Properties

Label 1050.3.h.a
Level $1050$
Weight $3$
Character orbit 1050.h
Analytic conductor $28.610$
Analytic rank $0$
Dimension $8$
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.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.6104277578\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{2} + ( -2 \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{3} -2 q^{4} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{6} + ( \zeta_{24} - 2 \zeta_{24}^{2} - 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} + 3 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{7} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{8} + 3 q^{9} +O(q^{10})\) \( q + ( -\zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{2} + ( -2 \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{3} -2 q^{4} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{6} + ( \zeta_{24} - 2 \zeta_{24}^{2} - 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} + 3 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{7} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{8} + 3 q^{9} + ( -6 - 3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{11} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{12} + ( -2 \zeta_{24} + 16 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 8 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{13} + ( 10 - 3 \zeta_{24} - \zeta_{24}^{3} - 8 \zeta_{24}^{4} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{14} + 4 q^{16} + ( 11 \zeta_{24} + 4 \zeta_{24}^{2} + 11 \zeta_{24}^{3} + 11 \zeta_{24}^{5} - 2 \zeta_{24}^{6} - 22 \zeta_{24}^{7} ) q^{17} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{18} + ( -2 - 8 \zeta_{24} + 8 \zeta_{24}^{3} + 4 \zeta_{24}^{4} - 8 \zeta_{24}^{5} - 16 \zeta_{24}^{7} ) q^{19} + ( 5 + 9 \zeta_{24} + 3 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 3 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{21} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - 6 \zeta_{24}^{6} ) q^{22} + ( -9 \zeta_{24} + 9 \zeta_{24}^{3} + 9 \zeta_{24}^{5} - 6 \zeta_{24}^{6} ) q^{23} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{24} + ( 4 + 8 \zeta_{24} - 8 \zeta_{24}^{3} - 8 \zeta_{24}^{4} + 8 \zeta_{24}^{5} + 16 \zeta_{24}^{7} ) q^{26} + ( -6 \zeta_{24}^{2} + 3 \zeta_{24}^{6} ) q^{27} + ( -2 \zeta_{24} + 4 \zeta_{24}^{2} + 10 \zeta_{24}^{3} + 10 \zeta_{24}^{5} - 6 \zeta_{24}^{6} - 8 \zeta_{24}^{7} ) q^{28} -30 q^{29} + ( 12 - 12 \zeta_{24} + 12 \zeta_{24}^{3} - 24 \zeta_{24}^{4} - 12 \zeta_{24}^{5} - 24 \zeta_{24}^{7} ) q^{31} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{32} + ( 3 \zeta_{24} + 12 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 6 \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{33} + ( -22 + 2 \zeta_{24} - 2 \zeta_{24}^{3} + 44 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{34} -6 q^{36} + ( -36 \zeta_{24} + 36 \zeta_{24}^{3} + 36 \zeta_{24}^{5} - 20 \zeta_{24}^{6} ) q^{37} + ( -2 \zeta_{24} + 32 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 16 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{38} + ( -24 + 6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{39} + ( -14 - 7 \zeta_{24} + 7 \zeta_{24}^{3} + 28 \zeta_{24}^{4} - 7 \zeta_{24}^{5} - 14 \zeta_{24}^{7} ) q^{41} + ( -\zeta_{24} - 12 \zeta_{24}^{2} + 5 \zeta_{24}^{3} + 5 \zeta_{24}^{5} + 18 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{42} + ( -30 \zeta_{24} + 30 \zeta_{24}^{3} + 30 \zeta_{24}^{5} + 32 \zeta_{24}^{6} ) q^{43} + ( 12 + 6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{44} + ( -18 + 6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{46} + ( -4 \zeta_{24} - 56 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 28 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{47} + ( -8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{48} + ( -15 + 22 \zeta_{24} + 26 \zeta_{24}^{3} + 40 \zeta_{24}^{4} - 26 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{49} + ( -6 - 33 \zeta_{24} - 33 \zeta_{24}^{3} + 33 \zeta_{24}^{5} ) q^{51} + ( 4 \zeta_{24} - 32 \zeta_{24}^{2} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 16 \zeta_{24}^{6} - 8 \zeta_{24}^{7} ) q^{52} + ( -12 \zeta_{24} + 12 \zeta_{24}^{3} + 12 \zeta_{24}^{5} + 54 \zeta_{24}^{6} ) q^{53} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{54} + ( -20 + 6 \zeta_{24} + 2 \zeta_{24}^{3} + 16 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{56} + ( -24 \zeta_{24} + 24 \zeta_{24}^{3} + 24 \zeta_{24}^{5} - 6 \zeta_{24}^{6} ) q^{57} + ( 30 \zeta_{24} - 30 \zeta_{24}^{3} - 30 \zeta_{24}^{5} ) q^{58} + ( -28 - 20 \zeta_{24} + 20 \zeta_{24}^{3} + 56 \zeta_{24}^{4} - 20 \zeta_{24}^{5} - 40 \zeta_{24}^{7} ) q^{59} + ( -4 - 4 \zeta_{24} + 4 \zeta_{24}^{3} + 8 \zeta_{24}^{4} - 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{61} + ( 12 \zeta_{24} + 48 \zeta_{24}^{2} + 12 \zeta_{24}^{3} + 12 \zeta_{24}^{5} - 24 \zeta_{24}^{6} - 24 \zeta_{24}^{7} ) q^{62} + ( 3 \zeta_{24} - 6 \zeta_{24}^{2} - 15 \zeta_{24}^{3} - 15 \zeta_{24}^{5} + 9 \zeta_{24}^{6} + 12 \zeta_{24}^{7} ) q^{63} -8 q^{64} + ( -6 + 6 \zeta_{24} - 6 \zeta_{24}^{3} + 12 \zeta_{24}^{4} + 6 \zeta_{24}^{5} + 12 \zeta_{24}^{7} ) q^{66} + ( 12 \zeta_{24} - 12 \zeta_{24}^{3} - 12 \zeta_{24}^{5} + 44 \zeta_{24}^{6} ) q^{67} + ( -22 \zeta_{24} - 8 \zeta_{24}^{2} - 22 \zeta_{24}^{3} - 22 \zeta_{24}^{5} + 4 \zeta_{24}^{6} + 44 \zeta_{24}^{7} ) q^{68} + ( -6 - 9 \zeta_{24} + 9 \zeta_{24}^{3} + 12 \zeta_{24}^{4} - 9 \zeta_{24}^{5} - 18 \zeta_{24}^{7} ) q^{69} + ( -30 + 57 \zeta_{24} + 57 \zeta_{24}^{3} - 57 \zeta_{24}^{5} ) q^{71} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{72} + ( 26 \zeta_{24} + 8 \zeta_{24}^{2} + 26 \zeta_{24}^{3} + 26 \zeta_{24}^{5} - 4 \zeta_{24}^{6} - 52 \zeta_{24}^{7} ) q^{73} + ( -72 + 20 \zeta_{24} + 20 \zeta_{24}^{3} - 20 \zeta_{24}^{5} ) q^{74} + ( 4 + 16 \zeta_{24} - 16 \zeta_{24}^{3} - 8 \zeta_{24}^{4} + 16 \zeta_{24}^{5} + 32 \zeta_{24}^{7} ) q^{76} + ( 3 \zeta_{24} + 36 \zeta_{24}^{2} + 27 \zeta_{24}^{3} + 27 \zeta_{24}^{5} - 12 \zeta_{24}^{6} - 30 \zeta_{24}^{7} ) q^{77} + ( 24 \zeta_{24} - 24 \zeta_{24}^{3} - 24 \zeta_{24}^{5} + 12 \zeta_{24}^{6} ) q^{78} + ( -32 - 72 \zeta_{24} - 72 \zeta_{24}^{3} + 72 \zeta_{24}^{5} ) q^{79} + 9 q^{81} + ( -14 \zeta_{24} + 28 \zeta_{24}^{2} - 14 \zeta_{24}^{3} - 14 \zeta_{24}^{5} - 14 \zeta_{24}^{6} + 28 \zeta_{24}^{7} ) q^{82} + ( 20 \zeta_{24} + 64 \zeta_{24}^{2} + 20 \zeta_{24}^{3} + 20 \zeta_{24}^{5} - 32 \zeta_{24}^{6} - 40 \zeta_{24}^{7} ) q^{83} + ( -10 - 18 \zeta_{24} - 6 \zeta_{24}^{3} + 8 \zeta_{24}^{4} + 6 \zeta_{24}^{5} - 12 \zeta_{24}^{7} ) q^{84} + ( -60 - 32 \zeta_{24} - 32 \zeta_{24}^{3} + 32 \zeta_{24}^{5} ) q^{86} + ( 60 \zeta_{24}^{2} - 30 \zeta_{24}^{6} ) q^{87} + ( -12 \zeta_{24} + 12 \zeta_{24}^{3} + 12 \zeta_{24}^{5} + 12 \zeta_{24}^{6} ) q^{88} + ( -54 + 21 \zeta_{24} - 21 \zeta_{24}^{3} + 108 \zeta_{24}^{4} + 21 \zeta_{24}^{5} + 42 \zeta_{24}^{7} ) q^{89} + ( -28 - 70 \zeta_{24} - 14 \zeta_{24}^{3} + 56 \zeta_{24}^{4} + 14 \zeta_{24}^{5} - 56 \zeta_{24}^{7} ) q^{91} + ( 18 \zeta_{24} - 18 \zeta_{24}^{3} - 18 \zeta_{24}^{5} + 12 \zeta_{24}^{6} ) q^{92} + ( -36 \zeta_{24} + 36 \zeta_{24}^{3} + 36 \zeta_{24}^{5} + 36 \zeta_{24}^{6} ) q^{93} + ( 8 - 28 \zeta_{24} + 28 \zeta_{24}^{3} - 16 \zeta_{24}^{4} - 28 \zeta_{24}^{5} - 56 \zeta_{24}^{7} ) q^{94} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{96} + ( -10 \zeta_{24} - 88 \zeta_{24}^{2} - 10 \zeta_{24}^{3} - 10 \zeta_{24}^{5} + 44 \zeta_{24}^{6} + 20 \zeta_{24}^{7} ) q^{97} + ( -25 \zeta_{24} + 8 \zeta_{24}^{2} - 15 \zeta_{24}^{3} - 15 \zeta_{24}^{5} + 44 \zeta_{24}^{6} + 40 \zeta_{24}^{7} ) q^{98} + ( -18 - 9 \zeta_{24} - 9 \zeta_{24}^{3} + 9 \zeta_{24}^{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 16q^{4} + 24q^{9} + O(q^{10}) \) \( 8q - 16q^{4} + 24q^{9} - 48q^{11} + 48q^{14} + 32q^{16} + 24q^{21} - 240q^{29} - 48q^{36} - 192q^{39} + 96q^{44} - 144q^{46} + 40q^{49} - 48q^{51} - 96q^{56} - 64q^{64} - 240q^{71} - 576q^{74} - 256q^{79} + 72q^{81} - 48q^{84} - 480q^{86} - 144q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1
0.965926 0.258819i
−0.965926 0.258819i
0.258819 + 0.965926i
−0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 + 0.258819i
0.258819 0.965926i
−0.258819 0.965926i
1.41421i −1.73205 −2.00000 0 2.44949i −6.63103 + 2.24264i 2.82843i 3.00000 0
349.2 1.41421i −1.73205 −2.00000 0 2.44949i 3.16693 + 6.24264i 2.82843i 3.00000 0
349.3 1.41421i 1.73205 −2.00000 0 2.44949i −3.16693 + 6.24264i 2.82843i 3.00000 0
349.4 1.41421i 1.73205 −2.00000 0 2.44949i 6.63103 + 2.24264i 2.82843i 3.00000 0
349.5 1.41421i −1.73205 −2.00000 0 2.44949i −6.63103 2.24264i 2.82843i 3.00000 0
349.6 1.41421i −1.73205 −2.00000 0 2.44949i 3.16693 6.24264i 2.82843i 3.00000 0
349.7 1.41421i 1.73205 −2.00000 0 2.44949i −3.16693 6.24264i 2.82843i 3.00000 0
349.8 1.41421i 1.73205 −2.00000 0 2.44949i 6.63103 2.24264i 2.82843i 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.h.a 8
5.b even 2 1 inner 1050.3.h.a 8
5.c odd 4 1 42.3.c.a 4
5.c odd 4 1 1050.3.f.a 4
7.b odd 2 1 inner 1050.3.h.a 8
15.e even 4 1 126.3.c.b 4
20.e even 4 1 336.3.f.c 4
35.c odd 2 1 inner 1050.3.h.a 8
35.f even 4 1 42.3.c.a 4
35.f even 4 1 1050.3.f.a 4
35.k even 12 1 294.3.g.b 4
35.k even 12 1 294.3.g.c 4
35.l odd 12 1 294.3.g.b 4
35.l odd 12 1 294.3.g.c 4
40.i odd 4 1 1344.3.f.f 4
40.k even 4 1 1344.3.f.e 4
60.l odd 4 1 1008.3.f.g 4
105.k odd 4 1 126.3.c.b 4
105.w odd 12 1 882.3.n.a 4
105.w odd 12 1 882.3.n.d 4
105.x even 12 1 882.3.n.a 4
105.x even 12 1 882.3.n.d 4
140.j odd 4 1 336.3.f.c 4
280.s even 4 1 1344.3.f.f 4
280.y odd 4 1 1344.3.f.e 4
420.w even 4 1 1008.3.f.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.c.a 4 5.c odd 4 1
42.3.c.a 4 35.f even 4 1
126.3.c.b 4 15.e even 4 1
126.3.c.b 4 105.k odd 4 1
294.3.g.b 4 35.k even 12 1
294.3.g.b 4 35.l odd 12 1
294.3.g.c 4 35.k even 12 1
294.3.g.c 4 35.l odd 12 1
336.3.f.c 4 20.e even 4 1
336.3.f.c 4 140.j odd 4 1
882.3.n.a 4 105.w odd 12 1
882.3.n.a 4 105.x even 12 1
882.3.n.d 4 105.w odd 12 1
882.3.n.d 4 105.x even 12 1
1008.3.f.g 4 60.l odd 4 1
1008.3.f.g 4 420.w even 4 1
1050.3.f.a 4 5.c odd 4 1
1050.3.f.a 4 35.f even 4 1
1050.3.h.a 8 1.a even 1 1 trivial
1050.3.h.a 8 5.b even 2 1 inner
1050.3.h.a 8 7.b odd 2 1 inner
1050.3.h.a 8 35.c odd 2 1 inner
1344.3.f.e 4 40.k even 4 1
1344.3.f.e 4 280.y odd 4 1
1344.3.f.f 4 40.i odd 4 1
1344.3.f.f 4 280.s even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{2} + 12 T_{11} + 18 \) acting on \(S_{3}^{\mathrm{new}}(1050, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + T^{2} )^{4} \)
$3$ \( ( -3 + T^{2} )^{4} \)
$5$ \( T^{8} \)
$7$ \( 5764801 - 48020 T^{2} + 294 T^{4} - 20 T^{6} + T^{8} \)
$11$ \( ( 18 + 12 T + T^{2} )^{4} \)
$13$ \( ( 28224 - 432 T^{2} + T^{4} )^{2} \)
$17$ \( ( 509796 - 1476 T^{2} + T^{4} )^{2} \)
$19$ \( ( 138384 + 792 T^{2} + T^{4} )^{2} \)
$23$ \( ( 15876 + 396 T^{2} + T^{4} )^{2} \)
$29$ \( ( 30 + T )^{8} \)
$31$ \( ( 186624 + 2592 T^{2} + T^{4} )^{2} \)
$37$ \( ( 4804864 + 5984 T^{2} + T^{4} )^{2} \)
$41$ \( ( 86436 + 1764 T^{2} + T^{4} )^{2} \)
$43$ \( ( 602176 + 5648 T^{2} + T^{4} )^{2} \)
$47$ \( ( 5089536 - 4896 T^{2} + T^{4} )^{2} \)
$53$ \( ( 6906384 + 6408 T^{2} + T^{4} )^{2} \)
$59$ \( ( 2304 + 9504 T^{2} + T^{4} )^{2} \)
$61$ \( ( 2304 + 288 T^{2} + T^{4} )^{2} \)
$67$ \( ( 2715904 + 4448 T^{2} + T^{4} )^{2} \)
$71$ \( ( -5598 + 60 T + T^{2} )^{4} \)
$73$ \( ( 16064064 - 8208 T^{2} + T^{4} )^{2} \)
$79$ \( ( -9344 + 64 T + T^{2} )^{4} \)
$83$ \( ( 451584 - 10944 T^{2} + T^{4} )^{2} \)
$89$ \( ( 37234404 + 22788 T^{2} + T^{4} )^{2} \)
$97$ \( ( 27123264 - 12816 T^{2} + T^{4} )^{2} \)
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