Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [325,2,Mod(49,325)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(325, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("325.49");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 325 = 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 325.m (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: | \(2.59513806569\) |
| 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: |
\( x^{20} + 16 x^{18} + 172 x^{16} + 1018 x^{14} + 4330 x^{12} + 9943 x^{10} + 16225 x^{8} + 14698 x^{6} + \cdots + 729 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 1 \) |
| 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.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{20} + 16 x^{18} + 172 x^{16} + 1018 x^{14} + 4330 x^{12} + 9943 x^{10} + 16225 x^{8} + 14698 x^{6} + \cdots + 729 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 21 \nu^{18} - 389 \nu^{16} - 4228 \nu^{14} - 26928 \nu^{12} - 108416 \nu^{10} - 241753 \nu^{8} + \cdots - 115236 ) / 345645 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 62301822 \nu^{18} - 468089890 \nu^{16} - 11053228346 \nu^{14} - 162813284061 \nu^{12} + \cdots - 1636187231925 ) / 694103895945 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 359598672 \nu^{18} + 5472475123 \nu^{16} + 57248165642 \nu^{14} + 316757341596 \nu^{12} + \cdots + 657482244777 ) / 694103895945 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 1274276258 \nu^{18} - 16769907491 \nu^{16} - 161883652190 \nu^{14} - 687153208895 \nu^{12} + \cdots + 3038729439465 ) / 2082311687835 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 1298117264 \nu^{18} - 20265092252 \nu^{16} - 215629354508 \nu^{14} - 1241099688002 \nu^{12} + \cdots - 3349745883504 ) / 2082311687835 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 1075501120 \nu^{18} - 19349305429 \nu^{16} - 217803254710 \nu^{14} - 1439834678185 \nu^{12} + \cdots - 6799982394600 ) / 694103895945 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 4191129944 \nu^{18} - 65262725027 \nu^{16} - 690040421723 \nu^{14} - 3925995900605 \nu^{12} + \cdots + 203109612426 ) / 2082311687835 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 5850111196 \nu^{18} + 92795389735 \nu^{16} + 989872032505 \nu^{14} + 5767125127426 \nu^{12} + \cdots + 6571581523398 ) / 2082311687835 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 4268 \nu^{19} - 67721 \nu^{17} - 723593 \nu^{15} - 4230668 \nu^{13} - 17753384 \nu^{11} + \cdots - 22640175 \nu ) / 9332415 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 1192547842 \nu^{18} - 18255628888 \nu^{16} - 191904050660 \nu^{14} - 1072318059621 \nu^{12} + \cdots + 658112116743 ) / 231367965315 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 1298117264 \nu^{19} + 20265092252 \nu^{17} + 215629354508 \nu^{15} + \cdots + 3349745883504 \nu ) / 2082311687835 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 15780448463 \nu^{19} + 243916627211 \nu^{17} + 2587571812238 \nu^{15} + \cdots - 3396821295672 \nu ) / 18740805190515 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 19334308303 \nu^{19} + 339177356107 \nu^{17} + 3762157943242 \nu^{15} + \cdots + 32224578589062 \nu ) / 18740805190515 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 22000955230 \nu^{19} - 378279332557 \nu^{17} - 4173199804117 \nu^{15} + \cdots - 68550172212186 \nu ) / 18740805190515 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 25884365029 \nu^{19} - 371751346507 \nu^{17} - 3786237280411 \nu^{15} + \cdots + 27516846408336 \nu ) / 18740805190515 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 15159515072 \nu^{19} + 211796278316 \nu^{17} + 2126989502825 \nu^{15} + \cdots - 66925603044873 \nu ) / 6246935063505 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 66168954425 \nu^{19} - 1038133259321 \nu^{17} - 11034537642299 \nu^{15} + \cdots - 59396843535624 \nu ) / 18740805190515 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 43389830780 \nu^{19} - 703477046918 \nu^{17} - 7593830983475 \nu^{15} + \cdots - 85847391115911 \nu ) / 6246935063505 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{11} + \beta_{9} - \beta_{8} + \beta_{7} - 4\beta_{6} + \beta_{5} - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{18} - \beta_{16} - \beta_{15} - 2\beta_{13} + 5\beta_{12} + \beta_{10} - 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{9} - \beta_{8} - 7\beta_{7} + 24\beta_{6} - 7\beta_{5} + 7\beta_{3} \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{19} - 7 \beta_{18} + \beta_{17} + 9 \beta_{16} - 2 \beta_{15} - 7 \beta_{14} + \cdots - 18 \beta_{10} \)
|
| \(\nu^{6}\) | \(=\) |
\( -46\beta_{11} - 45\beta_{9} + 55\beta_{8} - 10\beta_{5} - 2\beta_{4} - 55\beta_{3} + 2\beta_{2} + 155 \)
|
| \(\nu^{7}\) | \(=\) |
\( 12 \beta_{19} - 25 \beta_{18} - 24 \beta_{17} - 25 \beta_{16} + 68 \beta_{15} + 68 \beta_{14} + \cdots + 190 \beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 301 \beta_{11} + 368 \beta_{9} - 288 \beta_{8} + 301 \beta_{7} - 1018 \beta_{6} + 368 \beta_{5} + \cdots - 1018 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 222 \beta_{19} + 492 \beta_{18} + 111 \beta_{17} - 257 \beta_{16} - 257 \beta_{15} + \cdots - 1226 \beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 603 \beta_{9} - 603 \beta_{8} - 1975 \beta_{7} + 6735 \beta_{6} - 1851 \beta_{5} + \cdots + 328 \beta_{2} \)
|
| \(\nu^{11}\) | \(=\) |
\( 931 \beta_{19} - 1523 \beta_{18} + 931 \beta_{17} + 3509 \beta_{16} - 1986 \beta_{15} + \cdots - 8986 \beta_{10} \)
|
| \(\nu^{12}\) | \(=\) |
\( - 13015 \beta_{11} - 11960 \beta_{9} + 16400 \beta_{8} - 4440 \beta_{5} - 2970 \beta_{4} - 16400 \beta_{3} + \cdots + 44789 \)
|
| \(\nu^{13}\) | \(=\) |
\( 7410 \beta_{19} - 15875 \beta_{18} - 14820 \beta_{17} - 15875 \beta_{16} + 24865 \beta_{15} + \cdots + 52309 \beta_1 \)
|
| \(\nu^{14}\) | \(=\) |
\( 86164 \beta_{11} + 109974 \beta_{9} - 77699 \beta_{8} + 86164 \beta_{7} - 299176 \beta_{6} + \cdots - 299176 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 114120 \beta_{19} + 175499 \beta_{18} + 57060 \beta_{17} - 52914 \beta_{16} - 52914 \beta_{15} + \cdots - 344600 \beta_1 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 232559 \beta_{9} - 232559 \beta_{8} - 573013 \beta_{7} + 2006406 \beta_{6} - 507488 \beta_{5} + \cdots + 196940 \beta_{2} \)
|
| \(\nu^{17}\) | \(=\) |
\( 429499 \beta_{19} - 310548 \beta_{18} + 429499 \beta_{17} + 1235071 \beta_{16} + \cdots - 3651782 \beta_{10} \)
|
| \(\nu^{18}\) | \(=\) |
\( - 3826954 \beta_{11} - 3331930 \beta_{9} + 4996500 \beta_{8} - 1664570 \beta_{5} - 1515343 \beta_{4} + \cdots + 13505075 \)
|
| \(\nu^{19}\) | \(=\) |
\( 3179913 \beta_{19} - 6854850 \beta_{18} - 6359826 \beta_{17} - 6854850 \beta_{16} + \cdots + 15172435 \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) | \(301\) |
| \(\chi(n)\) | \(-1\) | \(-\beta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
−1.31626 | − | 2.27983i | −2.33745 | + | 1.34953i | −2.46508 | + | 4.26964i | 0 | 6.15337 | + | 3.55265i | −1.37037 | + | 2.37354i | 7.71370 | 2.14244 | − | 3.71081i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.2 | −1.22199 | − | 2.11655i | 0.105533 | − | 0.0609297i | −1.98653 | + | 3.44078i | 0 | −0.257922 | − | 0.148911i | 0.207056 | − | 0.358632i | 4.82215 | −1.49258 | + | 2.58522i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.3 | −0.682755 | − | 1.18257i | 0.346339 | − | 0.199959i | 0.0676905 | − | 0.117243i | 0 | −0.472929 | − | 0.273046i | 1.82183 | − | 3.15550i | −2.91589 | −1.42003 | + | 2.45957i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.4 | −0.443998 | − | 0.769027i | 2.65171 | − | 1.53097i | 0.605732 | − | 1.04916i | 0 | −2.35471 | − | 1.35949i | −1.78152 | + | 3.08568i | −2.85177 | 3.18771 | − | 5.52128i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.5 | −0.333024 | − | 0.576815i | −2.15028 | + | 1.24146i | 0.778190 | − | 1.34786i | 0 | 1.43219 | + | 0.826874i | 0.293872 | − | 0.509002i | −2.36872 | 1.58246 | − | 2.74090i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.6 | 0.333024 | + | 0.576815i | 2.15028 | − | 1.24146i | 0.778190 | − | 1.34786i | 0 | 1.43219 | + | 0.826874i | −0.293872 | + | 0.509002i | 2.36872 | 1.58246 | − | 2.74090i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.7 | 0.443998 | + | 0.769027i | −2.65171 | + | 1.53097i | 0.605732 | − | 1.04916i | 0 | −2.35471 | − | 1.35949i | 1.78152 | − | 3.08568i | 2.85177 | 3.18771 | − | 5.52128i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.8 | 0.682755 | + | 1.18257i | −0.346339 | + | 0.199959i | 0.0676905 | − | 0.117243i | 0 | −0.472929 | − | 0.273046i | −1.82183 | + | 3.15550i | 2.91589 | −1.42003 | + | 2.45957i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.9 | 1.22199 | + | 2.11655i | −0.105533 | + | 0.0609297i | −1.98653 | + | 3.44078i | 0 | −0.257922 | − | 0.148911i | −0.207056 | + | 0.358632i | −4.82215 | −1.49258 | + | 2.58522i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.10 | 1.31626 | + | 2.27983i | 2.33745 | − | 1.34953i | −2.46508 | + | 4.26964i | 0 | 6.15337 | + | 3.55265i | 1.37037 | − | 2.37354i | −7.71370 | 2.14244 | − | 3.71081i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 199.1 | −1.31626 | + | 2.27983i | −2.33745 | − | 1.34953i | −2.46508 | − | 4.26964i | 0 | 6.15337 | − | 3.55265i | −1.37037 | − | 2.37354i | 7.71370 | 2.14244 | + | 3.71081i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 199.2 | −1.22199 | + | 2.11655i | 0.105533 | + | 0.0609297i | −1.98653 | − | 3.44078i | 0 | −0.257922 | + | 0.148911i | 0.207056 | + | 0.358632i | 4.82215 | −1.49258 | − | 2.58522i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 199.3 | −0.682755 | + | 1.18257i | 0.346339 | + | 0.199959i | 0.0676905 | + | 0.117243i | 0 | −0.472929 | + | 0.273046i | 1.82183 | + | 3.15550i | −2.91589 | −1.42003 | − | 2.45957i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 199.4 | −0.443998 | + | 0.769027i | 2.65171 | + | 1.53097i | 0.605732 | + | 1.04916i | 0 | −2.35471 | + | 1.35949i | −1.78152 | − | 3.08568i | −2.85177 | 3.18771 | + | 5.52128i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 199.5 | −0.333024 | + | 0.576815i | −2.15028 | − | 1.24146i | 0.778190 | + | 1.34786i | 0 | 1.43219 | − | 0.826874i | 0.293872 | + | 0.509002i | −2.36872 | 1.58246 | + | 2.74090i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 199.6 | 0.333024 | − | 0.576815i | 2.15028 | + | 1.24146i | 0.778190 | + | 1.34786i | 0 | 1.43219 | − | 0.826874i | −0.293872 | − | 0.509002i | 2.36872 | 1.58246 | + | 2.74090i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 199.7 | 0.443998 | − | 0.769027i | −2.65171 | − | 1.53097i | 0.605732 | + | 1.04916i | 0 | −2.35471 | + | 1.35949i | 1.78152 | + | 3.08568i | 2.85177 | 3.18771 | + | 5.52128i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 199.8 | 0.682755 | − | 1.18257i | −0.346339 | − | 0.199959i | 0.0676905 | + | 0.117243i | 0 | −0.472929 | + | 0.273046i | −1.82183 | − | 3.15550i | 2.91589 | −1.42003 | − | 2.45957i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 199.9 | 1.22199 | − | 2.11655i | −0.105533 | − | 0.0609297i | −1.98653 | − | 3.44078i | 0 | −0.257922 | + | 0.148911i | −0.207056 | − | 0.358632i | −4.82215 | −1.49258 | − | 2.58522i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 199.10 | 1.31626 | − | 2.27983i | 2.33745 | + | 1.34953i | −2.46508 | − | 4.26964i | 0 | 6.15337 | − | 3.55265i | 1.37037 | + | 2.37354i | −7.71370 | 2.14244 | + | 3.71081i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 13.e | even | 6 | 1 | inner |
| 65.l | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 325.2.m.d | 20 | |
| 5.b | even | 2 | 1 | inner | 325.2.m.d | 20 | |
| 5.c | odd | 4 | 1 | 325.2.n.e | ✓ | 10 | |
| 5.c | odd | 4 | 1 | 325.2.n.f | yes | 10 | |
| 13.e | even | 6 | 1 | inner | 325.2.m.d | 20 | |
| 65.l | even | 6 | 1 | inner | 325.2.m.d | 20 | |
| 65.o | even | 12 | 1 | 4225.2.a.bu | 10 | ||
| 65.o | even | 12 | 1 | 4225.2.a.bv | 10 | ||
| 65.r | odd | 12 | 1 | 325.2.n.e | ✓ | 10 | |
| 65.r | odd | 12 | 1 | 325.2.n.f | yes | 10 | |
| 65.t | even | 12 | 1 | 4225.2.a.bu | 10 | ||
| 65.t | even | 12 | 1 | 4225.2.a.bv | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 325.2.m.d | 20 | 1.a | even | 1 | 1 | trivial | |
| 325.2.m.d | 20 | 5.b | even | 2 | 1 | inner | |
| 325.2.m.d | 20 | 13.e | even | 6 | 1 | inner | |
| 325.2.m.d | 20 | 65.l | even | 6 | 1 | inner | |
| 325.2.n.e | ✓ | 10 | 5.c | odd | 4 | 1 | |
| 325.2.n.e | ✓ | 10 | 65.r | odd | 12 | 1 | |
| 325.2.n.f | yes | 10 | 5.c | odd | 4 | 1 | |
| 325.2.n.f | yes | 10 | 65.r | odd | 12 | 1 | |
| 4225.2.a.bu | 10 | 65.o | even | 12 | 1 | ||
| 4225.2.a.bu | 10 | 65.t | even | 12 | 1 | ||
| 4225.2.a.bv | 10 | 65.o | even | 12 | 1 | ||
| 4225.2.a.bv | 10 | 65.t | even | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{20} + 16 T_{2}^{18} + 172 T_{2}^{16} + 1018 T_{2}^{14} + 4330 T_{2}^{12} + 9943 T_{2}^{10} + \cdots + 729 \)
acting on \(S_{2}^{\mathrm{new}}(325, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} + 16 T^{18} + \cdots + 729 \)
|
| $3$ |
\( T^{20} - 23 T^{18} + \cdots + 1 \)
|
| $5$ |
\( T^{20} \)
|
| $7$ |
\( T^{20} + 34 T^{18} + \cdots + 5625 \)
|
| $11$ |
\( (T^{10} + 9 T^{9} + \cdots + 6075)^{2} \)
|
| $13$ |
\( T^{20} + \cdots + 137858491849 \)
|
| $17$ |
\( T^{20} - 64 T^{18} + \cdots + 2313441 \)
|
| $19$ |
\( (T^{10} - 68 T^{8} + \cdots + 531723)^{2} \)
|
| $23$ |
\( T^{20} - 85 T^{18} + \cdots + 81 \)
|
| $29$ |
\( (T^{10} + 7 T^{9} + \cdots + 1766241)^{2} \)
|
| $31$ |
\( (T^{10} + 154 T^{8} + \cdots + 10546875)^{2} \)
|
| $37$ |
\( T^{20} + \cdots + 12\!\cdots\!25 \)
|
| $41$ |
\( (T^{10} - 12 T^{9} + \cdots + 45387)^{2} \)
|
| $43$ |
\( T^{20} + \cdots + 276281640625 \)
|
| $47$ |
\( (T^{10} - 160 T^{8} + \cdots - 771147)^{2} \)
|
| $53$ |
\( (T^{10} + 246 T^{8} + \cdots + 71723961)^{2} \)
|
| $59$ |
\( (T^{10} - 48 T^{9} + \cdots + 3102867)^{2} \)
|
| $61$ |
\( (T^{10} - 13 T^{9} + \cdots + 2238709225)^{2} \)
|
| $67$ |
\( T^{20} + \cdots + 284765625 \)
|
| $71$ |
\( (T^{10} + 27 T^{9} + \cdots + 116251875)^{2} \)
|
| $73$ |
\( (T^{10} - 490 T^{8} + \cdots - 408916875)^{2} \)
|
| $79$ |
\( (T^{5} + 2 T^{4} + \cdots + 9475)^{4} \)
|
| $83$ |
\( (T^{10} - 393 T^{8} + \cdots - 243)^{2} \)
|
| $89$ |
\( (T^{10} + 24 T^{9} + \cdots + 45139923)^{2} \)
|
| $97$ |
\( T^{20} + \cdots + 32925412896489 \)
|
| show more | |
| show less |