LMFDB - Newform orbit 7644.2.e.q

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7644,2,Mod(4705,7644)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7644, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7644.4705"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 7644 = 2^{2} \cdot 3 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7644.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,16,0,0,0,0,0,16,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$61.0376473051$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 46x^{14} + 825x^{12} + 7500x^{10} + 37308x^{8} + 101640x^{6} + 144580x^{4} + 100432x^{2} + 26896 $ x^16 + 46x^14 + 825x^12 + 7500x^10 + 37308x^8 + 101640x^6 + 144580x^4 + 100432*x^2 + 26896
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

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 + q^{3} + \beta_1 q^{5} + q^{9} + \beta_{7} q^{11} + \beta_{10} q^{13} + \beta_1 q^{15} + ( - \beta_{5} - \beta_{4}) q^{17} + ( - \beta_{13} - \beta_{7}) q^{19} + ( - \beta_{6} - \beta_{5}) q^{23}+ \cdots + \beta_{7} q^{99}+O(q^{100}) $ q + q^3 + b1 * q^5 + q^9 + b7 * q^11 + b10 * q^13 + b1 * q^15 + (-b5 - b4) * q^17 + (-b13 - b7) * q^19 + (-b6 - b5) * q^23 + (b2 - 1) * q^25 + q^27 + (b11 - b10 - b4 - b2) * q^29 + b8 * q^31 + b7 * q^33 + (b13 + b11 + b10 + b9 - b1) * q^37 + b10 * q^39 + (-b13 + b12 - b11 - b10 + b8 + 2b1) q^41 + (b11 - b10 + b6 - b4 - b3 - 1) * q^43 + b1 * q^45 + (b15 + b14 + b11 + b10 - b1) * q^47 + (-b5 - b4) * q^51 + (-b15 + b14 + b11 - b10 + b4 - b2) * q^53 + (b11 - b10 + b6 - b4 - b2) * q^55 + (-b13 - b7) * q^57 + (-b15 - b14 + b13 - 2b9 - b8) q^59 + (-b15 + b14 + b11 - b10 - 2b2 + 2) q^61 + (-b14 + b12 - b9 + b7 + b5 + b4 + b2 + b1 - 2) * q^65 + (b15 + b14 - b13 + b12 + b9 + b8 - b7) * q^67 + (-b6 - b5) * q^69 + (-b15 - b14 + 2b13 - b12 + b11 + b10 - b9 - b8 + b1) q^71 + (-b15 - b14 + 2b12 - b9 + b7 + b1) q^73 + (b2 - 1) * q^75 + (b15 - b14 + 2b5 + b2 - 2) q^79 + q^81 + (-b15 - b14 + 2b13 + b12 + b11 + b10 - 2b8 + 2b7 - b1) q^83 + (b13 - b12 + 2b11 + 2b10 - b9 - b8 - b7 - 2b1) q^85 + (b11 - b10 - b4 - b2) * q^87 + (-2b13 + b12 + 2b7 - b1) * q^89 + b8 * q^93 + (-b11 + b10 - 2b6 + b4 + b3 + b2 + 1) q^95 + (-b15 - b14 + b13 - 3b9 - b8 - b1) q^97 + b7 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{3} + 16 q^{9} - 4 q^{23} - 12 q^{25} + 16 q^{27} - 4 q^{29} - 12 q^{43} - 12 q^{53} + 16 q^{61} - 24 q^{65} - 4 q^{69} - 12 q^{75} - 20 q^{79} + 16 q^{81} - 4 q^{87} + 12 q^{95}+O(q^{100}) $ 16 * q + 16 * q^3 + 16 * q^9 - 4 * q^23 - 12 * q^25 + 16 * q^27 - 4 * q^29 - 12 * q^43 - 12 * q^53 + 16 * q^61 - 24 * q^65 - 4 * q^69 - 12 * q^75 - 20 * q^79 + 16 * q^81 - 4 * q^87 + 12 * q^95

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 46x^{14} + 825x^{12} + 7500x^{10} + 37308x^{8} + 101640x^{6} + 144580x^{4} + 100432x^{2} + 26896 $ x^16 + 46*x^14 + 825*x^12 + 7500*x^10 + 37308*x^8 + 101640*x^6 + 144580*x^4 + 100432*x^2 + 26896 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} + 6 $ v^2 + 6
$\beta_{3}$	$=$	$ ( 1029 \nu^{14} + 46524 \nu^{12} + 810588 \nu^{10} + 7012670 \nu^{8} + 31937661 \nu^{6} + \cdots + 26131976 ) / 8084 $ (1029v^14 + 46524v^12 + 810588v^10 + 7012670v^8 + 31937661v^6 + 73595466v^4 + 74150946*v^2 + 26131976) / 8084
$\beta_{4}$	$=$	$ ( 5779 \nu^{14} + 259588 \nu^{12} + 4487279 \nu^{10} + 38499406 \nu^{8} + 174091604 \nu^{6} + \cdots + 144572216 ) / 32336 $ (5779v^14 + 259588v^12 + 4487279v^10 + 38499406v^8 + 174091604v^6 + 399873800v^4 + 405297012*v^2 + 144572216) / 32336
$\beta_{5}$	$=$	$ ( - 10181 \nu^{14} - 457843 \nu^{12} - 7925901 \nu^{10} - 68119059 \nu^{8} - 308620618 \nu^{6} + \cdots - 258224456 ) / 16168 $ (-10181v^14 - 457843v^12 - 7925901v^10 - 68119059v^8 - 308620618v^6 - 710359350v^4 - 721746200*v^2 - 258224456) / 16168
$\beta_{6}$	$=$	$ ( 7107 \nu^{14} + 319283 \nu^{12} + 5520145 \nu^{10} + 47371277 \nu^{8} + 214261466 \nu^{6} + \cdots + 178048832 ) / 8084 $ (7107v^14 + 319283v^12 + 5520145v^10 + 47371277v^8 + 214261466v^6 + 492252998v^4 + 498997524*v^2 + 178048832) / 8084
$\beta_{7}$	$=$	$ ( - 69333 \nu^{15} - 3111090 \nu^{13} - 53685041 \nu^{11} - 459255180 \nu^{9} + \cdots - 1622011624 \nu ) / 1325776 $ (-69333v^15 - 3111090v^13 - 53685041v^11 - 459255180v^9 - 2066177776v^7 - 4701102396v^5 - 4674475300v^3 - 1622011624v) / 1325776
$\beta_{8}$	$=$	$ ( 35401 \nu^{15} + 1588471 \nu^{13} + 27419701 \nu^{11} + 234835195 \nu^{9} + 1059524506 \nu^{7} + \cdots + 872739288 \nu ) / 662888 $ (35401v^15 + 1588471v^13 + 27419701v^11 + 234835195v^9 + 1059524506v^7 + 2426075142v^5 + 2447880160v^3 + 872739288v) / 662888
$\beta_{9}$	$=$	$ ( 5779 \nu^{15} + 259588 \nu^{13} + 4487279 \nu^{11} + 38499406 \nu^{9} + 174091604 \nu^{7} + \cdots + 144572216 \nu ) / 32336 $ (5779v^15 + 259588v^13 + 4487279v^11 + 38499406v^9 + 174091604v^7 + 399873800v^5 + 405297012v^3 + 144572216v) / 32336
$\beta_{10}$	$=$	$ ( 628715 \nu^{15} + 850381 \nu^{14} + 28258576 \nu^{13} + 38204620 \nu^{12} + 488879103 \nu^{11} + \cdots + 21399812568 ) / 2651552 $ (628715v^15 + 850381v^14 + 28258576v^13 + 38204620v^12 + 488879103v^11 + 660583021v^10 + 4198731586v^9 + 5669915994v^8 + 19010533180v^7 + 25655220936v^6 + 43737622952v^5 + 58984976032v^4 + 44439690292v^3 + 59875850436v^2 + 15908425448*v + 21399812568) / 2651552
$\beta_{11}$	$=$	$ ( 628715 \nu^{15} - 850381 \nu^{14} + 28258576 \nu^{13} - 38204620 \nu^{12} + 488879103 \nu^{11} + \cdots - 21399812568 ) / 2651552 $ (628715v^15 - 850381v^14 + 28258576v^13 - 38204620v^12 + 488879103v^11 - 660583021v^10 + 4198731586v^9 - 5669915994v^8 + 19010533180v^7 - 25655220936v^6 + 43737622952v^5 - 58984976032v^4 + 44439690292v^3 - 59875850436v^2 + 15908425448*v - 21399812568) / 2651552
$\beta_{12}$	$=$	$ ( - 252577 \nu^{15} - 11351796 \nu^{13} - 196381393 \nu^{11} - 1686697514 \nu^{9} + \cdots - 6418142072 \nu ) / 662888 $ (-252577v^15 - 11351796v^13 - 196381393v^11 - 1686697514v^9 - 7638638544v^7 - 17585276192v^5 - 17892903964v^3 - 6418142072v) / 662888
$\beta_{13}$	$=$	$ ( - 926273 \nu^{15} - 41611766 \nu^{13} - 719442749 \nu^{11} - 6174680152 \nu^{9} + \cdots - 23352611144 \nu ) / 1325776 $ (-926273v^15 - 41611766v^13 - 719442749v^11 - 6174680152v^9 - 27938581536v^7 - 64245283700v^5 - 65260715092v^3 - 23352611144v) / 1325776
$\beta_{14}$	$=$	$ ( - 2473233 \nu^{15} - 533492 \nu^{14} - 111120200 \nu^{13} - 23989264 \nu^{12} + \cdots - 13573364224 ) / 2651552 $ (-2473233v^15 - 533492v^14 - 111120200v^13 - 23989264v^12 - 1921435553v^11 - 415259972v^10 - 16492167702v^9 - 3569008344v^8 - 74617642192v^7 - 16173300176v^6 - 171517937192v^5 - 37249357392v^4 - 174030771876v^3 - 37894690832v^2 - 62168674264*v - 13573364224) / 2651552
$\beta_{15}$	$=$	$ ( - 2473233 \nu^{15} + 533492 \nu^{14} - 111120200 \nu^{13} + 23989264 \nu^{12} + \cdots + 13573364224 ) / 2651552 $ (-2473233v^15 + 533492v^14 - 111120200v^13 + 23989264v^12 - 1921435553v^11 + 415259972v^10 - 16492167702v^9 + 3569008344v^8 - 74617642192v^7 + 16173300176v^6 - 171517937192v^5 + 37249357392v^4 - 174030771876v^3 + 37894690832v^2 - 62168674264*v + 13573364224) / 2651552

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} - 6 $ b2 - 6
$\nu^{3}$	$=$	$ \beta_{13} - 2\beta_{12} - \beta_{11} - \beta_{10} + 2\beta_{9} + \beta_{8} - 9\beta_1 $ b13 - 2b12 - b11 - b10 + 2b9 + b8 - 9*b1
$\nu^{4}$	$=$	$ - 2 \beta_{15} + 2 \beta_{14} + \beta_{11} - \beta_{10} + 3 \beta_{6} - 2 \beta_{5} - 13 \beta_{4} + \cdots + 63 $ -2b15 + 2b14 + b11 - b10 + 3b6 - 2b5 - 13b4 - b3 - 18b2 + 63
$\nu^{5}$	$=$	$ 2 \beta_{15} + 2 \beta_{14} - 23 \beta_{13} + 35 \beta_{12} + 21 \beta_{11} + 21 \beta_{10} + \cdots + 109 \beta_1 $ 2b15 + 2b14 - 23b13 + 35b12 + 21b11 + 21b10 - 48b9 - 15b8 - 8b7 + 109b1
$\nu^{6}$	$=$	$ 41 \beta_{15} - 41 \beta_{14} - 27 \beta_{11} + 27 \beta_{10} - 79 \beta_{6} + 42 \beta_{5} + 331 \beta_{4} + \cdots - 823 $ 41b15 - 41b14 - 27b11 + 27b10 - 79b6 + 42b5 + 331b4 + 23b3 + 296*b2 - 823
$\nu^{7}$	$=$	$ - 50 \beta_{15} - 50 \beta_{14} + 425 \beta_{13} - 545 \beta_{12} - 357 \beta_{11} - 357 \beta_{10} + \cdots - 1537 \beta_1 $ -50b15 - 50b14 + 425b13 - 545b12 - 357b11 - 357b10 + 922b9 + 233b8 + 228b7 - 1537b1
$\nu^{8}$	$=$	$ - 669 \beta_{15} + 669 \beta_{14} + 561 \beta_{11} - 561 \beta_{10} + 1575 \beta_{6} - 714 \beta_{5} + \cdots + 11981 $ -669b15 + 669b14 + 561b11 - 561b10 + 1575b6 - 714b5 - 6441b4 - 425b3 - 4824*b2 + 11981
$\nu^{9}$	$=$	$ 986 \beta_{15} + 986 \beta_{14} - 7385 \beta_{13} + 8425 \beta_{12} + 5827 \beta_{11} + \cdots + 23415 \beta_1 $ 986b15 + 986b14 - 7385b13 + 8425b12 + 5827b11 + 5827b10 - 16354b9 - 3775b8 - 4798b7 + 23415b1
$\nu^{10}$	$=$	$ 10477 \beta_{15} - 10477 \beta_{14} - 10545 \beta_{11} + 10545 \beta_{10} - 28537 \beta_{6} + \cdots - 184757 $ 10477b15 - 10477b14 - 10545b11 + 10545b10 - 28537b6 + 11654b5 + 114745b4 + 7385b3 + 78818*b2 - 184757
$\nu^{11}$	$=$	$ - 17930 \beta_{15} - 17930 \beta_{14} + 125241 \beta_{13} - 132079 \beta_{12} - 94741 \beta_{11} + \cdots - 370629 \beta_1 $ -17930b15 - 17930b14 + 125241b13 - 132079b12 - 94741b11 - 94741b10 + 280026b9 + 62133b8 + 90016b7 - 370629b1
$\nu^{12}$	$=$	$ - 164687 \beta_{15} + 164687 \beta_{14} + 188009 \beta_{11} - 188009 \beta_{10} + 495283 \beta_{6} + \cdots + 2937641 $ -164687b15 + 164687b14 + 188009b11 - 188009b10 + 495283b6 - 189482b5 - 1969241b4 - 125241b3 - 1292304*b2 + 2937641
$\nu^{13}$	$=$	$ 313250 \beta_{15} + 313250 \beta_{14} - 2099069 \beta_{13} + 2103681 \beta_{12} + 1546027 \beta_{11} + \cdots + 5980679 \beta_1 $ 313250b15 + 313250b14 - 2099069b13 + 2103681b12 + 1546027b11 + 1546027b10 - 4713782b9 - 1027171b8 - 1596734b7 + 5980679b1
$\nu^{14}$	$=$	$ 2622537 \beta_{15} - 2622537 \beta_{14} - 3250405 \beta_{11} + 3250405 \beta_{10} - 8409585 \beta_{6} + \cdots - 47483817 $ 2622537b15 - 2622537b14 - 3250405b11 + 3250405b10 - 8409585b6 + 3092054b5 + 33197309b4 + 2099069b3 + 21244282*b2 - 47483817
$\nu^{15}$	$=$	$ - 5349474 \beta_{15} - 5349474 \beta_{14} + 34958089 \beta_{13} - 33929327 \beta_{12} + \cdots - 97484809 \beta_1 $ -5349474b15 - 5349474b14 + 34958089b13 - 33929327b12 - 25329321b11 - 25329321b10 + 78660090b9 + 16992193b8 + 27477476b7 - 97484809b1

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7644\mathbb{Z}\right)^\times$.

$n$	$2549$	$3433$	$3823$	$5293$
$\chi(n)$	$1$	$1$	$1$	$-1$

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.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

4705.1

	−	4.06325i
	−	3.13069i
	−	2.49203i
	−	2.43216i
	−	2.13442i
	−	1.03592i
	−	1.03273i
	−	0.931518i
		0.931518i
		1.03273i
		1.03592i
		2.13442i
		2.43216i
		2.49203i
		3.13069i
		4.06325i

1.00000

−

4.06325i

1.00000

4705.2

1.00000

−

3.13069i

1.00000

4705.3

1.00000

−

2.49203i

1.00000

4705.4

1.00000

−

2.43216i

1.00000

4705.5

1.00000

−

2.13442i

1.00000

4705.6

1.00000

−

1.03592i

1.00000

4705.7

1.00000

−

1.03273i

1.00000

4705.8

1.00000

−

0.931518i

1.00000

4705.9

1.00000

0.931518i

1.00000

4705.10

1.00000

1.03273i

1.00000

4705.11

1.00000

1.03592i

1.00000

4705.12

1.00000

2.13442i

1.00000

4705.13

1.00000

2.43216i

1.00000

4705.14

1.00000

2.49203i

1.00000

4705.15

1.00000

3.13069i

1.00000

4705.16

1.00000

4.06325i

1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	7644.2.e.q	yes	16
7.b	odd	2	1		7644.2.e.p	✓	16
13.b	even	2	1	inner	7644.2.e.q	yes	16
91.b	odd	2	1		7644.2.e.p	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
7644.2.e.p	✓	16	7.b	odd	2	1
7644.2.e.p	✓	16	91.b	odd	2	1
7644.2.e.q	yes	16	1.a	even	1	1	trivial
7644.2.e.q	yes	16	13.b	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 $S_{2}^{\mathrm{new}}(7644, [\chi])$:

$ T_{5}^{16} + 46 T_{5}^{14} + 825 T_{5}^{12} + 7500 T_{5}^{10} + 37308 T_{5}^{8} + 101640 T_{5}^{6} + \cdots + 26896 $

T5^16 + 46*T5^14 + 825*T5^12 + 7500*T5^10 + 37308*T5^8 + 101640*T5^6 + 144580*T5^4 + 100432*T5^2 + 26896

$ T_{11}^{16} + 88 T_{11}^{14} + 2772 T_{11}^{12} + 42528 T_{11}^{10} + 355556 T_{11}^{8} + 1668832 T_{11}^{6} + \cdots + 2166784 $

T11^16 + 88*T11^14 + 2772*T11^12 + 42528*T11^10 + 355556*T11^8 + 1668832*T11^6 + 4250944*T11^4 + 5198848*T11^2 + 2166784

$ T_{17}^{8} - 92T_{17}^{6} + 16T_{17}^{5} + 2708T_{17}^{4} - 416T_{17}^{3} - 30464T_{17}^{2} + 1152T_{17} + 106048 $

T17^8 - 92*T17^6 + 16*T17^5 + 2708*T17^4 - 416*T17^3 - 30464*T17^2 + 1152*T17 + 106048

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T - 1)^{16} $ (T - 1)^16
$5$	$ T^{16} + 46 T^{14} + \cdots + 26896 $ T^16 + 46T^14 + 825T^12 + 7500T^10 + 37308T^8 + 101640T^6 + 144580T^4 + 100432*T^2 + 26896
$7$	$ T^{16} $ T^16
$11$	$ T^{16} + 88 T^{14} + \cdots + 2166784 $ T^16 + 88T^14 + 2772T^12 + 42528T^10 + 355556T^8 + 1668832T^6 + 4250944T^4 + 5198848*T^2 + 2166784
$13$	$ T^{16} + \cdots + 815730721 $ T^16 + 6T^14 + 112T^13 + 16T^12 + 1072T^11 + 4554T^10 + 5536T^9 + 74302T^8 + 71968T^7 + 769626T^6 + 2355184T^5 + 456976T^4 + 41584816T^3 + 28960854*T^2 + 815730721
$17$	$ (T^{8} - 92 T^{6} + \cdots + 106048)^{2} $ (T^8 - 92T^6 + 16T^5 + 2708T^4 - 416T^3 - 30464T^2 + 1152T + 106048)^2
$19$	$ T^{16} + \cdots + 928055296 $ T^16 + 182T^14 + 13705T^12 + 548432T^10 + 12416128T^8 + 155108352T^6 + 940921856T^4 + 1889026048*T^2 + 928055296
$23$	$ (T^{8} + 2 T^{7} + \cdots - 294784)^{2} $ (T^8 + 2T^7 - 117T^6 - 340T^5 + 4372T^4 + 17120T^3 - 44032T^2 - 267776*T - 294784)^2
$29$	$ (T^{8} + 2 T^{7} + \cdots + 199552)^{2} $ (T^8 + 2T^7 - 121T^6 - 168T^5 + 4224T^4 + 5280T^3 - 52048T^2 - 45184*T + 199552)^2
$31$	$ T^{16} + \cdots + 1494286336 $ T^16 + 246T^14 + 23977T^12 + 1182672T^10 + 31425024T^8 + 445128704T^6 + 3085497344T^4 + 8259223552*T^2 + 1494286336
$37$	$ T^{16} + \cdots + 13153337344 $ T^16 + 264T^14 + 28432T^12 + 1617920T^10 + 52436992T^8 + 965771264T^6 + 9368567808T^4 + 37857787904*T^2 + 13153337344
$41$	$ T^{16} + \cdots + 581937071104 $ T^16 + 380T^14 + 55464T^12 + 3994416T^10 + 154098420T^8 + 3317801840T^6 + 39573246608T^4 + 241624769792*T^2 + 581937071104
$43$	$ (T^{8} + 6 T^{7} + \cdots + 42752)^{2} $ (T^8 + 6T^7 - 199T^6 - 720T^5 + 10888T^4 + 7040T^3 - 104304T^2 + 27008*T + 42752)^2
$47$	$ T^{16} + \cdots + 4650958065664 $ T^16 + 482T^14 + 93597T^12 + 9460552T^10 + 537617416T^8 + 17260704200T^6 + 295515915012T^4 + 2286494229248*T^2 + 4650958065664
$53$	$ (T^{8} + 6 T^{7} + \cdots + 1984)^{2} $ (T^8 + 6T^7 - 189T^6 - 892T^5 + 6628T^4 + 34816T^3 + 49040T^2 + 22848*T + 1984)^2
$59$	$ T^{16} + \cdots + 34780758016 $ T^16 + 456T^14 + 84596T^12 + 8207232T^10 + 443053860T^8 + 12879966944T^6 + 170764866624T^4 + 518584379392*T^2 + 34780758016
$61$	$ (T^{8} - 8 T^{7} + \cdots + 192512)^{2} $ (T^8 - 8T^7 - 298T^6 + 1712T^5 + 29680T^4 - 85760T^3 - 1007104T^2 - 90112*T + 192512)^2
$67$	$ T^{16} + \cdots + 9487582756864 $ T^16 + 588T^14 + 124068T^12 + 12016640T^10 + 617530368T^8 + 17993908224T^6 + 298510467072T^4 + 2625214873600*T^2 + 9487582756864
$71$	$ T^{16} + \cdots + 10654113796096 $ T^16 + 648T^14 + 167540T^12 + 22154560T^10 + 1592412068T^8 + 61072850016T^6 + 1145945544000T^4 + 8253510011904*T^2 + 10654113796096
$73$	$ T^{16} + \cdots + 140388753227776 $ T^16 + 810T^14 + 255333T^12 + 40774204T^10 + 3614609188T^8 + 180704807168T^6 + 4828369224704T^4 + 57437227548672*T^2 + 140388753227776
$79$	$ (T^{8} + 10 T^{7} + \cdots - 5325824)^{2} $ (T^8 + 10T^7 - 363T^6 - 5032T^5 + 18776T^4 + 587720T^3 + 2982980T^2 + 3389568*T - 5325824)^2
$83$	$ T^{16} + \cdots + 169190997934336 $ T^16 + 898T^14 + 327133T^12 + 62227624T^10 + 6594421352T^8 + 378992135144T^6 + 10207922463172T^4 + 77553730238656*T^2 + 169190997934336
$89$	$ T^{16} + \cdots + 9364358895376 $ T^16 + 926T^14 + 321433T^12 + 51803196T^10 + 3946282524T^8 + 134711433384T^6 + 1691500968068T^4 + 7913405706064*T^2 + 9364358895376
$97$	$ T^{16} + \cdots + 19550829543424 $ T^16 + 826T^14 + 276309T^12 + 48346716T^10 + 4759614372T^8 + 262552165376T^6 + 7492951293952T^4 + 86040600641536*T^2 + 19550829543424
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7644 = 2^{2} \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7644.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + q^{3} + \beta_1 q^{5} + q^{9} + \beta_{7} q^{11} + \beta_{10} q^{13} + \beta_1 q^{15} + ( - \beta_{5} - \beta_{4}) q^{17} + ( - \beta_{13} - \beta_{7}) q^{19} + ( - \beta_{6} - \beta_{5}) q^{23}+ \cdots + \beta_{7} q^{99}+O(q^{100}) \) q + q^3 + b1 * q^5 + q^9 + b7 * q^11 + b10 * q^13 + b1 * q^15 + (-b5 - b4) * q^17 + (-b13 - b7) * q^19 + (-b6 - b5) * q^23 + (b2 - 1) * q^25 + q^27 + (b11 - b10 - b4 - b2) * q^29 + b8 * q^31 + b7 * q^33 + (b13 + b11 + b10 + b9 - b1) * q^37 + b10 * q^39 + (-b13 + b12 - b11 - b10 + b8 + 2b1) q^41 + (b11 - b10 + b6 - b4 - b3 - 1) * q^43 + b1 * q^45 + (b15 + b14 + b11 + b10 - b1) * q^47 + (-b5 - b4) * q^51 + (-b15 + b14 + b11 - b10 + b4 - b2) * q^53 + (b11 - b10 + b6 - b4 - b2) * q^55 + (-b13 - b7) * q^57 + (-b15 - b14 + b13 - 2b9 - b8) q^59 + (-b15 + b14 + b11 - b10 - 2b2 + 2) q^61 + (-b14 + b12 - b9 + b7 + b5 + b4 + b2 + b1 - 2) * q^65 + (b15 + b14 - b13 + b12 + b9 + b8 - b7) * q^67 + (-b6 - b5) * q^69 + (-b15 - b14 + 2b13 - b12 + b11 + b10 - b9 - b8 + b1) q^71 + (-b15 - b14 + 2b12 - b9 + b7 + b1) q^73 + (b2 - 1) * q^75 + (b15 - b14 + 2b5 + b2 - 2) q^79 + q^81 + (-b15 - b14 + 2b13 + b12 + b11 + b10 - 2b8 + 2b7 - b1) q^83 + (b13 - b12 + 2b11 + 2b10 - b9 - b8 - b7 - 2b1) q^85 + (b11 - b10 - b4 - b2) * q^87 + (-2b13 + b12 + 2b7 - b1) * q^89 + b8 * q^93 + (-b11 + b10 - 2b6 + b4 + b3 + b2 + 1) q^95 + (-b15 - b14 + b13 - 3b9 - b8 - b1) q^97 + b7 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{3} + 16 q^{9} - 4 q^{23} - 12 q^{25} + 16 q^{27} - 4 q^{29} - 12 q^{43} - 12 q^{53} + 16 q^{61} - 24 q^{65} - 4 q^{69} - 12 q^{75} - 20 q^{79} + 16 q^{81} - 4 q^{87} + 12 q^{95}+O(q^{100}) \) 16 * q + 16 * q^3 + 16 * q^9 - 4 * q^23 - 12 * q^25 + 16 * q^27 - 4 * q^29 - 12 * q^43 - 12 * q^53 + 16 * q^61 - 24 * q^65 - 4 * q^69 - 12 * q^75 - 20 * q^79 + 16 * q^81 - 4 * q^87 + 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	7644.2.e.q

Level	$7644$
Weight	$2$
Character orbit	7644.e
Analytic conductor	$61.038$
Analytic rank	$0$
Dimension	$16$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(61.0376473051\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} + 46x^{14} + 825x^{12} + 7500x^{10} + 37308x^{8} + 101640x^{6} + 144580x^{4} + 100432x^{2} + 26896 \) x^16 + 46x^14 + 825x^12 + 7500x^10 + 37308x^8 + 101640x^6 + 144580x^4 + 100432*x^2 + 26896
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2^{8} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( \nu^{2} + 6 \) v^2 + 6
\(\beta_{3}\)	\(=\)	\( ( 1029 \nu^{14} + 46524 \nu^{12} + 810588 \nu^{10} + 7012670 \nu^{8} + 31937661 \nu^{6} + \cdots + 26131976 ) / 8084 \) (1029v^14 + 46524v^12 + 810588v^10 + 7012670v^8 + 31937661v^6 + 73595466v^4 + 74150946*v^2 + 26131976) / 8084
\(\beta_{4}\)	\(=\)	\( ( 5779 \nu^{14} + 259588 \nu^{12} + 4487279 \nu^{10} + 38499406 \nu^{8} + 174091604 \nu^{6} + \cdots + 144572216 ) / 32336 \) (5779v^14 + 259588v^12 + 4487279v^10 + 38499406v^8 + 174091604v^6 + 399873800v^4 + 405297012*v^2 + 144572216) / 32336
\(\beta_{5}\)	\(=\)	\( ( - 10181 \nu^{14} - 457843 \nu^{12} - 7925901 \nu^{10} - 68119059 \nu^{8} - 308620618 \nu^{6} + \cdots - 258224456 ) / 16168 \) (-10181v^14 - 457843v^12 - 7925901v^10 - 68119059v^8 - 308620618v^6 - 710359350v^4 - 721746200*v^2 - 258224456) / 16168
\(\beta_{6}\)	\(=\)	\( ( 7107 \nu^{14} + 319283 \nu^{12} + 5520145 \nu^{10} + 47371277 \nu^{8} + 214261466 \nu^{6} + \cdots + 178048832 ) / 8084 \) (7107v^14 + 319283v^12 + 5520145v^10 + 47371277v^8 + 214261466v^6 + 492252998v^4 + 498997524*v^2 + 178048832) / 8084
\(\beta_{7}\)	\(=\)	\( ( - 69333 \nu^{15} - 3111090 \nu^{13} - 53685041 \nu^{11} - 459255180 \nu^{9} + \cdots - 1622011624 \nu ) / 1325776 \) (-69333v^15 - 3111090v^13 - 53685041v^11 - 459255180v^9 - 2066177776v^7 - 4701102396v^5 - 4674475300v^3 - 1622011624v) / 1325776
\(\beta_{8}\)	\(=\)	\( ( 35401 \nu^{15} + 1588471 \nu^{13} + 27419701 \nu^{11} + 234835195 \nu^{9} + 1059524506 \nu^{7} + \cdots + 872739288 \nu ) / 662888 \) (35401v^15 + 1588471v^13 + 27419701v^11 + 234835195v^9 + 1059524506v^7 + 2426075142v^5 + 2447880160v^3 + 872739288v) / 662888
\(\beta_{9}\)	\(=\)	\( ( 5779 \nu^{15} + 259588 \nu^{13} + 4487279 \nu^{11} + 38499406 \nu^{9} + 174091604 \nu^{7} + \cdots + 144572216 \nu ) / 32336 \) (5779v^15 + 259588v^13 + 4487279v^11 + 38499406v^9 + 174091604v^7 + 399873800v^5 + 405297012v^3 + 144572216v) / 32336
\(\beta_{10}\)	\(=\)	\( ( 628715 \nu^{15} + 850381 \nu^{14} + 28258576 \nu^{13} + 38204620 \nu^{12} + 488879103 \nu^{11} + \cdots + 21399812568 ) / 2651552 \) (628715v^15 + 850381v^14 + 28258576v^13 + 38204620v^12 + 488879103v^11 + 660583021v^10 + 4198731586v^9 + 5669915994v^8 + 19010533180v^7 + 25655220936v^6 + 43737622952v^5 + 58984976032v^4 + 44439690292v^3 + 59875850436v^2 + 15908425448*v + 21399812568) / 2651552
\(\beta_{11}\)	\(=\)	\( ( 628715 \nu^{15} - 850381 \nu^{14} + 28258576 \nu^{13} - 38204620 \nu^{12} + 488879103 \nu^{11} + \cdots - 21399812568 ) / 2651552 \) (628715v^15 - 850381v^14 + 28258576v^13 - 38204620v^12 + 488879103v^11 - 660583021v^10 + 4198731586v^9 - 5669915994v^8 + 19010533180v^7 - 25655220936v^6 + 43737622952v^5 - 58984976032v^4 + 44439690292v^3 - 59875850436v^2 + 15908425448*v - 21399812568) / 2651552
\(\beta_{12}\)	\(=\)	\( ( - 252577 \nu^{15} - 11351796 \nu^{13} - 196381393 \nu^{11} - 1686697514 \nu^{9} + \cdots - 6418142072 \nu ) / 662888 \) (-252577v^15 - 11351796v^13 - 196381393v^11 - 1686697514v^9 - 7638638544v^7 - 17585276192v^5 - 17892903964v^3 - 6418142072v) / 662888
\(\beta_{13}\)	\(=\)	\( ( - 926273 \nu^{15} - 41611766 \nu^{13} - 719442749 \nu^{11} - 6174680152 \nu^{9} + \cdots - 23352611144 \nu ) / 1325776 \) (-926273v^15 - 41611766v^13 - 719442749v^11 - 6174680152v^9 - 27938581536v^7 - 64245283700v^5 - 65260715092v^3 - 23352611144v) / 1325776
\(\beta_{14}\)	\(=\)	\( ( - 2473233 \nu^{15} - 533492 \nu^{14} - 111120200 \nu^{13} - 23989264 \nu^{12} + \cdots - 13573364224 ) / 2651552 \) (-2473233v^15 - 533492v^14 - 111120200v^13 - 23989264v^12 - 1921435553v^11 - 415259972v^10 - 16492167702v^9 - 3569008344v^8 - 74617642192v^7 - 16173300176v^6 - 171517937192v^5 - 37249357392v^4 - 174030771876v^3 - 37894690832v^2 - 62168674264*v - 13573364224) / 2651552
\(\beta_{15}\)	\(=\)	\( ( - 2473233 \nu^{15} + 533492 \nu^{14} - 111120200 \nu^{13} + 23989264 \nu^{12} + \cdots + 13573364224 ) / 2651552 \) (-2473233v^15 + 533492v^14 - 111120200v^13 + 23989264v^12 - 1921435553v^11 + 415259972v^10 - 16492167702v^9 + 3569008344v^8 - 74617642192v^7 + 16173300176v^6 - 171517937192v^5 + 37249357392v^4 - 174030771876v^3 + 37894690832v^2 - 62168674264*v + 13573364224) / 2651552

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{2} - 6 \) b2 - 6
\(\nu^{3}\)	\(=\)	\( \beta_{13} - 2\beta_{12} - \beta_{11} - \beta_{10} + 2\beta_{9} + \beta_{8} - 9\beta_1 \) b13 - 2b12 - b11 - b10 + 2b9 + b8 - 9*b1
\(\nu^{4}\)	\(=\)	\( - 2 \beta_{15} + 2 \beta_{14} + \beta_{11} - \beta_{10} + 3 \beta_{6} - 2 \beta_{5} - 13 \beta_{4} + \cdots + 63 \) -2b15 + 2b14 + b11 - b10 + 3b6 - 2b5 - 13b4 - b3 - 18b2 + 63
\(\nu^{5}\)	\(=\)	\( 2 \beta_{15} + 2 \beta_{14} - 23 \beta_{13} + 35 \beta_{12} + 21 \beta_{11} + 21 \beta_{10} + \cdots + 109 \beta_1 \) 2b15 + 2b14 - 23b13 + 35b12 + 21b11 + 21b10 - 48b9 - 15b8 - 8b7 + 109b1
\(\nu^{6}\)	\(=\)	\( 41 \beta_{15} - 41 \beta_{14} - 27 \beta_{11} + 27 \beta_{10} - 79 \beta_{6} + 42 \beta_{5} + 331 \beta_{4} + \cdots - 823 \) 41b15 - 41b14 - 27b11 + 27b10 - 79b6 + 42b5 + 331b4 + 23b3 + 296*b2 - 823
\(\nu^{7}\)	\(=\)	\( - 50 \beta_{15} - 50 \beta_{14} + 425 \beta_{13} - 545 \beta_{12} - 357 \beta_{11} - 357 \beta_{10} + \cdots - 1537 \beta_1 \) -50b15 - 50b14 + 425b13 - 545b12 - 357b11 - 357b10 + 922b9 + 233b8 + 228b7 - 1537b1
\(\nu^{8}\)	\(=\)	\( - 669 \beta_{15} + 669 \beta_{14} + 561 \beta_{11} - 561 \beta_{10} + 1575 \beta_{6} - 714 \beta_{5} + \cdots + 11981 \) -669b15 + 669b14 + 561b11 - 561b10 + 1575b6 - 714b5 - 6441b4 - 425b3 - 4824*b2 + 11981
\(\nu^{9}\)	\(=\)	\( 986 \beta_{15} + 986 \beta_{14} - 7385 \beta_{13} + 8425 \beta_{12} + 5827 \beta_{11} + \cdots + 23415 \beta_1 \) 986b15 + 986b14 - 7385b13 + 8425b12 + 5827b11 + 5827b10 - 16354b9 - 3775b8 - 4798b7 + 23415b1
\(\nu^{10}\)	\(=\)	\( 10477 \beta_{15} - 10477 \beta_{14} - 10545 \beta_{11} + 10545 \beta_{10} - 28537 \beta_{6} + \cdots - 184757 \) 10477b15 - 10477b14 - 10545b11 + 10545b10 - 28537b6 + 11654b5 + 114745b4 + 7385b3 + 78818*b2 - 184757
\(\nu^{11}\)	\(=\)	\( - 17930 \beta_{15} - 17930 \beta_{14} + 125241 \beta_{13} - 132079 \beta_{12} - 94741 \beta_{11} + \cdots - 370629 \beta_1 \) -17930b15 - 17930b14 + 125241b13 - 132079b12 - 94741b11 - 94741b10 + 280026b9 + 62133b8 + 90016b7 - 370629b1
\(\nu^{12}\)	\(=\)	\( - 164687 \beta_{15} + 164687 \beta_{14} + 188009 \beta_{11} - 188009 \beta_{10} + 495283 \beta_{6} + \cdots + 2937641 \) -164687b15 + 164687b14 + 188009b11 - 188009b10 + 495283b6 - 189482b5 - 1969241b4 - 125241b3 - 1292304*b2 + 2937641
\(\nu^{13}\)	\(=\)	\( 313250 \beta_{15} + 313250 \beta_{14} - 2099069 \beta_{13} + 2103681 \beta_{12} + 1546027 \beta_{11} + \cdots + 5980679 \beta_1 \) 313250b15 + 313250b14 - 2099069b13 + 2103681b12 + 1546027b11 + 1546027b10 - 4713782b9 - 1027171b8 - 1596734b7 + 5980679b1
\(\nu^{14}\)	\(=\)	\( 2622537 \beta_{15} - 2622537 \beta_{14} - 3250405 \beta_{11} + 3250405 \beta_{10} - 8409585 \beta_{6} + \cdots - 47483817 \) 2622537b15 - 2622537b14 - 3250405b11 + 3250405b10 - 8409585b6 + 3092054b5 + 33197309b4 + 2099069b3 + 21244282*b2 - 47483817
\(\nu^{15}\)	\(=\)	\( - 5349474 \beta_{15} - 5349474 \beta_{14} + 34958089 \beta_{13} - 33929327 \beta_{12} + \cdots - 97484809 \beta_1 \) -5349474b15 - 5349474b14 + 34958089b13 - 33929327b12 - 25329321b11 - 25329321b10 + 78660090b9 + 16992193b8 + 27477476b7 - 97484809b1

\(n\)	\(2549\)	\(3433\)	\(3823\)	\(5293\)
\(\chi(n)\)	\(1\)	\(1\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 4705.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T - 1)^{16} \) (T - 1)^16
$5$	\( T^{16} + 46 T^{14} + \cdots + 26896 \) T^16 + 46T^14 + 825T^12 + 7500T^10 + 37308T^8 + 101640T^6 + 144580T^4 + 100432*T^2 + 26896
$7$	\( T^{16} \) T^16
$11$	\( T^{16} + 88 T^{14} + \cdots + 2166784 \) T^16 + 88T^14 + 2772T^12 + 42528T^10 + 355556T^8 + 1668832T^6 + 4250944T^4 + 5198848*T^2 + 2166784
$13$	\( T^{16} + \cdots + 815730721 \) T^16 + 6T^14 + 112T^13 + 16T^12 + 1072T^11 + 4554T^10 + 5536T^9 + 74302T^8 + 71968T^7 + 769626T^6 + 2355184T^5 + 456976T^4 + 41584816T^3 + 28960854*T^2 + 815730721
$17$	\( (T^{8} - 92 T^{6} + \cdots + 106048)^{2} \) (T^8 - 92T^6 + 16T^5 + 2708T^4 - 416T^3 - 30464T^2 + 1152T + 106048)^2
$19$	\( T^{16} + \cdots + 928055296 \) T^16 + 182T^14 + 13705T^12 + 548432T^10 + 12416128T^8 + 155108352T^6 + 940921856T^4 + 1889026048*T^2 + 928055296
$23$	\( (T^{8} + 2 T^{7} + \cdots - 294784)^{2} \) (T^8 + 2T^7 - 117T^6 - 340T^5 + 4372T^4 + 17120T^3 - 44032T^2 - 267776*T - 294784)^2
$29$	\( (T^{8} + 2 T^{7} + \cdots + 199552)^{2} \) (T^8 + 2T^7 - 121T^6 - 168T^5 + 4224T^4 + 5280T^3 - 52048T^2 - 45184*T + 199552)^2
$31$	\( T^{16} + \cdots + 1494286336 \) T^16 + 246T^14 + 23977T^12 + 1182672T^10 + 31425024T^8 + 445128704T^6 + 3085497344T^4 + 8259223552*T^2 + 1494286336
$37$	\( T^{16} + \cdots + 13153337344 \) T^16 + 264T^14 + 28432T^12 + 1617920T^10 + 52436992T^8 + 965771264T^6 + 9368567808T^4 + 37857787904*T^2 + 13153337344
$41$	\( T^{16} + \cdots + 581937071104 \) T^16 + 380T^14 + 55464T^12 + 3994416T^10 + 154098420T^8 + 3317801840T^6 + 39573246608T^4 + 241624769792*T^2 + 581937071104
$43$	\( (T^{8} + 6 T^{7} + \cdots + 42752)^{2} \) (T^8 + 6T^7 - 199T^6 - 720T^5 + 10888T^4 + 7040T^3 - 104304T^2 + 27008*T + 42752)^2
$47$	\( T^{16} + \cdots + 4650958065664 \) T^16 + 482T^14 + 93597T^12 + 9460552T^10 + 537617416T^8 + 17260704200T^6 + 295515915012T^4 + 2286494229248*T^2 + 4650958065664
$53$	\( (T^{8} + 6 T^{7} + \cdots + 1984)^{2} \) (T^8 + 6T^7 - 189T^6 - 892T^5 + 6628T^4 + 34816T^3 + 49040T^2 + 22848*T + 1984)^2
$59$	\( T^{16} + \cdots + 34780758016 \) T^16 + 456T^14 + 84596T^12 + 8207232T^10 + 443053860T^8 + 12879966944T^6 + 170764866624T^4 + 518584379392*T^2 + 34780758016
$61$	\( (T^{8} - 8 T^{7} + \cdots + 192512)^{2} \) (T^8 - 8T^7 - 298T^6 + 1712T^5 + 29680T^4 - 85760T^3 - 1007104T^2 - 90112*T + 192512)^2
$67$	\( T^{16} + \cdots + 9487582756864 \) T^16 + 588T^14 + 124068T^12 + 12016640T^10 + 617530368T^8 + 17993908224T^6 + 298510467072T^4 + 2625214873600*T^2 + 9487582756864
$71$	\( T^{16} + \cdots + 10654113796096 \) T^16 + 648T^14 + 167540T^12 + 22154560T^10 + 1592412068T^8 + 61072850016T^6 + 1145945544000T^4 + 8253510011904*T^2 + 10654113796096
$73$	\( T^{16} + \cdots + 140388753227776 \) T^16 + 810T^14 + 255333T^12 + 40774204T^10 + 3614609188T^8 + 180704807168T^6 + 4828369224704T^4 + 57437227548672*T^2 + 140388753227776
$79$	\( (T^{8} + 10 T^{7} + \cdots - 5325824)^{2} \) (T^8 + 10T^7 - 363T^6 - 5032T^5 + 18776T^4 + 587720T^3 + 2982980T^2 + 3389568*T - 5325824)^2
$83$	\( T^{16} + \cdots + 169190997934336 \) T^16 + 898T^14 + 327133T^12 + 62227624T^10 + 6594421352T^8 + 378992135144T^6 + 10207922463172T^4 + 77553730238656*T^2 + 169190997934336
$89$	\( T^{16} + \cdots + 9364358895376 \) T^16 + 926T^14 + 321433T^12 + 51803196T^10 + 3946282524T^8 + 134711433384T^6 + 1691500968068T^4 + 7913405706064*T^2 + 9364358895376
$97$	\( T^{16} + \cdots + 19550829543424 \) T^16 + 826T^14 + 276309T^12 + 48346716T^10 + 4759614372T^8 + 262552165376T^6 + 7492951293952T^4 + 86040600641536*T^2 + 19550829543424
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