Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [115,2,Mod(22,115)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(115, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([1, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("115.22");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 115 = 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 115.e (of order \(4\), 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: | \(0.918279623245\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 6 x^{18} + 3 x^{16} + 80 x^{14} - 600 x^{12} + 3500 x^{10} - 15000 x^{8} + 50000 x^{6} + \cdots + 9765625 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 5 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{20} - 6 x^{18} + 3 x^{16} + 80 x^{14} - 600 x^{12} + 3500 x^{10} - 15000 x^{8} + 50000 x^{6} + \cdots + 9765625 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 301 \nu^{19} - 1894 \nu^{17} + 116922 \nu^{15} - 115180 \nu^{13} + 1787100 \nu^{11} + \cdots + 450000000 \nu ) / 10070312500 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 63 \nu^{18} - 7728 \nu^{16} + 15664 \nu^{14} - 116510 \nu^{12} + 556450 \nu^{10} + \cdots - 12500000 ) / 2014062500 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 531 \nu^{18} - 1886 \nu^{16} - 19082 \nu^{14} + 62380 \nu^{12} - 326350 \nu^{10} + \cdots - 1725000000 ) / 2014062500 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 423 \nu^{18} - 1712 \nu^{16} + 6231 \nu^{14} + 25785 \nu^{12} + 326675 \nu^{10} + \cdots + 869531250 ) / 1007031250 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 777 \nu^{18} - 938 \nu^{16} - 606 \nu^{14} + 118540 \nu^{12} - 555550 \nu^{10} + \cdots - 854687500 ) / 2014062500 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 31 \nu^{18} + 31 \nu^{16} - 1124 \nu^{14} + 6146 \nu^{12} - 13260 \nu^{10} + 30550 \nu^{8} + \cdots - 95578125 ) / 80562500 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 244 \nu^{19} - 2661 \nu^{17} + 9018 \nu^{15} - 91270 \nu^{13} + 5150 \nu^{11} + \cdots + 664453125 \nu ) / 2014062500 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 634 \nu^{18} - 2529 \nu^{16} - 16248 \nu^{14} + 70045 \nu^{12} - 24900 \nu^{10} + \cdots - 817578125 ) / 1007031250 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 968 \nu^{19} - 1458 \nu^{17} + 58054 \nu^{15} + 324865 \nu^{13} - 242175 \nu^{11} + \cdots + 1040625000 \nu ) / 5035156250 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 1616 \nu^{18} - 7971 \nu^{16} - 752 \nu^{14} + 89080 \nu^{12} - 716100 \nu^{10} + \cdots - 3238671875 ) / 2014062500 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 1711 \nu^{18} - 5966 \nu^{16} - 12542 \nu^{14} - 3970 \nu^{12} - 387600 \nu^{10} + \cdots - 927343750 ) / 2014062500 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 2188 \nu^{19} - 32553 \nu^{17} - 16886 \nu^{15} + 159890 \nu^{13} + 1650700 \nu^{11} + \cdots + 4543359375 \nu ) / 10070312500 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 3232 \nu^{19} - 18867 \nu^{17} + 144046 \nu^{15} - 411740 \nu^{13} - 3934700 \nu^{11} + \cdots - 1577734375 \nu ) / 10070312500 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 2947 \nu^{18} - 11232 \nu^{16} + 37266 \nu^{14} - 65140 \nu^{12} - 223950 \nu^{10} + \cdots - 2172656250 ) / 2014062500 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 777 \nu^{19} - 938 \nu^{17} - 606 \nu^{15} + 118540 \nu^{13} - 555550 \nu^{11} + \cdots + 1159375000 \nu ) / 2014062500 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 4416 \nu^{19} - 13221 \nu^{17} - 33902 \nu^{15} - 123770 \nu^{13} - 1090100 \nu^{11} + \cdots + 2330859375 \nu ) / 10070312500 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( \nu^{19} - 6 \nu^{17} + 3 \nu^{15} + 80 \nu^{13} - 600 \nu^{11} + 3500 \nu^{9} + \cdots - 2343750 \nu ) / 1953125 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 6117 \nu^{19} - 17327 \nu^{17} + 37726 \nu^{15} - 213140 \nu^{13} + 171050 \nu^{11} + \cdots - 3701953125 \nu ) / 10070312500 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 3374 \nu^{19} + 6556 \nu^{17} - 63178 \nu^{15} + 359695 \nu^{13} - 2058525 \nu^{11} + \cdots - 5507031250 \nu ) / 5035156250 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{19} + 2 \beta_{18} - 4 \beta_{17} - \beta_{16} + \beta_{15} + \beta_{13} + 2 \beta_{12} + \cdots - 3 \beta_1 ) / 5 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{14} - 2\beta_{10} + \beta_{5} + 2\beta_{3} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - \beta_{19} + 18 \beta_{18} - \beta_{17} + \beta_{16} + 14 \beta_{15} - 6 \beta_{13} - 7 \beta_{12} + \cdots + 8 \beta_1 ) / 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{11} - 7\beta_{8} + 12\beta_{6} - \beta_{4} + 7\beta_{2} + 9 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 11 \beta_{19} + 62 \beta_{18} - 34 \beta_{17} - 86 \beta_{16} + 11 \beta_{15} + 21 \beta_{13} + \cdots - 53 \beta_1 ) / 5 \)
|
| \(\nu^{6}\) | \(=\) |
\( 7 \beta_{14} + 4 \beta_{11} - 24 \beta_{10} + 18 \beta_{8} + 22 \beta_{6} + 32 \beta_{5} - 26 \beta_{4} + \cdots + 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 186 \beta_{19} + 58 \beta_{18} + 69 \beta_{17} + 186 \beta_{16} + 94 \beta_{15} + 14 \beta_{13} + \cdots + 48 \beta_1 ) / 5 \)
|
| \(\nu^{8}\) | \(=\) |
\( 62 \beta_{14} - 248 \beta_{11} - 34 \beta_{10} + 104 \beta_{8} + 46 \beta_{6} + 62 \beta_{5} + \cdots + 247 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 469 \beta_{19} + 1422 \beta_{18} - 54 \beta_{17} - 931 \beta_{16} + 11 \beta_{15} - 399 \beta_{13} + \cdots - 1993 \beta_1 ) / 5 \)
|
| \(\nu^{10}\) | \(=\) |
\( 701 \beta_{14} - 1170 \beta_{11} - 1332 \beta_{10} + 1130 \beta_{8} + 750 \beta_{6} - 749 \beta_{5} + \cdots - 1000 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 4111 \beta_{19} + 4698 \beta_{18} + 1839 \beta_{17} + 1711 \beta_{16} - 1196 \beta_{15} + \cdots + 4938 \beta_1 ) / 5 \)
|
| \(\nu^{12}\) | \(=\) |
\( 2460 \beta_{14} - 6571 \beta_{11} + 2280 \beta_{10} + 1453 \beta_{8} + 8552 \beta_{6} + 1760 \beta_{5} + \cdots + 7089 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 10471 \beta_{19} - 1518 \beta_{18} + 3776 \beta_{17} - 25046 \beta_{16} + 7871 \beta_{15} + \cdots - 10533 \beta_1 ) / 5 \)
|
| \(\nu^{14}\) | \(=\) |
\( 33147 \beta_{14} - 22676 \beta_{11} - 5254 \beta_{10} - 5192 \beta_{8} - 14668 \beta_{6} + 13922 \beta_{5} + \cdots - 22076 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 86796 \beta_{19} - 55412 \beta_{18} + 54359 \beta_{17} + 99796 \beta_{16} + 9084 \beta_{15} + \cdots + 253228 \beta_1 ) / 5 \)
|
| \(\nu^{16}\) | \(=\) |
\( 4372 \beta_{14} - 91168 \beta_{11} + 44296 \beta_{10} - 122136 \beta_{8} + 144436 \beta_{6} + \cdots + 7977 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( 705041 \beta_{19} + 663942 \beta_{18} - 1159944 \beta_{17} - 1184441 \beta_{16} - 387079 \beta_{15} + \cdots - 871623 \beta_1 ) / 5 \)
|
| \(\nu^{18}\) | \(=\) |
\( 382341 \beta_{14} + 322700 \beta_{11} - 66062 \beta_{10} + 100520 \beta_{8} + 245460 \beta_{6} + \cdots + 36220 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( 1878479 \beta_{19} + 3726878 \beta_{18} - 2959921 \beta_{17} + 3582121 \beta_{16} - 1188006 \beta_{15} + \cdots + 755268 \beta_1 ) / 5 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).
| \(n\) | \(47\) | \(51\) |
| \(\chi(n)\) | \(\beta_{6}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 22.1 |
|
−1.68447 | − | 1.68447i | −0.639677 | + | 0.639677i | 3.67489i | −1.36809 | − | 1.76871i | 2.15503 | −2.98466 | + | 2.98466i | 2.82130 | − | 2.82130i | 2.18163i | −0.674828 | + | 5.28385i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 22.2 | −1.68447 | − | 1.68447i | −0.639677 | + | 0.639677i | 3.67489i | 1.36809 | + | 1.76871i | 2.15503 | 2.98466 | − | 2.98466i | 2.82130 | − | 2.82130i | 2.18163i | 0.674828 | − | 5.28385i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 22.3 | −0.819998 | − | 0.819998i | 1.37823 | − | 1.37823i | − | 0.655207i | −2.23390 | + | 0.0985483i | −2.26029 | 1.78496 | − | 1.78496i | −2.17726 | + | 2.17726i | − | 0.799034i | 1.91260 | + | 1.75098i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 22.4 | −0.819998 | − | 0.819998i | 1.37823 | − | 1.37823i | − | 0.655207i | 2.23390 | − | 0.0985483i | −2.26029 | −1.78496 | + | 1.78496i | −2.17726 | + | 2.17726i | − | 0.799034i | −1.91260 | − | 1.75098i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 22.5 | −0.459187 | − | 0.459187i | −1.79404 | + | 1.79404i | − | 1.57829i | −0.735651 | − | 2.11159i | 1.64760 | 2.52854 | − | 2.52854i | −1.64311 | + | 1.64311i | − | 3.43715i | −0.631815 | + | 1.30742i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 22.6 | −0.459187 | − | 0.459187i | −1.79404 | + | 1.79404i | − | 1.57829i | 0.735651 | + | 2.11159i | 1.64760 | −2.52854 | + | 2.52854i | −1.64311 | + | 1.64311i | − | 3.43715i | 0.631815 | − | 1.30742i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 22.7 | 0.562704 | + | 0.562704i | 0.534388 | − | 0.534388i | − | 1.36673i | −0.233538 | − | 2.22384i | 0.601404 | −0.567230 | + | 0.567230i | 1.89447 | − | 1.89447i | 2.42886i | 1.11995 | − | 1.38278i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 22.8 | 0.562704 | + | 0.562704i | 0.534388 | − | 0.534388i | − | 1.36673i | 0.233538 | + | 2.22384i | 0.601404 | 0.567230 | − | 0.567230i | 1.89447 | − | 1.89447i | 2.42886i | −1.11995 | + | 1.38278i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 22.9 | 1.40095 | + | 1.40095i | −1.47890 | + | 1.47890i | 1.92534i | −1.88210 | + | 1.20735i | −4.14375 | 2.58659 | − | 2.58659i | 0.104596 | − | 0.104596i | − | 1.37431i | −4.32817 | − | 0.945306i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 22.10 | 1.40095 | + | 1.40095i | −1.47890 | + | 1.47890i | 1.92534i | 1.88210 | − | 1.20735i | −4.14375 | −2.58659 | + | 2.58659i | 0.104596 | − | 0.104596i | − | 1.37431i | 4.32817 | + | 0.945306i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 68.1 | −1.68447 | + | 1.68447i | −0.639677 | − | 0.639677i | − | 3.67489i | −1.36809 | + | 1.76871i | 2.15503 | −2.98466 | − | 2.98466i | 2.82130 | + | 2.82130i | − | 2.18163i | −0.674828 | − | 5.28385i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 68.2 | −1.68447 | + | 1.68447i | −0.639677 | − | 0.639677i | − | 3.67489i | 1.36809 | − | 1.76871i | 2.15503 | 2.98466 | + | 2.98466i | 2.82130 | + | 2.82130i | − | 2.18163i | 0.674828 | + | 5.28385i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 68.3 | −0.819998 | + | 0.819998i | 1.37823 | + | 1.37823i | 0.655207i | −2.23390 | − | 0.0985483i | −2.26029 | 1.78496 | + | 1.78496i | −2.17726 | − | 2.17726i | 0.799034i | 1.91260 | − | 1.75098i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 68.4 | −0.819998 | + | 0.819998i | 1.37823 | + | 1.37823i | 0.655207i | 2.23390 | + | 0.0985483i | −2.26029 | −1.78496 | − | 1.78496i | −2.17726 | − | 2.17726i | 0.799034i | −1.91260 | + | 1.75098i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 68.5 | −0.459187 | + | 0.459187i | −1.79404 | − | 1.79404i | 1.57829i | −0.735651 | + | 2.11159i | 1.64760 | 2.52854 | + | 2.52854i | −1.64311 | − | 1.64311i | 3.43715i | −0.631815 | − | 1.30742i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 68.6 | −0.459187 | + | 0.459187i | −1.79404 | − | 1.79404i | 1.57829i | 0.735651 | − | 2.11159i | 1.64760 | −2.52854 | − | 2.52854i | −1.64311 | − | 1.64311i | 3.43715i | 0.631815 | + | 1.30742i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 68.7 | 0.562704 | − | 0.562704i | 0.534388 | + | 0.534388i | 1.36673i | −0.233538 | + | 2.22384i | 0.601404 | −0.567230 | − | 0.567230i | 1.89447 | + | 1.89447i | − | 2.42886i | 1.11995 | + | 1.38278i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 68.8 | 0.562704 | − | 0.562704i | 0.534388 | + | 0.534388i | 1.36673i | 0.233538 | − | 2.22384i | 0.601404 | 0.567230 | + | 0.567230i | 1.89447 | + | 1.89447i | − | 2.42886i | −1.11995 | − | 1.38278i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 68.9 | 1.40095 | − | 1.40095i | −1.47890 | − | 1.47890i | − | 1.92534i | −1.88210 | − | 1.20735i | −4.14375 | 2.58659 | + | 2.58659i | 0.104596 | + | 0.104596i | 1.37431i | −4.32817 | + | 0.945306i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 68.10 | 1.40095 | − | 1.40095i | −1.47890 | − | 1.47890i | − | 1.92534i | 1.88210 | + | 1.20735i | −4.14375 | −2.58659 | − | 2.58659i | 0.104596 | + | 0.104596i | 1.37431i | 4.32817 | − | 0.945306i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
| 23.b | odd | 2 | 1 | inner |
| 115.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 115.2.e.a | ✓ | 20 |
| 5.b | even | 2 | 1 | 575.2.e.d | 20 | ||
| 5.c | odd | 4 | 1 | inner | 115.2.e.a | ✓ | 20 |
| 5.c | odd | 4 | 1 | 575.2.e.d | 20 | ||
| 23.b | odd | 2 | 1 | inner | 115.2.e.a | ✓ | 20 |
| 115.c | odd | 2 | 1 | 575.2.e.d | 20 | ||
| 115.e | even | 4 | 1 | inner | 115.2.e.a | ✓ | 20 |
| 115.e | even | 4 | 1 | 575.2.e.d | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 115.2.e.a | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 115.2.e.a | ✓ | 20 | 5.c | odd | 4 | 1 | inner |
| 115.2.e.a | ✓ | 20 | 23.b | odd | 2 | 1 | inner |
| 115.2.e.a | ✓ | 20 | 115.e | even | 4 | 1 | inner |
| 575.2.e.d | 20 | 5.b | even | 2 | 1 | ||
| 575.2.e.d | 20 | 5.c | odd | 4 | 1 | ||
| 575.2.e.d | 20 | 115.c | odd | 2 | 1 | ||
| 575.2.e.d | 20 | 115.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(115, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{10} + 2 T^{9} + 2 T^{8} + \cdots + 8)^{2} \)
|
| $3$ |
\( (T^{10} + 4 T^{9} + \cdots + 50)^{2} \)
|
| $5$ |
\( T^{20} + 6 T^{18} + \cdots + 9765625 \)
|
| $7$ |
\( T^{20} + \cdots + 156250000 \)
|
| $11$ |
\( (T^{10} + 68 T^{8} + \cdots + 25000)^{2} \)
|
| $13$ |
\( (T^{10} - 2 T^{9} + 2 T^{8} + \cdots + 8)^{2} \)
|
| $17$ |
\( T^{20} + \cdots + 156250000 \)
|
| $19$ |
\( (T^{10} - 118 T^{8} + \cdots - 25000)^{2} \)
|
| $23$ |
\( T^{20} + \cdots + 41426511213649 \)
|
| $29$ |
\( (T^{10} + 97 T^{8} + \cdots + 1545049)^{2} \)
|
| $31$ |
\( (T^{5} + T^{4} - 58 T^{3} + \cdots + 25)^{4} \)
|
| $37$ |
\( T^{20} + \cdots + 94\!\cdots\!00 \)
|
| $41$ |
\( (T^{5} + 9 T^{4} - 10 T^{3} + \cdots - 47)^{4} \)
|
| $43$ |
\( T^{20} + \cdots + 1562500000000 \)
|
| $47$ |
\( (T^{10} + 4 T^{9} + \cdots + 315218)^{2} \)
|
| $53$ |
\( T^{20} + \cdots + 25000000000000 \)
|
| $59$ |
\( (T^{10} + 313 T^{8} + \cdots + 53290000)^{2} \)
|
| $61$ |
\( (T^{10} + 322 T^{8} + \cdots + 235225000)^{2} \)
|
| $67$ |
\( T^{20} + \cdots + 156250000 \)
|
| $71$ |
\( (T^{5} - 11 T^{4} + \cdots + 13903)^{4} \)
|
| $73$ |
\( (T^{10} + 28 T^{9} + \cdots + 1086338)^{2} \)
|
| $79$ |
\( (T^{10} - 458 T^{8} + \cdots - 133225000)^{2} \)
|
| $83$ |
\( T^{20} + \cdots + 97656250000 \)
|
| $89$ |
\( (T^{10} - 422 T^{8} + \cdots - 2175625000)^{2} \)
|
| $97$ |
\( T^{20} + \cdots + 74\!\cdots\!00 \)
|
| show more | |
| show less |