Properties

Label 1134.2.d.b
Level $1134$
Weight $2$
Character orbit 1134.d
Analytic conductor $9.055$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.05503558921\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - 8 x^{15} + 56 x^{14} - 252 x^{13} + 962 x^{12} - 2860 x^{11} + 7240 x^{10} - 15036 x^{9} + 26533 x^{8} - 38796 x^{7} + 47500 x^{6} - 47396 x^{5} + 38144 x^{4} + \cdots + 457 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{6} \)
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.

\(f(q)\) \(=\) \( q - \beta_{11} q^{2} - q^{4} - \beta_{2} q^{5} + \beta_{14} q^{7} + \beta_{11} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{11} q^{2} - q^{4} - \beta_{2} q^{5} + \beta_{14} q^{7} + \beta_{11} q^{8} - \beta_{12} q^{10} + \beta_{9} q^{11} + (\beta_{15} + \beta_{10}) q^{13} + \beta_{7} q^{14} + q^{16} + ( - \beta_{5} + \beta_{2}) q^{17} + (\beta_{15} - \beta_{12} - \beta_{10}) q^{19} + \beta_{2} q^{20} - \beta_{4} q^{22} + \beta_{8} q^{23} + (\beta_{14} - \beta_{13} + \beta_{4} - \beta_1 + 1) q^{25} + ( - \beta_{5} + \beta_{3}) q^{26} - \beta_{14} q^{28} + ( - 3 \beta_{11} - \beta_{9} + \beta_{8}) q^{29} + (\beta_{15} + \beta_{14} + \beta_{13} + 2 \beta_{12} - 1) q^{31} - \beta_{11} q^{32} + ( - \beta_{15} + \beta_{12}) q^{34} + ( - 2 \beta_{11} + \beta_{8} - \beta_{6} - \beta_{3} - \beta_{2}) q^{35} + ( - 2 \beta_{14} + 2 \beta_{13} + \beta_{4} + 1) q^{37} + ( - \beta_{5} - \beta_{3} + \beta_{2}) q^{38} + \beta_{12} q^{40} + (\beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{3} + \beta_{2}) q^{41} + ( - 2 \beta_1 + 4) q^{43} - \beta_{9} q^{44} + \beta_1 q^{46} + (2 \beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2}) q^{47} + (\beta_{14} - \beta_{13} + \beta_{12} + \beta_{10} + \beta_{4} - \beta_1 - 2) q^{49} + (\beta_{9} + \beta_{8} + \beta_{7} + \beta_{6}) q^{50} + ( - \beta_{15} - \beta_{10}) q^{52} + ( - 4 \beta_{11} + \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6}) q^{53} + (\beta_{15} - 2 \beta_{12}) q^{55} - \beta_{7} q^{56} + (\beta_{4} + \beta_1 - 3) q^{58} - \beta_{5} q^{59} + ( - \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - 2 \beta_{10} - 1) q^{61} + (\beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{2}) q^{62} - q^{64} + (3 \beta_{11} + \beta_{9} + 2 \beta_{8} + 3 \beta_{7} + 3 \beta_{6}) q^{65} + ( - \beta_{14} + \beta_{13} - 2 \beta_{4} - \beta_1 - 1) q^{67} + (\beta_{5} - \beta_{2}) q^{68} + ( - \beta_{13} - \beta_{12} + \beta_{10} + \beta_1 - 1) q^{70} + ( - 2 \beta_{11} + 2 \beta_{9} + 2 \beta_{8} + \beta_{7} + \beta_{6}) q^{71} + ( - \beta_{14} - \beta_{13} - \beta_{10} + 1) q^{73} + ( - 3 \beta_{11} + \beta_{9} - 2 \beta_{7} - 2 \beta_{6}) q^{74} + ( - \beta_{15} + \beta_{12} + \beta_{10}) q^{76} + (\beta_{11} + \beta_{7} + \beta_{6} - 2 \beta_{5} + \beta_{2}) q^{77} + (\beta_{4} + 2 \beta_1 + 2) q^{79} - \beta_{2} q^{80} + ( - 2 \beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} - \beta_{10} + 1) q^{82} + ( - \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{3} - 2 \beta_{2}) q^{83} + ( - 2 \beta_{4} + \beta_1 - 3) q^{85} + ( - 4 \beta_{11} + 2 \beta_{8}) q^{86} + \beta_{4} q^{88} + ( - \beta_{5} + 5 \beta_{2}) q^{89} + (\beta_{15} - \beta_{14} + 2 \beta_{13} - 2 \beta_{12} + \beta_{10} - 2 \beta_{4} - 2 \beta_1 + 1) q^{91} - \beta_{8} q^{92} + ( - \beta_{15} - 2 \beta_{14} - 2 \beta_{13} - \beta_{12} + \beta_{10} + 2) q^{94} + ( - 4 \beta_{11} + 2 \beta_{9} - \beta_{8} - 4 \beta_{7} - 4 \beta_{6}) q^{95} + ( - \beta_{12} + \beta_{10}) q^{97} + (3 \beta_{11} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{3} - \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} + 8 q^{7} + 16 q^{16} + 16 q^{25} - 8 q^{28} + 16 q^{37} + 64 q^{43} - 32 q^{49} - 48 q^{58} - 16 q^{64} - 16 q^{67} - 24 q^{70} + 32 q^{79} - 48 q^{85} + 24 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 56 x^{14} - 252 x^{13} + 962 x^{12} - 2860 x^{11} + 7240 x^{10} - 15036 x^{9} + 26533 x^{8} - 38796 x^{7} + 47500 x^{6} - 47396 x^{5} + 38144 x^{4} + \cdots + 457 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 54 \nu^{14} - 378 \nu^{13} + 2506 \nu^{12} - 10122 \nu^{11} + 35295 \nu^{10} - 92699 \nu^{9} + 205329 \nu^{8} - 361020 \nu^{7} + 520549 \nu^{6} - 592287 \nu^{5} + 520258 \nu^{4} + \cdots + 1872 ) / 655 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 321 \nu^{14} + 2247 \nu^{13} - 15086 \nu^{12} + 61305 \nu^{11} - 217429 \nu^{10} + 578736 \nu^{9} - 1315127 \nu^{8} + 2370641 \nu^{7} - 3566011 \nu^{6} + 4253522 \nu^{5} + \cdots - 76759 ) / 1965 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 259 \nu^{14} - 1813 \nu^{13} + 12233 \nu^{12} - 49829 \nu^{11} + 177662 \nu^{10} - 474754 \nu^{9} + 1084910 \nu^{8} - 1965161 \nu^{7} + 2973244 \nu^{6} - 3565294 \nu^{5} + \cdots + 62645 ) / 393 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 588 \nu^{14} + 4116 \nu^{13} - 27797 \nu^{12} + 113274 \nu^{11} - 404410 \nu^{10} + 1081803 \nu^{9} - 2477718 \nu^{8} + 4497900 \nu^{7} - 6833518 \nu^{6} + \cdots - 164484 ) / 655 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 395 \nu^{14} - 2765 \nu^{13} + 18656 \nu^{12} - 75991 \nu^{11} + 270978 \nu^{10} - 724205 \nu^{9} + 1655774 \nu^{8} - 3000797 \nu^{7} + 4545742 \nu^{6} - 5458815 \nu^{5} + \cdots + 96267 ) / 393 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 325218 \nu^{15} - 2510469 \nu^{14} + 16986696 \nu^{13} - 73571231 \nu^{12} + 267766992 \nu^{11} - 755616592 \nu^{10} + 1787650380 \nu^{9} + \cdots - 30378190 ) / 2595765 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 325218 \nu^{15} - 2367801 \nu^{14} + 15988020 \nu^{13} - 66777328 \nu^{12} + 239986362 \nu^{11} - 655964315 \nu^{10} + 1520242992 \nu^{9} + \cdots - 49140353 ) / 2595765 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 143764 \nu^{15} + 1078230 \nu^{14} - 7240864 \nu^{13} + 30712461 \nu^{12} - 110181896 \nu^{11} + 304140298 \nu^{10} - 704002373 \nu^{9} + \cdots + 4929077 ) / 865255 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 144812 \nu^{15} - 1086090 \nu^{14} + 7295622 \nu^{13} - 30949178 \nu^{12} + 110985188 \nu^{11} - 306216779 \nu^{10} + 707350209 \nu^{9} - 1319456100 \nu^{8} + \cdots + 11072049 ) / 865255 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 96874 \nu^{15} + 726555 \nu^{14} - 4887955 \nu^{13} + 20752290 \nu^{12} - 74578887 \nu^{11} + 206151407 \nu^{10} - 477998494 \nu^{9} + 895321773 \nu^{8} + \cdots + 3067531 ) / 519153 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 32580 \nu^{15} - 244350 \nu^{14} + 1650786 \nu^{13} - 7024134 \nu^{12} + 25400574 \nu^{11} - 70590828 \nu^{10} + 165381778 \nu^{9} - 313213977 \nu^{8} + \cdots - 3363404 ) / 173051 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 847558 \nu^{15} + 6356685 \nu^{14} - 42971171 \nu^{13} + 182902889 \nu^{12} - 661951717 \nu^{11} + 1840904054 \nu^{10} - 4318389740 \nu^{9} + \cdots + 117878701 ) / 2595765 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1265468 \nu^{15} - 9990348 \nu^{14} + 67896127 \nu^{13} - 298263571 \nu^{12} + 1094893661 \nu^{11} - 3131539819 \nu^{10} + 7501129528 \nu^{9} + \cdots - 347093285 ) / 2595765 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1265468 \nu^{15} - 8991672 \nu^{14} + 60905395 \nu^{13} - 251052352 \nu^{12} + 902505863 \nu^{11} - 2445049144 \nu^{10} + 5665618522 \nu^{9} + \cdots - 99619787 ) / 2595765 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 312314 \nu^{15} + 2342355 \nu^{14} - 15898989 \nu^{13} + 67817711 \nu^{12} - 246733441 \nu^{11} + 689195683 \nu^{10} - 1628974479 \nu^{9} + \cdots + 64404140 ) / 519153 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{15} - \beta_{14} - \beta_{13} + 3\beta_{12} + 3\beta_{11} - \beta_{10} + 4 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{15} - 4 \beta_{14} + 2 \beta_{13} + 3 \beta_{12} + 3 \beta_{11} - \beta_{10} + \beta_{7} - \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + \beta_{3} + 3 \beta_{2} + 3 \beta _1 - 17 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4 \beta_{15} - 2 \beta_{14} + 4 \beta_{13} - 13 \beta_{12} - 14 \beta_{11} + 5 \beta_{10} + 3 \beta_{9} + 3 \beta_{8} + 4 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + \beta_{3} + 3 \beta_{2} + 3 \beta _1 - 20 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 14 \beta_{15} + 34 \beta_{14} - 26 \beta_{13} - 42 \beta_{12} - 45 \beta_{11} + 16 \beta_{10} + 9 \beta_{9} + 9 \beta_{8} + 7 \beta_{7} + 11 \beta_{6} + 10 \beta_{5} + 18 \beta_{4} - 14 \beta_{3} - 36 \beta_{2} - 24 \beta _1 + 89 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 58 \beta_{15} + 172 \beta_{14} - 158 \beta_{13} + 249 \beta_{12} + 246 \beta_{11} - 119 \beta_{10} - 75 \beta_{9} - 105 \beta_{8} - 150 \beta_{7} - 120 \beta_{6} + 60 \beta_{5} + 105 \beta_{4} - 75 \beta_{3} - 195 \beta_{2} + \cdots + 554 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 41 \beta_{15} - 77 \beta_{14} + 77 \beta_{13} + 160 \beta_{12} + 161 \beta_{11} - 73 \beta_{10} - 45 \beta_{9} - 60 \beta_{8} - 80 \beta_{7} - 70 \beta_{6} - 21 \beta_{5} - 34 \beta_{4} + 48 \beta_{3} + 95 \beta_{2} + 53 \beta _1 - 180 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 150 \beta_{15} - 1164 \beta_{14} + 1041 \beta_{13} - 708 \beta_{12} - 675 \beta_{11} + 399 \beta_{10} + 231 \beta_{9} + 378 \beta_{8} + 553 \beta_{7} + 602 \beta_{6} - 329 \beta_{5} - 546 \beta_{4} + 637 \beta_{3} + 1344 \beta_{2} + \cdots - 2844 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1208 \beta_{15} + 1066 \beta_{14} - 1538 \beta_{13} - 5172 \beta_{12} - 5061 \beta_{11} + 2656 \beta_{10} + 1575 \beta_{9} + 2373 \beta_{8} + 3553 \beta_{7} + 3209 \beta_{6} + 328 \beta_{5} + 450 \beta_{4} + \cdots + 2771 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 270 \beta_{15} + 8796 \beta_{14} - 8310 \beta_{13} + 1431 \beta_{12} + 1316 \beta_{11} - 1017 \beta_{10} - 525 \beta_{9} - 987 \beta_{8} - 1188 \beta_{7} - 2370 \beta_{6} + 2388 \beta_{5} + 3687 \beta_{4} + \cdots + 19808 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 12019 \beta_{15} + 2860 \beta_{14} + 4282 \beta_{13} + 53097 \beta_{12} + 51438 \beta_{11} - 29162 \beta_{10} - 16740 \beta_{9} - 26505 \beta_{8} - 40966 \beta_{7} - 39674 \beta_{6} + 41 \beta_{5} + \cdots - 1384 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 13518 \beta_{15} - 267924 \beta_{14} + 270372 \beta_{13} + 56313 \beta_{12} + 55698 \beta_{11} - 26361 \beta_{10} - 16467 \beta_{9} - 23397 \beta_{8} - 53548 \beta_{7} - 6116 \beta_{6} + \cdots - 598980 ) / 12 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 38178 \beta_{15} - 55687 \beta_{14} + 29399 \beta_{13} - 170578 \beta_{12} - 164538 \beta_{11} + 96870 \beta_{10} + 54615 \beta_{9} + 88209 \beta_{8} + 135537 \beta_{7} + 141069 \beta_{6} + \cdots - 87901 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 176273 \beta_{15} + 1221805 \beta_{14} - 1326845 \beta_{13} - 778395 \beta_{12} - 754452 \beta_{11} + 428080 \beta_{10} + 245583 \beta_{9} + 389103 \beta_{8} + 714870 \beta_{7} + \cdots + 2815751 ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1004014 \beta_{15} + 2973496 \beta_{14} - 2241272 \beta_{13} + 4505304 \beta_{12} + 4335321 \beta_{11} - 2597126 \beta_{10} - 1451268 \beta_{9} - 2362542 \beta_{8} + \cdots + 5415443 ) / 6 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 1855568 \beta_{15} - 6465538 \beta_{14} + 7738524 \beta_{13} + 8283611 \beta_{12} + 7991044 \beta_{11} - 4704505 \beta_{10} - 2651201 \beta_{9} - 4279555 \beta_{8} + \cdots - 15882042 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(-1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1133.1
0.500000 2.18553i
0.500000 1.24085i
0.500000 + 1.61238i
0.500000 + 1.22108i
0.500000 0.221083i
0.500000 0.612384i
0.500000 + 2.24085i
0.500000 + 3.18553i
0.500000 + 2.18553i
0.500000 + 1.24085i
0.500000 1.61238i
0.500000 1.22108i
0.500000 + 0.221083i
0.500000 + 0.612384i
0.500000 2.24085i
0.500000 3.18553i
1.00000i 0 −1.00000 −3.79792 0 2.59077 + 0.536572i 1.00000i 0 3.79792i
1133.2 1.00000i 0 −1.00000 −2.46193 0 −1.59077 + 2.11411i 1.00000i 0 2.46193i
1133.3 1.00000i 0 −1.00000 −1.57315 0 0.858719 2.50252i 1.00000i 0 1.57315i
1133.4 1.00000i 0 −1.00000 −1.01976 0 0.141281 + 2.64198i 1.00000i 0 1.01976i
1133.5 1.00000i 0 −1.00000 1.01976 0 0.141281 2.64198i 1.00000i 0 1.01976i
1133.6 1.00000i 0 −1.00000 1.57315 0 0.858719 + 2.50252i 1.00000i 0 1.57315i
1133.7 1.00000i 0 −1.00000 2.46193 0 −1.59077 2.11411i 1.00000i 0 2.46193i
1133.8 1.00000i 0 −1.00000 3.79792 0 2.59077 0.536572i 1.00000i 0 3.79792i
1133.9 1.00000i 0 −1.00000 −3.79792 0 2.59077 0.536572i 1.00000i 0 3.79792i
1133.10 1.00000i 0 −1.00000 −2.46193 0 −1.59077 2.11411i 1.00000i 0 2.46193i
1133.11 1.00000i 0 −1.00000 −1.57315 0 0.858719 + 2.50252i 1.00000i 0 1.57315i
1133.12 1.00000i 0 −1.00000 −1.01976 0 0.141281 2.64198i 1.00000i 0 1.01976i
1133.13 1.00000i 0 −1.00000 1.01976 0 0.141281 + 2.64198i 1.00000i 0 1.01976i
1133.14 1.00000i 0 −1.00000 1.57315 0 0.858719 2.50252i 1.00000i 0 1.57315i
1133.15 1.00000i 0 −1.00000 2.46193 0 −1.59077 + 2.11411i 1.00000i 0 2.46193i
1133.16 1.00000i 0 −1.00000 3.79792 0 2.59077 + 0.536572i 1.00000i 0 3.79792i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1133.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
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 1134.2.d.b 16
3.b odd 2 1 inner 1134.2.d.b 16
7.b odd 2 1 inner 1134.2.d.b 16
9.c even 3 1 1134.2.m.h 16
9.c even 3 1 1134.2.m.i 16
9.d odd 6 1 1134.2.m.h 16
9.d odd 6 1 1134.2.m.i 16
21.c even 2 1 inner 1134.2.d.b 16
63.l odd 6 1 1134.2.m.h 16
63.l odd 6 1 1134.2.m.i 16
63.o even 6 1 1134.2.m.h 16
63.o even 6 1 1134.2.m.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1134.2.d.b 16 1.a even 1 1 trivial
1134.2.d.b 16 3.b odd 2 1 inner
1134.2.d.b 16 7.b odd 2 1 inner
1134.2.d.b 16 21.c even 2 1 inner
1134.2.m.h 16 9.c even 3 1
1134.2.m.h 16 9.d odd 6 1
1134.2.m.h 16 63.l odd 6 1
1134.2.m.h 16 63.o even 6 1
1134.2.m.i 16 9.c even 3 1
1134.2.m.i 16 9.d odd 6 1
1134.2.m.i 16 63.l odd 6 1
1134.2.m.i 16 63.o even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 24T_{5}^{6} + 162T_{5}^{4} - 360T_{5}^{2} + 225 \) acting on \(S_{2}^{\mathrm{new}}(1134, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 24 T^{6} + 162 T^{4} - 360 T^{2} + \cdots + 225)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} - 4 T^{7} + 16 T^{6} - 52 T^{5} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 36 T^{6} + 288 T^{4} + 648 T^{2} + \cdots + 324)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 84 T^{6} + 2502 T^{4} + \cdots + 119025)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 60 T^{6} + 918 T^{4} - 3420 T^{2} + \cdots + 225)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 156 T^{6} + 7632 T^{4} + \cdots + 476100)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 36 T^{6} + 288 T^{4} + 648 T^{2} + \cdots + 324)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 108 T^{6} + 2502 T^{4} + \cdots + 42849)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 180 T^{6} + 9072 T^{4} + \cdots + 72900)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 4 T^{3} - 84 T^{2} - 148 T + 73)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 240 T^{6} + 20232 T^{4} + \cdots + 8643600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 16 T^{3} + 24 T^{2} + 608 T - 2336)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} - 348 T^{6} + 39888 T^{4} + \cdots + 476100)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 72 T^{2} + 324)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} - 60 T^{6} + 432 T^{4} - 1080 T^{2} + \cdots + 900)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 348 T^{6} + 34038 T^{4} + \cdots + 497025)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 4 T^{3} - 102 T^{2} - 572 T - 242)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + 360 T^{6} + 32256 T^{4} + \cdots + 11451456)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 168 T^{6} + 9378 T^{4} + \cdots + 225)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 8 T^{3} - 66 T^{2} + 4 T + 142)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} - 420 T^{6} + 46512 T^{4} + \cdots + 8468100)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 540 T^{6} + 79542 T^{4} + \cdots + 9641025)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 120 T^{6} + 1368 T^{4} + \cdots + 3600)^{2} \) Copy content Toggle raw display
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