# Properties

 Label 2394.2.e.b Level $2394$ Weight $2$ Character orbit 2394.e Analytic conductor $19.116$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2394.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.1161862439$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 2 x^{11} + 2 x^{10} + 54 x^{8} - 114 x^{7} + 120 x^{6} + 46 x^{5} + 9 x^{4} - 4 x^{3} + 8 x^{2} + 4 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{7}$$ Twist minimal: no (minimal twist has level 798) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{4} q^{2} - q^{4} -\beta_{2} q^{5} + \beta_{7} q^{7} + \beta_{4} q^{8} +O(q^{10})$$ $$q -\beta_{4} q^{2} - q^{4} -\beta_{2} q^{5} + \beta_{7} q^{7} + \beta_{4} q^{8} + \beta_{3} q^{10} + ( -1 + \beta_{5} ) q^{11} + ( 2 - \beta_{3} ) q^{13} -\beta_{10} q^{14} + q^{16} + ( -\beta_{6} + \beta_{7} + \beta_{11} ) q^{17} + ( \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{9} ) q^{19} + \beta_{2} q^{20} + ( \beta_{4} + \beta_{8} ) q^{22} + ( 1 + \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} ) q^{23} + ( -\beta_{1} - \beta_{5} - 2 \beta_{9} + 2 \beta_{10} ) q^{25} + ( -\beta_{2} - 2 \beta_{4} ) q^{26} -\beta_{7} q^{28} + ( -\beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{29} + ( 2 + \beta_{1} + \beta_{3} + \beta_{6} + \beta_{7} ) q^{31} -\beta_{4} q^{32} + ( \beta_{1} + \beta_{9} - \beta_{10} ) q^{34} + ( -\beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{35} + ( \beta_{2} - \beta_{11} ) q^{37} + ( -\beta_{2} - \beta_{3} - \beta_{6} - \beta_{8} + \beta_{9} ) q^{38} -\beta_{3} q^{40} + ( 1 - \beta_{1} - 2 \beta_{3} + \beta_{5} + \beta_{9} - \beta_{10} ) q^{41} + ( -1 - \beta_{1} - \beta_{5} + \beta_{9} - \beta_{10} ) q^{43} + ( 1 - \beta_{5} ) q^{44} + ( -\beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{46} + ( \beta_{2} - \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{11} ) q^{47} + ( -\beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{11} ) q^{49} + ( -2 \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{11} ) q^{50} + ( -2 + \beta_{3} ) q^{52} + ( 2 \beta_{2} - 2 \beta_{4} - \beta_{6} + \beta_{7} - 2 \beta_{9} - 2 \beta_{10} ) q^{53} + ( 2 \beta_{2} - 2 \beta_{8} - \beta_{9} - \beta_{10} ) q^{55} + \beta_{10} q^{56} + ( 2 \beta_{5} + \beta_{9} - \beta_{10} ) q^{58} + ( -3 - \beta_{1} + 2 \beta_{3} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{9} + 2 \beta_{10} ) q^{59} + ( 2 \beta_{2} + \beta_{4} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{61} + ( \beta_{2} - 2 \beta_{4} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{62} - q^{64} + ( -2 \beta_{2} - 5 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{11} ) q^{65} + ( -2 \beta_{2} - 3 \beta_{4} - \beta_{8} + 2 \beta_{9} + 2 \beta_{10} ) q^{67} + ( \beta_{6} - \beta_{7} - \beta_{11} ) q^{68} + ( -2 - 2 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{10} ) q^{70} + ( -2 \beta_{2} - 2 \beta_{4} - 2 \beta_{8} + \beta_{9} + \beta_{10} ) q^{71} + ( -3 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{73} + ( -\beta_{1} - \beta_{3} ) q^{74} + ( -\beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{9} ) q^{76} + ( -\beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} ) q^{77} + ( -3 \beta_{2} - 5 \beta_{4} + \beta_{8} + \beta_{11} ) q^{79} -\beta_{2} q^{80} + ( -2 \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{11} ) q^{82} + ( 2 \beta_{4} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{83} + ( 2 - 6 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + \beta_{9} - \beta_{10} ) q^{85} + ( \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{11} ) q^{86} + ( -\beta_{4} - \beta_{8} ) q^{88} + ( 2 - 4 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} ) q^{89} + ( 2 + 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{10} ) q^{91} + ( -1 - \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} + \beta_{10} ) q^{92} + ( \beta_{1} - \beta_{3} + 2 \beta_{5} + \beta_{9} - \beta_{10} ) q^{94} + ( 3 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{95} + ( -4 - \beta_{1} - \beta_{6} - \beta_{7} - 2 \beta_{9} + 2 \beta_{10} ) q^{97} + ( 1 + \beta_{1} + \beta_{5} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 12 q^{4} + O(q^{10})$$ $$12 q - 12 q^{4} + 4 q^{10} - 12 q^{11} + 20 q^{13} + 4 q^{14} + 12 q^{16} - 8 q^{19} + 12 q^{23} - 12 q^{25} + 24 q^{31} + 4 q^{34} - 4 q^{35} - 4 q^{40} + 16 q^{41} + 12 q^{44} + 4 q^{49} - 20 q^{52} - 4 q^{56} + 8 q^{58} - 40 q^{59} - 12 q^{64} - 24 q^{70} + 8 q^{76} - 8 q^{77} + 8 q^{85} + 16 q^{89} + 24 q^{91} - 12 q^{92} + 44 q^{95} - 60 q^{97} + 16 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{11} + 2 x^{10} + 54 x^{8} - 114 x^{7} + 120 x^{6} + 46 x^{5} + 9 x^{4} - 4 x^{3} + 8 x^{2} + 4 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$1204841 \nu^{11} - 108477401 \nu^{10} + 281608184 \nu^{9} - 368064334 \nu^{8} + 241501496 \nu^{7} - 5903566419 \nu^{6} + 15848202401 \nu^{5} - 21490137717 \nu^{4} + 5588841687 \nu^{3} - 209312110 \nu^{2} - 500214763 \nu - 921139278$$$$)/62914123$$ $$\beta_{2}$$ $$=$$ $$($$$$4581636 \nu^{11} - 8559599 \nu^{10} + 8749533 \nu^{9} - 720282 \nu^{8} + 249994355 \nu^{7} - 490503010 \nu^{6} + 523983773 \nu^{5} + 174081471 \nu^{4} + 227964888 \nu^{3} - 25486381 \nu^{2} + 97570007 \nu + 17917471$$$$)/62914123$$ $$\beta_{3}$$ $$=$$ $$($$$$-4581636 \nu^{11} + 8559599 \nu^{10} - 8749533 \nu^{9} + 720282 \nu^{8} - 249994355 \nu^{7} + 490503010 \nu^{6} - 523983773 \nu^{5} - 174081471 \nu^{4} - 227964888 \nu^{3} + 25486381 \nu^{2} + 28258239 \nu - 17917471$$$$)/62914123$$ $$\beta_{4}$$ $$=$$ $$($$$$17917471 \nu^{11} - 40416578 \nu^{10} + 44394541 \nu^{9} - 8749533 \nu^{8} + 968263716 \nu^{7} - 2292586049 \nu^{6} + 2640599530 \nu^{5} + 300219893 \nu^{4} - 12824232 \nu^{3} - 299634772 \nu^{2} + 168826149 \nu + 37014000$$$$)/62914123$$ $$\beta_{5}$$ $$=$$ $$($$$$20522401 \nu^{11} - 9775985 \nu^{10} - 35893384 \nu^{9} + 93036208 \nu^{8} + 1072759908 \nu^{7} - 644294087 \nu^{6} - 1878067783 \nu^{5} + 6432847185 \nu^{4} - 465780605 \nu^{3} - 59683238 \nu^{2} + 5943769 \nu + 230084943$$$$)/62914123$$ $$\beta_{6}$$ $$=$$ $$($$$$-41974538 \nu^{11} + 142257941 \nu^{10} - 216580592 \nu^{9} + 149544607 \nu^{8} - 2297873802 \nu^{7} + 7928723841 \nu^{6} - 12542798654 \nu^{5} + 6933652460 \nu^{4} + 420288932 \nu^{3} - 297472207 \nu^{2} - 290379430 \nu + 218567625$$$$)/62914123$$ $$\beta_{7}$$ $$=$$ $$($$$$82307490 \nu^{11} - 150483059 \nu^{10} + 117210810 \nu^{9} + 74172527 \nu^{8} + 4388653824 \nu^{7} - 8597681653 \nu^{6} + 7225262552 \nu^{5} + 8072273018 \nu^{4} - 1908017636 \nu^{3} + 201117637 \nu^{2} + 353056424 \nu + 393280907$$$$)/62914123$$ $$\beta_{8}$$ $$=$$ $$($$$$-113238774 \nu^{11} + 252452879 \nu^{10} - 283529283 \nu^{9} + 56703661 \nu^{8} - 6111978542 \nu^{7} + 14299745714 \nu^{6} - 16819275411 \nu^{5} - 1805551174 \nu^{4} + 296518457 \nu^{3} - 308770920 \nu^{2} - 1015647344 \nu - 225912062$$$$)/62914123$$ $$\beta_{9}$$ $$=$$ $$($$$$138418019 \nu^{11} - 293957423 \nu^{10} + 301705425 \nu^{9} - 5867022 \nu^{8} + 7435401174 \nu^{7} - 16680815036 \nu^{6} + 18053195120 \nu^{5} + 5896142065 \nu^{4} - 1829381190 \nu^{3} + 230288873 \nu^{2} + 1375723819 \nu + 377640732$$$$)/62914123$$ $$\beta_{10}$$ $$=$$ $$($$$$149885313 \nu^{11} - 353497855 \nu^{10} + 423842543 \nu^{9} - 140795074 \nu^{8} + 8124335370 \nu^{7} - 19980557836 \nu^{6} + 25001588644 \nu^{5} - 1445773037 \nu^{4} + 724989514 \nu^{3} + 93793995 \nu^{2} + 1128179249 \nu + 184817236$$$$)/62914123$$ $$\beta_{11}$$ $$=$$ $$($$$$-451390185 \nu^{11} + 1040017769 \nu^{10} - 1173084792 \nu^{9} + 251195486 \nu^{8} - 24326352374 \nu^{7} + 58815486947 \nu^{6} - 69558395473 \nu^{5} - 5583891755 \nu^{4} + 5017252753 \nu^{3} + 66395006 \nu^{2} - 3146649797 \nu - 760408626$$$$)/62914123$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{11} + \beta_{8} - 2 \beta_{7} + 2 \beta_{6} - 5 \beta_{4}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{10} - \beta_{8} - 2 \beta_{6} + \beta_{5} + \beta_{4} - 7 \beta_{3} + 7 \beta_{2} + 1$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$14 \beta_{10} - 14 \beta_{9} - \beta_{7} - \beta_{6} - 5 \beta_{5} + 2 \beta_{3} - 7 \beta_{1} - 31$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{11} - \beta_{10} + 18 \beta_{9} + 11 \beta_{8} - 18 \beta_{7} + \beta_{6} + 11 \beta_{5} - 14 \beta_{4} - 49 \beta_{3} - 49 \beta_{2} + \beta_{1} + 14$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$49 \beta_{11} + 9 \beta_{10} + 9 \beta_{9} - 29 \beta_{8} + 99 \beta_{7} - 99 \beta_{6} + 213 \beta_{4} + 26 \beta_{2}$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-13 \beta_{11} + 144 \beta_{10} - 16 \beta_{9} + 92 \beta_{8} - 16 \beta_{7} + 144 \beta_{6} - 92 \beta_{5} - 132 \beta_{4} + 348 \beta_{3} - 348 \beta_{2} - 13 \beta_{1} - 132$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$-712 \beta_{10} + 712 \beta_{9} + 63 \beta_{7} + 63 \beta_{6} + 196 \beta_{5} - 258 \beta_{3} + 348 \beta_{1} + 1510$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$129 \beta_{11} + 182 \beta_{10} - 1109 \beta_{9} - 708 \beta_{8} + 1109 \beta_{7} - 182 \beta_{6} - 708 \beta_{5} + 1119 \beta_{4} + 2492 \beta_{3} + 2492 \beta_{2} - 129 \beta_{1} - 1119$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-2492 \beta_{11} - 397 \beta_{10} - 397 \beta_{9} + 1440 \beta_{8} - 5166 \beta_{7} + 5166 \beta_{6} - 10864 \beta_{4} - 2324 \beta_{2}$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$1162 \beta_{11} - 8413 \beta_{10} + 1798 \beta_{9} - 5285 \beta_{8} + 1798 \beta_{7} - 8413 \beta_{6} + 5285 \beta_{5} + 9096 \beta_{4} - 17933 \beta_{3} + 17933 \beta_{2} + 1162 \beta_{1} + 9096$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2394\mathbb{Z}\right)^\times$$.

 $$n$$ $$533$$ $$1009$$ $$1711$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1063.1
 −1.93391 + 1.93391i −0.301222 + 0.301222i −0.242887 + 0.242887i 0.382656 − 0.382656i 1.25342 − 1.25342i 1.84195 − 1.84195i 1.84195 + 1.84195i 1.25342 + 1.25342i 0.382656 + 0.382656i −0.242887 − 0.242887i −0.301222 − 0.301222i −1.93391 − 1.93391i
1.00000i 0 −1.00000 3.86783i 0 1.61372 + 2.09664i 1.00000i 0 −3.86783
1063.2 1.00000i 0 −1.00000 0.602444i 0 −0.307956 2.62777i 1.00000i 0 −0.602444
1063.3 1.00000i 0 −1.00000 0.485775i 0 2.59339 0.523742i 1.00000i 0 −0.485775
1063.4 1.00000i 0 −1.00000 0.765312i 0 −2.29245 + 1.32086i 1.00000i 0 0.765312
1063.5 1.00000i 0 −1.00000 2.50684i 0 −2.53963 0.741811i 1.00000i 0 2.50684
1063.6 1.00000i 0 −1.00000 3.68390i 0 0.932914 + 2.47582i 1.00000i 0 3.68390
1063.7 1.00000i 0 −1.00000 3.68390i 0 0.932914 2.47582i 1.00000i 0 3.68390
1063.8 1.00000i 0 −1.00000 2.50684i 0 −2.53963 + 0.741811i 1.00000i 0 2.50684
1063.9 1.00000i 0 −1.00000 0.765312i 0 −2.29245 1.32086i 1.00000i 0 0.765312
1063.10 1.00000i 0 −1.00000 0.485775i 0 2.59339 + 0.523742i 1.00000i 0 −0.485775
1063.11 1.00000i 0 −1.00000 0.602444i 0 −0.307956 + 2.62777i 1.00000i 0 −0.602444
1063.12 1.00000i 0 −1.00000 3.86783i 0 1.61372 2.09664i 1.00000i 0 −3.86783
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1063.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
133.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2394.2.e.b 12
3.b odd 2 1 798.2.e.b yes 12
7.b odd 2 1 2394.2.e.a 12
19.b odd 2 1 2394.2.e.a 12
21.c even 2 1 798.2.e.a 12
57.d even 2 1 798.2.e.a 12
133.c even 2 1 inner 2394.2.e.b 12
399.h odd 2 1 798.2.e.b yes 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.e.a 12 21.c even 2 1
798.2.e.a 12 57.d even 2 1
798.2.e.b yes 12 3.b odd 2 1
798.2.e.b yes 12 399.h odd 2 1
2394.2.e.a 12 7.b odd 2 1
2394.2.e.a 12 19.b odd 2 1
2394.2.e.b 12 1.a even 1 1 trivial
2394.2.e.b 12 133.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2394, [\chi])$$:

 $$T_{5}^{12} + 36 T_{5}^{10} + 424 T_{5}^{8} + 1744 T_{5}^{6} + 1680 T_{5}^{4} + 576 T_{5}^{2} + 64$$ $$T_{13}^{6} - 10 T_{13}^{5} + 24 T_{13}^{4} + 16 T_{13}^{3} - 92 T_{13}^{2} + 32 T_{13} + 40$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{6}$$
$3$ $$T^{12}$$
$5$ $$64 + 576 T^{2} + 1680 T^{4} + 1744 T^{6} + 424 T^{8} + 36 T^{10} + T^{12}$$
$7$ $$117649 - 4802 T^{2} + 5488 T^{3} - 833 T^{4} - 784 T^{5} + 196 T^{6} - 112 T^{7} - 17 T^{8} + 16 T^{9} - 2 T^{10} + T^{12}$$
$11$ $$( 104 + 72 T - 116 T^{2} - 104 T^{3} - 10 T^{4} + 6 T^{5} + T^{6} )^{2}$$
$13$ $$( 40 + 32 T - 92 T^{2} + 16 T^{3} + 24 T^{4} - 10 T^{5} + T^{6} )^{2}$$
$17$ $$5125696 + 6048640 T^{2} + 1624048 T^{4} + 158528 T^{6} + 6876 T^{8} + 136 T^{10} + T^{12}$$
$19$ $$47045881 + 19808792 T + 260642 T^{2} - 713336 T^{3} + 60287 T^{4} + 41040 T^{5} + 5820 T^{6} + 2160 T^{7} + 167 T^{8} - 104 T^{9} + 2 T^{10} + 8 T^{11} + T^{12}$$
$23$ $$( -2056 - 2080 T + 532 T^{2} + 368 T^{3} - 64 T^{4} - 6 T^{5} + T^{6} )^{2}$$
$29$ $$11505664 + 7913472 T^{2} + 1758720 T^{4} + 171136 T^{6} + 7760 T^{8} + 152 T^{10} + T^{12}$$
$31$ $$( 5192 - 1072 T - 2184 T^{2} + 860 T^{3} - 42 T^{4} - 12 T^{5} + T^{6} )^{2}$$
$37$ $$12334144 + 8545664 T^{2} + 1894432 T^{4} + 164512 T^{6} + 6740 T^{8} + 132 T^{10} + T^{12}$$
$41$ $$( 12224 - 32256 T + 4208 T^{2} + 1088 T^{3} - 144 T^{4} - 8 T^{5} + T^{6} )^{2}$$
$43$ $$( 4928 - 11200 T + 5232 T^{2} + 160 T^{3} - 176 T^{4} + T^{6} )^{2}$$
$47$ $$3534400 + 8712704 T^{2} + 4669984 T^{4} + 543616 T^{6} + 20068 T^{8} + 252 T^{10} + T^{12}$$
$53$ $$545689600 + 286150656 T^{2} + 36689408 T^{4} + 1784960 T^{6} + 36880 T^{8} + 328 T^{10} + T^{12}$$
$59$ $$( -3520 - 12736 T - 12288 T^{2} - 2368 T^{3} - 28 T^{4} + 20 T^{5} + T^{6} )^{2}$$
$61$ $$10816000000 + 1511501824 T^{2} + 85262848 T^{4} + 2475392 T^{6} + 38864 T^{8} + 312 T^{10} + T^{12}$$
$67$ $$1894686784 + 473470336 T^{2} + 43149168 T^{4} + 1838976 T^{6} + 37916 T^{8} + 344 T^{10} + T^{12}$$
$71$ $$125440000 + 318976000 T^{2} + 36333056 T^{4} + 1572224 T^{6} + 31760 T^{8} + 296 T^{10} + T^{12}$$
$73$ $$8131710976 + 2004232192 T^{2} + 159424768 T^{4} + 4857216 T^{6} + 67680 T^{8} + 432 T^{10} + T^{12}$$
$79$ $$18496 + 2062418112 T^{2} + 271939600 T^{4} + 9204400 T^{6} + 112104 T^{8} + 564 T^{10} + T^{12}$$
$83$ $$203689984 + 266151936 T^{2} + 91336448 T^{4} + 10001152 T^{6} + 146160 T^{8} + 680 T^{10} + T^{12}$$
$89$ $$( -755200 - 79616 T + 28544 T^{2} + 1600 T^{3} - 312 T^{4} - 8 T^{5} + T^{6} )^{2}$$
$97$ $$( -5656 + 8456 T - 372 T^{2} - 1672 T^{3} + 126 T^{4} + 30 T^{5} + T^{6} )^{2}$$