Properties

Label 462.2.k.g
Level $462$
Weight $2$
Character orbit 462.k
Analytic conductor $3.689$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.k (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.68908857338\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \( x^{20} - 6 x^{19} + 19 x^{18} - 42 x^{17} + 62 x^{16} - 42 x^{15} - 25 x^{14} + 6 x^{13} + 445 x^{12} - 1764 x^{11} + 3864 x^{10} - 5292 x^{9} + 4005 x^{8} + 162 x^{7} - 2025 x^{6} + \cdots + 59049 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} - \beta_{12} q^{3} - \beta_{3} q^{4} + ( - \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{6}) q^{5} + (\beta_{9} + \beta_{5}) q^{6} + (\beta_{15} + \beta_{8} - \beta_1) q^{7} + \beta_{6} q^{8} + ( - \beta_{19} - \beta_{14} + \beta_{13} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{2} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} - \beta_{12} q^{3} - \beta_{3} q^{4} + ( - \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{6}) q^{5} + (\beta_{9} + \beta_{5}) q^{6} + (\beta_{15} + \beta_{8} - \beta_1) q^{7} + \beta_{6} q^{8} + ( - \beta_{19} - \beta_{14} + \beta_{13} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{2} - \beta_1) q^{9} + (\beta_{12} - \beta_{3} - \beta_{2} - 2) q^{10} + (\beta_{7} + \beta_{6}) q^{11} - \beta_1 q^{12} + (\beta_{17} + 2 \beta_{15} + \beta_{12} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{5} - 2 \beta_{3} - \beta_{2} + \cdots - 1) q^{13}+ \cdots + (\beta_{18} + \beta_{17} + \beta_{16} - \beta_{14} + \beta_{13} + \beta_{9} + \beta_{7} + \beta_{4} - \beta_{2} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 6 q^{3} + 10 q^{4} - 6 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 6 q^{3} + 10 q^{4} - 6 q^{7} - 2 q^{9} - 18 q^{10} - 6 q^{12} - 8 q^{15} - 10 q^{16} + 4 q^{18} + 36 q^{19} + 24 q^{21} - 20 q^{22} - 12 q^{25} - 22 q^{30} + 36 q^{31} - 4 q^{36} + 16 q^{37} + 4 q^{39} - 18 q^{40} + 32 q^{42} + 32 q^{43} + 24 q^{45} + 30 q^{46} - 42 q^{49} - 24 q^{52} - 36 q^{54} - 24 q^{57} + 32 q^{58} - 4 q^{60} + 42 q^{61} - 10 q^{63} - 20 q^{64} + 6 q^{66} - 10 q^{67} - 36 q^{70} - 4 q^{72} + 12 q^{73} - 108 q^{75} + 6 q^{79} + 42 q^{81} + 18 q^{82} + 18 q^{84} - 28 q^{85} + 36 q^{87} - 10 q^{88} - 112 q^{91} - 36 q^{93} + 42 q^{94} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 6 x^{19} + 19 x^{18} - 42 x^{17} + 62 x^{16} - 42 x^{15} - 25 x^{14} + 6 x^{13} + 445 x^{12} - 1764 x^{11} + 3864 x^{10} - 5292 x^{9} + 4005 x^{8} + 162 x^{7} - 2025 x^{6} + \cdots + 59049 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{19} - 6 \nu^{18} + 19 \nu^{17} - 42 \nu^{16} + 62 \nu^{15} - 42 \nu^{14} - 25 \nu^{13} + 6 \nu^{12} + 445 \nu^{11} - 1764 \nu^{10} + 3864 \nu^{9} - 5292 \nu^{8} + 4005 \nu^{7} + 162 \nu^{6} + \cdots - 118098 ) / 19683 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1501 \nu^{19} - 11100 \nu^{18} + 79886 \nu^{17} - 186672 \nu^{16} + 236878 \nu^{15} + 4362 \nu^{14} - 634019 \nu^{13} + 876018 \nu^{12} + 1843352 \nu^{11} + \cdots + 31768362 ) / 25351704 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 538 \nu^{19} + 1727 \nu^{18} - 21322 \nu^{17} + 102482 \nu^{16} - 220028 \nu^{15} + 259474 \nu^{14} + 17812 \nu^{13} - 637247 \nu^{12} + 636608 \nu^{11} + \cdots + 308714733 ) / 8450568 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3941 \nu^{19} + 57282 \nu^{18} - 277064 \nu^{17} + 662019 \nu^{16} - 914977 \nu^{15} + 320778 \nu^{14} + 1525880 \nu^{13} - 2317281 \nu^{12} + \cdots + 510911631 ) / 33802272 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1232 \nu^{19} + 5597 \nu^{18} - 13781 \nu^{17} + 29807 \nu^{16} - 38128 \nu^{15} + 20905 \nu^{14} + 9539 \nu^{13} + 48103 \nu^{12} - 377696 \nu^{11} + 1323236 \nu^{10} + \cdots + 71849511 ) / 4828896 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 25957 \nu^{19} + 143919 \nu^{18} - 321337 \nu^{17} + 259002 \nu^{16} + 376723 \nu^{15} - 1654737 \nu^{14} + 1611259 \nu^{13} + 4421898 \nu^{12} + \cdots - 958365270 ) / 101406816 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 27967 \nu^{19} + 12315 \nu^{18} + 252995 \nu^{17} - 657600 \nu^{16} + 920893 \nu^{15} - 97281 \nu^{14} - 2948021 \nu^{13} + 5266932 \nu^{12} + \cdots + 638910180 ) / 101406816 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1795 \nu^{19} + 9627 \nu^{18} - 21937 \nu^{17} + 38256 \nu^{16} - 30839 \nu^{15} - 21261 \nu^{14} + 55495 \nu^{13} + 170544 \nu^{12} - 850012 \nu^{11} + \cdots + 72748368 ) / 4828896 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 16230 \nu^{19} + 123337 \nu^{18} - 452289 \nu^{17} + 1002997 \nu^{16} - 1265262 \nu^{15} + 304937 \nu^{14} + 2060487 \nu^{13} - 1708639 \nu^{12} + \cdots + 1053650673 ) / 33802272 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 77773 \nu^{19} + 134496 \nu^{18} + 62150 \nu^{17} - 578403 \nu^{16} + 1140403 \nu^{15} - 583692 \nu^{14} - 3963446 \nu^{13} + 10593261 \nu^{12} + \cdots + 977831757 ) / 101406816 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2234 \nu^{19} + 12045 \nu^{18} - 27746 \nu^{17} + 36660 \nu^{16} - 6520 \nu^{15} - 74616 \nu^{14} + 98336 \nu^{13} + 279033 \nu^{12} - 1282736 \nu^{11} + \cdots + 9848061 ) / 2816856 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 85589 \nu^{19} + 304506 \nu^{18} - 485684 \nu^{17} - 119673 \nu^{16} + 1927079 \nu^{15} - 4048698 \nu^{14} + 701660 \nu^{13} + 14980239 \nu^{12} + \cdots - 2300135697 ) / 101406816 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 93578 \nu^{19} - 620469 \nu^{18} + 2151005 \nu^{17} - 4614381 \nu^{16} + 5525050 \nu^{15} - 826701 \nu^{14} - 9383507 \nu^{13} + 5032575 \nu^{12} + \cdots - 4479437457 ) / 101406816 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 125911 \nu^{19} - 566055 \nu^{18} + 1259389 \nu^{17} - 1667448 \nu^{16} + 437723 \nu^{15} + 3023385 \nu^{14} - 3231763 \nu^{13} - 13469688 \nu^{12} + \cdots - 1121931000 ) / 101406816 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 152435 \nu^{19} - 609981 \nu^{18} + 1552223 \nu^{17} - 2696802 \nu^{16} + 2474215 \nu^{15} + 796455 \nu^{14} - 2943185 \nu^{13} - 9418854 \nu^{12} + \cdots - 3856962582 ) / 101406816 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 160799 \nu^{19} + 572295 \nu^{18} - 946949 \nu^{17} + 498360 \nu^{16} + 1645421 \nu^{15} - 4756569 \nu^{14} + 356867 \nu^{13} + 22815672 \nu^{12} + \cdots - 523252872 ) / 101406816 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 17987 \nu^{19} - 75384 \nu^{18} + 186410 \nu^{17} - 310323 \nu^{16} + 249247 \nu^{15} + 185688 \nu^{14} - 415238 \nu^{13} - 1313943 \nu^{12} + 7216412 \nu^{11} + \cdots - 375510087 ) / 11267424 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 83249 \nu^{19} + 337060 \nu^{18} - 747026 \nu^{17} + 1038097 \nu^{16} - 458641 \nu^{15} - 1388020 \nu^{14} + 1361690 \nu^{13} + 8009669 \nu^{12} + \cdots + 1122121269 ) / 33802272 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{19} - \beta_{13} - \beta_{11} - \beta_{10} - \beta_{9} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{17} - 2 \beta_{15} - 2 \beta_{10} - \beta_{9} - \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{19} - \beta_{18} - 2 \beta_{15} - \beta_{14} + \beta_{13} - 2 \beta_{12} - \beta_{10} + \beta_{9} - \beta_{8} + 2 \beta_{7} + 2 \beta_{5} + 6 \beta_{3} + \beta_{2} + \beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 4 \beta_{18} - \beta_{17} - 2 \beta_{16} + \beta_{15} + 6 \beta_{14} - 5 \beta_{13} + 7 \beta_{12} + \beta_{10} - 4 \beta_{9} - \beta_{8} - 2 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} - \beta_{4} + 9 \beta_{2} + 3 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{18} + \beta_{17} - 2 \beta_{16} + 5 \beta_{14} - 2 \beta_{13} + 4 \beta_{12} - 6 \beta_{11} + 6 \beta_{10} - 9 \beta_{9} + 6 \beta_{8} - 2 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} + \beta_{4} - 4 \beta_{2} + 3 \beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4 \beta_{19} + 2 \beta_{18} - 2 \beta_{17} + 4 \beta_{16} - \beta_{15} + 5 \beta_{14} + 9 \beta_{13} - \beta_{12} - 8 \beta_{11} + 4 \beta_{10} - 12 \beta_{9} + 2 \beta_{8} + 12 \beta_{7} + 20 \beta_{6} - 12 \beta_{5} + 16 \beta_{4} - 14 \beta_{3} - 5 \beta_{2} - 12 \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 10 \beta_{19} - 8 \beta_{17} + 13 \beta_{16} - 8 \beta_{15} + 37 \beta_{13} - 20 \beta_{12} + 10 \beta_{11} + 22 \beta_{10} + 23 \beta_{9} - 6 \beta_{8} + 12 \beta_{7} + 12 \beta_{6} - 11 \beta_{5} + 16 \beta_{4} + 6 \beta_{3} - 4 \beta_{2} + 16 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 24 \beta_{19} - 24 \beta_{18} + 13 \beta_{17} + 24 \beta_{16} + 26 \beta_{15} + 22 \beta_{14} - 20 \beta_{13} + 19 \beta_{12} - 12 \beta_{11} + 21 \beta_{10} - 12 \beta_{9} - 9 \beta_{8} - 66 \beta_{7} - 21 \beta_{6} + 22 \beta_{5} - 17 \beta_{4} + 22 \beta_{3} + \cdots + 11 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 7 \beta_{19} - 24 \beta_{18} - 45 \beta_{15} + 61 \beta_{14} - 61 \beta_{13} + 67 \beta_{12} + 38 \beta_{10} - 108 \beta_{9} - 34 \beta_{8} - 42 \beta_{7} - 79 \beta_{5} + 36 \beta_{4} - 129 \beta_{3} + 4 \beta_{2} + 50 \beta _1 - 129 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 56 \beta_{19} + 64 \beta_{18} + 108 \beta_{17} + 32 \beta_{16} - 108 \beta_{15} - 68 \beta_{14} + 66 \beta_{13} - 100 \beta_{12} - 56 \beta_{11} - 2 \beta_{10} - 42 \beta_{9} - 76 \beta_{8} - 22 \beta_{7} + 86 \beta_{6} - 74 \beta_{5} + \cdots + 172 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 20 \beta_{18} + 80 \beta_{17} + 20 \beta_{16} + 38 \beta_{14} + 20 \beta_{13} + 72 \beta_{12} - 8 \beta_{11} - 96 \beta_{10} + 10 \beta_{9} - 310 \beta_{8} + 20 \beta_{7} + 90 \beta_{6} - 56 \beta_{5} + 248 \beta_{4} + 140 \beta_{2} - 8 \beta _1 + 69 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 72 \beta_{19} + 56 \beta_{18} + 216 \beta_{17} + 112 \beta_{16} + 108 \beta_{15} - 82 \beta_{14} + 30 \beta_{13} + 108 \beta_{12} - 144 \beta_{11} + 26 \beta_{10} + 64 \beta_{9} - 86 \beta_{8} - 330 \beta_{7} - 772 \beta_{6} + 62 \beta_{5} + \cdots - 510 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 159 \beta_{19} + 4 \beta_{17} + 64 \beta_{16} + 4 \beta_{15} - 189 \beta_{13} + 616 \beta_{12} - 159 \beta_{11} - 81 \beta_{10} - 989 \beta_{9} - 168 \beta_{8} - 90 \beta_{7} - 90 \beta_{6} - 398 \beta_{5} + 436 \beta_{4} + 42 \beta_{3} + \cdots - 979 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1224 \beta_{19} + 432 \beta_{18} - 651 \beta_{17} - 432 \beta_{16} - 1302 \beta_{15} - 950 \beta_{14} + 1468 \beta_{13} - 788 \beta_{12} + 612 \beta_{11} + 790 \beta_{10} + 419 \beta_{9} + 299 \beta_{8} + 50 \beta_{7} + \cdots + 523 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 514 \beta_{19} + 29 \beta_{18} + 1150 \beta_{15} - 1557 \beta_{14} + 1557 \beta_{13} - 490 \beta_{12} + 115 \beta_{10} + 1879 \beta_{9} + 1411 \beta_{8} - 2740 \beta_{7} + 508 \beta_{5} - 1836 \beta_{4} + 1596 \beta_{3} + \cdots + 1596 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 820 \beta_{19} - 372 \beta_{18} - 1879 \beta_{17} - 186 \beta_{16} + 1879 \beta_{15} - 74 \beta_{14} - 149 \beta_{13} + 225 \beta_{12} + 820 \beta_{11} + 1829 \beta_{10} + 1730 \beta_{9} + 3745 \beta_{8} + 780 \beta_{7} + \cdots - 9716 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 3390 \beta_{18} + 771 \beta_{17} + 3390 \beta_{16} - 4307 \beta_{14} + 3390 \beta_{13} - 5768 \beta_{12} + 1654 \beta_{11} + 558 \beta_{10} + 5011 \beta_{9} + 4652 \beta_{8} + 3390 \beta_{7} + 4929 \beta_{6} + \cdots - 4578 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 5768 \beta_{19} - 950 \beta_{18} - 3830 \beta_{17} - 1900 \beta_{16} - 1915 \beta_{15} + 757 \beta_{14} - 1143 \beta_{13} - 1915 \beta_{12} + 11536 \beta_{11} - 1158 \beta_{10} + 11368 \beta_{9} - 9012 \beta_{8} + \cdots + 2384 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/462\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(211\)
\(\chi(n)\) \(-1\) \(1 + \beta_{3}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1
−0.106512 + 1.72877i
1.68119 + 0.416664i
1.69321 0.364755i
−1.53534 + 0.801712i
−0.232547 1.71637i
0.530717 + 1.64874i
1.20144 + 1.24762i
1.44390 0.956629i
−1.60269 + 0.656793i
−0.0733649 1.73050i
−0.106512 1.72877i
1.68119 0.416664i
1.69321 + 0.364755i
−1.53534 0.801712i
−0.232547 + 1.71637i
0.530717 1.64874i
1.20144 1.24762i
1.44390 + 0.956629i
−1.60269 0.656793i
−0.0733649 + 1.73050i
−0.866025 0.500000i −1.44390 0.956629i 0.500000 + 0.866025i 0.0938814 0.162607i 0.772144 + 1.55042i −2.64498 0.0638828i 1.00000i 1.16972 + 2.76256i −0.162607 + 0.0938814i
89.2 −0.866025 0.500000i −1.20144 + 1.24762i 0.500000 + 0.866025i −0.798258 + 1.38262i 1.66428 0.479752i 0.157053 2.64109i 1.00000i −0.113106 2.99787i 1.38262 0.798258i
89.3 −0.866025 0.500000i −0.530717 + 1.64874i 0.500000 + 0.866025i −0.417958 + 0.723925i 1.28398 1.16249i −1.42670 + 2.22812i 1.00000i −2.43668 1.75003i 0.723925 0.417958i
89.4 −0.866025 0.500000i 0.0733649 1.73050i 0.500000 + 0.866025i 1.79481 3.10870i −0.928784 + 1.46197i 0.833981 2.51087i 1.00000i −2.98924 0.253915i −3.10870 + 1.79481i
89.5 −0.866025 0.500000i 1.60269 + 0.656793i 0.500000 + 0.866025i 1.92560 3.33524i −1.05958 1.37015i 1.58064 + 2.12169i 1.00000i 2.13725 + 2.10527i −3.33524 + 1.92560i
89.6 0.866025 + 0.500000i −1.69321 0.364755i 0.500000 + 0.866025i 0.417958 0.723925i −1.28398 1.16249i −1.42670 + 2.22812i 1.00000i 2.73391 + 1.23521i 0.723925 0.417958i
89.7 0.866025 + 0.500000i −1.68119 + 0.416664i 0.500000 + 0.866025i 0.798258 1.38262i −1.66428 0.479752i 0.157053 2.64109i 1.00000i 2.65278 1.40098i 1.38262 0.798258i
89.8 0.866025 + 0.500000i 0.106512 + 1.72877i 0.500000 + 0.866025i −0.0938814 + 0.162607i −0.772144 + 1.55042i −2.64498 0.0638828i 1.00000i −2.97731 + 0.368271i −0.162607 + 0.0938814i
89.9 0.866025 + 0.500000i 0.232547 1.71637i 0.500000 + 0.866025i −1.92560 + 3.33524i 1.05958 1.37015i 1.58064 + 2.12169i 1.00000i −2.89184 0.798273i −3.33524 + 1.92560i
89.10 0.866025 + 0.500000i 1.53534 + 0.801712i 0.500000 + 0.866025i −1.79481 + 3.10870i 0.928784 + 1.46197i 0.833981 2.51087i 1.00000i 1.71451 + 2.46180i −3.10870 + 1.79481i
353.1 −0.866025 + 0.500000i −1.44390 + 0.956629i 0.500000 0.866025i 0.0938814 + 0.162607i 0.772144 1.55042i −2.64498 + 0.0638828i 1.00000i 1.16972 2.76256i −0.162607 0.0938814i
353.2 −0.866025 + 0.500000i −1.20144 1.24762i 0.500000 0.866025i −0.798258 1.38262i 1.66428 + 0.479752i 0.157053 + 2.64109i 1.00000i −0.113106 + 2.99787i 1.38262 + 0.798258i
353.3 −0.866025 + 0.500000i −0.530717 1.64874i 0.500000 0.866025i −0.417958 0.723925i 1.28398 + 1.16249i −1.42670 2.22812i 1.00000i −2.43668 + 1.75003i 0.723925 + 0.417958i
353.4 −0.866025 + 0.500000i 0.0733649 + 1.73050i 0.500000 0.866025i 1.79481 + 3.10870i −0.928784 1.46197i 0.833981 + 2.51087i 1.00000i −2.98924 + 0.253915i −3.10870 1.79481i
353.5 −0.866025 + 0.500000i 1.60269 0.656793i 0.500000 0.866025i 1.92560 + 3.33524i −1.05958 + 1.37015i 1.58064 2.12169i 1.00000i 2.13725 2.10527i −3.33524 1.92560i
353.6 0.866025 0.500000i −1.69321 + 0.364755i 0.500000 0.866025i 0.417958 + 0.723925i −1.28398 + 1.16249i −1.42670 2.22812i 1.00000i 2.73391 1.23521i 0.723925 + 0.417958i
353.7 0.866025 0.500000i −1.68119 0.416664i 0.500000 0.866025i 0.798258 + 1.38262i −1.66428 + 0.479752i 0.157053 + 2.64109i 1.00000i 2.65278 + 1.40098i 1.38262 + 0.798258i
353.8 0.866025 0.500000i 0.106512 1.72877i 0.500000 0.866025i −0.0938814 0.162607i −0.772144 1.55042i −2.64498 + 0.0638828i 1.00000i −2.97731 0.368271i −0.162607 0.0938814i
353.9 0.866025 0.500000i 0.232547 + 1.71637i 0.500000 0.866025i −1.92560 3.33524i 1.05958 + 1.37015i 1.58064 2.12169i 1.00000i −2.89184 + 0.798273i −3.33524 1.92560i
353.10 0.866025 0.500000i 1.53534 0.801712i 0.500000 0.866025i −1.79481 3.10870i 0.928784 1.46197i 0.833981 + 2.51087i 1.00000i 1.71451 2.46180i −3.10870 1.79481i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 353.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.k.g 20
3.b odd 2 1 inner 462.2.k.g 20
7.d odd 6 1 inner 462.2.k.g 20
21.g even 6 1 inner 462.2.k.g 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.k.g 20 1.a even 1 1 trivial
462.2.k.g 20 3.b odd 2 1 inner
462.2.k.g 20 7.d odd 6 1 inner
462.2.k.g 20 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\):

\( T_{5}^{20} + 31 T_{5}^{18} + 677 T_{5}^{16} + 7444 T_{5}^{14} + 59212 T_{5}^{12} + 170564 T_{5}^{10} + 358652 T_{5}^{8} + 240704 T_{5}^{6} + 124336 T_{5}^{4} + 4368 T_{5}^{2} + 144 \) Copy content Toggle raw display
\( T_{17}^{20} + 59 T_{17}^{18} + 2561 T_{17}^{16} + 42632 T_{17}^{14} + 486784 T_{17}^{12} + 3485632 T_{17}^{10} + 18281408 T_{17}^{8} + 64568320 T_{17}^{6} + 165425152 T_{17}^{4} + 248832000 T_{17}^{2} + \cdots + 241864704 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{20} + 6 T^{19} + 19 T^{18} + \cdots + 59049 \) Copy content Toggle raw display
$5$ \( T^{20} + 31 T^{18} + 677 T^{16} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( (T^{10} + 3 T^{9} + 15 T^{8} + 60 T^{7} + \cdots + 16807)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - T^{2} + 1)^{5} \) Copy content Toggle raw display
$13$ \( (T^{10} + 68 T^{8} + 1694 T^{6} + \cdots + 9408)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + 59 T^{18} + \cdots + 241864704 \) Copy content Toggle raw display
$19$ \( (T^{10} - 18 T^{9} + 106 T^{8} + \cdots + 363312)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} - 141 T^{18} + \cdots + 2379503694096 \) Copy content Toggle raw display
$29$ \( (T^{10} + 196 T^{8} + 14718 T^{6} + \cdots + 52881984)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} - 18 T^{9} - 8 T^{8} + \cdots + 845107968)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} - 8 T^{9} + 110 T^{8} + 16 T^{7} + \cdots + 256)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} - 31 T^{8} + 284 T^{6} - 680 T^{4} + \cdots - 12)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} - 8 T^{4} - 84 T^{3} + 848 T^{2} + \cdots - 3712)^{4} \) Copy content Toggle raw display
$47$ \( T^{20} + 499 T^{18} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{20} - 400 T^{18} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 435854640545424 \) Copy content Toggle raw display
$61$ \( (T^{10} - 21 T^{9} + 61 T^{8} + \cdots + 29862075)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + 5 T^{9} + 169 T^{8} + \cdots + 6215049)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + 300 T^{8} + 24048 T^{6} + \cdots + 1679616)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} - 6 T^{9} - 214 T^{8} + \cdots + 119675568)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} - 3 T^{9} + 113 T^{8} + \cdots + 94848121)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} - 667 T^{8} + 147848 T^{6} + \cdots - 3378820800)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + 496 T^{18} + \cdots + 34\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( (T^{10} + 263 T^{8} + 22106 T^{6} + \cdots + 240267)^{2} \) Copy content Toggle raw display
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