Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [196,4,Mod(195,196)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(196, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("196.195");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 196 = 2^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 196.d (of order \(2\), degree \(1\), 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: | \(11.5643743611\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| 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: | \( 2^{34}\cdot 7^{2} \) |
| Twist minimal: | no (minimal twist has level 28) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{20} - 2 x^{18} - 24 x^{17} + 28 x^{16} + 56 x^{15} - 192 x^{14} + 352 x^{13} - 448 x^{12} + \cdots + 1073741824 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 647 \nu^{19} + 13656 \nu^{18} + 111550 \nu^{17} - 323784 \nu^{16} + 943772 \nu^{15} + \cdots - 3620321886208 ) / 1140112490496 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 23993 \nu^{19} - 6344 \nu^{18} - 14094 \nu^{17} + 12136 \nu^{16} - 12924 \nu^{15} + \cdots + 3730447532032 ) / 2280224980992 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 793 \nu^{19} + 7760 \nu^{18} + 70462 \nu^{17} - 82360 \nu^{16} + 36188 \nu^{15} + \cdots - 2080173457408 ) / 162873212928 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{19} + 30 \nu^{17} - 24 \nu^{16} - 36 \nu^{15} - 712 \nu^{14} + 704 \nu^{13} + \cdots - 1342177280 ) / 67108864 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 13897 \nu^{19} + 28016 \nu^{18} - 2418 \nu^{17} - 141240 \nu^{16} + 1971516 \nu^{15} + \cdots + 1506862432256 ) / 1140112490496 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 31255 \nu^{19} + 175232 \nu^{18} - 16402 \nu^{17} + 13480 \nu^{16} - 1137668 \nu^{15} + \cdots - 84422950912 ) / 2280224980992 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 1827 \nu^{19} - 12184 \nu^{18} + 26338 \nu^{17} + 23752 \nu^{16} + 154756 \nu^{15} + \cdots - 1009703190528 ) / 142514061312 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 21709 \nu^{19} - 80848 \nu^{18} - 680090 \nu^{17} - 450584 \nu^{16} + 2546668 \nu^{15} + \cdots + 35910624215040 ) / 1140112490496 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 39513 \nu^{19} - 150440 \nu^{18} + 112562 \nu^{17} - 15832 \nu^{16} + \cdots - 3687497859072 ) / 2280224980992 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 3119 \nu^{19} + 162512 \nu^{18} + 9170 \nu^{17} + 472632 \nu^{16} - 2265020 \nu^{15} + \cdots - 4703861604352 ) / 1140112490496 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 485 \nu^{19} + 4503 \nu^{18} - 3958 \nu^{17} + 874 \nu^{16} - 4612 \nu^{15} + \cdots - 53217329152 ) / 35628515328 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 11333 \nu^{19} + 9801 \nu^{18} - 77998 \nu^{17} - 275354 \nu^{16} + 448188 \nu^{15} + \cdots + 1503775424512 ) / 285028122624 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 49249 \nu^{19} - 164864 \nu^{18} - 592146 \nu^{17} + 199464 \nu^{16} + 2980924 \nu^{15} + \cdots + 35468578127872 ) / 1140112490496 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 109597 \nu^{19} + 800 \nu^{18} + 262758 \nu^{17} - 332024 \nu^{16} + 2330348 \nu^{15} + \cdots - 21151774408704 ) / 2280224980992 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 6773 \nu^{19} + 122270 \nu^{18} + 424210 \nu^{17} - 367396 \nu^{16} + \cdots - 21329344462848 ) / 570056245248 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 120103 \nu^{19} - 101960 \nu^{18} + 386354 \nu^{17} - 570008 \nu^{16} + \cdots - 10653800595456 ) / 2280224980992 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 80405 \nu^{19} - 286056 \nu^{18} - 322806 \nu^{17} + 265672 \nu^{16} + \cdots + 26576318103552 ) / 2280224980992 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 120529 \nu^{19} + 227552 \nu^{18} - 437634 \nu^{17} - 1349976 \nu^{16} + \cdots + 34701590921216 ) / 2280224980992 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 183231 \nu^{19} - 98024 \nu^{18} - 1540478 \nu^{17} - 1850008 \nu^{16} + \cdots + 71002251853824 ) / 2280224980992 \)
|
| \(\nu\) | \(=\) |
\( ( 3 \beta_{19} + 3 \beta_{17} - 3 \beta_{13} - 4 \beta_{12} + 7 \beta_{11} - 3 \beta_{10} + \cdots - 3 \beta_1 ) / 56 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3 \beta_{19} - 4 \beta_{17} - 7 \beta_{16} + 7 \beta_{15} + 4 \beta_{13} - 4 \beta_{12} - 3 \beta_{10} + \cdots + 7 ) / 28 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{11} + 2\beta_{9} - 2\beta_{2} + 8 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 10 \beta_{19} + 14 \beta_{18} + 11 \beta_{17} - 7 \beta_{16} + 21 \beta_{15} + 14 \beta_{14} + \cdots - 77 ) / 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 5 \beta_{19} + 19 \beta_{17} + 14 \beta_{16} - 14 \beta_{15} - 112 \beta_{14} - 33 \beta_{13} + \cdots - 70 ) / 14 \)
|
| \(\nu^{6}\) | \(=\) |
\( -12\beta_{18} + 10\beta_{16} - 12\beta_{14} + 12\beta_{9} - 20\beta_{7} + 20\beta_{5} - 86\beta_{2} + 110 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 63 \beta_{19} + 84 \beta_{18} + 483 \beta_{17} - 112 \beta_{16} - 168 \beta_{15} + 196 \beta_{14} + \cdots - 2464 ) / 7 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 452 \beta_{19} + 84 \beta_{18} - 802 \beta_{17} - 266 \beta_{16} + 98 \beta_{15} - 588 \beta_{14} + \cdots + 2898 ) / 7 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 160 \beta_{18} + 424 \beta_{16} + 288 \beta_{14} + 332 \beta_{11} + 232 \beta_{9} - 224 \beta_{7} + \cdots + 888 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 516 \beta_{19} + 3192 \beta_{18} + 6704 \beta_{17} - 308 \beta_{16} + 3388 \beta_{15} + 4088 \beta_{14} + \cdots + 49028 ) / 7 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 3068 \beta_{19} + 8624 \beta_{18} - 17204 \beta_{17} - 5376 \beta_{16} - 5600 \beta_{15} - 4368 \beta_{14} + \cdots - 52416 ) / 7 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 1696 \beta_{18} - 9264 \beta_{16} + 20064 \beta_{14} - 4432 \beta_{11} + 1344 \beta_{9} + \cdots + 75248 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 13352 \beta_{19} + 85568 \beta_{18} + 216808 \beta_{17} - 92624 \beta_{16} + 26544 \beta_{15} + \cdots + 437136 ) / 7 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 128240 \beta_{19} - 5600 \beta_{18} - 598080 \beta_{17} - 170800 \beta_{16} + 486640 \beta_{15} + \cdots + 1294832 ) / 7 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 84352 \beta_{18} + 48128 \beta_{16} + 31872 \beta_{14} + 64352 \beta_{11} + 90048 \beta_{9} + \cdots - 450048 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( - 3899328 \beta_{19} + 1037120 \beta_{18} - 2617824 \beta_{17} + 56672 \beta_{16} + 1829856 \beta_{15} + \cdots + 11332384 ) / 7 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 2506912 \beta_{19} + 2367232 \beta_{18} - 5478496 \beta_{17} + 1022784 \beta_{16} - 4130112 \beta_{15} + \cdots + 125963712 ) / 7 \)
|
| \(\nu^{18}\) | \(=\) |
\( - 1661184 \beta_{18} - 551552 \beta_{16} - 272640 \beta_{14} + 7832576 \beta_{11} + 509184 \beta_{9} + \cdots + 55624832 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( 54448192 \beta_{19} + 8424192 \beta_{18} + 32666432 \beta_{17} + 50376704 \beta_{16} + 47054336 \beta_{15} + \cdots + 480256 ) / 7 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/196\mathbb{Z}\right)^\times\).
| \(n\) | \(99\) | \(101\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 195.1 |
|
−2.64271 | − | 1.00800i | −4.22328 | 5.96785 | + | 5.32773i | − | 1.19347i | 11.1609 | + | 4.25709i | 0 | −10.4009 | − | 20.0953i | −9.16387 | −1.20303 | + | 3.15401i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 195.2 | −2.64271 | − | 1.00800i | 4.22328 | 5.96785 | + | 5.32773i | 1.19347i | −11.1609 | − | 4.25709i | 0 | −10.4009 | − | 20.0953i | −9.16387 | 1.20303 | − | 3.15401i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 195.3 | −2.64271 | + | 1.00800i | −4.22328 | 5.96785 | − | 5.32773i | 1.19347i | 11.1609 | − | 4.25709i | 0 | −10.4009 | + | 20.0953i | −9.16387 | −1.20303 | − | 3.15401i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 195.4 | −2.64271 | + | 1.00800i | 4.22328 | 5.96785 | − | 5.32773i | − | 1.19347i | −11.1609 | + | 4.25709i | 0 | −10.4009 | + | 20.0953i | −9.16387 | 1.20303 | + | 3.15401i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 195.5 | −1.33521 | − | 2.49344i | −9.30691 | −4.43445 | + | 6.65850i | 5.90673i | 12.4266 | + | 23.2062i | 0 | 22.5235 | + | 2.16656i | 59.6185 | 14.7281 | − | 7.88670i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 195.6 | −1.33521 | − | 2.49344i | 9.30691 | −4.43445 | + | 6.65850i | − | 5.90673i | −12.4266 | − | 23.2062i | 0 | 22.5235 | + | 2.16656i | 59.6185 | −14.7281 | + | 7.88670i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 195.7 | −1.33521 | + | 2.49344i | −9.30691 | −4.43445 | − | 6.65850i | − | 5.90673i | 12.4266 | − | 23.2062i | 0 | 22.5235 | − | 2.16656i | 59.6185 | 14.7281 | + | 7.88670i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 195.8 | −1.33521 | + | 2.49344i | 9.30691 | −4.43445 | − | 6.65850i | 5.90673i | −12.4266 | + | 23.2062i | 0 | 22.5235 | − | 2.16656i | 59.6185 | −14.7281 | − | 7.88670i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 195.9 | −0.334409 | − | 2.80859i | −3.34267 | −7.77634 | + | 1.87844i | 18.4271i | 1.11782 | + | 9.38820i | 0 | 7.87623 | + | 21.2124i | −15.8265 | 51.7541 | − | 6.16219i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 195.10 | −0.334409 | − | 2.80859i | 3.34267 | −7.77634 | + | 1.87844i | − | 18.4271i | −1.11782 | − | 9.38820i | 0 | 7.87623 | + | 21.2124i | −15.8265 | −51.7541 | + | 6.16219i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 195.11 | −0.334409 | + | 2.80859i | −3.34267 | −7.77634 | − | 1.87844i | − | 18.4271i | 1.11782 | − | 9.38820i | 0 | 7.87623 | − | 21.2124i | −15.8265 | 51.7541 | + | 6.16219i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 195.12 | −0.334409 | + | 2.80859i | 3.34267 | −7.77634 | − | 1.87844i | 18.4271i | −1.11782 | + | 9.38820i | 0 | 7.87623 | − | 21.2124i | −15.8265 | −51.7541 | − | 6.16219i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 195.13 | 1.51453 | − | 2.38877i | −0.0938614 | −3.41241 | − | 7.23571i | 14.4177i | −0.142156 | + | 0.224213i | 0 | −22.4526 | − | 2.80725i | −26.9912 | 34.4404 | + | 21.8360i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 195.14 | 1.51453 | − | 2.38877i | 0.0938614 | −3.41241 | − | 7.23571i | − | 14.4177i | 0.142156 | − | 0.224213i | 0 | −22.4526 | − | 2.80725i | −26.9912 | −34.4404 | − | 21.8360i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 195.15 | 1.51453 | + | 2.38877i | −0.0938614 | −3.41241 | + | 7.23571i | − | 14.4177i | −0.142156 | − | 0.224213i | 0 | −22.4526 | + | 2.80725i | −26.9912 | 34.4404 | − | 21.8360i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 195.16 | 1.51453 | + | 2.38877i | 0.0938614 | −3.41241 | + | 7.23571i | 14.4177i | 0.142156 | + | 0.224213i | 0 | −22.4526 | + | 2.80725i | −26.9912 | −34.4404 | + | 21.8360i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 195.17 | 2.79780 | − | 0.415122i | −6.88209 | 7.65535 | − | 2.32285i | 4.82284i | −19.2547 | + | 2.85690i | 0 | 20.4539 | − | 9.67677i | 20.3631 | 2.00207 | + | 13.4933i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 195.18 | 2.79780 | − | 0.415122i | 6.88209 | 7.65535 | − | 2.32285i | − | 4.82284i | 19.2547 | − | 2.85690i | 0 | 20.4539 | − | 9.67677i | 20.3631 | −2.00207 | − | 13.4933i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 195.19 | 2.79780 | + | 0.415122i | −6.88209 | 7.65535 | + | 2.32285i | − | 4.82284i | −19.2547 | − | 2.85690i | 0 | 20.4539 | + | 9.67677i | 20.3631 | 2.00207 | − | 13.4933i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 195.20 | 2.79780 | + | 0.415122i | 6.88209 | 7.65535 | + | 2.32285i | 4.82284i | 19.2547 | + | 2.85690i | 0 | 20.4539 | + | 9.67677i | 20.3631 | −2.00207 | + | 13.4933i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 28.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 196.4.d.b | 20 | |
| 4.b | odd | 2 | 1 | inner | 196.4.d.b | 20 | |
| 7.b | odd | 2 | 1 | inner | 196.4.d.b | 20 | |
| 7.c | even | 3 | 1 | 28.4.f.a | ✓ | 20 | |
| 7.c | even | 3 | 1 | 196.4.f.d | 20 | ||
| 7.d | odd | 6 | 1 | 28.4.f.a | ✓ | 20 | |
| 7.d | odd | 6 | 1 | 196.4.f.d | 20 | ||
| 28.d | even | 2 | 1 | inner | 196.4.d.b | 20 | |
| 28.f | even | 6 | 1 | 28.4.f.a | ✓ | 20 | |
| 28.f | even | 6 | 1 | 196.4.f.d | 20 | ||
| 28.g | odd | 6 | 1 | 28.4.f.a | ✓ | 20 | |
| 28.g | odd | 6 | 1 | 196.4.f.d | 20 | ||
| 56.j | odd | 6 | 1 | 448.4.p.h | 20 | ||
| 56.k | odd | 6 | 1 | 448.4.p.h | 20 | ||
| 56.m | even | 6 | 1 | 448.4.p.h | 20 | ||
| 56.p | 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 | 7.c | even | 3 | 1 | |
| 28.4.f.a | ✓ | 20 | 7.d | odd | 6 | 1 | |
| 28.4.f.a | ✓ | 20 | 28.f | even | 6 | 1 | |
| 28.4.f.a | ✓ | 20 | 28.g | odd | 6 | 1 | |
| 196.4.d.b | 20 | 1.a | even | 1 | 1 | trivial | |
| 196.4.d.b | 20 | 4.b | odd | 2 | 1 | inner | |
| 196.4.d.b | 20 | 7.b | odd | 2 | 1 | inner | |
| 196.4.d.b | 20 | 28.d | even | 2 | 1 | inner | |
| 196.4.f.d | 20 | 7.c | even | 3 | 1 | ||
| 196.4.f.d | 20 | 7.d | odd | 6 | 1 | ||
| 196.4.f.d | 20 | 28.f | even | 6 | 1 | ||
| 196.4.f.d | 20 | 28.g | odd | 6 | 1 | ||
| 448.4.p.h | 20 | 56.j | odd | 6 | 1 | ||
| 448.4.p.h | 20 | 56.k | odd | 6 | 1 | ||
| 448.4.p.h | 20 | 56.m | even | 6 | 1 | ||
| 448.4.p.h | 20 | 56.p | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} - 163T_{3}^{8} + 8190T_{3}^{6} - 145786T_{3}^{4} + 818881T_{3}^{2} - 7203 \)
acting on \(S_{4}^{\mathrm{new}}(196, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{10} + 2 T^{8} + \cdots + 32768)^{2} \)
|
| $3$ |
\( (T^{10} - 163 T^{8} + \cdots - 7203)^{2} \)
|
| $5$ |
\( (T^{10} + 607 T^{8} + \cdots + 81588675)^{2} \)
|
| $7$ |
\( T^{20} \)
|
| $11$ |
\( (T^{10} + \cdots + 450690682225)^{2} \)
|
| $13$ |
\( (T^{10} + \cdots + 10072189747200)^{2} \)
|
| $17$ |
\( (T^{10} + \cdots + 60\!\cdots\!75)^{2} \)
|
| $19$ |
\( (T^{10} + \cdots - 11\!\cdots\!75)^{2} \)
|
| $23$ |
\( (T^{10} + \cdots + 41\!\cdots\!81)^{2} \)
|
| $29$ |
\( (T^{5} + 88 T^{4} + \cdots + 5066614016)^{4} \)
|
| $31$ |
\( (T^{10} + \cdots - 65\!\cdots\!75)^{2} \)
|
| $37$ |
\( (T^{5} + 129 T^{4} + \cdots + 21966324285)^{4} \)
|
| $41$ |
\( (T^{10} + \cdots + 14\!\cdots\!72)^{2} \)
|
| $43$ |
\( (T^{10} + \cdots + 55\!\cdots\!56)^{2} \)
|
| $47$ |
\( (T^{10} + \cdots - 95\!\cdots\!75)^{2} \)
|
| $53$ |
\( (T^{5} + \cdots + 1529727834265)^{4} \)
|
| $59$ |
\( (T^{10} + \cdots - 73\!\cdots\!75)^{2} \)
|
| $61$ |
\( (T^{10} + \cdots + 25\!\cdots\!47)^{2} \)
|
| $67$ |
\( (T^{10} + \cdots + 12\!\cdots\!89)^{2} \)
|
| $71$ |
\( (T^{10} + \cdots + 16\!\cdots\!00)^{2} \)
|
| $73$ |
\( (T^{10} + \cdots + 39\!\cdots\!75)^{2} \)
|
| $79$ |
\( (T^{10} + \cdots + 38\!\cdots\!25)^{2} \)
|
| $83$ |
\( (T^{10} + \cdots - 41\!\cdots\!08)^{2} \)
|
| $89$ |
\( (T^{10} + \cdots + 99\!\cdots\!27)^{2} \)
|
| $97$ |
\( (T^{10} + \cdots + 12\!\cdots\!00)^{2} \)
|
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