Properties

Label 234.6.a.h
Level $234$
Weight $6$
Character orbit 234.a
Self dual yes
Analytic conductor $37.530$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [234,6,Mod(1,234)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(234, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("234.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 234.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: \(37.5298138362\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{849}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 212 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 26)
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 \(\beta = \frac{1}{2}(-1 + 3\sqrt{849})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 q^{2} + 16 q^{4} + ( - \beta - 37) q^{5} + (3 \beta + 79) q^{7} - 64 q^{8} + (4 \beta + 148) q^{10} + (8 \beta + 114) q^{11} - 169 q^{13} + ( - 12 \beta - 316) q^{14} + 256 q^{16} + ( - 43 \beta + 73) q^{17}+ \cdots + ( - 1860 \beta - 26496) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} + 32 q^{4} - 73 q^{5} + 155 q^{7} - 128 q^{8} + 292 q^{10} + 220 q^{11} - 338 q^{13} - 620 q^{14} + 512 q^{16} + 189 q^{17} - 2496 q^{19} - 1168 q^{20} - 880 q^{22} + 3044 q^{23} + 235 q^{25}+ \cdots - 51132 q^{98}+O(q^{100}) \) 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
15.0688
−14.0688
−4.00000 0 16.0000 −80.2064 0 208.619 −64.0000 0 320.826
1.2 −4.00000 0 16.0000 7.20641 0 −53.6192 −64.0000 0 −28.8256
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)
\(13\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 234.6.a.h 2
3.b odd 2 1 26.6.a.c 2
12.b even 2 1 208.6.a.g 2
15.d odd 2 1 650.6.a.b 2
15.e even 4 2 650.6.b.h 4
24.f even 2 1 832.6.a.m 2
24.h odd 2 1 832.6.a.k 2
39.d odd 2 1 338.6.a.f 2
39.f even 4 2 338.6.b.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.6.a.c 2 3.b odd 2 1
208.6.a.g 2 12.b even 2 1
234.6.a.h 2 1.a even 1 1 trivial
338.6.a.f 2 39.d odd 2 1
338.6.b.b 4 39.f even 4 2
650.6.a.b 2 15.d odd 2 1
650.6.b.h 4 15.e even 4 2
832.6.a.k 2 24.h odd 2 1
832.6.a.m 2 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(234))\):

\( T_{5}^{2} + 73T_{5} - 578 \) Copy content Toggle raw display
\( T_{7}^{2} - 155T_{7} - 11186 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 73T - 578 \) Copy content Toggle raw display
$7$ \( T^{2} - 155T - 11186 \) Copy content Toggle raw display
$11$ \( T^{2} - 220T - 110156 \) Copy content Toggle raw display
$13$ \( (T + 169)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 189 T - 3523122 \) Copy content Toggle raw display
$19$ \( T^{2} + 2496 T + 793404 \) Copy content Toggle raw display
$23$ \( T^{2} - 3044 T - 159200 \) Copy content Toggle raw display
$29$ \( T^{2} + 1900 T - 13890476 \) Copy content Toggle raw display
$31$ \( T^{2} - 2798 T - 28369928 \) Copy content Toggle raw display
$37$ \( T^{2} - 17805 T + 72604926 \) Copy content Toggle raw display
$41$ \( T^{2} + 11634 T - 11466000 \) Copy content Toggle raw display
$43$ \( T^{2} + 4069 T - 6040532 \) Copy content Toggle raw display
$47$ \( T^{2} - 25489 T + 127607974 \) Copy content Toggle raw display
$53$ \( T^{2} - 4614 T - 129839400 \) Copy content Toggle raw display
$59$ \( T^{2} - 23420 T + 92989684 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 2309711776 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 1295720044 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 1862988962 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 1462893860 \) Copy content Toggle raw display
$79$ \( T^{2} - 52024 T + 439936528 \) Copy content Toggle raw display
$83$ \( T^{2} - 37758 T + 83472480 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 1072134396 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 4229773940 \) Copy content Toggle raw display
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