Properties

Label 162.9.b.c
Level $162$
Weight $9$
Character orbit 162.b
Analytic conductor $65.995$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [162,9,Mod(161,162)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(162, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.161");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 162.b (of order \(2\), degree \(1\), 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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 5476 x^{14} - 38192 x^{13} + 11414542 x^{12} - 67991120 x^{11} + 11330952892 x^{10} - 56032421816 x^{9} + 5575009041895 x^{8} + \cdots + 19\!\cdots\!29 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{36}\cdot 3^{80} \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{8} q^{2} - 128 q^{4} - \beta_{10} q^{5} + ( - \beta_1 + 231) q^{7} + 128 \beta_{8} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{2} - 128 q^{4} - \beta_{10} q^{5} + ( - \beta_1 + 231) q^{7} + 128 \beta_{8} q^{8} + ( - \beta_{2} - \beta_1) q^{10} + ( - \beta_{10} + \beta_{9} + 60 \beta_{8}) q^{11} + (\beta_{3} - 3 \beta_1 + 422) q^{13} + (\beta_{13} + 8 \beta_{10} - 231 \beta_{8}) q^{14} + 16384 q^{16} + ( - \beta_{12} + 31 \beta_{10} - 215 \beta_{8}) q^{17} + (\beta_{5} + \beta_{3} - 2 \beta_{2} + 8 \beta_1 + 22634) q^{19} + 128 \beta_{10} q^{20} + ( - \beta_{6} - \beta_{3} - \beta_{2} + 2 \beta_1 + 7727) q^{22} + ( - 2 \beta_{15} + \beta_{14} + \beta_{13} - 2 \beta_{12} - \beta_{11} - 17 \beta_{10} + \cdots + 3620 \beta_{8}) q^{23}+ \cdots + (114 \beta_{15} - 140 \beta_{14} + 1110 \beta_{13} - 42 \beta_{12} + \cdots - 295668 \beta_{8}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2048 q^{4} + 3692 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2048 q^{4} + 3692 q^{7} + 6740 q^{13} + 262144 q^{16} + 362180 q^{19} + 123648 q^{22} - 1926788 q^{25} - 472576 q^{28} - 1084876 q^{31} - 440832 q^{34} + 3343328 q^{37} - 679024 q^{43} + 7417344 q^{46} + 4729308 q^{49} - 862720 q^{52} - 4584276 q^{55} + 15705600 q^{58} + 1683908 q^{61} - 33554432 q^{64} - 59893288 q^{67} + 68719104 q^{70} - 7547764 q^{73} - 46359040 q^{76} - 67626004 q^{79} - 137346048 q^{82} + 251393544 q^{85} - 15826944 q^{88} + 268578316 q^{91} + 23665152 q^{94} + 178830968 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 5476 x^{14} - 38192 x^{13} + 11414542 x^{12} - 67991120 x^{11} + 11330952892 x^{10} - 56032421816 x^{9} + 5575009041895 x^{8} + \cdots + 19\!\cdots\!29 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 30\!\cdots\!91 \nu^{14} + \cdots - 10\!\cdots\!31 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 87\!\cdots\!59 \nu^{14} + \cdots - 20\!\cdots\!19 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 94\!\cdots\!36 \nu^{14} + \cdots + 19\!\cdots\!76 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 25\!\cdots\!03 \nu^{14} + \cdots + 82\!\cdots\!73 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 69\!\cdots\!51 \nu^{14} + \cdots + 64\!\cdots\!21 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 25\!\cdots\!47 \nu^{14} + \cdots + 47\!\cdots\!02 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10\!\cdots\!25 \nu^{14} + \cdots + 20\!\cdots\!92 ) / 26\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 26\!\cdots\!18 \nu^{15} + \cdots + 29\!\cdots\!19 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 19\!\cdots\!98 \nu^{15} + \cdots - 15\!\cdots\!59 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 87\!\cdots\!46 \nu^{15} + \cdots - 83\!\cdots\!93 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 49\!\cdots\!86 \nu^{15} + \cdots - 20\!\cdots\!33 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 25\!\cdots\!98 \nu^{15} + \cdots + 46\!\cdots\!09 ) / 62\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 14\!\cdots\!14 \nu^{15} + \cdots - 85\!\cdots\!87 ) / 62\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 13\!\cdots\!94 \nu^{15} + \cdots - 20\!\cdots\!73 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 17\!\cdots\!86 \nu^{15} + \cdots + 17\!\cdots\!13 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( - 16 \beta_{15} - 10 \beta_{14} - 117 \beta_{13} - 100 \beta_{12} + 20 \beta_{11} - 518 \beta_{10} - 56 \beta_{9} + 66 \beta_{8} + 157464 ) / 314928 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 16 \beta_{15} - 10 \beta_{14} - 117 \beta_{13} - 100 \beta_{12} + 20 \beta_{11} - 518 \beta_{10} - 56 \beta_{9} + 66 \beta_{8} + 314 \beta_{7} - 403 \beta_{6} - 76 \beta_{5} - 278 \beta_{4} - 4227 \beta_{3} + \cdots - 214313849 ) / 314928 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 72508 \beta_{15} + 11806 \beta_{14} + 232097 \beta_{13} + 265768 \beta_{12} - 22536 \beta_{11} + 1337518 \beta_{10} + 205084 \beta_{9} - 499043510 \beta_{8} + 942 \beta_{7} + \cdots - 643099011 ) / 629856 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 36262 \beta_{15} + 5908 \beta_{14} + 116107 \beta_{13} + 132934 \beta_{12} - 11278 \beta_{11} + 669018 \beta_{10} + 102570 \beta_{9} - 249521788 \beta_{8} - 303946 \beta_{7} + \cdots + 139158208177 ) / 157464 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 121812248 \beta_{15} - 12256190 \beta_{14} - 344160507 \beta_{13} - 389341340 \beta_{12} + 33754756 \beta_{11} + 1319886182 \beta_{10} + \cdots + 1392653965943 ) / 629856 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 91449845 \beta_{15} - 9206915 \beta_{14} - 258410677 \beta_{13} - 292338365 \beta_{12} + 25344267 \beta_{11} + 988241962 \beta_{10} - 234314411 \beta_{9} + \cdots - 205976109643350 ) / 157464 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 100682915762 \beta_{15} + 9599456816 \beta_{14} + 280924740887 \beta_{13} + 290321294390 \beta_{12} - 31554061922 \beta_{11} - 2977513261590 \beta_{10} + \cdots - 14\!\cdots\!84 ) / 314928 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 100896341042 \beta_{15} + 9620946512 \beta_{14} + 281527834611 \beta_{13} + 291003572348 \beta_{12} - 31613211706 \beta_{11} - 2979818378894 \beta_{10} + \cdots + 15\!\cdots\!39 ) / 78732 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 332517498718712 \beta_{15} - 34070366075294 \beta_{14} - 938719408801003 \beta_{13} - 876421381607660 \beta_{12} + 116150616079620 \beta_{11} + \cdots + 57\!\cdots\!73 ) / 629856 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 834321917688514 \beta_{15} - 85464672539500 \beta_{14} + \cdots - 98\!\cdots\!07 ) / 314928 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 54\!\cdots\!36 \beta_{15} + \cdots - 10\!\cdots\!01 ) / 629856 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 41\!\cdots\!85 \beta_{15} + \cdots + 38\!\cdots\!21 ) / 78732 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 45\!\cdots\!60 \beta_{15} + \cdots + 10\!\cdots\!42 ) / 314928 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 31\!\cdots\!80 \beta_{15} + \cdots - 24\!\cdots\!91 ) / 314928 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 14\!\cdots\!92 \beta_{15} + \cdots - 37\!\cdots\!55 ) / 629856 \) Copy content Toggle raw display

Character values

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

\(n\) \(83\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
0.500000 22.4281i
0.500000 35.1177i
0.500000 + 40.7320i
0.500000 + 39.6902i
0.500000 + 9.67409i
0.500000 6.78035i
0.500000 14.2466i
0.500000 11.5235i
0.500000 + 11.5235i
0.500000 + 14.2466i
0.500000 + 6.78035i
0.500000 9.67409i
0.500000 39.6902i
0.500000 40.7320i
0.500000 + 35.1177i
0.500000 + 22.4281i
11.3137i 0 −128.000 1090.86i 0 −2501.39 1448.15i 0 −12341.6
161.2 11.3137i 0 −128.000 789.056i 0 −179.426 1448.15i 0 −8927.15
161.3 11.3137i 0 −128.000 549.748i 0 3262.72 1448.15i 0 −6219.69
161.4 11.3137i 0 −128.000 80.4359i 0 −741.697 1448.15i 0 910.029
161.5 11.3137i 0 −128.000 132.841i 0 2120.65 1448.15i 0 1502.93
161.6 11.3137i 0 −128.000 306.059i 0 −1540.21 1448.15i 0 3462.67
161.7 11.3137i 0 −128.000 830.391i 0 −2686.82 1448.15i 0 9394.80
161.8 11.3137i 0 −128.000 1079.93i 0 4112.18 1448.15i 0 12218.1
161.9 11.3137i 0 −128.000 1079.93i 0 4112.18 1448.15i 0 12218.1
161.10 11.3137i 0 −128.000 830.391i 0 −2686.82 1448.15i 0 9394.80
161.11 11.3137i 0 −128.000 306.059i 0 −1540.21 1448.15i 0 3462.67
161.12 11.3137i 0 −128.000 132.841i 0 2120.65 1448.15i 0 1502.93
161.13 11.3137i 0 −128.000 80.4359i 0 −741.697 1448.15i 0 910.029
161.14 11.3137i 0 −128.000 549.748i 0 3262.72 1448.15i 0 −6219.69
161.15 11.3137i 0 −128.000 789.056i 0 −179.426 1448.15i 0 −8927.15
161.16 11.3137i 0 −128.000 1090.86i 0 −2501.39 1448.15i 0 −12341.6
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.9.b.c 16
3.b odd 2 1 inner 162.9.b.c 16
9.c even 3 1 18.9.d.a 16
9.c even 3 1 54.9.d.a 16
9.d odd 6 1 18.9.d.a 16
9.d odd 6 1 54.9.d.a 16
36.f odd 6 1 144.9.q.b 16
36.f odd 6 1 432.9.q.c 16
36.h even 6 1 144.9.q.b 16
36.h even 6 1 432.9.q.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.9.d.a 16 9.c even 3 1
18.9.d.a 16 9.d odd 6 1
54.9.d.a 16 9.c even 3 1
54.9.d.a 16 9.d odd 6 1
144.9.q.b 16 36.f odd 6 1
144.9.q.b 16 36.h even 6 1
162.9.b.c 16 1.a even 1 1 trivial
162.9.b.c 16 3.b odd 2 1 inner
432.9.q.c 16 36.f odd 6 1
432.9.q.c 16 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 4088394 T_{5}^{14} + 6487601526567 T_{5}^{12} + \cdots + 19\!\cdots\!00 \) acting on \(S_{9}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 128)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 4088394 T^{14} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{8} - 1846 T^{7} + \cdots - 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 1882323504 T^{14} + \cdots + 52\!\cdots\!49 \) Copy content Toggle raw display
$13$ \( (T^{8} - 3370 T^{7} + \cdots - 47\!\cdots\!04)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 42263005590 T^{14} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{8} - 181090 T^{7} + \cdots - 11\!\cdots\!40)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 1006052775498 T^{14} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{16} + 4830634406538 T^{14} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{8} + 542438 T^{7} + \cdots + 44\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 1671664 T^{7} + \cdots - 17\!\cdots\!56)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + 62077768541496 T^{14} + \cdots + 24\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( (T^{8} + 339512 T^{7} + \cdots + 48\!\cdots\!65)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 133543775771946 T^{14} + \cdots + 18\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{16} + 337987454025744 T^{14} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( (T^{8} - 841954 T^{7} + \cdots + 10\!\cdots\!96)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 29946644 T^{7} + \cdots - 79\!\cdots\!95)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 26\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( (T^{8} + 3773882 T^{7} + \cdots + 79\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 33813002 T^{7} + \cdots - 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 27\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{8} - 89415484 T^{7} + \cdots + 15\!\cdots\!25)^{2} \) Copy content Toggle raw display
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