Properties

Label 195.2.i.a
Level $195$
Weight $2$
Character orbit 195.i
Analytic conductor $1.557$
Analytic rank $1$
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: \(1\)
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.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 2) q^{2} + (\zeta_{6} - 1) q^{3} - 2 \zeta_{6} q^{4} - q^{5} - 2 \zeta_{6} q^{6} - 5 \zeta_{6} q^{7} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} - 2) q^{2} + (\zeta_{6} - 1) q^{3} - 2 \zeta_{6} q^{4} - q^{5} - 2 \zeta_{6} q^{6} - 5 \zeta_{6} q^{7} - \zeta_{6} q^{9} + ( - 2 \zeta_{6} + 2) q^{10} + (2 \zeta_{6} - 2) q^{11} + 2 q^{12} + (3 \zeta_{6} - 4) q^{13} + 10 q^{14} + ( - \zeta_{6} + 1) q^{15} + ( - 4 \zeta_{6} + 4) q^{16} - 2 \zeta_{6} q^{17} + 2 q^{18} + 2 \zeta_{6} q^{20} + 5 q^{21} - 4 \zeta_{6} q^{22} + (6 \zeta_{6} - 6) q^{23} + q^{25} + ( - 8 \zeta_{6} + 2) q^{26} + q^{27} + (10 \zeta_{6} - 10) q^{28} + ( - 4 \zeta_{6} + 4) q^{29} + 2 \zeta_{6} q^{30} - 7 q^{31} + 8 \zeta_{6} q^{32} - 2 \zeta_{6} q^{33} + 4 q^{34} + 5 \zeta_{6} q^{35} + (2 \zeta_{6} - 2) q^{36} + ( - 2 \zeta_{6} + 2) q^{37} + ( - 4 \zeta_{6} + 1) q^{39} + (6 \zeta_{6} - 6) q^{41} + (10 \zeta_{6} - 10) q^{42} - \zeta_{6} q^{43} + 4 q^{44} + \zeta_{6} q^{45} - 12 \zeta_{6} q^{46} - 8 q^{47} + 4 \zeta_{6} q^{48} + (18 \zeta_{6} - 18) q^{49} + (2 \zeta_{6} - 2) q^{50} + 2 q^{51} + (2 \zeta_{6} + 6) q^{52} - 4 q^{53} + (2 \zeta_{6} - 2) q^{54} + ( - 2 \zeta_{6} + 2) q^{55} + 8 \zeta_{6} q^{58} - 12 \zeta_{6} q^{59} - 2 q^{60} + 13 \zeta_{6} q^{61} + ( - 14 \zeta_{6} + 14) q^{62} + (5 \zeta_{6} - 5) q^{63} - 8 q^{64} + ( - 3 \zeta_{6} + 4) q^{65} + 4 q^{66} + ( - 7 \zeta_{6} + 7) q^{67} + (4 \zeta_{6} - 4) q^{68} - 6 \zeta_{6} q^{69} - 10 q^{70} - 12 \zeta_{6} q^{71} + 15 q^{73} + 4 \zeta_{6} q^{74} + (\zeta_{6} - 1) q^{75} + 10 q^{77} + (2 \zeta_{6} + 6) q^{78} + 3 q^{79} + (4 \zeta_{6} - 4) q^{80} + (\zeta_{6} - 1) q^{81} - 12 \zeta_{6} q^{82} + 8 q^{83} - 10 \zeta_{6} q^{84} + 2 \zeta_{6} q^{85} + 2 q^{86} + 4 \zeta_{6} q^{87} + (14 \zeta_{6} - 14) q^{89} - 2 q^{90} + (5 \zeta_{6} + 15) q^{91} + 12 q^{92} + ( - 7 \zeta_{6} + 7) q^{93} + ( - 16 \zeta_{6} + 16) q^{94} - 8 q^{96} + 5 \zeta_{6} q^{97} - 36 \zeta_{6} q^{98} + 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - q^{3} - 2 q^{4} - 2 q^{5} - 2 q^{6} - 5 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - q^{3} - 2 q^{4} - 2 q^{5} - 2 q^{6} - 5 q^{7} - q^{9} + 2 q^{10} - 2 q^{11} + 4 q^{12} - 5 q^{13} + 20 q^{14} + q^{15} + 4 q^{16} - 2 q^{17} + 4 q^{18} + 2 q^{20} + 10 q^{21} - 4 q^{22} - 6 q^{23} + 2 q^{25} - 4 q^{26} + 2 q^{27} - 10 q^{28} + 4 q^{29} + 2 q^{30} - 14 q^{31} + 8 q^{32} - 2 q^{33} + 8 q^{34} + 5 q^{35} - 2 q^{36} + 2 q^{37} - 2 q^{39} - 6 q^{41} - 10 q^{42} - q^{43} + 8 q^{44} + q^{45} - 12 q^{46} - 16 q^{47} + 4 q^{48} - 18 q^{49} - 2 q^{50} + 4 q^{51} + 14 q^{52} - 8 q^{53} - 2 q^{54} + 2 q^{55} + 8 q^{58} - 12 q^{59} - 4 q^{60} + 13 q^{61} + 14 q^{62} - 5 q^{63} - 16 q^{64} + 5 q^{65} + 8 q^{66} + 7 q^{67} - 4 q^{68} - 6 q^{69} - 20 q^{70} - 12 q^{71} + 30 q^{73} + 4 q^{74} - q^{75} + 20 q^{77} + 14 q^{78} + 6 q^{79} - 4 q^{80} - q^{81} - 12 q^{82} + 16 q^{83} - 10 q^{84} + 2 q^{85} + 4 q^{86} + 4 q^{87} - 14 q^{89} - 4 q^{90} + 35 q^{91} + 24 q^{92} + 7 q^{93} + 16 q^{94} - 16 q^{96} + 5 q^{97} - 36 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
−1.00000 1.73205i −0.500000 0.866025i −1.00000 + 1.73205i −1.00000 −1.00000 + 1.73205i −2.50000 + 4.33013i 0 −0.500000 + 0.866025i 1.00000 + 1.73205i
61.1 −1.00000 + 1.73205i −0.500000 + 0.866025i −1.00000 1.73205i −1.00000 −1.00000 1.73205i −2.50000 4.33013i 0 −0.500000 0.866025i 1.00000 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
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.a 2
3.b odd 2 1 585.2.j.b 2
5.b even 2 1 975.2.i.i 2
5.c odd 4 2 975.2.bb.f 4
13.c even 3 1 inner 195.2.i.a 2
13.c even 3 1 2535.2.a.m 1
13.e even 6 1 2535.2.a.c 1
39.h odd 6 1 7605.2.a.s 1
39.i odd 6 1 585.2.j.b 2
39.i odd 6 1 7605.2.a.a 1
65.n even 6 1 975.2.i.i 2
65.q odd 12 2 975.2.bb.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.i.a 2 1.a even 1 1 trivial
195.2.i.a 2 13.c even 3 1 inner
585.2.j.b 2 3.b odd 2 1
585.2.j.b 2 39.i odd 6 1
975.2.i.i 2 5.b even 2 1
975.2.i.i 2 65.n even 6 1
975.2.bb.f 4 5.c odd 4 2
975.2.bb.f 4 65.q odd 12 2
2535.2.a.c 1 13.e even 6 1
2535.2.a.m 1 13.c even 3 1
7605.2.a.a 1 39.i odd 6 1
7605.2.a.s 1 39.h odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$11$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$13$ \( T^{2} + 5T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$31$ \( (T + 7)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$41$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$43$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$47$ \( (T + 8)^{2} \) Copy content Toggle raw display
$53$ \( (T + 4)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$61$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$67$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$71$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$73$ \( (T - 15)^{2} \) Copy content Toggle raw display
$79$ \( (T - 3)^{2} \) Copy content Toggle raw display
$83$ \( (T - 8)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$97$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
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