Properties

Label 195.2.bb.a
Level $195$
Weight $2$
Character orbit 195.bb
Analytic conductor $1.557$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [195,2,Mod(121,195)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("195.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 195 = 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 195.bb (of order \(6\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{2} + \zeta_{12} - 1) q^{2} + (\zeta_{12}^{2} - 1) q^{3} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12}) q^{4} - \zeta_{12}^{3} q^{5} + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12} + 2) q^{6} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - \zeta_{12} - 4) q^{7} + (6 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{8} - \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{2} + \zeta_{12} - 1) q^{2} + (\zeta_{12}^{2} - 1) q^{3} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12}) q^{4} - \zeta_{12}^{3} q^{5} + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12} + 2) q^{6} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - \zeta_{12} - 4) q^{7} + (6 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{8} - \zeta_{12}^{2} q^{9} + (2 \zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12} + 1) q^{10} + ( - 2 \zeta_{12}^{2} - 2) q^{11} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} - 2) q^{12} + (3 \zeta_{12}^{3} + \zeta_{12}) q^{13} + (\zeta_{12}^{3} - 2 \zeta_{12} + 5) q^{14} + \zeta_{12} q^{15} + ( - 8 \zeta_{12}^{3} + 8 \zeta_{12}^{2} + 4 \zeta_{12} - 8) q^{16} + ( - \zeta_{12}^{3} - 5 \zeta_{12}^{2} - \zeta_{12}) q^{17} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{18} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 4) q^{19} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 4) q^{20} + ( - \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{21} + ( - 2 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 2 \zeta_{12}) q^{22} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 2 \zeta_{12} - 4) q^{23} + (2 \zeta_{12}^{2} - 6 \zeta_{12} + 2) q^{24} - q^{25} + ( - 7 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 2 \zeta_{12} - 3) q^{26} + q^{27} + ( - 2 \zeta_{12}^{2} + 10 \zeta_{12} - 2) q^{28} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} + 1) q^{29} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{30} + (2 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 3) q^{31} + (8 \zeta_{12}^{3} - 8 \zeta_{12}^{2} - 8 \zeta_{12} + 16) q^{32} + ( - 2 \zeta_{12}^{2} + 4) q^{33} + ( - 2 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 4) q^{34} + (2 \zeta_{12}^{3} + \zeta_{12}^{2} + 2 \zeta_{12}) q^{35} + (4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 2) q^{36} + 4 \zeta_{12} q^{37} + 4 q^{38} + (\zeta_{12}^{3} - 4 \zeta_{12}) q^{39} + (2 \zeta_{12}^{3} - 4 \zeta_{12} + 6) q^{40} + ( - \zeta_{12}^{2} + 7 \zeta_{12} - 1) q^{41} + ( - 2 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + \zeta_{12} - 5) q^{42} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} + \zeta_{12}) q^{43} + (12 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 4) q^{44} + (\zeta_{12}^{3} - \zeta_{12}) q^{45} + (10 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 10 \zeta_{12} + 12) q^{46} + ( - 5 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 3) q^{47} + (4 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 4 \zeta_{12}) q^{48} + ( - 8 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 4 \zeta_{12} + 6) q^{49} + (\zeta_{12}^{2} - \zeta_{12} + 1) q^{50} + ( - \zeta_{12}^{3} + 2 \zeta_{12} + 5) q^{51} + (8 \zeta_{12}^{3} - 10 \zeta_{12}^{2} - 6 \zeta_{12} + 14) q^{52} + (4 \zeta_{12}^{3} - 8 \zeta_{12}) q^{53} + ( - \zeta_{12}^{2} + \zeta_{12} - 1) q^{54} + (4 \zeta_{12}^{3} - 2 \zeta_{12}) q^{55} + ( - 10 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 10 \zeta_{12}) q^{56} + ( - 2 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{57} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{58} + ( - 9 \zeta_{12}^{3} + \zeta_{12}^{2} + 9 \zeta_{12} - 2) q^{59} + (2 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{60} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} - 2 \zeta_{12}) q^{61} + ( - 10 \zeta_{12}^{3} + 11 \zeta_{12}^{2} + 5 \zeta_{12} - 11) q^{62} + (2 \zeta_{12}^{2} + \zeta_{12} + 2) q^{63} + ( - 8 \zeta_{12}^{3} + 16 \zeta_{12} - 16) q^{64} + ( - \zeta_{12}^{2} + 4) q^{65} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} - 6) q^{66} + ( - 2 \zeta_{12}^{2} - 9 \zeta_{12} - 2) q^{67} + (16 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 8 \zeta_{12} + 4) q^{68} + (2 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2 \zeta_{12}) q^{69} + ( - 5 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{70} + (11 \zeta_{12}^{3} + \zeta_{12}^{2} - 11 \zeta_{12} - 2) q^{71} + ( - 6 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 6 \zeta_{12} - 4) q^{72} + (5 \zeta_{12}^{3} + 12 \zeta_{12}^{2} - 6) q^{73} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 4 \zeta_{12}) q^{74} + ( - \zeta_{12}^{2} + 1) q^{75} + 8 \zeta_{12} q^{76} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} + 12) q^{77} + (2 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 5 \zeta_{12} - 1) q^{78} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12} - 5) q^{79} + ( - 4 \zeta_{12}^{2} + 8 \zeta_{12} - 4) q^{80} + (\zeta_{12}^{2} - 1) q^{81} + ( - 8 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 8 \zeta_{12}) q^{82} + ( - 6 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{83} + (10 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 10 \zeta_{12} + 4) q^{84} + (5 \zeta_{12}^{3} + \zeta_{12}^{2} - 5 \zeta_{12} - 2) q^{85} + ( - 5 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 3) q^{86} + (\zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{87} + ( - 24 \zeta_{12}^{3} + 12 \zeta_{12}^{2} + 12 \zeta_{12} - 12) q^{88} + (\zeta_{12}^{2} - 3 \zeta_{12} + 1) q^{89} + ( - \zeta_{12}^{3} + 2 \zeta_{12} - 1) q^{90} + ( - 4 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 10 \zeta_{12} - 1) q^{91} + ( - 12 \zeta_{12}^{3} + 24 \zeta_{12} - 20) q^{92} + (3 \zeta_{12}^{2} - 2 \zeta_{12} + 3) q^{93} + (16 \zeta_{12}^{3} - 14 \zeta_{12}^{2} - 8 \zeta_{12} + 14) q^{94} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12}) q^{95} + ( - 8 \zeta_{12}^{3} + 16 \zeta_{12}^{2} - 8) q^{96} + (13 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 13 \zeta_{12} - 4) q^{97} + (6 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 6 \zeta_{12} - 4) q^{98} + (4 \zeta_{12}^{2} - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{2} - 2 q^{3} + 4 q^{4} + 6 q^{6} - 12 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{2} - 2 q^{3} + 4 q^{4} + 6 q^{6} - 12 q^{7} - 2 q^{9} + 2 q^{10} - 12 q^{11} - 8 q^{12} + 20 q^{14} - 16 q^{16} - 10 q^{17} - 12 q^{19} - 12 q^{20} + 12 q^{22} - 8 q^{23} + 12 q^{24} - 4 q^{25} - 4 q^{26} + 4 q^{27} - 12 q^{28} + 2 q^{29} + 2 q^{30} + 48 q^{32} + 12 q^{33} + 2 q^{35} + 4 q^{36} + 16 q^{38} + 24 q^{40} - 6 q^{41} - 10 q^{42} - 4 q^{43} + 36 q^{46} - 16 q^{48} + 12 q^{49} + 6 q^{50} + 20 q^{51} + 36 q^{52} - 6 q^{54} + 12 q^{56} - 6 q^{59} - 2 q^{61} - 22 q^{62} + 12 q^{63} - 64 q^{64} + 14 q^{65} - 24 q^{66} - 12 q^{67} + 8 q^{68} - 8 q^{69} - 6 q^{71} - 12 q^{72} + 8 q^{74} + 2 q^{75} + 48 q^{77} - 10 q^{78} - 20 q^{79} - 24 q^{80} - 2 q^{81} + 20 q^{82} + 12 q^{84} - 6 q^{85} + 2 q^{87} - 24 q^{88} + 6 q^{89} - 4 q^{90} - 10 q^{91} - 80 q^{92} + 18 q^{93} + 28 q^{94} + 4 q^{95} - 12 q^{97} - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(106\) \(131\) \(157\)
\(\chi(n)\) \(\zeta_{12}^{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−2.36603 1.36603i −0.500000 + 0.866025i 2.73205 + 4.73205i 1.00000i 2.36603 1.36603i −2.13397 + 1.23205i 9.46410i −0.500000 0.866025i 1.36603 2.36603i
121.2 −0.633975 0.366025i −0.500000 + 0.866025i −0.732051 1.26795i 1.00000i 0.633975 0.366025i −3.86603 + 2.23205i 2.53590i −0.500000 0.866025i −0.366025 + 0.633975i
166.1 −2.36603 + 1.36603i −0.500000 0.866025i 2.73205 4.73205i 1.00000i 2.36603 + 1.36603i −2.13397 1.23205i 9.46410i −0.500000 + 0.866025i 1.36603 + 2.36603i
166.2 −0.633975 + 0.366025i −0.500000 0.866025i −0.732051 + 1.26795i 1.00000i 0.633975 + 0.366025i −3.86603 2.23205i 2.53590i −0.500000 + 0.866025i −0.366025 0.633975i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 195.2.bb.a 4
3.b odd 2 1 585.2.bu.a 4
5.b even 2 1 975.2.bc.h 4
5.c odd 4 1 975.2.w.a 4
5.c odd 4 1 975.2.w.f 4
13.e even 6 1 inner 195.2.bb.a 4
13.f odd 12 1 2535.2.a.n 2
13.f odd 12 1 2535.2.a.s 2
39.h odd 6 1 585.2.bu.a 4
39.k even 12 1 7605.2.a.y 2
39.k even 12 1 7605.2.a.bk 2
65.l even 6 1 975.2.bc.h 4
65.r odd 12 1 975.2.w.a 4
65.r odd 12 1 975.2.w.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.bb.a 4 1.a even 1 1 trivial
195.2.bb.a 4 13.e even 6 1 inner
585.2.bu.a 4 3.b odd 2 1
585.2.bu.a 4 39.h odd 6 1
975.2.w.a 4 5.c odd 4 1
975.2.w.a 4 65.r odd 12 1
975.2.w.f 4 5.c odd 4 1
975.2.w.f 4 65.r odd 12 1
975.2.bc.h 4 5.b even 2 1
975.2.bc.h 4 65.l even 6 1
2535.2.a.n 2 13.f odd 12 1
2535.2.a.s 2 13.f odd 12 1
7605.2.a.y 2 39.k even 12 1
7605.2.a.bk 2 39.k even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 6T_{2}^{3} + 14T_{2}^{2} + 12T_{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(195, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 6 T^{3} + 14 T^{2} + 12 T + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 12 T^{3} + 59 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$11$ \( (T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 23T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{4} + 10 T^{3} + 78 T^{2} + \cdots + 484 \) Copy content Toggle raw display
$19$ \( T^{4} + 12 T^{3} + 56 T^{2} + 96 T + 64 \) Copy content Toggle raw display
$23$ \( T^{4} + 8 T^{3} + 60 T^{2} + 32 T + 16 \) Copy content Toggle raw display
$29$ \( T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$31$ \( T^{4} + 62T^{2} + 529 \) Copy content Toggle raw display
$37$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$41$ \( T^{4} + 6 T^{3} - 34 T^{2} + \cdots + 2116 \) Copy content Toggle raw display
$43$ \( T^{4} + 4 T^{3} + 15 T^{2} + 4 T + 1 \) Copy content Toggle raw display
$47$ \( T^{4} + 104T^{2} + 4 \) Copy content Toggle raw display
$53$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 6 T^{3} - 66 T^{2} + \cdots + 6084 \) Copy content Toggle raw display
$61$ \( T^{4} + 2 T^{3} + 15 T^{2} - 22 T + 121 \) Copy content Toggle raw display
$67$ \( T^{4} + 12 T^{3} - 21 T^{2} + \cdots + 4761 \) Copy content Toggle raw display
$71$ \( T^{4} + 6 T^{3} - 106 T^{2} + \cdots + 13924 \) Copy content Toggle raw display
$73$ \( T^{4} + 266T^{2} + 6889 \) Copy content Toggle raw display
$79$ \( (T^{2} + 10 T - 23)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 96T^{2} + 576 \) Copy content Toggle raw display
$89$ \( T^{4} - 6 T^{3} + 6 T^{2} + 36 T + 36 \) Copy content Toggle raw display
$97$ \( T^{4} + 12 T^{3} - 109 T^{2} + \cdots + 24649 \) Copy content Toggle raw display
show more
show less