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Properties

Label 28.4.f.a
Level $28$
Weight $4$
Character orbit 28.f
Analytic conductor $1.652$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [28,4,Mod(3,28)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(28, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("28.3");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 28.f (of order \(6\), 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: \(1.65205348016\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{18} - 24 x^{17} + 28 x^{16} + 56 x^{15} - 192 x^{14} + 352 x^{13} - 448 x^{12} + \cdots + 1073741824 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{24} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} - \beta_1) q^{2} - \beta_{10} q^{3} - \beta_{14} q^{4} + ( - \beta_{7} + \beta_{4}) q^{5} + (\beta_{19} - \beta_{16} - \beta_{4} + \cdots - 1) q^{6}+ \cdots + ( - 2 \beta_{19} - \beta_{18} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - \beta_1) q^{2} - \beta_{10} q^{3} - \beta_{14} q^{4} + ( - \beta_{7} + \beta_{4}) q^{5} + (\beta_{19} - \beta_{16} - \beta_{4} + \cdots - 1) q^{6}+ \cdots + ( - \beta_{19} + 2 \beta_{18} + \cdots + 31) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{4} - 6 q^{5} + 72 q^{8} - 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 4 q^{4} - 6 q^{5} + 72 q^{8} - 56 q^{9} - 12 q^{10} - 168 q^{12} - 56 q^{14} - 104 q^{16} - 6 q^{17} + 68 q^{18} + 238 q^{21} - 184 q^{22} + 348 q^{24} - 36 q^{25} + 396 q^{26} + 448 q^{28} - 352 q^{29} + 644 q^{30} - 40 q^{32} + 30 q^{33} + 208 q^{36} + 258 q^{37} - 1620 q^{38} - 1548 q^{40} - 980 q^{42} - 1248 q^{44} - 504 q^{45} + 232 q^{46} - 644 q^{49} - 864 q^{50} + 2592 q^{52} + 570 q^{53} + 4572 q^{54} + 1904 q^{56} + 1452 q^{57} + 2244 q^{58} - 736 q^{60} + 294 q^{61} + 2560 q^{64} - 124 q^{65} - 4272 q^{66} - 6084 q^{68} - 4144 q^{70} - 4672 q^{72} + 966 q^{73} + 832 q^{74} - 378 q^{77} - 4056 q^{78} + 7032 q^{80} - 1262 q^{81} + 7692 q^{82} + 6188 q^{84} - 2980 q^{85} + 5696 q^{86} - 1396 q^{88} - 3186 q^{89} + 3312 q^{92} - 306 q^{93} - 6780 q^{94} - 11784 q^{96} - 4900 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{18} - 24 x^{17} + 28 x^{16} + 56 x^{15} - 192 x^{14} + 352 x^{13} - 448 x^{12} + \cdots + 1073741824 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 793 \nu^{19} + 7760 \nu^{18} + 70462 \nu^{17} - 82360 \nu^{16} + 36188 \nu^{15} + \cdots - 940060966912 ) / 2280224980992 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 647 \nu^{19} - 13656 \nu^{18} - 111550 \nu^{17} + 323784 \nu^{16} - 943772 \nu^{15} + \cdots + 3620321886208 ) / 1140112490496 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 485 \nu^{19} + 4503 \nu^{18} - 3958 \nu^{17} + 874 \nu^{16} - 4612 \nu^{15} + \cdots - 53217329152 ) / 142514061312 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3865 \nu^{19} - 96776 \nu^{18} + 881390 \nu^{17} + 176568 \nu^{16} - 2032324 \nu^{15} + \cdots - 46140162572288 ) / 2280224980992 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15263 \nu^{19} - 111128 \nu^{18} + 99154 \nu^{17} + 513800 \nu^{16} + 294276 \nu^{15} + \cdots - 3317191147520 ) / 2280224980992 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1751 \nu^{19} + 1586 \nu^{18} + 12018 \nu^{17} + 98900 \nu^{16} - 115692 \nu^{15} + \cdots - 932611883008 ) / 142514061312 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 15231 \nu^{19} - 44416 \nu^{18} + 27030 \nu^{17} + 237384 \nu^{16} + 2711148 \nu^{15} + \cdots + 2908095512576 ) / 1140112490496 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 39171 \nu^{19} + 88016 \nu^{18} + 123178 \nu^{17} - 159272 \nu^{16} + 596340 \nu^{15} + \cdots - 10618098679808 ) / 2280224980992 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 31255 \nu^{19} + 175232 \nu^{18} - 16402 \nu^{17} + 13480 \nu^{16} - 1137668 \nu^{15} + \cdots - 84422950912 ) / 2280224980992 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 11957 \nu^{19} - 10004 \nu^{18} - 13414 \nu^{17} + 144736 \nu^{16} - 1989964 \nu^{15} + \cdots - 1149675503616 ) / 570056245248 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 54759 \nu^{19} - 23200 \nu^{18} - 57294 \nu^{17} - 913128 \nu^{16} - 334268 \nu^{15} + \cdots + 4940420349952 ) / 2280224980992 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 4503 \nu^{19} - 2988 \nu^{18} + 12514 \nu^{17} - 18192 \nu^{16} + 81796 \nu^{15} + \cdots - 520764784640 ) / 142514061312 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 24923 \nu^{19} + 200112 \nu^{18} + 109250 \nu^{17} + 300344 \nu^{16} - 1935100 \nu^{15} + \cdots - 15698508120064 ) / 1140112490496 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 36627 \nu^{19} - 138788 \nu^{18} - 194058 \nu^{17} + 142160 \nu^{16} + \cdots + 13000597569536 ) / 1140112490496 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 22105 \nu^{19} - 251020 \nu^{18} + 371394 \nu^{17} + 18352 \nu^{16} + \cdots - 11577352781824 ) / 1140112490496 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 23993 \nu^{19} + 6344 \nu^{18} + 14094 \nu^{17} - 12136 \nu^{16} + 12924 \nu^{15} + \cdots - 3730447532032 ) / 570056245248 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 80381 \nu^{19} - 19988 \nu^{18} + 382666 \nu^{17} + 429344 \nu^{16} + \cdots - 19108577935360 ) / 1140112490496 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} + \beta_{18} - \beta_{17} - \beta_{15} - \beta_{11} + \beta_{10} + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{17} - 2 \beta_{16} - 2 \beta_{14} - 2 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} - 2 \beta_{9} + \cdots - 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{19} - 2 \beta_{18} - 4 \beta_{17} - 2 \beta_{16} - 2 \beta_{15} + 2 \beta_{14} + 2 \beta_{13} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12 \beta_{19} + 28 \beta_{18} + 12 \beta_{16} + 8 \beta_{14} - 4 \beta_{12} - 40 \beta_{7} - 8 \beta_{5} + \cdots + 144 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 28 \beta_{19} - 4 \beta_{17} + 12 \beta_{16} - 36 \beta_{14} + 68 \beta_{13} + 4 \beta_{12} - 72 \beta_{11} + \cdots - 700 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 32 \beta_{19} - 80 \beta_{17} - 16 \beta_{16} - 40 \beta_{15} + 40 \beta_{14} + 40 \beta_{13} + \cdots + 40 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 232 \beta_{19} - 360 \beta_{18} - 72 \beta_{17} + 288 \beta_{16} + 56 \beta_{15} + 688 \beta_{14} + \cdots + 1344 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1360 \beta_{19} - 448 \beta_{17} - 1504 \beta_{16} + 1184 \beta_{14} + 1216 \beta_{13} + 448 \beta_{12} + \cdots + 15232 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2016 \beta_{19} + 80 \beta_{18} - 6944 \beta_{17} - 1008 \beta_{16} - 1680 \beta_{15} + 3472 \beta_{14} + \cdots + 3472 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1344 \beta_{19} - 768 \beta_{18} + 352 \beta_{17} + 2720 \beta_{16} + 1376 \beta_{15} - 20224 \beta_{14} + \cdots + 70016 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 7264 \beta_{19} + 17184 \beta_{17} - 16864 \beta_{16} - 45664 \beta_{14} - 2336 \beta_{13} + \cdots + 60896 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 62848 \beta_{19} + 49984 \beta_{18} + 66048 \beta_{17} + 31424 \beta_{16} + 120064 \beta_{15} + \cdots - 33024 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 90048 \beta_{19} - 32448 \beta_{18} - 5696 \beta_{17} + 148864 \beta_{16} + 58816 \beta_{15} + \cdots - 216064 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 86784 \beta_{19} + 209536 \beta_{17} - 297088 \beta_{16} + 119168 \beta_{14} - 123520 \beta_{13} + \cdots + 3797888 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 220416 \beta_{19} - 1540480 \beta_{18} - 1132288 \beta_{17} + 110208 \beta_{16} - 101248 \beta_{15} + \cdots + 566144 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 509184 \beta_{19} - 1090304 \beta_{18} + 1152000 \beta_{17} - 349952 \beta_{16} - 859136 \beta_{15} + \cdots + 57238528 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 155392 \beta_{19} + 2562304 \beta_{17} + 3511552 \beta_{16} + 28631296 \beta_{14} + 3200768 \beta_{13} + \cdots + 5453568 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/28\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\)
\(\chi(n)\) \(-1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
1.31147 + 2.50600i
2.59951 + 1.11469i
−1.03939 + 2.63053i
2.82698 + 0.0903966i
−1.75840 + 2.21540i
2.19431 − 1.78465i
−2.82600 + 0.117237i
0.448398 − 2.79266i
−2.26510 − 1.69390i
−1.49178 − 2.40304i
1.31147 − 2.50600i
2.59951 − 1.11469i
−1.03939 − 2.63053i
2.82698 − 0.0903966i
−1.75840 − 2.21540i
2.19431 + 1.78465i
−2.82600 − 0.117237i
0.448398 + 2.79266i
−2.26510 + 1.69390i
−1.49178 + 2.40304i
−2.82600 − 0.117237i 0.0469307 − 0.0812864i 7.97251 + 0.662623i 12.4861 − 7.20883i −0.142156 + 0.224213i 15.0686 − 10.7674i −22.4526 − 2.80725i 13.4956 + 23.3751i −36.1307 + 18.9083i
3.2 −2.26510 + 1.69390i −1.67134 + 2.89484i 2.26140 − 7.67373i −15.9583 + 9.21354i −1.11782 − 9.38820i −15.4841 − 10.1609i 7.87623 + 21.2124i 7.91327 + 13.7062i 20.5404 − 47.9014i
3.3 −1.75840 − 2.21540i −3.44104 + 5.96006i −1.81602 + 7.79115i −4.17670 + 2.41142i 19.2547 − 2.85690i −5.03893 + 17.8216i 20.4539 − 9.67677i −10.1816 − 17.6350i 12.6866 + 5.01283i
3.4 −1.49178 + 2.40304i 4.65345 − 8.06002i −3.54920 − 7.16960i 5.11538 − 2.95337i 12.4266 + 23.2062i 1.92900 + 18.4195i 22.5235 + 2.16656i −29.8092 − 51.6311i −0.533949 + 16.6982i
3.5 −1.03939 − 2.63053i 3.44104 − 5.96006i −5.83932 + 5.46830i −4.17670 + 2.41142i −19.2547 − 2.85690i 5.03893 − 17.8216i 20.4539 + 9.67677i −10.1816 − 17.6350i 10.6845 + 8.48051i
3.6 0.448398 + 2.79266i −2.11164 + 3.65747i −7.59788 + 2.50445i 1.03358 − 0.596737i −11.1609 − 4.25709i 16.7651 + 7.86959i −10.4009 − 20.0953i 4.58193 + 7.93614i 2.12994 + 2.61886i
3.7 1.31147 − 2.50600i −0.0469307 + 0.0812864i −4.56010 − 6.57309i 12.4861 − 7.20883i 0.142156 + 0.224213i −15.0686 + 10.7674i −22.4526 + 2.80725i 13.4956 + 23.3751i −1.69029 − 40.7443i
3.8 2.19431 + 1.78465i 2.11164 − 3.65747i 1.63002 + 7.83218i 1.03358 − 0.596737i 11.1609 − 4.25709i −16.7651 − 7.86959i −10.4009 + 20.0953i 4.58193 + 7.93614i 3.33297 + 0.535152i
3.9 2.59951 − 1.11469i 1.67134 − 2.89484i 5.51494 − 5.79529i −15.9583 + 9.21354i 1.11782 − 9.38820i 15.4841 + 10.1609i 7.87623 − 21.2124i 7.91327 + 13.7062i −31.2137 + 41.7393i
3.10 2.82698 − 0.0903966i −4.65345 + 8.06002i 7.98366 − 0.511099i 5.11538 − 2.95337i −12.4266 + 23.2062i −1.92900 − 18.4195i 22.5235 − 2.16656i −29.8092 − 51.6311i 14.1941 − 8.81153i
19.1 −2.82600 + 0.117237i 0.0469307 + 0.0812864i 7.97251 − 0.662623i 12.4861 + 7.20883i −0.142156 − 0.224213i 15.0686 + 10.7674i −22.4526 + 2.80725i 13.4956 − 23.3751i −36.1307 − 18.9083i
19.2 −2.26510 − 1.69390i −1.67134 − 2.89484i 2.26140 + 7.67373i −15.9583 − 9.21354i −1.11782 + 9.38820i −15.4841 + 10.1609i 7.87623 − 21.2124i 7.91327 − 13.7062i 20.5404 + 47.9014i
19.3 −1.75840 + 2.21540i −3.44104 − 5.96006i −1.81602 − 7.79115i −4.17670 − 2.41142i 19.2547 + 2.85690i −5.03893 − 17.8216i 20.4539 + 9.67677i −10.1816 + 17.6350i 12.6866 − 5.01283i
19.4 −1.49178 − 2.40304i 4.65345 + 8.06002i −3.54920 + 7.16960i 5.11538 + 2.95337i 12.4266 − 23.2062i 1.92900 − 18.4195i 22.5235 − 2.16656i −29.8092 + 51.6311i −0.533949 − 16.6982i
19.5 −1.03939 + 2.63053i 3.44104 + 5.96006i −5.83932 − 5.46830i −4.17670 − 2.41142i −19.2547 + 2.85690i 5.03893 + 17.8216i 20.4539 − 9.67677i −10.1816 + 17.6350i 10.6845 − 8.48051i
19.6 0.448398 − 2.79266i −2.11164 − 3.65747i −7.59788 − 2.50445i 1.03358 + 0.596737i −11.1609 + 4.25709i 16.7651 − 7.86959i −10.4009 + 20.0953i 4.58193 − 7.93614i 2.12994 − 2.61886i
19.7 1.31147 + 2.50600i −0.0469307 − 0.0812864i −4.56010 + 6.57309i 12.4861 + 7.20883i 0.142156 − 0.224213i −15.0686 − 10.7674i −22.4526 − 2.80725i 13.4956 − 23.3751i −1.69029 + 40.7443i
19.8 2.19431 − 1.78465i 2.11164 + 3.65747i 1.63002 − 7.83218i 1.03358 + 0.596737i 11.1609 + 4.25709i −16.7651 + 7.86959i −10.4009 − 20.0953i 4.58193 − 7.93614i 3.33297 − 0.535152i
19.9 2.59951 + 1.11469i 1.67134 + 2.89484i 5.51494 + 5.79529i −15.9583 − 9.21354i 1.11782 + 9.38820i 15.4841 − 10.1609i 7.87623 + 21.2124i 7.91327 − 13.7062i −31.2137 − 41.7393i
19.10 2.82698 + 0.0903966i −4.65345 − 8.06002i 7.98366 + 0.511099i 5.11538 + 2.95337i −12.4266 − 23.2062i −1.92900 + 18.4195i 22.5235 + 2.16656i −29.8092 + 51.6311i 14.1941 + 8.81153i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.d odd 6 1 inner
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 28.4.f.a ✓ 20
4.b odd 2 1 inner 28.4.f.a ✓ 20
7.b odd 2 1 196.4.f.d 20
7.c even 3 1 196.4.d.b 20
7.c even 3 1 196.4.f.d 20
7.d odd 6 1 inner 28.4.f.a ✓ 20
7.d odd 6 1 196.4.d.b 20
8.b even 2 1 448.4.p.h 20
8.d odd 2 1 448.4.p.h 20
28.d even 2 1 196.4.f.d 20
28.f even 6 1 inner 28.4.f.a ✓ 20
28.f even 6 1 196.4.d.b 20
28.g odd 6 1 196.4.d.b 20
28.g odd 6 1 196.4.f.d 20
56.j odd 6 1 448.4.p.h 20
56.m even 6 1 448.4.p.h 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.f.a ✓ 20 1.a even 1 1 trivial
28.4.f.a ✓ 20 4.b odd 2 1 inner
28.4.f.a ✓ 20 7.d odd 6 1 inner
28.4.f.a ✓ 20 28.f even 6 1 inner
196.4.d.b 20 7.c even 3 1
196.4.d.b 20 7.d odd 6 1
196.4.d.b 20 28.f even 6 1
196.4.d.b 20 28.g odd 6 1
196.4.f.d 20 7.b odd 2 1
196.4.f.d 20 7.c even 3 1
196.4.f.d 20 28.d even 2 1
196.4.f.d 20 28.g odd 6 1
448.4.p.h 20 8.b even 2 1
448.4.p.h 20 8.d odd 2 1
448.4.p.h 20 56.j odd 6 1
448.4.p.h 20 56.m even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(28, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 1073741824 \) Copy content Toggle raw display
$3$ \( T^{20} + 163 T^{18} + \cdots + 51883209 \) Copy content Toggle raw display
$5$ \( (T^{10} + 3 T^{9} + \cdots + 81588675)^{2} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 22\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 20\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 10072189747200)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 60\!\cdots\!75)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 17\!\cdots\!61 \) Copy content Toggle raw display
$29$ \( (T^{5} + 88 T^{4} + \cdots + 5066614016)^{4} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 43\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 48\!\cdots\!25)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 14\!\cdots\!72)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 55\!\cdots\!56)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 90\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 23\!\cdots\!25)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 53\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 25\!\cdots\!47)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 16\!\cdots\!21 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 39\!\cdots\!75)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 41\!\cdots\!08)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 99\!\cdots\!27)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
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