Properties

Label 162.4.a
Level 162
Weight 4
Character orbit a
Rep. character \(\chi_{162}(1,\cdot)\)
Character field \(\Q\)
Dimension 12
Newform subspaces 8
Sturm bound 108
Trace bound 5

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 162.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(108\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(162))\).

Total New Old
Modular forms 93 12 81
Cusp forms 69 12 57
Eisenstein series 24 0 24

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

\(2\)\(3\)FrickeDim.
\(+\)\(+\)\(+\)\(3\)
\(+\)\(-\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(4\)
Plus space\(+\)\(7\)
Minus space\(-\)\(5\)

Trace form

\( 12q + 48q^{4} + 24q^{7} + O(q^{10}) \) \( 12q + 48q^{4} + 24q^{7} - 36q^{10} - 30q^{13} + 192q^{16} - 30q^{19} + 36q^{22} + 822q^{25} + 96q^{28} + 132q^{31} + 216q^{34} - 678q^{37} - 144q^{40} - 1434q^{43} + 504q^{46} - 72q^{49} - 120q^{52} - 324q^{55} + 1332q^{58} - 1002q^{61} + 768q^{64} - 2838q^{67} - 288q^{70} + 2184q^{73} - 120q^{76} + 780q^{79} - 1980q^{82} + 3294q^{85} + 144q^{88} - 1140q^{91} - 3096q^{94} - 6078q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(162))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3
162.4.a.a \(1\) \(9.558\) \(\Q\) None \(-2\) \(0\) \(9\) \(-31\) \(+\) \(-\) \(q-2q^{2}+4q^{4}+9q^{5}-31q^{7}-8q^{8}+\cdots\)
162.4.a.b \(1\) \(9.558\) \(\Q\) None \(-2\) \(0\) \(21\) \(8\) \(+\) \(+\) \(q-2q^{2}+4q^{4}+21q^{5}+8q^{7}-8q^{8}+\cdots\)
162.4.a.c \(1\) \(9.558\) \(\Q\) None \(2\) \(0\) \(-21\) \(8\) \(-\) \(+\) \(q+2q^{2}+4q^{4}-21q^{5}+8q^{7}+8q^{8}+\cdots\)
162.4.a.d \(1\) \(9.558\) \(\Q\) None \(2\) \(0\) \(-9\) \(-31\) \(-\) \(+\) \(q+2q^{2}+4q^{4}-9q^{5}-31q^{7}+8q^{8}+\cdots\)
162.4.a.e \(2\) \(9.558\) \(\Q(\sqrt{3}) \) None \(-4\) \(0\) \(-12\) \(16\) \(+\) \(-\) \(q-2q^{2}+4q^{4}+(-6+\beta )q^{5}+(8-2\beta )q^{7}+\cdots\)
162.4.a.f \(2\) \(9.558\) \(\Q(\sqrt{105}) \) None \(-4\) \(0\) \(-9\) \(19\) \(+\) \(+\) \(q-2q^{2}+4q^{4}+(-5-\beta )q^{5}+(9-\beta )q^{7}+\cdots\)
162.4.a.g \(2\) \(9.558\) \(\Q(\sqrt{105}) \) None \(4\) \(0\) \(9\) \(19\) \(-\) \(-\) \(q+2q^{2}+4q^{4}+(4-\beta )q^{5}+(10+\beta )q^{7}+\cdots\)
162.4.a.h \(2\) \(9.558\) \(\Q(\sqrt{3}) \) None \(4\) \(0\) \(12\) \(16\) \(-\) \(-\) \(q+2q^{2}+4q^{4}+(6+\beta )q^{5}+(8+2\beta )q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(162))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(162)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T \))(\( 1 + 2 T \))(\( 1 - 2 T \))(\( 1 - 2 T \))(\( ( 1 + 2 T )^{2} \))(\( ( 1 + 2 T )^{2} \))(\( ( 1 - 2 T )^{2} \))(\( ( 1 - 2 T )^{2} \))
$3$ 1
$5$ (\( 1 - 9 T + 125 T^{2} \))(\( 1 - 21 T + 125 T^{2} \))(\( 1 + 21 T + 125 T^{2} \))(\( 1 + 9 T + 125 T^{2} \))(\( 1 + 12 T + 259 T^{2} + 1500 T^{3} + 15625 T^{4} \))(\( 1 + 9 T + 34 T^{2} + 1125 T^{3} + 15625 T^{4} \))(\( 1 - 9 T + 34 T^{2} - 1125 T^{3} + 15625 T^{4} \))(\( 1 - 12 T + 259 T^{2} - 1500 T^{3} + 15625 T^{4} \))
$7$ (\( 1 + 31 T + 343 T^{2} \))(\( 1 - 8 T + 343 T^{2} \))(\( 1 - 8 T + 343 T^{2} \))(\( 1 + 31 T + 343 T^{2} \))(\( 1 - 16 T + 642 T^{2} - 5488 T^{3} + 117649 T^{4} \))(\( 1 - 19 T + 540 T^{2} - 6517 T^{3} + 117649 T^{4} \))(\( 1 - 19 T + 540 T^{2} - 6517 T^{3} + 117649 T^{4} \))(\( 1 - 16 T + 642 T^{2} - 5488 T^{3} + 117649 T^{4} \))
$11$ (\( 1 - 15 T + 1331 T^{2} \))(\( 1 - 36 T + 1331 T^{2} \))(\( 1 + 36 T + 1331 T^{2} \))(\( 1 + 15 T + 1331 T^{2} \))(\( 1 + 36 T + 1258 T^{2} + 47916 T^{3} + 1771561 T^{4} \))(\( 1 + 24 T + 1861 T^{2} + 31944 T^{3} + 1771561 T^{4} \))(\( 1 - 24 T + 1861 T^{2} - 31944 T^{3} + 1771561 T^{4} \))(\( 1 - 36 T + 1258 T^{2} - 47916 T^{3} + 1771561 T^{4} \))
$13$ (\( 1 + 37 T + 2197 T^{2} \))(\( 1 + 49 T + 2197 T^{2} \))(\( 1 + 49 T + 2197 T^{2} \))(\( 1 + 37 T + 2197 T^{2} \))(\( 1 - 10 T - 873 T^{2} - 21970 T^{3} + 4826809 T^{4} \))(\( 1 - 61 T + 5088 T^{2} - 134017 T^{3} + 4826809 T^{4} \))(\( 1 - 61 T + 5088 T^{2} - 134017 T^{3} + 4826809 T^{4} \))(\( 1 - 10 T - 873 T^{2} - 21970 T^{3} + 4826809 T^{4} \))
$17$ (\( 1 - 42 T + 4913 T^{2} \))(\( 1 - 21 T + 4913 T^{2} \))(\( 1 + 21 T + 4913 T^{2} \))(\( 1 + 42 T + 4913 T^{2} \))(\( 1 + 120 T + 10159 T^{2} + 589560 T^{3} + 24137569 T^{4} \))(\( 1 - 3 T + 9592 T^{2} - 14739 T^{3} + 24137569 T^{4} \))(\( 1 + 3 T + 9592 T^{2} + 14739 T^{3} + 24137569 T^{4} \))(\( 1 - 120 T + 10159 T^{2} - 589560 T^{3} + 24137569 T^{4} \))
$19$ (\( 1 + 28 T + 6859 T^{2} \))(\( 1 + 112 T + 6859 T^{2} \))(\( 1 + 112 T + 6859 T^{2} \))(\( 1 + 28 T + 6859 T^{2} \))(\( 1 + 8 T + 666 T^{2} + 54872 T^{3} + 47045881 T^{4} \))(\( 1 - 133 T + 12234 T^{2} - 912247 T^{3} + 47045881 T^{4} \))(\( 1 - 133 T + 12234 T^{2} - 912247 T^{3} + 47045881 T^{4} \))(\( 1 + 8 T + 666 T^{2} + 54872 T^{3} + 47045881 T^{4} \))
$23$ (\( 1 + 195 T + 12167 T^{2} \))(\( 1 - 180 T + 12167 T^{2} \))(\( 1 + 180 T + 12167 T^{2} \))(\( 1 - 195 T + 12167 T^{2} \))(\( 1 + 180 T + 32002 T^{2} + 2190060 T^{3} + 148035889 T^{4} \))(\( 1 - 69 T + 25288 T^{2} - 839523 T^{3} + 148035889 T^{4} \))(\( 1 + 69 T + 25288 T^{2} + 839523 T^{3} + 148035889 T^{4} \))(\( 1 - 180 T + 32002 T^{2} - 2190060 T^{3} + 148035889 T^{4} \))
$29$ (\( 1 + 111 T + 24389 T^{2} \))(\( 1 + 135 T + 24389 T^{2} \))(\( 1 - 135 T + 24389 T^{2} \))(\( 1 - 111 T + 24389 T^{2} \))(\( 1 + 324 T + 73699 T^{2} + 7902036 T^{3} + 594823321 T^{4} \))(\( 1 - 237 T + 51244 T^{2} - 5780193 T^{3} + 594823321 T^{4} \))(\( 1 + 237 T + 51244 T^{2} + 5780193 T^{3} + 594823321 T^{4} \))(\( 1 - 324 T + 73699 T^{2} - 7902036 T^{3} + 594823321 T^{4} \))
$31$ (\( 1 + 205 T + 29791 T^{2} \))(\( 1 - 308 T + 29791 T^{2} \))(\( 1 - 308 T + 29791 T^{2} \))(\( 1 + 205 T + 29791 T^{2} \))(\( 1 + 248 T + 39966 T^{2} + 7388168 T^{3} + 887503681 T^{4} \))(\( 1 - 211 T + 68586 T^{2} - 6285901 T^{3} + 887503681 T^{4} \))(\( 1 - 211 T + 68586 T^{2} - 6285901 T^{3} + 887503681 T^{4} \))(\( 1 + 248 T + 39966 T^{2} + 7388168 T^{3} + 887503681 T^{4} \))
$37$ (\( 1 + 166 T + 50653 T^{2} \))(\( 1 + T + 50653 T^{2} \))(\( 1 + T + 50653 T^{2} \))(\( 1 + 166 T + 50653 T^{2} \))(\( 1 + 434 T + 148287 T^{2} + 21983402 T^{3} + 2565726409 T^{4} \))(\( 1 - 262 T + 72162 T^{2} - 13271086 T^{3} + 2565726409 T^{4} \))(\( 1 - 262 T + 72162 T^{2} - 13271086 T^{3} + 2565726409 T^{4} \))(\( 1 + 434 T + 148287 T^{2} + 21983402 T^{3} + 2565726409 T^{4} \))
$41$ (\( 1 - 261 T + 68921 T^{2} \))(\( 1 + 42 T + 68921 T^{2} \))(\( 1 - 42 T + 68921 T^{2} \))(\( 1 + 261 T + 68921 T^{2} \))(\( 1 + 192 T + 94786 T^{2} + 13232832 T^{3} + 4750104241 T^{4} \))(\( 1 - 468 T + 191653 T^{2} - 32255028 T^{3} + 4750104241 T^{4} \))(\( 1 + 468 T + 191653 T^{2} + 32255028 T^{3} + 4750104241 T^{4} \))(\( 1 - 192 T + 94786 T^{2} - 13232832 T^{3} + 4750104241 T^{4} \))
$43$ (\( 1 + 43 T + 79507 T^{2} \))(\( 1 - 20 T + 79507 T^{2} \))(\( 1 - 20 T + 79507 T^{2} \))(\( 1 + 43 T + 79507 T^{2} \))(\( 1 + 608 T + 250458 T^{2} + 48340256 T^{3} + 6321363049 T^{4} \))(\( 1 + 86 T + 24783 T^{2} + 6837602 T^{3} + 6321363049 T^{4} \))(\( 1 + 86 T + 24783 T^{2} + 6837602 T^{3} + 6321363049 T^{4} \))(\( 1 + 608 T + 250458 T^{2} + 48340256 T^{3} + 6321363049 T^{4} \))
$47$ (\( 1 + 177 T + 103823 T^{2} \))(\( 1 - 84 T + 103823 T^{2} \))(\( 1 + 84 T + 103823 T^{2} \))(\( 1 - 177 T + 103823 T^{2} \))(\( 1 - 384 T + 209518 T^{2} - 39868032 T^{3} + 10779215329 T^{4} \))(\( 1 - 483 T + 212812 T^{2} - 50146509 T^{3} + 10779215329 T^{4} \))(\( 1 + 483 T + 212812 T^{2} + 50146509 T^{3} + 10779215329 T^{4} \))(\( 1 + 384 T + 209518 T^{2} + 39868032 T^{3} + 10779215329 T^{4} \))
$53$ (\( 1 + 114 T + 148877 T^{2} \))(\( 1 + 174 T + 148877 T^{2} \))(\( 1 - 174 T + 148877 T^{2} \))(\( 1 - 114 T + 148877 T^{2} \))(\( 1 - 408 T + 183418 T^{2} - 60741816 T^{3} + 22164361129 T^{4} \))(\( 1 - 150 T + 257074 T^{2} - 22331550 T^{3} + 22164361129 T^{4} \))(\( 1 + 150 T + 257074 T^{2} + 22331550 T^{3} + 22164361129 T^{4} \))(\( 1 + 408 T + 183418 T^{2} + 60741816 T^{3} + 22164361129 T^{4} \))
$59$ (\( 1 + 159 T + 205379 T^{2} \))(\( 1 - 504 T + 205379 T^{2} \))(\( 1 + 504 T + 205379 T^{2} \))(\( 1 - 159 T + 205379 T^{2} \))(\( 1 - 1008 T + 637126 T^{2} - 207022032 T^{3} + 42180533641 T^{4} \))(\( 1 - 168 T + 416869 T^{2} - 34503672 T^{3} + 42180533641 T^{4} \))(\( 1 + 168 T + 416869 T^{2} + 34503672 T^{3} + 42180533641 T^{4} \))(\( 1 + 1008 T + 637126 T^{2} + 207022032 T^{3} + 42180533641 T^{4} \))
$61$ (\( 1 - 191 T + 226981 T^{2} \))(\( 1 + 385 T + 226981 T^{2} \))(\( 1 + 385 T + 226981 T^{2} \))(\( 1 - 191 T + 226981 T^{2} \))(\( 1 - 742 T + 582855 T^{2} - 168419902 T^{3} + 51520374361 T^{4} \))(\( 1 + 1049 T + 675906 T^{2} + 238103069 T^{3} + 51520374361 T^{4} \))(\( 1 + 1049 T + 675906 T^{2} + 238103069 T^{3} + 51520374361 T^{4} \))(\( 1 - 742 T + 582855 T^{2} - 168419902 T^{3} + 51520374361 T^{4} \))
$67$ (\( 1 + 421 T + 300763 T^{2} \))(\( 1 - 272 T + 300763 T^{2} \))(\( 1 - 272 T + 300763 T^{2} \))(\( 1 + 421 T + 300763 T^{2} \))(\( 1 + 104 T + 90042 T^{2} + 31279352 T^{3} + 90458382169 T^{4} \))(\( 1 + 1166 T + 907395 T^{2} + 350689658 T^{3} + 90458382169 T^{4} \))(\( 1 + 1166 T + 907395 T^{2} + 350689658 T^{3} + 90458382169 T^{4} \))(\( 1 + 104 T + 90042 T^{2} + 31279352 T^{3} + 90458382169 T^{4} \))
$71$ (\( 1 + 156 T + 357911 T^{2} \))(\( 1 + 888 T + 357911 T^{2} \))(\( 1 - 888 T + 357911 T^{2} \))(\( 1 - 156 T + 357911 T^{2} \))(\( 1 - 1140 T + 1038994 T^{2} - 408018540 T^{3} + 128100283921 T^{4} \))(\( 1 + 312 T + 498238 T^{2} + 111668232 T^{3} + 128100283921 T^{4} \))(\( 1 - 312 T + 498238 T^{2} - 111668232 T^{3} + 128100283921 T^{4} \))(\( 1 + 1140 T + 1038994 T^{2} + 408018540 T^{3} + 128100283921 T^{4} \))
$73$ (\( 1 - 182 T + 389017 T^{2} \))(\( 1 - 371 T + 389017 T^{2} \))(\( 1 - 371 T + 389017 T^{2} \))(\( 1 - 182 T + 389017 T^{2} \))(\( 1 - 850 T + 709827 T^{2} - 330664450 T^{3} + 151334226289 T^{4} \))(\( 1 + 311 T + 698028 T^{2} + 120984287 T^{3} + 151334226289 T^{4} \))(\( 1 + 311 T + 698028 T^{2} + 120984287 T^{3} + 151334226289 T^{4} \))(\( 1 - 850 T + 709827 T^{2} - 330664450 T^{3} + 151334226289 T^{4} \))
$79$ (\( 1 - 1133 T + 493039 T^{2} \))(\( 1 + 652 T + 493039 T^{2} \))(\( 1 + 652 T + 493039 T^{2} \))(\( 1 - 1133 T + 493039 T^{2} \))(\( 1 + 440 T + 966978 T^{2} + 216937160 T^{3} + 243087455521 T^{4} \))(\( 1 - 349 T + 976602 T^{2} - 172070611 T^{3} + 243087455521 T^{4} \))(\( 1 - 349 T + 976602 T^{2} - 172070611 T^{3} + 243087455521 T^{4} \))(\( 1 + 440 T + 966978 T^{2} + 216937160 T^{3} + 243087455521 T^{4} \))
$83$ (\( 1 - 1083 T + 571787 T^{2} \))(\( 1 - 84 T + 571787 T^{2} \))(\( 1 + 84 T + 571787 T^{2} \))(\( 1 + 1083 T + 571787 T^{2} \))(\( 1 + 264 T + 1063798 T^{2} + 150951768 T^{3} + 326940373369 T^{4} \))(\( 1 - 1221 T + 1412098 T^{2} - 698151927 T^{3} + 326940373369 T^{4} \))(\( 1 + 1221 T + 1412098 T^{2} + 698151927 T^{3} + 326940373369 T^{4} \))(\( 1 - 264 T + 1063798 T^{2} - 150951768 T^{3} + 326940373369 T^{4} \))
$89$ (\( 1 - 1050 T + 704969 T^{2} \))(\( 1 - 21 T + 704969 T^{2} \))(\( 1 + 21 T + 704969 T^{2} \))(\( 1 + 1050 T + 704969 T^{2} \))(\( 1 + 768 T + 1343527 T^{2} + 541416192 T^{3} + 496981290961 T^{4} \))(\( 1 + 492 T + 1092454 T^{2} + 346844748 T^{3} + 496981290961 T^{4} \))(\( 1 - 492 T + 1092454 T^{2} - 346844748 T^{3} + 496981290961 T^{4} \))(\( 1 - 768 T + 1343527 T^{2} - 541416192 T^{3} + 496981290961 T^{4} \))
$97$ (\( 1 + 901 T + 912673 T^{2} \))(\( 1 + 1246 T + 912673 T^{2} \))(\( 1 + 1246 T + 912673 T^{2} \))(\( 1 + 901 T + 912673 T^{2} \))(\( 1 + 764 T + 1886598 T^{2} + 697282172 T^{3} + 832972004929 T^{4} \))(\( 1 + 128 T + 1715097 T^{2} + 116822144 T^{3} + 832972004929 T^{4} \))(\( 1 + 128 T + 1715097 T^{2} + 116822144 T^{3} + 832972004929 T^{4} \))(\( 1 + 764 T + 1886598 T^{2} + 697282172 T^{3} + 832972004929 T^{4} \))
show more
show less