Properties

Label 1075.2.b.j
Level $1075$
Weight $2$
Character orbit 1075.b
Analytic conductor $8.584$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1075,2,Mod(474,1075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1075, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1075.474");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1075 = 5^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1075.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.58391821729\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 23x^{10} + 205x^{8} + 890x^{6} + 1942x^{4} + 1980x^{2} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{4} q^{3} + (\beta_{2} - 2) q^{4} + ( - \beta_{8} + \beta_{5} + 2) q^{6} + ( - \beta_{9} + \beta_1) q^{7} + ( - \beta_{7} + \beta_{6} - \beta_1) q^{8} + ( - \beta_{11} + \beta_{10} + \beta_{8} - \beta_{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{4} q^{3} + (\beta_{2} - 2) q^{4} + ( - \beta_{8} + \beta_{5} + 2) q^{6} + ( - \beta_{9} + \beta_1) q^{7} + ( - \beta_{7} + \beta_{6} - \beta_1) q^{8} + ( - \beta_{11} + \beta_{10} + \beta_{8} - \beta_{2} - 2) q^{9} + (2 \beta_{10} + \beta_{5} - 1) q^{11} + ( - \beta_{7} + \beta_{3} + \beta_1) q^{12} + (\beta_{3} - \beta_1) q^{13} + (\beta_{11} + \beta_{10} + \beta_{2} - 3) q^{14} + (\beta_{8} - \beta_{5} - 2 \beta_{2} + 1) q^{16} + ( - 2 \beta_{9} + \beta_{7} + \beta_{4} + \beta_1) q^{17} + ( - 2 \beta_{9} + 3 \beta_{7} - \beta_{6} + 2 \beta_{4} - \beta_{3} + \beta_1) q^{18} + ( - \beta_{10} - \beta_{5} - 2 \beta_{2} + 1) q^{19} + ( - \beta_{10} - \beta_{5} + 3) q^{21} + (2 \beta_{9} + 6 \beta_{7} - \beta_{4} - \beta_{3} - \beta_1) q^{22} + (\beta_{7} - \beta_{6} + 2 \beta_{4} + \beta_{3} + \beta_1) q^{23} + (2 \beta_{10} + 3 \beta_{5} + \beta_{2} - 1) q^{24} + (\beta_{10} + 2 \beta_{8} + \beta_{5} - \beta_{2} + 3) q^{26} + (2 \beta_{9} - 5 \beta_{4} - \beta_{3} - 2 \beta_1) q^{27} + (2 \beta_{9} + 4 \beta_{7} + \beta_{6} - \beta_{4} - \beta_{3} - 3 \beta_1) q^{28} + (2 \beta_{10} - \beta_{8} + 2 \beta_{5} + \beta_{2} - 3) q^{29} + (3 \beta_{10} + \beta_{8} + \beta_{5} - 2) q^{31} + (\beta_{7} + 2 \beta_{4} - \beta_{3} + 2 \beta_1) q^{32} + ( - \beta_{9} + \beta_{7} - \beta_{6} - 3 \beta_{4} - \beta_{3} - \beta_1) q^{33} + (2 \beta_{11} + \beta_{10} - \beta_{8} + \beta_{5} + \beta_{2}) q^{34} + (\beta_{10} - 3 \beta_{8} + 2 \beta_{5} + 2 \beta_{2} - 2) q^{36} + ( - \beta_{9} + \beta_{7} + \beta_{4} + 2 \beta_{3}) q^{37} + ( - \beta_{9} - 2 \beta_{6} + \beta_{4} + \beta_{3} + 3 \beta_1) q^{38} + (\beta_{10} - \beta_{8} - \beta_{5} - \beta_{2}) q^{39} + ( - 3 \beta_{5} - 1) q^{41} + ( - \beta_{9} - 2 \beta_{7} + \beta_{4} + \beta_{3} + 3 \beta_1) q^{42} + \beta_{7} q^{43} + ( - 2 \beta_{11} - 5 \beta_{10} - \beta_{8} - \beta_{2} - 1) q^{44} + (\beta_{10} - \beta_{8} + 4 \beta_{5} + 4 \beta_{2} - 2) q^{46} + (\beta_{9} - \beta_{7} + \beta_{4} + 2 \beta_{3} + \beta_1) q^{47} + (2 \beta_{9} - \beta_{7} + \beta_{6} - 3 \beta_{4} - \beta_{3}) q^{48} + (2 \beta_{11} + 2 \beta_{10} - \beta_{8} + \beta_{5} + \beta_{2}) q^{49} + ( - \beta_{11} - \beta_{10} + 2 \beta_{8} - 4 \beta_{5} - \beta_{2} - 1) q^{51} + (\beta_{9} + \beta_{7} - \beta_{6} + \beta_{4} - 3 \beta_{3} + 4 \beta_1) q^{52} + (\beta_{9} + \beta_{7} + 2 \beta_{6} + \beta_1) q^{53} + ( - 2 \beta_{11} - 3 \beta_{10} + 3 \beta_{8} - 6 \beta_{5} - 2 \beta_{2} - 3) q^{54} + ( - 6 \beta_{10} - 3 \beta_{5} - 4 \beta_{2} + 4) q^{56} + (\beta_{9} - \beta_{7} + \beta_{6} - 2 \beta_{4} - 2 \beta_{3} - \beta_1) q^{57} + (2 \beta_{9} + 4 \beta_{7} + \beta_{6} - 3 \beta_{4} - 5 \beta_1) q^{58} + ( - \beta_{11} - \beta_{5} - 4) q^{59} + (\beta_{11} - \beta_{10} - 2 \beta_{8} - 2 \beta_{5} - 2) q^{61} + (3 \beta_{9} + 9 \beta_{7} - 3 \beta_{3} - \beta_1) q^{62} + ( - 2 \beta_{9} - 3 \beta_{7} + \beta_{6} + 4 \beta_{4} + 4 \beta_1) q^{63} + ( - 2 \beta_{10} - 2 \beta_{8} - \beta_{5} - 2 \beta_{2} - 1) q^{64} + (\beta_{11} - 3 \beta_{5} + 2 \beta_{2} - 1) q^{66} + ( - \beta_{9} + 2 \beta_{4} + 2 \beta_{3} + \beta_1) q^{67} + (3 \beta_{9} + 6 \beta_{7} + \beta_{6} - 2 \beta_{4} - \beta_{3} - 2 \beta_1) q^{68} + ( - 2 \beta_{11} + \beta_{10} - 3 \beta_{5} - 4 \beta_{2} - 7) q^{69} + ( - \beta_{11} + 2 \beta_{10} + 2 \beta_{5} + 3 \beta_{2} + 2) q^{71} + ( - 3 \beta_{9} + 7 \beta_{7} - \beta_{4} + 2 \beta_{3} - 5 \beta_1) q^{72} + ( - \beta_{9} + 2 \beta_{6} + 2 \beta_{4} - \beta_{3} + 3 \beta_1) q^{73} + (\beta_{11} + 2 \beta_{10} + 3 \beta_{8} + 3 \beta_{5} + 1) q^{74} + (\beta_{11} + 2 \beta_{10} - \beta_{8} + 2 \beta_{5} + 5 \beta_{2} - 10) q^{76} + (3 \beta_{9} + 3 \beta_{7} + 2 \beta_{6} + \beta_{4} - 2 \beta_1) q^{77} + (\beta_{9} + 8 \beta_{7} - \beta_{6} + 3 \beta_{3}) q^{78} + ( - \beta_{10} + 2 \beta_{8} + 2 \beta_{5} + 2 \beta_{2} + 6) q^{79} + (2 \beta_{11} - \beta_{10} + 2 \beta_{5} + 3 \beta_{2} + 11) q^{81} + (6 \beta_{7} + 3 \beta_{4} + 3 \beta_{3} - \beta_1) q^{82} + (3 \beta_{9} + 2 \beta_{7} - 2 \beta_{6} - 3 \beta_{4}) q^{83} + (\beta_{11} + 2 \beta_{10} + \beta_{8} + 3 \beta_{2} - 4) q^{84} - \beta_{10} q^{86} + ( - 3 \beta_{9} + 3 \beta_{7} - 2 \beta_{6} - 3 \beta_{4} - \beta_{3} - 2 \beta_1) q^{87} + ( - 7 \beta_{9} - 8 \beta_{7} - \beta_{6} - \beta_{4} + 2 \beta_{3} - \beta_1) q^{88} + ( - 3 \beta_{5} + 2 \beta_{2} - 8) q^{89} + ( - \beta_{11} - \beta_{10} + \beta_{8} + 2) q^{91} + (\beta_{9} - 5 \beta_{7} + 2 \beta_{6} - \beta_{4} - 5 \beta_1) q^{92} + (\beta_{7} - \beta_{6} - 5 \beta_{4}) q^{93} + ( - \beta_{11} + 2 \beta_{10} + 3 \beta_{8} + 3 \beta_{5} + \beta_{2} - 5) q^{94} + ( - 2 \beta_{11} + \beta_{10} + 2 \beta_{8} + \beta_{5} - \beta_{2} - 8) q^{96} + (2 \beta_{9} + 3 \beta_{7} - 2 \beta_{6} - \beta_{3} - 2 \beta_1) q^{97} + (8 \beta_{9} + 8 \beta_{7} + \beta_{6} - 4 \beta_{4} - \beta_{3} - 4 \beta_1) q^{98} + (3 \beta_{11} - \beta_{10} + 2 \beta_{8} - 4 \beta_{5} + 3 \beta_{2} + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 22 q^{4} + 22 q^{6} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 22 q^{4} + 22 q^{6} - 24 q^{9} - 12 q^{11} - 34 q^{14} + 10 q^{16} + 10 q^{19} + 38 q^{21} - 18 q^{24} + 28 q^{26} - 36 q^{29} - 24 q^{31} - 2 q^{34} - 20 q^{36} + 6 q^{39} - 18 q^{44} - 28 q^{46} - 2 q^{51} - 24 q^{54} + 40 q^{56} - 42 q^{59} - 16 q^{61} - 12 q^{64} + 2 q^{66} - 74 q^{69} + 28 q^{71} - 4 q^{74} - 114 q^{76} + 62 q^{79} + 124 q^{81} - 42 q^{84} - 2 q^{86} - 80 q^{89} + 22 q^{91} - 70 q^{94} - 100 q^{96} + 106 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 23x^{10} + 205x^{8} + 890x^{6} + 1942x^{4} + 1980x^{2} + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{11} - 107\nu^{9} - 436\nu^{7} + 142\nu^{5} + 2957\nu^{3} + 2961\nu ) / 405 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} + 23\nu^{9} + 205\nu^{7} + 890\nu^{5} + 1861\nu^{3} + 1332\nu ) / 81 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{10} + 74\nu^{8} + 487\nu^{6} + 1346\nu^{4} + 1396\nu^{2} + 333 ) / 45 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -17\nu^{11} - 337\nu^{9} - 2486\nu^{7} - 8353\nu^{5} - 12008\nu^{3} - 4284\nu ) / 405 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -17\nu^{11} - 337\nu^{9} - 2486\nu^{7} - 8353\nu^{5} - 12413\nu^{3} - 6309\nu ) / 405 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 4\nu^{10} + 74\nu^{8} + 487\nu^{6} + 1391\nu^{4} + 1756\nu^{2} + 828 ) / 45 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 14\nu^{11} + 349\nu^{9} + 3167\nu^{7} + 12811\nu^{5} + 22301\nu^{3} + 12843\nu ) / 405 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -2\nu^{10} - 37\nu^{8} - 251\nu^{6} - 763\nu^{4} - 1013\nu^{2} - 459 ) / 15 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{10} + 26\nu^{8} + 238\nu^{6} + 944\nu^{4} + 1564\nu^{2} + 822 ) / 15 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} - \beta_{5} - 8\beta_{2} + 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{7} - 8\beta_{6} + 2\beta_{4} - \beta_{3} + 30\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{10} - 12\beta_{8} + 9\beta_{5} + 54\beta_{2} - 123 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -2\beta_{9} - 68\beta_{7} + 54\beta_{6} - 21\beta_{4} + 15\beta_{3} - 189\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2\beta_{11} + 31\beta_{10} + 105\beta_{8} - 60\beta_{5} - 351\beta_{2} + 755 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 37\beta_{9} + 492\beta_{7} - 351\beta_{6} + 163\beta_{4} - 152\beta_{3} + 1209\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -37\beta_{11} - 330\beta_{10} - 818\beta_{8} + 362\beta_{5} + 2262\beta_{2} - 4746 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -441\beta_{9} - 3525\beta_{7} + 2262\beta_{6} - 1143\beta_{4} + 1311\beta_{3} - 7789\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1075\mathbb{Z}\right)^\times\).

\(n\) \(302\) \(476\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
474.1
2.60821i
2.47811i
2.22264i
1.66064i
1.22887i
0.920974i
0.920974i
1.22887i
1.66064i
2.22264i
2.47811i
2.60821i
2.60821i 0.327982i −4.80276 0 −0.855447 0.194547i 7.31018i 2.89243 0
474.2 2.47811i 0.974899i −4.14105 0 2.41591 4.61916i 5.30576i 2.04957 0
474.3 2.22264i 2.75173i −2.94015 0 6.11612 1.28250i 2.08961i −4.57203 0
474.4 1.66064i 3.28593i −0.757722 0 5.45675 0.418362i 2.06297i −7.79735 0
474.5 1.22887i 0.651450i 0.489869 0 0.800550 3.71874i 3.05974i 2.57561 0
474.6 0.920974i 3.18563i 1.15181 0 −2.93388 2.23083i 2.90273i −7.14823 0
474.7 0.920974i 3.18563i 1.15181 0 −2.93388 2.23083i 2.90273i −7.14823 0
474.8 1.22887i 0.651450i 0.489869 0 0.800550 3.71874i 3.05974i 2.57561 0
474.9 1.66064i 3.28593i −0.757722 0 5.45675 0.418362i 2.06297i −7.79735 0
474.10 2.22264i 2.75173i −2.94015 0 6.11612 1.28250i 2.08961i −4.57203 0
474.11 2.47811i 0.974899i −4.14105 0 2.41591 4.61916i 5.30576i 2.04957 0
474.12 2.60821i 0.327982i −4.80276 0 −0.855447 0.194547i 7.31018i 2.89243 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 474.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1075.2.b.j 12
5.b even 2 1 inner 1075.2.b.j 12
5.c odd 4 1 1075.2.a.q 6
5.c odd 4 1 1075.2.a.r yes 6
15.e even 4 1 9675.2.a.cj 6
15.e even 4 1 9675.2.a.ck 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1075.2.a.q 6 5.c odd 4 1
1075.2.a.r yes 6 5.c odd 4 1
1075.2.b.j 12 1.a even 1 1 trivial
1075.2.b.j 12 5.b even 2 1 inner
9675.2.a.cj 6 15.e even 4 1
9675.2.a.ck 6 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1075, [\chi])\):

\( T_{2}^{12} + 23T_{2}^{10} + 205T_{2}^{8} + 890T_{2}^{6} + 1942T_{2}^{4} + 1980T_{2}^{2} + 729 \) Copy content Toggle raw display
\( T_{3}^{12} + 30T_{3}^{10} + 311T_{3}^{8} + 1243T_{3}^{6} + 1379T_{3}^{4} + 469T_{3}^{2} + 36 \) Copy content Toggle raw display
\( T_{7}^{12} + 42T_{7}^{10} + 545T_{7}^{8} + 2356T_{7}^{6} + 2896T_{7}^{4} + 529T_{7}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 23 T^{10} + 205 T^{8} + \cdots + 729 \) Copy content Toggle raw display
$3$ \( T^{12} + 30 T^{10} + 311 T^{8} + \cdots + 36 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 42 T^{10} + 545 T^{8} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{6} + 6 T^{5} - 24 T^{4} - 133 T^{3} + \cdots - 79)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 66 T^{10} + 1365 T^{8} + \cdots + 19321 \) Copy content Toggle raw display
$17$ \( T^{12} + 107 T^{10} + 4139 T^{8} + \cdots + 1454436 \) Copy content Toggle raw display
$19$ \( (T^{6} - 5 T^{5} - 39 T^{4} + 145 T^{3} + \cdots - 1440)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 179 T^{10} + \cdots + 212838921 \) Copy content Toggle raw display
$29$ \( (T^{6} + 18 T^{5} + 68 T^{4} - 179 T^{3} + \cdots - 80)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 12 T^{5} - 43 T^{4} - 1061 T^{3} + \cdots + 13863)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 229 T^{10} + 16272 T^{8} + \cdots + 8282884 \) Copy content Toggle raw display
$41$ \( (T^{6} - 132 T^{4} + 113 T^{3} + \cdots - 5009)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$47$ \( T^{12} + 332 T^{10} + 30494 T^{8} + \cdots + 168921 \) Copy content Toggle raw display
$53$ \( T^{12} + 348 T^{10} + \cdots + 1232782321 \) Copy content Toggle raw display
$59$ \( (T^{6} + 21 T^{5} + 129 T^{4} + 143 T^{3} + \cdots - 75)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 8 T^{5} - 169 T^{4} + \cdots + 104304)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 242 T^{10} + \cdots + 194100624 \) Copy content Toggle raw display
$71$ \( (T^{6} - 14 T^{5} - 64 T^{4} + 427 T^{3} + \cdots - 474)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 383 T^{10} + 35603 T^{8} + \cdots + 2735716 \) Copy content Toggle raw display
$79$ \( (T^{6} - 31 T^{5} + 59 T^{4} + \cdots - 979905)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 701 T^{10} + \cdots + 69144280209 \) Copy content Toggle raw display
$89$ \( (T^{6} + 40 T^{5} + 427 T^{4} + \cdots - 46440)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 625 T^{10} + \cdots + 239200224561 \) Copy content Toggle raw display
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