Properties

Label 49.2.e.b
Level $49$
Weight $2$
Character orbit 49.e
Analytic conductor $0.391$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 49.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.391266969904\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{7})\)
Coefficient field: \(\Q(\zeta_{21})\)
Defining polynomial: \(x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 7 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{21}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{21}^{2} + 2 \zeta_{21}^{3} - \zeta_{21}^{5} + \zeta_{21}^{6} + \zeta_{21}^{7} - \zeta_{21}^{8} + 2 \zeta_{21}^{10} - \zeta_{21}^{11} ) q^{2} + ( \zeta_{21}^{2} - \zeta_{21}^{3} + \zeta_{21}^{9} - \zeta_{21}^{11} ) q^{3} + ( -1 + \zeta_{21}^{2} - \zeta_{21}^{3} + 2 \zeta_{21}^{5} - \zeta_{21}^{6} - \zeta_{21}^{7} + \zeta_{21}^{8} - \zeta_{21}^{10} + \zeta_{21}^{11} ) q^{4} + ( -2 - \zeta_{21}^{3} - \zeta_{21}^{4} + \zeta_{21}^{5} - \zeta_{21}^{6} - 2 \zeta_{21}^{7} + \zeta_{21}^{8} - \zeta_{21}^{10} + \zeta_{21}^{11} ) q^{5} + ( \zeta_{21} - 2 \zeta_{21}^{2} + \zeta_{21}^{3} + \zeta_{21}^{4} - \zeta_{21}^{5} - \zeta_{21}^{6} + 2 \zeta_{21}^{7} + \zeta_{21}^{8} - 2 \zeta_{21}^{9} + \zeta_{21}^{10} ) q^{6} + ( -1 + \zeta_{21} - 2 \zeta_{21}^{3} + 2 \zeta_{21}^{4} + \zeta_{21}^{7} + 2 \zeta_{21}^{8} - \zeta_{21}^{9} - \zeta_{21}^{10} + \zeta_{21}^{11} ) q^{7} + ( \zeta_{21} + \zeta_{21}^{4} + \zeta_{21}^{5} - \zeta_{21}^{9} + \zeta_{21}^{11} ) q^{8} + ( -\zeta_{21} + 2 \zeta_{21}^{3} - 2 \zeta_{21}^{4} - 2 \zeta_{21}^{5} + 2 \zeta_{21}^{6} - 2 \zeta_{21}^{7} - 2 \zeta_{21}^{8} + 2 \zeta_{21}^{9} - \zeta_{21}^{11} ) q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{21}^{2} + 2 \zeta_{21}^{3} - \zeta_{21}^{5} + \zeta_{21}^{6} + \zeta_{21}^{7} - \zeta_{21}^{8} + 2 \zeta_{21}^{10} - \zeta_{21}^{11} ) q^{2} + ( \zeta_{21}^{2} - \zeta_{21}^{3} + \zeta_{21}^{9} - \zeta_{21}^{11} ) q^{3} + ( -1 + \zeta_{21}^{2} - \zeta_{21}^{3} + 2 \zeta_{21}^{5} - \zeta_{21}^{6} - \zeta_{21}^{7} + \zeta_{21}^{8} - \zeta_{21}^{10} + \zeta_{21}^{11} ) q^{4} + ( -2 - \zeta_{21}^{3} - \zeta_{21}^{4} + \zeta_{21}^{5} - \zeta_{21}^{6} - 2 \zeta_{21}^{7} + \zeta_{21}^{8} - \zeta_{21}^{10} + \zeta_{21}^{11} ) q^{5} + ( \zeta_{21} - 2 \zeta_{21}^{2} + \zeta_{21}^{3} + \zeta_{21}^{4} - \zeta_{21}^{5} - \zeta_{21}^{6} + 2 \zeta_{21}^{7} + \zeta_{21}^{8} - 2 \zeta_{21}^{9} + \zeta_{21}^{10} ) q^{6} + ( -1 + \zeta_{21} - 2 \zeta_{21}^{3} + 2 \zeta_{21}^{4} + \zeta_{21}^{7} + 2 \zeta_{21}^{8} - \zeta_{21}^{9} - \zeta_{21}^{10} + \zeta_{21}^{11} ) q^{7} + ( \zeta_{21} + \zeta_{21}^{4} + \zeta_{21}^{5} - \zeta_{21}^{9} + \zeta_{21}^{11} ) q^{8} + ( -\zeta_{21} + 2 \zeta_{21}^{3} - 2 \zeta_{21}^{4} - 2 \zeta_{21}^{5} + 2 \zeta_{21}^{6} - 2 \zeta_{21}^{7} - 2 \zeta_{21}^{8} + 2 \zeta_{21}^{9} - \zeta_{21}^{11} ) q^{9} + ( 2 - 2 \zeta_{21} - \zeta_{21}^{4} - 2 \zeta_{21}^{5} - 2 \zeta_{21}^{8} - 2 \zeta_{21}^{10} - \zeta_{21}^{11} ) q^{10} + ( -3 \zeta_{21} + \zeta_{21}^{2} + \zeta_{21}^{3} - \zeta_{21}^{4} + 2 \zeta_{21}^{6} - \zeta_{21}^{7} - 2 \zeta_{21}^{8} + \zeta_{21}^{9} - 2 \zeta_{21}^{10} + \zeta_{21}^{11} ) q^{11} + ( -1 + 2 \zeta_{21}^{2} - \zeta_{21}^{3} - 2 \zeta_{21}^{8} + 2 \zeta_{21}^{9} ) q^{12} + ( 1 + \zeta_{21} + \zeta_{21}^{2} + \zeta_{21}^{7} + \zeta_{21}^{8} + \zeta_{21}^{9} ) q^{13} + ( -3 + \zeta_{21} - \zeta_{21}^{2} - \zeta_{21}^{3} - \zeta_{21}^{4} + \zeta_{21}^{5} - 2 \zeta_{21}^{9} + \zeta_{21}^{10} + \zeta_{21}^{11} ) q^{14} + ( -\zeta_{21} + \zeta_{21}^{2} - 2 \zeta_{21}^{4} + \zeta_{21}^{5} - \zeta_{21}^{6} + \zeta_{21}^{7} - 2 \zeta_{21}^{8} + \zeta_{21}^{10} - \zeta_{21}^{11} ) q^{15} + ( 2 - 2 \zeta_{21} + 2 \zeta_{21}^{4} + \zeta_{21}^{5} + 2 \zeta_{21}^{7} + \zeta_{21}^{10} + 2 \zeta_{21}^{11} ) q^{16} + ( 1 - 2 \zeta_{21} - 4 \zeta_{21}^{2} + \zeta_{21}^{3} + \zeta_{21}^{4} - \zeta_{21}^{5} + \zeta_{21}^{7} - \zeta_{21}^{9} + \zeta_{21}^{10} - \zeta_{21}^{11} ) q^{17} + ( 1 + 2 \zeta_{21} - \zeta_{21}^{3} + \zeta_{21}^{4} - 2 \zeta_{21}^{6} + 2 \zeta_{21}^{8} - 2 \zeta_{21}^{10} - \zeta_{21}^{11} ) q^{18} + ( 1 + 2 \zeta_{21} - \zeta_{21}^{2} + 3 \zeta_{21}^{3} - \zeta_{21}^{4} - 3 \zeta_{21}^{5} + 2 \zeta_{21}^{7} - 2 \zeta_{21}^{8} + \zeta_{21}^{9} + 2 \zeta_{21}^{10} - 3 \zeta_{21}^{11} ) q^{19} + ( 1 + 2 \zeta_{21}^{3} - 2 \zeta_{21}^{4} + 2 \zeta_{21}^{6} - \zeta_{21}^{7} + \zeta_{21}^{10} - 2 \zeta_{21}^{11} ) q^{20} + ( -2 + \zeta_{21} - 2 \zeta_{21}^{3} + \zeta_{21}^{4} + 3 \zeta_{21}^{5} - 5 \zeta_{21}^{7} + 4 \zeta_{21}^{8} - \zeta_{21}^{9} - 2 \zeta_{21}^{10} + 2 \zeta_{21}^{11} ) q^{21} + ( -3 + 3 \zeta_{21} + \zeta_{21}^{2} - 4 \zeta_{21}^{3} + 2 \zeta_{21}^{4} + 2 \zeta_{21}^{5} - \zeta_{21}^{6} - 4 \zeta_{21}^{7} + 4 \zeta_{21}^{8} - 2 \zeta_{21}^{9} - \zeta_{21}^{10} ) q^{22} + ( -1 + \zeta_{21} - \zeta_{21}^{3} + \zeta_{21}^{4} + 2 \zeta_{21}^{8} - \zeta_{21}^{9} + \zeta_{21}^{10} + \zeta_{21}^{11} ) q^{23} + ( \zeta_{21}^{2} + \zeta_{21}^{3} - \zeta_{21}^{6} + \zeta_{21}^{9} + \zeta_{21}^{10} ) q^{24} + ( 2 \zeta_{21} + 2 \zeta_{21}^{2} + 2 \zeta_{21}^{3} + 3 \zeta_{21}^{4} + \zeta_{21}^{5} - \zeta_{21}^{6} + \zeta_{21}^{7} + 3 \zeta_{21}^{8} + 2 \zeta_{21}^{9} + 2 \zeta_{21}^{10} + 2 \zeta_{21}^{11} ) q^{25} + ( -1 + \zeta_{21} - \zeta_{21}^{3} + \zeta_{21}^{4} + \zeta_{21}^{7} + 2 \zeta_{21}^{8} - \zeta_{21}^{9} + \zeta_{21}^{10} + 2 \zeta_{21}^{11} ) q^{26} + ( 4 - \zeta_{21} + \zeta_{21}^{3} - 4 \zeta_{21}^{4} + 3 \zeta_{21}^{7} - \zeta_{21}^{8} - \zeta_{21}^{11} ) q^{27} + ( 1 + \zeta_{21} - 2 \zeta_{21}^{3} + \zeta_{21}^{4} - \zeta_{21}^{5} - \zeta_{21}^{6} + 2 \zeta_{21}^{9} - 2 \zeta_{21}^{10} + \zeta_{21}^{11} ) q^{28} + ( \zeta_{21} - 3 \zeta_{21}^{2} + \zeta_{21}^{3} + 4 \zeta_{21}^{4} - 3 \zeta_{21}^{5} - \zeta_{21}^{6} + 2 \zeta_{21}^{7} - 2 \zeta_{21}^{8} - \zeta_{21}^{9} + 4 \zeta_{21}^{10} - 2 \zeta_{21}^{11} ) q^{29} + ( 2 + \zeta_{21}^{2} - 3 \zeta_{21}^{3} + 2 \zeta_{21}^{4} + \zeta_{21}^{5} - \zeta_{21}^{6} - \zeta_{21}^{7} + 2 \zeta_{21}^{8} - 3 \zeta_{21}^{10} + \zeta_{21}^{11} ) q^{30} + ( 1 - 5 \zeta_{21}^{2} + 4 \zeta_{21}^{3} - 2 \zeta_{21}^{4} + \zeta_{21}^{5} + 2 \zeta_{21}^{6} + 2 \zeta_{21}^{7} - 4 \zeta_{21}^{8} - 3 \zeta_{21}^{9} + 4 \zeta_{21}^{10} - 2 \zeta_{21}^{11} ) q^{31} + ( -2 - \zeta_{21} - \zeta_{21}^{2} - 3 \zeta_{21}^{4} + 3 \zeta_{21}^{6} - 2 \zeta_{21}^{7} + 2 \zeta_{21}^{10} - 3 \zeta_{21}^{11} ) q^{32} + ( 2 - 3 \zeta_{21} + 2 \zeta_{21}^{2} - \zeta_{21}^{3} + 3 \zeta_{21}^{4} + \zeta_{21}^{5} - 4 \zeta_{21}^{6} + 2 \zeta_{21}^{7} + \zeta_{21}^{8} - 3 \zeta_{21}^{9} - \zeta_{21}^{10} + 4 \zeta_{21}^{11} ) q^{33} + ( -6 \zeta_{21} + \zeta_{21}^{2} + 5 \zeta_{21}^{3} - 7 \zeta_{21}^{4} - 2 \zeta_{21}^{5} + 4 \zeta_{21}^{6} - 2 \zeta_{21}^{7} - 7 \zeta_{21}^{8} + 5 \zeta_{21}^{9} + \zeta_{21}^{10} - 6 \zeta_{21}^{11} ) q^{34} + ( 5 - \zeta_{21}^{2} + 2 \zeta_{21}^{3} - \zeta_{21}^{4} - \zeta_{21}^{5} + 3 \zeta_{21}^{7} - 5 \zeta_{21}^{8} + 2 \zeta_{21}^{9} - 5 \zeta_{21}^{11} ) q^{35} + ( 1 - 2 \zeta_{21} - \zeta_{21}^{2} + 5 \zeta_{21}^{3} - 2 \zeta_{21}^{4} - 2 \zeta_{21}^{5} + 5 \zeta_{21}^{6} - \zeta_{21}^{7} - 2 \zeta_{21}^{8} + \zeta_{21}^{9} ) q^{36} + ( -6 + 3 \zeta_{21} + 5 \zeta_{21}^{2} - 6 \zeta_{21}^{3} + 2 \zeta_{21}^{5} - 2 \zeta_{21}^{6} - 2 \zeta_{21}^{7} + 2 \zeta_{21}^{8} - 2 \zeta_{21}^{10} + 4 \zeta_{21}^{11} ) q^{37} + ( 2 + 5 \zeta_{21}^{2} - 2 \zeta_{21}^{3} + 2 \zeta_{21}^{5} + \zeta_{21}^{7} + 2 \zeta_{21}^{8} + 4 \zeta_{21}^{9} - 4 \zeta_{21}^{10} + 2 \zeta_{21}^{11} ) q^{38} + ( 1 - \zeta_{21} - \zeta_{21}^{4} + 2 \zeta_{21}^{6} - \zeta_{21}^{8} + 2 \zeta_{21}^{9} - \zeta_{21}^{11} ) q^{39} + ( -2 \zeta_{21} - \zeta_{21}^{2} - 2 \zeta_{21}^{4} - 2 \zeta_{21}^{5} - 2 \zeta_{21}^{7} - 2 \zeta_{21}^{8} - \zeta_{21}^{10} - 2 \zeta_{21}^{11} ) q^{40} + ( -3 + 3 \zeta_{21} + \zeta_{21}^{2} - 6 \zeta_{21}^{3} + 5 \zeta_{21}^{4} + 4 \zeta_{21}^{5} - 2 \zeta_{21}^{6} - 2 \zeta_{21}^{7} + \zeta_{21}^{8} - 5 \zeta_{21}^{9} - \zeta_{21}^{10} + 6 \zeta_{21}^{11} ) q^{41} + ( 1 - 2 \zeta_{21} - 5 \zeta_{21}^{2} + 7 \zeta_{21}^{3} - 4 \zeta_{21}^{5} + 4 \zeta_{21}^{7} - 5 \zeta_{21}^{8} - 2 \zeta_{21}^{9} + 4 \zeta_{21}^{10} - \zeta_{21}^{11} ) q^{42} + ( 3 - 5 \zeta_{21} + 4 \zeta_{21}^{2} - 2 \zeta_{21}^{3} - 6 \zeta_{21}^{4} - 2 \zeta_{21}^{5} + \zeta_{21}^{6} - \zeta_{21}^{7} - 2 \zeta_{21}^{8} + 3 \zeta_{21}^{9} - 6 \zeta_{21}^{10} - 4 \zeta_{21}^{11} ) q^{43} + ( 4 - \zeta_{21} - 5 \zeta_{21}^{2} + 6 \zeta_{21}^{3} - 3 \zeta_{21}^{4} - 3 \zeta_{21}^{5} + 2 \zeta_{21}^{6} + 4 \zeta_{21}^{7} - 2 \zeta_{21}^{8} + 2 \zeta_{21}^{10} - 2 \zeta_{21}^{11} ) q^{44} + ( -4 + 2 \zeta_{21} - \zeta_{21}^{2} - \zeta_{21}^{3} + 5 \zeta_{21}^{4} - 2 \zeta_{21}^{5} + \zeta_{21}^{6} + 2 \zeta_{21}^{7} + \zeta_{21}^{8} + \zeta_{21}^{10} + 5 \zeta_{21}^{11} ) q^{45} + ( -2 + \zeta_{21} - 2 \zeta_{21}^{2} - \zeta_{21}^{3} - \zeta_{21}^{4} + \zeta_{21}^{5} - 2 \zeta_{21}^{6} - \zeta_{21}^{9} + \zeta_{21}^{11} ) q^{46} + ( -4 + 3 \zeta_{21} + \zeta_{21}^{2} + \zeta_{21}^{3} + 2 \zeta_{21}^{4} + 4 \zeta_{21}^{5} + 3 \zeta_{21}^{6} - 3 \zeta_{21}^{7} + 5 \zeta_{21}^{8} - 2 \zeta_{21}^{9} - 4 \zeta_{21}^{10} + 2 \zeta_{21}^{11} ) q^{47} + ( -3 + 2 \zeta_{21} + 2 \zeta_{21}^{2} - 7 \zeta_{21}^{3} + 6 \zeta_{21}^{4} - 3 \zeta_{21}^{6} - \zeta_{21}^{7} + 4 \zeta_{21}^{8} + \zeta_{21}^{9} - 5 \zeta_{21}^{10} + 3 \zeta_{21}^{11} ) q^{48} + ( -3 - \zeta_{21} + \zeta_{21}^{3} + \zeta_{21}^{4} - \zeta_{21}^{5} - 4 \zeta_{21}^{6} - 4 \zeta_{21}^{7} + 3 \zeta_{21}^{8} - 4 \zeta_{21}^{9} + \zeta_{21}^{10} + 2 \zeta_{21}^{11} ) q^{49} + ( -8 + 4 \zeta_{21}^{2} - 4 \zeta_{21}^{3} + \zeta_{21}^{4} + 2 \zeta_{21}^{5} - 3 \zeta_{21}^{6} - 3 \zeta_{21}^{7} + 6 \zeta_{21}^{8} + \zeta_{21}^{9} + 7 \zeta_{21}^{11} ) q^{50} + ( -7 + 6 \zeta_{21} + 2 \zeta_{21}^{2} - 10 \zeta_{21}^{3} + 4 \zeta_{21}^{4} + 7 \zeta_{21}^{5} - 7 \zeta_{21}^{6} - 4 \zeta_{21}^{7} + 9 \zeta_{21}^{8} - 4 \zeta_{21}^{9} - 6 \zeta_{21}^{10} + 3 \zeta_{21}^{11} ) q^{51} + ( -3 + \zeta_{21} + \zeta_{21}^{2} - 2 \zeta_{21}^{3} + \zeta_{21}^{4} + 2 \zeta_{21}^{5} - 2 \zeta_{21}^{6} - 2 \zeta_{21}^{7} + \zeta_{21}^{8} - \zeta_{21}^{9} - \zeta_{21}^{10} + 2 \zeta_{21}^{11} ) q^{52} + ( 2 - 2 \zeta_{21}^{2} + 2 \zeta_{21}^{3} - 4 \zeta_{21}^{5} + 2 \zeta_{21}^{6} - 4 \zeta_{21}^{8} - 2 \zeta_{21}^{9} - 4 \zeta_{21}^{11} ) q^{53} + ( 4 + \zeta_{21}^{2} - 2 \zeta_{21}^{3} + 4 \zeta_{21}^{6} + \zeta_{21}^{9} - \zeta_{21}^{11} ) q^{54} + ( -1 + 4 \zeta_{21} - 6 \zeta_{21}^{2} + 3 \zeta_{21}^{3} + 3 \zeta_{21}^{4} - 3 \zeta_{21}^{5} + 3 \zeta_{21}^{7} + \zeta_{21}^{8} - 3 \zeta_{21}^{9} + 3 \zeta_{21}^{10} ) q^{55} + ( -2 + 2 \zeta_{21} + 2 \zeta_{21}^{2} - 3 \zeta_{21}^{3} - \zeta_{21}^{4} - 2 \zeta_{21}^{6} + \zeta_{21}^{7} + 4 \zeta_{21}^{8} + \zeta_{21}^{9} - \zeta_{21}^{10} + 2 \zeta_{21}^{11} ) q^{56} + ( 4 - 5 \zeta_{21}^{2} + 6 \zeta_{21}^{3} - 4 \zeta_{21}^{4} - \zeta_{21}^{5} + 3 \zeta_{21}^{6} + 2 \zeta_{21}^{7} - \zeta_{21}^{8} + \zeta_{21}^{10} - \zeta_{21}^{11} ) q^{57} + ( -3 \zeta_{21} + \zeta_{21}^{2} + 5 \zeta_{21}^{3} - 4 \zeta_{21}^{4} - 4 \zeta_{21}^{8} + 5 \zeta_{21}^{9} + \zeta_{21}^{10} - 3 \zeta_{21}^{11} ) q^{58} + ( -2 - \zeta_{21} + 6 \zeta_{21}^{2} - 3 \zeta_{21}^{3} - \zeta_{21}^{4} + 3 \zeta_{21}^{5} - 3 \zeta_{21}^{6} - 3 \zeta_{21}^{7} + 2 \zeta_{21}^{8} - 3 \zeta_{21}^{10} + 2 \zeta_{21}^{11} ) q^{59} + ( -\zeta_{21}^{2} + 2 \zeta_{21}^{3} + \zeta_{21}^{4} - 2 \zeta_{21}^{5} - \zeta_{21}^{6} + 5 \zeta_{21}^{7} - 3 \zeta_{21}^{8} - 3 \zeta_{21}^{9} + 6 \zeta_{21}^{10} - 3 \zeta_{21}^{11} ) q^{60} + ( 5 - 6 \zeta_{21} - 5 \zeta_{21}^{2} + 6 \zeta_{21}^{3} - 5 \zeta_{21}^{4} - 2 \zeta_{21}^{5} + 7 \zeta_{21}^{6} + \zeta_{21}^{7} - 5 \zeta_{21}^{8} + 3 \zeta_{21}^{9} + 3 \zeta_{21}^{10} - 9 \zeta_{21}^{11} ) q^{61} + ( 4 - 6 \zeta_{21} + 3 \zeta_{21}^{2} - 3 \zeta_{21}^{3} + 2 \zeta_{21}^{4} + 5 \zeta_{21}^{5} - 3 \zeta_{21}^{7} - 3 \zeta_{21}^{8} + 7 \zeta_{21}^{9} - 6 \zeta_{21}^{10} + 3 \zeta_{21}^{11} ) q^{62} + ( 4 - 2 \zeta_{21} + 2 \zeta_{21}^{2} + \zeta_{21}^{3} - 3 \zeta_{21}^{4} + 3 \zeta_{21}^{5} + 2 \zeta_{21}^{6} + 3 \zeta_{21}^{7} - 4 \zeta_{21}^{8} + 2 \zeta_{21}^{9} + 4 \zeta_{21}^{10} - 6 \zeta_{21}^{11} ) q^{63} + ( 3 \zeta_{21} - \zeta_{21}^{2} - 4 \zeta_{21}^{3} + 3 \zeta_{21}^{4} + 4 \zeta_{21}^{5} - 4 \zeta_{21}^{6} + 4 \zeta_{21}^{7} + 3 \zeta_{21}^{8} - 4 \zeta_{21}^{9} - \zeta_{21}^{10} + 3 \zeta_{21}^{11} ) q^{64} + ( 2 - \zeta_{21} - 2 \zeta_{21}^{2} + \zeta_{21}^{3} - 3 \zeta_{21}^{4} - 2 \zeta_{21}^{5} - \zeta_{21}^{7} - 4 \zeta_{21}^{8} - \zeta_{21}^{9} - 4 \zeta_{21}^{11} ) q^{65} + ( -4 + 2 \zeta_{21} + 3 \zeta_{21}^{2} + 2 \zeta_{21}^{3} - 5 \zeta_{21}^{4} + 5 \zeta_{21}^{6} - 6 \zeta_{21}^{7} - \zeta_{21}^{8} + \zeta_{21}^{9} + 6 \zeta_{21}^{10} - 5 \zeta_{21}^{11} ) q^{66} + ( 5 + \zeta_{21} + 3 \zeta_{21}^{3} - 2 \zeta_{21}^{4} - 2 \zeta_{21}^{5} + \zeta_{21}^{7} - \zeta_{21}^{8} + \zeta_{21}^{9} + 5 \zeta_{21}^{10} + \zeta_{21}^{11} ) q^{67} + ( 2 + 3 \zeta_{21} + \zeta_{21}^{2} - 3 \zeta_{21}^{3} + 2 \zeta_{21}^{4} + \zeta_{21}^{5} - 4 \zeta_{21}^{6} - \zeta_{21}^{7} + 5 \zeta_{21}^{8} - 2 \zeta_{21}^{10} + 2 \zeta_{21}^{11} ) q^{68} + ( -1 + \zeta_{21} - 2 \zeta_{21}^{2} + \zeta_{21}^{4} - \zeta_{21}^{5} + \zeta_{21}^{8} - 2 \zeta_{21}^{9} + 2 \zeta_{21}^{10} - \zeta_{21}^{11} ) q^{69} + ( 7 - 3 \zeta_{21} + 2 \zeta_{21}^{2} + 7 \zeta_{21}^{3} + 3 \zeta_{21}^{4} - 3 \zeta_{21}^{5} + 6 \zeta_{21}^{6} + 5 \zeta_{21}^{7} - 2 \zeta_{21}^{8} + 3 \zeta_{21}^{9} + 4 \zeta_{21}^{10} - 2 \zeta_{21}^{11} ) q^{70} + ( 5 - 6 \zeta_{21} - 3 \zeta_{21}^{2} + 9 \zeta_{21}^{3} - 2 \zeta_{21}^{4} - 6 \zeta_{21}^{5} + 3 \zeta_{21}^{6} + 6 \zeta_{21}^{7} - 8 \zeta_{21}^{8} + 8 \zeta_{21}^{9} + 4 \zeta_{21}^{10} - 2 \zeta_{21}^{11} ) q^{71} + ( 1 - 2 \zeta_{21} - \zeta_{21}^{2} + 3 \zeta_{21}^{3} - 2 \zeta_{21}^{5} + \zeta_{21}^{6} - \zeta_{21}^{7} - 4 \zeta_{21}^{8} + \zeta_{21}^{9} - 3 \zeta_{21}^{11} ) q^{72} + ( 6 \zeta_{21} - 4 \zeta_{21}^{3} + 5 \zeta_{21}^{4} - 2 \zeta_{21}^{6} + 5 \zeta_{21}^{8} - 4 \zeta_{21}^{9} + 6 \zeta_{21}^{11} ) q^{73} + ( 6 \zeta_{21} - 4 \zeta_{21}^{2} - 9 \zeta_{21}^{3} + 6 \zeta_{21}^{4} + \zeta_{21}^{5} - 7 \zeta_{21}^{6} + \zeta_{21}^{7} + 6 \zeta_{21}^{8} - 9 \zeta_{21}^{9} - 4 \zeta_{21}^{10} + 6 \zeta_{21}^{11} ) q^{74} + ( -1 + 3 \zeta_{21} + \zeta_{21}^{2} - 4 \zeta_{21}^{3} + 2 \zeta_{21}^{5} - \zeta_{21}^{6} + 4 \zeta_{21}^{8} + \zeta_{21}^{9} - \zeta_{21}^{10} + 2 \zeta_{21}^{11} ) q^{75} + ( -4 + 2 \zeta_{21} - \zeta_{21}^{2} - \zeta_{21}^{3} + 5 \zeta_{21}^{4} - 2 \zeta_{21}^{5} + \zeta_{21}^{6} - \zeta_{21}^{7} + \zeta_{21}^{8} - 3 \zeta_{21}^{9} + \zeta_{21}^{10} + 2 \zeta_{21}^{11} ) q^{76} + ( 4 - 4 \zeta_{21} + 7 \zeta_{21}^{2} + \zeta_{21}^{3} - 2 \zeta_{21}^{4} - 3 \zeta_{21}^{5} + 2 \zeta_{21}^{6} + \zeta_{21}^{7} - 5 \zeta_{21}^{8} + 5 \zeta_{21}^{9} + 3 \zeta_{21}^{10} - \zeta_{21}^{11} ) q^{77} + ( -3 \zeta_{21}^{2} + 2 \zeta_{21}^{4} - \zeta_{21}^{5} - \zeta_{21}^{6} + \zeta_{21}^{7} + \zeta_{21}^{8} - 2 \zeta_{21}^{9} + \zeta_{21}^{10} ) q^{78} + ( -6 \zeta_{21} + \zeta_{21}^{2} + \zeta_{21}^{3} - 2 \zeta_{21}^{4} + \zeta_{21}^{5} + 5 \zeta_{21}^{6} - \zeta_{21}^{7} - 4 \zeta_{21}^{8} ) q^{79} + ( 7 + 2 \zeta_{21} - 3 \zeta_{21}^{2} + 7 \zeta_{21}^{3} - 5 \zeta_{21}^{4} - \zeta_{21}^{5} + 2 \zeta_{21}^{7} - 2 \zeta_{21}^{8} - \zeta_{21}^{9} - 9 \zeta_{21}^{11} ) q^{80} + ( -3 - 2 \zeta_{21} + 3 \zeta_{21}^{2} - 4 \zeta_{21}^{3} - \zeta_{21}^{4} + 2 \zeta_{21}^{5} - \zeta_{21}^{6} - \zeta_{21}^{7} - \zeta_{21}^{8} + 3 \zeta_{21}^{9} - 3 \zeta_{21}^{10} + 3 \zeta_{21}^{11} ) q^{81} + ( 3 + \zeta_{21} - 8 \zeta_{21}^{2} + 4 \zeta_{21}^{3} + 5 \zeta_{21}^{4} - \zeta_{21}^{5} - \zeta_{21}^{6} + 2 \zeta_{21}^{7} - 5 \zeta_{21}^{8} - 5 \zeta_{21}^{9} + 7 \zeta_{21}^{10} + \zeta_{21}^{11} ) q^{82} + ( -3 \zeta_{21} + 5 \zeta_{21}^{2} + 6 \zeta_{21}^{3} - 2 \zeta_{21}^{4} - 3 \zeta_{21}^{5} - \zeta_{21}^{6} - 3 \zeta_{21}^{7} - 2 \zeta_{21}^{8} + 6 \zeta_{21}^{9} + 5 \zeta_{21}^{10} - 3 \zeta_{21}^{11} ) q^{83} + ( 2 - 4 \zeta_{21} + 5 \zeta_{21}^{2} - \zeta_{21}^{3} - 8 \zeta_{21}^{4} + 5 \zeta_{21}^{5} + 4 \zeta_{21}^{6} - 7 \zeta_{21}^{7} + 2 \zeta_{21}^{8} + 6 \zeta_{21}^{9} - 5 \zeta_{21}^{10} - \zeta_{21}^{11} ) q^{84} + ( 1 + 4 \zeta_{21} + 4 \zeta_{21}^{2} - 2 \zeta_{21}^{3} + 2 \zeta_{21}^{4} + 4 \zeta_{21}^{5} + 6 \zeta_{21}^{8} + 3 \zeta_{21}^{9} - 4 \zeta_{21}^{10} + 2 \zeta_{21}^{11} ) q^{85} + ( 4 + 8 \zeta_{21} - 3 \zeta_{21}^{2} + 3 \zeta_{21}^{3} - 4 \zeta_{21}^{5} + 4 \zeta_{21}^{6} + 5 \zeta_{21}^{7} + 3 \zeta_{21}^{8} - 7 \zeta_{21}^{9} + 3 \zeta_{21}^{10} - 8 \zeta_{21}^{11} ) q^{86} + ( -3 + 6 \zeta_{21} - 3 \zeta_{21}^{2} + \zeta_{21}^{3} - 2 \zeta_{21}^{4} + 3 \zeta_{21}^{6} - 7 \zeta_{21}^{7} + 8 \zeta_{21}^{8} - \zeta_{21}^{9} - 4 \zeta_{21}^{10} + 2 \zeta_{21}^{11} ) q^{87} + ( 3 - \zeta_{21} - 4 \zeta_{21}^{2} + 2 \zeta_{21}^{3} - 3 \zeta_{21}^{4} - 3 \zeta_{21}^{5} + 3 \zeta_{21}^{6} + 3 \zeta_{21}^{7} - 2 \zeta_{21}^{8} + \zeta_{21}^{9} + \zeta_{21}^{10} - 5 \zeta_{21}^{11} ) q^{88} + ( -2 \zeta_{21}^{2} + \zeta_{21}^{3} + 3 \zeta_{21}^{4} + 4 \zeta_{21}^{5} - 4 \zeta_{21}^{6} + 4 \zeta_{21}^{7} + 3 \zeta_{21}^{8} + \zeta_{21}^{9} - 2 \zeta_{21}^{10} ) q^{89} + ( -9 + \zeta_{21} + 2 \zeta_{21}^{2} - 11 \zeta_{21}^{3} - \zeta_{21}^{4} + 5 \zeta_{21}^{5} - 5 \zeta_{21}^{6} - 8 \zeta_{21}^{7} + 4 \zeta_{21}^{8} - \zeta_{21}^{9} - 4 \zeta_{21}^{10} + 5 \zeta_{21}^{11} ) q^{90} + ( -4 - 3 \zeta_{21} - 3 \zeta_{21}^{3} + \zeta_{21}^{4} + 2 \zeta_{21}^{5} - \zeta_{21}^{6} - 3 \zeta_{21}^{7} + 2 \zeta_{21}^{8} - 3 \zeta_{21}^{10} + 2 \zeta_{21}^{11} ) q^{91} + ( 1 - 2 \zeta_{21} + 2 \zeta_{21}^{2} - \zeta_{21}^{3} - \zeta_{21}^{7} - \zeta_{21}^{8} + \zeta_{21}^{9} - 2 \zeta_{21}^{10} + \zeta_{21}^{11} ) q^{92} + ( -2 + \zeta_{21} - 4 \zeta_{21}^{2} + 6 \zeta_{21}^{3} + 3 \zeta_{21}^{4} - 3 \zeta_{21}^{5} - 6 \zeta_{21}^{6} + 8 \zeta_{21}^{7} - 4 \zeta_{21}^{8} - 3 \zeta_{21}^{9} + 4 \zeta_{21}^{10} - \zeta_{21}^{11} ) q^{93} + ( -10 + 3 \zeta_{21} + 4 \zeta_{21}^{2} - 7 \zeta_{21}^{3} + 6 \zeta_{21}^{4} + 8 \zeta_{21}^{5} - 4 \zeta_{21}^{6} - \zeta_{21}^{7} + 7 \zeta_{21}^{8} - 3 \zeta_{21}^{9} - 4 \zeta_{21}^{10} + 13 \zeta_{21}^{11} ) q^{94} + ( -5 - 2 \zeta_{21} + \zeta_{21}^{2} - 9 \zeta_{21}^{3} + 4 \zeta_{21}^{4} - \zeta_{21}^{5} - 4 \zeta_{21}^{6} - 2 \zeta_{21}^{7} + \zeta_{21}^{8} - 4 \zeta_{21}^{9} - \zeta_{21}^{10} + 5 \zeta_{21}^{11} ) q^{95} + ( -2 - 2 \zeta_{21} - 4 \zeta_{21}^{2} + 7 \zeta_{21}^{3} - \zeta_{21}^{4} - 5 \zeta_{21}^{5} + 3 \zeta_{21}^{6} + 4 \zeta_{21}^{7} - 6 \zeta_{21}^{8} - 6 \zeta_{21}^{9} + 8 \zeta_{21}^{10} - 4 \zeta_{21}^{11} ) q^{96} + ( -7 - \zeta_{21} + 2 \zeta_{21}^{2} - 4 \zeta_{21}^{3} + 6 \zeta_{21}^{5} - 3 \zeta_{21}^{6} - 4 \zeta_{21}^{7} + 7 \zeta_{21}^{8} - 2 \zeta_{21}^{9} - 4 \zeta_{21}^{10} + 4 \zeta_{21}^{11} ) q^{97} + ( -1 + \zeta_{21} + 4 \zeta_{21}^{2} + 4 \zeta_{21}^{3} - 9 \zeta_{21}^{4} - 3 \zeta_{21}^{7} - 4 \zeta_{21}^{8} + 5 \zeta_{21}^{9} - 2 \zeta_{21}^{10} - 6 \zeta_{21}^{11} ) q^{98} + ( -5 + 6 \zeta_{21} - 3 \zeta_{21}^{2} - 9 \zeta_{21}^{3} + 10 \zeta_{21}^{4} + \zeta_{21}^{5} - 5 \zeta_{21}^{6} + \zeta_{21}^{7} + 4 \zeta_{21}^{8} - 2 \zeta_{21}^{9} - 5 \zeta_{21}^{10} + 3 \zeta_{21}^{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 2q^{2} + 2q^{4} - 7q^{5} - 7q^{6} - 7q^{7} + 6q^{8} - 8q^{9} + O(q^{10}) \) \( 12q - 2q^{2} + 2q^{4} - 7q^{5} - 7q^{6} - 7q^{7} + 6q^{8} - 8q^{9} + 14q^{10} - 8q^{11} - 14q^{12} + 7q^{13} - 28q^{14} - 7q^{15} + 16q^{16} + 20q^{18} - 14q^{19} + 7q^{20} + 21q^{21} + 13q^{22} - 2q^{23} + 3q^{25} - 7q^{26} + 21q^{27} + 14q^{28} - 11q^{29} + 42q^{30} - 14q^{31} - 24q^{32} + 35q^{33} - 42q^{34} + 21q^{35} - 13q^{36} - 30q^{37} + 21q^{38} + 21q^{41} - 35q^{42} + 17q^{43} - 6q^{44} - 49q^{45} - 16q^{46} - 21q^{47} + 7q^{49} - 46q^{50} + 7q^{51} - 7q^{52} + 6q^{53} + 42q^{54} - 28q^{55} - 14q^{56} + 7q^{57} - 32q^{58} + 14q^{59} - 28q^{60} - 7q^{61} + 56q^{62} + 14q^{63} + 14q^{64} + 14q^{65} - 28q^{66} + 48q^{67} + 56q^{68} - 7q^{69} + 21q^{70} - 39q^{71} - 4q^{72} + 42q^{73} + 61q^{74} + 7q^{75} - 28q^{76} + 21q^{77} - 16q^{79} + 42q^{80} - 25q^{81} + 28q^{82} - 7q^{83} + 42q^{84} + 28q^{85} + 17q^{86} + 7q^{87} - 11q^{88} - 14q^{89} - 14q^{90} - 21q^{91} + 16q^{92} - 70q^{93} - 49q^{94} - 7q^{95} - 70q^{96} - 28q^{97} - 28q^{98} - 18q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(\zeta_{21}^{4}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1
−0.988831 + 0.149042i
0.365341 0.930874i
−0.733052 0.680173i
0.955573 0.294755i
0.0747301 0.997204i
0.826239 + 0.563320i
0.0747301 + 0.997204i
0.826239 0.563320i
−0.733052 + 0.680173i
0.955573 + 0.294755i
−0.988831 0.149042i
0.365341 + 0.930874i
−1.19158 1.49419i 1.55929 0.750915i −0.367711 + 1.61105i −1.91647 + 0.922924i −2.98003 1.43511i 1.91879 1.82161i −0.598393 + 0.288171i −0.00295512 + 0.00370560i 3.66265 + 1.76384i
8.2 0.914101 + 1.14625i −0.880843 + 0.424191i −0.0332580 + 0.145713i −0.830509 + 0.399952i −1.29141 0.621909i −0.938402 2.47374i 2.44440 1.17716i −1.27452 + 1.59820i −1.21761 0.586371i
15.1 0.0332580 0.145713i −1.81507 + 2.27603i 1.78181 + 0.858075i 1.36967 1.71752i 0.271281 + 0.340175i −2.64558 + 0.0302261i 0.370666 0.464800i −1.21825 5.33750i −0.204712 0.256700i
15.2 0.367711 1.61105i 0.290611 0.364415i −0.658322 0.317031i −2.42463 + 3.04039i −0.480228 0.602187i −1.62586 2.08724i 1.30778 1.63991i 0.619220 + 2.71298i 4.00665 + 5.02418i
22.1 −1.78181 + 0.858075i 0.590232 2.58597i 1.19158 1.49419i 0.359497 1.57506i 1.16728 + 5.11418i 1.95991 + 1.77729i 0.0391023 0.171318i −3.63598 1.75100i 0.710963 + 3.11493i
22.2 0.658322 0.317031i 0.255779 1.12064i −0.914101 + 1.14625i −0.0575591 + 0.252183i −0.186893 0.818832i −2.16885 1.51528i −0.563561 + 2.46912i 1.51249 + 0.728379i 0.0420574 + 0.184265i
29.1 −1.78181 0.858075i 0.590232 + 2.58597i 1.19158 + 1.49419i 0.359497 + 1.57506i 1.16728 5.11418i 1.95991 1.77729i 0.0391023 + 0.171318i −3.63598 + 1.75100i 0.710963 3.11493i
29.2 0.658322 + 0.317031i 0.255779 + 1.12064i −0.914101 1.14625i −0.0575591 0.252183i −0.186893 + 0.818832i −2.16885 + 1.51528i −0.563561 2.46912i 1.51249 0.728379i 0.0420574 0.184265i
36.1 0.0332580 + 0.145713i −1.81507 2.27603i 1.78181 0.858075i 1.36967 + 1.71752i 0.271281 0.340175i −2.64558 0.0302261i 0.370666 + 0.464800i −1.21825 + 5.33750i −0.204712 + 0.256700i
36.2 0.367711 + 1.61105i 0.290611 + 0.364415i −0.658322 + 0.317031i −2.42463 3.04039i −0.480228 + 0.602187i −1.62586 + 2.08724i 1.30778 + 1.63991i 0.619220 2.71298i 4.00665 5.02418i
43.1 −1.19158 + 1.49419i 1.55929 + 0.750915i −0.367711 1.61105i −1.91647 0.922924i −2.98003 + 1.43511i 1.91879 + 1.82161i −0.598393 0.288171i −0.00295512 0.00370560i 3.66265 1.76384i
43.2 0.914101 1.14625i −0.880843 0.424191i −0.0332580 0.145713i −0.830509 0.399952i −1.29141 + 0.621909i −0.938402 + 2.47374i 2.44440 + 1.17716i −1.27452 1.59820i −1.21761 + 0.586371i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.2.e.b 12
3.b odd 2 1 441.2.u.b 12
4.b odd 2 1 784.2.u.b 12
7.b odd 2 1 343.2.e.b 12
7.c even 3 1 343.2.g.b 12
7.c even 3 1 343.2.g.d 12
7.d odd 6 1 343.2.g.a 12
7.d odd 6 1 343.2.g.c 12
49.e even 7 1 inner 49.2.e.b 12
49.e even 7 1 2401.2.a.c 6
49.f odd 14 1 343.2.e.b 12
49.f odd 14 1 2401.2.a.d 6
49.g even 21 1 343.2.g.b 12
49.g even 21 1 343.2.g.d 12
49.h odd 42 1 343.2.g.a 12
49.h odd 42 1 343.2.g.c 12
147.l odd 14 1 441.2.u.b 12
196.k odd 14 1 784.2.u.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.2.e.b 12 1.a even 1 1 trivial
49.2.e.b 12 49.e even 7 1 inner
343.2.e.b 12 7.b odd 2 1
343.2.e.b 12 49.f odd 14 1
343.2.g.a 12 7.d odd 6 1
343.2.g.a 12 49.h odd 42 1
343.2.g.b 12 7.c even 3 1
343.2.g.b 12 49.g even 21 1
343.2.g.c 12 7.d odd 6 1
343.2.g.c 12 49.h odd 42 1
343.2.g.d 12 7.c even 3 1
343.2.g.d 12 49.g even 21 1
441.2.u.b 12 3.b odd 2 1
441.2.u.b 12 147.l odd 14 1
784.2.u.b 12 4.b odd 2 1
784.2.u.b 12 196.k odd 14 1
2401.2.a.c 6 49.e even 7 1
2401.2.a.d 6 49.f odd 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{12} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(49, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 5 T + 52 T^{2} - 94 T^{3} + 54 T^{4} + 6 T^{5} + 7 T^{6} + 6 T^{7} + 12 T^{8} + 4 T^{9} + 3 T^{10} + 2 T^{11} + T^{12} \)
$3$ \( 49 - 98 T + 147 T^{2} + 147 T^{3} + 49 T^{5} + 112 T^{6} - 112 T^{7} + 42 T^{8} - 7 T^{9} + 7 T^{10} + T^{12} \)
$5$ \( 49 + 196 T + 1029 T^{2} + 1911 T^{3} + 1715 T^{4} + 1029 T^{5} + 749 T^{6} + 371 T^{7} + 126 T^{8} + 56 T^{9} + 28 T^{10} + 7 T^{11} + T^{12} \)
$7$ \( 117649 + 117649 T + 50421 T^{2} + 12005 T^{3} + 4802 T^{4} + 4116 T^{5} + 2107 T^{6} + 588 T^{7} + 98 T^{8} + 35 T^{9} + 21 T^{10} + 7 T^{11} + T^{12} \)
$11$ \( 1849 + 13115 T + 58903 T^{2} + 70564 T^{3} + 99816 T^{4} + 43102 T^{5} + 12005 T^{6} + 3918 T^{7} + 1049 T^{8} + 144 T^{9} + 27 T^{10} + 8 T^{11} + T^{12} \)
$13$ \( 49 - 98 T + 294 T^{2} - 637 T^{3} + 882 T^{4} - 735 T^{5} + 406 T^{6} - 210 T^{7} + 147 T^{8} - 77 T^{9} + 28 T^{10} - 7 T^{11} + T^{12} \)
$17$ \( 3087049 - 1365189 T + 79821 T^{2} - 200116 T^{3} + 120050 T^{4} + 103488 T^{5} + 30933 T^{6} + 4256 T^{7} + 1519 T^{8} + 140 T^{9} + 35 T^{10} + T^{12} \)
$19$ \( ( 2107 + 1568 T - 49 T^{2} - 210 T^{3} - 21 T^{4} + 7 T^{5} + T^{6} )^{2} \)
$23$ \( 1 - 5 T + 3 T^{2} - 10 T^{3} + 243 T^{4} - 246 T^{5} + 343 T^{6} - 99 T^{7} + 54 T^{8} + 46 T^{9} + 10 T^{10} + 2 T^{11} + T^{12} \)
$29$ \( 1681 + 51332 T + 451338 T^{2} - 759692 T^{3} + 822968 T^{4} - 214503 T^{5} + 62377 T^{6} + 1788 T^{7} + 2957 T^{8} + 508 T^{9} + 75 T^{10} + 11 T^{11} + T^{12} \)
$31$ \( ( -8183 + 7791 T + 1813 T^{2} - 574 T^{3} - 98 T^{4} + 7 T^{5} + T^{6} )^{2} \)
$37$ \( 295118041 + 76154507 T + 96270828 T^{2} + 49677870 T^{3} + 12650748 T^{4} + 2715782 T^{5} + 784105 T^{6} + 204574 T^{7} + 38736 T^{8} + 5212 T^{9} + 493 T^{10} + 30 T^{11} + T^{12} \)
$41$ \( 10413529 + 9171134 T + 7110831 T^{2} - 1782228 T^{3} + 923748 T^{4} - 315364 T^{5} + 99043 T^{6} - 41552 T^{7} + 15120 T^{8} - 1568 T^{9} + 196 T^{10} - 21 T^{11} + T^{12} \)
$43$ \( 200307409 + 20988899 T + 232868670 T^{2} - 86491919 T^{3} + 29767569 T^{4} - 2984424 T^{5} + 854119 T^{6} - 50250 T^{7} - 7683 T^{8} + 977 T^{9} + 82 T^{10} - 17 T^{11} + T^{12} \)
$47$ \( 3021810841 - 1423364103 T - 222607392 T^{2} + 87256309 T^{3} + 99204665 T^{4} + 25115538 T^{5} + 4568823 T^{6} + 527212 T^{7} + 63721 T^{8} + 5425 T^{9} + 350 T^{10} + 21 T^{11} + T^{12} \)
$53$ \( 2985984 + 1492992 T + 4478976 T^{2} + 4105728 T^{3} + 2426112 T^{4} + 756864 T^{5} + 157248 T^{6} + 21600 T^{7} + 2736 T^{8} + 144 T^{9} - 36 T^{10} - 6 T^{11} + T^{12} \)
$59$ \( 54125449 - 29096935 T + 15223761 T^{2} - 4564742 T^{3} + 1498371 T^{4} - 194040 T^{5} - 2975 T^{6} + 5061 T^{7} + 126 T^{8} - 364 T^{9} + 112 T^{10} - 14 T^{11} + T^{12} \)
$61$ \( 261242569 - 204672069 T + 50009106 T^{2} - 8810641 T^{3} + 8613955 T^{4} - 1702358 T^{5} + 354039 T^{6} - 95480 T^{7} + 14147 T^{8} - 455 T^{9} + 140 T^{10} + 7 T^{11} + T^{12} \)
$67$ \( ( -293 + 1486 T - 1886 T^{2} + 71 T^{3} + 142 T^{4} - 24 T^{5} + T^{6} )^{2} \)
$71$ \( 312925003609 + 180407210691 T + 48568012464 T^{2} + 8445075811 T^{3} + 1207844693 T^{4} + 163533123 T^{5} + 21143409 T^{6} + 2443507 T^{7} + 236029 T^{8} + 17644 T^{9} + 1013 T^{10} + 39 T^{11} + T^{12} \)
$73$ \( 460917961 - 251122893 T + 178502492 T^{2} - 93821770 T^{3} + 35843500 T^{4} - 11074882 T^{5} + 2884791 T^{6} - 603960 T^{7} + 95984 T^{8} - 10990 T^{9} + 861 T^{10} - 42 T^{11} + T^{12} \)
$79$ \( ( -21629 + 1933 T + 3480 T^{2} - 447 T^{3} - 97 T^{4} + 8 T^{5} + T^{6} )^{2} \)
$83$ \( 118178641 + 39189955 T + 266644770 T^{2} - 89724537 T^{3} + 34397951 T^{4} + 249312 T^{5} + 327677 T^{6} + 214004 T^{7} + 13755 T^{8} - 2821 T^{9} - 14 T^{10} + 7 T^{11} + T^{12} \)
$89$ \( 89755965649 + 30272374685 T + 9958060770 T^{2} + 2513769531 T^{3} + 482009423 T^{4} + 61622694 T^{5} + 5052698 T^{6} + 146125 T^{7} - 12642 T^{8} - 1589 T^{9} + 70 T^{10} + 14 T^{11} + T^{12} \)
$97$ \( ( 18571 + 9898 T - 1911 T^{2} - 966 T^{3} - 28 T^{4} + 14 T^{5} + T^{6} )^{2} \)
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