Properties

Label 2450.4.a.bb
Level $2450$
Weight $4$
Character orbit 2450.a
Self dual yes
Analytic conductor $144.555$
Analytic rank $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 10)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{6} + 8 q^{8} - 23 q^{9} + O(q^{10}) \) \( q + 2 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{6} + 8 q^{8} - 23 q^{9} - 28 q^{11} - 8 q^{12} - 12 q^{13} + 16 q^{16} + 64 q^{17} - 46 q^{18} + 60 q^{19} - 56 q^{22} - 58 q^{23} - 16 q^{24} - 24 q^{26} + 100 q^{27} + 90 q^{29} + 128 q^{31} + 32 q^{32} + 56 q^{33} + 128 q^{34} - 92 q^{36} + 236 q^{37} + 120 q^{38} + 24 q^{39} - 242 q^{41} + 362 q^{43} - 112 q^{44} - 116 q^{46} - 226 q^{47} - 32 q^{48} - 128 q^{51} - 48 q^{52} - 108 q^{53} + 200 q^{54} - 120 q^{57} + 180 q^{58} + 20 q^{59} - 542 q^{61} + 256 q^{62} + 64 q^{64} + 112 q^{66} - 434 q^{67} + 256 q^{68} + 116 q^{69} - 1128 q^{71} - 184 q^{72} - 632 q^{73} + 472 q^{74} + 240 q^{76} + 48 q^{78} - 720 q^{79} + 421 q^{81} - 484 q^{82} + 478 q^{83} + 724 q^{86} - 180 q^{87} - 224 q^{88} + 490 q^{89} - 232 q^{92} - 256 q^{93} - 452 q^{94} - 64 q^{96} - 1456 q^{97} + 644 q^{99} + O(q^{100}) \)

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 −2.00000 4.00000 0 −4.00000 0 8.00000 −23.0000 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.bb 1
5.b even 2 1 2450.4.a.o 1
5.c odd 4 2 490.4.c.b 2
7.b odd 2 1 50.4.a.d 1
21.c even 2 1 450.4.a.j 1
28.d even 2 1 400.4.a.h 1
35.c odd 2 1 50.4.a.b 1
35.f even 4 2 10.4.b.a 2
56.e even 2 1 1600.4.a.bg 1
56.h odd 2 1 1600.4.a.u 1
105.g even 2 1 450.4.a.k 1
105.k odd 4 2 90.4.c.b 2
140.c even 2 1 400.4.a.n 1
140.j odd 4 2 80.4.c.a 2
280.c odd 2 1 1600.4.a.bh 1
280.n even 2 1 1600.4.a.t 1
280.s even 4 2 320.4.c.d 2
280.y odd 4 2 320.4.c.c 2
420.w even 4 2 720.4.f.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.b.a 2 35.f even 4 2
50.4.a.b 1 35.c odd 2 1
50.4.a.d 1 7.b odd 2 1
80.4.c.a 2 140.j odd 4 2
90.4.c.b 2 105.k odd 4 2
320.4.c.c 2 280.y odd 4 2
320.4.c.d 2 280.s even 4 2
400.4.a.h 1 28.d even 2 1
400.4.a.n 1 140.c even 2 1
450.4.a.j 1 21.c even 2 1
450.4.a.k 1 105.g even 2 1
490.4.c.b 2 5.c odd 4 2
720.4.f.f 2 420.w even 4 2
1600.4.a.t 1 280.n even 2 1
1600.4.a.u 1 56.h odd 2 1
1600.4.a.bg 1 56.e even 2 1
1600.4.a.bh 1 280.c odd 2 1
2450.4.a.o 1 5.b even 2 1
2450.4.a.bb 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2450))\):

\( T_{3} + 2 \)
\( T_{11} + 28 \)
\( T_{19} - 60 \)
\( T_{23} + 58 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( 2 + T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( 28 + T \)
$13$ \( 12 + T \)
$17$ \( -64 + T \)
$19$ \( -60 + T \)
$23$ \( 58 + T \)
$29$ \( -90 + T \)
$31$ \( -128 + T \)
$37$ \( -236 + T \)
$41$ \( 242 + T \)
$43$ \( -362 + T \)
$47$ \( 226 + T \)
$53$ \( 108 + T \)
$59$ \( -20 + T \)
$61$ \( 542 + T \)
$67$ \( 434 + T \)
$71$ \( 1128 + T \)
$73$ \( 632 + T \)
$79$ \( 720 + T \)
$83$ \( -478 + T \)
$89$ \( -490 + T \)
$97$ \( 1456 + T \)
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