LMFDB - Newform orbit 875.2.f.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [875,2,Mod(307,875)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("875.307"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(875, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1, 2])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 875 = 5^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	875.f (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$6.98691017686$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
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} + 36x^{14} + 456x^{12} + 2616x^{10} + 7596x^{8} + 11600x^{6} + 9040x^{4} + 3200x^{2} + 400 $ x^16 + 36x^14 + 456x^12 + 2616x^10 + 7596x^8 + 11600x^6 + 9040x^4 + 3200*x^2 + 400
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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_{6} q^{2} + ( - \beta_{12} - \beta_{3}) q^{3} + (\beta_{9} - 3 \beta_{7}) q^{4} + ( - \beta_{15} + \beta_{14}) q^{6} + ( - \beta_{11} + \beta_{6}) q^{7} + (\beta_{5} + \beta_1) q^{8} + (4 \beta_{9} - 4 \beta_{7}) q^{9}+ \cdots + (8 \beta_{9} - 12 \beta_{7}) q^{99}+O(q^{100}) $ q - b6 * q^2 + (-b12 - b3) * q^3 + (b9 - 3b7) q^4 + (-b15 + b14) * q^6 + (-b11 + b6) * q^7 + (b5 + b1) * q^8 + (4b9 - 4b7) * q^9 + (b8 + 2) * q^11 + (3b11 - 4b2) * q^12 + (-2b12 + 2b3) * q^13 + (-b10 - b9 + 5b7) q^14 - 3b8 q^16 + b11 * q^17 + (4b5 + 4b1) * q^18 + (-b13 + b10) * q^19 + (b15 - b14 + b8 + 3) * q^21 + (-2b6 + b4) q^22 - 3b5 q^23 + (-2b13 + b10) q^24 + (-2b15 - 2b14) * q^26 + (b11 - 5b2) q^27 + (-2b12 - 3b5 - b3 - b1) * q^28 + (-6b9 + 5b7) * q^29 + (-2b15 - b14) q^31 + (-2b6 - b4) q^32 + (-2b12 - 3b3) * q^33 + b10 * q^34 + (-12b8 - 16) q^36 + 3b4 q^37 + (5b12 + 6b3) * q^38 + (-4b9 + 2b7) * q^39 + 2b15 q^41 + (-5b11 - 3b6 + b4 + 6b2) q^42 + b1 * q^43 + (4b9 - 7b7) * q^44 + (3b8 + 15) q^46 - 4b2 q^47 + 3b3 q^48 + (2b10 + 2b9 - 3b7) q^49 + (-b8 - 3) * q^51 + (2b11 + 4b2) * q^52 + (3b5 - b1) q^53 + (-5b13 + b10) q^54 + (b14 + 5b8 + 6) q^56 + (7b6 - 4b4) * q^57 + (-5b5 - 6b1) * q^58 + (3b13 + b10) q^59 + (-2b15 - 3b14) * q^61 + (7b11 + 3b2) * q^62 + (-4b5 - 4b3 - 4b1) q^63 + (4b9 - 9b7) * q^64 + (-2b15 + 3b14) * q^66 + (b6 + b4) * q^67 + (2b12 + b3) q^68 + (3b13 - 3b10) * q^69 + (-2b8 + 3) q^71 + (8b6 - 4b4) * q^72 + (2b12 - 9b3) * q^73 + (12b9 - 3b7) * q^74 + (3b15 - 4b14) * q^76 + (-b11 + 2b6 - b4 + b2) q^77 + (-2b5 - 4b1) * q^78 - 6b9 q^79 + (-4b8 - 11) q^81 + (-8b11 + 2b2) * q^82 + (-b12 + 3b3) q^83 + (4b13 - 3b10 + 5b9 - 10b7) * q^84 + (-4b8 - 1) q^86 + (-5b11 + 11b2) * q^87 + (3b5 + 2b1) * q^88 + (-2b13 + b10) q^89 + (2b15 + 2b14 - 6b8 + 2) q^91 + (-9b6 + 3b4) * q^92 + (2b5 - b1) q^93 - 4b13 q^94 + (-2b15 + b14) q^96 + (-2b11 - 2b2) * q^97 + (8b12 + 3b5 + 2b3 + 2b1) * q^98 + (8b9 - 12b7) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 24 q^{11} + 24 q^{16} + 40 q^{21} - 160 q^{36} + 216 q^{46} - 40 q^{51} + 56 q^{56} + 64 q^{71} - 144 q^{81} + 16 q^{86} + 80 q^{91}+O(q^{100}) $ 16 * q + 24 * q^11 + 24 * q^16 + 40 * q^21 - 160 * q^36 + 216 * q^46 - 40 * q^51 + 56 * q^56 + 64 * q^71 - 144 * q^81 + 16 * q^86 + 80 * q^91

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 36x^{14} + 456x^{12} + 2616x^{10} + 7596x^{8} + 11600x^{6} + 9040x^{4} + 3200x^{2} + 400 $ x^16 + 36*x^14 + 456*x^12 + 2616*x^10 + 7596*x^8 + 11600*x^6 + 9040*x^4 + 3200*x^2 + 400 :

$\beta_{1}$	$=$	$ ( - 1433 \nu^{15} - 3415 \nu^{14} - 39763 \nu^{13} - 126690 \nu^{12} - 251288 \nu^{11} + \cdots - 5551600 ) / 311200 $ (-1433v^15 - 3415v^14 - 39763v^13 - 126690v^12 - 251288v^11 - 1677990v^10 + 839582v^9 - 10165340v^8 + 10798892v^7 - 30424140v^6 + 28639660v^5 - 44043000v^4 + 27051040v^3 - 27588400v^2 + 6818400*v - 5551600) / 311200
$\beta_{2}$	$=$	$ ( 444 \nu^{15} + 99 \nu^{14} + 14997 \nu^{13} + 3888 \nu^{12} + 168582 \nu^{11} + 55888 \nu^{10} + \cdots + 151280 ) / 62240 $ (444v^15 + 99v^14 + 14997v^13 + 3888v^12 + 168582v^11 + 55888v^10 + 768452v^9 + 375548v^8 + 1460172v^7 + 1245648v^6 + 984408v^5 + 1945584v^4 + 99360v^3 + 1177800v^2 + 75400*v + 151280) / 62240
$\beta_{3}$	$=$	$ ( 444 \nu^{15} - 99 \nu^{14} + 14997 \nu^{13} - 3888 \nu^{12} + 168582 \nu^{11} - 55888 \nu^{10} + \cdots - 151280 ) / 62240 $ (444v^15 - 99v^14 + 14997v^13 - 3888v^12 + 168582v^11 - 55888v^10 + 768452v^9 - 375548v^8 + 1460172v^7 - 1245648v^6 + 984408v^5 - 1945584v^4 + 99360v^3 - 1177800v^2 + 75400*v - 151280) / 62240
$\beta_{4}$	$=$	$ ( - 1433 \nu^{15} + 3415 \nu^{14} - 39763 \nu^{13} + 126690 \nu^{12} - 251288 \nu^{11} + \cdots + 5551600 ) / 311200 $ (-1433v^15 + 3415v^14 - 39763v^13 + 126690v^12 - 251288v^11 + 1677990v^10 + 839582v^9 + 10165340v^8 + 10798892v^7 + 30424140v^6 + 28639660v^5 + 44043000v^4 + 27051040v^3 + 27588400v^2 + 6818400*v + 5551600) / 311200
$\beta_{5}$	$=$	$ ( 1891 \nu^{15} + 4935 \nu^{14} + 67581 \nu^{13} + 169410 \nu^{12} + 842856 \nu^{11} + 1965260 \nu^{10} + \cdots + 1199200 ) / 311200 $ (1891v^15 + 4935v^14 + 67581v^13 + 169410v^12 + 842856v^11 + 1965260v^10 + 4667416v^9 + 9562260v^8 + 12486296v^7 + 20829960v^6 + 15707360v^5 + 19572000v^4 + 7366720v^3 + 6882600v^2 + 317800*v + 1199200) / 311200
$\beta_{6}$	$=$	$ ( - 1891 \nu^{15} + 4935 \nu^{14} - 67581 \nu^{13} + 169410 \nu^{12} - 842856 \nu^{11} + \cdots + 1199200 ) / 311200 $ (-1891v^15 + 4935v^14 - 67581v^13 + 169410v^12 - 842856v^11 + 1965260v^10 - 4667416v^9 + 9562260v^8 - 12486296v^7 + 20829960v^6 - 15707360v^5 + 19572000v^4 - 7366720v^3 + 6882600v^2 - 317800*v + 1199200) / 311200
$\beta_{7}$	$=$	$ ( 4111 \nu^{15} + 142566 \nu^{13} + 1685766 \nu^{11} + 8509676 \nu^{9} + 19787156 \nu^{7} + \cdots + 383600 \nu ) / 311200 $ (4111v^15 + 142566v^13 + 1685766v^11 + 8509676v^9 + 19787156v^7 + 20629400v^5 + 7863520v^3 + 383600v) / 311200
$\beta_{8}$	$=$	$ ( - 279 \nu^{14} - 9684 \nu^{12} - 114854 \nu^{10} - 585869 \nu^{8} - 1410004 \nu^{6} - 1627100 \nu^{4} + \cdots - 144840 ) / 7780 $ (-279v^14 - 9684v^12 - 114854v^10 - 585869v^8 - 1410004v^6 - 1627100v^4 - 828240*v^2 - 144840) / 7780
$\beta_{9}$	$=$	$ ( - 2376 \nu^{15} - 83976 \nu^{13} - 1030501 \nu^{11} - 5611736 \nu^{9} - 15162566 \nu^{7} + \cdots - 3055000 \nu ) / 155600 $ (-2376v^15 - 83976v^13 - 1030501v^11 - 5611736v^9 - 15162566v^7 - 21111820v^5 - 14022020v^3 - 3055000v) / 155600
$\beta_{10}$	$=$	$ ( 708 \nu^{14} + 24587 \nu^{12} + 292202 \nu^{10} + 1502022 \nu^{8} + 3711372 \nu^{6} + 4611964 \nu^{4} + \cdots + 530680 ) / 15560 $ (708v^14 + 24587v^12 + 292202v^10 + 1502022v^8 + 3711372v^6 + 4611964v^4 + 2726680*v^2 + 530680) / 15560
$\beta_{11}$	$=$	$ ( 995 \nu^{15} + 925 \nu^{14} + 35929 \nu^{13} + 30952 \nu^{12} + 456374 \nu^{11} + 341682 \nu^{10} + \cdots - 331760 ) / 62240 $ (995v^15 + 925v^14 + 35929v^13 + 30952v^12 + 456374v^11 + 341682v^10 + 2610174v^9 + 1499672v^8 + 7374784v^7 + 2615292v^6 + 10300004v^5 + 1225392v^4 + 6427320v^3 - 711040v^2 + 1300400*v - 331760) / 62240
$\beta_{12}$	$=$	$ ( - 995 \nu^{15} + 925 \nu^{14} - 35929 \nu^{13} + 30952 \nu^{12} - 456374 \nu^{11} + 341682 \nu^{10} + \cdots - 331760 ) / 62240 $ (-995v^15 + 925v^14 - 35929v^13 + 30952v^12 - 456374v^11 + 341682v^10 - 2610174v^9 + 1499672v^8 - 7374784v^7 + 2615292v^6 - 10300004v^5 + 1225392v^4 - 6427320v^3 - 711040v^2 - 1300400*v - 331760) / 62240
$\beta_{13}$	$=$	$ ( - 1101 \nu^{14} - 38253 \nu^{12} - 454178 \nu^{10} - 2315738 \nu^{8} - 5525828 \nu^{6} + \cdots - 407440 ) / 15560 $ (-1101v^14 - 38253v^12 - 454178v^10 - 2315738v^8 - 5525828v^6 - 6184192v^4 - 2918080*v^2 - 407440) / 15560
$\beta_{14}$	$=$	$ ( - 2331 \nu^{15} - 81360 \nu^{13} - 974720 \nu^{11} - 5062500 \nu^{9} - 12525680 \nu^{7} + \cdots - 998800 \nu ) / 62240 $ (-2331v^15 - 81360v^13 - 974720v^11 - 5062500v^9 - 12525680v^7 - 14962384v^5 - 7772600v^3 - 998800v) / 62240
$\beta_{15}$	$=$	$ ( 6563 \nu^{15} + 229456 \nu^{13} + 2757666 \nu^{11} + 14411016 \nu^{9} + 36067636 \nu^{7} + \cdots + 4458000 \nu ) / 62240 $ (6563v^15 + 229456v^13 + 2757666v^11 + 14411016v^9 + 36067636v^7 + 44066128v^5 + 24222880v^3 + 4458000v) / 62240

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

$\nu$	$=$	$ ( -2\beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} ) / 2 $ (-2*b7 - b6 + b5 + b3 + b2) / 2
$\nu^{2}$	$=$	$ \beta_{13} - \beta_{8} + \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} - 5 $ b13 - b8 + b6 + b5 - b3 + b2 - 5
$\nu^{3}$	$=$	$ 3 \beta_{14} - \beta_{12} + \beta_{11} - 3 \beta_{9} + 13 \beta_{7} + 5 \beta_{6} - 5 \beta_{5} + \cdots - \beta_1 $ 3b14 - b12 + b11 - 3b9 + 13b7 + 5b6 - 5b5 - b4 - 6b3 - 6*b2 - b1
$\nu^{4}$	$=$	$ - 14 \beta_{13} + 4 \beta_{12} + 4 \beta_{11} + 2 \beta_{10} + 20 \beta_{8} - 16 \beta_{6} - 16 \beta_{5} + \cdots + 58 $ -14b13 + 4b12 + 4b11 + 2b10 + 20b8 - 16b6 - 16b5 + 4b4 + 20b3 - 20b2 - 4*b1 + 58
$\nu^{5}$	$=$	$ 10 \beta_{15} - 50 \beta_{14} + 25 \beta_{12} - 25 \beta_{11} + 80 \beta_{9} - 206 \beta_{7} + \cdots + 23 \beta_1 $ 10b15 - 50b14 + 25b12 - 25b11 + 80b9 - 206b7 - 67b6 + 67b5 + 23b4 + 85b3 + 85b2 + 23b1
$\nu^{6}$	$=$	$ 202 \beta_{13} - 110 \beta_{12} - 110 \beta_{11} - 58 \beta_{10} - 374 \beta_{8} + 250 \beta_{6} + \cdots - 844 $ 202b13 - 110b12 - 110b11 - 58b10 - 374b8 + 250b6 + 250b5 - 98b4 - 318b3 + 318b2 + 98*b1 - 844
$\nu^{7}$	$=$	$ - 266 \beta_{15} + 770 \beta_{14} - 514 \beta_{12} + 514 \beta_{11} - 1554 \beta_{9} + 3256 \beta_{7} + \cdots - 444 \beta_1 $ -266b15 + 770b14 - 514b12 + 514b11 - 1554b9 + 3256b7 + 1004b6 - 1004b5 - 444b4 - 1274b3 - 1274b2 - 444b1
$\nu^{8}$	$=$	$ - 3048 \beta_{13} + 2208 \beta_{12} + 2208 \beta_{11} + 1224 \beta_{10} + 6636 \beta_{8} - 3920 \beta_{6} + \cdots + 13056 $ -3048b13 + 2208b12 + 2208b11 + 1224b10 + 6636b8 - 3920b6 - 3920b5 + 1872b4 + 4960b3 - 4960b2 - 1872*b1 + 13056
$\nu^{9}$	$=$	$ 5304 \beta_{15} - 11928 \beta_{14} + 9498 \beta_{12} - 9498 \beta_{11} + 27420 \beta_{9} + \cdots + 7938 \beta_1 $ 5304b15 - 11928b14 + 9498b12 - 9498b11 + 27420b9 - 51616b7 - 15632b6 + 15632b5 + 7938b4 + 19720b3 + 19720b2 + 7938b1
$\nu^{10}$	$=$	$ 47280 \beta_{13} - 39780 \beta_{12} - 39780 \beta_{11} - 22740 \beta_{10} - 113520 \beta_{8} + \cdots - 206372 $ 47280b13 - 39780b12 - 39780b11 - 22740b10 - 113520b8 + 61984b6 + 61984b5 - 32916b4 - 78000b3 + 78000b2 + 32916*b1 - 206372
$\nu^{11}$	$=$	$ - 95436 \beta_{15} + 187264 \beta_{14} - 165660 \beta_{12} + 165660 \beta_{11} - 464728 \beta_{9} + \cdots - 136056 \beta_1 $ -95436b15 + 187264b14 - 165660b12 + 165660b11 - 464728b9 + 822988b7 + 247460b6 - 247460b5 - 136056b4 - 310756b3 - 310756b2 - 136056b1
$\nu^{12}$	$=$	$ - 745480 \beta_{13} + 682528 \beta_{12} + 682528 \beta_{11} + 397152 \beta_{10} + 1898488 \beta_{8} + \cdots + 3294272 $ -745480b13 + 682528b12 + 682528b11 + 397152b10 + 1898488b8 - 987568b6 - 987568b5 + 557488b4 + 1238128b3 - 1238128b2 - 557488*b1 + 3294272
$\nu^{13}$	$=$	$ 1637168 \beta_{15} - 2971176 \beta_{14} + 2798816 \beta_{12} - 2798816 \beta_{11} + 7719192 \beta_{9} + \cdots + 2276624 \beta_1 $ 1637168b15 - 2971176b14 + 2798816b12 - 2798816b11 + 7719192b9 - 13186512b7 - 3951500b6 + 3951500b5 + 2276624b4 + 4947540b3 + 4947540b2 + 2276624b1
$\nu^{14}$	$=$	$ 11870216 \beta_{13} - 11418336 \beta_{12} - 11418336 \beta_{11} - 6712608 \beta_{10} - 31322360 \beta_{8} + \cdots - 52862072 $ 11870216b13 - 11418336b12 - 11418336b11 - 6712608b10 - 31322360b8 + 15820104b6 + 15820104b5 - 9258976b4 - 19787320b3 + 19787320b2 + 9258976*b1 - 52862072
$\nu^{15}$	$=$	$ - 27389920 \beta_{15} + 47477640 \beta_{14} - 46439840 \beta_{12} + 46439840 \beta_{11} + \cdots - 37567872 \beta_1 $ -27389920b15 + 47477640b14 - 46439840b12 + 46439840b11 - 126802360b9 + 212001864b7 + 63412768b6 - 63412768b5 - 37567872b4 - 79250200b3 - 79250200b2 - 37567872b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/875\mathbb{Z}\right)^\times$.

$n$	$127$	$626$
$\chi(n)$	$-\beta_{7}$	$-1$

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

		0.668985i
	−	2.02101i
	−	1.13163i
		0.530876i
		1.46912i
		3.13163i
		4.02101i
		1.33102i
	−	0.668985i
		2.02101i
		1.13163i
	−	0.530876i
	−	1.46912i
	−	3.13163i
	−	4.02101i
	−	1.33102i

−1.67601

−

1.67601i

−2.17625

−

2.17625i

3.61803i

7.29485i

0.844758

2.50727i

2.71184

−

2.71184i

6.47214i

307.2

−1.67601

−

1.67601i

2.17625

2.17625i

3.61803i

−

7.29485i

2.50727

0.844758i

2.71184

−

2.71184i

6.47214i

307.3

−1.30038

−

1.30038i

−0.513743

−

0.513743i

1.38197i

1.33612i

−0.0446190

2.64538i

−0.803678

0.803678i

−

2.47214i

307.4

−1.30038

−

1.30038i

0.513743

0.513743i

1.38197i

−

1.33612i

2.64538

−

0.0446190i

−0.803678

0.803678i

−

2.47214i

307.5

1.30038

1.30038i

−0.513743

−

0.513743i

1.38197i

−

1.33612i

−2.64538

0.0446190i

0.803678

−

0.803678i

−

2.47214i

307.6

1.30038

1.30038i

0.513743

0.513743i

1.38197i

1.33612i

0.0446190

−

2.64538i

0.803678

−

0.803678i

−

2.47214i

307.7

1.67601

1.67601i

−2.17625

−

2.17625i

3.61803i

−

7.29485i

−2.50727

−

0.844758i

−2.71184

2.71184i

6.47214i

307.8

1.67601

1.67601i

2.17625

2.17625i

3.61803i

7.29485i

−0.844758

−

2.50727i

−2.71184

2.71184i

6.47214i

818.1

−1.67601

1.67601i

−2.17625

2.17625i

−

3.61803i

−

7.29485i

0.844758

−

2.50727i

2.71184

2.71184i

−

6.47214i

818.2

−1.67601

1.67601i

2.17625

−

2.17625i

−

3.61803i

7.29485i

2.50727

−

0.844758i

2.71184

2.71184i

−

6.47214i

818.3

−1.30038

1.30038i

−0.513743

0.513743i

−

1.38197i

−

1.33612i

−0.0446190

−

2.64538i

−0.803678

−

0.803678i

2.47214i

818.4

−1.30038

1.30038i

0.513743

−

0.513743i

−

1.38197i

1.33612i

2.64538

0.0446190i

−0.803678

−

0.803678i

2.47214i

818.5

1.30038

−

1.30038i

−0.513743

0.513743i

−

1.38197i

1.33612i

−2.64538

−

0.0446190i

0.803678

0.803678i

2.47214i

818.6

1.30038

−

1.30038i

0.513743

−

0.513743i

−

1.38197i

−

1.33612i

0.0446190

2.64538i

0.803678

0.803678i

2.47214i

818.7

1.67601

−

1.67601i

−2.17625

2.17625i

−

3.61803i

7.29485i

−2.50727

0.844758i

−2.71184

−

2.71184i

−

6.47214i

818.8

1.67601

−

1.67601i

2.17625

−

2.17625i

−

3.61803i

−

7.29485i

−0.844758

2.50727i

−2.71184

−

2.71184i

−

6.47214i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
5.c	odd	4	2	inner
7.b	odd	2	1	inner
35.c	odd	2	1	inner
35.f	even	4	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	875.2.f.b	✓	16
5.b	even	2	1	inner	875.2.f.b	✓	16
5.c	odd	4	2	inner	875.2.f.b	✓	16
7.b	odd	2	1	inner	875.2.f.b	✓	16
35.c	odd	2	1	inner	875.2.f.b	✓	16
35.f	even	4	2	inner	875.2.f.b	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
875.2.f.b	✓	16	1.a	even	1	1	trivial
875.2.f.b	✓	16	5.b	even	2	1	inner
875.2.f.b	✓	16	5.c	odd	4	2	inner
875.2.f.b	✓	16	7.b	odd	2	1	inner
875.2.f.b	✓	16	35.c	odd	2	1	inner
875.2.f.b	✓	16	35.f	even	4	2	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{8} + 43T_{2}^{4} + 361 $ T2^8 + 43*T2^4 + 361 acting on $S_{2}^{\mathrm{new}}(875, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{8} + 43 T^{4} + 361)^{2} $ (T^8 + 43*T^4 + 361)^2
$3$	$ (T^{8} + 90 T^{4} + 25)^{2} $ (T^8 + 90*T^4 + 25)^2
$5$	$ T^{16} $ T^16
$7$	$ T^{16} - 124 T^{12} + \cdots + 5764801 $ T^16 - 124T^12 + 7366T^8 - 297724*T^4 + 5764801
$11$	$ (T^{2} - 3 T + 1)^{8} $ (T^2 - 3*T + 1)^8
$13$	$ (T^{8} + 1440 T^{4} + 6400)^{2} $ (T^8 + 1440*T^4 + 6400)^2
$17$	$ (T^{8} + 15 T^{4} + 25)^{2} $ (T^8 + 15*T^4 + 25)^2
$19$	$ (T^{4} - 55 T^{2} + 95)^{4} $ (T^4 - 55*T^2 + 95)^4
$23$	$ (T^{8} + 3483 T^{4} + 2368521)^{2} $ (T^8 + 3483*T^4 + 2368521)^2
$29$	$ (T^{4} + 98 T^{2} + 1681)^{4} $ (T^4 + 98*T^2 + 1681)^4
$31$	$ (T^{4} + 85 T^{2} + 95)^{4} $ (T^4 + 85*T^2 + 95)^4
$37$	$ (T^{8} + 6723 T^{4} + 2368521)^{2} $ (T^8 + 6723*T^4 + 2368521)^2
$41$	$ (T^{4} + 80 T^{2} + 1520)^{4} $ (T^4 + 80*T^2 + 1520)^4
$43$	$ (T^{8} + 83 T^{4} + 361)^{2} $ (T^8 + 83*T^4 + 361)^2
$47$	$ (T^{8} + 3840 T^{4} + 1638400)^{2} $ (T^8 + 3840*T^4 + 1638400)^2
$53$	$ (T^{8} + 6218 T^{4} + 5285401)^{2} $ (T^8 + 6218*T^4 + 5285401)^2
$59$	$ (T^{4} - 215 T^{2} + 11495)^{4} $ (T^4 - 215*T^2 + 11495)^4
$61$	$ (T^{4} + 245 T^{2} + 11495)^{4} $ (T^4 + 245*T^2 + 11495)^4
$67$	$ (T^{8} + 538 T^{4} + 361)^{2} $ (T^8 + 538*T^4 + 361)^2
$71$	$ (T^{2} - 8 T + 11)^{8} $ (T^2 - 8*T + 11)^8
$73$	$ (T^{8} + 90375 T^{4} + 2036265625)^{2} $ (T^8 + 90375*T^4 + 2036265625)^2
$79$	$ (T^{4} + 108 T^{2} + 1296)^{4} $ (T^4 + 108*T^2 + 1296)^4
$83$	$ (T^{8} + 1290 T^{4} + 366025)^{2} $ (T^8 + 1290*T^4 + 366025)^2
$89$	$ (T^{4} - 140 T^{2} + 95)^{4} $ (T^4 - 140*T^2 + 95)^4
$97$	$ (T^{8} + 1440 T^{4} + 6400)^{2} $ (T^8 + 1440*T^4 + 6400)^2
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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:	\( N \)	\(=\)	\( 875 = 5^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	875.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{6} q^{2} + ( - \beta_{12} - \beta_{3}) q^{3} + (\beta_{9} - 3 \beta_{7}) q^{4} + ( - \beta_{15} + \beta_{14}) q^{6} + ( - \beta_{11} + \beta_{6}) q^{7} + (\beta_{5} + \beta_1) q^{8} + (4 \beta_{9} - 4 \beta_{7}) q^{9}+ \cdots + (8 \beta_{9} - 12 \beta_{7}) q^{99}+O(q^{100}) \) q - b6 * q^2 + (-b12 - b3) * q^3 + (b9 - 3b7) q^4 + (-b15 + b14) * q^6 + (-b11 + b6) * q^7 + (b5 + b1) * q^8 + (4b9 - 4b7) * q^9 + (b8 + 2) * q^11 + (3b11 - 4b2) * q^12 + (-2b12 + 2b3) * q^13 + (-b10 - b9 + 5b7) q^14 - 3b8 q^16 + b11 * q^17 + (4b5 + 4b1) * q^18 + (-b13 + b10) * q^19 + (b15 - b14 + b8 + 3) * q^21 + (-2b6 + b4) q^22 - 3b5 q^23 + (-2b13 + b10) q^24 + (-2b15 - 2b14) * q^26 + (b11 - 5b2) q^27 + (-2b12 - 3b5 - b3 - b1) * q^28 + (-6b9 + 5b7) * q^29 + (-2b15 - b14) q^31 + (-2b6 - b4) q^32 + (-2b12 - 3b3) * q^33 + b10 * q^34 + (-12b8 - 16) q^36 + 3b4 q^37 + (5b12 + 6b3) * q^38 + (-4b9 + 2b7) * q^39 + 2b15 q^41 + (-5b11 - 3b6 + b4 + 6b2) q^42 + b1 * q^43 + (4b9 - 7b7) * q^44 + (3b8 + 15) q^46 - 4b2 q^47 + 3b3 q^48 + (2b10 + 2b9 - 3b7) q^49 + (-b8 - 3) * q^51 + (2b11 + 4b2) * q^52 + (3b5 - b1) q^53 + (-5b13 + b10) q^54 + (b14 + 5b8 + 6) q^56 + (7b6 - 4b4) * q^57 + (-5b5 - 6b1) * q^58 + (3b13 + b10) q^59 + (-2b15 - 3b14) * q^61 + (7b11 + 3b2) * q^62 + (-4b5 - 4b3 - 4b1) q^63 + (4b9 - 9b7) * q^64 + (-2b15 + 3b14) * q^66 + (b6 + b4) * q^67 + (2b12 + b3) q^68 + (3b13 - 3b10) * q^69 + (-2b8 + 3) q^71 + (8b6 - 4b4) * q^72 + (2b12 - 9b3) * q^73 + (12b9 - 3b7) * q^74 + (3b15 - 4b14) * q^76 + (-b11 + 2b6 - b4 + b2) q^77 + (-2b5 - 4b1) * q^78 - 6b9 q^79 + (-4b8 - 11) q^81 + (-8b11 + 2b2) * q^82 + (-b12 + 3b3) q^83 + (4b13 - 3b10 + 5b9 - 10b7) * q^84 + (-4b8 - 1) q^86 + (-5b11 + 11b2) * q^87 + (3b5 + 2b1) * q^88 + (-2b13 + b10) q^89 + (2b15 + 2b14 - 6b8 + 2) q^91 + (-9b6 + 3b4) * q^92 + (2b5 - b1) q^93 - 4b13 q^94 + (-2b15 + b14) q^96 + (-2b11 - 2b2) * q^97 + (8b12 + 3b5 + 2b3 + 2b1) * q^98 + (8b9 - 12b7) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 24 q^{11} + 24 q^{16} + 40 q^{21} - 160 q^{36} + 216 q^{46} - 40 q^{51} + 56 q^{56} + 64 q^{71} - 144 q^{81} + 16 q^{86} + 80 q^{91}+O(q^{100}) \) 16 * q + 24 * q^11 + 24 * q^16 + 40 * q^21 - 160 * q^36 + 216 * q^46 - 40 * q^51 + 56 * q^56 + 64 * q^71 - 144 * q^81 + 16 * q^86 + 80 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Learn more

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	875.2.f.b

Level	$875$
Weight	$2$
Character orbit	875.f
Analytic conductor	$6.987$
Analytic rank	$0$
Dimension	$16$
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(6.98691017686\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} + 36x^{14} + 456x^{12} + 2616x^{10} + 7596x^{8} + 11600x^{6} + 9040x^{4} + 3200x^{2} + 400 \) x^16 + 36x^14 + 456x^12 + 2616x^10 + 7596x^8 + 11600x^6 + 9040x^4 + 3200*x^2 + 400
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{6} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( - 1433 \nu^{15} - 3415 \nu^{14} - 39763 \nu^{13} - 126690 \nu^{12} - 251288 \nu^{11} + \cdots - 5551600 ) / 311200 \) (-1433v^15 - 3415v^14 - 39763v^13 - 126690v^12 - 251288v^11 - 1677990v^10 + 839582v^9 - 10165340v^8 + 10798892v^7 - 30424140v^6 + 28639660v^5 - 44043000v^4 + 27051040v^3 - 27588400v^2 + 6818400*v - 5551600) / 311200
\(\beta_{2}\)	\(=\)	\( ( 444 \nu^{15} + 99 \nu^{14} + 14997 \nu^{13} + 3888 \nu^{12} + 168582 \nu^{11} + 55888 \nu^{10} + \cdots + 151280 ) / 62240 \) (444v^15 + 99v^14 + 14997v^13 + 3888v^12 + 168582v^11 + 55888v^10 + 768452v^9 + 375548v^8 + 1460172v^7 + 1245648v^6 + 984408v^5 + 1945584v^4 + 99360v^3 + 1177800v^2 + 75400*v + 151280) / 62240
\(\beta_{3}\)	\(=\)	\( ( 444 \nu^{15} - 99 \nu^{14} + 14997 \nu^{13} - 3888 \nu^{12} + 168582 \nu^{11} - 55888 \nu^{10} + \cdots - 151280 ) / 62240 \) (444v^15 - 99v^14 + 14997v^13 - 3888v^12 + 168582v^11 - 55888v^10 + 768452v^9 - 375548v^8 + 1460172v^7 - 1245648v^6 + 984408v^5 - 1945584v^4 + 99360v^3 - 1177800v^2 + 75400*v - 151280) / 62240
\(\beta_{4}\)	\(=\)	\( ( - 1433 \nu^{15} + 3415 \nu^{14} - 39763 \nu^{13} + 126690 \nu^{12} - 251288 \nu^{11} + \cdots + 5551600 ) / 311200 \) (-1433v^15 + 3415v^14 - 39763v^13 + 126690v^12 - 251288v^11 + 1677990v^10 + 839582v^9 + 10165340v^8 + 10798892v^7 + 30424140v^6 + 28639660v^5 + 44043000v^4 + 27051040v^3 + 27588400v^2 + 6818400*v + 5551600) / 311200
\(\beta_{5}\)	\(=\)	\( ( 1891 \nu^{15} + 4935 \nu^{14} + 67581 \nu^{13} + 169410 \nu^{12} + 842856 \nu^{11} + 1965260 \nu^{10} + \cdots + 1199200 ) / 311200 \) (1891v^15 + 4935v^14 + 67581v^13 + 169410v^12 + 842856v^11 + 1965260v^10 + 4667416v^9 + 9562260v^8 + 12486296v^7 + 20829960v^6 + 15707360v^5 + 19572000v^4 + 7366720v^3 + 6882600v^2 + 317800*v + 1199200) / 311200
\(\beta_{6}\)	\(=\)	\( ( - 1891 \nu^{15} + 4935 \nu^{14} - 67581 \nu^{13} + 169410 \nu^{12} - 842856 \nu^{11} + \cdots + 1199200 ) / 311200 \) (-1891v^15 + 4935v^14 - 67581v^13 + 169410v^12 - 842856v^11 + 1965260v^10 - 4667416v^9 + 9562260v^8 - 12486296v^7 + 20829960v^6 - 15707360v^5 + 19572000v^4 - 7366720v^3 + 6882600v^2 - 317800*v + 1199200) / 311200
\(\beta_{7}\)	\(=\)	\( ( 4111 \nu^{15} + 142566 \nu^{13} + 1685766 \nu^{11} + 8509676 \nu^{9} + 19787156 \nu^{7} + \cdots + 383600 \nu ) / 311200 \) (4111v^15 + 142566v^13 + 1685766v^11 + 8509676v^9 + 19787156v^7 + 20629400v^5 + 7863520v^3 + 383600v) / 311200
\(\beta_{8}\)	\(=\)	\( ( - 279 \nu^{14} - 9684 \nu^{12} - 114854 \nu^{10} - 585869 \nu^{8} - 1410004 \nu^{6} - 1627100 \nu^{4} + \cdots - 144840 ) / 7780 \) (-279v^14 - 9684v^12 - 114854v^10 - 585869v^8 - 1410004v^6 - 1627100v^4 - 828240*v^2 - 144840) / 7780
\(\beta_{9}\)	\(=\)	\( ( - 2376 \nu^{15} - 83976 \nu^{13} - 1030501 \nu^{11} - 5611736 \nu^{9} - 15162566 \nu^{7} + \cdots - 3055000 \nu ) / 155600 \) (-2376v^15 - 83976v^13 - 1030501v^11 - 5611736v^9 - 15162566v^7 - 21111820v^5 - 14022020v^3 - 3055000v) / 155600
\(\beta_{10}\)	\(=\)	\( ( 708 \nu^{14} + 24587 \nu^{12} + 292202 \nu^{10} + 1502022 \nu^{8} + 3711372 \nu^{6} + 4611964 \nu^{4} + \cdots + 530680 ) / 15560 \) (708v^14 + 24587v^12 + 292202v^10 + 1502022v^8 + 3711372v^6 + 4611964v^4 + 2726680*v^2 + 530680) / 15560
\(\beta_{11}\)	\(=\)	\( ( 995 \nu^{15} + 925 \nu^{14} + 35929 \nu^{13} + 30952 \nu^{12} + 456374 \nu^{11} + 341682 \nu^{10} + \cdots - 331760 ) / 62240 \) (995v^15 + 925v^14 + 35929v^13 + 30952v^12 + 456374v^11 + 341682v^10 + 2610174v^9 + 1499672v^8 + 7374784v^7 + 2615292v^6 + 10300004v^5 + 1225392v^4 + 6427320v^3 - 711040v^2 + 1300400*v - 331760) / 62240
\(\beta_{12}\)	\(=\)	\( ( - 995 \nu^{15} + 925 \nu^{14} - 35929 \nu^{13} + 30952 \nu^{12} - 456374 \nu^{11} + 341682 \nu^{10} + \cdots - 331760 ) / 62240 \) (-995v^15 + 925v^14 - 35929v^13 + 30952v^12 - 456374v^11 + 341682v^10 - 2610174v^9 + 1499672v^8 - 7374784v^7 + 2615292v^6 - 10300004v^5 + 1225392v^4 - 6427320v^3 - 711040v^2 - 1300400*v - 331760) / 62240
\(\beta_{13}\)	\(=\)	\( ( - 1101 \nu^{14} - 38253 \nu^{12} - 454178 \nu^{10} - 2315738 \nu^{8} - 5525828 \nu^{6} + \cdots - 407440 ) / 15560 \) (-1101v^14 - 38253v^12 - 454178v^10 - 2315738v^8 - 5525828v^6 - 6184192v^4 - 2918080*v^2 - 407440) / 15560
\(\beta_{14}\)	\(=\)	\( ( - 2331 \nu^{15} - 81360 \nu^{13} - 974720 \nu^{11} - 5062500 \nu^{9} - 12525680 \nu^{7} + \cdots - 998800 \nu ) / 62240 \) (-2331v^15 - 81360v^13 - 974720v^11 - 5062500v^9 - 12525680v^7 - 14962384v^5 - 7772600v^3 - 998800v) / 62240
\(\beta_{15}\)	\(=\)	\( ( 6563 \nu^{15} + 229456 \nu^{13} + 2757666 \nu^{11} + 14411016 \nu^{9} + 36067636 \nu^{7} + \cdots + 4458000 \nu ) / 62240 \) (6563v^15 + 229456v^13 + 2757666v^11 + 14411016v^9 + 36067636v^7 + 44066128v^5 + 24222880v^3 + 4458000v) / 62240

\(\nu\)	\(=\)	\( ( -2\beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} ) / 2 \) (-2*b7 - b6 + b5 + b3 + b2) / 2
\(\nu^{2}\)	\(=\)	\( \beta_{13} - \beta_{8} + \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} - 5 \) b13 - b8 + b6 + b5 - b3 + b2 - 5
\(\nu^{3}\)	\(=\)	\( 3 \beta_{14} - \beta_{12} + \beta_{11} - 3 \beta_{9} + 13 \beta_{7} + 5 \beta_{6} - 5 \beta_{5} + \cdots - \beta_1 \) 3b14 - b12 + b11 - 3b9 + 13b7 + 5b6 - 5b5 - b4 - 6b3 - 6*b2 - b1
\(\nu^{4}\)	\(=\)	\( - 14 \beta_{13} + 4 \beta_{12} + 4 \beta_{11} + 2 \beta_{10} + 20 \beta_{8} - 16 \beta_{6} - 16 \beta_{5} + \cdots + 58 \) -14b13 + 4b12 + 4b11 + 2b10 + 20b8 - 16b6 - 16b5 + 4b4 + 20b3 - 20b2 - 4*b1 + 58
\(\nu^{5}\)	\(=\)	\( 10 \beta_{15} - 50 \beta_{14} + 25 \beta_{12} - 25 \beta_{11} + 80 \beta_{9} - 206 \beta_{7} + \cdots + 23 \beta_1 \) 10b15 - 50b14 + 25b12 - 25b11 + 80b9 - 206b7 - 67b6 + 67b5 + 23b4 + 85b3 + 85b2 + 23b1
\(\nu^{6}\)	\(=\)	\( 202 \beta_{13} - 110 \beta_{12} - 110 \beta_{11} - 58 \beta_{10} - 374 \beta_{8} + 250 \beta_{6} + \cdots - 844 \) 202b13 - 110b12 - 110b11 - 58b10 - 374b8 + 250b6 + 250b5 - 98b4 - 318b3 + 318b2 + 98*b1 - 844
\(\nu^{7}\)	\(=\)	\( - 266 \beta_{15} + 770 \beta_{14} - 514 \beta_{12} + 514 \beta_{11} - 1554 \beta_{9} + 3256 \beta_{7} + \cdots - 444 \beta_1 \) -266b15 + 770b14 - 514b12 + 514b11 - 1554b9 + 3256b7 + 1004b6 - 1004b5 - 444b4 - 1274b3 - 1274b2 - 444b1
\(\nu^{8}\)	\(=\)	\( - 3048 \beta_{13} + 2208 \beta_{12} + 2208 \beta_{11} + 1224 \beta_{10} + 6636 \beta_{8} - 3920 \beta_{6} + \cdots + 13056 \) -3048b13 + 2208b12 + 2208b11 + 1224b10 + 6636b8 - 3920b6 - 3920b5 + 1872b4 + 4960b3 - 4960b2 - 1872*b1 + 13056
\(\nu^{9}\)	\(=\)	\( 5304 \beta_{15} - 11928 \beta_{14} + 9498 \beta_{12} - 9498 \beta_{11} + 27420 \beta_{9} + \cdots + 7938 \beta_1 \) 5304b15 - 11928b14 + 9498b12 - 9498b11 + 27420b9 - 51616b7 - 15632b6 + 15632b5 + 7938b4 + 19720b3 + 19720b2 + 7938b1
\(\nu^{10}\)	\(=\)	\( 47280 \beta_{13} - 39780 \beta_{12} - 39780 \beta_{11} - 22740 \beta_{10} - 113520 \beta_{8} + \cdots - 206372 \) 47280b13 - 39780b12 - 39780b11 - 22740b10 - 113520b8 + 61984b6 + 61984b5 - 32916b4 - 78000b3 + 78000b2 + 32916*b1 - 206372
\(\nu^{11}\)	\(=\)	\( - 95436 \beta_{15} + 187264 \beta_{14} - 165660 \beta_{12} + 165660 \beta_{11} - 464728 \beta_{9} + \cdots - 136056 \beta_1 \) -95436b15 + 187264b14 - 165660b12 + 165660b11 - 464728b9 + 822988b7 + 247460b6 - 247460b5 - 136056b4 - 310756b3 - 310756b2 - 136056b1
\(\nu^{12}\)	\(=\)	\( - 745480 \beta_{13} + 682528 \beta_{12} + 682528 \beta_{11} + 397152 \beta_{10} + 1898488 \beta_{8} + \cdots + 3294272 \) -745480b13 + 682528b12 + 682528b11 + 397152b10 + 1898488b8 - 987568b6 - 987568b5 + 557488b4 + 1238128b3 - 1238128b2 - 557488*b1 + 3294272
\(\nu^{13}\)	\(=\)	\( 1637168 \beta_{15} - 2971176 \beta_{14} + 2798816 \beta_{12} - 2798816 \beta_{11} + 7719192 \beta_{9} + \cdots + 2276624 \beta_1 \) 1637168b15 - 2971176b14 + 2798816b12 - 2798816b11 + 7719192b9 - 13186512b7 - 3951500b6 + 3951500b5 + 2276624b4 + 4947540b3 + 4947540b2 + 2276624b1
\(\nu^{14}\)	\(=\)	\( 11870216 \beta_{13} - 11418336 \beta_{12} - 11418336 \beta_{11} - 6712608 \beta_{10} - 31322360 \beta_{8} + \cdots - 52862072 \) 11870216b13 - 11418336b12 - 11418336b11 - 6712608b10 - 31322360b8 + 15820104b6 + 15820104b5 - 9258976b4 - 19787320b3 + 19787320b2 + 9258976*b1 - 52862072
\(\nu^{15}\)	\(=\)	\( - 27389920 \beta_{15} + 47477640 \beta_{14} - 46439840 \beta_{12} + 46439840 \beta_{11} + \cdots - 37567872 \beta_1 \) -27389920b15 + 47477640b14 - 46439840b12 + 46439840b11 - 126802360b9 + 212001864b7 + 63412768b6 - 63412768b5 - 37567872b4 - 79250200b3 - 79250200b2 - 37567872b1

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 307.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{8} + 43 T^{4} + 361)^{2} \) (T^8 + 43*T^4 + 361)^2
$3$	\( (T^{8} + 90 T^{4} + 25)^{2} \) (T^8 + 90*T^4 + 25)^2
$5$	\( T^{16} \) T^16
$7$	\( T^{16} - 124 T^{12} + \cdots + 5764801 \) T^16 - 124T^12 + 7366T^8 - 297724*T^4 + 5764801
$11$	\( (T^{2} - 3 T + 1)^{8} \) (T^2 - 3*T + 1)^8
$13$	\( (T^{8} + 1440 T^{4} + 6400)^{2} \) (T^8 + 1440*T^4 + 6400)^2
$17$	\( (T^{8} + 15 T^{4} + 25)^{2} \) (T^8 + 15*T^4 + 25)^2
$19$	\( (T^{4} - 55 T^{2} + 95)^{4} \) (T^4 - 55*T^2 + 95)^4
$23$	\( (T^{8} + 3483 T^{4} + 2368521)^{2} \) (T^8 + 3483*T^4 + 2368521)^2
$29$	\( (T^{4} + 98 T^{2} + 1681)^{4} \) (T^4 + 98*T^2 + 1681)^4
$31$	\( (T^{4} + 85 T^{2} + 95)^{4} \) (T^4 + 85*T^2 + 95)^4
$37$	\( (T^{8} + 6723 T^{4} + 2368521)^{2} \) (T^8 + 6723*T^4 + 2368521)^2
$41$	\( (T^{4} + 80 T^{2} + 1520)^{4} \) (T^4 + 80*T^2 + 1520)^4
$43$	\( (T^{8} + 83 T^{4} + 361)^{2} \) (T^8 + 83*T^4 + 361)^2
$47$	\( (T^{8} + 3840 T^{4} + 1638400)^{2} \) (T^8 + 3840*T^4 + 1638400)^2
$53$	\( (T^{8} + 6218 T^{4} + 5285401)^{2} \) (T^8 + 6218*T^4 + 5285401)^2
$59$	\( (T^{4} - 215 T^{2} + 11495)^{4} \) (T^4 - 215*T^2 + 11495)^4
$61$	\( (T^{4} + 245 T^{2} + 11495)^{4} \) (T^4 + 245*T^2 + 11495)^4
$67$	\( (T^{8} + 538 T^{4} + 361)^{2} \) (T^8 + 538*T^4 + 361)^2
$71$	\( (T^{2} - 8 T + 11)^{8} \) (T^2 - 8*T + 11)^8
$73$	\( (T^{8} + 90375 T^{4} + 2036265625)^{2} \) (T^8 + 90375*T^4 + 2036265625)^2
$79$	\( (T^{4} + 108 T^{2} + 1296)^{4} \) (T^4 + 108*T^2 + 1296)^4
$83$	\( (T^{8} + 1290 T^{4} + 366025)^{2} \) (T^8 + 1290*T^4 + 366025)^2
$89$	\( (T^{4} - 140 T^{2} + 95)^{4} \) (T^4 - 140*T^2 + 95)^4
$97$	\( (T^{8} + 1440 T^{4} + 6400)^{2} \) (T^8 + 1440*T^4 + 6400)^2
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\(n\)	\(127\)	\(626\)
\(\chi(n)\)	\(-\beta_{7}\)	\(-1\)