Properties

Label 441.2.f.e
Level $441$
Weight $2$
Character orbit 441.f
Analytic conductor $3.521$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.991381711347.1
Defining polynomial: \(x^{10} - 2 x^{9} + 9 x^{8} - 8 x^{7} + 40 x^{6} - 36 x^{5} + 90 x^{4} - 3 x^{3} + 36 x^{2} - 9 x + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 63)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} + 9 x^{8} - 8 x^{7} + 40 x^{6} - 36 x^{5} + 90 x^{4} - 3 x^{3} + 36 x^{2} - 9 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{9} + 9 \nu^{8} - 3 \nu^{7} + 95 \nu^{6} + 18 \nu^{5} + 402 \nu^{4} - 87 \nu^{3} + 936 \nu^{2} + 342 \nu + 72 \)\()/189\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{9} + \nu^{8} - 12 \nu^{7} - 8 \nu^{6} - 68 \nu^{5} - 30 \nu^{4} - 123 \nu^{3} - 204 \nu^{2} - 270 \nu - 63 \)\()/63\)
\(\beta_{4}\)\(=\)\((\)\( 17 \nu^{9} - 24 \nu^{8} + 159 \nu^{7} - 106 \nu^{6} + 786 \nu^{5} - 417 \nu^{4} + 1893 \nu^{3} - 27 \nu^{2} + 1395 \nu + 639 \)\()/567\)
\(\beta_{5}\)\(=\)\((\)\( 16 \nu^{9} - 39 \nu^{8} + 156 \nu^{7} - 176 \nu^{6} + 663 \nu^{5} - 780 \nu^{4} + 1680 \nu^{3} - 351 \nu^{2} + 684 \nu - 180 \)\()/567\)
\(\beta_{6}\)\(=\)\((\)\( 20 \nu^{9} - 24 \nu^{8} + 141 \nu^{7} - 4 \nu^{6} + 624 \nu^{5} - 57 \nu^{4} + 1020 \nu^{3} + 1620 \nu^{2} + 369 \nu + 504 \)\()/567\)
\(\beta_{7}\)\(=\)\((\)\( 8 \nu^{9} - 12 \nu^{8} + 69 \nu^{7} - 43 \nu^{6} + 330 \nu^{5} - 219 \nu^{4} + 732 \nu^{3} - 45 \nu^{2} + 477 \nu - 306 \)\()/189\)
\(\beta_{8}\)\(=\)\((\)\( -71 \nu^{9} + 123 \nu^{8} - 591 \nu^{7} + 403 \nu^{6} - 2604 \nu^{5} + 1794 \nu^{4} - 5214 \nu^{3} - 1458 \nu^{2} - 1476 \nu - 234 \)\()/567\)
\(\beta_{9}\)\(=\)\((\)\( -82 \nu^{9} + 165 \nu^{8} - 732 \nu^{7} + 632 \nu^{6} - 3264 \nu^{5} + 2850 \nu^{4} - 7260 \nu^{3} - 432 \nu^{2} - 2898 \nu + 720 \)\()/567\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} + 3 \beta_{6} + \beta_{4} - \beta_{2} - 3\)
\(\nu^{3}\)\(=\)\(\beta_{9} + 4 \beta_{5} - \beta_{3} - 4 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(-5 \beta_{8} - 5 \beta_{7} - 13 \beta_{6} - \beta_{4} - \beta_{3} + 4 \beta_{2} - \beta_{1}\)
\(\nu^{5}\)\(=\)\(-7 \beta_{9} + \beta_{8} - 2 \beta_{7} - 7 \beta_{6} - 19 \beta_{5} - \beta_{4} + \beta_{2} + 7\)
\(\nu^{6}\)\(=\)\(-10 \beta_{9} + 9 \beta_{8} + 15 \beta_{7} - 10 \beta_{5} - 15 \beta_{4} + 10 \beta_{3} + 9 \beta_{2} + 10 \beta_{1} + 61\)
\(\nu^{7}\)\(=\)\(11 \beta_{8} + 11 \beta_{7} + 46 \beta_{6} + 19 \beta_{4} + 43 \beta_{3} + 8 \beta_{2} + 94 \beta_{1}\)
\(\nu^{8}\)\(=\)\(73 \beta_{9} + 56 \beta_{8} + 62 \beta_{7} + 298 \beta_{6} + 76 \beta_{5} + 118 \beta_{4} - 118 \beta_{2} - 298\)
\(\nu^{9}\)\(=\)\(253 \beta_{9} - 135 \beta_{8} + 48 \beta_{7} + 478 \beta_{5} - 48 \beta_{4} - 253 \beta_{3} - 135 \beta_{2} - 478 \beta_{1} - 295\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(1\) \(-1 + \beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
148.1
−1.02682 1.77851i
−0.335166 0.580525i
0.247934 + 0.429435i
0.920620 + 1.59456i
1.19343 + 2.06709i
−1.02682 + 1.77851i
−0.335166 + 0.580525i
0.247934 0.429435i
0.920620 1.59456i
1.19343 2.06709i
−1.02682 1.77851i 0.608729 1.62156i −1.10873 + 1.92038i 0.0731228 0.126652i −3.50901 + 0.582422i 0 0.446582 −2.25890 1.97418i −0.300337
148.2 −0.335166 0.580525i −1.27533 + 1.17198i 0.775327 1.34291i −0.712469 + 1.23403i 1.10781 + 0.347551i 0 −2.38012 0.252918 2.98932i 0.955182
148.3 0.247934 + 0.429435i −1.37706 1.05058i 0.877057 1.51911i 1.84629 3.19787i 0.109735 0.851830i 0 1.86155 0.792574 + 2.89341i 1.83103
148.4 0.920620 + 1.59456i 0.195084 + 1.72103i −0.695084 + 1.20392i −0.667377 + 1.15593i −2.56469 + 1.89549i 0 1.12285 −2.92388 + 0.671489i −2.45760
148.5 1.19343 + 2.06709i 1.34857 1.08690i −1.84857 + 3.20182i 1.46043 2.52954i 3.85615 + 1.49047i 0 −4.05086 0.637290 2.93153i 6.97172
295.1 −1.02682 + 1.77851i 0.608729 + 1.62156i −1.10873 1.92038i 0.0731228 + 0.126652i −3.50901 0.582422i 0 0.446582 −2.25890 + 1.97418i −0.300337
295.2 −0.335166 + 0.580525i −1.27533 1.17198i 0.775327 + 1.34291i −0.712469 1.23403i 1.10781 0.347551i 0 −2.38012 0.252918 + 2.98932i 0.955182
295.3 0.247934 0.429435i −1.37706 + 1.05058i 0.877057 + 1.51911i 1.84629 + 3.19787i 0.109735 + 0.851830i 0 1.86155 0.792574 2.89341i 1.83103
295.4 0.920620 1.59456i 0.195084 1.72103i −0.695084 1.20392i −0.667377 1.15593i −2.56469 1.89549i 0 1.12285 −2.92388 0.671489i −2.45760
295.5 1.19343 2.06709i 1.34857 + 1.08690i −1.84857 3.20182i 1.46043 + 2.52954i 3.85615 1.49047i 0 −4.05086 0.637290 + 2.93153i 6.97172
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 295.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.f.e 10
3.b odd 2 1 1323.2.f.e 10
7.b odd 2 1 441.2.f.f 10
7.c even 3 1 63.2.g.b 10
7.c even 3 1 63.2.h.b yes 10
7.d odd 6 1 441.2.g.f 10
7.d odd 6 1 441.2.h.f 10
9.c even 3 1 inner 441.2.f.e 10
9.c even 3 1 3969.2.a.z 5
9.d odd 6 1 1323.2.f.e 10
9.d odd 6 1 3969.2.a.bc 5
21.c even 2 1 1323.2.f.f 10
21.g even 6 1 1323.2.g.f 10
21.g even 6 1 1323.2.h.f 10
21.h odd 6 1 189.2.g.b 10
21.h odd 6 1 189.2.h.b 10
28.g odd 6 1 1008.2.q.i 10
28.g odd 6 1 1008.2.t.i 10
63.g even 3 1 63.2.h.b yes 10
63.g even 3 1 567.2.e.f 10
63.h even 3 1 63.2.g.b 10
63.h even 3 1 567.2.e.f 10
63.i even 6 1 1323.2.g.f 10
63.j odd 6 1 189.2.g.b 10
63.j odd 6 1 567.2.e.e 10
63.k odd 6 1 441.2.h.f 10
63.l odd 6 1 441.2.f.f 10
63.l odd 6 1 3969.2.a.ba 5
63.n odd 6 1 189.2.h.b 10
63.n odd 6 1 567.2.e.e 10
63.o even 6 1 1323.2.f.f 10
63.o even 6 1 3969.2.a.bb 5
63.s even 6 1 1323.2.h.f 10
63.t odd 6 1 441.2.g.f 10
84.n even 6 1 3024.2.q.i 10
84.n even 6 1 3024.2.t.i 10
252.o even 6 1 3024.2.q.i 10
252.u odd 6 1 1008.2.t.i 10
252.bb even 6 1 3024.2.t.i 10
252.bl odd 6 1 1008.2.q.i 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.b 10 7.c even 3 1
63.2.g.b 10 63.h even 3 1
63.2.h.b yes 10 7.c even 3 1
63.2.h.b yes 10 63.g even 3 1
189.2.g.b 10 21.h odd 6 1
189.2.g.b 10 63.j odd 6 1
189.2.h.b 10 21.h odd 6 1
189.2.h.b 10 63.n odd 6 1
441.2.f.e 10 1.a even 1 1 trivial
441.2.f.e 10 9.c even 3 1 inner
441.2.f.f 10 7.b odd 2 1
441.2.f.f 10 63.l odd 6 1
441.2.g.f 10 7.d odd 6 1
441.2.g.f 10 63.t odd 6 1
441.2.h.f 10 7.d odd 6 1
441.2.h.f 10 63.k odd 6 1
567.2.e.e 10 63.j odd 6 1
567.2.e.e 10 63.n odd 6 1
567.2.e.f 10 63.g even 3 1
567.2.e.f 10 63.h even 3 1
1008.2.q.i 10 28.g odd 6 1
1008.2.q.i 10 252.bl odd 6 1
1008.2.t.i 10 28.g odd 6 1
1008.2.t.i 10 252.u odd 6 1
1323.2.f.e 10 3.b odd 2 1
1323.2.f.e 10 9.d odd 6 1
1323.2.f.f 10 21.c even 2 1
1323.2.f.f 10 63.o even 6 1
1323.2.g.f 10 21.g even 6 1
1323.2.g.f 10 63.i even 6 1
1323.2.h.f 10 21.g even 6 1
1323.2.h.f 10 63.s even 6 1
3024.2.q.i 10 84.n even 6 1
3024.2.q.i 10 252.o even 6 1
3024.2.t.i 10 84.n even 6 1
3024.2.t.i 10 252.bb even 6 1
3969.2.a.z 5 9.c even 3 1
3969.2.a.ba 5 63.l odd 6 1
3969.2.a.bb 5 63.o even 6 1
3969.2.a.bc 5 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\):

\(T_{2}^{10} - \cdots\)
\(T_{5}^{10} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T - T^{2} + 4 T^{3} - 2 T^{4} + 2 T^{5} + 6 T^{6} - 21 T^{7} + 6 T^{8} + 13 T^{9} - 5 T^{10} + 26 T^{11} + 24 T^{12} - 168 T^{13} + 96 T^{14} + 64 T^{15} - 128 T^{16} + 512 T^{17} - 256 T^{18} - 1024 T^{19} + 1024 T^{20} \)
$3$ \( 1 + T + 4 T^{2} + 6 T^{3} + 18 T^{4} + 9 T^{5} + 54 T^{6} + 54 T^{7} + 108 T^{8} + 81 T^{9} + 243 T^{10} \)
$5$ \( 1 - 4 T - 4 T^{2} + 44 T^{3} - 41 T^{4} - 119 T^{5} + 222 T^{6} - 456 T^{7} + 1623 T^{8} + 2021 T^{9} - 16541 T^{10} + 10105 T^{11} + 40575 T^{12} - 57000 T^{13} + 138750 T^{14} - 371875 T^{15} - 640625 T^{16} + 3437500 T^{17} - 1562500 T^{18} - 7812500 T^{19} + 9765625 T^{20} \)
$7$ 1
$11$ \( 1 - 4 T - 31 T^{2} + 134 T^{3} + 607 T^{4} - 2492 T^{5} - 8385 T^{6} + 27495 T^{7} + 98940 T^{8} - 135733 T^{9} - 1043873 T^{10} - 1493063 T^{11} + 11971740 T^{12} + 36595845 T^{13} - 122764785 T^{14} - 401339092 T^{15} + 1075337527 T^{16} + 2611280914 T^{17} - 6645125311 T^{18} - 9431790764 T^{19} + 25937424601 T^{20} \)
$13$ \( 1 + 8 T - 14 T^{2} - 182 T^{3} + 686 T^{4} + 4429 T^{5} - 12871 T^{6} - 43199 T^{7} + 305249 T^{8} + 358672 T^{9} - 3841969 T^{10} + 4662736 T^{11} + 51587081 T^{12} - 94908203 T^{13} - 367608631 T^{14} + 1644456697 T^{15} + 3311190974 T^{16} - 11420230094 T^{17} - 11420230094 T^{18} + 84835994984 T^{19} + 137858491849 T^{20} \)
$17$ \( ( 1 + 12 T + 130 T^{2} + 876 T^{3} + 5203 T^{4} + 22839 T^{5} + 88451 T^{6} + 253164 T^{7} + 638690 T^{8} + 1002252 T^{9} + 1419857 T^{10} )^{2} \)
$19$ \( ( 1 + T + 54 T^{2} + 122 T^{3} + 1532 T^{4} + 3483 T^{5} + 29108 T^{6} + 44042 T^{7} + 370386 T^{8} + 130321 T^{9} + 2476099 T^{10} )^{2} \)
$23$ \( 1 - 3 T - 43 T^{2} + 294 T^{3} + 6 T^{4} - 5127 T^{5} + 21792 T^{6} - 135027 T^{7} + 502362 T^{8} + 3271749 T^{9} - 33095343 T^{10} + 75250227 T^{11} + 265749498 T^{12} - 1642873509 T^{13} + 6098295072 T^{14} - 32999130561 T^{15} + 888215334 T^{16} + 1001018681418 T^{17} - 3367372367083 T^{18} - 5403457984389 T^{19} + 41426511213649 T^{20} \)
$29$ \( 1 - 7 T - 76 T^{2} + 419 T^{3} + 4561 T^{4} - 15146 T^{5} - 199563 T^{6} + 341373 T^{7} + 6918636 T^{8} - 2570041 T^{9} - 219913241 T^{10} - 74531189 T^{11} + 5818572876 T^{12} + 8325746097 T^{13} - 141147118203 T^{14} - 310661862754 T^{15} + 2712989167081 T^{16} + 7227698173471 T^{17} - 38018727385036 T^{18} - 101550021831083 T^{19} + 420707233300201 T^{20} \)
$31$ \( 1 + 3 T - 125 T^{2} - 214 T^{3} + 9282 T^{4} + 8387 T^{5} - 503981 T^{6} - 245082 T^{7} + 21459514 T^{8} + 3619498 T^{9} - 734820027 T^{10} + 112204438 T^{11} + 20622592954 T^{12} - 7301237862 T^{13} - 465437037101 T^{14} + 240112689437 T^{15} + 8237809167042 T^{16} - 5887699419754 T^{17} - 106611379680125 T^{18} + 79318866482013 T^{19} + 819628286980801 T^{20} \)
$37$ \( ( 1 + 89 T^{2} + 280 T^{3} + 3418 T^{4} + 20432 T^{5} + 126466 T^{6} + 383320 T^{7} + 4508117 T^{8} + 69343957 T^{10} )^{2} \)
$41$ \( 1 - 5 T - 136 T^{2} + 733 T^{3} + 10507 T^{4} - 54412 T^{5} - 554055 T^{6} + 2345451 T^{7} + 23706084 T^{8} - 41392439 T^{9} - 952045937 T^{10} - 1697089999 T^{11} + 39849927204 T^{12} + 161650828371 T^{13} - 1565627010855 T^{14} - 6303967608812 T^{15} + 49909345260187 T^{16} + 142754882754773 T^{17} - 1085949831160456 T^{18} - 1636909671969805 T^{19} + 13422659310152401 T^{20} \)
$43$ \( 1 + 7 T - 77 T^{2} - 66 T^{3} + 7014 T^{4} - 3843 T^{5} - 95427 T^{6} + 1632678 T^{7} - 3708600 T^{8} - 15416324 T^{9} + 670279801 T^{10} - 662901932 T^{11} - 6857201400 T^{12} + 129809329746 T^{13} - 326245923027 T^{14} - 564953446449 T^{15} + 44338040425686 T^{16} - 17940028333062 T^{17} - 899991421375277 T^{18} + 3518148283557901 T^{19} + 21611482313284249 T^{20} \)
$47$ \( 1 - 27 T + 281 T^{2} - 1758 T^{3} + 13050 T^{4} - 78783 T^{5} - 25248 T^{6} + 1518381 T^{7} + 9454350 T^{8} - 53043051 T^{9} - 242331903 T^{10} - 2493023397 T^{11} + 20884659150 T^{12} + 157642870563 T^{13} - 123202185888 T^{14} - 18068487686481 T^{15} + 140668760043450 T^{16} - 890643445773954 T^{17} + 6690971551954841 T^{18} - 30216522773774709 T^{19} + 52599132235830049 T^{20} \)
$53$ \( ( 1 - 21 T + 400 T^{2} - 4662 T^{3} + 49132 T^{4} - 375771 T^{5} + 2603996 T^{6} - 13095558 T^{7} + 59550800 T^{8} - 165700101 T^{9} + 418195493 T^{10} )^{2} \)
$59$ \( 1 - 30 T + 299 T^{2} - 1644 T^{3} + 26547 T^{4} - 344442 T^{5} + 1635267 T^{6} - 9620487 T^{7} + 170035344 T^{8} - 1056366303 T^{9} + 3109579647 T^{10} - 62325611877 T^{11} + 591893032464 T^{12} - 1975845999573 T^{13} + 19815120570387 T^{14} - 246249955396158 T^{15} + 1119766626567627 T^{16} - 4091343041042436 T^{17} + 43902300843691979 T^{18} - 259889874559648170 T^{19} + 511116753300641401 T^{20} \)
$61$ \( 1 + 14 T - 143 T^{2} - 2072 T^{3} + 23777 T^{4} + 251656 T^{5} - 2164351 T^{6} - 13562879 T^{7} + 202896254 T^{8} + 466067647 T^{9} - 12461386219 T^{10} + 28430126467 T^{11} + 754976961134 T^{12} - 3078515838299 T^{13} - 29967259814191 T^{14} + 212547726724456 T^{15} + 1224999941181497 T^{16} - 6511763156235512 T^{17} - 27414145758611183 T^{18} + 163718045299677974 T^{19} + 713342911662882601 T^{20} \)
$67$ \( 1 + 2 T - 128 T^{2} - 128 T^{3} + 6161 T^{4} - 2183 T^{5} + 29300 T^{6} + 394018 T^{7} - 17169907 T^{8} - 2850929 T^{9} + 1197895103 T^{10} - 191012243 T^{11} - 77075712523 T^{12} + 118506035734 T^{13} + 590427845300 T^{14} - 2947323108581 T^{15} + 557314092543209 T^{16} - 775771085481344 T^{17} - 51976662727250048 T^{18} + 54413068792589894 T^{19} + 1822837804551761449 T^{20} \)
$71$ \( ( 1 + 3 T + 187 T^{2} + 285 T^{3} + 15679 T^{4} + 10143 T^{5} + 1113209 T^{6} + 1436685 T^{7} + 66929357 T^{8} + 76235043 T^{9} + 1804229351 T^{10} )^{2} \)
$73$ \( ( 1 + 15 T + 359 T^{2} + 3943 T^{3} + 53173 T^{4} + 414929 T^{5} + 3881629 T^{6} + 21012247 T^{7} + 139657103 T^{8} + 425973615 T^{9} + 2073071593 T^{10} )^{2} \)
$79$ \( 1 + 4 T - 284 T^{2} - 1776 T^{3} + 44175 T^{4} + 312399 T^{5} - 4187754 T^{6} - 29772300 T^{7} + 295992489 T^{8} + 1067553919 T^{9} - 20151634301 T^{10} + 84336759601 T^{11} + 1847289123849 T^{12} - 14678905019700 T^{13} - 163113357508074 T^{14} + 961269341991201 T^{15} + 10738388347640175 T^{16} - 34106142359418384 T^{17} - 430858902013463324 T^{18} + 479406383930473276 T^{19} + 9468276082626847201 T^{20} \)
$83$ \( 1 - 9 T - 148 T^{2} - 297 T^{3} + 24654 T^{4} + 118125 T^{5} - 807174 T^{6} - 21382137 T^{7} - 37648479 T^{8} + 452536146 T^{9} + 15509586612 T^{10} + 37560500118 T^{11} - 259360371831 T^{12} - 12226027968819 T^{13} - 38307122794854 T^{14} + 465299175954375 T^{15} + 8060387965039326 T^{16} - 8059407143919219 T^{17} - 333339250356578068 T^{18} - 1682462297407863627 T^{19} + 15516041187205853449 T^{20} \)
$89$ \( ( 1 + 28 T + 680 T^{2} + 10388 T^{3} + 140263 T^{4} + 1402827 T^{5} + 12483407 T^{6} + 82283348 T^{7} + 479378920 T^{8} + 1756782748 T^{9} + 5584059449 T^{10} )^{2} \)
$97$ \( 1 + 12 T - 197 T^{2} - 1534 T^{3} + 27813 T^{4} + 14090 T^{5} - 4545035 T^{6} - 6881349 T^{7} + 472663750 T^{8} + 908843245 T^{9} - 38512186359 T^{10} + 88157794765 T^{11} + 4447293223750 T^{12} - 6280421435877 T^{13} - 402368680669835 T^{14} + 120995624221130 T^{15} + 23167450373090277 T^{16} - 123944568389425342 T^{17} - 1543974418092261317 T^{18} + 9122772703854782604 T^{19} + 73742412689492826049 T^{20} \)
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