# Properties

 Label 1600.2.l.f Level $1600$ Weight $2$ Character orbit 1600.l Analytic conductor $12.776$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1600.l (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.7760643234$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(i)$$ Coefficient field: 12.0.4767670494822400.1 Defining polynomial: $$x^{12} - 4 x^{11} + 7 x^{10} - 4 x^{9} - 8 x^{8} + 24 x^{7} - 38 x^{6} + 48 x^{5} - 32 x^{4} - 32 x^{3} + 112 x^{2} - 128 x + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{7}$$ Twist minimal: no (minimal twist has level 400) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{3} + ( \beta_{2} + \beta_{3} ) q^{7} + ( -1 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{9} +O(q^{10})$$ $$q + \beta_{4} q^{3} + ( \beta_{2} + \beta_{3} ) q^{7} + ( -1 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{9} + ( \beta_{2} + \beta_{5} - \beta_{8} + 2 \beta_{9} - \beta_{10} ) q^{11} + ( \beta_{1} - \beta_{4} ) q^{13} + ( -1 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{17} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{11} ) q^{19} + ( -2 - 2 \beta_{3} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{21} + ( -1 - \beta_{1} - \beta_{3} - \beta_{8} - 2 \beta_{11} ) q^{23} + ( 2 - \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{7} + 3 \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{27} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{7} - \beta_{11} ) q^{29} + ( -\beta_{4} + \beta_{6} - \beta_{9} - \beta_{10} ) q^{31} + ( 2 - \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{33} + ( 1 - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{8} - \beta_{9} + 2 \beta_{10} ) q^{37} + ( 1 + \beta_{1} - \beta_{2} + 4 \beta_{3} + \beta_{8} ) q^{39} + ( -1 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{8} - \beta_{9} ) q^{41} + ( \beta_{2} + \beta_{5} - \beta_{8} + 3 \beta_{9} + \beta_{10} ) q^{43} + ( -1 - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{9} + \beta_{10} ) q^{47} + ( -\beta_{1} + \beta_{3} - \beta_{4} - 4 \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{49} + ( 1 + 3 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{11} ) q^{51} + ( -1 + 2 \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{7} + 2 \beta_{9} - \beta_{11} ) q^{53} + ( 2 + 2 \beta_{1} - \beta_{2} + 4 \beta_{4} + \beta_{6} + 2 \beta_{8} - 4 \beta_{9} + \beta_{10} - \beta_{11} ) q^{57} + ( -3 + 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{5} - 3 \beta_{8} + \beta_{9} + \beta_{10} ) q^{59} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} + \beta_{11} ) q^{61} + ( -1 - 3 \beta_{1} + 3 \beta_{3} - 5 \beta_{4} - 4 \beta_{5} - \beta_{6} - 2 \beta_{7} + 3 \beta_{8} - 5 \beta_{9} + \beta_{10} ) q^{63} + ( -5 - \beta_{1} + \beta_{2} + 5 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{11} ) q^{67} + ( 1 + \beta_{3} - \beta_{7} + 4 \beta_{8} - 6 \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{69} + ( -\beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{9} - \beta_{10} ) q^{71} + ( -1 - \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{8} - \beta_{9} + \beta_{11} ) q^{73} + ( -4 - 5 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} - 5 \beta_{4} - 5 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{11} ) q^{77} + ( -2 - 5 \beta_{1} + 5 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - \beta_{6} - 2 \beta_{7} + 5 \beta_{8} - 3 \beta_{9} + \beta_{10} ) q^{79} + ( 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{9} - \beta_{10} ) q^{81} + ( 2 + 3 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{7} + \beta_{11} ) q^{83} + ( 3 \beta_{2} - 5 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{87} + ( -1 - \beta_{1} + \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + \beta_{6} - \beta_{8} + 3 \beta_{9} + \beta_{10} + \beta_{11} ) q^{89} + ( -\beta_{7} + \beta_{11} ) q^{91} + ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + \beta_{5} ) q^{93} + ( 1 + \beta_{1} - \beta_{3} + 3 \beta_{4} + 4 \beta_{5} - 2 \beta_{7} - \beta_{8} + 3 \beta_{9} ) q^{97} + ( \beta_{1} + 2 \beta_{2} + 3 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 2q^{3} + O(q^{10})$$ $$12q - 2q^{3} + 2q^{11} - 4q^{13} - 8q^{17} + 14q^{19} - 20q^{21} + 10q^{27} + 4q^{31} + 28q^{33} + 8q^{37} - 8q^{47} + 4q^{49} - 10q^{51} - 16q^{53} - 20q^{59} + 4q^{61} + 8q^{63} - 50q^{67} - 8q^{77} - 12q^{79} - 8q^{81} + 2q^{83} - 44q^{93} - 12q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 4 x^{11} + 7 x^{10} - 4 x^{9} - 8 x^{8} + 24 x^{7} - 38 x^{6} + 48 x^{5} - 32 x^{4} - 32 x^{3} + 112 x^{2} - 128 x + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{9} + \nu^{7} + 6 \nu^{3} - 8 \nu - 8$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{8} - 2 \nu^{7} + 3 \nu^{6} - 2 \nu^{5} + 4 \nu^{3} - 10 \nu^{2} + 20 \nu - 12$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{11} + 6 \nu^{10} - 11 \nu^{9} + 2 \nu^{8} + 12 \nu^{7} - 24 \nu^{6} + 38 \nu^{5} - 60 \nu^{4} + 40 \nu^{3} + 64 \nu^{2} - 144 \nu + 64$$$$)/32$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{11} - 2 \nu^{10} + 4 \nu^{8} - 5 \nu^{7} + 6 \nu^{6} - 10 \nu^{5} + 4 \nu^{4} + 18 \nu^{3} - 36 \nu^{2} + 16 \nu + 8$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{11} + 6 \nu^{10} + 7 \nu^{9} - 22 \nu^{8} + 12 \nu^{7} + 2 \nu^{5} + 20 \nu^{4} - 120 \nu^{3} + 144 \nu^{2} + 48 \nu - 192$$$$)/32$$ $$\beta_{6}$$ $$=$$ $$($$$$3 \nu^{11} - 14 \nu^{10} + 17 \nu^{9} + 6 \nu^{8} - 36 \nu^{7} + 56 \nu^{6} - 66 \nu^{5} + 92 \nu^{4} - 24 \nu^{3} - 160 \nu^{2} + 272 \nu - 96$$$$)/32$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{11} + 4 \nu^{10} - 5 \nu^{9} + 10 \nu^{7} - 20 \nu^{6} + 26 \nu^{5} - 32 \nu^{4} + 12 \nu^{3} + 56 \nu^{2} - 88 \nu + 64$$$$)/8$$ $$\beta_{8}$$ $$=$$ $$($$$$3 \nu^{11} - 8 \nu^{10} + 11 \nu^{9} - 22 \nu^{7} + 40 \nu^{6} - 58 \nu^{5} + 72 \nu^{4} - 20 \nu^{3} - 96 \nu^{2} + 208 \nu - 144$$$$)/16$$ $$\beta_{9}$$ $$=$$ $$($$$$9 \nu^{11} - 22 \nu^{10} + 15 \nu^{9} + 22 \nu^{8} - 56 \nu^{7} + 80 \nu^{6} - 118 \nu^{5} + 124 \nu^{4} + 80 \nu^{3} - 336 \nu^{2} + 304 \nu - 64$$$$)/32$$ $$\beta_{10}$$ $$=$$ $$($$$$5 \nu^{11} - 18 \nu^{10} + 31 \nu^{9} - 6 \nu^{8} - 52 \nu^{7} + 104 \nu^{6} - 142 \nu^{5} + 164 \nu^{4} - 56 \nu^{3} - 224 \nu^{2} + 496 \nu - 352$$$$)/32$$ $$\beta_{11}$$ $$=$$ $$($$$$-5 \nu^{11} + 22 \nu^{10} - 31 \nu^{9} + 2 \nu^{8} + 60 \nu^{7} - 104 \nu^{6} + 150 \nu^{5} - 188 \nu^{4} + 72 \nu^{3} + 288 \nu^{2} - 544 \nu + 352$$$$)/16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{9} + \beta_{8} - \beta_{5} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{1} + 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{10} + \beta_{7} - \beta_{5} - \beta_{4} + \beta_{1} - 1$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{10} + 3 \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta_{1} + 1$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{11} + 2 \beta_{8} - 2 \beta_{7} + \beta_{6} - 3 \beta_{5} - 5 \beta_{4} + 2 \beta_{3} - \beta_{1} - 3$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$2 \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} - 3 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \beta_{2} + 2 \beta_{1} + 7$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$3 \beta_{11} + 2 \beta_{10} - 6 \beta_{9} + 4 \beta_{8} - 4 \beta_{7} - \beta_{6} - 5 \beta_{5} + 3 \beta_{4} - 10 \beta_{3} + 3 \beta_{1} - 1$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$2 \beta_{11} - 3 \beta_{10} + 5 \beta_{9} - 5 \beta_{8} + 3 \beta_{7} + 4 \beta_{6} + 12 \beta_{5} + 10 \beta_{4} - 16 \beta_{3} + 5 \beta_{2} + 4 \beta_{1} - 11$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$3 \beta_{11} + 8 \beta_{10} + 2 \beta_{9} - 4 \beta_{8} + 2 \beta_{7} - \beta_{6} - 3 \beta_{5} - 3 \beta_{4} - 10 \beta_{3} - 7 \beta_{1} - 31$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$2 \beta_{11} - \beta_{10} + 23 \beta_{9} - 11 \beta_{8} + \beta_{7} - 12 \beta_{6} + 16 \beta_{5} - 10 \beta_{4} - 12 \beta_{3} + 11 \beta_{2} - 8 \beta_{1} + 3$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$9 \beta_{11} - 6 \beta_{9} + 32 \beta_{8} - 18 \beta_{7} - 11 \beta_{6} - 21 \beta_{5} - 17 \beta_{4} + 18 \beta_{3} - 8 \beta_{2} - 29 \beta_{1} - 9$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1151$$ $$\chi(n)$$ $$1$$ $$\beta_{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
401.1
 0.618969 − 1.27156i 1.35979 + 0.388551i −1.41313 + 0.0554252i −0.507829 + 1.31989i 1.22306 − 0.710021i 0.719139 + 1.21772i 0.618969 + 1.27156i 1.35979 − 0.388551i −1.41313 − 0.0554252i −0.507829 − 1.31989i 1.22306 + 0.710021i 0.719139 − 1.21772i
0 −2.16859 2.16859i 0 0 0 3.30519i 0 6.40553i 0
401.2 0 −1.03997 1.03997i 0 0 0 1.49668i 0 0.836925i 0
401.3 0 −0.488516 0.488516i 0 0 0 4.71540i 0 2.52270i 0
401.4 0 −0.0623209 0.0623209i 0 0 0 0.375877i 0 2.99223i 0
401.5 0 1.09156 + 1.09156i 0 0 0 0.973926i 0 0.616985i 0
401.6 0 1.66783 + 1.66783i 0 0 0 1.87372i 0 2.56332i 0
1201.1 0 −2.16859 + 2.16859i 0 0 0 3.30519i 0 6.40553i 0
1201.2 0 −1.03997 + 1.03997i 0 0 0 1.49668i 0 0.836925i 0
1201.3 0 −0.488516 + 0.488516i 0 0 0 4.71540i 0 2.52270i 0
1201.4 0 −0.0623209 + 0.0623209i 0 0 0 0.375877i 0 2.99223i 0
1201.5 0 1.09156 1.09156i 0 0 0 0.973926i 0 0.616985i 0
1201.6 0 1.66783 1.66783i 0 0 0 1.87372i 0 2.56332i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1201.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.l.f 12
4.b odd 2 1 400.2.l.g yes 12
5.b even 2 1 1600.2.l.g 12
5.c odd 4 1 1600.2.q.e 12
5.c odd 4 1 1600.2.q.f 12
16.e even 4 1 inner 1600.2.l.f 12
16.f odd 4 1 400.2.l.g yes 12
20.d odd 2 1 400.2.l.f 12
20.e even 4 1 400.2.q.e 12
20.e even 4 1 400.2.q.f 12
80.i odd 4 1 1600.2.q.f 12
80.j even 4 1 400.2.q.e 12
80.k odd 4 1 400.2.l.f 12
80.q even 4 1 1600.2.l.g 12
80.s even 4 1 400.2.q.f 12
80.t odd 4 1 1600.2.q.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.2.l.f 12 20.d odd 2 1
400.2.l.f 12 80.k odd 4 1
400.2.l.g yes 12 4.b odd 2 1
400.2.l.g yes 12 16.f odd 4 1
400.2.q.e 12 20.e even 4 1
400.2.q.e 12 80.j even 4 1
400.2.q.f 12 20.e even 4 1
400.2.q.f 12 80.s even 4 1
1600.2.l.f 12 1.a even 1 1 trivial
1600.2.l.f 12 16.e even 4 1 inner
1600.2.l.g 12 5.b even 2 1
1600.2.l.g 12 80.q even 4 1
1600.2.q.e 12 5.c odd 4 1
1600.2.q.e 12 80.t odd 4 1
1600.2.q.f 12 5.c odd 4 1
1600.2.q.f 12 80.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1600, [\chi])$$:

 $$T_{3}^{12} + \cdots$$ $$T_{7}^{12} + 40 T_{7}^{10} + 484 T_{7}^{8} + 2144 T_{7}^{6} + 3776 T_{7}^{4} + 2304 T_{7}^{2} + 256$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$1 + 18 T + 162 T^{2} + 282 T^{3} + 243 T^{4} - 20 T^{5} + 36 T^{6} + 60 T^{7} + 51 T^{8} - 6 T^{9} + 2 T^{10} + 2 T^{11} + T^{12}$$
$5$ $$T^{12}$$
$7$ $$256 + 2304 T^{2} + 3776 T^{4} + 2144 T^{6} + 484 T^{8} + 40 T^{10} + T^{12}$$
$11$ $$85849 + 294758 T + 506018 T^{2} + 383110 T^{3} + 162307 T^{4} + 30308 T^{5} + 2212 T^{6} + 484 T^{7} + 619 T^{8} + 94 T^{9} + 2 T^{10} - 2 T^{11} + T^{12}$$
$13$ $$256 - 3328 T + 21632 T^{2} - 30784 T^{3} + 22800 T^{4} - 5888 T^{5} + 832 T^{6} - 384 T^{7} + 488 T^{8} - 112 T^{9} + 8 T^{10} + 4 T^{11} + T^{12}$$
$17$ $$( -823 + 1412 T + 631 T^{2} - 152 T^{3} - 49 T^{4} + 4 T^{5} + T^{6} )^{2}$$
$19$ $$29997529 - 20494934 T + 7001282 T^{2} - 1036582 T^{3} + 792387 T^{4} - 486580 T^{5} + 165412 T^{6} - 25044 T^{7} + 2219 T^{8} - 254 T^{9} + 98 T^{10} - 14 T^{11} + T^{12}$$
$23$ $$8248384 + 7631232 T^{2} + 2431600 T^{4} + 304192 T^{6} + 11868 T^{8} + 184 T^{10} + T^{12}$$
$29$ $$428655616 + 33788928 T + 1331712 T^{2} - 4592512 T^{3} + 5399440 T^{4} - 92800 T^{5} + 512 T^{6} - 2208 T^{7} + 7960 T^{8} - 32 T^{9} + T^{12}$$
$31$ $$( 2152 - 3688 T + 1100 T^{2} + 248 T^{3} - 82 T^{4} - 2 T^{5} + T^{6} )^{2}$$
$37$ $$6801664 - 9847808 T + 7129088 T^{2} + 8940288 T^{3} + 4476736 T^{4} + 781568 T^{5} + 51840 T^{6} - 4704 T^{7} + 3652 T^{8} + 512 T^{9} + 32 T^{10} - 8 T^{11} + T^{12}$$
$41$ $$86397025 + 111297670 T^{2} + 17569359 T^{4} + 1016212 T^{6} + 24687 T^{8} + 262 T^{10} + T^{12}$$
$43$ $$26214400 - 52428800 T + 52428800 T^{2} - 26214400 T^{3} + 7356416 T^{4} - 806912 T^{5} + 8192 T^{6} - 3840 T^{7} + 7108 T^{8} - 128 T^{9} + T^{12}$$
$47$ $$( -22016 + 9792 T + 4068 T^{2} - 472 T^{3} - 136 T^{4} + 4 T^{5} + T^{6} )^{2}$$
$53$ $$7225000000 + 4216000000 T + 1230080000 T^{2} + 168787200 T^{3} + 15410224 T^{4} + 2509056 T^{5} + 812032 T^{6} + 106368 T^{7} + 7436 T^{8} + 352 T^{9} + 128 T^{10} + 16 T^{11} + T^{12}$$
$59$ $$56712564736 + 33926946816 T + 10147995648 T^{2} + 963770368 T^{3} + 71756800 T^{4} + 14528000 T^{5} + 4040192 T^{6} + 403136 T^{7} + 21380 T^{8} + 712 T^{9} + 200 T^{10} + 20 T^{11} + T^{12}$$
$61$ $$473344 + 374272 T + 147968 T^{2} - 236544 T^{3} + 628544 T^{4} + 249344 T^{5} + 59776 T^{6} - 33248 T^{7} + 9476 T^{8} + 72 T^{9} + 8 T^{10} - 4 T^{11} + T^{12}$$
$67$ $$38626225 + 6152850 T + 490050 T^{2} + 17786890 T^{3} + 37824659 T^{4} + 22084652 T^{5} + 7133348 T^{6} + 1460572 T^{7} + 203043 T^{8} + 19386 T^{9} + 1250 T^{10} + 50 T^{11} + T^{12}$$
$71$ $$95257600 + 132730880 T^{2} + 18877456 T^{4} + 1009152 T^{6} + 24008 T^{8} + 256 T^{10} + T^{12}$$
$73$ $$192626641 + 287556114 T^{2} + 55765023 T^{4} + 2400924 T^{6} + 42511 T^{8} + 338 T^{10} + T^{12}$$
$79$ $$( 1250320 + 571120 T + 45632 T^{2} - 3976 T^{3} - 450 T^{4} + 6 T^{5} + T^{6} )^{2}$$
$83$ $$4583881 + 12601926 T + 17322498 T^{2} + 13230166 T^{3} + 5853795 T^{4} + 1106836 T^{5} + 14084 T^{6} + 13044 T^{7} + 40923 T^{8} + 494 T^{9} + 2 T^{10} - 2 T^{11} + T^{12}$$
$89$ $$2165692369 + 2006433682 T^{2} + 240821727 T^{4} + 7595612 T^{6} + 96847 T^{8} + 530 T^{10} + T^{12}$$
$97$ $$( 37504 - 63488 T + 14336 T^{2} + 1088 T^{3} - 324 T^{4} + T^{6} )^{2}$$