Properties

Label 105.4.a
Level 105
Weight 4
Character orbit a
Rep. character \(\chi_{105}(1,\cdot)\)
Character field \(\Q\)
Dimension 12
Newform subspaces 7
Sturm bound 64
Trace bound 2

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Defining parameters

Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 105.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(64\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(105))\).

Total New Old
Modular forms 52 12 40
Cusp forms 44 12 32
Eisenstein series 8 0 8

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

\(3\)\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(2\)
\(+\)\(-\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(2\)
Plus space\(+\)\(8\)
Minus space\(-\)\(4\)

Trace form

\( 12q - 4q^{2} + 56q^{4} + 12q^{6} - 28q^{7} + 36q^{8} + 108q^{9} + O(q^{10}) \) \( 12q - 4q^{2} + 56q^{4} + 12q^{6} - 28q^{7} + 36q^{8} + 108q^{9} + 60q^{10} - 40q^{11} - 32q^{13} + 140q^{14} - 60q^{15} + 328q^{16} - 64q^{17} - 36q^{18} - 104q^{19} + 72q^{22} + 248q^{23} + 324q^{24} + 300q^{25} - 160q^{26} - 140q^{28} - 240q^{29} - 120q^{30} + 288q^{31} - 596q^{32} - 192q^{33} - 928q^{34} + 504q^{36} - 56q^{37} - 648q^{38} - 24q^{39} + 420q^{40} - 80q^{41} - 168q^{42} - 416q^{43} - 432q^{44} - 1040q^{46} + 272q^{47} - 1248q^{48} + 588q^{49} - 100q^{50} + 864q^{51} - 216q^{52} + 1824q^{53} + 108q^{54} + 360q^{55} + 1092q^{56} - 456q^{57} + 920q^{58} - 304q^{59} + 180q^{60} - 1408q^{61} - 696q^{62} - 252q^{63} - 488q^{64} - 320q^{65} + 624q^{66} - 976q^{67} - 3088q^{68} - 1008q^{69} - 280q^{70} + 1408q^{71} + 324q^{72} + 2592q^{73} + 688q^{74} - 3952q^{76} + 1904q^{77} - 2856q^{78} + 2544q^{79} - 1760q^{80} + 972q^{81} - 3096q^{82} + 2000q^{83} - 440q^{85} + 1800q^{86} + 240q^{87} + 6872q^{88} - 3392q^{89} + 540q^{90} + 840q^{91} + 4760q^{92} + 72q^{93} - 472q^{94} + 40q^{95} + 1644q^{96} - 3776q^{97} - 196q^{98} - 360q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(105))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 5 7
105.4.a.a \(1\) \(6.195\) \(\Q\) None \(0\) \(-3\) \(5\) \(7\) \(+\) \(-\) \(-\) \(q-3q^{3}-8q^{4}+5q^{5}+7q^{7}+9q^{9}+\cdots\)
105.4.a.b \(1\) \(6.195\) \(\Q\) None \(5\) \(-3\) \(5\) \(7\) \(+\) \(-\) \(-\) \(q+5q^{2}-3q^{3}+17q^{4}+5q^{5}-15q^{6}+\cdots\)
105.4.a.c \(2\) \(6.195\) \(\Q(\sqrt{17}) \) None \(-7\) \(-6\) \(-10\) \(-14\) \(+\) \(+\) \(+\) \(q+(-3-\beta )q^{2}-3q^{3}+(5+7\beta )q^{4}+\cdots\)
105.4.a.d \(2\) \(6.195\) \(\Q(\sqrt{5}) \) None \(-4\) \(6\) \(-10\) \(-14\) \(-\) \(+\) \(+\) \(q+(-2-\beta )q^{2}+3q^{3}+(1+4\beta )q^{4}+\cdots\)
105.4.a.e \(2\) \(6.195\) \(\Q(\sqrt{2}) \) None \(-2\) \(-6\) \(10\) \(-14\) \(+\) \(-\) \(+\) \(q+(-1+\beta )q^{2}-3q^{3}+(1-2\beta )q^{4}+\cdots\)
105.4.a.f \(2\) \(6.195\) \(\Q(\sqrt{65}) \) None \(1\) \(6\) \(10\) \(-14\) \(-\) \(-\) \(+\) \(q+\beta q^{2}+3q^{3}+(8+\beta )q^{4}+5q^{5}+3\beta q^{6}+\cdots\)
105.4.a.g \(2\) \(6.195\) \(\Q(\sqrt{41}) \) None \(3\) \(6\) \(-10\) \(14\) \(-\) \(+\) \(-\) \(q+(1+\beta )q^{2}+3q^{3}+(3+3\beta )q^{4}-5q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(105))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(105)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 8 T^{2} \))(\( 1 - 5 T + 8 T^{2} \))(\( 1 + 7 T + 24 T^{2} + 56 T^{3} + 64 T^{4} \))(\( 1 + 4 T + 15 T^{2} + 32 T^{3} + 64 T^{4} \))(\( 1 + 2 T + 9 T^{2} + 16 T^{3} + 64 T^{4} \))(\( 1 - T - 8 T^{3} + 64 T^{4} \))(\( 1 - 3 T + 8 T^{2} - 24 T^{3} + 64 T^{4} \))
$3$ (\( 1 + 3 T \))(\( 1 + 3 T \))(\( ( 1 + 3 T )^{2} \))(\( ( 1 - 3 T )^{2} \))(\( ( 1 + 3 T )^{2} \))(\( ( 1 - 3 T )^{2} \))(\( ( 1 - 3 T )^{2} \))
$5$ (\( 1 - 5 T \))(\( 1 - 5 T \))(\( ( 1 + 5 T )^{2} \))(\( ( 1 + 5 T )^{2} \))(\( ( 1 - 5 T )^{2} \))(\( ( 1 - 5 T )^{2} \))(\( ( 1 + 5 T )^{2} \))
$7$ (\( 1 - 7 T \))(\( 1 - 7 T \))(\( ( 1 + 7 T )^{2} \))(\( ( 1 + 7 T )^{2} \))(\( ( 1 + 7 T )^{2} \))(\( ( 1 + 7 T )^{2} \))(\( ( 1 - 7 T )^{2} \))
$11$ (\( 1 - 42 T + 1331 T^{2} \))(\( 1 - 12 T + 1331 T^{2} \))(\( 1 + 26 T + 2406 T^{2} + 34606 T^{3} + 1771561 T^{4} \))(\( 1 + 92 T + 4758 T^{2} + 122452 T^{3} + 1771561 T^{4} \))(\( 1 + 16 T - 474 T^{2} + 21296 T^{3} + 1771561 T^{4} \))(\( 1 + 22 T + 2718 T^{2} + 29282 T^{3} + 1771561 T^{4} \))(\( 1 - 62 T + 3582 T^{2} - 82522 T^{3} + 1771561 T^{4} \))
$13$ (\( 1 - 20 T + 2197 T^{2} \))(\( 1 - 30 T + 2197 T^{2} \))(\( 1 - 14 T + 2386 T^{2} - 30758 T^{3} + 4826809 T^{4} \))(\( 1 - 8 T - 2810 T^{2} - 17576 T^{3} + 4826809 T^{4} \))(\( 1 + 76 T + 5806 T^{2} + 166972 T^{3} + 4826809 T^{4} \))(\( 1 + 22 T + 4450 T^{2} + 48334 T^{3} + 4826809 T^{4} \))(\( 1 + 6 T + 3378 T^{2} + 13182 T^{3} + 4826809 T^{4} \))
$17$ (\( 1 - 66 T + 4913 T^{2} \))(\( 1 + 134 T + 4913 T^{2} \))(\( 1 - 16 T + 6558 T^{2} - 78608 T^{3} + 24137569 T^{4} \))(\( 1 + 44 T + 630 T^{2} + 216172 T^{3} + 24137569 T^{4} \))(\( 1 + 124 T + 13638 T^{2} + 609212 T^{3} + 24137569 T^{4} \))(\( 1 - 116 T + 9030 T^{2} - 569908 T^{3} + 24137569 T^{4} \))(\( 1 - 40 T + 8750 T^{2} - 196520 T^{3} + 24137569 T^{4} \))
$19$ (\( 1 - 38 T + 6859 T^{2} \))(\( 1 + 92 T + 6859 T^{2} \))(\( 1 - 174 T + 18414 T^{2} - 1193466 T^{3} + 47045881 T^{4} \))(\( 1 + 108 T + 13254 T^{2} + 740772 T^{3} + 47045881 T^{4} \))(\( 1 + 96 T + 13974 T^{2} + 658464 T^{3} + 47045881 T^{4} \))(\( 1 - 102 T + 13134 T^{2} - 699618 T^{3} + 47045881 T^{4} \))(\( 1 + 122 T + 17398 T^{2} + 836798 T^{3} + 47045881 T^{4} \))
$23$ (\( 1 - 12 T + 12167 T^{2} \))(\( 1 - 112 T + 12167 T^{2} \))(\( 1 - 184 T + 19470 T^{2} - 2238728 T^{3} + 148035889 T^{4} \))(\( 1 + 320 T + 47934 T^{2} + 3893440 T^{3} + 148035889 T^{4} \))(\( 1 + 16 T + 15150 T^{2} + 194672 T^{3} + 148035889 T^{4} \))(\( 1 - 260 T + 34734 T^{2} - 3163420 T^{3} + 148035889 T^{4} \))(\( 1 - 16 T + 782 T^{2} - 194672 T^{3} + 148035889 T^{4} \))
$29$ (\( 1 + 258 T + 24389 T^{2} \))(\( 1 + 58 T + 24389 T^{2} \))(\( 1 + 32 T + 19046 T^{2} + 780448 T^{3} + 594823321 T^{4} \))(\( 1 + 236 T + 61982 T^{2} + 5755804 T^{3} + 594823321 T^{4} \))(\( 1 - 188 T + 34286 T^{2} - 4585132 T^{3} + 594823321 T^{4} \))(\( 1 + 196 T + 20942 T^{2} + 4780244 T^{3} + 594823321 T^{4} \))(\( 1 - 352 T + 78278 T^{2} - 8584928 T^{3} + 594823321 T^{4} \))
$31$ (\( 1 - 146 T + 29791 T^{2} \))(\( 1 + 224 T + 29791 T^{2} \))(\( 1 - 330 T + 85430 T^{2} - 9831030 T^{3} + 887503681 T^{4} \))(\( 1 + 60 T + 8462 T^{2} + 1787460 T^{3} + 887503681 T^{4} \))(\( 1 + 120 T + 60590 T^{2} + 3574920 T^{3} + 887503681 T^{4} \))(\( 1 - 150 T + 36542 T^{2} - 4468650 T^{3} + 887503681 T^{4} \))(\( 1 - 66 T + 45870 T^{2} - 1966206 T^{3} + 887503681 T^{4} \))
$37$ (\( 1 - 434 T + 50653 T^{2} \))(\( 1 + 146 T + 50653 T^{2} \))(\( 1 + 132 T + 103214 T^{2} + 6686196 T^{3} + 2565726409 T^{4} \))(\( 1 - 204 T + 108830 T^{2} - 10333212 T^{3} + 2565726409 T^{4} \))(\( 1 + 132 T + 70814 T^{2} + 6686196 T^{3} + 2565726409 T^{4} \))(\( 1 + 96 T + 82550 T^{2} + 4862688 T^{3} + 2565726409 T^{4} \))(\( 1 + 188 T + 44542 T^{2} + 9522764 T^{3} + 2565726409 T^{4} \))
$41$ (\( 1 + 282 T + 68921 T^{2} \))(\( 1 - 18 T + 68921 T^{2} \))(\( 1 - 200 T + 64542 T^{2} - 13784200 T^{3} + 4750104241 T^{4} \))(\( 1 - 44 T + 106326 T^{2} - 3032524 T^{3} + 4750104241 T^{4} \))(\( 1 - 100 T + 89142 T^{2} - 6892100 T^{3} + 4750104241 T^{4} \))(\( 1 + 176 T - 16914 T^{2} + 12130096 T^{3} + 4750104241 T^{4} \))(\( 1 - 16 T + 18350 T^{2} - 1102736 T^{3} + 4750104241 T^{4} \))
$43$ (\( 1 - 20 T + 79507 T^{2} \))(\( 1 - 340 T + 79507 T^{2} \))(\( 1 - 364 T + 172486 T^{2} - 28940548 T^{3} + 6321363049 T^{4} \))(\( 1 - 136 T + 81718 T^{2} - 10812952 T^{3} + 6321363049 T^{4} \))(\( 1 + 536 T + 200086 T^{2} + 42615752 T^{3} + 6321363049 T^{4} \))(\( 1 + 344 T + 171958 T^{2} + 27350408 T^{3} + 6321363049 T^{4} \))(\( 1 + 396 T + 95718 T^{2} + 31484772 T^{3} + 6321363049 T^{4} \))
$47$ (\( 1 + 72 T + 103823 T^{2} \))(\( 1 - 208 T + 103823 T^{2} \))(\( 1 - 292 T + 179390 T^{2} - 30316316 T^{3} + 10779215329 T^{4} \))(\( 1 - 400 T + 106526 T^{2} - 41529200 T^{3} + 10779215329 T^{4} \))(\( 1 + 928 T + 408830 T^{2} + 96347744 T^{3} + 10779215329 T^{4} \))(\( 1 - 560 T + 248606 T^{2} - 58140880 T^{3} + 10779215329 T^{4} \))(\( 1 + 188 T + 15582 T^{2} + 19518724 T^{3} + 10779215329 T^{4} \))
$53$ (\( 1 - 336 T + 148877 T^{2} \))(\( 1 + 754 T + 148877 T^{2} \))(\( 1 - 34 T + 103410 T^{2} - 5061818 T^{3} + 22164361129 T^{4} \))(\( 1 - 16 T + 260838 T^{2} - 2382032 T^{3} + 22164361129 T^{4} \))(\( 1 - 884 T + 460350 T^{2} - 131607268 T^{3} + 22164361129 T^{4} \))(\( 1 - 326 T + 204138 T^{2} - 48533902 T^{3} + 22164361129 T^{4} \))(\( 1 - 982 T + 504354 T^{2} - 146197214 T^{3} + 22164361129 T^{4} \))
$59$ (\( 1 + 360 T + 205379 T^{2} \))(\( 1 - 380 T + 205379 T^{2} \))(\( 1 - 364 T + 438374 T^{2} - 74757956 T^{3} + 42180533641 T^{4} \))(\( 1 + 464 T + 458102 T^{2} + 95295856 T^{3} + 42180533641 T^{4} \))(\( 1 - 104 T + 80534 T^{2} - 21359416 T^{3} + 42180533641 T^{4} \))(\( 1 + 844 T + 474182 T^{2} + 173339876 T^{3} + 42180533641 T^{4} \))(\( 1 - 516 T + 418118 T^{2} - 105975564 T^{3} + 42180533641 T^{4} \))
$61$ (\( 1 + 682 T + 226981 T^{2} \))(\( 1 - 718 T + 226981 T^{2} \))(\( 1 - 792 T + 340886 T^{2} - 179768952 T^{3} + 51520374361 T^{4} \))(\( 1 + 684 T + 535646 T^{2} + 155255004 T^{3} + 51520374361 T^{4} \))(\( 1 + 468 T + 494606 T^{2} + 106227108 T^{3} + 51520374361 T^{4} \))(\( 1 + 204 T + 455006 T^{2} + 46304124 T^{3} + 51520374361 T^{4} \))(\( 1 + 880 T + 575238 T^{2} + 199743280 T^{3} + 51520374361 T^{4} \))
$67$ (\( 1 - 812 T + 300763 T^{2} \))(\( 1 - 412 T + 300763 T^{2} \))(\( 1 + 788 T + 753430 T^{2} + 237001244 T^{3} + 90458382169 T^{4} \))(\( 1 - 736 T + 602470 T^{2} - 221361568 T^{3} + 90458382169 T^{4} \))(\( 1 + 1688 T + 1302310 T^{2} + 507687944 T^{3} + 90458382169 T^{4} \))(\( 1 + 104 T + 537670 T^{2} + 31279352 T^{3} + 90458382169 T^{4} \))(\( 1 + 356 T + 100374 T^{2} + 107071628 T^{3} + 90458382169 T^{4} \))
$71$ (\( 1 - 810 T + 357911 T^{2} \))(\( 1 + 960 T + 357911 T^{2} \))(\( 1 - 454 T + 304526 T^{2} - 162491594 T^{3} + 128100283921 T^{4} \))(\( 1 + 740 T + 757502 T^{2} + 264854140 T^{3} + 128100283921 T^{4} \))(\( 1 + 136 T + 540446 T^{2} + 48675896 T^{3} + 128100283921 T^{4} \))(\( 1 - 1670 T + 1384382 T^{2} - 597711370 T^{3} + 128100283921 T^{4} \))(\( 1 - 310 T + 664038 T^{2} - 110952410 T^{3} + 128100283921 T^{4} \))
$73$ (\( 1 + 124 T + 389017 T^{2} \))(\( 1 - 1066 T + 389017 T^{2} \))(\( 1 - 778 T + 794698 T^{2} - 302655226 T^{3} + 151334226289 T^{4} \))(\( 1 - 424 T + 748558 T^{2} - 164943208 T^{3} + 151334226289 T^{4} \))(\( 1 - 508 T + 13078 T^{2} - 197620636 T^{3} + 151334226289 T^{4} \))(\( 1 + 386 T + 152218 T^{2} + 150160562 T^{3} + 151334226289 T^{4} \))(\( 1 - 326 T + 789802 T^{2} - 126819542 T^{3} + 151334226289 T^{4} \))
$79$ (\( 1 - 1136 T + 493039 T^{2} \))(\( 1 - 896 T + 493039 T^{2} \))(\( 1 - 408 T + 994782 T^{2} - 201159912 T^{3} + 243087455521 T^{4} \))(\( 1 + 408 T - 143586 T^{2} + 201159912 T^{3} + 243087455521 T^{4} \))(\( 1 + 432 T + 602142 T^{2} + 212992848 T^{3} + 243087455521 T^{4} \))(\( 1 + 888 T + 1007454 T^{2} + 437818632 T^{3} + 243087455521 T^{4} \))(\( 1 - 1832 T + 1824478 T^{2} - 903247448 T^{3} + 243087455521 T^{4} \))
$83$ (\( 1 - 156 T + 571787 T^{2} \))(\( 1 - 436 T + 571787 T^{2} \))(\( 1 - 1136 T + 1169990 T^{2} - 649550032 T^{3} + 326940373369 T^{4} \))(\( 1 - 608 T + 1200710 T^{2} - 347646496 T^{3} + 326940373369 T^{4} \))(\( 1 + 584 T + 1172390 T^{2} + 333923608 T^{3} + 326940373369 T^{4} \))(\( 1 - 928 T + 600710 T^{2} - 530618336 T^{3} + 326940373369 T^{4} \))(\( 1 + 680 T + 744870 T^{2} + 388815160 T^{3} + 326940373369 T^{4} \))
$89$ (\( 1 + 1038 T + 704969 T^{2} \))(\( 1 + 1038 T + 704969 T^{2} \))(\( 1 - 36 T - 842170 T^{2} - 25378884 T^{3} + 496981290961 T^{4} \))(\( 1 + 1332 T + 1790774 T^{2} + 939018708 T^{3} + 496981290961 T^{4} \))(\( 1 + 1404 T + 1802390 T^{2} + 989776476 T^{3} + 496981290961 T^{4} \))(\( 1 - 588 T + 1495334 T^{2} - 414521772 T^{3} + 496981290961 T^{4} \))(\( 1 - 796 T + 411158 T^{2} - 561155324 T^{3} + 496981290961 T^{4} \))
$97$ (\( 1 - 1208 T + 912673 T^{2} \))(\( 1 + 702 T + 912673 T^{2} \))(\( 1 + 498 T + 1796754 T^{2} + 454511154 T^{3} + 832972004929 T^{4} \))(\( 1 + 2448 T + 3286542 T^{2} + 2234223504 T^{3} + 832972004929 T^{4} \))(\( 1 + 1188 T + 2161254 T^{2} + 1084255524 T^{3} + 832972004929 T^{4} \))(\( 1 - 522 T + 291282 T^{2} - 476415306 T^{3} + 832972004929 T^{4} \))(\( 1 + 670 T + 1145410 T^{2} + 611490910 T^{3} + 832972004929 T^{4} \))
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