Properties

Label 414.4.a.b
Level $414$
Weight $4$
Character orbit 414.a
Self dual yes
Analytic conductor $24.427$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [414,4,Mod(1,414)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(414, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("414.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 414.a (trivial)

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: yes
Analytic conductor: \(24.4267907424\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 46)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 2 q^{2} + 4 q^{4} + 20 q^{5} + 2 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 4 q^{4} + 20 q^{5} + 2 q^{7} - 8 q^{8} - 40 q^{10} + 52 q^{11} + 43 q^{13} - 4 q^{14} + 16 q^{16} + 50 q^{17} - 74 q^{19} + 80 q^{20} - 104 q^{22} + 23 q^{23} + 275 q^{25} - 86 q^{26} + 8 q^{28} + 7 q^{29} - 273 q^{31} - 32 q^{32} - 100 q^{34} + 40 q^{35} - 4 q^{37} + 148 q^{38} - 160 q^{40} - 123 q^{41} - 152 q^{43} + 208 q^{44} - 46 q^{46} - 75 q^{47} - 339 q^{49} - 550 q^{50} + 172 q^{52} - 86 q^{53} + 1040 q^{55} - 16 q^{56} - 14 q^{58} + 444 q^{59} + 262 q^{61} + 546 q^{62} + 64 q^{64} + 860 q^{65} + 764 q^{67} + 200 q^{68} - 80 q^{70} + 21 q^{71} + 681 q^{73} + 8 q^{74} - 296 q^{76} + 104 q^{77} + 426 q^{79} + 320 q^{80} + 246 q^{82} - 902 q^{83} + 1000 q^{85} + 304 q^{86} - 416 q^{88} + 1272 q^{89} + 86 q^{91} + 92 q^{92} + 150 q^{94} - 1480 q^{95} - 342 q^{97} + 678 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−2.00000 0 4.00000 20.0000 0 2.00000 −8.00000 0 −40.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(23\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 414.4.a.b 1
3.b odd 2 1 46.4.a.b 1
12.b even 2 1 368.4.a.e 1
15.d odd 2 1 1150.4.a.d 1
15.e even 4 2 1150.4.b.a 2
21.c even 2 1 2254.4.a.b 1
24.f even 2 1 1472.4.a.a 1
24.h odd 2 1 1472.4.a.j 1
69.c even 2 1 1058.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
46.4.a.b 1 3.b odd 2 1
368.4.a.e 1 12.b even 2 1
414.4.a.b 1 1.a even 1 1 trivial
1058.4.a.b 1 69.c even 2 1
1150.4.a.d 1 15.d odd 2 1
1150.4.b.a 2 15.e even 4 2
1472.4.a.a 1 24.f even 2 1
1472.4.a.j 1 24.h odd 2 1
2254.4.a.b 1 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(414))\):

\( T_{5} - 20 \) Copy content Toggle raw display
\( T_{7} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 20 \) Copy content Toggle raw display
$7$ \( T - 2 \) Copy content Toggle raw display
$11$ \( T - 52 \) Copy content Toggle raw display
$13$ \( T - 43 \) Copy content Toggle raw display
$17$ \( T - 50 \) Copy content Toggle raw display
$19$ \( T + 74 \) Copy content Toggle raw display
$23$ \( T - 23 \) Copy content Toggle raw display
$29$ \( T - 7 \) Copy content Toggle raw display
$31$ \( T + 273 \) Copy content Toggle raw display
$37$ \( T + 4 \) Copy content Toggle raw display
$41$ \( T + 123 \) Copy content Toggle raw display
$43$ \( T + 152 \) Copy content Toggle raw display
$47$ \( T + 75 \) Copy content Toggle raw display
$53$ \( T + 86 \) Copy content Toggle raw display
$59$ \( T - 444 \) Copy content Toggle raw display
$61$ \( T - 262 \) Copy content Toggle raw display
$67$ \( T - 764 \) Copy content Toggle raw display
$71$ \( T - 21 \) Copy content Toggle raw display
$73$ \( T - 681 \) Copy content Toggle raw display
$79$ \( T - 426 \) Copy content Toggle raw display
$83$ \( T + 902 \) Copy content Toggle raw display
$89$ \( T - 1272 \) Copy content Toggle raw display
$97$ \( T + 342 \) Copy content Toggle raw display
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