Properties

Label 315.2.bf.c
Level 315
Weight 2
Character orbit 315.bf
Analytic conductor 2.515
Analytic rank 0
Dimension 16
CM no
Inner twists 8

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Newspace parameters

Level: \( N \) = \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 315.bf (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} + ( 1 + \beta_{5} - \beta_{8} ) q^{4} + ( \beta_{1} + \beta_{9} - \beta_{14} ) q^{5} -\beta_{4} q^{7} + ( -\beta_{1} + \beta_{3} - \beta_{9} - 2 \beta_{10} + \beta_{12} + \beta_{14} ) q^{8} +O(q^{10})\) \( q + \beta_{9} q^{2} + ( 1 + \beta_{5} - \beta_{8} ) q^{4} + ( \beta_{1} + \beta_{9} - \beta_{14} ) q^{5} -\beta_{4} q^{7} + ( -\beta_{1} + \beta_{3} - \beta_{9} - 2 \beta_{10} + \beta_{12} + \beta_{14} ) q^{8} + ( 1 - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{10} + ( \beta_{11} - 2 \beta_{15} ) q^{11} + ( \beta_{2} + \beta_{4} ) q^{13} + ( -\beta_{1} + \beta_{3} - \beta_{9} + \beta_{11} - \beta_{12} + \beta_{14} ) q^{14} + 2 \beta_{5} q^{16} + ( -2 \beta_{1} + 2 \beta_{3} - 3 \beta_{9} - 3 \beta_{10} + 2 \beta_{12} ) q^{17} + ( -2 \beta_{5} + \beta_{7} ) q^{19} + ( -\beta_{10} - \beta_{11} + 2 \beta_{12} - \beta_{15} ) q^{20} + ( -\beta_{2} - 2 \beta_{4} + \beta_{6} + \beta_{13} ) q^{22} + ( \beta_{9} - 2 \beta_{14} ) q^{23} + ( -2 - \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{8} - \beta_{13} ) q^{25} + ( 2 \beta_{1} + \beta_{9} - 2 \beta_{11} - \beta_{14} + \beta_{15} ) q^{26} + ( 3 \beta_{2} + \beta_{6} - \beta_{13} ) q^{28} + ( -\beta_{1} + \beta_{3} - \beta_{9} + \beta_{11} - \beta_{12} + \beta_{14} + \beta_{15} ) q^{29} + ( -5 - 5 \beta_{5} ) q^{31} + ( 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{12} ) q^{32} + ( -5 - 3 \beta_{7} + 3 \beta_{8} ) q^{34} + ( \beta_{1} - 2 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{14} - \beta_{15} ) q^{35} + ( \beta_{2} - \beta_{13} ) q^{37} + ( \beta_{1} - \beta_{3} + 5 \beta_{9} + 5 \beta_{10} - \beta_{12} ) q^{38} + ( -2 \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{13} ) q^{40} + ( 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{9} + \beta_{11} + 2 \beta_{12} - 2 \beta_{14} + \beta_{15} ) q^{41} + ( 2 \beta_{2} - \beta_{4} + 3 \beta_{6} + 3 \beta_{13} ) q^{43} + ( -6 \beta_{1} - 3 \beta_{9} + 3 \beta_{14} ) q^{44} + ( 1 + \beta_{5} + \beta_{8} ) q^{46} + ( \beta_{9} + \beta_{14} ) q^{47} + ( 2 \beta_{7} + \beta_{8} ) q^{49} + ( 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{9} + 5 \beta_{10} - 2 \beta_{14} ) q^{50} + ( -3 \beta_{2} + 3 \beta_{4} - 2 \beta_{6} - \beta_{13} ) q^{52} + ( 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{12} ) q^{53} + ( -2 \beta_{2} - \beta_{4} - \beta_{6} + 3 \beta_{7} - 3 \beta_{8} - \beta_{13} ) q^{55} + ( 2 \beta_{3} - \beta_{9} - 2 \beta_{11} - 2 \beta_{12} + \beta_{14} + 2 \beta_{15} ) q^{56} + ( 4 \beta_{2} + \beta_{4} + \beta_{6} - 4 \beta_{13} ) q^{58} + ( -\beta_{1} - \beta_{3} - \beta_{11} + \beta_{12} + 2 \beta_{15} ) q^{59} + ( 3 \beta_{5} - \beta_{7} ) q^{61} + 5 \beta_{10} q^{62} + ( 6 + 2 \beta_{7} - 2 \beta_{8} ) q^{64} + ( -\beta_{1} - 2 \beta_{9} - 2 \beta_{11} - 2 \beta_{14} + \beta_{15} ) q^{65} + ( -2 \beta_{2} + 2 \beta_{4} - 3 \beta_{6} + \beta_{13} ) q^{67} + ( -9 \beta_{9} + \beta_{14} ) q^{68} + ( 7 \beta_{5} - 2 \beta_{7} - \beta_{8} - \beta_{13} ) q^{70} + ( \beta_{1} - \beta_{3} + \beta_{9} - \beta_{11} + \beta_{12} - \beta_{14} - \beta_{15} ) q^{71} + ( \beta_{2} - \beta_{4} - \beta_{6} + 2 \beta_{13} ) q^{73} + ( 2 \beta_{1} + 2 \beta_{3} - \beta_{11} - 2 \beta_{12} + 2 \beta_{15} ) q^{74} + ( 9 + 3 \beta_{7} - 3 \beta_{8} ) q^{76} + ( 5 \beta_{1} - 5 \beta_{3} + \beta_{10} - 5 \beta_{12} - 4 \beta_{14} ) q^{77} + ( 2 \beta_{5} - 3 \beta_{7} ) q^{79} -2 \beta_{3} q^{80} + ( -2 \beta_{2} + \beta_{4} + \beta_{6} + 2 \beta_{13} ) q^{82} + ( -4 \beta_{1} + 4 \beta_{3} - 4 \beta_{9} - 3 \beta_{10} + 4 \beta_{12} + 4 \beta_{14} ) q^{83} + ( 3 - 2 \beta_{2} - 3 \beta_{4} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{13} ) q^{85} + ( -8 \beta_{1} - 4 \beta_{9} + 2 \beta_{11} + 4 \beta_{14} - \beta_{15} ) q^{86} + ( 4 \beta_{2} - 4 \beta_{4} + 2 \beta_{6} + 2 \beta_{13} ) q^{88} + ( 2 \beta_{11} - \beta_{15} ) q^{89} + ( 7 - 2 \beta_{7} - \beta_{8} ) q^{91} + ( -3 \beta_{1} + 3 \beta_{3} - 3 \beta_{9} + 3 \beta_{12} + 3 \beta_{14} ) q^{92} + ( 4 + 4 \beta_{5} - 2 \beta_{8} ) q^{94} + ( 2 \beta_{1} + \beta_{3} + \beta_{9} + \beta_{10} - \beta_{11} - 2 \beta_{12} + 2 \beta_{15} ) q^{95} + ( -\beta_{2} + 2 \beta_{4} - 3 \beta_{6} - 3 \beta_{13} ) q^{97} + ( 3 \beta_{1} - 3 \beta_{3} + 7 \beta_{9} + 9 \beta_{10} - 3 \beta_{12} - \beta_{14} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 8q^{4} + O(q^{10}) \) \( 16q + 8q^{4} + 8q^{10} - 16q^{16} + 16q^{19} - 16q^{25} - 40q^{31} - 80q^{34} + 8q^{40} + 8q^{46} - 24q^{61} + 96q^{64} - 56q^{70} + 144q^{76} - 16q^{79} + 48q^{85} + 112q^{91} + 32q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 8 x^{14} + 26 x^{12} - 96 x^{10} - 781 x^{8} - 2400 x^{6} + 16250 x^{4} + 125000 x^{2} + 390625\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 1547 \nu^{14} - 7449 \nu^{12} - 12128 \nu^{10} - 110837 \nu^{8} + 1254368 \nu^{6} + 226775 \nu^{4} + 27253125 \nu^{2} - 125000000 \)\()/29812500\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{15} + 8 \nu^{13} + 26 \nu^{11} - 96 \nu^{9} - 781 \nu^{7} - 2400 \nu^{5} + 16250 \nu^{3} + 125000 \nu \)\()/78125\)
\(\beta_{4}\)\(=\)\((\)\( -2269 \nu^{14} - 10152 \nu^{12} + 11881 \nu^{10} + 183949 \nu^{8} + 1464089 \nu^{6} - 321775 \nu^{4} - 38831250 \nu^{2} - 197046875 \)\()/29812500\)
\(\beta_{5}\)\(=\)\((\)\( -3848 \nu^{14} - 43134 \nu^{12} - 8848 \nu^{10} + 365183 \nu^{8} + 4255888 \nu^{6} - 359450 \nu^{4} - 64155000 \nu^{2} - 679046875 \)\()/22359375\)
\(\beta_{6}\)\(=\)\((\)\( 34369 \nu^{14} - 129048 \nu^{12} + 47219 \nu^{10} - 3327799 \nu^{8} + 12561811 \nu^{6} - 2653475 \nu^{4} + 514376250 \nu^{2} - 2057359375 \)\()/89437500\)
\(\beta_{7}\)\(=\)\((\)\( -26311 \nu^{14} - 69888 \nu^{12} - 2411 \nu^{10} + 2573956 \nu^{8} + 6311291 \nu^{6} - 6497200 \nu^{4} - 409053750 \nu^{2} - 1034187500 \)\()/44718750\)
\(\beta_{8}\)\(=\)\((\)\( 28067 \nu^{14} + 78936 \nu^{12} + 16192 \nu^{10} - 2932532 \nu^{8} - 7788352 \nu^{6} + 657800 \nu^{4} + 462961875 \nu^{2} + 1201750000 \)\()/44718750\)
\(\beta_{9}\)\(=\)\((\)\( -11438 \nu^{15} - 121929 \nu^{13} - 3913 \nu^{11} + 1158248 \nu^{9} + 11684503 \nu^{7} - 1505000 \nu^{5} - 183421875 \nu^{3} - 1885759375 \nu \)\()/89437500\)
\(\beta_{10}\)\(=\)\((\)\( -2267 \nu^{15} - 1911 \nu^{13} - 392 \nu^{11} + 210107 \nu^{9} + 188552 \nu^{7} - 15925 \nu^{5} - 35898125 \nu^{3} - 29093750 \nu \)\()/16562500\)
\(\beta_{11}\)\(=\)\((\)\( 14233 \nu^{15} - 15576 \nu^{13} + 413 \nu^{11} - 1507933 \nu^{9} + 2048017 \nu^{7} + 2093815 \nu^{5} + 240747000 \nu^{3} - 274553125 \nu \)\()/89437500\)
\(\beta_{12}\)\(=\)\((\)\( -3848 \nu^{15} - 43134 \nu^{13} - 8848 \nu^{11} + 365183 \nu^{9} + 4255888 \nu^{7} - 359450 \nu^{5} - 64155000 \nu^{3} - 656687500 \nu \)\()/22359375\)
\(\beta_{13}\)\(=\)\((\)\( -98366 \nu^{14} - 511053 \nu^{12} - 31141 \nu^{10} + 9670886 \nu^{8} + 50252971 \nu^{6} - 5836850 \nu^{4} - 1547979375 \nu^{2} - 7873140625 \)\()/89437500\)
\(\beta_{14}\)\(=\)\((\)\( 36974 \nu^{15} + 116817 \nu^{13} + 12649 \nu^{11} - 3744104 \nu^{9} - 12118219 \nu^{7} + 4865000 \nu^{5} + 592921875 \nu^{3} + 1872296875 \nu \)\()/ 149062500 \)
\(\beta_{15}\)\(=\)\((\)\( -122416 \nu^{15} - 333453 \nu^{13} - 86441 \nu^{11} + 11729686 \nu^{9} + 30344771 \nu^{7} - 13896850 \nu^{5} - 1838045625 \nu^{3} - 4889515625 \nu \)\()/ 447187500 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{13} + \beta_{8} - 2 \beta_{5} - \beta_{4} + \beta_{2} - 2\)
\(\nu^{3}\)\(=\)\(2 \beta_{15} - \beta_{14} - 2 \beta_{12} + 2 \beta_{11} - 2 \beta_{10} + \beta_{9} - \beta_{3} + \beta_{1}\)
\(\nu^{4}\)\(=\)\(4 \beta_{13} - 8 \beta_{7} - 2 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} - 4 \beta_{2}\)
\(\nu^{5}\)\(=\)\(-12 \beta_{15} - 17 \beta_{12} + 6 \beta_{11} + 26 \beta_{10} + 26 \beta_{9} - 22 \beta_{3} + 17 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-3 \beta_{13} - \beta_{8} + \beta_{7} - 3 \beta_{6} + 32 \beta_{4} + 29 \beta_{2} + 50\)
\(\nu^{7}\)\(=\)\(-28 \beta_{15} - 89 \beta_{14} + 56 \beta_{11} - 43 \beta_{9} + 77 \beta_{1}\)
\(\nu^{8}\)\(=\)\(44 \beta_{13} - 96 \beta_{8} + 104 \beta_{6} - 289 \beta_{5} - 148 \beta_{4} + 148 \beta_{2} - 289\)
\(\nu^{9}\)\(=\)\(-52 \beta_{15} - 324 \beta_{14} - 325 \beta_{12} - 52 \beta_{11} - 468 \beta_{10} + 324 \beta_{9} - 324 \beta_{3} + 324 \beta_{1}\)
\(\nu^{10}\)\(=\)\(416 \beta_{13} - 169 \beta_{7} + 429 \beta_{6} - 1438 \beta_{5} + 429 \beta_{4} - 416 \beta_{2}\)
\(\nu^{11}\)\(=\)\(-1196 \beta_{15} - 1594 \beta_{12} + 598 \beta_{11} + 2678 \beta_{10} + 2678 \beta_{9} + 949 \beta_{3} + 1594 \beta_{1}\)
\(\nu^{12}\)\(=\)\(-486 \beta_{13} - 200 \beta_{8} + 200 \beta_{7} - 486 \beta_{6} + 3276 \beta_{4} + 2790 \beta_{2} - 10611\)
\(\nu^{13}\)\(=\)\(-2590 \beta_{15} - 8770 \beta_{14} + 5180 \beta_{11} - 5020 \beta_{9} - 8107 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-11697 \beta_{13} - 24647 \beta_{8} + 11200 \beta_{6} + 3694 \beta_{5} + 497 \beta_{4} - 497 \beta_{2} + 3694\)
\(\nu^{15}\)\(=\)\(-36344 \beta_{15} - 14203 \beta_{14} + 1944 \beta_{12} - 36344 \beta_{11} - 19656 \beta_{10} + 14203 \beta_{9} - 14203 \beta_{3} + 14203 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(-1\) \(\beta_{5}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
−0.733576 2.11231i
1.46253 + 1.69145i
0.717291 2.11790i
2.19280 + 0.437757i
−2.19280 0.437757i
−0.717291 + 2.11790i
−1.46253 1.69145i
0.733576 + 2.11231i
−0.733576 + 2.11231i
1.46253 1.69145i
0.717291 + 2.11790i
2.19280 0.437757i
−2.19280 + 0.437757i
−0.717291 2.11790i
−1.46253 + 1.69145i
0.733576 2.11231i
−2.05774 + 1.18804i 0 1.82288 3.15731i −1.46253 1.69145i 0 −2.03996 1.68480i 3.91044i 0 5.01902 + 1.74303i
109.2 −2.05774 + 1.18804i 0 1.82288 3.15731i 0.733576 + 2.11231i 0 2.03996 + 1.68480i 3.91044i 0 −4.01902 3.47508i
109.3 −0.515448 + 0.297594i 0 −0.822876 + 1.42526i −2.19280 0.437757i 0 −1.68480 + 2.03996i 2.16991i 0 1.26055 0.426923i
109.4 −0.515448 + 0.297594i 0 −0.822876 + 1.42526i −0.717291 + 2.11790i 0 1.68480 2.03996i 2.16991i 0 −0.260548 1.30513i
109.5 0.515448 0.297594i 0 −0.822876 + 1.42526i 0.717291 2.11790i 0 1.68480 2.03996i 2.16991i 0 −0.260548 1.30513i
109.6 0.515448 0.297594i 0 −0.822876 + 1.42526i 2.19280 + 0.437757i 0 −1.68480 + 2.03996i 2.16991i 0 1.26055 0.426923i
109.7 2.05774 1.18804i 0 1.82288 3.15731i −0.733576 2.11231i 0 2.03996 + 1.68480i 3.91044i 0 −4.01902 3.47508i
109.8 2.05774 1.18804i 0 1.82288 3.15731i 1.46253 + 1.69145i 0 −2.03996 1.68480i 3.91044i 0 5.01902 + 1.74303i
289.1 −2.05774 1.18804i 0 1.82288 + 3.15731i −1.46253 + 1.69145i 0 −2.03996 + 1.68480i 3.91044i 0 5.01902 1.74303i
289.2 −2.05774 1.18804i 0 1.82288 + 3.15731i 0.733576 2.11231i 0 2.03996 1.68480i 3.91044i 0 −4.01902 + 3.47508i
289.3 −0.515448 0.297594i 0 −0.822876 1.42526i −2.19280 + 0.437757i 0 −1.68480 2.03996i 2.16991i 0 1.26055 + 0.426923i
289.4 −0.515448 0.297594i 0 −0.822876 1.42526i −0.717291 2.11790i 0 1.68480 + 2.03996i 2.16991i 0 −0.260548 + 1.30513i
289.5 0.515448 + 0.297594i 0 −0.822876 1.42526i 0.717291 + 2.11790i 0 1.68480 + 2.03996i 2.16991i 0 −0.260548 + 1.30513i
289.6 0.515448 + 0.297594i 0 −0.822876 1.42526i 2.19280 0.437757i 0 −1.68480 2.03996i 2.16991i 0 1.26055 + 0.426923i
289.7 2.05774 + 1.18804i 0 1.82288 + 3.15731i −0.733576 + 2.11231i 0 2.03996 1.68480i 3.91044i 0 −4.01902 + 3.47508i
289.8 2.05774 + 1.18804i 0 1.82288 + 3.15731i 1.46253 1.69145i 0 −2.03996 + 1.68480i 3.91044i 0 5.01902 1.74303i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.c even 3 1 inner
15.d odd 2 1 inner
21.h odd 6 1 inner
35.j even 6 1 inner
105.o odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.bf.c 16
3.b odd 2 1 inner 315.2.bf.c 16
5.b even 2 1 inner 315.2.bf.c 16
7.c even 3 1 inner 315.2.bf.c 16
7.c even 3 1 2205.2.d.p 8
7.d odd 6 1 2205.2.d.r 8
15.d odd 2 1 inner 315.2.bf.c 16
21.g even 6 1 2205.2.d.r 8
21.h odd 6 1 inner 315.2.bf.c 16
21.h odd 6 1 2205.2.d.p 8
35.i odd 6 1 2205.2.d.r 8
35.j even 6 1 inner 315.2.bf.c 16
35.j even 6 1 2205.2.d.p 8
105.o odd 6 1 inner 315.2.bf.c 16
105.o odd 6 1 2205.2.d.p 8
105.p even 6 1 2205.2.d.r 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bf.c 16 1.a even 1 1 trivial
315.2.bf.c 16 3.b odd 2 1 inner
315.2.bf.c 16 5.b even 2 1 inner
315.2.bf.c 16 7.c even 3 1 inner
315.2.bf.c 16 15.d odd 2 1 inner
315.2.bf.c 16 21.h odd 6 1 inner
315.2.bf.c 16 35.j even 6 1 inner
315.2.bf.c 16 105.o odd 6 1 inner
2205.2.d.p 8 7.c even 3 1
2205.2.d.p 8 21.h odd 6 1
2205.2.d.p 8 35.j even 6 1
2205.2.d.p 8 105.o odd 6 1
2205.2.d.r 8 7.d odd 6 1
2205.2.d.r 8 21.g even 6 1
2205.2.d.r 8 35.i odd 6 1
2205.2.d.r 8 105.p even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 6 T_{2}^{6} + 34 T_{2}^{4} - 12 T_{2}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T^{2} + 2 T^{4} - 12 T^{6} - 28 T^{8} - 48 T^{10} + 32 T^{12} + 128 T^{14} + 256 T^{16} )^{2} \)
$3$ \( \)
$5$ \( 1 + 8 T^{2} + 26 T^{4} - 96 T^{6} - 781 T^{8} - 2400 T^{10} + 16250 T^{12} + 125000 T^{14} + 390625 T^{16} \)
$7$ \( ( 1 + 91 T^{4} + 2401 T^{8} )^{2} \)
$11$ \( ( 1 - 2 T^{2} - 176 T^{4} + 124 T^{6} + 17275 T^{8} + 15004 T^{10} - 2576816 T^{12} - 3543122 T^{14} + 214358881 T^{16} )^{2} \)
$13$ \( ( 1 - 24 T^{2} + 475 T^{4} - 4056 T^{6} + 28561 T^{8} )^{4} \)
$17$ \( ( 1 + 14 T^{2} - 424 T^{4} + 588 T^{6} + 244235 T^{8} + 169932 T^{10} - 35412904 T^{12} + 337925966 T^{14} + 6975757441 T^{16} )^{2} \)
$19$ \( ( 1 - 4 T - 19 T^{2} + 12 T^{3} + 560 T^{4} + 228 T^{5} - 6859 T^{6} - 27436 T^{7} + 130321 T^{8} )^{4} \)
$23$ \( ( 1 + 54 T^{2} + 1472 T^{4} + 20844 T^{6} + 262731 T^{8} + 11026476 T^{10} + 411925952 T^{12} + 7993938006 T^{14} + 78310985281 T^{16} )^{2} \)
$29$ \( ( 1 + 46 T^{2} + 1364 T^{4} + 38686 T^{6} + 707281 T^{8} )^{4} \)
$31$ \( ( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} )^{8} \)
$37$ \( ( 1 + 120 T^{2} + 8237 T^{4} + 411000 T^{6} + 16394808 T^{8} + 562659000 T^{10} + 15437464157 T^{12} + 307887169080 T^{14} + 3512479453921 T^{16} )^{2} \)
$41$ \( ( 1 + 10 T^{2} + 860 T^{4} + 16810 T^{6} + 2825761 T^{8} )^{4} \)
$43$ \( ( 1 + 24 T^{2} + 3499 T^{4} + 44376 T^{6} + 3418801 T^{8} )^{4} \)
$47$ \( ( 1 + 168 T^{2} + 16778 T^{4} + 1180704 T^{6} + 63784419 T^{8} + 2608175136 T^{10} + 81871287818 T^{12} + 1810908175272 T^{14} + 23811286661761 T^{16} )^{2} \)
$53$ \( ( 1 + 108 T^{2} + 3242 T^{4} + 302832 T^{6} + 30275427 T^{8} + 850655088 T^{10} + 25580939402 T^{12} + 2393751001932 T^{14} + 62259690411361 T^{16} )^{2} \)
$59$ \( ( 1 - 166 T^{2} + 14888 T^{4} - 947196 T^{6} + 52438427 T^{8} - 3297189276 T^{10} + 180403270568 T^{12} - 7001968584406 T^{14} + 146830437604321 T^{16} )^{2} \)
$61$ \( ( 1 + 6 T - 88 T^{2} + 12 T^{3} + 9459 T^{4} + 732 T^{5} - 327448 T^{6} + 1361886 T^{7} + 13845841 T^{8} )^{4} \)
$67$ \( ( 1 + 72 T^{2} + 3485 T^{4} - 524088 T^{6} - 40535496 T^{8} - 2352631032 T^{10} + 70226656685 T^{12} + 6513003516168 T^{14} + 406067677556641 T^{16} )^{2} \)
$71$ \( ( 1 + 214 T^{2} + 20684 T^{4} + 1078774 T^{6} + 25411681 T^{8} )^{4} \)
$73$ \( ( 1 + 208 T^{2} + 23365 T^{4} + 1922128 T^{6} + 137018104 T^{8} + 10243020112 T^{10} + 663524900965 T^{12} + 31477519068112 T^{14} + 806460091894081 T^{16} )^{2} \)
$79$ \( ( 1 + 4 T - 83 T^{2} - 236 T^{3} + 2296 T^{4} - 18644 T^{5} - 518003 T^{6} + 1972156 T^{7} + 38950081 T^{8} )^{4} \)
$83$ \( ( 1 - 182 T^{2} + 18356 T^{4} - 1253798 T^{6} + 47458321 T^{8} )^{4} \)
$89$ \( ( 1 - 314 T^{2} + 58168 T^{4} - 7720004 T^{6} + 790702027 T^{8} - 61150151684 T^{10} + 3649590674488 T^{12} - 156052125361754 T^{14} + 3936588805702081 T^{16} )^{2} \)
$97$ \( ( 1 - 192 T^{2} + 25234 T^{4} - 1806528 T^{6} + 88529281 T^{8} )^{4} \)
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