Properties

Label 1050.4.a.g
Level $1050$
Weight $4$
Character orbit 1050.a
Self dual yes
Analytic conductor $61.952$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1050.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.9520055060\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{2} + 3q^{3} + 4q^{4} - 6q^{6} - 7q^{7} - 8q^{8} + 9q^{9} + O(q^{10}) \) \( q - 2q^{2} + 3q^{3} + 4q^{4} - 6q^{6} - 7q^{7} - 8q^{8} + 9q^{9} - 72q^{11} + 12q^{12} + 34q^{13} + 14q^{14} + 16q^{16} - 6q^{17} - 18q^{18} + 92q^{19} - 21q^{21} + 144q^{22} + 180q^{23} - 24q^{24} - 68q^{26} + 27q^{27} - 28q^{28} - 114q^{29} + 56q^{31} - 32q^{32} - 216q^{33} + 12q^{34} + 36q^{36} + 34q^{37} - 184q^{38} + 102q^{39} + 6q^{41} + 42q^{42} - 164q^{43} - 288q^{44} - 360q^{46} - 168q^{47} + 48q^{48} + 49q^{49} - 18q^{51} + 136q^{52} - 654q^{53} - 54q^{54} + 56q^{56} + 276q^{57} + 228q^{58} - 492q^{59} - 250q^{61} - 112q^{62} - 63q^{63} + 64q^{64} + 432q^{66} + 124q^{67} - 24q^{68} + 540q^{69} + 36q^{71} - 72q^{72} - 1010q^{73} - 68q^{74} + 368q^{76} + 504q^{77} - 204q^{78} + 56q^{79} + 81q^{81} - 12q^{82} - 228q^{83} - 84q^{84} + 328q^{86} - 342q^{87} + 576q^{88} + 390q^{89} - 238q^{91} + 720q^{92} + 168q^{93} + 336q^{94} - 96q^{96} + 70q^{97} - 98q^{98} - 648q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−2.00000 3.00000 4.00000 0 −6.00000 −7.00000 −8.00000 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.4.a.g 1
5.b even 2 1 42.4.a.a 1
5.c odd 4 2 1050.4.g.a 2
15.d odd 2 1 126.4.a.a 1
20.d odd 2 1 336.4.a.l 1
35.c odd 2 1 294.4.a.i 1
35.i odd 6 2 294.4.e.b 2
35.j even 6 2 294.4.e.c 2
40.e odd 2 1 1344.4.a.a 1
40.f even 2 1 1344.4.a.o 1
60.h even 2 1 1008.4.a.b 1
105.g even 2 1 882.4.a.g 1
105.o odd 6 2 882.4.g.w 2
105.p even 6 2 882.4.g.o 2
140.c even 2 1 2352.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.a.a 1 5.b even 2 1
126.4.a.a 1 15.d odd 2 1
294.4.a.i 1 35.c odd 2 1
294.4.e.b 2 35.i odd 6 2
294.4.e.c 2 35.j even 6 2
336.4.a.l 1 20.d odd 2 1
882.4.a.g 1 105.g even 2 1
882.4.g.o 2 105.p even 6 2
882.4.g.w 2 105.o odd 6 2
1008.4.a.b 1 60.h even 2 1
1050.4.a.g 1 1.a even 1 1 trivial
1050.4.g.a 2 5.c odd 4 2
1344.4.a.a 1 40.e odd 2 1
1344.4.a.o 1 40.f even 2 1
2352.4.a.a 1 140.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1050))\):

\( T_{11} + 72 \)
\( T_{13} - 34 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( -3 + T \)
$5$ \( T \)
$7$ \( 7 + T \)
$11$ \( 72 + T \)
$13$ \( -34 + T \)
$17$ \( 6 + T \)
$19$ \( -92 + T \)
$23$ \( -180 + T \)
$29$ \( 114 + T \)
$31$ \( -56 + T \)
$37$ \( -34 + T \)
$41$ \( -6 + T \)
$43$ \( 164 + T \)
$47$ \( 168 + T \)
$53$ \( 654 + T \)
$59$ \( 492 + T \)
$61$ \( 250 + T \)
$67$ \( -124 + T \)
$71$ \( -36 + T \)
$73$ \( 1010 + T \)
$79$ \( -56 + T \)
$83$ \( 228 + T \)
$89$ \( -390 + T \)
$97$ \( -70 + T \)
show more
show less