# Properties

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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.h 12
4.b odd 2 1 1344.4.b.g 12
7.b odd 2 1 1344.4.b.g 12
8.b even 2 1 84.4.b.a 12
8.d odd 2 1 84.4.b.b yes 12
24.f even 2 1 252.4.b.e 12
24.h odd 2 1 252.4.b.f 12
28.d even 2 1 inner 1344.4.b.h 12
56.e even 2 1 84.4.b.a 12
56.h odd 2 1 84.4.b.b yes 12
168.e odd 2 1 252.4.b.f 12
168.i even 2 1 252.4.b.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.b.a 12 8.b even 2 1
84.4.b.a 12 56.e even 2 1
84.4.b.b yes 12 8.d odd 2 1
84.4.b.b yes 12 56.h odd 2 1
252.4.b.e 12 24.f even 2 1
252.4.b.e 12 168.i even 2 1
252.4.b.f 12 24.h odd 2 1
252.4.b.f 12 168.e odd 2 1
1344.4.b.g 12 4.b odd 2 1
1344.4.b.g 12 7.b odd 2 1
1344.4.b.h 12 1.a even 1 1 trivial
1344.4.b.h 12 28.d 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 $$S_{4}^{\mathrm{new}}(1344, [\chi])$$:

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

## Hecke characteristic polynomials

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