Properties

Label 338.4.a.e
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)
 
\(f(q)\) \(=\) \( q + 2 q^{2} + 3 q^{3} + 4 q^{4} - 11 q^{5} + 6 q^{6} - 19 q^{7} + 8 q^{8} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 3 q^{3} + 4 q^{4} - 11 q^{5} + 6 q^{6} - 19 q^{7} + 8 q^{8} - 18 q^{9} - 22 q^{10} + 38 q^{11} + 12 q^{12} - 38 q^{14} - 33 q^{15} + 16 q^{16} - 51 q^{17} - 36 q^{18} - 90 q^{19} - 44 q^{20} - 57 q^{21} + 76 q^{22} - 52 q^{23} + 24 q^{24} - 4 q^{25} - 135 q^{27} - 76 q^{28} - 190 q^{29} - 66 q^{30} - 292 q^{31} + 32 q^{32} + 114 q^{33} - 102 q^{34} + 209 q^{35} - 72 q^{36} + 441 q^{37} - 180 q^{38} - 88 q^{40} - 312 q^{41} - 114 q^{42} + 373 q^{43} + 152 q^{44} + 198 q^{45} - 104 q^{46} + 41 q^{47} + 48 q^{48} + 18 q^{49} - 8 q^{50} - 153 q^{51} + 468 q^{53} - 270 q^{54} - 418 q^{55} - 152 q^{56} - 270 q^{57} - 380 q^{58} - 530 q^{59} - 132 q^{60} + 592 q^{61} - 584 q^{62} + 342 q^{63} + 64 q^{64} + 228 q^{66} + 206 q^{67} - 204 q^{68} - 156 q^{69} + 418 q^{70} + 863 q^{71} - 144 q^{72} + 322 q^{73} + 882 q^{74} - 12 q^{75} - 360 q^{76} - 722 q^{77} - 460 q^{79} - 176 q^{80} + 81 q^{81} - 624 q^{82} - 528 q^{83} - 228 q^{84} + 561 q^{85} + 746 q^{86} - 570 q^{87} + 304 q^{88} - 870 q^{89} + 396 q^{90} - 208 q^{92} - 876 q^{93} + 82 q^{94} + 990 q^{95} + 96 q^{96} + 346 q^{97} + 36 q^{98} - 684 q^{99}+O(q^{100}) \) 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 −11.0000 6.00000 −19.0000 8.00000 −18.0000 −22.0000
\(n\): 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.e 1
13.b even 2 1 26.4.a.a 1
13.c even 3 2 338.4.c.b 2
13.d odd 4 2 338.4.b.c 2
13.e even 6 2 338.4.c.f 2
13.f odd 12 4 338.4.e.b 4
39.d odd 2 1 234.4.a.g 1
52.b odd 2 1 208.4.a.c 1
65.d even 2 1 650.4.a.f 1
65.h odd 4 2 650.4.b.b 2
91.b odd 2 1 1274.4.a.b 1
104.e even 2 1 832.4.a.e 1
104.h odd 2 1 832.4.a.m 1
156.h even 2 1 1872.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.a.a 1 13.b even 2 1
208.4.a.c 1 52.b odd 2 1
234.4.a.g 1 39.d odd 2 1
338.4.a.e 1 1.a even 1 1 trivial
338.4.b.c 2 13.d odd 4 2
338.4.c.b 2 13.c even 3 2
338.4.c.f 2 13.e even 6 2
338.4.e.b 4 13.f odd 12 4
650.4.a.f 1 65.d even 2 1
650.4.b.b 2 65.h odd 4 2
832.4.a.e 1 104.e even 2 1
832.4.a.m 1 104.h odd 2 1
1274.4.a.b 1 91.b odd 2 1
1872.4.a.c 1 156.h even 2 1

Hecke kernels

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

\( T_{3} - 3 \) Copy content Toggle raw display
\( T_{5} + 11 \) 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
$5$ \( T + 11 \) Copy content Toggle raw display
$7$ \( T + 19 \) Copy content Toggle raw display
$11$ \( T - 38 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 51 \) Copy content Toggle raw display
$19$ \( T + 90 \) Copy content Toggle raw display
$23$ \( T + 52 \) Copy content Toggle raw display
$29$ \( T + 190 \) Copy content Toggle raw display
$31$ \( T + 292 \) Copy content Toggle raw display
$37$ \( T - 441 \) Copy content Toggle raw display
$41$ \( T + 312 \) Copy content Toggle raw display
$43$ \( T - 373 \) Copy content Toggle raw display
$47$ \( T - 41 \) Copy content Toggle raw display
$53$ \( T - 468 \) Copy content Toggle raw display
$59$ \( T + 530 \) Copy content Toggle raw display
$61$ \( T - 592 \) Copy content Toggle raw display
$67$ \( T - 206 \) Copy content Toggle raw display
$71$ \( T - 863 \) Copy content Toggle raw display
$73$ \( T - 322 \) Copy content Toggle raw display
$79$ \( T + 460 \) Copy content Toggle raw display
$83$ \( T + 528 \) Copy content Toggle raw display
$89$ \( T + 870 \) Copy content Toggle raw display
$97$ \( T - 346 \) Copy content Toggle raw display
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