Properties

Label 529.2.c.o
Level $529$
Weight $2$
Character orbit 529.c
Analytic conductor $4.224$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $10$

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Newspace parameters

Level: \( N \) \(=\) \( 529 = 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 529.c (of order \(11\), degree \(10\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.22408626693\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{11})\)
Coefficient field: 20.0.54296067514572573056640625.1
Defining polynomial: \(x^{20} - x^{19} + 2 x^{18} - 3 x^{17} + 5 x^{16} - 8 x^{15} + 13 x^{14} - 21 x^{13} + 34 x^{12} - 55 x^{11} + 89 x^{10} + 55 x^{9} + 34 x^{8} + 21 x^{7} + 13 x^{6} + 8 x^{5} + 5 x^{4} + 3 x^{3} + 2 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 23)
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{16} q^{2} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} + 2 \beta_{12} - \beta_{13} - 2 \beta_{14} + \beta_{15} + 2 \beta_{16} + \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{3} + ( \beta_{10} + \beta_{11} ) q^{4} -2 \beta_{8} q^{5} + ( \beta_{12} + 2 \beta_{13} ) q^{6} + ( -2 \beta_{18} + 2 \beta_{19} ) q^{7} + ( 2 \beta_{4} - \beta_{5} ) q^{8} + 2 \beta_{15} q^{9} +O(q^{10})\) \( q + \beta_{16} q^{2} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} + 2 \beta_{12} - \beta_{13} - 2 \beta_{14} + \beta_{15} + 2 \beta_{16} + \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{3} + ( \beta_{10} + \beta_{11} ) q^{4} -2 \beta_{8} q^{5} + ( \beta_{12} + 2 \beta_{13} ) q^{6} + ( -2 \beta_{18} + 2 \beta_{19} ) q^{7} + ( 2 \beta_{4} - \beta_{5} ) q^{8} + 2 \beta_{15} q^{9} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{10} + ( -2 \beta_{6} + 4 \beta_{7} ) q^{11} + ( \beta_{6} - 3 \beta_{7} ) q^{12} + 3 \beta_{3} q^{13} + 2 \beta_{15} q^{14} + ( -2 \beta_{4} - 4 \beta_{5} ) q^{15} -3 \beta_{18} q^{16} + ( -2 \beta_{12} - 2 \beta_{13} ) q^{17} -2 \beta_{8} q^{18} + 2 \beta_{11} q^{19} + ( 2 + 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{11} - 2 \beta_{13} + 2 \beta_{15} + 2 \beta_{17} + 2 \beta_{19} ) q^{20} + ( -2 \beta_{16} - 6 \beta_{17} ) q^{21} + ( -2 - 2 \beta_{2} ) q^{22} + 5 q^{24} + ( -4 \beta_{16} - \beta_{17} ) q^{25} + ( -3 \beta_{1} - 3 \beta_{2} + 3 \beta_{4} - 3 \beta_{6} + 3 \beta_{8} + 3 \beta_{10} - 3 \beta_{12} + 3 \beta_{14} - 3 \beta_{16} + 3 \beta_{18} ) q^{26} + ( -2 \beta_{10} - \beta_{11} ) q^{27} + ( 2 \beta_{8} + 4 \beta_{9} ) q^{28} + 3 \beta_{13} q^{29} + ( 6 \beta_{18} + 2 \beta_{19} ) q^{30} + ( -6 \beta_{4} + 3 \beta_{5} ) q^{31} + ( -\beta_{14} + 5 \beta_{15} ) q^{32} + ( 6 \beta_{1} + 8 \beta_{3} ) q^{33} + ( 4 \beta_{6} + 2 \beta_{7} ) q^{34} -4 \beta_{7} q^{35} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{36} + 2 \beta_{14} q^{37} + 2 \beta_{4} q^{38} + ( 6 \beta_{18} - 3 \beta_{19} ) q^{39} + ( 2 \beta_{12} + 4 \beta_{13} ) q^{40} + ( 4 \beta_{8} + \beta_{9} ) q^{41} + ( 4 \beta_{10} + 2 \beta_{11} ) q^{42} + ( 4 \beta_{16} + 6 \beta_{17} ) q^{44} + 4 \beta_{2} q^{45} + ( -1 - 2 \beta_{2} ) q^{47} + ( 3 \beta_{16} - 6 \beta_{17} ) q^{48} + ( -1 - 4 \beta_{1} - 4 \beta_{2} - \beta_{3} + 4 \beta_{4} - \beta_{5} - 4 \beta_{6} + \beta_{7} + 4 \beta_{8} + \beta_{9} + 4 \beta_{10} + \beta_{11} - 4 \beta_{12} + \beta_{13} + 4 \beta_{14} - \beta_{15} - 4 \beta_{16} - \beta_{17} + 4 \beta_{18} - \beta_{19} ) q^{49} + ( -3 \beta_{10} + 4 \beta_{11} ) q^{50} + ( 6 \beta_{8} - 2 \beta_{9} ) q^{51} + ( -3 \beta_{12} + 3 \beta_{13} ) q^{52} + ( -4 \beta_{18} - 2 \beta_{19} ) q^{53} + ( \beta_{4} + 2 \beta_{5} ) q^{54} + ( 4 \beta_{14} - 4 \beta_{15} ) q^{55} + ( -2 \beta_{1} - 6 \beta_{3} ) q^{56} + ( 4 \beta_{6} - 2 \beta_{7} ) q^{57} -3 \beta_{6} q^{58} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{59} + ( -4 \beta_{14} + 2 \beta_{15} ) q^{60} + ( 8 \beta_{4} - 2 \beta_{5} ) q^{61} + ( 3 \beta_{18} + 6 \beta_{19} ) q^{62} + ( 4 \beta_{12} - 4 \beta_{13} ) q^{63} + ( 2 \beta_{8} - \beta_{9} ) q^{64} -6 \beta_{10} q^{65} + ( -6 - 2 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} + 6 \beta_{7} + 2 \beta_{8} + 6 \beta_{9} + 2 \beta_{10} + 6 \beta_{11} - 2 \beta_{12} + 6 \beta_{13} + 2 \beta_{14} - 6 \beta_{15} - 2 \beta_{16} - 6 \beta_{17} + 2 \beta_{18} - 6 \beta_{19} ) q^{66} + ( 2 \beta_{16} - 4 \beta_{17} ) q^{67} -2 \beta_{2} q^{68} + 4 \beta_{2} q^{70} + ( 2 \beta_{16} + 11 \beta_{17} ) q^{71} + ( 2 + 4 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} - 4 \beta_{10} - 2 \beta_{11} + 4 \beta_{12} - 2 \beta_{13} - 4 \beta_{14} + 2 \beta_{15} + 4 \beta_{16} + 2 \beta_{17} - 4 \beta_{18} + 2 \beta_{19} ) q^{72} + ( 4 \beta_{10} - 9 \beta_{11} ) q^{73} + ( -2 \beta_{8} + 2 \beta_{9} ) q^{74} + ( -2 \beta_{12} - 9 \beta_{13} ) q^{75} + ( -2 \beta_{18} + 2 \beta_{19} ) q^{76} + ( 8 \beta_{4} - 12 \beta_{5} ) q^{77} + ( -3 \beta_{14} - 6 \beta_{15} ) q^{78} + ( -8 \beta_{1} - 6 \beta_{3} ) q^{79} + ( -6 \beta_{6} - 6 \beta_{7} ) q^{80} + 11 \beta_{7} q^{81} + ( 3 \beta_{1} - 4 \beta_{3} ) q^{82} + ( -2 \beta_{14} - 10 \beta_{15} ) q^{83} + ( -6 \beta_{4} + 8 \beta_{5} ) q^{84} + ( -8 \beta_{18} - 4 \beta_{19} ) q^{85} + ( -6 \beta_{8} - 3 \beta_{9} ) q^{87} + ( -6 \beta_{10} - 8 \beta_{11} ) q^{88} + ( 8 + 4 \beta_{1} + 4 \beta_{2} + 8 \beta_{3} - 4 \beta_{4} + 8 \beta_{5} + 4 \beta_{6} - 8 \beta_{7} - 4 \beta_{8} - 8 \beta_{9} - 4 \beta_{10} - 8 \beta_{11} + 4 \beta_{12} - 8 \beta_{13} - 4 \beta_{14} + 8 \beta_{15} + 4 \beta_{16} + 8 \beta_{17} - 4 \beta_{18} + 8 \beta_{19} ) q^{89} + ( -4 \beta_{16} + 4 \beta_{17} ) q^{90} + ( 6 + 6 \beta_{2} ) q^{91} -15 q^{93} + ( \beta_{16} - 2 \beta_{17} ) q^{94} + ( 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{4} + 4 \beta_{6} - 4 \beta_{8} - 4 \beta_{10} + 4 \beta_{12} - 4 \beta_{14} + 4 \beta_{16} - 4 \beta_{18} ) q^{95} + ( 9 \beta_{10} + 7 \beta_{11} ) q^{96} + ( -6 \beta_{8} - 14 \beta_{9} ) q^{97} + ( -3 \beta_{12} - 4 \beta_{13} ) q^{98} + ( 4 \beta_{18} - 8 \beta_{19} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + q^{2} + q^{4} + 2q^{5} + 5q^{6} - 2q^{7} - 4q^{9} + O(q^{10}) \) \( 20q + q^{2} + q^{4} + 2q^{5} + 5q^{6} - 2q^{7} - 4q^{9} - 6q^{10} + 6q^{11} - 5q^{12} - 6q^{13} - 4q^{14} + 10q^{15} + 3q^{16} - 6q^{17} + 2q^{18} + 4q^{19} + 4q^{20} + 10q^{21} - 20q^{22} + 100q^{24} - 2q^{25} + 3q^{26} + 6q^{28} + 6q^{29} - 10q^{30} - 9q^{32} - 10q^{33} + 8q^{34} - 8q^{35} + 2q^{36} - 2q^{37} - 2q^{38} + 10q^{40} - 2q^{41} - 8q^{44} - 40q^{45} + 15q^{48} + 2q^{49} + 11q^{50} - 10q^{51} + 3q^{52} + 8q^{53} - 5q^{54} + 4q^{55} + 10q^{56} - 3q^{58} - 4q^{59} - 4q^{61} - 15q^{62} - 4q^{63} - 4q^{64} + 6q^{65} - 10q^{66} + 10q^{67} + 20q^{68} - 40q^{70} - 20q^{71} - 22q^{73} + 6q^{74} - 20q^{75} - 2q^{76} + 16q^{77} + 15q^{78} + 4q^{79} - 18q^{80} + 22q^{81} + 11q^{82} + 22q^{83} - 10q^{84} + 16q^{85} - 10q^{88} + 12q^{89} - 12q^{90} + 60q^{91} - 300q^{93} + 5q^{94} - 4q^{95} + 5q^{96} - 22q^{97} - 11q^{98} + 12q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - x^{19} + 2 x^{18} - 3 x^{17} + 5 x^{16} - 8 x^{15} + 13 x^{14} - 21 x^{13} + 34 x^{12} - 55 x^{11} + 89 x^{10} + 55 x^{9} + 34 x^{8} + 21 x^{7} + 13 x^{6} + 8 x^{5} + 5 x^{4} + 3 x^{3} + 2 x^{2} + x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{11} + 55 \)\()/89\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{12} + 144 \nu \)\()/89\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{13} - 144 \nu^{2} \)\()/89\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{13} + 233 \nu^{2} \)\()/89\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{14} + 233 \nu^{3} \)\()/89\)
\(\beta_{7}\)\(=\)\((\)\( -2 \nu^{14} - 377 \nu^{3} \)\()/89\)
\(\beta_{8}\)\(=\)\((\)\( -2 \nu^{15} - 377 \nu^{4} \)\()/89\)
\(\beta_{9}\)\(=\)\((\)\( -3 \nu^{15} - 610 \nu^{4} \)\()/89\)
\(\beta_{10}\)\(=\)\((\)\( -3 \nu^{16} - 610 \nu^{5} \)\()/89\)
\(\beta_{11}\)\(=\)\((\)\( -5 \nu^{16} - 987 \nu^{5} \)\()/89\)
\(\beta_{12}\)\(=\)\((\)\( 5 \nu^{17} + 987 \nu^{6} \)\()/89\)
\(\beta_{13}\)\(=\)\((\)\( -8 \nu^{17} - 1597 \nu^{6} \)\()/89\)
\(\beta_{14}\)\(=\)\((\)\( -8 \nu^{18} - 1597 \nu^{7} \)\()/89\)
\(\beta_{15}\)\(=\)\((\)\( 13 \nu^{18} + 2584 \nu^{7} \)\()/89\)
\(\beta_{16}\)\(=\)\((\)\( 13 \nu^{19} + 2584 \nu^{8} \)\()/89\)
\(\beta_{17}\)\(=\)\((\)\( 21 \nu^{19} + 4181 \nu^{8} \)\()/89\)
\(\beta_{18}\)\(=\)\((\)\( 34 \nu^{19} - 34 \nu^{18} + 68 \nu^{17} - 102 \nu^{16} + 170 \nu^{15} - 272 \nu^{14} + 442 \nu^{13} - 714 \nu^{12} + 1156 \nu^{11} - 1869 \nu^{10} + 3026 \nu^{9} + 1870 \nu^{8} + 1156 \nu^{7} + 714 \nu^{6} + 442 \nu^{5} + 272 \nu^{4} + 170 \nu^{3} + 102 \nu^{2} + 68 \nu + 34 \)\()/89\)
\(\beta_{19}\)\(=\)\((\)\( -55 \nu^{19} + 55 \nu^{18} - 110 \nu^{17} + 165 \nu^{16} - 275 \nu^{15} + 440 \nu^{14} - 715 \nu^{13} + 1155 \nu^{12} - 1870 \nu^{11} + 3026 \nu^{10} - 4895 \nu^{9} - 3025 \nu^{8} - 1870 \nu^{7} - 1155 \nu^{6} - 715 \nu^{5} - 440 \nu^{4} - 275 \nu^{3} - 165 \nu^{2} - 110 \nu - 55 \)\()/89\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} + \beta_{4}\)
\(\nu^{3}\)\(=\)\(\beta_{7} + 2 \beta_{6}\)
\(\nu^{4}\)\(=\)\(-2 \beta_{9} + 3 \beta_{8}\)
\(\nu^{5}\)\(=\)\(3 \beta_{11} - 5 \beta_{10}\)
\(\nu^{6}\)\(=\)\(-5 \beta_{13} - 8 \beta_{12}\)
\(\nu^{7}\)\(=\)\(-8 \beta_{15} - 13 \beta_{14}\)
\(\nu^{8}\)\(=\)\(13 \beta_{17} - 21 \beta_{16}\)
\(\nu^{9}\)\(=\)\(21 \beta_{19} + 34 \beta_{18} + 21 \beta_{17} - 34 \beta_{16} + 21 \beta_{15} + 34 \beta_{14} - 21 \beta_{13} - 34 \beta_{12} - 21 \beta_{11} + 34 \beta_{10} - 21 \beta_{9} + 34 \beta_{8} - 21 \beta_{7} - 34 \beta_{6} + 21 \beta_{5} + 34 \beta_{4} + 21 \beta_{3} - 34 \beta_{2} - 34 \beta_{1} + 21\)
\(\nu^{10}\)\(=\)\(34 \beta_{19} + 55 \beta_{18}\)
\(\nu^{11}\)\(=\)\(89 \beta_{2} - 55\)
\(\nu^{12}\)\(=\)\(89 \beta_{3} - 144 \beta_{1}\)
\(\nu^{13}\)\(=\)\(-144 \beta_{5} - 233 \beta_{4}\)
\(\nu^{14}\)\(=\)\(-233 \beta_{7} - 377 \beta_{6}\)
\(\nu^{15}\)\(=\)\(377 \beta_{9} - 610 \beta_{8}\)
\(\nu^{16}\)\(=\)\(-610 \beta_{11} + 987 \beta_{10}\)
\(\nu^{17}\)\(=\)\(987 \beta_{13} + 1597 \beta_{12}\)
\(\nu^{18}\)\(=\)\(1597 \beta_{15} + 2584 \beta_{14}\)
\(\nu^{19}\)\(=\)\(-2584 \beta_{17} + 4181 \beta_{16}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/529\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(\beta_{17}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
118.1
0.519923 + 0.334134i
−1.36118 0.874775i
0.256741 + 0.562183i
−0.672156 1.47182i
−0.592999 0.174120i
1.55249 + 0.455853i
1.05959 + 1.22283i
−0.404726 0.467079i
−0.592999 + 0.174120i
1.55249 0.455853i
1.05959 1.22283i
−0.404726 + 0.467079i
0.519923 0.334134i
−1.36118 + 0.874775i
0.230270 + 1.60156i
−0.0879554 0.611743i
0.230270 1.60156i
−0.0879554 + 0.611743i
0.256741 0.562183i
−0.672156 + 1.47182i
−0.0879554 0.611743i −0.928896 + 2.03400i 1.55249 0.455853i −0.809452 + 0.934158i 1.32599 + 0.389345i 2.72235 1.74955i −0.928896 2.03400i −1.30972 1.51150i 0.642661 + 0.413013i
118.2 0.230270 + 1.60156i 0.928896 2.03400i −0.592999 + 0.174120i 2.11917 2.44566i 3.47148 + 1.01932i −1.03985 + 0.668269i 0.928896 + 2.03400i −1.30972 1.51150i 4.40486 + 2.83083i
170.1 −0.592999 + 0.174120i 1.46431 + 1.68991i −1.36118 + 0.874775i −0.175911 1.22349i −1.16258 0.747147i 1.34431 2.94363i 1.46431 1.68991i −0.284630 + 1.97964i 0.317349 + 0.694897i
170.2 1.55249 0.455853i −1.46431 1.68991i 0.519923 0.334134i 0.460540 + 3.20313i −3.04368 1.95606i −0.513481 + 1.12437i −1.46431 + 1.68991i −0.284630 + 1.97964i 2.17514 + 4.76289i
177.1 −0.404726 + 0.467079i −1.88110 + 1.20891i 0.230270 + 1.60156i 0.513481 + 1.12437i 0.196674 1.36790i −3.10498 + 0.911706i −1.88110 1.20891i 0.830830 1.81926i −0.732987 0.215225i
177.2 1.05959 1.22283i 1.88110 1.20891i −0.0879554 0.611743i −1.34431 2.94363i 0.514900 3.58121i 1.18600 0.348241i 1.88110 + 1.20891i 0.830830 1.81926i −5.02397 1.47517i
255.1 −1.36118 0.874775i −0.318226 2.21331i 0.256741 + 0.562183i 3.10498 + 0.911706i −1.50299 + 3.29108i 0.809452 0.934158i −0.318226 + 2.21331i −1.91899 + 0.563465i −3.42890 3.95716i
255.2 0.519923 + 0.334134i 0.318226 + 2.21331i −0.672156 1.47182i −1.18600 0.348241i −0.574089 + 1.25708i −2.11917 + 2.44566i 0.318226 2.21331i −1.91899 + 0.563465i −0.500269 0.577341i
266.1 −0.404726 0.467079i −1.88110 1.20891i 0.230270 1.60156i 0.513481 1.12437i 0.196674 + 1.36790i −3.10498 0.911706i −1.88110 + 1.20891i 0.830830 + 1.81926i −0.732987 + 0.215225i
266.2 1.05959 + 1.22283i 1.88110 + 1.20891i −0.0879554 + 0.611743i −1.34431 + 2.94363i 0.514900 + 3.58121i 1.18600 + 0.348241i 1.88110 1.20891i 0.830830 + 1.81926i −5.02397 + 1.47517i
334.1 −1.36118 + 0.874775i −0.318226 + 2.21331i 0.256741 0.562183i 3.10498 0.911706i −1.50299 3.29108i 0.809452 + 0.934158i −0.318226 2.21331i −1.91899 0.563465i −3.42890 + 3.95716i
334.2 0.519923 0.334134i 0.318226 2.21331i −0.672156 + 1.47182i −1.18600 + 0.348241i −0.574089 1.25708i −2.11917 2.44566i 0.318226 + 2.21331i −1.91899 0.563465i −0.500269 + 0.577341i
399.1 −0.0879554 + 0.611743i −0.928896 2.03400i 1.55249 + 0.455853i −0.809452 0.934158i 1.32599 0.389345i 2.72235 + 1.74955i −0.928896 + 2.03400i −1.30972 + 1.51150i 0.642661 0.413013i
399.2 0.230270 1.60156i 0.928896 + 2.03400i −0.592999 0.174120i 2.11917 + 2.44566i 3.47148 1.01932i −1.03985 0.668269i 0.928896 2.03400i −1.30972 + 1.51150i 4.40486 2.83083i
466.1 −0.672156 + 1.47182i −2.14549 0.629973i −0.404726 0.467079i −2.72235 + 1.74955i 2.36931 2.73433i 0.175911 1.22349i −2.14549 + 0.629973i 1.68251 + 1.08128i −0.745170 5.18277i
466.2 0.256741 0.562183i 2.14549 + 0.629973i 1.05959 + 1.22283i 1.03985 0.668269i 0.904995 1.04442i −0.460540 + 3.20313i 2.14549 0.629973i 1.68251 + 1.08128i −0.108719 0.756156i
487.1 −0.672156 1.47182i −2.14549 + 0.629973i −0.404726 + 0.467079i −2.72235 1.74955i 2.36931 + 2.73433i 0.175911 + 1.22349i −2.14549 0.629973i 1.68251 1.08128i −0.745170 + 5.18277i
487.2 0.256741 + 0.562183i 2.14549 0.629973i 1.05959 1.22283i 1.03985 + 0.668269i 0.904995 + 1.04442i −0.460540 3.20313i 2.14549 + 0.629973i 1.68251 1.08128i −0.108719 + 0.756156i
501.1 −0.592999 0.174120i 1.46431 1.68991i −1.36118 0.874775i −0.175911 + 1.22349i −1.16258 + 0.747147i 1.34431 + 2.94363i 1.46431 + 1.68991i −0.284630 1.97964i 0.317349 0.694897i
501.2 1.55249 + 0.455853i −1.46431 + 1.68991i 0.519923 + 0.334134i 0.460540 3.20313i −3.04368 + 1.95606i −0.513481 1.12437i −1.46431 1.68991i −0.284630 1.97964i 2.17514 4.76289i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 501.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 9 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 529.2.c.o 20
23.b odd 2 1 529.2.c.n 20
23.c even 11 1 23.2.a.a 2
23.c even 11 9 inner 529.2.c.o 20
23.d odd 22 1 529.2.a.a 2
23.d odd 22 9 529.2.c.n 20
69.g even 22 1 4761.2.a.w 2
69.h odd 22 1 207.2.a.d 2
92.g odd 22 1 368.2.a.h 2
92.h even 22 1 8464.2.a.bb 2
115.j even 22 1 575.2.a.f 2
115.k odd 44 2 575.2.b.d 4
161.l odd 22 1 1127.2.a.c 2
184.k odd 22 1 1472.2.a.s 2
184.p even 22 1 1472.2.a.t 2
253.k odd 22 1 2783.2.a.c 2
276.o even 22 1 3312.2.a.ba 2
299.p even 22 1 3887.2.a.i 2
345.p odd 22 1 5175.2.a.be 2
391.n even 22 1 6647.2.a.b 2
437.o odd 22 1 8303.2.a.e 2
460.n odd 22 1 9200.2.a.bt 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.2.a.a 2 23.c even 11 1
207.2.a.d 2 69.h odd 22 1
368.2.a.h 2 92.g odd 22 1
529.2.a.a 2 23.d odd 22 1
529.2.c.n 20 23.b odd 2 1
529.2.c.n 20 23.d odd 22 9
529.2.c.o 20 1.a even 1 1 trivial
529.2.c.o 20 23.c even 11 9 inner
575.2.a.f 2 115.j even 22 1
575.2.b.d 4 115.k odd 44 2
1127.2.a.c 2 161.l odd 22 1
1472.2.a.s 2 184.k odd 22 1
1472.2.a.t 2 184.p even 22 1
2783.2.a.c 2 253.k odd 22 1
3312.2.a.ba 2 276.o even 22 1
3887.2.a.i 2 299.p even 22 1
4761.2.a.w 2 69.g even 22 1
5175.2.a.be 2 345.p odd 22 1
6647.2.a.b 2 391.n even 22 1
8303.2.a.e 2 437.o odd 22 1
8464.2.a.bb 2 92.h even 22 1
9200.2.a.bt 2 460.n odd 22 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(529, [\chi])\):

\(T_{2}^{20} - \cdots\)
\(T_{5}^{20} - \cdots\)
\(T_{7}^{20} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 2 T^{2} + 3 T^{3} + 5 T^{4} + 8 T^{5} + 13 T^{6} + 21 T^{7} + 34 T^{8} + 55 T^{9} + 89 T^{10} - 55 T^{11} + 34 T^{12} - 21 T^{13} + 13 T^{14} - 8 T^{15} + 5 T^{16} - 3 T^{17} + 2 T^{18} - T^{19} + T^{20} \)
$3$ \( 9765625 + 1953125 T^{2} + 390625 T^{4} + 78125 T^{6} + 15625 T^{8} + 3125 T^{10} + 625 T^{12} + 125 T^{14} + 25 T^{16} + 5 T^{18} + T^{20} \)
$5$ \( 1048576 + 524288 T + 524288 T^{2} + 393216 T^{3} + 327680 T^{4} + 262144 T^{5} + 212992 T^{6} + 172032 T^{7} + 139264 T^{8} + 112640 T^{9} + 91136 T^{10} - 28160 T^{11} + 8704 T^{12} - 2688 T^{13} + 832 T^{14} - 256 T^{15} + 80 T^{16} - 24 T^{17} + 8 T^{18} - 2 T^{19} + T^{20} \)
$7$ \( 1048576 - 524288 T + 524288 T^{2} - 393216 T^{3} + 327680 T^{4} - 262144 T^{5} + 212992 T^{6} - 172032 T^{7} + 139264 T^{8} - 112640 T^{9} + 91136 T^{10} + 28160 T^{11} + 8704 T^{12} + 2688 T^{13} + 832 T^{14} + 256 T^{15} + 80 T^{16} + 24 T^{17} + 8 T^{18} + 2 T^{19} + T^{20} \)
$11$ \( 1048576 - 1572864 T + 2097152 T^{2} - 2752512 T^{3} + 3604480 T^{4} - 4718592 T^{5} + 6176768 T^{6} - 8085504 T^{7} + 10584064 T^{8} - 13854720 T^{9} + 18136064 T^{10} - 3463680 T^{11} + 661504 T^{12} - 126336 T^{13} + 24128 T^{14} - 4608 T^{15} + 880 T^{16} - 168 T^{17} + 32 T^{18} - 6 T^{19} + T^{20} \)
$13$ \( ( 59049 + 19683 T + 6561 T^{2} + 2187 T^{3} + 729 T^{4} + 243 T^{5} + 81 T^{6} + 27 T^{7} + 9 T^{8} + 3 T^{9} + T^{10} )^{2} \)
$17$ \( 1048576 + 1572864 T + 2097152 T^{2} + 2752512 T^{3} + 3604480 T^{4} + 4718592 T^{5} + 6176768 T^{6} + 8085504 T^{7} + 10584064 T^{8} + 13854720 T^{9} + 18136064 T^{10} + 3463680 T^{11} + 661504 T^{12} + 126336 T^{13} + 24128 T^{14} + 4608 T^{15} + 880 T^{16} + 168 T^{17} + 32 T^{18} + 6 T^{19} + T^{20} \)
$19$ \( ( 1024 - 512 T + 256 T^{2} - 128 T^{3} + 64 T^{4} - 32 T^{5} + 16 T^{6} - 8 T^{7} + 4 T^{8} - 2 T^{9} + T^{10} )^{2} \)
$23$ \( T^{20} \)
$29$ \( ( 59049 - 19683 T + 6561 T^{2} - 2187 T^{3} + 729 T^{4} - 243 T^{5} + 81 T^{6} - 27 T^{7} + 9 T^{8} - 3 T^{9} + T^{10} )^{2} \)
$31$ \( 34050628916015625 + 756680642578125 T^{2} + 16815125390625 T^{4} + 373669453125 T^{6} + 8303765625 T^{8} + 184528125 T^{10} + 4100625 T^{12} + 91125 T^{14} + 2025 T^{16} + 45 T^{18} + T^{20} \)
$37$ \( 1048576 - 524288 T + 524288 T^{2} - 393216 T^{3} + 327680 T^{4} - 262144 T^{5} + 212992 T^{6} - 172032 T^{7} + 139264 T^{8} - 112640 T^{9} + 91136 T^{10} + 28160 T^{11} + 8704 T^{12} + 2688 T^{13} + 832 T^{14} + 256 T^{15} + 80 T^{16} + 24 T^{17} + 8 T^{18} + 2 T^{19} + T^{20} \)
$41$ \( 6131066257801 - 645375395558 T + 390621949943 T^{2} - 75085226076 T^{3} + 28462758005 T^{4} - 6947933794 T^{5} + 2229401347 T^{6} - 600354552 T^{7} + 180532129 T^{8} - 50600990 T^{9} + 14828111 T^{10} + 2663210 T^{11} + 500089 T^{12} + 87528 T^{13} + 17107 T^{14} + 2806 T^{15} + 605 T^{16} + 84 T^{17} + 23 T^{18} + 2 T^{19} + T^{20} \)
$43$ \( T^{20} \)
$47$ \( ( -5 + T^{2} )^{10} \)
$53$ \( 1048576 + 2097152 T + 4456448 T^{2} + 9437184 T^{3} + 19988480 T^{4} + 42336256 T^{5} + 89669632 T^{6} + 189923328 T^{7} + 402264064 T^{8} + 852008960 T^{9} + 1804583936 T^{10} - 213002240 T^{11} + 25141504 T^{12} - 2967552 T^{13} + 350272 T^{14} - 41344 T^{15} + 4880 T^{16} - 576 T^{17} + 68 T^{18} - 8 T^{19} + T^{20} \)
$59$ \( 1099511627776 - 274877906944 T + 137438953472 T^{2} - 51539607552 T^{3} + 21474836480 T^{4} - 8589934592 T^{5} + 3489660928 T^{6} - 1409286144 T^{7} + 570425344 T^{8} - 230686720 T^{9} + 93323264 T^{10} + 14417920 T^{11} + 2228224 T^{12} + 344064 T^{13} + 53248 T^{14} + 8192 T^{15} + 1280 T^{16} + 192 T^{17} + 32 T^{18} + 4 T^{19} + T^{20} \)
$61$ \( 6428888932339941376 - 338362575386312704 T + 102399200445857792 T^{2} - 9841570752233472 T^{3} + 1865335308615680 T^{4} - 227669894561792 T^{5} + 36526511669248 T^{6} - 4918104489984 T^{7} + 739459600384 T^{8} - 103630827520 T^{9} + 15183985664 T^{10} + 1363563520 T^{11} + 128022784 T^{12} + 11203584 T^{13} + 1094848 T^{14} + 89792 T^{15} + 9680 T^{16} + 672 T^{17} + 92 T^{18} + 4 T^{19} + T^{20} \)
$67$ \( 10240000000000 - 5120000000000 T + 2048000000000 T^{2} - 768000000000 T^{3} + 281600000000 T^{4} - 102400000000 T^{5} + 37120000000 T^{6} - 13440000000 T^{7} + 4864000000 T^{8} - 1760000000 T^{9} + 636800000 T^{10} - 88000000 T^{11} + 12160000 T^{12} - 1680000 T^{13} + 232000 T^{14} - 32000 T^{15} + 4400 T^{16} - 600 T^{17} + 80 T^{18} - 10 T^{19} + T^{20} \)
$71$ \( 59873693923837890625 + 12604988194492187500 T + 2023432315431640625 T^{2} + 293301664359375000 T^{3} + 40448431281640625 T^{4} + 5428073276562500 T^{5} + 716979307890625 T^{6} + 93805398750000 T^{7} + 12201354390625 T^{8} + 1581280937500 T^{9} + 204465940625 T^{10} + 16645062500 T^{11} + 1351950625 T^{12} + 109410000 T^{13} + 8802625 T^{14} + 701500 T^{15} + 55025 T^{16} + 4200 T^{17} + 305 T^{18} + 20 T^{19} + T^{20} \)
$73$ \( \)\(11\!\cdots\!01\)\( + 24061075999055939822 T + 4147341182555546783 T^{2} + 665152772447189004 T^{3} + 103821978329530805 T^{4} + 16029017334678106 T^{5} + 2463528742904827 T^{6} + 377907079299288 T^{7} + 57925019818609 T^{8} + 8875676799110 T^{9} + 1359800690711 T^{10} + 87877988110 T^{11} + 5678366809 T^{12} + 366792888 T^{13} + 23674027 T^{14} + 1525106 T^{15} + 97805 T^{16} + 6204 T^{17} + 383 T^{18} + 22 T^{19} + T^{20} \)
$79$ \( 6428888932339941376 + 338362575386312704 T + 102399200445857792 T^{2} + 9841570752233472 T^{3} + 1865335308615680 T^{4} + 227669894561792 T^{5} + 36526511669248 T^{6} + 4918104489984 T^{7} + 739459600384 T^{8} + 103630827520 T^{9} + 15183985664 T^{10} - 1363563520 T^{11} + 128022784 T^{12} - 11203584 T^{13} + 1094848 T^{14} - 89792 T^{15} + 9680 T^{16} - 672 T^{17} + 92 T^{18} - 4 T^{19} + T^{20} \)
$83$ \( \)\(44\!\cdots\!76\)\( - 83665148043360468992 T + 12064566802490851328 T^{2} - 1566856220788260864 T^{3} + 193157500472852480 T^{4} - 23125937841504256 T^{5} + 2720802862415872 T^{6} - 316652802859008 T^{7} + 36599644831744 T^{8} - 4211546408960 T^{9} + 483227380736 T^{10} - 36306434560 T^{11} + 2719949824 T^{12} - 202866048 T^{13} + 15026752 T^{14} - 1101056 T^{15} + 79280 T^{16} - 5544 T^{17} + 368 T^{18} - 22 T^{19} + T^{20} \)
$89$ \( 1099511627776 - 824633720832 T + 549755813888 T^{2} - 360777252864 T^{3} + 236223201280 T^{4} - 154618822656 T^{5} + 101200166912 T^{6} - 66236448768 T^{7} + 43352326144 T^{8} - 28374466560 T^{9} + 18571329536 T^{10} - 1773404160 T^{11} + 169345024 T^{12} - 16171008 T^{13} + 1544192 T^{14} - 147456 T^{15} + 14080 T^{16} - 1344 T^{17} + 128 T^{18} - 12 T^{19} + T^{20} \)
$97$ \( 6428888932339941376 + 1860994164624719872 T + 454118193281630208 T^{2} + 106968501152251904 T^{3} + 24989326737735680 T^{4} + 5826272198393856 T^{5} + 1357745547722752 T^{6} + 316370129625088 T^{7} + 73715754000384 T^{8} + 17176006031360 T^{9} + 4002057614336 T^{10} + 226000079360 T^{11} + 12762422784 T^{12} + 720700288 T^{13} + 40697152 T^{14} + 2297856 T^{15} + 129680 T^{16} + 7304 T^{17} + 408 T^{18} + 22 T^{19} + T^{20} \)
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