Properties

Label 2450.4.a.bo
Level $2450$
Weight $4$
Character orbit 2450.a
Self dual yes
Analytic conductor $144.555$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(144.554679514\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{2} + 8q^{3} + 4q^{4} + 16q^{6} + 8q^{8} + 37q^{9} + O(q^{10}) \) \( q + 2q^{2} + 8q^{3} + 4q^{4} + 16q^{6} + 8q^{8} + 37q^{9} - 28q^{11} + 32q^{12} + 18q^{13} + 16q^{16} + 74q^{17} + 74q^{18} - 80q^{19} - 56q^{22} + 112q^{23} + 64q^{24} + 36q^{26} + 80q^{27} + 190q^{29} - 72q^{31} + 32q^{32} - 224q^{33} + 148q^{34} + 148q^{36} + 346q^{37} - 160q^{38} + 144q^{39} - 162q^{41} + 412q^{43} - 112q^{44} + 224q^{46} + 24q^{47} + 128q^{48} + 592q^{51} + 72q^{52} - 318q^{53} + 160q^{54} - 640q^{57} + 380q^{58} + 200q^{59} + 198q^{61} - 144q^{62} + 64q^{64} - 448q^{66} + 716q^{67} + 296q^{68} + 896q^{69} + 392q^{71} + 296q^{72} + 538q^{73} + 692q^{74} - 320q^{76} + 288q^{78} + 240q^{79} - 359q^{81} - 324q^{82} - 1072q^{83} + 824q^{86} + 1520q^{87} - 224q^{88} - 810q^{89} + 448q^{92} - 576q^{93} + 48q^{94} + 256q^{96} + 1354q^{97} - 1036q^{99} + O(q^{100}) \)

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 8.00000 4.00000 0 16.0000 0 8.00000 37.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 2450.4.a.bo 1
5.b even 2 1 98.4.a.a 1
7.b odd 2 1 350.4.a.l 1
15.d odd 2 1 882.4.a.i 1
20.d odd 2 1 784.4.a.s 1
35.c odd 2 1 14.4.a.a 1
35.f even 4 2 350.4.c.b 2
35.i odd 6 2 98.4.c.d 2
35.j even 6 2 98.4.c.f 2
105.g even 2 1 126.4.a.h 1
105.o odd 6 2 882.4.g.k 2
105.p even 6 2 882.4.g.b 2
140.c even 2 1 112.4.a.a 1
280.c odd 2 1 448.4.a.b 1
280.n even 2 1 448.4.a.o 1
385.h even 2 1 1694.4.a.g 1
420.o odd 2 1 1008.4.a.s 1
455.h odd 2 1 2366.4.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.a 1 35.c odd 2 1
98.4.a.a 1 5.b even 2 1
98.4.c.d 2 35.i odd 6 2
98.4.c.f 2 35.j even 6 2
112.4.a.a 1 140.c even 2 1
126.4.a.h 1 105.g even 2 1
350.4.a.l 1 7.b odd 2 1
350.4.c.b 2 35.f even 4 2
448.4.a.b 1 280.c odd 2 1
448.4.a.o 1 280.n even 2 1
784.4.a.s 1 20.d odd 2 1
882.4.a.i 1 15.d odd 2 1
882.4.g.b 2 105.p even 6 2
882.4.g.k 2 105.o odd 6 2
1008.4.a.s 1 420.o odd 2 1
1694.4.a.g 1 385.h even 2 1
2366.4.a.h 1 455.h odd 2 1
2450.4.a.bo 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2450))\):

\( T_{3} - 8 \)
\( T_{11} + 28 \)
\( T_{19} + 80 \)
\( T_{23} - 112 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( -8 + T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( 28 + T \)
$13$ \( -18 + T \)
$17$ \( -74 + T \)
$19$ \( 80 + T \)
$23$ \( -112 + T \)
$29$ \( -190 + T \)
$31$ \( 72 + T \)
$37$ \( -346 + T \)
$41$ \( 162 + T \)
$43$ \( -412 + T \)
$47$ \( -24 + T \)
$53$ \( 318 + T \)
$59$ \( -200 + T \)
$61$ \( -198 + T \)
$67$ \( -716 + T \)
$71$ \( -392 + T \)
$73$ \( -538 + T \)
$79$ \( -240 + T \)
$83$ \( 1072 + T \)
$89$ \( 810 + T \)
$97$ \( -1354 + T \)
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