LMFDB - Newform orbit 792.2.h.i

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [792,2,Mod(307,792)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(792, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 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("792.307"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 792 = 2^{3} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	792.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$6.32415184009$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.3342602057661458415616.4
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 20x^{14} + 164x^{12} - 666x^{10} + 1300x^{8} - 924x^{6} + 273x^{4} + 404x^{2} + 64 $ x^16 - 20x^14 + 164x^12 - 666x^10 + 1300x^8 - 924x^6 + 273x^4 + 404*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{14} $
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 - \beta_{2} q^{2} + ( - \beta_{6} + 1) q^{4} - \beta_1 q^{5} + \beta_{12} q^{7} + ( - \beta_{13} + \beta_{9} - \beta_{2}) q^{8} - \beta_{3} q^{10} + ( - \beta_{13} - \beta_{10}) q^{11} + (\beta_{5} - \beta_{3}) q^{13}+ \cdots + ( - 2 \beta_{11} + 2 \beta_{9} + 3 \beta_{2}) q^{98}+O(q^{100}) $ q - b2 * q^2 + (-b6 + 1) * q^4 - b1 * q^5 + b12 * q^7 + (-b13 + b9 - b2) * q^8 - b3 * q^10 + (-b13 - b10) * q^11 + (b5 - b3) * q^13 + (b14 + b1) * q^14 + (-b15 + b8 - b6 - 1) * q^16 + (b11 - b2) * q^17 + (b12 - 2b7 - b5 - b3) q^19 + (b14 - b10 + b4) * q^20 + (-b15 - b12 - b7 + b5 - b3) * q^22 + (-b13 + b11 - 2b4 - b1) q^23 + q^25 + (b14 - b10 + b4 + 2b1) q^26 + (2b12 + b3) q^28 + (b13 - 2b9 - b2) q^29 + (-2b15 - 2b6) * q^31 + (-b13 - 2b11 + b9 + b2) q^32 + (b15 - b6 - 1) * q^34 + (2b13 + 2b2) * q^35 + (2b15 - b8 - b6 + 1) q^37 + (2b13 - 2b11 + b10 + 3b4 - b1) q^38 + (2b12 - 2b7 - b3) * q^40 + (2b13 + b11 + b2) q^41 + (b12 - 2b7 + b5 + b3) q^43 + (-b14 - b11 - b9 + 2b4 - b2 + b1) q^44 + (-2b12 + 2b7 + 2b5 - b3) q^46 - 3b1 q^47 + (b8 - b6 - 2) * q^49 - b2 * q^50 + (2b12 - 2b7 + b3) * q^52 + (-2b13 + 2b11 - 4b4 - b1) q^53 + (-2b15 + 2b12 + b8 + b6 - b5 + b3 - 1) * q^55 + (b14 + b10 - b4 + 2b1) q^56 + (b15 - 2b8 - b6 + 3) q^58 + (-4b14 + b13 - b11 + 2b10 - 2b1) q^59 + (-4b12 + 3b5 - 3b3) q^61 + (-4b13 - 2b2) * q^62 + (-3b15 + b8 + b6 + 1) q^64 + (2b13 - 2b11 + 4b2) q^65 + (2b8 - 2b6 - 2) * q^67 + (2b9 + 2b2) * q^68 + (2b15 + 2b6 + 2) * q^70 + (-2b13 + 2b11 - 4b4 + b1) q^71 + (-2b12 + 4b7) * q^73 + (2b11 + 2b9) * q^74 + (4b12 - 2b7 - 4b5) q^76 + (-b13 + 2b11 - 2b9 - 2b4 + b2 + b1) q^77 + (5b12 - 2b5 + 2b3) q^79 + (b14 + 2b13 - 2b11 + b10 + 3b4 + 2b1) * q^80 + (3b15 + b6 + 1) q^82 + (b13 + b11) * q^83 + (-2b12 + 2b5 - 2b3) q^85 + (-2b14 + 2b13 - 2b11 + 3b10 + b4 + 3b1) q^86 + (-b15 - b8 - 2b7 - b6 - 2b5 + b3 - 1) * q^88 + (-4b14 - 2b1) * q^89 + 4 * q^91 + (b14 - 2b13 + 2b11 - 3b10 - b4 + 2b1) * q^92 - 3b3 q^94 + (2b11 - 4b9 + 2b2) q^95 + (-3b8 + 3b6 - 1) * q^97 + (-2b11 + 2b9 + 3b2) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 12 q^{4} - 12 q^{16} + 4 q^{22} + 16 q^{25} - 24 q^{34} - 32 q^{49} + 32 q^{58} + 36 q^{64} - 32 q^{67} + 32 q^{70} + 8 q^{82} - 20 q^{88} + 64 q^{91} - 16 q^{97}+O(q^{100}) $ 16 * q + 12 * q^4 - 12 * q^16 + 4 * q^22 + 16 * q^25 - 24 * q^34 - 32 * q^49 + 32 * q^58 + 36 * q^64 - 32 * q^67 + 32 * q^70 + 8 * q^82 - 20 * q^88 + 64 * q^91 - 16 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 20x^{14} + 164x^{12} - 666x^{10} + 1300x^{8} - 924x^{6} + 273x^{4} + 404x^{2} + 64 $ x^16 - 20*x^14 + 164*x^12 - 666*x^10 + 1300*x^8 - 924*x^6 + 273*x^4 + 404*x^2 + 64 :

$\beta_{1}$	$=$	$ ( 6891 \nu^{15} - 135786 \nu^{13} + 1092346 \nu^{11} - 4305788 \nu^{9} + 7949022 \nu^{7} + \cdots + 3564012 \nu ) / 695368 $ (6891v^15 - 135786v^13 + 1092346v^11 - 4305788v^9 + 7949022v^7 - 4808686v^5 + 1379637v^3 + 3564012v) / 695368
$\beta_{2}$	$=$	$ ( - 22179 \nu^{14} + 450427 \nu^{12} - 3763783 \nu^{10} + 15736045 \nu^{8} - 32550017 \nu^{6} + \cdots - 3884384 ) / 1390736 $ (-22179v^14 + 450427v^12 - 3763783v^10 + 15736045v^8 - 32550017v^6 + 27985781v^4 - 13959008*v^2 - 3884384) / 1390736
$\beta_{3}$	$=$	$ ( 36955 \nu^{15} - 751451 \nu^{13} + 6309143 \nu^{11} - 26657365 \nu^{9} + 56418201 \nu^{7} + \cdots + 6467816 \nu ) / 695368 $ (36955v^15 - 751451v^13 + 6309143v^11 - 26657365v^9 + 56418201v^7 - 51028693v^5 + 24644192v^3 + 6467816v) / 695368
$\beta_{4}$	$=$	$ ( - 16228 \nu^{15} + 24213 \nu^{14} + 328656 \nu^{13} - 488205 \nu^{12} - 2729424 \nu^{11} + \cdots + 7615568 ) / 1390736 $ (-16228v^15 + 24213v^14 + 328656v^13 - 488205v^12 - 2729424v^11 + 4047401v^10 + 11243808v^9 - 16745323v^8 - 22263712v^7 + 34108615v^6 + 15498736v^5 - 28487387v^4 - 1576924v^3 + 13348320v^2 - 8868624*v + 7615568) / 1390736
$\beta_{5}$	$=$	$ ( - 55594 \nu^{15} + 1121031 \nu^{13} - 9302563 \nu^{11} + 38565451 \nu^{9} - 78706669 \nu^{7} + \cdots - 16210140 \nu ) / 695368 $ (-55594v^15 + 1121031v^13 - 9302563v^11 + 38565451v^9 - 78706669v^7 + 64953145v^5 - 28003055v^3 - 16210140v) / 695368
$\beta_{6}$	$=$	$ ( 61033 \nu^{14} - 1229185 \nu^{12} + 10210845 \nu^{10} - 42548751 \nu^{8} + 88260163 \nu^{6} + \cdots + 14879696 ) / 1390736 $ (61033v^14 - 1229185v^12 + 10210845v^10 - 42548751v^8 + 88260163v^6 - 77306007v^4 + 38490952*v^2 + 14879696) / 1390736
$\beta_{7}$	$=$	$ ( - 104651 \nu^{15} + 2119985 \nu^{13} - 17712017 \nu^{11} + 74285219 \nu^{9} - 155121315 \nu^{7} + \cdots - 27237648 \nu ) / 695368 $ (-104651v^15 + 2119985v^13 - 17712017v^11 + 74285219v^9 - 155121315v^7 + 135501083v^5 - 60382778v^3 - 27237648v) / 695368
$\beta_{8}$	$=$	$ ( 73833 \nu^{14} - 1482689 \nu^{12} + 12232797 \nu^{10} - 50317679 \nu^{8} + 101452099 \nu^{6} + \cdots + 19843264 ) / 1390736 $ (73833v^14 - 1482689v^12 + 12232797v^10 - 50317679v^8 + 101452099v^6 - 81442871v^4 + 33432264*v^2 + 19843264) / 1390736
$\beta_{9}$	$=$	$ ( - 85055 \nu^{14} + 1721967 \nu^{12} - 14375099 \nu^{10} + 60142625 \nu^{8} - 124633021 \nu^{6} + \cdots - 22927136 ) / 1390736 $ (-85055v^14 + 1721967v^12 - 14375099v^10 + 60142625v^8 - 124633021v^6 + 105733441v^4 - 44784832*v^2 - 22927136) / 1390736
$\beta_{10}$	$=$	$ ( 310062 \nu^{15} + 24213 \nu^{14} - 6283584 \nu^{13} - 488205 \nu^{12} + 52517600 \nu^{11} + \cdots + 7615568 ) / 1390736 $ (310062v^15 + 24213v^14 - 6283584v^13 - 488205v^12 + 52517600v^11 + 4047401v^10 - 220414172v^9 - 16745323v^8 + 461289072v^7 + 34108615v^6 - 407488256v^5 - 28487387v^4 + 189274350v^3 + 13348320v^2 + 80507432*v + 7615568) / 1390736
$\beta_{11}$	$=$	$ ( - 134075 \nu^{14} + 2719907 \nu^{12} - 22767383 \nu^{10} + 95787157 \nu^{8} - 201277353 \nu^{6} + \cdots - 33067424 ) / 1390736 $ (-134075v^14 + 2719907v^12 - 22767383v^10 + 95787157v^8 - 201277353v^6 + 178699029v^4 - 81477728*v^2 - 33067424) / 1390736
$\beta_{12}$	$=$	$ ( - 204751 \nu^{15} + 4145926 \nu^{13} - 34610826 \nu^{11} + 144971388 \nu^{9} - 302181482 \nu^{7} + \cdots - 51983964 \nu ) / 695368 $ (-204751v^15 + 4145926v^13 - 34610826v^11 + 144971388v^9 - 302181482v^7 + 264139390v^5 - 121650617v^3 - 51983964v) / 695368
$\beta_{13}$	$=$	$ ( - 182501 \nu^{14} + 3696317 \nu^{12} - 30862185 \nu^{10} + 129277803 \nu^{8} - 269494583 \nu^{6} + \cdots - 48298560 ) / 1390736 $ (-182501v^14 + 3696317v^12 - 30862185v^10 + 129277803v^8 - 269494583v^6 + 235673803v^4 - 108174368*v^2 - 48298560) / 1390736
$\beta_{14}$	$=$	$ ( 277964 \nu^{15} - 5630141 \nu^{13} + 47017185 \nu^{11} - 197053317 \nu^{9} + 411261295 \nu^{7} + \cdots + 69700236 \nu ) / 695368 $ (277964v^15 - 5630141v^13 + 47017185v^11 - 197053317v^9 + 411261295v^7 - 360766643v^5 + 166620483v^3 + 69700236v) / 695368
$\beta_{15}$	$=$	$ ( 221653 \nu^{14} - 4488493 \nu^{12} + 37469369 \nu^{10} - 156947139 \nu^{8} + 327124903 \nu^{6} + \cdots + 56840416 ) / 1390736 $ (221653v^14 - 4488493v^12 + 37469369v^10 - 156947139v^8 + 327124903v^6 - 285557643v^4 + 129227560*v^2 + 56840416) / 1390736

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

$\nu$	$=$	$ ( \beta_{13} + 2\beta_{12} - \beta_{11} + 2\beta_{10} - \beta_{5} + \beta_{3} + \beta_1 ) / 4 $ (b13 + 2b12 - b11 + 2b10 - b5 + b3 + b1) / 4
$\nu^{2}$	$=$	$ ( -2\beta_{15} + \beta_{13} - 3\beta_{11} + \beta_{8} + \beta_{6} - 4\beta_{2} + 9 ) / 4 $ (-2b15 + b13 - 3b11 + b8 + b6 - 4*b2 + 9) / 4
$\nu^{3}$	$=$	$ ( 2\beta_{14} + 3\beta_{13} + 10\beta_{12} - 3\beta_{11} + 6\beta_{10} - 3\beta_{5} + 9\beta_{3} + 9\beta_1 ) / 4 $ (2b14 + 3b13 + 10b12 - 3b11 + 6b10 - 3b5 + 9b3 + 9b1) / 4
$\nu^{4}$	$=$	$ ( -18\beta_{15} - \beta_{13} - 21\beta_{11} + 9\beta_{8} + \beta_{6} - 12\beta_{2} + 29 ) / 4 $ (-18b15 - b13 - 21b11 + 9b8 + b6 - 12b2 + 29) / 4
$\nu^{5}$	$=$	$ ( 10 \beta_{14} + 3 \beta_{13} + 34 \beta_{12} - 3 \beta_{11} + 16 \beta_{10} + 7 \beta_{5} + \cdots + 61 \beta_1 ) / 4 $ (10b14 + 3b13 + 34b12 - 3b11 + 16b10 + 7b5 - 10b4 + 43b3 + 61*b1) / 4
$\nu^{6}$	$=$	$ ( -124\beta_{15} - 55\beta_{13} - 107\beta_{11} + 8\beta_{9} + 51\beta_{8} - 7\beta_{6} - 20\beta_{2} + 37 ) / 4 $ (-124b15 - 55b13 - 107b11 + 8b9 + 51b8 - 7b6 - 20*b2 + 37) / 4
$\nu^{7}$	$=$	$ ( 22 \beta_{14} - 49 \beta_{13} + 18 \beta_{12} + 49 \beta_{11} + 28 \beta_{7} + 157 \beta_{5} + \cdots + 339 \beta_1 ) / 4 $ (22b14 - 49b13 + 18b12 + 49b11 + 28b7 + 157b5 - 98b4 + 165b3 + 339*b1) / 4
$\nu^{8}$	$=$	$ ( -710\beta_{15} - 529\beta_{13} - 405\beta_{11} + 32\beta_{9} + 219\beta_{8} - 77\beta_{6} + 100\beta_{2} - 477 ) / 4 $ (-710b15 - 529b13 - 405b11 + 32b9 + 219b8 - 77b6 + 100*b2 - 477) / 4
$\nu^{9}$	$=$	$ ( - 162 \beta_{14} - 551 \beta_{13} - 890 \beta_{12} + 551 \beta_{11} - 418 \beta_{10} + \cdots + 1531 \beta_1 ) / 4 $ (-162b14 - 551b13 - 890b12 + 551b11 - 418b10 + 312b7 + 1291b5 - 684b4 + 431b3 + 1531b1) / 4
$\nu^{10}$	$=$	$ ( - 3386 \beta_{15} - 3543 \beta_{13} - 795 \beta_{11} - 192 \beta_{9} + 543 \beta_{8} - 397 \beta_{6} + \cdots - 5833 ) / 4 $ (-3386b15 - 3543b13 - 795b11 - 192b9 + 543b8 - 397b6 + 1572*b2 - 5833) / 4
$\nu^{11}$	$=$	$ ( - 2714 \beta_{14} - 3799 \beta_{13} - 9594 \beta_{12} + 3799 \beta_{11} - 3880 \beta_{10} + \cdots + 5011 \beta_1 ) / 4 $ (-2714b14 - 3799b13 - 9594b12 + 3799b11 - 3880b10 + 2112b7 + 7881b5 - 3718b4 - 379b3 + 5011b1) / 4
$\nu^{12}$	$=$	$ ( - 12432 \beta_{15} - 18657 \beta_{13} + 4139 \beta_{11} - 4048 \beta_{9} - 1567 \beta_{8} + \cdots - 44101 ) / 4 $ (-12432b15 - 18657b13 + 4139b11 - 4048b9 - 1567b8 - 337b6 + 13636*b2 - 44101) / 4
$\nu^{13}$	$=$	$ ( - 24126 \beta_{14} - 19823 \beta_{13} - 69514 \beta_{12} + 19823 \beta_{11} - 25164 \beta_{10} + \cdots + 3993 \beta_1 ) / 4 $ (-24126b14 - 19823b13 - 69514b12 + 19823b11 - 25164b10 + 9724b7 + 38707b5 - 14482b4 - 16269b3 + 3993b1) / 4
$\nu^{14}$	$=$	$ ( - 21090 \beta_{15} - 74151 \beta_{13} + 64045 \beta_{11} - 38664 \beta_{9} - 32711 \beta_{8} + \cdots - 268279 ) / 4 $ (-21090b15 - 74151b13 + 64045b11 - 38664b9 - 32711b8 + 16801b6 + 97588*b2 - 268279) / 4
$\nu^{15}$	$=$	$ ( - 165206 \beta_{14} - 75189 \beta_{13} - 405126 \beta_{12} + 75189 \beta_{11} - 133058 \beta_{10} + \cdots - 109415 \beta_1 ) / 4 $ (-165206b14 - 75189b13 - 405126b12 + 75189b11 - 133058b10 + 19472b7 + 145005b5 - 17320b4 - 153839b3 - 109415b1) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/792\mathbb{Z}\right)^\times$.

$n$	$145$	$199$	$353$	$397$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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} $

307.1

−0.0131233	+	0.500000i
0.0131233	−	0.500000i
0.0131233	+	0.500000i
−0.0131233	−	0.500000i
−0.946412	+	0.500000i
0.946412	−	0.500000i
0.946412	+	0.500000i
−0.946412	−	0.500000i
2.14576	+	0.500000i
−2.14576	−	0.500000i
−2.14576	+	0.500000i
2.14576	−	0.500000i
−2.34517	+	0.500000i
2.34517	−	0.500000i
2.34517	+	0.500000i
−2.34517	−	0.500000i

−1.37491

−

0.331077i

1.78078

0.910404i

−

2.00000i

−3.02045

−2.14700

−

1.84130i

−0.662153

2.74983i

307.2

−1.37491

−

0.331077i

1.78078

0.910404i

2.00000i

3.02045

−2.14700

−

1.84130i

0.662153

−

2.74983i

307.3

−1.37491

0.331077i

1.78078

−

0.910404i

−

2.00000i

3.02045

−2.14700

1.84130i

0.662153

2.74983i

307.4

−1.37491

0.331077i

1.78078

−

0.910404i

2.00000i

−3.02045

−2.14700

1.84130i

−0.662153

−

2.74983i

307.5

−0.927153

−

1.06789i

−0.280776

1.98019i

−

2.00000i

−0.936426

2.37495

−

1.53610i

−2.13578

1.85431i

307.6

−0.927153

−

1.06789i

−0.280776

1.98019i

2.00000i

0.936426

2.37495

−

1.53610i

2.13578

−

1.85431i

307.7

−0.927153

1.06789i

−0.280776

−

1.98019i

−

2.00000i

0.936426

2.37495

1.53610i

2.13578

1.85431i

307.8

−0.927153

1.06789i

−0.280776

−

1.98019i

2.00000i

−0.936426

2.37495

1.53610i

−2.13578

−

1.85431i

307.9

0.927153

−

1.06789i

−0.280776

−

1.98019i

−

2.00000i

−0.936426

−2.37495

−

1.53610i

−2.13578

−

1.85431i

307.10

0.927153

−

1.06789i

−0.280776

−

1.98019i

2.00000i

0.936426

−2.37495

−

1.53610i

2.13578

1.85431i

307.11

0.927153

1.06789i

−0.280776

1.98019i

−

2.00000i

0.936426

−2.37495

1.53610i

2.13578

−

1.85431i

307.12

0.927153

1.06789i

−0.280776

1.98019i

2.00000i

−0.936426

−2.37495

1.53610i

−2.13578

1.85431i

307.13

1.37491

−

0.331077i

1.78078

−

0.910404i

−

2.00000i

−3.02045

2.14700

−

1.84130i

−0.662153

−

2.74983i

307.14

1.37491

−

0.331077i

1.78078

−

0.910404i

2.00000i

3.02045

2.14700

−

1.84130i

0.662153

2.74983i

307.15

1.37491

0.331077i

1.78078

0.910404i

−

2.00000i

3.02045

2.14700

1.84130i

0.662153

−

2.74983i

307.16

1.37491

0.331077i

1.78078

0.910404i

2.00000i

−3.02045

2.14700

1.84130i

−0.662153

2.74983i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
8.d	odd	2	1	inner
11.b	odd	2	1	inner
24.f	even	2	1	inner
33.d	even	2	1	inner
88.g	even	2	1	inner
264.p	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	792.2.h.i	✓	16
3.b	odd	2	1	inner	792.2.h.i	✓	16
4.b	odd	2	1		3168.2.h.i		16
8.b	even	2	1		3168.2.h.i		16
8.d	odd	2	1	inner	792.2.h.i	✓	16
11.b	odd	2	1	inner	792.2.h.i	✓	16
12.b	even	2	1		3168.2.h.i		16
24.f	even	2	1	inner	792.2.h.i	✓	16
24.h	odd	2	1		3168.2.h.i		16
33.d	even	2	1	inner	792.2.h.i	✓	16
44.c	even	2	1		3168.2.h.i		16
88.b	odd	2	1		3168.2.h.i		16
88.g	even	2	1	inner	792.2.h.i	✓	16
132.d	odd	2	1		3168.2.h.i		16
264.m	even	2	1		3168.2.h.i		16
264.p	odd	2	1	inner	792.2.h.i	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
792.2.h.i	✓	16	1.a	even	1	1	trivial
792.2.h.i	✓	16	3.b	odd	2	1	inner
792.2.h.i	✓	16	8.d	odd	2	1	inner
792.2.h.i	✓	16	11.b	odd	2	1	inner
792.2.h.i	✓	16	24.f	even	2	1	inner
792.2.h.i	✓	16	33.d	even	2	1	inner
792.2.h.i	✓	16	88.g	even	2	1	inner
792.2.h.i	✓	16	264.p	odd	2	1	inner
3168.2.h.i		16	4.b	odd	2	1
3168.2.h.i		16	8.b	even	2	1
3168.2.h.i		16	12.b	even	2	1
3168.2.h.i		16	24.h	odd	2	1
3168.2.h.i		16	44.c	even	2	1
3168.2.h.i		16	88.b	odd	2	1
3168.2.h.i		16	132.d	odd	2	1
3168.2.h.i		16	264.m	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(792, [\chi])$:

$ T_{5}^{2} + 4 $

T5^2 + 4

$ T_{7}^{4} - 10T_{7}^{2} + 8 $

T7^4 - 10*T7^2 + 8

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{8} - 3 T^{6} + 6 T^{4} + \cdots + 16)^{2} $ (T^8 - 3T^6 + 6T^4 - 12*T^2 + 16)^2
$3$	$ T^{16} $ T^16
$5$	$ (T^{2} + 4)^{8} $ (T^2 + 4)^8
$7$	$ (T^{4} - 10 T^{2} + 8)^{4} $ (T^4 - 10*T^2 + 8)^4
$11$	$ (T^{8} - 16 T^{6} + \cdots + 14641)^{2} $ (T^8 - 16T^6 + 238T^4 - 1936*T^2 + 14641)^2
$13$	$ (T^{4} - 20 T^{2} + 32)^{4} $ (T^4 - 20*T^2 + 32)^4
$17$	$ (T^{4} + 14 T^{2} + 32)^{4} $ (T^4 + 14*T^2 + 32)^4
$19$	$ (T^{4} + 58 T^{2} + 416)^{4} $ (T^4 + 58*T^2 + 416)^4
$23$	$ (T^{4} + 36 T^{2} + 256)^{4} $ (T^4 + 36*T^2 + 256)^4
$29$	$ (T^{4} - 46 T^{2} + 104)^{4} $ (T^4 - 46*T^2 + 104)^4
$31$	$ (T^{4} + 72 T^{2} + 208)^{4} $ (T^4 + 72*T^2 + 208)^4
$37$	$ (T^{4} + 60 T^{2} + 832)^{4} $ (T^4 + 60*T^2 + 832)^4
$41$	$ (T^{4} + 62 T^{2} + 128)^{4} $ (T^4 + 62*T^2 + 128)^4
$43$	$ (T^{4} + 122 T^{2} + 1664)^{4} $ (T^4 + 122*T^2 + 1664)^4
$47$	$ (T^{2} + 36)^{8} $ (T^2 + 36)^8
$53$	$ (T^{2} + 68)^{8} $ (T^2 + 68)^8
$59$	$ (T^{4} - 236 T^{2} + 13312)^{4} $ (T^4 - 236*T^2 + 13312)^4
$61$	$ (T^{4} - 148 T^{2} + 5408)^{4} $ (T^4 - 148*T^2 + 5408)^4
$67$	$ (T^{2} + 4 T - 64)^{8} $ (T^2 + 4*T - 64)^8
$71$	$ (T^{4} + 168 T^{2} + 2704)^{4} $ (T^4 + 168*T^2 + 2704)^4
$73$	$ (T^{4} + 184 T^{2} + 1664)^{4} $ (T^4 + 184*T^2 + 1664)^4
$79$	$ (T^{4} - 170 T^{2} + 2312)^{4} $ (T^4 - 170*T^2 + 2312)^4
$83$	$ (T^{4} + 28 T^{2} + 128)^{4} $ (T^4 + 28*T^2 + 128)^4
$89$	$ (T^{4} - 288 T^{2} + 3328)^{4} $ (T^4 - 288*T^2 + 3328)^4
$97$	$ (T^{2} + 2 T - 152)^{8} $ (T^2 + 2*T - 152)^8
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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:	\( N \)	\(=\)	\( 792 = 2^{3} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	792.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{2} q^{2} + ( - \beta_{6} + 1) q^{4} - \beta_1 q^{5} + \beta_{12} q^{7} + ( - \beta_{13} + \beta_{9} - \beta_{2}) q^{8} - \beta_{3} q^{10} + ( - \beta_{13} - \beta_{10}) q^{11} + (\beta_{5} - \beta_{3}) q^{13}+ \cdots + ( - 2 \beta_{11} + 2 \beta_{9} + 3 \beta_{2}) q^{98}+O(q^{100}) \) q - b2 * q^2 + (-b6 + 1) * q^4 - b1 * q^5 + b12 * q^7 + (-b13 + b9 - b2) * q^8 - b3 * q^10 + (-b13 - b10) * q^11 + (b5 - b3) * q^13 + (b14 + b1) * q^14 + (-b15 + b8 - b6 - 1) * q^16 + (b11 - b2) * q^17 + (b12 - 2b7 - b5 - b3) q^19 + (b14 - b10 + b4) * q^20 + (-b15 - b12 - b7 + b5 - b3) * q^22 + (-b13 + b11 - 2b4 - b1) q^23 + q^25 + (b14 - b10 + b4 + 2b1) q^26 + (2b12 + b3) q^28 + (b13 - 2b9 - b2) q^29 + (-2b15 - 2b6) * q^31 + (-b13 - 2b11 + b9 + b2) q^32 + (b15 - b6 - 1) * q^34 + (2b13 + 2b2) * q^35 + (2b15 - b8 - b6 + 1) q^37 + (2b13 - 2b11 + b10 + 3b4 - b1) q^38 + (2b12 - 2b7 - b3) * q^40 + (2b13 + b11 + b2) q^41 + (b12 - 2b7 + b5 + b3) q^43 + (-b14 - b11 - b9 + 2b4 - b2 + b1) q^44 + (-2b12 + 2b7 + 2b5 - b3) q^46 - 3b1 q^47 + (b8 - b6 - 2) * q^49 - b2 * q^50 + (2b12 - 2b7 + b3) * q^52 + (-2b13 + 2b11 - 4b4 - b1) q^53 + (-2b15 + 2b12 + b8 + b6 - b5 + b3 - 1) * q^55 + (b14 + b10 - b4 + 2b1) q^56 + (b15 - 2b8 - b6 + 3) q^58 + (-4b14 + b13 - b11 + 2b10 - 2b1) q^59 + (-4b12 + 3b5 - 3b3) q^61 + (-4b13 - 2b2) * q^62 + (-3b15 + b8 + b6 + 1) q^64 + (2b13 - 2b11 + 4b2) q^65 + (2b8 - 2b6 - 2) * q^67 + (2b9 + 2b2) * q^68 + (2b15 + 2b6 + 2) * q^70 + (-2b13 + 2b11 - 4b4 + b1) q^71 + (-2b12 + 4b7) * q^73 + (2b11 + 2b9) * q^74 + (4b12 - 2b7 - 4b5) q^76 + (-b13 + 2b11 - 2b9 - 2b4 + b2 + b1) q^77 + (5b12 - 2b5 + 2b3) q^79 + (b14 + 2b13 - 2b11 + b10 + 3b4 + 2b1) * q^80 + (3b15 + b6 + 1) q^82 + (b13 + b11) * q^83 + (-2b12 + 2b5 - 2b3) q^85 + (-2b14 + 2b13 - 2b11 + 3b10 + b4 + 3b1) q^86 + (-b15 - b8 - 2b7 - b6 - 2b5 + b3 - 1) * q^88 + (-4b14 - 2b1) * q^89 + 4 * q^91 + (b14 - 2b13 + 2b11 - 3b10 - b4 + 2b1) * q^92 - 3b3 q^94 + (2b11 - 4b9 + 2b2) q^95 + (-3b8 + 3b6 - 1) * q^97 + (-2b11 + 2b9 + 3b2) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{4} - 12 q^{16} + 4 q^{22} + 16 q^{25} - 24 q^{34} - 32 q^{49} + 32 q^{58} + 36 q^{64} - 32 q^{67} + 32 q^{70} + 8 q^{82} - 20 q^{88} + 64 q^{91} - 16 q^{97}+O(q^{100}) \) 16 * q + 12 * q^4 - 12 * q^16 + 4 * q^22 + 16 * q^25 - 24 * q^34 - 32 * q^49 + 32 * q^58 + 36 * q^64 - 32 * q^67 + 32 * q^70 + 8 * q^82 - 20 * q^88 + 64 * q^91 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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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	792.2.h.i

Level	$792$
Weight	$2$
Character orbit	792.h
Analytic conductor	$6.324$
Analytic rank	$0$
Dimension	$16$
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(6.32415184009\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	16.0.3342602057661458415616.4
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 20x^{14} + 164x^{12} - 666x^{10} + 1300x^{8} - 924x^{6} + 273x^{4} + 404x^{2} + 64 \) x^16 - 20x^14 + 164x^12 - 666x^10 + 1300x^8 - 924x^6 + 273x^4 + 404*x^2 + 64
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{14} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 6891 \nu^{15} - 135786 \nu^{13} + 1092346 \nu^{11} - 4305788 \nu^{9} + 7949022 \nu^{7} + \cdots + 3564012 \nu ) / 695368 \) (6891v^15 - 135786v^13 + 1092346v^11 - 4305788v^9 + 7949022v^7 - 4808686v^5 + 1379637v^3 + 3564012v) / 695368
\(\beta_{2}\)	\(=\)	\( ( - 22179 \nu^{14} + 450427 \nu^{12} - 3763783 \nu^{10} + 15736045 \nu^{8} - 32550017 \nu^{6} + \cdots - 3884384 ) / 1390736 \) (-22179v^14 + 450427v^12 - 3763783v^10 + 15736045v^8 - 32550017v^6 + 27985781v^4 - 13959008*v^2 - 3884384) / 1390736
\(\beta_{3}\)	\(=\)	\( ( 36955 \nu^{15} - 751451 \nu^{13} + 6309143 \nu^{11} - 26657365 \nu^{9} + 56418201 \nu^{7} + \cdots + 6467816 \nu ) / 695368 \) (36955v^15 - 751451v^13 + 6309143v^11 - 26657365v^9 + 56418201v^7 - 51028693v^5 + 24644192v^3 + 6467816v) / 695368
\(\beta_{4}\)	\(=\)	\( ( - 16228 \nu^{15} + 24213 \nu^{14} + 328656 \nu^{13} - 488205 \nu^{12} - 2729424 \nu^{11} + \cdots + 7615568 ) / 1390736 \) (-16228v^15 + 24213v^14 + 328656v^13 - 488205v^12 - 2729424v^11 + 4047401v^10 + 11243808v^9 - 16745323v^8 - 22263712v^7 + 34108615v^6 + 15498736v^5 - 28487387v^4 - 1576924v^3 + 13348320v^2 - 8868624*v + 7615568) / 1390736
\(\beta_{5}\)	\(=\)	\( ( - 55594 \nu^{15} + 1121031 \nu^{13} - 9302563 \nu^{11} + 38565451 \nu^{9} - 78706669 \nu^{7} + \cdots - 16210140 \nu ) / 695368 \) (-55594v^15 + 1121031v^13 - 9302563v^11 + 38565451v^9 - 78706669v^7 + 64953145v^5 - 28003055v^3 - 16210140v) / 695368
\(\beta_{6}\)	\(=\)	\( ( 61033 \nu^{14} - 1229185 \nu^{12} + 10210845 \nu^{10} - 42548751 \nu^{8} + 88260163 \nu^{6} + \cdots + 14879696 ) / 1390736 \) (61033v^14 - 1229185v^12 + 10210845v^10 - 42548751v^8 + 88260163v^6 - 77306007v^4 + 38490952*v^2 + 14879696) / 1390736
\(\beta_{7}\)	\(=\)	\( ( - 104651 \nu^{15} + 2119985 \nu^{13} - 17712017 \nu^{11} + 74285219 \nu^{9} - 155121315 \nu^{7} + \cdots - 27237648 \nu ) / 695368 \) (-104651v^15 + 2119985v^13 - 17712017v^11 + 74285219v^9 - 155121315v^7 + 135501083v^5 - 60382778v^3 - 27237648v) / 695368
\(\beta_{8}\)	\(=\)	\( ( 73833 \nu^{14} - 1482689 \nu^{12} + 12232797 \nu^{10} - 50317679 \nu^{8} + 101452099 \nu^{6} + \cdots + 19843264 ) / 1390736 \) (73833v^14 - 1482689v^12 + 12232797v^10 - 50317679v^8 + 101452099v^6 - 81442871v^4 + 33432264*v^2 + 19843264) / 1390736
\(\beta_{9}\)	\(=\)	\( ( - 85055 \nu^{14} + 1721967 \nu^{12} - 14375099 \nu^{10} + 60142625 \nu^{8} - 124633021 \nu^{6} + \cdots - 22927136 ) / 1390736 \) (-85055v^14 + 1721967v^12 - 14375099v^10 + 60142625v^8 - 124633021v^6 + 105733441v^4 - 44784832*v^2 - 22927136) / 1390736
\(\beta_{10}\)	\(=\)	\( ( 310062 \nu^{15} + 24213 \nu^{14} - 6283584 \nu^{13} - 488205 \nu^{12} + 52517600 \nu^{11} + \cdots + 7615568 ) / 1390736 \) (310062v^15 + 24213v^14 - 6283584v^13 - 488205v^12 + 52517600v^11 + 4047401v^10 - 220414172v^9 - 16745323v^8 + 461289072v^7 + 34108615v^6 - 407488256v^5 - 28487387v^4 + 189274350v^3 + 13348320v^2 + 80507432*v + 7615568) / 1390736
\(\beta_{11}\)	\(=\)	\( ( - 134075 \nu^{14} + 2719907 \nu^{12} - 22767383 \nu^{10} + 95787157 \nu^{8} - 201277353 \nu^{6} + \cdots - 33067424 ) / 1390736 \) (-134075v^14 + 2719907v^12 - 22767383v^10 + 95787157v^8 - 201277353v^6 + 178699029v^4 - 81477728*v^2 - 33067424) / 1390736
\(\beta_{12}\)	\(=\)	\( ( - 204751 \nu^{15} + 4145926 \nu^{13} - 34610826 \nu^{11} + 144971388 \nu^{9} - 302181482 \nu^{7} + \cdots - 51983964 \nu ) / 695368 \) (-204751v^15 + 4145926v^13 - 34610826v^11 + 144971388v^9 - 302181482v^7 + 264139390v^5 - 121650617v^3 - 51983964v) / 695368
\(\beta_{13}\)	\(=\)	\( ( - 182501 \nu^{14} + 3696317 \nu^{12} - 30862185 \nu^{10} + 129277803 \nu^{8} - 269494583 \nu^{6} + \cdots - 48298560 ) / 1390736 \) (-182501v^14 + 3696317v^12 - 30862185v^10 + 129277803v^8 - 269494583v^6 + 235673803v^4 - 108174368*v^2 - 48298560) / 1390736
\(\beta_{14}\)	\(=\)	\( ( 277964 \nu^{15} - 5630141 \nu^{13} + 47017185 \nu^{11} - 197053317 \nu^{9} + 411261295 \nu^{7} + \cdots + 69700236 \nu ) / 695368 \) (277964v^15 - 5630141v^13 + 47017185v^11 - 197053317v^9 + 411261295v^7 - 360766643v^5 + 166620483v^3 + 69700236v) / 695368
\(\beta_{15}\)	\(=\)	\( ( 221653 \nu^{14} - 4488493 \nu^{12} + 37469369 \nu^{10} - 156947139 \nu^{8} + 327124903 \nu^{6} + \cdots + 56840416 ) / 1390736 \) (221653v^14 - 4488493v^12 + 37469369v^10 - 156947139v^8 + 327124903v^6 - 285557643v^4 + 129227560*v^2 + 56840416) / 1390736

\(\nu\)	\(=\)	\( ( \beta_{13} + 2\beta_{12} - \beta_{11} + 2\beta_{10} - \beta_{5} + \beta_{3} + \beta_1 ) / 4 \) (b13 + 2b12 - b11 + 2b10 - b5 + b3 + b1) / 4
\(\nu^{2}\)	\(=\)	\( ( -2\beta_{15} + \beta_{13} - 3\beta_{11} + \beta_{8} + \beta_{6} - 4\beta_{2} + 9 ) / 4 \) (-2b15 + b13 - 3b11 + b8 + b6 - 4*b2 + 9) / 4
\(\nu^{3}\)	\(=\)	\( ( 2\beta_{14} + 3\beta_{13} + 10\beta_{12} - 3\beta_{11} + 6\beta_{10} - 3\beta_{5} + 9\beta_{3} + 9\beta_1 ) / 4 \) (2b14 + 3b13 + 10b12 - 3b11 + 6b10 - 3b5 + 9b3 + 9b1) / 4
\(\nu^{4}\)	\(=\)	\( ( -18\beta_{15} - \beta_{13} - 21\beta_{11} + 9\beta_{8} + \beta_{6} - 12\beta_{2} + 29 ) / 4 \) (-18b15 - b13 - 21b11 + 9b8 + b6 - 12b2 + 29) / 4
\(\nu^{5}\)	\(=\)	\( ( 10 \beta_{14} + 3 \beta_{13} + 34 \beta_{12} - 3 \beta_{11} + 16 \beta_{10} + 7 \beta_{5} + \cdots + 61 \beta_1 ) / 4 \) (10b14 + 3b13 + 34b12 - 3b11 + 16b10 + 7b5 - 10b4 + 43b3 + 61*b1) / 4
\(\nu^{6}\)	\(=\)	\( ( -124\beta_{15} - 55\beta_{13} - 107\beta_{11} + 8\beta_{9} + 51\beta_{8} - 7\beta_{6} - 20\beta_{2} + 37 ) / 4 \) (-124b15 - 55b13 - 107b11 + 8b9 + 51b8 - 7b6 - 20*b2 + 37) / 4
\(\nu^{7}\)	\(=\)	\( ( 22 \beta_{14} - 49 \beta_{13} + 18 \beta_{12} + 49 \beta_{11} + 28 \beta_{7} + 157 \beta_{5} + \cdots + 339 \beta_1 ) / 4 \) (22b14 - 49b13 + 18b12 + 49b11 + 28b7 + 157b5 - 98b4 + 165b3 + 339*b1) / 4
\(\nu^{8}\)	\(=\)	\( ( -710\beta_{15} - 529\beta_{13} - 405\beta_{11} + 32\beta_{9} + 219\beta_{8} - 77\beta_{6} + 100\beta_{2} - 477 ) / 4 \) (-710b15 - 529b13 - 405b11 + 32b9 + 219b8 - 77b6 + 100*b2 - 477) / 4
\(\nu^{9}\)	\(=\)	\( ( - 162 \beta_{14} - 551 \beta_{13} - 890 \beta_{12} + 551 \beta_{11} - 418 \beta_{10} + \cdots + 1531 \beta_1 ) / 4 \) (-162b14 - 551b13 - 890b12 + 551b11 - 418b10 + 312b7 + 1291b5 - 684b4 + 431b3 + 1531b1) / 4
\(\nu^{10}\)	\(=\)	\( ( - 3386 \beta_{15} - 3543 \beta_{13} - 795 \beta_{11} - 192 \beta_{9} + 543 \beta_{8} - 397 \beta_{6} + \cdots - 5833 ) / 4 \) (-3386b15 - 3543b13 - 795b11 - 192b9 + 543b8 - 397b6 + 1572*b2 - 5833) / 4
\(\nu^{11}\)	\(=\)	\( ( - 2714 \beta_{14} - 3799 \beta_{13} - 9594 \beta_{12} + 3799 \beta_{11} - 3880 \beta_{10} + \cdots + 5011 \beta_1 ) / 4 \) (-2714b14 - 3799b13 - 9594b12 + 3799b11 - 3880b10 + 2112b7 + 7881b5 - 3718b4 - 379b3 + 5011b1) / 4
\(\nu^{12}\)	\(=\)	\( ( - 12432 \beta_{15} - 18657 \beta_{13} + 4139 \beta_{11} - 4048 \beta_{9} - 1567 \beta_{8} + \cdots - 44101 ) / 4 \) (-12432b15 - 18657b13 + 4139b11 - 4048b9 - 1567b8 - 337b6 + 13636*b2 - 44101) / 4
\(\nu^{13}\)	\(=\)	\( ( - 24126 \beta_{14} - 19823 \beta_{13} - 69514 \beta_{12} + 19823 \beta_{11} - 25164 \beta_{10} + \cdots + 3993 \beta_1 ) / 4 \) (-24126b14 - 19823b13 - 69514b12 + 19823b11 - 25164b10 + 9724b7 + 38707b5 - 14482b4 - 16269b3 + 3993b1) / 4
\(\nu^{14}\)	\(=\)	\( ( - 21090 \beta_{15} - 74151 \beta_{13} + 64045 \beta_{11} - 38664 \beta_{9} - 32711 \beta_{8} + \cdots - 268279 ) / 4 \) (-21090b15 - 74151b13 + 64045b11 - 38664b9 - 32711b8 + 16801b6 + 97588*b2 - 268279) / 4
\(\nu^{15}\)	\(=\)	\( ( - 165206 \beta_{14} - 75189 \beta_{13} - 405126 \beta_{12} + 75189 \beta_{11} - 133058 \beta_{10} + \cdots - 109415 \beta_1 ) / 4 \) (-165206b14 - 75189b13 - 405126b12 + 75189b11 - 133058b10 + 19472b7 + 145005b5 - 17320b4 - 153839b3 - 109415b1) / 4

\(n\)	\(145\)	\(199\)	\(353\)	\(397\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 307.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{8} - 3 T^{6} + 6 T^{4} + \cdots + 16)^{2} \) (T^8 - 3T^6 + 6T^4 - 12*T^2 + 16)^2
$3$	\( T^{16} \) T^16
$5$	\( (T^{2} + 4)^{8} \) (T^2 + 4)^8
$7$	\( (T^{4} - 10 T^{2} + 8)^{4} \) (T^4 - 10*T^2 + 8)^4
$11$	\( (T^{8} - 16 T^{6} + \cdots + 14641)^{2} \) (T^8 - 16T^6 + 238T^4 - 1936*T^2 + 14641)^2
$13$	\( (T^{4} - 20 T^{2} + 32)^{4} \) (T^4 - 20*T^2 + 32)^4
$17$	\( (T^{4} + 14 T^{2} + 32)^{4} \) (T^4 + 14*T^2 + 32)^4
$19$	\( (T^{4} + 58 T^{2} + 416)^{4} \) (T^4 + 58*T^2 + 416)^4
$23$	\( (T^{4} + 36 T^{2} + 256)^{4} \) (T^4 + 36*T^2 + 256)^4
$29$	\( (T^{4} - 46 T^{2} + 104)^{4} \) (T^4 - 46*T^2 + 104)^4
$31$	\( (T^{4} + 72 T^{2} + 208)^{4} \) (T^4 + 72*T^2 + 208)^4
$37$	\( (T^{4} + 60 T^{2} + 832)^{4} \) (T^4 + 60*T^2 + 832)^4
$41$	\( (T^{4} + 62 T^{2} + 128)^{4} \) (T^4 + 62*T^2 + 128)^4
$43$	\( (T^{4} + 122 T^{2} + 1664)^{4} \) (T^4 + 122*T^2 + 1664)^4
$47$	\( (T^{2} + 36)^{8} \) (T^2 + 36)^8
$53$	\( (T^{2} + 68)^{8} \) (T^2 + 68)^8
$59$	\( (T^{4} - 236 T^{2} + 13312)^{4} \) (T^4 - 236*T^2 + 13312)^4
$61$	\( (T^{4} - 148 T^{2} + 5408)^{4} \) (T^4 - 148*T^2 + 5408)^4
$67$	\( (T^{2} + 4 T - 64)^{8} \) (T^2 + 4*T - 64)^8
$71$	\( (T^{4} + 168 T^{2} + 2704)^{4} \) (T^4 + 168*T^2 + 2704)^4
$73$	\( (T^{4} + 184 T^{2} + 1664)^{4} \) (T^4 + 184*T^2 + 1664)^4
$79$	\( (T^{4} - 170 T^{2} + 2312)^{4} \) (T^4 - 170*T^2 + 2312)^4
$83$	\( (T^{4} + 28 T^{2} + 128)^{4} \) (T^4 + 28*T^2 + 128)^4
$89$	\( (T^{4} - 288 T^{2} + 3328)^{4} \) (T^4 - 288*T^2 + 3328)^4
$97$	\( (T^{2} + 2 T - 152)^{8} \) (T^2 + 2*T - 152)^8
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