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Properties

Label 184.5.e.d
Level $184$
Weight $5$
Character orbit 184.e
Analytic conductor $19.020$
Analytic rank $0$
Dimension $6$
CM discriminant -23
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [184,5,Mod(45,184)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(184, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("184.45");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 184 = 2^{3} \cdot 23 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 184.e (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: \(19.0200732074\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.8869743.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{3} + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_{4} + \beta_1) q^{2} + (6 \beta_{5} - 3 \beta_{4} + \cdots - 4 \beta_1) q^{3}+ \cdots + (22 \beta_{5} - 36 \beta_{4} + \cdots - 81) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - \beta_{4} + \beta_1) q^{2} + (6 \beta_{5} - 3 \beta_{4} + \cdots - 4 \beta_1) q^{3}+ \cdots + ( - 2401 \beta_{5} - 2401 \beta_{4} + 2401 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 147 q^{6} + 237 q^{8} - 486 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 147 q^{6} + 237 q^{8} - 486 q^{9} - 861 q^{12} + 1437 q^{18} - 3174 q^{23} - 3750 q^{25} + 1533 q^{26} - 573 q^{36} - 17076 q^{39} - 2163 q^{48} + 14406 q^{49} - 15603 q^{52} - 11907 q^{54} + 4659 q^{58} + 7539 q^{62} - 5853 q^{64} - 19197 q^{72} + 16563 q^{78} + 39366 q^{81} - 34797 q^{82} + 50484 q^{87} + 51933 q^{94} - 30387 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{3} + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + \nu^{2} + 4\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{4} + 3\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} - \nu^{2} + 4\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + 3\nu^{2} ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} - \beta_{4} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{4} - 4\beta_{3} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -2\beta_{5} - 3\beta_{4} + 3\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/184\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(93\) \(97\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1
−0.261988 − 1.38973i
−0.261988 + 1.38973i
1.33454 − 0.467979i
1.33454 + 0.467979i
−1.07255 + 0.921756i
−1.07255 − 0.921756i
−2.93948 − 2.71283i 17.9999i 1.28113 + 15.9486i 0 48.8306 − 52.9104i 0 39.5000 − 50.3562i −242.997 0
45.2 −2.93948 + 2.71283i − 17.9999i 1.28113 − 15.9486i 0 48.8306 + 52.9104i 0 39.5000 + 50.3562i −242.997 0
45.3 −0.879635 − 3.90208i 9.04985i −14.4525 + 6.86482i 0 35.3133 − 7.96057i 0 39.5000 + 50.3562i −0.899768 0
45.4 −0.879635 + 3.90208i − 9.04985i −14.4525 − 6.86482i 0 35.3133 + 7.96057i 0 39.5000 − 50.3562i −0.899768 0
45.5 3.81912 − 1.18925i − 8.95006i 13.1713 − 9.08381i 0 −10.6439 − 34.1813i 0 39.5000 − 50.3562i 0.896448 0
45.6 3.81912 + 1.18925i 8.95006i 13.1713 + 9.08381i 0 −10.6439 + 34.1813i 0 39.5000 + 50.3562i 0.896448 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 45.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 CM by \(\Q(\sqrt{-23}) \)
8.b even 2 1 inner
184.e odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 184.5.e.d ✓ 6
4.b odd 2 1 736.5.e.d 6
8.b even 2 1 inner 184.5.e.d ✓ 6
8.d odd 2 1 736.5.e.d 6
23.b odd 2 1 CM 184.5.e.d ✓ 6
92.b even 2 1 736.5.e.d 6
184.e odd 2 1 inner 184.5.e.d ✓ 6
184.h even 2 1 736.5.e.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
184.5.e.d ✓ 6 1.a even 1 1 trivial
184.5.e.d ✓ 6 8.b even 2 1 inner
184.5.e.d ✓ 6 23.b odd 2 1 CM
184.5.e.d ✓ 6 184.e odd 2 1 inner
736.5.e.d 6 4.b odd 2 1
736.5.e.d 6 8.d odd 2 1
736.5.e.d 6 92.b even 2 1
736.5.e.d 6 184.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(184, [\chi])\):

\( T_{3}^{6} + 486T_{3}^{4} + 59049T_{3}^{2} + 2125568 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 79T^{3} + 4096 \) Copy content Toggle raw display
$3$ \( T^{6} + 486 T^{4} + \cdots + 2125568 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 21232849874688 \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( (T + 529)^{6} \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 13\!\cdots\!88 \) Copy content Toggle raw display
$31$ \( (T^{3} - 2770563 T + 1677025154)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} \) Copy content Toggle raw display
$41$ \( (T^{3} - 8477283 T - 7596282526)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} \) Copy content Toggle raw display
$47$ \( (T^{3} - 14639043 T - 20606906306)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} \) Copy content Toggle raw display
$59$ \( (T^{2} + 8955648)^{3} \) Copy content Toggle raw display
$61$ \( T^{6} \) Copy content Toggle raw display
$67$ \( T^{6} \) Copy content Toggle raw display
$71$ \( (T^{3} - 76235043 T - 188893891874)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 85194723 T + 223017449186)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} \) Copy content Toggle raw display
$83$ \( T^{6} \) Copy content Toggle raw display
$89$ \( T^{6} \) Copy content Toggle raw display
$97$ \( T^{6} \) Copy content Toggle raw display
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