Properties

Label 1050.2.i.l
Level $1050$
Weight $2$
Character orbit 1050.i
Analytic conductor $8.384$
Analytic rank $1$
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: \(1\)
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 42)
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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} - q^{6} + ( 1 - 3 \zeta_{6} ) q^{7} - q^{8} -\zeta_{6} q^{9} +O(q^{10})\) \( q + \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} - q^{6} + ( 1 - 3 \zeta_{6} ) q^{7} - q^{8} -\zeta_{6} q^{9} + ( -5 + 5 \zeta_{6} ) q^{11} -\zeta_{6} q^{12} + ( 3 - 2 \zeta_{6} ) q^{14} -\zeta_{6} q^{16} + ( -4 + 4 \zeta_{6} ) q^{17} + ( 1 - \zeta_{6} ) q^{18} -8 \zeta_{6} q^{19} + ( 2 + \zeta_{6} ) q^{21} -5 q^{22} -4 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{24} + q^{27} + ( 2 + \zeta_{6} ) q^{28} -5 q^{29} + ( -3 + 3 \zeta_{6} ) q^{31} + ( 1 - \zeta_{6} ) q^{32} -5 \zeta_{6} q^{33} -4 q^{34} + q^{36} -4 \zeta_{6} q^{37} + ( 8 - 8 \zeta_{6} ) q^{38} + ( -1 + 3 \zeta_{6} ) q^{42} -2 q^{43} -5 \zeta_{6} q^{44} + ( 4 - 4 \zeta_{6} ) q^{46} -6 \zeta_{6} q^{47} + q^{48} + ( -8 + 3 \zeta_{6} ) q^{49} -4 \zeta_{6} q^{51} + ( -9 + 9 \zeta_{6} ) q^{53} + \zeta_{6} q^{54} + ( -1 + 3 \zeta_{6} ) q^{56} + 8 q^{57} -5 \zeta_{6} q^{58} + ( 11 - 11 \zeta_{6} ) q^{59} + 6 \zeta_{6} q^{61} -3 q^{62} + ( -3 + 2 \zeta_{6} ) q^{63} + q^{64} + ( 5 - 5 \zeta_{6} ) q^{66} + ( -2 + 2 \zeta_{6} ) q^{67} -4 \zeta_{6} q^{68} + 4 q^{69} + 2 q^{71} + \zeta_{6} q^{72} + ( 10 - 10 \zeta_{6} ) q^{73} + ( 4 - 4 \zeta_{6} ) q^{74} + 8 q^{76} + ( 10 + 5 \zeta_{6} ) q^{77} -3 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + 7 q^{83} + ( -3 + 2 \zeta_{6} ) q^{84} -2 \zeta_{6} q^{86} + ( 5 - 5 \zeta_{6} ) q^{87} + ( 5 - 5 \zeta_{6} ) q^{88} + 6 \zeta_{6} q^{89} + 4 q^{92} -3 \zeta_{6} q^{93} + ( 6 - 6 \zeta_{6} ) q^{94} + \zeta_{6} q^{96} -7 q^{97} + ( -3 - 5 \zeta_{6} ) q^{98} + 5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{3} - q^{4} - 2q^{6} - q^{7} - 2q^{8} - q^{9} + O(q^{10}) \) \( 2q + q^{2} - q^{3} - q^{4} - 2q^{6} - q^{7} - 2q^{8} - q^{9} - 5q^{11} - q^{12} + 4q^{14} - q^{16} - 4q^{17} + q^{18} - 8q^{19} + 5q^{21} - 10q^{22} - 4q^{23} + q^{24} + 2q^{27} + 5q^{28} - 10q^{29} - 3q^{31} + q^{32} - 5q^{33} - 8q^{34} + 2q^{36} - 4q^{37} + 8q^{38} + q^{42} - 4q^{43} - 5q^{44} + 4q^{46} - 6q^{47} + 2q^{48} - 13q^{49} - 4q^{51} - 9q^{53} + q^{54} + q^{56} + 16q^{57} - 5q^{58} + 11q^{59} + 6q^{61} - 6q^{62} - 4q^{63} + 2q^{64} + 5q^{66} - 2q^{67} - 4q^{68} + 8q^{69} + 4q^{71} + q^{72} + 10q^{73} + 4q^{74} + 16q^{76} + 25q^{77} - 3q^{79} - q^{81} + 14q^{83} - 4q^{84} - 2q^{86} + 5q^{87} + 5q^{88} + 6q^{89} + 8q^{92} - 3q^{93} + 6q^{94} + q^{96} - 14q^{97} - 11q^{98} + 10q^{99} + O(q^{100}) \)

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 −0.500000 2.59808i −1.00000 −0.500000 0.866025i 0
751.1 0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0 −1.00000 −0.500000 + 2.59808i −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.l 2
5.b even 2 1 42.2.e.a 2
5.c odd 4 2 1050.2.o.a 4
7.c even 3 1 inner 1050.2.i.l 2
7.c even 3 1 7350.2.a.bl 1
7.d odd 6 1 7350.2.a.q 1
15.d odd 2 1 126.2.g.c 2
20.d odd 2 1 336.2.q.b 2
35.c odd 2 1 294.2.e.b 2
35.i odd 6 1 294.2.a.f 1
35.i odd 6 1 294.2.e.b 2
35.j even 6 1 42.2.e.a 2
35.j even 6 1 294.2.a.e 1
35.l odd 12 2 1050.2.o.a 4
40.e odd 2 1 1344.2.q.s 2
40.f even 2 1 1344.2.q.g 2
45.h odd 6 1 1134.2.e.e 2
45.h odd 6 1 1134.2.h.l 2
45.j even 6 1 1134.2.e.l 2
45.j even 6 1 1134.2.h.e 2
60.h even 2 1 1008.2.s.k 2
105.g even 2 1 882.2.g.i 2
105.o odd 6 1 126.2.g.c 2
105.o odd 6 1 882.2.a.c 1
105.p even 6 1 882.2.a.d 1
105.p even 6 1 882.2.g.i 2
140.c even 2 1 2352.2.q.u 2
140.p odd 6 1 336.2.q.b 2
140.p odd 6 1 2352.2.a.t 1
140.s even 6 1 2352.2.a.f 1
140.s even 6 1 2352.2.q.u 2
280.ba even 6 1 9408.2.a.cr 1
280.bf even 6 1 1344.2.q.g 2
280.bf even 6 1 9408.2.a.ce 1
280.bi odd 6 1 1344.2.q.s 2
280.bi odd 6 1 9408.2.a.q 1
280.bk odd 6 1 9408.2.a.z 1
315.r even 6 1 1134.2.h.e 2
315.v odd 6 1 1134.2.e.e 2
315.bo even 6 1 1134.2.e.l 2
315.br odd 6 1 1134.2.h.l 2
420.ba even 6 1 1008.2.s.k 2
420.ba even 6 1 7056.2.a.w 1
420.be odd 6 1 7056.2.a.bl 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.e.a 2 5.b even 2 1
42.2.e.a 2 35.j even 6 1
126.2.g.c 2 15.d odd 2 1
126.2.g.c 2 105.o odd 6 1
294.2.a.e 1 35.j even 6 1
294.2.a.f 1 35.i odd 6 1
294.2.e.b 2 35.c odd 2 1
294.2.e.b 2 35.i odd 6 1
336.2.q.b 2 20.d odd 2 1
336.2.q.b 2 140.p odd 6 1
882.2.a.c 1 105.o odd 6 1
882.2.a.d 1 105.p even 6 1
882.2.g.i 2 105.g even 2 1
882.2.g.i 2 105.p even 6 1
1008.2.s.k 2 60.h even 2 1
1008.2.s.k 2 420.ba even 6 1
1050.2.i.l 2 1.a even 1 1 trivial
1050.2.i.l 2 7.c even 3 1 inner
1050.2.o.a 4 5.c odd 4 2
1050.2.o.a 4 35.l odd 12 2
1134.2.e.e 2 45.h odd 6 1
1134.2.e.e 2 315.v odd 6 1
1134.2.e.l 2 45.j even 6 1
1134.2.e.l 2 315.bo even 6 1
1134.2.h.e 2 45.j even 6 1
1134.2.h.e 2 315.r even 6 1
1134.2.h.l 2 45.h odd 6 1
1134.2.h.l 2 315.br odd 6 1
1344.2.q.g 2 40.f even 2 1
1344.2.q.g 2 280.bf even 6 1
1344.2.q.s 2 40.e odd 2 1
1344.2.q.s 2 280.bi odd 6 1
2352.2.a.f 1 140.s even 6 1
2352.2.a.t 1 140.p odd 6 1
2352.2.q.u 2 140.c even 2 1
2352.2.q.u 2 140.s even 6 1
7056.2.a.w 1 420.ba even 6 1
7056.2.a.bl 1 420.be odd 6 1
7350.2.a.q 1 7.d odd 6 1
7350.2.a.bl 1 7.c even 3 1
9408.2.a.q 1 280.bi odd 6 1
9408.2.a.z 1 280.bk odd 6 1
9408.2.a.ce 1 280.bf even 6 1
9408.2.a.cr 1 280.ba even 6 1

Hecke kernels

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

\( T_{11}^{2} + 5 T_{11} + 25 \)
\( T_{13} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 7 + T + T^{2} \)
$11$ \( 25 + 5 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( 16 + 4 T + T^{2} \)
$19$ \( 64 + 8 T + T^{2} \)
$23$ \( 16 + 4 T + T^{2} \)
$29$ \( ( 5 + T )^{2} \)
$31$ \( 9 + 3 T + T^{2} \)
$37$ \( 16 + 4 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 2 + T )^{2} \)
$47$ \( 36 + 6 T + T^{2} \)
$53$ \( 81 + 9 T + T^{2} \)
$59$ \( 121 - 11 T + T^{2} \)
$61$ \( 36 - 6 T + T^{2} \)
$67$ \( 4 + 2 T + T^{2} \)
$71$ \( ( -2 + T )^{2} \)
$73$ \( 100 - 10 T + T^{2} \)
$79$ \( 9 + 3 T + T^{2} \)
$83$ \( ( -7 + T )^{2} \)
$89$ \( 36 - 6 T + T^{2} \)
$97$ \( ( 7 + T )^{2} \)
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