Properties

Label 161.2.a
Level 161
Weight 2
Character orbit a
Rep. character \(\chi_{161}(1,\cdot)\)
Character field \(\Q\)
Dimension 11
Newform subspaces 4
Sturm bound 32
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 161 = 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 161.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(32\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(161))\).

Total New Old
Modular forms 18 11 7
Cusp forms 15 11 4
Eisenstein series 3 0 3

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

\(7\)\(23\)FrickeDim.
\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(6\)
Plus space\(+\)\(2\)
Minus space\(-\)\(9\)

Trace form

\( 11q - q^{2} + 15q^{4} - 2q^{5} + 2q^{6} + q^{7} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 11q - q^{2} + 15q^{4} - 2q^{5} + 2q^{6} + q^{7} - 3q^{8} + 3q^{9} - 10q^{10} + 4q^{11} - 6q^{12} - 2q^{13} + 3q^{14} + 20q^{15} + 19q^{16} - 10q^{17} - 15q^{18} + 8q^{19} - 6q^{20} + 8q^{22} - 5q^{23} - 24q^{24} + 13q^{25} - 8q^{26} + 12q^{27} + 7q^{28} - 2q^{29} - 48q^{30} + 24q^{31} - 17q^{32} - 24q^{33} + 2q^{34} - 2q^{35} - 15q^{36} - 2q^{37} - 44q^{38} + 8q^{39} - 22q^{40} - 14q^{41} - 8q^{42} - 12q^{43} - 24q^{44} - 6q^{45} - q^{46} + 16q^{47} + 18q^{48} + 11q^{49} + 9q^{50} - 8q^{51} + 30q^{53} + 6q^{54} + 4q^{55} + 15q^{56} + 4q^{57} + 4q^{58} + 20q^{60} - 6q^{61} - 6q^{62} + 13q^{63} + 49q^{64} - 32q^{65} + 4q^{66} + 24q^{67} + 34q^{68} + 4q^{69} - 10q^{70} - 31q^{72} - 22q^{73} + 38q^{74} - 28q^{75} + 64q^{76} - 4q^{77} + 50q^{78} + 44q^{79} - 34q^{80} + 3q^{81} - 20q^{82} + 28q^{83} + 12q^{84} - 20q^{85} + 44q^{86} - 28q^{87} - 36q^{88} - 10q^{89} + 38q^{90} + 2q^{91} - 5q^{92} + 4q^{93} - 22q^{94} + 24q^{95} + 2q^{96} - 22q^{97} - q^{98} - 20q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(161))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 7 23
161.2.a.a \(1\) \(1.286\) \(\Q\) None \(-1\) \(0\) \(2\) \(1\) \(-\) \(+\) \(q-q^{2}-q^{4}+2q^{5}+q^{7}+3q^{8}-3q^{9}+\cdots\)
161.2.a.b \(2\) \(1.286\) \(\Q(\sqrt{5}) \) None \(-1\) \(-2\) \(-2\) \(-2\) \(+\) \(+\) \(q-\beta q^{2}-q^{3}+(-1+\beta )q^{4}+(-2+2\beta )q^{5}+\cdots\)
161.2.a.c \(3\) \(1.286\) 3.3.148.1 None \(-1\) \(2\) \(2\) \(-3\) \(+\) \(-\) \(q+(-\beta _{1}-\beta _{2})q^{2}+(1-\beta _{1})q^{3}+(1+2\beta _{1}+\cdots)q^{4}+\cdots\)
161.2.a.d \(5\) \(1.286\) 5.5.2147108.1 None \(2\) \(0\) \(-4\) \(5\) \(-\) \(+\) \(q+\beta _{1}q^{2}+\beta _{3}q^{3}+(2+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(161))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(161)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + 2 T^{2} \))(\( 1 + T + 3 T^{2} + 2 T^{3} + 4 T^{4} \))(\( 1 + T + T^{2} + 3 T^{3} + 2 T^{4} + 4 T^{5} + 8 T^{6} \))(\( 1 - 2 T + T^{2} + T^{3} + 2 T^{4} - 7 T^{5} + 4 T^{6} + 4 T^{7} + 8 T^{8} - 32 T^{9} + 32 T^{10} \))
$3$ (\( 1 + 3 T^{2} \))(\( ( 1 + T + 3 T^{2} )^{2} \))(\( 1 - 2 T + 7 T^{2} - 10 T^{3} + 21 T^{4} - 18 T^{5} + 27 T^{6} \))(\( 1 + 2 T^{2} + 11 T^{4} + 10 T^{5} + 33 T^{6} + 54 T^{8} + 243 T^{10} \))
$5$ (\( 1 - 2 T + 5 T^{2} \))(\( 1 + 2 T + 6 T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 - 2 T + 13 T^{2} - 18 T^{3} + 65 T^{4} - 50 T^{5} + 125 T^{6} \))(\( 1 + 4 T + 11 T^{2} + 26 T^{3} + 92 T^{4} + 228 T^{5} + 460 T^{6} + 650 T^{7} + 1375 T^{8} + 2500 T^{9} + 3125 T^{10} \))
$7$ (\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{5} \))
$11$ (\( 1 - 4 T + 11 T^{2} \))(\( 1 + 2 T^{2} + 121 T^{4} \))(\( 1 - 4 T + 33 T^{2} - 84 T^{3} + 363 T^{4} - 484 T^{5} + 1331 T^{6} \))(\( 1 + 4 T + 27 T^{2} + 28 T^{3} + 126 T^{4} - 400 T^{5} + 1386 T^{6} + 3388 T^{7} + 35937 T^{8} + 58564 T^{9} + 161051 T^{10} \))
$13$ (\( 1 - 6 T + 13 T^{2} \))(\( 1 + 4 T + 25 T^{2} + 52 T^{3} + 169 T^{4} \))(\( 1 - 2 T + 27 T^{2} - 44 T^{3} + 351 T^{4} - 338 T^{5} + 2197 T^{6} \))(\( 1 + 6 T + 56 T^{2} + 266 T^{3} + 1351 T^{4} + 4944 T^{5} + 17563 T^{6} + 44954 T^{7} + 123032 T^{8} + 171366 T^{9} + 371293 T^{10} \))
$17$ (\( 1 + 2 T + 17 T^{2} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( 1 - 4 T + 53 T^{2} - 134 T^{3} + 901 T^{4} - 1156 T^{5} + 4913 T^{6} \))(\( 1 + 12 T + 91 T^{2} + 430 T^{3} + 1692 T^{4} + 6148 T^{5} + 28764 T^{6} + 124270 T^{7} + 447083 T^{8} + 1002252 T^{9} + 1419857 T^{10} \))
$19$ (\( 1 - 4 T + 19 T^{2} \))(\( 1 + 10 T + 58 T^{2} + 190 T^{3} + 361 T^{4} \))(\( 1 - 8 T + 41 T^{2} - 144 T^{3} + 779 T^{4} - 2888 T^{5} + 6859 T^{6} \))(\( 1 - 6 T + 67 T^{2} - 360 T^{3} + 2334 T^{4} - 9220 T^{5} + 44346 T^{6} - 129960 T^{7} + 459553 T^{8} - 781926 T^{9} + 2476099 T^{10} \))
$23$ (\( 1 + T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{5} \))
$29$ (\( 1 + 2 T + 29 T^{2} \))(\( 1 - 6 T + 47 T^{2} - 174 T^{3} + 841 T^{4} \))(\( 1 + 2 T + 79 T^{2} + 120 T^{3} + 2291 T^{4} + 1682 T^{5} + 24389 T^{6} \))(\( 1 + 4 T + 34 T^{2} + 214 T^{3} + 1217 T^{4} + 4232 T^{5} + 35293 T^{6} + 179974 T^{7} + 829226 T^{8} + 2829124 T^{9} + 20511149 T^{10} \))
$31$ (\( 1 + 4 T + 31 T^{2} \))(\( ( 1 + 9 T + 31 T^{2} )^{2} \))(\( 1 - 16 T + 119 T^{2} - 654 T^{3} + 3689 T^{4} - 15376 T^{5} + 29791 T^{6} \))(\( 1 - 30 T + 502 T^{2} - 5646 T^{3} + 46995 T^{4} - 297598 T^{5} + 1456845 T^{6} - 5425806 T^{7} + 14955082 T^{8} - 27705630 T^{9} + 28629151 T^{10} \))
$37$ (\( 1 + 2 T + 37 T^{2} \))(\( 1 - 2 T + 30 T^{2} - 74 T^{3} + 1369 T^{4} \))(\( 1 + 6 T + 107 T^{2} + 404 T^{3} + 3959 T^{4} + 8214 T^{5} + 50653 T^{6} \))(\( 1 - 4 T + 109 T^{2} - 216 T^{3} + 5126 T^{4} - 5064 T^{5} + 189662 T^{6} - 295704 T^{7} + 5521177 T^{8} - 7496644 T^{9} + 69343957 T^{10} \))
$41$ (\( 1 + 6 T + 41 T^{2} \))(\( 1 + 77 T^{2} + 1681 T^{4} \))(\( 1 + 14 T + 135 T^{2} + 996 T^{3} + 5535 T^{4} + 23534 T^{5} + 68921 T^{6} \))(\( 1 - 6 T + 176 T^{2} - 838 T^{3} + 13295 T^{4} - 49000 T^{5} + 545095 T^{6} - 1408678 T^{7} + 12130096 T^{8} - 16954566 T^{9} + 115856201 T^{10} \))
$43$ (\( 1 - 12 T + 43 T^{2} \))(\( 1 + 4 T + 70 T^{2} + 172 T^{3} + 1849 T^{4} \))(\( 1 + 8 T + 89 T^{2} + 384 T^{3} + 3827 T^{4} + 14792 T^{5} + 79507 T^{6} \))(\( 1 + 12 T + 191 T^{2} + 1696 T^{3} + 16226 T^{4} + 101736 T^{5} + 697718 T^{6} + 3135904 T^{7} + 15185837 T^{8} + 41025612 T^{9} + 147008443 T^{10} \))
$47$ (\( 1 + 12 T + 47 T^{2} \))(\( 1 - 2 T + 75 T^{2} - 94 T^{3} + 2209 T^{4} \))(\( 1 - 16 T + 203 T^{2} - 1514 T^{3} + 9541 T^{4} - 35344 T^{5} + 103823 T^{6} \))(\( 1 - 10 T + 110 T^{2} + 38 T^{3} - 3609 T^{4} + 58894 T^{5} - 169623 T^{6} + 83942 T^{7} + 11420530 T^{8} - 48796810 T^{9} + 229345007 T^{10} \))
$53$ (\( 1 + 10 T + 53 T^{2} \))(\( 1 - 18 T + 182 T^{2} - 954 T^{3} + 2809 T^{4} \))(\( 1 - 6 T + 107 T^{2} - 388 T^{3} + 5671 T^{4} - 16854 T^{5} + 148877 T^{6} \))(\( 1 - 16 T + 317 T^{2} - 3240 T^{3} + 35942 T^{4} - 254032 T^{5} + 1904926 T^{6} - 9101160 T^{7} + 47194009 T^{8} - 126247696 T^{9} + 418195493 T^{10} \))
$59$ (\( 1 + 59 T^{2} \))(\( 1 + 12 T + 134 T^{2} + 708 T^{3} + 3481 T^{4} \))(\( 1 + 10 T + 75 T^{2} + 210 T^{3} + 4425 T^{4} + 34810 T^{5} + 205379 T^{6} \))(\( 1 - 22 T + 413 T^{2} - 5194 T^{3} + 54600 T^{4} - 458288 T^{5} + 3221400 T^{6} - 18080314 T^{7} + 84821527 T^{8} - 266581942 T^{9} + 714924299 T^{10} \))
$61$ (\( 1 - 2 T + 61 T^{2} \))(\( 1 - 58 T^{2} + 3721 T^{4} \))(\( 1 - 10 T + 137 T^{2} - 726 T^{3} + 8357 T^{4} - 37210 T^{5} + 226981 T^{6} \))(\( 1 + 18 T + 339 T^{2} + 3954 T^{3} + 43744 T^{4} + 348488 T^{5} + 2668384 T^{6} + 14712834 T^{7} + 76946559 T^{8} + 249225138 T^{9} + 844596301 T^{10} \))
$67$ (\( 1 - 12 T + 67 T^{2} \))(\( 1 + 2 T + 10 T^{2} + 134 T^{3} + 4489 T^{4} \))(\( 1 - 16 T + 193 T^{2} - 1468 T^{3} + 12931 T^{4} - 71824 T^{5} + 300763 T^{6} \))(\( 1 + 2 T + 35 T^{2} - 204 T^{3} + 6790 T^{4} + 16644 T^{5} + 454930 T^{6} - 915756 T^{7} + 10526705 T^{8} + 40302242 T^{9} + 1350125107 T^{10} \))
$71$ (\( 1 - 8 T + 71 T^{2} \))(\( 1 + 16 T + 201 T^{2} + 1136 T^{3} + 5041 T^{4} \))(\( 1 - 4 T + 133 T^{2} - 504 T^{3} + 9443 T^{4} - 20164 T^{5} + 357911 T^{6} \))(\( 1 - 4 T + 254 T^{2} - 860 T^{3} + 30833 T^{4} - 86976 T^{5} + 2189143 T^{6} - 4335260 T^{7} + 90909394 T^{8} - 101646724 T^{9} + 1804229351 T^{10} \))
$73$ (\( 1 + 14 T + 73 T^{2} \))(\( 1 + 101 T^{2} + 5329 T^{4} \))(\( 1 + 6 T + 111 T^{2} + 660 T^{3} + 8103 T^{4} + 31974 T^{5} + 389017 T^{6} \))(\( 1 + 2 T + 168 T^{2} + 946 T^{3} + 18175 T^{4} + 89144 T^{5} + 1326775 T^{6} + 5041234 T^{7} + 65354856 T^{8} + 56796482 T^{9} + 2073071593 T^{10} \))
$79$ (\( 1 - 8 T + 79 T^{2} \))(\( 1 + 10 T + 178 T^{2} + 790 T^{3} + 6241 T^{4} \))(\( 1 - 16 T + 245 T^{2} - 2100 T^{3} + 19355 T^{4} - 99856 T^{5} + 493039 T^{6} \))(\( 1 - 30 T + 703 T^{2} - 10676 T^{3} + 136374 T^{4} - 1310412 T^{5} + 10773546 T^{6} - 66628916 T^{7} + 346606417 T^{8} - 1168502430 T^{9} + 3077056399 T^{10} \))
$83$ (\( 1 + 4 T + 83 T^{2} \))(\( 1 - 4 T + 150 T^{2} - 332 T^{3} + 6889 T^{4} \))(\( 1 - 20 T + 145 T^{2} - 648 T^{3} + 12035 T^{4} - 137780 T^{5} + 571787 T^{6} \))(\( 1 - 8 T + 359 T^{2} - 2224 T^{3} + 55778 T^{4} - 264336 T^{5} + 4629574 T^{6} - 15321136 T^{7} + 205271533 T^{8} - 379666568 T^{9} + 3939040643 T^{10} \))
$89$ (\( 1 - 6 T + 89 T^{2} \))(\( 1 + 98 T^{2} + 7921 T^{4} \))(\( 1 - 4 T + 153 T^{2} - 990 T^{3} + 13617 T^{4} - 31684 T^{5} + 704969 T^{6} \))(\( 1 + 20 T + 511 T^{2} + 6422 T^{3} + 92672 T^{4} + 821572 T^{5} + 8247808 T^{6} + 50868662 T^{7} + 360239159 T^{8} + 1254844820 T^{9} + 5584059449 T^{10} \))
$97$ (\( 1 + 10 T + 97 T^{2} \))(\( 1 + 6 T + 158 T^{2} + 582 T^{3} + 9409 T^{4} \))(\( 1 - 6 T + 269 T^{2} - 1166 T^{3} + 26093 T^{4} - 56454 T^{5} + 912673 T^{6} \))(\( 1 + 12 T + 371 T^{2} + 3294 T^{3} + 61432 T^{4} + 417340 T^{5} + 5958904 T^{6} + 30993246 T^{7} + 338601683 T^{8} + 1062351372 T^{9} + 8587340257 T^{10} \))
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