Properties

Label 40.6.a.a
Level 40
Weight 6
Character orbit 40.a
Self dual yes
Analytic conductor 6.415
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 40.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.41535279252\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 18q^{3} - 25q^{5} + 242q^{7} + 81q^{9} + O(q^{10}) \) \( q - 18q^{3} - 25q^{5} + 242q^{7} + 81q^{9} + 656q^{11} - 206q^{13} + 450q^{15} + 1690q^{17} - 1364q^{19} - 4356q^{21} + 2198q^{23} + 625q^{25} + 2916q^{27} - 2218q^{29} - 1700q^{31} - 11808q^{33} - 6050q^{35} - 846q^{37} + 3708q^{39} - 1818q^{41} + 10534q^{43} - 2025q^{45} + 12074q^{47} + 41757q^{49} - 30420q^{51} + 32586q^{53} - 16400q^{55} + 24552q^{57} + 8668q^{59} - 34670q^{61} + 19602q^{63} + 5150q^{65} - 47566q^{67} - 39564q^{69} + 948q^{71} - 63102q^{73} - 11250q^{75} + 158752q^{77} + 46536q^{79} - 72171q^{81} - 88778q^{83} - 42250q^{85} + 39924q^{87} - 104934q^{89} - 49852q^{91} + 30600q^{93} + 34100q^{95} - 36254q^{97} + 53136q^{99} + O(q^{100}) \)

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
0 −18.0000 0 −25.0000 0 242.000 0 81.0000 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 40.6.a.a 1
3.b odd 2 1 360.6.a.i 1
4.b odd 2 1 80.6.a.g 1
5.b even 2 1 200.6.a.d 1
5.c odd 4 2 200.6.c.b 2
8.b even 2 1 320.6.a.m 1
8.d odd 2 1 320.6.a.d 1
12.b even 2 1 720.6.a.k 1
20.d odd 2 1 400.6.a.b 1
20.e even 4 2 400.6.c.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.a.a 1 1.a even 1 1 trivial
80.6.a.g 1 4.b odd 2 1
200.6.a.d 1 5.b even 2 1
200.6.c.b 2 5.c odd 4 2
320.6.a.d 1 8.d odd 2 1
320.6.a.m 1 8.b even 2 1
360.6.a.i 1 3.b odd 2 1
400.6.a.b 1 20.d odd 2 1
400.6.c.e 2 20.e even 4 2
720.6.a.k 1 12.b 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 \( T_{3} + 18 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(40))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 18 T + 243 T^{2} \)
$5$ \( 1 + 25 T \)
$7$ \( 1 - 242 T + 16807 T^{2} \)
$11$ \( 1 - 656 T + 161051 T^{2} \)
$13$ \( 1 + 206 T + 371293 T^{2} \)
$17$ \( 1 - 1690 T + 1419857 T^{2} \)
$19$ \( 1 + 1364 T + 2476099 T^{2} \)
$23$ \( 1 - 2198 T + 6436343 T^{2} \)
$29$ \( 1 + 2218 T + 20511149 T^{2} \)
$31$ \( 1 + 1700 T + 28629151 T^{2} \)
$37$ \( 1 + 846 T + 69343957 T^{2} \)
$41$ \( 1 + 1818 T + 115856201 T^{2} \)
$43$ \( 1 - 10534 T + 147008443 T^{2} \)
$47$ \( 1 - 12074 T + 229345007 T^{2} \)
$53$ \( 1 - 32586 T + 418195493 T^{2} \)
$59$ \( 1 - 8668 T + 714924299 T^{2} \)
$61$ \( 1 + 34670 T + 844596301 T^{2} \)
$67$ \( 1 + 47566 T + 1350125107 T^{2} \)
$71$ \( 1 - 948 T + 1804229351 T^{2} \)
$73$ \( 1 + 63102 T + 2073071593 T^{2} \)
$79$ \( 1 - 46536 T + 3077056399 T^{2} \)
$83$ \( 1 + 88778 T + 3939040643 T^{2} \)
$89$ \( 1 + 104934 T + 5584059449 T^{2} \)
$97$ \( 1 + 36254 T + 8587340257 T^{2} \)
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