Properties

Label 1155.2.i.c
Level 1155
Weight 2
Character orbit 1155.i
Analytic conductor 9.223
Analytic rank 0
Dimension 16
CM no
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1155.i (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.22272143346\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{8} q^{2} + \beta_{9} q^{3} + ( -2 - \beta_{3} ) q^{4} -\beta_{9} q^{5} -\beta_{14} q^{6} + ( -\beta_{7} + \beta_{14} ) q^{7} + ( -\beta_{4} + \beta_{8} ) q^{8} - q^{9} +O(q^{10})\) \( q -\beta_{8} q^{2} + \beta_{9} q^{3} + ( -2 - \beta_{3} ) q^{4} -\beta_{9} q^{5} -\beta_{14} q^{6} + ( -\beta_{7} + \beta_{14} ) q^{7} + ( -\beta_{4} + \beta_{8} ) q^{8} - q^{9} + \beta_{14} q^{10} + ( 1 + \beta_{1} - \beta_{10} ) q^{11} + ( \beta_{5} - 2 \beta_{9} ) q^{12} + ( -\beta_{11} + 2 \beta_{15} ) q^{13} + ( 1 - \beta_{1} - \beta_{5} + 4 \beta_{9} ) q^{14} + q^{15} + ( 2 + 2 \beta_{1} + \beta_{3} ) q^{16} + ( 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{17} + \beta_{8} q^{18} + ( -2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{19} + ( -\beta_{5} + 2 \beta_{9} ) q^{20} + ( -\beta_{8} - \beta_{13} ) q^{21} + ( -\beta_{1} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} ) q^{22} + ( 3 + \beta_{3} ) q^{23} + ( -\beta_{11} + \beta_{14} ) q^{24} - q^{25} + ( -2 \beta_{2} - 2 \beta_{5} + 2 \beta_{9} - 4 \beta_{12} ) q^{26} -\beta_{9} q^{27} + ( -\beta_{4} - \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} - 3 \beta_{14} ) q^{28} + ( -\beta_{4} + \beta_{8} - \beta_{10} ) q^{29} -\beta_{8} q^{30} + ( -\beta_{2} - \beta_{5} - 2 \beta_{9} - \beta_{12} ) q^{31} + ( \beta_{4} - 2 \beta_{7} - \beta_{8} - 2 \beta_{10} ) q^{32} + ( \beta_{9} - \beta_{12} - \beta_{15} ) q^{33} + ( -\beta_{2} - \beta_{5} + 2 \beta_{9} - 3 \beta_{12} ) q^{34} + ( \beta_{8} + \beta_{13} ) q^{35} + ( 2 + \beta_{3} ) q^{36} + ( -2 + \beta_{1} + \beta_{3} - \beta_{6} ) q^{37} + ( -\beta_{2} + \beta_{5} - 2 \beta_{9} + \beta_{12} ) q^{38} + ( \beta_{4} - 2 \beta_{10} ) q^{39} + ( \beta_{11} - \beta_{14} ) q^{40} + ( \beta_{11} - 2 \beta_{14} + 2 \beta_{15} ) q^{41} + ( -4 - \beta_{3} + \beta_{9} + \beta_{12} ) q^{42} + ( -3 \beta_{4} - \beta_{8} + \beta_{10} ) q^{43} + ( -3 - 2 \beta_{1} - 3 \beta_{3} - \beta_{4} + \beta_{6} - \beta_{8} + 2 \beta_{10} ) q^{44} + \beta_{9} q^{45} + ( \beta_{4} - 4 \beta_{8} ) q^{46} + ( -\beta_{2} - \beta_{5} + 8 \beta_{9} - \beta_{12} ) q^{47} + ( -\beta_{5} + 2 \beta_{9} - 2 \beta_{12} ) q^{48} + ( 1 + 2 \beta_{3} - 2 \beta_{9} - 2 \beta_{12} ) q^{49} + \beta_{8} q^{50} + ( -2 \beta_{7} - \beta_{8} - \beta_{10} ) q^{51} + ( 4 \beta_{11} - 2 \beta_{13} - 2 \beta_{14} - 6 \beta_{15} ) q^{52} + ( -1 - 2 \beta_{1} - 3 \beta_{3} + 2 \beta_{6} ) q^{53} + \beta_{14} q^{54} + ( -\beta_{9} + \beta_{12} + \beta_{15} ) q^{55} + ( 1 + \beta_{3} + 3 \beta_{5} - \beta_{6} - 6 \beta_{9} + 2 \beta_{12} ) q^{56} + ( 2 \beta_{7} + \beta_{8} - \beta_{10} ) q^{57} + ( 6 + \beta_{1} + 3 \beta_{3} + \beta_{6} ) q^{58} + ( \beta_{2} - 3 \beta_{9} + \beta_{12} ) q^{59} + ( -2 - \beta_{3} ) q^{60} + ( -2 \beta_{11} - 2 \beta_{13} - \beta_{14} + 3 \beta_{15} ) q^{61} + ( 2 \beta_{11} + 2 \beta_{14} - 4 \beta_{15} ) q^{62} + ( \beta_{7} - \beta_{14} ) q^{63} + ( -2 \beta_{1} - \beta_{3} + 2 \beta_{6} ) q^{64} + ( -\beta_{4} + 2 \beta_{10} ) q^{65} + ( \beta_{2} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{66} + ( 4 + 2 \beta_{1} - 2 \beta_{3} ) q^{67} + ( 4 \beta_{11} + 2 \beta_{13} - 4 \beta_{15} ) q^{68} + ( -\beta_{5} + 3 \beta_{9} ) q^{69} + ( 4 + \beta_{3} - \beta_{9} - \beta_{12} ) q^{70} + ( -2 - 3 \beta_{1} - \beta_{3} + \beta_{6} ) q^{71} + ( \beta_{4} - \beta_{8} ) q^{72} + ( -2 \beta_{11} - 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{73} + ( 2 \beta_{4} + 2 \beta_{8} - 4 \beta_{10} ) q^{74} -\beta_{9} q^{75} + ( -2 \beta_{11} - 2 \beta_{13} + 2 \beta_{14} ) q^{76} + ( -\beta_{2} - \beta_{4} - \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{77} + ( -2 - 4 \beta_{1} - 2 \beta_{3} + 2 \beta_{6} ) q^{78} + ( \beta_{4} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{10} ) q^{79} + ( \beta_{5} - 2 \beta_{9} + 2 \beta_{12} ) q^{80} + q^{81} + ( -2 \beta_{2} + 4 \beta_{5} - 10 \beta_{9} ) q^{82} + ( 2 \beta_{11} + 2 \beta_{13} + \beta_{14} - 5 \beta_{15} ) q^{83} + ( -\beta_{4} + 3 \beta_{8} - \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{84} + ( 2 \beta_{7} + \beta_{8} + \beta_{10} ) q^{85} + ( 2 + 7 \beta_{1} + 5 \beta_{3} - \beta_{6} ) q^{86} + ( -\beta_{11} + \beta_{14} - \beta_{15} ) q^{87} + ( -2 + 2 \beta_{1} + \beta_{3} - 3 \beta_{4} - \beta_{7} + 3 \beta_{8} + 3 \beta_{10} ) q^{88} + ( \beta_{2} - 3 \beta_{9} - \beta_{12} ) q^{89} -\beta_{14} q^{90} + ( 2 + 4 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{5} - 2 \beta_{6} + 2 \beta_{9} + \beta_{12} ) q^{91} + ( -12 - 2 \beta_{1} - 4 \beta_{3} ) q^{92} + ( 2 - \beta_{1} - \beta_{3} + \beta_{6} ) q^{93} + ( 2 \beta_{11} - 8 \beta_{14} - 4 \beta_{15} ) q^{94} + ( -2 \beta_{7} - \beta_{8} + \beta_{10} ) q^{95} + ( \beta_{11} - 2 \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{96} + ( \beta_{2} + 2 \beta_{5} + 5 \beta_{9} + \beta_{12} ) q^{97} + ( 2 \beta_{4} - 3 \beta_{8} + 2 \beta_{11} - 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{98} + ( -1 - \beta_{1} + \beta_{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 24q^{4} - 16q^{9} + O(q^{10}) \) \( 16q - 24q^{4} - 16q^{9} + 16q^{11} + 16q^{14} + 16q^{15} + 24q^{16} + 40q^{23} - 16q^{25} + 24q^{36} - 40q^{37} - 56q^{42} - 24q^{44} + 8q^{53} + 8q^{56} + 72q^{58} - 24q^{60} + 8q^{64} + 80q^{67} + 56q^{70} - 24q^{71} - 16q^{78} + 16q^{81} - 8q^{86} - 40q^{88} + 16q^{91} - 160q^{92} + 40q^{93} - 16q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + x^{14} - 4 x^{12} - 49 x^{10} + 11 x^{8} + 395 x^{6} + 900 x^{4} + 1125 x^{2} + 625\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -4976 \nu^{14} + 6064 \nu^{12} - 53806 \nu^{10} + 1664 \nu^{8} + 584304 \nu^{6} + 1397920 \nu^{4} + 1900400 \nu^{2} - 33766125 \)\()/15616375\)
\(\beta_{2}\)\(=\)\((\)\( 52907 \nu^{15} + 3769822 \nu^{13} + 7571087 \nu^{11} - 23218303 \nu^{9} - 220485308 \nu^{7} - 61141995 \nu^{5} + 2011459400 \nu^{3} + 5720425875 \nu \)\()/ 858900625 \)
\(\beta_{3}\)\(=\)\((\)\( 75631 \nu^{14} + 26336 \nu^{12} - 423219 \nu^{10} - 3371514 \nu^{8} + 2776621 \nu^{6} + 34025725 \nu^{4} + 47891850 \nu^{2} + 32055875 \)\()/15616375\)
\(\beta_{4}\)\(=\)\((\)\( 1098448 \nu^{14} - 1820672 \nu^{12} - 3507762 \nu^{10} - 38951322 \nu^{8} + 145662708 \nu^{6} + 254483540 \nu^{4} - 92662550 \nu^{2} - 101116250 \)\()/ 171780125 \)
\(\beta_{5}\)\(=\)\((\)\( 739864 \nu^{15} - 254886 \nu^{13} - 6858931 \nu^{11} - 26926811 \nu^{9} + 104639904 \nu^{7} + 341075305 \nu^{5} - 72330250 \nu^{3} - 1014860875 \nu \)\()/ 858900625 \)
\(\beta_{6}\)\(=\)\((\)\( 64321 \nu^{14} + 44136 \nu^{12} - 365249 \nu^{10} - 2994144 \nu^{8} + 2128291 \nu^{6} + 28658755 \nu^{4} + 40690320 \nu^{2} + 30499350 \)\()/3123275\)
\(\beta_{7}\)\(=\)\((\)\( -3708119 \nu^{14} + 2186986 \nu^{12} + 13453081 \nu^{10} + 157291286 \nu^{8} - 302484579 \nu^{6} - 1049398975 \nu^{4} - 1461240500 \nu^{2} - 1320364000 \)\()/ 171780125 \)
\(\beta_{8}\)\(=\)\((\)\( 1073735 \nu^{14} - 529596 \nu^{12} - 3783171 \nu^{10} - 46979616 \nu^{8} + 81866729 \nu^{6} + 310606859 \nu^{4} + 477661630 \nu^{2} + 434987125 \)\()/34356025\)
\(\beta_{9}\)\(=\)\((\)\( 2293 \nu^{15} + 486 \nu^{13} - 12284 \nu^{11} - 101834 \nu^{9} + 117086 \nu^{7} + 906528 \nu^{5} + 1102430 \nu^{3} + 955700 \nu \)\()/633875\)
\(\beta_{10}\)\(=\)\((\)\( -1446284 \nu^{14} + 355734 \nu^{12} + 4966964 \nu^{10} + 65045254 \nu^{8} - 95756546 \nu^{6} - 437603572 \nu^{4} - 798277730 \nu^{2} - 730658725 \)\()/34356025\)
\(\beta_{11}\)\(=\)\((\)\( 515378 \nu^{15} + 520818 \nu^{13} - 2369422 \nu^{11} - 27002382 \nu^{9} + 8880998 \nu^{7} + 226234550 \nu^{5} + 509713400 \nu^{3} + 379881750 \nu \)\()/78081875\)
\(\beta_{12}\)\(=\)\((\)\( 599 \nu^{15} + 90 \nu^{13} - 3000 \nu^{11} - 26110 \nu^{9} + 31110 \nu^{7} + 237486 \nu^{5} + 287550 \nu^{3} + 249500 \nu \)\()/74525\)
\(\beta_{13}\)\(=\)\((\)\( -1468861 \nu^{15} + 201274 \nu^{13} + 5426879 \nu^{11} + 66369199 \nu^{9} - 90901286 \nu^{7} - 466969935 \nu^{5} - 843562400 \nu^{3} - 722601875 \nu \)\()/78081875\)
\(\beta_{14}\)\(=\)\((\)\( 2100604 \nu^{15} - 547346 \nu^{13} - 7374841 \nu^{11} - 94005021 \nu^{9} + 141150094 \nu^{7} + 639251405 \nu^{5} + 1106504500 \nu^{3} + 977629125 \nu \)\()/78081875\)
\(\beta_{15}\)\(=\)\((\)\( -2699561 \nu^{15} + 1648844 \nu^{13} + 8983874 \nu^{11} + 116269094 \nu^{9} - 219278916 \nu^{7} - 742870540 \nu^{5} - 1128269700 \nu^{3} - 1087173250 \nu \)\()/78081875\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{15} - \beta_{14} + \beta_{13} + \beta_{11} + \beta_{9} + \beta_{5} + \beta_{2}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{10} - 3 \beta_{8} - 3 \beta_{7} - \beta_{6} - \beta_{4} + 3 \beta_{3} + 2 \beta_{1} + 1\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{15} - 7 \beta_{14} - 9 \beta_{13} + \beta_{12} - 7 \beta_{9} + 2 \beta_{5}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-3 \beta_{10} + \beta_{8} + 11 \beta_{7} - \beta_{6} + 9 \beta_{4} + 9 \beta_{3} + 2 \beta_{1} + 9\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-3 \beta_{15} - \beta_{14} + 7 \beta_{13} + 10 \beta_{12} + 6 \beta_{11} - 18 \beta_{9}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-15 \beta_{10} - 31 \beta_{8} - 13 \beta_{7} + 2 \beta_{6} + 16 \beta_{4} - 16 \beta_{3} + 29 \beta_{1} + 59\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-12 \beta_{15} + 4 \beta_{14} + 34 \beta_{13} + 11 \beta_{12} + 23 \beta_{11} - 58 \beta_{9} + 99 \beta_{5} + 23 \beta_{2}\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(-53 \beta_{10} - 191 \beta_{8} - 167 \beta_{7} - 15 \beta_{6} + 15 \beta_{4} + 69 \beta_{3} - 38 \beta_{1} - 61\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(39 \beta_{15} - 153 \beta_{14} - 343 \beta_{13} + 191 \beta_{12} - 152 \beta_{11} - 457 \beta_{9} + 152 \beta_{5} + 38 \beta_{2}\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(-124 \beta_{10} - 28 \beta_{8} + 276 \beta_{7} + 248 \beta_{4} - 85 \beta_{1} - 199\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(150 \beta_{15} + 838 \beta_{14} + 954 \beta_{13} + 857 \beta_{12} + 97 \beta_{11} - 1942 \beta_{9} + 97 \beta_{5} + 19 \beta_{2}\)\()/4\)
\(\nu^{12}\)\(=\)\((\)\(-571 \beta_{10} - 1453 \beta_{8} - 843 \beta_{7} + 494 \beta_{6} + 494 \beta_{4} - 2092 \beta_{3} + 77 \beta_{1} - 299\)\()/4\)
\(\nu^{13}\)\(=\)\((\)\(1104 \beta_{15} + 2808 \beta_{14} + 1666 \beta_{13} - 169 \beta_{12} - 935 \beta_{11} - 542 \beta_{9} + 3947 \beta_{5} + 935 \beta_{2}\)\()/4\)
\(\nu^{14}\)\(=\)\((\)\(-1018 \beta_{10} - 5734 \beta_{8} - 6634 \beta_{7} + 169 \beta_{6} - 731 \beta_{4} - 731 \beta_{3} - 5903 \beta_{1} - 13386\)\()/4\)
\(\nu^{15}\)\(=\)\(1546 \beta_{15} + 367 \beta_{14} - 3459 \beta_{13} + 1040 \beta_{12} - 3092 \beta_{11} - 2334 \beta_{9}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1155\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(232\) \(386\) \(661\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
−1.86824 + 0.357358i
1.86824 0.357358i
−0.0566033 1.17421i
0.0566033 + 1.17421i
−0.644389 + 0.983224i
0.644389 0.983224i
0.917186 1.66637i
−0.917186 + 1.66637i
−0.917186 1.66637i
0.917186 + 1.66637i
0.644389 + 0.983224i
−0.644389 0.983224i
0.0566033 1.17421i
−0.0566033 + 1.17421i
1.86824 + 0.357358i
−1.86824 0.357358i
2.60278i 1.00000i −4.77447 1.00000i −2.60278 2.60278 0.474903i 7.22133i −1.00000 2.60278
76.2 2.60278i 1.00000i −4.77447 1.00000i 2.60278 −2.60278 0.474903i 7.22133i −1.00000 −2.60278
76.3 2.19849i 1.00000i −2.83337 1.00000i −2.19849 2.19849 + 1.47195i 1.83215i −1.00000 2.19849
76.4 2.19849i 1.00000i −2.83337 1.00000i 2.19849 −2.19849 + 1.47195i 1.83215i −1.00000 −2.19849
76.5 1.47195i 1.00000i −0.166634 1.00000i −1.47195 1.47195 + 2.19849i 2.69862i −1.00000 1.47195
76.6 1.47195i 1.00000i −0.166634 1.00000i 1.47195 −1.47195 + 2.19849i 2.69862i −1.00000 −1.47195
76.7 0.474903i 1.00000i 1.77447 1.00000i −0.474903 0.474903 2.60278i 1.79251i −1.00000 0.474903
76.8 0.474903i 1.00000i 1.77447 1.00000i 0.474903 −0.474903 2.60278i 1.79251i −1.00000 −0.474903
76.9 0.474903i 1.00000i 1.77447 1.00000i 0.474903 −0.474903 + 2.60278i 1.79251i −1.00000 −0.474903
76.10 0.474903i 1.00000i 1.77447 1.00000i −0.474903 0.474903 + 2.60278i 1.79251i −1.00000 0.474903
76.11 1.47195i 1.00000i −0.166634 1.00000i 1.47195 −1.47195 2.19849i 2.69862i −1.00000 −1.47195
76.12 1.47195i 1.00000i −0.166634 1.00000i −1.47195 1.47195 2.19849i 2.69862i −1.00000 1.47195
76.13 2.19849i 1.00000i −2.83337 1.00000i 2.19849 −2.19849 1.47195i 1.83215i −1.00000 −2.19849
76.14 2.19849i 1.00000i −2.83337 1.00000i −2.19849 2.19849 1.47195i 1.83215i −1.00000 2.19849
76.15 2.60278i 1.00000i −4.77447 1.00000i 2.60278 −2.60278 + 0.474903i 7.22133i −1.00000 −2.60278
76.16 2.60278i 1.00000i −4.77447 1.00000i −2.60278 2.60278 + 0.474903i 7.22133i −1.00000 2.60278
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.b odd 2 1 inner
77.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1155.2.i.c 16
7.b odd 2 1 inner 1155.2.i.c 16
11.b odd 2 1 inner 1155.2.i.c 16
77.b even 2 1 inner 1155.2.i.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.i.c 16 1.a even 1 1 trivial
1155.2.i.c 16 7.b odd 2 1 inner
1155.2.i.c 16 11.b odd 2 1 inner
1155.2.i.c 16 77.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 14 T_{2}^{6} + 61 T_{2}^{4} + 84 T_{2}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(1155, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T^{2} + 5 T^{4} - 12 T^{6} + 24 T^{8} - 48 T^{10} + 80 T^{12} - 128 T^{14} + 256 T^{16} )^{2} \)
$3$ \( ( 1 + T^{2} )^{8} \)
$5$ \( ( 1 + T^{2} )^{8} \)
$7$ \( 1 - 4 T^{4} - 314 T^{8} - 9604 T^{12} + 5764801 T^{16} \)
$11$ \( ( 1 - 4 T + 6 T^{2} - 44 T^{3} + 121 T^{4} )^{4} \)
$13$ \( ( 1 + 20 T^{2} + 116 T^{4} + 3340 T^{6} + 82486 T^{8} + 564460 T^{10} + 3313076 T^{12} + 96536180 T^{14} + 815730721 T^{16} )^{2} \)
$17$ \( ( 1 + 62 T^{2} + 2165 T^{4} + 53542 T^{6} + 1011484 T^{8} + 15473638 T^{10} + 180822965 T^{12} + 1496529278 T^{14} + 6975757441 T^{16} )^{2} \)
$19$ \( ( 1 + 78 T^{2} + 3413 T^{4} + 101206 T^{6} + 2229420 T^{8} + 36535366 T^{10} + 444785573 T^{12} + 3669578718 T^{14} + 16983563041 T^{16} )^{2} \)
$23$ \( ( 1 - 10 T + 117 T^{2} - 690 T^{3} + 4304 T^{4} - 15870 T^{5} + 61893 T^{6} - 121670 T^{7} + 279841 T^{8} )^{4} \)
$29$ \( ( 1 - 126 T^{2} + 8965 T^{4} - 422054 T^{6} + 14346764 T^{8} - 354947414 T^{10} + 6340774165 T^{12} - 74947738446 T^{14} + 500246412961 T^{16} )^{2} \)
$31$ \( ( 1 - 148 T^{2} + 10868 T^{4} - 529836 T^{6} + 18939030 T^{8} - 509172396 T^{10} + 10036826228 T^{12} - 131350544788 T^{14} + 852891037441 T^{16} )^{2} \)
$37$ \( ( 1 + 10 T + 148 T^{2} + 990 T^{3} + 8294 T^{4} + 36630 T^{5} + 202612 T^{6} + 506530 T^{7} + 1874161 T^{8} )^{4} \)
$41$ \( ( 1 + 92 T^{2} + 7508 T^{4} + 417444 T^{6} + 19059990 T^{8} + 701723364 T^{10} + 21215813588 T^{12} + 437009590172 T^{14} + 7984925229121 T^{16} )^{2} \)
$43$ \( ( 1 + 2 T^{2} + 3245 T^{4} + 20322 T^{6} + 8864604 T^{8} + 37575378 T^{10} + 11094009245 T^{12} + 12642726098 T^{14} + 11688200277601 T^{16} )^{2} \)
$47$ \( ( 1 - 76 T^{2} + 2612 T^{4} - 154868 T^{6} + 10677910 T^{8} - 342103412 T^{10} + 12745726772 T^{12} - 819220365004 T^{14} + 23811286661761 T^{16} )^{2} \)
$53$ \( ( 1 - 2 T + 21 T^{2} + 214 T^{3} + 2384 T^{4} + 11342 T^{5} + 58989 T^{6} - 297754 T^{7} + 7890481 T^{8} )^{4} \)
$59$ \( ( 1 - 366 T^{2} + 63505 T^{4} - 6759614 T^{6} + 481830644 T^{8} - 23530216334 T^{10} + 769513010305 T^{12} - 15438075312606 T^{14} + 146830437604321 T^{16} )^{2} \)
$61$ \( ( 1 + 262 T^{2} + 39293 T^{4} + 3890814 T^{6} + 278453340 T^{8} + 14477718894 T^{10} + 544044630413 T^{12} + 13498338082582 T^{14} + 191707312997281 T^{16} )^{2} \)
$67$ \( ( 1 - 20 T + 328 T^{2} - 3940 T^{3} + 34654 T^{4} - 263980 T^{5} + 1472392 T^{6} - 6015260 T^{7} + 20151121 T^{8} )^{4} \)
$71$ \( ( 1 + 6 T + 200 T^{2} + 974 T^{3} + 18254 T^{4} + 69154 T^{5} + 1008200 T^{6} + 2147466 T^{7} + 25411681 T^{8} )^{4} \)
$73$ \( ( 1 + 408 T^{2} + 81340 T^{4} + 10280488 T^{6} + 894736454 T^{8} + 54784720552 T^{10} + 2309912922940 T^{12} + 61744364325912 T^{14} + 806460091894081 T^{16} )^{2} \)
$79$ \( ( 1 - 332 T^{2} + 59108 T^{4} - 7101044 T^{6} + 639039030 T^{8} - 44317615604 T^{10} + 2302261387748 T^{12} - 80705035232972 T^{14} + 1517108809906561 T^{16} )^{2} \)
$83$ \( ( 1 + 198 T^{2} + 31805 T^{4} + 3108638 T^{6} + 297770844 T^{8} + 21415407182 T^{10} + 1509411899405 T^{12} + 64734193927062 T^{14} + 2252292232139041 T^{16} )^{2} \)
$89$ \( ( 1 - 566 T^{2} + 144105 T^{4} - 22231174 T^{6} + 2346778004 T^{8} - 176093129254 T^{10} + 9041470639305 T^{12} - 281291410683926 T^{14} + 3936588805702081 T^{16} )^{2} \)
$97$ \( ( 1 - 502 T^{2} + 125065 T^{4} - 20242182 T^{6} + 2314322804 T^{8} - 190458690438 T^{10} + 11071914528265 T^{12} - 418151946474358 T^{14} + 7837433594376961 T^{16} )^{2} \)
show more
show less