{3} + 41488 T^{4} + 27302 T^{5} + 3692432 T^{6} + 4847652 T^{7} + 222065235 T^{8} + 313711205 T^{9} + 5584059449 T^{10} \))
$97$ (\( 1 + 6 T + 97 T^{2} \))(\( 1 - 12 T + 97 T^{2} \))(\( 1 + 18 T + 97 T^{2} \))(\( 1 + 6 T + 97 T^{2} \))(\( 1 - 14 T + 339 T^{2} - 2712 T^{3} + 32883 T^{4} - 131726 T^{5} + 912673 T^{6} \))(\( 1 - 8 T + 137 T^{2} - 852 T^{3} + 5470 T^{4} - 41656 T^{5} + 530590 T^{6} - 8016468 T^{7} + 125036201 T^{8} - 708234248 T^{9} + 8587340257 T^{10} \))
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