Properties

Label 1050.3.q.c
Level 1050
Weight 3
Character orbit 1050.q
Analytic conductor 28.610
Analytic rank 0
Dimension 16
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1050.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.6104277578\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: 16.0.11007531417600000000.1
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{13} q^{2} + ( \beta_{1} + 2 \beta_{8} ) q^{3} -2 \beta_{6} q^{4} + ( -\beta_{4} - 2 \beta_{12} ) q^{6} + ( -2 \beta_{2} + 2 \beta_{5} + 3 \beta_{10} - 3 \beta_{13} ) q^{7} -2 \beta_{5} q^{8} + ( -3 - 3 \beta_{6} ) q^{9} +O(q^{10})\) \( q -\beta_{13} q^{2} + ( \beta_{1} + 2 \beta_{8} ) q^{3} -2 \beta_{6} q^{4} + ( -\beta_{4} - 2 \beta_{12} ) q^{6} + ( -2 \beta_{2} + 2 \beta_{5} + 3 \beta_{10} - 3 \beta_{13} ) q^{7} -2 \beta_{5} q^{8} + ( -3 - 3 \beta_{6} ) q^{9} + ( -\beta_{3} - 3 \beta_{4} + \beta_{6} + 2 \beta_{9} - 2 \beta_{11} - 3 \beta_{12} - 2 \beta_{15} ) q^{11} + ( 4 \beta_{1} + 2 \beta_{8} ) q^{12} + ( 2 \beta_{1} + \beta_{2} - 4 \beta_{5} + \beta_{7} - 2 \beta_{8} + 8 \beta_{13} ) q^{13} + ( 4 - 2 \beta_{6} + 3 \beta_{9} + \beta_{15} ) q^{14} + ( -4 - 4 \beta_{6} ) q^{16} + ( -7 \beta_{1} - 3 \beta_{2} + \beta_{5} - 14 \beta_{8} + 3 \beta_{10} + \beta_{13} - 2 \beta_{14} ) q^{17} + ( -3 \beta_{5} + 3 \beta_{13} ) q^{18} + ( -9 + 2 \beta_{3} + 14 \beta_{4} + 9 \beta_{6} + 2 \beta_{11} + 7 \beta_{12} - \beta_{15} ) q^{19} + ( 4 \beta_{3} - 5 \beta_{4} - \beta_{11} - 4 \beta_{12} ) q^{21} + ( 6 \beta_{1} - 4 \beta_{2} + \beta_{5} + \beta_{7} + 6 \beta_{8} + 8 \beta_{10} - 2 \beta_{14} ) q^{22} + ( -6 \beta_{2} - 2 \beta_{7} + 3 \beta_{8} + 3 \beta_{10} - \beta_{13} - 2 \beta_{14} ) q^{23} + ( 2 \beta_{4} - 2 \beta_{12} ) q^{24} + ( -8 + 2 \beta_{3} + 4 \beta_{4} + 8 \beta_{6} + 2 \beta_{11} + 2 \beta_{12} + \beta_{15} ) q^{26} + ( 3 \beta_{1} - 3 \beta_{8} ) q^{27} + ( 2 \beta_{2} - 2 \beta_{5} + 4 \beta_{10} - 4 \beta_{13} ) q^{28} + ( -9 + 2 \beta_{3} - 13 \beta_{4} - 2 \beta_{9} + \beta_{11} - 4 \beta_{15} ) q^{29} + ( -22 - 2 \beta_{3} - \beta_{4} - 11 \beta_{6} + 5 \beta_{9} + \beta_{12} + 5 \beta_{15} ) q^{31} + ( -4 \beta_{5} + 4 \beta_{13} ) q^{32} + ( -2 \beta_{1} - 6 \beta_{5} + 6 \beta_{7} - \beta_{8} + 3 \beta_{10} + 3 \beta_{13} - 6 \beta_{14} ) q^{33} + ( 2 + 7 \beta_{4} + 4 \beta_{6} + 3 \beta_{9} - 4 \beta_{11} + 14 \beta_{12} ) q^{34} -6 q^{36} + ( -10 \beta_{2} - \beta_{7} + 24 \beta_{8} + 5 \beta_{10} - 2 \beta_{13} - \beta_{14} ) q^{37} + ( -14 \beta_{1} - 2 \beta_{2} + 9 \beta_{5} - 28 \beta_{8} + 2 \beta_{10} + 9 \beta_{13} + 2 \beta_{14} ) q^{38} + ( \beta_{3} + 12 \beta_{4} + 6 \beta_{6} - \beta_{9} + 2 \beta_{11} + 12 \beta_{12} + \beta_{15} ) q^{39} + ( 3 + \beta_{4} + 6 \beta_{6} + 4 \beta_{9} + 19 \beta_{11} + 2 \beta_{12} ) q^{41} + ( 8 \beta_{1} + 5 \beta_{7} + 10 \beta_{8} - \beta_{14} ) q^{42} + ( -14 \beta_{1} + 9 \beta_{2} - 5 \beta_{5} - 4 \beta_{7} - 14 \beta_{8} - 18 \beta_{10} + 8 \beta_{14} ) q^{43} + ( 2 + 2 \beta_{3} + 2 \beta_{6} + 8 \beta_{9} - 2 \beta_{11} - 6 \beta_{12} + 4 \beta_{15} ) q^{44} + ( -4 \beta_{3} - 3 \beta_{4} - 2 \beta_{6} + 3 \beta_{9} - 8 \beta_{11} - 3 \beta_{12} - 3 \beta_{15} ) q^{46} + ( 4 \beta_{1} - 40 \beta_{5} - 2 \beta_{7} + 2 \beta_{8} - 8 \beta_{10} + 20 \beta_{13} + 2 \beta_{14} ) q^{47} + ( 4 \beta_{1} - 4 \beta_{8} ) q^{48} + ( -24 - 9 \beta_{6} + 10 \beta_{9} - 6 \beta_{15} ) q^{49} + ( 21 + 3 \beta_{3} + 21 \beta_{6} + 4 \beta_{9} - 3 \beta_{11} + 3 \beta_{12} + 2 \beta_{15} ) q^{51} + ( -4 \beta_{1} + 2 \beta_{2} + 8 \beta_{5} - 8 \beta_{8} - 2 \beta_{10} + 8 \beta_{13} + 2 \beta_{14} ) q^{52} + ( 8 \beta_{1} - 4 \beta_{2} + 26 \beta_{5} + 16 \beta_{7} - 4 \beta_{10} - 26 \beta_{13} - 8 \beta_{14} ) q^{53} + ( 6 \beta_{4} + 3 \beta_{12} ) q^{54} + ( -4 - 12 \beta_{6} + 4 \beta_{9} + 6 \beta_{15} ) q^{56} + ( -27 \beta_{1} + 2 \beta_{2} + 21 \beta_{5} + \beta_{7} - 27 \beta_{8} - 4 \beta_{10} - 2 \beta_{14} ) q^{57} + ( -8 \beta_{2} + \beta_{7} + 26 \beta_{8} + 4 \beta_{10} + 9 \beta_{13} + \beta_{14} ) q^{58} + ( -22 + 21 \beta_{3} - 7 \beta_{4} - 11 \beta_{6} + 8 \beta_{9} + 7 \beta_{12} + 8 \beta_{15} ) q^{59} + ( 8 - 16 \beta_{3} - 4 \beta_{4} - 8 \beta_{6} - 16 \beta_{11} - 2 \beta_{12} - 16 \beta_{15} ) q^{61} + ( -2 \beta_{1} + 10 \beta_{2} - 11 \beta_{5} - 2 \beta_{7} + 2 \beta_{8} + 22 \beta_{13} ) q^{62} + ( 9 \beta_{2} - 9 \beta_{5} - 3 \beta_{10} + 3 \beta_{13} ) q^{63} -8 q^{64} + ( -12 + 12 \beta_{3} - \beta_{4} - 6 \beta_{6} + 3 \beta_{9} + \beta_{12} + 3 \beta_{15} ) q^{66} + ( 30 \beta_{1} - 3 \beta_{2} - 9 \beta_{5} - 16 \beta_{7} - 3 \beta_{10} + 9 \beta_{13} + 8 \beta_{14} ) q^{67} + ( -28 \beta_{1} + 4 \beta_{5} + 4 \beta_{7} - 14 \beta_{8} + 6 \beta_{10} - 2 \beta_{13} - 4 \beta_{14} ) q^{68} + ( -3 - \beta_{4} - 6 \beta_{6} + 6 \beta_{9} - 9 \beta_{11} - 2 \beta_{12} ) q^{69} + ( 1 - 14 \beta_{3} + 29 \beta_{4} + 6 \beta_{9} - 7 \beta_{11} + 12 \beta_{15} ) q^{71} + 6 \beta_{13} q^{72} + ( 2 \beta_{1} - 25 \beta_{2} - 10 \beta_{5} + 4 \beta_{8} + 25 \beta_{10} - 10 \beta_{13} + 15 \beta_{14} ) q^{73} + ( -2 \beta_{3} - 24 \beta_{4} - 4 \beta_{6} + 5 \beta_{9} - 4 \beta_{11} - 24 \beta_{12} - 5 \beta_{15} ) q^{74} + ( 18 + 14 \beta_{4} + 36 \beta_{6} + 2 \beta_{9} + 4 \beta_{11} + 28 \beta_{12} ) q^{76} + ( 31 \beta_{1} - 21 \beta_{2} - 49 \beta_{5} + 8 \beta_{7} + 23 \beta_{8} + 14 \beta_{10} + 42 \beta_{13} - 10 \beta_{14} ) q^{77} + ( -24 \beta_{1} + 2 \beta_{2} + 6 \beta_{5} - \beta_{7} - 24 \beta_{8} - 4 \beta_{10} + 2 \beta_{14} ) q^{78} + ( -3 + 2 \beta_{3} - 3 \beta_{6} + 38 \beta_{9} - 2 \beta_{11} - 13 \beta_{12} + 19 \beta_{15} ) q^{79} + 9 \beta_{6} q^{81} + ( -4 \beta_{1} + 6 \beta_{5} - 19 \beta_{7} - 2 \beta_{8} + 8 \beta_{10} - 3 \beta_{13} + 19 \beta_{14} ) q^{82} + ( -3 \beta_{1} - 7 \beta_{2} + 15 \beta_{5} + 34 \beta_{7} + 3 \beta_{8} - 30 \beta_{13} ) q^{83} + ( 10 \beta_{3} - 2 \beta_{4} + 8 \beta_{11} - 10 \beta_{12} ) q^{84} + ( -10 - 8 \beta_{3} - 10 \beta_{6} - 18 \beta_{9} + 8 \beta_{11} + 14 \beta_{12} - 9 \beta_{15} ) q^{86} + ( -9 \beta_{1} + 3 \beta_{2} - 13 \beta_{5} - 18 \beta_{8} - 3 \beta_{10} - 13 \beta_{13} - 6 \beta_{14} ) q^{87} + ( 12 \beta_{1} + 8 \beta_{2} + 2 \beta_{5} + 4 \beta_{7} + 8 \beta_{10} - 2 \beta_{13} - 2 \beta_{14} ) q^{88} + ( -41 + 19 \beta_{3} - 22 \beta_{4} + 41 \beta_{6} + 19 \beta_{11} - 11 \beta_{12} + 4 \beta_{15} ) q^{89} + ( -35 + 8 \beta_{3} + 18 \beta_{4} + 7 \beta_{6} - 14 \beta_{9} + 12 \beta_{11} + 13 \beta_{12} + 7 \beta_{15} ) q^{91} + ( 6 \beta_{1} - 6 \beta_{2} - 2 \beta_{5} + 4 \beta_{7} + 6 \beta_{8} + 12 \beta_{10} - 8 \beta_{14} ) q^{92} + ( -4 \beta_{2} + 5 \beta_{7} - 33 \beta_{8} + 2 \beta_{10} - 3 \beta_{13} + 5 \beta_{14} ) q^{93} + ( -80 - 4 \beta_{3} + 2 \beta_{4} - 40 \beta_{6} - 8 \beta_{9} - 2 \beta_{12} - 8 \beta_{15} ) q^{94} + ( 8 \beta_{4} + 4 \beta_{12} ) q^{96} + ( -36 \beta_{1} + 8 \beta_{2} + 20 \beta_{5} - 18 \beta_{7} + 36 \beta_{8} - 40 \beta_{13} ) q^{97} + ( -12 \beta_{2} - 9 \beta_{5} + 32 \beta_{10} + 24 \beta_{13} ) q^{98} + ( 3 + 6 \beta_{3} + 9 \beta_{4} + 6 \beta_{9} + 3 \beta_{11} + 12 \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 16q^{4} - 24q^{9} + O(q^{10}) \) \( 16q + 16q^{4} - 24q^{9} - 8q^{11} + 80q^{14} - 32q^{16} - 216q^{19} - 192q^{26} - 144q^{29} - 264q^{31} - 96q^{36} - 48q^{39} + 16q^{44} + 16q^{46} - 312q^{49} + 168q^{51} + 32q^{56} - 264q^{59} + 192q^{61} - 128q^{64} - 144q^{66} + 16q^{71} + 32q^{74} - 24q^{79} - 72q^{81} - 80q^{86} - 984q^{89} - 616q^{91} - 960q^{94} + 48q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 7 x^{12} + 48 x^{8} - 7 x^{4} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{14} + 377 \nu^{2} \)\()/144\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{12} - 161 \)\()/72\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{14} + 281 \nu^{2} \)\()/48\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{15} + 17 \nu^{13} - 120 \nu^{9} + 816 \nu^{5} - 305 \nu^{3} - 119 \nu \)\()/72\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{15} - 17 \nu^{13} + 120 \nu^{9} - 816 \nu^{5} - 305 \nu^{3} + 119 \nu \)\()/72\)
\(\beta_{6}\)\(=\)\((\)\( 7 \nu^{12} - 48 \nu^{8} + 336 \nu^{4} - 49 \)\()/48\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{15} + 19 \nu^{13} - 132 \nu^{9} + 912 \nu^{5} - 341 \nu^{3} - 133 \nu \)\()/36\)
\(\beta_{8}\)\(=\)\((\)\( 55 \nu^{14} - 384 \nu^{10} + 2640 \nu^{6} - 385 \nu^{2} \)\()/144\)
\(\beta_{9}\)\(=\)\((\)\( \nu^{15} + 19 \nu^{13} - 132 \nu^{9} + 912 \nu^{5} + 341 \nu^{3} - 133 \nu \)\()/36\)
\(\beta_{10}\)\(=\)\((\)\( -49 \nu^{12} + 336 \nu^{8} - 2256 \nu^{4} + 7 \)\()/144\)
\(\beta_{11}\)\(=\)\((\)\( -7 \nu^{14} + 48 \nu^{10} - 328 \nu^{6} + \nu^{2} \)\()/8\)
\(\beta_{12}\)\(=\)\((\)\( 91 \nu^{15} + \nu^{13} - 624 \nu^{11} + 4272 \nu^{7} - 13 \nu^{3} + 233 \nu \)\()/144\)
\(\beta_{13}\)\(=\)\((\)\( -91 \nu^{15} + \nu^{13} + 624 \nu^{11} - 4272 \nu^{7} + 13 \nu^{3} + 233 \nu \)\()/144\)
\(\beta_{14}\)\(=\)\((\)\( 199 \nu^{15} + 77 \nu^{13} - 1392 \nu^{11} - 528 \nu^{9} + 9552 \nu^{7} + 3648 \nu^{5} - 1393 \nu^{3} - 11 \nu \)\()/144\)
\(\beta_{15}\)\(=\)\((\)\( -203 \nu^{15} + \nu^{13} + 1392 \nu^{11} - 9552 \nu^{7} + 29 \nu^{3} + 521 \nu \)\()/144\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} - \beta_{7}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 3 \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{9} - \beta_{7} + 2 \beta_{5} + 2 \beta_{4}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{10} + 7 \beta_{6} + 7\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(5 \beta_{15} + 5 \beta_{14} - 11 \beta_{13} - 11 \beta_{12} + 5 \beta_{9} + 11 \beta_{5} - 11 \beta_{4}\)\()/4\)
\(\nu^{6}\)\(=\)\(4 \beta_{11} + 9 \beta_{8} + 9 \beta_{1}\)
\(\nu^{7}\)\(=\)\((\)\(-13 \beta_{15} + 13 \beta_{14} + 29 \beta_{13} - 29 \beta_{12} - 13 \beta_{7}\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(21 \beta_{10} + 47 \beta_{6} - 21 \beta_{2}\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(17 \beta_{9} + 17 \beta_{7} + 38 \beta_{5} - 38 \beta_{4}\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(55 \beta_{11} + 123 \beta_{8} + 55 \beta_{3}\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(-89 \beta_{15} + 89 \beta_{14} + 199 \beta_{13} - 199 \beta_{12} - 89 \beta_{9} - 199 \beta_{5} - 199 \beta_{4}\)\()/4\)
\(\nu^{12}\)\(=\)\(-72 \beta_{2} - 161\)
\(\nu^{13}\)\(=\)\((\)\(-233 \beta_{15} - 233 \beta_{14} + 521 \beta_{13} + 521 \beta_{12} + 233 \beta_{7}\)\()/4\)
\(\nu^{14}\)\(=\)\((\)\(377 \beta_{3} - 843 \beta_{1}\)\()/2\)
\(\nu^{15}\)\(=\)\((\)\(-305 \beta_{9} + 305 \beta_{7} - 682 \beta_{5} - 682 \beta_{4}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(-1\) \(1 + \beta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
−1.56290 0.418778i
0.596975 + 0.159959i
−0.418778 + 1.56290i
0.159959 0.596975i
1.56290 + 0.418778i
−0.596975 0.159959i
0.418778 1.56290i
−0.159959 + 0.596975i
−1.56290 + 0.418778i
0.596975 0.159959i
−0.418778 1.56290i
0.159959 + 0.596975i
1.56290 0.418778i
−0.596975 + 0.159959i
0.418778 + 1.56290i
−0.159959 0.596975i
−1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i −4.79227 + 5.10237i 2.82843i −1.50000 2.59808i 0
199.2 −1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i −2.55620 6.51658i 2.82843i −1.50000 2.59808i 0
199.3 −1.22474 + 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i −4.79227 + 5.10237i 2.82843i −1.50000 2.59808i 0
199.4 −1.22474 + 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i −2.55620 6.51658i 2.82843i −1.50000 2.59808i 0
199.5 1.22474 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i 2.55620 + 6.51658i 2.82843i −1.50000 2.59808i 0
199.6 1.22474 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i 4.79227 5.10237i 2.82843i −1.50000 2.59808i 0
199.7 1.22474 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i 2.55620 + 6.51658i 2.82843i −1.50000 2.59808i 0
199.8 1.22474 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i 4.79227 5.10237i 2.82843i −1.50000 2.59808i 0
649.1 −1.22474 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 0 2.44949i −4.79227 5.10237i 2.82843i −1.50000 + 2.59808i 0
649.2 −1.22474 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 0 2.44949i −2.55620 + 6.51658i 2.82843i −1.50000 + 2.59808i 0
649.3 −1.22474 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i −4.79227 5.10237i 2.82843i −1.50000 + 2.59808i 0
649.4 −1.22474 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i −2.55620 + 6.51658i 2.82843i −1.50000 + 2.59808i 0
649.5 1.22474 + 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 0 2.44949i 2.55620 6.51658i 2.82843i −1.50000 + 2.59808i 0
649.6 1.22474 + 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 0 2.44949i 4.79227 + 5.10237i 2.82843i −1.50000 + 2.59808i 0
649.7 1.22474 + 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i 2.55620 6.51658i 2.82843i −1.50000 + 2.59808i 0
649.8 1.22474 + 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i 4.79227 + 5.10237i 2.82843i −1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.d odd 6 1 inner
35.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.q.c 16
5.b even 2 1 inner 1050.3.q.c 16
5.c odd 4 1 210.3.o.a 8
5.c odd 4 1 1050.3.p.b 8
7.d odd 6 1 inner 1050.3.q.c 16
15.e even 4 1 630.3.v.b 8
35.i odd 6 1 inner 1050.3.q.c 16
35.k even 12 1 210.3.o.a 8
35.k even 12 1 1050.3.p.b 8
35.k even 12 1 1470.3.f.a 8
35.l odd 12 1 1470.3.f.a 8
105.w odd 12 1 630.3.v.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.o.a 8 5.c odd 4 1
210.3.o.a 8 35.k even 12 1
630.3.v.b 8 15.e even 4 1
630.3.v.b 8 105.w odd 12 1
1050.3.p.b 8 5.c odd 4 1
1050.3.p.b 8 35.k even 12 1
1050.3.q.c 16 1.a even 1 1 trivial
1050.3.q.c 16 5.b even 2 1 inner
1050.3.q.c 16 7.d odd 6 1 inner
1050.3.q.c 16 35.i odd 6 1 inner
1470.3.f.a 8 35.k even 12 1
1470.3.f.a 8 35.l odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{11}^{8} + \cdots\) acting on \(S_{3}^{\mathrm{new}}(1050, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T^{2} + 4 T^{4} )^{4} \)
$3$ \( ( 1 + 3 T^{2} + 9 T^{4} )^{4} \)
$5$ 1
$7$ \( ( 1 + 78 T^{2} + 5243 T^{4} + 187278 T^{6} + 5764801 T^{8} )^{2} \)
$11$ \( ( 1 + 4 T - 168 T^{2} + 1928 T^{3} + 16250 T^{4} - 341700 T^{5} + 2353408 T^{6} + 32486668 T^{7} - 296735661 T^{8} + 3930886828 T^{9} + 34456246528 T^{10} - 605342393700 T^{11} + 3483331816250 T^{12} + 50007354630728 T^{13} - 527255967289128 T^{14} + 1518999334332964 T^{15} + 45949729863572161 T^{16} )^{2} \)
$13$ \( ( 1 + 860 T^{2} + 357498 T^{4} + 95438800 T^{6} + 18541152803 T^{8} + 2725827566800 T^{10} + 291622101296058 T^{12} + 20036353205333660 T^{14} + 665416609183179841 T^{16} )^{2} \)
$17$ \( 1 - 1360 T^{2} + 1007892 T^{4} - 484918880 T^{6} + 158004387626 T^{8} - 28335498542640 T^{10} - 2581588059123248 T^{12} + 4022303846530970960 T^{14} - \)\(15\!\cdots\!61\)\( T^{16} + \)\(33\!\cdots\!60\)\( T^{18} - \)\(18\!\cdots\!68\)\( T^{20} - \)\(16\!\cdots\!40\)\( T^{22} + \)\(76\!\cdots\!06\)\( T^{24} - \)\(19\!\cdots\!80\)\( T^{26} + \)\(34\!\cdots\!32\)\( T^{28} - \)\(38\!\cdots\!60\)\( T^{30} + \)\(23\!\cdots\!61\)\( T^{32} \)
$19$ \( ( 1 + 108 T + 6142 T^{2} + 243432 T^{3} + 7548345 T^{4} + 198492408 T^{5} + 4660679966 T^{6} + 100240172052 T^{7} + 1986343374068 T^{8} + 36186702110772 T^{9} + 607384473849086 T^{10} + 9338250206171448 T^{11} + 128197793162717145 T^{12} + 1492497721269013032 T^{13} + 13594180232904360862 T^{14} + 86292722064551485068 T^{15} + \)\(28\!\cdots\!81\)\( T^{16} )^{2} \)
$23$ \( 1 + 3168 T^{2} + 5331380 T^{4} + 6217399872 T^{6} + 5630654580586 T^{8} + 4267434643483680 T^{10} + 2845042190919890768 T^{12} + \)\(17\!\cdots\!00\)\( T^{14} + \)\(94\!\cdots\!31\)\( T^{16} + \)\(48\!\cdots\!00\)\( T^{18} + \)\(22\!\cdots\!08\)\( T^{20} + \)\(93\!\cdots\!80\)\( T^{22} + \)\(34\!\cdots\!46\)\( T^{24} + \)\(10\!\cdots\!72\)\( T^{26} + \)\(25\!\cdots\!80\)\( T^{28} + \)\(42\!\cdots\!08\)\( T^{30} + \)\(37\!\cdots\!21\)\( T^{32} \)
$29$ \( ( 1 + 36 T + 2904 T^{2} + 82956 T^{3} + 3490070 T^{4} + 69765996 T^{5} + 2053944024 T^{6} + 21413639556 T^{7} + 500246412961 T^{8} )^{4} \)
$31$ \( ( 1 + 132 T + 11278 T^{2} + 722040 T^{3} + 38602473 T^{4} + 1778529960 T^{5} + 72524765390 T^{6} + 2637318955548 T^{7} + 86278260959732 T^{8} + 2534463516281628 T^{9} + 66978143857738190 T^{10} + 1578451886268782760 T^{11} + 32923703244758191593 T^{12} + \)\(59\!\cdots\!40\)\( T^{13} + \)\(88\!\cdots\!58\)\( T^{14} + \)\(99\!\cdots\!72\)\( T^{15} + \)\(72\!\cdots\!81\)\( T^{16} )^{2} \)
$37$ \( 1 + 6996 T^{2} + 25457990 T^{4} + 61371124392 T^{6} + 109756301878513 T^{8} + 161223035589295992 T^{10} + \)\(21\!\cdots\!38\)\( T^{12} + \)\(30\!\cdots\!40\)\( T^{14} + \)\(41\!\cdots\!52\)\( T^{16} + \)\(56\!\cdots\!40\)\( T^{18} + \)\(76\!\cdots\!98\)\( T^{20} + \)\(10\!\cdots\!52\)\( T^{22} + \)\(13\!\cdots\!33\)\( T^{24} + \)\(14\!\cdots\!92\)\( T^{26} + \)\(11\!\cdots\!90\)\( T^{28} + \)\(56\!\cdots\!16\)\( T^{30} + \)\(15\!\cdots\!81\)\( T^{32} \)
$41$ \( ( 1 - 5456 T^{2} + 19963884 T^{4} - 49821420976 T^{6} + 96897341176550 T^{8} - 140783428358562736 T^{10} + \)\(15\!\cdots\!64\)\( T^{12} - \)\(12\!\cdots\!36\)\( T^{14} + \)\(63\!\cdots\!41\)\( T^{16} )^{2} \)
$43$ \( ( 1 - 7028 T^{2} + 27746538 T^{4} - 77283436816 T^{6} + 162320522170835 T^{8} - 264216691069977616 T^{10} + \)\(32\!\cdots\!38\)\( T^{12} - \)\(28\!\cdots\!28\)\( T^{14} + \)\(13\!\cdots\!01\)\( T^{16} )^{2} \)
$47$ \( 1 - 6584 T^{2} + 15775396 T^{4} - 19188298768 T^{6} + 24343433903882 T^{8} + 18647543618687416 T^{10} - \)\(26\!\cdots\!68\)\( T^{12} + \)\(40\!\cdots\!52\)\( T^{14} - \)\(15\!\cdots\!69\)\( T^{16} + \)\(19\!\cdots\!12\)\( T^{18} - \)\(62\!\cdots\!48\)\( T^{20} + \)\(21\!\cdots\!56\)\( T^{22} + \)\(13\!\cdots\!22\)\( T^{24} - \)\(53\!\cdots\!68\)\( T^{26} + \)\(21\!\cdots\!76\)\( T^{28} - \)\(43\!\cdots\!24\)\( T^{30} + \)\(32\!\cdots\!41\)\( T^{32} \)
$53$ \( 1 + 8168 T^{2} + 48567460 T^{4} + 206177098672 T^{6} + 777590092376906 T^{8} + 2702263686540102680 T^{10} + \)\(88\!\cdots\!08\)\( T^{12} + \)\(27\!\cdots\!40\)\( T^{14} + \)\(79\!\cdots\!91\)\( T^{16} + \)\(21\!\cdots\!40\)\( T^{18} + \)\(54\!\cdots\!88\)\( T^{20} + \)\(13\!\cdots\!80\)\( T^{22} + \)\(30\!\cdots\!26\)\( T^{24} + \)\(63\!\cdots\!72\)\( T^{26} + \)\(11\!\cdots\!60\)\( T^{28} + \)\(15\!\cdots\!48\)\( T^{30} + \)\(15\!\cdots\!41\)\( T^{32} \)
$59$ \( ( 1 + 132 T + 15632 T^{2} + 1296768 T^{3} + 88851370 T^{4} + 4629146772 T^{5} + 218386477856 T^{6} + 7743785317668 T^{7} + 393482793133843 T^{8} + 26956116690802308 T^{9} + 2646267789699658016 T^{10} + \)\(19\!\cdots\!52\)\( T^{11} + \)\(13\!\cdots\!70\)\( T^{12} + \)\(66\!\cdots\!68\)\( T^{13} + \)\(27\!\cdots\!92\)\( T^{14} + \)\(81\!\cdots\!52\)\( T^{15} + \)\(21\!\cdots\!41\)\( T^{16} )^{2} \)
$61$ \( ( 1 - 96 T + 11380 T^{2} - 797568 T^{3} + 60840138 T^{4} - 3862327392 T^{5} + 197844384848 T^{6} - 13588740778464 T^{7} + 649379857320947 T^{8} - 50563704436664544 T^{9} + 2739321895348217168 T^{10} - \)\(19\!\cdots\!12\)\( T^{11} + \)\(11\!\cdots\!78\)\( T^{12} - \)\(56\!\cdots\!68\)\( T^{13} + \)\(30\!\cdots\!80\)\( T^{14} - \)\(94\!\cdots\!36\)\( T^{15} + \)\(36\!\cdots\!61\)\( T^{16} )^{2} \)
$67$ \( 1 + 23444 T^{2} + 292939206 T^{4} + 2427630494248 T^{6} + 14602519355979377 T^{8} + 65507724741505354104 T^{10} + \)\(22\!\cdots\!22\)\( T^{12} + \)\(58\!\cdots\!48\)\( T^{14} + \)\(18\!\cdots\!16\)\( T^{16} + \)\(11\!\cdots\!08\)\( T^{18} + \)\(89\!\cdots\!02\)\( T^{20} + \)\(53\!\cdots\!44\)\( T^{22} + \)\(24\!\cdots\!37\)\( T^{24} + \)\(80\!\cdots\!48\)\( T^{26} + \)\(19\!\cdots\!26\)\( T^{28} + \)\(31\!\cdots\!04\)\( T^{30} + \)\(27\!\cdots\!61\)\( T^{32} \)
$71$ \( ( 1 - 4 T + 13176 T^{2} - 338828 T^{3} + 79144886 T^{4} - 1708031948 T^{5} + 334824308856 T^{6} - 512401135684 T^{7} + 645753531245761 T^{8} )^{4} \)
$73$ \( 1 - 18684 T^{2} + 176413046 T^{4} - 952750589688 T^{6} + 2174868174650977 T^{8} + 10696706165275097976 T^{10} - \)\(13\!\cdots\!38\)\( T^{12} + \)\(84\!\cdots\!52\)\( T^{14} - \)\(44\!\cdots\!04\)\( T^{16} + \)\(23\!\cdots\!32\)\( T^{18} - \)\(11\!\cdots\!78\)\( T^{20} + \)\(24\!\cdots\!96\)\( T^{22} + \)\(14\!\cdots\!97\)\( T^{24} - \)\(17\!\cdots\!88\)\( T^{26} + \)\(92\!\cdots\!86\)\( T^{28} - \)\(27\!\cdots\!04\)\( T^{30} + \)\(42\!\cdots\!21\)\( T^{32} \)
$79$ \( ( 1 + 12 T - 2418 T^{2} - 386472 T^{3} - 58917335 T^{4} + 324907032 T^{5} + 66125106126 T^{6} + 10581169939908 T^{7} + 1991860414252788 T^{8} + 66037081594965828 T^{9} + 2575578239741296206 T^{10} + 78980823689760123672 T^{11} - \)\(89\!\cdots\!35\)\( T^{12} - \)\(36\!\cdots\!72\)\( T^{13} - \)\(14\!\cdots\!38\)\( T^{14} + \)\(44\!\cdots\!72\)\( T^{15} + \)\(23\!\cdots\!21\)\( T^{16} )^{2} \)
$83$ \( ( 1 + 2384 T^{2} + 54393900 T^{4} + 170213772976 T^{6} + 3204332319622694 T^{8} + 8078059876516133296 T^{10} + \)\(12\!\cdots\!00\)\( T^{12} + \)\(25\!\cdots\!24\)\( T^{14} + \)\(50\!\cdots\!81\)\( T^{16} )^{2} \)
$89$ \( ( 1 + 492 T + 137248 T^{2} + 27827520 T^{3} + 4512312618 T^{4} + 615183153660 T^{5} + 72837035721440 T^{6} + 7634393359602348 T^{7} + 716618506839255347 T^{8} + 60472029801410198508 T^{9} + \)\(45\!\cdots\!40\)\( T^{10} + \)\(30\!\cdots\!60\)\( T^{11} + \)\(17\!\cdots\!58\)\( T^{12} + \)\(86\!\cdots\!20\)\( T^{13} + \)\(33\!\cdots\!08\)\( T^{14} + \)\(96\!\cdots\!72\)\( T^{15} + \)\(15\!\cdots\!61\)\( T^{16} )^{2} \)
$97$ \( ( 1 + 35880 T^{2} + 724155484 T^{4} + 9811041724440 T^{6} + 103976225413471686 T^{8} + \)\(86\!\cdots\!40\)\( T^{10} + \)\(56\!\cdots\!24\)\( T^{12} + \)\(24\!\cdots\!80\)\( T^{14} + \)\(61\!\cdots\!21\)\( T^{16} )^{2} \)
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