Properties

Label 190.2.b.b
Level $190$
Weight $2$
Character orbit 190.b
Analytic conductor $1.517$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [190,2,Mod(39,190)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(190, 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("190.39");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 190 = 2 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 190.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: \(1.51715763840\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5161984.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \) 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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + (\beta_{5} + \beta_{4}) q^{3} - q^{4} + ( - \beta_{3} - \beta_1) q^{5} + (\beta_{3} + 1) q^{6} + (\beta_{5} - \beta_{4}) q^{7} + \beta_{4} q^{8} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + (\beta_{5} + \beta_{4}) q^{3} - q^{4} + ( - \beta_{3} - \beta_1) q^{5} + (\beta_{3} + 1) q^{6} + (\beta_{5} - \beta_{4}) q^{7} + \beta_{4} q^{8} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{9} + (\beta_{5} + \beta_{2}) q^{10} + ( - \beta_{2} - \beta_1) q^{11} + ( - \beta_{5} - \beta_{4}) q^{12} + (\beta_{5} + 3 \beta_{4} + \cdots - \beta_1) q^{13}+ \cdots + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} + 2 q^{5} + 4 q^{6} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{4} + 2 q^{5} + 4 q^{6} - 10 q^{9} - 8 q^{14} + 8 q^{15} + 6 q^{16} - 6 q^{19} - 2 q^{20} - 20 q^{21} - 4 q^{24} + 10 q^{25} + 16 q^{26} + 16 q^{29} - 16 q^{30} + 8 q^{31} + 4 q^{34} + 8 q^{35} + 10 q^{36} - 20 q^{39} + 4 q^{41} + 18 q^{45} + 6 q^{49} - 4 q^{50} - 12 q^{51} - 4 q^{54} + 4 q^{55} + 8 q^{56} - 4 q^{59} - 8 q^{60} - 60 q^{61} - 6 q^{64} - 12 q^{65} + 16 q^{66} + 44 q^{69} - 20 q^{70} - 16 q^{71} - 28 q^{74} + 44 q^{75} + 6 q^{76} + 2 q^{80} - 34 q^{81} + 20 q^{84} - 12 q^{85} + 36 q^{86} + 28 q^{89} - 12 q^{90} + 12 q^{91} + 28 q^{94} - 2 q^{95} + 4 q^{96} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{5} - 2\nu^{4} - 25\nu^{3} + 10\nu^{2} - 121\nu + 100 ) / 121 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{5} + 27\nu^{4} + 35\nu^{3} - 14\nu^{2} + 223 ) / 121 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -25\nu^{5} - 10\nu^{4} - 4\nu^{3} + 50\nu^{2} - 605\nu + 258 ) / 242 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -65\nu^{5} - 26\nu^{4} + 38\nu^{3} + 372\nu^{2} - 1331\nu + 574 ) / 242 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 3\beta_{4} + \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{4} - 5\beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{3} + 7\beta_{2} + 7\beta _1 - 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} - 16\beta_{4} - 2\beta_{3} - 29\beta _1 + 16 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/190\mathbb{Z}\right)^\times\).

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

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
1.32001 1.32001i
−1.75233 + 1.75233i
0.432320 0.432320i
0.432320 + 0.432320i
−1.75233 1.75233i
1.32001 + 1.32001i
1.00000i 2.12489i −1.00000 1.80487 + 1.32001i −2.12489 4.12489i 1.00000i −1.51514 1.32001 1.80487i
39.2 1.00000i 1.36333i −1.00000 1.38900 1.75233i 1.36333 0.636672i 1.00000i 1.14134 −1.75233 1.38900i
39.3 1.00000i 2.76156i −1.00000 −2.19388 + 0.432320i 2.76156 0.761557i 1.00000i −4.62620 0.432320 + 2.19388i
39.4 1.00000i 2.76156i −1.00000 −2.19388 0.432320i 2.76156 0.761557i 1.00000i −4.62620 0.432320 2.19388i
39.5 1.00000i 1.36333i −1.00000 1.38900 + 1.75233i 1.36333 0.636672i 1.00000i 1.14134 −1.75233 + 1.38900i
39.6 1.00000i 2.12489i −1.00000 1.80487 1.32001i −2.12489 4.12489i 1.00000i −1.51514 1.32001 + 1.80487i
\(n\): 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 190.2.b.b 6
3.b odd 2 1 1710.2.d.d 6
4.b odd 2 1 1520.2.d.j 6
5.b even 2 1 inner 190.2.b.b 6
5.c odd 4 1 950.2.a.i 3
5.c odd 4 1 950.2.a.n 3
15.d odd 2 1 1710.2.d.d 6
15.e even 4 1 8550.2.a.ck 3
15.e even 4 1 8550.2.a.cl 3
20.d odd 2 1 1520.2.d.j 6
20.e even 4 1 7600.2.a.bi 3
20.e even 4 1 7600.2.a.cd 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.b.b 6 1.a even 1 1 trivial
190.2.b.b 6 5.b even 2 1 inner
950.2.a.i 3 5.c odd 4 1
950.2.a.n 3 5.c odd 4 1
1520.2.d.j 6 4.b odd 2 1
1520.2.d.j 6 20.d odd 2 1
1710.2.d.d 6 3.b odd 2 1
1710.2.d.d 6 15.d odd 2 1
7600.2.a.bi 3 20.e even 4 1
7600.2.a.cd 3 20.e even 4 1
8550.2.a.ck 3 15.e even 4 1
8550.2.a.cl 3 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 14T_{3}^{4} + 57T_{3}^{2} + 64 \) acting on \(S_{2}^{\mathrm{new}}(190, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} + 14 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{6} - 2 T^{5} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 18 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( (T^{3} - 10 T - 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 38 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$17$ \( T^{6} + 18 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( (T + 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + 98 T^{4} + \cdots + 14884 \) Copy content Toggle raw display
$29$ \( (T^{3} - 8 T^{2} + \cdots + 410)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 4 T^{2} + \cdots + 232)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 116 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$41$ \( (T^{3} - 2 T^{2} + \cdots - 100)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 128 T^{4} + \cdots + 21904 \) Copy content Toggle raw display
$47$ \( T^{6} + 132 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$53$ \( T^{6} + 102 T^{4} + \cdots + 11236 \) Copy content Toggle raw display
$59$ \( (T^{3} + 2 T^{2} - 29 T - 80)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 30 T^{2} + \cdots + 892)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 126 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$71$ \( (T^{3} + 8 T^{2} + \cdots - 1016)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 290 T^{4} + \cdots + 26896 \) Copy content Toggle raw display
$79$ \( (T^{3} - 228 T + 880)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 92 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$89$ \( (T^{3} - 14 T^{2} + \cdots + 20)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 300 T^{4} + \cdots + 238144 \) Copy content Toggle raw display
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