Properties

Label 338.4.a.d
Level $338$
Weight $4$
Character orbit 338.a
Self dual yes
Analytic conductor $19.943$
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] = [338,4,Mod(1,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 338.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: \(19.9426455819\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
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)
 
\(f(q)\) \(=\) \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} - 2 q^{5} - 6 q^{6} + 5 q^{7} + 8 q^{8} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} - 2 q^{5} - 6 q^{6} + 5 q^{7} + 8 q^{8} - 18 q^{9} - 4 q^{10} - 13 q^{11} - 12 q^{12} + 10 q^{14} + 6 q^{15} + 16 q^{16} + 27 q^{17} - 36 q^{18} - 75 q^{19} - 8 q^{20} - 15 q^{21} - 26 q^{22} - 187 q^{23} - 24 q^{24} - 121 q^{25} + 135 q^{27} + 20 q^{28} - 13 q^{29} + 12 q^{30} + 104 q^{31} + 32 q^{32} + 39 q^{33} + 54 q^{34} - 10 q^{35} - 72 q^{36} - 423 q^{37} - 150 q^{38} - 16 q^{40} - 195 q^{41} - 30 q^{42} + 199 q^{43} - 52 q^{44} + 36 q^{45} - 374 q^{46} - 388 q^{47} - 48 q^{48} - 318 q^{49} - 242 q^{50} - 81 q^{51} + 618 q^{53} + 270 q^{54} + 26 q^{55} + 40 q^{56} + 225 q^{57} - 26 q^{58} - 491 q^{59} + 24 q^{60} + 175 q^{61} + 208 q^{62} - 90 q^{63} + 64 q^{64} + 78 q^{66} - 817 q^{67} + 108 q^{68} + 561 q^{69} - 20 q^{70} - 79 q^{71} - 144 q^{72} - 230 q^{73} - 846 q^{74} + 363 q^{75} - 300 q^{76} - 65 q^{77} + 764 q^{79} - 32 q^{80} + 81 q^{81} - 390 q^{82} + 732 q^{83} - 60 q^{84} - 54 q^{85} + 398 q^{86} + 39 q^{87} - 104 q^{88} + 1041 q^{89} + 72 q^{90} - 748 q^{92} - 312 q^{93} - 776 q^{94} + 150 q^{95} - 96 q^{96} + 97 q^{97} - 636 q^{98} + 234 q^{99}+O(q^{100}) \) 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 −3.00000 4.00000 −2.00000 −6.00000 5.00000 8.00000 −18.0000 −4.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(13\) \(1\)

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 338.4.a.d 1
13.b even 2 1 338.4.a.a 1
13.c even 3 2 338.4.c.d 2
13.d odd 4 2 338.4.b.a 2
13.e even 6 2 26.4.c.a 2
13.f odd 12 4 338.4.e.d 4
39.h odd 6 2 234.4.h.b 2
52.i odd 6 2 208.4.i.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.c.a 2 13.e even 6 2
208.4.i.a 2 52.i odd 6 2
234.4.h.b 2 39.h odd 6 2
338.4.a.a 1 13.b even 2 1
338.4.a.d 1 1.a even 1 1 trivial
338.4.b.a 2 13.d odd 4 2
338.4.c.d 2 13.c even 3 2
338.4.e.d 4 13.f odd 12 4

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(338))\):

\( T_{3} + 3 \) Copy content Toggle raw display
\( T_{5} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T + 2 \) Copy content Toggle raw display
$7$ \( T - 5 \) Copy content Toggle raw display
$11$ \( T + 13 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 27 \) Copy content Toggle raw display
$19$ \( T + 75 \) Copy content Toggle raw display
$23$ \( T + 187 \) Copy content Toggle raw display
$29$ \( T + 13 \) Copy content Toggle raw display
$31$ \( T - 104 \) Copy content Toggle raw display
$37$ \( T + 423 \) Copy content Toggle raw display
$41$ \( T + 195 \) Copy content Toggle raw display
$43$ \( T - 199 \) Copy content Toggle raw display
$47$ \( T + 388 \) Copy content Toggle raw display
$53$ \( T - 618 \) Copy content Toggle raw display
$59$ \( T + 491 \) Copy content Toggle raw display
$61$ \( T - 175 \) Copy content Toggle raw display
$67$ \( T + 817 \) Copy content Toggle raw display
$71$ \( T + 79 \) Copy content Toggle raw display
$73$ \( T + 230 \) Copy content Toggle raw display
$79$ \( T - 764 \) Copy content Toggle raw display
$83$ \( T - 732 \) Copy content Toggle raw display
$89$ \( T - 1041 \) Copy content Toggle raw display
$97$ \( T - 97 \) Copy content Toggle raw display
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