Properties

Label 3.9.b.a
Level $3$
Weight $9$
Character orbit 3.b
Analytic conductor $1.222$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3,9,Mod(2,3)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3, 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("3.2");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 3.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: \(1.22213583018\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-14}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3 \)
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 \(\beta = 6\sqrt{-14}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( - 3 \beta + 45) q^{3} - 248 q^{4} - 10 \beta q^{5} + (45 \beta + 1512) q^{6} - 1750 q^{7} + 8 \beta q^{8} + ( - 270 \beta - 2511) q^{9} + 5040 q^{10} + 310 \beta q^{11} + (744 \beta - 11160) q^{12} + \cdots + ( - 778410 \beta + 42184800) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 90 q^{3} - 496 q^{4} + 3024 q^{6} - 3500 q^{7} - 5022 q^{9} + 10080 q^{10} - 22320 q^{12} + 51460 q^{13} - 30240 q^{15} - 135040 q^{16} + 272160 q^{18} + 37876 q^{19} - 157500 q^{21} - 312480 q^{22}+ \cdots + 84369600 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
3.74166i
3.74166i
22.4499i 45.0000 + 67.3498i −248.000 224.499i 1512.00 1010.25i −1750.00 179.600i −2511.00 + 6061.48i 5040.00
2.2 22.4499i 45.0000 67.3498i −248.000 224.499i 1512.00 + 1010.25i −1750.00 179.600i −2511.00 6061.48i 5040.00
\(n\): 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.9.b.a 2
3.b odd 2 1 inner 3.9.b.a 2
4.b odd 2 1 48.9.e.b 2
5.b even 2 1 75.9.c.c 2
5.c odd 4 2 75.9.d.b 4
8.b even 2 1 192.9.e.e 2
8.d odd 2 1 192.9.e.f 2
9.c even 3 2 81.9.d.d 4
9.d odd 6 2 81.9.d.d 4
12.b even 2 1 48.9.e.b 2
15.d odd 2 1 75.9.c.c 2
15.e even 4 2 75.9.d.b 4
24.f even 2 1 192.9.e.f 2
24.h odd 2 1 192.9.e.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.9.b.a 2 1.a even 1 1 trivial
3.9.b.a 2 3.b odd 2 1 inner
48.9.e.b 2 4.b odd 2 1
48.9.e.b 2 12.b even 2 1
75.9.c.c 2 5.b even 2 1
75.9.c.c 2 15.d odd 2 1
75.9.d.b 4 5.c odd 4 2
75.9.d.b 4 15.e even 4 2
81.9.d.d 4 9.c even 3 2
81.9.d.d 4 9.d odd 6 2
192.9.e.e 2 8.b even 2 1
192.9.e.e 2 24.h odd 2 1
192.9.e.f 2 8.d odd 2 1
192.9.e.f 2 24.f even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(3, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 504 \) Copy content Toggle raw display
$3$ \( T^{2} - 90T + 6561 \) Copy content Toggle raw display
$5$ \( T^{2} + 50400 \) Copy content Toggle raw display
$7$ \( (T + 1750)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 48434400 \) Copy content Toggle raw display
$13$ \( (T - 25730)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 5608963584 \) Copy content Toggle raw display
$19$ \( (T - 18938)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 221333583744 \) Copy content Toggle raw display
$29$ \( T^{2} + 212426373600 \) Copy content Toggle raw display
$31$ \( (T + 351478)^{2} \) Copy content Toggle raw display
$37$ \( (T - 1335170)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 3517381526400 \) Copy content Toggle raw display
$43$ \( (T + 3526150)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 16654893018624 \) Copy content Toggle raw display
$53$ \( T^{2} + 43583305427424 \) Copy content Toggle raw display
$59$ \( T^{2} + 188098358162400 \) Copy content Toggle raw display
$61$ \( (T - 753602)^{2} \) Copy content Toggle raw display
$67$ \( (T - 2268890)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 289748374473600 \) Copy content Toggle raw display
$73$ \( (T - 27672770)^{2} \) Copy content Toggle raw display
$79$ \( (T + 22980982)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 21\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{2} + 52\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T - 147271010)^{2} \) Copy content Toggle raw display
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