Properties

Label 45.4.e.c
Level $45$
Weight $4$
Character orbit 45.e
Analytic conductor $2.655$
Analytic rank $0$
Dimension $14$
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] = [45,4,Mod(16,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.16");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 45.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: \(2.65508595026\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} + 48 x^{12} - 60 x^{11} + 1605 x^{10} - 1800 x^{9} + 23232 x^{8} - 2346 x^{7} + 209529 x^{6} - 55412 x^{5} + 765088 x^{4} + 276096 x^{3} + 1572480 x^{2} + \cdots + 82944 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{6} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - \beta_1) q^{2} + ( - \beta_{10} + \beta_{7}) q^{3} + ( - \beta_{11} + \beta_{10} - \beta_{5} + 5 \beta_{4} - \beta_1 - 5) q^{4} + ( - 5 \beta_{4} + 5) q^{5} + ( - \beta_{11} - \beta_{9} + \beta_{6} - \beta_{5} + 4 \beta_{4} - \beta_{2} + 2 \beta_1 - 4) q^{6} + (\beta_{13} + \beta_{11} - \beta_{10} + \beta_{9} - \beta_{6} - 3 \beta_{4} - 1) q^{7} + ( - \beta_{12} - \beta_{11} - \beta_{10} - 2 \beta_{9} - \beta_{8} + \beta_{5} + \beta_{4} - \beta_{3} - 5 \beta_{2} + \cdots - 1) q^{8}+ \cdots + (\beta_{13} - \beta_{11} + \beta_{8} - \beta_{7} + \beta_{6} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - \beta_1) q^{2} + ( - \beta_{10} + \beta_{7}) q^{3} + ( - \beta_{11} + \beta_{10} - \beta_{5} + 5 \beta_{4} - \beta_1 - 5) q^{4} + ( - 5 \beta_{4} + 5) q^{5} + ( - \beta_{11} - \beta_{9} + \beta_{6} - \beta_{5} + 4 \beta_{4} - \beta_{2} + 2 \beta_1 - 4) q^{6} + (\beta_{13} + \beta_{11} - \beta_{10} + \beta_{9} - \beta_{6} - 3 \beta_{4} - 1) q^{7} + ( - \beta_{12} - \beta_{11} - \beta_{10} - 2 \beta_{9} - \beta_{8} + \beta_{5} + \beta_{4} - \beta_{3} - 5 \beta_{2} + \cdots - 1) q^{8}+ \cdots + (5 \beta_{13} + 48 \beta_{12} + 33 \beta_{11} + 73 \beta_{10} + 17 \beta_{9} + \cdots + 301) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 2 q^{2} - 5 q^{3} - 36 q^{4} + 35 q^{5} - 31 q^{6} - 22 q^{7} - 36 q^{8} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 2 q^{2} - 5 q^{3} - 36 q^{4} + 35 q^{5} - 31 q^{6} - 22 q^{7} - 36 q^{8} + 17 q^{9} + 20 q^{10} + 23 q^{11} + 287 q^{12} - 96 q^{13} - 21 q^{14} - 20 q^{15} - 324 q^{16} - 322 q^{17} - 89 q^{18} + 558 q^{19} + 180 q^{20} + 180 q^{21} - 311 q^{22} + 96 q^{23} + 48 q^{24} - 175 q^{25} + 716 q^{26} - 470 q^{27} + 674 q^{28} - 296 q^{29} + 80 q^{30} - 244 q^{31} - 314 q^{32} - 211 q^{33} - 125 q^{34} - 220 q^{35} - 2399 q^{36} + 808 q^{37} + 305 q^{38} + 634 q^{39} - 90 q^{40} - 47 q^{41} + 1941 q^{42} - 525 q^{43} - 110 q^{44} + 185 q^{45} + 1434 q^{46} + 164 q^{47} + 2051 q^{48} - 1225 q^{49} + 50 q^{50} + 1517 q^{51} - 1682 q^{52} - 1012 q^{53} - 4066 q^{54} + 230 q^{55} - 981 q^{56} + 337 q^{57} - 1183 q^{58} - 85 q^{59} + 65 q^{60} - 828 q^{61} + 1572 q^{62} - 828 q^{63} + 4472 q^{64} + 480 q^{65} + 4930 q^{66} - 1093 q^{67} + 2473 q^{68} - 822 q^{69} + 105 q^{70} - 656 q^{71} - 4626 q^{72} + 4170 q^{73} - 1316 q^{74} + 25 q^{75} - 2789 q^{76} + 24 q^{77} - 5314 q^{78} - 2110 q^{79} - 3240 q^{80} - 2167 q^{81} - 124 q^{82} + 1290 q^{83} + 5775 q^{84} - 805 q^{85} - 2569 q^{86} + 3604 q^{87} - 2271 q^{88} + 6096 q^{89} + 730 q^{90} + 6676 q^{91} + 2763 q^{92} - 696 q^{93} + 517 q^{94} + 1395 q^{95} - 593 q^{96} - 1787 q^{97} - 2558 q^{98} + 2320 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} + 48 x^{12} - 60 x^{11} + 1605 x^{10} - 1800 x^{9} + 23232 x^{8} - 2346 x^{7} + 209529 x^{6} - 55412 x^{5} + 765088 x^{4} + 276096 x^{3} + 1572480 x^{2} + \cdots + 82944 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 20061126179 \nu^{13} + 306116042626 \nu^{12} - 1952416155384 \nu^{11} + 13978725434868 \nu^{10} - 67214635202751 \nu^{9} + \cdots + 81\!\cdots\!72 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 102404815617 \nu^{13} + 5266027591948 \nu^{12} - 8326481101332 \nu^{11} + 268246879929564 \nu^{10} + \cdots + 63\!\cdots\!56 ) / 68\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2939472898789 \nu^{13} + 5818762419041 \nu^{12} - 140176351013994 \nu^{11} + 170511125461188 \nu^{10} + \cdots + 93\!\cdots\!52 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 376371909409 \nu^{13} + 779547299216 \nu^{12} - 17633627499780 \nu^{11} + 23331293206188 \nu^{10} + \cdots - 63\!\cdots\!28 ) / 81\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2181076654577 \nu^{13} + 4927484464048 \nu^{12} - 102520174110372 \nu^{11} + 135970911492204 \nu^{10} + \cdots - 22\!\cdots\!24 ) / 40\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 11125627233907 \nu^{13} + 27262271328908 \nu^{12} - 529205531611572 \nu^{11} + 877253848876644 \nu^{10} + \cdots + 11\!\cdots\!76 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11877650203069 \nu^{13} - 84950703264836 \nu^{12} + 708621055239924 \nu^{11} + \cdots - 58\!\cdots\!92 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10816153989491 \nu^{13} - 13376805115054 \nu^{12} + 497974097019636 \nu^{11} - 253186185774972 \nu^{10} + \cdots - 56\!\cdots\!88 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4781648360579 \nu^{13} - 8884383807886 \nu^{12} + 231881123650104 \nu^{11} - 261650594664228 \nu^{10} + \cdots + 41\!\cdots\!28 ) / 40\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 6918798848131 \nu^{13} + 13573388130414 \nu^{12} - 327948078905676 \nu^{11} + 388609132067052 \nu^{10} + \cdots + 26\!\cdots\!08 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2499552103127 \nu^{13} + 3909850857688 \nu^{12} - 117122204938752 \nu^{11} + 101188546669524 \nu^{10} + \cdots + 88\!\cdots\!96 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 47810889109763 \nu^{13} + 95353418254572 \nu^{12} + \cdots + 18\!\cdots\!84 ) / 68\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - \beta_{10} + \beta_{7} - 13\beta_{4} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{12} - \beta_{11} - \beta_{10} - 2\beta_{9} - \beta_{8} + \beta_{5} + \beta_{4} - \beta_{3} - 21\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{13} - 3 \beta_{12} - 30 \beta_{11} + 24 \beta_{10} + 3 \beta_{8} + 9 \beta_{7} - 27 \beta_{5} + 294 \beta_{4} - 25 \beta _1 - 291 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 30 \beta_{13} + 66 \beta_{12} + 80 \beta_{11} + 4 \beta_{10} + 78 \beta_{9} + 26 \beta_{7} - 78 \beta_{6} + 36 \beta_{5} - 62 \beta_{4} - 36 \beta_{3} + 529 \beta_{2} - 529 \beta _1 + 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 130 \beta_{12} + 130 \beta_{11} + 382 \beta_{10} + 8 \beta_{9} - 122 \beta_{8} - 1029 \beta_{7} + 851 \beta_{5} - 106 \beta_{4} - 146 \beta_{3} + 585 \beta_{2} + 7591 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 819 \beta_{13} - 771 \beta_{12} - 1515 \beta_{11} + 687 \beta_{10} + 819 \beta_{8} - 939 \beta_{7} + 2490 \beta_{6} - 2658 \beta_{5} + 3033 \beta_{4} + 1914 \beta_{3} + 14197 \beta _1 - 3600 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 4053 \beta_{13} + 840 \beta_{12} + 19675 \beta_{11} - 30877 \beta_{10} - 480 \beta_{9} + 18262 \beta_{7} + 480 \beta_{6} - 4365 \beta_{5} - 202939 \beta_{4} + 4365 \beta_{3} - 13585 \beta_{2} + \cdots - 3045 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 34738 \beta_{12} - 34738 \beta_{11} - 29248 \beta_{10} - 74966 \beta_{9} - 22408 \beta_{8} + 9672 \beta_{7} + 48166 \beta_{5} + 14278 \beta_{4} - 19768 \beta_{3} - 391569 \beta_{2} + \cdots + 132320 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 125772 \beta_{13} - 166692 \beta_{12} - 681957 \beta_{11} + 522429 \beta_{10} + 125772 \beta_{8} + 385404 \beta_{7} - 18264 \beta_{6} - 547401 \beta_{5} + 5722041 \beta_{4} + 32136 \beta_{3} + \cdots - 5637885 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 622773 \beta_{13} + 1551186 \beta_{12} + 2507858 \beta_{11} + 304264 \beta_{10} + 2204082 \beta_{9} + 579293 \beta_{7} - 2204082 \beta_{6} + 1029213 \beta_{5} - 5182736 \beta_{4} + \cdots + 376317 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3988855 \beta_{12} + 3988855 \beta_{11} + 11399653 \beta_{10} + 566912 \beta_{9} - 3783527 \beta_{8} - 26396937 \beta_{7} + 19434914 \beta_{5} - 2321479 \beta_{4} + \cdots + 161007274 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 17562210 \beta_{13} - 14227458 \beta_{12} - 44588544 \beta_{11} + 17387382 \beta_{10} + 17562210 \beta_{8} - 25965810 \beta_{7} + 64059678 \beta_{6} - 74597040 \beta_{5} + \cdots - 170775582 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(-1 + \beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
−2.69252 4.66357i
−1.52087 2.63422i
−0.785104 1.35984i
0.112625 + 0.195072i
1.09722 + 1.90044i
2.13089 + 3.69081i
2.65775 + 4.60336i
−2.69252 + 4.66357i
−1.52087 + 2.63422i
−0.785104 + 1.35984i
0.112625 0.195072i
1.09722 1.90044i
2.13089 3.69081i
2.65775 4.60336i
−2.69252 + 4.66357i −4.09553 + 3.19791i −10.4993 18.1853i 2.50000 + 4.33013i −3.88642 27.7102i −6.28510 + 10.8861i 69.9976 6.54673 26.1943i −26.9252
16.2 −1.52087 + 2.63422i 4.01867 + 3.29398i −0.626094 1.08443i 2.50000 + 4.33013i −14.7890 + 5.57636i 6.85611 11.8751i −20.5251 5.29939 + 26.4748i −15.2087
16.3 −0.785104 + 1.35984i −3.89934 3.43441i 2.76722 + 4.79297i 2.50000 + 4.33013i 7.73164 2.60611i −17.1199 + 29.6525i −21.2519 3.40971 + 26.7838i −7.85104
16.4 0.112625 0.195072i 2.06755 4.76710i 3.97463 + 6.88426i 2.50000 + 4.33013i −0.697071 0.940215i 15.5970 27.0148i 3.59257 −18.4505 19.7124i 1.12625
16.5 1.09722 1.90044i 0.206141 + 5.19206i 1.59221 + 2.75778i 2.50000 + 4.33013i 10.0934 + 5.30509i −1.38302 + 2.39547i 24.5436 −26.9150 + 2.14059i 10.9722
16.6 2.13089 3.69081i 4.39640 2.76977i −5.08138 8.80120i 2.50000 + 4.33013i −0.854448 22.1284i −15.3820 + 26.6423i −9.21718 11.6567 24.3541i 21.3089
16.7 2.65775 4.60336i −5.19389 + 0.153351i −10.1273 17.5410i 2.50000 + 4.33013i −13.0981 + 24.3169i 6.71686 11.6339i −65.1396 26.9530 1.59298i 26.5775
31.1 −2.69252 4.66357i −4.09553 3.19791i −10.4993 + 18.1853i 2.50000 4.33013i −3.88642 + 27.7102i −6.28510 10.8861i 69.9976 6.54673 + 26.1943i −26.9252
31.2 −1.52087 2.63422i 4.01867 3.29398i −0.626094 + 1.08443i 2.50000 4.33013i −14.7890 5.57636i 6.85611 + 11.8751i −20.5251 5.29939 26.4748i −15.2087
31.3 −0.785104 1.35984i −3.89934 + 3.43441i 2.76722 4.79297i 2.50000 4.33013i 7.73164 + 2.60611i −17.1199 29.6525i −21.2519 3.40971 26.7838i −7.85104
31.4 0.112625 + 0.195072i 2.06755 + 4.76710i 3.97463 6.88426i 2.50000 4.33013i −0.697071 + 0.940215i 15.5970 + 27.0148i 3.59257 −18.4505 + 19.7124i 1.12625
31.5 1.09722 + 1.90044i 0.206141 5.19206i 1.59221 2.75778i 2.50000 4.33013i 10.0934 5.30509i −1.38302 2.39547i 24.5436 −26.9150 2.14059i 10.9722
31.6 2.13089 + 3.69081i 4.39640 + 2.76977i −5.08138 + 8.80120i 2.50000 4.33013i −0.854448 + 22.1284i −15.3820 26.6423i −9.21718 11.6567 + 24.3541i 21.3089
31.7 2.65775 + 4.60336i −5.19389 0.153351i −10.1273 + 17.5410i 2.50000 4.33013i −13.0981 24.3169i 6.71686 + 11.6339i −65.1396 26.9530 + 1.59298i 26.5775
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.7
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.4.e.c 14
3.b odd 2 1 135.4.e.c 14
5.b even 2 1 225.4.e.d 14
5.c odd 4 2 225.4.k.d 28
9.c even 3 1 inner 45.4.e.c 14
9.c even 3 1 405.4.a.m 7
9.d odd 6 1 135.4.e.c 14
9.d odd 6 1 405.4.a.n 7
45.h odd 6 1 2025.4.a.ba 7
45.j even 6 1 225.4.e.d 14
45.j even 6 1 2025.4.a.bb 7
45.k odd 12 2 225.4.k.d 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.e.c 14 1.a even 1 1 trivial
45.4.e.c 14 9.c even 3 1 inner
135.4.e.c 14 3.b odd 2 1
135.4.e.c 14 9.d odd 6 1
225.4.e.d 14 5.b even 2 1
225.4.e.d 14 45.j even 6 1
225.4.k.d 28 5.c odd 4 2
225.4.k.d 28 45.k odd 12 2
405.4.a.m 7 9.c even 3 1
405.4.a.n 7 9.d odd 6 1
2025.4.a.ba 7 45.h odd 6 1
2025.4.a.bb 7 45.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} - 2 T_{2}^{13} + 48 T_{2}^{12} - 60 T_{2}^{11} + 1605 T_{2}^{10} - 1800 T_{2}^{9} + 23232 T_{2}^{8} - 2346 T_{2}^{7} + 209529 T_{2}^{6} - 55412 T_{2}^{5} + 765088 T_{2}^{4} + 276096 T_{2}^{3} + 1572480 T_{2}^{2} + \cdots + 82944 \) acting on \(S_{4}^{\mathrm{new}}(45, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - 2 T^{13} + 48 T^{12} + \cdots + 82944 \) Copy content Toggle raw display
$3$ \( T^{14} + 5 T^{13} + \cdots + 10460353203 \) Copy content Toggle raw display
$5$ \( (T^{2} - 5 T + 25)^{7} \) Copy content Toggle raw display
$7$ \( T^{14} + 22 T^{13} + \cdots + 44\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( T^{14} - 23 T^{13} + \cdots + 40\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( T^{14} + 96 T^{13} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( (T^{7} + 161 T^{6} + \cdots - 60588009792)^{2} \) Copy content Toggle raw display
$19$ \( (T^{7} - 279 T^{6} + \cdots - 1377989598400)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} - 96 T^{13} + \cdots + 16\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{14} + 296 T^{13} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( T^{14} + 244 T^{13} + \cdots + 86\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( (T^{7} - 404 T^{6} + \cdots - 83646911884544)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + 47 T^{13} + \cdots + 23\!\cdots\!21 \) Copy content Toggle raw display
$43$ \( T^{14} + 525 T^{13} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{14} - 164 T^{13} + \cdots + 42\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{7} + 506 T^{6} + \cdots - 11\!\cdots\!92)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + 85 T^{13} + \cdots + 35\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{14} + 828 T^{13} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{14} + 1093 T^{13} + \cdots + 85\!\cdots\!09 \) Copy content Toggle raw display
$71$ \( (T^{7} + 328 T^{6} + \cdots - 32\!\cdots\!28)^{2} \) Copy content Toggle raw display
$73$ \( (T^{7} - 2085 T^{6} + \cdots + 82\!\cdots\!12)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + 2110 T^{13} + \cdots + 26\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{14} - 1290 T^{13} + \cdots + 43\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{7} - 3048 T^{6} + \cdots - 32\!\cdots\!50)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + 1787 T^{13} + \cdots + 22\!\cdots\!76 \) Copy content Toggle raw display
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