Properties

Label 740.2.bc.a
Level $740$
Weight $2$
Character orbit 740.bc
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(81,740)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(740, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("740.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 740.bc (of order \(9\), degree \(6\), 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: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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 \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{18}^{3} - \zeta_{18}^{2} - \zeta_{18}) q^{3} + (\zeta_{18}^{5} - \zeta_{18}^{2}) q^{5} + ( - 2 \zeta_{18}^{5} + \cdots + 2 \zeta_{18}^{2}) q^{7}+ \cdots + (2 \zeta_{18}^{5} + 3 \zeta_{18}^{3} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{18}^{3} - \zeta_{18}^{2} - \zeta_{18}) q^{3} + (\zeta_{18}^{5} - \zeta_{18}^{2}) q^{5} + ( - 2 \zeta_{18}^{5} + \cdots + 2 \zeta_{18}^{2}) q^{7}+ \cdots + (9 \zeta_{18}^{5} - 5 \zeta_{18}^{4} + \cdots - 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{3} + 3 q^{7} + 3 q^{9} + 12 q^{11} - 9 q^{13} + 6 q^{15} - 15 q^{17} + 9 q^{19} - 12 q^{21} - 3 q^{23} + 6 q^{27} - 3 q^{29} + 42 q^{31} + 12 q^{33} + 3 q^{35} + 6 q^{37} + 12 q^{39} - 6 q^{41} + 30 q^{43} + 27 q^{47} - 15 q^{49} - 15 q^{51} + 12 q^{53} + 3 q^{55} - 6 q^{57} + 30 q^{59} + 9 q^{61} + 3 q^{63} - 9 q^{65} + 6 q^{69} - 27 q^{71} - 6 q^{75} - 21 q^{77} - 21 q^{79} + 6 q^{81} - 9 q^{83} + 3 q^{85} - 3 q^{87} + 6 q^{89} + 15 q^{91} - 39 q^{93} - 18 q^{95} + 6 q^{97} - 33 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(261\) \(297\) \(371\)
\(\chi(n)\) \(-\zeta_{18}^{5}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
−0.173648 + 0.984808i
−0.766044 0.642788i
−0.173648 0.984808i
0.939693 0.342020i
−0.766044 + 0.642788i
0.939693 + 0.342020i
0 0.613341 + 0.223238i 0 0.173648 + 0.984808i 0 −0.613341 3.47843i 0 −1.97178 1.65452i 0
181.1 0 0.0923963 + 0.524005i 0 0.766044 0.642788i 0 −0.0923963 + 0.0775297i 0 2.55303 0.929228i 0
201.1 0 0.613341 0.223238i 0 0.173648 0.984808i 0 −0.613341 + 3.47843i 0 −1.97178 + 1.65452i 0
441.1 0 −2.20574 + 1.85083i 0 −0.939693 0.342020i 0 2.20574 + 0.802823i 0 0.918748 5.21048i 0
601.1 0 0.0923963 0.524005i 0 0.766044 + 0.642788i 0 −0.0923963 0.0775297i 0 2.55303 + 0.929228i 0
641.1 0 −2.20574 1.85083i 0 −0.939693 + 0.342020i 0 2.20574 0.802823i 0 0.918748 + 5.21048i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.bc.a 6
37.f even 9 1 inner 740.2.bc.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.bc.a 6 1.a even 1 1 trivial
740.2.bc.a 6 37.f even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 3T_{3}^{5} + 3T_{3}^{4} - 8T_{3}^{3} + 6T_{3}^{2} - 3T_{3} + 1 \) 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
$3$ \( T^{6} + 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$7$ \( T^{6} - 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{6} - 12 T^{5} + \cdots + 2601 \) Copy content Toggle raw display
$13$ \( T^{6} + 9 T^{5} + \cdots + 289 \) Copy content Toggle raw display
$17$ \( T^{6} + 15 T^{5} + \cdots + 3249 \) Copy content Toggle raw display
$19$ \( T^{6} - 9 T^{5} + \cdots + 361 \) Copy content Toggle raw display
$23$ \( T^{6} + 3 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$29$ \( T^{6} + 3 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$31$ \( (T^{3} - 21 T^{2} + \cdots - 251)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} - 6 T^{5} + \cdots + 50653 \) Copy content Toggle raw display
$41$ \( T^{6} + 6 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$43$ \( (T^{3} - 15 T^{2} + \cdots + 181)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} - 27 T^{5} + \cdots + 110889 \) Copy content Toggle raw display
$53$ \( T^{6} - 12 T^{5} + \cdots + 3249 \) Copy content Toggle raw display
$59$ \( T^{6} - 30 T^{5} + \cdots + 140625 \) Copy content Toggle raw display
$61$ \( T^{6} - 9 T^{5} + \cdots + 39601 \) Copy content Toggle raw display
$67$ \( T^{6} + 54 T^{4} + \cdots + 104329 \) Copy content Toggle raw display
$71$ \( T^{6} + 27 T^{5} + \cdots + 210681 \) Copy content Toggle raw display
$73$ \( (T^{3} - 57 T - 107)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 21 T^{5} + \cdots + 292681 \) Copy content Toggle raw display
$83$ \( T^{6} + 9 T^{5} + \cdots + 29241 \) Copy content Toggle raw display
$89$ \( T^{6} - 6 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$97$ \( T^{6} - 6 T^{5} + \cdots + 11881 \) Copy content Toggle raw display
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