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Properties

Label 168.3.z.b
Level $168$
Weight $3$
Character orbit 168.z
Analytic conductor $4.578$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [168,3,Mod(73,168)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(168, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("168.73");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 168.z (of order \(6\), degree \(2\), minimal)

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: no
Analytic conductor: \(4.57766844125\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.35911766016.9
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 7x^{6} - 2x^{5} + 78x^{4} - 18x^{3} - 153x^{2} - 230x + 529 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 7 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{3} + (\beta_{7} - 1) q^{5} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \cdots + 1) q^{7}+ \cdots - 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 1) q^{3} + (\beta_{7} - 1) q^{5} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \cdots + 1) q^{7}+ \cdots + (3 \beta_{7} - 3 \beta_{6} - 3 \beta_{4} + \cdots - 15) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{3} - 6 q^{5} + 8 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{3} - 6 q^{5} + 8 q^{7} + 12 q^{9} - 22 q^{11} - 12 q^{15} + 36 q^{17} + 42 q^{19} + 6 q^{21} + 48 q^{23} + 42 q^{25} + 68 q^{29} - 60 q^{31} - 66 q^{33} - 12 q^{35} - 118 q^{37} - 18 q^{39} - 92 q^{43} - 18 q^{45} - 12 q^{47} - 20 q^{49} + 36 q^{51} + 10 q^{53} + 84 q^{57} - 54 q^{59} + 24 q^{61} - 6 q^{63} - 148 q^{65} + 22 q^{67} - 392 q^{71} - 138 q^{73} + 126 q^{75} - 126 q^{77} + 164 q^{79} - 36 q^{81} + 200 q^{85} + 102 q^{87} - 60 q^{89} + 90 q^{91} - 60 q^{93} - 132 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 7x^{6} - 2x^{5} + 78x^{4} - 18x^{3} - 153x^{2} - 230x + 529 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1844 \nu^{7} - 5827 \nu^{6} + 5814 \nu^{5} + 40633 \nu^{4} + 110804 \nu^{3} - 500966 \nu^{2} + \cdots + 876346 ) / 329245 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4244 \nu^{7} + 873 \nu^{6} - 33756 \nu^{5} - 71462 \nu^{4} + 213594 \nu^{3} + 469674 \nu^{2} + \cdots - 1693329 ) / 658490 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4146 \nu^{7} - 4888 \nu^{6} - 30494 \nu^{5} - 61123 \nu^{4} + 340546 \nu^{3} + 333461 \nu^{2} + \cdots - 2591226 ) / 329245 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 33\nu^{7} + 41\nu^{6} - 222\nu^{5} - 414\nu^{4} + 488\nu^{3} + 1608\nu^{2} - 207\nu + 3637 ) / 2045 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6546 \nu^{7} + 1812 \nu^{6} - 70064 \nu^{5} - 173218 \nu^{4} + 443336 \nu^{3} + 974856 \nu^{2} + \cdots - 3514676 ) / 329245 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11533 \nu^{7} + 30909 \nu^{6} + 121832 \nu^{5} - 29696 \nu^{4} - 997968 \nu^{3} - 162453 \nu^{2} + \cdots + 2603393 ) / 329245 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 45244 \nu^{7} - 5583 \nu^{6} + 215876 \nu^{5} + 916372 \nu^{4} - 1365974 \nu^{3} + \cdots + 10829159 ) / 658490 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{5} + 2\beta_{4} + 4\beta_{3} - 8\beta_{2} - 2\beta _1 + 2 ) / 14 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -5\beta_{5} - 4\beta_{4} - \beta_{3} + 30\beta_{2} - 3\beta _1 + 31 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{7} + 7\beta_{6} + 9\beta_{5} + 3\beta_{4} + 27\beta_{3} + 9\beta_{2} + 18\beta _1 + 108 ) / 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 7\beta_{7} - 33\beta_{5} + 10\beta_{4} + 20\beta_{3} + 121\beta_{2} - 10\beta _1 + 10 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 49\beta_{6} - 108\beta_{5} - 141\beta_{4} + 33\beta_{3} + 802\beta_{2} + 99\beta _1 + 769 ) / 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 91\beta_{7} + 91\beta_{6} + 75\beta_{5} + 235\beta_{4} + 225\beta_{3} + 75\beta_{2} + 150\beta _1 - 269 ) / 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -22\beta_{7} - 199\beta_{5} - 8\beta_{4} - 16\beta_{3} + 734\beta_{2} + 8\beta _1 - 8 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(85\) \(113\) \(127\)
\(\chi(n)\) \(-\beta_{2}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1
1.83172 − 0.480194i
−1.90015 + 1.67440i
2.40015 − 0.808379i
−1.33172 + 1.34622i
1.83172 + 0.480194i
−1.90015 − 1.67440i
2.40015 + 0.808379i
−1.33172 − 1.34622i
0 1.50000 + 0.866025i 0 −6.80550 + 3.92916i 0 6.99187 − 0.337312i 0 1.50000 + 2.59808i 0
73.2 0 1.50000 + 0.866025i 0 −4.68140 + 2.70281i 0 −6.12873 + 3.38210i 0 1.50000 + 2.59808i 0
73.3 0 1.50000 + 0.866025i 0 3.18140 − 1.83678i 0 2.47188 − 6.54903i 0 1.50000 + 2.59808i 0
73.4 0 1.50000 + 0.866025i 0 5.30550 − 3.06313i 0 0.664986 + 6.96834i 0 1.50000 + 2.59808i 0
145.1 0 1.50000 − 0.866025i 0 −6.80550 − 3.92916i 0 6.99187 + 0.337312i 0 1.50000 − 2.59808i 0
145.2 0 1.50000 − 0.866025i 0 −4.68140 − 2.70281i 0 −6.12873 − 3.38210i 0 1.50000 − 2.59808i 0
145.3 0 1.50000 − 0.866025i 0 3.18140 + 1.83678i 0 2.47188 + 6.54903i 0 1.50000 − 2.59808i 0
145.4 0 1.50000 − 0.866025i 0 5.30550 + 3.06313i 0 0.664986 − 6.96834i 0 1.50000 − 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.3.z.b ✓ 8
3.b odd 2 1 504.3.by.c 8
4.b odd 2 1 336.3.bh.g 8
7.b odd 2 1 1176.3.z.c 8
7.c even 3 1 1176.3.f.c 8
7.c even 3 1 1176.3.z.c 8
7.d odd 6 1 inner 168.3.z.b ✓ 8
7.d odd 6 1 1176.3.f.c 8
12.b even 2 1 1008.3.cg.p 8
21.g even 6 1 504.3.by.c 8
21.g even 6 1 3528.3.f.b 8
21.h odd 6 1 3528.3.f.b 8
28.f even 6 1 336.3.bh.g 8
28.f even 6 1 2352.3.f.g 8
28.g odd 6 1 2352.3.f.g 8
84.j odd 6 1 1008.3.cg.p 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.3.z.b ✓ 8 1.a even 1 1 trivial
168.3.z.b ✓ 8 7.d odd 6 1 inner
336.3.bh.g 8 4.b odd 2 1
336.3.bh.g 8 28.f even 6 1
504.3.by.c 8 3.b odd 2 1
504.3.by.c 8 21.g even 6 1
1008.3.cg.p 8 12.b even 2 1
1008.3.cg.p 8 84.j odd 6 1
1176.3.f.c 8 7.c even 3 1
1176.3.f.c 8 7.d odd 6 1
1176.3.z.c 8 7.b odd 2 1
1176.3.z.c 8 7.c even 3 1
2352.3.f.g 8 28.f even 6 1
2352.3.f.g 8 28.g odd 6 1
3528.3.f.b 8 21.g even 6 1
3528.3.f.b 8 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 6T_{5}^{7} - 53T_{5}^{6} - 390T_{5}^{5} + 2861T_{5}^{4} + 13260T_{5}^{3} - 48268T_{5}^{2} - 195024T_{5} + 913936 \) acting on \(S_{3}^{\mathrm{new}}(168, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + 6 T^{7} + \cdots + 913936 \) Copy content Toggle raw display
$7$ \( T^{8} - 8 T^{7} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{8} + 22 T^{7} + \cdots + 17875984 \) Copy content Toggle raw display
$13$ \( T^{8} + 262 T^{6} + \cdots + 2408704 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 3470623744 \) Copy content Toggle raw display
$19$ \( T^{8} - 42 T^{7} + \cdots + 17272336 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 22620160000 \) Copy content Toggle raw display
$29$ \( (T^{4} - 34 T^{3} + \cdots + 224128)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 25912950625 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 17069945104 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 580010189056 \) Copy content Toggle raw display
$43$ \( (T^{4} + 46 T^{3} + \cdots + 1658308)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 52408029184 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 1491466217536 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 1048985640000 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 43785853599744 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 95387087104 \) Copy content Toggle raw display
$71$ \( (T^{4} + 196 T^{3} + \cdots - 13209344)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 622497709609216 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 26\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 839297841424 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 723343446016 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
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