Properties

Label 8820.2.d.c
Level $8820$
Weight $2$
Character orbit 8820.d
Analytic conductor $70.428$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8820.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(70.4280545828\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 8 x^{15} + 60 x^{14} - 280 x^{13} + 1134 x^{12} - 3528 x^{11} + 9316 x^{10} - 19960 x^{9} + 35421 x^{8} - 51008 x^{7} + 58384 x^{6} - 51840 x^{5} + 34228 x^{4} - 15904 x^{3} + 4848 x^{2} - 864 x + 68\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{17} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q - q^{5} +O(q^{10})\) \( q - q^{5} + ( -\beta_{10} + \beta_{11} - \beta_{12} ) q^{11} -\beta_{15} q^{13} + ( \beta_{4} + \beta_{6} ) q^{17} + ( -\beta_{8} - \beta_{13} - \beta_{14} ) q^{19} + ( \beta_{8} + \beta_{9} + 2 \beta_{11} ) q^{23} + q^{25} + ( \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} - \beta_{15} ) q^{29} + ( \beta_{8} - \beta_{9} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{31} + ( 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{37} + ( -2 + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{41} + ( -2 + \beta_{1} + \beta_{5} - 2 \beta_{6} ) q^{43} + ( 2 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} ) q^{47} + ( -2 \beta_{9} + \beta_{10} - \beta_{12} - \beta_{14} + \beta_{15} ) q^{53} + ( \beta_{10} - \beta_{11} + \beta_{12} ) q^{55} + ( 2 + \beta_{2} + 2 \beta_{3} - \beta_{5} + \beta_{6} ) q^{59} + ( \beta_{8} + 3 \beta_{9} + \beta_{10} + 2 \beta_{11} - 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{61} + \beta_{15} q^{65} + ( -2 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{67} + ( -\beta_{8} + 3 \beta_{9} + \beta_{10} - 3 \beta_{11} + \beta_{13} + \beta_{15} ) q^{71} + ( -\beta_{8} - \beta_{9} - 4 \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{14} ) q^{73} + ( \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{79} + ( -\beta_{1} + 2 \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{83} + ( -\beta_{4} - \beta_{6} ) q^{85} + ( 4 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{5} + \beta_{6} ) q^{89} + ( \beta_{8} + \beta_{13} + \beta_{14} ) q^{95} + ( -\beta_{8} - \beta_{9} - \beta_{10} - 3 \beta_{11} - 3 \beta_{12} + 2 \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{5} + O(q^{10}) \) \( 16 q - 16 q^{5} + 16 q^{25} - 32 q^{41} - 32 q^{43} + 32 q^{47} + 32 q^{59} - 32 q^{67} + 64 q^{89} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 8 x^{15} + 60 x^{14} - 280 x^{13} + 1134 x^{12} - 3528 x^{11} + 9316 x^{10} - 19960 x^{9} + 35421 x^{8} - 51008 x^{7} + 58384 x^{6} - 51840 x^{5} + 34228 x^{4} - 15904 x^{3} + 4848 x^{2} - 864 x + 68\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(563 \nu^{14} - 3941 \nu^{13} + 29695 \nu^{12} - 126937 \nu^{11} + 503911 \nu^{10} - 1449893 \nu^{9} + 3666175 \nu^{8} - 7200631 \nu^{7} + 11804248 \nu^{6} - 15072848 \nu^{5} + 14780744 \nu^{4} - 10553790 \nu^{3} + 4940444 \nu^{2} - 1317740 \nu + 150044\)\()/56\)
\(\beta_{2}\)\(=\)\((\)\(-31 \nu^{14} + 217 \nu^{13} - 1635 \nu^{12} + 6989 \nu^{11} - 27743 \nu^{10} + 79821 \nu^{9} - 201813 \nu^{8} + 396339 \nu^{7} - 649602 \nu^{6} + 829298 \nu^{5} - 812840 \nu^{4} + 580008 \nu^{3} - 271104 \nu^{2} + 72096 \nu - 8164\)\()/2\)
\(\beta_{3}\)\(=\)\((\)\(-64 \nu^{14} + 448 \nu^{13} - 3376 \nu^{12} + 14432 \nu^{11} - 57300 \nu^{10} + 164884 \nu^{9} - 417015 \nu^{8} + 819204 \nu^{7} - 1343486 \nu^{6} + 1716224 \nu^{5} - 1684475 \nu^{4} + 1204216 \nu^{3} - 565228 \nu^{2} + 151536 \nu - 17398\)\()/4\)
\(\beta_{4}\)\(=\)\((\)\(933 \nu^{14} - 6531 \nu^{13} + 49213 \nu^{12} - 210375 \nu^{11} + 835197 \nu^{10} - 2403203 \nu^{9} + 6077301 \nu^{8} - 11937273 \nu^{7} + 19572612 \nu^{6} - 24996888 \nu^{5} + 24522096 \nu^{4} - 17518682 \nu^{3} + 8210868 \nu^{2} - 2195268 \nu + 251108\)\()/56\)
\(\beta_{5}\)\(=\)\((\)\(-477 \nu^{14} + 3339 \nu^{13} - 25163 \nu^{12} + 107571 \nu^{11} - 427122 \nu^{10} + 1229122 \nu^{9} - 3108941 \nu^{8} + 6107891 \nu^{7} - 10018741 \nu^{6} + 12800879 \nu^{5} - 12569384 \nu^{4} + 8990894 \nu^{3} - 4225470 \nu^{2} + 1135602 \nu - 130804\)\()/28\)
\(\beta_{6}\)\(=\)\((\)\(1203 \nu^{14} - 8421 \nu^{13} + 63457 \nu^{12} - 271269 \nu^{11} + 1077003 \nu^{10} - 3099083 \nu^{9} + 7837701 \nu^{8} - 15396207 \nu^{7} + 25247728 \nu^{6} - 32249910 \nu^{5} + 31647880 \nu^{4} - 22619510 \nu^{3} + 10611548 \nu^{2} - 2842120 \nu + 325876\)\()/56\)
\(\beta_{7}\)\(=\)\((\)\(911 \nu^{14} - 6377 \nu^{13} + 48053 \nu^{12} - 205417 \nu^{11} + 815524 \nu^{10} - 2346616 \nu^{9} + 5934323 \nu^{8} - 11656637 \nu^{7} + 19113169 \nu^{6} - 24411051 \nu^{5} + 23949172 \nu^{4} - 17111062 \nu^{3} + 8021234 \nu^{2} - 2145226 \nu + 245380\)\()/28\)
\(\beta_{8}\)\(=\)\((\)\(-9578 \nu^{15} + 71835 \nu^{14} - 538751 \nu^{13} + 2412384 \nu^{12} - 9654653 \nu^{11} + 28961262 \nu^{10} - 74737703 \nu^{9} + 153778332 \nu^{8} - 262297961 \nu^{7} + 357257717 \nu^{6} - 380329884 \nu^{5} + 306047064 \nu^{4} - 174504810 \nu^{3} + 64854694 \nu^{2} - 13901100 \nu + 1295576\)\()/56\)
\(\beta_{9}\)\(=\)\((\)\(41198 \nu^{15} - 308985 \nu^{14} + 2317365 \nu^{13} - 10376600 \nu^{12} + 41529045 \nu^{11} - 124576947 \nu^{10} + 321491944 \nu^{9} - 661508118 \nu^{8} + 1128373599 \nu^{7} - 1536952540 \nu^{6} + 1636354831 \nu^{5} - 1316922472 \nu^{4} + 751070018 \nu^{3} - 279252788 \nu^{2} + 59901234 \nu - 5590392\)\()/56\)
\(\beta_{10}\)\(=\)\((\)\(-45046 \nu^{15} + 337845 \nu^{14} - 2533879 \nu^{13} + 11346231 \nu^{12} - 45411237 \nu^{11} + 136226024 \nu^{10} - 351572654 \nu^{9} + 723438657 \nu^{8} - 1234125131 \nu^{7} + 1681174263 \nu^{6} - 1790248231 \nu^{5} + 1441171232 \nu^{4} - 822338834 \nu^{3} + 306021290 \nu^{2} - 65738314 \nu + 6148892\)\()/56\)
\(\beta_{11}\)\(=\)\((\)\(3856 \nu^{15} - 28920 \nu^{14} + 216902 \nu^{13} - 971243 \nu^{12} + 3887188 \nu^{11} - 11660825 \nu^{10} + 30093874 \nu^{9} - 61923975 \nu^{8} + 105634416 \nu^{7} - 143895227 \nu^{6} + 153223032 \nu^{5} - 123337072 \nu^{4} + 70367120 \nu^{3} - 26179718 \nu^{2} + 5621528 \nu - 525468\)\()/4\)
\(\beta_{12}\)\(=\)\((\)\(-58612 \nu^{15} + 439590 \nu^{14} - 3296928 \nu^{13} + 14762917 \nu^{12} - 59084708 \nu^{11} + 177241460 \nu^{10} - 457411359 \nu^{9} + 941197611 \nu^{8} - 1605516550 \nu^{7} + 2186961412 \nu^{6} - 2328586171 \nu^{5} + 1874234912 \nu^{4} - 1069135048 \nu^{3} + 397656120 \nu^{2} - 85350718 \nu + 7973036\)\()/56\)
\(\beta_{13}\)\(=\)\((\)\(-4336 \nu^{15} + 32520 \nu^{14} - 243898 \nu^{13} + 1092117 \nu^{12} - 4370860 \nu^{11} + 13111527 \nu^{10} - 33836622 \nu^{9} + 69623097 \nu^{8} - 118760880 \nu^{7} + 161764661 \nu^{6} - 172228752 \nu^{5} + 138610316 \nu^{4} - 79054960 \nu^{3} + 29394806 \nu^{2} - 6306008 \nu + 588636\)\()/4\)
\(\beta_{14}\)\(=\)\((\)\(-18462 \nu^{15} + 138465 \nu^{14} - 1038487 \nu^{13} + 4650113 \nu^{12} - 18610825 \nu^{11} + 55828410 \nu^{10} - 144077334 \nu^{9} + 296461545 \nu^{8} - 505708399 \nu^{7} + 688848917 \nu^{6} - 733450047 \nu^{5} + 590330982 \nu^{4} - 336737754 \nu^{3} + 125240438 \nu^{2} - 26878490 \nu + 2510464\)\()/8\)
\(\beta_{15}\)\(=\)\((\)\(-198424 \nu^{15} + 1488180 \nu^{14} - 11161258 \nu^{13} + 49977447 \nu^{12} - 200019240 \nu^{11} + 600009509 \nu^{10} - 1548430882 \nu^{9} + 3186090657 \nu^{8} - 5434732004 \nu^{7} + 7402668495 \nu^{6} - 7881522940 \nu^{5} + 6343074326 \nu^{4} - 3617702120 \nu^{3} + 1345157438 \nu^{2} - 288572112 \nu + 26936464\)\()/56\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{12} + \beta_{11} + 2 \beta_{9} + 2 \beta_{8} + 2\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{12} + \beta_{11} + 2 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + 4 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{1} - 14\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{15} - \beta_{14} + 6 \beta_{13} - 17 \beta_{12} - 11 \beta_{11} - \beta_{10} - 13 \beta_{9} - 13 \beta_{8} - 3 \beta_{7} + 6 \beta_{6} + 3 \beta_{5} + 3 \beta_{4} - 3 \beta_{1} - 22\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-2 \beta_{15} - 2 \beta_{14} + 12 \beta_{13} - 36 \beta_{12} - 23 \beta_{11} - 2 \beta_{10} - 28 \beta_{9} - 28 \beta_{8} + 20 \beta_{7} - 32 \beta_{6} - 12 \beta_{5} - 16 \beta_{4} + 4 \beta_{2} + 16 \beta_{1} + 90\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(5 \beta_{15} + 5 \beta_{14} - 60 \beta_{13} + 133 \beta_{12} + 89 \beta_{11} + 13 \beta_{10} + 57 \beta_{9} + 81 \beta_{8} + 55 \beta_{7} - 90 \beta_{6} - 35 \beta_{5} - 45 \beta_{4} + 10 \beta_{2} + 45 \beta_{1} + 262\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(20 \beta_{15} + 20 \beta_{14} - 210 \beta_{13} + 490 \beta_{12} + 325 \beta_{11} + 44 \beta_{10} + 242 \beta_{9} + 314 \beta_{8} - 142 \beta_{7} + 224 \beta_{6} + 50 \beta_{5} + 112 \beta_{4} + 48 \beta_{3} - 34 \beta_{2} - 96 \beta_{1} - 506\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-19 \beta_{15} + 13 \beta_{14} + 406 \beta_{13} - 789 \beta_{12} - 539 \beta_{11} - 129 \beta_{10} - 103 \beta_{9} - 401 \beta_{8} - 693 \beta_{7} + 1106 \beta_{6} + 301 \beta_{5} + 553 \beta_{4} + 168 \beta_{3} - 154 \beta_{2} - 497 \beta_{1} - 2714\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(-174 \beta_{15} - 46 \beta_{14} + 2632 \beta_{13} - 5528 \beta_{12} - 3727 \beta_{11} - 726 \beta_{10} - 1608 \beta_{9} - 3136 \beta_{8} + 664 \beta_{7} - 1152 \beta_{6} - 40 \beta_{5} - 512 \beta_{4} - 624 \beta_{3} + 184 \beta_{2} + 384 \beta_{1} + 1834\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(29 \beta_{15} - 411 \beta_{14} - 1416 \beta_{13} + 1957 \beta_{12} + 1415 \beta_{11} + 769 \beta_{10} - 1331 \beta_{9} + 577 \beta_{8} + 7383 \beta_{7} - 12210 \beta_{6} - 2139 \beta_{5} - 5817 \beta_{4} - 3816 \beta_{3} + 1794 \beta_{2} + 4905 \beta_{1} + 25682\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(1600 \beta_{15} - 1560 \beta_{14} - 28350 \beta_{13} + 54858 \beta_{12} + 37419 \beta_{11} + 9608 \beta_{10} + 7242 \beta_{9} + 28746 \beta_{8} + 1042 \beta_{7} - 488 \beta_{6} - 2022 \beta_{5} - 996 \beta_{4} + 3320 \beta_{3} + 58 \beta_{2} + 1268 \beta_{1} + 8026\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(911 \beta_{15} + 2791 \beta_{14} - 14498 \beta_{13} + 36049 \beta_{12} + 24079 \beta_{11} + 2029 \beta_{10} + 18907 \beta_{9} + 22253 \beta_{8} - 70191 \beta_{7} + 122430 \beta_{6} + 12199 \beta_{5} + 54439 \beta_{4} + 55088 \beta_{3} - 17930 \beta_{2} - 43967 \beta_{1} - 224134\)\()/4\)
\(\nu^{12}\)\(=\)\((\)\(-15478 \beta_{15} + 32674 \beta_{14} + 273108 \beta_{13} - 489740 \beta_{12} - 336165 \beta_{11} - 106494 \beta_{10} + 1452 \beta_{9} - 241812 \beta_{8} - 79860 \beta_{7} + 127152 \beta_{6} + 27516 \beta_{5} + 63824 \beta_{4} + 29544 \beta_{3} - 19284 \beta_{2} - 55104 \beta_{1} - 286698\)\()/4\)
\(\nu^{13}\)\(=\)\((\)\(-22361 \beta_{15} + 18215 \beta_{14} + 422188 \beta_{13} - 830061 \beta_{12} - 567603 \beta_{11} - 141273 \beta_{10} - 121721 \beta_{9} - 437081 \beta_{8} + 600093 \beta_{7} - 1105286 \beta_{6} - 44577 \beta_{5} - 456079 \beta_{4} - 621920 \beta_{3} + 157222 \beta_{2} + 352599 \beta_{1} + 1784902\)\()/4\)
\(\nu^{14}\)\(=\)\((\)\(146204 \beta_{15} - 414660 \beta_{14} - 2342886 \beta_{13} + 3894802 \beta_{12} + 2674609 \beta_{11} + 1003300 \beta_{10} - 491062 \beta_{9} + 1824898 \beta_{8} + 1366778 \beta_{7} - 2388864 \beta_{6} - 233126 \beta_{5} - 1061316 \beta_{4} - 1084496 \beta_{3} + 351026 \beta_{2} + 860252 \beta_{1} + 4357318\)\()/4\)
\(\nu^{15}\)\(=\)\((\)\(377749 \beta_{15} - 821099 \beta_{14} - 6628866 \beta_{13} + 11837823 \beta_{12} + 8131025 \beta_{11} + 2610843 \beta_{10} - 125567 \beta_{9} + 5834595 \beta_{8} - 4485797 \beta_{7} + 8710018 \beta_{6} - 85523 \beta_{5} + 3320881 \beta_{4} + 5747160 \beta_{3} - 1183598 \beta_{2} - 2420793 \beta_{1} - 12458762\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8820\mathbb{Z}\right)^\times\).

\(n\) \(1081\) \(4411\) \(7057\) \(7841\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
0.500000 0.0420372i
0.500000 0.00906270i
0.500000 + 3.11247i
0.500000 + 2.03007i
0.500000 2.65516i
0.500000 2.81224i
0.500000 0.199114i
0.500000 1.09145i
0.500000 + 1.09145i
0.500000 + 0.199114i
0.500000 + 2.81224i
0.500000 + 2.65516i
0.500000 2.03007i
0.500000 3.11247i
0.500000 + 0.00906270i
0.500000 + 0.0420372i
0 0 0 −1.00000 0 0 0 0 0
881.2 0 0 0 −1.00000 0 0 0 0 0
881.3 0 0 0 −1.00000 0 0 0 0 0
881.4 0 0 0 −1.00000 0 0 0 0 0
881.5 0 0 0 −1.00000 0 0 0 0 0
881.6 0 0 0 −1.00000 0 0 0 0 0
881.7 0 0 0 −1.00000 0 0 0 0 0
881.8 0 0 0 −1.00000 0 0 0 0 0
881.9 0 0 0 −1.00000 0 0 0 0 0
881.10 0 0 0 −1.00000 0 0 0 0 0
881.11 0 0 0 −1.00000 0 0 0 0 0
881.12 0 0 0 −1.00000 0 0 0 0 0
881.13 0 0 0 −1.00000 0 0 0 0 0
881.14 0 0 0 −1.00000 0 0 0 0 0
881.15 0 0 0 −1.00000 0 0 0 0 0
881.16 0 0 0 −1.00000 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8820.2.d.c 16
3.b odd 2 1 8820.2.d.d yes 16
7.b odd 2 1 8820.2.d.d yes 16
21.c even 2 1 inner 8820.2.d.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8820.2.d.c 16 1.a even 1 1 trivial
8820.2.d.c 16 21.c even 2 1 inner
8820.2.d.d yes 16 3.b odd 2 1
8820.2.d.d yes 16 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(8820, [\chi])\):

\(T_{11}^{16} + \cdots\)
\( T_{17}^{8} - 72 T_{17}^{6} + 32 T_{17}^{5} + 1288 T_{17}^{4} - 128 T_{17}^{3} - 5088 T_{17}^{2} - 2176 T_{17} + 1552 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( T^{16} \)
$5$ \( ( 1 + T )^{16} \)
$7$ \( T^{16} \)
$11$ \( 73984 + 1339392 T^{2} + 5820672 T^{4} + 6144000 T^{6} + 1442400 T^{8} + 137984 T^{10} + 6160 T^{12} + 128 T^{14} + T^{16} \)
$13$ \( 4096 + 7806976 T^{2} + 12099584 T^{4} + 5319680 T^{6} + 1015936 T^{8} + 96896 T^{10} + 4736 T^{12} + 112 T^{14} + T^{16} \)
$17$ \( ( 1552 - 2176 T - 5088 T^{2} - 128 T^{3} + 1288 T^{4} + 32 T^{5} - 72 T^{6} + T^{8} )^{2} \)
$19$ \( 1993744 + 21631104 T^{2} + 36676000 T^{4} + 14494912 T^{6} + 2357656 T^{8} + 187232 T^{10} + 7528 T^{12} + 144 T^{14} + T^{16} \)
$23$ \( 570063376 + 637810816 T^{2} + 270097376 T^{4} + 55904832 T^{6} + 6170776 T^{8} + 369504 T^{10} + 11640 T^{12} + 176 T^{14} + T^{16} \)
$29$ \( 31899904 + 508228608 T^{2} + 391677696 T^{4} + 98278144 T^{6} + 10780768 T^{8} + 584384 T^{10} + 15984 T^{12} + 208 T^{14} + T^{16} \)
$31$ \( 49005562384 + 44613499264 T^{2} + 11325998240 T^{4} + 1241076800 T^{6} + 69749272 T^{8} + 2145824 T^{10} + 36008 T^{12} + 304 T^{14} + T^{16} \)
$37$ \( ( -291776 - 233728 T - 28160 T^{2} + 19200 T^{3} + 4688 T^{4} - 288 T^{5} - 128 T^{6} + T^{8} )^{2} \)
$41$ \( ( 344576 + 363520 T + 101632 T^{2} - 14336 T^{3} - 12000 T^{4} - 1920 T^{5} - 32 T^{6} + 16 T^{7} + T^{8} )^{2} \)
$43$ \( ( -32512 - 44032 T + 112128 T^{2} + 33024 T^{3} - 3328 T^{4} - 1472 T^{5} - 32 T^{6} + 16 T^{7} + T^{8} )^{2} \)
$47$ \( ( -6896 + 1792 T + 17952 T^{2} - 11968 T^{3} - 1464 T^{4} + 896 T^{5} - 8 T^{6} - 16 T^{7} + T^{8} )^{2} \)
$53$ \( 613546357264 + 252392719232 T^{2} + 39593138144 T^{4} + 3071144384 T^{6} + 131768984 T^{8} + 3255200 T^{10} + 45688 T^{12} + 336 T^{14} + T^{16} \)
$59$ \( ( 477248 + 431104 T - 16000 T^{2} - 57472 T^{3} + 1360 T^{4} + 1792 T^{5} - 80 T^{6} - 16 T^{7} + T^{8} )^{2} \)
$61$ \( 193258710544 + 362008502144 T^{2} + 198112002208 T^{4} + 27852541760 T^{6} + 1059031320 T^{8} + 17829024 T^{10} + 150696 T^{12} + 624 T^{14} + T^{16} \)
$67$ \( ( 53312 + 1792 T - 84864 T^{2} + 31616 T^{3} + 6384 T^{4} - 2144 T^{5} - 144 T^{6} + 16 T^{7} + T^{8} )^{2} \)
$71$ \( 7539813040384 + 3614912671744 T^{2} + 458930923776 T^{4} + 25123559424 T^{6} + 721027168 T^{8} + 11665152 T^{10} + 106640 T^{12} + 512 T^{14} + T^{16} \)
$73$ \( 6105722392576 + 12232456200192 T^{2} + 2059693244416 T^{4} + 120348660736 T^{6} + 2963589248 T^{8} + 36635008 T^{10} + 239360 T^{12} + 784 T^{14} + T^{16} \)
$79$ \( ( 156944 + 111360 T - 79456 T^{2} - 23808 T^{3} + 10712 T^{4} + 192 T^{5} - 216 T^{6} + T^{8} )^{2} \)
$83$ \( ( 75536 + 53376 T - 64864 T^{2} - 2944 T^{3} + 6216 T^{4} - 32 T^{5} - 168 T^{6} + T^{8} )^{2} \)
$89$ \( ( -12224 - 242176 T + 152064 T^{2} + 6400 T^{3} - 13584 T^{4} + 832 T^{5} + 256 T^{6} - 32 T^{7} + T^{8} )^{2} \)
$97$ \( 82218758311936 + 60634605592576 T^{2} + 7271503568896 T^{4} + 304186326016 T^{6} + 6043691136 T^{8} + 62926976 T^{10} + 346240 T^{12} + 944 T^{14} + T^{16} \)
show more
show less