Properties

Label 1050.2.i.j
Level 1050
Weight 2
Character orbit 1050.i
Analytic conductor 8.384
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1050.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.38429221223\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 0 −1.00000 2.50000 + 0.866025i 1.00000 −0.500000 0.866025i 0
751.1 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 0 −1.00000 2.50000 0.866025i 1.00000 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.i.j 2
5.b even 2 1 1050.2.i.k yes 2
5.c odd 4 2 1050.2.o.f 4
7.c even 3 1 inner 1050.2.i.j 2
7.c even 3 1 7350.2.a.bt 1
7.d odd 6 1 7350.2.a.cn 1
35.i odd 6 1 7350.2.a.h 1
35.j even 6 1 1050.2.i.k yes 2
35.j even 6 1 7350.2.a.z 1
35.l odd 12 2 1050.2.o.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.2.i.j 2 1.a even 1 1 trivial
1050.2.i.j 2 7.c even 3 1 inner
1050.2.i.k yes 2 5.b even 2 1
1050.2.i.k yes 2 35.j even 6 1
1050.2.o.f 4 5.c odd 4 2
1050.2.o.f 4 35.l odd 12 2
7350.2.a.h 1 35.i odd 6 1
7350.2.a.z 1 35.j even 6 1
7350.2.a.bt 1 7.c even 3 1
7350.2.a.cn 1 7.d odd 6 1

Hecke kernels

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

\( T_{11}^{2} - 2 T_{11} + 4 \)
\( T_{13} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ 1
$7$ \( 1 - 5 T + 7 T^{2} \)
$11$ \( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 4 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 5 T + 8 T^{2} + 85 T^{3} + 289 T^{4} \)
$19$ \( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \)
$23$ \( 1 - 5 T + 2 T^{2} - 115 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 7 T + 31 T^{2} )( 1 - 4 T + 31 T^{2} ) \)
$37$ \( 1 - 8 T + 27 T^{2} - 296 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 - 5 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 43 T^{2} )^{2} \)
$47$ \( 1 + T - 46 T^{2} + 47 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 12 T + 91 T^{2} - 636 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 2 T - 55 T^{2} - 118 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 67 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + T + 71 T^{2} )^{2} \)
$73$ \( 1 - 2 T - 69 T^{2} - 146 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 9 T + 2 T^{2} + 711 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 6 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 11 T + 32 T^{2} + 979 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 - T + 97 T^{2} )^{2} \)
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