Properties

Label 630.2.bv.a
Level $630$
Weight $2$
Character orbit 630.bv
Analytic conductor $5.031$
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] = [630,2,Mod(73,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 9, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.73");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 630.bv (of order \(12\), degree \(4\), 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: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 12 x^{14} - 48 x^{13} + 67 x^{12} - 24 x^{11} + 118 x^{10} - 176 x^{9} + 351 x^{8} - 180 x^{7} + 358 x^{6} - 336 x^{5} + 390 x^{4} - 344 x^{3} + 164 x^{2} - 40 x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{15} q^{2} + ( - \beta_{13} + \beta_{5}) q^{4} + (\beta_{15} + \beta_{14} + \beta_{12} + \beta_{11} - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_1 - 1) q^{5} + ( - \beta_{15} + \beta_{12} - \beta_{10} + \beta_{9} + \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2}) q^{7} + ( - \beta_{12} - \beta_{6}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{15} q^{2} + ( - \beta_{13} + \beta_{5}) q^{4} + (\beta_{15} + \beta_{14} + \beta_{12} + \beta_{11} - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_1 - 1) q^{5} + ( - \beta_{15} + \beta_{12} - \beta_{10} + \beta_{9} + \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2}) q^{7} + ( - \beta_{12} - \beta_{6}) q^{8} + ( - \beta_{15} - \beta_{12} + \beta_{10} - \beta_{8} - \beta_{7} + \beta_{5} + 1) q^{10} + (2 \beta_{15} + 2 \beta_{12} + \beta_{11} + \beta_{10} - 2 \beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{3} + \cdots - 1) q^{11}+ \cdots + (3 \beta_{15} - 4 \beta_{14} + 3 \beta_{12} + \beta_{10} - 2 \beta_{9} - \beta_{8} - 3 \beta_{7} + \cdots - 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 12 q^{5} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 12 q^{5} - 8 q^{7} + 8 q^{10} - 4 q^{11} - 16 q^{13} - 16 q^{14} + 8 q^{16} + 12 q^{17} - 8 q^{19} + 8 q^{20} + 4 q^{22} - 32 q^{23} - 32 q^{25} + 12 q^{26} - 8 q^{28} - 24 q^{31} + 16 q^{34} - 4 q^{35} - 8 q^{37} + 28 q^{38} - 24 q^{43} - 4 q^{46} + 24 q^{47} + 52 q^{49} - 8 q^{52} - 44 q^{53} - 56 q^{55} - 8 q^{56} + 48 q^{58} - 8 q^{59} + 24 q^{61} - 8 q^{62} - 16 q^{65} + 36 q^{67} + 12 q^{68} + 32 q^{70} + 32 q^{71} - 40 q^{73} + 24 q^{74} + 44 q^{77} + 12 q^{79} - 12 q^{80} + 12 q^{82} + 16 q^{83} + 8 q^{85} + 8 q^{86} + 8 q^{88} + 16 q^{89} + 8 q^{91} - 8 q^{92} + 8 q^{94} + 48 q^{95} + 44 q^{97} + 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 12 x^{14} - 48 x^{13} + 67 x^{12} - 24 x^{11} + 118 x^{10} - 176 x^{9} + 351 x^{8} - 180 x^{7} + 358 x^{6} - 336 x^{5} + 390 x^{4} - 344 x^{3} + 164 x^{2} - 40 x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1434138866353 \nu^{15} - 74224360591111 \nu^{14} + 217326371958939 \nu^{13} - 648847305484636 \nu^{12} + \cdots + 46\!\cdots\!94 ) / 51\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 123434729989149 \nu^{15} + 492304781090243 \nu^{14} + \cdots + 195044141171618 ) / 51\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 137809854283897 \nu^{15} + 666808683532056 \nu^{14} + \cdots + 69\!\cdots\!44 ) / 10\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3909951128714 \nu^{15} - 13126607646508 \nu^{14} + 38145193554947 \nu^{13} - 162143292202288 \nu^{12} + \cdots - 16575048540328 ) / 28585593772190 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4143762135082 \nu^{15} + 12665097411614 \nu^{14} - 36598537974476 \nu^{13} + 160755388928989 \nu^{12} + \cdots + 3631882693804 ) / 28585593772190 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 14\!\cdots\!27 \nu^{15} + \cdots + 35\!\cdots\!44 ) / 51\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6100180981 \nu^{15} - 22259671750 \nu^{14} + 65510725326 \nu^{13} - 270219726184 \nu^{12} + 314950092835 \nu^{11} - 40514369326 \nu^{10} + \cdots - 77929949288 ) / 18265555126 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6072104849199 \nu^{15} + 21780610045393 \nu^{14} - 63731636791632 \nu^{13} + 264606089742658 \nu^{12} + \cdots + 65633529574178 ) / 16530327389030 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 20\!\cdots\!87 \nu^{15} + \cdots - 28\!\cdots\!14 ) / 51\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 30320067 \nu^{15} - 111020724 \nu^{14} + 326080511 \nu^{13} - 1344589864 \nu^{12} + 1575363211 \nu^{11} - 188871116 \nu^{10} + 3515753069 \nu^{9} + \cdots - 330526984 ) / 63923990 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 567889621787826 \nu^{15} + \cdots - 77\!\cdots\!96 ) / 10\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 52\!\cdots\!58 \nu^{15} + \cdots - 60\!\cdots\!06 ) / 51\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 19482487322 \nu^{15} + 71829768307 \nu^{14} - 211530176114 \nu^{13} + 869648666130 \nu^{12} - 1035106924390 \nu^{11} + \cdots + 258618081490 ) / 18265555126 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 82631746 \nu^{15} + 300206917 \nu^{14} - 880560228 \nu^{13} + 3640243297 \nu^{12} - 4191737118 \nu^{11} + 407798693 \nu^{10} + \cdots + 791533522 ) / 63923990 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 69\!\cdots\!34 \nu^{15} + \cdots - 83\!\cdots\!68 ) / 51\!\cdots\!90 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{6} - \beta_{5} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + \beta_{6} + \beta_{5} + \beta_{4} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{15} + 4\beta_{14} + 2\beta_{13} + 3\beta_{12} + 2\beta_{11} - 2\beta_{8} + \beta_{6} - 5\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{15} - 6 \beta_{14} - 11 \beta_{12} - 4 \beta_{11} - 4 \beta_{10} + 9 \beta_{9} - 10 \beta_{8} - 9 \beta_{7} - 3 \beta_{6} - 15 \beta_{5} + 9 \beta_{4} - 10 \beta_{3} - 20 \beta_{2} - 9 \beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 23 \beta_{15} - 16 \beta_{14} - 22 \beta_{13} - 21 \beta_{12} - 2 \beta_{11} + 21 \beta_{10} - 2 \beta_{9} + 5 \beta_{8} + 22 \beta_{6} - 2 \beta_{5} + 39 \beta_{4} - 21 \beta_{3} + 14 \beta_{2} + 7 \beta _1 - 37 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 28 \beta_{15} + 154 \beta_{14} + 23 \beta_{13} + 140 \beta_{12} + 89 \beta_{11} + 95 \beta_{10} - 56 \beta_{9} + 12 \beta_{8} + 89 \beta_{7} + 84 \beta_{6} + 121 \beta_{5} - 12 \beta_{4} + 56 \beta_{3} + 173 \beta_{2} - 39 \beta _1 + 93 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 163 \beta_{15} + 184 \beta_{14} + 240 \beta_{13} + 50 \beta_{12} - 94 \beta_{10} + 200 \beta_{9} - 351 \beta_{8} - 35 \beta_{7} - 219 \beta_{6} - 128 \beta_{5} - 94 \beta_{4} + 35 \beta_{3} - 440 \beta_{2} - 400 \beta _1 + 985 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 645 \beta_{15} - 1488 \beta_{14} - 329 \beta_{13} - 1794 \beta_{12} - 914 \beta_{11} - 242 \beta_{10} + 914 \beta_{9} - 672 \beta_{8} - 663 \beta_{7} - 645 \beta_{6} - 1243 \beta_{5} + 914 \beta_{4} - 914 \beta_{3} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1828 \beta_{15} - 186 \beta_{14} - 1819 \beta_{13} - 195 \beta_{12} + 477 \beta_{11} + 3325 \beta_{10} - 1299 \beta_{9} + 1776 \beta_{8} + 1915 \beta_{7} + 2296 \beta_{6} + 2491 \beta_{5} + 1915 \beta_{4} + \cdots - 5566 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 6400 \beta_{15} + 16412 \beta_{14} + 6539 \beta_{13} + 14863 \beta_{12} + 7481 \beta_{11} + 6042 \beta_{10} - 3465 \beta_{9} - 1493 \beta_{8} + 8974 \beta_{7} + 1665 \beta_{6} + 11899 \beta_{5} - 7481 \beta_{4} + \cdots + 17408 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 8014 \beta_{15} - 12665 \beta_{14} + 18481 \beta_{13} - 24749 \beta_{12} - 18707 \beta_{11} - 21849 \beta_{10} + 32416 \beta_{9} - 37414 \beta_{8} - 15992 \beta_{7} - 38812 \beta_{6} - 32352 \beta_{5} + \cdots + 77085 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 91679 \beta_{15} - 170862 \beta_{14} - 70257 \beta_{13} - 181429 \beta_{12} - 89750 \beta_{11} + 7240 \beta_{10} + 55995 \beta_{9} - 7240 \beta_{8} - 43472 \beta_{7} - 31804 \beta_{6} + \cdots - 166538 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 55995 \beta_{15} + 241852 \beta_{14} - 92227 \beta_{13} + 278084 \beta_{12} + 185857 \beta_{11} + 372602 \beta_{10} - 253893 \beta_{9} + 253893 \beta_{8} + 322709 \beta_{7} + 281000 \beta_{6} + \cdots - 469625 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 880388 \beta_{15} + 1459906 \beta_{14} + 949204 \beta_{13} + 1273161 \beta_{12} + 510702 \beta_{11} + 13265 \beta_{10} + 13265 \beta_{9} - 533645 \beta_{8} + 533645 \beta_{7} - 400341 \beta_{6} + \cdots + 2650257 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 533645 \beta_{15} - 4295841 \beta_{14} + 533645 \beta_{13} - 5220149 \beta_{12} - 3264759 \beta_{11} - 2855121 \beta_{10} + 3769764 \beta_{9} - 3264759 \beta_{8} - 2855121 \beta_{7} + \cdots + 3840272 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(\beta_{13}\) \(1\) \(1 + \beta_{14}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1
−0.709944 + 0.925217i
0.792206 1.03242i
−1.09227 0.838128i
0.277956 + 0.213283i
−0.709944 0.925217i
0.792206 + 1.03242i
−1.09227 + 0.838128i
0.277956 0.213283i
−0.424637 + 3.22544i
0.117630 0.893490i
0.339278 + 0.0446668i
2.69978 + 0.355433i
−0.424637 3.22544i
0.117630 + 0.893490i
0.339278 0.0446668i
2.69978 0.355433i
−0.965926 0.258819i 0 0.866025 + 0.500000i −1.55103 1.61069i 0 −1.38658 2.25331i −0.707107 0.707107i 0 1.08130 + 1.95724i
73.2 −0.965926 0.258819i 0 0.866025 + 0.500000i −0.0488750 + 2.23553i 0 2.15951 + 1.52856i −0.707107 0.707107i 0 0.625808 2.14671i
73.3 0.965926 + 0.258819i 0 0.866025 + 0.500000i −0.519137 + 2.17497i 0 −2.64131 0.153213i 0.707107 + 0.707107i 0 −1.06437 + 1.96650i
73.4 0.965926 + 0.258819i 0 0.866025 + 0.500000i 0.851088 2.06776i 0 −1.86367 + 1.87796i 0.707107 + 0.707107i 0 1.35727 1.77703i
397.1 −0.965926 + 0.258819i 0 0.866025 0.500000i −1.55103 + 1.61069i 0 −1.38658 + 2.25331i −0.707107 + 0.707107i 0 1.08130 1.95724i
397.2 −0.965926 + 0.258819i 0 0.866025 0.500000i −0.0488750 2.23553i 0 2.15951 1.52856i −0.707107 + 0.707107i 0 0.625808 + 2.14671i
397.3 0.965926 0.258819i 0 0.866025 0.500000i −0.519137 2.17497i 0 −2.64131 + 0.153213i 0.707107 0.707107i 0 −1.06437 1.96650i
397.4 0.965926 0.258819i 0 0.866025 0.500000i 0.851088 + 2.06776i 0 −1.86367 1.87796i 0.707107 0.707107i 0 1.35727 + 1.77703i
523.1 −0.258819 0.965926i 0 −0.866025 + 0.500000i −1.58356 + 1.57872i 0 2.22701 1.42843i 0.707107 + 0.707107i 0 1.93478 + 1.12100i
523.2 −0.258819 0.965926i 0 −0.866025 + 0.500000i −1.04129 1.97882i 0 −2.55046 + 0.703686i 0.707107 + 0.707107i 0 −1.64189 + 1.51796i
523.3 0.258819 + 0.965926i 0 −0.866025 + 0.500000i −2.23385 0.0994727i 0 2.52756 + 0.781940i −0.707107 0.707107i 0 −0.482081 2.18348i
523.4 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0.126648 2.23248i 0 −2.47207 + 0.942805i −0.707107 0.707107i 0 2.18919 0.455475i
577.1 −0.258819 + 0.965926i 0 −0.866025 0.500000i −1.58356 1.57872i 0 2.22701 + 1.42843i 0.707107 0.707107i 0 1.93478 1.12100i
577.2 −0.258819 + 0.965926i 0 −0.866025 0.500000i −1.04129 + 1.97882i 0 −2.55046 0.703686i 0.707107 0.707107i 0 −1.64189 1.51796i
577.3 0.258819 0.965926i 0 −0.866025 0.500000i −2.23385 + 0.0994727i 0 2.52756 0.781940i −0.707107 + 0.707107i 0 −0.482081 + 2.18348i
577.4 0.258819 0.965926i 0 −0.866025 0.500000i 0.126648 + 2.23248i 0 −2.47207 0.942805i −0.707107 + 0.707107i 0 2.18919 + 0.455475i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.bv.a 16
3.b odd 2 1 210.2.u.a 16
5.c odd 4 1 630.2.bv.b 16
7.d odd 6 1 630.2.bv.b 16
15.d odd 2 1 1050.2.bc.h 16
15.e even 4 1 210.2.u.b yes 16
15.e even 4 1 1050.2.bc.g 16
21.g even 6 1 210.2.u.b yes 16
21.g even 6 1 1470.2.m.e 16
21.h odd 6 1 1470.2.m.d 16
35.k even 12 1 inner 630.2.bv.a 16
105.p even 6 1 1050.2.bc.g 16
105.w odd 12 1 210.2.u.a 16
105.w odd 12 1 1050.2.bc.h 16
105.w odd 12 1 1470.2.m.d 16
105.x even 12 1 1470.2.m.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.u.a 16 3.b odd 2 1
210.2.u.a 16 105.w odd 12 1
210.2.u.b yes 16 15.e even 4 1
210.2.u.b yes 16 21.g even 6 1
630.2.bv.a 16 1.a even 1 1 trivial
630.2.bv.a 16 35.k even 12 1 inner
630.2.bv.b 16 5.c odd 4 1
630.2.bv.b 16 7.d odd 6 1
1050.2.bc.g 16 15.e even 4 1
1050.2.bc.g 16 105.p even 6 1
1050.2.bc.h 16 15.d odd 2 1
1050.2.bc.h 16 105.w odd 12 1
1470.2.m.d 16 21.h odd 6 1
1470.2.m.d 16 105.w odd 12 1
1470.2.m.e 16 21.g even 6 1
1470.2.m.e 16 105.x even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\):

\( T_{11}^{16} + 4 T_{11}^{15} + 46 T_{11}^{14} + 192 T_{11}^{13} + 1530 T_{11}^{12} + 5596 T_{11}^{11} + 23088 T_{11}^{10} + 57924 T_{11}^{9} + 172687 T_{11}^{8} + 365284 T_{11}^{7} + 799216 T_{11}^{6} + 1104284 T_{11}^{5} + \cdots + 146689 \) Copy content Toggle raw display
\( T_{13}^{16} + 16 T_{13}^{15} + 128 T_{13}^{14} + 536 T_{13}^{13} + 1442 T_{13}^{12} + 4424 T_{13}^{11} + 29856 T_{13}^{10} + 126936 T_{13}^{9} + 304225 T_{13}^{8} + 370872 T_{13}^{7} + 1764384 T_{13}^{6} + \cdots + 171295744 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 12 T^{15} + 88 T^{14} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} + 8 T^{15} + 6 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} + 4 T^{15} + 46 T^{14} + \cdots + 146689 \) Copy content Toggle raw display
$13$ \( T^{16} + 16 T^{15} + \cdots + 171295744 \) Copy content Toggle raw display
$17$ \( T^{16} - 12 T^{15} + 144 T^{14} + \cdots + 16384 \) Copy content Toggle raw display
$19$ \( T^{16} + 8 T^{15} + \cdots + 5893325824 \) Copy content Toggle raw display
$23$ \( T^{16} + 32 T^{15} + \cdots + 82925569024 \) Copy content Toggle raw display
$29$ \( T^{16} + 216 T^{14} + \cdots + 939790336 \) Copy content Toggle raw display
$31$ \( T^{16} + 24 T^{15} + 86 T^{14} + \cdots + 85229824 \) Copy content Toggle raw display
$37$ \( T^{16} + 8 T^{15} + \cdots + 1586310022144 \) Copy content Toggle raw display
$41$ \( T^{16} + 484 T^{14} + \cdots + 729780649984 \) Copy content Toggle raw display
$43$ \( T^{16} + 24 T^{15} + \cdots + 396169216 \) Copy content Toggle raw display
$47$ \( T^{16} - 24 T^{15} + \cdots + 405330862336 \) Copy content Toggle raw display
$53$ \( T^{16} + 44 T^{15} + \cdots + 7800599041 \) Copy content Toggle raw display
$59$ \( T^{16} + 8 T^{15} + \cdots + 1714622464 \) Copy content Toggle raw display
$61$ \( T^{16} - 24 T^{15} + \cdots + 4676942565376 \) Copy content Toggle raw display
$67$ \( T^{16} - 36 T^{15} + \cdots + 13679819493376 \) Copy content Toggle raw display
$71$ \( (T^{8} - 16 T^{7} - 176 T^{6} + \cdots - 17030912)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + 40 T^{15} + \cdots + 34279432458496 \) Copy content Toggle raw display
$79$ \( T^{16} - 12 T^{15} + \cdots + 31950847504 \) Copy content Toggle raw display
$83$ \( T^{16} - 16 T^{15} + \cdots + 8632854701584 \) Copy content Toggle raw display
$89$ \( T^{16} - 16 T^{15} + 272 T^{14} + \cdots + 67108864 \) Copy content Toggle raw display
$97$ \( T^{16} - 44 T^{15} + 968 T^{14} + \cdots + 67108864 \) Copy content Toggle raw display
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