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Properties

Label 495.2.c.d
Level $495$
Weight $2$
Character orbit 495.c
Analytic conductor $3.953$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [495,2,Mod(199,495)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage: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")
 
Copy content magma://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

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,-10,-2,0,0,0,0,16,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.95259490005\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_{3} - \beta_1) q^{2} + ( - \beta_{5} + \beta_{2} + \beta_1 - 2) q^{4} + (\beta_{5} + \beta_{3}) q^{5} + (\beta_{5} + \beta_{4} - \beta_{3} + \beta_1) q^{7} + (2 \beta_{5} + 3 \beta_{4} + 2 \beta_1) q^{8}+ \cdots + ( - 2 \beta_{5} + 2 \beta_{4} + \cdots - 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{4} - 2 q^{5} + 16 q^{10} + 6 q^{11} + 4 q^{14} + 10 q^{16} + 4 q^{19} - 14 q^{20} - 2 q^{25} + 16 q^{26} - 16 q^{29} + 16 q^{31} - 52 q^{34} + 12 q^{35} - 12 q^{40} + 16 q^{41} - 10 q^{44} + 24 q^{46}+ \cdots + 44 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( -2\nu^{5} - 7\nu^{4} + 15\nu^{3} - 25\nu^{2} - 42\nu - 16 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} - 4\nu^{2} + 8\nu - 9 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{5} + 10\nu^{4} - 5\nu^{3} - 7\nu^{2} - 32\nu + 13 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -12\nu^{5} + 27\nu^{4} - 25\nu^{3} - 35\nu^{2} - 22\nu + 42 ) / 23 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{4} + \beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + 2\beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} - 2\beta_{4} + 4\beta_{3} - 4\beta_{2} - \beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} - 5\beta_{2} - 2\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{5} + 3\beta_{4} - 8\beta_{3} - 8\beta_{2} - 2\beta _1 - 9 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
0.403032 − 0.403032i
−0.854638 + 0.854638i
1.45161 + 1.45161i
1.45161 − 1.45161i
−0.854638 − 0.854638i
0.403032 + 0.403032i
− 2.67513i 0 −5.15633 1.48119 + 1.67513i 0 2.80606i 8.44358i 0 4.48119 − 3.96239i
199.2 − 1.53919i 0 −0.369102 −2.17009 + 0.539189i 0 0.290725i − 2.51026i 0 0.829914 + 3.34017i
199.3 − 1.21432i 0 0.525428 −0.311108 + 2.21432i 0 − 4.90321i − 3.06668i 0 2.68889 + 0.377784i
199.4 1.21432i 0 0.525428 −0.311108 − 2.21432i 0 4.90321i 3.06668i 0 2.68889 − 0.377784i
199.5 1.53919i 0 −0.369102 −2.17009 − 0.539189i 0 − 0.290725i 2.51026i 0 0.829914 − 3.34017i
199.6 2.67513i 0 −5.15633 1.48119 − 1.67513i 0 − 2.80606i − 8.44358i 0 4.48119 + 3.96239i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.6
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.d 6
3.b odd 2 1 165.2.c.a ✓ 6
5.b even 2 1 inner 495.2.c.d 6
5.c odd 4 1 2475.2.a.y 3
5.c odd 4 1 2475.2.a.be 3
12.b even 2 1 2640.2.d.i 6
15.d odd 2 1 165.2.c.a ✓ 6
15.e even 4 1 825.2.a.h 3
15.e even 4 1 825.2.a.n 3
33.d even 2 1 1815.2.c.d 6
60.h even 2 1 2640.2.d.i 6
165.d even 2 1 1815.2.c.d 6
165.l odd 4 1 9075.2.a.cc 3
165.l odd 4 1 9075.2.a.ck 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.c.a ✓ 6 3.b odd 2 1
165.2.c.a ✓ 6 15.d odd 2 1
495.2.c.d 6 1.a even 1 1 trivial
495.2.c.d 6 5.b even 2 1 inner
825.2.a.h 3 15.e even 4 1
825.2.a.n 3 15.e even 4 1
1815.2.c.d 6 33.d even 2 1
1815.2.c.d 6 165.d even 2 1
2475.2.a.y 3 5.c odd 4 1
2475.2.a.be 3 5.c odd 4 1
2640.2.d.i 6 12.b even 2 1
2640.2.d.i 6 60.h even 2 1
9075.2.a.cc 3 165.l odd 4 1
9075.2.a.ck 3 165.l odd 4 1

Hecke kernels

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

\( T_{2}^{6} + 11T_{2}^{4} + 31T_{2}^{2} + 25 \) Copy content Toggle raw display
\( T_{29}^{3} + 8T_{29}^{2} - 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 11 T^{4} + \cdots + 25 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 2 T^{5} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 32 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T - 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 92 T^{4} + \cdots + 21904 \) Copy content Toggle raw display
$17$ \( T^{6} + 72 T^{4} + \cdots + 13456 \) Copy content Toggle raw display
$19$ \( (T^{3} - 2 T^{2} + \cdots + 184)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 16)^{3} \) Copy content Toggle raw display
$29$ \( (T^{3} + 8 T^{2} - 32)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 8 T^{2} + 8 T + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 48 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$41$ \( (T^{3} - 8 T^{2} + 32)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 32 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$47$ \( T^{6} + 96 T^{4} + \cdots + 25600 \) Copy content Toggle raw display
$53$ \( T^{6} + 128 T^{4} + \cdots + 73984 \) Copy content Toggle raw display
$59$ \( (T^{3} + 8 T^{2} - 64 T + 80)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 2 T^{2} - 52 T + 40)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 176 T^{4} + \cdots + 102400 \) Copy content Toggle raw display
$71$ \( (T^{3} - 12 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 204 T^{4} + \cdots + 8464 \) Copy content Toggle raw display
$79$ \( (T^{3} - 6 T^{2} - 4 T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 12 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$89$ \( (T^{3} - 2 T^{2} - 12 T + 8)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 128 T^{4} + \cdots + 16384 \) Copy content Toggle raw display
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