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Properties

Label 195.2.i.b
Level $195$
Weight $2$
Character orbit 195.i
Analytic conductor $1.557$
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] = [195,2,Mod(16,195)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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

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: \(1.55708283941\)
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: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{3} + 2 \zeta_{6} q^{4} - q^{5} + \zeta_{6} q^{7} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{3} + 2 \zeta_{6} q^{4} - q^{5} + \zeta_{6} q^{7} - \zeta_{6} q^{9} + (6 \zeta_{6} - 6) q^{11} - 2 q^{12} + ( - 3 \zeta_{6} + 4) q^{13} + ( - \zeta_{6} + 1) q^{15} + (4 \zeta_{6} - 4) q^{16} + 4 \zeta_{6} q^{19} - 2 \zeta_{6} q^{20} - q^{21} + ( - 6 \zeta_{6} + 6) q^{23} + q^{25} + q^{27} + (2 \zeta_{6} - 2) q^{28} + ( - 6 \zeta_{6} + 6) q^{29} + 5 q^{31} - 6 \zeta_{6} q^{33} - \zeta_{6} q^{35} + ( - 2 \zeta_{6} + 2) q^{36} + (2 \zeta_{6} - 2) q^{37} + (4 \zeta_{6} - 1) q^{39} - 11 \zeta_{6} q^{43} - 12 q^{44} + \zeta_{6} q^{45} + 6 q^{47} - 4 \zeta_{6} q^{48} + ( - 6 \zeta_{6} + 6) q^{49} + (2 \zeta_{6} + 6) q^{52} + ( - 6 \zeta_{6} + 6) q^{55} - 4 q^{57} - 6 \zeta_{6} q^{59} + 2 q^{60} + \zeta_{6} q^{61} + ( - \zeta_{6} + 1) q^{63} - 8 q^{64} + (3 \zeta_{6} - 4) q^{65} + (11 \zeta_{6} - 11) q^{67} + 6 \zeta_{6} q^{69} + 6 \zeta_{6} q^{71} + 5 q^{73} + (\zeta_{6} - 1) q^{75} + (8 \zeta_{6} - 8) q^{76} - 6 q^{77} + 11 q^{79} + ( - 4 \zeta_{6} + 4) q^{80} + (\zeta_{6} - 1) q^{81} + 12 q^{83} - 2 \zeta_{6} q^{84} + 6 \zeta_{6} q^{87} + (12 \zeta_{6} - 12) q^{89} + (\zeta_{6} + 3) q^{91} + 12 q^{92} + (5 \zeta_{6} - 5) q^{93} - 4 \zeta_{6} q^{95} - 17 \zeta_{6} q^{97} + 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 2 q^{4} - 2 q^{5} + q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + 2 q^{4} - 2 q^{5} + q^{7} - q^{9} - 6 q^{11} - 4 q^{12} + 5 q^{13} + q^{15} - 4 q^{16} + 4 q^{19} - 2 q^{20} - 2 q^{21} + 6 q^{23} + 2 q^{25} + 2 q^{27} - 2 q^{28} + 6 q^{29} + 10 q^{31} - 6 q^{33} - q^{35} + 2 q^{36} - 2 q^{37} + 2 q^{39} - 11 q^{43} - 24 q^{44} + q^{45} + 12 q^{47} - 4 q^{48} + 6 q^{49} + 14 q^{52} + 6 q^{55} - 8 q^{57} - 6 q^{59} + 4 q^{60} + q^{61} + q^{63} - 16 q^{64} - 5 q^{65} - 11 q^{67} + 6 q^{69} + 6 q^{71} + 10 q^{73} - q^{75} - 8 q^{76} - 12 q^{77} + 22 q^{79} + 4 q^{80} - q^{81} + 24 q^{83} - 2 q^{84} + 6 q^{87} - 12 q^{89} + 7 q^{91} + 24 q^{92} - 5 q^{93} - 4 q^{95} - 17 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(106\) \(131\) \(157\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 −0.500000 − 0.866025i 1.00000 − 1.73205i −1.00000 0 0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0
61.1 0 −0.500000 + 0.866025i 1.00000 + 1.73205i −1.00000 0 0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
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.b ✓ 2
3.b odd 2 1 585.2.j.a 2
5.b even 2 1 975.2.i.d 2
5.c odd 4 2 975.2.bb.b 4
13.c even 3 1 inner 195.2.i.b ✓ 2
13.c even 3 1 2535.2.a.h 1
13.e even 6 1 2535.2.a.i 1
39.h odd 6 1 7605.2.a.k 1
39.i odd 6 1 585.2.j.a 2
39.i odd 6 1 7605.2.a.l 1
65.n even 6 1 975.2.i.d 2
65.q odd 12 2 975.2.bb.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.i.b ✓ 2 1.a even 1 1 trivial
195.2.i.b ✓ 2 13.c even 3 1 inner
585.2.j.a 2 3.b odd 2 1
585.2.j.a 2 39.i odd 6 1
975.2.i.d 2 5.b even 2 1
975.2.i.d 2 65.n even 6 1
975.2.bb.b 4 5.c odd 4 2
975.2.bb.b 4 65.q odd 12 2
2535.2.a.h 1 13.c even 3 1
2535.2.a.i 1 13.e even 6 1
7605.2.a.k 1 39.h odd 6 1
7605.2.a.l 1 39.i odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$13$ \( T^{2} - 5T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$23$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$31$ \( (T - 5)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$47$ \( (T - 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$73$ \( (T - 5)^{2} \) Copy content Toggle raw display
$79$ \( (T - 11)^{2} \) Copy content Toggle raw display
$83$ \( (T - 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$97$ \( T^{2} + 17T + 289 \) Copy content Toggle raw display
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