Properties

Label 103.2.a
Level 103
Weight 2
Character orbit a
Rep. character \(\chi_{103}(1,\cdot)\)
Character field \(\Q\)
Dimension 8
Newform subspaces 2
Sturm bound 17
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 103.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(17\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(103))\).

Total New Old
Modular forms 9 9 0
Cusp forms 8 8 0
Eisenstein series 1 1 0

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

\(103\)Dim.
\(+\)\(2\)
\(-\)\(6\)

Trace form

\( 8q + q^{2} - 2q^{3} + 9q^{4} - 4q^{7} + 3q^{8} + 4q^{9} + O(q^{10}) \) \( 8q + q^{2} - 2q^{3} + 9q^{4} - 4q^{7} + 3q^{8} + 4q^{9} - 8q^{10} - 4q^{11} - 16q^{12} - 4q^{13} - 6q^{14} - 6q^{15} + 15q^{16} + 12q^{17} + 3q^{18} - 2q^{19} - 6q^{20} - 12q^{21} - 4q^{22} + 12q^{23} - 30q^{24} - 2q^{25} + 7q^{26} + 10q^{27} - 13q^{28} + 6q^{29} - 16q^{31} + 12q^{32} + 18q^{33} + 26q^{34} + 8q^{35} - 9q^{36} - 23q^{38} + 8q^{39} - 2q^{40} + 14q^{41} + 22q^{42} - 10q^{43} - 16q^{44} + 14q^{45} + 9q^{46} - 2q^{47} - 54q^{48} - 14q^{49} + 13q^{50} + 4q^{51} - 32q^{52} + 10q^{53} + 8q^{54} - 8q^{55} - 7q^{56} + 18q^{57} + 18q^{58} + 18q^{59} + 14q^{60} + 16q^{61} + 38q^{62} - 16q^{63} + 65q^{64} + 20q^{65} - 6q^{66} - 10q^{67} - 7q^{68} - 22q^{69} - 16q^{70} - 24q^{71} + 43q^{72} - 22q^{73} - 14q^{75} - 8q^{76} + 30q^{77} - 32q^{78} - 14q^{79} - 40q^{80} + 33q^{82} - 6q^{83} + 64q^{84} + 2q^{85} + 10q^{86} - 6q^{87} - 12q^{88} - 32q^{89} - 26q^{90} - 30q^{91} + 60q^{92} - 32q^{93} - 8q^{94} - 6q^{95} - 18q^{96} + 2q^{97} + 20q^{98} - 48q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(103))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 103
103.2.a.a \(2\) \(0.822\) \(\Q(\sqrt{5}) \) None \(-3\) \(-2\) \(-3\) \(-2\) \(+\) \(q+(-1-\beta )q^{2}-q^{3}+3\beta q^{4}+(-2+\cdots)q^{5}+\cdots\)
103.2.a.b \(6\) \(0.822\) 6.6.6999257.1 None \(4\) \(0\) \(3\) \(-2\) \(-\) \(q+(1-\beta _{1})q^{2}+(-\beta _{2}-\beta _{3}+\beta _{5})q^{3}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4} \))(\( 1 - 4 T + 11 T^{2} - 23 T^{3} + 43 T^{4} - 74 T^{5} + 111 T^{6} - 148 T^{7} + 172 T^{8} - 184 T^{9} + 176 T^{10} - 128 T^{11} + 64 T^{12} \))
$3$ (\( ( 1 + T + 3 T^{2} )^{2} \))(\( 1 + 5 T^{2} + 19 T^{4} - 8 T^{5} + 62 T^{6} - 24 T^{7} + 171 T^{8} + 405 T^{10} + 729 T^{12} \))
$5$ (\( 1 + 3 T + 11 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 - 3 T + 19 T^{2} - 41 T^{3} + 167 T^{4} - 280 T^{5} + 954 T^{6} - 1400 T^{7} + 4175 T^{8} - 5125 T^{9} + 11875 T^{10} - 9375 T^{11} + 15625 T^{12} \))
$7$ (\( ( 1 + T + 7 T^{2} )^{2} \))(\( 1 + 2 T + 24 T^{2} + 44 T^{3} + 305 T^{4} + 500 T^{5} + 2605 T^{6} + 3500 T^{7} + 14945 T^{8} + 15092 T^{9} + 57624 T^{10} + 33614 T^{11} + 117649 T^{12} \))
$11$ (\( 1 + 3 T + 23 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 + T + 25 T^{2} - 13 T^{3} + 427 T^{4} - 66 T^{5} + 6278 T^{6} - 726 T^{7} + 51667 T^{8} - 17303 T^{9} + 366025 T^{10} + 161051 T^{11} + 1771561 T^{12} \))
$13$ (\( 1 + 3 T + 17 T^{2} + 39 T^{3} + 169 T^{4} \))(\( 1 + T + 50 T^{2} + 118 T^{3} + 1099 T^{4} + 3654 T^{5} + 16123 T^{6} + 47502 T^{7} + 185731 T^{8} + 259246 T^{9} + 1428050 T^{10} + 371293 T^{11} + 4826809 T^{12} \))
$17$ (\( 1 + 9 T + 53 T^{2} + 153 T^{3} + 289 T^{4} \))(\( 1 - 21 T + 246 T^{2} - 2038 T^{3} + 13215 T^{4} - 70382 T^{5} + 315203 T^{6} - 1196494 T^{7} + 3819135 T^{8} - 10012694 T^{9} + 20546166 T^{10} - 29816997 T^{11} + 24137569 T^{12} \))
$19$ (\( 1 - 5 T + 33 T^{2} - 95 T^{3} + 361 T^{4} \))(\( 1 + 7 T + 106 T^{2} + 492 T^{3} + 4299 T^{4} + 14820 T^{5} + 100307 T^{6} + 281580 T^{7} + 1551939 T^{8} + 3374628 T^{9} + 13814026 T^{10} + 17332693 T^{11} + 47045881 T^{12} \))
$23$ (\( 1 + 26 T^{2} + 529 T^{4} \))(\( 1 - 12 T + 115 T^{2} - 740 T^{3} + 4872 T^{4} - 25912 T^{5} + 139044 T^{6} - 595976 T^{7} + 2577288 T^{8} - 9003580 T^{9} + 32181715 T^{10} - 77236116 T^{11} + 148035889 T^{12} \))
$29$ (\( 1 + 6 T + 62 T^{2} + 174 T^{3} + 841 T^{4} \))(\( 1 - 12 T + 201 T^{2} - 1712 T^{3} + 15708 T^{4} - 98482 T^{5} + 621764 T^{6} - 2855978 T^{7} + 13210428 T^{8} - 41753968 T^{9} + 142163481 T^{10} - 246133788 T^{11} + 594823321 T^{12} \))
$31$ (\( 1 + 17 T^{2} + 961 T^{4} \))(\( 1 + 16 T + 243 T^{2} + 2330 T^{3} + 20463 T^{4} + 138538 T^{5} + 860842 T^{6} + 4294678 T^{7} + 19664943 T^{8} + 69413030 T^{9} + 224415603 T^{10} + 458066416 T^{11} + 887503681 T^{12} \))
$37$ (\( 1 + 29 T^{2} + 1369 T^{4} \))(\( 1 + 139 T^{2} - 322 T^{3} + 7915 T^{4} - 35678 T^{5} + 306610 T^{6} - 1320086 T^{7} + 10835635 T^{8} - 16310266 T^{9} + 260508379 T^{10} + 2565726409 T^{12} \))
$41$ (\( 1 + 2 T^{2} + 1681 T^{4} \))(\( 1 - 14 T + 209 T^{2} - 1296 T^{3} + 9460 T^{4} - 19394 T^{5} + 195752 T^{6} - 795154 T^{7} + 15902260 T^{8} - 89321616 T^{9} + 590584049 T^{10} - 1621986814 T^{11} + 4750104241 T^{12} \))
$43$ (\( 1 + 4 T + 45 T^{2} + 172 T^{3} + 1849 T^{4} \))(\( 1 + 6 T + 87 T^{2} + 130 T^{3} + 2043 T^{4} - 19180 T^{5} - 10998 T^{6} - 824740 T^{7} + 3777507 T^{8} + 10335910 T^{9} + 297435687 T^{10} + 882050658 T^{11} + 6321363049 T^{12} \))
$47$ (\( 1 + 3 T + 65 T^{2} + 141 T^{3} + 2209 T^{4} \))(\( 1 - T + 139 T^{2} - 587 T^{3} + 9299 T^{4} - 66266 T^{5} + 445266 T^{6} - 3114502 T^{7} + 20541491 T^{8} - 60944101 T^{9} + 678275659 T^{10} - 229345007 T^{11} + 10779215329 T^{12} \))
$53$ (\( 1 + 9 T + 95 T^{2} + 477 T^{3} + 2809 T^{4} \))(\( 1 - 19 T + 427 T^{2} - 5229 T^{3} + 65155 T^{4} - 564172 T^{5} + 4805218 T^{6} - 29901116 T^{7} + 183020395 T^{8} - 778477833 T^{9} + 3369235387 T^{10} - 7945714367 T^{11} + 22164361129 T^{12} \))
$59$ (\( 1 - 15 T + 173 T^{2} - 885 T^{3} + 3481 T^{4} \))(\( 1 - 3 T + 190 T^{2} - 604 T^{3} + 21143 T^{4} - 56860 T^{5} + 1504679 T^{6} - 3354740 T^{7} + 73598783 T^{8} - 124048916 T^{9} + 2302298590 T^{10} - 2144772897 T^{11} + 42180533641 T^{12} \))
$61$ (\( 1 - 15 T + 167 T^{2} - 915 T^{3} + 3721 T^{4} \))(\( 1 - T + 172 T^{2} - 578 T^{3} + 12081 T^{4} - 85710 T^{5} + 645325 T^{6} - 5228310 T^{7} + 44953401 T^{8} - 131195018 T^{9} + 2381484652 T^{10} - 844596301 T^{11} + 51520374361 T^{12} \))
$67$ (\( 1 - 2 T - 45 T^{2} - 134 T^{3} + 4489 T^{4} \))(\( 1 + 12 T + 369 T^{2} + 3268 T^{3} + 57475 T^{4} + 397320 T^{5} + 5012870 T^{6} + 26620440 T^{7} + 258005275 T^{8} + 982893484 T^{9} + 7435763649 T^{10} + 16201501284 T^{11} + 90458382169 T^{12} \))
$71$ (\( 1 - 3 T + 113 T^{2} - 213 T^{3} + 5041 T^{4} \))(\( 1 + 27 T + 565 T^{2} + 8239 T^{3} + 104135 T^{4} + 1073500 T^{5} + 9890294 T^{6} + 76218500 T^{7} + 524944535 T^{8} + 2948828729 T^{9} + 14357599765 T^{10} + 48714192477 T^{11} + 128100283921 T^{12} \))
$73$ (\( 1 + 15 T + 191 T^{2} + 1095 T^{3} + 5329 T^{4} \))(\( 1 + 7 T + 377 T^{2} + 2127 T^{3} + 62883 T^{4} + 284026 T^{5} + 5936262 T^{6} + 20733898 T^{7} + 335103507 T^{8} + 827439159 T^{9} + 10706136857 T^{10} + 14511501151 T^{11} + 151334226289 T^{12} \))
$79$ (\( 1 - 7 T + 69 T^{2} - 553 T^{3} + 6241 T^{4} \))(\( 1 + 21 T + 462 T^{2} + 6312 T^{3} + 83999 T^{4} + 849672 T^{5} + 8497015 T^{6} + 67124088 T^{7} + 524237759 T^{8} + 3112062168 T^{9} + 17994937422 T^{10} + 64618184379 T^{11} + 243087455521 T^{12} \))
$83$ (\( 1 - 3 T + 107 T^{2} - 249 T^{3} + 6889 T^{4} \))(\( 1 + 9 T + 432 T^{2} + 2916 T^{3} + 79961 T^{4} + 420324 T^{5} + 8474641 T^{6} + 34886892 T^{7} + 550851329 T^{8} + 1667330892 T^{9} + 20501994672 T^{10} + 35451365787 T^{11} + 326940373369 T^{12} \))
$89$ (\( 1 + 18 T + 214 T^{2} + 1602 T^{3} + 7921 T^{4} \))(\( 1 + 14 T + 162 T^{2} + 510 T^{3} + 2607 T^{4} + 72260 T^{5} + 975356 T^{6} + 6431140 T^{7} + 20650047 T^{8} + 359534190 T^{9} + 10164243042 T^{10} + 78176832286 T^{11} + 496981290961 T^{12} \))
$97$ (\( 1 - 10 T + 174 T^{2} - 970 T^{3} + 9409 T^{4} \))(\( 1 + 8 T + 245 T^{2} + 2588 T^{3} + 39320 T^{4} + 435662 T^{5} + 4282548 T^{6} + 42259214 T^{7} + 369961880 T^{8} + 2361997724 T^{9} + 21689673845 T^{10} + 68698722056 T^{11} + 832972004929 T^{12} \))
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