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Properties

Label 162.7.d.e
Level $162$
Weight $7$
Character orbit 162.d
Analytic conductor $37.269$
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] = [162,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.53");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 7 \)
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: \(37.2687615464\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{12} \)
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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + 32 \beta_{2} q^{4} + ( - 15 \beta_{5} - 15 \beta_{4} - 5 \beta_{3} + 5 \beta_1) q^{5} + (\beta_{7} - \beta_{6} + 209 \beta_{2} - 209) q^{7} - 32 \beta_{4} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + 32 \beta_{2} q^{4} + ( - 15 \beta_{5} - 15 \beta_{4} - 5 \beta_{3} + 5 \beta_1) q^{5} + (\beta_{7} - \beta_{6} + 209 \beta_{2} - 209) q^{7} - 32 \beta_{4} q^{8} + (5 \beta_{7} - 480) q^{10} + ( - 150 \beta_{5} - 29 \beta_1) q^{11} + ( - 22 \beta_{6} - 110 \beta_{2}) q^{13} + ( - 209 \beta_{5} - 209 \beta_{4} - 32 \beta_{3} + 32 \beta_1) q^{14} + (1024 \beta_{2} - 1024) q^{16} + (612 \beta_{4} + 106 \beta_{3}) q^{17} + (46 \beta_{7} - 1672) q^{19} + ( - 480 \beta_{5} + 160 \beta_1) q^{20} + ( - 29 \beta_{6} - 4800 \beta_{2}) q^{22} + (1074 \beta_{5} + 1074 \beta_{4} + 260 \beta_{3} - 260 \beta_1) q^{23} + ( - 150 \beta_{7} + 150 \beta_{6} - 9800 \beta_{2} + 9800) q^{25} + (110 \beta_{4} - 704 \beta_{3}) q^{26} + (32 \beta_{7} - 6688) q^{28} + ( - 672 \beta_{5} - 422 \beta_1) q^{29} + ( - 15 \beta_{6} - 40043 \beta_{2}) q^{31} + ( - 1024 \beta_{5} - 1024 \beta_{4}) q^{32} + ( - 106 \beta_{7} + 106 \beta_{6} - 19584 \beta_{2} + \cdots + 19584) q^{34}+ \cdots + ( - 50640 \beta_{4} + 13376 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 128 q^{4} - 836 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 128 q^{4} - 836 q^{7} - 3840 q^{10} - 440 q^{13} - 4096 q^{16} - 13376 q^{19} - 19200 q^{22} + 39200 q^{25} - 53504 q^{28} - 160172 q^{31} + 78336 q^{34} - 301088 q^{37} - 61440 q^{40} - 90152 q^{43} + 274944 q^{46} + 202560 q^{49} + 14080 q^{52} - 269640 q^{55} - 86016 q^{58} - 584144 q^{61} - 262144 q^{64} + 766792 q^{67} + 867840 q^{70} + 3149512 q^{73} - 214016 q^{76} - 323000 q^{79} + 2081280 q^{82} + 2720520 q^{85} + 614400 q^{88} - 3921808 q^{91} - 1970688 q^{94} - 4432940 q^{97}+O(q^{100}) \) 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
\(\beta_{4}\)\(=\) \( -4\zeta_{24}^{5} - 4\zeta_{24}^{3} + 4\zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -4\zeta_{24}^{7} + 4\zeta_{24}^{5} + 4\zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 108\zeta_{24}^{7} + 108\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -108\zeta_{24}^{5} + 108\zeta_{24}^{3} + 108\zeta_{24} \) Copy content Toggle raw display

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

\(\zeta_{24}\)\(=\) \( ( \beta_{6} + 27\beta_{5} + 27\beta_{4} ) / 216 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_1 ) / 27 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} - 27\beta_{4} ) / 216 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + \beta_{6} + 27\beta_{5} ) / 216 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( \beta_{3} ) / 27 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{6} - 27\beta_{5} - 27\beta_{4} ) / 216 \) 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
−4.89898 + 2.82843i 0 16.0000 − 27.7128i −43.4287 − 25.0736i 0 −28.1325 − 48.7269i 181.019i 0 283.675
53.2 −4.89898 + 2.82843i 0 16.0000 − 27.7128i 190.398 + 109.926i 0 −180.868 − 313.272i 181.019i 0 −1243.68
53.3 4.89898 − 2.82843i 0 16.0000 − 27.7128i −190.398 − 109.926i 0 −180.868 − 313.272i − 181.019i 0 −1243.68
53.4 4.89898 − 2.82843i 0 16.0000 − 27.7128i 43.4287 + 25.0736i 0 −28.1325 − 48.7269i − 181.019i 0 283.675
107.1 −4.89898 − 2.82843i 0 16.0000 + 27.7128i −43.4287 + 25.0736i 0 −28.1325 + 48.7269i − 181.019i 0 283.675
107.2 −4.89898 − 2.82843i 0 16.0000 + 27.7128i 190.398 − 109.926i 0 −180.868 + 313.272i − 181.019i 0 −1243.68
107.3 4.89898 + 2.82843i 0 16.0000 + 27.7128i −190.398 + 109.926i 0 −180.868 + 313.272i 181.019i 0 −1243.68
107.4 4.89898 + 2.82843i 0 16.0000 + 27.7128i 43.4287 − 25.0736i 0 −28.1325 + 48.7269i 181.019i 0 283.675
\(n\): 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 162.7.d.e 8
3.b odd 2 1 inner 162.7.d.e 8
9.c even 3 1 54.7.b.c ✓ 4
9.c even 3 1 inner 162.7.d.e 8
9.d odd 6 1 54.7.b.c ✓ 4
9.d odd 6 1 inner 162.7.d.e 8
36.f odd 6 1 432.7.e.h 4
36.h even 6 1 432.7.e.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.7.b.c ✓ 4 9.c even 3 1
54.7.b.c ✓ 4 9.d odd 6 1
162.7.d.e 8 1.a even 1 1 trivial
162.7.d.e 8 3.b odd 2 1 inner
162.7.d.e 8 9.c even 3 1 inner
162.7.d.e 8 9.d odd 6 1 inner
432.7.e.h 4 36.f odd 6 1
432.7.e.h 4 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 50850T_{5}^{6} + 2464171875T_{5}^{4} - 6180849281250T_{5}^{2} + 14774554437890625 \) acting on \(S_{7}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 32 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 50850 T^{6} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{4} + 418 T^{3} + 154371 T^{2} + \cdots + 414244609)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 2666178 T^{6} + \cdots + 13\!\cdots\!41 \) Copy content Toggle raw display
$13$ \( (T^{4} + 220 T^{3} + \cdots + 127207990937104)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 40352904 T^{2} + \cdots + 14397198164496)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 3344 T - 46566464)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} - 172383264 T^{6} + \cdots + 23\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{8} - 288547848 T^{6} + \cdots + 17\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( (T^{4} + 80086 T^{3} + \cdots + 25\!\cdots\!01)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 75272 T + 1154355088)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} - 13110914592 T^{6} + \cdots + 29\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( (T^{4} + 45076 T^{3} + \cdots + 89\!\cdots\!16)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 36334830024 T^{6} + \cdots + 80\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( (T^{4} + 117509051106 T^{2} + \cdots + 34\!\cdots\!21)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 12634340616 T^{6} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{4} + 292072 T^{3} + \cdots + 36\!\cdots\!96)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 383396 T^{3} + \cdots + 11\!\cdots\!96)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 238720256136 T^{2} + \cdots + 11\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 787378 T + 154378808689)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 161500 T^{3} + \cdots + 33\!\cdots\!04)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 574584155154 T^{6} + \cdots + 61\!\cdots\!41 \) Copy content Toggle raw display
$89$ \( (T^{4} + 992009774088 T^{2} + \cdots + 24\!\cdots\!64)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 2216470 T^{3} + \cdots + 65\!\cdots\!09)^{2} \) Copy content Toggle raw display
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