Properties

Label 162.6.a.f
Level $162$
Weight $6$
Character orbit 162.a
Self dual yes
Analytic conductor $25.982$
Analytic rank $1$
Dimension $2$
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] = [162,6,Mod(1,162)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(162, 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("162.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 162.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: \(25.9821788097\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 18)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 6\sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 q^{2} + 16 q^{4} + (2 \beta - 27) q^{5} + ( - \beta - 37) q^{7} + 64 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + 16 q^{4} + (2 \beta - 27) q^{5} + ( - \beta - 37) q^{7} + 64 q^{8} + (8 \beta - 108) q^{10} + ( - 43 \beta - 39) q^{11} + ( - 20 \beta - 553) q^{13} + ( - 4 \beta - 148) q^{14} + 256 q^{16} + (94 \beta - 246) q^{17} + (124 \beta - 820) q^{19} + (32 \beta - 432) q^{20} + ( - 172 \beta - 156) q^{22} + (25 \beta - 2769) q^{23} + ( - 108 \beta - 1532) q^{25} + ( - 80 \beta - 2212) q^{26} + ( - 16 \beta - 592) q^{28} + ( - 124 \beta - 1947) q^{29} + (435 \beta - 2359) q^{31} + 1024 q^{32} + (376 \beta - 984) q^{34} + ( - 47 \beta + 567) q^{35} + ( - 826 \beta - 2398) q^{37} + (496 \beta - 3280) q^{38} + (128 \beta - 1728) q^{40} + (14 \beta + 7677) q^{41} + ( - 267 \beta - 16429) q^{43} + ( - 688 \beta - 624) q^{44} + (100 \beta - 11076) q^{46} + (249 \beta + 12477) q^{47} + (74 \beta - 15222) q^{49} + ( - 432 \beta - 6128) q^{50} + ( - 320 \beta - 8848) q^{52} + ( - 178 \beta + 8166) q^{53} + (1083 \beta - 17523) q^{55} + ( - 64 \beta - 2368) q^{56} + ( - 496 \beta - 7788) q^{58} + ( - 311 \beta + 10983) q^{59} + (708 \beta - 1525) q^{61} + (1740 \beta - 9436) q^{62} + 4096 q^{64} + ( - 566 \beta + 6291) q^{65} + ( - 1671 \beta - 18379) q^{67} + (1504 \beta - 3936) q^{68} + ( - 188 \beta + 2268) q^{70} + (1798 \beta + 36924) q^{71} + (870 \beta - 25594) q^{73} + ( - 3304 \beta - 9592) q^{74} + (1984 \beta - 13120) q^{76} + (1630 \beta + 10731) q^{77} + ( - 707 \beta + 7463) q^{79} + (512 \beta - 6912) q^{80} + (56 \beta + 30708) q^{82} + (147 \beta + 45381) q^{83} + ( - 3030 \beta + 47250) q^{85} + ( - 1068 \beta - 65716) q^{86} + ( - 2752 \beta - 2496) q^{88} + ( - 6086 \beta + 4650) q^{89} + (1293 \beta + 24781) q^{91} + (400 \beta - 44304) q^{92} + (996 \beta + 49908) q^{94} + ( - 4988 \beta + 75708) q^{95} + (6574 \beta + 15131) q^{97} + (296 \beta - 60888) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{2} + 32 q^{4} - 54 q^{5} - 74 q^{7} + 128 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{2} + 32 q^{4} - 54 q^{5} - 74 q^{7} + 128 q^{8} - 216 q^{10} - 78 q^{11} - 1106 q^{13} - 296 q^{14} + 512 q^{16} - 492 q^{17} - 1640 q^{19} - 864 q^{20} - 312 q^{22} - 5538 q^{23} - 3064 q^{25} - 4424 q^{26} - 1184 q^{28} - 3894 q^{29} - 4718 q^{31} + 2048 q^{32} - 1968 q^{34} + 1134 q^{35} - 4796 q^{37} - 6560 q^{38} - 3456 q^{40} + 15354 q^{41} - 32858 q^{43} - 1248 q^{44} - 22152 q^{46} + 24954 q^{47} - 30444 q^{49} - 12256 q^{50} - 17696 q^{52} + 16332 q^{53} - 35046 q^{55} - 4736 q^{56} - 15576 q^{58} + 21966 q^{59} - 3050 q^{61} - 18872 q^{62} + 8192 q^{64} + 12582 q^{65} - 36758 q^{67} - 7872 q^{68} + 4536 q^{70} + 73848 q^{71} - 51188 q^{73} - 19184 q^{74} - 26240 q^{76} + 21462 q^{77} + 14926 q^{79} - 13824 q^{80} + 61416 q^{82} + 90762 q^{83} + 94500 q^{85} - 131432 q^{86} - 4992 q^{88} + 9300 q^{89} + 49562 q^{91} - 88608 q^{92} + 99816 q^{94} + 151416 q^{95} + 30262 q^{97} - 121776 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
−2.44949
2.44949
4.00000 0 16.0000 −56.3939 0 −22.3031 64.0000 0 −225.576
1.2 4.00000 0 16.0000 2.39388 0 −51.6969 64.0000 0 9.57551
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)

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 162.6.a.f 2
3.b odd 2 1 162.6.a.e 2
9.c even 3 2 54.6.c.a 4
9.d odd 6 2 18.6.c.a 4
36.f odd 6 2 432.6.i.a 4
36.h even 6 2 144.6.i.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.6.c.a 4 9.d odd 6 2
54.6.c.a 4 9.c even 3 2
144.6.i.a 4 36.h even 6 2
162.6.a.e 2 3.b odd 2 1
162.6.a.f 2 1.a even 1 1 trivial
432.6.i.a 4 36.f odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 54T_{5} - 135 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(162))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 54T - 135 \) Copy content Toggle raw display
$7$ \( T^{2} + 74T + 1153 \) Copy content Toggle raw display
$11$ \( T^{2} + 78T - 397863 \) Copy content Toggle raw display
$13$ \( T^{2} + 1106 T + 219409 \) Copy content Toggle raw display
$17$ \( T^{2} + 492 T - 1848060 \) Copy content Toggle raw display
$19$ \( T^{2} + 1640 T - 2648816 \) Copy content Toggle raw display
$23$ \( T^{2} + 5538 T + 7532361 \) Copy content Toggle raw display
$29$ \( T^{2} + 3894 T + 469593 \) Copy content Toggle raw display
$31$ \( T^{2} + 4718 T - 35307719 \) Copy content Toggle raw display
$37$ \( T^{2} + 4796 T - 141621212 \) Copy content Toggle raw display
$41$ \( T^{2} - 15354 T + 58893993 \) Copy content Toggle raw display
$43$ \( T^{2} + 32858 T + 254513617 \) Copy content Toggle raw display
$47$ \( T^{2} - 24954 T + 142283313 \) Copy content Toggle raw display
$53$ \( T^{2} - 16332 T + 59839812 \) Copy content Toggle raw display
$59$ \( T^{2} - 21966 T + 99734553 \) Copy content Toggle raw display
$61$ \( T^{2} + 3050 T - 105947399 \) Copy content Toggle raw display
$67$ \( T^{2} + 36758 T - 265336415 \) Copy content Toggle raw display
$71$ \( T^{2} - 73848 T + 665096112 \) Copy content Toggle raw display
$73$ \( T^{2} + 51188 T + 491562436 \) Copy content Toggle raw display
$79$ \( T^{2} - 14926 T - 52271015 \) Copy content Toggle raw display
$83$ \( T^{2} - 90762 T + 2054767617 \) Copy content Toggle raw display
$89$ \( T^{2} - 9300 T - 7978887036 \) Copy content Toggle raw display
$97$ \( T^{2} - 30262 T - 9106027655 \) Copy content Toggle raw display
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