Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [740,2,Mod(221,740)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(740, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("740.221");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 740.h (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: \(5.90892974957\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) 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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + i q^{5} - q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + i q^{5} - q^{7} + 6 q^{9} + 3 q^{11} - 4 i q^{13} + 3 i q^{15} + 2 i q^{17} + 2 i q^{19} - 3 q^{21} - 6 i q^{23} - q^{25} + 9 q^{27} + 2 i q^{29} + 10 i q^{31} + 9 q^{33} - i q^{35} + (6 i + 1) q^{37} - 12 i q^{39} + 5 q^{41} - 2 i q^{43} + 6 i q^{45} - 5 q^{47} - 6 q^{49} + 6 i q^{51} - 9 q^{53} + 3 i q^{55} + 6 i q^{57} - 12 i q^{59} - 10 i q^{61} - 6 q^{63} + 4 q^{65} - 4 q^{67} - 18 i q^{69} - 13 q^{71} + 3 q^{73} - 3 q^{75} - 3 q^{77} + 9 q^{81} - 3 q^{83} - 2 q^{85} + 6 i q^{87} + 4 i q^{89} + 4 i q^{91} + 30 i q^{93} - 2 q^{95} - 2 i q^{97} + 18 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 2 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} - 2 q^{7} + 12 q^{9} + 6 q^{11} - 6 q^{21} - 2 q^{25} + 18 q^{27} + 18 q^{33} + 2 q^{37} + 10 q^{41} - 10 q^{47} - 12 q^{49} - 18 q^{53} - 12 q^{63} + 8 q^{65} - 8 q^{67} - 26 q^{71} + 6 q^{73} - 6 q^{75} - 6 q^{77} + 18 q^{81} - 6 q^{83} - 4 q^{85} - 4 q^{95} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(261\) \(297\) \(371\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
221.1
1.00000i
1.00000i
0 3.00000 0 1.00000i 0 −1.00000 0 6.00000 0
221.2 0 3.00000 0 1.00000i 0 −1.00000 0 6.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.h.a 2
3.b odd 2 1 6660.2.n.d 2
4.b odd 2 1 2960.2.p.a 2
5.b even 2 1 3700.2.h.a 2
5.c odd 4 1 3700.2.e.a 2
5.c odd 4 1 3700.2.e.b 2
37.b even 2 1 inner 740.2.h.a 2
111.d odd 2 1 6660.2.n.d 2
148.b odd 2 1 2960.2.p.a 2
185.d even 2 1 3700.2.h.a 2
185.h odd 4 1 3700.2.e.a 2
185.h odd 4 1 3700.2.e.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.h.a 2 1.a even 1 1 trivial
740.2.h.a 2 37.b even 2 1 inner
2960.2.p.a 2 4.b odd 2 1
2960.2.p.a 2 148.b odd 2 1
3700.2.e.a 2 5.c odd 4 1
3700.2.e.a 2 185.h odd 4 1
3700.2.e.b 2 5.c odd 4 1
3700.2.e.b 2 185.h odd 4 1
3700.2.h.a 2 5.b even 2 1
3700.2.h.a 2 185.d even 2 1
6660.2.n.d 2 3.b odd 2 1
6660.2.n.d 2 111.d odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 1 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{2} + 4 \) Copy content Toggle raw display
$19$ \( T^{2} + 4 \) Copy content Toggle raw display
$23$ \( T^{2} + 36 \) Copy content Toggle raw display
$29$ \( T^{2} + 4 \) Copy content Toggle raw display
$31$ \( T^{2} + 100 \) Copy content Toggle raw display
$37$ \( T^{2} - 2T + 37 \) Copy content Toggle raw display
$41$ \( (T - 5)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 4 \) Copy content Toggle raw display
$47$ \( (T + 5)^{2} \) Copy content Toggle raw display
$53$ \( (T + 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 144 \) Copy content Toggle raw display
$61$ \( T^{2} + 100 \) Copy content Toggle raw display
$67$ \( (T + 4)^{2} \) Copy content Toggle raw display
$71$ \( (T + 13)^{2} \) Copy content Toggle raw display
$73$ \( (T - 3)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( (T + 3)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 16 \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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