Properties

Label 162.9.d.d
Level $162$
Weight $9$
Character orbit 162.d
Analytic conductor $65.995$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 18)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (8 \beta_{3} - 8 \beta_1) q^{2} + ( - 128 \beta_{2} + 128) q^{4} + 165 \beta_1 q^{5} + 3532 \beta_{2} q^{7} + 1024 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (8 \beta_{3} - 8 \beta_1) q^{2} + ( - 128 \beta_{2} + 128) q^{4} + 165 \beta_1 q^{5} + 3532 \beta_{2} q^{7} + 1024 \beta_{3} q^{8} - 2640 q^{10} + ( - 14268 \beta_{3} + 14268 \beta_1) q^{11} + ( - 41824 \beta_{2} + 41824) q^{13} - 28256 \beta_1 q^{14} - 16384 \beta_{2} q^{16} - 67023 \beta_{3} q^{17} - 36304 q^{19} + ( - 21120 \beta_{3} + 21120 \beta_1) q^{20} + (228288 \beta_{2} - 228288) q^{22} + 292476 \beta_1 q^{23} - 336175 \beta_{2} q^{25} + 334592 \beta_{3} q^{26} + 452096 q^{28} + (190347 \beta_{3} - 190347 \beta_1) q^{29} + ( - 471196 \beta_{2} + 471196) q^{31} + 131072 \beta_1 q^{32} + 1072368 \beta_{2} q^{34} + 582780 \beta_{3} q^{35} - 3007402 q^{37} + ( - 290432 \beta_{3} + 290432 \beta_1) q^{38} + (337920 \beta_{2} - 337920) q^{40} - 1212927 \beta_1 q^{41} - 3623720 \beta_{2} q^{43} - 1826304 \beta_{3} q^{44} - 4679616 q^{46} + ( - 4252980 \beta_{3} + 4252980 \beta_1) q^{47} + (6710223 \beta_{2} - 6710223) q^{49} + 2689400 \beta_1 q^{50} - 5353472 \beta_{2} q^{52} - 7266699 \beta_{3} q^{53} + 4708440 q^{55} + (3616768 \beta_{3} - 3616768 \beta_1) q^{56} + ( - 3045552 \beta_{2} + 3045552) q^{58} + 1900776 \beta_1 q^{59} + 5440630 \beta_{2} q^{61} + 3769568 \beta_{3} q^{62} - 2097152 q^{64} + ( - 6900960 \beta_{3} + 6900960 \beta_1) q^{65} + ( - 6121576 \beta_{2} + 6121576) q^{67} - 8578944 \beta_1 q^{68} - 9324480 \beta_{2} q^{70} + 14986476 \beta_{3} q^{71} - 49031152 q^{73} + ( - 24059216 \beta_{3} + 24059216 \beta_1) q^{74} + (4646912 \beta_{2} - 4646912) q^{76} + 50394576 \beta_1 q^{77} - 8357756 \beta_{2} q^{79} - 2703360 \beta_{3} q^{80} + 19406832 q^{82} + ( - 36339492 \beta_{3} + 36339492 \beta_1) q^{83} + ( - 22117590 \beta_{2} + 22117590) q^{85} + 28989760 \beta_1 q^{86} + 29220864 \beta_{2} q^{88} - 75898881 \beta_{3} q^{89} + 147722368 q^{91} + ( - 37436928 \beta_{3} + 37436928 \beta_1) q^{92} + (68047680 \beta_{2} - 68047680) q^{94} - 5990160 \beta_1 q^{95} - 20431328 \beta_{2} q^{97} - 53681784 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 256 q^{4} + 7064 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 256 q^{4} + 7064 q^{7} - 10560 q^{10} + 83648 q^{13} - 32768 q^{16} - 145216 q^{19} - 456576 q^{22} - 672350 q^{25} + 1808384 q^{28} + 942392 q^{31} + 2144736 q^{34} - 12029608 q^{37} - 675840 q^{40} - 7247440 q^{43} - 18718464 q^{46} - 13420446 q^{49} - 10706944 q^{52} + 18833760 q^{55} + 6091104 q^{58} + 10881260 q^{61} - 8388608 q^{64} + 12243152 q^{67} - 18648960 q^{70} - 196124608 q^{73} - 9293824 q^{76} - 16715512 q^{79} + 77627328 q^{82} + 44235180 q^{85} + 58441728 q^{88} + 590889472 q^{91} - 136095360 q^{94} - 40862656 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(83\)
\(\chi(n)\) \(1 - \beta_{2}\)

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
1.22474 + 0.707107i
−1.22474 0.707107i
1.22474 0.707107i
−1.22474 + 0.707107i
−9.79796 + 5.65685i 0 64.0000 110.851i 202.083 + 116.673i 0 1766.00 + 3058.80i 1448.15i 0 −2640.00
53.2 9.79796 5.65685i 0 64.0000 110.851i −202.083 116.673i 0 1766.00 + 3058.80i 1448.15i 0 −2640.00
107.1 −9.79796 5.65685i 0 64.0000 + 110.851i 202.083 116.673i 0 1766.00 3058.80i 1448.15i 0 −2640.00
107.2 9.79796 + 5.65685i 0 64.0000 + 110.851i −202.083 + 116.673i 0 1766.00 3058.80i 1448.15i 0 −2640.00
\(n\): e.g. 2-40 or 990-1000
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.d 4
3.b odd 2 1 inner 162.9.d.d 4
9.c even 3 1 18.9.b.a 2
9.c even 3 1 inner 162.9.d.d 4
9.d odd 6 1 18.9.b.a 2
9.d odd 6 1 inner 162.9.d.d 4
36.f odd 6 1 144.9.e.d 2
36.h even 6 1 144.9.e.d 2
45.h odd 6 1 450.9.d.b 2
45.j even 6 1 450.9.d.b 2
45.k odd 12 2 450.9.b.a 4
45.l even 12 2 450.9.b.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.9.b.a 2 9.c even 3 1
18.9.b.a 2 9.d odd 6 1
144.9.e.d 2 36.f odd 6 1
144.9.e.d 2 36.h even 6 1
162.9.d.d 4 1.a even 1 1 trivial
162.9.d.d 4 3.b odd 2 1 inner
162.9.d.d 4 9.c even 3 1 inner
162.9.d.d 4 9.d odd 6 1 inner
450.9.b.a 4 45.k odd 12 2
450.9.b.a 4 45.l even 12 2
450.9.d.b 2 45.h odd 6 1
450.9.d.b 2 45.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 54450T_{5}^{2} + 2964802500 \) acting on \(S_{9}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 128 T^{2} + 16384 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 54450 T^{2} + \cdots + 2964802500 \) Copy content Toggle raw display
$7$ \( (T^{2} - 3532 T + 12475024)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 407151648 T^{2} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( (T^{2} - 41824 T + 1749246976)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 8984165058)^{2} \) Copy content Toggle raw display
$19$ \( (T + 36304)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 171084421152 T^{2} + \cdots + 29\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{4} - 72463960818 T^{2} + \cdots + 52\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( (T^{2} - 471196 T + 222025670416)^{2} \) Copy content Toggle raw display
$37$ \( (T + 3007402)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 2942383814658 T^{2} + \cdots + 86\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( (T^{2} + 3623720 T + 13131346638400)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 36175677760800 T^{2} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{2} + 105609828713202)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 7225898804352 T^{2} + \cdots + 52\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( (T^{2} - 5440630 T + 29600454796900)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 6121576 T + 37473692723776)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 449188925797152)^{2} \) Copy content Toggle raw display
$73$ \( (T + 49031152)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 8357756 T + 69852085355536)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 69\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{2} + 11\!\cdots\!22)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 20431328 T + 417439163843584)^{2} \) Copy content Toggle raw display
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