Properties

Label 912.2.bn.o
Level $912$
Weight $2$
Character orbit 912.bn
Analytic conductor $7.282$
Analytic rank $0$
Dimension $16$
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] = [912,2,Mod(65,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.65");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.bn (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: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} - 6 x^{14} + 5 x^{13} + 21 x^{12} - 4 x^{11} - 94 x^{10} - 6 x^{9} + 364 x^{8} - 18 x^{7} - 846 x^{6} - 108 x^{5} + 1701 x^{4} + 1215 x^{3} - 4374 x^{2} - 2187 x + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 456)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} + ( - \beta_{14} + \beta_{11}) q^{5} + (\beta_{13} - \beta_{5}) q^{7} + (\beta_{8} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{3} + ( - \beta_{14} + \beta_{11}) q^{5} + (\beta_{13} - \beta_{5}) q^{7} + (\beta_{8} + 1) q^{9} + ( - \beta_{14} + \beta_{12} - \beta_{11} - \beta_{10} - \beta_{8} + \beta_{7} + \beta_{5} + 2 \beta_{3} + \beta_{2} + \cdots + 1) q^{11}+ \cdots + (\beta_{15} + 3 \beta_{14} + \beta_{13} + \beta_{12} - 2 \beta_{11} + 2 \beta_{10} - \beta_{7} - \beta_{6} - 4 \beta_{5} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + q^{3} - 3 q^{5} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + q^{3} - 3 q^{5} + 13 q^{9} - 3 q^{13} - 15 q^{15} + 3 q^{17} - 11 q^{19} - 6 q^{21} - 3 q^{23} + 11 q^{25} + 4 q^{27} - 5 q^{29} + q^{33} + 24 q^{35} + 9 q^{39} - 6 q^{41} - 13 q^{43} + 33 q^{45} + 27 q^{47} + 8 q^{49} - 15 q^{51} + 7 q^{53} + 12 q^{55} + 23 q^{57} - 10 q^{59} - q^{61} + 8 q^{63} + 30 q^{65} + 24 q^{67} + 41 q^{69} + 27 q^{71} + 2 q^{73} + 21 q^{75} + 21 q^{79} - 7 q^{81} - 5 q^{85} + 23 q^{87} - 25 q^{89} + 78 q^{91} - 56 q^{93} + 13 q^{95} - 60 q^{97} + 35 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} - 6 x^{14} + 5 x^{13} + 21 x^{12} - 4 x^{11} - 94 x^{10} - 6 x^{9} + 364 x^{8} - 18 x^{7} - 846 x^{6} - 108 x^{5} + 1701 x^{4} + 1215 x^{3} - 4374 x^{2} - 2187 x + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3 \nu^{15} - 4 \nu^{14} + 16 \nu^{13} + 33 \nu^{12} - 50 \nu^{11} - 30 \nu^{10} + 316 \nu^{9} + 106 \nu^{8} - 1050 \nu^{7} - 1012 \nu^{6} + 2850 \nu^{5} + 2862 \nu^{4} - 7911 \nu^{3} - 1782 \nu^{2} + \cdots + 2187 ) / 23328 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{15} + 34 \nu^{14} + 18 \nu^{13} - 113 \nu^{12} - 18 \nu^{11} + 598 \nu^{10} + 124 \nu^{9} - 2142 \nu^{8} - 958 \nu^{7} + 5388 \nu^{6} + 3186 \nu^{5} - 13014 \nu^{4} - 5427 \nu^{3} + \cdots - 19683 ) / 23328 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4 \nu^{15} - 53 \nu^{14} + 153 \nu^{13} + 385 \nu^{12} - 612 \nu^{11} - 1226 \nu^{10} + 1576 \nu^{9} + 5904 \nu^{8} - 4534 \nu^{7} - 19056 \nu^{6} + 13932 \nu^{5} + 42066 \nu^{4} + \cdots + 203391 ) / 34992 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 25 \nu^{15} - 76 \nu^{14} + 467 \nu^{12} - 54 \nu^{11} - 1162 \nu^{10} - 364 \nu^{9} + 4734 \nu^{8} + 2962 \nu^{7} - 20796 \nu^{6} - 4122 \nu^{5} + 12474 \nu^{4} + 8667 \nu^{3} + \cdots + 111537 ) / 69984 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{15} + \nu^{14} + 6 \nu^{13} - 5 \nu^{12} - 21 \nu^{11} + 4 \nu^{10} + 94 \nu^{9} + 6 \nu^{8} - 364 \nu^{7} + 18 \nu^{6} + 846 \nu^{5} + 108 \nu^{4} - 1701 \nu^{3} - 1215 \nu^{2} + 4374 \nu + 2187 ) / 2187 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{15} - 2 \nu^{14} + 9 \nu^{13} + 13 \nu^{12} - 36 \nu^{11} - 59 \nu^{10} + 106 \nu^{9} + 288 \nu^{8} - 346 \nu^{7} - 1074 \nu^{6} + 900 \nu^{5} + 2646 \nu^{4} - 1377 \nu^{3} - 6318 \nu^{2} + \cdots + 13122 ) / 2187 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 35 \nu^{15} + 62 \nu^{14} + 174 \nu^{13} - 31 \nu^{12} - 438 \nu^{11} - 310 \nu^{10} + 3020 \nu^{9} + 3054 \nu^{8} - 11786 \nu^{7} - 8820 \nu^{6} + 20502 \nu^{5} + 29430 \nu^{4} + \cdots + 247131 ) / 69984 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2 \nu^{15} + 13 \nu^{14} + 31 \nu^{13} - 25 \nu^{12} - 86 \nu^{11} - 16 \nu^{10} + 450 \nu^{9} + 154 \nu^{8} - 1508 \nu^{7} - 514 \nu^{6} + 3018 \nu^{5} + 2844 \nu^{4} - 3564 \nu^{3} + \cdots + 22599 ) / 5832 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 5 \nu^{15} - 2 \nu^{14} - 24 \nu^{13} - 2 \nu^{12} + 66 \nu^{11} + 88 \nu^{10} - 293 \nu^{9} - 348 \nu^{8} + 956 \nu^{7} + 948 \nu^{6} - 1008 \nu^{5} - 3240 \nu^{4} + 2754 \nu^{3} + 10206 \nu^{2} + \cdots - 13122 ) / 4374 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 15 \nu^{15} - 8 \nu^{14} + 62 \nu^{13} + 15 \nu^{12} - 322 \nu^{11} - 30 \nu^{10} + 1232 \nu^{9} + 350 \nu^{8} - 3210 \nu^{7} - 1676 \nu^{6} + 7986 \nu^{5} + 3294 \nu^{4} - 20601 \nu^{3} + \cdots + 9477 ) / 11664 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 52 \nu^{15} + 23 \nu^{14} - 171 \nu^{13} - \nu^{12} + 252 \nu^{11} + 746 \nu^{10} - 1300 \nu^{9} - 3852 \nu^{8} + 6274 \nu^{7} + 6924 \nu^{6} - 2196 \nu^{5} - 17874 \nu^{4} - 7128 \nu^{3} + \cdots - 107163 ) / 34992 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 53 \nu^{15} - 73 \nu^{14} + 327 \nu^{13} + 176 \nu^{12} - 1014 \nu^{11} - 508 \nu^{10} + 3056 \nu^{9} + 5178 \nu^{8} - 10184 \nu^{7} - 13644 \nu^{6} + 25254 \nu^{5} + 16956 \nu^{4} + \cdots + 183708 ) / 34992 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 289 \nu^{15} + 86 \nu^{14} - 1398 \nu^{13} - 679 \nu^{12} + 3570 \nu^{11} + 5738 \nu^{10} - 15868 \nu^{9} - 27354 \nu^{8} + 49774 \nu^{7} + 67308 \nu^{6} - 83970 \nu^{5} + \cdots - 925101 ) / 69984 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} - 2\beta_{13} + \beta_{12} + 2\beta_{11} + \beta_{10} - \beta_{9} - \beta_{7} + \beta_{5} + \beta_{4} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} - 2\beta_{7} - 2\beta_{6} + 2\beta_{3} + 2\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{15} - 2 \beta_{14} + 6 \beta_{11} - 2 \beta_{9} + 2 \beta_{8} + 3 \beta_{7} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{14} + 4 \beta_{13} - 2 \beta_{12} - 4 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} - 6 \beta_{8} + 2 \beta_{7} - 6 \beta_{6} + 4 \beta_{5} - 4 \beta_{4} + 2 \beta_{2} - 10 \beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 6 \beta_{15} + 4 \beta_{14} + 14 \beta_{13} - 2 \beta_{12} + 4 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} + 4 \beta_{8} - 10 \beta_{7} - 2 \beta_{6} - 10 \beta_{5} - 4 \beta_{4} - 6 \beta_{3} - 2 \beta_{2} + 13 \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 4 \beta_{15} + 24 \beta_{14} + 26 \beta_{13} - 12 \beta_{12} - 20 \beta_{11} - 4 \beta_{10} + 34 \beta_{9} - 18 \beta_{8} - 2 \beta_{7} - 10 \beta_{6} - 2 \beta_{5} - 30 \beta_{4} - 22 \beta_{3} + 17 \beta_{2} - 32 \beta _1 - 17 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 8 \beta_{15} + \beta_{14} - 2 \beta_{13} + 23 \beta_{12} + 32 \beta_{11} + 31 \beta_{10} + 5 \beta_{9} - 8 \beta_{8} - 69 \beta_{7} - 21 \beta_{5} - 21 \beta_{4} + 40 \beta_{3} + 4 \beta _1 + 40 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 20 \beta_{15} + 60 \beta_{14} + 24 \beta_{13} - 24 \beta_{11} + 100 \beta_{9} - 7 \beta_{8} - 120 \beta_{7} - 40 \beta_{6} - 16 \beta_{5} - 20 \beta_{4} + 28 \beta_{3} + 20 \beta_{2} + 4 \beta _1 - 83 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 56 \beta_{15} - 8 \beta_{14} + 8 \beta_{13} - 24 \beta_{12} + 112 \beta_{11} + 8 \beta_{10} + 24 \beta_{9} + 40 \beta_{8} - 77 \beta_{7} + 16 \beta_{6} - 108 \beta_{5} + 4 \beta_{4} + 220 \beta_{3} - 12 \beta_{2} - 8 \beta _1 - 24 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 16 \beta_{15} - 64 \beta_{14} + 12 \beta_{13} + 16 \beta_{12} - 60 \beta_{11} - 144 \beta_{10} + 116 \beta_{9} - 160 \beta_{8} - 20 \beta_{7} - 20 \beta_{6} + 216 \beta_{5} + 176 \beta_{4} + 48 \beta_{3} - 32 \beta_{2} - 64 \beta _1 - 15 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 104 \beta_{15} + 176 \beta_{14} + 220 \beta_{13} - 256 \beta_{12} - 160 \beta_{11} + 16 \beta_{10} - 260 \beta_{9} + 84 \beta_{8} - 128 \beta_{7} - 24 \beta_{6} - 232 \beta_{5} + 96 \beta_{4} + 500 \beta_{3} - 168 \beta_{2} + \cdots - 164 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 68 \beta_{15} - 16 \beta_{14} + 596 \beta_{13} - 252 \beta_{12} - 524 \beta_{11} - 292 \beta_{10} + 548 \beta_{9} - 292 \beta_{8} + 388 \beta_{7} + 156 \beta_{6} + 260 \beta_{5} + 224 \beta_{4} - 1188 \beta_{3} + 129 \beta_{2} + \cdots - 1127 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 408 \beta_{15} + 595 \beta_{14} + 130 \beta_{13} - 283 \beta_{12} - 598 \beta_{11} + 781 \beta_{10} - 249 \beta_{9} - 480 \beta_{8} - 129 \beta_{7} - 40 \beta_{6} + 17 \beta_{5} - 1023 \beta_{4} + 1400 \beta_{3} - 520 \beta_{2} + \cdots + 416 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(-\beta_{3}\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−1.72340 0.172852i
−1.63023 + 0.585107i
−0.909329 1.47415i
−0.809132 + 1.53144i
0.956703 + 1.44386i
1.25083 1.19809i
1.66415 + 0.480229i
1.70042 0.329508i
−1.72340 + 0.172852i
−1.63023 0.585107i
−0.909329 + 1.47415i
−0.809132 1.53144i
0.956703 1.44386i
1.25083 + 1.19809i
1.66415 0.480229i
1.70042 + 0.329508i
0 −1.72340 + 0.172852i 0 −1.02997 0.594652i 0 −1.04533 0 2.94024 0.595786i 0
65.2 0 −1.63023 0.585107i 0 2.85312 + 1.64725i 0 4.86833 0 2.31530 + 1.90772i 0
65.3 0 −0.909329 + 1.47415i 0 2.16072 + 1.24749i 0 −3.30744 0 −1.34624 2.68098i 0
65.4 0 −0.809132 1.53144i 0 −3.15813 1.82335i 0 −1.32651 0 −1.69061 + 2.47827i 0
65.5 0 0.956703 1.44386i 0 −1.67415 0.966572i 0 1.30064 0 −1.16944 2.76268i 0
65.6 0 1.25083 + 1.19809i 0 −1.30557 0.753769i 0 3.02808 0 0.129140 + 2.99722i 0
65.7 0 1.66415 0.480229i 0 2.52758 + 1.45930i 0 −0.106684 0 2.53876 1.59834i 0
65.8 0 1.70042 + 0.329508i 0 −1.87360 1.08173i 0 −3.41109 0 2.78285 + 1.12060i 0
449.1 0 −1.72340 0.172852i 0 −1.02997 + 0.594652i 0 −1.04533 0 2.94024 + 0.595786i 0
449.2 0 −1.63023 + 0.585107i 0 2.85312 1.64725i 0 4.86833 0 2.31530 1.90772i 0
449.3 0 −0.909329 1.47415i 0 2.16072 1.24749i 0 −3.30744 0 −1.34624 + 2.68098i 0
449.4 0 −0.809132 + 1.53144i 0 −3.15813 + 1.82335i 0 −1.32651 0 −1.69061 2.47827i 0
449.5 0 0.956703 + 1.44386i 0 −1.67415 + 0.966572i 0 1.30064 0 −1.16944 + 2.76268i 0
449.6 0 1.25083 1.19809i 0 −1.30557 + 0.753769i 0 3.02808 0 0.129140 2.99722i 0
449.7 0 1.66415 + 0.480229i 0 2.52758 1.45930i 0 −0.106684 0 2.53876 + 1.59834i 0
449.8 0 1.70042 0.329508i 0 −1.87360 + 1.08173i 0 −3.41109 0 2.78285 1.12060i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.bn.o 16
3.b odd 2 1 912.2.bn.n 16
4.b odd 2 1 456.2.bf.c 16
12.b even 2 1 456.2.bf.d yes 16
19.d odd 6 1 912.2.bn.n 16
57.f even 6 1 inner 912.2.bn.o 16
76.f even 6 1 456.2.bf.d yes 16
228.n odd 6 1 456.2.bf.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.bf.c 16 4.b odd 2 1
456.2.bf.c 16 228.n odd 6 1
456.2.bf.d yes 16 12.b even 2 1
456.2.bf.d yes 16 76.f even 6 1
912.2.bn.n 16 3.b odd 2 1
912.2.bn.n 16 19.d odd 6 1
912.2.bn.o 16 1.a even 1 1 trivial
912.2.bn.o 16 57.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{16} + 3 T_{5}^{15} - 21 T_{5}^{14} - 72 T_{5}^{13} + 326 T_{5}^{12} + 1188 T_{5}^{11} - 2224 T_{5}^{10} - 10632 T_{5}^{9} + 9624 T_{5}^{8} + 72960 T_{5}^{7} + 25760 T_{5}^{6} - 242688 T_{5}^{5} - 252000 T_{5}^{4} + \cdots + 430336 \) Copy content Toggle raw display
\( T_{7}^{8} - 30T_{7}^{6} - 22T_{7}^{5} + 223T_{7}^{4} + 234T_{7}^{3} - 278T_{7}^{2} - 332T_{7} - 32 \) Copy content Toggle raw display
\( T_{17}^{16} - 3 T_{17}^{15} - 69 T_{17}^{14} + 216 T_{17}^{13} + 3260 T_{17}^{12} - 13344 T_{17}^{11} - 73808 T_{17}^{10} + 402432 T_{17}^{9} + 1128960 T_{17}^{8} - 9556992 T_{17}^{7} + 2434048 T_{17}^{6} + \cdots + 2945449984 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - T^{15} - 6 T^{14} + 5 T^{13} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{16} + 3 T^{15} - 21 T^{14} + \cdots + 430336 \) Copy content Toggle raw display
$7$ \( (T^{8} - 30 T^{6} - 22 T^{5} + 223 T^{4} + \cdots - 32)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 121 T^{14} + 5883 T^{12} + \cdots + 2637376 \) Copy content Toggle raw display
$13$ \( T^{16} + 3 T^{15} - 61 T^{14} - 192 T^{13} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( T^{16} - 3 T^{15} + \cdots + 2945449984 \) Copy content Toggle raw display
$19$ \( T^{16} + 11 T^{15} + \cdots + 16983563041 \) Copy content Toggle raw display
$23$ \( T^{16} + 3 T^{15} + \cdots + 1328456704 \) Copy content Toggle raw display
$29$ \( T^{16} + 5 T^{15} + 153 T^{14} + \cdots + 554696704 \) Copy content Toggle raw display
$31$ \( T^{16} + 300 T^{14} + \cdots + 5554422784 \) Copy content Toggle raw display
$37$ \( T^{16} + 80 T^{14} + 2386 T^{12} + \cdots + 262144 \) Copy content Toggle raw display
$41$ \( T^{16} + 6 T^{15} + 188 T^{14} + \cdots + 27541504 \) Copy content Toggle raw display
$43$ \( T^{16} + 13 T^{15} + \cdots + 334805733376 \) Copy content Toggle raw display
$47$ \( T^{16} - 27 T^{15} + \cdots + 18653553688576 \) Copy content Toggle raw display
$53$ \( T^{16} - 7 T^{15} + 139 T^{14} + \cdots + 50176 \) Copy content Toggle raw display
$59$ \( T^{16} + 10 T^{15} + \cdots + 933720229264 \) Copy content Toggle raw display
$61$ \( T^{16} + T^{15} + \cdots + 215225477776 \) Copy content Toggle raw display
$67$ \( T^{16} - 24 T^{15} + \cdots + 5605723904881 \) Copy content Toggle raw display
$71$ \( T^{16} - 27 T^{15} + \cdots + 59895709696 \) Copy content Toggle raw display
$73$ \( T^{16} - 2 T^{15} + \cdots + 24812542725961 \) Copy content Toggle raw display
$79$ \( T^{16} - 21 T^{15} + \cdots + 3046449086464 \) Copy content Toggle raw display
$83$ \( T^{16} + 585 T^{14} + \cdots + 8981961048064 \) Copy content Toggle raw display
$89$ \( T^{16} + 25 T^{15} + 627 T^{14} + \cdots + 6885376 \) Copy content Toggle raw display
$97$ \( T^{16} + 60 T^{15} + \cdots + 14736876810496 \) Copy content Toggle raw display
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