Properties

Label 3.6.a.a
Level $3$
Weight $6$
Character orbit 3.a
Self dual yes
Analytic conductor $0.481$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3,6,Mod(1,3)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.481151459439\)
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 6 q^{2} + 9 q^{3} + 4 q^{4} + 6 q^{5} - 54 q^{6} - 40 q^{7} + 168 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 6 q^{2} + 9 q^{3} + 4 q^{4} + 6 q^{5} - 54 q^{6} - 40 q^{7} + 168 q^{8} + 81 q^{9} - 36 q^{10} - 564 q^{11} + 36 q^{12} + 638 q^{13} + 240 q^{14} + 54 q^{15} - 1136 q^{16} + 882 q^{17} - 486 q^{18} - 556 q^{19} + 24 q^{20} - 360 q^{21} + 3384 q^{22} - 840 q^{23} + 1512 q^{24} - 3089 q^{25} - 3828 q^{26} + 729 q^{27} - 160 q^{28} + 4638 q^{29} - 324 q^{30} + 4400 q^{31} + 1440 q^{32} - 5076 q^{33} - 5292 q^{34} - 240 q^{35} + 324 q^{36} - 2410 q^{37} + 3336 q^{38} + 5742 q^{39} + 1008 q^{40} - 6870 q^{41} + 2160 q^{42} + 9644 q^{43} - 2256 q^{44} + 486 q^{45} + 5040 q^{46} - 18672 q^{47} - 10224 q^{48} - 15207 q^{49} + 18534 q^{50} + 7938 q^{51} + 2552 q^{52} + 33750 q^{53} - 4374 q^{54} - 3384 q^{55} - 6720 q^{56} - 5004 q^{57} - 27828 q^{58} - 18084 q^{59} + 216 q^{60} + 39758 q^{61} - 26400 q^{62} - 3240 q^{63} + 27712 q^{64} + 3828 q^{65} + 30456 q^{66} - 23068 q^{67} + 3528 q^{68} - 7560 q^{69} + 1440 q^{70} - 4248 q^{71} + 13608 q^{72} - 41110 q^{73} + 14460 q^{74} - 27801 q^{75} - 2224 q^{76} + 22560 q^{77} - 34452 q^{78} + 21920 q^{79} - 6816 q^{80} + 6561 q^{81} + 41220 q^{82} + 82452 q^{83} - 1440 q^{84} + 5292 q^{85} - 57864 q^{86} + 41742 q^{87} - 94752 q^{88} - 94086 q^{89} - 2916 q^{90} - 25520 q^{91} - 3360 q^{92} + 39600 q^{93} + 112032 q^{94} - 3336 q^{95} + 12960 q^{96} + 49442 q^{97} + 91242 q^{98} - 45684 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Expression as an eta quotient

\(f(z) = \eta(z)^{6}\eta(3z)^{6}=q\prod_{n=1}^\infty(1 - q^{n})^{6}(1 - q^{3n})^{6}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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
−6.00000 9.00000 4.00000 6.00000 −54.0000 −40.0000 168.000 81.0000 −36.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)

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 3.6.a.a 1
3.b odd 2 1 9.6.a.a 1
4.b odd 2 1 48.6.a.a 1
5.b even 2 1 75.6.a.e 1
5.c odd 4 2 75.6.b.b 2
7.b odd 2 1 147.6.a.a 1
7.c even 3 2 147.6.e.h 2
7.d odd 6 2 147.6.e.k 2
8.b even 2 1 192.6.a.d 1
8.d odd 2 1 192.6.a.l 1
9.c even 3 2 81.6.c.c 2
9.d odd 6 2 81.6.c.a 2
11.b odd 2 1 363.6.a.d 1
12.b even 2 1 144.6.a.f 1
13.b even 2 1 507.6.a.b 1
15.d odd 2 1 225.6.a.a 1
15.e even 4 2 225.6.b.b 2
16.e even 4 2 768.6.d.k 2
16.f odd 4 2 768.6.d.h 2
17.b even 2 1 867.6.a.a 1
19.b odd 2 1 1083.6.a.c 1
21.c even 2 1 441.6.a.i 1
24.f even 2 1 576.6.a.t 1
24.h odd 2 1 576.6.a.s 1
33.d even 2 1 1089.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.6.a.a 1 1.a even 1 1 trivial
9.6.a.a 1 3.b odd 2 1
48.6.a.a 1 4.b odd 2 1
75.6.a.e 1 5.b even 2 1
75.6.b.b 2 5.c odd 4 2
81.6.c.a 2 9.d odd 6 2
81.6.c.c 2 9.c even 3 2
144.6.a.f 1 12.b even 2 1
147.6.a.a 1 7.b odd 2 1
147.6.e.h 2 7.c even 3 2
147.6.e.k 2 7.d odd 6 2
192.6.a.d 1 8.b even 2 1
192.6.a.l 1 8.d odd 2 1
225.6.a.a 1 15.d odd 2 1
225.6.b.b 2 15.e even 4 2
363.6.a.d 1 11.b odd 2 1
441.6.a.i 1 21.c even 2 1
507.6.a.b 1 13.b even 2 1
576.6.a.s 1 24.h odd 2 1
576.6.a.t 1 24.f even 2 1
768.6.d.h 2 16.f odd 4 2
768.6.d.k 2 16.e even 4 2
867.6.a.a 1 17.b even 2 1
1083.6.a.c 1 19.b odd 2 1
1089.6.a.b 1 33.d even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 6 \) Copy content Toggle raw display
$3$ \( T - 9 \) Copy content Toggle raw display
$5$ \( T - 6 \) Copy content Toggle raw display
$7$ \( T + 40 \) Copy content Toggle raw display
$11$ \( T + 564 \) Copy content Toggle raw display
$13$ \( T - 638 \) Copy content Toggle raw display
$17$ \( T - 882 \) Copy content Toggle raw display
$19$ \( T + 556 \) Copy content Toggle raw display
$23$ \( T + 840 \) Copy content Toggle raw display
$29$ \( T - 4638 \) Copy content Toggle raw display
$31$ \( T - 4400 \) Copy content Toggle raw display
$37$ \( T + 2410 \) Copy content Toggle raw display
$41$ \( T + 6870 \) Copy content Toggle raw display
$43$ \( T - 9644 \) Copy content Toggle raw display
$47$ \( T + 18672 \) Copy content Toggle raw display
$53$ \( T - 33750 \) Copy content Toggle raw display
$59$ \( T + 18084 \) Copy content Toggle raw display
$61$ \( T - 39758 \) Copy content Toggle raw display
$67$ \( T + 23068 \) Copy content Toggle raw display
$71$ \( T + 4248 \) Copy content Toggle raw display
$73$ \( T + 41110 \) Copy content Toggle raw display
$79$ \( T - 21920 \) Copy content Toggle raw display
$83$ \( T - 82452 \) Copy content Toggle raw display
$89$ \( T + 94086 \) Copy content Toggle raw display
$97$ \( T - 49442 \) Copy content Toggle raw display
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