Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [120,2,Mod(59,120)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(120, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("120.59");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 120 = 2^{3} \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 120.m (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: | \(0.958204824255\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} + 24x^{14} + 192x^{12} + 672x^{10} + 1092x^{8} + 880x^{6} + 352x^{4} + 64x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{13} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} + 24x^{14} + 192x^{12} + 672x^{10} + 1092x^{8} + 880x^{6} + 352x^{4} + 64x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 3\nu^{13} + 69\nu^{11} + 506\nu^{9} + 1488\nu^{7} + 1638\nu^{5} + 594\nu^{3} + 44\nu ) / 8 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 7 \nu^{14} + 3 \nu^{13} - 164 \nu^{12} + 69 \nu^{11} - 1250 \nu^{10} + 506 \nu^{9} - 3984 \nu^{8} + \cdots - 16 ) / 16 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7 \nu^{14} + 3 \nu^{13} + 164 \nu^{12} + 69 \nu^{11} + 1250 \nu^{10} + 506 \nu^{9} + 3984 \nu^{8} + \cdots + 16 ) / 16 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5 \nu^{15} - 20 \nu^{14} + 107 \nu^{13} - 474 \nu^{12} + 657 \nu^{11} - 3698 \nu^{10} + 1074 \nu^{9} + \cdots - 300 ) / 16 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 5 \nu^{15} - 20 \nu^{14} - 107 \nu^{13} - 474 \nu^{12} - 657 \nu^{11} - 3698 \nu^{10} - 1074 \nu^{9} + \cdots - 300 ) / 16 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 8 \nu^{15} - 3 \nu^{14} + 190 \nu^{13} - 70 \nu^{12} + 1487 \nu^{11} - 530 \nu^{10} + 4970 \nu^{9} + \cdots - 20 ) / 16 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 19 \nu^{15} + 6 \nu^{14} - 448 \nu^{13} + 138 \nu^{12} - 3460 \nu^{11} + 1012 \nu^{10} - 11326 \nu^{9} + \cdots + 16 ) / 16 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 19 \nu^{15} + 6 \nu^{14} + 454 \nu^{13} + 138 \nu^{12} + 3598 \nu^{11} + 1012 \nu^{10} + 12338 \nu^{9} + \cdots + 16 ) / 16 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 11\nu^{15} + 261\nu^{13} + 2042\nu^{11} + 6862\nu^{9} + 10334\nu^{7} + 7410\nu^{5} + 2412\nu^{3} + 252\nu ) / 8 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 11 \nu^{15} + 12 \nu^{14} + 258 \nu^{13} + 282 \nu^{12} + 1971 \nu^{11} + 2164 \nu^{10} + 6311 \nu^{9} + \cdots + 136 ) / 8 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 11 \nu^{15} - 12 \nu^{14} + 258 \nu^{13} - 282 \nu^{12} + 1971 \nu^{11} - 2164 \nu^{10} + 6311 \nu^{9} + \cdots - 136 ) / 8 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 38 \nu^{15} + \nu^{14} - 896 \nu^{13} + 20 \nu^{12} - 6921 \nu^{11} + 100 \nu^{10} - 22674 \nu^{9} + \cdots - 72 ) / 16 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 38 \nu^{15} - \nu^{14} - 896 \nu^{13} - 20 \nu^{12} - 6921 \nu^{11} - 100 \nu^{10} - 22674 \nu^{9} + \cdots + 72 ) / 16 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 38 \nu^{15} + 15 \nu^{14} + 891 \nu^{13} + 352 \nu^{12} + 6803 \nu^{11} + 2692 \nu^{10} + 21762 \nu^{9} + \cdots + 76 ) / 16 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 38 \nu^{15} - 15 \nu^{14} + 891 \nu^{13} - 352 \nu^{12} + 6803 \nu^{11} - 2692 \nu^{10} + 21762 \nu^{9} + \cdots - 76 ) / 16 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} - \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} - \beta_{9} + 2\beta_{8} - 2\beta_{6} - 2\beta_{2} - 6 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 3 \beta_{15} - 3 \beta_{14} + 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} + \cdots - 8 \beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 8 \beta_{15} + 8 \beta_{14} - 20 \beta_{13} + 20 \beta_{12} + 5 \beta_{11} - 5 \beta_{10} + 12 \beta_{9} + \cdots + 48 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 40 \beta_{15} + 40 \beta_{14} - 10 \beta_{13} - 10 \beta_{12} - 31 \beta_{11} - 31 \beta_{10} + \cdots + 83 \beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 37 \beta_{15} - 37 \beta_{14} + 140 \beta_{13} - 140 \beta_{12} - 42 \beta_{11} + 42 \beta_{10} + \cdots - 252 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 497 \beta_{15} - 497 \beta_{14} + 175 \beta_{13} + 175 \beta_{12} + 404 \beta_{11} + 404 \beta_{10} + \cdots - 956 \beta_1 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 396 \beta_{15} + 396 \beta_{14} - 1792 \beta_{13} + 1792 \beta_{12} + 564 \beta_{11} - 564 \beta_{10} + \cdots + 2910 \)
|
| \(\nu^{9}\) | \(=\) |
\( 3054 \beta_{15} + 3054 \beta_{14} - 1194 \beta_{13} - 1194 \beta_{12} - 2520 \beta_{11} + \cdots + 5723 \beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( 4587 \beta_{15} - 4587 \beta_{14} + 22287 \beta_{13} - 22287 \beta_{12} - 7128 \beta_{11} + 7128 \beta_{10} + \cdots - 34858 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 37433 \beta_{15} - 37433 \beta_{14} + 15180 \beta_{13} + 15180 \beta_{12} + 31030 \beta_{11} + \cdots - 69508 \beta_1 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 54976 \beta_{15} + 54976 \beta_{14} - 274428 \beta_{13} + 274428 \beta_{12} + 88283 \beta_{11} + \cdots + 423384 \)
|
| \(\nu^{13}\) | \(=\) |
\( 458484 \beta_{15} + 458484 \beta_{14} - 188422 \beta_{13} - 188422 \beta_{12} - 380577 \beta_{11} + \cdots + 848621 \beta_1 \)
|
| \(\nu^{14}\) | \(=\) |
\( 667774 \beta_{15} - 667774 \beta_{14} + 3367136 \beta_{13} - 3367136 \beta_{12} - 1085560 \beta_{11} + \cdots - 5168936 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 5614479 \beta_{15} - 5614479 \beta_{14} + 2318785 \beta_{13} + 2318785 \beta_{12} + \cdots - 10380392 \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/120\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(41\) | \(61\) | \(97\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 59.1 |
|
−1.30656 | − | 0.541196i | 0.541196 | − | 1.64533i | 1.41421 | + | 1.41421i | 1.25928 | + | 1.84776i | −1.59755 | + | 1.85683i | 3.29066 | −1.08239 | − | 2.61313i | −2.41421 | − | 1.78089i | −0.645329 | − | 3.09573i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.2 | −1.30656 | − | 0.541196i | 0.541196 | + | 1.64533i | 1.41421 | + | 1.41421i | −1.25928 | + | 1.84776i | 0.183339 | − | 2.44262i | −3.29066 | −1.08239 | − | 2.61313i | −2.41421 | + | 1.78089i | 2.64533 | − | 1.73270i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.3 | −1.30656 | + | 0.541196i | 0.541196 | − | 1.64533i | 1.41421 | − | 1.41421i | −1.25928 | − | 1.84776i | 0.183339 | + | 2.44262i | −3.29066 | −1.08239 | + | 2.61313i | −2.41421 | − | 1.78089i | 2.64533 | + | 1.73270i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.4 | −1.30656 | + | 0.541196i | 0.541196 | + | 1.64533i | 1.41421 | − | 1.41421i | 1.25928 | − | 1.84776i | −1.59755 | − | 1.85683i | 3.29066 | −1.08239 | + | 2.61313i | −2.41421 | + | 1.78089i | −0.645329 | + | 3.09573i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.5 | −0.541196 | − | 1.30656i | −1.30656 | − | 1.13705i | −1.41421 | + | 1.41421i | −2.10100 | + | 0.765367i | −0.778527 | + | 2.32248i | −2.27411 | 2.61313 | + | 1.08239i | 0.414214 | + | 2.97127i | 2.13705 | + | 2.33088i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.6 | −0.541196 | − | 1.30656i | −1.30656 | + | 1.13705i | −1.41421 | + | 1.41421i | 2.10100 | + | 0.765367i | 2.19274 | + | 1.09174i | 2.27411 | 2.61313 | + | 1.08239i | 0.414214 | − | 2.97127i | −0.137055 | − | 3.15931i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.7 | −0.541196 | + | 1.30656i | −1.30656 | − | 1.13705i | −1.41421 | − | 1.41421i | 2.10100 | − | 0.765367i | 2.19274 | − | 1.09174i | 2.27411 | 2.61313 | − | 1.08239i | 0.414214 | + | 2.97127i | −0.137055 | + | 3.15931i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.8 | −0.541196 | + | 1.30656i | −1.30656 | + | 1.13705i | −1.41421 | − | 1.41421i | −2.10100 | − | 0.765367i | −0.778527 | − | 2.32248i | −2.27411 | 2.61313 | − | 1.08239i | 0.414214 | − | 2.97127i | 2.13705 | − | 2.33088i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.9 | 0.541196 | − | 1.30656i | 1.30656 | − | 1.13705i | −1.41421 | − | 1.41421i | −2.10100 | + | 0.765367i | −0.778527 | − | 2.32248i | 2.27411 | −2.61313 | + | 1.08239i | 0.414214 | − | 2.97127i | −0.137055 | + | 3.15931i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.10 | 0.541196 | − | 1.30656i | 1.30656 | + | 1.13705i | −1.41421 | − | 1.41421i | 2.10100 | + | 0.765367i | 2.19274 | − | 1.09174i | −2.27411 | −2.61313 | + | 1.08239i | 0.414214 | + | 2.97127i | 2.13705 | − | 2.33088i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.11 | 0.541196 | + | 1.30656i | 1.30656 | − | 1.13705i | −1.41421 | + | 1.41421i | 2.10100 | − | 0.765367i | 2.19274 | + | 1.09174i | −2.27411 | −2.61313 | − | 1.08239i | 0.414214 | − | 2.97127i | 2.13705 | + | 2.33088i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.12 | 0.541196 | + | 1.30656i | 1.30656 | + | 1.13705i | −1.41421 | + | 1.41421i | −2.10100 | − | 0.765367i | −0.778527 | + | 2.32248i | 2.27411 | −2.61313 | − | 1.08239i | 0.414214 | + | 2.97127i | −0.137055 | − | 3.15931i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.13 | 1.30656 | − | 0.541196i | −0.541196 | − | 1.64533i | 1.41421 | − | 1.41421i | 1.25928 | + | 1.84776i | −1.59755 | − | 1.85683i | −3.29066 | 1.08239 | − | 2.61313i | −2.41421 | + | 1.78089i | 2.64533 | + | 1.73270i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.14 | 1.30656 | − | 0.541196i | −0.541196 | + | 1.64533i | 1.41421 | − | 1.41421i | −1.25928 | + | 1.84776i | 0.183339 | + | 2.44262i | 3.29066 | 1.08239 | − | 2.61313i | −2.41421 | − | 1.78089i | −0.645329 | + | 3.09573i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.15 | 1.30656 | + | 0.541196i | −0.541196 | − | 1.64533i | 1.41421 | + | 1.41421i | −1.25928 | − | 1.84776i | 0.183339 | − | 2.44262i | 3.29066 | 1.08239 | + | 2.61313i | −2.41421 | + | 1.78089i | −0.645329 | − | 3.09573i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 59.16 | 1.30656 | + | 0.541196i | −0.541196 | + | 1.64533i | 1.41421 | + | 1.41421i | 1.25928 | − | 1.84776i | −1.59755 | + | 1.85683i | −3.29066 | 1.08239 | + | 2.61313i | −2.41421 | − | 1.78089i | 2.64533 | − | 1.73270i | |||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 5.b | even | 2 | 1 | inner |
| 8.d | odd | 2 | 1 | inner |
| 15.d | odd | 2 | 1 | inner |
| 24.f | even | 2 | 1 | inner |
| 40.e | odd | 2 | 1 | inner |
| 120.m | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 120.2.m.b | ✓ | 16 |
| 3.b | odd | 2 | 1 | inner | 120.2.m.b | ✓ | 16 |
| 4.b | odd | 2 | 1 | 480.2.m.b | 16 | ||
| 5.b | even | 2 | 1 | inner | 120.2.m.b | ✓ | 16 |
| 5.c | odd | 4 | 2 | 600.2.b.i | 16 | ||
| 8.b | even | 2 | 1 | 480.2.m.b | 16 | ||
| 8.d | odd | 2 | 1 | inner | 120.2.m.b | ✓ | 16 |
| 12.b | even | 2 | 1 | 480.2.m.b | 16 | ||
| 15.d | odd | 2 | 1 | inner | 120.2.m.b | ✓ | 16 |
| 15.e | even | 4 | 2 | 600.2.b.i | 16 | ||
| 20.d | odd | 2 | 1 | 480.2.m.b | 16 | ||
| 20.e | even | 4 | 2 | 2400.2.b.i | 16 | ||
| 24.f | even | 2 | 1 | inner | 120.2.m.b | ✓ | 16 |
| 24.h | odd | 2 | 1 | 480.2.m.b | 16 | ||
| 40.e | odd | 2 | 1 | inner | 120.2.m.b | ✓ | 16 |
| 40.f | even | 2 | 1 | 480.2.m.b | 16 | ||
| 40.i | odd | 4 | 2 | 2400.2.b.i | 16 | ||
| 40.k | even | 4 | 2 | 600.2.b.i | 16 | ||
| 60.h | even | 2 | 1 | 480.2.m.b | 16 | ||
| 60.l | odd | 4 | 2 | 2400.2.b.i | 16 | ||
| 120.i | odd | 2 | 1 | 480.2.m.b | 16 | ||
| 120.m | even | 2 | 1 | inner | 120.2.m.b | ✓ | 16 |
| 120.q | odd | 4 | 2 | 600.2.b.i | 16 | ||
| 120.w | even | 4 | 2 | 2400.2.b.i | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 120.2.m.b | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 120.2.m.b | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
| 120.2.m.b | ✓ | 16 | 5.b | even | 2 | 1 | inner |
| 120.2.m.b | ✓ | 16 | 8.d | odd | 2 | 1 | inner |
| 120.2.m.b | ✓ | 16 | 15.d | odd | 2 | 1 | inner |
| 120.2.m.b | ✓ | 16 | 24.f | even | 2 | 1 | inner |
| 120.2.m.b | ✓ | 16 | 40.e | odd | 2 | 1 | inner |
| 120.2.m.b | ✓ | 16 | 120.m | even | 2 | 1 | inner |
| 480.2.m.b | 16 | 4.b | odd | 2 | 1 | ||
| 480.2.m.b | 16 | 8.b | even | 2 | 1 | ||
| 480.2.m.b | 16 | 12.b | even | 2 | 1 | ||
| 480.2.m.b | 16 | 20.d | odd | 2 | 1 | ||
| 480.2.m.b | 16 | 24.h | odd | 2 | 1 | ||
| 480.2.m.b | 16 | 40.f | even | 2 | 1 | ||
| 480.2.m.b | 16 | 60.h | even | 2 | 1 | ||
| 480.2.m.b | 16 | 120.i | odd | 2 | 1 | ||
| 600.2.b.i | 16 | 5.c | odd | 4 | 2 | ||
| 600.2.b.i | 16 | 15.e | even | 4 | 2 | ||
| 600.2.b.i | 16 | 40.k | even | 4 | 2 | ||
| 600.2.b.i | 16 | 120.q | odd | 4 | 2 | ||
| 2400.2.b.i | 16 | 20.e | even | 4 | 2 | ||
| 2400.2.b.i | 16 | 40.i | odd | 4 | 2 | ||
| 2400.2.b.i | 16 | 60.l | odd | 4 | 2 | ||
| 2400.2.b.i | 16 | 120.w | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} - 16T_{7}^{2} + 56 \)
acting on \(S_{2}^{\mathrm{new}}(120, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{8} + 16)^{2} \)
|
| $3$ |
\( (T^{8} + 4 T^{6} + 14 T^{4} + \cdots + 81)^{2} \)
|
| $5$ |
\( (T^{8} - 4 T^{6} + \cdots + 625)^{2} \)
|
| $7$ |
\( (T^{4} - 16 T^{2} + 56)^{4} \)
|
| $11$ |
\( (T^{4} + 24 T^{2} + 112)^{4} \)
|
| $13$ |
\( (T^{4} - 32 T^{2} + 224)^{4} \)
|
| $17$ |
\( (T^{4} - 16 T^{2} + 32)^{4} \)
|
| $19$ |
\( (T^{2} + 4 T - 4)^{8} \)
|
| $23$ |
\( (T^{4} + 8 T^{2} + 8)^{4} \)
|
| $29$ |
\( (T^{4} - 40 T^{2} + 112)^{4} \)
|
| $31$ |
\( (T^{4} + 48 T^{2} + 64)^{4} \)
|
| $37$ |
\( (T^{4} - 64 T^{2} + 224)^{4} \)
|
| $41$ |
\( (T^{4} + 80 T^{2} + 448)^{4} \)
|
| $43$ |
\( (T^{4} + 112 T^{2} + 2744)^{4} \)
|
| $47$ |
\( (T^{4} + 8 T^{2} + 8)^{4} \)
|
| $53$ |
\( (T^{4} + 144 T^{2} + 2592)^{4} \)
|
| $59$ |
\( (T^{4} + 24 T^{2} + 112)^{4} \)
|
| $61$ |
\( (T^{2} + 72)^{8} \)
|
| $67$ |
\( (T^{4} + 16 T^{2} + 56)^{4} \)
|
| $71$ |
\( (T^{4} - 192 T^{2} + 7168)^{4} \)
|
| $73$ |
\( (T^{4} + 64 T^{2} + 896)^{4} \)
|
| $79$ |
\( (T^{4} + 272 T^{2} + 64)^{4} \)
|
| $83$ |
\( (T^{4} - 136 T^{2} + 4232)^{4} \)
|
| $89$ |
\( (T^{4} + 96 T^{2} + 1792)^{4} \)
|
| $97$ |
\( (T^{4} + 128 T^{2} + 896)^{4} \)
|
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