Properties

Label 13.3.f.a
Level $13$
Weight $3$
Character orbit 13.f
Analytic conductor $0.354$
Analytic rank $0$
Dimension $4$
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] = [13,3,Mod(2,13)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(13, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.2");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 13.f (of order \(12\), degree \(4\), 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: \(0.354224343668\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - 1) q^{2} + (\zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{3} + ( - \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{4} + (3 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} - 3) q^{5} + (2 \zeta_{12}^{3} - \zeta_{12}^{2} - 3 \zeta_{12} + 3) q^{6} + ( - 6 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 2 \zeta_{12} + 2) q^{7} + ( - 4 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 5 \zeta_{12} - 4) q^{8} + ( - 4 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 2 \zeta_{12} + 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - 1) q^{2} + (\zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{3} + ( - \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{4} + (3 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} - 3) q^{5} + (2 \zeta_{12}^{3} - \zeta_{12}^{2} - 3 \zeta_{12} + 3) q^{6} + ( - 6 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 2 \zeta_{12} + 2) q^{7} + ( - 4 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 5 \zeta_{12} - 4) q^{8} + ( - 4 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 2 \zeta_{12} + 5) q^{9} + (5 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 5 \zeta_{12} + 8) q^{10} + (10 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 6 \zeta_{12} - 4) q^{11} + ( - \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{12} - 13 \zeta_{12}^{2} q^{13} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12} - 10) q^{14} + ( - 9 \zeta_{12}^{3} + 9 \zeta_{12}^{2} + \zeta_{12} - 8) q^{15} + (8 \zeta_{12}^{3} - \zeta_{12}^{2} + 8 \zeta_{12}) q^{16} + ( - 2 \zeta_{12}^{2} - 15 \zeta_{12} - 2) q^{17} + (2 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 3 \zeta_{12} - 2) q^{18} + (4 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - \zeta_{12} + 1) q^{19} + ( - 5 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 8 \zeta_{12} + 8) q^{20} + (14 \zeta_{12}^{3} - 8 \zeta_{12}^{2} - 8 \zeta_{12} + 14) q^{21} + ( - 12 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 6 \zeta_{12} - 2) q^{22} + ( - 15 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 15 \zeta_{12} + 6) q^{23} + (5 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 13 \zeta_{12} + 8) q^{24} + ( - \zeta_{12}^{3} + 14 \zeta_{12}^{2} - 7) q^{25} + (13 \zeta_{12}^{3} - 13 \zeta_{12} + 13) q^{26} + ( - 12 \zeta_{12}^{3} + 24 \zeta_{12} - 8) q^{27} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 12 \zeta_{12} - 10) q^{28} + ( - 6 \zeta_{12}^{3} + \zeta_{12}^{2} - 6 \zeta_{12}) q^{29} + ( - 9 \zeta_{12}^{2} + 17 \zeta_{12} - 9) q^{30} + (9 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 8 \zeta_{12} - 9) q^{31} + ( - 11 \zeta_{12}^{3} - 12 \zeta_{12}^{2} - \zeta_{12} + 1) q^{32} + (12 \zeta_{12}^{3} - 8 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{33} + ( - 11 \zeta_{12}^{3} + 13 \zeta_{12}^{2} + 13 \zeta_{12} - 11) q^{34} + (44 \zeta_{12}^{3} - 16 \zeta_{12}^{2} - 22 \zeta_{12} + 16) q^{35} + (16 \zeta_{12}^{3} + 9 \zeta_{12}^{2} - 16 \zeta_{12} - 18) q^{36} + ( - 24 \zeta_{12}^{3} - 24 \zeta_{12}^{2} + 29 \zeta_{12} - 5) q^{37} + ( - 5 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{38} + ( - 26 \zeta_{12}^{3} + 13 \zeta_{12}^{2} + 13 \zeta_{12} - 13) q^{39} + (19 \zeta_{12}^{3} - 38 \zeta_{12} + 18) q^{40} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 23 \zeta_{12} + 27) q^{41} + ( - 14 \zeta_{12}^{3} + 22 \zeta_{12}^{2} - 14 \zeta_{12}) q^{42} + (30 \zeta_{12}^{2} + 30) q^{43} + ( - 34 \zeta_{12}^{3} - 24 \zeta_{12}^{2} + 24 \zeta_{12} + 34) q^{44} + (9 \zeta_{12}^{3} + 19 \zeta_{12}^{2} + 10 \zeta_{12} - 10) q^{45} + (12 \zeta_{12}^{3} - 9 \zeta_{12}^{2} - 3 \zeta_{12} - 3) q^{46} + ( - 7 \zeta_{12}^{3} - 20 \zeta_{12}^{2} - 20 \zeta_{12} - 7) q^{47} + ( - 18 \zeta_{12}^{3} + 25 \zeta_{12}^{2} + 9 \zeta_{12} - 25) q^{48} + ( - 7 \zeta_{12}^{3} + 12 \zeta_{12}^{2} + 7 \zeta_{12} - 24) q^{49} + ( - 7 \zeta_{12}^{3} - 7 \zeta_{12}^{2} + 15 \zeta_{12} - 8) q^{50} + (9 \zeta_{12}^{3} - 26 \zeta_{12}^{2} + 13) q^{51} + (26 \zeta_{12}^{3} + 26 \zeta_{12}^{2} - 13) q^{52} + (27 \zeta_{12}^{3} - 54 \zeta_{12} + 32) q^{53} + (32 \zeta_{12}^{3} - 32 \zeta_{12}^{2} - 12 \zeta_{12} + 20) q^{54} + ( - 6 \zeta_{12}^{3} - 50 \zeta_{12}^{2} - 6 \zeta_{12}) q^{55} + ( - 14 \zeta_{12}^{2} + 48 \zeta_{12} - 14) q^{56} + (4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} - 4) q^{57} + ( - 7 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 13 \zeta_{12} - 13) q^{58} + (22 \zeta_{12}^{3} + 38 \zeta_{12}^{2} - 60 \zeta_{12} - 60) q^{59} + ( - \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 6 \zeta_{12} - 1) q^{60} + ( - 42 \zeta_{12}^{3} + 62 \zeta_{12}^{2} + 21 \zeta_{12} - 62) q^{61} + ( - 7 \zeta_{12}^{3} - \zeta_{12}^{2} + 7 \zeta_{12} + 2) q^{62} + ( - 26 \zeta_{12}^{3} - 26 \zeta_{12}^{2} + 26) q^{63} + (14 \zeta_{12}^{3} - 62 \zeta_{12}^{2} + 31) q^{64} + ( - 52 \zeta_{12}^{3} + 52 \zeta_{12}^{2} + 39 \zeta_{12} - 13) q^{65} + (8 \zeta_{12}^{3} - 16 \zeta_{12} + 20) q^{66} + (37 \zeta_{12}^{3} - 37 \zeta_{12}^{2} + 11 \zeta_{12} + 48) q^{67} + (19 \zeta_{12}^{3} + 36 \zeta_{12}^{2} + 19 \zeta_{12}) q^{68} + (12 \zeta_{12}^{2} - 6 \zeta_{12} + 12) q^{69} + ( - 22 \zeta_{12}^{3} + 38 \zeta_{12}^{2} - 38 \zeta_{12} + 22) q^{70} + ( - 32 \zeta_{12}^{3} - 21 \zeta_{12}^{2} + 11 \zeta_{12} - 11) q^{71} + ( - 19 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 17 \zeta_{12} + 17) q^{72} + (34 \zeta_{12}^{3} - 39 \zeta_{12}^{2} - 39 \zeta_{12} + 34) q^{73} + (58 \zeta_{12}^{3} - 34 \zeta_{12}^{2} - 29 \zeta_{12} + 34) q^{74} + (22 \zeta_{12}^{3} - 8 \zeta_{12}^{2} - 22 \zeta_{12} + 16) q^{75} + ( - 13 \zeta_{12}^{3} - 13 \zeta_{12}^{2} + 3 \zeta_{12} + 10) q^{76} + (20 \zeta_{12}^{3} + 88 \zeta_{12}^{2} - 44) q^{77} + (13 \zeta_{12}^{3} - 26 \zeta_{12}^{2} + 26 \zeta_{12} - 13) q^{78} + ( - 42 \zeta_{12}^{3} + 84 \zeta_{12} - 10) q^{79} + ( - 44 \zeta_{12}^{3} + 44 \zeta_{12}^{2} - 13 \zeta_{12} - 57) q^{80} + ( - 38 \zeta_{12}^{3} - \zeta_{12}^{2} - 38 \zeta_{12}) q^{81} + (4 \zeta_{12}^{2} - 31 \zeta_{12} + 4) q^{82} + (34 \zeta_{12}^{3} - 26 \zeta_{12}^{2} + 26 \zeta_{12} - 34) q^{83} + ( - 4 \zeta_{12}^{3} - 10 \zeta_{12}^{2} - 6 \zeta_{12} + 6) q^{84} + (\zeta_{12}^{3} - 50 \zeta_{12}^{2} + 49 \zeta_{12} + 49) q^{85} + ( - 60 \zeta_{12}^{3} + 30 \zeta_{12}^{2} + 30 \zeta_{12} - 60) q^{86} + (14 \zeta_{12}^{3} - 19 \zeta_{12}^{2} - 7 \zeta_{12} + 19) q^{87} + (6 \zeta_{12}^{3} + 34 \zeta_{12}^{2} - 6 \zeta_{12} - 68) q^{88} + (45 \zeta_{12}^{3} + 45 \zeta_{12}^{2} + 5 \zeta_{12} - 50) q^{89} + (\zeta_{12}^{3} - 20 \zeta_{12}^{2} + 10) q^{90} + (52 \zeta_{12}^{3} - 78 \zeta_{12}^{2} - 78 \zeta_{12} + 52) q^{91} + (21 \zeta_{12}^{3} - 42 \zeta_{12} - 39) q^{92} + (6 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 8 \zeta_{12} - 2) q^{93} + (7 \zeta_{12}^{3} + 13 \zeta_{12}^{2} + 7 \zeta_{12}) q^{94} + ( - 13 \zeta_{12}^{2} - \zeta_{12} - 13) q^{95} + ( - 11 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 11) q^{96} + (54 \zeta_{12}^{3} + 19 \zeta_{12}^{2} - 35 \zeta_{12} + 35) q^{97} + (19 \zeta_{12}^{3} - 31 \zeta_{12}^{2} + 12 \zeta_{12} + 12) q^{98} + (26 \zeta_{12}^{3} + 52 \zeta_{12}^{2} + 52 \zeta_{12} + 26) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{3} - 6 q^{4} - 14 q^{5} + 10 q^{6} + 16 q^{7} - 6 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 2 q^{3} - 6 q^{4} - 14 q^{5} + 10 q^{6} + 16 q^{7} - 6 q^{8} + 10 q^{9} + 24 q^{10} + 4 q^{11} - 26 q^{13} - 40 q^{14} - 14 q^{15} - 2 q^{16} - 12 q^{17} - 2 q^{18} + 10 q^{19} + 26 q^{20} + 40 q^{21} - 4 q^{22} + 18 q^{23} + 42 q^{24} + 52 q^{26} - 32 q^{27} - 44 q^{28} + 2 q^{29} - 54 q^{30} - 20 q^{31} - 20 q^{32} - 32 q^{33} - 18 q^{34} + 32 q^{35} - 54 q^{36} - 68 q^{37} - 26 q^{39} + 72 q^{40} + 100 q^{41} + 44 q^{42} + 180 q^{43} + 88 q^{44} - 2 q^{45} - 30 q^{46} - 68 q^{47} - 50 q^{48} - 72 q^{49} - 46 q^{50} + 128 q^{53} + 16 q^{54} - 100 q^{55} - 84 q^{56} - 20 q^{57} - 40 q^{58} - 164 q^{59} + 8 q^{60} - 124 q^{61} + 6 q^{62} + 52 q^{63} + 52 q^{65} + 80 q^{66} + 118 q^{67} + 72 q^{68} + 72 q^{69} + 164 q^{70} - 86 q^{71} + 72 q^{72} + 58 q^{73} + 68 q^{74} + 48 q^{75} + 14 q^{76} - 104 q^{78} - 40 q^{79} - 140 q^{80} - 2 q^{81} + 24 q^{82} - 188 q^{83} + 4 q^{84} + 96 q^{85} - 180 q^{86} + 38 q^{87} - 204 q^{88} - 110 q^{89} + 52 q^{91} - 156 q^{92} - 20 q^{93} + 26 q^{94} - 78 q^{95} + 40 q^{96} + 178 q^{97} - 14 q^{98} + 208 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{12}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
−0.500000 0.133975i 0.366025 + 0.633975i −3.23205 1.86603i −2.63397 + 2.63397i −0.0980762 0.366025i 5.73205 1.53590i 2.83013 + 2.83013i 4.23205 7.33013i 1.66987 0.964102i
6.1 −0.500000 1.86603i −1.36603 + 2.36603i 0.232051 0.133975i −4.36603 + 4.36603i 5.09808 + 1.36603i 2.26795 8.46410i −5.83013 5.83013i 0.767949 + 1.33013i 10.3301 + 5.96410i
7.1 −0.500000 + 0.133975i 0.366025 0.633975i −3.23205 + 1.86603i −2.63397 2.63397i −0.0980762 + 0.366025i 5.73205 + 1.53590i 2.83013 2.83013i 4.23205 + 7.33013i 1.66987 + 0.964102i
11.1 −0.500000 + 1.86603i −1.36603 2.36603i 0.232051 + 0.133975i −4.36603 4.36603i 5.09808 1.36603i 2.26795 + 8.46410i −5.83013 + 5.83013i 0.767949 1.33013i 10.3301 5.96410i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.f odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.3.f.a 4
3.b odd 2 1 117.3.bd.b 4
4.b odd 2 1 208.3.bd.d 4
5.b even 2 1 325.3.t.a 4
5.c odd 4 1 325.3.w.a 4
5.c odd 4 1 325.3.w.b 4
13.b even 2 1 169.3.f.b 4
13.c even 3 1 169.3.d.a 4
13.c even 3 1 169.3.f.c 4
13.d odd 4 1 169.3.f.a 4
13.d odd 4 1 169.3.f.c 4
13.e even 6 1 169.3.d.c 4
13.e even 6 1 169.3.f.a 4
13.f odd 12 1 inner 13.3.f.a 4
13.f odd 12 1 169.3.d.a 4
13.f odd 12 1 169.3.d.c 4
13.f odd 12 1 169.3.f.b 4
39.k even 12 1 117.3.bd.b 4
52.l even 12 1 208.3.bd.d 4
65.o even 12 1 325.3.w.b 4
65.s odd 12 1 325.3.t.a 4
65.t even 12 1 325.3.w.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.3.f.a 4 1.a even 1 1 trivial
13.3.f.a 4 13.f odd 12 1 inner
117.3.bd.b 4 3.b odd 2 1
117.3.bd.b 4 39.k even 12 1
169.3.d.a 4 13.c even 3 1
169.3.d.a 4 13.f odd 12 1
169.3.d.c 4 13.e even 6 1
169.3.d.c 4 13.f odd 12 1
169.3.f.a 4 13.d odd 4 1
169.3.f.a 4 13.e even 6 1
169.3.f.b 4 13.b even 2 1
169.3.f.b 4 13.f odd 12 1
169.3.f.c 4 13.c even 3 1
169.3.f.c 4 13.d odd 4 1
208.3.bd.d 4 4.b odd 2 1
208.3.bd.d 4 52.l even 12 1
325.3.t.a 4 5.b even 2 1
325.3.t.a 4 65.s odd 12 1
325.3.w.a 4 5.c odd 4 1
325.3.w.a 4 65.t even 12 1
325.3.w.b 4 5.c odd 4 1
325.3.w.b 4 65.o even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(13, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + 5 T^{2} + 4 T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + 6 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$5$ \( T^{4} + 14 T^{3} + 98 T^{2} + \cdots + 529 \) Copy content Toggle raw display
$7$ \( T^{4} - 16 T^{3} + 164 T^{2} + \cdots + 2704 \) Copy content Toggle raw display
$11$ \( T^{4} - 4 T^{3} + 200 T^{2} + \cdots + 10816 \) Copy content Toggle raw display
$13$ \( (T^{2} + 13 T + 169)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 12 T^{3} - 165 T^{2} + \cdots + 45369 \) Copy content Toggle raw display
$19$ \( T^{4} - 10 T^{3} + 74 T^{2} + \cdots + 484 \) Copy content Toggle raw display
$23$ \( T^{4} - 18 T^{3} - 90 T^{2} + \cdots + 39204 \) Copy content Toggle raw display
$29$ \( T^{4} - 2 T^{3} + 111 T^{2} + \cdots + 11449 \) Copy content Toggle raw display
$31$ \( T^{4} + 20 T^{3} + 200 T^{2} + \cdots + 2116 \) Copy content Toggle raw display
$37$ \( T^{4} + 68 T^{3} + 1517 T^{2} + \cdots + 1868689 \) Copy content Toggle raw display
$41$ \( T^{4} - 100 T^{3} + 3461 T^{2} + \cdots + 833569 \) Copy content Toggle raw display
$43$ \( (T^{2} - 90 T + 2700)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 68 T^{3} + 2312 T^{2} + \cdots + 484 \) Copy content Toggle raw display
$53$ \( (T^{2} - 64 T - 1163)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 164 T^{3} + \cdots + 16613776 \) Copy content Toggle raw display
$61$ \( T^{4} + 124 T^{3} + 12855 T^{2} + \cdots + 6355441 \) Copy content Toggle raw display
$67$ \( T^{4} - 118 T^{3} + 10706 T^{2} + \cdots + 9721924 \) Copy content Toggle raw display
$71$ \( T^{4} + 86 T^{3} + 4658 T^{2} + \cdots + 2208196 \) Copy content Toggle raw display
$73$ \( T^{4} - 58 T^{3} + 1682 T^{2} + \cdots + 3463321 \) Copy content Toggle raw display
$79$ \( (T^{2} + 20 T - 5192)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 188 T^{3} + \cdots + 11587216 \) Copy content Toggle raw display
$89$ \( T^{4} + 110 T^{3} + 12050 T^{2} + \cdots + 8702500 \) Copy content Toggle raw display
$97$ \( T^{4} - 178 T^{3} + \cdots + 18028516 \) Copy content Toggle raw display
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