Properties

Label 403.2.k
Level 403
Weight 2
Character orbit k
Rep. character \(\chi_{403}(66,\cdot)\)
Character field \(\Q(\zeta_{5})\)
Dimension 128
Newform subspaces 5
Sturm bound 74
Trace bound 3

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Defining parameters

Level: \( N \) \(=\) \( 403 = 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 403.k (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 31 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 5 \)
Sturm bound: \(74\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(403, [\chi])\).

Total New Old
Modular forms 152 128 24
Cusp forms 136 128 8
Eisenstein series 16 0 16

Trace form

\( 128q - 4q^{3} - 40q^{4} - 4q^{5} + 8q^{6} + 12q^{7} - 6q^{8} - 36q^{9} + O(q^{10}) \) \( 128q - 4q^{3} - 40q^{4} - 4q^{5} + 8q^{6} + 12q^{7} - 6q^{8} - 36q^{9} - 22q^{10} - 8q^{11} - 16q^{12} - 2q^{13} - 26q^{15} - 16q^{16} + 6q^{17} + 24q^{19} - 22q^{20} - 20q^{21} - 8q^{22} - 10q^{23} + 4q^{24} + 72q^{25} + 24q^{26} + 38q^{27} - 30q^{28} + 4q^{29} - 68q^{30} + 24q^{31} + 8q^{32} + 2q^{33} - 48q^{34} - 32q^{35} + 40q^{36} + 40q^{37} - 42q^{38} - 12q^{40} - 8q^{41} + 26q^{42} - 28q^{43} + 18q^{44} + 44q^{45} + 8q^{46} + 36q^{47} + 16q^{48} + 22q^{49} - 66q^{50} + 26q^{51} - 6q^{52} - 10q^{53} + 42q^{54} - 28q^{55} - 48q^{56} + 2q^{58} + 24q^{59} - 50q^{60} - 28q^{61} + 4q^{62} - 60q^{63} - 20q^{64} - 2q^{65} + 2q^{66} + 68q^{67} + 16q^{68} - 28q^{69} + 4q^{70} + 186q^{72} - 22q^{74} + 12q^{75} - 110q^{76} - 4q^{77} - 68q^{79} + 60q^{80} - 42q^{81} + 52q^{82} - 36q^{83} + 40q^{84} - 40q^{85} - 12q^{86} - 40q^{87} + 64q^{88} - 58q^{89} - 110q^{90} - 8q^{91} - 32q^{92} + 76q^{93} + 112q^{94} + 18q^{95} - 68q^{96} - 46q^{97} - 36q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(403, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
403.2.k.a \(4\) \(3.218\) \(\Q(\zeta_{10})\) None \(-2\) \(-1\) \(-6\) \(-7\) \(q+(-1+\zeta_{10}-\zeta_{10}^{2})q^{2}+(1-2\zeta_{10}+\cdots)q^{3}+\cdots\)
403.2.k.b \(4\) \(3.218\) \(\Q(\zeta_{10})\) None \(-1\) \(-2\) \(-4\) \(1\) \(q-\zeta_{10}q^{2}+(-2+2\zeta_{10}-2\zeta_{10}^{2}+\cdots)q^{3}+\cdots\)
403.2.k.c \(4\) \(3.218\) \(\Q(\zeta_{10})\) None \(-1\) \(3\) \(6\) \(-9\) \(q+(-1+2\zeta_{10}-\zeta_{10}^{2})q^{2}+(1-\zeta_{10}^{3})q^{3}+\cdots\)
403.2.k.d \(48\) \(3.218\) None \(7\) \(-2\) \(-12\) \(25\)
403.2.k.e \(68\) \(3.218\) None \(-3\) \(-2\) \(12\) \(2\)

Decomposition of \(S_{2}^{\mathrm{old}}(403, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(403, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(31, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} + 5 T^{3} + 11 T^{4} + 10 T^{5} + 8 T^{6} + 16 T^{7} + 16 T^{8} \))(\( 1 + T - T^{2} - 3 T^{3} - T^{4} - 6 T^{5} - 4 T^{6} + 8 T^{7} + 16 T^{8} \))(\( 1 + T + 4 T^{2} + 2 T^{3} + 9 T^{4} + 4 T^{5} + 16 T^{6} + 8 T^{7} + 16 T^{8} \))
$3$ (\( 1 + T + 3 T^{2} + 5 T^{3} + 16 T^{4} + 15 T^{5} + 27 T^{6} + 27 T^{7} + 81 T^{8} \))(\( 1 + 2 T + T^{2} - 4 T^{3} - 11 T^{4} - 12 T^{5} + 9 T^{6} + 54 T^{7} + 81 T^{8} \))(\( 1 - 3 T + T^{2} + T^{3} + 4 T^{4} + 3 T^{5} + 9 T^{6} - 81 T^{7} + 81 T^{8} \))
$5$ (\( ( 1 + 3 T + 11 T^{2} + 15 T^{3} + 25 T^{4} )^{2} \))(\( ( 1 + 2 T + 6 T^{2} + 10 T^{3} + 25 T^{4} )^{2} \))(\( ( 1 - 3 T + 11 T^{2} - 15 T^{3} + 25 T^{4} )^{2} \))
$7$ (\( 1 + 7 T + 27 T^{2} + 95 T^{3} + 296 T^{4} + 665 T^{5} + 1323 T^{6} + 2401 T^{7} + 2401 T^{8} \))(\( 1 - T - T^{2} - 17 T^{3} + 64 T^{4} - 119 T^{5} - 49 T^{6} - 343 T^{7} + 2401 T^{8} \))(\( 1 + 9 T + 29 T^{2} + 33 T^{3} + 4 T^{4} + 231 T^{5} + 1421 T^{6} + 3087 T^{7} + 2401 T^{8} \))
$11$ (\( 1 - 5 T + 4 T^{2} - 25 T^{3} + 201 T^{4} - 275 T^{5} + 484 T^{6} - 6655 T^{7} + 14641 T^{8} \))(\( ( 1 - 9 T + 41 T^{2} - 99 T^{3} + 121 T^{4} )( 1 + 11 T + 51 T^{2} + 121 T^{3} + 121 T^{4} ) \))(\( 1 - 3 T + 8 T^{2} - 51 T^{3} + 265 T^{4} - 561 T^{5} + 968 T^{6} - 3993 T^{7} + 14641 T^{8} \))
$13$ (\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))
$17$ (\( 1 + 15 T + 118 T^{2} + 675 T^{3} + 3079 T^{4} + 11475 T^{5} + 34102 T^{6} + 73695 T^{7} + 83521 T^{8} \))(\( 1 + T - T^{2} - 53 T^{3} + 104 T^{4} - 901 T^{5} - 289 T^{6} + 4913 T^{7} + 83521 T^{8} \))(\( 1 + T - 16 T^{2} - 33 T^{3} + 239 T^{4} - 561 T^{5} - 4624 T^{6} + 4913 T^{7} + 83521 T^{8} \))
$19$ (\( 1 - 2 T - 15 T^{2} + 68 T^{3} + 149 T^{4} + 1292 T^{5} - 5415 T^{6} - 13718 T^{7} + 130321 T^{8} \))(\( 1 - 19 T^{2} + 361 T^{4} - 6859 T^{6} + 130321 T^{8} \))(\( 1 + 10 T + 21 T^{2} - 190 T^{3} - 1519 T^{4} - 3610 T^{5} + 7581 T^{6} + 68590 T^{7} + 130321 T^{8} \))
$23$ (\( 1 + 8 T + T^{2} - 146 T^{3} - 651 T^{4} - 3358 T^{5} + 529 T^{6} + 97336 T^{7} + 279841 T^{8} \))(\( 1 - 2 T + T^{2} - 106 T^{3} + 729 T^{4} - 2438 T^{5} + 529 T^{6} - 24334 T^{7} + 279841 T^{8} \))(\( 1 - 12 T + 41 T^{2} - 36 T^{3} + 49 T^{4} - 828 T^{5} + 21689 T^{6} - 146004 T^{7} + 279841 T^{8} \))
$29$ (\( 1 + 8 T + 5 T^{2} - 232 T^{3} - 1711 T^{4} - 6728 T^{5} + 4205 T^{6} + 195112 T^{7} + 707281 T^{8} \))(\( 1 + 10 T + 71 T^{2} + 520 T^{3} + 3641 T^{4} + 15080 T^{5} + 59711 T^{6} + 243890 T^{7} + 707281 T^{8} \))(\( 1 - 10 T + 11 T^{2} + 120 T^{3} - 439 T^{4} + 3480 T^{5} + 9251 T^{6} - 243890 T^{7} + 707281 T^{8} \))
$31$ (\( 1 + 4 T + 46 T^{2} + 124 T^{3} + 961 T^{4} \))(\( 1 - 4 T + 46 T^{2} - 124 T^{3} + 961 T^{4} \))(\( 1 - 4 T + 46 T^{2} - 124 T^{3} + 961 T^{4} \))
$37$ (\( ( 1 + 13 T + 105 T^{2} + 481 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 - 2 T + 70 T^{2} - 74 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 13 T + 115 T^{2} + 481 T^{3} + 1369 T^{4} )^{2} \))
$41$ (\( 1 + 19 T + 95 T^{2} - 599 T^{3} - 8776 T^{4} - 24559 T^{5} + 159695 T^{6} + 1309499 T^{7} + 2825761 T^{8} \))(\( 1 - 41 T^{2} + 1681 T^{4} - 68921 T^{6} + 2825761 T^{8} \))(\( 1 - 5 T - 31 T^{2} + 205 T^{3} + 476 T^{4} + 8405 T^{5} - 52111 T^{6} - 344605 T^{7} + 2825761 T^{8} \))
$43$ (\( 1 - 6 T + 93 T^{2} - 350 T^{3} + 4851 T^{4} - 15050 T^{5} + 171957 T^{6} - 477042 T^{7} + 3418801 T^{8} \))(\( 1 - 18 T + 101 T^{2} - 174 T^{3} + 49 T^{4} - 7482 T^{5} + 186749 T^{6} - 1431126 T^{7} + 3418801 T^{8} \))(\( 1 + 2 T - 19 T^{2} - 254 T^{3} + 619 T^{4} - 10922 T^{5} - 35131 T^{6} + 159014 T^{7} + 3418801 T^{8} \))
$47$ (\( 1 - 11 T + 49 T^{2} - 547 T^{3} + 5964 T^{4} - 25709 T^{5} + 108241 T^{6} - 1142053 T^{7} + 4879681 T^{8} \))(\( 1 - 16 T + 59 T^{2} + 738 T^{3} - 9721 T^{4} + 34686 T^{5} + 130331 T^{6} - 1661168 T^{7} + 4879681 T^{8} \))(\( 1 - T + 59 T^{2} - 177 T^{3} + 2264 T^{4} - 8319 T^{5} + 130331 T^{6} - 103823 T^{7} + 4879681 T^{8} \))
$53$ (\( 1 + 6 T + 83 T^{2} + 480 T^{3} + 6481 T^{4} + 25440 T^{5} + 233147 T^{6} + 893262 T^{7} + 7890481 T^{8} \))(\( 1 - 2 T - 49 T^{2} + 204 T^{3} + 2189 T^{4} + 10812 T^{5} - 137641 T^{6} - 297754 T^{7} + 7890481 T^{8} \))(\( 1 + 28 T + 401 T^{2} + 4024 T^{3} + 32129 T^{4} + 213272 T^{5} + 1126409 T^{6} + 4168556 T^{7} + 7890481 T^{8} \))
$59$ (\( 1 + 14 T + 37 T^{2} + 142 T^{3} + 3555 T^{4} + 8378 T^{5} + 128797 T^{6} + 2875306 T^{7} + 12117361 T^{8} \))(\( 1 + 15 T + 41 T^{2} - 45 T^{3} + 1156 T^{4} - 2655 T^{5} + 142721 T^{6} + 3080685 T^{7} + 12117361 T^{8} \))(\( 1 - 49 T^{2} + 270 T^{3} + 3211 T^{4} + 15930 T^{5} - 170569 T^{6} + 12117361 T^{8} \))
$61$ (\( ( 1 + 8 T + 133 T^{2} + 488 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 23 T + 253 T^{2} + 1403 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 11 T + 61 T^{2} )^{4} \))
$67$ (\( ( 1 - 3 T + 35 T^{2} - 201 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 11 T + 163 T^{2} - 737 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 9 T + 153 T^{2} + 603 T^{3} + 4489 T^{4} )^{2} \))
$71$ (\( 1 - 34 T + 665 T^{2} - 8926 T^{3} + 86729 T^{4} - 633746 T^{5} + 3352265 T^{6} - 12168974 T^{7} + 25411681 T^{8} \))(\( 1 - 2 T - 67 T^{2} - 74 T^{3} + 5255 T^{4} - 5254 T^{5} - 337747 T^{6} - 715822 T^{7} + 25411681 T^{8} \))(\( 1 - 2 T - 7 T^{2} + 466 T^{3} + 1865 T^{4} + 33086 T^{5} - 35287 T^{6} - 715822 T^{7} + 25411681 T^{8} \))
$73$ (\( 1 - 5 T + 12 T^{2} - 655 T^{3} + 8639 T^{4} - 47815 T^{5} + 63948 T^{6} - 1945085 T^{7} + 28398241 T^{8} \))(\( 1 + 14 T + 63 T^{2} + 850 T^{3} + 12521 T^{4} + 62050 T^{5} + 335727 T^{6} + 5446238 T^{7} + 28398241 T^{8} \))(\( 1 - 11 T - 22 T^{2} + 845 T^{3} - 5609 T^{4} + 61685 T^{5} - 117238 T^{6} - 4279187 T^{7} + 28398241 T^{8} \))
$79$ (\( 1 + 21 T + 162 T^{2} + 1343 T^{3} + 15405 T^{4} + 106097 T^{5} + 1011042 T^{6} + 10353819 T^{7} + 38950081 T^{8} \))(\( 1 - 24 T + 297 T^{2} - 3382 T^{3} + 35205 T^{4} - 267178 T^{5} + 1853577 T^{6} - 11832936 T^{7} + 38950081 T^{8} \))(\( 1 - 9 T - 48 T^{2} + 643 T^{3} - 195 T^{4} + 50797 T^{5} - 299568 T^{6} - 4437351 T^{7} + 38950081 T^{8} \))
$83$ (\( 1 + 12 T + 61 T^{2} + 1146 T^{3} + 17149 T^{4} + 95118 T^{5} + 420229 T^{6} + 6861444 T^{7} + 47458321 T^{8} \))(\( 1 - T - 67 T^{2} + 515 T^{3} + 5516 T^{4} + 42745 T^{5} - 461563 T^{6} - 571787 T^{7} + 47458321 T^{8} \))(\( 1 + 14 T + 53 T^{2} + 870 T^{3} + 14801 T^{4} + 72210 T^{5} + 365117 T^{6} + 8005018 T^{7} + 47458321 T^{8} \))
$89$ (\( 1 + T - 38 T^{2} - 727 T^{3} + 5055 T^{4} - 64703 T^{5} - 300998 T^{6} + 704969 T^{7} + 62742241 T^{8} \))(\( 1 - 4 T - 73 T^{2} + 648 T^{3} + 3905 T^{4} + 57672 T^{5} - 578233 T^{6} - 2819876 T^{7} + 62742241 T^{8} \))(\( 1 + 11 T - 38 T^{2} - 1037 T^{3} - 5625 T^{4} - 92293 T^{5} - 300998 T^{6} + 7754659 T^{7} + 62742241 T^{8} \))
$97$ (\( 1 + 8 T + 87 T^{2} + 1270 T^{3} + 19901 T^{4} + 123190 T^{5} + 818583 T^{6} + 7301384 T^{7} + 88529281 T^{8} \))(\( 1 - 28 T + 287 T^{2} - 1970 T^{3} + 17821 T^{4} - 191090 T^{5} + 2700383 T^{6} - 25554844 T^{7} + 88529281 T^{8} \))(\( 1 + 12 T + 47 T^{2} - 600 T^{3} - 11759 T^{4} - 58200 T^{5} + 442223 T^{6} + 10952076 T^{7} + 88529281 T^{8} \))
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