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Properties

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

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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: \( 2^{6}\cdot 3^{12} \)
Twist minimal: no (minimal twist has level 54)
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + 8 \beta_{2} q^{4} + ( - 3 \beta_{5} - 3 \beta_{4} - \beta_{3} + \beta_1) q^{5} + (\beta_{7} - \beta_{6} + 17 \beta_{2} - 17) q^{7} + 8 \beta_{4} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + 8 \beta_{2} q^{4} + ( - 3 \beta_{5} - 3 \beta_{4} - \beta_{3} + \beta_1) q^{5} + (\beta_{7} - \beta_{6} + 17 \beta_{2} - 17) q^{7} + 8 \beta_{4} q^{8} + ( - \beta_{7} + 24) q^{10} + ( - 30 \beta_{5} - \beta_1) q^{11} + (2 \beta_{6} + 130 \beta_{2}) q^{13} + (17 \beta_{5} + 17 \beta_{4} + 8 \beta_{3} - 8 \beta_1) q^{14} + (64 \beta_{2} - 64) q^{16} + (108 \beta_{4} + 2 \beta_{3}) q^{17} + ( - 2 \beta_{7} + 272) q^{19} + ( - 24 \beta_{5} + 8 \beta_1) q^{20} + (\beta_{6} + 240 \beta_{2}) q^{22} + (114 \beta_{5} + 114 \beta_{4} - 20 \beta_{3} + 20 \beta_1) q^{23} + ( - 6 \beta_{7} + 6 \beta_{6} - 176 \beta_{2} + 176) q^{25} + (130 \beta_{4} - 16 \beta_{3}) q^{26} + (8 \beta_{7} - 136) q^{28} + ( - 264 \beta_{5} - 22 \beta_1) q^{29} + ( - 15 \beta_{6} - 203 \beta_{2}) q^{31} + (64 \beta_{5} + 64 \beta_{4}) q^{32} + (2 \beta_{7} - 2 \beta_{6} + 864 \beta_{2} - 864) q^{34} + (780 \beta_{4} + 41 \beta_{3}) q^{35} + (14 \beta_{7} - 268) q^{37} + ( - 272 \beta_{5} + 16 \beta_1) q^{38} + ( - 8 \beta_{6} + 192 \beta_{2}) q^{40} + (330 \beta_{5} + 330 \beta_{4} + 68 \beta_{3} - 68 \beta_1) q^{41} + (6 \beta_{7} - 6 \beta_{6} - 1774 \beta_{2} + 1774) q^{43} + (240 \beta_{4} - 8 \beta_{3}) q^{44} + ( - 20 \beta_{7} - 912) q^{46} + ( - 804 \beta_{5} + 30 \beta_1) q^{47} + (34 \beta_{6} - 3720 \beta_{2}) q^{49} + ( - 176 \beta_{5} - 176 \beta_{4} - 48 \beta_{3} + 48 \beta_1) q^{50} + ( - 16 \beta_{7} + 16 \beta_{6} + 1040 \beta_{2} - 1040) q^{52} + (549 \beta_{4} - 119 \beta_{3}) q^{53} + ( - 27 \beta_{7} - 9) q^{55} + (136 \beta_{5} - 64 \beta_1) q^{56} + (22 \beta_{6} + 2112 \beta_{2}) q^{58} + ( - 228 \beta_{5} - 228 \beta_{4} - 98 \beta_{3} + 98 \beta_1) q^{59} + (24 \beta_{7} - 24 \beta_{6} - 3484 \beta_{2} + 3484) q^{61} + ( - 203 \beta_{4} + 120 \beta_{3}) q^{62} - 512 q^{64} + (1068 \beta_{5} + 82 \beta_1) q^{65} + (6 \beta_{6} - 806 \beta_{2}) q^{67} + (864 \beta_{5} + 864 \beta_{4} + 16 \beta_{3} - 16 \beta_1) q^{68} + (41 \beta_{7} - 41 \beta_{6} + 6240 \beta_{2} - 6240) q^{70} + ( - 1998 \beta_{4} + 62 \beta_{3}) q^{71} + ( - 18 \beta_{7} + 2153) q^{73} + (268 \beta_{5} - 112 \beta_1) q^{74} + ( - 16 \beta_{6} + 2176 \beta_{2}) q^{76} + ( - 219 \beta_{5} - 219 \beta_{4} + 223 \beta_{3} - 223 \beta_1) q^{77} + ( - 28 \beta_{7} + 28 \beta_{6} - 2890 \beta_{2} + 2890) q^{79} + (192 \beta_{4} + 64 \beta_{3}) q^{80} + (68 \beta_{7} - 2640) q^{82} + (1698 \beta_{5} - 27 \beta_1) q^{83} + ( - 114 \beta_{6} + 4050 \beta_{2}) q^{85} + ( - 1774 \beta_{5} - 1774 \beta_{4} + 48 \beta_{3} - 48 \beta_1) q^{86} + ( - 8 \beta_{7} + 8 \beta_{6} + 1920 \beta_{2} - 1920) q^{88} + ( - 2826 \beta_{4} - 262 \beta_{3}) q^{89} + (96 \beta_{7} + 9454) q^{91} + (912 \beta_{5} + 160 \beta_1) q^{92} + ( - 30 \beta_{6} + 6432 \beta_{2}) q^{94} + ( - 2274 \beta_{5} - 2274 \beta_{4} - 320 \beta_{3} + 320 \beta_1) q^{95} + ( - 88 \beta_{7} + 88 \beta_{6} + 395 \beta_{2} - 395) q^{97} + ( - 3720 \beta_{4} - 272 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 32 q^{4} - 68 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 32 q^{4} - 68 q^{7} + 192 q^{10} + 520 q^{13} - 256 q^{16} + 2176 q^{19} + 960 q^{22} + 704 q^{25} - 1088 q^{28} - 812 q^{31} - 3456 q^{34} - 2144 q^{37} + 768 q^{40} + 7096 q^{43} - 7296 q^{46} - 14880 q^{49} - 4160 q^{52} - 72 q^{55} + 8448 q^{58} + 13936 q^{61} - 4096 q^{64} - 3224 q^{67} - 24960 q^{70} + 17224 q^{73} + 8704 q^{76} + 11560 q^{79} - 21120 q^{82} + 16200 q^{85} - 7680 q^{88} + 75632 q^{91} + 25728 q^{94} - 1580 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 27\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 27\zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\zeta_{24}^{5} - 2\zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -2\zeta_{24}^{7} + 2\zeta_{24}^{5} + 2\zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 54\zeta_{24}^{7} + 54\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -54\zeta_{24}^{5} + 54\zeta_{24}^{3} + 54\zeta_{24} \) Copy content Toggle raw display

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

\(\zeta_{24}\)\(=\) \( ( \beta_{6} + 27\beta_{5} + 27\beta_{4} ) / 108 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_1 ) / 27 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} - 27\beta_{4} ) / 108 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + \beta_{6} + 27\beta_{5} ) / 108 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( \beta_{3} ) / 27 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{6} - 27\beta_{5} - 27\beta_{4} ) / 108 \) 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 \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(\beta_{2}\)

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
−0.258819 − 0.965926i
−0.965926 + 0.258819i
0.258819 − 0.965926i
0.965926 + 0.258819i
−0.258819 + 0.965926i
−0.965926 − 0.258819i
−2.44949 + 1.41421i 0 4.00000 − 6.92820i −30.7312 − 17.7426i 0 −46.6838 − 80.8587i 22.6274i 0 100.368
53.2 −2.44949 + 1.41421i 0 4.00000 − 6.92820i 16.0342 + 9.25736i 0 29.6838 + 51.4138i 22.6274i 0 −52.3675
53.3 2.44949 − 1.41421i 0 4.00000 − 6.92820i −16.0342 − 9.25736i 0 29.6838 + 51.4138i − 22.6274i 0 −52.3675
53.4 2.44949 − 1.41421i 0 4.00000 − 6.92820i 30.7312 + 17.7426i 0 −46.6838 − 80.8587i − 22.6274i 0 100.368
107.1 −2.44949 − 1.41421i 0 4.00000 + 6.92820i −30.7312 + 17.7426i 0 −46.6838 + 80.8587i − 22.6274i 0 100.368
107.2 −2.44949 − 1.41421i 0 4.00000 + 6.92820i 16.0342 − 9.25736i 0 29.6838 − 51.4138i − 22.6274i 0 −52.3675
107.3 2.44949 + 1.41421i 0 4.00000 + 6.92820i −16.0342 + 9.25736i 0 29.6838 − 51.4138i 22.6274i 0 −52.3675
107.4 2.44949 + 1.41421i 0 4.00000 + 6.92820i 30.7312 − 17.7426i 0 −46.6838 + 80.8587i 22.6274i 0 100.368
\(n\): 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.c 8
3.b odd 2 1 inner 162.5.d.c 8
9.c even 3 1 54.5.b.b ✓ 4
9.c even 3 1 inner 162.5.d.c 8
9.d odd 6 1 54.5.b.b ✓ 4
9.d odd 6 1 inner 162.5.d.c 8
36.f odd 6 1 432.5.e.h 4
36.h even 6 1 432.5.e.h 4
45.h odd 6 1 1350.5.d.c 4
45.j even 6 1 1350.5.d.c 4
45.k odd 12 1 1350.5.b.a 4
45.k odd 12 1 1350.5.b.c 4
45.l even 12 1 1350.5.b.a 4
45.l even 12 1 1350.5.b.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.5.b.b ✓ 4 9.c even 3 1
54.5.b.b ✓ 4 9.d odd 6 1
162.5.d.c 8 1.a even 1 1 trivial
162.5.d.c 8 3.b odd 2 1 inner
162.5.d.c 8 9.c even 3 1 inner
162.5.d.c 8 9.d odd 6 1 inner
432.5.e.h 4 36.f odd 6 1
432.5.e.h 4 36.h even 6 1
1350.5.b.a 4 45.k odd 12 1
1350.5.b.a 4 45.l even 12 1
1350.5.b.c 4 45.k odd 12 1
1350.5.b.c 4 45.l even 12 1
1350.5.d.c 4 45.h odd 6 1
1350.5.d.c 4 45.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 1602T_{5}^{6} + 2134755T_{5}^{4} - 691501698T_{5}^{2} + 186320859201 \) 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
$5$ \( T^{8} - 1602 T^{6} + \cdots + 186320859201 \) Copy content Toggle raw display
$7$ \( (T^{4} + 34 T^{3} + 6699 T^{2} + \cdots + 30724849)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 15858 T^{6} + \cdots + 17\!\cdots\!81 \) Copy content Toggle raw display
$13$ \( (T^{4} - 260 T^{3} + 74028 T^{2} + \cdots + 41319184)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 192456 T^{2} + \cdots + 8171436816)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 544 T + 50656)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} - 791136 T^{6} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{8} - 1820808 T^{6} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( (T^{4} + 406 T^{3} + \cdots + 1615418122081)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 536 T - 1071248)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} - 8484192 T^{6} + \cdots + 39\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( (T^{4} - 3548 T^{3} + \cdots + 8626697391376)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 11654856 T^{6} + \cdots + 41\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( (T^{4} + 25469154 T^{2} + \cdots + 62602291689921)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 14834376 T^{6} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{4} - 6968 T^{3} + \cdots + 77071262392576)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 1612 T^{3} + \cdots + 193322019856)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 69476616 T^{2} + \cdots + 848775738667536)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 4306 T + 2745841)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 5780 T^{3} + \cdots + 14286978755344)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 47194146 T^{6} + \cdots + 25\!\cdots\!61 \) Copy content Toggle raw display
$89$ \( (T^{4} + 227863368 T^{2} + \cdots + 191787378007824)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 790 T^{3} + \cdots + 20\!\cdots\!89)^{2} \) Copy content Toggle raw display
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