Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [448,4,Mod(255,448)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(448, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("448.255");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 448 = 2^{6} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 448.p (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: | \(26.4328556826\) |
| 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: |
\( x^{20} - 2 x^{18} - 24 x^{17} + 28 x^{16} + 56 x^{15} - 192 x^{14} + 352 x^{13} - 448 x^{12} + \cdots + 1073741824 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{44} \) |
| Twist minimal: | no (minimal twist has level 28) |
| 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
\( x^{20} - 2 x^{18} - 24 x^{17} + 28 x^{16} + 56 x^{15} - 192 x^{14} + 352 x^{13} - 448 x^{12} + \cdots + 1073741824 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 793 \nu^{19} - 7760 \nu^{18} - 70462 \nu^{17} + 82360 \nu^{16} - 36188 \nu^{15} + \cdots + 940060966912 ) / 2280224980992 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 253 \nu^{19} + 720 \nu^{18} - 11414 \nu^{17} - 65832 \nu^{16} - 109324 \nu^{15} + \cdots + 1201651318784 ) / 325746425856 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 647 \nu^{19} - 13656 \nu^{18} - 111550 \nu^{17} + 323784 \nu^{16} - 943772 \nu^{15} + \cdots + 3620321886208 ) / 1140112490496 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 13969 \nu^{19} + 83816 \nu^{18} - 322254 \nu^{17} + 133768 \nu^{16} + \cdots + 10557834919936 ) / 2280224980992 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 7197 \nu^{19} + 68984 \nu^{18} + 122498 \nu^{17} + 1668776 \nu^{16} + \cdots - 9654952263680 ) / 1140112490496 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 4809 \nu^{19} + 488 \nu^{18} - 47870 \nu^{17} + 103368 \nu^{16} - 318364 \nu^{15} + \cdots + 562506498048 ) / 325746425856 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 39171 \nu^{19} + 88016 \nu^{18} + 123178 \nu^{17} - 159272 \nu^{16} + 596340 \nu^{15} + \cdots - 10618098679808 ) / 2280224980992 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 1751 \nu^{19} + 1586 \nu^{18} + 12018 \nu^{17} + 98900 \nu^{16} - 115692 \nu^{15} + \cdots - 932611883008 ) / 71257030656 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 1894 \nu^{19} + 58567 \nu^{18} + 387356 \nu^{17} - 54686 \nu^{16} - 826352 \nu^{15} + \cdots - 20322543730688 ) / 285028122624 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 35213 \nu^{19} + 43608 \nu^{18} - 69790 \nu^{17} + 86376 \nu^{16} - 867004 \nu^{15} + \cdots + 5266837864448 ) / 1140112490496 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 7973 \nu^{19} - 27196 \nu^{18} + 132966 \nu^{17} - 40640 \nu^{16} + 118860 \nu^{15} + \cdots - 4866399272960 ) / 190018748416 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 14861 \nu^{19} + 21795 \nu^{18} + 209574 \nu^{17} - 243230 \nu^{16} - 861276 \nu^{15} + \cdots - 10412292571136 ) / 285028122624 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 4095 \nu^{19} - 10250 \nu^{18} + 27770 \nu^{17} + 32460 \nu^{16} - 150956 \nu^{15} + \cdots - 1487602188288 ) / 81436606464 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 9811 \nu^{19} - 11616 \nu^{18} + 66782 \nu^{17} - 1880 \nu^{16} - 190468 \nu^{15} + \cdots - 2344381054976 ) / 162873212928 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 74377 \nu^{19} - 27452 \nu^{18} - 793194 \nu^{17} + 594432 \nu^{16} + 1813708 \nu^{15} + \cdots + 34979287400448 ) / 1140112490496 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 16233 \nu^{19} + 65704 \nu^{18} - 77422 \nu^{17} + 26120 \nu^{16} - 86716 \nu^{15} + \cdots + 2878970265600 ) / 285028122624 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 151077 \nu^{19} + 463976 \nu^{18} - 229926 \nu^{17} - 371992 \nu^{16} + \cdots - 7711211126784 ) / 2280224980992 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 158821 \nu^{19} - 488368 \nu^{18} + 901626 \nu^{17} + 2990744 \nu^{16} + \cdots - 3806280548352 ) / 2280224980992 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 25722 \nu^{19} - 28761 \nu^{18} + 6684 \nu^{17} - 453598 \nu^{16} + 507664 \nu^{15} + \cdots + 1721912786944 ) / 285028122624 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{10} - \beta_{8} + \beta_{5} ) / 16 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2 \beta_{19} + \beta_{16} + \beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} + \cdots - 1 ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - \beta_{18} - \beta_{16} + 4 \beta_{14} - 4 \beta_{11} - 5 \beta_{10} + 5 \beta_{7} + \beta_{5} + \cdots + 28 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - \beta_{19} + \beta_{17} + 8 \beta_{14} - 8 \beta_{13} - \beta_{12} + \beta_{11} - 15 \beta_{10} + \cdots - 37 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 12 \beta_{19} + \beta_{18} + 11 \beta_{16} - 10 \beta_{15} - 10 \beta_{14} - 6 \beta_{13} - 6 \beta_{12} + \cdots - 6 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 3 \beta_{19} - 8 \beta_{18} - 13 \beta_{17} - 11 \beta_{16} - 13 \beta_{15} - \beta_{14} + 11 \beta_{13} + \cdots + 274 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 8 \beta_{19} + 24 \beta_{17} - 44 \beta_{14} + 44 \beta_{13} - 8 \beta_{12} + 8 \beta_{11} + \cdots - 1436 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 26 \beta_{19} - 55 \beta_{18} + 46 \beta_{16} + 51 \beta_{15} + 27 \beta_{14} + 59 \beta_{13} + \cdots + 13 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 114 \beta_{19} + 97 \beta_{18} - 98 \beta_{17} + 59 \beta_{16} - 98 \beta_{15} - 46 \beta_{14} + \cdots + 1128 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 6 \beta_{19} + 38 \beta_{17} - 152 \beta_{14} + 152 \beta_{13} - 6 \beta_{12} + 6 \beta_{11} + \cdots + 14250 \)
|
| \(\nu^{11}\) | \(=\) |
\( 288 \beta_{19} + 1138 \beta_{18} - 334 \beta_{16} + 624 \beta_{15} - 1840 \beta_{14} + 232 \beta_{13} + \cdots - 144 \)
|
| \(\nu^{12}\) | \(=\) |
\( 2316 \beta_{19} + 1892 \beta_{18} + 2316 \beta_{17} + 1400 \beta_{16} + 2316 \beta_{15} + 2476 \beta_{14} + \cdots + 75160 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 184 \beta_{19} + 13048 \beta_{17} + 13392 \beta_{14} - 13392 \beta_{13} - 184 \beta_{12} + \cdots + 76120 \)
|
| \(\nu^{14}\) | \(=\) |
\( 8144 \beta_{19} + 960 \beta_{18} - 7976 \beta_{16} + 20328 \beta_{15} + 21928 \beta_{14} - 38904 \beta_{13} + \cdots - 4072 \)
|
| \(\nu^{15}\) | \(=\) |
\( 65856 \beta_{19} + 30664 \beta_{18} - 89920 \beta_{17} + 90056 \beta_{16} - 89920 \beta_{15} + \cdots - 388832 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 94256 \beta_{19} - 102352 \beta_{17} + 209920 \beta_{14} - 209920 \beta_{13} - 94256 \beta_{12} + \cdots + 3601040 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 209344 \beta_{19} + 616944 \beta_{18} + 137296 \beta_{16} - 41440 \beta_{15} - 717792 \beta_{14} + \cdots + 104672 \)
|
| \(\nu^{18}\) | \(=\) |
\( 789408 \beta_{19} - 2061056 \beta_{18} - 513632 \beta_{17} + 3020128 \beta_{16} - 513632 \beta_{15} + \cdots + 58623424 \)
|
| \(\nu^{19}\) | \(=\) |
\( 5736192 \beta_{19} - 1460480 \beta_{17} - 396928 \beta_{14} + 396928 \beta_{13} + 5736192 \beta_{12} + \cdots - 6007424 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(129\) | \(197\) |
| \(\chi(n)\) | \(-1\) | \(1 + \beta_{1}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 255.1 |
|
0 | −4.65345 | + | 8.06002i | 0 | −5.11538 | + | 2.95337i | 0 | 1.92900 | + | 18.4195i | 0 | −29.8092 | − | 51.6311i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 255.2 | 0 | −3.44104 | + | 5.96006i | 0 | 4.17670 | − | 2.41142i | 0 | 5.03893 | − | 17.8216i | 0 | −10.1816 | − | 17.6350i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 255.3 | 0 | −2.11164 | + | 3.65747i | 0 | −1.03358 | + | 0.596737i | 0 | −16.7651 | − | 7.86959i | 0 | 4.58193 | + | 7.93614i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 255.4 | 0 | −1.67134 | + | 2.89484i | 0 | 15.9583 | − | 9.21354i | 0 | 15.4841 | + | 10.1609i | 0 | 7.91327 | + | 13.7062i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 255.5 | 0 | −0.0469307 | + | 0.0812864i | 0 | −12.4861 | + | 7.20883i | 0 | 15.0686 | − | 10.7674i | 0 | 13.4956 | + | 23.3751i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 255.6 | 0 | 0.0469307 | − | 0.0812864i | 0 | −12.4861 | + | 7.20883i | 0 | −15.0686 | + | 10.7674i | 0 | 13.4956 | + | 23.3751i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 255.7 | 0 | 1.67134 | − | 2.89484i | 0 | 15.9583 | − | 9.21354i | 0 | −15.4841 | − | 10.1609i | 0 | 7.91327 | + | 13.7062i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 255.8 | 0 | 2.11164 | − | 3.65747i | 0 | −1.03358 | + | 0.596737i | 0 | 16.7651 | + | 7.86959i | 0 | 4.58193 | + | 7.93614i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 255.9 | 0 | 3.44104 | − | 5.96006i | 0 | 4.17670 | − | 2.41142i | 0 | −5.03893 | + | 17.8216i | 0 | −10.1816 | − | 17.6350i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 255.10 | 0 | 4.65345 | − | 8.06002i | 0 | −5.11538 | + | 2.95337i | 0 | −1.92900 | − | 18.4195i | 0 | −29.8092 | − | 51.6311i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 383.1 | 0 | −4.65345 | − | 8.06002i | 0 | −5.11538 | − | 2.95337i | 0 | 1.92900 | − | 18.4195i | 0 | −29.8092 | + | 51.6311i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 383.2 | 0 | −3.44104 | − | 5.96006i | 0 | 4.17670 | + | 2.41142i | 0 | 5.03893 | + | 17.8216i | 0 | −10.1816 | + | 17.6350i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 383.3 | 0 | −2.11164 | − | 3.65747i | 0 | −1.03358 | − | 0.596737i | 0 | −16.7651 | + | 7.86959i | 0 | 4.58193 | − | 7.93614i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 383.4 | 0 | −1.67134 | − | 2.89484i | 0 | 15.9583 | + | 9.21354i | 0 | 15.4841 | − | 10.1609i | 0 | 7.91327 | − | 13.7062i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 383.5 | 0 | −0.0469307 | − | 0.0812864i | 0 | −12.4861 | − | 7.20883i | 0 | 15.0686 | + | 10.7674i | 0 | 13.4956 | − | 23.3751i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 383.6 | 0 | 0.0469307 | + | 0.0812864i | 0 | −12.4861 | − | 7.20883i | 0 | −15.0686 | − | 10.7674i | 0 | 13.4956 | − | 23.3751i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 383.7 | 0 | 1.67134 | + | 2.89484i | 0 | 15.9583 | + | 9.21354i | 0 | −15.4841 | + | 10.1609i | 0 | 7.91327 | − | 13.7062i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 383.8 | 0 | 2.11164 | + | 3.65747i | 0 | −1.03358 | − | 0.596737i | 0 | 16.7651 | − | 7.86959i | 0 | 4.58193 | − | 7.93614i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 383.9 | 0 | 3.44104 | + | 5.96006i | 0 | 4.17670 | + | 2.41142i | 0 | −5.03893 | − | 17.8216i | 0 | −10.1816 | + | 17.6350i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 383.10 | 0 | 4.65345 | + | 8.06002i | 0 | −5.11538 | − | 2.95337i | 0 | −1.92900 | + | 18.4195i | 0 | −29.8092 | + | 51.6311i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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 | 448.4.p.h | 20 | |
| 4.b | odd | 2 | 1 | inner | 448.4.p.h | 20 | |
| 7.d | odd | 6 | 1 | inner | 448.4.p.h | 20 | |
| 8.b | even | 2 | 1 | 28.4.f.a | ✓ | 20 | |
| 8.d | odd | 2 | 1 | 28.4.f.a | ✓ | 20 | |
| 28.f | even | 6 | 1 | inner | 448.4.p.h | 20 | |
| 56.e | even | 2 | 1 | 196.4.f.d | 20 | ||
| 56.h | odd | 2 | 1 | 196.4.f.d | 20 | ||
| 56.j | odd | 6 | 1 | 28.4.f.a | ✓ | 20 | |
| 56.j | odd | 6 | 1 | 196.4.d.b | 20 | ||
| 56.k | odd | 6 | 1 | 196.4.d.b | 20 | ||
| 56.k | odd | 6 | 1 | 196.4.f.d | 20 | ||
| 56.m | even | 6 | 1 | 28.4.f.a | ✓ | 20 | |
| 56.m | even | 6 | 1 | 196.4.d.b | 20 | ||
| 56.p | even | 6 | 1 | 196.4.d.b | 20 | ||
| 56.p | even | 6 | 1 | 196.4.f.d | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 28.4.f.a | ✓ | 20 | 8.b | even | 2 | 1 | |
| 28.4.f.a | ✓ | 20 | 8.d | odd | 2 | 1 | |
| 28.4.f.a | ✓ | 20 | 56.j | odd | 6 | 1 | |
| 28.4.f.a | ✓ | 20 | 56.m | even | 6 | 1 | |
| 196.4.d.b | 20 | 56.j | odd | 6 | 1 | ||
| 196.4.d.b | 20 | 56.k | odd | 6 | 1 | ||
| 196.4.d.b | 20 | 56.m | even | 6 | 1 | ||
| 196.4.d.b | 20 | 56.p | even | 6 | 1 | ||
| 196.4.f.d | 20 | 56.e | even | 2 | 1 | ||
| 196.4.f.d | 20 | 56.h | odd | 2 | 1 | ||
| 196.4.f.d | 20 | 56.k | odd | 6 | 1 | ||
| 196.4.f.d | 20 | 56.p | even | 6 | 1 | ||
| 448.4.p.h | 20 | 1.a | even | 1 | 1 | trivial | |
| 448.4.p.h | 20 | 4.b | odd | 2 | 1 | inner | |
| 448.4.p.h | 20 | 7.d | odd | 6 | 1 | inner | |
| 448.4.p.h | 20 | 28.f | even | 6 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{20} + 163 T_{3}^{18} + 18379 T_{3}^{16} + 1043398 T_{3}^{14} + 42494101 T_{3}^{12} + \cdots + 51883209 \)
acting on \(S_{4}^{\mathrm{new}}(448, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( T^{20} + 163 T^{18} + \cdots + 51883209 \)
|
| $5$ |
\( (T^{10} - 3 T^{9} + \cdots + 81588675)^{2} \)
|
| $7$ |
\( T^{20} + \cdots + 22\!\cdots\!49 \)
|
| $11$ |
\( T^{20} + \cdots + 20\!\cdots\!25 \)
|
| $13$ |
\( (T^{10} + \cdots + 10072189747200)^{2} \)
|
| $17$ |
\( (T^{10} + \cdots + 60\!\cdots\!75)^{2} \)
|
| $19$ |
\( T^{20} + \cdots + 12\!\cdots\!25 \)
|
| $23$ |
\( T^{20} + \cdots + 17\!\cdots\!61 \)
|
| $29$ |
\( (T^{5} - 88 T^{4} + \cdots - 5066614016)^{4} \)
|
| $31$ |
\( T^{20} + \cdots + 43\!\cdots\!25 \)
|
| $37$ |
\( (T^{10} + \cdots + 48\!\cdots\!25)^{2} \)
|
| $41$ |
\( (T^{10} + \cdots + 14\!\cdots\!72)^{2} \)
|
| $43$ |
\( (T^{10} + \cdots + 55\!\cdots\!56)^{2} \)
|
| $47$ |
\( T^{20} + \cdots + 90\!\cdots\!25 \)
|
| $53$ |
\( (T^{10} + \cdots + 23\!\cdots\!25)^{2} \)
|
| $59$ |
\( T^{20} + \cdots + 53\!\cdots\!25 \)
|
| $61$ |
\( (T^{10} + \cdots + 25\!\cdots\!47)^{2} \)
|
| $67$ |
\( T^{20} + \cdots + 16\!\cdots\!21 \)
|
| $71$ |
\( (T^{10} + \cdots + 16\!\cdots\!00)^{2} \)
|
| $73$ |
\( (T^{10} + \cdots + 39\!\cdots\!75)^{2} \)
|
| $79$ |
\( T^{20} + \cdots + 14\!\cdots\!25 \)
|
| $83$ |
\( (T^{10} + \cdots - 41\!\cdots\!08)^{2} \)
|
| $89$ |
\( (T^{10} + \cdots + 99\!\cdots\!27)^{2} \)
|
| $97$ |
\( (T^{10} + \cdots + 12\!\cdots\!00)^{2} \)
|
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