Properties

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

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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

\(f(q)\) \(=\) \( q - 2q^{2} - 3q^{3} + 4q^{4} + 6q^{6} + 7q^{7} - 8q^{8} + 9q^{9} + O(q^{10}) \) \( q - 2q^{2} - 3q^{3} + 4q^{4} + 6q^{6} + 7q^{7} - 8q^{8} + 9q^{9} - 8q^{11} - 12q^{12} + 42q^{13} - 14q^{14} + 16q^{16} + 2q^{17} - 18q^{18} - 124q^{19} - 21q^{21} + 16q^{22} - 76q^{23} + 24q^{24} - 84q^{26} - 27q^{27} + 28q^{28} + 254q^{29} - 72q^{31} - 32q^{32} + 24q^{33} - 4q^{34} + 36q^{36} - 398q^{37} + 248q^{38} - 126q^{39} + 462q^{41} + 42q^{42} - 212q^{43} - 32q^{44} + 152q^{46} + 264q^{47} - 48q^{48} + 49q^{49} - 6q^{51} + 168q^{52} + 162q^{53} + 54q^{54} - 56q^{56} + 372q^{57} - 508q^{58} - 772q^{59} + 30q^{61} + 144q^{62} + 63q^{63} + 64q^{64} - 48q^{66} + 764q^{67} + 8q^{68} + 228q^{69} - 236q^{71} - 72q^{72} - 418q^{73} + 796q^{74} - 496q^{76} - 56q^{77} + 252q^{78} + 552q^{79} + 81q^{81} - 924q^{82} - 1036q^{83} - 84q^{84} + 424q^{86} - 762q^{87} + 64q^{88} + 30q^{89} + 294q^{91} - 304q^{92} + 216q^{93} - 528q^{94} + 96q^{96} + 1190q^{97} - 98q^{98} - 72q^{99} + O(q^{100}) \)

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.d 1
5.b even 2 1 42.4.a.b 1
5.c odd 4 2 1050.4.g.n 2
15.d odd 2 1 126.4.a.c 1
20.d odd 2 1 336.4.a.d 1
35.c odd 2 1 294.4.a.h 1
35.i odd 6 2 294.4.e.d 2
35.j even 6 2 294.4.e.a 2
40.e odd 2 1 1344.4.a.t 1
40.f even 2 1 1344.4.a.f 1
60.h even 2 1 1008.4.a.j 1
105.g even 2 1 882.4.a.d 1
105.o odd 6 2 882.4.g.s 2
105.p even 6 2 882.4.g.r 2
140.c even 2 1 2352.4.a.ba 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.a.b 1 5.b even 2 1
126.4.a.c 1 15.d odd 2 1
294.4.a.h 1 35.c odd 2 1
294.4.e.a 2 35.j even 6 2
294.4.e.d 2 35.i odd 6 2
336.4.a.d 1 20.d odd 2 1
882.4.a.d 1 105.g even 2 1
882.4.g.r 2 105.p even 6 2
882.4.g.s 2 105.o odd 6 2
1008.4.a.j 1 60.h even 2 1
1050.4.a.d 1 1.a even 1 1 trivial
1050.4.g.n 2 5.c odd 4 2
1344.4.a.f 1 40.f even 2 1
1344.4.a.t 1 40.e odd 2 1
2352.4.a.ba 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1050))\):

\( T_{11} + 8 \)
\( T_{13} - 42 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( 3 + T \)
$5$ \( T \)
$7$ \( -7 + T \)
$11$ \( 8 + T \)
$13$ \( -42 + T \)
$17$ \( -2 + T \)
$19$ \( 124 + T \)
$23$ \( 76 + T \)
$29$ \( -254 + T \)
$31$ \( 72 + T \)
$37$ \( 398 + T \)
$41$ \( -462 + T \)
$43$ \( 212 + T \)
$47$ \( -264 + T \)
$53$ \( -162 + T \)
$59$ \( 772 + T \)
$61$ \( -30 + T \)
$67$ \( -764 + T \)
$71$ \( 236 + T \)
$73$ \( 418 + T \)
$79$ \( -552 + T \)
$83$ \( 1036 + T \)
$89$ \( -30 + T \)
$97$ \( -1190 + T \)
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