Properties

Label 4.6.a.a
Level 4
Weight 6
Character orbit 4.a
Self dual yes
Analytic conductor 0.642
Analytic rank 0
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 4 = 2^{2} \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 4.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.641535279252\)
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

\(f(q)\) \(=\) \( q - 12q^{3} + 54q^{5} - 88q^{7} - 99q^{9} + O(q^{10}) \) \( q - 12q^{3} + 54q^{5} - 88q^{7} - 99q^{9} + 540q^{11} - 418q^{13} - 648q^{15} + 594q^{17} + 836q^{19} + 1056q^{21} - 4104q^{23} - 209q^{25} + 4104q^{27} - 594q^{29} + 4256q^{31} - 6480q^{33} - 4752q^{35} - 298q^{37} + 5016q^{39} + 17226q^{41} - 12100q^{43} - 5346q^{45} - 1296q^{47} - 9063q^{49} - 7128q^{51} + 19494q^{53} + 29160q^{55} - 10032q^{57} - 7668q^{59} - 34738q^{61} + 8712q^{63} - 22572q^{65} + 21812q^{67} + 49248q^{69} - 46872q^{71} + 67562q^{73} + 2508q^{75} - 47520q^{77} - 76912q^{79} - 25191q^{81} + 67716q^{83} + 32076q^{85} + 7128q^{87} + 29754q^{89} + 36784q^{91} - 51072q^{93} + 45144q^{95} - 122398q^{97} - 53460q^{99} + O(q^{100}) \)

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 −12.0000 0 54.0000 0 −88.0000 0 −99.0000 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 4.6.a.a 1
3.b odd 2 1 36.6.a.a 1
4.b odd 2 1 16.6.a.b 1
5.b even 2 1 100.6.a.b 1
5.c odd 4 2 100.6.c.b 2
7.b odd 2 1 196.6.a.e 1
7.c even 3 2 196.6.e.g 2
7.d odd 6 2 196.6.e.d 2
8.b even 2 1 64.6.a.f 1
8.d odd 2 1 64.6.a.b 1
9.c even 3 2 324.6.e.a 2
9.d odd 6 2 324.6.e.d 2
11.b odd 2 1 484.6.a.a 1
12.b even 2 1 144.6.a.c 1
13.b even 2 1 676.6.a.a 1
13.d odd 4 2 676.6.d.a 2
15.d odd 2 1 900.6.a.h 1
15.e even 4 2 900.6.d.a 2
16.e even 4 2 256.6.b.g 2
16.f odd 4 2 256.6.b.c 2
20.d odd 2 1 400.6.a.d 1
20.e even 4 2 400.6.c.f 2
24.f even 2 1 576.6.a.bd 1
24.h odd 2 1 576.6.a.bc 1
28.d even 2 1 784.6.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.6.a.a 1 1.a even 1 1 trivial
16.6.a.b 1 4.b odd 2 1
36.6.a.a 1 3.b odd 2 1
64.6.a.b 1 8.d odd 2 1
64.6.a.f 1 8.b even 2 1
100.6.a.b 1 5.b even 2 1
100.6.c.b 2 5.c odd 4 2
144.6.a.c 1 12.b even 2 1
196.6.a.e 1 7.b odd 2 1
196.6.e.d 2 7.d odd 6 2
196.6.e.g 2 7.c even 3 2
256.6.b.c 2 16.f odd 4 2
256.6.b.g 2 16.e even 4 2
324.6.e.a 2 9.c even 3 2
324.6.e.d 2 9.d odd 6 2
400.6.a.d 1 20.d odd 2 1
400.6.c.f 2 20.e even 4 2
484.6.a.a 1 11.b odd 2 1
576.6.a.bc 1 24.h odd 2 1
576.6.a.bd 1 24.f even 2 1
676.6.a.a 1 13.b even 2 1
676.6.d.a 2 13.d odd 4 2
784.6.a.d 1 28.d even 2 1
900.6.a.h 1 15.d odd 2 1
900.6.d.a 2 15.e even 4 2

Atkin-Lehner signs

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

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 + 12 T + 243 T^{2} \)
$5$ \( 1 - 54 T + 3125 T^{2} \)
$7$ \( 1 + 88 T + 16807 T^{2} \)
$11$ \( 1 - 540 T + 161051 T^{2} \)
$13$ \( 1 + 418 T + 371293 T^{2} \)
$17$ \( 1 - 594 T + 1419857 T^{2} \)
$19$ \( 1 - 836 T + 2476099 T^{2} \)
$23$ \( 1 + 4104 T + 6436343 T^{2} \)
$29$ \( 1 + 594 T + 20511149 T^{2} \)
$31$ \( 1 - 4256 T + 28629151 T^{2} \)
$37$ \( 1 + 298 T + 69343957 T^{2} \)
$41$ \( 1 - 17226 T + 115856201 T^{2} \)
$43$ \( 1 + 12100 T + 147008443 T^{2} \)
$47$ \( 1 + 1296 T + 229345007 T^{2} \)
$53$ \( 1 - 19494 T + 418195493 T^{2} \)
$59$ \( 1 + 7668 T + 714924299 T^{2} \)
$61$ \( 1 + 34738 T + 844596301 T^{2} \)
$67$ \( 1 - 21812 T + 1350125107 T^{2} \)
$71$ \( 1 + 46872 T + 1804229351 T^{2} \)
$73$ \( 1 - 67562 T + 2073071593 T^{2} \)
$79$ \( 1 + 76912 T + 3077056399 T^{2} \)
$83$ \( 1 - 67716 T + 3939040643 T^{2} \)
$89$ \( 1 - 29754 T + 5584059449 T^{2} \)
$97$ \( 1 + 122398 T + 8587340257 T^{2} \)
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