Properties

Label 10.6.a.b
Level 10
Weight 6
Character orbit 10.a
Self dual yes
Analytic conductor 1.604
Analytic rank 0
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 10.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.60383819813\)
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

\(f(q)\) \(=\) \( q - 4q^{2} + 24q^{3} + 16q^{4} + 25q^{5} - 96q^{6} - 172q^{7} - 64q^{8} + 333q^{9} + O(q^{10}) \) \( q - 4q^{2} + 24q^{3} + 16q^{4} + 25q^{5} - 96q^{6} - 172q^{7} - 64q^{8} + 333q^{9} - 100q^{10} + 132q^{11} + 384q^{12} - 946q^{13} + 688q^{14} + 600q^{15} + 256q^{16} - 222q^{17} - 1332q^{18} + 500q^{19} + 400q^{20} - 4128q^{21} - 528q^{22} + 3564q^{23} - 1536q^{24} + 625q^{25} + 3784q^{26} + 2160q^{27} - 2752q^{28} + 2190q^{29} - 2400q^{30} + 2312q^{31} - 1024q^{32} + 3168q^{33} + 888q^{34} - 4300q^{35} + 5328q^{36} - 11242q^{37} - 2000q^{38} - 22704q^{39} - 1600q^{40} + 1242q^{41} + 16512q^{42} + 20624q^{43} + 2112q^{44} + 8325q^{45} - 14256q^{46} + 6588q^{47} + 6144q^{48} + 12777q^{49} - 2500q^{50} - 5328q^{51} - 15136q^{52} - 21066q^{53} - 8640q^{54} + 3300q^{55} + 11008q^{56} + 12000q^{57} - 8760q^{58} + 7980q^{59} + 9600q^{60} + 16622q^{61} - 9248q^{62} - 57276q^{63} + 4096q^{64} - 23650q^{65} - 12672q^{66} + 1808q^{67} - 3552q^{68} + 85536q^{69} + 17200q^{70} - 24528q^{71} - 21312q^{72} + 20474q^{73} + 44968q^{74} + 15000q^{75} + 8000q^{76} - 22704q^{77} + 90816q^{78} - 46240q^{79} + 6400q^{80} - 29079q^{81} - 4968q^{82} - 51576q^{83} - 66048q^{84} - 5550q^{85} - 82496q^{86} + 52560q^{87} - 8448q^{88} - 110310q^{89} - 33300q^{90} + 162712q^{91} + 57024q^{92} + 55488q^{93} - 26352q^{94} + 12500q^{95} - 24576q^{96} - 78382q^{97} - 51108q^{98} + 43956q^{99} + O(q^{100}) \)

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 24.0000 16.0000 25.0000 −96.0000 −172.000 −64.0000 333.000 −100.000
\(n\): 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 10.6.a.b 1
3.b odd 2 1 90.6.a.d 1
4.b odd 2 1 80.6.a.a 1
5.b even 2 1 50.6.a.d 1
5.c odd 4 2 50.6.b.a 2
7.b odd 2 1 490.6.a.a 1
8.b even 2 1 320.6.a.b 1
8.d odd 2 1 320.6.a.o 1
12.b even 2 1 720.6.a.j 1
15.d odd 2 1 450.6.a.l 1
15.e even 4 2 450.6.c.h 2
20.d odd 2 1 400.6.a.n 1
20.e even 4 2 400.6.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.b 1 1.a even 1 1 trivial
50.6.a.d 1 5.b even 2 1
50.6.b.a 2 5.c odd 4 2
80.6.a.a 1 4.b odd 2 1
90.6.a.d 1 3.b odd 2 1
320.6.a.b 1 8.b even 2 1
320.6.a.o 1 8.d odd 2 1
400.6.a.n 1 20.d odd 2 1
400.6.c.b 2 20.e even 4 2
450.6.a.l 1 15.d odd 2 1
450.6.c.h 2 15.e even 4 2
490.6.a.a 1 7.b odd 2 1
720.6.a.j 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 \( T_{3} - 24 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(10))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T \)
$3$ \( 1 - 24 T + 243 T^{2} \)
$5$ \( 1 - 25 T \)
$7$ \( 1 + 172 T + 16807 T^{2} \)
$11$ \( 1 - 132 T + 161051 T^{2} \)
$13$ \( 1 + 946 T + 371293 T^{2} \)
$17$ \( 1 + 222 T + 1419857 T^{2} \)
$19$ \( 1 - 500 T + 2476099 T^{2} \)
$23$ \( 1 - 3564 T + 6436343 T^{2} \)
$29$ \( 1 - 2190 T + 20511149 T^{2} \)
$31$ \( 1 - 2312 T + 28629151 T^{2} \)
$37$ \( 1 + 11242 T + 69343957 T^{2} \)
$41$ \( 1 - 1242 T + 115856201 T^{2} \)
$43$ \( 1 - 20624 T + 147008443 T^{2} \)
$47$ \( 1 - 6588 T + 229345007 T^{2} \)
$53$ \( 1 + 21066 T + 418195493 T^{2} \)
$59$ \( 1 - 7980 T + 714924299 T^{2} \)
$61$ \( 1 - 16622 T + 844596301 T^{2} \)
$67$ \( 1 - 1808 T + 1350125107 T^{2} \)
$71$ \( 1 + 24528 T + 1804229351 T^{2} \)
$73$ \( 1 - 20474 T + 2073071593 T^{2} \)
$79$ \( 1 + 46240 T + 3077056399 T^{2} \)
$83$ \( 1 + 51576 T + 3939040643 T^{2} \)
$89$ \( 1 + 110310 T + 5584059449 T^{2} \)
$97$ \( 1 + 78382 T + 8587340257 T^{2} \)
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