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Properties

Label 162.9.d.h
Level $162$
Weight $9$
Character orbit 162.d
Analytic conductor $65.995$
Analytic rank $0$
Dimension $16$
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] = [162,9,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, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.53");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 9 \)
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: \(65.9953348299\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3364 x^{14} + 7128063 x^{12} - 9514897636 x^{10} + 9385153250593 x^{8} + \cdots + 21\!\cdots\!96 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{36} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 8 \beta_{9} q^{2} + 128 \beta_1 q^{4} + (\beta_{14} - \beta_{12} + 67 \beta_{8}) q^{5} + ( - 7 \beta_{4} - 7 \beta_{3} - 7 \beta_{2} - 1106 \beta_1 + 1106) q^{7} + (1024 \beta_{9} - 1024 \beta_{8}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 \beta_{9} q^{2} + 128 \beta_1 q^{4} + (\beta_{14} - \beta_{12} + 67 \beta_{8}) q^{5} + ( - 7 \beta_{4} - 7 \beta_{3} - 7 \beta_{2} - 1106 \beta_1 + 1106) q^{7} + (1024 \beta_{9} - 1024 \beta_{8}) q^{8} + ( - 8 \beta_{7} + 8 \beta_{6} - 8 \beta_{3} + 1056) q^{10} + ( - 62 \beta_{15} - 62 \beta_{14} - \beta_{13} + 47 \beta_{11} - 47 \beta_{10} + \cdots - 15 \beta_{8}) q^{11}+ \cdots + ( - 183456 \beta_{15} + 33712 \beta_{13} + \cdots - 13714904 \beta_{8}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 1024 q^{4} + 8876 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 1024 q^{4} + 8876 q^{7} + 16896 q^{10} + 117380 q^{13} - 131072 q^{16} + 540440 q^{19} + 210816 q^{22} + 1801672 q^{25} + 2272256 q^{28} + 393344 q^{31} + 691968 q^{34} + 3661976 q^{37} + 1081344 q^{40} - 11135236 q^{43} + 10592640 q^{46} + 13586328 q^{49} - 15024640 q^{52} - 3159432 q^{55} + 32988672 q^{58} - 12184204 q^{61} - 33554432 q^{64} + 80355716 q^{67} + 18723264 q^{70} + 394171520 q^{73} + 34588160 q^{76} - 84451852 q^{79} + 289278336 q^{82} + 582634548 q^{85} - 26984448 q^{88} + 746159176 q^{91} + 210121536 q^{94} - 341136928 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3364 x^{14} + 7128063 x^{12} - 9514897636 x^{10} + 9385153250593 x^{8} + \cdots + 21\!\cdots\!96 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 19\!\cdots\!55 \nu^{14} + \cdots - 70\!\cdots\!88 ) / 71\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 45\!\cdots\!33 \nu^{14} + \cdots + 26\!\cdots\!92 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 112682355396425 \nu^{14} + \cdots - 17\!\cdots\!84 ) / 14\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\!\cdots\!91 \nu^{14} + \cdots + 68\!\cdots\!64 ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 16\!\cdots\!43 \nu^{14} + \cdots + 17\!\cdots\!32 ) / 16\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 58\!\cdots\!41 \nu^{14} + \cdots - 49\!\cdots\!40 ) / 40\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 30\!\cdots\!23 \nu^{14} + \cdots - 13\!\cdots\!12 ) / 80\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 13\!\cdots\!69 \nu^{15} + \cdots - 11\!\cdots\!28 \nu ) / 20\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 24\!\cdots\!41 \nu^{15} + \cdots + 22\!\cdots\!96 \nu ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 28\!\cdots\!73 \nu^{15} + \cdots - 58\!\cdots\!60 \nu ) / 40\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 14\!\cdots\!71 \nu^{15} + \cdots - 54\!\cdots\!08 \nu ) / 40\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 31\!\cdots\!07 \nu^{15} + \cdots + 41\!\cdots\!12 \nu ) / 81\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 28\!\cdots\!99 \nu^{15} + \cdots - 46\!\cdots\!80 \nu ) / 40\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 14\!\cdots\!73 \nu^{15} + \cdots - 85\!\cdots\!04 \nu ) / 20\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 51\!\cdots\!71 \nu^{15} + \cdots + 23\!\cdots\!60 \nu ) / 37\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{15} - 2\beta_{14} - 6\beta_{13} - 3\beta_{11} + 3\beta_{10} - 35\beta_{9} - 5\beta_{8} ) / 162 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{7} + 30\beta_{5} + 30\beta_{4} - 178\beta_{2} + 136227\beta_1 ) / 162 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -250\beta_{15} - 1684\beta_{13} + 1684\beta_{12} - 1679\beta_{11} - 43459\beta_{9} + 43459\beta_{8} ) / 54 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6711\beta_{6} + 57120\beta_{4} - 299596\beta_{3} - 299596\beta_{2} + 119026455\beta _1 - 119026455 ) / 162 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -498074\beta_{14} + 4421202\beta_{12} - 6466773\beta_{10} + 243713768\beta_{8} ) / 162 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -3604147\beta_{7} + 3604147\beta_{6} - 27621430\beta_{5} - 127821642\beta_{3} - 35916617107 ) / 54 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1776403958 \beta_{15} - 1776403958 \beta_{14} + 4009684620 \beta_{13} + 7495086327 \beta_{11} - 7495086327 \beta_{10} + 351792378619 \beta_{9} + 5718682369 \beta_{8} ) / 162 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 15126964125 \beta_{7} - 108137043240 \beta_{5} - 108137043240 \beta_{4} + 442312434860 \beta_{2} - 100600182123837 \beta_1 ) / 162 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1033139189066 \beta_{15} + 1250333457002 \beta_{13} - 1250333457002 \beta_{12} + 2753963656861 \beta_{11} + 156564388670725 \beta_{9} - 156564388670725 \beta_{8} ) / 54 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 19563256071723 \beta_{6} - 133482741278430 \beta_{4} + 484788818349938 \beta_{3} + 484788818349938 \beta_{2} + \cdots + 96\!\cdots\!87 ) / 162 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 44\!\cdots\!70 \beta_{14} + \cdots - 59\!\cdots\!24 \beta_{8} ) / 162 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 80\!\cdots\!17 \beta_{7} + \cdots + 31\!\cdots\!25 ) / 54 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 58\!\cdots\!14 \beta_{15} + \cdots - 34\!\cdots\!79 \beta_{8} ) / 162 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 28\!\cdots\!81 \beta_{7} + \cdots + 93\!\cdots\!41 \beta_1 ) / 162 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 24\!\cdots\!66 \beta_{15} + \cdots + 27\!\cdots\!13 \beta_{8} ) / 54 \) Copy content Toggle raw display

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

Character values

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

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

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
−22.8192 + 13.1747i
−26.5535 + 15.3307i
22.3710 − 12.9159i
28.2265 − 16.2966i
−28.2265 + 16.2966i
−22.3710 + 12.9159i
26.5535 − 15.3307i
22.8192 − 13.1747i
−22.8192 − 13.1747i
−26.5535 − 15.3307i
22.3710 + 12.9159i
28.2265 + 16.2966i
−28.2265 − 16.2966i
−22.3710 − 12.9159i
26.5535 + 15.3307i
22.8192 + 13.1747i
−9.79796 + 5.65685i 0 64.0000 − 110.851i −919.678 − 530.976i 0 1616.80 + 2800.38i 1448.15i 0 12014.6
53.2 −9.79796 + 5.65685i 0 64.0000 − 110.851i −566.671 − 327.168i 0 −649.702 − 1125.32i 1448.15i 0 7402.96
53.3 −9.79796 + 5.65685i 0 64.0000 − 110.851i 535.273 + 309.040i 0 274.720 + 475.829i 1448.15i 0 −6992.78
53.4 −9.79796 + 5.65685i 0 64.0000 − 110.851i 627.743 + 362.428i 0 977.181 + 1692.53i 1448.15i 0 −8200.80
53.5 9.79796 − 5.65685i 0 64.0000 − 110.851i −627.743 − 362.428i 0 977.181 + 1692.53i − 1448.15i 0 −8200.80
53.6 9.79796 − 5.65685i 0 64.0000 − 110.851i −535.273 − 309.040i 0 274.720 + 475.829i − 1448.15i 0 −6992.78
53.7 9.79796 − 5.65685i 0 64.0000 − 110.851i 566.671 + 327.168i 0 −649.702 − 1125.32i − 1448.15i 0 7402.96
53.8 9.79796 − 5.65685i 0 64.0000 − 110.851i 919.678 + 530.976i 0 1616.80 + 2800.38i − 1448.15i 0 12014.6
107.1 −9.79796 − 5.65685i 0 64.0000 + 110.851i −919.678 + 530.976i 0 1616.80 − 2800.38i − 1448.15i 0 12014.6
107.2 −9.79796 − 5.65685i 0 64.0000 + 110.851i −566.671 + 327.168i 0 −649.702 + 1125.32i − 1448.15i 0 7402.96
107.3 −9.79796 − 5.65685i 0 64.0000 + 110.851i 535.273 − 309.040i 0 274.720 − 475.829i − 1448.15i 0 −6992.78
107.4 −9.79796 − 5.65685i 0 64.0000 + 110.851i 627.743 − 362.428i 0 977.181 − 1692.53i − 1448.15i 0 −8200.80
107.5 9.79796 + 5.65685i 0 64.0000 + 110.851i −627.743 + 362.428i 0 977.181 − 1692.53i 1448.15i 0 −8200.80
107.6 9.79796 + 5.65685i 0 64.0000 + 110.851i −535.273 + 309.040i 0 274.720 − 475.829i 1448.15i 0 −6992.78
107.7 9.79796 + 5.65685i 0 64.0000 + 110.851i 566.671 − 327.168i 0 −649.702 + 1125.32i 1448.15i 0 7402.96
107.8 9.79796 + 5.65685i 0 64.0000 + 110.851i 919.678 − 530.976i 0 1616.80 − 2800.38i 1448.15i 0 12014.6
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.8
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 162.9.d.h 16
3.b odd 2 1 inner 162.9.d.h 16
9.c even 3 1 162.9.b.a ✓ 8
9.c even 3 1 inner 162.9.d.h 16
9.d odd 6 1 162.9.b.a ✓ 8
9.d odd 6 1 inner 162.9.d.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.9.b.a ✓ 8 9.c even 3 1
162.9.b.a ✓ 8 9.d odd 6 1
162.9.d.h 16 1.a even 1 1 trivial
162.9.d.h 16 3.b odd 2 1 inner
162.9.d.h 16 9.c even 3 1 inner
162.9.d.h 16 9.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} - 2463336 T_{5}^{14} + 3972574081422 T_{5}^{12} + \cdots + 93\!\cdots\!25 \) acting on \(S_{9}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 128 T^{2} + 16384)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 2463336 T^{14} + \cdots + 93\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{8} - 4438 T^{7} + \cdots + 20\!\cdots\!16)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} - 1083686796 T^{14} + \cdots + 80\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( (T^{8} - 58690 T^{7} + \cdots + 28\!\cdots\!29)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 40156998768 T^{6} + \cdots + 42\!\cdots\!25)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 135110 T^{3} + \cdots + 27\!\cdots\!24)^{4} \) Copy content Toggle raw display
$23$ \( T^{16} - 283534373268 T^{14} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{16} - 1677572170848 T^{14} + \cdots + 99\!\cdots\!21 \) Copy content Toggle raw display
$31$ \( (T^{8} - 196672 T^{7} + \cdots + 29\!\cdots\!16)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 915494 T^{3} + \cdots + 16\!\cdots\!21)^{4} \) Copy content Toggle raw display
$41$ \( T^{16} - 42402792363120 T^{14} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( (T^{8} + 5567618 T^{7} + \cdots + 18\!\cdots\!96)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} - 62658471632112 T^{14} + \cdots + 45\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{8} + 288880676034192 T^{6} + \cdots + 47\!\cdots\!24)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} - 85968765926928 T^{14} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{8} + 6092102 T^{7} + \cdots + 15\!\cdots\!01)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 40177858 T^{7} + \cdots + 19\!\cdots\!24)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 34\!\cdots\!76)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 98542880 T^{3} + \cdots - 56\!\cdots\!27)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + 42225926 T^{7} + \cdots + 27\!\cdots\!64)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 22\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 39\!\cdots\!61)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 170568464 T^{7} + \cdots + 17\!\cdots\!24)^{2} \) Copy content Toggle raw display
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