Properties

Label 162.9.d.b
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.

\(f(q)\) \(=\) \( q + ( - 8 \beta_{3} + 8 \beta_1) q^{2} + ( - 128 \beta_{2} + 128) q^{4} + 645 \beta_1 q^{5} - 1652 \beta_{2} q^{7} - 1024 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 8 \beta_{3} + 8 \beta_1) q^{2} + ( - 128 \beta_{2} + 128) q^{4} + 645 \beta_1 q^{5} - 1652 \beta_{2} q^{7} - 1024 \beta_{3} q^{8} + 10320 q^{10} + (9732 \beta_{3} - 9732 \beta_1) q^{11} + (46304 \beta_{2} - 46304) q^{13} - 13216 \beta_1 q^{14} - 16384 \beta_{2} q^{16} - 77967 \beta_{3} q^{17} - 243664 q^{19} + ( - 82560 \beta_{3} + 82560 \beta_1) q^{20} + (155712 \beta_{2} - 155712) q^{22} - 101316 \beta_1 q^{23} + 441425 \beta_{2} q^{25} + 370432 \beta_{3} q^{26} - 211456 q^{28} + (215787 \beta_{3} - 215787 \beta_1) q^{29} + (384164 \beta_{2} - 384164) q^{31} - 131072 \beta_1 q^{32} - 1247472 \beta_{2} q^{34} - 1065540 \beta_{3} q^{35} + 496982 q^{37} + (1949312 \beta_{3} - 1949312 \beta_1) q^{38} + ( - 1320960 \beta_{2} + 1320960) q^{40} - 712767 \beta_1 q^{41} - 5334440 \beta_{2} q^{43} + 1245696 \beta_{3} q^{44} - 1621056 q^{46} + ( - 4563060 \beta_{3} + 4563060 \beta_1) q^{47} + ( - 3035697 \beta_{2} + 3035697) q^{49} + 3531400 \beta_1 q^{50} + 5926912 \beta_{2} q^{52} + 1915029 \beta_{3} q^{53} - 12554280 q^{55} + (1691648 \beta_{3} - 1691648 \beta_1) q^{56} + (3452592 \beta_{2} - 3452592) q^{58} - 8799384 \beta_1 q^{59} - 2335370 \beta_{2} q^{61} + 3073312 \beta_{3} q^{62} - 2097152 q^{64} + (29866080 \beta_{3} - 29866080 \beta_1) q^{65} + (30674456 \beta_{2} - 30674456) q^{67} - 9979776 \beta_1 q^{68} - 17048640 \beta_{2} q^{70} + 8613036 \beta_{3} q^{71} - 11519728 q^{73} + ( - 3975856 \beta_{3} + 3975856 \beta_1) q^{74} + (31188992 \beta_{2} - 31188992) q^{76} + 16077264 \beta_1 q^{77} + 2658244 \beta_{2} q^{79} - 10567680 \beta_{3} q^{80} - 11404272 q^{82} + (36660252 \beta_{3} - 36660252 \beta_1) q^{83} + ( - 100577430 \beta_{2} + 100577430) q^{85} - 42675520 \beta_1 q^{86} + 19931136 \beta_{2} q^{88} + 27337599 \beta_{3} q^{89} + 76494208 q^{91} + (12968448 \beta_{3} - 12968448 \beta_1) q^{92} + ( - 73008960 \beta_{2} + 73008960) q^{94} - 157163280 \beta_1 q^{95} + 51595168 \beta_{2} q^{97} - 24285576 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 256 q^{4} - 3304 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 256 q^{4} - 3304 q^{7} + 41280 q^{10} - 92608 q^{13} - 32768 q^{16} - 974656 q^{19} - 311424 q^{22} + 882850 q^{25} - 845824 q^{28} - 768328 q^{31} - 2494944 q^{34} + 1987928 q^{37} + 2641920 q^{40} - 10668880 q^{43} - 6484224 q^{46} + 6071394 q^{49} + 11853824 q^{52} - 50217120 q^{55} - 6905184 q^{58} - 4670740 q^{61} - 8388608 q^{64} - 61348912 q^{67} - 34097280 q^{70} - 46078912 q^{73} - 62377984 q^{76} + 5316488 q^{79} - 45617088 q^{82} + 201154860 q^{85} + 39862272 q^{88} + 305976832 q^{91} + 146017920 q^{94} + 103190336 q^{97}+O(q^{100}) \) 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 −789.960 456.084i 0 −826.000 1430.67i 1448.15i 0 10320.0
53.2 9.79796 5.65685i 0 64.0000 110.851i 789.960 + 456.084i 0 −826.000 1430.67i 1448.15i 0 10320.0
107.1 −9.79796 5.65685i 0 64.0000 + 110.851i −789.960 + 456.084i 0 −826.000 + 1430.67i 1448.15i 0 10320.0
107.2 9.79796 + 5.65685i 0 64.0000 + 110.851i 789.960 456.084i 0 −826.000 + 1430.67i 1448.15i 0 10320.0
\(n\): 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.b 4
3.b odd 2 1 inner 162.9.d.b 4
9.c even 3 1 18.9.b.b 2
9.c even 3 1 inner 162.9.d.b 4
9.d odd 6 1 18.9.b.b 2
9.d odd 6 1 inner 162.9.d.b 4
36.f odd 6 1 144.9.e.b 2
36.h even 6 1 144.9.e.b 2
45.h odd 6 1 450.9.d.a 2
45.j even 6 1 450.9.d.a 2
45.k odd 12 2 450.9.b.b 4
45.l even 12 2 450.9.b.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.9.b.b 2 9.c even 3 1
18.9.b.b 2 9.d odd 6 1
144.9.e.b 2 36.f odd 6 1
144.9.e.b 2 36.h even 6 1
162.9.d.b 4 1.a even 1 1 trivial
162.9.d.b 4 3.b odd 2 1 inner
162.9.d.b 4 9.c even 3 1 inner
162.9.d.b 4 9.d odd 6 1 inner
450.9.b.b 4 45.k odd 12 2
450.9.b.b 4 45.l even 12 2
450.9.d.a 2 45.h odd 6 1
450.9.d.a 2 45.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 832050T_{5}^{2} + 692307202500 \) 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
$5$ \( T^{4} - 832050 T^{2} + \cdots + 692307202500 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1652 T + 2729104)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 189423648 T^{2} + \cdots + 35\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( (T^{2} + 46304 T + 2144060416)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 12157706178)^{2} \) Copy content Toggle raw display
$19$ \( (T + 243664)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 20529863712 T^{2} + \cdots + 42\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{4} - 93128058738 T^{2} + \cdots + 86\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( (T^{2} + 384164 T + 147581978896)^{2} \) Copy content Toggle raw display
$37$ \( (T - 496982)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 1016073592578 T^{2} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( (T^{2} + 5334440 T + 28456250113600)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 41643033127200 T^{2} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{2} + 7334672141682)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 154858317558912 T^{2} + \cdots + 23\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( (T^{2} + 2335370 T + 5453953036900)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 30674456 T + 940922250895936)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 148368778274592)^{2} \) Copy content Toggle raw display
$73$ \( (T + 11519728)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 2658244 T + 7066261163536)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 72\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{2} + 14\!\cdots\!02)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 51595168 T + 26\!\cdots\!24)^{2} \) Copy content Toggle raw display
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