Properties

Label 1050.2.i.i
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: no (minimal twist has level 210)
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.

\(f(q)\) \(=\) \( q -\zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} - q^{6} + ( 3 - 2 \zeta_{6} ) q^{7} + q^{8} -\zeta_{6} q^{9} +O(q^{10})\) \( q -\zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} - q^{6} + ( 3 - 2 \zeta_{6} ) q^{7} + q^{8} -\zeta_{6} q^{9} + ( -3 + 3 \zeta_{6} ) q^{11} + \zeta_{6} q^{12} -5 q^{13} + ( -2 - \zeta_{6} ) q^{14} -\zeta_{6} q^{16} + ( -1 + \zeta_{6} ) q^{18} -5 \zeta_{6} q^{19} + ( 1 - 3 \zeta_{6} ) q^{21} + 3 q^{22} -9 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{24} + 5 \zeta_{6} q^{26} - q^{27} + ( -1 + 3 \zeta_{6} ) q^{28} + ( 10 - 10 \zeta_{6} ) q^{31} + ( -1 + \zeta_{6} ) q^{32} + 3 \zeta_{6} q^{33} + q^{36} -\zeta_{6} q^{37} + ( -5 + 5 \zeta_{6} ) q^{38} + ( -5 + 5 \zeta_{6} ) q^{39} + 9 q^{41} + ( -3 + 2 \zeta_{6} ) q^{42} -8 q^{43} -3 \zeta_{6} q^{44} + ( -9 + 9 \zeta_{6} ) q^{46} + 3 \zeta_{6} q^{47} - q^{48} + ( 5 - 8 \zeta_{6} ) q^{49} + ( 5 - 5 \zeta_{6} ) q^{52} + ( -3 + 3 \zeta_{6} ) q^{53} + \zeta_{6} q^{54} + ( 3 - 2 \zeta_{6} ) q^{56} -5 q^{57} + ( -12 + 12 \zeta_{6} ) q^{59} -8 \zeta_{6} q^{61} -10 q^{62} + ( -2 - \zeta_{6} ) q^{63} + q^{64} + ( 3 - 3 \zeta_{6} ) q^{66} + ( 8 - 8 \zeta_{6} ) q^{67} -9 q^{69} -6 q^{71} -\zeta_{6} q^{72} + ( 2 - 2 \zeta_{6} ) q^{73} + ( -1 + \zeta_{6} ) q^{74} + 5 q^{76} + ( -3 + 9 \zeta_{6} ) q^{77} + 5 q^{78} -8 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -9 \zeta_{6} q^{82} + ( 2 + \zeta_{6} ) q^{84} + 8 \zeta_{6} q^{86} + ( -3 + 3 \zeta_{6} ) q^{88} -6 \zeta_{6} q^{89} + ( -15 + 10 \zeta_{6} ) q^{91} + 9 q^{92} -10 \zeta_{6} q^{93} + ( 3 - 3 \zeta_{6} ) q^{94} + \zeta_{6} q^{96} -8 q^{97} + ( -8 + 3 \zeta_{6} ) q^{98} + 3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + q^{3} - q^{4} - 2q^{6} + 4q^{7} + 2q^{8} - q^{9} + O(q^{10}) \) \( 2q - q^{2} + q^{3} - q^{4} - 2q^{6} + 4q^{7} + 2q^{8} - q^{9} - 3q^{11} + q^{12} - 10q^{13} - 5q^{14} - q^{16} - q^{18} - 5q^{19} - q^{21} + 6q^{22} - 9q^{23} + q^{24} + 5q^{26} - 2q^{27} + q^{28} + 10q^{31} - q^{32} + 3q^{33} + 2q^{36} - q^{37} - 5q^{38} - 5q^{39} + 18q^{41} - 4q^{42} - 16q^{43} - 3q^{44} - 9q^{46} + 3q^{47} - 2q^{48} + 2q^{49} + 5q^{52} - 3q^{53} + q^{54} + 4q^{56} - 10q^{57} - 12q^{59} - 8q^{61} - 20q^{62} - 5q^{63} + 2q^{64} + 3q^{66} + 8q^{67} - 18q^{69} - 12q^{71} - q^{72} + 2q^{73} - q^{74} + 10q^{76} + 3q^{77} + 10q^{78} - 8q^{79} - q^{81} - 9q^{82} + 5q^{84} + 8q^{86} - 3q^{88} - 6q^{89} - 20q^{91} + 18q^{92} - 10q^{93} + 3q^{94} + q^{96} - 16q^{97} - 13q^{98} + 6q^{99} + O(q^{100}) \)

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 2.00000 1.73205i 1.00000 −0.500000 0.866025i 0
751.1 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 0 −1.00000 2.00000 + 1.73205i 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.i 2
5.b even 2 1 210.2.i.c 2
5.c odd 4 2 1050.2.o.c 4
7.c even 3 1 inner 1050.2.i.i 2
7.c even 3 1 7350.2.a.cd 1
7.d odd 6 1 7350.2.a.cx 1
15.d odd 2 1 630.2.k.a 2
20.d odd 2 1 1680.2.bg.n 2
35.c odd 2 1 1470.2.i.p 2
35.i odd 6 1 1470.2.a.e 1
35.i odd 6 1 1470.2.i.p 2
35.j even 6 1 210.2.i.c 2
35.j even 6 1 1470.2.a.f 1
35.l odd 12 2 1050.2.o.c 4
105.o odd 6 1 630.2.k.a 2
105.o odd 6 1 4410.2.a.bh 1
105.p even 6 1 4410.2.a.w 1
140.p odd 6 1 1680.2.bg.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.i.c 2 5.b even 2 1
210.2.i.c 2 35.j even 6 1
630.2.k.a 2 15.d odd 2 1
630.2.k.a 2 105.o odd 6 1
1050.2.i.i 2 1.a even 1 1 trivial
1050.2.i.i 2 7.c even 3 1 inner
1050.2.o.c 4 5.c odd 4 2
1050.2.o.c 4 35.l odd 12 2
1470.2.a.e 1 35.i odd 6 1
1470.2.a.f 1 35.j even 6 1
1470.2.i.p 2 35.c odd 2 1
1470.2.i.p 2 35.i odd 6 1
1680.2.bg.n 2 20.d odd 2 1
1680.2.bg.n 2 140.p odd 6 1
4410.2.a.w 1 105.p even 6 1
4410.2.a.bh 1 105.o odd 6 1
7350.2.a.cd 1 7.c even 3 1
7350.2.a.cx 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 \(S_{2}^{\mathrm{new}}(1050, [\chi])\):

\( T_{11}^{2} + 3 T_{11} + 9 \)
\( T_{13} + 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 7 - 4 T + T^{2} \)
$11$ \( 9 + 3 T + T^{2} \)
$13$ \( ( 5 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( 25 + 5 T + T^{2} \)
$23$ \( 81 + 9 T + T^{2} \)
$29$ \( T^{2} \)
$31$ \( 100 - 10 T + T^{2} \)
$37$ \( 1 + T + T^{2} \)
$41$ \( ( -9 + T )^{2} \)
$43$ \( ( 8 + T )^{2} \)
$47$ \( 9 - 3 T + T^{2} \)
$53$ \( 9 + 3 T + T^{2} \)
$59$ \( 144 + 12 T + T^{2} \)
$61$ \( 64 + 8 T + T^{2} \)
$67$ \( 64 - 8 T + T^{2} \)
$71$ \( ( 6 + T )^{2} \)
$73$ \( 4 - 2 T + T^{2} \)
$79$ \( 64 + 8 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( 36 + 6 T + T^{2} \)
$97$ \( ( 8 + T )^{2} \)
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