Properties

Label 210.4.a
Level 210
Weight 4
Character orbit a
Rep. character \(\chi_{210}(1,\cdot)\)
Character field \(\Q\)
Dimension 12
Newform subspaces 11
Sturm bound 192
Trace bound 7

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Defining parameters

Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 210.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(192\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(210))\).

Total New Old
Modular forms 152 12 140
Cusp forms 136 12 124
Eisenstein series 16 0 16

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

\(2\)\(3\)\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(1\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(1\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(2\)
Plus space\(+\)\(8\)
Minus space\(-\)\(4\)

Trace form

\( 12q + 48q^{4} + 108q^{9} + O(q^{10}) \) \( 12q + 48q^{4} + 108q^{9} + 160q^{11} + 64q^{13} + 192q^{16} + 128q^{17} + 160q^{19} + 176q^{22} - 144q^{23} + 300q^{25} + 160q^{26} - 320q^{29} + 120q^{30} - 184q^{31} + 384q^{33} + 144q^{34} + 432q^{36} - 216q^{37} + 128q^{38} + 24q^{39} - 240q^{41} + 168q^{42} + 1104q^{43} + 640q^{44} + 256q^{46} - 1184q^{47} + 588q^{49} - 360q^{51} + 256q^{52} - 1968q^{53} - 360q^{55} + 456q^{57} - 96q^{58} - 32q^{59} + 272q^{61} + 128q^{62} + 768q^{64} + 640q^{65} - 144q^{66} - 704q^{67} + 512q^{68} + 960q^{69} + 280q^{70} - 1032q^{71} - 4192q^{73} + 624q^{74} + 640q^{76} + 96q^{78} + 16q^{79} + 972q^{81} + 768q^{82} - 4640q^{83} + 680q^{85} - 2992q^{86} - 1296q^{87} + 704q^{88} - 1184q^{89} - 840q^{91} - 576q^{92} - 72q^{93} + 752q^{94} - 40q^{95} + 1376q^{97} + 1440q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(210))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 5 7
210.4.a.a \(1\) \(12.390\) \(\Q\) None \(-2\) \(-3\) \(-5\) \(7\) \(+\) \(+\) \(+\) \(-\) \(q-2q^{2}-3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots\)
210.4.a.b \(1\) \(12.390\) \(\Q\) None \(-2\) \(-3\) \(5\) \(-7\) \(+\) \(+\) \(-\) \(+\) \(q-2q^{2}-3q^{3}+4q^{4}+5q^{5}+6q^{6}+\cdots\)
210.4.a.c \(1\) \(12.390\) \(\Q\) None \(-2\) \(-3\) \(5\) \(7\) \(+\) \(+\) \(-\) \(-\) \(q-2q^{2}-3q^{3}+4q^{4}+5q^{5}+6q^{6}+\cdots\)
210.4.a.d \(1\) \(12.390\) \(\Q\) None \(-2\) \(3\) \(-5\) \(-7\) \(+\) \(-\) \(+\) \(+\) \(q-2q^{2}+3q^{3}+4q^{4}-5q^{5}-6q^{6}+\cdots\)
210.4.a.e \(1\) \(12.390\) \(\Q\) None \(-2\) \(3\) \(-5\) \(7\) \(+\) \(-\) \(+\) \(-\) \(q-2q^{2}+3q^{3}+4q^{4}-5q^{5}-6q^{6}+\cdots\)
210.4.a.f \(1\) \(12.390\) \(\Q\) None \(-2\) \(3\) \(5\) \(-7\) \(+\) \(-\) \(-\) \(+\) \(q-2q^{2}+3q^{3}+4q^{4}+5q^{5}-6q^{6}+\cdots\)
210.4.a.g \(1\) \(12.390\) \(\Q\) None \(2\) \(-3\) \(-5\) \(-7\) \(-\) \(+\) \(+\) \(+\) \(q+2q^{2}-3q^{3}+4q^{4}-5q^{5}-6q^{6}+\cdots\)
210.4.a.h \(1\) \(12.390\) \(\Q\) None \(2\) \(-3\) \(-5\) \(7\) \(-\) \(+\) \(+\) \(-\) \(q+2q^{2}-3q^{3}+4q^{4}-5q^{5}-6q^{6}+\cdots\)
210.4.a.i \(1\) \(12.390\) \(\Q\) None \(2\) \(-3\) \(5\) \(-7\) \(-\) \(+\) \(-\) \(+\) \(q+2q^{2}-3q^{3}+4q^{4}+5q^{5}-6q^{6}+\cdots\)
210.4.a.j \(1\) \(12.390\) \(\Q\) None \(2\) \(3\) \(-5\) \(-7\) \(-\) \(-\) \(+\) \(+\) \(q+2q^{2}+3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots\)
210.4.a.k \(2\) \(12.390\) \(\Q(\sqrt{106}) \) None \(4\) \(6\) \(10\) \(14\) \(-\) \(-\) \(-\) \(-\) \(q+2q^{2}+3q^{3}+4q^{4}+5q^{5}+6q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(210))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(210)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T \))(\( 1 + 2 T \))(\( 1 + 2 T \))(\( 1 + 2 T \))(\( 1 + 2 T \))(\( 1 + 2 T \))(\( 1 - 2 T \))(\( 1 - 2 T \))(\( 1 - 2 T \))(\( 1 - 2 T \))(\( ( 1 - 2 T )^{2} \))
$3$ (\( 1 + 3 T \))(\( 1 + 3 T \))(\( 1 + 3 T \))(\( 1 - 3 T \))(\( 1 - 3 T \))(\( 1 - 3 T \))(\( 1 + 3 T \))(\( 1 + 3 T \))(\( 1 + 3 T \))(\( 1 - 3 T \))(\( ( 1 - 3 T )^{2} \))
$5$ (\( 1 + 5 T \))(\( 1 - 5 T \))(\( 1 - 5 T \))(\( 1 + 5 T \))(\( 1 + 5 T \))(\( 1 - 5 T \))(\( 1 + 5 T \))(\( 1 + 5 T \))(\( 1 - 5 T \))(\( 1 + 5 T \))(\( ( 1 - 5 T )^{2} \))
$7$ (\( 1 - 7 T \))(\( 1 + 7 T \))(\( 1 - 7 T \))(\( 1 + 7 T \))(\( 1 - 7 T \))(\( 1 + 7 T \))(\( 1 + 7 T \))(\( 1 - 7 T \))(\( 1 + 7 T \))(\( 1 + 7 T \))(\( ( 1 - 7 T )^{2} \))
$11$ (\( 1 - 12 T + 1331 T^{2} \))(\( 1 + 44 T + 1331 T^{2} \))(\( 1 - 12 T + 1331 T^{2} \))(\( 1 - 28 T + 1331 T^{2} \))(\( 1 + 1331 T^{2} \))(\( 1 - 28 T + 1331 T^{2} \))(\( 1 + 4 T + 1331 T^{2} \))(\( 1 - 24 T + 1331 T^{2} \))(\( 1 - 16 T + 1331 T^{2} \))(\( 1 - 56 T + 1331 T^{2} \))(\( 1 - 32 T + 1222 T^{2} - 42592 T^{3} + 1771561 T^{4} \))
$13$ (\( 1 - 2 T + 2197 T^{2} \))(\( 1 - 54 T + 2197 T^{2} \))(\( 1 + 58 T + 2197 T^{2} \))(\( 1 + 86 T + 2197 T^{2} \))(\( 1 - 26 T + 2197 T^{2} \))(\( 1 - 54 T + 2197 T^{2} \))(\( 1 + 42 T + 2197 T^{2} \))(\( 1 - 14 T + 2197 T^{2} \))(\( 1 - 58 T + 2197 T^{2} \))(\( 1 - 54 T + 2197 T^{2} \))(\( 1 + 12 T - 2354 T^{2} + 26364 T^{3} + 4826809 T^{4} \))
$17$ (\( 1 + 18 T + 4913 T^{2} \))(\( 1 - 98 T + 4913 T^{2} \))(\( 1 - 42 T + 4913 T^{2} \))(\( 1 + 66 T + 4913 T^{2} \))(\( 1 - 18 T + 4913 T^{2} \))(\( 1 + 46 T + 4913 T^{2} \))(\( 1 + 86 T + 4913 T^{2} \))(\( 1 - 54 T + 4913 T^{2} \))(\( 1 - 34 T + 4913 T^{2} \))(\( 1 - 94 T + 4913 T^{2} \))(\( 1 - 4 T + 8134 T^{2} - 19652 T^{3} + 24137569 T^{4} \))
$19$ (\( 1 - 56 T + 6859 T^{2} \))(\( 1 + 60 T + 6859 T^{2} \))(\( 1 + 4 T + 6859 T^{2} \))(\( 1 + 48 T + 6859 T^{2} \))(\( 1 - 92 T + 6859 T^{2} \))(\( 1 - 12 T + 6859 T^{2} \))(\( 1 + 96 T + 6859 T^{2} \))(\( 1 - 44 T + 6859 T^{2} \))(\( 1 - 64 T + 6859 T^{2} \))(\( 1 - 36 T + 6859 T^{2} \))(\( 1 - 64 T - 522 T^{2} - 438976 T^{3} + 47045881 T^{4} \))
$23$ (\( 1 + 156 T + 12167 T^{2} \))(\( 1 + 144 T + 12167 T^{2} \))(\( 1 - 24 T + 12167 T^{2} \))(\( 1 - 140 T + 12167 T^{2} \))(\( 1 + 12167 T^{2} \))(\( 1 + 12167 T^{2} \))(\( 1 + 96 T + 12167 T^{2} \))(\( 1 - 156 T + 12167 T^{2} \))(\( 1 + 16 T + 12167 T^{2} \))(\( 1 + 84 T + 12167 T^{2} \))(\( 1 - 32 T + 22894 T^{2} - 389344 T^{3} + 148035889 T^{4} \))
$29$ (\( 1 + 186 T + 24389 T^{2} \))(\( 1 + 210 T + 24389 T^{2} \))(\( 1 - 294 T + 24389 T^{2} \))(\( 1 + 34 T + 24389 T^{2} \))(\( 1 + 6 T + 24389 T^{2} \))(\( 1 - 6 T + 24389 T^{2} \))(\( 1 + 78 T + 24389 T^{2} \))(\( 1 - 174 T + 24389 T^{2} \))(\( 1 - 62 T + 24389 T^{2} \))(\( 1 + 258 T + 24389 T^{2} \))(\( 1 + 84 T + 35278 T^{2} + 2048676 T^{3} + 594823321 T^{4} \))
$31$ (\( 1 + 52 T + 29791 T^{2} \))(\( 1 + 208 T + 29791 T^{2} \))(\( 1 - 128 T + 29791 T^{2} \))(\( 1 + 284 T + 29791 T^{2} \))(\( 1 + 4 T + 29791 T^{2} \))(\( 1 - 296 T + 29791 T^{2} \))(\( 1 - 80 T + 29791 T^{2} \))(\( 1 + 88 T + 29791 T^{2} \))(\( 1 - 60 T + 29791 T^{2} \))(\( 1 + 40 T + 29791 T^{2} \))(\( 1 + 72 T + 54094 T^{2} + 2144952 T^{3} + 887503681 T^{4} \))
$37$ (\( 1 + 178 T + 50653 T^{2} \))(\( 1 + 226 T + 50653 T^{2} \))(\( 1 + 58 T + 50653 T^{2} \))(\( 1 + 346 T + 50653 T^{2} \))(\( 1 - 410 T + 50653 T^{2} \))(\( 1 - 134 T + 50653 T^{2} \))(\( 1 - 50 T + 50653 T^{2} \))(\( 1 + 34 T + 50653 T^{2} \))(\( 1 - 150 T + 50653 T^{2} \))(\( 1 + 178 T + 50653 T^{2} \))(\( 1 - 60 T + 100510 T^{2} - 3039180 T^{3} + 2565726409 T^{4} \))
$41$ (\( 1 + 138 T + 68921 T^{2} \))(\( 1 + 502 T + 68921 T^{2} \))(\( 1 - 282 T + 68921 T^{2} \))(\( 1 + 274 T + 68921 T^{2} \))(\( 1 - 174 T + 68921 T^{2} \))(\( 1 - 146 T + 68921 T^{2} \))(\( 1 + 26 T + 68921 T^{2} \))(\( 1 + 138 T + 68921 T^{2} \))(\( 1 - 474 T + 68921 T^{2} \))(\( 1 + 146 T + 68921 T^{2} \))(\( 1 + 92 T + 31414 T^{2} + 6340732 T^{3} + 4750104241 T^{4} \))
$43$ (\( 1 + 412 T + 79507 T^{2} \))(\( 1 - 484 T + 79507 T^{2} \))(\( 1 - 428 T + 79507 T^{2} \))(\( 1 + 4 T + 79507 T^{2} \))(\( 1 - 248 T + 79507 T^{2} \))(\( 1 - 556 T + 79507 T^{2} \))(\( 1 + 32 T + 79507 T^{2} \))(\( 1 - 164 T + 79507 T^{2} \))(\( 1 + 292 T + 79507 T^{2} \))(\( 1 - 148 T + 79507 T^{2} \))(\( 1 + 184 T + 165782 T^{2} + 14629288 T^{3} + 6321363049 T^{4} \))
$47$ (\( 1 + 456 T + 103823 T^{2} \))(\( 1 + 232 T + 103823 T^{2} \))(\( 1 - 384 T + 103823 T^{2} \))(\( 1 + 448 T + 103823 T^{2} \))(\( 1 - 420 T + 103823 T^{2} \))(\( 1 + 448 T + 103823 T^{2} \))(\( 1 + 20 T + 103823 T^{2} \))(\( 1 + 216 T + 103823 T^{2} \))(\( 1 - 240 T + 103823 T^{2} \))(\( 1 + 200 T + 103823 T^{2} \))(\( 1 + 208 T + 176062 T^{2} + 21595184 T^{3} + 10779215329 T^{4} \))
$53$ (\( 1 + 198 T + 148877 T^{2} \))(\( 1 + 530 T + 148877 T^{2} \))(\( 1 + 138 T + 148877 T^{2} \))(\( 1 + 94 T + 148877 T^{2} \))(\( 1 - 102 T + 148877 T^{2} \))(\( 1 - 46 T + 148877 T^{2} \))(\( 1 + 382 T + 148877 T^{2} \))(\( 1 - 318 T + 148877 T^{2} \))(\( 1 + 662 T + 148877 T^{2} \))(\( 1 + 130 T + 148877 T^{2} \))(\( 1 + 300 T + 304990 T^{2} + 44663100 T^{3} + 22164361129 T^{4} \))
$59$ (\( 1 - 348 T + 205379 T^{2} \))(\( 1 + 764 T + 205379 T^{2} \))(\( 1 - 468 T + 205379 T^{2} \))(\( 1 - 308 T + 205379 T^{2} \))(\( 1 + 588 T + 205379 T^{2} \))(\( 1 - 748 T + 205379 T^{2} \))(\( 1 - 356 T + 205379 T^{2} \))(\( 1 + 204 T + 205379 T^{2} \))(\( 1 + 324 T + 205379 T^{2} \))(\( 1 - 188 T + 205379 T^{2} \))(\( 1 + 568 T + 464278 T^{2} + 116655272 T^{3} + 42180533641 T^{4} \))
$61$ (\( 1 - 110 T + 226981 T^{2} \))(\( 1 - 814 T + 226981 T^{2} \))(\( 1 + 250 T + 226981 T^{2} \))(\( 1 - 510 T + 226981 T^{2} \))(\( 1 - 650 T + 226981 T^{2} \))(\( 1 + 50 T + 226981 T^{2} \))(\( 1 + 134 T + 226981 T^{2} \))(\( 1 + 442 T + 226981 T^{2} \))(\( 1 + 514 T + 226981 T^{2} \))(\( 1 - 94 T + 226981 T^{2} \))(\( 1 + 516 T + 478126 T^{2} + 117122196 T^{3} + 51520374361 T^{4} \))
$67$ (\( 1 + 196 T + 300763 T^{2} \))(\( 1 - 60 T + 300763 T^{2} \))(\( 1 + 556 T + 300763 T^{2} \))(\( 1 + 156 T + 300763 T^{2} \))(\( 1 - 152 T + 300763 T^{2} \))(\( 1 + 156 T + 300763 T^{2} \))(\( 1 - 888 T + 300763 T^{2} \))(\( 1 + 316 T + 300763 T^{2} \))(\( 1 + 372 T + 300763 T^{2} \))(\( 1 + 444 T + 300763 T^{2} \))(\( 1 - 392 T + 597542 T^{2} - 117899096 T^{3} + 90458382169 T^{4} \))
$71$ (\( 1 + 936 T + 357911 T^{2} \))(\( 1 - 848 T + 357911 T^{2} \))(\( 1 - 624 T + 357911 T^{2} \))(\( 1 - 336 T + 357911 T^{2} \))(\( 1 + 168 T + 357911 T^{2} \))(\( 1 + 1024 T + 357911 T^{2} \))(\( 1 - 868 T + 357911 T^{2} \))(\( 1 + 252 T + 357911 T^{2} \))(\( 1 + 412 T + 357911 T^{2} \))(\( 1 - 532 T + 357911 T^{2} \))(\( 1 + 1448 T + 1212862 T^{2} + 518255128 T^{3} + 128100283921 T^{4} \))
$73$ (\( 1 - 542 T + 389017 T^{2} \))(\( 1 + 958 T + 389017 T^{2} \))(\( 1 + 958 T + 389017 T^{2} \))(\( 1 + 1170 T + 389017 T^{2} \))(\( 1 + 610 T + 389017 T^{2} \))(\( 1 + 310 T + 389017 T^{2} \))(\( 1 + 70 T + 389017 T^{2} \))(\( 1 - 98 T + 389017 T^{2} \))(\( 1 + 770 T + 389017 T^{2} \))(\( 1 - 770 T + 389017 T^{2} \))(\( 1 + 756 T + 751318 T^{2} + 294096852 T^{3} + 151334226289 T^{4} \))
$79$ (\( 1 - 992 T + 493039 T^{2} \))(\( 1 + 152 T + 493039 T^{2} \))(\( 1 - 632 T + 493039 T^{2} \))(\( 1 - 16 T + 493039 T^{2} \))(\( 1 + 1048 T + 493039 T^{2} \))(\( 1 - 856 T + 493039 T^{2} \))(\( 1 - 400 T + 493039 T^{2} \))(\( 1 + 1000 T + 493039 T^{2} \))(\( 1 + 560 T + 493039 T^{2} \))(\( 1 + 536 T + 493039 T^{2} \))(\( 1 - 416 T + 1022558 T^{2} - 205104224 T^{3} + 243087455521 T^{4} \))
$83$ (\( 1 + 276 T + 571787 T^{2} \))(\( 1 - 308 T + 571787 T^{2} \))(\( 1 - 84 T + 571787 T^{2} \))(\( 1 - 772 T + 571787 T^{2} \))(\( 1 + 684 T + 571787 T^{2} \))(\( 1 + 628 T + 571787 T^{2} \))(\( 1 + 1052 T + 571787 T^{2} \))(\( 1 - 516 T + 571787 T^{2} \))(\( 1 + 852 T + 571787 T^{2} \))(\( 1 + 1076 T + 571787 T^{2} \))(\( 1 + 1752 T + 1666726 T^{2} + 1001770824 T^{3} + 326940373369 T^{4} \))
$89$ (\( 1 - 630 T + 704969 T^{2} \))(\( 1 + 1094 T + 704969 T^{2} \))(\( 1 - 810 T + 704969 T^{2} \))(\( 1 - 1630 T + 704969 T^{2} \))(\( 1 + 834 T + 704969 T^{2} \))(\( 1 + 590 T + 704969 T^{2} \))(\( 1 + 634 T + 704969 T^{2} \))(\( 1 + 522 T + 704969 T^{2} \))(\( 1 - 1466 T + 704969 T^{2} \))(\( 1 + 1090 T + 704969 T^{2} \))(\( 1 + 956 T + 1529878 T^{2} + 673950364 T^{3} + 496981290961 T^{4} \))
$97$ (\( 1 - 110 T + 912673 T^{2} \))(\( 1 - 554 T + 912673 T^{2} \))(\( 1 + 790 T + 912673 T^{2} \))(\( 1 - 110 T + 912673 T^{2} \))(\( 1 - 110 T + 912673 T^{2} \))(\( 1 + 1390 T + 912673 T^{2} \))(\( 1 - 1202 T + 912673 T^{2} \))(\( 1 + 310 T + 912673 T^{2} \))(\( 1 + 178 T + 912673 T^{2} \))(\( 1 - 1274 T + 912673 T^{2} \))(\( 1 - 684 T + 795814 T^{2} - 624268332 T^{3} + 832972004929 T^{4} \))
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