↑

Properties

Label 95.2.e.c
Level $95$
Weight $2$
Character orbit 95.e
Analytic conductor $0.759$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [95,2,Mod(11,95)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(95, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("95.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 95 = 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 95.e (of order \(3\), degree \(2\), 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: \(0.758578819202\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.4601315889.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 6x^{6} - 3x^{5} + 26x^{4} - 14x^{3} + 31x^{2} + 12x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} + ( - \beta_{5} + \beta_1) q^{3} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - 1) q^{4} - \beta_{5} q^{5} + ( - \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} + \beta_1) q^{6} + ( - \beta_{6} + \beta_{2} - 1) q^{7} + (2 \beta_{6} - \beta_{4} + \beta_{3} - 2 \beta_{2} + 3) q^{8} + (\beta_{7} + \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} + ( - \beta_{5} + \beta_1) q^{3} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - 1) q^{4} - \beta_{5} q^{5} + ( - \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} + \beta_1) q^{6} + ( - \beta_{6} + \beta_{2} - 1) q^{7} + (2 \beta_{6} - \beta_{4} + \beta_{3} - 2 \beta_{2} + 3) q^{8} + (\beta_{7} + \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 1) q^{9} + ( - \beta_{7} + \beta_{4}) q^{10} + ( - \beta_{6} + 2 \beta_{3} + \beta_{2}) q^{11} + (\beta_{6} - 2 \beta_{4} - \beta_{2} + 1) q^{12} + (\beta_{6} + 2 \beta_{5} - \beta_{3} - \beta_1 - 2) q^{13} + ( - 2 \beta_{7} - \beta_{2} - \beta_1) q^{14} + (\beta_{5} - \beta_{3} - \beta_1 - 1) q^{15} + (4 \beta_{7} - \beta_{5} + 2 \beta_{2} + \beta_1) q^{16} + (\beta_{7} + 2 \beta_1) q^{17} + (\beta_{6} - 2 \beta_{3} - \beta_{2} - 3) q^{18} + (\beta_{4} + 2 \beta_{2} - \beta_1 + 1) q^{19} + (\beta_{6} - \beta_{4} - \beta_{2} + 1) q^{20} + ( - \beta_{7} + \beta_{5} - \beta_1) q^{21} + ( - \beta_{7} + \beta_{2} - 3 \beta_1) q^{22} + ( - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_1) q^{23} + (2 \beta_{7} - 6 \beta_{5} + \beta_{2} + 3 \beta_1) q^{24} + (\beta_{5} - 1) q^{25} + ( - \beta_{4} - 2 \beta_{3}) q^{26} + ( - \beta_{6} + 3 \beta_{4} + \beta_{3} + \beta_{2} + 4) q^{27} + (3 \beta_{7} + 2 \beta_{6} - 4 \beta_{5} - 3 \beta_{4} + 4) q^{28} + (2 \beta_{7} + 2 \beta_{6} - 2 \beta_{4} + \beta_{3} + \beta_1) q^{29} + (\beta_{6} - \beta_{4} - \beta_{3} - \beta_{2}) q^{30} + ( - \beta_{6} - 3 \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{31} + ( - 5 \beta_{7} - 3 \beta_{6} + 6 \beta_{5} + 5 \beta_{4} + \beta_{3} + \beta_1 - 6) q^{32} + ( - 3 \beta_{7} - 6 \beta_{5} + 2 \beta_1) q^{33} + ( - \beta_{7} - 3 \beta_{6} + 3 \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 3) q^{34} + (\beta_{5} - \beta_{2}) q^{35} + (2 \beta_{5} - \beta_{2} - \beta_1) q^{36} + (\beta_{6} - 2 \beta_{4} - \beta_{2} - 1) q^{37} + ( - 2 \beta_{7} - \beta_{6} + 3 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} - \beta_{2} - 3 \beta_1) q^{38} + (2 \beta_{4} + 3 \beta_{3} + 5) q^{39} + (\beta_{7} - 3 \beta_{5} + 2 \beta_{2} + \beta_1) q^{40} + (2 \beta_{7} + 3 \beta_{5} - 3 \beta_{2} - 2 \beta_1) q^{41} + (2 \beta_{7} + 2 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_1 + 3) q^{42} + (\beta_{7} + \beta_{5} - \beta_{2} - 4 \beta_1) q^{43} + (\beta_{6} - 3 \beta_{5} + 3) q^{44} + (\beta_{4} + 2 \beta_{3} + 1) q^{45} + ( - \beta_{6} + \beta_{2} + 3) q^{46} + (2 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1 + 3) q^{47} + ( - 5 \beta_{7} - 4 \beta_{6} + 4 \beta_{5} + 5 \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 4) q^{48} + (\beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} - 3) q^{49} - \beta_{4} q^{50} + (\beta_{7} - \beta_{6} + 6 \beta_{5} - \beta_{4} - \beta_{3} - \beta_1 - 6) q^{51} + (\beta_{7} + \beta_{5} + \beta_{2}) q^{52} + (3 \beta_{7} + 2 \beta_{6} - 3 \beta_{4} - 2 \beta_{3} - 2 \beta_1) q^{53} + (9 \beta_{5} - 3 \beta_{2} - 2 \beta_1) q^{54} + ( - \beta_{2} + 2 \beta_1) q^{55} + ( - 3 \beta_{6} + 5 \beta_{4} + 3 \beta_{2} - 9) q^{56} + ( - 4 \beta_{7} - 4 \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1 + 3) q^{57} + ( - 5 \beta_{6} + 4 \beta_{4} - \beta_{3} + 5 \beta_{2} - 6) q^{58} + 5 \beta_1 q^{59} + (2 \beta_{7} - \beta_{5} + \beta_{2}) q^{60} + ( - 4 \beta_{7} + 2 \beta_{6} - \beta_{5} + 4 \beta_{4} + 1) q^{61} + (\beta_{7} - 9 \beta_{5} + \beta_{2}) q^{62} + ( - 2 \beta_{6} - \beta_{5} + \beta_{3} + \beta_1 + 1) q^{63} + (3 \beta_{6} - 6 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 13) q^{64} + ( - \beta_{6} + \beta_{3} + \beta_{2} + 2) q^{65} + ( - 3 \beta_{7} + \beta_{6} - 9 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + 2 \beta_1 + 9) q^{66} + (2 \beta_{7} + 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 2) q^{67} + (2 \beta_{6} - 5 \beta_{4} + \beta_{3} - 2 \beta_{2} + 3) q^{68} + (\beta_{6} - \beta_{4} - 2 \beta_{3} - \beta_{2} - 3) q^{69} + (2 \beta_{7} + \beta_{6} - 2 \beta_{4} + \beta_{3} + \beta_1) q^{70} + ( - 3 \beta_{7} - 6 \beta_{5} + \beta_1) q^{71} + (3 \beta_{7} + 4 \beta_{6} + 6 \beta_{5} - 3 \beta_{4} - 4 \beta_{3} - 4 \beta_1 - 6) q^{72} + ( - \beta_{7} + 4 \beta_{5} - \beta_{2} + 3 \beta_1) q^{73} + (2 \beta_{7} - 6 \beta_{5} + 3 \beta_{2} + \beta_1) q^{74} + (\beta_{3} + 1) q^{75} + (2 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} - 5 \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_1 + 8) q^{76} + (2 \beta_{6} + \beta_{4} - 3 \beta_{3} - 2 \beta_{2} + 3) q^{77} + (3 \beta_{7} + 6 \beta_{5} + \beta_{2} - 3 \beta_1) q^{78} + ( - 2 \beta_{7} - 5 \beta_{5} + \beta_{2} + \beta_1) q^{79} + ( - 4 \beta_{7} - 2 \beta_{6} + \beta_{5} + 4 \beta_{4} - \beta_{3} - \beta_1 - 1) q^{80} + ( - 2 \beta_{7} - 4 \beta_{5} - 3 \beta_{2} + 2 \beta_1) q^{81} + (4 \beta_{7} + 3 \beta_{6} + 6 \beta_{5} - 4 \beta_{4} + \beta_{3} + \beta_1 - 6) q^{82} + ( - 2 \beta_{6} + \beta_{4} - 2 \beta_{3} + 2 \beta_{2}) q^{83} + ( - 3 \beta_{6} + 5 \beta_{4} - \beta_{3} + 3 \beta_{2} - 4) q^{84} + ( - \beta_{7} + \beta_{4} - 2 \beta_{3} - 2 \beta_1) q^{85} + (\beta_{7} + 4 \beta_{6} + 3 \beta_{5} - \beta_{4} - 3 \beta_{3} - 3 \beta_1 - 3) q^{86} + ( - 2 \beta_{6} + 3 \beta_{4} + \beta_{3} + 2 \beta_{2} - 3) q^{87} + ( - 3 \beta_{6} + 2 \beta_{4} + 5 \beta_{3} + 3 \beta_{2}) q^{88} + ( - 2 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} - 3 \beta_1 - 3) q^{89} + (3 \beta_{5} + \beta_{2} - 2 \beta_1) q^{90} + (\beta_{6} + \beta_{5} + 2 \beta_{3} + 2 \beta_1 - 1) q^{91} + (\beta_{2} + \beta_1) q^{92} + (3 \beta_{7} + 4 \beta_{5} + 3 \beta_{2} - 5 \beta_1) q^{93} + ( - \beta_{6} + 2 \beta_{4} + 5 \beta_{3} + \beta_{2} - 6) q^{94} + ( - \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{3} + \beta_1) q^{95} + (5 \beta_{6} - 9 \beta_{4} - 5 \beta_{2} + 3) q^{96} + (\beta_{5} - 4 \beta_{2} - 5 \beta_1) q^{97} + ( - \beta_{7} - 3 \beta_{5} + \beta_{2} + 2 \beta_1) q^{98} + (5 \beta_{7} + 12 \beta_{5} - 5 \beta_{4} - 5 \beta_{3} - 5 \beta_1 - 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{2} - 3 q^{3} - 5 q^{4} - 4 q^{5} - 2 q^{6} - 8 q^{7} + 24 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - q^{2} - 3 q^{3} - 5 q^{4} - 4 q^{5} - 2 q^{6} - 8 q^{7} + 24 q^{8} - q^{9} - q^{10} - 4 q^{11} + 12 q^{12} - 7 q^{13} + q^{14} - 3 q^{15} - 7 q^{16} + q^{17} - 20 q^{18} + 5 q^{19} + 10 q^{20} + 4 q^{21} - 2 q^{22} - 2 q^{23} - 23 q^{24} - 4 q^{25} + 6 q^{26} + 24 q^{27} + 19 q^{28} + q^{29} + 4 q^{30} - 30 q^{32} - 19 q^{33} - 15 q^{34} + 4 q^{35} + 7 q^{36} - 4 q^{37} + 13 q^{38} + 30 q^{39} - 12 q^{40} + 8 q^{41} + 15 q^{42} - q^{43} + 12 q^{44} + 2 q^{45} + 24 q^{46} + 12 q^{47} - 23 q^{48} - 20 q^{49} + 2 q^{50} - 22 q^{51} + 3 q^{52} + 5 q^{53} + 34 q^{54} + 2 q^{55} - 82 q^{56} + 7 q^{57} - 54 q^{58} + 5 q^{59} - 6 q^{60} - 37 q^{62} + 3 q^{63} + 112 q^{64} + 14 q^{65} + 31 q^{66} - 4 q^{67} + 32 q^{68} - 18 q^{69} + q^{70} - 20 q^{71} - 17 q^{72} + 20 q^{73} - 25 q^{74} + 6 q^{75} + 63 q^{76} + 28 q^{77} + 18 q^{78} - 17 q^{79} - 7 q^{80} - 12 q^{81} - 21 q^{82} + 2 q^{83} - 40 q^{84} + q^{85} - 8 q^{86} - 32 q^{87} - 14 q^{88} - 11 q^{89} + 10 q^{90} - 6 q^{91} + q^{92} + 8 q^{93} - 62 q^{94} - 4 q^{95} + 42 q^{96} - q^{97} - 9 q^{98} - 38 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 6x^{6} - 3x^{5} + 26x^{4} - 14x^{3} + 31x^{2} + 12x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -26\nu^{7} - 189\nu^{6} + 729\nu^{5} - 911\nu^{4} + 3051\nu^{3} - 3618\nu^{2} + 14317\nu - 1215 ) / 4243 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 115\nu^{7} + 20\nu^{6} + 529\nu^{5} + 276\nu^{4} + 3314\nu^{3} + 989\nu^{2} + 483\nu + 1947 ) / 4243 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -135\nu^{7} + 161\nu^{6} - 621\nu^{5} - 324\nu^{4} - 2599\nu^{3} - 1161\nu^{2} - 567\nu - 11694 ) / 4243 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -649\nu^{7} + 994\nu^{6} - 3834\nu^{5} + 3534\nu^{4} - 16046\nu^{3} + 19028\nu^{2} - 17152\nu + 6390 ) / 12729 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 434\nu^{7} - 109\nu^{6} + 2845\nu^{5} + 193\nu^{4} + 12064\nu^{3} + 338\nu^{2} + 16249\nu + 6573 ) / 4243 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 514\nu^{7} - 833\nu^{6} + 3213\nu^{5} - 3858\nu^{4} + 13447\nu^{3} - 15946\nu^{2} + 16585\nu - 5355 ) / 4243 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + 3\beta_{5} - \beta_{4} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} + 4\beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{7} - 12\beta_{5} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{7} + 6\beta_{6} + \beta_{4} - 17\beta_{3} - 17\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7\beta_{6} + 23\beta_{4} - \beta_{3} - 7\beta_{2} + 51 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{7} + 3\beta_{5} - 30\beta_{2} + 74\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\)
\(\chi(n)\) \(-1 + \beta_{5}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−0.245959 + 0.426014i
0.689667 − 1.19454i
−1.02359 + 1.77290i
1.07988 − 1.87040i
−0.245959 − 0.426014i
0.689667 + 1.19454i
−1.02359 − 1.77290i
1.07988 + 1.87040i
−1.37901 + 2.38851i −0.745959 + 1.29204i −2.80333 − 4.85550i −0.500000 + 0.866025i −2.05737 − 3.56347i −2.84864 9.94721 0.387090 + 0.670459i −1.37901 − 2.38851i
11.2 −0.548719 + 0.950409i 0.189667 − 0.328513i 0.397815 + 0.689035i −0.500000 + 0.866025i 0.208148 + 0.360522i 1.89307 −3.06803 1.42805 + 2.47346i −0.548719 − 0.950409i
11.3 0.595455 − 1.03136i −1.52359 + 2.63893i 0.290867 + 0.503797i −0.500000 + 0.866025i 1.81445 + 3.14272i −0.609175 3.07461 −3.14263 − 5.44319i 0.595455 + 1.03136i
11.4 0.832272 − 1.44154i 0.579878 − 1.00438i −0.385355 − 0.667454i −0.500000 + 0.866025i −0.965233 − 1.67183i −2.43525 2.04621 0.827483 + 1.43324i 0.832272 + 1.44154i
26.1 −1.37901 − 2.38851i −0.745959 − 1.29204i −2.80333 + 4.85550i −0.500000 − 0.866025i −2.05737 + 3.56347i −2.84864 9.94721 0.387090 − 0.670459i −1.37901 + 2.38851i
26.2 −0.548719 − 0.950409i 0.189667 + 0.328513i 0.397815 − 0.689035i −0.500000 − 0.866025i 0.208148 − 0.360522i 1.89307 −3.06803 1.42805 − 2.47346i −0.548719 + 0.950409i
26.3 0.595455 + 1.03136i −1.52359 − 2.63893i 0.290867 − 0.503797i −0.500000 − 0.866025i 1.81445 − 3.14272i −0.609175 3.07461 −3.14263 + 5.44319i 0.595455 − 1.03136i
26.4 0.832272 + 1.44154i 0.579878 + 1.00438i −0.385355 + 0.667454i −0.500000 − 0.866025i −0.965233 + 1.67183i −2.43525 2.04621 0.827483 − 1.43324i 0.832272 − 1.44154i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 95.2.e.c ✓ 8
3.b odd 2 1 855.2.k.h 8
4.b odd 2 1 1520.2.q.o 8
5.b even 2 1 475.2.e.e 8
5.c odd 4 2 475.2.j.c 16
19.c even 3 1 inner 95.2.e.c ✓ 8
19.c even 3 1 1805.2.a.o 4
19.d odd 6 1 1805.2.a.i 4
57.h odd 6 1 855.2.k.h 8
76.g odd 6 1 1520.2.q.o 8
95.h odd 6 1 9025.2.a.bp 4
95.i even 6 1 475.2.e.e 8
95.i even 6 1 9025.2.a.bg 4
95.m odd 12 2 475.2.j.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.e.c ✓ 8 1.a even 1 1 trivial
95.2.e.c ✓ 8 19.c even 3 1 inner
475.2.e.e 8 5.b even 2 1
475.2.e.e 8 95.i even 6 1
475.2.j.c 16 5.c odd 4 2
475.2.j.c 16 95.m odd 12 2
855.2.k.h 8 3.b odd 2 1
855.2.k.h 8 57.h odd 6 1
1520.2.q.o 8 4.b odd 2 1
1520.2.q.o 8 76.g odd 6 1
1805.2.a.i 4 19.d odd 6 1
1805.2.a.o 4 19.c even 3 1
9025.2.a.bg 4 95.i even 6 1
9025.2.a.bp 4 95.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + T_{2}^{7} + 7T_{2}^{6} - 4T_{2}^{5} + 31T_{2}^{4} - 6T_{2}^{3} + 37T_{2}^{2} - 6T_{2} + 36 \) acting on \(S_{2}^{\mathrm{new}}(95, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{7} + 7 T^{6} - 4 T^{5} + \cdots + 36 \) Copy content Toggle raw display
$3$ \( T^{8} + 3 T^{7} + 11 T^{6} + 4 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 4 T^{3} - T^{2} - 15 T - 8)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 2 T^{3} - 25 T^{2} - 19 T + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 7 T^{7} + 42 T^{6} + 95 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{8} - T^{7} + 31 T^{6} + 64 T^{5} + \cdots + 11664 \) Copy content Toggle raw display
$19$ \( T^{8} - 5 T^{7} + 31 T^{6} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( T^{8} + 2 T^{7} + 21 T^{6} - 48 T^{5} + \cdots + 36 \) Copy content Toggle raw display
$29$ \( T^{8} - T^{7} + 64 T^{6} + 451 T^{5} + \cdots + 19881 \) Copy content Toggle raw display
$31$ \( (T^{4} - 67 T^{2} - 5 T + 1063)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2 T^{3} - 31 T^{2} - 123 T - 118)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 8 T^{7} + 151 T^{6} + \cdots + 5008644 \) Copy content Toggle raw display
$43$ \( T^{8} + T^{7} + 99 T^{6} + \cdots + 630436 \) Copy content Toggle raw display
$47$ \( T^{8} - 12 T^{7} + 199 T^{6} + \cdots + 5363856 \) Copy content Toggle raw display
$53$ \( T^{8} - 5 T^{7} + 111 T^{6} + \cdots + 2916 \) Copy content Toggle raw display
$59$ \( T^{8} - 5 T^{7} + 150 T^{6} + \cdots + 3515625 \) Copy content Toggle raw display
$61$ \( T^{8} + 130 T^{6} + 176 T^{5} + \cdots + 9296401 \) Copy content Toggle raw display
$67$ \( T^{8} + 4 T^{7} + 52 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$71$ \( T^{8} + 20 T^{7} + 309 T^{6} + \cdots + 59049 \) Copy content Toggle raw display
$73$ \( T^{8} - 20 T^{7} + 305 T^{6} + \cdots + 2979076 \) Copy content Toggle raw display
$79$ \( T^{8} + 17 T^{7} + 217 T^{6} + \cdots + 33856 \) Copy content Toggle raw display
$83$ \( (T^{4} - T^{3} - 62 T^{2} + 55 T + 366)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 11 T^{7} + 211 T^{6} + \cdots + 14561856 \) Copy content Toggle raw display
$97$ \( T^{8} + T^{7} + 267 T^{6} + \cdots + 55383364 \) Copy content Toggle raw display
show more
show less