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Properties

Label 324.6.e.j
Level $324$
Weight $6$
Character orbit 324.e
Analytic conductor $51.964$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [324,6,Mod(109,324)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(324, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("324.109");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 324.e (of order \(3\), degree \(2\), not 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: \(51.9643576194\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.1421970391296.10
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 45x^{6} + 1496x^{4} + 23805x^{2} + 279841 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \beta_{5} - \beta_{4}) q^{5} + ( - \beta_{3} + 44 \beta_1 - 44) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_{5} - \beta_{4}) q^{5} + ( - \beta_{3} + 44 \beta_1 - 44) q^{7} + ( - 5 \beta_{7} + 11 \beta_{5} + 11 \beta_{2}) q^{11} + (7 \beta_{6} - 7 \beta_{3} + 343 \beta_1) q^{13} + (22 \beta_{7} - 22 \beta_{4} - 33 \beta_{2}) q^{17} + ( - 13 \beta_{6} - 736) q^{19} + ( - 67 \beta_{5} + 25 \beta_{4}) q^{23} + ( - 30 \beta_{3} + 826 \beta_1 - 826) q^{25} + ( - 29 \beta_{7} + \cdots + 142 \beta_{2}) q^{29}+ \cdots + (76 \beta_{3} - 82966 \beta_1 + 82966) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 176 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 176 q^{7} + 1372 q^{13} - 5888 q^{19} - 3304 q^{25} + 4144 q^{31} + 3352 q^{37} + 38320 q^{43} + 20172 q^{49} - 161280 q^{55} + 120196 q^{61} + 107296 q^{67} - 184760 q^{73} - 5024 q^{79} + 329868 q^{85} - 671104 q^{91} + 331864 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 45x^{6} + 1496x^{4} + 23805x^{2} + 279841 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 45\nu^{6} + 1496\nu^{4} + 67320\nu^{2} + 1071225 ) / 791384 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9\nu^{7} - 4356\nu^{5} - 91278\nu^{3} - 2090079\nu ) / 267674 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 273\nu^{6} - 201960\nu^{4} - 4339896\nu^{2} - 100338075 ) / 395692 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1179\nu^{7} + 197472\nu^{5} + 4137936\nu^{3} + 298719423\nu ) / 18201832 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13059\nu^{7} + 592416\nu^{5} + 12413808\nu^{3} + 154175463\nu ) / 18201832 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -12\nu^{6} - 540\nu^{4} - 11604\nu^{2} - 142830 ) / 529 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19743\nu^{7} + 709104\nu^{5} + 24787224\nu^{3} + 365519427\nu ) / 18201832 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{5} + 9\beta_{4} + 4\beta_{2} ) / 108 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{6} - \beta_{3} + 810\beta _1 - 810 ) / 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 99\beta_{7} - 136\beta_{5} - 99\beta_{4} - 101\beta_{2} ) / 54 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{6} - 5\beta_{3} - 1934\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -4149\beta_{7} + 6445\beta_{5} - 3679\beta_{2} ) / 108 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -374\beta_{6} + 748\beta_{3} + 88695 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 128525\beta_{5} + 81963\beta_{4} + 311692\beta_{2} ) / 108 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\) \(163\) \(245\)
\(\chi(n)\) \(1\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
2.81786 − 3.88068i
1.95184 − 4.38068i
−1.95184 + 4.38068i
−2.81786 + 3.88068i
2.81786 + 3.88068i
1.95184 + 4.38068i
−1.95184 − 4.38068i
−2.81786 − 3.88068i
0 0 0 −41.6086 + 72.0681i 0 −71.5681 − 123.960i 0 0 0
109.2 0 0 0 −15.6278 + 27.0681i 0 27.5681 + 47.7494i 0 0 0
109.3 0 0 0 15.6278 − 27.0681i 0 27.5681 + 47.7494i 0 0 0
109.4 0 0 0 41.6086 − 72.0681i 0 −71.5681 − 123.960i 0 0 0
217.1 0 0 0 −41.6086 − 72.0681i 0 −71.5681 + 123.960i 0 0 0
217.2 0 0 0 −15.6278 − 27.0681i 0 27.5681 − 47.7494i 0 0 0
217.3 0 0 0 15.6278 + 27.0681i 0 27.5681 − 47.7494i 0 0 0
217.4 0 0 0 41.6086 + 72.0681i 0 −71.5681 + 123.960i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.6.e.j 8
3.b odd 2 1 inner 324.6.e.j 8
9.c even 3 1 324.6.a.c ✓ 4
9.c even 3 1 inner 324.6.e.j 8
9.d odd 6 1 324.6.a.c ✓ 4
9.d odd 6 1 inner 324.6.e.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.6.a.c ✓ 4 9.c even 3 1
324.6.a.c ✓ 4 9.d odd 6 1
324.6.e.j 8 1.a even 1 1 trivial
324.6.e.j 8 3.b odd 2 1 inner
324.6.e.j 8 9.c even 3 1 inner
324.6.e.j 8 9.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(324, [\chi])\):

\( T_{5}^{8} + 7902T_{5}^{6} + 55676403T_{5}^{4} + 53458618302T_{5}^{2} + 45767944570401 \) Copy content Toggle raw display
\( T_{7}^{4} + 88T_{7}^{3} + 15636T_{7}^{2} - 694496T_{7} + 62283664 \) Copy content Toggle raw display
\( T_{11}^{8} + 206136T_{11}^{6} + 38803674672T_{11}^{4} + 760307038856064T_{11}^{2} + 13604116219067678976 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 45767944570401 \) Copy content Toggle raw display
$7$ \( (T^{4} + 88 T^{3} + \cdots + 62283664)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( (T^{4} - 686 T^{3} + \cdots + 132439949929)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 2032053101001)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 1472 T - 1119236)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 21\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 50\!\cdots\!41 \) Copy content Toggle raw display
$31$ \( (T^{4} - 2072 T^{3} + \cdots + 517662982144)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 838 T - 52197851)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 33\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 20682739517584)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 51\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 21\!\cdots\!64)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 25\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 16\!\cdots\!01)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 48\!\cdots\!04)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 11\!\cdots\!36)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 46190 T - 281794607)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 66\!\cdots\!04)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 13\!\cdots\!69)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 46\!\cdots\!84)^{2} \) Copy content Toggle raw display
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