Properties

Label 925.2.b.g
Level $925$
Weight $2$
Character orbit 925.b
Analytic conductor $7.386$
Analytic rank $0$
Dimension $10$
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] = [925,2,Mod(149,925)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(925, 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("925.149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.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: \(7.38616218697\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: 10.0.8689006034944.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + 2x^{8} + 10x^{7} + 26x^{6} - 4x^{5} + 6x^{4} + 22x^{3} + 25x^{2} + 10x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 185)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{8} q^{2} - \beta_{7} q^{3} + ( - \beta_{6} + \beta_1 - 2) q^{4} + \beta_{6} q^{6} + (\beta_{4} - \beta_{3}) q^{7} + ( - 2 \beta_{9} + \beta_{8} - 2 \beta_{3}) q^{8} + (2 \beta_{6} + \beta_{5} - \beta_{2} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{2} - \beta_{7} q^{3} + ( - \beta_{6} + \beta_1 - 2) q^{4} + \beta_{6} q^{6} + (\beta_{4} - \beta_{3}) q^{7} + ( - 2 \beta_{9} + \beta_{8} - 2 \beta_{3}) q^{8} + (2 \beta_{6} + \beta_{5} - \beta_{2} - \beta_1 + 1) q^{9} + ( - \beta_{6} - 2 \beta_{5} - \beta_1 + 1) q^{11} + (\beta_{9} - \beta_{7} + \beta_{4} + \beta_{3}) q^{12} + (\beta_{9} - \beta_{8} - \beta_{7} + \beta_{3}) q^{13} + ( - \beta_{6} - \beta_{5} - \beta_1 - 1) q^{14} + (\beta_{6} - 2 \beta_{5} + 2 \beta_{2} - \beta_1 + 2) q^{16} + ( - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{3}) q^{17} + (\beta_{9} - 2 \beta_{8} + \beta_{7} - \beta_{3}) q^{18} + ( - \beta_{6} - \beta_{5} - 2 \beta_{2} - \beta_1 - 3) q^{19} + (2 \beta_{6} + 3 \beta_{5} - \beta_{2} - \beta_1) q^{21} + (2 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} - 2 \beta_{4}) q^{22} + ( - 2 \beta_{8} + 2 \beta_{7}) q^{23} + (\beta_{6} - 2 \beta_{2} - 2) q^{24} + ( - \beta_{6} + \beta_{5} - \beta_{2} + 2 \beta_1 - 5) q^{26} + ( - \beta_{9} + \beta_{7} - 2 \beta_{4}) q^{27} + (\beta_{9} - 2 \beta_{7} - 2 \beta_{3}) q^{28} + (2 \beta_{6} + 2 \beta_{5} + 2 \beta_1) q^{29} + ( - 3 \beta_{6} - 2 \beta_{5}) q^{31} + (2 \beta_{9} - \beta_{8} + 2 \beta_{4} + 6 \beta_{3}) q^{32} + (\beta_{9} - 2 \beta_{7} + \beta_{4} - 3 \beta_{3}) q^{33} + ( - \beta_{6} - \beta_{5} - \beta_{2} - 3) q^{34} + (3 \beta_{5} - \beta_{2} + \beta_1 - 7) q^{36} + \beta_{3} q^{37} + ( - \beta_{9} + 2 \beta_{8} - 2 \beta_{7} - 4 \beta_{4} - 8 \beta_{3}) q^{38} + (2 \beta_{6} + \beta_{5} - \beta_1 - 1) q^{39} + (3 \beta_{6} + \beta_1 - 1) q^{41} + ( - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{3}) q^{42} + (2 \beta_{9} + 2 \beta_{7} + 2 \beta_{4} + 4 \beta_{3}) q^{43} + ( - 2 \beta_{6} + 2 \beta_{2} + 4 \beta_1 - 6) q^{44} + ( - 4 \beta_{6} + 2 \beta_1 - 8) q^{46} + ( - 4 \beta_{9} - 2 \beta_{8} - \beta_{4} - \beta_{3}) q^{47} + (\beta_{9} + 2 \beta_{8} - \beta_{7} + \beta_{4} - 5 \beta_{3}) q^{48} + (\beta_{5} + \beta_{2} - 3 \beta_1 + 1) q^{49} + ( - 3 \beta_{5} + \beta_1 + 1) q^{51} + ( - 3 \beta_{9} + 5 \beta_{8} - \beta_{7} - 5 \beta_{3}) q^{52} + (\beta_{9} - \beta_{4} - 3 \beta_{3}) q^{53} + (2 \beta_{6} + \beta_{5} + 2 \beta_{2} + \beta_1 + 3) q^{54} + ( - \beta_{6} - \beta_{5} + 2 \beta_{2} - \beta_1 - 3) q^{56} + (\beta_{9} - \beta_{8} + 3 \beta_{7} - \beta_{3}) q^{57} + ( - 2 \beta_{9} + 2 \beta_{8} + 4 \beta_{7} + 4 \beta_{4}) q^{58} + ( - 3 \beta_{6} - 2 \beta_{5} + 4 \beta_{2} - 4) q^{59} + ( - 2 \beta_{6} + 2 \beta_{5} + 2) q^{61} + ( - \beta_{9} - 3 \beta_{7} - 3 \beta_{4} - 3 \beta_{3}) q^{62} + ( - 5 \beta_{9} - 2 \beta_{8} + 7 \beta_{7} - \beta_{4} + \beta_{3}) q^{63} + ( - 3 \beta_{6} - 4 \beta_{5} - 4 \beta_{2} - \beta_1 - 4) q^{64} + (2 \beta_{2} - 2) q^{66} + ( - \beta_{9} - 4 \beta_{8} + 3 \beta_{7} - \beta_{4} - \beta_{3}) q^{67} + ( - 3 \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{4} - 3 \beta_{3}) q^{68} + ( - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{2} + 2 \beta_1 + 4) q^{69} + ( - \beta_{6} + 6 \beta_{5} - 4 \beta_{2} + \beta_1 + 3) q^{71} + ( - 3 \beta_{9} + 4 \beta_{8} + 3 \beta_{7} - 7 \beta_{3}) q^{72} + ( - 3 \beta_{9} + 2 \beta_{7} - \beta_{4} - 3 \beta_{3}) q^{73} - \beta_{2} q^{74} + (7 \beta_{6} + \beta_{5} + 8 \beta_{2} - \beta_1 + 7) q^{76} + (5 \beta_{9} + 4 \beta_{8} - 6 \beta_{7} + \beta_{4} + 5 \beta_{3}) q^{77} + (2 \beta_{9} + \beta_{7} + \beta_{4} + 3 \beta_{3}) q^{78} + ( - \beta_{6} + 2 \beta_{5} + 2 \beta_{2} + 2 \beta_1 - 8) q^{79} + (\beta_{6} - 3 \beta_{5} - \beta_{2} + 4) q^{81} + (2 \beta_{9} + 2 \beta_{8} + 4 \beta_{7} + 4 \beta_{4} + 2 \beta_{3}) q^{82} + (2 \beta_{9} - 2 \beta_{8} - \beta_{7} + 2 \beta_{4}) q^{83} + (3 \beta_{6} + 5 \beta_{5} - \beta_{2} - 2 \beta_1 - 3) q^{84} + ( - 6 \beta_{6} - 6 \beta_{2} - 4) q^{86} + (2 \beta_{9} + 2 \beta_{8} + 2 \beta_{3}) q^{87} + (6 \beta_{8} - 2 \beta_{7} + 2 \beta_{3}) q^{88} + ( - 4 \beta_{6} - 2 \beta_{2} + 4 \beta_1) q^{89} + (2 \beta_{6} + 3 \beta_{5} - 2 \beta_{2} - \beta_1 + 1) q^{91} + ( - 6 \beta_{9} + 6 \beta_{8} + 2 \beta_{7} - 2 \beta_{4} - 6 \beta_{3}) q^{92} + (\beta_{9} - \beta_{8} - 5 \beta_{7} + 2 \beta_{4} - \beta_{3}) q^{93} + (3 \beta_{6} - 3 \beta_{5} + 2 \beta_{2} - \beta_1 - 3) q^{94} + (3 \beta_{6} - 2 \beta_1 + 2) q^{96} + (4 \beta_{9} + 2 \beta_{8} + 6 \beta_{4} + 4 \beta_{3}) q^{97} + (3 \beta_{9} - 4 \beta_{8} - 3 \beta_{7} - 2 \beta_{4} + 7 \beta_{3}) q^{98} + (2 \beta_{6} + 2 \beta_{5} - 2 \beta_{2} - 6 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 12 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 12 q^{4} - 4 q^{6} - 4 q^{9} + 14 q^{11} - 8 q^{14} + 16 q^{16} - 28 q^{19} - 18 q^{21} - 24 q^{24} - 40 q^{26} - 4 q^{29} + 16 q^{31} - 24 q^{34} - 72 q^{36} - 24 q^{39} - 18 q^{41} - 36 q^{44} - 56 q^{46} - 4 q^{49} + 20 q^{51} + 24 q^{54} - 28 q^{56} - 24 q^{59} + 24 q^{61} - 24 q^{64} - 20 q^{66} + 60 q^{69} + 26 q^{71} + 36 q^{76} - 72 q^{79} + 42 q^{81} - 60 q^{84} - 16 q^{86} + 32 q^{89} - 8 q^{91} - 40 q^{94} - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} + 2x^{8} + 10x^{7} + 26x^{6} - 4x^{5} + 6x^{4} + 22x^{3} + 25x^{2} + 10x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1951 \nu^{9} - 5395 \nu^{8} + 7880 \nu^{7} + 14244 \nu^{6} + 39189 \nu^{5} - 42633 \nu^{4} + 49020 \nu^{3} + 23923 \nu^{2} + 6478 \nu - 30880 ) / 22589 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 28772 \nu^{9} + 98191 \nu^{8} - 148234 \nu^{7} - 183373 \nu^{6} - 375106 \nu^{5} + 1099606 \nu^{4} - 552112 \nu^{3} - 294087 \nu^{2} - 84858 \nu + 446968 ) / 158123 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 40647 \nu^{9} - 90690 \nu^{8} + 104347 \nu^{7} + 372966 \nu^{6} + 984518 \nu^{5} - 379480 \nu^{4} + 338897 \nu^{3} + 634442 \nu^{2} + 892811 \nu + 215667 ) / 158123 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 71898 \nu^{9} - 158327 \nu^{8} + 175190 \nu^{7} + 673628 \nu^{6} + 1752144 \nu^{5} - 663945 \nu^{4} + 418002 \nu^{3} + 1303643 \nu^{2} + 1436696 \nu + 350040 ) / 158123 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 76155 \nu^{9} - 231324 \nu^{8} + 350310 \nu^{7} + 520395 \nu^{6} + 1259115 \nu^{5} - 1911242 \nu^{4} + 1685580 \nu^{3} + 855480 \nu^{2} + 238620 \nu - 159289 ) / 158123 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 114323 \nu^{9} + 352568 \nu^{8} - 532048 \nu^{7} - 776072 \nu^{6} - 1851113 \nu^{5} + 3105863 \nu^{4} - 2497484 \nu^{3} - 1272931 \nu^{2} + \cdots + 366840 ) / 158123 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 126781 \nu^{9} - 270078 \nu^{8} + 258003 \nu^{7} + 1321805 \nu^{6} + 2996078 \nu^{5} - 1128954 \nu^{4} + 376525 \nu^{3} + 3311692 \nu^{2} + 2192289 \nu + 540245 ) / 158123 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 146248 \nu^{9} + 323747 \nu^{8} - 360133 \nu^{7} - 1391637 \nu^{6} - 3501786 \nu^{5} + 1352618 \nu^{4} - 1161953 \nu^{3} - 3138351 \nu^{2} + \cdots - 760472 ) / 158123 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 222986 \nu^{9} - 480082 \nu^{8} + 480285 \nu^{7} + 2268172 \nu^{6} + 5296454 \nu^{5} - 2007077 \nu^{4} + 939789 \nu^{3} + 5692237 \nu^{2} + 4095674 \nu + 1003756 ) / 158123 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} - \beta_{8} + \beta_{7} + \beta_{4} - 3\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 7 \beta_{9} - 5 \beta_{8} + 8 \beta_{7} - 7 \beta_{6} - 8 \beta_{5} + 3 \beta_{4} - 10 \beta_{3} + 5 \beta_{2} - 3 \beta _1 - 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -12\beta_{6} - 13\beta_{5} + 9\beta_{2} - 9\beta _1 - 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 59 \beta_{9} + 41 \beta_{8} - 67 \beta_{7} - 59 \beta_{6} - 67 \beta_{5} - 34 \beta_{4} + 93 \beta_{3} + 41 \beta_{2} - 34 \beta _1 - 93 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 118\beta_{9} + 84\beta_{8} - 131\beta_{7} - 80\beta_{4} + 205\beta_{3} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 529 \beta_{9} + 369 \beta_{8} - 596 \beta_{7} + 529 \beta_{6} + 596 \beta_{5} - 329 \beta_{4} + 866 \beta_{3} - 369 \beta_{2} + 329 \beta _1 + 866 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1111\beta_{6} + 1242\beta_{5} - 782\beta_{2} + 727\beta _1 + 1878 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 4841 \beta_{9} - 3387 \beta_{8} + 5437 \beta_{7} + 4841 \beta_{6} + 5437 \beta_{5} + 3080 \beta_{4} - 8031 \beta_{3} - 3387 \beta_{2} + 3080 \beta _1 + 8031 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(852\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
−0.272959 0.272959i
0.786830 0.786830i
2.14978 2.14978i
−0.443916 0.443916i
−1.21974 + 1.21974i
−1.21974 1.21974i
−0.443916 + 0.443916i
2.14978 + 2.14978i
0.786830 + 0.786830i
−0.272959 + 0.272959i
2.66356i 0.671838i −5.09455 0 1.78948 0.305070i 8.24252i 2.54863 0
149.2 2.27092i 0.518894i −3.15709 0 −1.17837 1.33546i 2.62766i 2.73075 0
149.3 1.46516i 0.744131i −0.146703 0 1.09027 3.94357i 2.71538i 2.44627 0
149.4 1.25268i 2.51369i 0.430798 0 −3.14884 0.281955i 3.04501i −3.31863 0
149.5 0.180152i 3.06709i 1.96755 0 −0.552543 4.41500i 0.714762i −6.40702 0
149.6 0.180152i 3.06709i 1.96755 0 −0.552543 4.41500i 0.714762i −6.40702 0
149.7 1.25268i 2.51369i 0.430798 0 −3.14884 0.281955i 3.04501i −3.31863 0
149.8 1.46516i 0.744131i −0.146703 0 1.09027 3.94357i 2.71538i 2.44627 0
149.9 2.27092i 0.518894i −3.15709 0 −1.17837 1.33546i 2.62766i 2.73075 0
149.10 2.66356i 0.671838i −5.09455 0 1.78948 0.305070i 8.24252i 2.54863 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.10
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 925.2.b.g 10
5.b even 2 1 inner 925.2.b.g 10
5.c odd 4 1 185.2.a.d 5
5.c odd 4 1 925.2.a.h 5
15.e even 4 1 1665.2.a.q 5
15.e even 4 1 8325.2.a.cc 5
20.e even 4 1 2960.2.a.ba 5
35.f even 4 1 9065.2.a.j 5
185.h odd 4 1 6845.2.a.g 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.a.d 5 5.c odd 4 1
925.2.a.h 5 5.c odd 4 1
925.2.b.g 10 1.a even 1 1 trivial
925.2.b.g 10 5.b even 2 1 inner
1665.2.a.q 5 15.e even 4 1
2960.2.a.ba 5 20.e even 4 1
6845.2.a.g 5 185.h odd 4 1
8325.2.a.cc 5 15.e even 4 1
9065.2.a.j 5 35.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(925, [\chi])\):

\( T_{2}^{10} + 16T_{2}^{8} + 86T_{2}^{6} + 180T_{2}^{4} + 129T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{3}^{10} + 17T_{3}^{8} + 80T_{3}^{6} + 84T_{3}^{4} + 32T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{10} + 37T_{7}^{8} + 372T_{7}^{6} + 604T_{7}^{4} + 96T_{7}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 16 T^{8} + 86 T^{6} + 180 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{10} + 17 T^{8} + 80 T^{6} + 84 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + 37 T^{8} + 372 T^{6} + 604 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( (T^{5} - 7 T^{4} - 12 T^{3} + 144 T^{2} + \cdots - 32)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 44 T^{8} + 632 T^{6} + \cdots + 7744 \) Copy content Toggle raw display
$17$ \( T^{10} + 40 T^{8} + 536 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$19$ \( (T^{5} + 14 T^{4} + 26 T^{3} - 362 T^{2} + \cdots - 2224)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 116 T^{8} + 4896 T^{6} + \cdots + 2262016 \) Copy content Toggle raw display
$29$ \( (T^{5} + 2 T^{4} - 80 T^{3} - 272 T^{2} + \cdots - 32)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 8 T^{4} - 38 T^{3} + 314 T^{2} + \cdots - 3016)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$41$ \( (T^{5} + 9 T^{4} - 64 T^{3} - 304 T^{2} + \cdots - 928)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 228 T^{8} + 15072 T^{6} + \cdots + 1721344 \) Copy content Toggle raw display
$47$ \( T^{10} + 253 T^{8} + \cdots + 15952036 \) Copy content Toggle raw display
$53$ \( T^{10} + 97 T^{8} + 2672 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$59$ \( (T^{5} + 12 T^{4} - 94 T^{3} - 526 T^{2} + \cdots - 5456)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 12 T^{4} - 32 T^{3} + 176 T^{2} + \cdots - 512)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 352 T^{8} + \cdots + 57274624 \) Copy content Toggle raw display
$71$ \( (T^{5} - 13 T^{4} - 348 T^{3} + \cdots - 291136)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 161 T^{8} + 8016 T^{6} + \cdots + 30976 \) Copy content Toggle raw display
$79$ \( (T^{5} + 36 T^{4} + 430 T^{3} + 1938 T^{2} + \cdots - 5912)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 289 T^{8} + 25880 T^{6} + \cdots + 98596 \) Copy content Toggle raw display
$89$ \( (T^{5} - 16 T^{4} - 240 T^{3} + 5136 T^{2} + \cdots - 1856)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 928 T^{8} + \cdots + 37406654464 \) Copy content Toggle raw display
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