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Properties

Label 2184.2.bj.k
Level $2184$
Weight $2$
Character orbit 2184.bj
Analytic conductor $17.439$
Analytic rank $0$
Dimension $8$
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] = [2184,2,Mod(841,2184)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2184, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2184.841");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2184 = 2^{3} \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2184.bj (of order \(3\), degree \(2\), 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: \(17.4393278014\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.8248090761.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{6} - 2x^{5} + 41x^{4} - 7x^{3} + 57x^{2} + 8x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} - \beta_{5} q^{5} + (\beta_{2} - 1) q^{7} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} - \beta_{5} q^{5} + (\beta_{2} - 1) q^{7} + (\beta_{2} - 1) q^{9} + (\beta_{6} + \beta_{4} + \beta_1) q^{11} + (\beta_{7} + \beta_{5} - \beta_{4} + \beta_1) q^{13} + \beta_1 q^{15} + \beta_{7} q^{17} + (2 \beta_{5} - \beta_{2} + 2 \beta_1 + 1) q^{19} - q^{21} + (\beta_{6} + \beta_{2} - \beta_1) q^{23} + (\beta_{3} - 2) q^{25} - q^{27} + (\beta_{4} - \beta_{2} - 2 \beta_1) q^{29} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \cdots - 4) q^{31}+ \cdots + (\beta_{7} - \beta_{6} + \beta_{5} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} - 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} - 4 q^{7} - 4 q^{9} - 3 q^{11} + 3 q^{13} + q^{17} + 4 q^{19} - 8 q^{21} + 3 q^{23} - 12 q^{25} - 8 q^{27} - 6 q^{29} - 38 q^{31} + 3 q^{33} - 12 q^{37} - 9 q^{41} + q^{43} + 28 q^{47} - 4 q^{49} + 2 q^{51} - 16 q^{53} + 4 q^{55} + 8 q^{57} + 5 q^{59} - 8 q^{61} - 4 q^{63} - 4 q^{65} + 15 q^{67} - 3 q^{69} + 20 q^{71} + 48 q^{73} - 6 q^{75} + 6 q^{77} - 4 q^{79} - 4 q^{81} - 8 q^{83} + 7 q^{85} + 6 q^{87} + 30 q^{89} - 3 q^{91} - 19 q^{93} - 28 q^{95} + 9 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 7x^{6} - 2x^{5} + 41x^{4} - 7x^{3} + 57x^{2} + 8x + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -343\nu^{7} + 2296\nu^{6} - 2009\nu^{5} + 14134\nu^{4} - 16359\nu^{3} + 96145\nu^{2} - 16927\nu + 128576 ) / 110328 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -49\nu^{7} + 328\nu^{6} - 287\nu^{5} + 49\nu^{4} - 2337\nu^{3} - 56\nu^{2} - 448\nu - 23005 ) / 13791 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 245\nu^{7} - 1640\nu^{6} + 1435\nu^{5} - 14036\nu^{4} + 11685\nu^{3} - 68675\nu^{2} + 16031\nu - 91840 ) / 27582 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 287\nu^{7} + 49\nu^{6} + 1681\nu^{5} - 287\nu^{4} + 11718\nu^{3} + 328\nu^{2} + 2624\nu + 2744 ) / 13791 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1087\nu^{7} - 1272\nu^{6} + 1113\nu^{5} - 14878\nu^{4} + 9063\nu^{3} - 53265\nu^{2} - 118121\nu - 71232 ) / 36776 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2609 \nu^{7} - 2800 \nu^{6} - 20535 \nu^{5} - 11182 \nu^{4} - 104169 \nu^{3} - 57489 \nu^{2} + \cdots - 46472 ) / 36776 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} + 4\beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{6} + 5\beta_{5} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{4} - 20\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{7} - 27\beta_{5} + 8\beta_{4} - 8\beta_{3} + 11\beta_{2} - 27\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{7} - \beta_{6} + 12\beta_{5} + 42\beta_{3} + 67 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 41\beta_{6} - 55\beta_{4} - 89\beta_{2} + 150\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2184\mathbb{Z}\right)^\times\).

\(n\) \(1093\) \(1249\) \(1457\) \(1639\) \(2017\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1\) \(-1 + \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
841.1
−1.23409 + 2.13751i
−0.545220 + 0.944349i
0.669692 − 1.15994i
1.10962 − 1.92192i
−1.23409 − 2.13751i
−0.545220 − 0.944349i
0.669692 + 1.15994i
1.10962 + 1.92192i
0 0.500000 − 0.866025i 0 −2.46819 0 −0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0
841.2 0 0.500000 − 0.866025i 0 −1.09044 0 −0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0
841.3 0 0.500000 − 0.866025i 0 1.33938 0 −0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0
841.4 0 0.500000 − 0.866025i 0 2.21924 0 −0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0
1849.1 0 0.500000 + 0.866025i 0 −2.46819 0 −0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0
1849.2 0 0.500000 + 0.866025i 0 −1.09044 0 −0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0
1849.3 0 0.500000 + 0.866025i 0 1.33938 0 −0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0
1849.4 0 0.500000 + 0.866025i 0 2.21924 0 −0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 841.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2184.2.bj.k ✓ 8
13.c even 3 1 inner 2184.2.bj.k ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2184.2.bj.k ✓ 8 1.a even 1 1 trivial
2184.2.bj.k ✓ 8 13.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2184, [\chi])\):

\( T_{5}^{4} - 7T_{5}^{2} + T_{5} + 8 \) Copy content Toggle raw display
\( T_{11}^{8} + 3T_{11}^{7} + 38T_{11}^{6} + 9T_{11}^{5} + 984T_{11}^{4} + 1386T_{11}^{3} + 2333T_{11}^{2} - 48T_{11} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} - 7 T^{2} + T + 8)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} + 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{8} - 3 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} - T^{7} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{8} - 4 T^{7} + \cdots + 11881 \) Copy content Toggle raw display
$23$ \( T^{8} - 3 T^{7} + \cdots + 2916 \) Copy content Toggle raw display
$29$ \( T^{8} + 6 T^{7} + \cdots + 68121 \) Copy content Toggle raw display
$31$ \( (T^{4} + 19 T^{3} + \cdots - 324)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 12 T^{7} + \cdots + 1296 \) Copy content Toggle raw display
$41$ \( T^{8} + 9 T^{7} + \cdots + 97344 \) Copy content Toggle raw display
$43$ \( T^{8} - T^{7} + \cdots + 44944 \) Copy content Toggle raw display
$47$ \( (T^{4} - 14 T^{3} + \cdots - 7659)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 8 T^{3} + \cdots - 2551)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 5 T^{7} + \cdots + 334084 \) Copy content Toggle raw display
$61$ \( T^{8} + 8 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$67$ \( T^{8} - 15 T^{7} + \cdots + 487204 \) Copy content Toggle raw display
$71$ \( T^{8} - 20 T^{7} + \cdots + 236196 \) Copy content Toggle raw display
$73$ \( (T^{4} - 24 T^{3} + \cdots - 912)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 2 T^{3} + \cdots + 403)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 4 T^{3} + \cdots + 18642)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 30 T^{7} + \cdots + 400680289 \) Copy content Toggle raw display
$97$ \( T^{8} - 9 T^{7} + \cdots + 5683456 \) Copy content Toggle raw display
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