Properties

Label 30.6.e.a
Level $30$
Weight $6$
Character orbit 30.e
Analytic conductor $4.812$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [30,6,Mod(17,30)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(30, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("30.17");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 30.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: \(4.81151459439\)
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: \( x^{20} + 1126 x^{18} + 420245 x^{16} + 66878446 x^{14} + 5652274660 x^{12} + 280235806770 x^{10} + 8434804524517 x^{8} + 152000977499746 x^{6} + \cdots + 87\!\cdots\!24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{25}\cdot 3^{14}\cdot 5^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{9} q^{3} + 16 \beta_{5} q^{4} + (\beta_{8} - \beta_{3} - 2 \beta_{2}) q^{5} + (\beta_{7} + 4) q^{6} + (\beta_{19} + \beta_{13} + \beta_{10} + \beta_{7} - \beta_{6} - 4 \beta_{5} - \beta_{4} + \beta_1 + 4) q^{7} - 16 \beta_{2} q^{8} + ( - \beta_{19} - \beta_{16} - \beta_{14} + \beta_{11} + 2 \beta_{10} + \beta_{8} - \beta_{6} + 42 \beta_{5} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{9} q^{3} + 16 \beta_{5} q^{4} + (\beta_{8} - \beta_{3} - 2 \beta_{2}) q^{5} + (\beta_{7} + 4) q^{6} + (\beta_{19} + \beta_{13} + \beta_{10} + \beta_{7} - \beta_{6} - 4 \beta_{5} - \beta_{4} + \beta_1 + 4) q^{7} - 16 \beta_{2} q^{8} + ( - \beta_{19} - \beta_{16} - \beta_{14} + \beta_{11} + 2 \beta_{10} + \beta_{8} - \beta_{6} + 42 \beta_{5} + \cdots - 1) q^{9}+ \cdots + ( - 484 \beta_{19} + 75 \beta_{18} - 317 \beta_{17} - 92 \beta_{16} + \cdots - 131) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 4 q^{3} + 80 q^{6} + 76 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 4 q^{3} + 80 q^{6} + 76 q^{7} - 496 q^{10} + 64 q^{12} + 2640 q^{13} + 3128 q^{15} - 5120 q^{16} - 2272 q^{18} - 760 q^{21} - 464 q^{22} - 15776 q^{25} + 16556 q^{27} + 1216 q^{28} + 3408 q^{30} + 23960 q^{31} - 7684 q^{33} - 13120 q^{36} - 60696 q^{37} - 512 q^{40} + 46864 q^{42} + 48024 q^{43} + 31912 q^{45} + 58880 q^{46} + 1024 q^{48} - 155240 q^{51} - 42240 q^{52} - 132228 q^{55} + 167744 q^{57} + 38000 q^{58} + 19136 q^{60} + 160680 q^{61} - 219364 q^{63} - 158560 q^{66} - 118736 q^{67} - 55216 q^{70} + 36352 q^{72} + 379100 q^{73} + 365548 q^{75} + 78080 q^{76} - 25440 q^{78} - 282260 q^{81} - 262976 q^{82} - 227432 q^{85} + 249380 q^{87} - 7424 q^{88} - 128992 q^{90} + 499200 q^{91} - 304072 q^{93} - 20480 q^{96} - 288228 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 1126 x^{18} + 420245 x^{16} + 66878446 x^{14} + 5652274660 x^{12} + 280235806770 x^{10} + 8434804524517 x^{8} + 152000977499746 x^{6} + \cdots + 87\!\cdots\!24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 27\!\cdots\!34 \nu^{19} + \cdots - 31\!\cdots\!76 ) / 36\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 27\!\cdots\!34 \nu^{19} + \cdots - 31\!\cdots\!76 ) / 36\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\!\cdots\!52 \nu^{19} + \cdots - 54\!\cdots\!80 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 16\!\cdots\!60 \nu^{19} + \cdots + 21\!\cdots\!40 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 15\!\cdots\!95 \nu^{19} + \cdots - 37\!\cdots\!36 \nu ) / 83\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 16\!\cdots\!31 \nu^{19} + \cdots + 50\!\cdots\!92 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 65\!\cdots\!41 \nu^{19} + \cdots + 40\!\cdots\!80 ) / 36\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 13\!\cdots\!85 \nu^{19} + \cdots - 21\!\cdots\!48 ) / 70\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 14\!\cdots\!61 \nu^{19} + \cdots - 83\!\cdots\!28 ) / 70\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 34\!\cdots\!12 \nu^{19} + \cdots + 17\!\cdots\!40 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 11\!\cdots\!53 \nu^{19} + \cdots - 40\!\cdots\!12 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 16\!\cdots\!61 \nu^{19} + \cdots - 27\!\cdots\!00 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 26\!\cdots\!23 \nu^{19} + \cdots + 64\!\cdots\!24 ) / 35\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 30\!\cdots\!03 \nu^{19} + \cdots - 32\!\cdots\!16 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 15\!\cdots\!26 \nu^{19} + \cdots + 16\!\cdots\!72 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 39\!\cdots\!41 \nu^{19} + \cdots - 39\!\cdots\!16 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 42\!\cdots\!67 \nu^{19} + \cdots - 25\!\cdots\!36 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 11\!\cdots\!36 \nu^{19} + \cdots - 17\!\cdots\!48 ) / 53\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 13\!\cdots\!11 \nu^{19} + \cdots - 54\!\cdots\!00 ) / 53\!\cdots\!20 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( - 15 \beta_{19} + 15 \beta_{18} - \beta_{17} + \beta_{16} - 5 \beta_{15} + 3 \beta_{14} + 25 \beta_{13} - 34 \beta_{12} - 10 \beta_{11} - 3 \beta_{10} + 9 \beta_{9} + 80 \beta_{8} - 15 \beta_{7} - 43 \beta_{5} - 15 \beta_{4} + 11 \beta_{3} - 34 \beta_{2} + \cdots - 20 ) / 360 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 7 \beta_{19} + 75 \beta_{18} - 371 \beta_{17} + 289 \beta_{16} - 55 \beta_{15} + 23 \beta_{14} - 15 \beta_{13} - 358 \beta_{12} - 170 \beta_{11} + 371 \beta_{10} - 873 \beta_{9} - 1304 \beta_{8} + 565 \beta_{7} + 40 \beta_{6} + \cdots - 40614 ) / 360 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 74 \beta_{19} - 1770 \beta_{18} - 102 \beta_{17} - 1594 \beta_{16} + 505 \beta_{15} - 2824 \beta_{14} - 5560 \beta_{13} + 7720 \beta_{12} + 2400 \beta_{11} + 1191 \beta_{10} + 3352 \beta_{9} - 14292 \beta_{8} + \cdots + 4043 ) / 180 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 3821 \beta_{19} - 16245 \beta_{18} + 200291 \beta_{17} - 187867 \beta_{16} + 10345 \beta_{15} - 16763 \beta_{14} - 27765 \beta_{13} + 176122 \beta_{12} + 74300 \beta_{11} - 166785 \beta_{10} + \cdots + 15476018 ) / 360 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 875797 \beta_{19} + 1330365 \beta_{18} + 164379 \beta_{17} + 2041783 \beta_{16} - 489825 \beta_{15} + 3613003 \beta_{14} + 6487745 \beta_{13} - 7716610 \beta_{12} - 2311260 \beta_{11} + \cdots - 4725386 ) / 360 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1355321 \beta_{19} - 90750 \beta_{18} - 26431856 \beta_{17} + 27877927 \beta_{16} - 229855 \beta_{15} + 2113838 \beta_{14} + 6916425 \beta_{13} - 21165217 \beta_{12} + \cdots - 1894151458 ) / 90 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 722371461 \beta_{19} - 565938735 \beta_{18} - 114568099 \beta_{17} - 1173742097 \beta_{16} + 274117205 \beta_{15} - 2131685583 \beta_{14} - 3765205465 \beta_{13} + \cdots + 2724326188 ) / 360 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1318697253 \beta_{19} + 1096342725 \beta_{18} + 18975899455 \beta_{17} - 21390939433 \beta_{16} - 264334545 \beta_{15} - 1423473555 \beta_{14} + \cdots + 1310151864334 ) / 120 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 237693334476 \beta_{19} + 129802629180 \beta_{18} + 36168855028 \beta_{17} + 331327108628 \beta_{16} - 77361600145 \beta_{15} + 612574383486 \beta_{14} + \cdots - 773758636153 ) / 180 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 2494434333269 \beta_{19} - 2840463386655 \beta_{18} - 31079297497211 \beta_{17} + 36414195217135 \beta_{16} + 846596615795 \beta_{15} + \cdots - 20\!\cdots\!42 ) / 360 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 288583873836311 \beta_{19} - 126532403005395 \beta_{18} - 43174948560249 \beta_{17} - 371941328281457 \beta_{16} + 86979160095995 \beta_{15} + \cdots + 872033855220418 ) / 360 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 370957099791512 \beta_{19} + 478122052964355 \beta_{18} + \cdots + 28\!\cdots\!96 ) / 90 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 16\!\cdots\!57 \beta_{19} + \cdots - 48\!\cdots\!76 ) / 360 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 85\!\cdots\!57 \beta_{19} + \cdots - 62\!\cdots\!66 ) / 360 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 48\!\cdots\!10 \beta_{19} + \cdots + 13\!\cdots\!35 ) / 180 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 16\!\cdots\!87 \beta_{19} + \cdots + 11\!\cdots\!66 ) / 120 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 54\!\cdots\!33 \beta_{19} + \cdots - 15\!\cdots\!74 ) / 360 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 68\!\cdots\!67 \beta_{19} + \cdots - 47\!\cdots\!76 ) / 90 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 30\!\cdots\!77 \beta_{19} + \cdots + 85\!\cdots\!96 ) / 360 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/30\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(-\beta_{5}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
6.04149i
18.0817i
5.20628i
3.30354i
23.6117i
5.27572i
1.34618i
6.07365i
6.17029i
7.90019i
6.04149i
18.0817i
5.20628i
3.30354i
23.6117i
5.27572i
1.34618i
6.07365i
6.17029i
7.90019i
−2.82843 + 2.82843i −15.3235 + 2.86199i 16.0000i 7.18151 55.4385i 35.2464 51.4363i 137.897 + 137.897i 45.2548 + 45.2548i 226.618 87.7113i 136.491 + 177.116i
17.2 −2.82843 + 2.82843i −9.52925 + 12.3367i 16.0000i 12.9164 + 54.3890i −7.94059 61.8462i −141.804 141.804i 45.2548 + 45.2548i −61.3868 235.118i −190.369 117.302i
17.3 −2.82843 + 2.82843i −5.94397 14.4107i 16.0000i −4.92923 + 55.6840i 57.5718 + 23.9476i 93.4247 + 93.4247i 45.2548 + 45.2548i −172.338 + 171.314i −143.556 171.440i
17.4 −2.82843 + 2.82843i 12.0966 9.83224i 16.0000i −50.1686 24.6600i −6.40449 + 62.0240i −72.4506 72.4506i 45.2548 + 45.2548i 49.6540 237.873i 211.647 72.1491i
17.5 −2.82843 + 2.82843i 14.1646 + 6.50878i 16.0000i 55.5060 6.63996i −58.4731 + 21.6539i 1.93288 + 1.93288i 45.2548 + 45.2548i 158.272 + 184.389i −138.214 + 175.775i
17.6 2.82843 2.82843i −12.3367 + 9.52925i 16.0000i −12.9164 54.3890i −7.94059 + 61.8462i −141.804 141.804i −45.2548 45.2548i 61.3868 235.118i −190.369 117.302i
17.7 2.82843 2.82843i −6.50878 14.1646i 16.0000i −55.5060 + 6.63996i −58.4731 21.6539i 1.93288 + 1.93288i −45.2548 45.2548i −158.272 + 184.389i −138.214 + 175.775i
17.8 2.82843 2.82843i −2.86199 + 15.3235i 16.0000i −7.18151 + 55.4385i 35.2464 + 51.4363i 137.897 + 137.897i −45.2548 45.2548i −226.618 87.7113i 136.491 + 177.116i
17.9 2.82843 2.82843i 9.83224 12.0966i 16.0000i 50.1686 + 24.6600i −6.40449 62.0240i −72.4506 72.4506i −45.2548 45.2548i −49.6540 237.873i 211.647 72.1491i
17.10 2.82843 2.82843i 14.4107 + 5.94397i 16.0000i 4.92923 55.6840i 57.5718 23.9476i 93.4247 + 93.4247i −45.2548 45.2548i 172.338 + 171.314i −143.556 171.440i
23.1 −2.82843 2.82843i −15.3235 2.86199i 16.0000i 7.18151 + 55.4385i 35.2464 + 51.4363i 137.897 137.897i 45.2548 45.2548i 226.618 + 87.7113i 136.491 177.116i
23.2 −2.82843 2.82843i −9.52925 12.3367i 16.0000i 12.9164 54.3890i −7.94059 + 61.8462i −141.804 + 141.804i 45.2548 45.2548i −61.3868 + 235.118i −190.369 + 117.302i
23.3 −2.82843 2.82843i −5.94397 + 14.4107i 16.0000i −4.92923 55.6840i 57.5718 23.9476i 93.4247 93.4247i 45.2548 45.2548i −172.338 171.314i −143.556 + 171.440i
23.4 −2.82843 2.82843i 12.0966 + 9.83224i 16.0000i −50.1686 + 24.6600i −6.40449 62.0240i −72.4506 + 72.4506i 45.2548 45.2548i 49.6540 + 237.873i 211.647 + 72.1491i
23.5 −2.82843 2.82843i 14.1646 6.50878i 16.0000i 55.5060 + 6.63996i −58.4731 21.6539i 1.93288 1.93288i 45.2548 45.2548i 158.272 184.389i −138.214 175.775i
23.6 2.82843 + 2.82843i −12.3367 9.52925i 16.0000i −12.9164 + 54.3890i −7.94059 61.8462i −141.804 + 141.804i −45.2548 + 45.2548i 61.3868 + 235.118i −190.369 + 117.302i
23.7 2.82843 + 2.82843i −6.50878 + 14.1646i 16.0000i −55.5060 6.63996i −58.4731 + 21.6539i 1.93288 1.93288i −45.2548 + 45.2548i −158.272 184.389i −138.214 175.775i
23.8 2.82843 + 2.82843i −2.86199 15.3235i 16.0000i −7.18151 55.4385i 35.2464 51.4363i 137.897 137.897i −45.2548 + 45.2548i −226.618 + 87.7113i 136.491 177.116i
23.9 2.82843 + 2.82843i 9.83224 + 12.0966i 16.0000i 50.1686 24.6600i −6.40449 + 62.0240i −72.4506 + 72.4506i −45.2548 + 45.2548i −49.6540 + 237.873i 211.647 + 72.1491i
23.10 2.82843 + 2.82843i 14.4107 5.94397i 16.0000i 4.92923 + 55.6840i 57.5718 + 23.9476i 93.4247 93.4247i −45.2548 + 45.2548i 172.338 171.314i −143.556 + 171.440i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 30.6.e.a 20
3.b odd 2 1 inner 30.6.e.a 20
5.b even 2 1 150.6.e.b 20
5.c odd 4 1 inner 30.6.e.a 20
5.c odd 4 1 150.6.e.b 20
15.d odd 2 1 150.6.e.b 20
15.e even 4 1 inner 30.6.e.a 20
15.e even 4 1 150.6.e.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.6.e.a 20 1.a even 1 1 trivial
30.6.e.a 20 3.b odd 2 1 inner
30.6.e.a 20 5.c odd 4 1 inner
30.6.e.a 20 15.e even 4 1 inner
150.6.e.b 20 5.b even 2 1
150.6.e.b 20 5.c odd 4 1
150.6.e.b 20 15.d odd 2 1
150.6.e.b 20 15.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(30, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 256)^{5} \) Copy content Toggle raw display
$3$ \( T^{20} + 4 T^{19} + \cdots + 71\!\cdots\!49 \) Copy content Toggle raw display
$5$ \( T^{20} + 7888 T^{18} + \cdots + 88\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{10} - 38 T^{9} + \cdots + 20\!\cdots\!32)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + 1075810 T^{8} + \cdots + 26\!\cdots\!32)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} - 1320 T^{9} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + 20962068675680 T^{16} + \cdots + 38\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{10} + 16394480 T^{8} + \cdots + 15\!\cdots\!76)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + 418345665868880 T^{16} + \cdots + 96\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{10} - 82947250 T^{8} + \cdots - 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 5990 T^{4} + \cdots - 17\!\cdots\!68)^{4} \) Copy content Toggle raw display
$37$ \( (T^{10} + 30348 T^{9} + \cdots + 14\!\cdots\!32)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + 620239360 T^{8} + \cdots + 27\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} - 24012 T^{9} + \cdots + 76\!\cdots\!32)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{10} - 6125230690 T^{8} + \cdots - 16\!\cdots\!68)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 40170 T^{4} + \cdots - 16\!\cdots\!24)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + 59368 T^{9} + \cdots + 24\!\cdots\!32)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + 2699614600 T^{8} + \cdots + 69\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} - 189550 T^{9} + \cdots + 95\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + 14343013380 T^{8} + \cdots + 17\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 46\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{10} - 27233821240 T^{8} + \cdots - 69\!\cdots\!68)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + 144114 T^{9} + \cdots + 20\!\cdots\!68)^{2} \) Copy content Toggle raw display
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