Properties

Label 430.2.a
Level 430
Weight 2
Character orbit a
Rep. character \(\chi_{430}(1,\cdot)\)
Character field \(\Q\)
Dimension 13
Newform subspaces 8
Sturm bound 132
Trace bound 5

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Defining parameters

Level: \( N \) = \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 430.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(132\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(430))\).

Total New Old
Modular forms 70 13 57
Cusp forms 63 13 50
Eisenstein series 7 0 7

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

\(2\)\(5\)\(43\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(1\)
\(+\)\(+\)\(-\)\(-\)\(3\)
\(+\)\(-\)\(+\)\(-\)\(2\)
\(+\)\(-\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(2\)
Plus space\(+\)\(4\)
Minus space\(-\)\(9\)

Trace form

\( 13q - q^{2} - 4q^{3} + 13q^{4} - q^{5} - 4q^{6} - 8q^{7} - q^{8} + 9q^{9} + O(q^{10}) \) \( 13q - q^{2} - 4q^{3} + 13q^{4} - q^{5} - 4q^{6} - 8q^{7} - q^{8} + 9q^{9} + q^{10} + 4q^{11} - 4q^{12} - 6q^{13} + 4q^{14} + 4q^{15} + 13q^{16} + 10q^{17} + 3q^{18} - 8q^{19} - q^{20} + 16q^{21} - 4q^{22} + 12q^{23} - 4q^{24} + 13q^{25} + 6q^{26} + 8q^{27} - 8q^{28} + 18q^{29} - 4q^{30} - 28q^{31} - q^{32} + 16q^{33} - 18q^{34} + 9q^{36} - 30q^{37} + 8q^{39} + q^{40} - 10q^{41} + 8q^{42} + q^{43} + 4q^{44} - 13q^{45} - 20q^{46} + 28q^{47} - 4q^{48} + q^{49} - q^{50} + 16q^{51} - 6q^{52} + 34q^{53} + 8q^{54} + 4q^{55} + 4q^{56} + 8q^{57} - 10q^{58} - 20q^{59} + 4q^{60} - 22q^{61} + 8q^{62} - 24q^{63} + 13q^{64} - 2q^{65} - 16q^{66} - 20q^{67} + 10q^{68} - 8q^{70} + 3q^{72} - 10q^{73} - 2q^{74} - 4q^{75} - 8q^{76} + 24q^{77} - 24q^{78} + 20q^{79} - q^{80} - 27q^{81} - 10q^{82} + 20q^{83} + 16q^{84} + 6q^{85} - q^{86} - 40q^{87} - 4q^{88} + 2q^{89} - 3q^{90} - 8q^{91} + 12q^{92} - 40q^{93} + 4q^{94} - 4q^{95} - 4q^{96} - 30q^{97} - 25q^{98} - 28q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(430))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5 43
430.2.a.a \(1\) \(3.434\) \(\Q\) None \(-1\) \(0\) \(-1\) \(1\) \(+\) \(+\) \(+\) \(q-q^{2}+q^{4}-q^{5}+q^{7}-q^{8}-3q^{9}+\cdots\)
430.2.a.b \(1\) \(3.434\) \(\Q\) None \(-1\) \(0\) \(1\) \(-3\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{4}+q^{5}-3q^{7}-q^{8}-3q^{9}+\cdots\)
430.2.a.c \(1\) \(3.434\) \(\Q\) None \(1\) \(-2\) \(-1\) \(-1\) \(-\) \(+\) \(-\) \(q+q^{2}-2q^{3}+q^{4}-q^{5}-2q^{6}-q^{7}+\cdots\)
430.2.a.d \(1\) \(3.434\) \(\Q\) None \(1\) \(-2\) \(1\) \(-5\) \(-\) \(-\) \(+\) \(q+q^{2}-2q^{3}+q^{4}+q^{5}-2q^{6}-5q^{7}+\cdots\)
430.2.a.e \(2\) \(3.434\) \(\Q(\sqrt{3}) \) None \(-2\) \(2\) \(2\) \(2\) \(+\) \(-\) \(+\) \(q-q^{2}+(1+\beta )q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
430.2.a.f \(2\) \(3.434\) \(\Q(\sqrt{6}) \) None \(2\) \(0\) \(-2\) \(2\) \(-\) \(+\) \(+\) \(q+q^{2}+\beta q^{3}+q^{4}-q^{5}+\beta q^{6}+q^{7}+\cdots\)
430.2.a.g \(2\) \(3.434\) \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(2\) \(2\) \(-\) \(-\) \(-\) \(q+q^{2}+\beta q^{3}+q^{4}+q^{5}+\beta q^{6}+q^{7}+\cdots\)
430.2.a.h \(3\) \(3.434\) 3.3.316.1 None \(-3\) \(-2\) \(-3\) \(-6\) \(+\) \(+\) \(-\) \(q-q^{2}+(-1+\beta _{1}-\beta _{2})q^{3}+q^{4}-q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(430))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(430)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(43))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(86))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(215))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{3} \))
$3$ (\( 1 + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 + 2 T + 3 T^{2} \))(\( 1 + 2 T + 3 T^{2} \))(\( 1 - 2 T + 4 T^{2} - 6 T^{3} + 9 T^{4} \))(\( 1 + 9 T^{4} \))(\( 1 + 4 T^{2} + 9 T^{4} \))(\( 1 + 2 T + 3 T^{2} + 4 T^{3} + 9 T^{4} + 18 T^{5} + 27 T^{6} \))
$5$ (\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{3} \))
$7$ (\( 1 - T + 7 T^{2} \))(\( 1 + 3 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 - 2 T + 3 T^{2} - 14 T^{3} + 49 T^{4} \))(\( ( 1 - T + 7 T^{2} )^{2} \))(\( ( 1 - T + 7 T^{2} )^{2} \))(\( 1 + 6 T + 26 T^{2} + 76 T^{3} + 182 T^{4} + 294 T^{5} + 343 T^{6} \))
$11$ (\( 1 + 4 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 6 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 - 2 T + 20 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 - 4 T + 20 T^{2} - 44 T^{3} + 121 T^{4} \))(\( 1 - 4 T + 24 T^{2} - 44 T^{3} + 121 T^{4} \))(\( 1 - 6 T + 11 T^{2} + 4 T^{3} + 121 T^{4} - 726 T^{5} + 1331 T^{6} \))
$13$ (\( 1 + T + 13 T^{2} \))(\( 1 + 3 T + 13 T^{2} \))(\( 1 - 5 T + 13 T^{2} \))(\( 1 + 5 T + 13 T^{2} \))(\( 1 - 2 T + 15 T^{2} - 26 T^{3} + 169 T^{4} \))(\( ( 1 + T + 13 T^{2} )^{2} \))(\( 1 - 2 T + 19 T^{2} - 26 T^{3} + 169 T^{4} \))(\( 1 + 4 T + 12 T^{2} - 2 T^{3} + 156 T^{4} + 676 T^{5} + 2197 T^{6} \))
$17$ (\( 1 + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 - 10 T + 56 T^{2} - 170 T^{3} + 289 T^{4} \))(\( 1 + 28 T^{2} + 289 T^{4} \))(\( 1 + 32 T^{2} + 289 T^{4} \))(\( 1 - 8 T + 57 T^{2} - 228 T^{3} + 969 T^{4} - 2312 T^{5} + 4913 T^{6} \))
$19$ (\( 1 - T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 + 7 T + 19 T^{2} \))(\( 1 - 3 T + 19 T^{2} \))(\( 1 + 6 T + 35 T^{2} + 114 T^{3} + 361 T^{4} \))(\( 1 - 2 T + 15 T^{2} - 38 T^{3} + 361 T^{4} \))(\( ( 1 + T + 19 T^{2} )^{2} \))(\( 1 - 2 T + 42 T^{2} - 44 T^{3} + 798 T^{4} - 722 T^{5} + 6859 T^{6} \))
$23$ (\( 1 + 4 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 - 6 T + 28 T^{2} - 138 T^{3} + 529 T^{4} \))(\( 1 - 4 T + 44 T^{2} - 92 T^{3} + 529 T^{4} \))(\( 1 - 4 T - 92 T^{3} + 529 T^{4} \))(\( 1 - 14 T + 119 T^{2} - 652 T^{3} + 2737 T^{4} - 7406 T^{5} + 12167 T^{6} \))
$29$ (\( 1 + 5 T + 29 T^{2} \))(\( 1 + 3 T + 29 T^{2} \))(\( 1 + 3 T + 29 T^{2} \))(\( 1 + T + 29 T^{2} \))(\( 1 - 16 T + 119 T^{2} - 464 T^{3} + 841 T^{4} \))(\( 1 - 14 T + 101 T^{2} - 406 T^{3} + 841 T^{4} \))(\( 1 + 6 T + 49 T^{2} + 174 T^{3} + 841 T^{4} \))(\( 1 - 6 T - 4 T^{2} + 194 T^{3} - 116 T^{4} - 5046 T^{5} + 24389 T^{6} \))
$31$ (\( 1 + 9 T + 31 T^{2} \))(\( 1 - 7 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 + 11 T + 31 T^{2} \))(\( 1 + 16 T + 123 T^{2} + 496 T^{3} + 961 T^{4} \))(\( 1 + 6 T + 65 T^{2} + 186 T^{3} + 961 T^{4} \))(\( 1 - 2 T + 61 T^{2} - 62 T^{3} + 961 T^{4} \))(\( 1 + 54 T^{2} - 16 T^{3} + 1674 T^{4} + 29791 T^{6} \))
$37$ (\( 1 - 4 T + 37 T^{2} \))(\( 1 + 8 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 10 T + 37 T^{2} \))(\( 1 + 2 T + 48 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( 1 + 4 T + 72 T^{2} + 148 T^{3} + 1369 T^{4} \))(\( 1 + 4 T + 60 T^{2} + 148 T^{3} + 1369 T^{4} \))(\( 1 + 8 T + 53 T^{2} + 300 T^{3} + 1961 T^{4} + 10952 T^{5} + 50653 T^{6} \))
$41$ (\( 1 + 7 T + 41 T^{2} \))(\( 1 + 7 T + 41 T^{2} \))(\( 1 + 3 T + 41 T^{2} \))(\( 1 - 5 T + 41 T^{2} \))(\( 1 + 2 T + 71 T^{2} + 82 T^{3} + 1681 T^{4} \))(\( 1 + 14 T + 107 T^{2} + 574 T^{3} + 1681 T^{4} \))(\( 1 - 2 T + 75 T^{2} - 82 T^{3} + 1681 T^{4} \))(\( 1 - 16 T + 148 T^{2} - 978 T^{3} + 6068 T^{4} - 26896 T^{5} + 68921 T^{6} \))
$43$ (\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{3} \))
$47$ (\( 1 - 6 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 - 4 T + 47 T^{2} \))(\( 1 - 14 T + 140 T^{2} - 658 T^{3} + 2209 T^{4} \))(\( 1 + 88 T^{2} + 2209 T^{4} \))(\( 1 + 44 T^{2} + 2209 T^{4} \))(\( 1 + 2 T + 135 T^{2} + 180 T^{3} + 6345 T^{4} + 4418 T^{5} + 103823 T^{6} \))
$53$ (\( 1 + 2 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 10 T + 53 T^{2} \))(\( 1 + 4 T + 98 T^{2} + 212 T^{3} + 2809 T^{4} \))(\( 1 - 8 T + 26 T^{2} - 424 T^{3} + 2809 T^{4} \))(\( 1 + 74 T^{2} + 2809 T^{4} \))(\( 1 - 22 T + 223 T^{2} - 1644 T^{3} + 11819 T^{4} - 61798 T^{5} + 148877 T^{6} \))
$59$ (\( 1 + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 - 8 T + 59 T^{2} \))(\( 1 + 20 T + 206 T^{2} + 1180 T^{3} + 3481 T^{4} \))(\( 1 + 94 T^{2} + 3481 T^{4} \))(\( 1 + 8 T + 126 T^{2} + 472 T^{3} + 3481 T^{4} \))(\( 1 - 16 T + 201 T^{2} - 1536 T^{3} + 11859 T^{4} - 55696 T^{5} + 205379 T^{6} \))
$61$ (\( 1 + 7 T + 61 T^{2} \))(\( 1 - 7 T + 61 T^{2} \))(\( 1 + T + 61 T^{2} \))(\( 1 + 3 T + 61 T^{2} \))(\( 1 + 119 T^{2} + 3721 T^{4} \))(\( 1 + 2 T + 117 T^{2} + 122 T^{3} + 3721 T^{4} \))(\( 1 + 14 T + 121 T^{2} + 854 T^{3} + 3721 T^{4} \))(\( 1 + 2 T + 84 T^{2} - 118 T^{3} + 5124 T^{4} + 7442 T^{5} + 226981 T^{6} \))
$67$ (\( 1 - 15 T + 67 T^{2} \))(\( 1 - 5 T + 67 T^{2} \))(\( 1 + 13 T + 67 T^{2} \))(\( 1 + 3 T + 67 T^{2} \))(\( 1 + 107 T^{2} + 4489 T^{4} \))(\( 1 + 6 T + 137 T^{2} + 402 T^{3} + 4489 T^{4} \))(\( 1 - 6 T + 125 T^{2} - 402 T^{3} + 4489 T^{4} \))(\( 1 + 24 T + 350 T^{2} + 3340 T^{3} + 23450 T^{4} + 107736 T^{5} + 300763 T^{6} \))
$71$ (\( 1 + 6 T + 71 T^{2} \))(\( 1 - 2 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 + 10 T + 164 T^{2} + 710 T^{3} + 5041 T^{4} \))(\( 1 - 12 T + 124 T^{2} - 852 T^{3} + 5041 T^{4} \))(\( 1 - 4 T + 48 T^{2} - 284 T^{3} + 5041 T^{4} \))(\( 1 + 6 T + 191 T^{2} + 868 T^{3} + 13561 T^{4} + 30246 T^{5} + 357911 T^{6} \))
$73$ (\( 1 + 5 T + 73 T^{2} \))(\( 1 + T + 73 T^{2} \))(\( 1 - 11 T + 73 T^{2} \))(\( 1 - 7 T + 73 T^{2} \))(\( 1 - 8 T + 135 T^{2} - 584 T^{3} + 5329 T^{4} \))(\( 1 + 10 T + 165 T^{2} + 730 T^{3} + 5329 T^{4} \))(\( 1 + 10 T + 153 T^{2} + 730 T^{3} + 5329 T^{4} \))(\( 1 + 10 T + 68 T^{2} - 42 T^{3} + 4964 T^{4} + 53290 T^{5} + 389017 T^{6} \))
$79$ (\( 1 - 9 T + 79 T^{2} \))(\( 1 - 9 T + 79 T^{2} \))(\( 1 + T + 79 T^{2} \))(\( 1 - 7 T + 79 T^{2} \))(\( 1 + 4 T + 15 T^{2} + 316 T^{3} + 6241 T^{4} \))(\( 1 - 18 T + 233 T^{2} - 1422 T^{3} + 6241 T^{4} \))(\( 1 + 14 T + 205 T^{2} + 1106 T^{3} + 6241 T^{4} \))(\( 1 + 4 T + 202 T^{2} + 640 T^{3} + 15958 T^{4} + 24964 T^{5} + 493039 T^{6} \))
$83$ (\( 1 + 83 T^{2} \))(\( 1 - 8 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 154 T^{2} + 6889 T^{4} \))(\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( 1 + 4 T + 138 T^{2} + 332 T^{3} + 6889 T^{4} \))(\( 1 - 4 T + 157 T^{2} - 168 T^{3} + 13031 T^{4} - 27556 T^{5} + 571787 T^{6} \))
$89$ (\( 1 + 89 T^{2} \))(\( 1 - 4 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 6 T + 40 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 - 12 T + 64 T^{2} - 1068 T^{3} + 7921 T^{4} \))(\( 1 + 4 T + 132 T^{2} + 356 T^{3} + 7921 T^{4} \))(\( 1 + 16 T + 121 T^{2} + 444 T^{3} + 10769 T^{4} + 126736 T^{5} + 704969 T^{6} \))
$97$ (\( 1 + 2 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 8 T + 97 T^{2} \))(\( 1 - 12 T + 97 T^{2} \))(\( 1 + 6 T - 40 T^{2} + 582 T^{3} + 9409 T^{4} \))(\( 1 + 28 T + 384 T^{2} + 2716 T^{3} + 9409 T^{4} \))(\( 1 + 4 T - 44 T^{2} + 388 T^{3} + 9409 T^{4} \))(\( 1 + 8 T + 233 T^{2} + 1260 T^{3} + 22601 T^{4} + 75272 T^{5} + 912673 T^{6} \))
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