Properties

Label 9.8.a.b
Level $9$
Weight $8$
Character orbit 9.a
Self dual yes
Analytic conductor $2.811$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9,8,Mod(1,9)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

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: yes
Analytic conductor: \(2.81146522936\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{10}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = 6\sqrt{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 232 q^{4} - 16 \beta q^{5} + 260 q^{7} + 104 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + 232 q^{4} - 16 \beta q^{5} + 260 q^{7} + 104 \beta q^{8} - 5760 q^{10} - 320 \beta q^{11} + 6890 q^{13} + 260 \beta q^{14} + 7744 q^{16} + 1248 \beta q^{17} + 33176 q^{19} - 3712 \beta q^{20} - 115200 q^{22} + 1664 \beta q^{23} + 14035 q^{25} + 6890 \beta q^{26} + 60320 q^{28} - 7280 \beta q^{29} + 1508 q^{31} - 5568 \beta q^{32} + 449280 q^{34} - 4160 \beta q^{35} - 380770 q^{37} + 33176 \beta q^{38} - 599040 q^{40} + 4640 \beta q^{41} + 7640 q^{43} - 74240 \beta q^{44} + 599040 q^{46} + 29824 \beta q^{47} - 755943 q^{49} + 14035 \beta q^{50} + 1598480 q^{52} + 54288 \beta q^{53} + 1843200 q^{55} + 27040 \beta q^{56} - 2620800 q^{58} - 142720 \beta q^{59} - 988858 q^{61} + 1508 \beta q^{62} - 2995712 q^{64} - 110240 \beta q^{65} + 3857360 q^{67} + 289536 \beta q^{68} - 1497600 q^{70} + 222720 \beta q^{71} - 2004730 q^{73} - 380770 \beta q^{74} + 7696832 q^{76} - 83200 \beta q^{77} + 2699684 q^{79} - 123904 \beta q^{80} + 1670400 q^{82} + 142912 \beta q^{83} - 7188480 q^{85} + 7640 \beta q^{86} - 11980800 q^{88} + 408000 \beta q^{89} + 1791400 q^{91} + 386048 \beta q^{92} + 10736640 q^{94} - 530816 \beta q^{95} - 12957490 q^{97} - 755943 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 464 q^{4} + 520 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 464 q^{4} + 520 q^{7} - 11520 q^{10} + 13780 q^{13} + 15488 q^{16} + 66352 q^{19} - 230400 q^{22} + 28070 q^{25} + 120640 q^{28} + 3016 q^{31} + 898560 q^{34} - 761540 q^{37} - 1198080 q^{40} + 15280 q^{43} + 1198080 q^{46} - 1511886 q^{49} + 3196960 q^{52} + 3686400 q^{55} - 5241600 q^{58} - 1977716 q^{61} - 5991424 q^{64} + 7714720 q^{67} - 2995200 q^{70} - 4009460 q^{73} + 15393664 q^{76} + 5399368 q^{79} + 3340800 q^{82} - 14376960 q^{85} - 23961600 q^{88} + 3582800 q^{91} + 21473280 q^{94} - 25914980 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.16228
3.16228
−18.9737 0 232.000 303.579 0 260.000 −1973.26 0 −5760.00
1.2 18.9737 0 232.000 −303.579 0 260.000 1973.26 0 −5760.00
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)

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 9.8.a.b 2
3.b odd 2 1 inner 9.8.a.b 2
4.b odd 2 1 144.8.a.m 2
5.b even 2 1 225.8.a.q 2
5.c odd 4 2 225.8.b.k 4
7.b odd 2 1 441.8.a.k 2
8.b even 2 1 576.8.a.bj 2
8.d odd 2 1 576.8.a.bi 2
9.c even 3 2 81.8.c.f 4
9.d odd 6 2 81.8.c.f 4
12.b even 2 1 144.8.a.m 2
15.d odd 2 1 225.8.a.q 2
15.e even 4 2 225.8.b.k 4
21.c even 2 1 441.8.a.k 2
24.f even 2 1 576.8.a.bi 2
24.h odd 2 1 576.8.a.bj 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.8.a.b 2 1.a even 1 1 trivial
9.8.a.b 2 3.b odd 2 1 inner
81.8.c.f 4 9.c even 3 2
81.8.c.f 4 9.d odd 6 2
144.8.a.m 2 4.b odd 2 1
144.8.a.m 2 12.b even 2 1
225.8.a.q 2 5.b even 2 1
225.8.a.q 2 15.d odd 2 1
225.8.b.k 4 5.c odd 4 2
225.8.b.k 4 15.e even 4 2
441.8.a.k 2 7.b odd 2 1
441.8.a.k 2 21.c even 2 1
576.8.a.bi 2 8.d odd 2 1
576.8.a.bi 2 24.f even 2 1
576.8.a.bj 2 8.b even 2 1
576.8.a.bj 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 360 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(9))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 360 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 92160 \) Copy content Toggle raw display
$7$ \( (T - 260)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 36864000 \) Copy content Toggle raw display
$13$ \( (T - 6890)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 560701440 \) Copy content Toggle raw display
$19$ \( (T - 33176)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 996802560 \) Copy content Toggle raw display
$29$ \( T^{2} - 19079424000 \) Copy content Toggle raw display
$31$ \( (T - 1508)^{2} \) Copy content Toggle raw display
$37$ \( (T + 380770)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 7750656000 \) Copy content Toggle raw display
$43$ \( (T - 7640)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 320209551360 \) Copy content Toggle raw display
$53$ \( T^{2} - 1060987299840 \) Copy content Toggle raw display
$59$ \( T^{2} - 7332839424000 \) Copy content Toggle raw display
$61$ \( (T + 988858)^{2} \) Copy content Toggle raw display
$67$ \( (T - 3857360)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 17857511424000 \) Copy content Toggle raw display
$73$ \( (T + 2004730)^{2} \) Copy content Toggle raw display
$79$ \( (T - 2699684)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 7352582307840 \) Copy content Toggle raw display
$89$ \( T^{2} - 59927040000000 \) Copy content Toggle raw display
$97$ \( (T + 12957490)^{2} \) Copy content Toggle raw display
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