Properties

Label 670.2.c
Level 670
Weight 2
Character orbit c
Rep. character \(\chi_{670}(269,\cdot)\)
Character field \(\Q\)
Dimension 34
Newform subspaces 2
Sturm bound 204
Trace bound 1

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Defining parameters

Level: \( N \) = \( 670 = 2 \cdot 5 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 670.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(204\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(670, [\chi])\).

Total New Old
Modular forms 106 34 72
Cusp forms 98 34 64
Eisenstein series 8 0 8

Trace form

\( 34q - 34q^{4} - 4q^{5} + 4q^{6} - 30q^{9} + O(q^{10}) \) \( 34q - 34q^{4} - 4q^{5} + 4q^{6} - 30q^{9} - 4q^{11} - 4q^{14} - 8q^{15} + 34q^{16} + 4q^{20} + 24q^{21} - 4q^{24} + 4q^{25} - 12q^{29} + 8q^{31} + 4q^{35} + 30q^{36} - 24q^{39} - 28q^{41} + 4q^{44} + 40q^{45} + 12q^{46} - 86q^{49} + 16q^{50} - 8q^{51} - 40q^{54} + 48q^{55} + 4q^{56} - 16q^{59} + 8q^{60} + 8q^{61} - 34q^{64} + 32q^{65} - 16q^{66} + 32q^{69} + 4q^{70} + 16q^{71} + 12q^{74} + 36q^{75} - 56q^{79} - 4q^{80} + 2q^{81} - 24q^{84} - 4q^{85} + 20q^{86} - 52q^{89} - 4q^{90} + 8q^{91} + 20q^{94} + 40q^{95} + 4q^{96} - 12q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(670, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
670.2.c.a \(10\) \(5.350\) 10.0.\(\cdots\).1 None \(0\) \(0\) \(-2\) \(0\) \(q+\beta _{3}q^{2}-\beta _{8}q^{3}-q^{4}+(\beta _{1}-\beta _{7})q^{5}+\cdots\)
670.2.c.b \(24\) \(5.350\) None \(0\) \(0\) \(-2\) \(0\)

Decomposition of \(S_{2}^{\mathrm{old}}(670, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(670, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(335, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + T^{2} )^{5} \))
$3$ (\( 1 - 18 T^{2} + 165 T^{4} - 1004 T^{6} + 4474 T^{8} - 15248 T^{10} + 40266 T^{12} - 81324 T^{14} + 120285 T^{16} - 118098 T^{18} + 59049 T^{20} \))
$5$ (\( 1 + 2 T + T^{2} + 8 T^{3} + 2 T^{4} - 52 T^{5} + 10 T^{6} + 200 T^{7} + 125 T^{8} + 1250 T^{9} + 3125 T^{10} \))
$7$ (\( 1 - 45 T^{2} + 1011 T^{4} - 14862 T^{6} + 157941 T^{8} - 1265687 T^{10} + 7739109 T^{12} - 35683662 T^{14} + 118943139 T^{16} - 259416045 T^{18} + 282475249 T^{20} \))
$11$ (\( ( 1 + 5 T + 41 T^{2} + 158 T^{3} + 777 T^{4} + 2275 T^{5} + 8547 T^{6} + 19118 T^{7} + 54571 T^{8} + 73205 T^{9} + 161051 T^{10} )^{2} \))
$13$ (\( 1 - 94 T^{2} + 4253 T^{4} - 122540 T^{6} + 2494726 T^{8} - 37505152 T^{10} + 421608694 T^{12} - 3499864940 T^{14} + 20528418677 T^{16} - 76678687774 T^{18} + 137858491849 T^{20} \))
$17$ (\( 1 - 82 T^{2} + 3789 T^{4} - 123128 T^{6} + 3017618 T^{8} - 57758216 T^{10} + 872091602 T^{12} - 10283773688 T^{14} + 91457248941 T^{16} - 572012110162 T^{18} + 2015993900449 T^{20} \))
$19$ (\( ( 1 - 16 T + 187 T^{2} - 1440 T^{3} + 9054 T^{4} - 43192 T^{5} + 172026 T^{6} - 519840 T^{7} + 1282633 T^{8} - 2085136 T^{9} + 2476099 T^{10} )^{2} \))
$23$ (\( 1 - 62 T^{2} + 1437 T^{4} - 23000 T^{6} + 937394 T^{8} - 31808148 T^{10} + 495881426 T^{12} - 6436343000 T^{14} + 212727572493 T^{16} - 4855281087422 T^{18} + 41426511213649 T^{20} \))
$29$ (\( ( 1 - 6 T + 101 T^{2} - 478 T^{3} + 4940 T^{4} - 18130 T^{5} + 143260 T^{6} - 401998 T^{7} + 2463289 T^{8} - 4243686 T^{9} + 20511149 T^{10} )^{2} \))
$31$ (\( ( 1 + 22 T + 303 T^{2} + 2872 T^{3} + 21804 T^{4} + 132454 T^{5} + 675924 T^{6} + 2759992 T^{7} + 9026673 T^{8} + 20317462 T^{9} + 28629151 T^{10} )^{2} \))
$37$ (\( 1 - 313 T^{2} + 45847 T^{4} - 4132514 T^{6} + 253562625 T^{8} - 11058842659 T^{10} + 347127233625 T^{12} - 7744996570754 T^{14} + 117630858673423 T^{16} - 1099406069077273 T^{18} + 4808584372417849 T^{20} \))
$41$ (\( ( 1 + 14 T + 191 T^{2} + 1564 T^{3} + 13928 T^{4} + 86898 T^{5} + 571048 T^{6} + 2629084 T^{7} + 13163911 T^{8} + 39560654 T^{9} + 115856201 T^{10} )^{2} \))
$43$ (\( 1 - 354 T^{2} + 58877 T^{4} - 6058604 T^{6} + 427212514 T^{8} - 21571299984 T^{10} + 789915938386 T^{12} - 20713161413804 T^{14} + 372182892235973 T^{16} - 4137622898270754 T^{18} + 21611482313284249 T^{20} \))
$47$ (\( 1 - 262 T^{2} + 33957 T^{4} - 2943524 T^{6} + 192523542 T^{8} - 10045496272 T^{10} + 425284504278 T^{12} - 14363458135844 T^{14} + 366029814926853 T^{16} - 6238557105381382 T^{18} + 52599132235830049 T^{20} \))
$53$ (\( 1 - 170 T^{2} + 12485 T^{4} - 655208 T^{6} + 35704230 T^{8} - 1969785216 T^{10} + 100293182070 T^{12} - 5169906275048 T^{14} + 276722048695565 T^{16} - 10584147369931370 T^{18} + 174887470365513049 T^{20} \))
$59$ (\( ( 1 - 14 T + 257 T^{2} - 2624 T^{3} + 26124 T^{4} - 210486 T^{5} + 1541316 T^{6} - 9134144 T^{7} + 52782403 T^{8} - 169643054 T^{9} + 714924299 T^{10} )^{2} \))
$61$ (\( ( 1 + 17 T + 343 T^{2} + 3856 T^{3} + 43731 T^{4} + 344465 T^{5} + 2667591 T^{6} + 14348176 T^{7} + 77854483 T^{8} + 235379297 T^{9} + 844596301 T^{10} )^{2} \))
$67$ (\( ( 1 + T^{2} )^{5} \))
$71$ (\( ( 1 - 21 T + 419 T^{2} - 5154 T^{3} + 59679 T^{4} - 516319 T^{5} + 4237209 T^{6} - 25981314 T^{7} + 149964709 T^{8} - 533645301 T^{9} + 1804229351 T^{10} )^{2} \))
$73$ (\( 1 - 562 T^{2} + 150869 T^{4} - 25463572 T^{6} + 2988190594 T^{8} - 254492780640 T^{10} + 15924067675426 T^{12} - 723120654376852 T^{14} + 22831643385995141 T^{16} - 453230571644473522 T^{18} + 4297625829703557649 T^{20} \))
$79$ (\( ( 1 - 16 T + 375 T^{2} - 3532 T^{3} + 49326 T^{4} - 346096 T^{5} + 3896754 T^{6} - 22043212 T^{7} + 184889625 T^{8} - 623201296 T^{9} + 3077056399 T^{10} )^{2} \))
$83$ (\( 1 + 43 T^{2} + 14155 T^{4} + 472430 T^{6} + 146527317 T^{8} + 6307431145 T^{10} + 1009426686813 T^{12} + 22420734590030 T^{14} + 4627840985038195 T^{16} + 96848565981978763 T^{18} + 15516041187205853449 T^{20} \))
$89$ (\( ( 1 - 11 T + 303 T^{2} - 3546 T^{3} + 46941 T^{4} - 445557 T^{5} + 4177749 T^{6} - 28087866 T^{7} + 213605607 T^{8} - 690164651 T^{9} + 5584059449 T^{10} )^{2} \))
$97$ (\( 1 - 677 T^{2} + 227007 T^{4} - 48829758 T^{6} + 7431063213 T^{8} - 834105913299 T^{10} + 69918873771117 T^{12} - 4322863367143998 T^{14} + 189090475922917503 T^{16} - 5305942543393202597 T^{18} + 73742412689492826049 T^{20} \))
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