Properties

Label 95.4.e.a
Level $95$
Weight $4$
Character orbit 95.e
Analytic conductor $5.605$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [95,4,Mod(11,95)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(95, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("95.11");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 95 = 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 95.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: \(5.60518145055\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} + ( - 5 \zeta_{6} + 5) q^{3} + 7 \zeta_{6} q^{4} + (5 \zeta_{6} - 5) q^{5} - 5 \zeta_{6} q^{6} + 22 q^{7} + 15 q^{8} + 2 \zeta_{6} q^{9} + 5 \zeta_{6} q^{10} + 9 q^{11} + 35 q^{12} + \cdots + 18 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 5 q^{3} + 7 q^{4} - 5 q^{5} - 5 q^{6} + 44 q^{7} + 30 q^{8} + 2 q^{9} + 5 q^{10} + 18 q^{11} + 70 q^{12} - 54 q^{13} + 22 q^{14} + 25 q^{15} - 41 q^{16} + 54 q^{17} + 4 q^{18} - 133 q^{19}+ \cdots + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i 2.50000 4.33013i 3.50000 + 6.06218i −2.50000 + 4.33013i −2.50000 4.33013i 22.0000 15.0000 1.00000 + 1.73205i 2.50000 + 4.33013i
26.1 0.500000 + 0.866025i 2.50000 + 4.33013i 3.50000 6.06218i −2.50000 4.33013i −2.50000 + 4.33013i 22.0000 15.0000 1.00000 1.73205i 2.50000 4.33013i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 95.4.e.a 2
19.c even 3 1 inner 95.4.e.a 2
19.c even 3 1 1805.4.a.e 1
19.d odd 6 1 1805.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.4.e.a 2 1.a even 1 1 trivial
95.4.e.a 2 19.c even 3 1 inner
1805.4.a.e 1 19.c even 3 1
1805.4.a.g 1 19.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} + 1 \) acting on \(S_{4}^{\mathrm{new}}(95, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$5$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$7$ \( (T - 22)^{2} \) Copy content Toggle raw display
$11$ \( (T - 9)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 54T + 2916 \) Copy content Toggle raw display
$17$ \( T^{2} - 54T + 2916 \) Copy content Toggle raw display
$19$ \( T^{2} + 133T + 6859 \) Copy content Toggle raw display
$23$ \( T^{2} - 92T + 8464 \) Copy content Toggle raw display
$29$ \( T^{2} - 134T + 17956 \) Copy content Toggle raw display
$31$ \( (T + 252)^{2} \) Copy content Toggle raw display
$37$ \( (T + 236)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 243T + 59049 \) Copy content Toggle raw display
$43$ \( T^{2} + 496T + 246016 \) Copy content Toggle raw display
$47$ \( T^{2} + 502T + 252004 \) Copy content Toggle raw display
$53$ \( T^{2} + 62T + 3844 \) Copy content Toggle raw display
$59$ \( T^{2} + 681T + 463761 \) Copy content Toggle raw display
$61$ \( T^{2} - 142T + 20164 \) Copy content Toggle raw display
$67$ \( T^{2} + 55T + 3025 \) Copy content Toggle raw display
$71$ \( T^{2} - 974T + 948676 \) Copy content Toggle raw display
$73$ \( T^{2} + 695T + 483025 \) Copy content Toggle raw display
$79$ \( T^{2} - 736T + 541696 \) Copy content Toggle raw display
$83$ \( (T + 63)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 726T + 527076 \) Copy content Toggle raw display
$97$ \( T^{2} - 1167 T + 1361889 \) Copy content Toggle raw display
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