Properties

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

$p$ $F_p(T)$
$2$ \( 9 - 9 T + 36 T^{2} - 3 T^{3} + 90 T^{4} - 36 T^{5} + 40 T^{6} - 8 T^{7} + 9 T^{8} - 2 T^{9} + T^{10} \)
$3$ \( 243 - 81 T + 108 T^{2} - 54 T^{3} + 54 T^{4} - 9 T^{5} + 18 T^{6} - 6 T^{7} + 4 T^{8} - T^{9} + T^{10} \)
$5$ \( 9 + 54 T + 378 T^{2} - 294 T^{3} + 402 T^{4} - 51 T^{5} + 79 T^{6} + 16 T^{7} + 21 T^{8} + 4 T^{9} + T^{10} \)
$7$ \( T^{10} \)
$11$ \( 225 - 180 T + 369 T^{2} - 60 T^{3} + 261 T^{4} - 39 T^{5} + 112 T^{6} + 2 T^{7} + 24 T^{8} - 4 T^{9} + T^{10} \)
$13$ \( 25 + 115 T + 594 T^{2} - 169 T^{3} + 428 T^{4} - 204 T^{5} + 296 T^{6} - 130 T^{7} + 51 T^{8} - 8 T^{9} + T^{10} \)
$17$ \( ( 9 + 18 T - 60 T^{2} + 45 T^{3} - 12 T^{4} + T^{5} )^{2} \)
$19$ \( ( 431 + 259 T - 46 T^{2} - 41 T^{3} - T^{4} + T^{5} )^{2} \)
$23$ \( 2595321 + 1739880 T + 1084239 T^{2} + 258066 T^{3} + 75474 T^{4} + 4878 T^{5} + 3042 T^{6} + 87 T^{7} + 72 T^{8} - 3 T^{9} + T^{10} \)
$29$ \( 81 + 1215 T + 19710 T^{2} - 22635 T^{3} + 24462 T^{4} - 5199 T^{5} + 1690 T^{6} - 190 T^{7} + 69 T^{8} - 7 T^{9} + T^{10} \)
$31$ \( 81225 - 26505 T + 26889 T^{2} - 6018 T^{3} + 5194 T^{4} - 1071 T^{5} + 540 T^{6} - 65 T^{7} + 30 T^{8} - 3 T^{9} + T^{10} \)
$37$ \( ( -288 + 384 T + 280 T^{2} - 96 T^{3} + T^{5} )^{2} \)
$41$ \( 2025 + 7695 T + 31536 T^{2} - 4761 T^{3} + 9900 T^{4} + 579 T^{5} + 2020 T^{6} - 118 T^{7} + 69 T^{8} + 5 T^{9} + T^{10} \)
$43$ \( 687241 + 1214485 T + 1541055 T^{2} + 921888 T^{3} + 408318 T^{4} + 84651 T^{5} + 14496 T^{6} + 837 T^{7} + 138 T^{8} + 7 T^{9} + T^{10} \)
$47$ \( 43758225 + 26671680 T + 16872219 T^{2} + 2443014 T^{3} + 1046070 T^{4} + 230922 T^{5} + 46890 T^{6} + 5565 T^{7} + 516 T^{8} + 27 T^{9} + T^{10} \)
$53$ \( ( 423 - 423 T - 210 T^{2} + 135 T^{3} - 21 T^{4} + T^{5} )^{2} \)
$59$ \( 31640625 + 607500 T + 6272289 T^{2} + 3322296 T^{3} + 1440567 T^{4} + 341433 T^{5} + 60354 T^{6} + 6954 T^{7} + 594 T^{8} + 30 T^{9} + T^{10} \)
$61$ \( 1 - 14 T + 189 T^{2} - 166 T^{3} + 539 T^{4} - 153 T^{5} + 1268 T^{6} - 490 T^{7} + 162 T^{8} - 14 T^{9} + T^{10} \)
$67$ \( 50708641 + 48308864 T + 48443796 T^{2} + 584566 T^{3} + 1478510 T^{4} + 49005 T^{5} + 35105 T^{6} + 274 T^{7} + 207 T^{8} + 2 T^{9} + T^{10} \)
$71$ \( ( -81 + 1053 T - 567 T^{2} - 168 T^{3} + 3 T^{4} + T^{5} )^{2} \)
$73$ \( ( 879 + 1197 T + 437 T^{2} - 6 T^{3} - 15 T^{4} + T^{5} )^{2} \)
$79$ \( 37249 + 4246 T + 43716 T^{2} - 41598 T^{3} + 48858 T^{4} - 21297 T^{5} + 8151 T^{6} - 828 T^{7} + 111 T^{8} + 4 T^{9} + T^{10} \)
$83$ \( 218123361 - 40275063 T + 38540043 T^{2} + 11237130 T^{3} + 3795093 T^{4} + 455571 T^{5} + 56277 T^{6} + 2538 T^{7} + 267 T^{8} + 9 T^{9} + T^{10} \)
$89$ \( ( 2661 - 1692 T - 420 T^{2} + 235 T^{3} - 28 T^{4} + T^{5} )^{2} \)
$97$ \( 2307745521 - 121346514 T + 94916553 T^{2} - 9179814 T^{3} + 3183925 T^{4} - 252807 T^{5} + 40326 T^{6} - 1958 T^{7} + 288 T^{8} - 12 T^{9} + T^{10} \)
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