Properties

Label 2007.1.v.a
Level 2007
Weight 1
Character orbit 2007.v
Analytic conductor 1.002
Analytic rank 0
Dimension 36
Projective image \(D_{74}\)
CM discriminant -3
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 2007 = 3^{2} \cdot 223 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2007.v (of order \(74\), degree \(36\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.00162348035\)
Analytic rank: \(0\)
Dimension: \(36\)
Coefficient field: \(\Q(\zeta_{74})\)
Defining polynomial: \(x^{36} - x^{35} + x^{34} - x^{33} + x^{32} - x^{31} + x^{30} - x^{29} + x^{28} - x^{27} + x^{26} - x^{25} + x^{24} - x^{23} + x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{74}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{74} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{74}^{20} q^{4} + ( \zeta_{74}^{21} + \zeta_{74}^{27} ) q^{7} +O(q^{10})\) \( q -\zeta_{74}^{20} q^{4} + ( \zeta_{74}^{21} + \zeta_{74}^{27} ) q^{7} + ( \zeta_{74}^{9} + \zeta_{74}^{32} ) q^{13} -\zeta_{74}^{3} q^{16} + ( \zeta_{74}^{24} + \zeta_{74}^{28} ) q^{19} + \zeta_{74}^{14} q^{25} + ( \zeta_{74}^{4} + \zeta_{74}^{10} ) q^{28} + ( -\zeta_{74}^{6} - \zeta_{74}^{36} ) q^{31} + ( \zeta_{74}^{8} - \zeta_{74}^{13} ) q^{37} + ( \zeta_{74}^{12} + \zeta_{74}^{34} ) q^{43} + ( -\zeta_{74}^{5} - \zeta_{74}^{11} - \zeta_{74}^{17} ) q^{49} + ( \zeta_{74}^{15} - \zeta_{74}^{29} ) q^{52} + ( -\zeta_{74}^{15} - \zeta_{74}^{26} ) q^{61} + \zeta_{74}^{23} q^{64} + ( \zeta_{74} + \zeta_{74}^{2} ) q^{67} + ( -\zeta_{74}^{2} + \zeta_{74}^{5} ) q^{73} + ( \zeta_{74}^{7} + \zeta_{74}^{11} ) q^{76} + ( \zeta_{74}^{13} + \zeta_{74}^{22} ) q^{79} + ( -\zeta_{74}^{16} - \zeta_{74}^{22} + \zeta_{74}^{30} + \zeta_{74}^{36} ) q^{91} + ( \zeta_{74}^{29} - \zeta_{74}^{31} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36q + q^{4} + 2q^{7} + O(q^{10}) \) \( 36q + q^{4} + 2q^{7} - q^{16} - 2q^{19} - q^{25} - 2q^{28} + 2q^{31} - 2q^{37} - 2q^{43} - 3q^{49} + q^{64} + 2q^{73} + 2q^{76} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2007\mathbb{Z}\right)^\times\).

\(n\) \(226\) \(893\)
\(\chi(n)\) \(\zeta_{74}^{33}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
−0.985616 + 0.169001i
−0.524307 0.851529i
−0.0424412 0.999099i
0.721956 0.691939i
0.594633 + 0.803997i
0.292823 + 0.956167i
−0.985616 0.169001i
0.828510 + 0.559975i
−0.660675 + 0.750672i
−0.873014 0.487695i
0.594633 0.803997i
0.828510 0.559975i
0.292823 0.956167i
−0.942877 0.333140i
0.721956 + 0.691939i
−0.873014 + 0.487695i
−0.210679 0.977555i
−0.660675 0.750672i
−0.372856 0.927889i
0.996397 + 0.0848059i
0 0 0.967733 0.251978i 0 0 1.03825 1.40380i 0 0 0
118.1 0 0 −0.0424412 0.999099i 0 0 1.55047 1.25191i 0 0 0
163.1 0 0 −0.660675 + 0.750672i 0 0 0.133193 0.216319i 0 0 0
190.1 0 0 0.911228 + 0.411901i 0 0 −1.15356 0.644415i 0 0 0
208.1 0 0 −0.985616 + 0.169001i 0 0 1.71835 + 0.776745i 0 0 0
334.1 0 0 −0.942877 0.333140i 0 0 −1.02806 + 1.16810i 0 0 0
397.1 0 0 0.967733 + 0.251978i 0 0 1.03825 + 1.40380i 0 0 0
442.1 0 0 −0.778036 + 0.628220i 0 0 0.0535200 0.417946i 0 0 0
505.1 0 0 0.292823 0.956167i 0 0 0.0703259 1.65553i 0 0 0
541.1 0 0 0.721956 + 0.691939i 0 0 −0.0800337 + 0.0282777i 0 0 0
550.1 0 0 −0.985616 0.169001i 0 0 1.71835 0.776745i 0 0 0
613.1 0 0 −0.778036 0.628220i 0 0 0.0535200 + 0.417946i 0 0 0
667.1 0 0 −0.942877 + 0.333140i 0 0 −1.02806 1.16810i 0 0 0
721.1 0 0 −0.873014 0.487695i 0 0 0.307058 1.00265i 0 0 0
919.1 0 0 0.911228 0.411901i 0 0 −1.15356 + 0.644415i 0 0 0
946.1 0 0 0.721956 0.691939i 0 0 −0.0800337 0.0282777i 0 0 0
1000.1 0 0 0.450204 0.892926i 0 0 0.443426 + 1.10351i 0 0 0
1081.1 0 0 0.292823 + 0.956167i 0 0 0.0703259 + 1.65553i 0 0 0
1099.1 0 0 −0.210679 + 0.977555i 0 0 −1.76365 0.459219i 0 0 0
1108.1 0 0 0.127018 0.991900i 0 0 −0.871354 + 1.72823i 0 0 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1999.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
223.f odd 74 1 inner
669.k even 74 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2007.1.v.a 36
3.b odd 2 1 CM 2007.1.v.a 36
223.f odd 74 1 inner 2007.1.v.a 36
669.k even 74 1 inner 2007.1.v.a 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2007.1.v.a 36 1.a even 1 1 trivial
2007.1.v.a 36 3.b odd 2 1 CM
2007.1.v.a 36 223.f odd 74 1 inner
2007.1.v.a 36 669.k even 74 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2007, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} - T^{34} + T^{36} - T^{38} + T^{40} - T^{42} + T^{44} - T^{46} + T^{48} - T^{50} + T^{52} - T^{54} + T^{56} - T^{58} + T^{60} - T^{62} + T^{64} - T^{66} + T^{68} - T^{70} + T^{72} \)
$3$ 1
$5$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} - T^{19} + T^{20} - T^{21} + T^{22} - T^{23} + T^{24} - T^{25} + T^{26} - T^{27} + T^{28} - T^{29} + T^{30} - T^{31} + T^{32} - T^{33} + T^{34} - T^{35} + T^{36} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} + T^{23} + T^{24} + T^{25} + T^{26} + T^{27} + T^{28} + T^{29} + T^{30} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} ) \)
$7$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} - T^{19} + T^{20} - T^{21} + T^{22} - T^{23} + T^{24} - T^{25} + T^{26} - T^{27} + T^{28} - T^{29} + T^{30} - T^{31} + T^{32} - T^{33} + T^{34} - T^{35} + T^{36} )^{2} \)
$11$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} - T^{19} + T^{20} - T^{21} + T^{22} - T^{23} + T^{24} - T^{25} + T^{26} - T^{27} + T^{28} - T^{29} + T^{30} - T^{31} + T^{32} - T^{33} + T^{34} - T^{35} + T^{36} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} + T^{23} + T^{24} + T^{25} + T^{26} + T^{27} + T^{28} + T^{29} + T^{30} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} ) \)
$13$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} - T^{19} + T^{20} - T^{21} + T^{22} - T^{23} + T^{24} - T^{25} + T^{26} - T^{27} + T^{28} - T^{29} + T^{30} - T^{31} + T^{32} - T^{33} + T^{34} - T^{35} + T^{36} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} + T^{23} + T^{24} + T^{25} + T^{26} + T^{27} + T^{28} + T^{29} + T^{30} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} ) \)
$17$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} - T^{34} + T^{36} - T^{38} + T^{40} - T^{42} + T^{44} - T^{46} + T^{48} - T^{50} + T^{52} - T^{54} + T^{56} - T^{58} + T^{60} - T^{62} + T^{64} - T^{66} + T^{68} - T^{70} + T^{72} \)
$19$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} + T^{23} + T^{24} + T^{25} + T^{26} + T^{27} + T^{28} + T^{29} + T^{30} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} )^{2} \)
$23$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} - T^{19} + T^{20} - T^{21} + T^{22} - T^{23} + T^{24} - T^{25} + T^{26} - T^{27} + T^{28} - T^{29} + T^{30} - T^{31} + T^{32} - T^{33} + T^{34} - T^{35} + T^{36} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} + T^{23} + T^{24} + T^{25} + T^{26} + T^{27} + T^{28} + T^{29} + T^{30} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} ) \)
$29$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} - T^{34} + T^{36} - T^{38} + T^{40} - T^{42} + T^{44} - T^{46} + T^{48} - T^{50} + T^{52} - T^{54} + T^{56} - T^{58} + T^{60} - T^{62} + T^{64} - T^{66} + T^{68} - T^{70} + T^{72} \)
$31$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} - T^{19} + T^{20} - T^{21} + T^{22} - T^{23} + T^{24} - T^{25} + T^{26} - T^{27} + T^{28} - T^{29} + T^{30} - T^{31} + T^{32} - T^{33} + T^{34} - T^{35} + T^{36} )^{2} \)
$37$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} + T^{23} + T^{24} + T^{25} + T^{26} + T^{27} + T^{28} + T^{29} + T^{30} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} )^{2} \)
$41$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} - T^{34} + T^{36} - T^{38} + T^{40} - T^{42} + T^{44} - T^{46} + T^{48} - T^{50} + T^{52} - T^{54} + T^{56} - T^{58} + T^{60} - T^{62} + T^{64} - T^{66} + T^{68} - T^{70} + T^{72} \)
$43$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} + T^{23} + T^{24} + T^{25} + T^{26} + T^{27} + T^{28} + T^{29} + T^{30} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} )^{2} \)
$47$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} - T^{34} + T^{36} - T^{38} + T^{40} - T^{42} + T^{44} - T^{46} + T^{48} - T^{50} + T^{52} - T^{54} + T^{56} - T^{58} + T^{60} - T^{62} + T^{64} - T^{66} + T^{68} - T^{70} + T^{72} \)
$53$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} - T^{34} + T^{36} - T^{38} + T^{40} - T^{42} + T^{44} - T^{46} + T^{48} - T^{50} + T^{52} - T^{54} + T^{56} - T^{58} + T^{60} - T^{62} + T^{64} - T^{66} + T^{68} - T^{70} + T^{72} \)
$59$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} - T^{19} + T^{20} - T^{21} + T^{22} - T^{23} + T^{24} - T^{25} + T^{26} - T^{27} + T^{28} - T^{29} + T^{30} - T^{31} + T^{32} - T^{33} + T^{34} - T^{35} + T^{36} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} + T^{23} + T^{24} + T^{25} + T^{26} + T^{27} + T^{28} + T^{29} + T^{30} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} ) \)
$61$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} - T^{19} + T^{20} - T^{21} + T^{22} - T^{23} + T^{24} - T^{25} + T^{26} - T^{27} + T^{28} - T^{29} + T^{30} - T^{31} + T^{32} - T^{33} + T^{34} - T^{35} + T^{36} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} + T^{23} + T^{24} + T^{25} + T^{26} + T^{27} + T^{28} + T^{29} + T^{30} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} ) \)
$67$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} - T^{19} + T^{20} - T^{21} + T^{22} - T^{23} + T^{24} - T^{25} + T^{26} - T^{27} + T^{28} - T^{29} + T^{30} - T^{31} + T^{32} - T^{33} + T^{34} - T^{35} + T^{36} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} + T^{23} + T^{24} + T^{25} + T^{26} + T^{27} + T^{28} + T^{29} + T^{30} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} ) \)
$71$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} - T^{19} + T^{20} - T^{21} + T^{22} - T^{23} + T^{24} - T^{25} + T^{26} - T^{27} + T^{28} - T^{29} + T^{30} - T^{31} + T^{32} - T^{33} + T^{34} - T^{35} + T^{36} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} + T^{23} + T^{24} + T^{25} + T^{26} + T^{27} + T^{28} + T^{29} + T^{30} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} ) \)
$73$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} - T^{19} + T^{20} - T^{21} + T^{22} - T^{23} + T^{24} - T^{25} + T^{26} - T^{27} + T^{28} - T^{29} + T^{30} - T^{31} + T^{32} - T^{33} + T^{34} - T^{35} + T^{36} )^{2} \)
$79$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} - T^{19} + T^{20} - T^{21} + T^{22} - T^{23} + T^{24} - T^{25} + T^{26} - T^{27} + T^{28} - T^{29} + T^{30} - T^{31} + T^{32} - T^{33} + T^{34} - T^{35} + T^{36} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} + T^{23} + T^{24} + T^{25} + T^{26} + T^{27} + T^{28} + T^{29} + T^{30} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} ) \)
$83$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} - T^{34} + T^{36} - T^{38} + T^{40} - T^{42} + T^{44} - T^{46} + T^{48} - T^{50} + T^{52} - T^{54} + T^{56} - T^{58} + T^{60} - T^{62} + T^{64} - T^{66} + T^{68} - T^{70} + T^{72} \)
$89$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} - T^{34} + T^{36} - T^{38} + T^{40} - T^{42} + T^{44} - T^{46} + T^{48} - T^{50} + T^{52} - T^{54} + T^{56} - T^{58} + T^{60} - T^{62} + T^{64} - T^{66} + T^{68} - T^{70} + T^{72} \)
$97$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} - T^{19} + T^{20} - T^{21} + T^{22} - T^{23} + T^{24} - T^{25} + T^{26} - T^{27} + T^{28} - T^{29} + T^{30} - T^{31} + T^{32} - T^{33} + T^{34} - T^{35} + T^{36} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} + T^{23} + T^{24} + T^{25} + T^{26} + T^{27} + T^{28} + T^{29} + T^{30} + T^{31} + T^{32} + T^{33} + T^{34} + T^{35} + T^{36} ) \)
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