Properties

 Label 1050.2.b.f Level 1050 Weight 2 Character orbit 1050.b Analytic conductor 8.384 Analytic rank 0 Dimension 16 CM no Inner twists 8

Related objects

Newspace parameters

 Level: $$N$$ = $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1050.b (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$8.38429221223$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: 16.0.101415451701035401216.7 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{16}$$ Twist minimal: no (minimal twist has level 210) 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 18 x^{12} + 145 x^{8} - 72 x^{4} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{12} + 10 \nu^{8} - 5 \nu^{4} - 508$$$$)/180$$ $$\beta_{2}$$ $$=$$ $$($$$$7 \nu^{13} - 130 \nu^{9} + 1055 \nu^{5} - 764 \nu$$$$)/240$$ $$\beta_{3}$$ $$=$$ $$($$$$49 \nu^{15} - 910 \nu^{11} + 7625 \nu^{7} - 7268 \nu^{3}$$$$)/1440$$ $$\beta_{4}$$ $$=$$ $$($$$$-11 \nu^{13} + 194 \nu^{9} - 1531 \nu^{5} + 412 \nu$$$$)/144$$ $$\beta_{5}$$ $$=$$ $$($$$$11 \nu^{12} - 190 \nu^{8} + 1495 \nu^{4} - 212$$$$)/120$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{14} - 16 \nu^{10} + 113 \nu^{6} + 154 \nu^{2}$$$$)/36$$ $$\beta_{7}$$ $$=$$ $$($$$$9 \nu^{15} - 158 \nu^{11} + 1233 \nu^{7} - 52 \nu^{3}$$$$)/96$$ $$\beta_{8}$$ $$=$$ $$($$$$35 \nu^{15} + 51 \nu^{14} - 30 \nu^{13} - 2 \nu^{12} - 650 \nu^{11} - 930 \nu^{10} + 540 \nu^{9} + 20 \nu^{8} + 5395 \nu^{7} + 7635 \nu^{6} - 4470 \nu^{5} - 10 \nu^{4} - 5140 \nu^{3} - 6252 \nu^{2} + 3240 \nu - 1016$$$$)/1440$$ $$\beta_{9}$$ $$=$$ $$($$$$35 \nu^{15} - 51 \nu^{14} - 30 \nu^{13} + 2 \nu^{12} - 650 \nu^{11} + 930 \nu^{10} + 540 \nu^{9} - 20 \nu^{8} + 5395 \nu^{7} - 7635 \nu^{6} - 4470 \nu^{5} + 10 \nu^{4} - 5140 \nu^{3} + 6252 \nu^{2} + 3240 \nu + 1016$$$$)/1440$$ $$\beta_{10}$$ $$=$$ $$($$$$-13 \nu^{14} + 230 \nu^{10} - 1805 \nu^{6} + 316 \nu^{2}$$$$)/240$$ $$\beta_{11}$$ $$=$$ $$($$$$96 \nu^{15} + 55 \nu^{14} - 76 \nu^{13} + 90 \nu^{12} - 1680 \nu^{11} - 970 \nu^{10} + 1360 \nu^{9} - 1620 \nu^{8} + 13080 \nu^{7} + 7655 \nu^{6} - 10820 \nu^{5} + 12690 \nu^{4} - 552 \nu^{3} - 1340 \nu^{2} + 2912 \nu - 3240$$$$)/1440$$ $$\beta_{12}$$ $$=$$ $$($$$$-41 \nu^{15} + 10 \nu^{13} + 734 \nu^{11} - 172 \nu^{9} - 5857 \nu^{7} + 1274 \nu^{5} + 2212 \nu^{3} + 472 \nu$$$$)/288$$ $$\beta_{13}$$ $$=$$ $$($$$$-96 \nu^{15} + 55 \nu^{14} + 76 \nu^{13} + 90 \nu^{12} + 1680 \nu^{11} - 970 \nu^{10} - 1360 \nu^{9} - 1620 \nu^{8} - 13080 \nu^{7} + 7655 \nu^{6} + 10820 \nu^{5} + 12690 \nu^{4} + 552 \nu^{3} - 1340 \nu^{2} - 2912 \nu - 3240$$$$)/1440$$ $$\beta_{14}$$ $$=$$ $$($$$$241 \nu^{15} + 110 \nu^{13} - 4270 \nu^{11} - 1940 \nu^{9} + 33785 \nu^{7} + 15310 \nu^{5} - 8372 \nu^{3} - 1240 \nu$$$$)/1440$$ $$\beta_{15}$$ $$=$$ $$($$$$11 \nu^{14} - 194 \nu^{10} + 1531 \nu^{6} - 268 \nu^{2}$$$$)/72$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{14} + \beta_{13} - \beta_{11} + 2 \beta_{4} - \beta_{3} - \beta_{2}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{15} + 2 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} - 2 \beta_{6} + \beta_{1}$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{14} + 2 \beta_{13} + \beta_{12} - 2 \beta_{11} - \beta_{9} - \beta_{8} + 7 \beta_{7} + \beta_{4} + 4 \beta_{3} - 2 \beta_{2}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{15} - 4 \beta_{13} - 4 \beta_{11} + 6 \beta_{5} + 9 \beta_{1} + 18$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$8 \beta_{14} + 8 \beta_{13} - 9 \beta_{12} - 8 \beta_{11} - 9 \beta_{9} - 9 \beta_{8} - 9 \beta_{7} + 13 \beta_{4} - 8 \beta_{3} - 18 \beta_{2}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$32 \beta_{15} + 70 \beta_{10} + 10 \beta_{9} - 10 \beta_{8} - 14 \beta_{6} + 5 \beta_{1}$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-2 \beta_{14} + 2 \beta_{13} + 29 \beta_{12} - 2 \beta_{11} - 29 \beta_{9} - 29 \beta_{8} + 35 \beta_{7} + 29 \beta_{4} + 84 \beta_{3} - 2 \beta_{2}$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$36 \beta_{15} - 72 \beta_{13} - 72 \beta_{11} + 102 \beta_{5} + 63 \beta_{1} + 34$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$22 \beta_{14} + 22 \beta_{13} - 111 \beta_{12} - 22 \beta_{11} - 111 \beta_{9} - 111 \beta_{8} - 111 \beta_{7} - 49 \beta_{4} - 22 \beta_{3} - 292 \beta_{2}$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$278 \beta_{15} + 902 \beta_{10} - 58 \beta_{9} + 58 \beta_{8} + 82 \beta_{6} - 29 \beta_{1}$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$244 \beta_{14} - 244 \beta_{13} + 391 \beta_{12} + 244 \beta_{11} - 391 \beta_{9} - 391 \beta_{8} - 299 \beta_{7} + 391 \beta_{4} + 862 \beta_{3} + 244 \beta_{2}$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$($$$$350 \beta_{15} - 700 \beta_{13} - 700 \beta_{11} + 990 \beta_{5} - 135 \beta_{1} - 1782$$$$)/4$$ $$\nu^{13}$$ $$=$$ $$($$$$-688 \beta_{14} - 688 \beta_{13} - 705 \beta_{12} + 688 \beta_{11} - 705 \beta_{9} - 705 \beta_{8} - 705 \beta_{7} - 2651 \beta_{4} + 688 \beta_{3} - 2682 \beta_{2}$$$$)/4$$ $$\nu^{14}$$ $$=$$ $$($$$$524 \beta_{15} + 6214 \beta_{10} - 2366 \beta_{9} + 2366 \beta_{8} + 3346 \beta_{6} - 1183 \beta_{1}$$$$)/4$$ $$\nu^{15}$$ $$=$$ $$($$$$4546 \beta_{14} - 4546 \beta_{13} + 2897 \beta_{12} + 4546 \beta_{11} - 2897 \beta_{9} - 2897 \beta_{8} - 9961 \beta_{7} + 2897 \beta_{4} + 3648 \beta_{3} + 4546 \beta_{2}$$$$)/4$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$701$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 1.81768 + 0.332046i −0.137538 + 0.752908i −0.332046 − 1.81768i 0.752908 − 0.137538i −0.752908 + 0.137538i 0.332046 + 1.81768i 0.137538 − 0.752908i −1.81768 − 0.332046i −0.137538 − 0.752908i 1.81768 − 0.332046i 0.752908 + 0.137538i −0.332046 + 1.81768i 0.332046 − 1.81768i −0.752908 − 0.137538i −1.81768 + 0.332046i 0.137538 + 0.752908i
1.00000i −1.68014 0.420861i −1.00000 0 −0.420861 + 1.68014i 1.16372 2.37608i 1.00000i 2.64575 + 1.41421i 0
251.2 1.00000i −1.68014 + 0.420861i −1.00000 0 0.420861 + 1.68014i −1.16372 + 2.37608i 1.00000i 2.64575 1.41421i 0
251.3 1.00000i −0.420861 1.68014i −1.00000 0 −1.68014 + 0.420861i −2.57794 0.595188i 1.00000i −2.64575 + 1.41421i 0
251.4 1.00000i −0.420861 + 1.68014i −1.00000 0 1.68014 + 0.420861i 2.57794 + 0.595188i 1.00000i −2.64575 1.41421i 0
251.5 1.00000i 0.420861 1.68014i −1.00000 0 −1.68014 0.420861i 2.57794 0.595188i 1.00000i −2.64575 1.41421i 0
251.6 1.00000i 0.420861 + 1.68014i −1.00000 0 1.68014 0.420861i −2.57794 + 0.595188i 1.00000i −2.64575 + 1.41421i 0
251.7 1.00000i 1.68014 0.420861i −1.00000 0 −0.420861 1.68014i −1.16372 2.37608i 1.00000i 2.64575 1.41421i 0
251.8 1.00000i 1.68014 + 0.420861i −1.00000 0 0.420861 1.68014i 1.16372 + 2.37608i 1.00000i 2.64575 + 1.41421i 0
251.9 1.00000i −1.68014 0.420861i −1.00000 0 0.420861 1.68014i −1.16372 2.37608i 1.00000i 2.64575 + 1.41421i 0
251.10 1.00000i −1.68014 + 0.420861i −1.00000 0 −0.420861 1.68014i 1.16372 + 2.37608i 1.00000i 2.64575 1.41421i 0
251.11 1.00000i −0.420861 1.68014i −1.00000 0 1.68014 0.420861i 2.57794 0.595188i 1.00000i −2.64575 + 1.41421i 0
251.12 1.00000i −0.420861 + 1.68014i −1.00000 0 −1.68014 0.420861i −2.57794 + 0.595188i 1.00000i −2.64575 1.41421i 0
251.13 1.00000i 0.420861 1.68014i −1.00000 0 1.68014 + 0.420861i −2.57794 0.595188i 1.00000i −2.64575 1.41421i 0
251.14 1.00000i 0.420861 + 1.68014i −1.00000 0 −1.68014 + 0.420861i 2.57794 + 0.595188i 1.00000i −2.64575 + 1.41421i 0
251.15 1.00000i 1.68014 0.420861i −1.00000 0 0.420861 + 1.68014i 1.16372 2.37608i 1.00000i 2.64575 1.41421i 0
251.16 1.00000i 1.68014 + 0.420861i −1.00000 0 −0.420861 + 1.68014i −1.16372 + 2.37608i 1.00000i 2.64575 + 1.41421i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 251.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

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.b odd 2 1 inner
15.d odd 2 1 inner
21.c even 2 1 inner
35.c odd 2 1 inner
105.g even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.b.f 16
3.b odd 2 1 inner 1050.2.b.f 16
5.b even 2 1 inner 1050.2.b.f 16
5.c odd 4 1 210.2.d.a 8
5.c odd 4 1 210.2.d.b yes 8
7.b odd 2 1 inner 1050.2.b.f 16
15.d odd 2 1 inner 1050.2.b.f 16
15.e even 4 1 210.2.d.a 8
15.e even 4 1 210.2.d.b yes 8
20.e even 4 1 1680.2.k.e 8
20.e even 4 1 1680.2.k.f 8
21.c even 2 1 inner 1050.2.b.f 16
35.c odd 2 1 inner 1050.2.b.f 16
35.f even 4 1 210.2.d.a 8
35.f even 4 1 210.2.d.b yes 8
60.l odd 4 1 1680.2.k.e 8
60.l odd 4 1 1680.2.k.f 8
105.g even 2 1 inner 1050.2.b.f 16
105.k odd 4 1 210.2.d.a 8
105.k odd 4 1 210.2.d.b yes 8
140.j odd 4 1 1680.2.k.e 8
140.j odd 4 1 1680.2.k.f 8
420.w even 4 1 1680.2.k.e 8
420.w even 4 1 1680.2.k.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.d.a 8 5.c odd 4 1
210.2.d.a 8 15.e even 4 1
210.2.d.a 8 35.f even 4 1
210.2.d.a 8 105.k odd 4 1
210.2.d.b yes 8 5.c odd 4 1
210.2.d.b yes 8 15.e even 4 1
210.2.d.b yes 8 35.f even 4 1
210.2.d.b yes 8 105.k odd 4 1
1050.2.b.f 16 1.a even 1 1 trivial
1050.2.b.f 16 3.b odd 2 1 inner
1050.2.b.f 16 5.b even 2 1 inner
1050.2.b.f 16 7.b odd 2 1 inner
1050.2.b.f 16 15.d odd 2 1 inner
1050.2.b.f 16 21.c even 2 1 inner
1050.2.b.f 16 35.c odd 2 1 inner
1050.2.b.f 16 105.g even 2 1 inner
1680.2.k.e 8 20.e even 4 1
1680.2.k.e 8 60.l odd 4 1
1680.2.k.e 8 140.j odd 4 1
1680.2.k.e 8 420.w even 4 1
1680.2.k.f 8 20.e even 4 1
1680.2.k.f 8 60.l odd 4 1
1680.2.k.f 8 140.j odd 4 1
1680.2.k.f 8 420.w even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1050, [\chi])$$:

 $$T_{11}^{2} + 8$$ $$T_{17}^{4} - 24 T_{17}^{2} + 32$$ $$T_{37}^{4} - 128 T_{37}^{2} + 1296$$

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{8}$$
$3$ $$( 1 - 10 T^{4} + 81 T^{8} )^{2}$$
$5$ 1
$7$ $$( 1 - 4 T^{2} - 10 T^{4} - 196 T^{6} + 2401 T^{8} )^{2}$$
$11$ $$( 1 - 6 T + 11 T^{2} )^{8}( 1 + 6 T + 11 T^{2} )^{8}$$
$13$ $$( 1 - 40 T^{2} + 710 T^{4} - 6760 T^{6} + 28561 T^{8} )^{4}$$
$17$ $$( 1 + 44 T^{2} + 950 T^{4} + 12716 T^{6} + 83521 T^{8} )^{4}$$
$19$ $$( 1 - 24 T^{2} + 838 T^{4} - 8664 T^{6} + 130321 T^{8} )^{4}$$
$23$ $$( 1 - 28 T^{2} + 806 T^{4} - 14812 T^{6} + 279841 T^{8} )^{4}$$
$29$ $$( 1 - 52 T^{2} + 1350 T^{4} - 43732 T^{6} + 707281 T^{8} )^{4}$$
$31$ $$( 1 - 84 T^{2} + 3574 T^{4} - 80724 T^{6} + 923521 T^{8} )^{4}$$
$37$ $$( 1 + 20 T^{2} + 38 T^{4} + 27380 T^{6} + 1874161 T^{8} )^{4}$$
$41$ $$( 1 + 84 T^{2} + 4678 T^{4} + 141204 T^{6} + 2825761 T^{8} )^{4}$$
$43$ $$( 1 + 44 T^{2} + 2390 T^{4} + 81356 T^{6} + 3418801 T^{8} )^{4}$$
$47$ $$( 1 + 108 T^{2} + 6886 T^{4} + 238572 T^{6} + 4879681 T^{8} )^{4}$$
$53$ $$( 1 + 20 T^{2} + 3926 T^{4} + 56180 T^{6} + 7890481 T^{8} )^{4}$$
$59$ $$( 1 + 216 T^{2} + 18598 T^{4} + 751896 T^{6} + 12117361 T^{8} )^{4}$$
$61$ $$( 1 - 136 T^{2} + 9798 T^{4} - 506056 T^{6} + 13845841 T^{8} )^{4}$$
$67$ $$( 1 + 140 T^{2} + 12086 T^{4} + 628460 T^{6} + 20151121 T^{8} )^{4}$$
$71$ $$( 1 - 28 T^{2} + 9270 T^{4} - 141148 T^{6} + 25411681 T^{8} )^{4}$$
$73$ $$( 1 - 252 T^{2} + 26422 T^{4} - 1342908 T^{6} + 28398241 T^{8} )^{4}$$
$79$ $$( 1 - 4 T + 134 T^{2} - 316 T^{3} + 6241 T^{4} )^{8}$$
$83$ $$( 1 + 224 T^{2} + 26294 T^{4} + 1543136 T^{6} + 47458321 T^{8} )^{4}$$
$89$ $$( 1 - 60 T^{2} + 14950 T^{4} - 475260 T^{6} + 62742241 T^{8} )^{4}$$
$97$ $$( 1 - 108 T^{2} + 16246 T^{4} - 1016172 T^{6} + 88529281 T^{8} )^{4}$$