Properties

Label 430.2.e.a
Level $430$
Weight $2$
Character orbit 430.e
Analytic conductor $3.434$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [430,2,Mod(221,430)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(430, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("430.221");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 430.e (of order \(3\), 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: \(3.43356728692\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + (3 \zeta_{6} - 3) q^{3} + q^{4} + (\zeta_{6} - 1) q^{5} + ( - 3 \zeta_{6} + 3) q^{6} - q^{8} - 6 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + (3 \zeta_{6} - 3) q^{3} + q^{4} + (\zeta_{6} - 1) q^{5} + ( - 3 \zeta_{6} + 3) q^{6} - q^{8} - 6 \zeta_{6} q^{9} + ( - \zeta_{6} + 1) q^{10} - 6 q^{11} + (3 \zeta_{6} - 3) q^{12} - 3 \zeta_{6} q^{15} + q^{16} + 4 \zeta_{6} q^{17} + 6 \zeta_{6} q^{18} + ( - 4 \zeta_{6} + 4) q^{19} + (\zeta_{6} - 1) q^{20} + 6 q^{22} + ( - 3 \zeta_{6} + 3) q^{24} - \zeta_{6} q^{25} + 9 q^{27} - 9 \zeta_{6} q^{29} + 3 \zeta_{6} q^{30} + (4 \zeta_{6} - 4) q^{31} - q^{32} + ( - 18 \zeta_{6} + 18) q^{33} - 4 \zeta_{6} q^{34} - 6 \zeta_{6} q^{36} + ( - 8 \zeta_{6} + 8) q^{37} + (4 \zeta_{6} - 4) q^{38} + ( - \zeta_{6} + 1) q^{40} + 5 q^{41} + (\zeta_{6} - 7) q^{43} - 6 q^{44} + 6 q^{45} - 3 q^{47} + (3 \zeta_{6} - 3) q^{48} + ( - 7 \zeta_{6} + 7) q^{49} + \zeta_{6} q^{50} - 12 q^{51} + (6 \zeta_{6} - 6) q^{53} - 9 q^{54} + ( - 6 \zeta_{6} + 6) q^{55} + 12 \zeta_{6} q^{57} + 9 \zeta_{6} q^{58} + 2 q^{59} - 3 \zeta_{6} q^{60} + 2 \zeta_{6} q^{61} + ( - 4 \zeta_{6} + 4) q^{62} + q^{64} + (18 \zeta_{6} - 18) q^{66} + (11 \zeta_{6} - 11) q^{67} + 4 \zeta_{6} q^{68} - 8 \zeta_{6} q^{71} + 6 \zeta_{6} q^{72} - 8 \zeta_{6} q^{73} + (8 \zeta_{6} - 8) q^{74} + 3 q^{75} + ( - 4 \zeta_{6} + 4) q^{76} + (\zeta_{6} - 1) q^{80} + (9 \zeta_{6} - 9) q^{81} - 5 q^{82} + (11 \zeta_{6} - 11) q^{83} - 4 q^{85} + ( - \zeta_{6} + 7) q^{86} + 27 q^{87} + 6 q^{88} + (13 \zeta_{6} - 13) q^{89} - 6 q^{90} - 12 \zeta_{6} q^{93} + 3 q^{94} + 4 \zeta_{6} q^{95} + ( - 3 \zeta_{6} + 3) q^{96} - 8 q^{97} + (7 \zeta_{6} - 7) q^{98} + 36 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 3 q^{3} + 2 q^{4} - q^{5} + 3 q^{6} - 2 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 3 q^{3} + 2 q^{4} - q^{5} + 3 q^{6} - 2 q^{8} - 6 q^{9} + q^{10} - 12 q^{11} - 3 q^{12} - 3 q^{15} + 2 q^{16} + 4 q^{17} + 6 q^{18} + 4 q^{19} - q^{20} + 12 q^{22} + 3 q^{24} - q^{25} + 18 q^{27} - 9 q^{29} + 3 q^{30} - 4 q^{31} - 2 q^{32} + 18 q^{33} - 4 q^{34} - 6 q^{36} + 8 q^{37} - 4 q^{38} + q^{40} + 10 q^{41} - 13 q^{43} - 12 q^{44} + 12 q^{45} - 6 q^{47} - 3 q^{48} + 7 q^{49} + q^{50} - 24 q^{51} - 6 q^{53} - 18 q^{54} + 6 q^{55} + 12 q^{57} + 9 q^{58} + 4 q^{59} - 3 q^{60} + 2 q^{61} + 4 q^{62} + 2 q^{64} - 18 q^{66} - 11 q^{67} + 4 q^{68} - 8 q^{71} + 6 q^{72} - 8 q^{73} - 8 q^{74} + 6 q^{75} + 4 q^{76} - q^{80} - 9 q^{81} - 10 q^{82} - 11 q^{83} - 8 q^{85} + 13 q^{86} + 54 q^{87} + 12 q^{88} - 13 q^{89} - 12 q^{90} - 12 q^{93} + 6 q^{94} + 4 q^{95} + 3 q^{96} - 16 q^{97} - 7 q^{98} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/430\mathbb{Z}\right)^\times\).

\(n\) \(87\) \(261\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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
0.500000 + 0.866025i
0.500000 0.866025i
−1.00000 −1.50000 + 2.59808i 1.00000 −0.500000 + 0.866025i 1.50000 2.59808i 0 −1.00000 −3.00000 5.19615i 0.500000 0.866025i
251.1 −1.00000 −1.50000 2.59808i 1.00000 −0.500000 0.866025i 1.50000 + 2.59808i 0 −1.00000 −3.00000 + 5.19615i 0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.e.a 2
43.c even 3 1 inner 430.2.e.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.e.a 2 1.a even 1 1 trivial
430.2.e.a 2 43.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 3T_{3} + 9 \) acting on \(S_{2}^{\mathrm{new}}(430, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$19$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$31$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$37$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$41$ \( (T - 5)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 13T + 43 \) Copy content Toggle raw display
$47$ \( (T + 3)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$59$ \( (T - 2)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$71$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$73$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$89$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$97$ \( (T + 8)^{2} \) Copy content Toggle raw display
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