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Properties

Label 495.2.k.b
Level $495$
Weight $2$
Character orbit 495.k
Analytic conductor $3.953$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [495,2,Mod(208,495)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(495, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("495.208");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 495.k (of order \(4\), 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: \(3.95259490005\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 3 \beta_{2} q^{4} + (\beta_{2} + 2) q^{5} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 3 \beta_{2} q^{4} + (\beta_{2} + 2) q^{5} + \beta_{3} q^{8} + (\beta_{3} + 2 \beta_1) q^{10} + (\beta_{3} + \beta_1 - 1) q^{11} - 2 \beta_{3} q^{13} + q^{16} - 2 \beta_1 q^{17} + (2 \beta_{3} - 2 \beta_1) q^{19} + (6 \beta_{2} - 3) q^{20} + (5 \beta_{2} - \beta_1 - 5) q^{22} + (\beta_{2} + 1) q^{23} + (4 \beta_{2} + 3) q^{25} + 10 q^{26} + (2 \beta_{3} - 2 \beta_1) q^{29} + 2 q^{31} + 3 \beta_1 q^{32} - 10 \beta_{2} q^{34} + ( - 3 \beta_{2} + 3) q^{37} + ( - 10 \beta_{2} - 10) q^{38} + (2 \beta_{3} - \beta_1) q^{40} + ( - 2 \beta_{3} - 2 \beta_1) q^{41} + (3 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{44} + (\beta_{3} + \beta_1) q^{46} + (3 \beta_{2} - 3) q^{47} - 7 \beta_{2} q^{49} + (4 \beta_{3} + 3 \beta_1) q^{50} + 6 \beta_1 q^{52} + (\beta_{2} + 1) q^{53} + (3 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{55} + ( - 10 \beta_{2} - 10) q^{58} - 6 \beta_{2} q^{59} + ( - 2 \beta_{3} - 2 \beta_1) q^{61} + 2 \beta_1 q^{62} + 13 \beta_{2} q^{64} + ( - 4 \beta_{3} + 2 \beta_1) q^{65} + ( - 3 \beta_{2} + 3) q^{67} - 6 \beta_{3} q^{68} + 8 q^{71} + 2 \beta_{3} q^{73} + ( - 3 \beta_{3} + 3 \beta_1) q^{74} + ( - 6 \beta_{3} - 6 \beta_1) q^{76} + ( - 2 \beta_{3} + 2 \beta_1) q^{79} + (\beta_{2} + 2) q^{80} + ( - 10 \beta_{2} + 10) q^{82} + 4 \beta_{3} q^{83} + ( - 2 \beta_{3} - 4 \beta_1) q^{85} + ( - \beta_{3} - 5 \beta_{2} - 5) q^{88} - 6 \beta_{2} q^{89} + (3 \beta_{2} - 3) q^{92} + (3 \beta_{3} - 3 \beta_1) q^{94} + (2 \beta_{3} - 6 \beta_1) q^{95} + (7 \beta_{2} - 7) q^{97} - 7 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{5} - 4 q^{11} + 4 q^{16} - 12 q^{20} - 20 q^{22} + 4 q^{23} + 12 q^{25} + 40 q^{26} + 8 q^{31} + 12 q^{37} - 40 q^{38} - 12 q^{47} + 4 q^{53} - 8 q^{55} - 40 q^{58} + 12 q^{67} + 32 q^{71} + 8 q^{80} + 40 q^{82} - 20 q^{88} - 12 q^{92} - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 25 \) : Copy content Toggle raw display

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

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(46\) \(56\) \(397\)
\(\chi(n)\) \(-1\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
208.1
−1.58114 − 1.58114i
1.58114 + 1.58114i
−1.58114 + 1.58114i
1.58114 − 1.58114i
−1.58114 − 1.58114i 0 3.00000i 2.00000 + 1.00000i 0 0 1.58114 − 1.58114i 0 −1.58114 − 4.74342i
208.2 1.58114 + 1.58114i 0 3.00000i 2.00000 + 1.00000i 0 0 −1.58114 + 1.58114i 0 1.58114 + 4.74342i
307.1 −1.58114 + 1.58114i 0 − 3.00000i 2.00000 − 1.00000i 0 0 1.58114 + 1.58114i 0 −1.58114 + 4.74342i
307.2 1.58114 − 1.58114i 0 − 3.00000i 2.00000 − 1.00000i 0 0 −1.58114 − 1.58114i 0 1.58114 − 4.74342i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
11.b odd 2 1 inner
55.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.2.k.b 4
3.b odd 2 1 55.2.e.a ✓ 4
5.c odd 4 1 inner 495.2.k.b 4
11.b odd 2 1 inner 495.2.k.b 4
12.b even 2 1 880.2.bd.e 4
15.d odd 2 1 275.2.e.b 4
15.e even 4 1 55.2.e.a ✓ 4
15.e even 4 1 275.2.e.b 4
33.d even 2 1 55.2.e.a ✓ 4
33.f even 10 4 605.2.m.b 16
33.h odd 10 4 605.2.m.b 16
55.e even 4 1 inner 495.2.k.b 4
60.l odd 4 1 880.2.bd.e 4
132.d odd 2 1 880.2.bd.e 4
165.d even 2 1 275.2.e.b 4
165.l odd 4 1 55.2.e.a ✓ 4
165.l odd 4 1 275.2.e.b 4
165.u odd 20 4 605.2.m.b 16
165.v even 20 4 605.2.m.b 16
660.q even 4 1 880.2.bd.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.e.a ✓ 4 3.b odd 2 1
55.2.e.a ✓ 4 15.e even 4 1
55.2.e.a ✓ 4 33.d even 2 1
55.2.e.a ✓ 4 165.l odd 4 1
275.2.e.b 4 15.d odd 2 1
275.2.e.b 4 15.e even 4 1
275.2.e.b 4 165.d even 2 1
275.2.e.b 4 165.l odd 4 1
495.2.k.b 4 1.a even 1 1 trivial
495.2.k.b 4 5.c odd 4 1 inner
495.2.k.b 4 11.b odd 2 1 inner
495.2.k.b 4 55.e even 4 1 inner
605.2.m.b 16 33.f even 10 4
605.2.m.b 16 33.h odd 10 4
605.2.m.b 16 165.u odd 20 4
605.2.m.b 16 165.v even 20 4
880.2.bd.e 4 12.b even 2 1
880.2.bd.e 4 60.l odd 4 1
880.2.bd.e 4 132.d odd 2 1
880.2.bd.e 4 660.q even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 25 \) acting on \(S_{2}^{\mathrm{new}}(495, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 25 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 4 T + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T + 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 400 \) Copy content Toggle raw display
$17$ \( T^{4} + 400 \) Copy content Toggle raw display
$19$ \( (T^{2} - 40)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 40)^{2} \) Copy content Toggle raw display
$31$ \( (T - 2)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 6 T + 18)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 40)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 6 T + 18)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 40)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 6 T + 18)^{2} \) Copy content Toggle raw display
$71$ \( (T - 8)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 400 \) Copy content Toggle raw display
$79$ \( (T^{2} - 40)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 6400 \) Copy content Toggle raw display
$89$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 14 T + 98)^{2} \) Copy content Toggle raw display
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