Properties

Label 845.2.m.i
Level $845$
Weight $2$
Character orbit 845.m
Analytic conductor $6.747$
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] = [845,2,Mod(316,845)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(845, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("845.316");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.m (of order \(6\), degree \(2\), 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: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.17213603549184.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{9} - 2 \beta_{7} + 2) q^{3} + (\beta_{9} - \beta_{7} + \beta_{4} - \beta_{3} + 1) q^{4} + ( - \beta_{10} + \beta_{6}) q^{5} + (\beta_{10} - \beta_{8} + \beta_1) q^{6} + (\beta_{11} + 2 \beta_{10}) q^{7} + (\beta_{11} - \beta_{8} + \beta_{2}) q^{8} + (4 \beta_{9} - 3 \beta_{7} - 3 \beta_{5} + \beta_{4} - 4 \beta_{3} + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{9} - 2 \beta_{7} + 2) q^{3} + (\beta_{9} - \beta_{7} + \beta_{4} - \beta_{3} + 1) q^{4} + ( - \beta_{10} + \beta_{6}) q^{5} + (\beta_{10} - \beta_{8} + \beta_1) q^{6} + (\beta_{11} + 2 \beta_{10}) q^{7} + (\beta_{11} - \beta_{8} + \beta_{2}) q^{8} + (4 \beta_{9} - 3 \beta_{7} - 3 \beta_{5} + \beta_{4} - 4 \beta_{3} + 4) q^{9} + \beta_{4} q^{10} + ( - \beta_{6} - \beta_{2} + \beta_1) q^{11} + ( - \beta_{5} - 3 \beta_{3}) q^{12} + (\beta_{5} + 1) q^{14} + (2 \beta_{6} - \beta_{2}) q^{15} + ( - 3 \beta_{9} - 2 \beta_{4}) q^{16} + ( - 3 \beta_{9} + 3 \beta_{5} + 3 \beta_{3} - 3) q^{17} + (\beta_{11} + 3 \beta_{10} + 2 \beta_{8} - 3 \beta_{6} + \beta_{2}) q^{18} + ( - \beta_{11} + \beta_{10} - 4 \beta_{8} + 4 \beta_1) q^{19} + (\beta_{11} + \beta_{10} + \beta_{8} - \beta_1) q^{20} + (4 \beta_{11} + 6 \beta_{10} + \beta_{8} - 6 \beta_{6} + 4 \beta_{2}) q^{21} + (\beta_{9} + 2 \beta_{7} - 2 \beta_{5} - \beta_{4} - \beta_{3} + 1) q^{22} + ( - \beta_{7} + 5 \beta_{4} + 1) q^{23} + ( - 3 \beta_{6} + 2 \beta_{2}) q^{24} - q^{25} + ( - 3 \beta_{5} - 8 \beta_{3} + 1) q^{27} + ( - 3 \beta_{6} + 3 \beta_{2} + 2 \beta_1) q^{28} + (3 \beta_{9} - 5 \beta_{7} + 6 \beta_{4} + 5) q^{29} + (\beta_{7} + \beta_{5} + \beta_{4}) q^{30} + ( - 5 \beta_{11} - 4 \beta_{10} - 4 \beta_{8} + 4 \beta_{6} - 5 \beta_{2}) q^{31} + ( - 4 \beta_{11} - \beta_{10} - 3 \beta_{8} + 3 \beta_1) q^{32} + (\beta_{11} + \beta_{10}) q^{33} + ( - 3 \beta_{10} - 3 \beta_{8} + 3 \beta_{6}) q^{34} + ( - \beta_{9} + 2 \beta_{7} + \beta_{5} + \beta_{3} - 1) q^{35} + ( - 6 \beta_{9} + 7 \beta_{7} - 2 \beta_{4} - 7) q^{36} + (\beta_{6} + 2 \beta_1) q^{37} + (2 \beta_{5} - 4 \beta_{3} + 7) q^{38} + (2 \beta_{5} + \beta_{3} - 1) q^{40} + ( - 5 \beta_{6} - 4 \beta_{2} - \beta_1) q^{41} + (\beta_{9} - 3 \beta_{7} - \beta_{4} + 3) q^{42} + ( - \beta_{9} - 7 \beta_{7} - \beta_{4} + \beta_{3} - 1) q^{43} + (\beta_{11} + \beta_{2}) q^{44} + (4 \beta_{11} + 3 \beta_{10} + \beta_{8} - \beta_1) q^{45} + (5 \beta_{11} - 5 \beta_{10} + 4 \beta_{8} - 4 \beta_1) q^{46} + ( - 4 \beta_{11} - 7 \beta_{10} - \beta_{8} + 7 \beta_{6} - 4 \beta_{2}) q^{47} + ( - 6 \beta_{9} + 4 \beta_{7} + \beta_{5} - 5 \beta_{4} + 6 \beta_{3} - 6) q^{48} + (4 \beta_{9} + \beta_{7} + \beta_{4} - 1) q^{49} - \beta_1 q^{50} + (3 \beta_{5} + 6 \beta_{3}) q^{51} + (\beta_{3} - 1) q^{53} + ( - 3 \beta_{6} + 5 \beta_{2} + 6 \beta_1) q^{54} + ( - \beta_{9} - \beta_{7} + \beta_{4} + 1) q^{55} + (2 \beta_{9} - 3 \beta_{7} - 2 \beta_{5} - 2 \beta_{3} + 2) q^{56} + ( - \beta_{11} + 4 \beta_{10} - 5 \beta_{8} - 4 \beta_{6} - \beta_{2}) q^{57} + (6 \beta_{11} - 3 \beta_{10} + 4 \beta_{8} - 4 \beta_1) q^{58} + (2 \beta_{11} + 5 \beta_{10} + 2 \beta_{8} - 2 \beta_1) q^{59} + (3 \beta_{11} + 3 \beta_{10} + 2 \beta_{8} - 3 \beta_{6} + 3 \beta_{2}) q^{60} + (\beta_{9} + 5 \beta_{7} + \beta_{4} - \beta_{3} + 1) q^{61} + ( - 4 \beta_{9} + \beta_{7} - 5 \beta_{4} - 1) q^{62} + ( - 13 \beta_{6} + 11 \beta_{2} + 5 \beta_1) q^{63} + (5 \beta_{5} + 3 \beta_{3} - 4) q^{64} + q^{66} + ( - 9 \beta_{6} + 6 \beta_{2} + 6 \beta_1) q^{67} + (3 \beta_{9} - 3 \beta_{7} + 3) q^{68} + (\beta_{9} + 3 \beta_{7} + 4 \beta_{5} + 5 \beta_{4} - \beta_{3} + 1) q^{69} + ( - \beta_{10} + \beta_{8} + \beta_{6}) q^{70} + (4 \beta_{11} + 8 \beta_{10} + 7 \beta_{8} - 7 \beta_1) q^{71} + ( - 4 \beta_{11} - 10 \beta_{10} - 5 \beta_{8} + 5 \beta_1) q^{72} + ( - 2 \beta_{11} - 2 \beta_{10} + 8 \beta_{8} + 2 \beta_{6} - 2 \beta_{2}) q^{73} + (2 \beta_{9} + 2 \beta_{7} + \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + 2) q^{74} + ( - \beta_{9} + 2 \beta_{7} - 2) q^{75} + (4 \beta_{2} + 5 \beta_1) q^{76} + (\beta_{5} + \beta_{3}) q^{77} + (2 \beta_{5} - 3 \beta_{3} - 12) q^{79} + (3 \beta_{2} + 2 \beta_1) q^{80} + ( - 11 \beta_{9} + 16 \beta_{7} - 10 \beta_{4} - 16) q^{81} + ( - \beta_{9} + 3 \beta_{7} - 9 \beta_{5} - 10 \beta_{4} + \beta_{3} - 1) q^{82} + ( - 2 \beta_{11} + 10 \beta_{10} + 3 \beta_{8} - 10 \beta_{6} - 2 \beta_{2}) q^{83} + ( - 9 \beta_{11} - 10 \beta_{10} - 5 \beta_{8} + 5 \beta_1) q^{84} - 3 \beta_{11} q^{85} + ( - \beta_{11} - 9 \beta_{8} - \beta_{2}) q^{86} + (11 \beta_{9} - 10 \beta_{7} - 2 \beta_{5} + 9 \beta_{4} - 11 \beta_{3} + 11) q^{87} + ( - 2 \beta_{9} - 5 \beta_{7} + 3 \beta_{4} + 5) q^{88} + (5 \beta_{6} - 8 \beta_{2} - 10 \beta_1) q^{89} + ( - \beta_{5} + \beta_{3} + 2) q^{90} + (4 \beta_{3} - 5) q^{92} + (14 \beta_{6} - 14 \beta_{2} - 9 \beta_1) q^{93} + ( - \beta_{9} + 3 \beta_{7} + 2 \beta_{4} - 3) q^{94} + (\beta_{9} + \beta_{7} + 3 \beta_{5} + 4 \beta_{4} - \beta_{3} + 1) q^{95} + ( - 9 \beta_{11} - 7 \beta_{10} - 7 \beta_{8} + 7 \beta_{6} - 9 \beta_{2}) q^{96} + (5 \beta_{11} - \beta_{10} + 4 \beta_{8} - 4 \beta_1) q^{97} + (\beta_{11} + 3 \beta_{10} + 6 \beta_{8} - 6 \beta_1) q^{98} + (7 \beta_{10} + 4 \beta_{8} - 7 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 10 q^{3} - 2 q^{4} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 10 q^{3} - 2 q^{4} - 8 q^{9} - 2 q^{10} - 16 q^{12} + 16 q^{14} + 10 q^{16} - 6 q^{17} + 12 q^{22} - 4 q^{23} - 12 q^{25} - 32 q^{27} + 12 q^{29} + 8 q^{30} + 10 q^{35} - 26 q^{36} + 76 q^{38} + 18 q^{42} - 46 q^{43} + 2 q^{48} - 16 q^{49} + 36 q^{51} - 8 q^{53} + 6 q^{55} - 14 q^{56} + 34 q^{61} + 12 q^{62} - 16 q^{64} + 12 q^{66} + 12 q^{68} + 30 q^{69} + 22 q^{74} - 10 q^{75} + 8 q^{77} - 148 q^{79} - 54 q^{81} - 4 q^{82} - 20 q^{87} + 28 q^{88} + 24 q^{90} - 44 q^{92} - 20 q^{94} + 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -25\nu^{11} + 95\nu^{9} - 361\nu^{7} + 155\nu^{5} - 30\nu^{3} - 1563\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 25\nu^{10} - 95\nu^{8} + 361\nu^{6} - 155\nu^{4} + 30\nu^{2} + 1004 ) / 559 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{10} - 20\nu^{8} + 76\nu^{6} - 139\nu^{4} + 124\nu^{2} - 24 ) / 43 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 45\nu^{10} - 171\nu^{8} + 538\nu^{6} - 279\nu^{4} + 54\nu^{2} + 242 ) / 559 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 70\nu^{11} - 266\nu^{9} + 899\nu^{7} - 434\nu^{5} + 84\nu^{3} + 1246\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 114\nu^{10} - 545\nu^{8} + 2071\nu^{6} - 2831\nu^{4} + 3379\nu^{2} - 95 ) / 559 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 114\nu^{11} - 545\nu^{9} + 2071\nu^{7} - 2831\nu^{5} + 3379\nu^{3} - 95\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -128\nu^{10} + 710\nu^{8} - 2698\nu^{6} + 4483\nu^{4} - 4402\nu^{2} + 852 ) / 559 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -242\nu^{11} + 1255\nu^{9} - 4769\nu^{7} + 7314\nu^{5} - 7781\nu^{3} + 1506\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -317\nu^{11} + 1540\nu^{9} - 5852\nu^{7} + 8338\nu^{5} - 9548\nu^{3} + 1848\nu ) / 559 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{7} + \beta_{4} - \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + 3\beta_{8} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{9} + 2\beta_{7} + 4\beta_{4} - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{11} - \beta_{10} + 9\beta_{8} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -5\beta_{5} + 9\beta_{3} - 14 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -5\beta_{6} - 14\beta_{2} - 28\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -28\beta_{9} - 14\beta_{7} - 19\beta_{5} - 47\beta_{4} + 28\beta_{3} - 28 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -47\beta_{11} + 19\beta_{10} - 89\beta_{8} - 19\beta_{6} - 47\beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -89\beta_{9} - 42\beta_{7} - 155\beta_{4} + 42 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -155\beta_{11} + 66\beta_{10} - 286\beta_{8} + 286\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\) \(171\) \(677\)
\(\chi(n)\) \(\beta_{7}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
316.1
−1.56052 0.900969i
−1.07992 0.623490i
−0.385418 0.222521i
0.385418 + 0.222521i
1.07992 + 0.623490i
1.56052 + 0.900969i
−1.56052 + 0.900969i
−1.07992 + 0.623490i
−0.385418 + 0.222521i
0.385418 0.222521i
1.07992 0.623490i
1.56052 0.900969i
−1.56052 0.900969i 0.777479 1.34663i 0.623490 + 1.07992i 1.00000i −2.42655 + 1.40097i −1.34663 + 0.777479i 1.35690i 0.291053 + 0.504118i −0.900969 + 1.56052i
316.2 −1.07992 0.623490i 0.0990311 0.171527i −0.222521 0.385418i 1.00000i −0.213891 + 0.123490i 0.171527 0.0990311i 3.04892i 1.48039 + 2.56410i 0.623490 1.07992i
316.3 −0.385418 0.222521i 1.62349 2.81197i −0.900969 1.56052i 1.00000i −1.25144 + 0.722521i −2.81197 + 1.62349i 1.69202i −3.77144 6.53232i −0.222521 + 0.385418i
316.4 0.385418 + 0.222521i 1.62349 2.81197i −0.900969 1.56052i 1.00000i 1.25144 0.722521i 2.81197 1.62349i 1.69202i −3.77144 6.53232i −0.222521 + 0.385418i
316.5 1.07992 + 0.623490i 0.0990311 0.171527i −0.222521 0.385418i 1.00000i 0.213891 0.123490i −0.171527 + 0.0990311i 3.04892i 1.48039 + 2.56410i 0.623490 1.07992i
316.6 1.56052 + 0.900969i 0.777479 1.34663i 0.623490 + 1.07992i 1.00000i 2.42655 1.40097i 1.34663 0.777479i 1.35690i 0.291053 + 0.504118i −0.900969 + 1.56052i
361.1 −1.56052 + 0.900969i 0.777479 + 1.34663i 0.623490 1.07992i 1.00000i −2.42655 1.40097i −1.34663 0.777479i 1.35690i 0.291053 0.504118i −0.900969 1.56052i
361.2 −1.07992 + 0.623490i 0.0990311 + 0.171527i −0.222521 + 0.385418i 1.00000i −0.213891 0.123490i 0.171527 + 0.0990311i 3.04892i 1.48039 2.56410i 0.623490 + 1.07992i
361.3 −0.385418 + 0.222521i 1.62349 + 2.81197i −0.900969 + 1.56052i 1.00000i −1.25144 0.722521i −2.81197 1.62349i 1.69202i −3.77144 + 6.53232i −0.222521 0.385418i
361.4 0.385418 0.222521i 1.62349 + 2.81197i −0.900969 + 1.56052i 1.00000i 1.25144 + 0.722521i 2.81197 + 1.62349i 1.69202i −3.77144 + 6.53232i −0.222521 0.385418i
361.5 1.07992 0.623490i 0.0990311 + 0.171527i −0.222521 + 0.385418i 1.00000i 0.213891 + 0.123490i −0.171527 0.0990311i 3.04892i 1.48039 2.56410i 0.623490 + 1.07992i
361.6 1.56052 0.900969i 0.777479 + 1.34663i 0.623490 1.07992i 1.00000i 2.42655 + 1.40097i 1.34663 + 0.777479i 1.35690i 0.291053 0.504118i −0.900969 1.56052i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 316.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.m.i 12
13.b even 2 1 inner 845.2.m.i 12
13.c even 3 1 845.2.c.f 6
13.c even 3 1 inner 845.2.m.i 12
13.d odd 4 1 845.2.e.j 6
13.d odd 4 1 845.2.e.l 6
13.e even 6 1 845.2.c.f 6
13.e even 6 1 inner 845.2.m.i 12
13.f odd 12 1 845.2.a.h 3
13.f odd 12 1 845.2.a.j yes 3
13.f odd 12 1 845.2.e.j 6
13.f odd 12 1 845.2.e.l 6
39.k even 12 1 7605.2.a.br 3
39.k even 12 1 7605.2.a.by 3
65.s odd 12 1 4225.2.a.bd 3
65.s odd 12 1 4225.2.a.bf 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
845.2.a.h 3 13.f odd 12 1
845.2.a.j yes 3 13.f odd 12 1
845.2.c.f 6 13.c even 3 1
845.2.c.f 6 13.e even 6 1
845.2.e.j 6 13.d odd 4 1
845.2.e.j 6 13.f odd 12 1
845.2.e.l 6 13.d odd 4 1
845.2.e.l 6 13.f odd 12 1
845.2.m.i 12 1.a even 1 1 trivial
845.2.m.i 12 13.b even 2 1 inner
845.2.m.i 12 13.c even 3 1 inner
845.2.m.i 12 13.e even 6 1 inner
4225.2.a.bd 3 65.s odd 12 1
4225.2.a.bf 3 65.s odd 12 1
7605.2.a.br 3 39.k even 12 1
7605.2.a.by 3 39.k even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 5T_{2}^{10} + 19T_{2}^{8} - 28T_{2}^{6} + 31T_{2}^{4} - 6T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 5 T^{10} + 19 T^{8} - 28 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{6} - 5 T^{5} + 19 T^{4} - 28 T^{3} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{12} - 13 T^{10} + 143 T^{8} - 336 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{12} - 17 T^{10} + 279 T^{8} - 168 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( (T^{6} + 3 T^{5} + 27 T^{4} + 405 T^{2} + \cdots + 729)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} - 82 T^{10} + 6179 T^{8} + \cdots + 707281 \) Copy content Toggle raw display
$23$ \( (T^{6} + 2 T^{5} + 61 T^{4} + 28 T^{3} + \cdots + 5041)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 6 T^{5} + 87 T^{4} - 308 T^{3} + \cdots + 94249)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 101 T^{4} + 2110 T^{2} + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} - 27 T^{10} + 598 T^{8} + \cdots + 28561 \) Copy content Toggle raw display
$41$ \( T^{12} - 194 T^{10} + 28747 T^{8} + \cdots + 3418801 \) Copy content Toggle raw display
$43$ \( (T^{6} + 23 T^{5} + 355 T^{4} + \cdots + 187489)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 146 T^{4} + 1713 T^{2} + \cdots + 1681)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 2 T^{2} - T - 1)^{4} \) Copy content Toggle raw display
$59$ \( T^{12} - 59 T^{10} + 2806 T^{8} + \cdots + 28561 \) Copy content Toggle raw display
$61$ \( (T^{6} - 17 T^{5} + 195 T^{4} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} - 243 T^{10} + \cdots + 15178486401 \) Copy content Toggle raw display
$71$ \( T^{12} - 229 T^{10} + \cdots + 130466162401 \) Copy content Toggle raw display
$73$ \( (T^{6} + 440 T^{4} + 56656 T^{2} + \cdots + 2096704)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 37 T^{2} + 412 T + 1217)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 369 T^{4} + 21650 T^{2} + \cdots + 344569)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} - 395 T^{10} + \cdots + 80706559921 \) Copy content Toggle raw display
$97$ \( T^{12} - 146 T^{10} + \cdots + 1073283121 \) Copy content Toggle raw display
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