Properties

Label 50.4.a.b
Level $50$
Weight $4$
Character orbit 50.a
Self dual yes
Analytic conductor $2.950$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 50.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.95009550029\)
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

\(f(q)\) \(=\) \( q - 2q^{2} - 2q^{3} + 4q^{4} + 4q^{6} - 26q^{7} - 8q^{8} - 23q^{9} + O(q^{10}) \) \( q - 2q^{2} - 2q^{3} + 4q^{4} + 4q^{6} - 26q^{7} - 8q^{8} - 23q^{9} - 28q^{11} - 8q^{12} - 12q^{13} + 52q^{14} + 16q^{16} + 64q^{17} + 46q^{18} - 60q^{19} + 52q^{21} + 56q^{22} + 58q^{23} + 16q^{24} + 24q^{26} + 100q^{27} - 104q^{28} + 90q^{29} - 128q^{31} - 32q^{32} + 56q^{33} - 128q^{34} - 92q^{36} - 236q^{37} + 120q^{38} + 24q^{39} + 242q^{41} - 104q^{42} - 362q^{43} - 112q^{44} - 116q^{46} - 226q^{47} - 32q^{48} + 333q^{49} - 128q^{51} - 48q^{52} + 108q^{53} - 200q^{54} + 208q^{56} + 120q^{57} - 180q^{58} - 20q^{59} + 542q^{61} + 256q^{62} + 598q^{63} + 64q^{64} - 112q^{66} + 434q^{67} + 256q^{68} - 116q^{69} - 1128q^{71} + 184q^{72} - 632q^{73} + 472q^{74} - 240q^{76} + 728q^{77} - 48q^{78} - 720q^{79} + 421q^{81} - 484q^{82} + 478q^{83} + 208q^{84} + 724q^{86} - 180q^{87} + 224q^{88} - 490q^{89} + 312q^{91} + 232q^{92} + 256q^{93} + 452q^{94} + 64q^{96} - 1456q^{97} - 666q^{98} + 644q^{99} + O(q^{100}) \)

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 −26.0000 −8.00000 −23.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 50.4.a.b 1
3.b odd 2 1 450.4.a.k 1
4.b odd 2 1 400.4.a.n 1
5.b even 2 1 50.4.a.d 1
5.c odd 4 2 10.4.b.a 2
7.b odd 2 1 2450.4.a.o 1
8.b even 2 1 1600.4.a.bh 1
8.d odd 2 1 1600.4.a.t 1
15.d odd 2 1 450.4.a.j 1
15.e even 4 2 90.4.c.b 2
20.d odd 2 1 400.4.a.h 1
20.e even 4 2 80.4.c.a 2
35.c odd 2 1 2450.4.a.bb 1
35.f even 4 2 490.4.c.b 2
40.e odd 2 1 1600.4.a.bg 1
40.f even 2 1 1600.4.a.u 1
40.i odd 4 2 320.4.c.d 2
40.k even 4 2 320.4.c.c 2
60.l odd 4 2 720.4.f.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.b.a 2 5.c odd 4 2
50.4.a.b 1 1.a even 1 1 trivial
50.4.a.d 1 5.b even 2 1
80.4.c.a 2 20.e even 4 2
90.4.c.b 2 15.e even 4 2
320.4.c.c 2 40.k even 4 2
320.4.c.d 2 40.i odd 4 2
400.4.a.h 1 20.d odd 2 1
400.4.a.n 1 4.b odd 2 1
450.4.a.j 1 15.d odd 2 1
450.4.a.k 1 3.b odd 2 1
490.4.c.b 2 35.f even 4 2
720.4.f.f 2 60.l odd 4 2
1600.4.a.t 1 8.d odd 2 1
1600.4.a.u 1 40.f even 2 1
1600.4.a.bg 1 40.e odd 2 1
1600.4.a.bh 1 8.b even 2 1
2450.4.a.o 1 7.b odd 2 1
2450.4.a.bb 1 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 2 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(50))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( 2 + T \)
$5$ \( T \)
$7$ \( 26 + T \)
$11$ \( 28 + T \)
$13$ \( 12 + T \)
$17$ \( -64 + T \)
$19$ \( 60 + T \)
$23$ \( -58 + T \)
$29$ \( -90 + T \)
$31$ \( 128 + T \)
$37$ \( 236 + T \)
$41$ \( -242 + T \)
$43$ \( 362 + T \)
$47$ \( 226 + T \)
$53$ \( -108 + T \)
$59$ \( 20 + T \)
$61$ \( -542 + T \)
$67$ \( -434 + T \)
$71$ \( 1128 + T \)
$73$ \( 632 + T \)
$79$ \( 720 + T \)
$83$ \( -478 + T \)
$89$ \( 490 + T \)
$97$ \( 1456 + T \)
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