Properties

Label 725.2.r.b
Level $725$
Weight $2$
Character orbit 725.r
Analytic conductor $5.789$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [725,2,Mod(24,725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(725, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("725.24");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 725.r (of order \(14\), degree \(6\), 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: \(5.78915414654\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{14})\)
Coefficient field: \(\Q(\zeta_{28})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 29)
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-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 \(\zeta_{28}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{28}^{9} - \zeta_{28}^{3}) q^{2} + ( - \zeta_{28}^{11} - \zeta_{28}^{7}) q^{3} + (\zeta_{28}^{6} - \zeta_{28}^{4}) q^{4} + (\zeta_{28}^{10} + \zeta_{28}^{6} + \zeta_{28}^{2} - 1) q^{6} + (\zeta_{28}^{11} - 2 \zeta_{28}^{9} + \zeta_{28}^{7} + 2 \zeta_{28}^{3} - 2 \zeta_{28}) q^{7} + ( - \zeta_{28}^{11} - 2 \zeta_{28}^{7} + \zeta_{28}^{5} - \zeta_{28}^{3} + 2 \zeta_{28}) q^{8} + ( - \zeta_{28}^{8} + \zeta_{28}^{4} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{28}^{9} - \zeta_{28}^{3}) q^{2} + ( - \zeta_{28}^{11} - \zeta_{28}^{7}) q^{3} + (\zeta_{28}^{6} - \zeta_{28}^{4}) q^{4} + (\zeta_{28}^{10} + \zeta_{28}^{6} + \zeta_{28}^{2} - 1) q^{6} + (\zeta_{28}^{11} - 2 \zeta_{28}^{9} + \zeta_{28}^{7} + 2 \zeta_{28}^{3} - 2 \zeta_{28}) q^{7} + ( - \zeta_{28}^{11} - 2 \zeta_{28}^{7} + \zeta_{28}^{5} - \zeta_{28}^{3} + 2 \zeta_{28}) q^{8} + ( - \zeta_{28}^{8} + \zeta_{28}^{4} - 1) q^{9} + (3 \zeta_{28}^{6} - 2 \zeta_{28}^{4} + 2 \zeta_{28}^{2} - 3) q^{11} + (\zeta_{28}^{9} - \zeta_{28}^{7} + \zeta_{28}^{5}) q^{12} + (3 \zeta_{28}^{11} - 3 \zeta_{28}^{9} + \zeta_{28}^{5} + \zeta_{28}^{3} + \zeta_{28}) q^{13} + (\zeta_{28}^{10} - 4 \zeta_{28}^{8} + \zeta_{28}^{6} + 3 \zeta_{28}^{2} - 3) q^{14} + (3 \zeta_{28}^{10} + \zeta_{28}^{6} - 3 \zeta_{28}^{4} + 3 \zeta_{28}^{2} - 1) q^{16} + (2 \zeta_{28}^{9} - 2 \zeta_{28}^{7} + 2 \zeta_{28}^{5}) q^{17} + (2 \zeta_{28}^{11} - 2 \zeta_{28}^{9} - \zeta_{28}^{5} + 3 \zeta_{28}^{3} - \zeta_{28}) q^{18} + (\zeta_{28}^{10} + 2 \zeta_{28}^{6} - 2 \zeta_{28}^{4} - 1) q^{19} + (\zeta_{28}^{8} + 1) q^{21} + ( - 4 \zeta_{28}^{9} + \zeta_{28}^{3} - \zeta_{28}) q^{22} + ( - 2 \zeta_{28}^{11} + 3 \zeta_{28}^{9} - 3 \zeta_{28}^{7} + 2 \zeta_{28}^{5} + 4 \zeta_{28}) q^{23} + ( - 2 \zeta_{28}^{10} - 3 \zeta_{28}^{6} - 2 \zeta_{28}^{2}) q^{24} + (5 \zeta_{28}^{10} - 5 \zeta_{28}^{8} - 2 \zeta_{28}^{4} + 4 \zeta_{28}^{2} - 2) q^{26} + ( - 3 \zeta_{28}^{11} + \zeta_{28}^{7} - \zeta_{28}^{5} + 3 \zeta_{28}) q^{27} + (2 \zeta_{28}^{11} - \zeta_{28}^{9} - \zeta_{28}^{7} - \zeta_{28}^{5} + 2 \zeta_{28}^{3}) q^{28} + (2 \zeta_{28}^{10} - 2 \zeta_{28}^{8} + 6 \zeta_{28}^{6} - 2 \zeta_{28}^{4} - 3) q^{29} + ( - \zeta_{28}^{10} - \zeta_{28}^{8} + 5 \zeta_{28}^{6} - \zeta_{28}^{4} - \zeta_{28}^{2}) q^{31} + (\zeta_{28}^{11} + 2 \zeta_{28}^{9} + \zeta_{28}^{7} - 3 \zeta_{28}^{3} + 3 \zeta_{28}) q^{32} + (3 \zeta_{28}^{9} - 2 \zeta_{28}^{7} + 5 \zeta_{28}^{5} - 2 \zeta_{28}^{3} + 3 \zeta_{28}) q^{33} - 2 \zeta_{28}^{6} q^{34} + (2 \zeta_{28}^{10} - 2 \zeta_{28}^{8} + \zeta_{28}^{2}) q^{36} + ( - 3 \zeta_{28}^{7} + 2 \zeta_{28}^{5} - 2 \zeta_{28}^{3} + 3 \zeta_{28}) q^{37} + ( - 3 \zeta_{28}^{11} - \zeta_{28}^{7} + 2 \zeta_{28}^{5} - 2 \zeta_{28}^{3} + \zeta_{28}) q^{38} + ( - 3 \zeta_{28}^{10} + 4 \zeta_{28}^{8} - 5 \zeta_{28}^{6} + 5 \zeta_{28}^{4} - 4 \zeta_{28}^{2} + 3) q^{39} + ( - 2 \zeta_{28}^{10} + 2 \zeta_{28}^{4} + 4) q^{41} + ( - \zeta_{28}^{11} + \zeta_{28}^{9} - 2 \zeta_{28}^{3}) q^{42} + ( - 3 \zeta_{28}^{9} + 3 \zeta_{28}^{7} - 2 \zeta_{28}) q^{43} + ( - 2 \zeta_{28}^{10} + \zeta_{28}^{8} - 2 \zeta_{28}^{6} + 3 \zeta_{28}^{2} - 3) q^{44} + (4 \zeta_{28}^{10} + \zeta_{28}^{8} - \zeta_{28}^{6} - 4 \zeta_{28}^{4} - 1) q^{46} + (\zeta_{28}^{11} - \zeta_{28}^{9} - \zeta_{28}^{5} + 7 \zeta_{28}^{3} - \zeta_{28}) q^{47} + (\zeta_{28}^{9} + 4 \zeta_{28}^{5} + \zeta_{28}) q^{48} + (7 \zeta_{28}^{8} - 4 \zeta_{28}^{6} + 5 \zeta_{28}^{4} - 4 \zeta_{28}^{2} + 7) q^{49} + ( - 2 \zeta_{28}^{10} + 2 \zeta_{28}^{8} + 2 \zeta_{28}^{2}) q^{51} + (4 \zeta_{28}^{11} - 3 \zeta_{28}^{9} + 3 \zeta_{28}^{7} - 4 \zeta_{28}^{5} + 3 \zeta_{28}) q^{52} + (2 \zeta_{28}^{11} - 4 \zeta_{28}^{9} + 3 \zeta_{28}^{7} - 3 \zeta_{28}^{5} + 4 \zeta_{28}^{3} - 2 \zeta_{28}) q^{53} + (2 \zeta_{28}^{10} + \zeta_{28}^{8} + 3 \zeta_{28}^{6} - 3 \zeta_{28}^{4} - \zeta_{28}^{2} - 2) q^{54} + ( - 2 \zeta_{28}^{10} + 2 \zeta_{28}^{8} + \zeta_{28}^{6} + 2 \zeta_{28}^{4} - 2 \zeta_{28}^{2}) q^{56} + (\zeta_{28}^{11} + 2 \zeta_{28}^{9} + 2 \zeta_{28}^{5} + \zeta_{28}^{3}) q^{57} + ( - 2 \zeta_{28}^{11} - 5 \zeta_{28}^{9} - 2 \zeta_{28}^{7} + 2 \zeta_{28}^{5} + \zeta_{28}^{3} - 2 \zeta_{28}) q^{58} + (2 \zeta_{28}^{10} - 2 \zeta_{28}^{8} + 2 \zeta_{28}^{6} - 2 \zeta_{28}^{4} + 8) q^{59} + (2 \zeta_{28}^{10} - 2 \zeta_{28}^{8} - \zeta_{28}^{4} + 4 \zeta_{28}^{2} - 1) q^{61} + ( - 5 \zeta_{28}^{9} + \zeta_{28}^{7} + 2 \zeta_{28}^{5} + \zeta_{28}^{3} - 5 \zeta_{28}) q^{62} + ( - 4 \zeta_{28}^{11} + 6 \zeta_{28}^{9} - \zeta_{28}^{7} + \zeta_{28}^{5} - 6 \zeta_{28}^{3} + 4 \zeta_{28}) q^{63} + ( - 5 \zeta_{28}^{10} + \zeta_{28}^{8} - 5 \zeta_{28}^{6} - 2 \zeta_{28}^{2} + 2) q^{64} + ( - 3 \zeta_{28}^{6} - \zeta_{28}^{4} - 3 \zeta_{28}^{2}) q^{66} + ( - 8 \zeta_{28}^{11} - 7 \zeta_{28}^{7} + 4 \zeta_{28}^{5} - 4 \zeta_{28}^{3} + 7 \zeta_{28}) q^{67} + (2 \zeta_{28}^{9} - 4 \zeta_{28}^{7} + 4 \zeta_{28}^{5} - 4 \zeta_{28}^{3} + 2 \zeta_{28}) q^{68} + ( - 6 \zeta_{28}^{10} - 3 \zeta_{28}^{6} + \zeta_{28}^{4} - \zeta_{28}^{2} + 3) q^{69} + (6 \zeta_{28}^{10} - 3 \zeta_{28}^{6} + 3) q^{71} + ( - \zeta_{28}^{11} + \zeta_{28}^{5} - 2 \zeta_{28}) q^{72} + (4 \zeta_{28}^{11} + 3 \zeta_{28}^{9} - 3 \zeta_{28}^{7} - 4 \zeta_{28}^{5} - 2 \zeta_{28}) q^{73} + (4 \zeta_{28}^{10} - \zeta_{28}^{4} + \zeta_{28}^{2}) q^{74} + ( - 2 \zeta_{28}^{10} + \zeta_{28}^{6} - \zeta_{28}^{4} + \zeta_{28}^{2} - 1) q^{76} + (5 \zeta_{28}^{9} - 4 \zeta_{28}^{7} - \zeta_{28}^{5} - 4 \zeta_{28}^{3} + 5 \zeta_{28}) q^{77} + (3 \zeta_{28}^{7} - \zeta_{28}^{5} + \zeta_{28}^{3} - 3 \zeta_{28}) q^{78} + (3 \zeta_{28}^{8} + 6 \zeta_{28}^{4} + 3) q^{79} + ( - 5 \zeta_{28}^{10} + 2 \zeta_{28}^{8} - 5 \zeta_{28}^{6} - 6 \zeta_{28}^{2} + 6) q^{81} + (4 \zeta_{28}^{11} + 2 \zeta_{28}^{7} - 2 \zeta_{28}^{5} - 4 \zeta_{28}) q^{82} + ( - 4 \zeta_{28}^{9} + 5 \zeta_{28}^{7} + 5 \zeta_{28}^{3} - 4 \zeta_{28}) q^{83} + ( - \zeta_{28}^{10} + \zeta_{28}^{8} - \zeta_{28}^{2}) q^{84} + ( - 2 \zeta_{28}^{10} - 3 \zeta_{28}^{8} + 3 \zeta_{28}^{6} + 2 \zeta_{28}^{4} - 3) q^{86} + ( - \zeta_{28}^{11} + 6 \zeta_{28}^{9} - \zeta_{28}^{7} + 4 \zeta_{28}^{5} + 2 \zeta_{28}^{3} + 2 \zeta_{28}) q^{87} + (2 \zeta_{28}^{11} - \zeta_{28}^{9} + 8 \zeta_{28}^{7} - \zeta_{28}^{5} + 2 \zeta_{28}^{3}) q^{88} + (\zeta_{28}^{10} + 5 \zeta_{28}^{8} + \zeta_{28}^{6} + 5 \zeta_{28}^{4} + \zeta_{28}^{2}) q^{89} + ( - 7 \zeta_{28}^{10} + 12 \zeta_{28}^{8} - 3 \zeta_{28}^{6} + 3 \zeta_{28}^{4} - 12 \zeta_{28}^{2} + \cdots + 7) q^{91} + \cdots + (3 \zeta_{28}^{10} - \zeta_{28}^{8} + \zeta_{28}^{6} - 3 \zeta_{28}^{4} + 4) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{4} - 6 q^{6} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{4} - 6 q^{6} - 12 q^{9} - 22 q^{11} - 18 q^{14} + 8 q^{16} - 2 q^{19} + 10 q^{21} - 14 q^{24} + 8 q^{26} - 12 q^{29} + 10 q^{31} - 4 q^{34} + 10 q^{36} - 6 q^{39} + 40 q^{41} - 40 q^{44} + 44 q^{49} - 4 q^{51} - 12 q^{54} - 14 q^{56} + 112 q^{59} + 6 q^{61} - 2 q^{64} - 10 q^{66} + 14 q^{69} + 42 q^{71} + 12 q^{74} - 10 q^{76} + 18 q^{79} + 36 q^{81} - 6 q^{84} - 32 q^{86} - 14 q^{89} + 10 q^{91} - 16 q^{94} - 4 q^{96} + 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/725\mathbb{Z}\right)^\times\).

\(n\) \(176\) \(552\)
\(\chi(n)\) \(-1 + \zeta_{28}^{2} - \zeta_{28}^{4} + \zeta_{28}^{6} - \zeta_{28}^{8} + \zeta_{28}^{10}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1
−0.781831 0.623490i
0.781831 + 0.623490i
−0.433884 + 0.900969i
0.433884 0.900969i
−0.433884 0.900969i
0.433884 + 0.900969i
0.974928 0.222521i
−0.974928 + 0.222521i
−0.781831 + 0.623490i
0.781831 0.623490i
0.974928 + 0.222521i
−0.974928 0.222521i
−1.40881 + 1.12349i 0.433884 0.0990311i 0.277479 1.21572i 0 −0.500000 + 0.626980i 3.94740 0.900969i −0.588735 1.22252i −2.52446 + 1.21572i 0
24.2 1.40881 1.12349i −0.433884 + 0.0990311i 0.277479 1.21572i 0 −0.500000 + 0.626980i −3.94740 + 0.900969i 0.588735 + 1.22252i −2.52446 + 1.21572i 0
49.1 −0.193096 0.400969i 0.974928 0.777479i 1.12349 1.40881i 0 −0.500000 0.240787i 0.279032 0.222521i −1.64960 0.376510i −0.321552 + 1.40881i 0
49.2 0.193096 + 0.400969i −0.974928 + 0.777479i 1.12349 1.40881i 0 −0.500000 0.240787i −0.279032 + 0.222521i 1.64960 + 0.376510i −0.321552 + 1.40881i 0
74.1 −0.193096 + 0.400969i 0.974928 + 0.777479i 1.12349 + 1.40881i 0 −0.500000 + 0.240787i 0.279032 + 0.222521i −1.64960 + 0.376510i −0.321552 1.40881i 0
74.2 0.193096 0.400969i −0.974928 0.777479i 1.12349 + 1.40881i 0 −0.500000 + 0.240787i −0.279032 0.222521i 1.64960 0.376510i −0.321552 1.40881i 0
199.1 −1.21572 0.277479i 0.781831 + 1.62349i −0.400969 0.193096i 0 −0.500000 2.19064i −0.300257 0.623490i 2.38374 + 1.90097i −0.153989 + 0.193096i 0
199.2 1.21572 + 0.277479i −0.781831 1.62349i −0.400969 0.193096i 0 −0.500000 2.19064i 0.300257 + 0.623490i −2.38374 1.90097i −0.153989 + 0.193096i 0
574.1 −1.40881 1.12349i 0.433884 + 0.0990311i 0.277479 + 1.21572i 0 −0.500000 0.626980i 3.94740 + 0.900969i −0.588735 + 1.22252i −2.52446 1.21572i 0
574.2 1.40881 + 1.12349i −0.433884 0.0990311i 0.277479 + 1.21572i 0 −0.500000 0.626980i −3.94740 0.900969i 0.588735 1.22252i −2.52446 1.21572i 0
674.1 −1.21572 + 0.277479i 0.781831 1.62349i −0.400969 + 0.193096i 0 −0.500000 + 2.19064i −0.300257 + 0.623490i 2.38374 1.90097i −0.153989 0.193096i 0
674.2 1.21572 0.277479i −0.781831 + 1.62349i −0.400969 + 0.193096i 0 −0.500000 + 2.19064i 0.300257 0.623490i −2.38374 + 1.90097i −0.153989 0.193096i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
29.d even 7 1 inner
145.n even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 725.2.r.b 12
5.b even 2 1 inner 725.2.r.b 12
5.c odd 4 1 29.2.d.a 6
5.c odd 4 1 725.2.l.b 6
15.e even 4 1 261.2.k.a 6
20.e even 4 1 464.2.u.f 6
29.d even 7 1 inner 725.2.r.b 12
145.e even 4 1 841.2.e.d 12
145.h odd 4 1 841.2.d.d 6
145.j even 4 1 841.2.e.d 12
145.n even 14 1 inner 725.2.r.b 12
145.o even 28 1 841.2.b.c 6
145.o even 28 2 841.2.e.b 12
145.o even 28 2 841.2.e.c 12
145.o even 28 1 841.2.e.d 12
145.p odd 28 1 29.2.d.a 6
145.p odd 28 1 725.2.l.b 6
145.p odd 28 1 841.2.a.e 3
145.p odd 28 2 841.2.d.b 6
145.p odd 28 2 841.2.d.e 6
145.q odd 28 1 841.2.a.f 3
145.q odd 28 2 841.2.d.a 6
145.q odd 28 2 841.2.d.c 6
145.q odd 28 1 841.2.d.d 6
145.t even 28 1 841.2.b.c 6
145.t even 28 2 841.2.e.b 12
145.t even 28 2 841.2.e.c 12
145.t even 28 1 841.2.e.d 12
435.bg even 28 1 7569.2.a.p 3
435.bj even 28 1 261.2.k.a 6
435.bj even 28 1 7569.2.a.r 3
580.bi even 28 1 464.2.u.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.d.a 6 5.c odd 4 1
29.2.d.a 6 145.p odd 28 1
261.2.k.a 6 15.e even 4 1
261.2.k.a 6 435.bj even 28 1
464.2.u.f 6 20.e even 4 1
464.2.u.f 6 580.bi even 28 1
725.2.l.b 6 5.c odd 4 1
725.2.l.b 6 145.p odd 28 1
725.2.r.b 12 1.a even 1 1 trivial
725.2.r.b 12 5.b even 2 1 inner
725.2.r.b 12 29.d even 7 1 inner
725.2.r.b 12 145.n even 14 1 inner
841.2.a.e 3 145.p odd 28 1
841.2.a.f 3 145.q odd 28 1
841.2.b.c 6 145.o even 28 1
841.2.b.c 6 145.t even 28 1
841.2.d.a 6 145.q odd 28 2
841.2.d.b 6 145.p odd 28 2
841.2.d.c 6 145.q odd 28 2
841.2.d.d 6 145.h odd 4 1
841.2.d.d 6 145.q odd 28 1
841.2.d.e 6 145.p odd 28 2
841.2.e.b 12 145.o even 28 2
841.2.e.b 12 145.t even 28 2
841.2.e.c 12 145.o even 28 2
841.2.e.c 12 145.t even 28 2
841.2.e.d 12 145.e even 4 1
841.2.e.d 12 145.j even 4 1
841.2.e.d 12 145.o even 28 1
841.2.e.d 12 145.t even 28 1
7569.2.a.p 3 435.bg even 28 1
7569.2.a.r 3 435.bj even 28 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 4T_{2}^{10} + 16T_{2}^{8} - 29T_{2}^{6} + 18T_{2}^{4} + 5T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(725, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 4 T^{10} + 16 T^{8} - 29 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} + 3 T^{10} + 9 T^{8} - T^{6} + 25 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} - 29 T^{10} + 253 T^{8} + 139 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( (T^{6} + 11 T^{5} + 79 T^{4} + 365 T^{3} + \cdots + 1681)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} - 25 T^{10} - 159 T^{8} + \cdots + 3418801 \) Copy content Toggle raw display
$17$ \( (T^{6} + 24 T^{4} + 80 T^{2} + 64)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + T^{5} + T^{4} + 15 T^{3} + 29 T^{2} + \cdots + 169)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 49 T^{10} + 1029 T^{8} + \cdots + 5764801 \) Copy content Toggle raw display
$29$ \( (T^{6} + 6 T^{5} - 13 T^{4} - 316 T^{3} + \cdots + 24389)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 5 T^{5} - 3 T^{4} - 139 T^{3} + \cdots + 6889)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} - 37 T^{10} + 389 T^{8} + \cdots + 2825761 \) Copy content Toggle raw display
$41$ \( (T^{3} - 10 T^{2} + 24 T - 8)^{4} \) Copy content Toggle raw display
$43$ \( T^{12} - T^{10} + 1009 T^{8} + \cdots + 28561 \) Copy content Toggle raw display
$47$ \( T^{12} - 9 T^{10} + \cdots + 3262808641 \) Copy content Toggle raw display
$53$ \( T^{12} + 19 T^{10} + 613 T^{8} + 5795 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( (T^{3} - 28 T^{2} + 252 T - 728)^{4} \) Copy content Toggle raw display
$61$ \( (T^{6} - 3 T^{5} + 37 T^{4} - 13 T^{3} + \cdots + 169)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} - 109 T^{10} + 50913 T^{8} + \cdots + 28561 \) Copy content Toggle raw display
$71$ \( (T^{6} - 21 T^{5} + 189 T^{4} + \cdots + 35721)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} - 121 T^{10} + \cdots + 30821664721 \) Copy content Toggle raw display
$79$ \( (T^{6} - 9 T^{5} + 81 T^{4} + 27 T^{3} + \cdots + 729)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} - 149 T^{10} + \cdots + 815730721 \) Copy content Toggle raw display
$89$ \( (T^{6} + 7 T^{5} + 119 T^{4} + 1337 T^{3} + \cdots + 8281)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} - 421 T^{10} + 50345 T^{8} + \cdots + 28561 \) Copy content Toggle raw display
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