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Properties

Label 25.2.d
Level 25
Weight 2
Character orbit d
Rep. character \(\chi_{25}(6,\cdot)\)
Character field \(\Q(\zeta_{5})\)
Dimension 4
Newform subspaces 1
Sturm bound 5
Trace bound 0

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Defining parameters

Level: \( N \) = \( 25 = 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 25.d (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 25 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 1 \)
Sturm bound: \(5\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(25, [\chi])\).

Total New Old
Modular forms 12 12 0
Cusp forms 4 4 0
Eisenstein series 8 8 0

Trace form

\( 4q - 2q^{2} - q^{3} - 2q^{4} - 5q^{5} + 3q^{6} - 2q^{7} - 5q^{8} + 2q^{9} + O(q^{10}) \) \( 4q - 2q^{2} - q^{3} - 2q^{4} - 5q^{5} + 3q^{6} - 2q^{7} - 5q^{8} + 2q^{9} + 10q^{10} - 2q^{11} + 3q^{12} + 9q^{13} + q^{14} - 5q^{15} - 6q^{16} + 8q^{17} + 4q^{18} - 5q^{19} + 5q^{20} - 2q^{21} - 14q^{22} - 11q^{23} - 5q^{25} - 12q^{26} + 5q^{27} + q^{28} + 5q^{29} - 5q^{30} + 3q^{31} + 18q^{32} + 8q^{33} + 6q^{34} + 5q^{35} - 6q^{36} - 7q^{37} + 5q^{38} + 9q^{39} + 5q^{40} + 8q^{41} + q^{42} - 6q^{43} + 6q^{44} - 10q^{45} + 13q^{46} - 2q^{47} - 6q^{48} - 22q^{49} - 10q^{50} - 12q^{51} - 12q^{52} + 9q^{53} - 15q^{54} + 20q^{55} - 10q^{57} - 10q^{58} + 13q^{61} + 6q^{62} - 6q^{63} + 3q^{64} + 6q^{66} - 2q^{67} - 4q^{68} - q^{69} + 8q^{71} + 10q^{72} + 9q^{73} + 6q^{74} + 20q^{75} + 20q^{76} - 4q^{77} + 3q^{78} + 15q^{79} - 15q^{80} - q^{81} - 4q^{82} + 9q^{83} - 4q^{84} - 20q^{85} + 3q^{86} - 10q^{88} - 20q^{89} - 12q^{91} - 12q^{92} - 12q^{93} + q^{94} - 5q^{95} - 2q^{96} + 8q^{97} + 11q^{98} + 24q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(25, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
25.2.d.a \(4\) \(0.200\) \(\Q(\zeta_{10})\) None \(-2\) \(-1\) \(-5\) \(-2\) \(q+(-\zeta_{10}+\zeta_{10}^{2})q^{2}-\zeta_{10}^{3}q^{3}+(-1+\cdots)q^{4}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 2 T^{2} + 5 T^{3} + 11 T^{4} + 10 T^{5} + 8 T^{6} + 16 T^{7} + 16 T^{8} \)
$3$ \( 1 + T - 2 T^{2} - 5 T^{3} + T^{4} - 15 T^{5} - 18 T^{6} + 27 T^{7} + 81 T^{8} \)
$5$ \( 1 + 5 T + 15 T^{2} + 25 T^{3} + 25 T^{4} \)
$7$ \( ( 1 + T + 13 T^{2} + 7 T^{3} + 49 T^{4} )^{2} \)
$11$ \( 1 + 2 T + 13 T^{2} + 34 T^{3} + 225 T^{4} + 374 T^{5} + 1573 T^{6} + 2662 T^{7} + 14641 T^{8} \)
$13$ \( 1 - 9 T + 23 T^{2} - 15 T^{3} + 16 T^{4} - 195 T^{5} + 3887 T^{6} - 19773 T^{7} + 28561 T^{8} \)
$17$ \( 1 - 8 T + 7 T^{2} + 110 T^{3} - 579 T^{4} + 1870 T^{5} + 2023 T^{6} - 39304 T^{7} + 83521 T^{8} \)
$19$ \( 1 + 5 T + 21 T^{2} + 145 T^{3} + 956 T^{4} + 2755 T^{5} + 7581 T^{6} + 34295 T^{7} + 130321 T^{8} \)
$23$ \( 1 + 11 T + 28 T^{2} - 245 T^{3} - 2259 T^{4} - 5635 T^{5} + 14812 T^{6} + 133837 T^{7} + 279841 T^{8} \)
$29$ \( 1 - 5 T - 19 T^{2} + 145 T^{3} - 4 T^{4} + 4205 T^{5} - 15979 T^{6} - 121945 T^{7} + 707281 T^{8} \)
$31$ \( 1 - 3 T - 22 T^{2} + 159 T^{3} + 205 T^{4} + 4929 T^{5} - 21142 T^{6} - 89373 T^{7} + 923521 T^{8} \)
$37$ \( 1 + 7 T - 18 T^{2} - 145 T^{3} + 371 T^{4} - 5365 T^{5} - 24642 T^{6} + 354571 T^{7} + 1874161 T^{8} \)
$41$ \( 1 - 8 T - 17 T^{2} + 254 T^{3} - 435 T^{4} + 10414 T^{5} - 28577 T^{6} - 551368 T^{7} + 2825761 T^{8} \)
$43$ \( ( 1 + 3 T + 77 T^{2} + 129 T^{3} + 1849 T^{4} )^{2} \)
$47$ \( 1 + 2 T - 43 T^{2} + 50 T^{3} + 2351 T^{4} + 2350 T^{5} - 94987 T^{6} + 207646 T^{7} + 4879681 T^{8} \)
$53$ \( 1 - 9 T + 8 T^{2} - 315 T^{3} + 5131 T^{4} - 16695 T^{5} + 22472 T^{6} - 1339893 T^{7} + 7890481 T^{8} \)
$59$ \( 1 + 31 T^{2} + 210 T^{3} + 2851 T^{4} + 12390 T^{5} + 107911 T^{6} + 12117361 T^{8} \)
$61$ \( 1 - 13 T + 78 T^{2} - 941 T^{3} + 11075 T^{4} - 57401 T^{5} + 290238 T^{6} - 2950753 T^{7} + 13845841 T^{8} \)
$67$ \( 1 + 2 T - 3 T^{2} - 410 T^{3} + 1601 T^{4} - 27470 T^{5} - 13467 T^{6} + 601526 T^{7} + 20151121 T^{8} \)
$71$ \( 1 - 8 T - 37 T^{2} + 694 T^{3} - 2425 T^{4} + 49274 T^{5} - 186517 T^{6} - 2863288 T^{7} + 25411681 T^{8} \)
$73$ \( 1 - 9 T + 8 T^{2} + 585 T^{3} - 5849 T^{4} + 42705 T^{5} + 42632 T^{6} - 3501153 T^{7} + 28398241 T^{8} \)
$79$ \( 1 - 15 T + 21 T^{2} + 145 T^{3} + 2916 T^{4} + 11455 T^{5} + 131061 T^{6} - 7395585 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 9 T - 52 T^{2} + 675 T^{3} + 121 T^{4} + 56025 T^{5} - 358228 T^{6} - 5146083 T^{7} + 47458321 T^{8} \)
$89$ \( 1 + 20 T + 151 T^{2} + 1600 T^{3} + 21441 T^{4} + 142400 T^{5} + 1196071 T^{6} + 14099380 T^{7} + 62742241 T^{8} \)
$97$ \( 1 - 8 T - 63 T^{2} + 20 T^{3} + 9821 T^{4} + 1940 T^{5} - 592767 T^{6} - 7301384 T^{7} + 88529281 T^{8} \)
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