Properties

Label 65.2.t.a
Level $65$
Weight $2$
Character orbit 65.t
Analytic conductor $0.519$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 65 = 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 65.t (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.519027613138\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial: \( x^{20} + 26 x^{18} + 279 x^{16} + 1604 x^{14} + 5353 x^{12} + 10466 x^{10} + 11441 x^{8} + 6176 x^{6} + 1263 x^{4} + 78 x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{17} + \beta_{16} - \beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} - \beta_{8} + \beta_{6} - \beta_{4} + \cdots - \beta_1) q^{3}+ \cdots + ( - \beta_{18} + \beta_{17} + 2 \beta_{16} + \beta_{14} - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{6} - \beta_{5} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{17} + \beta_{16} - \beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} - \beta_{8} + \beta_{6} - \beta_{4} + \cdots - \beta_1) q^{3}+ \cdots + ( - \beta_{19} + 2 \beta_{18} - 2 \beta_{15} + \beta_{14} - \beta_{13} + \beta_{12} - \beta_{10} - 2 \beta_{7} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 8 q^{6} - 2 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 8 q^{6} - 2 q^{7} + 12 q^{9} - 2 q^{10} - 16 q^{11} - 24 q^{12} - 4 q^{13} - 20 q^{15} - 2 q^{16} + 4 q^{17} - 20 q^{19} + 4 q^{21} + 16 q^{22} - 10 q^{23} + 32 q^{24} + 18 q^{25} - 24 q^{26} + 4 q^{27} + 18 q^{28} - 26 q^{30} + 48 q^{32} + 18 q^{33} + 2 q^{34} + 40 q^{35} + 36 q^{36} - 4 q^{37} - 8 q^{38} + 4 q^{39} - 16 q^{40} + 10 q^{41} + 40 q^{42} + 10 q^{43} - 36 q^{44} + 4 q^{46} - 40 q^{47} - 56 q^{48} + 18 q^{49} + 36 q^{50} - 30 q^{52} - 10 q^{53} - 48 q^{54} - 10 q^{55} - 16 q^{59} + 28 q^{60} - 16 q^{61} - 44 q^{62} - 36 q^{63} + 20 q^{64} - 14 q^{65} - 32 q^{66} + 18 q^{67} + 22 q^{68} - 16 q^{69} - 12 q^{70} - 16 q^{71} + 4 q^{72} + 18 q^{74} - 38 q^{75} - 64 q^{76} - 28 q^{77} + 68 q^{78} - 2 q^{80} - 14 q^{81} + 56 q^{82} + 48 q^{83} - 40 q^{84} - 26 q^{85} + 60 q^{86} - 34 q^{87} + 82 q^{88} - 6 q^{89} + 46 q^{90} + 8 q^{91} - 8 q^{92} + 32 q^{93} - 48 q^{94} - 26 q^{95} + 56 q^{96} + 66 q^{97} - 30 q^{98} + 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 26 x^{18} + 279 x^{16} + 1604 x^{14} + 5353 x^{12} + 10466 x^{10} + 11441 x^{8} + 6176 x^{6} + 1263 x^{4} + 78 x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 20 \nu^{18} - 389 \nu^{16} - 2695 \nu^{14} - 7125 \nu^{12} + 1214 \nu^{10} + 39860 \nu^{8} + 68102 \nu^{6} + 46015 \nu^{4} + 16571 \nu^{2} + 2996 \nu - 409 ) / 5992 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 20 \nu^{18} + 389 \nu^{16} + 2695 \nu^{14} + 7125 \nu^{12} - 1214 \nu^{10} - 39860 \nu^{8} - 68102 \nu^{6} - 46015 \nu^{4} - 16571 \nu^{2} + 2996 \nu + 409 ) / 5992 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 69 \nu^{18} - 1647 \nu^{16} - 15798 \nu^{14} - 78322 \nu^{12} - 214723 \nu^{10} - 324081 \nu^{8} - 257858 \nu^{6} - 105030 \nu^{4} - 19961 \nu^{2} - 539 ) / 1712 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 321 \nu^{19} - 483 \nu^{18} + 9951 \nu^{17} - 11529 \nu^{16} + 127330 \nu^{15} - 110586 \nu^{14} + 869910 \nu^{13} - 548254 \nu^{12} + 3425391 \nu^{11} - 1503061 \nu^{10} + \cdots - 3773 ) / 23968 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 700 \nu^{19} + 617 \nu^{18} + 17360 \nu^{17} + 16607 \nu^{16} + 174468 \nu^{15} + 185192 \nu^{14} + 912240 \nu^{13} + 1108682 \nu^{12} + 2625448 \nu^{11} + 3846937 \nu^{10} + \cdots + 26961 ) / 11984 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 409 \nu^{19} - 10614 \nu^{17} - 113722 \nu^{15} - 653341 \nu^{13} - 2182252 \nu^{11} - 4281808 \nu^{9} - 4719229 \nu^{7} - 2594086 \nu^{5} - 562582 \nu^{3} + \cdots - 2996 ) / 5992 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1635 \nu^{19} - 309 \nu^{18} + 41725 \nu^{17} - 7171 \nu^{16} + 436590 \nu^{15} - 66542 \nu^{14} + 2423698 \nu^{13} - 319614 \nu^{12} + 7689981 \nu^{11} - 870831 \nu^{10} + \cdots - 30699 ) / 23968 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 700 \nu^{19} - 617 \nu^{18} + 17360 \nu^{17} - 16607 \nu^{16} + 174468 \nu^{15} - 185192 \nu^{14} + 912240 \nu^{13} - 1108682 \nu^{12} + 2625448 \nu^{11} - 3846937 \nu^{10} + \cdots - 26961 ) / 11984 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 419 \nu^{19} + 4593 \nu^{18} - 12681 \nu^{17} + 118807 \nu^{16} - 158886 \nu^{15} + 1266230 \nu^{14} - 1064734 \nu^{13} + 7212374 \nu^{12} - 4108073 \nu^{11} + \cdots + 83703 ) / 23968 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3773 \nu^{19} - 565 \nu^{18} - 97615 \nu^{17} - 11551 \nu^{16} - 1041138 \nu^{15} - 83062 \nu^{14} - 5941306 \nu^{13} - 198098 \nu^{12} - 19648615 \nu^{11} + \cdots + 81509 ) / 23968 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3773 \nu^{19} + 321 \nu^{18} + 97615 \nu^{17} + 9951 \nu^{16} + 1041138 \nu^{15} + 127330 \nu^{14} + 5941306 \nu^{13} + 869910 \nu^{12} + 19648615 \nu^{11} + 3425391 \nu^{10} + \cdots + 39483 ) / 23968 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3773 \nu^{19} - 321 \nu^{18} + 97615 \nu^{17} - 9951 \nu^{16} + 1041138 \nu^{15} - 127330 \nu^{14} + 5941306 \nu^{13} - 869910 \nu^{12} + 19648615 \nu^{11} - 3425391 \nu^{10} + \cdots - 39483 ) / 23968 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 2753 \nu^{19} - 867 \nu^{18} - 71035 \nu^{17} - 22593 \nu^{16} - 754642 \nu^{15} - 243222 \nu^{14} - 4279914 \nu^{13} - 1404094 \nu^{12} - 14010639 \nu^{11} + \cdots - 15221 ) / 11984 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 2753 \nu^{19} - 867 \nu^{18} + 71035 \nu^{17} - 22593 \nu^{16} + 754642 \nu^{15} - 243222 \nu^{14} + 4279914 \nu^{13} - 1404094 \nu^{12} + 14010639 \nu^{11} - 4705845 \nu^{10} + \cdots - 15221 ) / 11984 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 6965 \nu^{19} + 1863 \nu^{18} - 180971 \nu^{17} + 49605 \nu^{16} - 1940134 \nu^{15} + 545958 \nu^{14} - 11139534 \nu^{13} + 3217222 \nu^{12} - 37111487 \nu^{11} + \cdots + 1285 ) / 23968 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 8297 \nu^{19} - 175 \nu^{18} + 216915 \nu^{17} - 2093 \nu^{16} + 2342074 \nu^{15} + 9562 \nu^{14} + 13551722 \nu^{13} + 273770 \nu^{12} + 45488043 \nu^{11} + 1753171 \nu^{10} + \cdots + 23947 ) / 23968 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 4979 \nu^{19} - 129835 \nu^{17} - 1398012 \nu^{15} - 8068552 \nu^{13} - 27041293 \nu^{11} - 53105203 \nu^{9} - 58320582 \nu^{7} - 31671424 \nu^{5} + \cdots + 11984 ) / 11984 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 10417 \nu^{19} - 483 \nu^{18} + 268635 \nu^{17} - 11529 \nu^{16} + 2853942 \nu^{15} - 110586 \nu^{14} + 16210930 \nu^{13} - 548254 \nu^{12} + 53334711 \nu^{11} + \cdots + 20195 ) / 23968 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 16083 \nu^{19} - 483 \nu^{18} - 419809 \nu^{17} - 11529 \nu^{16} - 4525598 \nu^{15} - 110586 \nu^{14} - 26151990 \nu^{13} - 548254 \nu^{12} - 87742737 \nu^{11} + \cdots - 15757 ) / 23968 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} + \beta_{10} - \beta_{3} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} + 2 \beta_{18} - \beta_{13} - 3 \beta_{12} - 2 \beta_{11} - 2 \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} - 6 \beta_{2} - 7 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{19} - \beta_{18} + \beta_{17} + 2 \beta_{16} - 2 \beta_{15} + 3 \beta_{14} - 6 \beta_{12} - 2 \beta_{11} - 7 \beta_{10} + \beta_{9} - 3 \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + 7 \beta_{3} - \beta_{2} - 7 \beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 9 \beta_{19} - 17 \beta_{18} - \beta_{17} - 3 \beta_{14} + 11 \beta_{13} + 26 \beta_{12} + 17 \beta_{11} + 17 \beta_{10} + 8 \beta_{9} + 10 \beta_{8} + 8 \beta_{7} + 5 \beta_{6} - 6 \beta_{5} - 18 \beta_{4} + 9 \beta_{3} + 37 \beta_{2} + 46 \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 13 \beta_{19} + 13 \beta_{18} - 13 \beta_{17} - 26 \beta_{16} + 26 \beta_{15} - 32 \beta_{14} + 4 \beta_{13} + 38 \beta_{12} + 25 \beta_{11} + 53 \beta_{10} - 10 \beta_{9} + 37 \beta_{8} + 10 \beta_{7} - 13 \beta_{6} - 11 \beta_{5} + 16 \beta_{4} + \cdots - 39 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 66 \beta_{19} + 122 \beta_{18} + 14 \beta_{17} + 37 \beta_{14} - 89 \beta_{13} - 189 \beta_{12} - 121 \beta_{11} - 120 \beta_{10} - 52 \beta_{9} - 76 \beta_{8} - 52 \beta_{7} - 24 \beta_{6} + 28 \beta_{5} + 138 \beta_{4} - 69 \beta_{3} - 236 \beta_{2} + \cdots + 21 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 120 \beta_{19} - 116 \beta_{18} + 116 \beta_{17} + 232 \beta_{16} - 240 \beta_{15} + 260 \beta_{14} - 56 \beta_{13} - 252 \beta_{12} - 229 \beta_{11} - 397 \beta_{10} + 76 \beta_{9} - 332 \beta_{8} - 84 \beta_{7} + 112 \beta_{6} + \cdots + 213 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 462 \beta_{19} - 840 \beta_{18} - 136 \beta_{17} - 332 \beta_{14} + 657 \beta_{13} + 1306 \beta_{12} + 818 \beta_{11} + 813 \beta_{10} + 325 \beta_{9} + 537 \beta_{8} + 325 \beta_{7} + 122 \beta_{6} - 113 \beta_{5} + \cdots - 170 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 974 \beta_{19} + 914 \beta_{18} - 914 \beta_{17} - 1828 \beta_{16} + 1948 \beta_{15} - 1941 \beta_{14} + 546 \beta_{13} + 1707 \beta_{12} + 1862 \beta_{11} + 2910 \beta_{10} - 539 \beta_{9} + 2648 \beta_{8} + \cdots - 1255 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3210 \beta_{19} + 5733 \beta_{18} + 1135 \beta_{17} + 2648 \beta_{14} - 4690 \beta_{13} - 8890 \beta_{12} - 5456 \beta_{11} - 5476 \beta_{10} - 2042 \beta_{9} - 3716 \beta_{8} - 2042 \beta_{7} - 676 \beta_{6} + \cdots + 1267 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 7438 \beta_{19} - 6832 \beta_{18} + 6832 \beta_{17} + 13664 \beta_{16} - 14876 \beta_{15} + 14032 \beta_{14} - 4610 \beta_{13} - 11688 \beta_{12} - 14278 \beta_{11} - 20988 \beta_{10} + 3766 \beta_{9} + \cdots + 7797 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 22356 \beta_{19} - 39188 \beta_{18} - 8780 \beta_{17} - 19984 \beta_{14} + 33058 \beta_{13} + 60422 \beta_{12} + 36438 \beta_{11} + 37058 \beta_{10} + 13074 \beta_{9} + 25620 \beta_{8} + 13074 \beta_{7} + \cdots - 9130 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 54982 \beta_{19} + 49778 \beta_{18} - 49778 \beta_{17} - 99556 \beta_{16} + 109964 \beta_{15} - 100084 \beta_{14} + 36202 \beta_{13} + 80555 \beta_{12} + 105990 \beta_{11} + 149815 \beta_{10} + \cdots - 50290 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 156271 \beta_{19} + 269080 \beta_{18} + 65190 \beta_{17} + 146494 \beta_{14} - 231975 \beta_{13} - 412141 \beta_{12} - 245006 \beta_{11} - 252616 \beta_{10} - 85481 \beta_{9} - 176993 \beta_{8} + \cdots + 64782 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 398897 \beta_{19} - 357747 \beta_{18} + 357747 \beta_{17} + 715494 \beta_{16} - 797794 \beta_{15} + 709463 \beta_{14} - 272912 \beta_{13} - 557750 \beta_{12} - 771768 \beta_{11} - 1062637 \beta_{10} + \cdots + 333034 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 1095413 \beta_{19} - 1856511 \beta_{18} - 472963 \beta_{17} - 1056441 \beta_{14} + 1625819 \beta_{13} + 2825568 \beta_{12} + 1660831 \beta_{11} + 1734115 \beta_{10} + 569378 \beta_{9} + \cdots - 456258 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 2861437 \beta_{19} + 2550751 \beta_{18} - 2550751 \beta_{17} - 5101502 \beta_{16} + 5722874 \beta_{15} - 5013336 \beta_{14} + 2007386 \beta_{13} + 3875098 \beta_{12} + \cdots - 2246355 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 7691530 \beta_{19} + 12862874 \beta_{18} + 3385870 \beta_{17} + 7545155 \beta_{14} - 11394317 \beta_{13} - 19469181 \beta_{12} - 11344809 \beta_{11} - 11973534 \beta_{10} + \cdots + 3202109 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/65\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(41\)
\(\chi(n)\) \(\beta_{11} + \beta_{12}\) \(-\beta_{11}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
2.64975i
1.83163i
0.493902i
0.274809i
1.51805i
2.64975i
1.83163i
0.493902i
0.274809i
1.51805i
2.25081i
1.02262i
0.131303i
1.58474i
2.08794i
2.25081i
1.02262i
0.131303i
1.58474i
2.08794i
−2.29475 1.32488i 0.335680 + 1.25278i 2.51060 + 4.34849i 1.81654 1.30391i 0.889471 3.31955i 0.0561740 + 0.0972962i 8.00544i 1.14131 0.658935i −5.89604 + 0.585458i
7.2 −1.58624 0.915816i −0.512942 1.91432i 0.677439 + 1.17336i −1.69810 + 1.45480i −0.939520 + 3.50634i −1.76945 3.06478i 1.18163i −0.803451 + 0.463873i 4.02593 0.752519i
7.3 −0.427732 0.246951i −0.243392 0.908353i −0.878030 1.52079i −0.284413 2.21791i −0.120212 + 0.448637i 1.83775 + 3.18307i 1.85513i 1.83221 1.05783i −0.426062 + 1.01890i
7.4 −0.237991 0.137404i 0.611610 + 2.28256i −0.962240 1.66665i 1.45395 + 1.69883i 0.168076 0.627267i −0.193052 0.334376i 1.07848i −2.23793 + 1.29207i −0.112600 0.604086i
7.5 1.31467 + 0.759023i 0.175069 + 0.653367i 0.152233 + 0.263675i −2.15400 + 0.600231i −0.265763 + 0.991842i −1.29744 2.24723i 2.57390i 2.20184 1.27123i −3.28738 0.845834i
28.1 −2.29475 + 1.32488i 0.335680 1.25278i 2.51060 4.34849i 1.81654 + 1.30391i 0.889471 + 3.31955i 0.0561740 0.0972962i 8.00544i 1.14131 + 0.658935i −5.89604 0.585458i
28.2 −1.58624 + 0.915816i −0.512942 + 1.91432i 0.677439 1.17336i −1.69810 1.45480i −0.939520 3.50634i −1.76945 + 3.06478i 1.18163i −0.803451 0.463873i 4.02593 + 0.752519i
28.3 −0.427732 + 0.246951i −0.243392 + 0.908353i −0.878030 + 1.52079i −0.284413 + 2.21791i −0.120212 0.448637i 1.83775 3.18307i 1.85513i 1.83221 + 1.05783i −0.426062 1.01890i
28.4 −0.237991 + 0.137404i 0.611610 2.28256i −0.962240 + 1.66665i 1.45395 1.69883i 0.168076 + 0.627267i −0.193052 + 0.334376i 1.07848i −2.23793 1.29207i −0.112600 + 0.604086i
28.5 1.31467 0.759023i 0.175069 0.653367i 0.152233 0.263675i −2.15400 0.600231i −0.265763 0.991842i −1.29744 + 2.24723i 2.57390i 2.20184 + 1.27123i −3.28738 + 0.845834i
37.1 −1.94926 + 1.12540i −1.91913 0.514229i 1.53307 2.65535i 0.247944 2.22228i 4.31958 1.15743i 0.638592 1.10607i 2.39966i 0.820542 + 0.473740i 2.01765 + 4.61083i
37.2 −0.885613 + 0.511309i 2.69193 + 0.721300i −0.477126 + 0.826407i −1.45744 1.69584i −2.75281 + 0.737614i −0.481787 + 0.834479i 3.02107i 4.12812 + 2.38337i 2.15782 + 0.756660i
37.3 −0.113711 + 0.0656513i 0.332179 + 0.0890070i −0.991380 + 1.71712i 2.08297 + 0.813169i −0.0436159 + 0.0116869i 1.39069 2.40874i 0.522947i −2.49566 1.44087i −0.290243 + 0.0442830i
37.4 1.37242 0.792369i 0.190588 + 0.0510678i 0.255697 0.442881i −2.23506 + 0.0672627i 0.302032 0.0809291i 0.274164 0.474866i 2.35905i −2.56436 1.48053i −3.01415 + 1.86330i
37.5 1.80821 1.04397i −2.66159 0.713171i 1.17974 2.04338i 2.22760 0.194361i −5.55724 + 1.48906i −1.45563 + 2.52122i 0.750585i 3.97738 + 2.29634i 3.82506 2.67700i
58.1 −1.94926 1.12540i −1.91913 + 0.514229i 1.53307 + 2.65535i 0.247944 + 2.22228i 4.31958 + 1.15743i 0.638592 + 1.10607i 2.39966i 0.820542 0.473740i 2.01765 4.61083i
58.2 −0.885613 0.511309i 2.69193 0.721300i −0.477126 0.826407i −1.45744 + 1.69584i −2.75281 0.737614i −0.481787 0.834479i 3.02107i 4.12812 2.38337i 2.15782 0.756660i
58.3 −0.113711 0.0656513i 0.332179 0.0890070i −0.991380 1.71712i 2.08297 0.813169i −0.0436159 0.0116869i 1.39069 + 2.40874i 0.522947i −2.49566 + 1.44087i −0.290243 0.0442830i
58.4 1.37242 + 0.792369i 0.190588 0.0510678i 0.255697 + 0.442881i −2.23506 0.0672627i 0.302032 + 0.0809291i 0.274164 + 0.474866i 2.35905i −2.56436 + 1.48053i −3.01415 1.86330i
58.5 1.80821 + 1.04397i −2.66159 + 0.713171i 1.17974 + 2.04338i 2.22760 + 0.194361i −5.55724 1.48906i −1.45563 2.52122i 0.750585i 3.97738 2.29634i 3.82506 + 2.67700i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 58.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.t even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 65.2.t.a yes 20
3.b odd 2 1 585.2.dp.a 20
5.b even 2 1 325.2.x.b 20
5.c odd 4 1 65.2.o.a 20
5.c odd 4 1 325.2.s.b 20
13.b even 2 1 845.2.t.g 20
13.c even 3 1 845.2.f.e 20
13.c even 3 1 845.2.t.f 20
13.d odd 4 1 845.2.o.e 20
13.d odd 4 1 845.2.o.f 20
13.e even 6 1 845.2.f.d 20
13.e even 6 1 845.2.t.e 20
13.f odd 12 1 65.2.o.a 20
13.f odd 12 1 845.2.k.d 20
13.f odd 12 1 845.2.k.e 20
13.f odd 12 1 845.2.o.g 20
15.e even 4 1 585.2.cf.a 20
39.k even 12 1 585.2.cf.a 20
65.f even 4 1 845.2.t.f 20
65.h odd 4 1 845.2.o.g 20
65.k even 4 1 845.2.t.e 20
65.o even 12 1 325.2.x.b 20
65.o even 12 1 845.2.f.d 20
65.o even 12 1 845.2.t.g 20
65.q odd 12 1 845.2.k.e 20
65.q odd 12 1 845.2.o.e 20
65.r odd 12 1 845.2.k.d 20
65.r odd 12 1 845.2.o.f 20
65.s odd 12 1 325.2.s.b 20
65.t even 12 1 inner 65.2.t.a yes 20
65.t even 12 1 845.2.f.e 20
195.bc odd 12 1 585.2.dp.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.o.a 20 5.c odd 4 1
65.2.o.a 20 13.f odd 12 1
65.2.t.a yes 20 1.a even 1 1 trivial
65.2.t.a yes 20 65.t even 12 1 inner
325.2.s.b 20 5.c odd 4 1
325.2.s.b 20 65.s odd 12 1
325.2.x.b 20 5.b even 2 1
325.2.x.b 20 65.o even 12 1
585.2.cf.a 20 15.e even 4 1
585.2.cf.a 20 39.k even 12 1
585.2.dp.a 20 3.b odd 2 1
585.2.dp.a 20 195.bc odd 12 1
845.2.f.d 20 13.e even 6 1
845.2.f.d 20 65.o even 12 1
845.2.f.e 20 13.c even 3 1
845.2.f.e 20 65.t even 12 1
845.2.k.d 20 13.f odd 12 1
845.2.k.d 20 65.r odd 12 1
845.2.k.e 20 13.f odd 12 1
845.2.k.e 20 65.q odd 12 1
845.2.o.e 20 13.d odd 4 1
845.2.o.e 20 65.q odd 12 1
845.2.o.f 20 13.d odd 4 1
845.2.o.f 20 65.r odd 12 1
845.2.o.g 20 13.f odd 12 1
845.2.o.g 20 65.h odd 4 1
845.2.t.e 20 13.e even 6 1
845.2.t.e 20 65.k even 4 1
845.2.t.f 20 13.c even 3 1
845.2.t.f 20 65.f even 4 1
845.2.t.g 20 13.b even 2 1
845.2.t.g 20 65.o even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(65, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 6 T^{19} + 5 T^{18} - 42 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{20} + 2 T^{19} - 4 T^{18} - 4 T^{17} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{20} - 9 T^{18} + 33 T^{16} + \cdots + 9765625 \) Copy content Toggle raw display
$7$ \( T^{20} + 2 T^{19} + 28 T^{18} + 44 T^{17} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{20} + 16 T^{19} + 140 T^{18} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{20} + 4 T^{19} + \cdots + 137858491849 \) Copy content Toggle raw display
$17$ \( T^{20} - 4 T^{19} + 65 T^{18} + \cdots + 1168561 \) Copy content Toggle raw display
$19$ \( T^{20} + 20 T^{19} + \cdots + 1583721616 \) Copy content Toggle raw display
$23$ \( T^{20} + 10 T^{19} + 104 T^{18} + \cdots + 144 \) Copy content Toggle raw display
$29$ \( T^{20} - 173 T^{18} + \cdots + 206213167449 \) Copy content Toggle raw display
$31$ \( T^{20} - 104 T^{17} + 6072 T^{16} + \cdots + 2166784 \) Copy content Toggle raw display
$37$ \( T^{20} + 4 T^{19} + \cdots + 4508182449 \) Copy content Toggle raw display
$41$ \( T^{20} - 10 T^{19} + \cdots + 3748255729 \) Copy content Toggle raw display
$43$ \( T^{20} - 10 T^{19} + \cdots + 1370772640000 \) Copy content Toggle raw display
$47$ \( (T^{10} + 20 T^{9} + 16 T^{8} - 1912 T^{7} + \cdots - 28416)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + 10 T^{19} + \cdots + 2978634160384 \) Copy content Toggle raw display
$59$ \( T^{20} + 16 T^{19} + 320 T^{18} + \cdots + 33856 \) Copy content Toggle raw display
$61$ \( T^{20} + 16 T^{19} + \cdots + 826457355409 \) Copy content Toggle raw display
$67$ \( T^{20} - 18 T^{19} + \cdots + 15478905336976 \) Copy content Toggle raw display
$71$ \( T^{20} + 16 T^{19} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{20} + 798 T^{18} + \cdots + 64\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{20} + 808 T^{18} + \cdots + 75\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( (T^{10} - 24 T^{9} + 64 T^{8} + \cdots + 3393024)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + 6 T^{19} + \cdots + 329648222500 \) Copy content Toggle raw display
$97$ \( T^{20} - 66 T^{19} + \cdots + 28\!\cdots\!36 \) Copy content Toggle raw display
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