Properties

Label 18.9.b.b
Level $18$
Weight $9$
Character orbit 18.b
Analytic conductor $7.333$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [18,9,Mod(17,18)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(18, 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("18.17");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 18.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: \(7.33281498110\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 8 \beta q^{2} - 128 q^{4} - 645 \beta q^{5} + 1652 q^{7} - 1024 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + 8 \beta q^{2} - 128 q^{4} - 645 \beta q^{5} + 1652 q^{7} - 1024 \beta q^{8} + 10320 q^{10} - 9732 \beta q^{11} + 46304 q^{13} + 13216 \beta q^{14} + 16384 q^{16} - 77967 \beta q^{17} - 243664 q^{19} + 82560 \beta q^{20} + 155712 q^{22} + 101316 \beta q^{23} - 441425 q^{25} + 370432 \beta q^{26} - 211456 q^{28} - 215787 \beta q^{29} + 384164 q^{31} + 131072 \beta q^{32} + 1247472 q^{34} - 1065540 \beta q^{35} + 496982 q^{37} - 1949312 \beta q^{38} - 1320960 q^{40} + 712767 \beta q^{41} + 5334440 q^{43} + 1245696 \beta q^{44} - 1621056 q^{46} + 4563060 \beta q^{47} - 3035697 q^{49} - 3531400 \beta q^{50} - 5926912 q^{52} + 1915029 \beta q^{53} - 12554280 q^{55} - 1691648 \beta q^{56} + 3452592 q^{58} + 8799384 \beta q^{59} + 2335370 q^{61} + 3073312 \beta q^{62} - 2097152 q^{64} - 29866080 \beta q^{65} + 30674456 q^{67} + 9979776 \beta q^{68} + 17048640 q^{70} + 8613036 \beta q^{71} - 11519728 q^{73} + 3975856 \beta q^{74} + 31188992 q^{76} - 16077264 \beta q^{77} - 2658244 q^{79} - 10567680 \beta q^{80} - 11404272 q^{82} - 36660252 \beta q^{83} - 100577430 q^{85} + 42675520 \beta q^{86} - 19931136 q^{88} + 27337599 \beta q^{89} + 76494208 q^{91} - 12968448 \beta q^{92} - 73008960 q^{94} + 157163280 \beta q^{95} - 51595168 q^{97} - 24285576 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 256 q^{4} + 3304 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 256 q^{4} + 3304 q^{7} + 20640 q^{10} + 92608 q^{13} + 32768 q^{16} - 487328 q^{19} + 311424 q^{22} - 882850 q^{25} - 422912 q^{28} + 768328 q^{31} + 2494944 q^{34} + 993964 q^{37} - 2641920 q^{40} + 10668880 q^{43} - 3242112 q^{46} - 6071394 q^{49} - 11853824 q^{52} - 25108560 q^{55} + 6905184 q^{58} + 4670740 q^{61} - 4194304 q^{64} + 61348912 q^{67} + 34097280 q^{70} - 23039456 q^{73} + 62377984 q^{76} - 5316488 q^{79} - 22808544 q^{82} - 201154860 q^{85} - 39862272 q^{88} + 152988416 q^{91} - 146017920 q^{94} - 103190336 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).

\(n\) \(11\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
1.41421i
1.41421i
11.3137i 0 −128.000 912.168i 0 1652.00 1448.15i 0 10320.0
17.2 11.3137i 0 −128.000 912.168i 0 1652.00 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 18.9.b.b 2
3.b odd 2 1 inner 18.9.b.b 2
4.b odd 2 1 144.9.e.b 2
5.b even 2 1 450.9.d.a 2
5.c odd 4 2 450.9.b.b 4
9.c even 3 2 162.9.d.b 4
9.d odd 6 2 162.9.d.b 4
12.b even 2 1 144.9.e.b 2
15.d odd 2 1 450.9.d.a 2
15.e even 4 2 450.9.b.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.9.b.b 2 1.a even 1 1 trivial
18.9.b.b 2 3.b odd 2 1 inner
144.9.e.b 2 4.b odd 2 1
144.9.e.b 2 12.b even 2 1
162.9.d.b 4 9.c even 3 2
162.9.d.b 4 9.d odd 6 2
450.9.b.b 4 5.c odd 4 2
450.9.b.b 4 15.e even 4 2
450.9.d.a 2 5.b even 2 1
450.9.d.a 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 832050 \) acting on \(S_{9}^{\mathrm{new}}(18, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 128 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 832050 \) Copy content Toggle raw display
$7$ \( (T - 1652)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 189423648 \) Copy content Toggle raw display
$13$ \( (T - 46304)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 12157706178 \) Copy content Toggle raw display
$19$ \( (T + 243664)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 20529863712 \) Copy content Toggle raw display
$29$ \( T^{2} + 93128058738 \) Copy content Toggle raw display
$31$ \( (T - 384164)^{2} \) Copy content Toggle raw display
$37$ \( (T - 496982)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 1016073592578 \) Copy content Toggle raw display
$43$ \( (T - 5334440)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 41643033127200 \) Copy content Toggle raw display
$53$ \( T^{2} + 7334672141682 \) Copy content Toggle raw display
$59$ \( T^{2} + 154858317558912 \) Copy content Toggle raw display
$61$ \( (T - 2335370)^{2} \) Copy content Toggle raw display
$67$ \( (T - 30674456)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 148368778274592 \) Copy content Toggle raw display
$73$ \( (T + 11519728)^{2} \) Copy content Toggle raw display
$79$ \( (T + 2658244)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 26\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{2} + 14\!\cdots\!02 \) Copy content Toggle raw display
$97$ \( (T + 51595168)^{2} \) Copy content Toggle raw display
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