Properties

Label 847.2.p
Level 847
Weight 2
Character orbit p
Rep. character \(\chi_{847}(76,\cdot)\)
Character field \(\Q(\zeta_{22})\)
Dimension 860
Newform subspaces 2
Sturm bound 176
Trace bound 1

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Defining parameters

Level: \( N \) = \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 847.p (of order \(22\) and degree \(10\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 847 \)
Character field: \(\Q(\zeta_{22})\)
Newform subspaces: \( 2 \)
Sturm bound: \(176\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(847, [\chi])\).

Total New Old
Modular forms 900 900 0
Cusp forms 860 860 0
Eisenstein series 40 40 0

Trace form

\( 860q - 22q^{2} + 68q^{4} - 11q^{7} - 22q^{8} - 864q^{9} + O(q^{10}) \) \( 860q - 22q^{2} + 68q^{4} - 11q^{7} - 22q^{8} - 864q^{9} - 36q^{11} - 27q^{14} + 46q^{15} - 100q^{16} + 22q^{21} + 28q^{23} + 48q^{25} - 11q^{28} - 22q^{29} + 44q^{30} - 22q^{32} - 11q^{35} - 114q^{36} + 52q^{37} + 44q^{39} - 4q^{42} - 22q^{43} - 13q^{44} - 22q^{46} - 39q^{49} - 22q^{50} + 132q^{51} - 82q^{53} - 71q^{56} + 44q^{57} - 11q^{58} + 32q^{60} + 198q^{64} - 22q^{65} - 2q^{67} + 21q^{70} - 10q^{71} - 132q^{72} - 22q^{74} - 145q^{77} + 108q^{78} - 110q^{79} + 812q^{81} + 22q^{84} + 88q^{85} - 18q^{86} + 26q^{88} + 70q^{91} + 362q^{92} - 74q^{93} + 66q^{95} + 77q^{98} - 80q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(847, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
847.2.p.a \(20\) \(6.763\) 20.0.\(\cdots\).2 \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) \(q+(\beta _{2}-\beta _{10}+\beta _{19})q^{2}+(-\beta _{5}-2\beta _{8}+\cdots)q^{4}+\cdots\)
847.2.p.b \(840\) \(6.763\) None \(-22\) \(0\) \(0\) \(-11\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - T - T^{2} + 3 T^{3} - T^{4} - 5 T^{5} + 7 T^{6} + 3 T^{7} - 17 T^{8} + 11 T^{9} + 23 T^{10} + 22 T^{11} - 68 T^{12} + 24 T^{13} + 112 T^{14} - 160 T^{15} - 64 T^{16} + 384 T^{17} - 256 T^{18} - 512 T^{19} + 1024 T^{20} )( 1 + T - T^{2} - 3 T^{3} - T^{4} + 5 T^{5} + 7 T^{6} - 3 T^{7} - 17 T^{8} - 11 T^{9} + 23 T^{10} - 22 T^{11} - 68 T^{12} - 24 T^{13} + 112 T^{14} + 160 T^{15} - 64 T^{16} - 384 T^{17} - 256 T^{18} + 512 T^{19} + 1024 T^{20} ) \))
$3$ (\( ( 1 - 3 T^{2} )^{20} \))
$5$ (\( ( 1 + 5 T^{2} + 25 T^{4} + 125 T^{6} + 625 T^{8} + 3125 T^{10} + 15625 T^{12} + 78125 T^{14} + 390625 T^{16} + 1953125 T^{18} + 9765625 T^{20} )^{2} \))
$7$ (\( 1 - 7 T^{2} + 49 T^{4} - 343 T^{6} + 2401 T^{8} - 16807 T^{10} + 117649 T^{12} - 823543 T^{14} + 5764801 T^{16} - 40353607 T^{18} + 282475249 T^{20} \))
$11$ (\( 1 + 4 T + 5 T^{2} - 24 T^{3} - 151 T^{4} - 340 T^{5} + 301 T^{6} + 4944 T^{7} + 16465 T^{8} + 11476 T^{9} - 135211 T^{10} + 126236 T^{11} + 1992265 T^{12} + 6580464 T^{13} + 4406941 T^{14} - 54757340 T^{15} - 267505711 T^{16} - 467692104 T^{17} + 1071794405 T^{18} + 9431790764 T^{19} + 25937424601 T^{20} \))
$13$ (\( ( 1 - 13 T^{2} + 169 T^{4} - 2197 T^{6} + 28561 T^{8} - 371293 T^{10} + 4826809 T^{12} - 62748517 T^{14} + 815730721 T^{16} - 10604499373 T^{18} + 137858491849 T^{20} )^{2} \))
$17$ (\( ( 1 - 17 T^{2} + 289 T^{4} - 4913 T^{6} + 83521 T^{8} - 1419857 T^{10} + 24137569 T^{12} - 410338673 T^{14} + 6975757441 T^{16} - 118587876497 T^{18} + 2015993900449 T^{20} )^{2} \))
$19$ (\( ( 1 - 19 T^{2} + 361 T^{4} - 6859 T^{6} + 130321 T^{8} - 2476099 T^{10} + 47045881 T^{12} - 893871739 T^{14} + 16983563041 T^{16} - 322687697779 T^{18} + 6131066257801 T^{20} )^{2} \))
$23$ (\( ( 1 + 8 T + 23 T^{2} )^{10}( 1 - 8 T + 41 T^{2} - 144 T^{3} + 209 T^{4} + 1640 T^{5} - 17927 T^{6} + 105696 T^{7} - 433247 T^{8} + 1034968 T^{9} + 1684937 T^{10} + 23804264 T^{11} - 229187663 T^{12} + 1286003232 T^{13} - 5016709607 T^{14} + 10555602520 T^{15} + 30939500801 T^{16} - 490294864368 T^{17} + 3210750396521 T^{18} - 14409221291704 T^{19} + 41426511213649 T^{20} ) \))
$29$ (\( ( 1 - 2 T - 25 T^{2} + 108 T^{3} + 509 T^{4} - 4150 T^{5} - 6461 T^{6} + 133272 T^{7} - 79175 T^{8} - 3706538 T^{9} + 9709151 T^{10} - 107489602 T^{11} - 66586175 T^{12} + 3250370808 T^{13} - 4569742541 T^{14} - 85121268350 T^{15} + 302765070389 T^{16} + 1862986641372 T^{17} - 12506160324025 T^{18} - 29014291951738 T^{19} + 420707233300201 T^{20} )( 1 + 2 T - 25 T^{2} - 108 T^{3} + 509 T^{4} + 4150 T^{5} - 6461 T^{6} - 133272 T^{7} - 79175 T^{8} + 3706538 T^{9} + 9709151 T^{10} + 107489602 T^{11} - 66586175 T^{12} - 3250370808 T^{13} - 4569742541 T^{14} + 85121268350 T^{15} + 302765070389 T^{16} - 1862986641372 T^{17} - 12506160324025 T^{18} + 29014291951738 T^{19} + 420707233300201 T^{20} ) \))
$31$ (\( ( 1 + 31 T^{2} + 961 T^{4} + 29791 T^{6} + 923521 T^{8} + 28629151 T^{10} + 887503681 T^{12} + 27512614111 T^{14} + 852891037441 T^{16} + 26439622160671 T^{18} + 819628286980801 T^{20} )^{2} \))
$37$ (\( ( 1 - 6 T - T^{2} + 228 T^{3} - 1331 T^{4} - 450 T^{5} + 51947 T^{6} - 295032 T^{7} - 151847 T^{8} + 11827266 T^{9} - 65345257 T^{10} + 437608842 T^{11} - 207878543 T^{12} - 14944255896 T^{13} + 97357041467 T^{14} - 31204780650 T^{15} - 3414981850379 T^{16} + 21644467986324 T^{17} - 3512479453921 T^{18} - 779770438770462 T^{19} + 4808584372417849 T^{20} )^{2} \))
$41$ (\( ( 1 - 41 T^{2} + 1681 T^{4} - 68921 T^{6} + 2825761 T^{8} - 115856201 T^{10} + 4750104241 T^{12} - 194754273881 T^{14} + 7984925229121 T^{16} - 327381934393961 T^{18} + 13422659310152401 T^{20} )^{2} \))
$43$ (\( ( 1 - 12 T + 101 T^{2} - 696 T^{3} + 4009 T^{4} - 18180 T^{5} + 45773 T^{6} + 232464 T^{7} - 4757807 T^{8} + 47097732 T^{9} - 360587083 T^{10} + 2025202476 T^{11} - 8797185143 T^{12} + 18482515248 T^{13} + 156488778173 T^{14} - 2672613493740 T^{15} + 25342344463441 T^{16} - 189185753330472 T^{17} + 1180508228037701 T^{18} - 6031111343242116 T^{19} + 21611482313284249 T^{20} )( 1 + 12 T + 101 T^{2} + 696 T^{3} + 4009 T^{4} + 18180 T^{5} + 45773 T^{6} - 232464 T^{7} - 4757807 T^{8} - 47097732 T^{9} - 360587083 T^{10} - 2025202476 T^{11} - 8797185143 T^{12} - 18482515248 T^{13} + 156488778173 T^{14} + 2672613493740 T^{15} + 25342344463441 T^{16} + 189185753330472 T^{17} + 1180508228037701 T^{18} + 6031111343242116 T^{19} + 21611482313284249 T^{20} ) \))
$47$ (\( ( 1 + 47 T^{2} + 2209 T^{4} + 103823 T^{6} + 4879681 T^{8} + 229345007 T^{10} + 10779215329 T^{12} + 506623120463 T^{14} + 23811286661761 T^{16} + 1119130473102767 T^{18} + 52599132235830049 T^{20} )^{2} \))
$53$ (\( ( 1 + 10 T + 47 T^{2} - 60 T^{3} - 3091 T^{4} - 27730 T^{5} - 113477 T^{6} + 334920 T^{7} + 9363481 T^{8} + 75884050 T^{9} + 262576007 T^{10} + 4021854650 T^{11} + 26302018129 T^{12} + 49861884840 T^{13} - 895388112437 T^{14} - 11596561020890 T^{15} - 68510040249739 T^{16} - 70482668390220 T^{17} + 2926205449333967 T^{18} + 32997635918021330 T^{19} + 174887470365513049 T^{20} )^{2} \))
$59$ (\( ( 1 + 59 T^{2} + 3481 T^{4} + 205379 T^{6} + 12117361 T^{8} + 714924299 T^{10} + 42180533641 T^{12} + 2488651484819 T^{14} + 146830437604321 T^{16} + 8662995818654939 T^{18} + 511116753300641401 T^{20} )^{2} \))
$61$ (\( ( 1 - 61 T^{2} + 3721 T^{4} - 226981 T^{6} + 13845841 T^{8} - 844596301 T^{10} + 51520374361 T^{12} - 3142742836021 T^{14} + 191707312997281 T^{16} - 11694146092834141 T^{18} + 713342911662882601 T^{20} )^{2} \))
$67$ (\( ( 1 - 4 T - 51 T^{2} + 472 T^{3} + 1529 T^{4} - 37740 T^{5} + 48517 T^{6} + 2334512 T^{7} - 12588687 T^{8} - 106057556 T^{9} + 1267672253 T^{10} - 7105856252 T^{11} - 56510615943 T^{12} + 702134832656 T^{13} + 977671937557 T^{14} - 50953721538180 T^{15} + 138310866336401 T^{16} + 2860655877712456 T^{17} - 20709451555388691 T^{18} - 108826137585179788 T^{19} + 1822837804551761449 T^{20} )^{2} \))
$71$ (\( ( 1 - 16 T + 185 T^{2} - 1824 T^{3} + 16049 T^{4} - 127280 T^{5} + 897001 T^{6} - 5315136 T^{7} + 21355105 T^{8} + 35692976 T^{9} - 2087300071 T^{10} + 2534201296 T^{11} + 107651084305 T^{12} - 1902345640896 T^{13} + 22794303268681 T^{14} - 229642311795280 T^{15} + 2055881456648129 T^{16} - 16589499168905184 T^{17} + 119464403280465785 T^{18} - 733576011495184496 T^{19} + 3255243551009881201 T^{20} )^{2} \))
$73$ (\( ( 1 - 73 T^{2} + 5329 T^{4} - 389017 T^{6} + 28398241 T^{8} - 2073071593 T^{10} + 151334226289 T^{12} - 11047398519097 T^{14} + 806460091894081 T^{16} - 58871586708267913 T^{18} + 4297625829703557649 T^{20} )^{2} \))
$79$ (\( ( 1 - 8 T - 15 T^{2} + 752 T^{3} - 4831 T^{4} - 20760 T^{5} + 547729 T^{6} - 2741792 T^{7} - 21336255 T^{8} + 387291608 T^{9} - 1412768719 T^{10} + 30596037032 T^{11} - 133159567455 T^{12} - 1351810385888 T^{13} + 21334088916049 T^{14} - 63879690843240 T^{15} - 1174355497621951 T^{16} + 14441339557591568 T^{17} - 22756632148598415 T^{18} - 958812767860946552 T^{19} + 9468276082626847201 T^{20} )( 1 + 8 T - 15 T^{2} - 752 T^{3} - 4831 T^{4} + 20760 T^{5} + 547729 T^{6} + 2741792 T^{7} - 21336255 T^{8} - 387291608 T^{9} - 1412768719 T^{10} - 30596037032 T^{11} - 133159567455 T^{12} + 1351810385888 T^{13} + 21334088916049 T^{14} + 63879690843240 T^{15} - 1174355497621951 T^{16} - 14441339557591568 T^{17} - 22756632148598415 T^{18} + 958812767860946552 T^{19} + 9468276082626847201 T^{20} ) \))
$83$ (\( ( 1 - 83 T^{2} + 6889 T^{4} - 571787 T^{6} + 47458321 T^{8} - 3939040643 T^{10} + 326940373369 T^{12} - 27136050989627 T^{14} + 2252292232139041 T^{16} - 186940255267540403 T^{18} + 15516041187205853449 T^{20} )^{2} \))
$89$ (\( ( 1 + 89 T^{2} + 7921 T^{4} + 704969 T^{6} + 62742241 T^{8} + 5584059449 T^{10} + 496981290961 T^{12} + 44231334895529 T^{14} + 3936588805702081 T^{16} + 350356403707485209 T^{18} + 31181719929966183601 T^{20} )^{2} \))
$97$ (\( ( 1 + 97 T^{2} + 9409 T^{4} + 912673 T^{6} + 88529281 T^{8} + 8587340257 T^{10} + 832972004929 T^{12} + 80798284478113 T^{14} + 7837433594376961 T^{16} + 760231058654565217 T^{18} + 73742412689492826049 T^{20} )^{2} \))
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