Properties

Label 1050.2.m.d
Level 1050
Weight 2
Character orbit 1050.m
Analytic conductor 8.384
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: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1050.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.38429221223\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

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

Character values

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

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(\zeta_{24}^{3}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
−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
0.965926 0.258819i
−0.707107 0.707107i −0.707107 0.707107i 1.00000i 0 1.00000i −2.63896 0.189469i 0.707107 0.707107i 1.00000i 0
307.2 −0.707107 0.707107i −0.707107 0.707107i 1.00000i 0 1.00000i −0.189469 2.63896i 0.707107 0.707107i 1.00000i 0
307.3 0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 0 1.00000i 0.189469 + 2.63896i −0.707107 + 0.707107i 1.00000i 0
307.4 0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 0 1.00000i 2.63896 + 0.189469i −0.707107 + 0.707107i 1.00000i 0
643.1 −0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 0 1.00000i −2.63896 + 0.189469i 0.707107 + 0.707107i 1.00000i 0
643.2 −0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 0 1.00000i −0.189469 + 2.63896i 0.707107 + 0.707107i 1.00000i 0
643.3 0.707107 0.707107i 0.707107 0.707107i 1.00000i 0 1.00000i 0.189469 2.63896i −0.707107 0.707107i 1.00000i 0
643.4 0.707107 0.707107i 0.707107 0.707107i 1.00000i 0 1.00000i 2.63896 0.189469i −0.707107 0.707107i 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 643.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
35.f even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.m.d yes 8
5.b even 2 1 inner 1050.2.m.d yes 8
5.c odd 4 2 1050.2.m.c 8
7.b odd 2 1 1050.2.m.c 8
35.c odd 2 1 1050.2.m.c 8
35.f even 4 2 inner 1050.2.m.d yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.2.m.c 8 5.c odd 4 2
1050.2.m.c 8 7.b odd 2 1
1050.2.m.c 8 35.c odd 2 1
1050.2.m.d yes 8 1.a even 1 1 trivial
1050.2.m.d yes 8 5.b even 2 1 inner
1050.2.m.d yes 8 35.f even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1050, [\chi])\):

\( T_{11}^{2} + 4 T_{11} - 8 \)
\( T_{13}^{8} + 194 T_{13}^{4} + 1 \)
\( T_{19}^{2} - 8 T_{19} + 4 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{4} )^{2} \)
$3$ \( ( 1 + T^{4} )^{2} \)
$5$ 1
$7$ \( 1 - 94 T^{4} + 2401 T^{8} \)
$11$ \( ( 1 + 4 T + 14 T^{2} + 44 T^{3} + 121 T^{4} )^{4} \)
$13$ \( 1 + 142 T^{4} - 7149 T^{8} + 4055662 T^{12} + 815730721 T^{16} \)
$17$ \( ( 1 - 497 T^{4} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 8 T + 42 T^{2} - 152 T^{3} + 361 T^{4} )^{4} \)
$23$ \( ( 1 + 967 T^{4} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 - 38 T^{2} + 1611 T^{4} - 31958 T^{6} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 - 26 T^{2} + 1899 T^{4} - 24986 T^{6} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 - 2062 T^{4} + 1874161 T^{8} )^{2} \)
$41$ \( ( 1 - 126 T^{2} + 7139 T^{4} - 211806 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( 1 + 5182 T^{4} + 12352611 T^{8} + 17716226782 T^{12} + 11688200277601 T^{16} \)
$47$ \( 1 - 4732 T^{4} + 13117830 T^{8} - 23090650492 T^{12} + 23811286661761 T^{16} \)
$53$ \( ( 1 - 56 T^{2} + 327 T^{4} - 157304 T^{6} + 7890481 T^{8} )( 1 + 56 T^{2} + 327 T^{4} + 157304 T^{6} + 7890481 T^{8} ) \)
$59$ \( ( 1 - 4 T + 95 T^{2} - 236 T^{3} + 3481 T^{4} )^{4} \)
$61$ \( ( 1 - 218 T^{2} + 19275 T^{4} - 811178 T^{6} + 13845841 T^{8} )^{2} \)
$67$ \( 1 - 3932 T^{4} + 41402598 T^{8} - 79234207772 T^{12} + 406067677556641 T^{16} \)
$71$ \( ( 1 + 16 T + 194 T^{2} + 1136 T^{3} + 5041 T^{4} )^{4} \)
$73$ \( 1 - 15548 T^{4} + 109241286 T^{8} - 441535851068 T^{12} + 806460091894081 T^{16} \)
$79$ \( ( 1 - 92 T^{2} + 12870 T^{4} - 574172 T^{6} + 38950081 T^{8} )^{2} \)
$83$ \( 1 + 15358 T^{4} + 117552483 T^{8} + 728864893918 T^{12} + 2252292232139041 T^{16} \)
$89$ \( ( 1 - 12 T + 89 T^{2} )^{8} \)
$97$ \( 1 + 4996 T^{4} + 20789766 T^{8} + 442292287876 T^{12} + 7837433594376961 T^{16} \)
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