Properties

Label 169.2.e.b
Level $169$
Weight $2$
Character orbit 169.e
Analytic conductor $1.349$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 169.e (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.34947179416\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.17213603549184.1
Defining polynomial: \(x^{12} - 5 x^{10} + 19 x^{8} - 28 x^{6} + 31 x^{4} - 6 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 5 x^{10} + 19 x^{8} - 28 x^{6} + 31 x^{4} - 6 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -25 \nu^{11} + 95 \nu^{9} - 361 \nu^{7} + 155 \nu^{5} - 30 \nu^{3} - 1563 \nu \)\()/559\)
\(\beta_{3}\)\(=\)\((\)\( 25 \nu^{10} - 95 \nu^{8} + 361 \nu^{6} - 155 \nu^{4} + 30 \nu^{2} + 1004 \)\()/559\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{10} - 20 \nu^{8} + 76 \nu^{6} - 139 \nu^{4} + 124 \nu^{2} - 24 \)\()/43\)
\(\beta_{5}\)\(=\)\((\)\( 45 \nu^{10} - 171 \nu^{8} + 538 \nu^{6} - 279 \nu^{4} + 54 \nu^{2} + 242 \)\()/559\)
\(\beta_{6}\)\(=\)\((\)\( 70 \nu^{11} - 266 \nu^{9} + 899 \nu^{7} - 434 \nu^{5} + 84 \nu^{3} + 1246 \nu \)\()/559\)
\(\beta_{7}\)\(=\)\((\)\( 114 \nu^{10} - 545 \nu^{8} + 2071 \nu^{6} - 2831 \nu^{4} + 3379 \nu^{2} - 95 \)\()/559\)
\(\beta_{8}\)\(=\)\((\)\( 114 \nu^{11} - 545 \nu^{9} + 2071 \nu^{7} - 2831 \nu^{5} + 3379 \nu^{3} - 95 \nu \)\()/559\)
\(\beta_{9}\)\(=\)\((\)\( -128 \nu^{10} + 710 \nu^{8} - 2698 \nu^{6} + 4483 \nu^{4} - 4402 \nu^{2} + 852 \)\()/559\)
\(\beta_{10}\)\(=\)\((\)\( -242 \nu^{11} + 1255 \nu^{9} - 4769 \nu^{7} + 7314 \nu^{5} - 7781 \nu^{3} + 1506 \nu \)\()/559\)
\(\beta_{11}\)\(=\)\((\)\( -317 \nu^{11} + 1540 \nu^{9} - 5852 \nu^{7} + 8338 \nu^{5} - 9548 \nu^{3} + 1848 \nu \)\()/559\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{9} + \beta_{7} + \beta_{4} - \beta_{3} + 1\)
\(\nu^{3}\)\(=\)\(\beta_{11} + 3 \beta_{8} + \beta_{2}\)
\(\nu^{4}\)\(=\)\(3 \beta_{9} + 2 \beta_{7} + 4 \beta_{4} - 2\)
\(\nu^{5}\)\(=\)\(4 \beta_{11} - \beta_{10} + 9 \beta_{8} - 9 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-5 \beta_{5} + 9 \beta_{3} - 14\)
\(\nu^{7}\)\(=\)\(-5 \beta_{6} - 14 \beta_{2} - 28 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-28 \beta_{9} - 14 \beta_{7} - 19 \beta_{5} - 47 \beta_{4} + 28 \beta_{3} - 28\)
\(\nu^{9}\)\(=\)\(-47 \beta_{11} + 19 \beta_{10} - 89 \beta_{8} - 19 \beta_{6} - 47 \beta_{2}\)
\(\nu^{10}\)\(=\)\(-89 \beta_{9} - 42 \beta_{7} - 155 \beta_{4} + 42\)
\(\nu^{11}\)\(=\)\(-155 \beta_{11} + 66 \beta_{10} - 286 \beta_{8} + 286 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(1 - \beta_{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
1.07992 + 0.623490i
1.56052 + 0.900969i
−0.385418 0.222521i
0.385418 + 0.222521i
−1.56052 0.900969i
−1.07992 0.623490i
1.07992 0.623490i
1.56052 0.900969i
−0.385418 + 0.222521i
0.385418 0.222521i
−1.56052 + 0.900969i
−1.07992 + 0.623490i
−1.94594 + 1.12349i 0.277479 + 0.480608i 1.52446 2.64044i 1.44504i −1.07992 0.623490i 1.77441 + 1.02446i 2.35690i 1.34601 2.33136i 1.62349 + 2.81197i
23.2 −0.694498 + 0.400969i 1.12349 + 1.94594i −0.678448 + 1.17511i 0.246980i −1.56052 0.900969i 2.04113 + 1.17845i 2.69202i −1.02446 + 1.77441i 0.0990311 + 0.171527i
23.3 −0.480608 + 0.277479i −0.400969 0.694498i −0.846011 + 1.46533i 2.80194i 0.385418 + 0.222521i −2.33136 1.34601i 2.04892i 1.17845 2.04113i 0.777479 + 1.34663i
23.4 0.480608 0.277479i −0.400969 0.694498i −0.846011 + 1.46533i 2.80194i −0.385418 0.222521i 2.33136 + 1.34601i 2.04892i 1.17845 2.04113i 0.777479 + 1.34663i
23.5 0.694498 0.400969i 1.12349 + 1.94594i −0.678448 + 1.17511i 0.246980i 1.56052 + 0.900969i −2.04113 1.17845i 2.69202i −1.02446 + 1.77441i 0.0990311 + 0.171527i
23.6 1.94594 1.12349i 0.277479 + 0.480608i 1.52446 2.64044i 1.44504i 1.07992 + 0.623490i −1.77441 1.02446i 2.35690i 1.34601 2.33136i 1.62349 + 2.81197i
147.1 −1.94594 1.12349i 0.277479 0.480608i 1.52446 + 2.64044i 1.44504i −1.07992 + 0.623490i 1.77441 1.02446i 2.35690i 1.34601 + 2.33136i 1.62349 2.81197i
147.2 −0.694498 0.400969i 1.12349 1.94594i −0.678448 1.17511i 0.246980i −1.56052 + 0.900969i 2.04113 1.17845i 2.69202i −1.02446 1.77441i 0.0990311 0.171527i
147.3 −0.480608 0.277479i −0.400969 + 0.694498i −0.846011 1.46533i 2.80194i 0.385418 0.222521i −2.33136 + 1.34601i 2.04892i 1.17845 + 2.04113i 0.777479 1.34663i
147.4 0.480608 + 0.277479i −0.400969 + 0.694498i −0.846011 1.46533i 2.80194i −0.385418 + 0.222521i 2.33136 1.34601i 2.04892i 1.17845 + 2.04113i 0.777479 1.34663i
147.5 0.694498 + 0.400969i 1.12349 1.94594i −0.678448 1.17511i 0.246980i 1.56052 0.900969i −2.04113 + 1.17845i 2.69202i −1.02446 1.77441i 0.0990311 0.171527i
147.6 1.94594 + 1.12349i 0.277479 0.480608i 1.52446 + 2.64044i 1.44504i 1.07992 0.623490i −1.77441 + 1.02446i 2.35690i 1.34601 + 2.33136i 1.62349 2.81197i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 147.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.2.e.b 12
13.b even 2 1 inner 169.2.e.b 12
13.c even 3 1 169.2.b.b 6
13.c even 3 1 inner 169.2.e.b 12
13.d odd 4 1 169.2.c.b 6
13.d odd 4 1 169.2.c.c 6
13.e even 6 1 169.2.b.b 6
13.e even 6 1 inner 169.2.e.b 12
13.f odd 12 1 169.2.a.b 3
13.f odd 12 1 169.2.a.c yes 3
13.f odd 12 1 169.2.c.b 6
13.f odd 12 1 169.2.c.c 6
39.h odd 6 1 1521.2.b.l 6
39.i odd 6 1 1521.2.b.l 6
39.k even 12 1 1521.2.a.o 3
39.k even 12 1 1521.2.a.r 3
52.i odd 6 1 2704.2.f.o 6
52.j odd 6 1 2704.2.f.o 6
52.l even 12 1 2704.2.a.z 3
52.l even 12 1 2704.2.a.ba 3
65.s odd 12 1 4225.2.a.bb 3
65.s odd 12 1 4225.2.a.bg 3
91.bc even 12 1 8281.2.a.bf 3
91.bc even 12 1 8281.2.a.bj 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.2.a.b 3 13.f odd 12 1
169.2.a.c yes 3 13.f odd 12 1
169.2.b.b 6 13.c even 3 1
169.2.b.b 6 13.e even 6 1
169.2.c.b 6 13.d odd 4 1
169.2.c.b 6 13.f odd 12 1
169.2.c.c 6 13.d odd 4 1
169.2.c.c 6 13.f odd 12 1
169.2.e.b 12 1.a even 1 1 trivial
169.2.e.b 12 13.b even 2 1 inner
169.2.e.b 12 13.c even 3 1 inner
169.2.e.b 12 13.e even 6 1 inner
1521.2.a.o 3 39.k even 12 1
1521.2.a.r 3 39.k even 12 1
1521.2.b.l 6 39.h odd 6 1
1521.2.b.l 6 39.i odd 6 1
2704.2.a.z 3 52.l even 12 1
2704.2.a.ba 3 52.l even 12 1
2704.2.f.o 6 52.i odd 6 1
2704.2.f.o 6 52.j odd 6 1
4225.2.a.bb 3 65.s odd 12 1
4225.2.a.bg 3 65.s odd 12 1
8281.2.a.bf 3 91.bc even 12 1
8281.2.a.bj 3 91.bc even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 6 T_{2}^{10} + 31 T_{2}^{8} - 28 T_{2}^{6} + 19 T_{2}^{4} - 5 T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(169, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 5 T^{2} + 19 T^{4} - 28 T^{6} + 31 T^{8} - 6 T^{10} + T^{12} \)
$3$ \( ( 1 - T + 3 T^{2} + 5 T^{4} - 2 T^{5} + T^{6} )^{2} \)
$5$ \( ( 1 + 17 T^{2} + 10 T^{4} + T^{6} )^{2} \)
$7$ \( 28561 - 15886 T^{2} + 5963 T^{4} - 1260 T^{6} + 195 T^{8} - 17 T^{10} + T^{12} \)
$11$ \( 28561 - 25857 T^{2} + 19015 T^{4} - 3640 T^{6} + 523 T^{8} - 26 T^{10} + T^{12} \)
$13$ \( T^{12} \)
$17$ \( ( 169 - 195 T + 199 T^{2} - 56 T^{3} + 19 T^{4} + 2 T^{5} + T^{6} )^{2} \)
$19$ \( 1 - 129 T^{2} + 16603 T^{4} - 4900 T^{6} + 1315 T^{8} - 38 T^{10} + T^{12} \)
$23$ \( ( 169 + 13 T + 66 T^{2} + 21 T^{3} + 26 T^{4} + 5 T^{5} + T^{6} )^{2} \)
$29$ \( ( 6889 + 3652 T + 1853 T^{2} + 210 T^{3} + 45 T^{4} - T^{5} + T^{6} )^{2} \)
$31$ \( ( 27889 + 2966 T^{2} + 97 T^{4} + T^{6} )^{2} \)
$37$ \( 707281 - 828385 T^{2} + 918083 T^{4} - 59388 T^{6} + 2859 T^{8} - 62 T^{10} + T^{12} \)
$41$ \( 5764801 - 4117715 T^{2} + 2588278 T^{4} - 247303 T^{6} + 19894 T^{8} - 147 T^{10} + T^{12} \)
$43$ \( ( 169 + 520 T + 1769 T^{2} - 546 T^{3} + 129 T^{4} - 13 T^{5} + T^{6} )^{2} \)
$47$ \( ( 27889 + 4189 T^{2} + 122 T^{4} + T^{6} )^{2} \)
$53$ \( ( 337 - 86 T - T^{2} + T^{3} )^{4} \)
$59$ \( 1 - 6851 T^{2} + 46936006 T^{4} - 1335943 T^{6} + 31174 T^{8} - 195 T^{10} + T^{12} \)
$61$ \( ( 57121 + 16013 T + 5445 T^{2} + 210 T^{3} + 83 T^{4} + 4 T^{5} + T^{6} )^{2} \)
$67$ \( 2825761 - 8576462 T^{2} + 25786659 T^{4} - 736428 T^{6} + 15923 T^{8} - 145 T^{10} + T^{12} \)
$71$ \( 89526025681 - 5908180914 T^{2} + 304629951 T^{4} - 5029192 T^{6} + 61479 T^{8} - 285 T^{10} + T^{12} \)
$73$ \( ( 829921 + 30798 T^{2} + 321 T^{4} + T^{6} )^{2} \)
$79$ \( ( 127 - 162 T + 5 T^{2} + T^{3} )^{4} \)
$83$ \( ( 41209 + 16758 T^{2} + 329 T^{4} + T^{6} )^{2} \)
$89$ \( 6234839521 - 920527338 T^{2} + 114668455 T^{4} - 2978080 T^{6} + 60703 T^{8} - 269 T^{10} + T^{12} \)
$97$ \( 8208541201 - 1021073270 T^{2} + 107352483 T^{4} - 2264388 T^{6} + 35819 T^{8} - 217 T^{10} + T^{12} \)
show more
show less