Properties

Label 273.2.i.e
Level $273$
Weight $2$
Character orbit 273.i
Analytic conductor $2.180$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - x^{9} + 7 x^{8} - 8 x^{7} + 41 x^{6} - 40 x^{5} + 59 x^{4} - 10 x^{3} + 18 x^{2} - 4 x + 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - x^{9} + 7 x^{8} - 8 x^{7} + 41 x^{6} - 40 x^{5} + 59 x^{4} - 10 x^{3} + 18 x^{2} - 4 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2640 \nu^{9} + 11641 \nu^{8} + 15246 \nu^{7} + 63880 \nu^{6} + 75479 \nu^{5} + 368907 \nu^{4} + 65127 \nu^{3} + 240781 \nu^{2} + 615246 \nu + 66894 \)\()/101089\)
\(\beta_{2}\)\(=\)\((\)\( -14281 \nu^{9} + 3234 \nu^{8} - 85000 \nu^{7} + 32761 \nu^{6} - 474507 \nu^{5} + 90633 \nu^{4} - 267181 \nu^{3} - 668815 \nu^{2} - 77454 \nu + 10560 \)\()/202178\)
\(\beta_{3}\)\(=\)\((\)\( 8527 \nu^{9} - 12294 \nu^{8} + 66934 \nu^{7} - 107661 \nu^{6} + 394774 \nu^{5} - 573725 \nu^{4} + 781623 \nu^{3} - 652973 \nu^{2} + 240426 \nu - 56648 \)\()/101089\)
\(\beta_{4}\)\(=\)\((\)\( 19323 \nu^{9} + 15246 \nu^{8} + 119172 \nu^{7} + 67797 \nu^{6} + 622413 \nu^{5} + 485035 \nu^{4} + 386739 \nu^{3} + 1092753 \nu^{2} + 559102 \nu + 309726 \)\()/202178\)
\(\beta_{5}\)\(=\)\((\)\( -19323 \nu^{9} - 15246 \nu^{8} - 119172 \nu^{7} - 67797 \nu^{6} - 622413 \nu^{5} - 485035 \nu^{4} - 386739 \nu^{3} - 1092753 \nu^{2} - 154746 \nu - 309726 \)\()/202178\)
\(\beta_{6}\)\(=\)\((\)\( -18327 \nu^{9} + 22689 \nu^{8} - 123529 \nu^{7} + 176861 \nu^{6} - 728569 \nu^{5} + 872266 \nu^{4} - 954458 \nu^{3} + 272268 \nu^{2} - 70588 \nu - 53822 \)\()/101089\)
\(\beta_{7}\)\(=\)\((\)\( -37803 \nu^{9} + 4356 \nu^{8} - 225894 \nu^{7} + 91577 \nu^{6} - 1251855 \nu^{5} + 268553 \nu^{4} - 741539 \nu^{3} - 857529 \nu^{2} - 215730 \nu + 30728 \)\()/202178\)
\(\beta_{8}\)\(=\)\((\)\( -20287 \nu^{9} + 24768 \nu^{8} - 134848 \nu^{7} + 190701 \nu^{6} - 795328 \nu^{5} + 952192 \nu^{4} - 989025 \nu^{3} + 297216 \nu^{2} - 77056 \nu + 227351 \)\()/101089\)
\(\beta_{9}\)\(=\)\((\)\( 45021 \nu^{9} - 43061 \nu^{8} + 313068 \nu^{7} - 348849 \nu^{6} + 1832021 \nu^{5} - 1734081 \nu^{4} + 2576313 \nu^{3} - 415643 \nu^{2} + 785430 \nu - 173616 \)\()/101089\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + \beta_{4}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{8} - \beta_{4} + \beta_{3} - 3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(-\beta_{9} - 3 \beta_{8} + 2 \beta_{7} - 2 \beta_{5} + \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-6 \beta_{6} + \beta_{5} + 6 \beta_{4} - 6 \beta_{3} + 15 \beta_{2} - 15\)
\(\nu^{5}\)\(=\)\(6 \beta_{9} + 18 \beta_{8} - 10 \beta_{7} - 18 \beta_{4} + \beta_{3} - 10 \beta_{2}\)
\(\nu^{6}\)\(=\)\(-\beta_{9} - 36 \beta_{8} + 9 \beta_{7} + 33 \beta_{6} - 9 \beta_{5} + \beta_{1} + 81\)
\(\nu^{7}\)\(=\)\(-13 \beta_{6} + 54 \beta_{5} + 106 \beta_{4} - 13 \beta_{3} + 76 \beta_{2} - 33 \beta_{1} - 76\)
\(\nu^{8}\)\(=\)\(13 \beta_{9} + 218 \beta_{8} - 64 \beta_{7} - 218 \beta_{4} + 180 \beta_{3} - 448 \beta_{2}\)
\(\nu^{9}\)\(=\)\(-180 \beta_{9} - 622 \beta_{8} + 301 \beta_{7} + 115 \beta_{6} - 301 \beta_{5} + 180 \beta_{1} + 525\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.253637 + 0.439313i
1.08681 + 1.88241i
−0.281188 0.487032i
0.660865 + 1.14465i
−1.22012 2.11332i
0.253637 0.439313i
1.08681 1.88241i
−0.281188 + 0.487032i
0.660865 1.14465i
−1.22012 + 2.11332i
−1.26840 + 2.19693i 0.500000 + 0.866025i −2.21768 3.84114i 1.26113 2.18434i −2.53680 1.47427 2.19693i 6.17804 −0.500000 + 0.866025i 3.19923 + 5.54123i
79.2 −0.660777 + 1.14450i 0.500000 + 0.866025i 0.126747 + 0.219533i −1.01284 + 1.75429i −1.32155 −2.38540 1.14450i −2.97812 −0.500000 + 0.866025i −1.33853 2.31839i
79.3 −0.0388377 + 0.0672688i 0.500000 + 0.866025i 0.996983 + 1.72683i 1.10121 1.90736i −0.0776754 2.64490 0.0672688i −0.310233 −0.500000 + 0.866025i 0.0855372 + 0.148155i
79.4 0.893230 1.54712i 0.500000 + 0.866025i −0.595718 1.03181i −1.71496 + 2.97040i 1.78646 2.14626 + 1.54712i 1.44447 −0.500000 + 0.866025i 3.06371 + 5.30650i
79.5 1.07479 1.86158i 0.500000 + 0.866025i −1.31033 2.26956i 1.86546 3.23108i 2.14957 −1.88003 + 1.86158i −1.33415 −0.500000 + 0.866025i −4.00995 6.94543i
235.1 −1.26840 2.19693i 0.500000 0.866025i −2.21768 + 3.84114i 1.26113 + 2.18434i −2.53680 1.47427 + 2.19693i 6.17804 −0.500000 0.866025i 3.19923 5.54123i
235.2 −0.660777 1.14450i 0.500000 0.866025i 0.126747 0.219533i −1.01284 1.75429i −1.32155 −2.38540 + 1.14450i −2.97812 −0.500000 0.866025i −1.33853 + 2.31839i
235.3 −0.0388377 0.0672688i 0.500000 0.866025i 0.996983 1.72683i 1.10121 + 1.90736i −0.0776754 2.64490 + 0.0672688i −0.310233 −0.500000 0.866025i 0.0855372 0.148155i
235.4 0.893230 + 1.54712i 0.500000 0.866025i −0.595718 + 1.03181i −1.71496 2.97040i 1.78646 2.14626 1.54712i 1.44447 −0.500000 0.866025i 3.06371 5.30650i
235.5 1.07479 + 1.86158i 0.500000 0.866025i −1.31033 + 2.26956i 1.86546 + 3.23108i 2.14957 −1.88003 1.86158i −1.33415 −0.500000 0.866025i −4.00995 + 6.94543i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 235.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.i.e 10
3.b odd 2 1 819.2.j.g 10
7.c even 3 1 inner 273.2.i.e 10
7.c even 3 1 1911.2.a.t 5
7.d odd 6 1 1911.2.a.u 5
21.g even 6 1 5733.2.a.bp 5
21.h odd 6 1 819.2.j.g 10
21.h odd 6 1 5733.2.a.bq 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.i.e 10 1.a even 1 1 trivial
273.2.i.e 10 7.c even 3 1 inner
819.2.j.g 10 3.b odd 2 1
819.2.j.g 10 21.h odd 6 1
1911.2.a.t 5 7.c even 3 1
1911.2.a.u 5 7.d odd 6 1
5733.2.a.bp 5 21.g even 6 1
5733.2.a.bq 5 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{10} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 13 T + 168 T^{2} + 29 T^{3} + 105 T^{4} - 7 T^{5} + 51 T^{6} - 2 T^{7} + 8 T^{8} + T^{10} \)
$3$ \( ( 1 - T + T^{2} )^{5} \)
$5$ \( 20736 - 6912 T + 9072 T^{2} - 2352 T^{3} + 2545 T^{4} - 608 T^{5} + 349 T^{6} - 46 T^{7} + 25 T^{8} - 3 T^{9} + T^{10} \)
$7$ \( 16807 - 9604 T - 1029 T^{2} + 1519 T^{3} + 245 T^{4} - 375 T^{5} + 35 T^{6} + 31 T^{7} - 3 T^{8} - 4 T^{9} + T^{10} \)
$11$ \( 1 - 12 T + 182 T^{2} + 502 T^{3} + 1169 T^{4} + 851 T^{5} + 503 T^{6} + 99 T^{7} + 24 T^{8} - T^{9} + T^{10} \)
$13$ \( ( 1 + T )^{10} \)
$17$ \( 1 + 22 T + 566 T^{2} - 1914 T^{3} + 5527 T^{4} - 3937 T^{5} + 1937 T^{6} - 551 T^{7} + 114 T^{8} - 13 T^{9} + T^{10} \)
$19$ \( 361 + 2641 T + 21335 T^{2} - 14962 T^{3} + 10269 T^{4} - 2601 T^{5} + 917 T^{6} - 170 T^{7} + 55 T^{8} - 7 T^{9} + T^{10} \)
$23$ \( 1227664 - 452064 T + 324908 T^{2} - 45808 T^{3} + 35193 T^{4} - 4565 T^{5} + 2373 T^{6} - 98 T^{7} + 63 T^{8} - 4 T^{9} + T^{10} \)
$29$ \( ( -251 - 537 T - 231 T^{2} + 2 T^{3} + 12 T^{4} + T^{5} )^{2} \)
$31$ \( 1893376 - 165120 T + 359776 T^{2} - 77208 T^{3} + 59425 T^{4} - 9725 T^{5} + 2907 T^{6} - 268 T^{7} + 75 T^{8} - 6 T^{9} + T^{10} \)
$37$ \( 55591936 + 42051840 T + 21736544 T^{2} + 6516152 T^{3} + 1489857 T^{4} + 216598 T^{5} + 25977 T^{6} + 1888 T^{7} + 195 T^{8} + 11 T^{9} + T^{10} \)
$41$ \( ( 224 - 392 T - 279 T^{2} - 13 T^{3} + 10 T^{4} + T^{5} )^{2} \)
$43$ \( ( -76 - 688 T + 413 T^{2} - 37 T^{3} - 10 T^{4} + T^{5} )^{2} \)
$47$ \( 338265664 - 138528544 T + 53457248 T^{2} - 7630760 T^{3} + 1393224 T^{4} - 48210 T^{5} + 22421 T^{6} - 328 T^{7} + 187 T^{8} + 4 T^{9} + T^{10} \)
$53$ \( 7060872841 - 732900938 T + 240770124 T^{2} - 18029002 T^{3} + 4908237 T^{4} - 336673 T^{5} + 52599 T^{6} - 2039 T^{7} + 290 T^{8} - 9 T^{9} + T^{10} \)
$59$ \( 5156245249 - 825349658 T + 232641836 T^{2} - 16221550 T^{3} + 4043501 T^{4} - 225891 T^{5} + 48931 T^{6} - 1225 T^{7} + 274 T^{8} - 7 T^{9} + T^{10} \)
$61$ \( 1104601 - 1582806 T + 1423032 T^{2} - 792526 T^{3} + 322549 T^{4} - 91771 T^{5} + 19603 T^{6} - 2969 T^{7} + 330 T^{8} - 23 T^{9} + T^{10} \)
$67$ \( 127757809 + 48828960 T + 18345916 T^{2} + 4122222 T^{3} + 1047999 T^{4} + 199741 T^{5} + 36349 T^{6} + 4481 T^{7} + 448 T^{8} + 25 T^{9} + T^{10} \)
$71$ \( ( 761 - 1624 T + 206 T^{2} + 205 T^{3} + 27 T^{4} + T^{5} )^{2} \)
$73$ \( 55771024 - 31963040 T + 14054172 T^{2} - 3354976 T^{3} + 721545 T^{4} - 111781 T^{5} + 18279 T^{6} - 2240 T^{7} + 263 T^{8} - 18 T^{9} + T^{10} \)
$79$ \( 344919184 - 213355136 T + 112306396 T^{2} - 20671768 T^{3} + 3900809 T^{4} + 40131 T^{5} + 49425 T^{6} + 286 T^{7} + 293 T^{8} + 8 T^{9} + T^{10} \)
$83$ \( ( 17248 - 2856 T - 2103 T^{2} - 145 T^{3} + 12 T^{4} + T^{5} )^{2} \)
$89$ \( 4596736 - 6397696 T + 5857632 T^{2} - 2945288 T^{3} + 1055897 T^{4} - 258214 T^{5} + 47011 T^{6} - 5916 T^{7} + 539 T^{8} - 29 T^{9} + T^{10} \)
$97$ \( ( 14308 - 6804 T - 3039 T^{2} - 186 T^{3} + 13 T^{4} + T^{5} )^{2} \)
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