LMFDB - Newform orbit 900.2.w.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [900,2,Mod(109,900)] 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("900.109"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(900, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 7])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	900.w (of order $10$, degree $4$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$7.18653618192$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{10})$
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} - 2x^{14} - 16x^{12} + 2x^{10} + 161x^{8} - 160x^{6} + 225x^{4} - 375x^{2} + 625 $ x^16 - 2x^14 - 16x^12 + 2x^10 + 161x^8 - 160x^6 + 225x^4 - 375*x^2 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-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_{12} q^{5} + (\beta_{9} - \beta_{4} - \beta_{3}) q^{7} + (\beta_{13} - \beta_{11}) q^{11} + (\beta_{9} - \beta_{6} - \beta_{5} + \cdots - \beta_1) q^{13} + (\beta_{15} + \beta_{12} + \cdots + \beta_{2}) q^{17}+ \cdots + ( - 2 \beta_{9} - 2 \beta_{8} + \cdots + 2) q^{97}+O(q^{100}) $ q + b12 * q^5 + (b9 - b4 - b3) * q^7 + (b13 - b11) * q^11 + (b9 - b6 - b5 - b3 - b1) * q^13 + (b15 + b12 + b11 - b10 + b7 + b2) * q^17 + (2b9 + b8 - b6 - b5 - b4 - b3 - 2b1) * q^19 + (-b12 - b7) * q^23 + (2b9 - 2b6 - b5 - b1) * q^25 + (-b15 - b14 - b13 - b12 + b11 + b7) * q^29 + (2b9 + b8 - b6 - b5 - b1 + 1) q^31 + (b14 - b13 - b12 + b10 - 2b7 - 2b2) * q^35 + (2b9 - 2b6 - b5 - b4 - b1 + 1) * q^37 + (-2b15 - b13 - 2b12 - 2b11 + 2b10 - b2) * q^41 + (2b9 - b3 + b1 + 1) q^43 + (b15 + b14 + b13 - b12 + b11 + b7) * q^47 + (b9 - 2b6 - 2b5 - b4 - 2b3 - 3b1) * q^49 + (b14 - 3b13 - 2b12 + b11 + b10 - 2b7 + b2) q^53 + (b9 + b8 + b6 + b5 + b1 + 5) * q^55 + (-2b15 - 3b14 + b13 + b12 + b11 + b7) * q^59 + (2b9 - 2b8 - 4b4 - 5b3 - 2) * q^61 + (2b14 + b13 - 3b11 - 2b7 - 3b2) * q^65 + (2b8 - 2b5 + 2b4 - b3 - 2b1 + 1) * q^67 + (4b15 + b13 + 3b12 - b11 - 3b10 + 2b7 + 3b2) q^71 + (-5b4 - 2b3 - 4b1 + 1) q^73 + (2b13 + 3b12 + 2b11 - 2b10 + 3b7 + 2b2) * q^77 + (b9 - b8 + 2b6 + 3b5 + 2b4 + b3 + 5b1 + 4) * q^79 + (-2b14 + 2b13 + 4b12 + b11 - b10 + 4b2) * q^83 + (b9 - b6 - 3b5 - 5b4 - 5b3 - 3b1 - 5) * q^85 + (4b15 - 4b14 + 2b13 + 2b12 + 2b11 - 3b10 + 2b7 + b2) q^89 + (2b8 - 2b6 - b5 + 5b4 + 4b3 - 2b1 - 1) q^91 + (-b15 + 3b14 - b13 - b12 - b11 - 2b7 - 4b2) q^95 + (-2b9 - 2b8 + 2b5 + b4 + 4b3 + 2b1 + 2) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{19} - 10 q^{25} + 6 q^{31} + 10 q^{37} + 16 q^{49} + 70 q^{55} - 16 q^{61} + 20 q^{67} + 60 q^{73} + 24 q^{79} - 40 q^{85} - 38 q^{91} + 20 q^{97}+O(q^{100}) $ 16 * q + 2 * q^19 - 10 * q^25 + 6 * q^31 + 10 * q^37 + 16 * q^49 + 70 * q^55 - 16 * q^61 + 20 * q^67 + 60 * q^73 + 24 * q^79 - 40 * q^85 - 38 * q^91 + 20 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 2x^{14} - 16x^{12} + 2x^{10} + 161x^{8} - 160x^{6} + 225x^{4} - 375x^{2} + 625 $ x^16 - 2*x^14 - 16*x^12 + 2*x^10 + 161*x^8 - 160*x^6 + 225*x^4 - 375*x^2 + 625 :

$\beta_{1}$	$=$	$ ( 424 \nu^{14} - 27053 \nu^{12} - 475624 \nu^{10} - 1162872 \nu^{8} + 7126104 \nu^{6} + \cdots - 65470000 ) / 88991375 $ (424v^14 - 27053v^12 - 475624v^10 - 1162872v^8 + 7126104v^6 + 23271280v^4 - 8261800*v^2 - 65470000) / 88991375
$\beta_{2}$	$=$	$ ( 424 \nu^{15} - 27053 \nu^{13} - 475624 \nu^{11} - 1162872 \nu^{9} + 7126104 \nu^{7} + \cdots - 154461375 \nu ) / 88991375 $ (424v^15 - 27053v^13 - 475624v^11 - 1162872v^9 + 7126104v^7 + 23271280v^5 - 8261800v^3 - 154461375v) / 88991375
$\beta_{3}$	$=$	$ ( - 33864 \nu^{14} - 55722 \nu^{12} + 1114449 \nu^{10} + 2743022 \nu^{8} - 8400729 \nu^{6} + \cdots - 70231500 ) / 88991375 $ (-33864v^14 - 55722v^12 + 1114449v^10 + 2743022v^8 - 8400729v^6 - 21242385v^4 + 6018075*v^2 - 70231500) / 88991375
$\beta_{4}$	$=$	$ ( 129364 \nu^{14} + 175022 \nu^{12} - 2459824 \nu^{10} - 6084772 \nu^{8} + 14968229 \nu^{6} + \cdots + 16012750 ) / 88991375 $ (129364v^14 + 175022v^12 - 2459824v^10 - 6084772v^8 + 14968229v^6 + 32426760v^4 - 7455600*v^2 + 16012750) / 88991375
$\beta_{5}$	$=$	$ ( 2721 \nu^{14} - 1887 \nu^{12} - 29946 \nu^{10} - 74713 \nu^{8} + 60391 \nu^{6} - 96005 \nu^{4} + \cdots - 380000 ) / 1618025 $ (2721v^14 - 1887v^12 - 29946v^10 - 74713v^8 + 60391v^6 - 96005v^4 + 2695300*v^2 - 380000) / 1618025
$\beta_{6}$	$=$	$ ( 188171 \nu^{14} - 378462 \nu^{12} - 2875471 \nu^{10} + 2754462 \nu^{8} + 36109891 \nu^{6} + \cdots - 29255125 ) / 88991375 $ (188171v^14 - 378462v^12 - 2875471v^10 + 2754462v^8 + 36109891v^6 - 65737880v^4 - 74017925*v^2 - 29255125) / 88991375
$\beta_{7}$	$=$	$ ( 33864 \nu^{15} + 55722 \nu^{13} - 1114449 \nu^{11} - 2743022 \nu^{9} + 8400729 \nu^{7} + \cdots + 159222875 \nu ) / 88991375 $ (33864v^15 + 55722v^13 - 1114449v^11 - 2743022v^9 + 8400729v^7 + 21242385v^5 - 6018075v^3 + 159222875v) / 88991375
$\beta_{8}$	$=$	$ ( 278181 \nu^{14} - 1033862 \nu^{12} - 5047396 \nu^{10} + 7283237 \nu^{8} + 61495891 \nu^{6} + \cdots - 97130250 ) / 88991375 $ (278181v^14 - 1033862v^12 - 5047396v^10 + 7283237v^8 + 61495891v^6 - 77346460v^4 + 6668850*v^2 - 97130250) / 88991375
$\beta_{9}$	$=$	$ ( 3470 \nu^{14} - 3120 \nu^{12} - 50748 \nu^{10} - 46875 \nu^{8} + 425000 \nu^{6} - 292500 \nu^{4} + \cdots - 1358751 ) / 711931 $ (3470v^14 - 3120v^12 - 50748v^10 - 46875v^8 + 425000v^6 - 292500v^4 + 1228125*v^2 - 1358751) / 711931
$\beta_{10}$	$=$	$ ( - 273046 \nu^{15} - 1535348 \nu^{13} + 1839391 \nu^{11} + 34311773 \nu^{9} + \cdots + 202542125 \nu ) / 444956875 $ (-273046v^15 - 1535348v^13 + 1839391v^11 + 34311773v^9 + 42404064v^7 - 165296805v^5 - 507691175v^3 + 202542125v) / 444956875
$\beta_{11}$	$=$	$ ( - 96348 \nu^{15} - 65194 \nu^{13} + 2296623 \nu^{11} + 5667494 \nu^{9} - 20819708 \nu^{7} + \cdots + 96167375 \nu ) / 88991375 $ (-96348v^15 - 65194v^13 + 2296623v^11 + 5667494v^9 - 20819708v^7 - 57726935v^5 + 17961125v^3 + 96167375v) / 88991375
$\beta_{12}$	$=$	$ ( 913556 \nu^{15} - 2979772 \nu^{13} - 16463101 \nu^{11} + 28464847 \nu^{9} + 212837346 \nu^{7} + \cdots - 256343375 \nu ) / 444956875 $ (913556v^15 - 2979772v^13 - 16463101v^11 + 28464847v^9 + 212837346v^7 - 311390520v^5 - 158863075v^3 - 256343375v) / 444956875
$\beta_{13}$	$=$	$ ( - 954849 \nu^{15} + 1593038 \nu^{13} + 15500754 \nu^{11} + 8757412 \nu^{9} + \cdots + 500401625 \nu ) / 444956875 $ (-954849v^15 + 1593038v^13 + 15500754v^11 + 8757412v^9 - 127479609v^7 + 19419905v^5 - 629976575v^3 + 500401625v) / 444956875
$\beta_{14}$	$=$	$ ( - 225288 \nu^{15} - 267269 \nu^{13} + 4280823 \nu^{11} + 10589394 \nu^{9} - 28661833 \nu^{7} + \cdots + 14684625 \nu ) / 88991375 $ (-225288v^15 - 267269v^13 + 4280823v^11 + 10589394v^9 - 28661833v^7 - 66882415v^5 + 17154925v^3 + 14684625v) / 88991375
$\beta_{15}$	$=$	$ ( - 1895704 \nu^{15} + 3485348 \nu^{13} + 29878109 \nu^{11} - 5014898 \nu^{9} + \cdots + 646677250 \nu ) / 444956875 $ (-1895704v^15 + 3485348v^13 + 29878109v^11 - 5014898v^9 - 308029064v^7 + 348109305v^5 - 259886950v^3 + 646677250v) / 444956875

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

$\nu$	$=$	$ ( \beta_{14} - 2\beta_{11} + \beta_{7} - 3\beta_{2} ) / 5 $ (b14 - 2b11 + b7 - 3b2) / 5
$\nu^{2}$	$=$	$ \beta_{9} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_1 $ b9 - b6 - b5 - b4 - b3 - b1
$\nu^{3}$	$=$	$ ( 5\beta_{15} + 3\beta_{14} + 10\beta_{12} - \beta_{11} - 10\beta_{10} + 3\beta_{7} + \beta_{2} ) / 5 $ (5b15 + 3b14 + 10b12 - b11 - 10b10 + 3*b7 + b2) / 5
$\nu^{4}$	$=$	$ \beta_{8} - 2\beta_{6} - \beta_{5} + 4\beta_{4} + 8\beta_{3} + 3\beta _1 + 8 $ b8 - 2b6 - b5 + 4b4 + 8b3 + 3b1 + 8
$\nu^{5}$	$=$	$ ( 10\beta_{15} + 4\beta_{14} - 10\beta_{13} + 5\beta_{12} - 28\beta_{11} + 5\beta_{10} - 16\beta_{7} - 32\beta_{2} ) / 5 $ (10b15 + 4b14 - 10b13 + 5b12 - 28b11 + 5b10 - 16b7 - 32b2) / 5
$\nu^{6}$	$=$	$ 8\beta_{9} - 4\beta_{8} - 8\beta_{5} - 9\beta_{4} + 10\beta _1 + 18 $ 8b9 - 4b8 - 8b5 - 9b4 + 10*b1 + 18
$\nu^{7}$	$=$	$ ( 20 \beta_{15} + 72 \beta_{14} + 60 \beta_{13} + 100 \beta_{12} - 99 \beta_{11} - 120 \beta_{10} + \cdots - 36 \beta_{2} ) / 5 $ (20b15 + 72b14 + 60b13 + 100b12 - 99b11 - 120b10 + 27b7 - 36b2) / 5
$\nu^{8}$	$=$	$ 18\beta_{9} + 18\beta_{8} - 45\beta_{6} - 45\beta_{5} + 33\beta_{4} + 55\beta_{3} - 45\beta _1 + 33 $ 18b9 + 18b8 - 45b6 - 45b5 + 33b4 + 55b3 - 45*b1 + 33
$\nu^{9}$	$=$	$ ( 270 \beta_{15} - 44 \beta_{14} - 90 \beta_{13} + 360 \beta_{12} - 77 \beta_{11} - 135 \beta_{10} + \cdots - 198 \beta_{2} ) / 5 $ (270b15 - 44b14 - 90b13 + 360b12 - 77b11 - 135b10 - 154b7 - 198b2) / 5
$\nu^{10}$	$=$	$ 33\beta_{9} - 55\beta_{8} + 55\beta_{6} + 280\beta_{3} + 280\beta _1 + 448 $ 33b9 - 55b8 + 55b6 + 280b3 + 280*b1 + 448
$\nu^{11}$	$=$	$ ( - 165 \beta_{15} + 1008 \beta_{14} + 550 \beta_{13} + 275 \beta_{12} - 2016 \beta_{11} + \cdots - 1624 \beta_{2} ) / 5 $ (-165b15 + 1008b14 + 550b13 + 275b12 - 2016b11 - 715b10 - 392b7 - 1624b2) / 5
$\nu^{12}$	$=$	$ 448\beta_{9} - 448\beta_{6} - 728\beta_{5} - 8\beta_{4} - 8\beta_{3} - 723\beta_1 $ 448b9 - 448b6 - 728b5 - 8b4 - 8b3 - 723b1
$\nu^{13}$	$=$	$ ( 3640 \beta_{15} + 29 \beta_{14} + 1400 \beta_{13} + 7280 \beta_{12} - 18 \beta_{11} + \cdots + 18 \beta_{2} ) / 5 $ (3640b15 + 29b14 + 1400b13 + 7280b12 - 18b11 - 5880b10 + 29b7 + 18b2) / 5
$\nu^{14}$	$=$	$ 13\beta_{8} - 21\beta_{6} - 13\beta_{5} + 2232\beta_{4} + 5859\beta_{3} + 2219\beta _1 + 5859 $ 13b8 - 21b6 - 13b5 + 2232b4 + 5859b3 + 2219b1 + 5859
$\nu^{15}$	$=$	$ ( 105 \beta_{15} + 5022 \beta_{14} - 105 \beta_{13} + 65 \beta_{12} - 21204 \beta_{11} + \cdots - 26226 \beta_{2} ) / 5 $ (105b15 + 5022b14 - 105b13 + 65b12 - 21204b11 + 65b10 - 13113b7 - 26226b2) / 5

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$1$	$1$	$-\beta_{4}$

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} $

109.1

1.10084	+	0.412453i
−0.648165	+	0.980739i
0.648165	−	0.980739i
−1.10084	−	0.412453i
1.10084	−	0.412453i
−0.648165	−	0.980739i
0.648165	+	0.980739i
−1.10084	+	0.412453i
−1.90024	−	0.0842796i
0.507053	−	1.83328i
−0.507053	+	1.83328i
1.90024	+	0.0842796i
−1.90024	+	0.0842796i
0.507053	+	1.83328i
−0.507053	−	1.83328i
1.90024	−	0.0842796i

−1.86548

−

1.23288i

1.81618i

109.2

−0.784532

2.09392i

−

2.54272i

109.3

0.784532

−

2.09392i

−

2.54272i

109.4

1.86548

1.23288i

1.81618i

289.1

−1.86548

1.23288i

−

1.81618i

289.2

−0.784532

−

2.09392i

2.54272i

289.3

0.784532

2.09392i

2.54272i

289.4

1.86548

−

1.23288i

−

1.81618i

469.1

−2.15515

−

0.596077i

0.640608i

469.2

−0.0990766

2.23387i

−

3.71829i

469.3

0.0990766

−

2.23387i

−

3.71829i

469.4

2.15515

0.596077i

0.640608i

829.1

−2.15515

0.596077i

−

0.640608i

829.2

−0.0990766

−

2.23387i

3.71829i

829.3

0.0990766

2.23387i

3.71829i

829.4

2.15515

−

0.596077i

−

0.640608i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
25.e	even	10	1	inner
75.h	odd	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	900.2.w.b	✓	16
3.b	odd	2	1	inner	900.2.w.b	✓	16
25.e	even	10	1	inner	900.2.w.b	✓	16
75.h	odd	10	1	inner	900.2.w.b	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
900.2.w.b	✓	16	1.a	even	1	1	trivial
900.2.w.b	✓	16	3.b	odd	2	1	inner
900.2.w.b	✓	16	25.e	even	10	1	inner
900.2.w.b	✓	16	75.h	odd	10	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{8} + 24T_{7}^{6} + 166T_{7}^{4} + 359T_{7}^{2} + 121 $ T7^8 + 24*T7^6 + 166*T7^4 + 359*T7^2 + 121 acting on $S_{2}^{\mathrm{new}}(900, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} $ T^16
$5$	$ T^{16} + 5 T^{14} + \cdots + 390625 $ T^16 + 5T^14 - 10T^12 + 25T^10 + 775T^8 + 625T^6 - 6250T^4 + 78125*T^2 + 390625
$7$	$ (T^{8} + 24 T^{6} + \cdots + 121)^{2} $ (T^8 + 24T^6 + 166T^4 + 359*T^2 + 121)^2
$11$	$ T^{16} + 20 T^{14} + \cdots + 625 $ T^16 + 20T^14 + 255T^12 + 2575T^10 + 64400T^8 - 80375T^6 + 400125T^4 + 25625*T^2 + 625
$13$	$ (T^{8} - 19 T^{6} + \cdots + 121)^{2} $ (T^8 - 19T^6 + 25T^5 + 136T^4 - 475T^3 + 541T^2 + 275T + 121)^2
$17$	$ T^{16} + 15 T^{14} + \cdots + 625 $ T^16 + 15T^14 + 1205T^12 - 9225T^10 + 57400T^8 - 321375T^6 + 2786375T^4 - 67500*T^2 + 625
$19$	$ (T^{8} - T^{7} + \cdots + 10201)^{2} $ (T^8 - T^7 + 27T^6 + 47T^5 + 180T^4 - 147T^3 + 327T^2 + 101T + 10201)^2
$23$	$ T^{16} - 15 T^{14} + \cdots + 625 $ T^16 - 15T^14 + 330T^12 - 6725T^10 + 90525T^8 - 297875T^6 + 398250T^4 + 9375*T^2 + 625
$29$	$ T^{16} + 65 T^{14} + \cdots + 625 $ T^16 + 65T^14 + 1730T^12 + 11075T^10 + 83525T^8 + 578125T^6 + 2773250T^4 - 25625*T^2 + 625
$31$	$ (T^{8} - 3 T^{7} + \cdots + 10201)^{2} $ (T^8 - 3T^7 + 28T^6 - 51T^5 + 275T^4 - 351T^3 + 778T^2 + 4747*T + 10201)^2
$37$	$ (T^{8} - 5 T^{7} + \cdots + 255025)^{2} $ (T^8 - 5T^7 - 15T^6 - 5T^5 + 860T^4 - 475T^3 - 9625T^2 + 60600*T + 255025)^2
$41$	$ T^{16} + \cdots + 442050625 $ T^16 + 190T^14 + 13955T^12 - 19400T^10 + 13067525T^8 + 88751000T^6 + 231693875T^4 - 68331250*T^2 + 442050625
$43$	$ (T^{8} + 86 T^{6} + \cdots + 73441)^{2} $ (T^8 + 86T^6 + 2391T^4 + 23686*T^2 + 73441)^2
$47$	$ T^{16} + \cdots + 8653650625 $ T^16 - 115T^14 + 5530T^12 - 7025T^10 + 14589525T^8 + 100147625T^6 + 3843453250T^4 - 9388548125*T^2 + 8653650625
$53$	$ T^{16} + \cdots + 6472063200625 $ T^16 - 245T^14 + 24105T^12 + 224375T^10 + 225942900T^8 + 3653709625T^6 + 356342423875T^4 - 2473301105000*T^2 + 6472063200625
$59$	$ T^{16} + \cdots + 276281640625 $ T^16 + 120T^14 + 12425T^12 + 1191500T^10 + 173128125T^8 - 1878662500T^6 + 166568390625T^4 + 347963750000*T^2 + 276281640625
$61$	$ (T^{8} + 8 T^{7} + \cdots + 326041)^{2} $ (T^8 + 8T^7 + 203T^6 - 414T^5 - 2245T^4 + 11346T^3 + 143663T^2 - 121052*T + 326041)^2
$67$	$ (T^{8} - 10 T^{7} + \cdots + 73441)^{2} $ (T^8 - 10T^7 - 41T^6 + 340T^5 + 2436T^4 + 5510T^3 + 3514T^2 + 2710*T + 73441)^2
$71$	$ T^{16} + \cdots + 26\!\cdots\!25 $ T^16 + 95T^14 + 34205T^12 + 6180575T^10 + 937603400T^8 + 39434904625T^6 + 11692790126375T^4 + 282751775747500*T^2 + 2679928412400625
$73$	$ (T^{4} - 15 T^{3} + \cdots + 605)^{4} $ (T^4 - 15T^3 + 120T^2 - 440*T + 605)^4
$79$	$ (T^{8} - 12 T^{7} + \cdots + 3500641)^{2} $ (T^8 - 12T^7 + 83T^6 + 211T^5 + 1930T^4 + 18821T^3 + 519793T^2 + 922403*T + 3500641)^2
$83$	$ T^{16} + \cdots + 24\!\cdots\!25 $ T^16 - 270T^14 + 65355T^12 - 15737200T^10 + 3447570525T^8 - 171240544000T^6 + 3987386818875T^4 - 32913477378750*T^2 + 2463337905600625
$89$	$ T^{16} + \cdots + 36\!\cdots\!25 $ T^16 + 20T^14 + 54255T^12 + 8282875T^10 + 1255249400T^8 + 197753054125T^6 + 26270197735125T^4 + 1475326429383125*T^2 + 36319983612900625
$97$	$ (T^{8} - 10 T^{7} + \cdots + 755161)^{2} $ (T^8 - 10T^7 - 11T^6 + 790T^5 - 3609T^4 - 19390T^3 + 209809T^2 - 530090*T + 755161)^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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	900.w (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{12} q^{5} + (\beta_{9} - \beta_{4} - \beta_{3}) q^{7} + (\beta_{13} - \beta_{11}) q^{11} + (\beta_{9} - \beta_{6} - \beta_{5} + \cdots - \beta_1) q^{13} + (\beta_{15} + \beta_{12} + \cdots + \beta_{2}) q^{17}+ \cdots + ( - 2 \beta_{9} - 2 \beta_{8} + \cdots + 2) q^{97}+O(q^{100}) \) q + b12 * q^5 + (b9 - b4 - b3) * q^7 + (b13 - b11) * q^11 + (b9 - b6 - b5 - b3 - b1) * q^13 + (b15 + b12 + b11 - b10 + b7 + b2) * q^17 + (2b9 + b8 - b6 - b5 - b4 - b3 - 2b1) * q^19 + (-b12 - b7) * q^23 + (2b9 - 2b6 - b5 - b1) * q^25 + (-b15 - b14 - b13 - b12 + b11 + b7) * q^29 + (2b9 + b8 - b6 - b5 - b1 + 1) q^31 + (b14 - b13 - b12 + b10 - 2b7 - 2b2) * q^35 + (2b9 - 2b6 - b5 - b4 - b1 + 1) * q^37 + (-2b15 - b13 - 2b12 - 2b11 + 2b10 - b2) * q^41 + (2b9 - b3 + b1 + 1) q^43 + (b15 + b14 + b13 - b12 + b11 + b7) * q^47 + (b9 - 2b6 - 2b5 - b4 - 2b3 - 3b1) * q^49 + (b14 - 3b13 - 2b12 + b11 + b10 - 2b7 + b2) q^53 + (b9 + b8 + b6 + b5 + b1 + 5) * q^55 + (-2b15 - 3b14 + b13 + b12 + b11 + b7) * q^59 + (2b9 - 2b8 - 4b4 - 5b3 - 2) * q^61 + (2b14 + b13 - 3b11 - 2b7 - 3b2) * q^65 + (2b8 - 2b5 + 2b4 - b3 - 2b1 + 1) * q^67 + (4b15 + b13 + 3b12 - b11 - 3b10 + 2b7 + 3b2) q^71 + (-5b4 - 2b3 - 4b1 + 1) q^73 + (2b13 + 3b12 + 2b11 - 2b10 + 3b7 + 2b2) * q^77 + (b9 - b8 + 2b6 + 3b5 + 2b4 + b3 + 5b1 + 4) * q^79 + (-2b14 + 2b13 + 4b12 + b11 - b10 + 4b2) * q^83 + (b9 - b6 - 3b5 - 5b4 - 5b3 - 3b1 - 5) * q^85 + (4b15 - 4b14 + 2b13 + 2b12 + 2b11 - 3b10 + 2b7 + b2) q^89 + (2b8 - 2b6 - b5 + 5b4 + 4b3 - 2b1 - 1) q^91 + (-b15 + 3b14 - b13 - b12 - b11 - 2b7 - 4b2) q^95 + (-2b9 - 2b8 + 2b5 + b4 + 4b3 + 2b1 + 2) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{19} - 10 q^{25} + 6 q^{31} + 10 q^{37} + 16 q^{49} + 70 q^{55} - 16 q^{61} + 20 q^{67} + 60 q^{73} + 24 q^{79} - 40 q^{85} - 38 q^{91} + 20 q^{97}+O(q^{100}) \) 16 * q + 2 * q^19 - 10 * q^25 + 6 * q^31 + 10 * q^37 + 16 * q^49 + 70 * q^55 - 16 * q^61 + 20 * q^67 + 60 * q^73 + 24 * q^79 - 40 * q^85 - 38 * q^91 + 20 * q^97

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	900.2.w.b

Level	$900$
Weight	$2$
Character orbit	900.w
Analytic conductor	$7.187$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(7.18653618192\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{10})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 2x^{14} - 16x^{12} + 2x^{10} + 161x^{8} - 160x^{6} + 225x^{4} - 375x^{2} + 625 \) x^16 - 2x^14 - 16x^12 + 2x^10 + 161x^8 - 160x^6 + 225x^4 - 375*x^2 + 625
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 5^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

\(\beta_{1}\)	\(=\)	\( ( 424 \nu^{14} - 27053 \nu^{12} - 475624 \nu^{10} - 1162872 \nu^{8} + 7126104 \nu^{6} + \cdots - 65470000 ) / 88991375 \) (424v^14 - 27053v^12 - 475624v^10 - 1162872v^8 + 7126104v^6 + 23271280v^4 - 8261800*v^2 - 65470000) / 88991375
\(\beta_{2}\)	\(=\)	\( ( 424 \nu^{15} - 27053 \nu^{13} - 475624 \nu^{11} - 1162872 \nu^{9} + 7126104 \nu^{7} + \cdots - 154461375 \nu ) / 88991375 \) (424v^15 - 27053v^13 - 475624v^11 - 1162872v^9 + 7126104v^7 + 23271280v^5 - 8261800v^3 - 154461375v) / 88991375
\(\beta_{3}\)	\(=\)	\( ( - 33864 \nu^{14} - 55722 \nu^{12} + 1114449 \nu^{10} + 2743022 \nu^{8} - 8400729 \nu^{6} + \cdots - 70231500 ) / 88991375 \) (-33864v^14 - 55722v^12 + 1114449v^10 + 2743022v^8 - 8400729v^6 - 21242385v^4 + 6018075*v^2 - 70231500) / 88991375
\(\beta_{4}\)	\(=\)	\( ( 129364 \nu^{14} + 175022 \nu^{12} - 2459824 \nu^{10} - 6084772 \nu^{8} + 14968229 \nu^{6} + \cdots + 16012750 ) / 88991375 \) (129364v^14 + 175022v^12 - 2459824v^10 - 6084772v^8 + 14968229v^6 + 32426760v^4 - 7455600*v^2 + 16012750) / 88991375
\(\beta_{5}\)	\(=\)	\( ( 2721 \nu^{14} - 1887 \nu^{12} - 29946 \nu^{10} - 74713 \nu^{8} + 60391 \nu^{6} - 96005 \nu^{4} + \cdots - 380000 ) / 1618025 \) (2721v^14 - 1887v^12 - 29946v^10 - 74713v^8 + 60391v^6 - 96005v^4 + 2695300*v^2 - 380000) / 1618025
\(\beta_{6}\)	\(=\)	\( ( 188171 \nu^{14} - 378462 \nu^{12} - 2875471 \nu^{10} + 2754462 \nu^{8} + 36109891 \nu^{6} + \cdots - 29255125 ) / 88991375 \) (188171v^14 - 378462v^12 - 2875471v^10 + 2754462v^8 + 36109891v^6 - 65737880v^4 - 74017925*v^2 - 29255125) / 88991375
\(\beta_{7}\)	\(=\)	\( ( 33864 \nu^{15} + 55722 \nu^{13} - 1114449 \nu^{11} - 2743022 \nu^{9} + 8400729 \nu^{7} + \cdots + 159222875 \nu ) / 88991375 \) (33864v^15 + 55722v^13 - 1114449v^11 - 2743022v^9 + 8400729v^7 + 21242385v^5 - 6018075v^3 + 159222875v) / 88991375
\(\beta_{8}\)	\(=\)	\( ( 278181 \nu^{14} - 1033862 \nu^{12} - 5047396 \nu^{10} + 7283237 \nu^{8} + 61495891 \nu^{6} + \cdots - 97130250 ) / 88991375 \) (278181v^14 - 1033862v^12 - 5047396v^10 + 7283237v^8 + 61495891v^6 - 77346460v^4 + 6668850*v^2 - 97130250) / 88991375
\(\beta_{9}\)	\(=\)	\( ( 3470 \nu^{14} - 3120 \nu^{12} - 50748 \nu^{10} - 46875 \nu^{8} + 425000 \nu^{6} - 292500 \nu^{4} + \cdots - 1358751 ) / 711931 \) (3470v^14 - 3120v^12 - 50748v^10 - 46875v^8 + 425000v^6 - 292500v^4 + 1228125*v^2 - 1358751) / 711931
\(\beta_{10}\)	\(=\)	\( ( - 273046 \nu^{15} - 1535348 \nu^{13} + 1839391 \nu^{11} + 34311773 \nu^{9} + \cdots + 202542125 \nu ) / 444956875 \) (-273046v^15 - 1535348v^13 + 1839391v^11 + 34311773v^9 + 42404064v^7 - 165296805v^5 - 507691175v^3 + 202542125v) / 444956875
\(\beta_{11}\)	\(=\)	\( ( - 96348 \nu^{15} - 65194 \nu^{13} + 2296623 \nu^{11} + 5667494 \nu^{9} - 20819708 \nu^{7} + \cdots + 96167375 \nu ) / 88991375 \) (-96348v^15 - 65194v^13 + 2296623v^11 + 5667494v^9 - 20819708v^7 - 57726935v^5 + 17961125v^3 + 96167375v) / 88991375
\(\beta_{12}\)	\(=\)	\( ( 913556 \nu^{15} - 2979772 \nu^{13} - 16463101 \nu^{11} + 28464847 \nu^{9} + 212837346 \nu^{7} + \cdots - 256343375 \nu ) / 444956875 \) (913556v^15 - 2979772v^13 - 16463101v^11 + 28464847v^9 + 212837346v^7 - 311390520v^5 - 158863075v^3 - 256343375v) / 444956875
\(\beta_{13}\)	\(=\)	\( ( - 954849 \nu^{15} + 1593038 \nu^{13} + 15500754 \nu^{11} + 8757412 \nu^{9} + \cdots + 500401625 \nu ) / 444956875 \) (-954849v^15 + 1593038v^13 + 15500754v^11 + 8757412v^9 - 127479609v^7 + 19419905v^5 - 629976575v^3 + 500401625v) / 444956875
\(\beta_{14}\)	\(=\)	\( ( - 225288 \nu^{15} - 267269 \nu^{13} + 4280823 \nu^{11} + 10589394 \nu^{9} - 28661833 \nu^{7} + \cdots + 14684625 \nu ) / 88991375 \) (-225288v^15 - 267269v^13 + 4280823v^11 + 10589394v^9 - 28661833v^7 - 66882415v^5 + 17154925v^3 + 14684625v) / 88991375
\(\beta_{15}\)	\(=\)	\( ( - 1895704 \nu^{15} + 3485348 \nu^{13} + 29878109 \nu^{11} - 5014898 \nu^{9} + \cdots + 646677250 \nu ) / 444956875 \) (-1895704v^15 + 3485348v^13 + 29878109v^11 - 5014898v^9 - 308029064v^7 + 348109305v^5 - 259886950v^3 + 646677250v) / 444956875

\(\nu\)	\(=\)	\( ( \beta_{14} - 2\beta_{11} + \beta_{7} - 3\beta_{2} ) / 5 \) (b14 - 2b11 + b7 - 3b2) / 5
\(\nu^{2}\)	\(=\)	\( \beta_{9} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_1 \) b9 - b6 - b5 - b4 - b3 - b1
\(\nu^{3}\)	\(=\)	\( ( 5\beta_{15} + 3\beta_{14} + 10\beta_{12} - \beta_{11} - 10\beta_{10} + 3\beta_{7} + \beta_{2} ) / 5 \) (5b15 + 3b14 + 10b12 - b11 - 10b10 + 3*b7 + b2) / 5
\(\nu^{4}\)	\(=\)	\( \beta_{8} - 2\beta_{6} - \beta_{5} + 4\beta_{4} + 8\beta_{3} + 3\beta _1 + 8 \) b8 - 2b6 - b5 + 4b4 + 8b3 + 3b1 + 8
\(\nu^{5}\)	\(=\)	\( ( 10\beta_{15} + 4\beta_{14} - 10\beta_{13} + 5\beta_{12} - 28\beta_{11} + 5\beta_{10} - 16\beta_{7} - 32\beta_{2} ) / 5 \) (10b15 + 4b14 - 10b13 + 5b12 - 28b11 + 5b10 - 16b7 - 32b2) / 5
\(\nu^{6}\)	\(=\)	\( 8\beta_{9} - 4\beta_{8} - 8\beta_{5} - 9\beta_{4} + 10\beta _1 + 18 \) 8b9 - 4b8 - 8b5 - 9b4 + 10*b1 + 18
\(\nu^{7}\)	\(=\)	\( ( 20 \beta_{15} + 72 \beta_{14} + 60 \beta_{13} + 100 \beta_{12} - 99 \beta_{11} - 120 \beta_{10} + \cdots - 36 \beta_{2} ) / 5 \) (20b15 + 72b14 + 60b13 + 100b12 - 99b11 - 120b10 + 27b7 - 36b2) / 5
\(\nu^{8}\)	\(=\)	\( 18\beta_{9} + 18\beta_{8} - 45\beta_{6} - 45\beta_{5} + 33\beta_{4} + 55\beta_{3} - 45\beta _1 + 33 \) 18b9 + 18b8 - 45b6 - 45b5 + 33b4 + 55b3 - 45*b1 + 33
\(\nu^{9}\)	\(=\)	\( ( 270 \beta_{15} - 44 \beta_{14} - 90 \beta_{13} + 360 \beta_{12} - 77 \beta_{11} - 135 \beta_{10} + \cdots - 198 \beta_{2} ) / 5 \) (270b15 - 44b14 - 90b13 + 360b12 - 77b11 - 135b10 - 154b7 - 198b2) / 5
\(\nu^{10}\)	\(=\)	\( 33\beta_{9} - 55\beta_{8} + 55\beta_{6} + 280\beta_{3} + 280\beta _1 + 448 \) 33b9 - 55b8 + 55b6 + 280b3 + 280*b1 + 448
\(\nu^{11}\)	\(=\)	\( ( - 165 \beta_{15} + 1008 \beta_{14} + 550 \beta_{13} + 275 \beta_{12} - 2016 \beta_{11} + \cdots - 1624 \beta_{2} ) / 5 \) (-165b15 + 1008b14 + 550b13 + 275b12 - 2016b11 - 715b10 - 392b7 - 1624b2) / 5
\(\nu^{12}\)	\(=\)	\( 448\beta_{9} - 448\beta_{6} - 728\beta_{5} - 8\beta_{4} - 8\beta_{3} - 723\beta_1 \) 448b9 - 448b6 - 728b5 - 8b4 - 8b3 - 723b1
\(\nu^{13}\)	\(=\)	\( ( 3640 \beta_{15} + 29 \beta_{14} + 1400 \beta_{13} + 7280 \beta_{12} - 18 \beta_{11} + \cdots + 18 \beta_{2} ) / 5 \) (3640b15 + 29b14 + 1400b13 + 7280b12 - 18b11 - 5880b10 + 29b7 + 18b2) / 5
\(\nu^{14}\)	\(=\)	\( 13\beta_{8} - 21\beta_{6} - 13\beta_{5} + 2232\beta_{4} + 5859\beta_{3} + 2219\beta _1 + 5859 \) 13b8 - 21b6 - 13b5 + 2232b4 + 5859b3 + 2219b1 + 5859
\(\nu^{15}\)	\(=\)	\( ( 105 \beta_{15} + 5022 \beta_{14} - 105 \beta_{13} + 65 \beta_{12} - 21204 \beta_{11} + \cdots - 26226 \beta_{2} ) / 5 \) (105b15 + 5022b14 - 105b13 + 65b12 - 21204b11 + 65b10 - 13113b7 - 26226b2) / 5

\(n\)	\(101\)	\(451\)	\(577\)
\(\chi(n)\)	\(1\)	\(1\)	\(-\beta_{4}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 109.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} \) T^16
$5$	\( T^{16} + 5 T^{14} + \cdots + 390625 \) T^16 + 5T^14 - 10T^12 + 25T^10 + 775T^8 + 625T^6 - 6250T^4 + 78125*T^2 + 390625
$7$	\( (T^{8} + 24 T^{6} + \cdots + 121)^{2} \) (T^8 + 24T^6 + 166T^4 + 359*T^2 + 121)^2
$11$	\( T^{16} + 20 T^{14} + \cdots + 625 \) T^16 + 20T^14 + 255T^12 + 2575T^10 + 64400T^8 - 80375T^6 + 400125T^4 + 25625*T^2 + 625
$13$	\( (T^{8} - 19 T^{6} + \cdots + 121)^{2} \) (T^8 - 19T^6 + 25T^5 + 136T^4 - 475T^3 + 541T^2 + 275T + 121)^2
$17$	\( T^{16} + 15 T^{14} + \cdots + 625 \) T^16 + 15T^14 + 1205T^12 - 9225T^10 + 57400T^8 - 321375T^6 + 2786375T^4 - 67500*T^2 + 625
$19$	\( (T^{8} - T^{7} + \cdots + 10201)^{2} \) (T^8 - T^7 + 27T^6 + 47T^5 + 180T^4 - 147T^3 + 327T^2 + 101T + 10201)^2
$23$	\( T^{16} - 15 T^{14} + \cdots + 625 \) T^16 - 15T^14 + 330T^12 - 6725T^10 + 90525T^8 - 297875T^6 + 398250T^4 + 9375*T^2 + 625
$29$	\( T^{16} + 65 T^{14} + \cdots + 625 \) T^16 + 65T^14 + 1730T^12 + 11075T^10 + 83525T^8 + 578125T^6 + 2773250T^4 - 25625*T^2 + 625
$31$	\( (T^{8} - 3 T^{7} + \cdots + 10201)^{2} \) (T^8 - 3T^7 + 28T^6 - 51T^5 + 275T^4 - 351T^3 + 778T^2 + 4747*T + 10201)^2
$37$	\( (T^{8} - 5 T^{7} + \cdots + 255025)^{2} \) (T^8 - 5T^7 - 15T^6 - 5T^5 + 860T^4 - 475T^3 - 9625T^2 + 60600*T + 255025)^2
$41$	\( T^{16} + \cdots + 442050625 \) T^16 + 190T^14 + 13955T^12 - 19400T^10 + 13067525T^8 + 88751000T^6 + 231693875T^4 - 68331250*T^2 + 442050625
$43$	\( (T^{8} + 86 T^{6} + \cdots + 73441)^{2} \) (T^8 + 86T^6 + 2391T^4 + 23686*T^2 + 73441)^2
$47$	\( T^{16} + \cdots + 8653650625 \) T^16 - 115T^14 + 5530T^12 - 7025T^10 + 14589525T^8 + 100147625T^6 + 3843453250T^4 - 9388548125*T^2 + 8653650625
$53$	\( T^{16} + \cdots + 6472063200625 \) T^16 - 245T^14 + 24105T^12 + 224375T^10 + 225942900T^8 + 3653709625T^6 + 356342423875T^4 - 2473301105000*T^2 + 6472063200625
$59$	\( T^{16} + \cdots + 276281640625 \) T^16 + 120T^14 + 12425T^12 + 1191500T^10 + 173128125T^8 - 1878662500T^6 + 166568390625T^4 + 347963750000*T^2 + 276281640625
$61$	\( (T^{8} + 8 T^{7} + \cdots + 326041)^{2} \) (T^8 + 8T^7 + 203T^6 - 414T^5 - 2245T^4 + 11346T^3 + 143663T^2 - 121052*T + 326041)^2
$67$	\( (T^{8} - 10 T^{7} + \cdots + 73441)^{2} \) (T^8 - 10T^7 - 41T^6 + 340T^5 + 2436T^4 + 5510T^3 + 3514T^2 + 2710*T + 73441)^2
$71$	\( T^{16} + \cdots + 26\!\cdots\!25 \) T^16 + 95T^14 + 34205T^12 + 6180575T^10 + 937603400T^8 + 39434904625T^6 + 11692790126375T^4 + 282751775747500*T^2 + 2679928412400625
$73$	\( (T^{4} - 15 T^{3} + \cdots + 605)^{4} \) (T^4 - 15T^3 + 120T^2 - 440*T + 605)^4
$79$	\( (T^{8} - 12 T^{7} + \cdots + 3500641)^{2} \) (T^8 - 12T^7 + 83T^6 + 211T^5 + 1930T^4 + 18821T^3 + 519793T^2 + 922403*T + 3500641)^2
$83$	\( T^{16} + \cdots + 24\!\cdots\!25 \) T^16 - 270T^14 + 65355T^12 - 15737200T^10 + 3447570525T^8 - 171240544000T^6 + 3987386818875T^4 - 32913477378750*T^2 + 2463337905600625
$89$	\( T^{16} + \cdots + 36\!\cdots\!25 \) T^16 + 20T^14 + 54255T^12 + 8282875T^10 + 1255249400T^8 + 197753054125T^6 + 26270197735125T^4 + 1475326429383125*T^2 + 36319983612900625
$97$	\( (T^{8} - 10 T^{7} + \cdots + 755161)^{2} \) (T^8 - 10T^7 - 11T^6 + 790T^5 - 3609T^4 - 19390T^3 + 209809T^2 - 530090*T + 755161)^2
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