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Properties

Label 495.2.c.a
Level $495$
Weight $2$
Character orbit 495.c
Analytic conductor $3.953$
Analytic rank $0$
Dimension $4$
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] = [495,2,Mod(199,495)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(495, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("495.199");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 495.c (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: \(3.95259490005\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_{2} - \beta_1) q^{2} + (\beta_{3} + \beta_1 - 1) q^{4} + (\beta_{3} - \beta_{2} + 1) q^{5} - 2 \beta_{2} q^{7} + ( - \beta_{3} + 3 \beta_{2} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_{2} - \beta_1) q^{2} + (\beta_{3} + \beta_1 - 1) q^{4} + (\beta_{3} - \beta_{2} + 1) q^{5} - 2 \beta_{2} q^{7} + ( - \beta_{3} + 3 \beta_{2} + \beta_1) q^{8} + (\beta_{3} + \beta_{2} + \beta_1 - 2) q^{10} + q^{11} + (2 \beta_{3} + 2 \beta_1 - 2) q^{14} + ( - \beta_{3} - \beta_1 + 3) q^{16} + ( - 2 \beta_{3} + 2 \beta_1) q^{17} - 4 q^{19} + ( - 3 \beta_{3} + 4 \beta_{2} + \cdots + 3) q^{20}+ \cdots + ( - 5 \beta_{3} + 5 \beta_{2} + 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{4} + 3 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{4} + 3 q^{5} - 10 q^{10} + 4 q^{11} - 12 q^{14} + 14 q^{16} - 16 q^{19} + 12 q^{20} + q^{25} + 12 q^{29} + 2 q^{31} + 16 q^{34} - 18 q^{35} + 28 q^{40} - 12 q^{41} - 6 q^{44} - 8 q^{46} - 20 q^{49} + 18 q^{50} + 3 q^{55} + 60 q^{56} - 18 q^{59} + 20 q^{61} - 2 q^{64} + 24 q^{70} + 6 q^{71} - 12 q^{74} + 24 q^{76} - 28 q^{79} - 6 q^{80} + 2 q^{85} + 12 q^{86} - 6 q^{89} - 44 q^{94} - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 2\nu^{2} + 4\nu - 9 ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} - 2\nu^{2} + 2\nu + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu + 3 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + 2\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} - 3\beta_{2} + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{3} + \beta_{2} + 2\beta _1 + 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/495\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\) \(397\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
−1.18614 + 1.26217i
1.68614 + 0.396143i
1.68614 − 0.396143i
−1.18614 − 1.26217i
− 2.52434i 0 −4.37228 −0.686141 − 2.12819i 0 − 3.46410i 5.98844i 0 −5.37228 + 1.73205i
199.2 − 0.792287i 0 1.37228 2.18614 + 0.469882i 0 3.46410i − 2.67181i 0 0.372281 − 1.73205i
199.3 0.792287i 0 1.37228 2.18614 − 0.469882i 0 − 3.46410i 2.67181i 0 0.372281 + 1.73205i
199.4 2.52434i 0 −4.37228 −0.686141 + 2.12819i 0 3.46410i − 5.98844i 0 −5.37228 − 1.73205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.2.c.a 4
3.b odd 2 1 55.2.b.a ✓ 4
5.b even 2 1 inner 495.2.c.a 4
5.c odd 4 2 2475.2.a.bi 4
12.b even 2 1 880.2.b.h 4
15.d odd 2 1 55.2.b.a ✓ 4
15.e even 4 2 275.2.a.h 4
33.d even 2 1 605.2.b.c 4
33.f even 10 4 605.2.j.j 16
33.h odd 10 4 605.2.j.i 16
60.h even 2 1 880.2.b.h 4
60.l odd 4 2 4400.2.a.cc 4
165.d even 2 1 605.2.b.c 4
165.l odd 4 2 3025.2.a.ba 4
165.o odd 10 4 605.2.j.i 16
165.r even 10 4 605.2.j.j 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.b.a ✓ 4 3.b odd 2 1
55.2.b.a ✓ 4 15.d odd 2 1
275.2.a.h 4 15.e even 4 2
495.2.c.a 4 1.a even 1 1 trivial
495.2.c.a 4 5.b even 2 1 inner
605.2.b.c 4 33.d even 2 1
605.2.b.c 4 165.d even 2 1
605.2.j.i 16 33.h odd 10 4
605.2.j.i 16 165.o odd 10 4
605.2.j.j 16 33.f even 10 4
605.2.j.j 16 165.r even 10 4
880.2.b.h 4 12.b even 2 1
880.2.b.h 4 60.h even 2 1
2475.2.a.bi 4 5.c odd 4 2
3025.2.a.ba 4 165.l odd 4 2
4400.2.a.cc 4 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(495, [\chi])\):

\( T_{2}^{4} + 7T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{29}^{2} - 6T_{29} - 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 7T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 3 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 28T^{2} + 64 \) Copy content Toggle raw display
$19$ \( (T + 4)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 7T^{2} + 4 \) Copy content Toggle raw display
$29$ \( (T^{2} - 6 T - 24)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - T - 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 123T^{2} + 144 \) Copy content Toggle raw display
$41$ \( (T^{2} + 6 T - 24)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 44)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 112T^{2} + 1024 \) Copy content Toggle raw display
$59$ \( (T^{2} + 9 T + 12)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 10 T - 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 87T^{2} + 36 \) Copy content Toggle raw display
$71$ \( (T^{2} - 3 T - 72)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 14 T + 16)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 44)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 3 T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 51T^{2} + 576 \) Copy content Toggle raw display
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