↑

Properties

Label 162.5.d.e
Level $162$
Weight $5$
Character orbit 162.d
Analytic conductor $16.746$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [162,5,Mod(53,162)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(162, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.53");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 162.d (of order \(6\), degree \(2\), not 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: \(16.7459340196\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{10} \)
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.

\(f(q)\) \(=\) \( q + (2 \beta_{5} + 2 \beta_{3}) q^{2} + ( - 8 \beta_1 + 8) q^{4} + ( - \beta_{6} + 7 \beta_{5}) q^{5} + ( - \beta_{2} + 13 \beta_1) q^{7} + 16 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_{5} + 2 \beta_{3}) q^{2} + ( - 8 \beta_1 + 8) q^{4} + ( - \beta_{6} + 7 \beta_{5}) q^{5} + ( - \beta_{2} + 13 \beta_1) q^{7} + 16 \beta_{3} q^{8} + ( - 2 \beta_{4} + 30) q^{10} + (2 \beta_{7} - 2 \beta_{6} + 53 \beta_{5} + 55 \beta_{3}) q^{11} + (10 \beta_{4} - 10 \beta_{2} - 73 \beta_1 + 73) q^{13} + ( - 4 \beta_{6} + 24 \beta_{5}) q^{14} - 64 \beta_1 q^{16} + ( - 37 \beta_{7} - 38 \beta_{3}) q^{17} + ( - 31 \beta_{4} + 185) q^{19} + (8 \beta_{7} - 8 \beta_{6} + 56 \beta_{5} + 64 \beta_{3}) q^{20} + ( - 4 \beta_{4} + 4 \beta_{2} - 216 \beta_1 + 216) q^{22} + (40 \beta_{6} + 155 \beta_{5}) q^{23} + ( - 15 \beta_{2} - 391 \beta_1) q^{25} + ( - 40 \beta_{7} + 126 \beta_{3}) q^{26} + ( - 8 \beta_{4} + 104) q^{28} + (59 \beta_{7} - 59 \beta_{6} - 34 \beta_{5} + 25 \beta_{3}) q^{29} + (18 \beta_{4} - 18 \beta_{2} - 1264 \beta_1 + 1264) q^{31} - 128 \beta_{5} q^{32} + ( - 74 \beta_{2} + 78 \beta_1) q^{34} + ( - 28 \beta_{7} - 233 \beta_{3}) q^{35} + ( - 83 \beta_{4} + 1106) q^{37} + (124 \beta_{7} - 124 \beta_{6} + 308 \beta_{5} + 432 \beta_{3}) q^{38} + ( - 16 \beta_{4} + 16 \beta_{2} - 240 \beta_1 + 240) q^{40} + (206 \beta_{6} + 7 \beta_{5}) q^{41} + ( - 141 \beta_{2} - 107 \beta_1) q^{43} + (16 \beta_{7} + 440 \beta_{3}) q^{44} + (80 \beta_{4} + 540) q^{46} + ( - 54 \beta_{7} + 54 \beta_{6} - 1812 \beta_{5} - 1866 \beta_{3}) q^{47} + (26 \beta_{4} - 26 \beta_{2} - 1989 \beta_1 + 1989) q^{49} + ( - 60 \beta_{6} - 812 \beta_{5}) q^{50} + ( - 80 \beta_{2} - 584 \beta_1) q^{52} + (34 \beta_{7} - 2047 \beta_{3}) q^{53} + ( - 69 \beta_{4} + 1053) q^{55} + (32 \beta_{7} - 32 \beta_{6} + 192 \beta_{5} + 224 \beta_{3}) q^{56} + ( - 118 \beta_{4} + 118 \beta_{2} + 18 \beta_1 - 18) q^{58} + (274 \beta_{6} + 272 \beta_{5}) q^{59} + (189 \beta_{2} - 1124 \beta_1) q^{61} + ( - 72 \beta_{7} + 2492 \beta_{3}) q^{62} - 512 q^{64} + ( - 77 \beta_{7} + 77 \beta_{6} - 629 \beta_{5} - 706 \beta_{3}) q^{65} + (129 \beta_{4} - 129 \beta_{2} + 4109 \beta_1 - 4109) q^{67} + ( - 296 \beta_{6} + 8 \beta_{5}) q^{68} + ( - 56 \beta_{2} + 876 \beta_1) q^{70} + (32 \beta_{7} - 1277 \beta_{3}) q^{71} + ( - 21 \beta_{4} - 5200) q^{73} + (332 \beta_{7} - 332 \beta_{6} + 2046 \beta_{5} + 2378 \beta_{3}) q^{74} + ( - 248 \beta_{4} + 248 \beta_{2} - 1480 \beta_1 + 1480) q^{76} + ( - 134 \beta_{6} + 878 \beta_{5}) q^{77} + (439 \beta_{2} + 4183 \beta_1) q^{79} + (64 \beta_{7} + 512 \beta_{3}) q^{80} + (412 \beta_{4} - 384) q^{82} + ( - 696 \beta_{7} + 696 \beta_{6} + 3450 \beta_{5} + 2754 \beta_{3}) q^{83} + (297 \beta_{4} - 297 \beta_{2} + 4788 \beta_1 - 4788) q^{85} + ( - 564 \beta_{6} - 496 \beta_{5}) q^{86} + (32 \beta_{2} - 1728 \beta_1) q^{88} + (365 \beta_{7} + 973 \beta_{3}) q^{89} + (57 \beta_{4} - 1481) q^{91} + ( - 320 \beta_{7} + 320 \beta_{6} + 1240 \beta_{5} + 920 \beta_{3}) q^{92} + (108 \beta_{4} - 108 \beta_{2} + 7356 \beta_1 - 7356) q^{94} + ( - 650 \beta_{6} + 4829 \beta_{5}) q^{95} + (766 \beta_{2} + 952 \beta_1) q^{97} + ( - 104 \beta_{7} + 3926 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 32 q^{4} + 52 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 32 q^{4} + 52 q^{7} + 240 q^{10} + 292 q^{13} - 256 q^{16} + 1480 q^{19} + 864 q^{22} - 1564 q^{25} + 832 q^{28} + 5056 q^{31} + 312 q^{34} + 8848 q^{37} + 960 q^{40} - 428 q^{43} + 4320 q^{46} + 7956 q^{49} - 2336 q^{52} + 8424 q^{55} - 72 q^{58} - 4496 q^{61} - 4096 q^{64} - 16436 q^{67} + 3504 q^{70} - 41600 q^{73} + 5920 q^{76} + 16732 q^{79} - 3072 q^{82} - 19152 q^{85} - 6912 q^{88} - 11848 q^{91} - 29424 q^{94} + 3808 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 9\zeta_{24}^{6} + 9\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -9\zeta_{24}^{6} + 18\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 5\zeta_{24}^{7} - 5\zeta_{24}^{5} + 4\zeta_{24}^{3} + 9\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 9\zeta_{24}^{7} + 5\zeta_{24}^{5} - 4\zeta_{24}^{3} + 4\zeta_{24} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{7} + \beta_{6} + 14\beta_{5} + 14\beta_{3} ) / 27 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{4} + \beta_{2} ) / 27 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -\beta_{7} + 2\beta_{6} + \beta_{5} - 14\beta_{3} ) / 27 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( 2\beta_{7} - \beta_{6} + 13\beta_{5} + \beta_{3} ) / 27 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( -\beta_{4} + 2\beta_{2} ) / 27 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} + \beta_{6} - 13\beta_{5} - 13\beta_{3} ) / 27 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(1 - \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−0.258819 + 0.965926i
−0.965926 − 0.258819i
0.965926 + 0.258819i
0.258819 − 0.965926i
−0.258819 − 0.965926i
−0.965926 + 0.258819i
0.965926 − 0.258819i
0.258819 + 0.965926i
−2.44949 + 1.41421i 0 4.00000 − 6.92820i −18.7315 − 10.8147i 0 14.2942 + 24.7583i 22.6274i 0 61.1769
53.2 −2.44949 + 1.41421i 0 4.00000 − 6.92820i 0.360355 + 0.208051i 0 −1.29423 − 2.24167i 22.6274i 0 −1.17691
53.3 2.44949 − 1.41421i 0 4.00000 − 6.92820i −0.360355 − 0.208051i 0 −1.29423 − 2.24167i − 22.6274i 0 −1.17691
53.4 2.44949 − 1.41421i 0 4.00000 − 6.92820i 18.7315 + 10.8147i 0 14.2942 + 24.7583i − 22.6274i 0 61.1769
107.1 −2.44949 − 1.41421i 0 4.00000 + 6.92820i −18.7315 + 10.8147i 0 14.2942 − 24.7583i − 22.6274i 0 61.1769
107.2 −2.44949 − 1.41421i 0 4.00000 + 6.92820i 0.360355 − 0.208051i 0 −1.29423 + 2.24167i − 22.6274i 0 −1.17691
107.3 2.44949 + 1.41421i 0 4.00000 + 6.92820i −0.360355 + 0.208051i 0 −1.29423 + 2.24167i 22.6274i 0 −1.17691
107.4 2.44949 + 1.41421i 0 4.00000 + 6.92820i 18.7315 − 10.8147i 0 14.2942 − 24.7583i 22.6274i 0 61.1769
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.5.d.e 8
3.b odd 2 1 inner 162.5.d.e 8
9.c even 3 1 162.5.b.b ✓ 4
9.c even 3 1 inner 162.5.d.e 8
9.d odd 6 1 162.5.b.b ✓ 4
9.d odd 6 1 inner 162.5.d.e 8
36.f odd 6 1 1296.5.e.a 4
36.h even 6 1 1296.5.e.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.5.b.b ✓ 4 9.c even 3 1
162.5.b.b ✓ 4 9.d odd 6 1
162.5.d.e 8 1.a even 1 1 trivial
162.5.d.e 8 3.b odd 2 1 inner
162.5.d.e 8 9.c even 3 1 inner
162.5.d.e 8 9.d odd 6 1 inner
1296.5.e.a 4 36.f odd 6 1
1296.5.e.a 4 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 468T_{5}^{6} + 218943T_{5}^{4} - 37908T_{5}^{2} + 6561 \) acting on \(S_{5}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 468 T^{6} + 218943 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$7$ \( (T^{4} - 26 T^{3} + 750 T^{2} + 1924 T + 5476)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 816800166640656 \) Copy content Toggle raw display
$13$ \( (T^{4} - 146 T^{3} + 40287 T^{2} + \cdots + 359898841)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 334188 T^{2} + \cdots + 27414418329)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 370 T - 199298)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} - 461700 T^{6} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} - 845964 T^{6} + \cdots + 31\!\cdots\!01 \) Copy content Toggle raw display
$31$ \( (T^{4} - 2528 T^{3} + \cdots + 2307251633296)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 2212 T - 450791)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} - 10348812 T^{6} + \cdots + 69\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( (T^{4} + 214 T^{3} + \cdots + 23228871893956)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 14236272 T^{6} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{4} + 17321292 T^{2} + \cdots + 70220008948644)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 18316368 T^{6} + \cdots + 68\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( (T^{4} + 2248 T^{3} + \cdots + 55009322747929)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 8218 T^{3} + \cdots + 164868630253924)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 6936228 T^{2} + \cdots + 10363776595524)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 10400 T + 26932837)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 8366 T^{3} + \cdots + 860466777033796)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 156202704 T^{6} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{4} + 34873236 T^{2} + \cdots + 223115671821249)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 1904 T^{3} + \cdots + 20\!\cdots\!16)^{2} \) Copy content Toggle raw display
show more
show less