# Properties

 Label 1900.2.i.f Level $1900$ Weight $2$ Character orbit 1900.i Analytic conductor $15.172$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1900.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.1715763840$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 3 x^{11} + 15 x^{10} - 16 x^{9} + 79 x^{8} - 65 x^{7} + 298 x^{6} + 13 x^{5} + 233 x^{4} + 14 x^{3} + 145 x^{2} + 33 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} + \beta_{6} ) q^{3} -\beta_{4} q^{7} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} + \beta_{10} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{1} + \beta_{6} ) q^{3} -\beta_{4} q^{7} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} + \beta_{10} ) q^{9} -\beta_{5} q^{11} + ( -1 + \beta_{1} - \beta_{2} - 2 \beta_{6} - \beta_{8} - \beta_{11} ) q^{13} + ( -\beta_{7} - \beta_{9} + \beta_{10} ) q^{17} + ( 1 - \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{19} + ( \beta_{7} + \beta_{9} + \beta_{10} ) q^{21} + ( -\beta_{3} + \beta_{4} + \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} ) q^{23} + ( -2 - 2 \beta_{3} + \beta_{4} ) q^{27} + ( -1 + \beta_{1} - \beta_{2} - \beta_{6} - \beta_{8} - \beta_{11} ) q^{29} + ( 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{31} + ( 2 - \beta_{1} + 2 \beta_{6} + \beta_{7} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{33} + ( 2 + 2 \beta_{5} ) q^{37} + ( 2 - 3 \beta_{2} + \beta_{3} + \beta_{5} - \beta_{8} ) q^{39} + ( -2 + \beta_{1} - 2 \beta_{6} + 2 \beta_{9} - 3 \beta_{10} ) q^{41} + ( 5 - 3 \beta_{1} + 5 \beta_{6} - \beta_{7} - 2 \beta_{9} + 2 \beta_{10} ) q^{43} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{5} + 3 \beta_{6} - \beta_{9} - 2 \beta_{10} ) q^{47} + ( -2 \beta_{2} - \beta_{3} - \beta_{8} ) q^{49} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} ) q^{51} + ( -1 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{53} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{57} + ( -1 + \beta_{1} - \beta_{6} + 2 \beta_{7} - \beta_{9} - 2 \beta_{11} ) q^{59} + ( 3 \beta_{4} + \beta_{5} + \beta_{6} + 3 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} ) q^{61} + ( 3 - 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{6} - \beta_{8} + 3 \beta_{10} - \beta_{11} ) q^{63} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{67} + ( \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{8} ) q^{69} + ( 1 + \beta_{1} + \beta_{6} + 2 \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{71} + ( -2 - 2 \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{73} + ( -1 - 3 \beta_{2} - \beta_{8} ) q^{77} + ( 2 - 4 \beta_{1} + 2 \beta_{6} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{79} + ( -1 + 3 \beta_{1} - \beta_{6} - 3 \beta_{7} - \beta_{9} ) q^{81} + ( -2 - \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{8} ) q^{83} + ( 2 - 2 \beta_{2} + \beta_{3} + \beta_{5} - \beta_{8} ) q^{87} + ( \beta_{3} - 2 \beta_{4} + \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{89} + ( 1 - \beta_{1} + \beta_{2} - 3 \beta_{4} - 2 \beta_{5} + \beta_{6} - 3 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{11} ) q^{91} + ( 4 - 5 \beta_{1} + 4 \beta_{6} + 2 \beta_{7} - 2 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{93} + ( -3 + \beta_{1} - 3 \beta_{6} - 2 \beta_{7} + \beta_{10} - 2 \beta_{11} ) q^{97} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 4 \beta_{6} + \beta_{7} + 2 \beta_{10} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 3 q^{3} - 3 q^{9} + O(q^{10})$$ $$12 q + 3 q^{3} - 3 q^{9} + 2 q^{11} + 7 q^{13} - q^{17} + q^{21} - 2 q^{23} - 24 q^{27} + q^{29} + 2 q^{31} + 10 q^{33} + 20 q^{37} + 36 q^{39} - 7 q^{41} + 19 q^{43} - 14 q^{47} + 8 q^{49} + 11 q^{51} + 6 q^{53} - 28 q^{57} - 5 q^{61} - 11 q^{63} + 14 q^{67} - 14 q^{69} + 8 q^{71} - 9 q^{73} + 2 q^{77} + q^{79} + 2 q^{81} - 26 q^{83} + 30 q^{87} - 8 q^{89} + 3 q^{91} + 9 q^{93} - 11 q^{97} - 18 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3 x^{11} + 15 x^{10} - 16 x^{9} + 79 x^{8} - 65 x^{7} + 298 x^{6} + 13 x^{5} + 233 x^{4} + 14 x^{3} + 145 x^{2} + 33 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-638495 \nu^{11} - 5059670 \nu^{10} + 2185864 \nu^{9} - 61632615 \nu^{8} - 86473310 \nu^{7} - 313073128 \nu^{6} - 445497855 \nu^{5} - 1233567830 \nu^{4} - 3200881200 \nu^{3} - 892972135 \nu^{2} - 207837960 \nu - 2681192682$$$$)/ 2135068398$$ $$\beta_{3}$$ $$=$$ $$($$$$-2112220 \nu^{11} + 5607653 \nu^{10} - 24678133 \nu^{9} + 8533470 \nu^{8} - 89367331 \nu^{7} + 19282261 \nu^{6} - 259923180 \nu^{5} - 606181345 \nu^{4} + 772282665 \nu^{3} - 452450816 \nu^{2} - 105738663 \nu - 1951111485$$$$)/ 711689466$$ $$\beta_{4}$$ $$=$$ $$($$$$3389210 \nu^{11} + 4511687 \nu^{10} + 20306405 \nu^{9} + 114731760 \nu^{8} + 262313951 \nu^{7} + 606863995 \nu^{6} + 1150918890 \nu^{5} + 3073317005 \nu^{4} + 4917790269 \nu^{3} + 2238395086 \nu^{2} + 521414583 \nu + 1619981121$$$$)/ 711689466$$ $$\beta_{5}$$ $$=$$ $$($$$$-35281678 \nu^{11} + 70497029 \nu^{10} - 385595812 \nu^{9} - 77727516 \nu^{8} - 1696717843 \nu^{7} - 1136318216 \nu^{6} - 5600322462 \nu^{5} - 13728352537 \nu^{4} - 422891238 \nu^{3} - 10151561138 \nu^{2} - 2369519571 \nu - 5196018312$$$$)/ 2135068398$$ $$\beta_{6}$$ $$=$$ $$($$$$-60680476 \nu^{11} + 182679923 \nu^{10} - 905147470 \nu^{9} + 968701752 \nu^{8} - 4732124989 \nu^{7} + 4030704250 \nu^{6} - 17769708720 \nu^{5} - 343348333 \nu^{4} - 12904983078 \nu^{3} + 2351354536 \nu^{2} - 7905696885 \nu - 1794617748$$$$)/ 2135068398$$ $$\beta_{7}$$ $$=$$ $$($$$$62867497 \nu^{11} - 203695424 \nu^{10} + 980933620 \nu^{9} - 1201195899 \nu^{8} + 5086161106 \nu^{7} - 5139259390 \nu^{6} + 19505876175 \nu^{5} - 3449941442 \nu^{4} + 14173622286 \nu^{3} - 2952954247 \nu^{2} + 17820058356 \nu - 397480590$$$$)/ 2135068398$$ $$\beta_{8}$$ $$=$$ $$($$$$102677425 \nu^{11} - 177003953 \nu^{10} + 1095480130 \nu^{9} + 493877715 \nu^{8} + 5185680031 \nu^{7} + 4686279104 \nu^{6} + 17827730625 \nu^{5} + 44330893465 \nu^{4} + 9517460142 \nu^{3} + 32695561541 \nu^{2} + 7628969433 \nu + 17785843362$$$$)/ 2135068398$$ $$\beta_{9}$$ $$=$$ $$($$$$-131028322 \nu^{11} + 444681245 \nu^{10} - 2136716515 \nu^{9} + 2925836544 \nu^{8} - 11360311729 \nu^{7} + 12770142025 \nu^{6} - 42996086940 \nu^{5} + 14406379883 \nu^{4} - 31258077591 \nu^{3} + 7259257192 \nu^{2} - 16532556735 \nu + 924096825$$$$)/ 2135068398$$ $$\beta_{10}$$ $$=$$ $$($$$$175066273 \nu^{11} - 536276480 \nu^{10} + 2643593875 \nu^{9} - 2942137461 \nu^{8} + 13841799664 \nu^{7} - 12347339095 \nu^{6} + 52083858765 \nu^{5} - 2022066866 \nu^{4} + 37830916029 \nu^{3} - 7169319793 \nu^{2} + 21056968308 \nu - 1015605495$$$$)/ 2135068398$$ $$\beta_{11}$$ $$=$$ $$($$$$137992489 \nu^{11} - 469201302 \nu^{10} + 2243299983 \nu^{9} - 3049379221 \nu^{8} + 11871320512 \nu^{7} - 13293030539 \nu^{6} + 45104060701 \nu^{5} - 14256401492 \nu^{4} + 32789448329 \nu^{3} - 7561467697 \nu^{2} + 19342466968 \nu - 965971179$$$$)/ 711689466$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{10} + 3 \beta_{6} - \beta_{3} - \beta_{2} + \beta_{1} - 1$$ $$\nu^{3}$$ $$=$$ $$-\beta_{4} - \beta_{3} - 6 \beta_{2} - 8$$ $$\nu^{4}$$ $$=$$ $$-7 \beta_{10} - \beta_{9} + \beta_{7} - 17 \beta_{6} - 10 \beta_{1} - 17$$ $$\nu^{5}$$ $$=$$ $$-\beta_{11} - 10 \beta_{10} - 3 \beta_{9} - \beta_{8} + 8 \beta_{7} - 21 \beta_{6} - 3 \beta_{5} + 8 \beta_{4} + 10 \beta_{3} + 42 \beta_{2} - 42 \beta_{1} + 42$$ $$\nu^{6}$$ $$=$$ $$-3 \beta_{8} - 15 \beta_{5} + 13 \beta_{4} + 47 \beta_{3} + 87 \beta_{2} + 197$$ $$\nu^{7}$$ $$=$$ $$15 \beta_{11} + 85 \beta_{10} + 46 \beta_{9} - 62 \beta_{7} + 184 \beta_{6} + 309 \beta_{1} + 184$$ $$\nu^{8}$$ $$=$$ $$46 \beta_{11} + 325 \beta_{10} + 169 \beta_{9} + 46 \beta_{8} - 131 \beta_{7} + 750 \beta_{6} + 169 \beta_{5} - 131 \beta_{4} - 325 \beta_{3} - 724 \beta_{2} + 724 \beta_{1} - 724$$ $$\nu^{9}$$ $$=$$ $$169 \beta_{8} + 515 \beta_{5} - 494 \beta_{4} - 686 \beta_{3} - 2339 \beta_{2} - 3848$$ $$\nu^{10}$$ $$=$$ $$-515 \beta_{11} - 2318 \beta_{10} - 1693 \beta_{9} + 1201 \beta_{7} - 5301 \beta_{6} - 5904 \beta_{1} - 5301$$ $$\nu^{11}$$ $$=$$ $$-1693 \beta_{11} - 5412 \beta_{10} - 5102 \beta_{9} - 1693 \beta_{8} + 4011 \beta_{7} - 12008 \beta_{6} - 5102 \beta_{5} + 4011 \beta_{4} + 5412 \beta_{3} + 18049 \beta_{2} - 18049 \beta_{1} + 18049$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$951$$ $$\chi(n)$$ $$1$$ $$\beta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
201.1
 1.42323 − 2.46510i 1.28913 − 2.23284i 0.430479 − 0.745611i −0.126563 + 0.219213i −0.432807 + 0.749643i −1.08347 + 1.87662i 1.42323 + 2.46510i 1.28913 + 2.23284i 0.430479 + 0.745611i −0.126563 − 0.219213i −0.432807 − 0.749643i −1.08347 − 1.87662i
0 −0.923228 + 1.59908i 0 0 0 −1.72830 0 −0.204702 0.354554i 0
201.2 0 −0.789132 + 1.36682i 0 0 0 1.39989 0 0.254541 + 0.440878i 0
201.3 0 0.0695210 0.120414i 0 0 0 3.40785 0 1.49033 + 2.58133i 0
201.4 0 0.626563 1.08524i 0 0 0 −2.18534 0 0.714838 + 1.23814i 0
201.5 0 0.932807 1.61567i 0 0 0 −3.93019 0 −0.240257 0.416137i 0
201.6 0 1.58347 2.74265i 0 0 0 3.03607 0 −3.51475 6.08773i 0
501.1 0 −0.923228 1.59908i 0 0 0 −1.72830 0 −0.204702 + 0.354554i 0
501.2 0 −0.789132 1.36682i 0 0 0 1.39989 0 0.254541 0.440878i 0
501.3 0 0.0695210 + 0.120414i 0 0 0 3.40785 0 1.49033 2.58133i 0
501.4 0 0.626563 + 1.08524i 0 0 0 −2.18534 0 0.714838 1.23814i 0
501.5 0 0.932807 + 1.61567i 0 0 0 −3.93019 0 −0.240257 + 0.416137i 0
501.6 0 1.58347 + 2.74265i 0 0 0 3.03607 0 −3.51475 + 6.08773i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 501.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.2.i.f yes 12
5.b even 2 1 1900.2.i.e 12
5.c odd 4 2 1900.2.s.e 24
19.c even 3 1 inner 1900.2.i.f yes 12
95.i even 6 1 1900.2.i.e 12
95.m odd 12 2 1900.2.s.e 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1900.2.i.e 12 5.b even 2 1
1900.2.i.e 12 95.i even 6 1
1900.2.i.f yes 12 1.a even 1 1 trivial
1900.2.i.f yes 12 19.c even 3 1 inner
1900.2.s.e 24 5.c odd 4 2
1900.2.s.e 24 95.m odd 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(1900, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$9 - 69 T + 505 T^{2} - 286 T^{3} + 473 T^{4} - 149 T^{5} + 274 T^{6} - 77 T^{7} + 79 T^{8} - 16 T^{9} + 15 T^{10} - 3 T^{11} + T^{12}$$
$5$ $$T^{12}$$
$7$ $$( -215 + 10 T + 141 T^{2} + 2 T^{3} - 23 T^{4} + T^{6} )^{2}$$
$11$ $$( 27 - 159 T + 66 T^{2} + 43 T^{3} - 22 T^{4} - T^{5} + T^{6} )^{2}$$
$13$ $$199809 - 169413 T + 238405 T^{2} - 95770 T^{3} + 128547 T^{4} - 60053 T^{5} + 32810 T^{6} - 7287 T^{7} + 1991 T^{8} - 254 T^{9} + 69 T^{10} - 7 T^{11} + T^{12}$$
$17$ $$762129 - 112617 T + 435681 T^{2} + 63666 T^{3} + 188367 T^{4} + 11991 T^{5} + 21166 T^{6} - 879 T^{7} + 1823 T^{8} - 50 T^{9} + 49 T^{10} + T^{11} + T^{12}$$
$19$ $$47045881 + 4300593 T^{2} + 864234 T^{3} + 238260 T^{4} + 93024 T^{5} + 10519 T^{6} + 4896 T^{7} + 660 T^{8} + 126 T^{9} + 33 T^{10} + T^{12}$$
$23$ $$455625 - 486000 T + 702675 T^{2} - 103140 T^{3} + 186444 T^{4} + 40284 T^{5} + 65877 T^{6} + 15390 T^{7} + 5212 T^{8} + 302 T^{9} + 75 T^{10} + 2 T^{11} + T^{12}$$
$29$ $$729 - 8667 T + 97533 T^{2} - 69102 T^{3} + 64203 T^{4} - 12039 T^{5} + 12382 T^{6} - 2593 T^{7} + 1463 T^{8} - 94 T^{9} + 41 T^{10} - T^{11} + T^{12}$$
$31$ $$( 2213 + 4937 T + 2880 T^{2} + 53 T^{3} - 120 T^{4} - T^{5} + T^{6} )^{2}$$
$37$ $$( -2880 - 480 T + 1088 T^{2} + 280 T^{3} - 48 T^{4} - 10 T^{5} + T^{6} )^{2}$$
$41$ $$303282225 + 610482825 T + 1070655165 T^{2} + 355394250 T^{3} + 123056091 T^{4} + 3944901 T^{5} + 2659294 T^{6} + 111591 T^{7} + 35207 T^{8} + 778 T^{9} + 241 T^{10} + 7 T^{11} + T^{12}$$
$43$ $$1154300625 + 923950125 T + 687042675 T^{2} + 149064720 T^{3} + 44542741 T^{4} - 2877225 T^{5} + 1865050 T^{6} - 117443 T^{7} + 31871 T^{8} - 2770 T^{9} + 381 T^{10} - 19 T^{11} + T^{12}$$
$47$ $$2546514369 + 1011884076 T + 483984153 T^{2} + 78272352 T^{3} + 26518356 T^{4} + 3972384 T^{5} + 965751 T^{6} + 106122 T^{7} + 18364 T^{8} + 1678 T^{9} + 233 T^{10} + 14 T^{11} + T^{12}$$
$53$ $$19210689 + 24167862 T + 50824593 T^{2} - 22604094 T^{3} + 22994142 T^{4} + 23094 T^{5} + 776245 T^{6} - 2054 T^{7} + 19654 T^{8} + 190 T^{9} + 185 T^{10} - 6 T^{11} + T^{12}$$
$59$ $$624650049 - 1078048062 T + 1609287327 T^{2} - 464717394 T^{3} + 133565568 T^{4} - 13329870 T^{5} + 2718901 T^{6} - 184328 T^{7} + 41476 T^{8} - 1244 T^{9} + 227 T^{10} + T^{12}$$
$61$ $$3932289 - 4071099 T + 36454423 T^{2} + 30732352 T^{3} + 265196165 T^{4} - 9827501 T^{5} + 4495158 T^{6} + 6223 T^{7} + 49577 T^{8} + 84 T^{9} + 275 T^{10} + 5 T^{11} + T^{12}$$
$67$ $$5089536 - 11334144 T + 18147712 T^{2} - 12709248 T^{3} + 6439296 T^{4} - 2078720 T^{5} + 521104 T^{6} - 85744 T^{7} + 12736 T^{8} - 1312 T^{9} + 200 T^{10} - 14 T^{11} + T^{12}$$
$71$ $$531441 + 2690010 T + 14716161 T^{2} - 5157054 T^{3} + 3246948 T^{4} - 284490 T^{5} + 194919 T^{6} - 6900 T^{7} + 10156 T^{8} + 212 T^{9} + 161 T^{10} - 8 T^{11} + T^{12}$$
$73$ $$352125225 + 193373325 T + 216643815 T^{2} + 34633440 T^{3} + 64524861 T^{4} + 19194219 T^{5} + 5225878 T^{6} + 618975 T^{7} + 67941 T^{8} + 3296 T^{9} + 279 T^{10} + 9 T^{11} + T^{12}$$
$79$ $$526105969 + 105900129 T + 150360251 T^{2} - 19598756 T^{3} + 28164979 T^{4} - 857169 T^{5} + 981510 T^{6} - 18385 T^{7} + 26913 T^{8} - 98 T^{9} + 181 T^{10} - T^{11} + T^{12}$$
$83$ $$( 27345 + 17355 T - 5378 T^{2} - 2185 T^{3} - 122 T^{4} + 13 T^{5} + T^{6} )^{2}$$
$89$ $$3908529 + 12217860 T + 47600943 T^{2} - 30754980 T^{3} + 20232538 T^{4} - 3599116 T^{5} + 917767 T^{6} - 28264 T^{7} + 23242 T^{8} - 592 T^{9} + 223 T^{10} + 8 T^{11} + T^{12}$$
$97$ $$22325625 + 38059875 T + 78179175 T^{2} - 4702320 T^{3} + 24015501 T^{4} + 8075649 T^{5} + 3067522 T^{6} + 385529 T^{7} + 51281 T^{8} + 1976 T^{9} + 287 T^{10} + 11 T^{11} + T^{12}$$