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Properties

Label 195.2.i.e
Level $195$
Weight $2$
Character orbit 195.i
Analytic conductor $1.557$
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] = [195,2,Mod(16,195)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(195, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) 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("195.16"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 195 = 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 195.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,3,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.55708283941\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.591408.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 4x^{4} + x^{3} + 10x^{2} - 3x + 1 \) 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

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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + ( - \beta_{4} + 1) q^{3} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{4} - q^{5} + ( - \beta_{5} + \beta_{3}) q^{6} + ( - \beta_{5} + \beta_{4} + \cdots + 2 \beta_1) q^{7}+ \cdots + (2 \beta_{3} - 2 \beta_{2} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} - 2 q^{4} - 6 q^{5} + 5 q^{7} + 12 q^{8} - 3 q^{9} - 4 q^{11} - 4 q^{12} + 3 q^{13} - 24 q^{14} - 3 q^{15} + 4 q^{16} + 4 q^{17} + 2 q^{20} + 10 q^{21} - 12 q^{22} - 8 q^{23} + 6 q^{24} + 6 q^{25}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 4x^{4} + x^{3} + 10x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} - 16\nu^{3} + 10\nu^{2} - 3\nu + 12 ) / 37 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{5} + 16\nu^{4} - 27\nu^{3} + 40\nu^{2} - 12\nu + 85 ) / 37 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\nu^{5} - 11\nu^{4} + 44\nu^{3} + 28\nu^{2} + 110\nu + 4 ) / 37 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -25\nu^{5} + 26\nu^{4} - 104\nu^{3} - 9\nu^{2} - 260\nu + 78 ) / 37 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{4} - \beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 4\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -4\beta_{5} - 7\beta_{4} + 4\beta_{3} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -6\beta_{5} - 8\beta_{4} + 17\beta_{2} - 17\beta _1 + 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/195\mathbb{Z}\right)^\times\).

\(n\) \(106\) \(131\) \(157\)
\(\chi(n)\) \(-\beta_{4}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
0.155554 − 0.269427i
1.08504 − 1.87935i
−0.740597 + 1.28275i
0.155554 + 0.269427i
1.08504 + 1.87935i
−0.740597 − 1.28275i
−1.10716 − 1.91766i 0.500000 + 0.866025i −1.45161 + 2.51426i −1.00000 1.10716 − 1.91766i 1.91827 − 3.32254i 2.00000 −0.500000 + 0.866025i 1.10716 + 1.91766i
16.2 0.269594 + 0.466951i 0.500000 + 0.866025i 0.854638 − 1.48028i −1.00000 −0.269594 + 0.466951i 2.40049 − 4.15777i 2.00000 −0.500000 + 0.866025i −0.269594 − 0.466951i
16.3 0.837565 + 1.45071i 0.500000 + 0.866025i −0.403032 + 0.698071i −1.00000 −0.837565 + 1.45071i −1.81876 + 3.15018i 2.00000 −0.500000 + 0.866025i −0.837565 − 1.45071i
61.1 −1.10716 + 1.91766i 0.500000 − 0.866025i −1.45161 − 2.51426i −1.00000 1.10716 + 1.91766i 1.91827 + 3.32254i 2.00000 −0.500000 − 0.866025i 1.10716 − 1.91766i
61.2 0.269594 − 0.466951i 0.500000 − 0.866025i 0.854638 + 1.48028i −1.00000 −0.269594 − 0.466951i 2.40049 + 4.15777i 2.00000 −0.500000 − 0.866025i −0.269594 + 0.466951i
61.3 0.837565 − 1.45071i 0.500000 − 0.866025i −0.403032 − 0.698071i −1.00000 −0.837565 − 1.45071i −1.81876 − 3.15018i 2.00000 −0.500000 − 0.866025i −0.837565 + 1.45071i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.3
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 195.2.i.e ✓ 6
3.b odd 2 1 585.2.j.g 6
5.b even 2 1 975.2.i.m 6
5.c odd 4 2 975.2.bb.j 12
13.c even 3 1 inner 195.2.i.e ✓ 6
13.c even 3 1 2535.2.a.y 3
13.e even 6 1 2535.2.a.z 3
39.h odd 6 1 7605.2.a.bt 3
39.i odd 6 1 585.2.j.g 6
39.i odd 6 1 7605.2.a.bu 3
65.n even 6 1 975.2.i.m 6
65.q odd 12 2 975.2.bb.j 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.i.e ✓ 6 1.a even 1 1 trivial
195.2.i.e ✓ 6 13.c even 3 1 inner
585.2.j.g 6 3.b odd 2 1
585.2.j.g 6 39.i odd 6 1
975.2.i.m 6 5.b even 2 1
975.2.i.m 6 65.n even 6 1
975.2.bb.j 12 5.c odd 4 2
975.2.bb.j 12 65.q odd 12 2
2535.2.a.y 3 13.c even 3 1
2535.2.a.z 3 13.e even 6 1
7605.2.a.bt 3 39.h odd 6 1
7605.2.a.bu 3 39.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 4T_{2}^{4} - 4T_{2}^{3} + 16T_{2}^{2} - 8T_{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(195, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 4 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T + 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 5 T^{5} + \cdots + 4489 \) Copy content Toggle raw display
$11$ \( T^{6} + 4 T^{5} + \cdots + 1024 \) Copy content Toggle raw display
$13$ \( T^{6} - 3 T^{5} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( T^{6} - 4 T^{5} + \cdots + 1156 \) Copy content Toggle raw display
$19$ \( T^{6} + 40 T^{4} + \cdots + 5776 \) Copy content Toggle raw display
$23$ \( T^{6} + 8 T^{5} + \cdots + 92416 \) Copy content Toggle raw display
$29$ \( T^{6} + 6 T^{5} + \cdots + 2916 \) Copy content Toggle raw display
$31$ \( (T^{3} + 9 T^{2} + \cdots - 109)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} - 14 T^{5} + \cdots + 23104 \) Copy content Toggle raw display
$41$ \( T^{6} - 20 T^{5} + \cdots + 45796 \) Copy content Toggle raw display
$43$ \( T^{6} - 15 T^{5} + \cdots + 11449 \) Copy content Toggle raw display
$47$ \( (T^{3} - 4 T - 2)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 16 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 10 T^{5} + \cdots + 940900 \) Copy content Toggle raw display
$61$ \( T^{6} + 9 T^{5} + \cdots + 11881 \) Copy content Toggle raw display
$67$ \( T^{6} - 11 T^{5} + \cdots + 61009 \) Copy content Toggle raw display
$71$ \( T^{6} + 4 T^{5} + \cdots + 2116 \) Copy content Toggle raw display
$73$ \( (T^{3} + 15 T^{2} + \cdots + 103)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 17 T^{2} + \cdots - 67)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} - 2 T^{2} - 12 T + 8)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 22 T^{5} + \cdots + 17956 \) Copy content Toggle raw display
$97$ \( T^{6} - T^{5} + \cdots + 299209 \) Copy content Toggle raw display
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