Properties

Label 50.6.a.g
Level 50
Weight 6
Character orbit 50.a
Self dual yes
Analytic conductor 8.019
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 50.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.01919099065\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q + 4q^{2} + 26q^{3} + 16q^{4} + 104q^{6} + 22q^{7} + 64q^{8} + 433q^{9} + O(q^{10}) \) \( q + 4q^{2} + 26q^{3} + 16q^{4} + 104q^{6} + 22q^{7} + 64q^{8} + 433q^{9} - 768q^{11} + 416q^{12} + 46q^{13} + 88q^{14} + 256q^{16} - 378q^{17} + 1732q^{18} + 1100q^{19} + 572q^{21} - 3072q^{22} + 1986q^{23} + 1664q^{24} + 184q^{26} + 4940q^{27} + 352q^{28} - 5610q^{29} - 3988q^{31} + 1024q^{32} - 19968q^{33} - 1512q^{34} + 6928q^{36} + 142q^{37} + 4400q^{38} + 1196q^{39} + 1542q^{41} + 2288q^{42} + 5026q^{43} - 12288q^{44} + 7944q^{46} - 24738q^{47} + 6656q^{48} - 16323q^{49} - 9828q^{51} + 736q^{52} + 14166q^{53} + 19760q^{54} + 1408q^{56} + 28600q^{57} - 22440q^{58} + 28380q^{59} + 5522q^{61} - 15952q^{62} + 9526q^{63} + 4096q^{64} - 79872q^{66} + 24742q^{67} - 6048q^{68} + 51636q^{69} + 42372q^{71} + 27712q^{72} + 52126q^{73} + 568q^{74} + 17600q^{76} - 16896q^{77} + 4784q^{78} - 39640q^{79} + 23221q^{81} + 6168q^{82} + 59826q^{83} + 9152q^{84} + 20104q^{86} - 145860q^{87} - 49152q^{88} + 57690q^{89} + 1012q^{91} + 31776q^{92} - 103688q^{93} - 98952q^{94} + 26624q^{96} + 144382q^{97} - 65292q^{98} - 332544q^{99} + O(q^{100}) \)

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
4.00000 26.0000 16.0000 0 104.000 22.0000 64.0000 433.000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 50.6.a.g 1
3.b odd 2 1 450.6.a.h 1
4.b odd 2 1 400.6.a.a 1
5.b even 2 1 10.6.a.a 1
5.c odd 4 2 50.6.b.d 2
15.d odd 2 1 90.6.a.f 1
15.e even 4 2 450.6.c.o 2
20.d odd 2 1 80.6.a.h 1
20.e even 4 2 400.6.c.a 2
35.c odd 2 1 490.6.a.j 1
40.e odd 2 1 320.6.a.a 1
40.f even 2 1 320.6.a.p 1
60.h even 2 1 720.6.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.a 1 5.b even 2 1
50.6.a.g 1 1.a even 1 1 trivial
50.6.b.d 2 5.c odd 4 2
80.6.a.h 1 20.d odd 2 1
90.6.a.f 1 15.d odd 2 1
320.6.a.a 1 40.e odd 2 1
320.6.a.p 1 40.f even 2 1
400.6.a.a 1 4.b odd 2 1
400.6.c.a 2 20.e even 4 2
450.6.a.h 1 3.b odd 2 1
450.6.c.o 2 15.e even 4 2
490.6.a.j 1 35.c odd 2 1
720.6.a.r 1 60.h even 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 26 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(50))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T \)
$3$ \( 1 - 26 T + 243 T^{2} \)
$5$ 1
$7$ \( 1 - 22 T + 16807 T^{2} \)
$11$ \( 1 + 768 T + 161051 T^{2} \)
$13$ \( 1 - 46 T + 371293 T^{2} \)
$17$ \( 1 + 378 T + 1419857 T^{2} \)
$19$ \( 1 - 1100 T + 2476099 T^{2} \)
$23$ \( 1 - 1986 T + 6436343 T^{2} \)
$29$ \( 1 + 5610 T + 20511149 T^{2} \)
$31$ \( 1 + 3988 T + 28629151 T^{2} \)
$37$ \( 1 - 142 T + 69343957 T^{2} \)
$41$ \( 1 - 1542 T + 115856201 T^{2} \)
$43$ \( 1 - 5026 T + 147008443 T^{2} \)
$47$ \( 1 + 24738 T + 229345007 T^{2} \)
$53$ \( 1 - 14166 T + 418195493 T^{2} \)
$59$ \( 1 - 28380 T + 714924299 T^{2} \)
$61$ \( 1 - 5522 T + 844596301 T^{2} \)
$67$ \( 1 - 24742 T + 1350125107 T^{2} \)
$71$ \( 1 - 42372 T + 1804229351 T^{2} \)
$73$ \( 1 - 52126 T + 2073071593 T^{2} \)
$79$ \( 1 + 39640 T + 3077056399 T^{2} \)
$83$ \( 1 - 59826 T + 3939040643 T^{2} \)
$89$ \( 1 - 57690 T + 5584059449 T^{2} \)
$97$ \( 1 - 144382 T + 8587340257 T^{2} \)
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