Properties

Label 1045.2.b.c
Level $1045$
Weight $2$
Character orbit 1045.b
Analytic conductor $8.344$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1045.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.34436701122\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial: \(x^{20} + 26 x^{18} + 281 x^{16} + 1640 x^{14} + 5623 x^{12} + 11551 x^{10} + 13894 x^{8} + 9095 x^{6} + 2753 x^{4} + 276 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{7} q^{3} + ( -1 + \beta_{2} ) q^{4} -\beta_{11} q^{5} + ( -1 + \beta_{2} + \beta_{12} ) q^{6} + \beta_{3} q^{7} + ( \beta_{6} - \beta_{7} ) q^{8} + ( -1 + \beta_{8} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{7} q^{3} + ( -1 + \beta_{2} ) q^{4} -\beta_{11} q^{5} + ( -1 + \beta_{2} + \beta_{12} ) q^{6} + \beta_{3} q^{7} + ( \beta_{6} - \beta_{7} ) q^{8} + ( -1 + \beta_{8} ) q^{9} + ( \beta_{1} + \beta_{3} - \beta_{7} - \beta_{14} - \beta_{17} ) q^{10} + q^{11} + ( -\beta_{1} + \beta_{4} ) q^{12} + ( \beta_{3} - \beta_{7} + \beta_{19} ) q^{13} + ( 1 - \beta_{2} - \beta_{8} + \beta_{14} - \beta_{15} + \beta_{18} ) q^{14} + ( \beta_{4} + \beta_{7} + \beta_{8} + \beta_{11} + \beta_{15} - \beta_{18} ) q^{15} + ( -\beta_{2} + \beta_{5} - \beta_{12} + \beta_{16} + \beta_{18} ) q^{16} + ( -\beta_{1} - \beta_{4} + \beta_{10} - \beta_{11} - \beta_{13} ) q^{17} + ( \beta_{4} - \beta_{6} - \beta_{17} - \beta_{19} ) q^{18} + q^{19} + ( \beta_{3} - \beta_{8} + \beta_{14} - \beta_{15} - \beta_{16} ) q^{20} + ( -2 - \beta_{2} - \beta_{12} + \beta_{14} - \beta_{15} + \beta_{18} ) q^{21} + \beta_{1} q^{22} + ( 2 \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{6} - 2 \beta_{7} + \beta_{10} - \beta_{11} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{23} + ( 2 - \beta_{2} - \beta_{8} + \beta_{9} + \beta_{18} ) q^{24} + ( \beta_{4} - \beta_{9} + \beta_{11} + \beta_{16} - \beta_{17} - \beta_{19} ) q^{25} + ( -\beta_{2} - \beta_{5} - \beta_{8} + \beta_{9} - \beta_{12} + \beta_{14} - \beta_{15} - \beta_{16} + \beta_{18} ) q^{26} + ( \beta_{1} - \beta_{4} + \beta_{6} - 2 \beta_{7} - \beta_{10} - \beta_{14} - \beta_{15} - \beta_{17} + \beta_{19} ) q^{27} + ( -2 \beta_{3} + \beta_{4} - \beta_{6} + 3 \beta_{7} - \beta_{10} + \beta_{14} + \beta_{15} - \beta_{19} ) q^{28} + ( 3 + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} - \beta_{18} ) q^{29} + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} + \beta_{17} + \beta_{18} + \beta_{19} ) q^{30} + ( -4 + 3 \beta_{2} - 2 \beta_{5} + \beta_{8} + \beta_{11} + 2 \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} - 2 \beta_{18} ) q^{31} + ( \beta_{4} - \beta_{7} - \beta_{10} + \beta_{11} + \beta_{13} - 2 \beta_{17} - \beta_{19} ) q^{32} + \beta_{7} q^{33} + ( 2 + \beta_{5} - \beta_{9} + \beta_{12} ) q^{34} + ( 1 - \beta_{2} - \beta_{3} - \beta_{5} + 2 \beta_{7} + \beta_{11} - \beta_{13} + \beta_{15} + \beta_{17} - \beta_{18} ) q^{35} + ( \beta_{8} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{16} - \beta_{18} ) q^{36} + ( -2 \beta_{1} - \beta_{3} + 3 \beta_{4} - 2 \beta_{6} + 3 \beta_{7} - \beta_{10} + 2 \beta_{11} + 2 \beta_{13} + \beta_{14} + \beta_{15} - \beta_{17} - \beta_{19} ) q^{37} + \beta_{1} q^{38} + ( 3 - \beta_{2} + \beta_{5} - 2 \beta_{8} + \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} + \beta_{18} ) q^{39} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{8} - \beta_{11} - \beta_{13} + \beta_{17} + \beta_{18} + \beta_{19} ) q^{40} + ( -1 + 2 \beta_{5} - \beta_{9} + \beta_{14} - \beta_{15} ) q^{41} + ( -3 \beta_{1} - 4 \beta_{3} + \beta_{4} - \beta_{6} + 4 \beta_{7} - \beta_{10} + \beta_{14} + \beta_{15} - \beta_{17} - 2 \beta_{19} ) q^{42} + ( \beta_{4} - \beta_{6} - \beta_{7} - \beta_{10} + \beta_{11} + \beta_{13} - 2 \beta_{17} - 2 \beta_{19} ) q^{43} + ( -1 + \beta_{2} ) q^{44} + ( -1 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{6} - 4 \beta_{7} + \beta_{8} - \beta_{11} - \beta_{13} - \beta_{14} - \beta_{15} + \beta_{19} ) q^{45} + ( -2 + \beta_{2} - 2 \beta_{8} - \beta_{9} - \beta_{12} + 2 \beta_{14} - 2 \beta_{15} - \beta_{16} + \beta_{18} ) q^{46} + ( \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 4 \beta_{7} + \beta_{10} - 2 \beta_{11} - 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} + \beta_{17} + 3 \beta_{19} ) q^{47} + ( -\beta_{3} + \beta_{4} - \beta_{6} - \beta_{19} ) q^{48} + ( -3 + 4 \beta_{2} - 2 \beta_{5} + \beta_{8} - \beta_{9} + 3 \beta_{12} - \beta_{14} + \beta_{15} + \beta_{16} - \beta_{18} ) q^{49} + ( 3 - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{14} - \beta_{17} ) q^{50} + ( -3 + \beta_{2} - \beta_{5} + 2 \beta_{8} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} - \beta_{18} ) q^{51} + ( -2 \beta_{1} - 3 \beta_{3} - 2 \beta_{4} + \beta_{6} + 2 \beta_{7} - \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} + \beta_{17} ) q^{52} + ( -\beta_{1} - 3 \beta_{4} + 2 \beta_{6} - 4 \beta_{7} + \beta_{10} - 3 \beta_{11} - 3 \beta_{13} - \beta_{14} - \beta_{15} + 2 \beta_{19} ) q^{53} + ( -1 + 2 \beta_{2} - \beta_{5} + 2 \beta_{8} + \beta_{11} + 2 \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} - 2 \beta_{18} ) q^{54} -\beta_{11} q^{55} + ( -3 + 3 \beta_{2} - \beta_{5} + 2 \beta_{12} - \beta_{14} + \beta_{15} - \beta_{18} ) q^{56} + \beta_{7} q^{57} + ( 5 \beta_{1} + 3 \beta_{3} - \beta_{4} - \beta_{6} - 3 \beta_{7} + \beta_{10} - \beta_{11} - \beta_{13} - \beta_{14} - \beta_{15} + \beta_{17} + 2 \beta_{19} ) q^{58} + ( 3 - \beta_{2} + \beta_{5} - \beta_{11} + \beta_{13} + \beta_{14} - \beta_{15} - 3 \beta_{16} ) q^{59} + ( -2 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{7} - \beta_{12} + 2 \beta_{14} + \beta_{18} - \beta_{19} ) q^{60} + ( -\beta_{2} + \beta_{5} + \beta_{8} - \beta_{12} - \beta_{14} + \beta_{15} + \beta_{18} ) q^{61} + ( -2 \beta_{1} + 2 \beta_{3} - \beta_{4} + 2 \beta_{6} - 5 \beta_{7} + 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{13} - \beta_{14} - \beta_{15} + \beta_{17} + \beta_{19} ) q^{62} + ( -\beta_{3} + \beta_{6} - 3 \beta_{7} - 2 \beta_{10} - \beta_{17} ) q^{63} + ( 4 - 3 \beta_{2} + 2 \beta_{5} + 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} + \beta_{16} ) q^{64} + ( 2 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} - \beta_{17} - \beta_{19} ) q^{65} + ( -1 + \beta_{2} + \beta_{12} ) q^{66} + ( 2 \beta_{1} + 3 \beta_{3} - 5 \beta_{4} - 5 \beta_{7} + \beta_{10} - 3 \beta_{11} - 3 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} + \beta_{17} + 2 \beta_{19} ) q^{67} + ( -\beta_{1} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{10} + \beta_{14} + \beta_{15} + \beta_{17} + \beta_{19} ) q^{68} + ( -2 + \beta_{2} - \beta_{5} - \beta_{9} - \beta_{11} + \beta_{13} + 2 \beta_{14} - 2 \beta_{15} - \beta_{16} ) q^{69} + ( -4 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{7} + \beta_{8} + \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{14} + \beta_{15} + \beta_{16} + 2 \beta_{17} + \beta_{19} ) q^{70} + ( -2 + \beta_{5} + 2 \beta_{8} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} + 2 \beta_{16} - \beta_{18} ) q^{71} + ( -3 \beta_{1} - \beta_{3} + 3 \beta_{7} + \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} + 2 \beta_{17} - \beta_{19} ) q^{72} + ( -2 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} + 4 \beta_{17} + \beta_{19} ) q^{73} + ( 3 + \beta_{2} - 2 \beta_{5} - \beta_{8} + 2 \beta_{9} + \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{16} - 2 \beta_{18} ) q^{74} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{14} - \beta_{16} - 2 \beta_{17} - \beta_{19} ) q^{75} + ( -1 + \beta_{2} ) q^{76} + \beta_{3} q^{77} + ( -\beta_{1} - 4 \beta_{3} - \beta_{4} + 3 \beta_{7} - \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} + 2 \beta_{17} + \beta_{19} ) q^{78} + ( 6 - 2 \beta_{2} - \beta_{5} - 2 \beta_{8} + \beta_{9} + \beta_{12} - \beta_{16} + \beta_{18} ) q^{79} + ( 1 - 2 \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{7} - \beta_{10} + \beta_{11} + \beta_{12} + 3 \beta_{13} + \beta_{14} + \beta_{15} + \beta_{18} - \beta_{19} ) q^{80} + ( 2 \beta_{5} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{11} + 2 \beta_{13} + \beta_{16} + 2 \beta_{18} ) q^{81} + ( -4 \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{7} - \beta_{10} + \beta_{14} + \beta_{15} + 2 \beta_{17} + 2 \beta_{19} ) q^{82} + ( 3 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} - 4 \beta_{7} - \beta_{14} - \beta_{15} - \beta_{17} - \beta_{19} ) q^{83} + ( -1 + 3 \beta_{2} + 4 \beta_{8} - \beta_{9} + \beta_{11} + 2 \beta_{12} - \beta_{13} - 3 \beta_{14} + 3 \beta_{15} - 4 \beta_{18} ) q^{84} + ( \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{8} - \beta_{9} - \beta_{12} - \beta_{15} + \beta_{17} ) q^{85} + ( 5 - \beta_{9} + 2 \beta_{11} - 2 \beta_{13} - \beta_{16} - 3 \beta_{18} ) q^{86} + ( 2 \beta_{3} - 2 \beta_{6} + 4 \beta_{7} + \beta_{10} + \beta_{11} + \beta_{13} - 2 \beta_{19} ) q^{87} + ( \beta_{6} - \beta_{7} ) q^{88} + ( 1 + 3 \beta_{2} + \beta_{5} - 2 \beta_{9} + \beta_{12} + \beta_{14} - \beta_{15} + \beta_{16} - \beta_{18} ) q^{89} + ( -4 + \beta_{1} + \beta_{4} - 2 \beta_{6} + \beta_{7} - \beta_{9} + \beta_{14} - \beta_{15} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{90} + ( -4 + 4 \beta_{2} + \beta_{5} + \beta_{8} - \beta_{9} - \beta_{11} + 3 \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} - \beta_{18} ) q^{91} + ( -6 \beta_{1} - 4 \beta_{3} - \beta_{4} + 2 \beta_{6} + 3 \beta_{7} - \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} + 3 \beta_{17} ) q^{92} + ( 2 \beta_{3} + \beta_{6} - 5 \beta_{7} - \beta_{10} + \beta_{11} + \beta_{13} - \beta_{14} - \beta_{15} - \beta_{17} + \beta_{19} ) q^{93} + ( 1 - 4 \beta_{2} - \beta_{8} + \beta_{9} - \beta_{11} - 2 \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} + 4 \beta_{18} ) q^{94} -\beta_{11} q^{95} + ( 5 - 2 \beta_{2} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{12} - \beta_{14} + \beta_{15} + \beta_{18} ) q^{96} + ( -2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{10} - \beta_{11} - \beta_{13} + \beta_{17} ) q^{97} + ( \beta_{1} + 2 \beta_{3} + 4 \beta_{4} - 6 \beta_{7} + \beta_{11} + \beta_{13} - \beta_{14} - \beta_{15} - 4 \beta_{17} - \beta_{19} ) q^{98} + ( -1 + \beta_{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 12 q^{4} - 8 q^{6} - 10 q^{9} + O(q^{10}) \) \( 20 q - 12 q^{4} - 8 q^{6} - 10 q^{9} - 6 q^{10} + 20 q^{11} + 24 q^{14} - 6 q^{15} - 4 q^{16} + 20 q^{19} - 6 q^{20} - 30 q^{21} + 38 q^{24} + 2 q^{25} + 8 q^{26} + 50 q^{29} - 20 q^{30} - 50 q^{31} + 28 q^{34} + 6 q^{35} - 12 q^{36} + 48 q^{39} + 12 q^{40} - 34 q^{41} - 12 q^{44} - 18 q^{45} - 36 q^{46} - 6 q^{49} + 26 q^{50} - 40 q^{51} - 6 q^{54} - 40 q^{56} + 30 q^{59} - 30 q^{60} - 14 q^{61} + 36 q^{64} + 30 q^{65} - 8 q^{66} - 12 q^{69} - 54 q^{70} - 40 q^{71} + 50 q^{74} - 8 q^{75} - 12 q^{76} + 106 q^{79} + 8 q^{80} - 30 q^{84} - 22 q^{85} + 56 q^{86} + 36 q^{89} - 64 q^{90} - 56 q^{91} + 28 q^{94} + 66 q^{96} - 10 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} + 26 x^{18} + 281 x^{16} + 1640 x^{14} + 5623 x^{12} + 11551 x^{10} + 13894 x^{8} + 9095 x^{6} + 2753 x^{4} + 276 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 3 \)
\(\beta_{3}\)\(=\)\((\)\( 61 \nu^{19} + 1299 \nu^{17} + 10262 \nu^{15} + 33228 \nu^{13} + 5459 \nu^{11} - 240718 \nu^{9} - 592650 \nu^{7} - 522039 \nu^{5} - 131958 \nu^{3} + 1604 \nu \)\()/1076\)
\(\beta_{4}\)\(=\)\((\)\( 57 \nu^{19} + 1205 \nu^{17} + 9104 \nu^{15} + 23376 \nu^{13} - 49969 \nu^{11} - 428112 \nu^{9} - 946122 \nu^{7} - 866947 \nu^{5} - 288912 \nu^{3} - 27024 \nu \)\()/1076\)
\(\beta_{5}\)\(=\)\((\)\( -127 \nu^{18} - 3119 \nu^{16} - 31252 \nu^{14} - 164582 \nu^{12} - 489083 \nu^{10} - 819526 \nu^{8} - 735502 \nu^{6} - 319949 \nu^{4} - 59110 \nu^{2} - 1602 \)\()/538\)
\(\beta_{6}\)\(=\)\((\)\( 424 \nu^{19} + 10233 \nu^{17} + 99883 \nu^{15} + 504698 \nu^{13} + 1398132 \nu^{11} + 2051391 \nu^{9} + 1360162 \nu^{7} + 189834 \nu^{5} - 78267 \nu^{3} - 8898 \nu \)\()/1076\)
\(\beta_{7}\)\(=\)\((\)\( 424 \nu^{19} + 10233 \nu^{17} + 99883 \nu^{15} + 504698 \nu^{13} + 1398132 \nu^{11} + 2051391 \nu^{9} + 1360162 \nu^{7} + 189834 \nu^{5} - 79343 \nu^{3} - 13202 \nu \)\()/1076\)
\(\beta_{8}\)\(=\)\((\)\( -119 \nu^{18} - 2931 \nu^{16} - 29474 \nu^{14} - 155907 \nu^{12} - 465652 \nu^{10} - 783140 \nu^{8} - 697023 \nu^{6} - 282189 \nu^{4} - 36260 \nu^{2} - 29 \)\()/269\)
\(\beta_{9}\)\(=\)\((\)\( -284 \nu^{18} - 6943 \nu^{16} - 69037 \nu^{14} - 358938 \nu^{12} - 1043638 \nu^{10} - 1682291 \nu^{8} - 1400840 \nu^{6} - 516802 \nu^{4} - 68197 \nu^{2} - 1100 \)\()/538\)
\(\beta_{10}\)\(=\)\((\)\( 169 \nu^{19} + 4644 \nu^{17} + 53633 \nu^{15} + 338506 \nu^{13} + 1269599 \nu^{11} + 2874857 \nu^{9} + 3799744 \nu^{7} + 2655663 \nu^{5} + 772083 \nu^{3} + 44494 \nu \)\()/1076\)
\(\beta_{11}\)\(=\)\((\)\(-563 \nu^{19} - 1152 \nu^{18} - 13634 \nu^{17} - 28148 \nu^{16} - 133533 \nu^{15} - 279704 \nu^{14} - 676778 \nu^{13} - 1452564 \nu^{12} - 1878649 \nu^{11} - 4211192 \nu^{10} - 2756833 \nu^{9} - 6729844 \nu^{8} - 1827220 \nu^{7} - 5449516 \nu^{6} - 280745 \nu^{5} - 1815624 \nu^{4} + 55629 \nu^{3} - 151708 \nu^{2} - 2730 \nu + 2676\)\()/2152\)
\(\beta_{12}\)\(=\)\((\)\( -791 \nu^{18} - 19261 \nu^{16} - 190662 \nu^{14} - 986020 \nu^{12} - 2846233 \nu^{10} - 4530894 \nu^{8} - 3666446 \nu^{6} - 1246615 \nu^{4} - 131302 \nu^{2} - 3848 \)\()/1076\)
\(\beta_{13}\)\(=\)\((\)\(-563 \nu^{19} + 1152 \nu^{18} - 13634 \nu^{17} + 28148 \nu^{16} - 133533 \nu^{15} + 279704 \nu^{14} - 676778 \nu^{13} + 1452564 \nu^{12} - 1878649 \nu^{11} + 4211192 \nu^{10} - 2756833 \nu^{9} + 6729844 \nu^{8} - 1827220 \nu^{7} + 5449516 \nu^{6} - 280745 \nu^{5} + 1815624 \nu^{4} + 55629 \nu^{3} + 151708 \nu^{2} - 2730 \nu - 2676\)\()/2152\)
\(\beta_{14}\)\(=\)\((\)\(269 \nu^{19} + 502 \nu^{18} + 6994 \nu^{17} + 12335 \nu^{16} + 75589 \nu^{15} + 123271 \nu^{14} + 441160 \nu^{13} + 643550 \nu^{12} + 1512587 \nu^{11} + 1873190 \nu^{10} + 3107219 \nu^{9} + 2997551 \nu^{8} + 3737486 \nu^{7} + 2419870 \nu^{6} + 2446555 \nu^{5} + 802784 \nu^{4} + 740557 \nu^{3} + 76329 \nu^{2} + 74244 \nu + 1126\)\()/1076\)
\(\beta_{15}\)\(=\)\((\)\(269 \nu^{19} - 502 \nu^{18} + 6994 \nu^{17} - 12335 \nu^{16} + 75589 \nu^{15} - 123271 \nu^{14} + 441160 \nu^{13} - 643550 \nu^{12} + 1512587 \nu^{11} - 1873190 \nu^{10} + 3107219 \nu^{9} - 2997551 \nu^{8} + 3737486 \nu^{7} - 2419870 \nu^{6} + 2446555 \nu^{5} - 802784 \nu^{4} + 740557 \nu^{3} - 76329 \nu^{2} + 74244 \nu - 1126\)\()/1076\)
\(\beta_{16}\)\(=\)\((\)\( 615 \nu^{18} + 15125 \nu^{16} + 151546 \nu^{14} + 795708 \nu^{12} + 2343125 \nu^{10} + 3838002 \nu^{8} + 3254612 \nu^{6} + 1214287 \nu^{4} + 153152 \nu^{2} + 3674 \)\()/538\)
\(\beta_{17}\)\(=\)\((\)\( -604 \nu^{19} - 15001 \nu^{17} - 152531 \nu^{15} - 819456 \nu^{13} - 2505428 \nu^{11} - 4382125 \nu^{9} - 4207528 \nu^{7} - 2033120 \nu^{5} - 434513 \nu^{3} - 28512 \nu \)\()/538\)
\(\beta_{18}\)\(=\)\((\)\( -1767 \nu^{18} - 43273 \nu^{16} - 431250 \nu^{14} - 2248272 \nu^{12} - 6554317 \nu^{10} - 10567846 \nu^{8} - 8704666 \nu^{6} - 3034215 \nu^{4} - 311854 \nu^{2} - 460 \)\()/1076\)
\(\beta_{19}\)\(=\)\((\)\( 1317 \nu^{19} + 32698 \nu^{17} + 332179 \nu^{15} + 1781218 \nu^{13} + 5425363 \nu^{11} + 9417307 \nu^{9} + 8896864 \nu^{7} + 4138215 \nu^{5} + 803421 \nu^{3} + 40090 \nu \)\()/1076\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 3\)
\(\nu^{3}\)\(=\)\(-\beta_{7} + \beta_{6} - 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{18} + \beta_{16} - \beta_{12} + \beta_{5} - 7 \beta_{2} + 14\)
\(\nu^{5}\)\(=\)\(-\beta_{19} - 2 \beta_{17} + \beta_{13} + \beta_{11} - \beta_{10} + 7 \beta_{7} - 8 \beta_{6} + \beta_{4} + 20 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-10 \beta_{18} - 9 \beta_{16} - 2 \beta_{13} + 8 \beta_{12} + 2 \beta_{11} - 8 \beta_{5} + 43 \beta_{2} - 72\)
\(\nu^{7}\)\(=\)\(12 \beta_{19} + 24 \beta_{17} + \beta_{15} + \beta_{14} - 11 \beta_{13} - 11 \beta_{11} + 10 \beta_{10} - 41 \beta_{7} + 54 \beta_{6} - 12 \beta_{4} - \beta_{3} - 111 \beta_{1}\)
\(\nu^{8}\)\(=\)\(75 \beta_{18} + 66 \beta_{16} + 2 \beta_{15} - 2 \beta_{14} + 24 \beta_{13} - 51 \beta_{12} - 24 \beta_{11} + 3 \beta_{8} + 52 \beta_{5} - 257 \beta_{2} + 389\)
\(\nu^{9}\)\(=\)\(-101 \beta_{19} - 206 \beta_{17} - 15 \beta_{15} - 15 \beta_{14} + 88 \beta_{13} + 88 \beta_{11} - 75 \beta_{10} + 226 \beta_{7} - 350 \beta_{6} + 102 \beta_{4} + 20 \beta_{3} + 651 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-509 \beta_{18} - 455 \beta_{16} - 33 \beta_{15} + 33 \beta_{14} - 206 \beta_{13} + 303 \beta_{12} + 206 \beta_{11} + \beta_{9} - 47 \beta_{8} - 324 \beta_{5} + 1533 \beta_{2} - 2175\)
\(\nu^{11}\)\(=\)\(741 \beta_{19} + 1554 \beta_{17} + 151 \beta_{15} + 151 \beta_{14} - 631 \beta_{13} - 631 \beta_{11} + 510 \beta_{10} - 1211 \beta_{7} + 2241 \beta_{6} - 764 \beta_{4} - 238 \beta_{3} - 3940 \beta_{1}\)
\(\nu^{12}\)\(=\)\(3303 \beta_{18} + 3054 \beta_{16} + 359 \beta_{15} - 359 \beta_{14} + 1554 \beta_{13} - 1747 \beta_{12} - 1554 \beta_{11} - 23 \beta_{9} + 492 \beta_{8} + 2010 \beta_{5} - 9190 \beta_{2} + 12486\)
\(\nu^{13}\)\(=\)\(-5088 \beta_{19} - 11001 \beta_{17} - 1282 \beta_{15} - 1282 \beta_{14} + 4316 \beta_{13} + 4316 \beta_{11} - 3326 \beta_{10} + 6386 \beta_{7} - 14292 \beta_{6} + 5397 \beta_{4} + 2261 \beta_{3} + 24301 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-20961 \beta_{18} - 20205 \beta_{16} - 3251 \beta_{15} + 3251 \beta_{14} - 11001 \beta_{13} + 9919 \beta_{12} + 11001 \beta_{11} + 309 \beta_{9} - 4332 \beta_{8} - 12530 \beta_{5} + 55455 \beta_{2} - 73183\)
\(\nu^{15}\)\(=\)\(33724 \beta_{19} + 75175 \beta_{17} + 9936 \beta_{15} + 9936 \beta_{14} - 28837 \beta_{13} - 28837 \beta_{11} + 21270 \beta_{10} - 33281 \beta_{7} + 91034 \beta_{6} - 36953 \beta_{4} - 19008 \beta_{3} - 151663 \beta_{1}\)
\(\nu^{16}\)\(=\)\(131518 \beta_{18} + 132485 \beta_{16} + 26575 \beta_{15} - 26575 \beta_{14} + 75175 \beta_{13} - 55820 \beta_{12} - 75175 \beta_{11} - 3229 \beta_{9} + 34691 \beta_{8} + 78580 \beta_{5} - 336819 \beta_{2} + 436023\)
\(\nu^{17}\)\(=\)\(-219147 \beta_{19} - 502949 \beta_{17} - 72913 \beta_{15} - 72913 \beta_{14} + 190159 \beta_{13} + 190159 \beta_{11} - 134747 \beta_{10} + 171462 \beta_{7} - 579693 \beta_{6} + 248365 \beta_{4} + 148300 \beta_{3} + 953706 \beta_{1}\)
\(\nu^{18}\)\(=\)\(-820724 \beta_{18} - 863495 \beta_{16} - 203712 \beta_{15} + 203712 \beta_{14} - 502949 \beta_{13} + 312428 \beta_{12} + 502949 \beta_{11} + 29218 \beta_{9} - 261918 \beta_{8} - 495293 \beta_{5} + 2058073 \beta_{2} - 2631488\)
\(\nu^{19}\)\(=\)\(1408073 \beta_{19} + 3320237 \beta_{17} + 516502 \beta_{15} + 516502 \beta_{14} - 1243813 \beta_{13} - 1243813 \beta_{11} + 849942 \beta_{10} - 871422 \beta_{7} + 3691782 \beta_{6} - 1649374 \beta_{4} - 1101444 \beta_{3} - 6026704 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1045\mathbb{Z}\right)^\times\).

\(n\) \(496\) \(761\) \(837\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
419.1
2.52711i
2.33380i
2.04146i
2.00304i
1.51095i
1.23709i
1.22468i
0.712399i
0.386431i
0.131596i
0.131596i
0.386431i
0.712399i
1.22468i
1.23709i
1.51095i
2.00304i
2.04146i
2.33380i
2.52711i
2.52711i 0.473551i −4.38628 2.21995 0.267979i −1.19672 1.17543i 6.03039i 2.77575 −0.677212 5.61006i
419.2 2.33380i 1.97214i −3.44661 −1.75107 1.39059i −4.60257 4.79282i 3.37609i −0.889329 −3.24536 + 4.08665i
419.3 2.04146i 1.84076i −2.16755 −2.00596 0.987985i 3.75784 2.14151i 0.342047i −0.388411 −2.01693 + 4.09509i
419.4 2.00304i 2.62978i −2.01218 0.741511 + 2.10954i −5.26757 4.24477i 0.0243959i −3.91575 4.22550 1.48528i
419.5 1.51095i 0.900785i −0.282982 0.672383 2.13258i 1.36104 0.603482i 2.59434i 2.18859 −3.22223 1.01594i
419.6 1.23709i 2.07277i 0.469598 2.13362 0.669071i −2.56421 0.314194i 3.05513i −1.29637 −0.827703 2.63949i
419.7 1.22468i 1.51450i 0.500160 −1.12399 + 1.93304i 1.85478 0.966830i 3.06189i 0.706295 2.36736 + 1.37652i
419.8 0.712399i 3.32227i 1.49249 1.31011 + 1.81207i 2.36678 2.70450i 2.48804i −8.03747 1.29092 0.933320i
419.9 0.386431i 0.261531i 1.85067 0.0276061 2.23590i 0.101064 4.13791i 1.48802i 2.93160 −0.864019 0.0106678i
419.10 0.131596i 1.44045i 1.98268 −2.22416 0.230433i 0.189557 0.456875i 0.524103i 0.925103 −0.0303239 + 0.292690i
419.11 0.131596i 1.44045i 1.98268 −2.22416 + 0.230433i 0.189557 0.456875i 0.524103i 0.925103 −0.0303239 0.292690i
419.12 0.386431i 0.261531i 1.85067 0.0276061 + 2.23590i 0.101064 4.13791i 1.48802i 2.93160 −0.864019 + 0.0106678i
419.13 0.712399i 3.32227i 1.49249 1.31011 1.81207i 2.36678 2.70450i 2.48804i −8.03747 1.29092 + 0.933320i
419.14 1.22468i 1.51450i 0.500160 −1.12399 1.93304i 1.85478 0.966830i 3.06189i 0.706295 2.36736 1.37652i
419.15 1.23709i 2.07277i 0.469598 2.13362 + 0.669071i −2.56421 0.314194i 3.05513i −1.29637 −0.827703 + 2.63949i
419.16 1.51095i 0.900785i −0.282982 0.672383 + 2.13258i 1.36104 0.603482i 2.59434i 2.18859 −3.22223 + 1.01594i
419.17 2.00304i 2.62978i −2.01218 0.741511 2.10954i −5.26757 4.24477i 0.0243959i −3.91575 4.22550 + 1.48528i
419.18 2.04146i 1.84076i −2.16755 −2.00596 + 0.987985i 3.75784 2.14151i 0.342047i −0.388411 −2.01693 4.09509i
419.19 2.33380i 1.97214i −3.44661 −1.75107 + 1.39059i −4.60257 4.79282i 3.37609i −0.889329 −3.24536 4.08665i
419.20 2.52711i 0.473551i −4.38628 2.21995 + 0.267979i −1.19672 1.17543i 6.03039i 2.77575 −0.677212 + 5.61006i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 419.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1045.2.b.c 20
5.b even 2 1 inner 1045.2.b.c 20
5.c odd 4 2 5225.2.a.ba 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.b.c 20 1.a even 1 1 trivial
1045.2.b.c 20 5.b even 2 1 inner
5225.2.a.ba 20 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{20} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(1045, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + 276 T^{2} + 2753 T^{4} + 9095 T^{6} + 13894 T^{8} + 11551 T^{10} + 5623 T^{12} + 1640 T^{14} + 281 T^{16} + 26 T^{18} + T^{20} \)
$3$ \( 256 + 5696 T^{2} + 33184 T^{4} + 72805 T^{6} + 78507 T^{8} + 47677 T^{10} + 17342 T^{12} + 3833 T^{14} + 500 T^{16} + 35 T^{18} + T^{20} \)
$5$ \( 9765625 - 390625 T^{2} + 468750 T^{3} - 234375 T^{4} - 243750 T^{5} - 12500 T^{6} + 5750 T^{7} - 4650 T^{8} + 4850 T^{9} + 3658 T^{10} + 970 T^{11} - 186 T^{12} + 46 T^{13} - 20 T^{14} - 78 T^{15} - 15 T^{16} + 6 T^{17} - T^{18} + T^{20} \)
$7$ \( 2304 + 46016 T^{2} + 304176 T^{4} + 856532 T^{6} + 1099880 T^{8} + 670233 T^{10} + 194005 T^{12} + 28024 T^{14} + 2053 T^{16} + 73 T^{18} + T^{20} \)
$11$ \( ( -1 + T )^{20} \)
$13$ \( 33732864 + 632022464 T^{2} + 1203719920 T^{4} + 739024900 T^{6} + 218996548 T^{8} + 36371889 T^{10} + 3599588 T^{12} + 214755 T^{14} + 7471 T^{16} + 137 T^{18} + T^{20} \)
$17$ \( 7054336 + 197432576 T^{2} + 698557056 T^{4} + 713687508 T^{6} + 285942072 T^{8} + 54391641 T^{10} + 5502316 T^{12} + 311179 T^{14} + 9767 T^{16} + 157 T^{18} + T^{20} \)
$19$ \( ( -1 + T )^{20} \)
$23$ \( 172764736 + 3101339248 T^{2} + 12311375020 T^{4} + 10039877237 T^{6} + 2685610562 T^{8} + 340694984 T^{10} + 23320564 T^{12} + 900965 T^{14} + 19519 T^{16} + 220 T^{18} + T^{20} \)
$29$ \( ( 4194192 + 1058552 T - 1402100 T^{2} - 194274 T^{3} + 179116 T^{4} + 3347 T^{5} - 9816 T^{6} + 773 T^{7} + 153 T^{8} - 25 T^{9} + T^{10} )^{2} \)
$31$ \( ( -7456 - 111504 T - 49976 T^{2} + 161615 T^{3} + 104238 T^{4} - 17956 T^{5} - 13575 T^{6} - 1350 T^{7} + 124 T^{8} + 25 T^{9} + T^{10} )^{2} \)
$37$ \( 61309721664 + 216751516976 T^{2} + 278071511884 T^{4} + 159108327209 T^{6} + 41465234119 T^{8} + 4495023337 T^{10} + 219561439 T^{12} + 5307491 T^{14} + 66317 T^{16} + 411 T^{18} + T^{20} \)
$41$ \( ( 166848 + 1861184 T + 3177512 T^{2} + 1952284 T^{3} + 487094 T^{4} + 15085 T^{5} - 15327 T^{6} - 2515 T^{7} - 44 T^{8} + 17 T^{9} + T^{10} )^{2} \)
$43$ \( 7783474176 + 1165458322688 T^{2} + 1929486894720 T^{4} + 589283731872 T^{6} + 76704661796 T^{8} + 5210165145 T^{10} + 200888495 T^{12} + 4504430 T^{14} + 57337 T^{16} + 379 T^{18} + T^{20} \)
$47$ \( 47457277067935744 + 15145302020294656 T^{2} + 1883937049449536 T^{4} + 125044907626096 T^{6} + 5012521307372 T^{8} + 128479792173 T^{10} + 2150767840 T^{12} + 23357469 T^{14} + 158220 T^{16} + 606 T^{18} + T^{20} \)
$53$ \( 58092830810407936 + 16918646609043712 T^{2} + 2037460361054080 T^{4} + 134794472735904 T^{6} + 5455102457380 T^{8} + 141577353693 T^{10} + 2390610524 T^{12} + 25947019 T^{14} + 173159 T^{16} + 641 T^{18} + T^{20} \)
$59$ \( ( 86423136 - 74502304 T - 16109652 T^{2} + 12168583 T^{3} + 637538 T^{4} - 517674 T^{5} + 18053 T^{6} + 5204 T^{7} - 296 T^{8} - 15 T^{9} + T^{10} )^{2} \)
$61$ \( ( -26701648 + 15671464 T + 4573252 T^{2} - 2030406 T^{3} - 379232 T^{4} + 83033 T^{5} + 13690 T^{6} - 1311 T^{7} - 203 T^{8} + 7 T^{9} + T^{10} )^{2} \)
$67$ \( 345472702914816 + 520408067604800 T^{2} + 260091159602176 T^{4} + 50523697290397 T^{6} + 4447439480336 T^{8} + 190375580699 T^{10} + 4146259377 T^{12} + 47228392 T^{14} + 283664 T^{16} + 850 T^{18} + T^{20} \)
$71$ \( ( -400032 + 6412880 T - 3032032 T^{2} - 1455235 T^{3} + 345152 T^{4} + 126732 T^{5} - 4066 T^{6} - 2929 T^{7} - 73 T^{8} + 20 T^{9} + T^{10} )^{2} \)
$73$ \( 27513477920784384 + 16215000063352832 T^{2} + 3712915646425344 T^{4} + 416354004159040 T^{6} + 23689439568384 T^{8} + 661341488853 T^{10} + 9770191054 T^{12} + 81273128 T^{14} + 384017 T^{16} + 964 T^{18} + T^{20} \)
$79$ \( ( -204893824 + 51728544 T + 30845000 T^{2} - 9602354 T^{3} - 883924 T^{4} + 509451 T^{5} - 31091 T^{6} - 6084 T^{7} + 990 T^{8} - 53 T^{9} + T^{10} )^{2} \)
$83$ \( 5917659525376 + 8357630438592 T^{2} + 4250865733872 T^{4} + 1009056635844 T^{6} + 123681216676 T^{8} + 8166843033 T^{10} + 301122568 T^{12} + 6263439 T^{14} + 72051 T^{16} + 425 T^{18} + T^{20} \)
$89$ \( ( 14189256 - 11234524 T - 24863770 T^{2} + 11072527 T^{3} + 97000 T^{4} - 397375 T^{5} + 15271 T^{6} + 4720 T^{7} - 234 T^{8} - 18 T^{9} + T^{10} )^{2} \)
$97$ \( 19285658703936 + 210128144766704 T^{2} + 391339491891292 T^{4} + 192044109078861 T^{6} + 24790141231190 T^{8} + 1030245989845 T^{10} + 17595489615 T^{12} + 144667532 T^{14} + 606332 T^{16} + 1248 T^{18} + T^{20} \)
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