Properties

Label 195.2.i.c
Level $195$
Weight $2$
Character orbit 195.i
Analytic conductor $1.557$
Analytic rank $0$
Dimension $4$
CM no
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
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: \( 3 \)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{2} - \beta_1 q^{3} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{4} + q^{5} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{6} + (\beta_{3} - \beta_{2}) q^{7} + (2 \beta_{3} - 6) q^{8} + (\beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{2} - \beta_1 q^{3} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{4} + q^{5} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{6} + (\beta_{3} - \beta_{2}) q^{7} + (2 \beta_{3} - 6) q^{8} + (\beta_1 - 1) q^{9} + ( - \beta_{2} + \beta_1) q^{10} + 2 \beta_1 q^{11} + ( - 2 \beta_{3} + 2) q^{12} + ( - \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{13} + (\beta_{3} - 3) q^{14} - \beta_1 q^{15} + (4 \beta_{2} - 8 \beta_1) q^{16} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{17} + (\beta_{3} - 1) q^{18} + ( - 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 4) q^{19} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{20} - \beta_{3} q^{21} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{22} - 2 \beta_1 q^{23} + ( - 2 \beta_{2} + 6 \beta_1) q^{24} + q^{25} + ( - \beta_{3} - 2 \beta_{2} + 6 \beta_1 - 1) q^{26} + q^{27} + (2 \beta_{2} - 6 \beta_1) q^{28} + ( - \beta_{2} + 5 \beta_1) q^{29} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{30} + (2 \beta_{3} + 1) q^{31} + ( - 8 \beta_{3} + 8 \beta_{2} - 8 \beta_1 + 8) q^{32} + ( - 2 \beta_1 + 2) q^{33} + ( - 2 \beta_{3} + 4) q^{34} + (\beta_{3} - \beta_{2}) q^{35} + (2 \beta_{2} - 2 \beta_1) q^{36} + 6 \beta_{2} q^{37} + ( - 6 \beta_{3} + 10) q^{38} + ( - \beta_{3} + 2 \beta_{2} - 2) q^{39} + (2 \beta_{3} - 6) q^{40} + (\beta_{2} + 9 \beta_1) q^{41} + ( - \beta_{2} + 3 \beta_1) q^{42} + (3 \beta_{3} - 3 \beta_{2} - 4 \beta_1 + 4) q^{43} + (4 \beta_{3} - 4) q^{44} + (\beta_1 - 1) q^{45} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{46} + (3 \beta_{3} - 5) q^{47} + (4 \beta_{3} - 4 \beta_{2} + 8 \beta_1 - 8) q^{48} + 4 \beta_1 q^{49} + ( - \beta_{2} + \beta_1) q^{50} + (\beta_{3} - 1) q^{51} + (4 \beta_{3} - 6 \beta_{2} + 10 \beta_1 - 12) q^{52} + ( - 2 \beta_{3} - 6) q^{53} + ( - \beta_{2} + \beta_1) q^{54} + 2 \beta_1 q^{55} + ( - 6 \beta_{3} + 6 \beta_{2} - 6 \beta_1 + 6) q^{56} + (2 \beta_{3} - 4) q^{57} + (6 \beta_{3} - 6 \beta_{2} + 8 \beta_1 - 8) q^{58} + (5 \beta_{3} - 5 \beta_{2} - 3 \beta_1 + 3) q^{59} + ( - 2 \beta_{3} + 2) q^{60} + ( - 6 \beta_{3} + 6 \beta_{2} - 5 \beta_1 + 5) q^{61} + (\beta_{2} - 5 \beta_1) q^{62} + \beta_{2} q^{63} + ( - 8 \beta_{3} + 16) q^{64} + ( - \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{65} + ( - 2 \beta_{3} + 2) q^{66} + (3 \beta_{2} + 4 \beta_1) q^{67} + ( - 4 \beta_{2} + 8 \beta_1) q^{68} + (2 \beta_1 - 2) q^{69} + (\beta_{3} - 3) q^{70} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 3) q^{71} + ( - 2 \beta_{3} + 2 \beta_{2} - 6 \beta_1 + 6) q^{72} + \beta_{3} q^{73} + ( - 6 \beta_{3} + 6 \beta_{2} - 18 \beta_1 + 18) q^{74} - \beta_1 q^{75} + ( - 12 \beta_{2} + 20 \beta_1) q^{76} + 2 \beta_{3} q^{77} + ( - 2 \beta_{3} + 3 \beta_{2} - 5 \beta_1 + 6) q^{78} - 11 q^{79} + (4 \beta_{2} - 8 \beta_1) q^{80} - \beta_1 q^{81} + (8 \beta_{3} - 8 \beta_{2} + 6 \beta_1 - 6) q^{82} + (4 \beta_{3} - 4) q^{83} + (2 \beta_{3} - 2 \beta_{2} + 6 \beta_1 - 6) q^{84} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{85} + ( - \beta_{3} - 5) q^{86} + ( - \beta_{3} + \beta_{2} - 5 \beta_1 + 5) q^{87} + (4 \beta_{2} - 12 \beta_1) q^{88} + (\beta_{2} - 7 \beta_1) q^{89} + (\beta_{3} - 1) q^{90} + ( - 2 \beta_{2} + 3 \beta_1 - 6) q^{91} + ( - 4 \beta_{3} + 4) q^{92} + ( - 2 \beta_{2} - \beta_1) q^{93} + (8 \beta_{2} - 14 \beta_1) q^{94} + ( - 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 4) q^{95} + (8 \beta_{3} - 8) q^{96} + ( - 3 \beta_{3} + 3 \beta_{2}) q^{97} + (4 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 4) q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{3} - 4 q^{4} + 4 q^{5} + 2 q^{6} - 24 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{3} - 4 q^{4} + 4 q^{5} + 2 q^{6} - 24 q^{8} - 2 q^{9} + 2 q^{10} + 4 q^{11} + 8 q^{12} + 4 q^{13} - 12 q^{14} - 2 q^{15} - 16 q^{16} + 2 q^{17} - 4 q^{18} + 8 q^{19} - 4 q^{20} - 4 q^{22} - 4 q^{23} + 12 q^{24} + 4 q^{25} + 8 q^{26} + 4 q^{27} - 12 q^{28} + 10 q^{29} + 2 q^{30} + 4 q^{31} + 16 q^{32} + 4 q^{33} + 16 q^{34} - 4 q^{36} + 40 q^{38} - 8 q^{39} - 24 q^{40} + 18 q^{41} + 6 q^{42} + 8 q^{43} - 16 q^{44} - 2 q^{45} + 4 q^{46} - 20 q^{47} - 16 q^{48} + 8 q^{49} + 2 q^{50} - 4 q^{51} - 28 q^{52} - 24 q^{53} + 2 q^{54} + 4 q^{55} + 12 q^{56} - 16 q^{57} - 16 q^{58} + 6 q^{59} + 8 q^{60} + 10 q^{61} - 10 q^{62} + 64 q^{64} + 4 q^{65} + 8 q^{66} + 8 q^{67} + 16 q^{68} - 4 q^{69} - 12 q^{70} + 6 q^{71} + 12 q^{72} + 36 q^{74} - 2 q^{75} + 40 q^{76} + 14 q^{78} - 44 q^{79} - 16 q^{80} - 2 q^{81} - 12 q^{82} - 16 q^{83} - 12 q^{84} + 2 q^{85} - 20 q^{86} + 10 q^{87} - 24 q^{88} - 14 q^{89} - 4 q^{90} - 18 q^{91} + 16 q^{92} - 2 q^{93} - 28 q^{94} + 8 q^{95} - 32 q^{96} - 8 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

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

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

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.

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
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.366025 0.633975i −0.500000 0.866025i 0.732051 1.26795i 1.00000 −0.366025 + 0.633975i 0.866025 1.50000i −2.53590 −0.500000 + 0.866025i −0.366025 0.633975i
16.2 1.36603 + 2.36603i −0.500000 0.866025i −2.73205 + 4.73205i 1.00000 1.36603 2.36603i −0.866025 + 1.50000i −9.46410 −0.500000 + 0.866025i 1.36603 + 2.36603i
61.1 −0.366025 + 0.633975i −0.500000 + 0.866025i 0.732051 + 1.26795i 1.00000 −0.366025 0.633975i 0.866025 + 1.50000i −2.53590 −0.500000 0.866025i −0.366025 + 0.633975i
61.2 1.36603 2.36603i −0.500000 + 0.866025i −2.73205 4.73205i 1.00000 1.36603 + 2.36603i −0.866025 1.50000i −9.46410 −0.500000 0.866025i 1.36603 2.36603i
\(n\): 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.c 4
3.b odd 2 1 585.2.j.c 4
5.b even 2 1 975.2.i.j 4
5.c odd 4 1 975.2.bb.a 4
5.c odd 4 1 975.2.bb.h 4
13.c even 3 1 inner 195.2.i.c 4
13.c even 3 1 2535.2.a.o 2
13.e even 6 1 2535.2.a.r 2
39.h odd 6 1 7605.2.a.z 2
39.i odd 6 1 585.2.j.c 4
39.i odd 6 1 7605.2.a.bj 2
65.n even 6 1 975.2.i.j 4
65.q odd 12 1 975.2.bb.a 4
65.q odd 12 1 975.2.bb.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.i.c 4 1.a even 1 1 trivial
195.2.i.c 4 13.c even 3 1 inner
585.2.j.c 4 3.b odd 2 1
585.2.j.c 4 39.i odd 6 1
975.2.i.j 4 5.b even 2 1
975.2.i.j 4 65.n even 6 1
975.2.bb.a 4 5.c odd 4 1
975.2.bb.a 4 65.q odd 12 1
975.2.bb.h 4 5.c odd 4 1
975.2.bb.h 4 65.q odd 12 1
2535.2.a.o 2 13.c even 3 1
2535.2.a.r 2 13.e even 6 1
7605.2.a.z 2 39.h odd 6 1
7605.2.a.bj 2 39.i odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 4 T^{3} + 3 T^{2} - 52 T + 169 \) Copy content Toggle raw display
$17$ \( T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$19$ \( T^{4} - 8 T^{3} + 60 T^{2} - 32 T + 16 \) Copy content Toggle raw display
$23$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 10 T^{3} + 78 T^{2} + \cdots + 484 \) Copy content Toggle raw display
$31$ \( (T^{2} - 2 T - 11)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 108 T^{2} + 11664 \) Copy content Toggle raw display
$41$ \( T^{4} - 18 T^{3} + 246 T^{2} + \cdots + 6084 \) Copy content Toggle raw display
$43$ \( T^{4} - 8 T^{3} + 75 T^{2} + 88 T + 121 \) Copy content Toggle raw display
$47$ \( (T^{2} + 10 T - 2)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 12 T + 24)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 6 T^{3} + 102 T^{2} + \cdots + 4356 \) Copy content Toggle raw display
$61$ \( T^{4} - 10 T^{3} + 183 T^{2} + \cdots + 6889 \) Copy content Toggle raw display
$67$ \( T^{4} - 8 T^{3} + 75 T^{2} + 88 T + 121 \) Copy content Toggle raw display
$71$ \( T^{4} - 6 T^{3} + 30 T^{2} - 36 T + 36 \) Copy content Toggle raw display
$73$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$79$ \( (T + 11)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 8 T - 32)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 14 T^{3} + 150 T^{2} + \cdots + 2116 \) Copy content Toggle raw display
$97$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
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