Properties

Label 168.6.a.i
Level $168$
Weight $6$
Character orbit 168.a
Self dual yes
Analytic conductor $26.944$
Analytic rank $0$
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] = [168,6,Mod(1,168)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(168, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("168.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 168.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: \(26.9444817286\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{4281}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1070 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{4281}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 9 q^{3} + ( - \beta - 5) q^{5} + 49 q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 q^{3} + ( - \beta - 5) q^{5} + 49 q^{7} + 81 q^{9} + ( - 3 \beta + 27) q^{11} + ( - 2 \beta + 120) q^{13} + ( - 9 \beta - 45) q^{15} + (21 \beta + 953) q^{17} + ( - 20 \beta - 200) q^{19} + 441 q^{21} + (35 \beta + 1465) q^{23} + (10 \beta + 1181) q^{25} + 729 q^{27} + (44 \beta + 2770) q^{29} + (28 \beta + 2452) q^{31} + ( - 27 \beta + 243) q^{33} + ( - 49 \beta - 245) q^{35} + ( - 22 \beta + 1476) q^{37} + ( - 18 \beta + 1080) q^{39} + ( - 45 \beta + 3035) q^{41} + ( - 152 \beta + 1652) q^{43} + ( - 81 \beta - 405) q^{45} + (154 \beta + 8494) q^{47} + 2401 q^{49} + (189 \beta + 8577) q^{51} + ( - 234 \beta + 3448) q^{53} + ( - 12 \beta + 12708) q^{55} + ( - 180 \beta - 1800) q^{57} + (326 \beta + 26910) q^{59} + (308 \beta + 2074) q^{61} + 3969 q^{63} + ( - 110 \beta + 7962) q^{65} + ( - 410 \beta + 16758) q^{67} + (315 \beta + 13185) q^{69} + (77 \beta + 36759) q^{71} + (1026 \beta + 64) q^{73} + (90 \beta + 10629) q^{75} + ( - 147 \beta + 1323) q^{77} + (450 \beta + 28870) q^{79} + 6561 q^{81} + ( - 1696 \beta + 11508) q^{83} + ( - 1058 \beta - 94666) q^{85} + (396 \beta + 24930) q^{87} + ( - 565 \beta - 70765) q^{89} + ( - 98 \beta + 5880) q^{91} + (252 \beta + 22068) q^{93} + (300 \beta + 86620) q^{95} + (254 \beta - 113108) q^{97} + ( - 243 \beta + 2187) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 18 q^{3} - 10 q^{5} + 98 q^{7} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 18 q^{3} - 10 q^{5} + 98 q^{7} + 162 q^{9} + 54 q^{11} + 240 q^{13} - 90 q^{15} + 1906 q^{17} - 400 q^{19} + 882 q^{21} + 2930 q^{23} + 2362 q^{25} + 1458 q^{27} + 5540 q^{29} + 4904 q^{31} + 486 q^{33} - 490 q^{35} + 2952 q^{37} + 2160 q^{39} + 6070 q^{41} + 3304 q^{43} - 810 q^{45} + 16988 q^{47} + 4802 q^{49} + 17154 q^{51} + 6896 q^{53} + 25416 q^{55} - 3600 q^{57} + 53820 q^{59} + 4148 q^{61} + 7938 q^{63} + 15924 q^{65} + 33516 q^{67} + 26370 q^{69} + 73518 q^{71} + 128 q^{73} + 21258 q^{75} + 2646 q^{77} + 57740 q^{79} + 13122 q^{81} + 23016 q^{83} - 189332 q^{85} + 49860 q^{87} - 141530 q^{89} + 11760 q^{91} + 44136 q^{93} + 173240 q^{95} - 226216 q^{97} + 4374 q^{99}+O(q^{100}) \) 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
33.2147
−32.2147
0 9.00000 0 −70.4294 0 49.0000 0 81.0000 0
1.2 0 9.00000 0 60.4294 0 49.0000 0 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(7\) \( -1 \)

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 168.6.a.i 2
3.b odd 2 1 504.6.a.p 2
4.b odd 2 1 336.6.a.s 2
12.b even 2 1 1008.6.a.bp 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.6.a.i 2 1.a even 1 1 trivial
336.6.a.s 2 4.b odd 2 1
504.6.a.p 2 3.b odd 2 1
1008.6.a.bp 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 10T_{5} - 4256 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(168))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 10T - 4256 \) Copy content Toggle raw display
$7$ \( (T - 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 54T - 37800 \) Copy content Toggle raw display
$13$ \( T^{2} - 240T - 2724 \) Copy content Toggle raw display
$17$ \( T^{2} - 1906 T - 979712 \) Copy content Toggle raw display
$19$ \( T^{2} + 400 T - 1672400 \) Copy content Toggle raw display
$23$ \( T^{2} - 2930 T - 3098000 \) Copy content Toggle raw display
$29$ \( T^{2} - 5540 T - 615116 \) Copy content Toggle raw display
$31$ \( T^{2} - 4904 T + 2656000 \) Copy content Toggle raw display
$37$ \( T^{2} - 2952 T + 106572 \) Copy content Toggle raw display
$41$ \( T^{2} - 6070 T + 542200 \) Copy content Toggle raw display
$43$ \( T^{2} - 3304 T - 96179120 \) Copy content Toggle raw display
$47$ \( T^{2} - 16988 T - 29380160 \) Copy content Toggle raw display
$53$ \( T^{2} - 6896 T - 222521732 \) Copy content Toggle raw display
$59$ \( T^{2} - 53820 T + 269180544 \) Copy content Toggle raw display
$61$ \( T^{2} - 4148 T - 401811308 \) Copy content Toggle raw display
$67$ \( T^{2} - 33516 T - 438805536 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 1325842032 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 4506501860 \) Copy content Toggle raw display
$79$ \( T^{2} - 57740 T - 33425600 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 12181502832 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 3641083000 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 12517226668 \) Copy content Toggle raw display
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