Properties

Label 1050.2.bc.d
Level $1050$
Weight $2$
Character orbit 1050.bc
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.bc (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.38429221223\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

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

Character values

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

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(-\zeta_{24}^{6}\) \(\zeta_{24}^{4}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1
0.258819 0.965926i
−0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
0.965926 0.258819i
−0.965926 + 0.258819i
0.258819 + 0.965926i
−0.258819 0.965926i
−0.258819 + 0.965926i −0.965926 + 0.258819i −0.866025 0.500000i 0 1.00000i −0.189469 2.63896i 0.707107 0.707107i 0.866025 0.500000i 0
157.2 0.258819 0.965926i 0.965926 0.258819i −0.866025 0.500000i 0 1.00000i 0.189469 + 2.63896i −0.707107 + 0.707107i 0.866025 0.500000i 0
493.1 −0.965926 0.258819i −0.258819 0.965926i 0.866025 + 0.500000i 0 1.00000i 2.63896 0.189469i −0.707107 0.707107i −0.866025 + 0.500000i 0
493.2 0.965926 + 0.258819i 0.258819 + 0.965926i 0.866025 + 0.500000i 0 1.00000i −2.63896 + 0.189469i 0.707107 + 0.707107i −0.866025 + 0.500000i 0
607.1 −0.965926 + 0.258819i −0.258819 + 0.965926i 0.866025 0.500000i 0 1.00000i 2.63896 + 0.189469i −0.707107 + 0.707107i −0.866025 0.500000i 0
607.2 0.965926 0.258819i 0.258819 0.965926i 0.866025 0.500000i 0 1.00000i −2.63896 0.189469i 0.707107 0.707107i −0.866025 0.500000i 0
943.1 −0.258819 0.965926i −0.965926 0.258819i −0.866025 + 0.500000i 0 1.00000i −0.189469 + 2.63896i 0.707107 + 0.707107i 0.866025 + 0.500000i 0
943.2 0.258819 + 0.965926i 0.965926 + 0.258819i −0.866025 + 0.500000i 0 1.00000i 0.189469 2.63896i −0.707107 0.707107i 0.866025 + 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 943.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
35.k even 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.bc.d yes 8
5.b even 2 1 inner 1050.2.bc.d yes 8
5.c odd 4 2 1050.2.bc.a 8
7.d odd 6 1 1050.2.bc.a 8
35.i odd 6 1 1050.2.bc.a 8
35.k even 12 2 inner 1050.2.bc.d yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.2.bc.a 8 5.c odd 4 2
1050.2.bc.a 8 7.d odd 6 1
1050.2.bc.a 8 35.i odd 6 1
1050.2.bc.d yes 8 1.a even 1 1 trivial
1050.2.bc.d yes 8 5.b even 2 1 inner
1050.2.bc.d yes 8 35.k even 12 2 inner

Hecke kernels

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

\( T_{11}^{2} + T_{11} + 1 \)
\( T_{13}^{8} + 1106 T_{13}^{4} + 28561 \)
\( T_{17}^{8} + 96 T_{17}^{6} + 3056 T_{17}^{4} - 1536 T_{17}^{2} + 256 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{4} + T^{8} \)
$3$ \( 1 - T^{4} + T^{8} \)
$5$ \( T^{8} \)
$7$ \( 2401 - 94 T^{4} + T^{8} \)
$11$ \( ( 1 + T + T^{2} )^{4} \)
$13$ \( 28561 + 1106 T^{4} + T^{8} \)
$17$ \( 256 - 1536 T^{2} + 3056 T^{4} + 96 T^{6} + T^{8} \)
$19$ \( ( 169 - 104 T + 51 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$23$ \( 4879681 - 106032 T^{2} - 1441 T^{4} + 48 T^{6} + T^{8} \)
$29$ \( ( 48 + T^{2} )^{4} \)
$31$ \( ( 16 + 48 T + 44 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$37$ \( 2313441 + 219024 T^{2} + 5391 T^{4} - 144 T^{6} + T^{8} \)
$41$ \( ( 169 + 38 T^{2} + T^{4} )^{2} \)
$43$ \( 3748096 + 6944 T^{4} + T^{8} \)
$47$ \( 14641 - 17424 T^{2} + 6791 T^{4} + 144 T^{6} + T^{8} \)
$53$ \( 1 - T^{4} + T^{8} \)
$59$ \( ( 2704 - 832 T + 204 T^{2} - 16 T^{3} + T^{4} )^{2} \)
$61$ \( ( 2704 - 624 T - 4 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$67$ \( 256 - 16 T^{4} + T^{8} \)
$71$ \( ( -44 + 4 T + T^{2} )^{4} \)
$73$ \( 65536 - 256 T^{4} + T^{8} \)
$79$ \( ( 2704 + 1248 T + 140 T^{2} - 24 T^{3} + T^{4} )^{2} \)
$83$ \( ( 11664 + T^{4} )^{2} \)
$89$ \( ( 2304 + 48 T^{2} + T^{4} )^{2} \)
$97$ \( 1048576 + 14336 T^{4} + T^{8} \)
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