Properties

Label 414.6.a.d
Level $414$
Weight $6$
Character orbit 414.a
Self dual yes
Analytic conductor $66.399$
Analytic rank $1$
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] = [414,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("414.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(66.3989014026\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 138)
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)
 
\(f(q)\) \(=\) \( q + 4 q^{2} + 16 q^{4} + 44 q^{5} - 70 q^{7} + 64 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + 16 q^{4} + 44 q^{5} - 70 q^{7} + 64 q^{8} + 176 q^{10} + 136 q^{11} - 1022 q^{13} - 280 q^{14} + 256 q^{16} - 484 q^{17} - 1046 q^{19} + 704 q^{20} + 544 q^{22} - 529 q^{23} - 1189 q^{25} - 4088 q^{26} - 1120 q^{28} - 2618 q^{29} - 4860 q^{31} + 1024 q^{32} - 1936 q^{34} - 3080 q^{35} + 14918 q^{37} - 4184 q^{38} + 2816 q^{40} - 7530 q^{41} + 16186 q^{43} + 2176 q^{44} - 2116 q^{46} - 29160 q^{47} - 11907 q^{49} - 4756 q^{50} - 16352 q^{52} - 9896 q^{53} + 5984 q^{55} - 4480 q^{56} - 10472 q^{58} + 2004 q^{59} - 2570 q^{61} - 19440 q^{62} + 4096 q^{64} - 44968 q^{65} + 46118 q^{67} - 7744 q^{68} - 12320 q^{70} + 32688 q^{71} - 46830 q^{73} + 59672 q^{74} - 16736 q^{76} - 9520 q^{77} - 34338 q^{79} + 11264 q^{80} - 30120 q^{82} - 31736 q^{83} - 21296 q^{85} + 64744 q^{86} + 8704 q^{88} + 60792 q^{89} + 71540 q^{91} - 8464 q^{92} - 116640 q^{94} - 46024 q^{95} - 19218 q^{97} - 47628 q^{98}+O(q^{100}) \) 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
4.00000 0 16.0000 44.0000 0 −70.0000 64.0000 0 176.000
\(n\): 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 414.6.a.d 1
3.b odd 2 1 138.6.a.a 1
12.b even 2 1 1104.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.6.a.a 1 3.b odd 2 1
414.6.a.d 1 1.a even 1 1 trivial
1104.6.a.c 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 44 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(414))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 4 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 44 \) Copy content Toggle raw display
$7$ \( T + 70 \) Copy content Toggle raw display
$11$ \( T - 136 \) Copy content Toggle raw display
$13$ \( T + 1022 \) Copy content Toggle raw display
$17$ \( T + 484 \) Copy content Toggle raw display
$19$ \( T + 1046 \) Copy content Toggle raw display
$23$ \( T + 529 \) Copy content Toggle raw display
$29$ \( T + 2618 \) Copy content Toggle raw display
$31$ \( T + 4860 \) Copy content Toggle raw display
$37$ \( T - 14918 \) Copy content Toggle raw display
$41$ \( T + 7530 \) Copy content Toggle raw display
$43$ \( T - 16186 \) Copy content Toggle raw display
$47$ \( T + 29160 \) Copy content Toggle raw display
$53$ \( T + 9896 \) Copy content Toggle raw display
$59$ \( T - 2004 \) Copy content Toggle raw display
$61$ \( T + 2570 \) Copy content Toggle raw display
$67$ \( T - 46118 \) Copy content Toggle raw display
$71$ \( T - 32688 \) Copy content Toggle raw display
$73$ \( T + 46830 \) Copy content Toggle raw display
$79$ \( T + 34338 \) Copy content Toggle raw display
$83$ \( T + 31736 \) Copy content Toggle raw display
$89$ \( T - 60792 \) Copy content Toggle raw display
$97$ \( T + 19218 \) Copy content Toggle raw display
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