↑

Properties

Label 168.4.q.f
Level $168$
Weight $4$
Character orbit 168.q
Analytic conductor $9.912$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [168,4,Mod(25,168)]
 
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, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("168.25");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 168.q (of order \(3\), 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: \(9.91232088096\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 173x^{6} + 9457x^{4} + 168048x^{2} + 746496 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 7 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + 3 \beta_1 q^{3} + (\beta_{6} - \beta_1 - 1) q^{5} + (\beta_{5} + 2) q^{7} + ( - 9 \beta_1 - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 \beta_1 q^{3} + (\beta_{6} - \beta_1 - 1) q^{5} + (\beta_{5} + 2) q^{7} + ( - 9 \beta_1 - 9) q^{9} + (\beta_{7} - 2 \beta_{6} - \beta_{5} + \cdots + 1) q^{11}+ \cdots + (9 \beta_{5} + 9 \beta_{4} - 18 \beta_{2} + 27) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{3} - 4 q^{5} + 18 q^{7} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{3} - 4 q^{5} + 18 q^{7} - 36 q^{9} - 14 q^{11} + 44 q^{13} + 24 q^{15} - 96 q^{17} + 26 q^{19} - 36 q^{21} - 96 q^{23} - 110 q^{25} + 216 q^{27} - 152 q^{29} - 238 q^{31} - 42 q^{33} + 152 q^{35} - 562 q^{37} - 66 q^{39} + 856 q^{41} - 516 q^{43} - 36 q^{45} + 80 q^{47} + 156 q^{49} - 288 q^{51} + 2952 q^{55} - 156 q^{57} - 262 q^{59} + 276 q^{61} - 54 q^{63} - 2196 q^{65} - 150 q^{67} + 576 q^{69} - 1696 q^{71} + 218 q^{73} - 330 q^{75} - 764 q^{77} - 1762 q^{79} - 324 q^{81} + 6900 q^{83} + 2904 q^{85} + 228 q^{87} + 344 q^{89} - 2806 q^{91} - 714 q^{93} - 2004 q^{95} - 1244 q^{97} + 252 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 173x^{6} + 9457x^{4} + 168048x^{2} + 746496 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 691\nu^{5} + 67439\nu^{3} + 869616\nu - 673920 ) / 1347840 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 19\nu^{6} + 911\nu^{4} - 31781\nu^{2} - 208224 ) / 42120 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 283 \nu^{7} + 768 \nu^{6} + 29087 \nu^{5} + 143232 \nu^{4} + 233803 \nu^{3} + 8185728 \nu^{2} + \cdots + 109164672 ) / 4043520 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 135 \nu^{7} - 2384 \nu^{6} + 19035 \nu^{5} - 149776 \nu^{4} + 892215 \nu^{3} + 3526576 \nu^{2} + \cdots + 104194944 ) / 1347840 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 27 \nu^{7} - 560 \nu^{6} - 3807 \nu^{5} - 62320 \nu^{4} - 178443 \nu^{3} - 1680944 \nu^{2} + \cdots - 9334656 ) / 269568 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 43\nu^{7} + 57\nu^{6} + 5387\nu^{5} + 2733\nu^{4} + 181903\nu^{3} - 95343\nu^{2} + 1055052\nu - 624672 ) / 252720 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1087 \nu^{7} - 4272 \nu^{6} + 147443 \nu^{5} - 417648 \nu^{4} + 5991727 \nu^{3} - 7751472 \nu^{2} + \cdots - 12451968 ) / 2021760 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{7} - 11\beta_{6} - \beta_{5} - 2\beta_{4} + 5\beta_{3} + 5\beta_{2} - 12\beta _1 - 5 ) / 28 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{7} - 3\beta_{6} + \beta_{5} + 16\beta_{4} + 9\beta_{3} + 51\beta_{2} + 12\beta _1 - 1213 ) / 28 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -41\beta_{7} + 155\beta_{6} + 2\beta_{5} + 39\beta_{4} - 115\beta_{3} - 59\beta_{2} - 18\beta _1 - 11 ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 191\beta_{7} + 191\beta_{6} - 241\beta_{5} - 1196\beta_{4} - 573\beta_{3} - 4759\beta_{2} - 764\beta _1 + 77321 ) / 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 11113 \beta_{7} - 42185 \beta_{6} + 393 \beta_{5} - 11506 \beta_{4} + 34911 \beta_{3} + 15143 \beta_{2} + \cdots + 32457 ) / 28 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 3544 \beta_{7} - 3544 \beta_{6} + 3307 \beta_{5} + 21027 \beta_{4} + 10632 \beta_{3} + 93890 \beta_{2} + \cdots - 1357362 ) / 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 110295 \beta_{7} + 442367 \beta_{6} - 8363 \beta_{5} + 118658 \beta_{4} - 364337 \beta_{3} + \cdots - 536767 ) / 4 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
8.67551i
− 8.34231i
2.57353i
− 4.63878i
− 8.67551i
8.34231i
− 2.57353i
4.63878i
0 −1.50000 + 2.59808i 0 −9.47901 − 16.4181i 0 12.8033 + 13.3819i 0 −4.50000 − 7.79423i 0
25.2 0 −1.50000 + 2.59808i 0 −0.363171 − 0.629031i 0 −18.1420 − 3.72380i 0 −4.50000 − 7.79423i 0
25.3 0 −1.50000 + 2.59808i 0 −0.0642956 − 0.111363i 0 −0.866259 − 18.5000i 0 −4.50000 − 7.79423i 0
25.4 0 −1.50000 + 2.59808i 0 7.90648 + 13.6944i 0 15.2050 + 10.5739i 0 −4.50000 − 7.79423i 0
121.1 0 −1.50000 − 2.59808i 0 −9.47901 + 16.4181i 0 12.8033 − 13.3819i 0 −4.50000 + 7.79423i 0
121.2 0 −1.50000 − 2.59808i 0 −0.363171 + 0.629031i 0 −18.1420 + 3.72380i 0 −4.50000 + 7.79423i 0
121.3 0 −1.50000 − 2.59808i 0 −0.0642956 + 0.111363i 0 −0.866259 + 18.5000i 0 −4.50000 + 7.79423i 0
121.4 0 −1.50000 − 2.59808i 0 7.90648 − 13.6944i 0 15.2050 − 10.5739i 0 −4.50000 + 7.79423i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.4.q.f ✓ 8
3.b odd 2 1 504.4.s.j 8
4.b odd 2 1 336.4.q.m 8
7.c even 3 1 inner 168.4.q.f ✓ 8
7.c even 3 1 1176.4.a.bd 4
7.d odd 6 1 1176.4.a.ba 4
21.h odd 6 1 504.4.s.j 8
28.f even 6 1 2352.4.a.cp 4
28.g odd 6 1 336.4.q.m 8
28.g odd 6 1 2352.4.a.cm 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.q.f ✓ 8 1.a even 1 1 trivial
168.4.q.f ✓ 8 7.c even 3 1 inner
336.4.q.m 8 4.b odd 2 1
336.4.q.m 8 28.g odd 6 1
504.4.s.j 8 3.b odd 2 1
504.4.s.j 8 21.h odd 6 1
1176.4.a.ba 4 7.d odd 6 1
1176.4.a.bd 4 7.c even 3 1
2352.4.a.cm 4 28.g odd 6 1
2352.4.a.cp 4 28.f even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 4T_{5}^{7} + 313T_{5}^{6} - 676T_{5}^{5} + 89261T_{5}^{4} + 76256T_{5}^{3} + 57220T_{5}^{2} + 7168T_{5} + 784 \) acting on \(S_{4}^{\mathrm{new}}(168, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 9)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + 4 T^{7} + \cdots + 784 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 33244957032336 \) Copy content Toggle raw display
$13$ \( (T^{4} - 22 T^{3} + \cdots + 13795008)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 30110417391616 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 9663676416 \) Copy content Toggle raw display
$29$ \( (T^{4} + 76 T^{3} + \cdots - 802800)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 27\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 21\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( (T^{4} - 428 T^{3} + \cdots - 3866949120)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 258 T^{3} + \cdots - 2654719484)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 56\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 25753595040000 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 97\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 70\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{4} + 848 T^{3} + \cdots + 1940742864)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 78\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 86\!\cdots\!01 \) Copy content Toggle raw display
$83$ \( (T^{4} - 3450 T^{3} + \cdots + 352673538780)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 20\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( (T^{4} + 622 T^{3} + \cdots + 79407506004)^{2} \) Copy content Toggle raw display
show more
show less