Properties

Label 95.2.b.b
Level $95$
Weight $2$
Character orbit 95.b
Analytic conductor $0.759$
Analytic rank $0$
Dimension $6$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [95,2,Mod(39,95)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(95, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("95.39");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 95 = 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 95.b (of order \(2\), degree \(1\), 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: \(0.758578819202\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.16516096.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 9x^{4} + 13x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + ( - \beta_{5} + \beta_{4} + \cdots - \beta_1) q^{3}+ \cdots + ( - 2 \beta_{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + ( - \beta_{5} + \beta_{4} + \cdots - \beta_1) q^{3}+ \cdots + ( - \beta_{4} + \beta_{3} - 4 \beta_{2} + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 8 q^{4} - q^{5} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 8 q^{4} - q^{5} - 14 q^{9} + 12 q^{10} + 2 q^{11} - 16 q^{14} + 10 q^{15} - 4 q^{16} + 6 q^{19} + 10 q^{20} + 20 q^{21} - 8 q^{24} + 3 q^{25} + 8 q^{26} - 36 q^{29} + 24 q^{30} + 8 q^{34} - 3 q^{35} - 32 q^{36} + 8 q^{39} + 8 q^{40} + 12 q^{41} - 20 q^{44} - 15 q^{45} - 8 q^{46} + 4 q^{49} - 4 q^{50} + 4 q^{51} + 16 q^{54} - 33 q^{55} + 40 q^{56} - 20 q^{59} + 20 q^{60} - 14 q^{61} + 12 q^{64} - 20 q^{65} - 48 q^{66} + 24 q^{69} + 20 q^{70} + 52 q^{71} + 40 q^{74} - 34 q^{75} - 8 q^{76} + 24 q^{79} - 32 q^{80} + 38 q^{81} + 24 q^{84} + 13 q^{85} - 16 q^{86} - 24 q^{89} - 28 q^{90} - 24 q^{91} + 48 q^{94} - q^{95} - 64 q^{96} + 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 9x^{4} + 13x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 8\nu^{2} + 5 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + \nu^{4} + 10\nu^{3} + 6\nu^{2} + 19\nu - 1 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - \nu^{4} + 10\nu^{3} - 6\nu^{2} + 19\nu + 1 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} - 8\nu^{3} - 7\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} + \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{4} + 8\beta_{3} - 6\beta_{2} + 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -10\beta_{5} - 8\beta_{4} - 8\beta_{3} + 41\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
39.1
1.30397i
2.68667i
0.285442i
0.285442i
2.68667i
1.30397i
2.41987i 0.537080i −3.85577 −2.07772 + 0.826491i −1.29966 3.18676i 4.49073i 2.71155 2.00000 + 5.02781i
39.2 1.82254i 2.31446i −1.32164 1.94827 + 1.09737i 4.21819 1.45033i 1.23634i −2.35673 2.00000 3.55080i
39.3 0.906968i 3.21789i 1.17741 −0.370556 + 2.20515i −2.91852 2.59637i 2.88181i −7.35482 2.00000 + 0.336083i
39.4 0.906968i 3.21789i 1.17741 −0.370556 2.20515i −2.91852 2.59637i 2.88181i −7.35482 2.00000 0.336083i
39.5 1.82254i 2.31446i −1.32164 1.94827 1.09737i 4.21819 1.45033i 1.23634i −2.35673 2.00000 + 3.55080i
39.6 2.41987i 0.537080i −3.85577 −2.07772 0.826491i −1.29966 3.18676i 4.49073i 2.71155 2.00000 5.02781i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 39.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 95.2.b.b 6
3.b odd 2 1 855.2.c.d 6
4.b odd 2 1 1520.2.d.h 6
5.b even 2 1 inner 95.2.b.b 6
5.c odd 4 2 475.2.a.j 6
15.d odd 2 1 855.2.c.d 6
15.e even 4 2 4275.2.a.br 6
19.b odd 2 1 1805.2.b.e 6
20.d odd 2 1 1520.2.d.h 6
20.e even 4 2 7600.2.a.ck 6
95.d odd 2 1 1805.2.b.e 6
95.g even 4 2 9025.2.a.bx 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.b.b 6 1.a even 1 1 trivial
95.2.b.b 6 5.b even 2 1 inner
475.2.a.j 6 5.c odd 4 2
855.2.c.d 6 3.b odd 2 1
855.2.c.d 6 15.d odd 2 1
1520.2.d.h 6 4.b odd 2 1
1520.2.d.h 6 20.d odd 2 1
1805.2.b.e 6 19.b odd 2 1
1805.2.b.e 6 95.d odd 2 1
4275.2.a.br 6 15.e even 4 2
7600.2.a.ck 6 20.e even 4 2
9025.2.a.bx 6 95.g even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 10T_{2}^{4} + 27T_{2}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(95, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 10 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{6} + 16 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{6} + T^{5} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 19 T^{4} + \cdots + 144 \) Copy content Toggle raw display
$11$ \( (T^{3} - T^{2} - 16 T + 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 28 T^{4} + \cdots + 576 \) Copy content Toggle raw display
$17$ \( T^{6} + 59 T^{4} + \cdots + 5184 \) Copy content Toggle raw display
$19$ \( (T - 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + 36 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( (T + 6)^{6} \) Copy content Toggle raw display
$31$ \( (T^{3} - 56 T + 128)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 56 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$41$ \( (T^{3} - 6 T^{2} - 44 T - 24)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 19 T^{4} + \cdots + 144 \) Copy content Toggle raw display
$47$ \( T^{6} + 187 T^{4} + \cdots + 85264 \) Copy content Toggle raw display
$53$ \( T^{6} + 156 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$59$ \( (T^{3} + 10 T^{2} + \cdots - 48)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 7 T^{2} + \cdots - 776)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 340 T^{4} + \cdots + 484416 \) Copy content Toggle raw display
$71$ \( (T^{3} - 26 T^{2} + \cdots - 432)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 131 T^{4} + \cdots + 5184 \) Copy content Toggle raw display
$79$ \( (T^{3} - 12 T^{2} + \cdots + 32)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 228 T^{4} + \cdots + 141376 \) Copy content Toggle raw display
$89$ \( (T^{3} + 12 T^{2} + \cdots - 3456)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 28 T^{4} + \cdots + 576 \) Copy content Toggle raw display
show more
show less