Properties

Label 334.2.a
Level 334
Weight 2
Character orbit a
Rep. character \(\chi_{334}(1,\cdot)\)
Character field \(\Q\)
Dimension 13
Newform subspaces 6
Sturm bound 84
Trace bound 3

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Defining parameters

Level: \( N \) \(=\) \( 334 = 2 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 334.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(84\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(334))\).

Total New Old
Modular forms 44 13 31
Cusp forms 41 13 28
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.

\(2\)\(167\)FrickeDim.
\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(5\)
\(-\)\(+\)\(-\)\(4\)
\(-\)\(-\)\(+\)\(2\)
Plus space\(+\)\(4\)
Minus space\(-\)\(9\)

Trace form

\( 13q - q^{2} - 4q^{3} + 13q^{4} - 4q^{5} - 8q^{7} - q^{8} + 13q^{9} + O(q^{10}) \) \( 13q - q^{2} - 4q^{3} + 13q^{4} - 4q^{5} - 8q^{7} - q^{8} + 13q^{9} - 4q^{10} + 4q^{11} - 4q^{12} + 4q^{14} + 4q^{15} + 13q^{16} + 10q^{17} - 5q^{18} - 4q^{19} - 4q^{20} + 16q^{21} - 8q^{22} + 4q^{23} - q^{25} + 4q^{26} - 4q^{27} - 8q^{28} - 2q^{29} - 4q^{30} - 12q^{31} - q^{32} + 8q^{33} - 6q^{34} + 32q^{35} + 13q^{36} - 12q^{37} - 4q^{38} + 4q^{39} - 4q^{40} + 26q^{41} + 4q^{42} - 6q^{43} + 4q^{44} - 4q^{45} + 8q^{46} - 8q^{47} - 4q^{48} - 15q^{49} + q^{50} - 4q^{51} + 12q^{53} - 12q^{54} - 12q^{55} + 4q^{56} - 4q^{57} - 30q^{58} + 18q^{59} + 4q^{60} - 10q^{61} + 8q^{62} - 40q^{63} + 13q^{64} - 28q^{65} - 16q^{66} - 14q^{67} + 10q^{68} - 16q^{69} - 8q^{70} - 4q^{71} - 5q^{72} - 38q^{73} - 8q^{74} - 44q^{75} - 4q^{76} + 24q^{77} - 4q^{78} - 4q^{80} - 19q^{81} - 14q^{82} + 10q^{83} + 16q^{84} + 8q^{85} - 10q^{86} + 28q^{87} - 8q^{88} + 22q^{89} - 16q^{91} + 4q^{92} - 40q^{93} + 20q^{94} + 32q^{95} - 10q^{97} - 25q^{98} + 28q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(334))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 167
334.2.a.a \(1\) \(2.667\) \(\Q\) None \(1\) \(0\) \(3\) \(1\) \(-\) \(+\) \(q+q^{2}+q^{4}+3q^{5}+q^{7}+q^{8}-3q^{9}+\cdots\)
334.2.a.b \(2\) \(2.667\) \(\Q(\sqrt{5}) \) None \(-2\) \(-1\) \(-2\) \(0\) \(+\) \(+\) \(q-q^{2}-\beta q^{3}+q^{4}-q^{5}+\beta q^{6}+(-1+\cdots)q^{7}+\cdots\)
334.2.a.c \(2\) \(2.667\) \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(2\) \(-6\) \(+\) \(-\) \(q-q^{2}+2\beta q^{3}+q^{4}+(1-\beta )q^{5}-2\beta q^{6}+\cdots\)
334.2.a.d \(2\) \(2.667\) \(\Q(\sqrt{5}) \) None \(2\) \(-3\) \(-4\) \(-6\) \(-\) \(-\) \(q+q^{2}+(-1-\beta )q^{3}+q^{4}+(-3+2\beta )q^{5}+\cdots\)
334.2.a.e \(3\) \(2.667\) 3.3.469.1 None \(-3\) \(-1\) \(0\) \(0\) \(+\) \(-\) \(q-q^{2}-\beta _{1}q^{3}+q^{4}+(-\beta _{1}-\beta _{2})q^{5}+\cdots\)
334.2.a.f \(3\) \(2.667\) 3.3.733.1 None \(3\) \(1\) \(-3\) \(3\) \(-\) \(+\) \(q+q^{2}+\beta _{1}q^{3}+q^{4}-q^{5}+\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(334))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(334)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(167))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{3} \))
$3$ (\( 1 + 3 T^{2} \))(\( 1 + T + 5 T^{2} + 3 T^{3} + 9 T^{4} \))(\( 1 - 2 T^{2} + 9 T^{4} \))(\( 1 + 3 T + 7 T^{2} + 9 T^{3} + 9 T^{4} \))(\( 1 + T + 4 T^{2} + 2 T^{3} + 12 T^{4} + 9 T^{5} + 27 T^{6} \))(\( 1 - T + 2 T^{2} + 2 T^{3} + 6 T^{4} - 9 T^{5} + 27 T^{6} \))
$5$ (\( 1 - 3 T + 5 T^{2} \))(\( ( 1 + T + 5 T^{2} )^{2} \))(\( 1 - 2 T + 9 T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 + 4 T + 9 T^{2} + 20 T^{3} + 25 T^{4} \))(\( 1 + 2 T^{2} + 16 T^{3} + 10 T^{4} + 125 T^{6} \))(\( ( 1 + T + 5 T^{2} )^{3} \))
$7$ (\( 1 - T + 7 T^{2} \))(\( 1 + 9 T^{2} + 49 T^{4} \))(\( ( 1 + 3 T + 7 T^{2} )^{2} \))(\( ( 1 + 3 T + 7 T^{2} )^{2} \))(\( 1 + 8 T^{2} + 16 T^{3} + 56 T^{4} + 343 T^{6} \))(\( ( 1 - T + 7 T^{2} )^{3} \))
$11$ (\( 1 + 11 T^{2} \))(\( 1 + 5 T + 27 T^{2} + 55 T^{3} + 121 T^{4} \))(\( 1 + 14 T^{2} + 121 T^{4} \))(\( 1 + 9 T + 41 T^{2} + 99 T^{3} + 121 T^{4} \))(\( 1 - 11 T + 66 T^{2} - 258 T^{3} + 726 T^{4} - 1331 T^{5} + 1331 T^{6} \))(\( 1 - 7 T + 24 T^{2} - 66 T^{3} + 264 T^{4} - 847 T^{5} + 1331 T^{6} \))
$13$ (\( 1 + 2 T + 13 T^{2} \))(\( 1 + 21 T^{2} + 169 T^{4} \))(\( 1 - 8 T + 34 T^{2} - 104 T^{3} + 169 T^{4} \))(\( 1 - 2 T + 7 T^{2} - 26 T^{3} + 169 T^{4} \))(\( 1 + 10 T + 60 T^{2} + 256 T^{3} + 780 T^{4} + 1690 T^{5} + 2197 T^{6} \))(\( 1 - 2 T + 30 T^{2} - 38 T^{3} + 390 T^{4} - 338 T^{5} + 2197 T^{6} \))
$17$ (\( 1 + 2 T + 17 T^{2} \))(\( 1 + 5 T + 29 T^{2} + 85 T^{3} + 289 T^{4} \))(\( 1 - 4 T + 30 T^{2} - 68 T^{3} + 289 T^{4} \))(\( 1 - T + 3 T^{2} - 17 T^{3} + 289 T^{4} \))(\( 1 - 9 T + 50 T^{2} - 200 T^{3} + 850 T^{4} - 2601 T^{5} + 4913 T^{6} \))(\( 1 - 3 T + 44 T^{2} - 104 T^{3} + 748 T^{4} - 867 T^{5} + 4913 T^{6} \))
$19$ (\( 1 - 2 T + 19 T^{2} \))(\( 1 + 33 T^{2} + 361 T^{4} \))(\( 1 - 4 T + 10 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 + 2 T + 19 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 + 4 T + 8 T^{2} - 72 T^{3} + 152 T^{4} + 1444 T^{5} + 6859 T^{6} \))(\( 1 + 4 T + 38 T^{2} + 102 T^{3} + 722 T^{4} + 1444 T^{5} + 6859 T^{6} \))
$23$ (\( 1 - 2 T + 23 T^{2} \))(\( 1 + 4 T + 45 T^{2} + 92 T^{3} + 529 T^{4} \))(\( 1 - 12 T + 74 T^{2} - 276 T^{3} + 529 T^{4} \))(\( 1 + 41 T^{2} + 529 T^{4} \))(\( 1 + 10 T + 90 T^{2} + 456 T^{3} + 2070 T^{4} + 5290 T^{5} + 12167 T^{6} \))(\( 1 - 4 T - 2 T^{2} + 150 T^{3} - 46 T^{4} - 2116 T^{5} + 12167 T^{6} \))
$29$ (\( 1 + 4 T + 29 T^{2} \))(\( 1 + 12 T + 89 T^{2} + 348 T^{3} + 841 T^{4} \))(\( 1 - 8 T + 66 T^{2} - 232 T^{3} + 841 T^{4} \))(\( 1 + 16 T + 117 T^{2} + 464 T^{3} + 841 T^{4} \))(\( 1 - 18 T + 182 T^{2} - 1166 T^{3} + 5278 T^{4} - 15138 T^{5} + 24389 T^{6} \))(\( 1 - 4 T + 82 T^{2} - 228 T^{3} + 2378 T^{4} - 3364 T^{5} + 24389 T^{6} \))
$31$ (\( 1 - T + 31 T^{2} \))(\( 1 + 4 T + 21 T^{2} + 124 T^{3} + 961 T^{4} \))(\( 1 + 10 T + 79 T^{2} + 310 T^{3} + 961 T^{4} \))(\( 1 - 2 T + 43 T^{2} - 62 T^{3} + 961 T^{4} \))(\( 1 - 4 T + 28 T^{2} - 376 T^{3} + 868 T^{4} - 3844 T^{5} + 29791 T^{6} \))(\( 1 + 5 T + 60 T^{2} + 305 T^{3} + 1860 T^{4} + 4805 T^{5} + 29791 T^{6} \))
$37$ (\( 1 + 3 T + 37 T^{2} \))(\( 1 - 3 T - 25 T^{2} - 111 T^{3} + 1369 T^{4} \))(\( 1 + 6 T + 65 T^{2} + 222 T^{3} + 1369 T^{4} \))(\( 1 + 3 T + 65 T^{2} + 111 T^{3} + 1369 T^{4} \))(\( 1 - T + 106 T^{2} - 70 T^{3} + 3922 T^{4} - 1369 T^{5} + 50653 T^{6} \))(\( 1 + 4 T + 91 T^{2} + 223 T^{3} + 3367 T^{4} + 5476 T^{5} + 50653 T^{6} \))
$41$ (\( 1 - 2 T + 41 T^{2} \))(\( 1 + 8 T + 93 T^{2} + 328 T^{3} + 1681 T^{4} \))(\( 1 - 12 T + 86 T^{2} - 492 T^{3} + 1681 T^{4} \))(\( 1 - 6 T + 71 T^{2} - 246 T^{3} + 1681 T^{4} \))(\( 1 - 16 T + 196 T^{2} - 1398 T^{3} + 8036 T^{4} - 26896 T^{5} + 68921 T^{6} \))(\( 1 + 2 T + 114 T^{2} + 150 T^{3} + 4674 T^{4} + 3362 T^{5} + 68921 T^{6} \))
$43$ (\( 1 - 4 T + 43 T^{2} \))(\( 1 - 9 T + 75 T^{2} - 387 T^{3} + 1849 T^{4} \))(\( ( 1 - 2 T + 43 T^{2} )^{2} \))(\( 1 + 5 T - 9 T^{2} + 215 T^{3} + 1849 T^{4} \))(\( 1 + 11 T + 94 T^{2} + 572 T^{3} + 4042 T^{4} + 20339 T^{5} + 79507 T^{6} \))(\( 1 + 7 T + 84 T^{2} + 502 T^{3} + 3612 T^{4} + 12943 T^{5} + 79507 T^{6} \))
$47$ (\( 1 + T + 47 T^{2} \))(\( 1 + 6 T + 58 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 + 6 T + 31 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 - 6 T + 58 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 + 2 T + 113 T^{2} + 220 T^{3} + 5311 T^{4} + 4418 T^{5} + 103823 T^{6} \))(\( 1 - T + 23 T^{2} - 266 T^{3} + 1081 T^{4} - 2209 T^{5} + 103823 T^{6} \))
$53$ (\( 1 + 11 T + 53 T^{2} \))(\( 1 - 5 T + 111 T^{2} - 265 T^{3} + 2809 T^{4} \))(\( 1 + 22 T + 225 T^{2} + 1166 T^{3} + 2809 T^{4} \))(\( 1 + 7 T + 117 T^{2} + 371 T^{3} + 2809 T^{4} \))(\( 1 - 21 T + 278 T^{2} - 2422 T^{3} + 14734 T^{4} - 58989 T^{5} + 148877 T^{6} \))(\( 1 - 26 T + 377 T^{2} - 3349 T^{3} + 19981 T^{4} - 73034 T^{5} + 148877 T^{6} \))
$59$ (\( 1 + 3 T + 59 T^{2} \))(\( 1 + 2 T + 99 T^{2} + 118 T^{3} + 3481 T^{4} \))(\( 1 - 10 T + 141 T^{2} - 590 T^{3} + 3481 T^{4} \))(\( 1 - 10 T + 123 T^{2} - 590 T^{3} + 3481 T^{4} \))(\( 1 + 6 T + 176 T^{2} + 706 T^{3} + 10384 T^{4} + 20886 T^{5} + 205379 T^{6} \))(\( 1 - 9 T + 164 T^{2} - 881 T^{3} + 9676 T^{4} - 31329 T^{5} + 205379 T^{6} \))
$61$ (\( 1 + 4 T + 61 T^{2} \))(\( 1 + 5 T + 127 T^{2} + 305 T^{3} + 3721 T^{4} \))(\( 1 + 50 T^{2} + 3721 T^{4} \))(\( 1 - 9 T + 131 T^{2} - 549 T^{3} + 3721 T^{4} \))(\( 1 + 11 T + 62 T^{2} + 480 T^{3} + 3782 T^{4} + 40931 T^{5} + 226981 T^{6} \))(\( 1 - T + 44 T^{2} + 546 T^{3} + 2684 T^{4} - 3721 T^{5} + 226981 T^{6} \))
$67$ (\( 1 + 3 T + 67 T^{2} \))(\( 1 - 11 T + 153 T^{2} - 737 T^{3} + 4489 T^{4} \))(\( 1 + 2 T + 133 T^{2} + 134 T^{3} + 4489 T^{4} \))(\( 1 - T + 73 T^{2} - 67 T^{3} + 4489 T^{4} \))(\( 1 + 29 T + 476 T^{2} + 4740 T^{3} + 31892 T^{4} + 130181 T^{5} + 300763 T^{6} \))(\( 1 - 8 T + 61 T^{2} - 605 T^{3} + 4087 T^{4} - 35912 T^{5} + 300763 T^{6} \))
$71$ (\( 1 - 6 T + 71 T^{2} \))(\( 1 - 4 T - 34 T^{2} - 284 T^{3} + 5041 T^{4} \))(\( 1 - 12 T + 106 T^{2} - 852 T^{3} + 5041 T^{4} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 4 T + 71 T^{2} )^{3} \))(\( 1 - 10 T + 205 T^{2} - 1260 T^{3} + 14555 T^{4} - 50410 T^{5} + 357911 T^{6} \))
$73$ (\( 1 + 4 T + 73 T^{2} \))(\( 1 - T + 145 T^{2} - 73 T^{3} + 5329 T^{4} \))(\( 1 + 16 T + 178 T^{2} + 1168 T^{3} + 5329 T^{4} \))(\( 1 - 5 T + 151 T^{2} - 365 T^{3} + 5329 T^{4} \))(\( 1 + 7 T + 66 T^{2} + 100 T^{3} + 4818 T^{4} + 37303 T^{5} + 389017 T^{6} \))(\( 1 + 17 T + 254 T^{2} + 2286 T^{3} + 18542 T^{4} + 90593 T^{5} + 389017 T^{6} \))
$79$ (\( 1 + 12 T + 79 T^{2} \))(\( 1 - 12 T + 189 T^{2} - 948 T^{3} + 6241 T^{4} \))(\( 1 + 86 T^{2} + 6241 T^{4} \))(\( 1 + 12 T + 149 T^{2} + 948 T^{3} + 6241 T^{4} \))(\( 1 + 2 T + 166 T^{2} + 308 T^{3} + 13114 T^{4} + 12482 T^{5} + 493039 T^{6} \))(\( 1 - 14 T + 226 T^{2} - 2192 T^{3} + 17854 T^{4} - 87374 T^{5} + 493039 T^{6} \))
$83$ (\( 1 - 7 T + 83 T^{2} \))(\( 1 - 23 T + 287 T^{2} - 1909 T^{3} + 6889 T^{4} \))(\( 1 - 10 T + 93 T^{2} - 830 T^{3} + 6889 T^{4} \))(\( 1 - T + 15 T^{2} - 83 T^{3} + 6889 T^{4} \))(\( 1 + 19 T + 362 T^{2} + 3368 T^{3} + 30046 T^{4} + 130891 T^{5} + 571787 T^{6} \))(\( 1 + 12 T + 231 T^{2} + 1943 T^{3} + 19173 T^{4} + 82668 T^{5} + 571787 T^{6} \))
$89$ (\( 1 - 3 T + 89 T^{2} \))(\( 1 + 2 T + 54 T^{2} + 178 T^{3} + 7921 T^{4} \))(\( 1 - 6 T + 155 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 - 6 T + 182 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 - 12 T + 179 T^{2} - 2144 T^{3} + 15931 T^{4} - 95052 T^{5} + 704969 T^{6} \))(\( 1 + 3 T + 173 T^{2} + 74 T^{3} + 15397 T^{4} + 23763 T^{5} + 704969 T^{6} \))
$97$ (\( 1 - 7 T + 97 T^{2} \))(\( 1 + 6 T + 158 T^{2} + 582 T^{3} + 9409 T^{4} \))(\( 1 - 2 T + 187 T^{2} - 194 T^{3} + 9409 T^{4} \))(\( 1 - 2 T + 70 T^{2} - 194 T^{3} + 9409 T^{4} \))(\( 1 - 12 T + 75 T^{2} + 48 T^{3} + 7275 T^{4} - 112908 T^{5} + 912673 T^{6} \))(\( 1 + 27 T + 441 T^{2} + 4930 T^{3} + 42777 T^{4} + 254043 T^{5} + 912673 T^{6} \))
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