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Properties

Label 168.6.q.c
Level $168$
Weight $6$
Character orbit 168.q
Analytic conductor $26.944$
Analytic rank $0$
Dimension $10$
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] = [168,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("168.25");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(26.9444817286\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 564x^{8} + 117814x^{6} + 11067780x^{4} + 427918225x^{2} + 3489248448 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{3}\cdot 7^{2} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 9 \beta_1 q^{3} + ( - \beta_{6} + \beta_1 - 1) q^{5} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \cdots - 14) q^{7}+ \cdots + (81 \beta_1 - 81) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 \beta_1 q^{3} + ( - \beta_{6} + \beta_1 - 1) q^{5} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \cdots - 14) q^{7}+ \cdots + ( - 243 \beta_{9} + 162 \beta_{7} + \cdots - 6804) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 45 q^{3} - 6 q^{5} - 97 q^{7} - 405 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 45 q^{3} - 6 q^{5} - 97 q^{7} - 405 q^{9} + 424 q^{11} + 374 q^{13} - 108 q^{15} - 952 q^{17} + 139 q^{19} - 1044 q^{21} + 4288 q^{23} - 5605 q^{25} - 7290 q^{27} - 4216 q^{29} + 8131 q^{31} - 3816 q^{33} + 20106 q^{35} - 5425 q^{37} + 1683 q^{39} + 29364 q^{41} - 46862 q^{43} - 486 q^{45} - 17190 q^{47} + 23255 q^{49} + 8568 q^{51} + 15064 q^{53} - 1176 q^{55} + 2502 q^{57} - 83242 q^{59} + 14954 q^{61} - 1539 q^{63} - 23250 q^{65} + 39501 q^{67} + 77184 q^{69} - 56020 q^{71} - 90395 q^{73} + 50445 q^{75} + 63448 q^{77} - 43067 q^{79} - 32805 q^{81} - 75672 q^{83} - 75272 q^{85} - 18972 q^{87} - 72608 q^{89} + 288287 q^{91} - 73179 q^{93} + 190138 q^{95} + 183000 q^{97} - 68688 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 564x^{8} + 117814x^{6} + 11067780x^{4} + 427918225x^{2} + 3489248448 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -625\nu^{9} - 281450\nu^{7} - 41638514\nu^{5} - 2184633898\nu^{3} - 19307731905\nu + 221300856 ) / 442601712 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1520 \nu^{9} - 22127 \nu^{8} - 671941 \nu^{7} - 9881431 \nu^{6} - 97198651 \nu^{5} + \cdots - 661674519576 ) / 21076272 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 29349 \nu^{9} + 923650 \nu^{8} + 13163751 \nu^{7} + 413303534 \nu^{6} + 1937609757 \nu^{5} + \cdots + 27626932442592 ) / 221300856 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 92943 \nu^{9} + 1382633 \nu^{8} + 41485257 \nu^{7} + 619097017 \nu^{6} + 6068335779 \nu^{5} + \cdots + 41458506660144 ) / 442601712 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 29927 \nu^{9} - 75719 \nu^{8} + 13393927 \nu^{7} - 33593455 \nu^{6} + 1966231681 \nu^{5} + \cdots - 2191308184800 ) / 126457632 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 29927 \nu^{9} + 75719 \nu^{8} + 13393927 \nu^{7} + 33593455 \nu^{6} + 1966231681 \nu^{5} + \cdots + 2191308184800 ) / 126457632 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 249 \nu^{9} - 763 \nu^{8} + 111351 \nu^{7} - 340739 \nu^{6} + 16327677 \nu^{5} - 49889441 \nu^{4} + \cdots - 22652476560 ) / 726768 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 249 \nu^{9} + 763 \nu^{8} + 111351 \nu^{7} + 340739 \nu^{6} + 16327677 \nu^{5} + 49889441 \nu^{4} + \cdots + 22652476560 ) / 726768 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 8741 \nu^{9} - 44254 \nu^{8} + 3901120 \nu^{7} - 19762862 \nu^{6} + 570701284 \nu^{5} + \cdots - 1318596339816 ) / 21076272 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + \beta_{8} + 4\beta_{7} - 4\beta_{6} - 4\beta_{5} - 4\beta_{4} + 4\beta_{3} - \beta_{2} - 2\beta _1 + 1 ) / 72 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} - \beta_{8} + 2\beta_{7} - \beta_{2} - 451 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 101 \beta_{9} - 29 \beta_{8} - 656 \beta_{7} + 404 \beta_{6} + 404 \beta_{5} + 728 \beta_{4} + \cdots + 12859 ) / 72 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 75 \beta_{9} + 158 \beta_{8} - 217 \beta_{7} - 34 \beta_{6} + 34 \beta_{5} - 16 \beta_{4} - 16 \beta_{3} + \cdots + 32979 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 14737 \beta_{9} - 16367 \beta_{8} + 108412 \beta_{7} - 37348 \beta_{6} - 37348 \beta_{5} + \cdots - 3625295 ) / 72 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 22521 \beta_{9} - 77665 \beta_{8} + 89430 \beta_{7} + 21712 \beta_{6} - 21712 \beta_{5} + \cdots - 10614395 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2418317 \beta_{9} + 5917627 \beta_{8} - 19082264 \beta_{7} + 2430356 \beta_{6} + 2430356 \beta_{5} + \cdots + 855464563 ) / 72 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1820969 \beta_{9} + 8707940 \beta_{8} - 9173387 \beta_{7} - 2624930 \beta_{6} + 2624930 \beta_{5} + \cdots + 919333837 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 429358129 \beta_{9} - 1503955055 \beta_{8} + 3539964052 \beta_{7} + 105168572 \beta_{6} + \cdots - 188662697615 ) / 72 \) 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
− 10.6668i
10.7938i
3.29825i
− 14.3017i
10.8764i
10.6668i
− 10.7938i
− 3.29825i
14.3017i
− 10.8764i
0 4.50000 − 7.79423i 0 −50.8171 − 88.0179i 0 −119.572 − 50.0952i 0 −40.5000 − 70.1481i 0
25.2 0 4.50000 − 7.79423i 0 −19.4585 − 33.7032i 0 106.179 + 74.3848i 0 −40.5000 − 70.1481i 0
25.3 0 4.50000 − 7.79423i 0 8.04054 + 13.9266i 0 −77.3451 − 104.042i 0 −40.5000 − 70.1481i 0
25.4 0 4.50000 − 7.79423i 0 13.3782 + 23.1717i 0 −66.5893 + 111.233i 0 −40.5000 − 70.1481i 0
25.5 0 4.50000 − 7.79423i 0 45.8569 + 79.4265i 0 108.828 − 70.4522i 0 −40.5000 − 70.1481i 0
121.1 0 4.50000 + 7.79423i 0 −50.8171 + 88.0179i 0 −119.572 + 50.0952i 0 −40.5000 + 70.1481i 0
121.2 0 4.50000 + 7.79423i 0 −19.4585 + 33.7032i 0 106.179 − 74.3848i 0 −40.5000 + 70.1481i 0
121.3 0 4.50000 + 7.79423i 0 8.04054 − 13.9266i 0 −77.3451 + 104.042i 0 −40.5000 + 70.1481i 0
121.4 0 4.50000 + 7.79423i 0 13.3782 − 23.1717i 0 −66.5893 − 111.233i 0 −40.5000 + 70.1481i 0
121.5 0 4.50000 + 7.79423i 0 45.8569 − 79.4265i 0 108.828 + 70.4522i 0 −40.5000 + 70.1481i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.5
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 168.6.q.c ✓ 10
3.b odd 2 1 504.6.s.c 10
4.b odd 2 1 336.6.q.l 10
7.c even 3 1 inner 168.6.q.c ✓ 10
21.h odd 6 1 504.6.s.c 10
28.g odd 6 1 336.6.q.l 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.6.q.c ✓ 10 1.a even 1 1 trivial
168.6.q.c ✓ 10 7.c even 3 1 inner
336.6.q.l 10 4.b odd 2 1
336.6.q.l 10 28.g odd 6 1
504.6.s.c 10 3.b odd 2 1
504.6.s.c 10 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} + 6 T_{5}^{9} + 10633 T_{5}^{8} - 145618 T_{5}^{7} + 100355305 T_{5}^{6} + \cdots + 24\!\cdots\!76 \) acting on \(S_{6}^{\mathrm{new}}(168, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( (T^{2} - 9 T + 81)^{5} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 13\!\cdots\!07 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 68\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( (T^{5} - 187 T^{4} + \cdots + 735027021216)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 15\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 26\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots + 12\!\cdots\!76)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 68\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots + 25\!\cdots\!44)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots - 33\!\cdots\!72)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 40\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 22\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 36\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 50\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 29\!\cdots\!68)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 49\!\cdots\!89 \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots + 10\!\cdots\!96)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 64\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 14\!\cdots\!48)^{2} \) Copy content Toggle raw display
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