Properties

Label 1050.3.q.b
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: 16.0.22986704741655040229376.1
Defining polynomial: \(x^{16} - 31 x^{12} + 880 x^{8} - 2511 x^{4} + 6561\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q -\beta_{11} q^{2} + ( -\beta_{1} - \beta_{9} ) q^{3} + ( 2 + 2 \beta_{3} ) q^{4} + ( -\beta_{6} - 2 \beta_{12} ) q^{6} + ( -2 \beta_{7} - 2 \beta_{8} + \beta_{15} ) q^{7} + 2 \beta_{7} q^{8} + 3 \beta_{3} q^{9} +O(q^{10})\) \( q -\beta_{11} q^{2} + ( -\beta_{1} - \beta_{9} ) q^{3} + ( 2 + 2 \beta_{3} ) q^{4} + ( -\beta_{6} - 2 \beta_{12} ) q^{6} + ( -2 \beta_{7} - 2 \beta_{8} + \beta_{15} ) q^{7} + 2 \beta_{7} q^{8} + 3 \beta_{3} q^{9} + ( -1 - \beta_{3} + \beta_{10} - \beta_{12} - \beta_{14} ) q^{11} + ( 2 \beta_{1} - 4 \beta_{9} ) q^{12} + ( -2 \beta_{1} - 2 \beta_{2} + 4 \beta_{7} + 2 \beta_{8} + \beta_{9} + 6 \beta_{11} ) q^{13} + ( 1 - 2 \beta_{3} + 3 \beta_{5} + \beta_{13} ) q^{14} + 4 \beta_{3} q^{16} + ( 2 \beta_{1} - 5 \beta_{2} - 5 \beta_{4} - \beta_{7} + 2 \beta_{9} - 2 \beta_{11} - 3 \beta_{15} ) q^{17} + ( 3 \beta_{7} + 3 \beta_{11} ) q^{18} + ( -12 - 6 \beta_{3} + 2 \beta_{5} + 8 \beta_{6} - 7 \beta_{12} - \beta_{14} ) q^{19} + ( \beta_{6} + 4 \beta_{10} + \beta_{12} - \beta_{14} ) q^{21} + ( -\beta_{2} - 2 \beta_{4} - \beta_{7} - 3 \beta_{9} ) q^{22} + ( 15 \beta_{1} - 2 \beta_{2} - \beta_{4} + 2 \beta_{8} - 5 \beta_{11} - 2 \beta_{15} ) q^{23} + ( -4 \beta_{6} - 2 \beta_{12} ) q^{24} + ( -12 - 6 \beta_{3} - 2 \beta_{5} + 3 \beta_{6} + \beta_{12} - 4 \beta_{14} ) q^{26} + ( 6 \beta_{1} - 3 \beta_{9} ) q^{27} + ( 2 \beta_{7} + 2 \beta_{8} + 6 \beta_{15} ) q^{28} + ( 3 + \beta_{5} - 7 \beta_{6} - 4 \beta_{10} + 4 \beta_{12} - \beta_{13} - 8 \beta_{14} ) q^{29} + ( 8 - 8 \beta_{3} - 8 \beta_{6} - 2 \beta_{10} - 4 \beta_{12} + 4 \beta_{13} - 2 \beta_{14} ) q^{31} + ( 4 \beta_{7} + 4 \beta_{11} ) q^{32} + ( -\beta_{1} + 6 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} + 3 \beta_{11} + 3 \beta_{15} ) q^{33} + ( 1 + 2 \beta_{3} - 3 \beta_{5} - 3 \beta_{6} + 10 \beta_{10} + 4 \beta_{12} - 3 \beta_{13} ) q^{34} -6 q^{36} + ( 6 \beta_{1} - 10 \beta_{2} - 5 \beta_{4} - 2 \beta_{11} ) q^{37} + ( -15 \beta_{1} - \beta_{2} - \beta_{4} - 4 \beta_{7} - 15 \beta_{9} + 8 \beta_{11} + 4 \beta_{15} ) q^{38} + ( 3 + 3 \beta_{3} + 4 \beta_{5} - 2 \beta_{10} + 8 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} ) q^{39} + ( -1 - 2 \beta_{3} + 8 \beta_{5} + 6 \beta_{6} - 7 \beta_{10} + 5 \beta_{12} + 8 \beta_{13} ) q^{41} + ( -5 \beta_{1} - \beta_{2} - 5 \beta_{4} + \beta_{9} ) q^{42} + ( -4 \beta_{2} - 8 \beta_{4} - 10 \beta_{7} + 8 \beta_{8} - 14 \beta_{9} + 8 \beta_{11} + 16 \beta_{15} ) q^{43} + ( -2 \beta_{3} - 6 \beta_{6} + 4 \beta_{10} - 4 \beta_{12} + 2 \beta_{14} ) q^{44} + ( 8 + 8 \beta_{3} - 4 \beta_{5} + 2 \beta_{10} + 16 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{46} + ( 7 \beta_{1} - 10 \beta_{4} - 10 \beta_{7} - 13 \beta_{8} - 14 \beta_{9} - 5 \beta_{11} - 13 \beta_{15} ) q^{47} + ( 8 \beta_{1} - 4 \beta_{9} ) q^{48} + ( 48 + 30 \beta_{3} - 3 \beta_{5} - 8 \beta_{13} ) q^{49} + ( -6 \beta_{3} + 5 \beta_{5} + 3 \beta_{6} - 6 \beta_{10} + 10 \beta_{13} - 3 \beta_{14} ) q^{51} + ( -2 \beta_{1} - 4 \beta_{2} - 4 \beta_{4} - 8 \beta_{7} - 2 \beta_{9} + 4 \beta_{11} - 4 \beta_{15} ) q^{52} + ( -11 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + 4 \beta_{7} + 18 \beta_{8} + 11 \beta_{9} - 5 \beta_{11} + 9 \beta_{15} ) q^{53} + ( -3 \beta_{6} + 3 \beta_{12} ) q^{54} + ( 6 + 2 \beta_{3} + 4 \beta_{5} + 6 \beta_{13} ) q^{56} + ( -2 \beta_{2} - 4 \beta_{4} + 24 \beta_{7} + \beta_{8} + 18 \beta_{9} + \beta_{11} + 2 \beta_{15} ) q^{57} + ( 26 \beta_{1} - 8 \beta_{2} - 4 \beta_{4} - 2 \beta_{8} - 2 \beta_{11} + 2 \beta_{15} ) q^{58} + ( -26 + 26 \beta_{3} + 24 \beta_{6} - 21 \beta_{10} + 12 \beta_{12} - 5 \beta_{13} - 21 \beta_{14} ) q^{59} + ( -34 - 17 \beta_{3} - 10 \beta_{5} + 2 \beta_{6} + 8 \beta_{12} - 10 \beta_{14} ) q^{61} + ( 20 \beta_{1} - 2 \beta_{2} - 4 \beta_{7} + 8 \beta_{8} - 10 \beta_{9} - 16 \beta_{11} ) q^{62} + ( 9 \beta_{7} + 9 \beta_{8} + 6 \beta_{15} ) q^{63} -8 q^{64} + ( -3 + 3 \beta_{3} + 2 \beta_{6} + \beta_{12} + 3 \beta_{13} ) q^{66} + ( 30 \beta_{1} - 21 \beta_{7} - 6 \beta_{8} - 30 \beta_{9} - 18 \beta_{11} - 3 \beta_{15} ) q^{67} + ( -4 \beta_{1} - 10 \beta_{4} - 4 \beta_{7} - 6 \beta_{8} + 8 \beta_{9} - 2 \beta_{11} - 6 \beta_{15} ) q^{68} + ( -15 - 30 \beta_{3} + 3 \beta_{5} - \beta_{6} - 6 \beta_{10} - 8 \beta_{12} + 3 \beta_{13} ) q^{69} + ( -8 - 12 \beta_{5} - 4 \beta_{6} + 4 \beta_{10} - 4 \beta_{12} + 12 \beta_{13} + 8 \beta_{14} ) q^{71} + 6 \beta_{11} q^{72} + ( 7 \beta_{1} - 24 \beta_{2} - 24 \beta_{4} + 7 \beta_{9} - 10 \beta_{11} - 10 \beta_{15} ) q^{73} + ( 4 + 4 \beta_{3} + 10 \beta_{10} + 11 \beta_{12} - 10 \beta_{14} ) q^{74} + ( -12 - 24 \beta_{3} + 4 \beta_{5} - 16 \beta_{6} + 2 \beta_{10} - 30 \beta_{12} + 4 \beta_{13} ) q^{76} + ( -11 \beta_{1} - 5 \beta_{2} - 4 \beta_{4} - \beta_{7} - \beta_{8} - 23 \beta_{9} - 3 \beta_{15} ) q^{77} + ( 2 \beta_{2} + 4 \beta_{4} + 9 \beta_{7} + 4 \beta_{8} + 18 \beta_{9} + 4 \beta_{11} + 8 \beta_{15} ) q^{78} + ( -36 \beta_{3} - 12 \beta_{5} + 7 \beta_{6} - 2 \beta_{10} + 6 \beta_{12} - 24 \beta_{13} - \beta_{14} ) q^{79} + ( -9 - 9 \beta_{3} ) q^{81} + ( -5 \beta_{1} + 7 \beta_{4} + 14 \beta_{7} + 16 \beta_{8} + 10 \beta_{9} + 7 \beta_{11} + 16 \beta_{15} ) q^{82} + ( 32 \beta_{1} + \beta_{2} + 20 \beta_{7} + 7 \beta_{8} - 16 \beta_{9} + 33 \beta_{11} ) q^{83} + ( -8 \beta_{6} + 10 \beta_{10} - 8 \beta_{12} + 8 \beta_{14} ) q^{84} + ( -44 \beta_{3} + 8 \beta_{5} - 26 \beta_{6} + 16 \beta_{10} - 18 \beta_{12} + 16 \beta_{13} + 8 \beta_{14} ) q^{86} + ( -3 \beta_{1} - 3 \beta_{2} - 3 \beta_{4} - 7 \beta_{7} - 3 \beta_{9} + 19 \beta_{11} + 12 \beta_{15} ) q^{87} + ( 6 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{7} - 6 \beta_{9} - 2 \beta_{11} ) q^{88} + ( 56 + 28 \beta_{3} + \beta_{5} + 21 \beta_{6} - 10 \beta_{12} - 11 \beta_{14} ) q^{89} + ( -36 + 2 \beta_{3} - 10 \beta_{5} + 35 \beta_{6} - 7 \beta_{10} - 7 \beta_{12} + 6 \beta_{13} - 7 \beta_{14} ) q^{91} + ( -2 \beta_{2} - 4 \beta_{4} + 2 \beta_{7} - 4 \beta_{8} + 30 \beta_{9} - 4 \beta_{11} - 8 \beta_{15} ) q^{92} + ( -24 \beta_{1} + 8 \beta_{2} + 4 \beta_{4} - 2 \beta_{8} + 16 \beta_{11} + 2 \beta_{15} ) q^{93} + ( -3 + 3 \beta_{3} - 34 \beta_{6} + 20 \beta_{10} - 17 \beta_{12} - 13 \beta_{13} + 20 \beta_{14} ) q^{94} + ( -4 \beta_{6} + 4 \beta_{12} ) q^{96} + ( -48 \beta_{1} + 3 \beta_{2} - 26 \beta_{8} + 24 \beta_{9} + 26 \beta_{11} ) q^{97} + ( 19 \beta_{7} - 16 \beta_{8} - 21 \beta_{11} - 6 \beta_{15} ) q^{98} + ( 3 - 9 \beta_{6} + 3 \beta_{10} - 3 \beta_{12} + 6 \beta_{14} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{4} - 24 q^{9} + O(q^{10}) \) \( 16 q + 16 q^{4} - 24 q^{9} - 8 q^{11} + 32 q^{14} - 32 q^{16} - 144 q^{19} - 144 q^{26} + 48 q^{29} + 192 q^{31} - 96 q^{36} + 24 q^{39} + 16 q^{44} + 64 q^{46} + 528 q^{49} + 48 q^{51} + 80 q^{56} - 624 q^{59} - 408 q^{61} - 128 q^{64} - 72 q^{66} - 128 q^{71} + 32 q^{74} + 288 q^{79} - 72 q^{81} + 352 q^{86} + 672 q^{89} - 592 q^{91} - 72 q^{94} + 48 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 31 x^{12} + 880 x^{8} - 2511 x^{4} + 6561\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{14} + 32689 \nu^{2} \)\()/55440\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{12} + 11129 \)\()/3080\)
\(\beta_{3}\)\(=\)\((\)\( 31 \nu^{12} - 880 \nu^{8} + 27280 \nu^{4} - 77841 \)\()/71280\)
\(\beta_{4}\)\(=\)\((\)\( 799 \nu^{12} - 27280 \nu^{8} + 703120 \nu^{4} - 2006289 \)\()/498960\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{14} + 16849 \nu^{2} \)\()/7920\)
\(\beta_{6}\)\(=\)\((\)\( -9 \nu^{15} + 127 \nu^{13} - 4180 \nu^{9} + 111760 \nu^{5} - 169461 \nu^{3} - 318897 \nu \)\()/374220\)
\(\beta_{7}\)\(=\)\((\)\( 9 \nu^{15} + 127 \nu^{13} - 4180 \nu^{9} + 111760 \nu^{5} + 169461 \nu^{3} - 318897 \nu \)\()/374220\)
\(\beta_{8}\)\(=\)\((\)\( -3 \nu^{15} + 31 \nu^{13} - 880 \nu^{9} + 27280 \nu^{5} - 74307 \nu^{3} - 77841 \nu \)\()/71280\)
\(\beta_{9}\)\(=\)\((\)\( 31 \nu^{14} - 880 \nu^{10} + 25498 \nu^{6} - 6561 \nu^{2} \)\()/112266\)
\(\beta_{10}\)\(=\)\((\)\( -3 \nu^{15} - 31 \nu^{13} + 880 \nu^{9} - 27280 \nu^{5} - 74307 \nu^{3} + 77841 \nu \)\()/71280\)
\(\beta_{11}\)\(=\)\((\)\( 2683 \nu^{15} - 5301 \nu^{13} - 85360 \nu^{11} + 150480 \nu^{9} + 2361040 \nu^{7} - 4023360 \nu^{5} - 6737013 \nu^{3} + 1121931 \nu \)\()/13471920\)
\(\beta_{12}\)\(=\)\((\)\( 3007 \nu^{15} + 729 \nu^{13} - 85360 \nu^{11} + 2361040 \nu^{7} - 636417 \nu^{3} + 10358361 \nu \)\()/13471920\)
\(\beta_{13}\)\(=\)\((\)\( 601 \nu^{14} - 19360 \nu^{10} + 528880 \nu^{6} - 1509111 \nu^{2} \)\()/641520\)
\(\beta_{14}\)\(=\)\((\)\( -1159 \nu^{15} + 3720 \nu^{13} + 35200 \nu^{11} - 105600 \nu^{9} - 1019920 \nu^{7} + 3059760 \nu^{5} + 2910249 \nu^{3} - 787320 \nu \)\()/4490640\)
\(\beta_{15}\)\(=\)\((\)\( 1240 \nu^{15} + 243 \nu^{13} - 35200 \nu^{11} + 1019920 \nu^{7} - 262440 \nu^{3} + 7943427 \nu \)\()/4490640\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{15} + \beta_{14} - \beta_{12} + \beta_{11} + \beta_{10} + \beta_{7} - \beta_{6}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{5} + 7 \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-4 \beta_{10} - 4 \beta_{8} - 7 \beta_{7} + 7 \beta_{6}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-7 \beta_{4} + 31 \beta_{3} - 7 \beta_{2} + 31\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(19 \beta_{15} + 19 \beta_{14} - 40 \beta_{12} + 40 \beta_{11} + 19 \beta_{8} + 19 \beta_{7} - 40 \beta_{6}\)\()/2\)
\(\nu^{6}\)\(=\)\(-20 \beta_{13} + 77 \beta_{9} - 20 \beta_{5}\)
\(\nu^{7}\)\(=\)\((\)\(97 \beta_{15} - 97 \beta_{14} - 120 \beta_{12} - 120 \beta_{11} - 97 \beta_{10} - 120 \beta_{7} + 97 \beta_{6}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(-217 \beta_{4} + 799 \beta_{3}\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-508 \beta_{10} + 508 \beta_{8} - 651 \beta_{7} - 651 \beta_{6}\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(-1159 \beta_{13} + 4207 \beta_{9} - 4207 \beta_{1}\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(2683 \beta_{15} - 2683 \beta_{14} - 3477 \beta_{12} - 3477 \beta_{11} + 2683 \beta_{8} + 2683 \beta_{7} - 3477 \beta_{6}\)\()/2\)
\(\nu^{12}\)\(=\)\(3080 \beta_{2} - 11129\)
\(\nu^{13}\)\(=\)\((\)\(-14209 \beta_{15} - 14209 \beta_{14} + 32689 \beta_{12} - 32689 \beta_{11} - 14209 \beta_{10} - 32689 \beta_{7} + 14209 \beta_{6}\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(32689 \beta_{5} - 117943 \beta_{1}\)\()/2\)
\(\nu^{15}\)\(=\)\((\)\(75316 \beta_{10} + 75316 \beta_{8} + 173383 \beta_{7} - 173383 \beta_{6}\)\()/2\)

Character values

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

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(-1\) \(-\beta_{3}\) \(1\)

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
−1.25838 + 0.337183i
2.22431 0.596002i
0.596002 + 2.22431i
−0.337183 1.25838i
−2.22431 + 0.596002i
1.25838 0.337183i
0.337183 + 1.25838i
−0.596002 2.22431i
−1.25838 0.337183i
2.22431 + 0.596002i
0.596002 2.22431i
−0.337183 + 1.25838i
−2.22431 0.596002i
1.25838 + 0.337183i
0.337183 1.25838i
−0.596002 + 2.22431i
−1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i −6.98615 + 0.440173i 2.82843i −1.50000 2.59808i 0
199.2 −1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i 5.76140 3.97571i 2.82843i −1.50000 2.59808i 0
199.3 −1.22474 + 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i −6.98615 + 0.440173i 2.82843i −1.50000 2.59808i 0
199.4 −1.22474 + 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i 5.76140 3.97571i 2.82843i −1.50000 2.59808i 0
199.5 1.22474 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i −5.76140 + 3.97571i 2.82843i −1.50000 2.59808i 0
199.6 1.22474 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i 6.98615 0.440173i 2.82843i −1.50000 2.59808i 0
199.7 1.22474 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i −5.76140 + 3.97571i 2.82843i −1.50000 2.59808i 0
199.8 1.22474 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i 6.98615 0.440173i 2.82843i −1.50000 2.59808i 0
649.1 −1.22474 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 0 2.44949i −6.98615 0.440173i 2.82843i −1.50000 + 2.59808i 0
649.2 −1.22474 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 0 2.44949i 5.76140 + 3.97571i 2.82843i −1.50000 + 2.59808i 0
649.3 −1.22474 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i −6.98615 0.440173i 2.82843i −1.50000 + 2.59808i 0
649.4 −1.22474 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i 5.76140 + 3.97571i 2.82843i −1.50000 + 2.59808i 0
649.5 1.22474 + 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 0 2.44949i −5.76140 3.97571i 2.82843i −1.50000 + 2.59808i 0
649.6 1.22474 + 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 0 2.44949i 6.98615 + 0.440173i 2.82843i −1.50000 + 2.59808i 0
649.7 1.22474 + 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i −5.76140 3.97571i 2.82843i −1.50000 + 2.59808i 0
649.8 1.22474 + 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i 6.98615 + 0.440173i 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.b 16
5.b even 2 1 inner 1050.3.q.b 16
5.c odd 4 1 1050.3.p.c 8
5.c odd 4 1 1050.3.p.d yes 8
7.d odd 6 1 inner 1050.3.q.b 16
35.i odd 6 1 inner 1050.3.q.b 16
35.k even 12 1 1050.3.p.c 8
35.k even 12 1 1050.3.p.d yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.3.p.c 8 5.c odd 4 1
1050.3.p.c 8 35.k even 12 1
1050.3.p.d yes 8 5.c odd 4 1
1050.3.p.d yes 8 35.k even 12 1
1050.3.q.b 16 1.a even 1 1 trivial
1050.3.q.b 16 5.b even 2 1 inner
1050.3.q.b 16 7.d odd 6 1 inner
1050.3.q.b 16 35.i odd 6 1 inner

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

$p$ $F_p(T)$
$2$ \( ( 4 - 2 T^{2} + T^{4} )^{4} \)
$3$ \( ( 9 + 3 T^{2} + T^{4} )^{4} \)
$5$ \( T^{16} \)
$7$ \( ( 5764801 - 316932 T^{2} + 8183 T^{4} - 132 T^{6} + T^{8} )^{2} \)
$11$ \( ( 31684 - 16376 T + 15940 T^{2} + 2440 T^{3} + 1954 T^{4} + 16 T^{5} + 58 T^{6} + 4 T^{7} + T^{8} )^{2} \)
$13$ \( ( 3500641 - 686940 T^{2} + 39206 T^{4} - 540 T^{6} + T^{8} )^{2} \)
$17$ \( 18359080540098055696 + 453978732501277024 T^{2} + 8124090304153888 T^{4} + 63091715338528 T^{6} + 351511714972 T^{8} + 937667824 T^{10} + 1797832 T^{12} + 1588 T^{14} + T^{16} \)
$19$ \( ( 648364369 + 647167608 T + 218991024 T^{2} + 3659904 T^{3} - 563785 T^{4} - 10368 T^{5} + 1584 T^{6} + 72 T^{7} + T^{8} )^{2} \)
$23$ \( 66723634538745856 - 4125623362015232 T^{2} + 179634194532352 T^{4} - 3892904508416 T^{6} + 61186873792 T^{8} - 405080192 T^{10} + 1945888 T^{12} - 1496 T^{14} + T^{16} \)
$29$ \( ( -109496 + 24384 T - 1324 T^{2} - 12 T^{3} + T^{4} )^{4} \)
$31$ \( ( 641507584 - 150752256 T + 2082816 T^{2} + 2285568 T^{3} - 17680 T^{4} - 36864 T^{5} + 3456 T^{6} - 96 T^{7} + T^{8} )^{2} \)
$37$ \( \)\(56\!\cdots\!61\)\( - \)\(26\!\cdots\!04\)\( T^{2} + 8166234915161391034 T^{4} - 14760028113880496 T^{6} + 19425683272579 T^{8} - 16893495344 T^{10} + 10729114 T^{12} - 4076 T^{14} + T^{16} \)
$41$ \( ( 16384512004 + 1721127432 T^{2} + 6279368 T^{4} + 4764 T^{6} + T^{8} )^{2} \)
$43$ \( ( 41573196361984 + 77130981120 T^{2} + 48574304 T^{4} + 12144 T^{6} + T^{8} )^{2} \)
$47$ \( \)\(16\!\cdots\!76\)\( + \)\(15\!\cdots\!84\)\( T^{2} + \)\(10\!\cdots\!40\)\( T^{4} + 302798287757861472 T^{6} + 622084524264156 T^{8} + 255611120208 T^{10} + 77449480 T^{12} + 10236 T^{14} + T^{16} \)
$53$ \( \)\(24\!\cdots\!16\)\( - \)\(31\!\cdots\!32\)\( T^{2} + \)\(39\!\cdots\!84\)\( T^{4} - 438593213201466144 T^{6} + 324855692269788 T^{8} - 140002684656 T^{10} + 44304424 T^{12} - 8148 T^{14} + T^{16} \)
$59$ \( ( 14372454374404 + 1578478393128 T + 35957161516 T^{2} - 2397423912 T^{3} - 6356190 T^{4} + 1796496 T^{5} + 38206 T^{6} + 312 T^{7} + T^{8} )^{2} \)
$61$ \( ( 16806974336161 + 1334626672788 T + 30432207354 T^{2} - 388704312 T^{3} - 16611997 T^{4} + 243576 T^{5} + 15066 T^{6} + 204 T^{7} + T^{8} )^{2} \)
$67$ \( \)\(14\!\cdots\!41\)\( - \)\(39\!\cdots\!80\)\( T^{2} + 9280871921114078238 T^{4} - 30132525002083200 T^{6} + 74899282169763 T^{8} - 57015964800 T^{10} + 32192478 T^{12} - 6480 T^{14} + T^{16} \)
$71$ \( ( 27042304 - 217856 T - 11488 T^{2} + 32 T^{3} + T^{4} )^{4} \)
$73$ \( \)\(39\!\cdots\!21\)\( + \)\(28\!\cdots\!40\)\( T^{2} + \)\(13\!\cdots\!18\)\( T^{4} + \)\(37\!\cdots\!00\)\( T^{6} + 74176042946018083 T^{8} + 9040596071600 T^{10} + 785666938 T^{12} + 33740 T^{14} + T^{16} \)
$79$ \( ( 334870712077729 + 11217723344784 T + 441949702432 T^{2} + 3053611296 T^{3} + 83049135 T^{4} - 705312 T^{5} + 24352 T^{6} - 144 T^{7} + T^{8} )^{2} \)
$83$ \( ( 29997547204 - 12752786136 T^{2} + 24918152 T^{4} - 10932 T^{6} + T^{8} )^{2} \)
$89$ \( ( 9916238788036 + 284959850952 T - 27595327092 T^{2} - 871437960 T^{3} + 106021010 T^{4} - 3235680 T^{5} + 47262 T^{6} - 336 T^{7} + T^{8} )^{2} \)
$97$ \( ( 2745758363089 - 203590226420 T^{2} + 152995446 T^{4} - 29012 T^{6} + T^{8} )^{2} \)
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