Properties

Label 1344.4.b.g
Level $1344$
Weight $4$
Character orbit 1344.b
Analytic conductor $79.299$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - x^{11} - 2 x^{10} - 6 x^{9} + 56 x^{7} - 448 x^{6} + 448 x^{5} - 3072 x^{3} - 8192 x^{2} - 32768 x + 262144\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{30} \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -3 q^{3} -\beta_{7} q^{5} + ( -1 - \beta_{2} ) q^{7} + 9 q^{9} +O(q^{10})\) \( q -3 q^{3} -\beta_{7} q^{5} + ( -1 - \beta_{2} ) q^{7} + 9 q^{9} + \beta_{11} q^{11} -\beta_{9} q^{13} + 3 \beta_{7} q^{15} + ( \beta_{2} + \beta_{3} + \beta_{7} + \beta_{9} + \beta_{10} ) q^{17} + ( -7 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{19} + ( 3 + 3 \beta_{2} ) q^{21} + ( -\beta_{2} - \beta_{3} - \beta_{4} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{23} + ( -19 + \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{25} -27 q^{27} + ( -17 - 2 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{29} + ( 32 + 3 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{8} ) q^{31} -3 \beta_{11} q^{33} + ( 7 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + 4 \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{35} + ( 19 - \beta_{2} + 2 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{8} ) q^{37} + 3 \beta_{9} q^{39} + ( \beta_{2} + \beta_{3} - 3 \beta_{4} - 5 \beta_{7} - 3 \beta_{8} - \beta_{9} + \beta_{10} ) q^{41} + ( -3 \beta_{2} - 3 \beta_{3} - 9 \beta_{7} + 3 \beta_{9} + \beta_{10} - 3 \beta_{11} ) q^{43} -9 \beta_{7} q^{45} + ( -23 + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{8} ) q^{47} + ( -36 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} - 6 \beta_{7} + 2 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} - 3 \beta_{11} ) q^{49} + ( -3 \beta_{2} - 3 \beta_{3} - 3 \beta_{7} - 3 \beta_{9} - 3 \beta_{10} ) q^{51} + ( 2 + 2 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{6} + 2 \beta_{8} ) q^{53} + ( -12 + 8 \beta_{2} - 12 \beta_{3} - \beta_{4} + 4 \beta_{5} + \beta_{6} + 5 \beta_{8} ) q^{55} + ( 21 - 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{57} + ( 103 - 4 \beta_{1} + 7 \beta_{2} - 8 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{8} ) q^{59} + ( 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 18 \beta_{7} - 3 \beta_{8} - 5 \beta_{9} + 8 \beta_{10} - 6 \beta_{11} ) q^{61} + ( -9 - 9 \beta_{2} ) q^{63} + ( 23 - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{8} ) q^{65} + ( -3 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 15 \beta_{7} - 3 \beta_{8} - \beta_{9} + 9 \beta_{10} - 3 \beta_{11} ) q^{67} + ( 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} ) q^{69} + ( -3 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 23 \beta_{7} + 2 \beta_{8} + 9 \beta_{9} - 2 \beta_{10} ) q^{71} + ( 10 \beta_{2} + 10 \beta_{3} + 6 \beta_{4} + 6 \beta_{7} + 6 \beta_{8} - 2 \beta_{10} + 12 \beta_{11} ) q^{73} + ( 57 - 3 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{75} + ( -86 + 4 \beta_{1} + 3 \beta_{2} + 15 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} - 19 \beta_{7} - 3 \beta_{8} + 10 \beta_{9} - 4 \beta_{10} - 6 \beta_{11} ) q^{77} + ( -11 \beta_{2} - 11 \beta_{3} + 3 \beta_{4} - 24 \beta_{7} + 3 \beta_{8} - 16 \beta_{9} - 2 \beta_{10} ) q^{79} + 81 q^{81} + ( -87 - 2 \beta_{1} - 3 \beta_{2} + 12 \beta_{3} - 9 \beta_{5} - 9 \beta_{8} ) q^{83} + ( 67 - 6 \beta_{1} - 15 \beta_{2} + 18 \beta_{3} + \beta_{4} - 3 \beta_{5} - 5 \beta_{6} - 4 \beta_{8} ) q^{85} + ( 51 + 6 \beta_{2} - 3 \beta_{3} - 3 \beta_{5} - 3 \beta_{6} - 3 \beta_{8} ) q^{87} + ( 21 \beta_{2} + 21 \beta_{3} + 3 \beta_{4} + 37 \beta_{7} + 3 \beta_{8} + 7 \beta_{9} - 15 \beta_{10} ) q^{89} + ( 139 - \beta_{1} - 4 \beta_{2} - 16 \beta_{3} + 5 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - 9 \beta_{11} ) q^{91} + ( -96 - 9 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 6 \beta_{5} - 3 \beta_{6} + 3 \beta_{8} ) q^{93} + ( 8 \beta_{2} + 8 \beta_{3} + 5 \beta_{4} + 46 \beta_{7} + 5 \beta_{8} + 6 \beta_{10} + 2 \beta_{11} ) q^{95} + ( 8 \beta_{2} + 8 \beta_{3} - 9 \beta_{4} - 42 \beta_{7} - 9 \beta_{8} - 8 \beta_{9} + 2 \beta_{10} + 6 \beta_{11} ) q^{97} + 9 \beta_{11} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 36q^{3} - 10q^{7} + 108q^{9} + O(q^{10}) \) \( 12q - 36q^{3} - 10q^{7} + 108q^{9} - 84q^{19} + 30q^{21} - 216q^{25} - 324q^{27} - 200q^{29} + 384q^{31} + 84q^{35} + 244q^{37} - 280q^{47} - 424q^{49} + 16q^{53} - 212q^{55} + 252q^{57} + 1168q^{59} - 90q^{63} + 280q^{65} + 648q^{75} - 968q^{77} + 972q^{81} - 968q^{83} + 852q^{85} + 600q^{87} + 1648q^{91} - 1152q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - x^{11} - 2 x^{10} - 6 x^{9} + 56 x^{7} - 448 x^{6} + 448 x^{5} - 3072 x^{3} - 8192 x^{2} - 32768 x + 262144\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{11} - \nu^{10} - 2 \nu^{9} + 122 \nu^{8} - 128 \nu^{7} - 200 \nu^{6} - 1216 \nu^{5} - 7744 \nu^{4} + 15360 \nu^{3} - 44032 \nu^{2} + 65536 \nu - 24576 \)\()/8192\)
\(\beta_{2}\)\(=\)\((\)\( -13 \nu^{11} - 47 \nu^{10} - 74 \nu^{9} - 26 \nu^{8} + 40 \nu^{7} - 1048 \nu^{6} + 5792 \nu^{5} + 3904 \nu^{4} + 8960 \nu^{3} + 74752 \nu^{2} + 372736 \nu + 1671168 \)\()/49152\)
\(\beta_{3}\)\(=\)\((\)\( -7 \nu^{11} - 11 \nu^{10} - 32 \nu^{9} - 50 \nu^{8} + 172 \nu^{7} - 904 \nu^{6} + 1232 \nu^{5} - 2240 \nu^{4} - 1408 \nu^{3} + 31744 \nu^{2} + 120832 \nu + 786432 \)\()/24576\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{11} - 8 \nu^{10} - 49 \nu^{9} - 88 \nu^{8} - 130 \nu^{7} - 184 \nu^{6} - 1880 \nu^{5} + 640 \nu^{4} + 9536 \nu^{3} + 44032 \nu^{2} - 142336 \nu + 241664 \)\()/12288\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{11} - 85 \nu^{10} + 190 \nu^{9} + 226 \nu^{8} + 1744 \nu^{7} - 488 \nu^{6} - 9472 \nu^{5} + 38336 \nu^{4} - 95744 \nu^{3} + 195584 \nu^{2} - 401408 \nu + 360448 \)\()/49152\)
\(\beta_{6}\)\(=\)\((\)\( -7 \nu^{11} - \nu^{10} - 74 \nu^{9} - 102 \nu^{8} + 1008 \nu^{7} + 184 \nu^{6} - 896 \nu^{5} + 12480 \nu^{4} + 66048 \nu^{3} + 97280 \nu^{2} - 262144 \nu + 425984 \)\()/16384\)
\(\beta_{7}\)\(=\)\((\)\( 6 \nu^{11} + 11 \nu^{10} + 19 \nu^{9} + 10 \nu^{8} + 58 \nu^{7} + 688 \nu^{6} - 1480 \nu^{5} - 1984 \nu^{4} - 4160 \nu^{3} - 22528 \nu^{2} - 115712 \nu - 598016 \)\()/12288\)
\(\beta_{8}\)\(=\)\((\)\( -9 \nu^{11} + 4 \nu^{10} - 25 \nu^{9} - 16 \nu^{8} - 130 \nu^{7} - 856 \nu^{6} + 3496 \nu^{5} - 4736 \nu^{4} + 9536 \nu^{3} + 80896 \nu^{2} + 349184 \nu + 561152 \)\()/12288\)
\(\beta_{9}\)\(=\)\((\)\( -7 \nu^{11} - 13 \nu^{10} - 30 \nu^{9} + 18 \nu^{8} - 136 \nu^{7} - 264 \nu^{6} + 736 \nu^{5} + 3264 \nu^{4} + 2304 \nu^{3} + 13312 \nu^{2} + 200704 \nu + 589824 \)\()/8192\)
\(\beta_{10}\)\(=\)\((\)\( -27 \nu^{11} - 29 \nu^{10} - 114 \nu^{9} - 14 \nu^{8} - 240 \nu^{7} - 1704 \nu^{6} + 6912 \nu^{5} - 2624 \nu^{4} + 13824 \nu^{3} + 150528 \nu^{2} + 393216 \nu + 2260992 \)\()/16384\)
\(\beta_{11}\)\(=\)\((\)\( 31 \nu^{11} + 61 \nu^{10} + 70 \nu^{9} + 174 \nu^{8} + 280 \nu^{7} + 1928 \nu^{6} - 6176 \nu^{5} - 4544 \nu^{4} - 14080 \nu^{3} - 158720 \nu^{2} - 708608 \nu - 2686976 \)\()/16384\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{3} + \beta_{2} + 3\)\()/32\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{10} - \beta_{9} + 3 \beta_{8} + 3 \beta_{7} + 2 \beta_{5} + 2 \beta_{4} - 5 \beta_{3} + \beta_{2} + 15\)\()/32\)
\(\nu^{3}\)\(=\)\((\)\(-4 \beta_{11} - 3 \beta_{10} - 5 \beta_{9} + \beta_{8} - 9 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} - 17 \beta_{3} + \beta_{2} + 4 \beta_{1} + 67\)\()/32\)
\(\nu^{4}\)\(=\)\((\)\(12 \beta_{11} - 11 \beta_{10} + 19 \beta_{9} + 5 \beta_{8} - 57 \beta_{7} + 4 \beta_{6} + 2 \beta_{5} - 33 \beta_{3} - 3 \beta_{2} - 12 \beta_{1} + 119\)\()/32\)
\(\nu^{5}\)\(=\)\((\)\(20 \beta_{11} + 21 \beta_{10} - 61 \beta_{9} - 7 \beta_{8} - 57 \beta_{7} - 4 \beta_{6} - 22 \beta_{5} - 36 \beta_{4} - 97 \beta_{3} + 197 \beta_{2} - 20 \beta_{1} - 361\)\()/32\)
\(\nu^{6}\)\(=\)\((\)\(-108 \beta_{11} + 101 \beta_{10} + 83 \beta_{9} - 79 \beta_{8} + 807 \beta_{7} + 28 \beta_{6} - 30 \beta_{5} - 68 \beta_{4} - 257 \beta_{3} + 205 \beta_{2} - 20 \beta_{1} + 7279\)\()/32\)
\(\nu^{7}\)\(=\)\((\)\(132 \beta_{11} - 171 \beta_{10} + 67 \beta_{9} + 89 \beta_{8} + 631 \beta_{7} + 172 \beta_{6} + 82 \beta_{5} - 252 \beta_{4} + 1359 \beta_{3} + 717 \beta_{2} - 4 \beta_{1} - 929\)\()/32\)
\(\nu^{8}\)\(=\)\((\)\(1284 \beta_{11} + 965 \beta_{10} + 531 \beta_{9} + 473 \beta_{8} - 537 \beta_{7} - 20 \beta_{6} + 898 \beta_{5} - 220 \beta_{4} - 2241 \beta_{3} + 1485 \beta_{2} + 636 \beta_{1} + 12831\)\()/32\)
\(\nu^{9}\)\(=\)\((\)\(-3612 \beta_{11} - 971 \beta_{10} - 4765 \beta_{9} + 25 \beta_{8} - 3785 \beta_{7} + 12 \beta_{6} + 1026 \beta_{5} - 2028 \beta_{4} - 7665 \beta_{3} - 4899 \beta_{2} + 924 \beta_{1} + 146031\)\()/32\)
\(\nu^{10}\)\(=\)\((\)\(5796 \beta_{11} + 293 \beta_{10} + 5747 \beta_{9} + 6761 \beta_{8} - 8121 \beta_{7} + 1228 \beta_{6} - 1854 \beta_{5} - 1964 \beta_{4} + 5535 \beta_{3} - 23763 \beta_{2} - 3876 \beta_{1} + 431839\)\()/32\)
\(\nu^{11}\)\(=\)\((\)\(15908 \beta_{11} - 25067 \beta_{10} - 3197 \beta_{9} + 26105 \beta_{8} - 13929 \beta_{7} - 4020 \beta_{6} + 2658 \beta_{5} + 19252 \beta_{4} - 11921 \beta_{3} + 97501 \beta_{2} - 676 \beta_{1} + 1199151\)\()/32\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
895.1
0.965027 + 2.65871i
2.82801 + 0.0488466i
−0.951271 2.66366i
−1.72458 + 2.24184i
−2.78362 0.501431i
2.16644 1.81839i
2.16644 + 1.81839i
−2.78362 + 0.501431i
−1.72458 2.24184i
−0.951271 + 2.66366i
2.82801 0.0488466i
0.965027 2.65871i
0 −3.00000 0 19.4608i 0 −1.95109 18.4172i 0 9.00000 0
895.2 0 −3.00000 0 16.6517i 0 −15.0420 + 10.8045i 0 9.00000 0
895.3 0 −3.00000 0 10.8462i 0 15.1344 + 10.6748i 0 9.00000 0
895.4 0 −3.00000 0 6.58775i 0 −15.1925 + 10.5918i 0 9.00000 0
895.5 0 −3.00000 0 4.57514i 0 −2.93118 + 18.2868i 0 9.00000 0
895.6 0 −3.00000 0 4.47531i 0 14.9825 10.8869i 0 9.00000 0
895.7 0 −3.00000 0 4.47531i 0 14.9825 + 10.8869i 0 9.00000 0
895.8 0 −3.00000 0 4.57514i 0 −2.93118 18.2868i 0 9.00000 0
895.9 0 −3.00000 0 6.58775i 0 −15.1925 10.5918i 0 9.00000 0
895.10 0 −3.00000 0 10.8462i 0 15.1344 10.6748i 0 9.00000 0
895.11 0 −3.00000 0 16.6517i 0 −15.0420 10.8045i 0 9.00000 0
895.12 0 −3.00000 0 19.4608i 0 −1.95109 + 18.4172i 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 895.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.4.b.g 12
4.b odd 2 1 1344.4.b.h 12
7.b odd 2 1 1344.4.b.h 12
8.b even 2 1 84.4.b.b yes 12
8.d odd 2 1 84.4.b.a 12
24.f even 2 1 252.4.b.f 12
24.h odd 2 1 252.4.b.e 12
28.d even 2 1 inner 1344.4.b.g 12
56.e even 2 1 84.4.b.b yes 12
56.h odd 2 1 84.4.b.a 12
168.e odd 2 1 252.4.b.e 12
168.i even 2 1 252.4.b.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.b.a 12 8.d odd 2 1
84.4.b.a 12 56.h odd 2 1
84.4.b.b yes 12 8.b even 2 1
84.4.b.b yes 12 56.e even 2 1
252.4.b.e 12 24.h odd 2 1
252.4.b.e 12 168.e odd 2 1
252.4.b.f 12 24.f even 2 1
252.4.b.f 12 168.i even 2 1
1344.4.b.g 12 1.a even 1 1 trivial
1344.4.b.g 12 28.d even 2 1 inner
1344.4.b.h 12 4.b odd 2 1
1344.4.b.h 12 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1344, [\chi])\):

\( T_{5}^{12} + 858 T_{5}^{10} + 249644 T_{5}^{8} + 29440120 T_{5}^{6} + 1456436096 T_{5}^{4} + 30453516288 T_{5}^{2} + 224760987648 \)
\( T_{19}^{6} + 42 T_{19}^{5} - 27084 T_{19}^{4} - 1835880 T_{19}^{3} + 99824480 T_{19}^{2} + 8618904192 T_{19} + 111463534080 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( ( 3 + T )^{12} \)
$5$ \( 224760987648 + 30453516288 T^{2} + 1456436096 T^{4} + 29440120 T^{6} + 249644 T^{8} + 858 T^{10} + T^{12} \)
$7$ \( 1628413597910449 + 47475615099430 T + 3626417246662 T^{2} - 33977737094 T^{3} + 10113931583 T^{4} + 460310116 T^{5} + 96999028 T^{6} + 1342012 T^{7} + 85967 T^{8} - 842 T^{9} + 262 T^{10} + 10 T^{11} + T^{12} \)
$11$ \( 584217371305280000 + 12525766859191168 T^{2} + 45887475687456 T^{4} + 64285126792 T^{6} + 39015876 T^{8} + 10414 T^{10} + T^{12} \)
$13$ \( 62601055759761408 + 2779995917647872 T^{2} + 19645357400064 T^{4} + 39003959808 T^{6} + 30771776 T^{8} + 9976 T^{10} + T^{12} \)
$17$ \( \)\(59\!\cdots\!12\)\( + 9486366618786324480 T^{2} + 8294186258955776 T^{4} + 2811705354936 T^{6} + 454587148 T^{8} + 34826 T^{10} + T^{12} \)
$19$ \( ( 111463534080 + 8618904192 T + 99824480 T^{2} - 1835880 T^{3} - 27084 T^{4} + 42 T^{5} + T^{6} )^{2} \)
$23$ \( \)\(17\!\cdots\!12\)\( + \)\(16\!\cdots\!32\)\( T^{2} + 61948036290738720 T^{4} + 11431676066248 T^{6} + 1104167940 T^{8} + 53182 T^{10} + T^{12} \)
$29$ \( ( -3698977199040 - 95119425984 T + 2688214256 T^{2} - 2852896 T^{3} - 98436 T^{4} + 100 T^{5} + T^{6} )^{2} \)
$31$ \( ( 34547086393344 - 622925807616 T + 1743943680 T^{2} + 21663744 T^{3} - 99232 T^{4} - 192 T^{5} + T^{6} )^{2} \)
$37$ \( ( 11485380096 + 1692626304 T + 29975392 T^{2} - 3382616 T^{3} - 73420 T^{4} - 122 T^{5} + T^{6} )^{2} \)
$41$ \( \)\(18\!\cdots\!00\)\( + \)\(23\!\cdots\!32\)\( T^{2} + 8929551139378138112 T^{4} + 958649263640824 T^{6} + 35274441548 T^{8} + 373146 T^{10} + T^{12} \)
$43$ \( \)\(47\!\cdots\!12\)\( + \)\(10\!\cdots\!64\)\( T^{2} + 24122851474400416256 T^{4} + 1464388464157760 T^{6} + 34667163920 T^{8} + 328092 T^{10} + T^{12} \)
$47$ \( ( 118618190512128 - 3408906829824 T + 24842032128 T^{2} + 536640 T^{3} - 302608 T^{4} + 140 T^{5} + T^{6} )^{2} \)
$53$ \( ( -24896048217408 + 584664527232 T + 5115119536 T^{2} - 37315648 T^{3} - 272924 T^{4} - 8 T^{5} + T^{6} )^{2} \)
$59$ \( ( -1915391252459520 - 29557612136448 T + 64356818688 T^{2} + 311343360 T^{3} - 572048 T^{4} - 584 T^{5} + T^{6} )^{2} \)
$61$ \( \)\(19\!\cdots\!00\)\( + \)\(41\!\cdots\!28\)\( T^{2} + \)\(13\!\cdots\!56\)\( T^{4} + 165531877369638912 T^{6} + 808904152832 T^{8} + 1590928 T^{10} + T^{12} \)
$67$ \( \)\(64\!\cdots\!68\)\( + \)\(50\!\cdots\!44\)\( T^{2} + \)\(13\!\cdots\!28\)\( T^{4} + 154614504282475968 T^{6} + 710502378800 T^{8} + 1407700 T^{10} + T^{12} \)
$71$ \( \)\(62\!\cdots\!00\)\( + \)\(22\!\cdots\!72\)\( T^{2} + \)\(30\!\cdots\!24\)\( T^{4} + 208738353017380424 T^{6} + 753608341380 T^{8} + 1375838 T^{10} + T^{12} \)
$73$ \( \)\(22\!\cdots\!32\)\( + \)\(48\!\cdots\!36\)\( T^{2} + \)\(49\!\cdots\!40\)\( T^{4} + 1948491776553389568 T^{6} + 3675179437760 T^{8} + 3228968 T^{10} + T^{12} \)
$79$ \( \)\(96\!\cdots\!88\)\( + \)\(18\!\cdots\!12\)\( T^{2} + \)\(11\!\cdots\!08\)\( T^{4} + 3351704066729440704 T^{6} + 4957994738864 T^{8} + 3584020 T^{10} + T^{12} \)
$83$ \( ( -178662844658221056 + 77003384168448 T + 1331374132224 T^{2} - 508306752 T^{3} - 2160016 T^{4} + 484 T^{5} + T^{6} )^{2} \)
$89$ \( \)\(18\!\cdots\!72\)\( + \)\(12\!\cdots\!92\)\( T^{2} + \)\(73\!\cdots\!16\)\( T^{4} + 14989058094622396408 T^{6} + 14072024042060 T^{8} + 6134874 T^{10} + T^{12} \)
$97$ \( \)\(29\!\cdots\!88\)\( + \)\(30\!\cdots\!32\)\( T^{2} + \)\(31\!\cdots\!56\)\( T^{4} + 9820780041931402752 T^{6} + 11706611210432 T^{8} + 5772328 T^{10} + T^{12} \)
show more
show less