Properties

Label 740.2.o.a
Level $740$
Weight $2$
Character orbit 740.o
Analytic conductor $5.909$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [740,2,Mod(117,740)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(740, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("740.117");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 740.o (of order \(4\), 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: \(5.90892974957\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: 6.0.79423744.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 10x^{3} + 36x^{2} - 12x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + (\beta_{4} + 2) q^{5} + (\beta_{4} + \beta_{2} - 1) q^{7} + ( - \beta_{5} - \beta_{4} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + (\beta_{4} + 2) q^{5} + (\beta_{4} + \beta_{2} - 1) q^{7} + ( - \beta_{5} - \beta_{4} + \cdots - \beta_1) q^{9}+ \cdots + ( - 2 \beta_{3} - 5 \beta_{2} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} + 12 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{3} + 12 q^{5} - 4 q^{7} - 4 q^{13} - 6 q^{15} - 2 q^{19} - 16 q^{23} + 18 q^{25} + 22 q^{27} + 18 q^{29} - 20 q^{31} + 28 q^{33} - 12 q^{35} - 8 q^{37} + 30 q^{39} + 4 q^{43} + 10 q^{45} - 12 q^{47} + 26 q^{51} + 22 q^{53} + 4 q^{55} + 24 q^{57} - 18 q^{61} - 18 q^{63} - 8 q^{65} + 10 q^{67} + 32 q^{69} - 12 q^{71} + 2 q^{73} - 14 q^{75} - 24 q^{77} + 24 q^{79} - 34 q^{81} + 14 q^{83} + 4 q^{85} + 14 q^{89} - 26 q^{91} + 44 q^{93} - 2 q^{95} - 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 10x^{3} + 36x^{2} - 12x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -8\nu^{5} + 21\nu^{4} - 64\nu^{3} - 40\nu^{2} - 263\nu + 86 ) / 279 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13\nu^{5} - 69\nu^{4} + 104\nu^{3} + 65\nu^{2} - 26\nu - 1186 ) / 279 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 43\nu^{5} - 78\nu^{4} + 65\nu^{3} + 494\nu^{2} + 1588\nu - 253 ) / 279 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -20\nu^{5} + 37\nu^{4} - 36\nu^{3} - 193\nu^{2} - 766\nu + 122 ) / 31 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 4\beta_{4} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + 3\beta_{4} - \beta_{3} - 8\beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{3} - 13\beta_{2} - 13\beta _1 - 30 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -13\beta_{5} - 44\beta_{4} - 13\beta_{3} - 72\beta _1 - 44 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/740\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(297\) \(371\)
\(\chi(n)\) \(\beta_{4}\) \(-\beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
117.1
2.23976 2.23976i
0.159536 0.159536i
−1.39930 + 1.39930i
2.23976 + 2.23976i
0.159536 + 0.159536i
−1.39930 1.39930i
0 −2.23976 2.23976i 0 2.00000 1.00000i 0 1.23976 + 1.23976i 0 7.03304i 0
117.2 0 −0.159536 0.159536i 0 2.00000 1.00000i 0 −0.840464 0.840464i 0 2.94910i 0
117.3 0 1.39930 + 1.39930i 0 2.00000 1.00000i 0 −2.39930 2.39930i 0 0.916055i 0
253.1 0 −2.23976 + 2.23976i 0 2.00000 + 1.00000i 0 1.23976 1.23976i 0 7.03304i 0
253.2 0 −0.159536 + 0.159536i 0 2.00000 + 1.00000i 0 −0.840464 + 0.840464i 0 2.94910i 0
253.3 0 1.39930 1.39930i 0 2.00000 + 1.00000i 0 −2.39930 + 2.39930i 0 0.916055i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 117.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.o.a yes 6
5.c odd 4 1 740.2.l.a 6
37.d odd 4 1 740.2.l.a 6
185.k even 4 1 inner 740.2.o.a yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.l.a 6 5.c odd 4 1
740.2.l.a 6 37.d odd 4 1
740.2.o.a yes 6 1.a even 1 1 trivial
740.2.o.a yes 6 185.k even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 2T_{3}^{5} + 2T_{3}^{4} - 10T_{3}^{3} + 36T_{3}^{2} + 12T_{3} + 2 \) acting on \(S_{2}^{\mathrm{new}}(740, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 2 T^{5} + \cdots + 2 \) Copy content Toggle raw display
$5$ \( (T^{2} - 4 T + 5)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} + 4 T^{5} + \cdots + 50 \) Copy content Toggle raw display
$11$ \( T^{6} + 28 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( (T^{3} + 2 T^{2} - 28 T - 74)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 52 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$19$ \( T^{6} + 2 T^{5} + \cdots + 3362 \) Copy content Toggle raw display
$23$ \( (T^{3} + 8 T^{2} + 8 T - 12)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 6 T + 18)^{3} \) Copy content Toggle raw display
$31$ \( T^{6} + 20 T^{5} + \cdots + 7442 \) Copy content Toggle raw display
$37$ \( T^{6} + 8 T^{5} + \cdots + 50653 \) Copy content Toggle raw display
$41$ \( T^{6} + 152 T^{4} + \cdots + 76176 \) Copy content Toggle raw display
$43$ \( (T^{3} - 2 T^{2} + \cdots + 100)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 12 T^{5} + \cdots + 1250 \) Copy content Toggle raw display
$53$ \( T^{6} - 22 T^{5} + \cdots + 102152 \) Copy content Toggle raw display
$59$ \( T^{6} - 494 T^{3} + \cdots + 122018 \) Copy content Toggle raw display
$61$ \( T^{6} + 18 T^{5} + \cdots + 649800 \) Copy content Toggle raw display
$67$ \( T^{6} - 10 T^{5} + \cdots + 3042 \) Copy content Toggle raw display
$71$ \( (T^{3} + 6 T^{2} + \cdots - 248)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 2 T^{5} + \cdots + 16200 \) Copy content Toggle raw display
$79$ \( T^{6} - 24 T^{5} + \cdots + 40898 \) Copy content Toggle raw display
$83$ \( T^{6} - 14 T^{5} + \cdots + 5491298 \) Copy content Toggle raw display
$89$ \( T^{6} - 14 T^{5} + \cdots + 33800 \) Copy content Toggle raw display
$97$ \( T^{6} + 212 T^{4} + \cdots + 85264 \) Copy content Toggle raw display
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