Properties

Label 121.2.c
Level 121
Weight 2
Character orbit c
Rep. character \(\chi_{121}(3,\cdot)\)
Character field \(\Q(\zeta_{5})\)
Dimension 20
Newform subspaces 5
Sturm bound 22
Trace bound 2

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 121 = 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 121.c (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 11 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 5 \)
Sturm bound: \(22\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(121, [\chi])\).

Total New Old
Modular forms 68 52 16
Cusp forms 20 20 0
Eisenstein series 48 32 16

Trace form

\( 20q - q^{3} - q^{5} + 4q^{9} + O(q^{10}) \) \( 20q - q^{3} - q^{5} + 4q^{9} - 24q^{12} - 4q^{14} - 5q^{15} + 6q^{16} - 8q^{20} - 28q^{23} + 12q^{25} + 14q^{26} - 7q^{27} - 5q^{31} - 8q^{34} + 6q^{36} - 7q^{37} - 12q^{38} + 16q^{42} + 16q^{45} - 8q^{47} + 19q^{49} - 12q^{53} + 48q^{56} - 18q^{58} - 11q^{59} + 14q^{60} + 10q^{64} + 12q^{67} - 19q^{69} - 4q^{70} - 15q^{71} + 12q^{75} + 80q^{78} + 22q^{80} + 19q^{81} - 22q^{82} - 24q^{86} + 12q^{89} + 20q^{91} - 10q^{92} + 17q^{93} + 23q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(121, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
121.2.c.a \(4\) \(0.966\) \(\Q(\zeta_{10})\) None \(-2\) \(1\) \(-1\) \(-2\) \(q+(-2+2\zeta_{10}-2\zeta_{10}^{2}+2\zeta_{10}^{3})q^{2}+\cdots\)
121.2.c.b \(4\) \(0.966\) \(\Q(\zeta_{10})\) None \(-1\) \(-2\) \(-1\) \(2\) \(q+(-1+\zeta_{10}-\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
121.2.c.c \(4\) \(0.966\) \(\Q(\zeta_{10})\) \(\Q(\sqrt{-11}) \) \(0\) \(1\) \(3\) \(0\) \(q-\zeta_{10}^{2}q^{3}+2\zeta_{10}^{3}q^{4}+3\zeta_{10}q^{5}+\cdots\)
121.2.c.d \(4\) \(0.966\) \(\Q(\zeta_{10})\) None \(1\) \(-2\) \(-1\) \(-2\) \(q+(1-\zeta_{10}+\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+2\zeta_{10}^{2}q^{3}+\cdots\)
121.2.c.e \(4\) \(0.966\) \(\Q(\zeta_{10})\) None \(2\) \(1\) \(-1\) \(2\) \(q+(2-2\zeta_{10}+2\zeta_{10}^{2}-2\zeta_{10}^{3})q^{2}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} - 4 T^{4} + 8 T^{6} + 16 T^{7} + 16 T^{8} \))(\( 1 + T - T^{2} - 3 T^{3} - T^{4} - 6 T^{5} - 4 T^{6} + 8 T^{7} + 16 T^{8} \))(\( 1 - 2 T^{2} + 4 T^{4} - 8 T^{6} + 16 T^{8} \))(\( 1 - T - T^{2} + 3 T^{3} - T^{4} + 6 T^{5} - 4 T^{6} - 8 T^{7} + 16 T^{8} \))(\( 1 - 2 T + 2 T^{2} - 4 T^{4} + 8 T^{6} - 16 T^{7} + 16 T^{8} \))
$3$ (\( 1 - T - 2 T^{2} + 5 T^{3} + T^{4} + 15 T^{5} - 18 T^{6} - 27 T^{7} + 81 T^{8} \))(\( 1 + 2 T + T^{2} - 4 T^{3} - 11 T^{4} - 12 T^{5} + 9 T^{6} + 54 T^{7} + 81 T^{8} \))(\( 1 - T - 2 T^{2} + 5 T^{3} + T^{4} + 15 T^{5} - 18 T^{6} - 27 T^{7} + 81 T^{8} \))(\( 1 + 2 T + T^{2} - 4 T^{3} - 11 T^{4} - 12 T^{5} + 9 T^{6} + 54 T^{7} + 81 T^{8} \))(\( 1 - T - 2 T^{2} + 5 T^{3} + T^{4} + 15 T^{5} - 18 T^{6} - 27 T^{7} + 81 T^{8} \))
$5$ (\( 1 + T - 4 T^{2} - 9 T^{3} + 11 T^{4} - 45 T^{5} - 100 T^{6} + 125 T^{7} + 625 T^{8} \))(\( 1 + T - 4 T^{2} - 9 T^{3} + 11 T^{4} - 45 T^{5} - 100 T^{6} + 125 T^{7} + 625 T^{8} \))(\( 1 - 3 T + 4 T^{2} + 3 T^{3} - 29 T^{4} + 15 T^{5} + 100 T^{6} - 375 T^{7} + 625 T^{8} \))(\( 1 + T - 4 T^{2} - 9 T^{3} + 11 T^{4} - 45 T^{5} - 100 T^{6} + 125 T^{7} + 625 T^{8} \))(\( 1 + T - 4 T^{2} - 9 T^{3} + 11 T^{4} - 45 T^{5} - 100 T^{6} + 125 T^{7} + 625 T^{8} \))
$7$ (\( 1 + 2 T - 3 T^{2} - 20 T^{3} - 19 T^{4} - 140 T^{5} - 147 T^{6} + 686 T^{7} + 2401 T^{8} \))(\( 1 - 2 T - 3 T^{2} + 20 T^{3} - 19 T^{4} + 140 T^{5} - 147 T^{6} - 686 T^{7} + 2401 T^{8} \))(\( 1 - 7 T^{2} + 49 T^{4} - 343 T^{6} + 2401 T^{8} \))(\( 1 + 2 T - 3 T^{2} - 20 T^{3} - 19 T^{4} - 140 T^{5} - 147 T^{6} + 686 T^{7} + 2401 T^{8} \))(\( 1 - 2 T - 3 T^{2} + 20 T^{3} - 19 T^{4} + 140 T^{5} - 147 T^{6} - 686 T^{7} + 2401 T^{8} \))
$11$ (\( \))(\( \))(\( \))(\( \))(\( \))
$13$ (\( 1 - 4 T + 3 T^{2} + 40 T^{3} - 199 T^{4} + 520 T^{5} + 507 T^{6} - 8788 T^{7} + 28561 T^{8} \))(\( 1 + T - 12 T^{2} - 25 T^{3} + 131 T^{4} - 325 T^{5} - 2028 T^{6} + 2197 T^{7} + 28561 T^{8} \))(\( 1 - 13 T^{2} + 169 T^{4} - 2197 T^{6} + 28561 T^{8} \))(\( 1 - T - 12 T^{2} + 25 T^{3} + 131 T^{4} + 325 T^{5} - 2028 T^{6} - 2197 T^{7} + 28561 T^{8} \))(\( 1 + 4 T + 3 T^{2} - 40 T^{3} - 199 T^{4} - 520 T^{5} + 507 T^{6} + 8788 T^{7} + 28561 T^{8} \))
$17$ (\( 1 + 2 T - 13 T^{2} - 60 T^{3} + 101 T^{4} - 1020 T^{5} - 3757 T^{6} + 9826 T^{7} + 83521 T^{8} \))(\( 1 - 5 T + 8 T^{2} + 45 T^{3} - 361 T^{4} + 765 T^{5} + 2312 T^{6} - 24565 T^{7} + 83521 T^{8} \))(\( 1 - 17 T^{2} + 289 T^{4} - 4913 T^{6} + 83521 T^{8} \))(\( 1 + 5 T + 8 T^{2} - 45 T^{3} - 361 T^{4} - 765 T^{5} + 2312 T^{6} + 24565 T^{7} + 83521 T^{8} \))(\( 1 - 2 T - 13 T^{2} + 60 T^{3} + 101 T^{4} + 1020 T^{5} - 3757 T^{6} - 9826 T^{7} + 83521 T^{8} \))
$19$ (\( 1 - 19 T^{2} + 361 T^{4} - 6859 T^{6} + 130321 T^{8} \))(\( 1 + 6 T + 17 T^{2} - 12 T^{3} - 395 T^{4} - 228 T^{5} + 6137 T^{6} + 41154 T^{7} + 130321 T^{8} \))(\( 1 - 19 T^{2} + 361 T^{4} - 6859 T^{6} + 130321 T^{8} \))(\( 1 - 6 T + 17 T^{2} + 12 T^{3} - 395 T^{4} + 228 T^{5} + 6137 T^{6} - 41154 T^{7} + 130321 T^{8} \))(\( 1 - 19 T^{2} + 361 T^{4} - 6859 T^{6} + 130321 T^{8} \))
$23$ (\( ( 1 + T + 23 T^{2} )^{4} \))(\( ( 1 - 2 T + 23 T^{2} )^{4} \))(\( ( 1 + 9 T + 23 T^{2} )^{4} \))(\( ( 1 - 2 T + 23 T^{2} )^{4} \))(\( ( 1 + T + 23 T^{2} )^{4} \))
$29$ (\( 1 - 29 T^{2} + 841 T^{4} - 24389 T^{6} + 707281 T^{8} \))(\( 1 + 9 T + 52 T^{2} + 207 T^{3} + 355 T^{4} + 6003 T^{5} + 43732 T^{6} + 219501 T^{7} + 707281 T^{8} \))(\( 1 - 29 T^{2} + 841 T^{4} - 24389 T^{6} + 707281 T^{8} \))(\( 1 - 9 T + 52 T^{2} - 207 T^{3} + 355 T^{4} - 6003 T^{5} + 43732 T^{6} - 219501 T^{7} + 707281 T^{8} \))(\( 1 - 29 T^{2} + 841 T^{4} - 24389 T^{6} + 707281 T^{8} \))
$31$ (\( 1 + 7 T + 18 T^{2} - 91 T^{3} - 1195 T^{4} - 2821 T^{5} + 17298 T^{6} + 208537 T^{7} + 923521 T^{8} \))(\( 1 - 2 T - 27 T^{2} + 116 T^{3} + 605 T^{4} + 3596 T^{5} - 25947 T^{6} - 59582 T^{7} + 923521 T^{8} \))(\( 1 - 5 T - 6 T^{2} + 185 T^{3} - 739 T^{4} + 5735 T^{5} - 5766 T^{6} - 148955 T^{7} + 923521 T^{8} \))(\( 1 - 2 T - 27 T^{2} + 116 T^{3} + 605 T^{4} + 3596 T^{5} - 25947 T^{6} - 59582 T^{7} + 923521 T^{8} \))(\( 1 + 7 T + 18 T^{2} - 91 T^{3} - 1195 T^{4} - 2821 T^{5} + 17298 T^{6} + 208537 T^{7} + 923521 T^{8} \))
$37$ (\( 1 + 3 T - 28 T^{2} - 195 T^{3} + 451 T^{4} - 7215 T^{5} - 38332 T^{6} + 151959 T^{7} + 1874161 T^{8} \))(\( 1 - 3 T - 28 T^{2} + 195 T^{3} + 451 T^{4} + 7215 T^{5} - 38332 T^{6} - 151959 T^{7} + 1874161 T^{8} \))(\( 1 + 7 T + 12 T^{2} - 175 T^{3} - 1669 T^{4} - 6475 T^{5} + 16428 T^{6} + 354571 T^{7} + 1874161 T^{8} \))(\( 1 - 3 T - 28 T^{2} + 195 T^{3} + 451 T^{4} + 7215 T^{5} - 38332 T^{6} - 151959 T^{7} + 1874161 T^{8} \))(\( 1 + 3 T - 28 T^{2} - 195 T^{3} + 451 T^{4} - 7215 T^{5} - 38332 T^{6} + 151959 T^{7} + 1874161 T^{8} \))
$41$ (\( 1 + 8 T + 23 T^{2} - 144 T^{3} - 2095 T^{4} - 5904 T^{5} + 38663 T^{6} + 551368 T^{7} + 2825761 T^{8} \))(\( 1 - 5 T - 16 T^{2} + 285 T^{3} - 769 T^{4} + 11685 T^{5} - 26896 T^{6} - 344605 T^{7} + 2825761 T^{8} \))(\( 1 - 41 T^{2} + 1681 T^{4} - 68921 T^{6} + 2825761 T^{8} \))(\( 1 + 5 T - 16 T^{2} - 285 T^{3} - 769 T^{4} - 11685 T^{5} - 26896 T^{6} + 344605 T^{7} + 2825761 T^{8} \))(\( 1 - 8 T + 23 T^{2} + 144 T^{3} - 2095 T^{4} + 5904 T^{5} + 38663 T^{6} - 551368 T^{7} + 2825761 T^{8} \))
$43$ (\( ( 1 - 6 T + 43 T^{2} )^{4} \))(\( ( 1 + 43 T^{2} )^{4} \))(\( ( 1 + 43 T^{2} )^{4} \))(\( ( 1 + 43 T^{2} )^{4} \))(\( ( 1 + 6 T + 43 T^{2} )^{4} \))
$47$ (\( 1 + 8 T + 17 T^{2} - 240 T^{3} - 2719 T^{4} - 11280 T^{5} + 37553 T^{6} + 830584 T^{7} + 4879681 T^{8} \))(\( 1 + 2 T - 43 T^{2} - 180 T^{3} + 1661 T^{4} - 8460 T^{5} - 94987 T^{6} + 207646 T^{7} + 4879681 T^{8} \))(\( 1 - 12 T + 97 T^{2} - 600 T^{3} + 2641 T^{4} - 28200 T^{5} + 214273 T^{6} - 1245876 T^{7} + 4879681 T^{8} \))(\( 1 + 2 T - 43 T^{2} - 180 T^{3} + 1661 T^{4} - 8460 T^{5} - 94987 T^{6} + 207646 T^{7} + 4879681 T^{8} \))(\( 1 + 8 T + 17 T^{2} - 240 T^{3} - 2719 T^{4} - 11280 T^{5} + 37553 T^{6} + 830584 T^{7} + 4879681 T^{8} \))
$53$ (\( 1 - 6 T - 17 T^{2} + 420 T^{3} - 1619 T^{4} + 22260 T^{5} - 47753 T^{6} - 893262 T^{7} + 7890481 T^{8} \))(\( 1 + 9 T + 28 T^{2} - 225 T^{3} - 3509 T^{4} - 11925 T^{5} + 78652 T^{6} + 1339893 T^{7} + 7890481 T^{8} \))(\( 1 + 6 T - 17 T^{2} - 420 T^{3} - 1619 T^{4} - 22260 T^{5} - 47753 T^{6} + 893262 T^{7} + 7890481 T^{8} \))(\( 1 + 9 T + 28 T^{2} - 225 T^{3} - 3509 T^{4} - 11925 T^{5} + 78652 T^{6} + 1339893 T^{7} + 7890481 T^{8} \))(\( 1 - 6 T - 17 T^{2} + 420 T^{3} - 1619 T^{4} + 22260 T^{5} - 47753 T^{6} - 893262 T^{7} + 7890481 T^{8} \))
$59$ (\( 1 + 5 T - 34 T^{2} - 465 T^{3} - 319 T^{4} - 27435 T^{5} - 118354 T^{6} + 1026895 T^{7} + 12117361 T^{8} \))(\( 1 + 8 T + 5 T^{2} - 432 T^{3} - 3751 T^{4} - 25488 T^{5} + 17405 T^{6} + 1643032 T^{7} + 12117361 T^{8} \))(\( 1 - 15 T + 166 T^{2} - 1605 T^{3} + 14281 T^{4} - 94695 T^{5} + 577846 T^{6} - 3080685 T^{7} + 12117361 T^{8} \))(\( 1 + 8 T + 5 T^{2} - 432 T^{3} - 3751 T^{4} - 25488 T^{5} + 17405 T^{6} + 1643032 T^{7} + 12117361 T^{8} \))(\( 1 + 5 T - 34 T^{2} - 465 T^{3} - 319 T^{4} - 27435 T^{5} - 118354 T^{6} + 1026895 T^{7} + 12117361 T^{8} \))
$61$ (\( 1 - 12 T + 83 T^{2} - 264 T^{3} - 1895 T^{4} - 16104 T^{5} + 308843 T^{6} - 2723772 T^{7} + 13845841 T^{8} \))(\( 1 + 6 T - 25 T^{2} - 516 T^{3} - 1571 T^{4} - 31476 T^{5} - 93025 T^{6} + 1361886 T^{7} + 13845841 T^{8} \))(\( 1 - 61 T^{2} + 3721 T^{4} - 226981 T^{6} + 13845841 T^{8} \))(\( 1 - 6 T - 25 T^{2} + 516 T^{3} - 1571 T^{4} + 31476 T^{5} - 93025 T^{6} - 1361886 T^{7} + 13845841 T^{8} \))(\( 1 + 12 T + 83 T^{2} + 264 T^{3} - 1895 T^{4} + 16104 T^{5} + 308843 T^{6} + 2723772 T^{7} + 13845841 T^{8} \))
$67$ (\( ( 1 + 7 T + 67 T^{2} )^{4} \))(\( ( 1 - 2 T + 67 T^{2} )^{4} \))(\( ( 1 - 13 T + 67 T^{2} )^{4} \))(\( ( 1 - 2 T + 67 T^{2} )^{4} \))(\( ( 1 + 7 T + 67 T^{2} )^{4} \))
$71$ (\( 1 - 3 T - 62 T^{2} + 399 T^{3} + 3205 T^{4} + 28329 T^{5} - 312542 T^{6} - 1073733 T^{7} + 25411681 T^{8} \))(\( 1 + 12 T + 73 T^{2} + 24 T^{3} - 4895 T^{4} + 1704 T^{5} + 367993 T^{6} + 4294932 T^{7} + 25411681 T^{8} \))(\( 1 - 3 T - 62 T^{2} + 399 T^{3} + 3205 T^{4} + 28329 T^{5} - 312542 T^{6} - 1073733 T^{7} + 25411681 T^{8} \))(\( 1 + 12 T + 73 T^{2} + 24 T^{3} - 4895 T^{4} + 1704 T^{5} + 367993 T^{6} + 4294932 T^{7} + 25411681 T^{8} \))(\( 1 - 3 T - 62 T^{2} + 399 T^{3} + 3205 T^{4} + 28329 T^{5} - 312542 T^{6} - 1073733 T^{7} + 25411681 T^{8} \))
$73$ (\( 1 - 4 T - 57 T^{2} + 520 T^{3} + 2081 T^{4} + 37960 T^{5} - 303753 T^{6} - 1556068 T^{7} + 28398241 T^{8} \))(\( 1 - 2 T - 69 T^{2} + 284 T^{3} + 4469 T^{4} + 20732 T^{5} - 367701 T^{6} - 778034 T^{7} + 28398241 T^{8} \))(\( 1 - 73 T^{2} + 5329 T^{4} - 389017 T^{6} + 28398241 T^{8} \))(\( 1 + 2 T - 69 T^{2} - 284 T^{3} + 4469 T^{4} - 20732 T^{5} - 367701 T^{6} + 778034 T^{7} + 28398241 T^{8} \))(\( 1 + 4 T - 57 T^{2} - 520 T^{3} + 2081 T^{4} - 37960 T^{5} - 303753 T^{6} + 1556068 T^{7} + 28398241 T^{8} \))
$79$ (\( 1 + 10 T + 21 T^{2} - 580 T^{3} - 7459 T^{4} - 45820 T^{5} + 131061 T^{6} + 4930390 T^{7} + 38950081 T^{8} \))(\( 1 - 10 T + 21 T^{2} + 580 T^{3} - 7459 T^{4} + 45820 T^{5} + 131061 T^{6} - 4930390 T^{7} + 38950081 T^{8} \))(\( 1 - 79 T^{2} + 6241 T^{4} - 493039 T^{6} + 38950081 T^{8} \))(\( 1 + 10 T + 21 T^{2} - 580 T^{3} - 7459 T^{4} - 45820 T^{5} + 131061 T^{6} + 4930390 T^{7} + 38950081 T^{8} \))(\( 1 - 10 T + 21 T^{2} + 580 T^{3} - 7459 T^{4} + 45820 T^{5} + 131061 T^{6} - 4930390 T^{7} + 38950081 T^{8} \))
$83$ (\( 1 + 6 T - 47 T^{2} - 780 T^{3} - 779 T^{4} - 64740 T^{5} - 323783 T^{6} + 3430722 T^{7} + 47458321 T^{8} \))(\( 1 + 6 T - 47 T^{2} - 780 T^{3} - 779 T^{4} - 64740 T^{5} - 323783 T^{6} + 3430722 T^{7} + 47458321 T^{8} \))(\( 1 - 83 T^{2} + 6889 T^{4} - 571787 T^{6} + 47458321 T^{8} \))(\( 1 - 6 T - 47 T^{2} + 780 T^{3} - 779 T^{4} + 64740 T^{5} - 323783 T^{6} - 3430722 T^{7} + 47458321 T^{8} \))(\( 1 - 6 T - 47 T^{2} + 780 T^{3} - 779 T^{4} + 64740 T^{5} - 323783 T^{6} - 3430722 T^{7} + 47458321 T^{8} \))
$89$ (\( ( 1 - 15 T + 89 T^{2} )^{4} \))(\( ( 1 + 9 T + 89 T^{2} )^{4} \))(\( ( 1 + 9 T + 89 T^{2} )^{4} \))(\( ( 1 + 9 T + 89 T^{2} )^{4} \))(\( ( 1 - 15 T + 89 T^{2} )^{4} \))
$97$ (\( 1 - 7 T - 48 T^{2} + 1015 T^{3} - 2449 T^{4} + 98455 T^{5} - 451632 T^{6} - 6388711 T^{7} + 88529281 T^{8} \))(\( 1 - 13 T + 72 T^{2} + 325 T^{3} - 11209 T^{4} + 31525 T^{5} + 677448 T^{6} - 11864749 T^{7} + 88529281 T^{8} \))(\( 1 + 17 T + 192 T^{2} + 1615 T^{3} + 8831 T^{4} + 156655 T^{5} + 1806528 T^{6} + 15515441 T^{7} + 88529281 T^{8} \))(\( 1 - 13 T + 72 T^{2} + 325 T^{3} - 11209 T^{4} + 31525 T^{5} + 677448 T^{6} - 11864749 T^{7} + 88529281 T^{8} \))(\( 1 - 7 T - 48 T^{2} + 1015 T^{3} - 2449 T^{4} + 98455 T^{5} - 451632 T^{6} - 6388711 T^{7} + 88529281 T^{8} \))
show more
show less