Properties

Label 155.2.e.d
Level $155$
Weight $2$
Character orbit 155.e
Analytic conductor $1.238$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [155,2,Mod(36,155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(155, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("155.36");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 155 = 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 155.e (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: \(1.23768123133\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.42575625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 4x^{6} + x^{5} + 9x^{4} - x^{3} + 4x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_{4} + 1) q^{2} + (\beta_{7} - \beta_{4} - \beta_{3} - \beta_1) q^{3} + (\beta_{7} + \beta_{4} - \beta_{2}) q^{4} + \beta_{6} q^{5} + ( - \beta_{5} - \beta_{2} - 1) q^{6} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \cdots - 1) q^{7}+ \cdots + (2 \beta_{7} + 3 \beta_{5} + 9 \beta_{4} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - q^{3} - 2 q^{4} - 4 q^{5} - 2 q^{6} + q^{7} + 6 q^{8} - 5 q^{9} - q^{10} + 4 q^{11} - q^{13} - 8 q^{14} + 2 q^{15} - 6 q^{16} - 8 q^{17} + q^{18} + 3 q^{19} + q^{20} - 3 q^{21} - 8 q^{22}+ \cdots - 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 4x^{6} + x^{5} + 9x^{4} - x^{3} + 4x^{2} + x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{7} + \nu^{6} + 10\nu^{5} + 8\nu^{4} + 46\nu^{3} + 5\nu^{2} + 2\nu - 36 ) / 25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{7} - 2\nu^{6} + 5\nu^{5} + 19\nu^{4} + 8\nu^{3} - 10\nu^{2} - 39\nu - 3 ) / 25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6\nu^{7} - 3\nu^{6} + 20\nu^{5} + 16\nu^{4} + 62\nu^{3} + 10\nu^{2} + 4\nu + 8 ) / 25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - 5\nu^{5} + 4\nu^{4} - 8\nu^{3} + 10\nu^{2} + \nu + 3 ) / 5 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8\nu^{7} - 14\nu^{6} + 35\nu^{5} - 12\nu^{4} + 56\nu^{3} - 70\nu^{2} + 22\nu - 21 ) / 25 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{7} - 2\nu^{6} + 10\nu^{5} + 8\nu^{4} + 23\nu^{3} + 5\nu^{2} + 2\nu + 2 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + 3\beta_{4} - \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{6} + 4\beta_{5} - \beta_{3} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{7} + 5\beta_{6} + 6\beta_{5} - 12\beta_{4} - 4\beta_{3} + 6\beta_{2} - 12\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 6\beta_{7} - 23\beta_{4} + 16\beta_{2} + 28 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -23\beta_{6} - 29\beta_{5} + 16\beta_{3} + 51\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/155\mathbb{Z}\right)^\times\).

\(n\) \(32\) \(96\)
\(\chi(n)\) \(1\) \(\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
36.1
0.368820 + 0.638815i
1.04765 + 1.81458i
−0.677837 1.17405i
−0.238630 0.413319i
0.368820 0.638815i
1.04765 1.81458i
−0.677837 + 1.17405i
−0.238630 + 0.413319i
−1.35567 −0.440197 + 0.762443i −0.162147 −0.500000 0.866025i 0.596764 1.03362i 1.77460 3.07370i 2.93117 1.11245 + 1.92683i 0.677837 + 1.17405i
36.2 −0.477260 1.35666 2.34981i −1.77222 −0.500000 0.866025i −0.647481 + 1.12147i 0.0911485 0.157874i 1.80033 −2.18107 3.77773i 0.238630 + 0.413319i
36.3 0.737640 −1.48685 + 2.57531i −1.45589 −0.500000 0.866025i −1.09676 + 1.89965i −0.965584 + 1.67244i −2.54920 −2.92147 5.06014i −0.368820 0.638815i
36.4 2.09529 0.0703870 0.121914i 2.39026 −0.500000 0.866025i 0.147481 0.255445i −0.400166 + 0.693107i 0.817703 1.49009 + 2.58091i −1.04765 1.81458i
56.1 −1.35567 −0.440197 0.762443i −0.162147 −0.500000 + 0.866025i 0.596764 + 1.03362i 1.77460 + 3.07370i 2.93117 1.11245 1.92683i 0.677837 1.17405i
56.2 −0.477260 1.35666 + 2.34981i −1.77222 −0.500000 + 0.866025i −0.647481 1.12147i 0.0911485 + 0.157874i 1.80033 −2.18107 + 3.77773i 0.238630 0.413319i
56.3 0.737640 −1.48685 2.57531i −1.45589 −0.500000 + 0.866025i −1.09676 1.89965i −0.965584 1.67244i −2.54920 −2.92147 + 5.06014i −0.368820 + 0.638815i
56.4 2.09529 0.0703870 + 0.121914i 2.39026 −0.500000 + 0.866025i 0.147481 + 0.255445i −0.400166 0.693107i 0.817703 1.49009 2.58091i −1.04765 + 1.81458i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 36.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 155.2.e.d 8
5.b even 2 1 775.2.e.f 8
5.c odd 4 2 775.2.o.f 16
31.c even 3 1 inner 155.2.e.d 8
31.c even 3 1 4805.2.a.o 4
31.e odd 6 1 4805.2.a.m 4
155.j even 6 1 775.2.e.f 8
155.o odd 12 2 775.2.o.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.e.d 8 1.a even 1 1 trivial
155.2.e.d 8 31.c even 3 1 inner
775.2.e.f 8 5.b even 2 1
775.2.e.f 8 155.j even 6 1
775.2.o.f 16 5.c odd 4 2
775.2.o.f 16 155.o odd 12 2
4805.2.a.m 4 31.e odd 6 1
4805.2.a.o 4 31.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - T_{2}^{3} - 3T_{2}^{2} + T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(155, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} - 3 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + T^{7} + 9 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} - T^{7} + 9 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{8} - 4 T^{7} + \cdots + 6241 \) Copy content Toggle raw display
$13$ \( T^{8} + T^{7} + 11 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{8} + 8 T^{7} + \cdots + 6241 \) Copy content Toggle raw display
$19$ \( T^{8} - 3 T^{7} + \cdots + 841 \) Copy content Toggle raw display
$23$ \( (T^{4} - 19 T^{3} + \cdots + 211)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 19 T^{3} + \cdots - 1421)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 5 T^{7} + \cdots + 923521 \) Copy content Toggle raw display
$37$ \( T^{8} - 6 T^{7} + \cdots + 546121 \) Copy content Toggle raw display
$41$ \( T^{8} + 4 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$43$ \( T^{8} + 11 T^{7} + \cdots + 121 \) Copy content Toggle raw display
$47$ \( (T^{4} + 4 T^{3} + \cdots + 271)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 7 T^{7} + \cdots + 4289041 \) Copy content Toggle raw display
$59$ \( (T^{4} + 9 T^{3} + \cdots + 1681)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 5 T^{3} + \cdots + 589)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 9 T^{7} + \cdots + 6561 \) Copy content Toggle raw display
$71$ \( T^{8} + 12 T^{7} + \cdots + 183358681 \) Copy content Toggle raw display
$73$ \( T^{8} + 4 T^{7} + \cdots + 430336 \) Copy content Toggle raw display
$79$ \( T^{8} - 14 T^{7} + \cdots + 11566801 \) Copy content Toggle raw display
$83$ \( T^{8} + 18 T^{7} + \cdots + 495018001 \) Copy content Toggle raw display
$89$ \( (T^{4} + T^{3} - 38 T^{2} + \cdots + 121)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 7 T^{3} + \cdots + 121)^{2} \) Copy content Toggle raw display
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