Properties

Label 75.4.b
Level 75
Weight 4
Character orbit b
Rep. character \(\chi_{75}(49,\cdot)\)
Character field \(\Q\)
Dimension 8
Newform subspaces 3
Sturm bound 40
Trace bound 4

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Defining parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(40\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(75, [\chi])\).

Total New Old
Modular forms 36 8 28
Cusp forms 24 8 16
Eisenstein series 12 0 12

Trace form

\( 8q - 36q^{4} - 24q^{6} - 72q^{9} + O(q^{10}) \) \( 8q - 36q^{4} - 24q^{6} - 72q^{9} + 112q^{11} - 324q^{14} + 180q^{16} + 276q^{19} - 108q^{21} + 468q^{24} - 764q^{26} - 488q^{29} + 68q^{31} + 1088q^{34} + 324q^{36} + 204q^{39} + 1688q^{41} + 712q^{44} - 2232q^{46} - 1100q^{49} - 816q^{51} + 216q^{54} - 480q^{56} - 496q^{59} + 2340q^{61} - 3284q^{64} + 1488q^{66} - 504q^{69} + 416q^{71} - 1904q^{74} + 784q^{76} + 3280q^{79} + 648q^{81} - 936q^{84} - 1508q^{86} - 984q^{89} - 2492q^{91} + 824q^{94} - 2748q^{96} - 1008q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(75, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
75.4.b.a \(2\) \(4.425\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+3iq^{2}+3iq^{3}-q^{4}-9q^{6}+20iq^{7}+\cdots\)
75.4.b.b \(2\) \(4.425\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+iq^{2}-3iq^{3}+7q^{4}+3q^{6}-24iq^{7}+\cdots\)
75.4.b.c \(4\) \(4.425\) \(\Q(i, \sqrt{19})\) None \(0\) \(0\) \(0\) \(0\) \(q+(-\beta _{1}+\beta _{3})q^{2}-3\beta _{1}q^{3}+(-12+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(75, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(75, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 7 T^{2} + 64 T^{4} \))(\( 1 - 15 T^{2} + 64 T^{4} \))(\( 1 + 8 T^{2} + 68 T^{4} + 512 T^{6} + 4096 T^{8} \))
$3$ (\( 1 + 9 T^{2} \))(\( 1 + 9 T^{2} \))(\( ( 1 + 9 T^{2} )^{2} \))
$5$ 1
$7$ (\( 1 - 286 T^{2} + 117649 T^{4} \))(\( 1 - 110 T^{2} + 117649 T^{4} \))(\( 1 - 426 T^{2} + 75163 T^{4} - 50118474 T^{6} + 13841287201 T^{8} \))
$11$ (\( ( 1 + 24 T + 1331 T^{2} )^{2} \))(\( ( 1 - 52 T + 1331 T^{2} )^{2} \))(\( ( 1 - 28 T + 2554 T^{2} - 37268 T^{3} + 1771561 T^{4} )^{2} \))
$13$ (\( 1 + 1082 T^{2} + 4826809 T^{4} \))(\( 1 - 3910 T^{2} + 4826809 T^{4} \))(\( 1 + 1102 T^{2} + 8381283 T^{4} + 5319143518 T^{6} + 23298085122481 T^{8} \))
$17$ (\( 1 - 6910 T^{2} + 24137569 T^{4} \))(\( 1 - 9630 T^{2} + 24137569 T^{4} \))(\( 1 - 2140 T^{2} + 14277638 T^{4} - 51654397660 T^{6} + 582622237229761 T^{8} \))
$19$ (\( ( 1 - 124 T + 6859 T^{2} )^{2} \))(\( ( 1 - 20 T + 6859 T^{2} )^{2} \))(\( ( 1 + 6 T + 13423 T^{2} + 41154 T^{3} + 47045881 T^{4} )^{2} \))
$23$ (\( 1 - 9934 T^{2} + 148035889 T^{4} \))(\( 1 + 3890 T^{2} + 148035889 T^{4} \))(\( 1 - 34484 T^{2} + 545686278 T^{4} - 5104869596276 T^{6} + 21914624432020321 T^{8} \))
$29$ (\( ( 1 - 78 T + 24389 T^{2} )^{2} \))(\( ( 1 + 230 T + 24389 T^{2} )^{2} \))(\( ( 1 + 92 T + 35998 T^{2} + 2243788 T^{3} + 594823321 T^{4} )^{2} \))
$31$ (\( ( 1 - 200 T + 29791 T^{2} )^{2} \))(\( ( 1 + 288 T + 29791 T^{2} )^{2} \))(\( ( 1 - 122 T + 48407 T^{2} - 3634502 T^{3} + 887503681 T^{4} )^{2} \))
$37$ (\( 1 - 96406 T^{2} + 2565726409 T^{4} \))(\( 1 - 100150 T^{2} + 2565726409 T^{4} \))(\( 1 - 140396 T^{2} + 9176512758 T^{4} - 360217724917964 T^{6} + 6582952005840035281 T^{8} \))
$41$ (\( ( 1 - 330 T + 68921 T^{2} )^{2} \))(\( ( 1 - 122 T + 68921 T^{2} )^{2} \))(\( ( 1 - 392 T + 124882 T^{2} - 27017032 T^{3} + 4750104241 T^{4} )^{2} \))
$43$ (\( 1 - 150550 T^{2} + 6321363049 T^{4} \))(\( 1 - 123670 T^{2} + 6321363049 T^{4} \))(\( 1 - 79370 T^{2} + 14072890923 T^{4} - 501726585199130 T^{6} + 39959630797262576401 T^{8} \))
$47$ (\( 1 - 207070 T^{2} + 10779215329 T^{4} \))(\( 1 - 142110 T^{2} + 10779215329 T^{4} \))(\( 1 - 184180 T^{2} + 22560112358 T^{4} - 1985315879295220 T^{6} + \)\(11\!\cdots\!41\)\( T^{8} \))
$53$ (\( 1 - 95254 T^{2} + 22164361129 T^{4} \))(\( 1 - 183510 T^{2} + 22164361129 T^{4} \))(\( 1 - 206164 T^{2} + 44242847798 T^{4} - 4569493347799156 T^{6} + \)\(49\!\cdots\!41\)\( T^{8} \))
$59$ (\( ( 1 + 24 T + 205379 T^{2} )^{2} \))(\( ( 1 + 100 T + 205379 T^{2} )^{2} \))(\( ( 1 + 124 T + 336778 T^{2} + 25466996 T^{3} + 42180533641 T^{4} )^{2} \))
$61$ (\( ( 1 + 322 T + 226981 T^{2} )^{2} \))(\( ( 1 - 742 T + 226981 T^{2} )^{2} \))(\( ( 1 - 750 T + 535003 T^{2} - 170235750 T^{3} + 51520374361 T^{4} )^{2} \))
$67$ (\( 1 - 563110 T^{2} + 90458382169 T^{4} \))(\( 1 - 594470 T^{2} + 90458382169 T^{4} \))(\( 1 - 758970 T^{2} + 300574469563 T^{4} - 68655198314805930 T^{6} + \)\(81\!\cdots\!61\)\( T^{8} \))
$71$ (\( ( 1 + 288 T + 357911 T^{2} )^{2} \))(\( ( 1 + 328 T + 357911 T^{2} )^{2} \))(\( ( 1 - 824 T + 877966 T^{2} - 294918664 T^{3} + 128100283921 T^{4} )^{2} \))
$73$ (\( 1 - 593134 T^{2} + 151334226289 T^{4} \))(\( 1 - 776590 T^{2} + 151334226289 T^{4} \))(\( 1 - 1547804 T^{2} + 901578574758 T^{4} - 234235720787019356 T^{6} + \)\(22\!\cdots\!21\)\( T^{8} \))
$79$ (\( ( 1 - 520 T + 493039 T^{2} )^{2} \))(\( ( 1 - 240 T + 493039 T^{2} )^{2} \))(\( ( 1 - 880 T + 693278 T^{2} - 433874320 T^{3} + 243087455521 T^{4} )^{2} \))
$83$ (\( 1 - 1119238 T^{2} + 326940373369 T^{4} \))(\( 1 + 325370 T^{2} + 326940373369 T^{4} \))(\( 1 - 874148 T^{2} + 827869104438 T^{4} - 285794273499764612 T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \))
$89$ (\( ( 1 + 1026 T + 704969 T^{2} )^{2} \))(\( ( 1 + 330 T + 704969 T^{2} )^{2} \))(\( ( 1 - 864 T + 1202578 T^{2} - 609093216 T^{3} + 496981290961 T^{4} )^{2} \))
$97$ (\( 1 - 1743550 T^{2} + 832972004929 T^{4} \))(\( 1 - 1075390 T^{2} + 832972004929 T^{4} \))(\( ( 1 - 1553905 T^{2} + 832972004929 T^{4} )^{2} \))
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