Properties

Label 338.2.a.h
Level $338$
Weight $2$
Character orbit 338.a
Self dual yes
Analytic conductor $2.699$
Analytic rank $0$
Dimension $3$
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] = [338,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
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: \(2.69894358832\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + ( - \beta_{2} - \beta_1 + 1) q^{3} + q^{4} + 2 \beta_1 q^{5} + ( - \beta_{2} - \beta_1 + 1) q^{6} + (2 \beta_{2} - 2 \beta_1) q^{7} + q^{8} + ( - \beta_1 + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + ( - \beta_{2} - \beta_1 + 1) q^{3} + q^{4} + 2 \beta_1 q^{5} + ( - \beta_{2} - \beta_1 + 1) q^{6} + (2 \beta_{2} - 2 \beta_1) q^{7} + q^{8} + ( - \beta_1 + 3) q^{9} + 2 \beta_1 q^{10} + ( - \beta_{2} + 2 \beta_1) q^{11} + ( - \beta_{2} - \beta_1 + 1) q^{12} + (2 \beta_{2} - 2 \beta_1) q^{14} + ( - 4 \beta_{2} + 2 \beta_1 - 6) q^{15} + q^{16} + ( - \beta_{2} + 4 \beta_1) q^{17} + ( - \beta_1 + 3) q^{18} + (2 \beta_{2} - 3 \beta_1 + 2) q^{19} + 2 \beta_1 q^{20} + (6 \beta_{2} - 4 \beta_1 + 2) q^{21} + ( - \beta_{2} + 2 \beta_1) q^{22} + (4 \beta_{2} - 2 \beta_1 + 2) q^{23} + ( - \beta_{2} - \beta_1 + 1) q^{24} + (4 \beta_{2} + 3) q^{25} + (2 \beta_{2} - \beta_1 + 3) q^{27} + (2 \beta_{2} - 2 \beta_1) q^{28} + ( - 4 \beta_{2} + 4 \beta_1 - 6) q^{29} + ( - 4 \beta_{2} + 2 \beta_1 - 6) q^{30} + ( - 2 \beta_{2} - 6) q^{31} + q^{32} + ( - 5 \beta_{2} + 3 \beta_1 - 4) q^{33} + ( - \beta_{2} + 4 \beta_1) q^{34} - 4 q^{35} + ( - \beta_1 + 3) q^{36} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{37} + (2 \beta_{2} - 3 \beta_1 + 2) q^{38} + 2 \beta_1 q^{40} + (4 \beta_{2} - \beta_1 + 4) q^{41} + (6 \beta_{2} - 4 \beta_1 + 2) q^{42} + (2 \beta_{2} - 5 \beta_1 + 6) q^{43} + ( - \beta_{2} + 2 \beta_1) q^{44} + ( - 2 \beta_{2} + 6 \beta_1 - 4) q^{45} + (4 \beta_{2} - 2 \beta_1 + 2) q^{46} + ( - 4 \beta_{2} - 2) q^{47} + ( - \beta_{2} - \beta_1 + 1) q^{48} + ( - 8 \beta_{2} + 4 \beta_1 - 3) q^{49} + (4 \beta_{2} + 3) q^{50} + ( - 9 \beta_{2} + 5 \beta_1 - 10) q^{51} + ( - 4 \beta_{2} + 2 \beta_1 - 2) q^{53} + (2 \beta_{2} - \beta_1 + 3) q^{54} + (2 \beta_{2} + 6) q^{55} + (2 \beta_{2} - 2 \beta_1) q^{56} + (6 \beta_{2} - 7 \beta_1 + 7) q^{57} + ( - 4 \beta_{2} + 4 \beta_1 - 6) q^{58} + ( - 3 \beta_{2} - 3 \beta_1 - 1) q^{59} + ( - 4 \beta_{2} + 2 \beta_1 - 6) q^{60} + (8 \beta_{2} - 8 \beta_1 + 4) q^{61} + ( - 2 \beta_{2} - 6) q^{62} + (6 \beta_{2} - 6 \beta_1 + 2) q^{63} + q^{64} + ( - 5 \beta_{2} + 3 \beta_1 - 4) q^{66} + ( - 3 \beta_{2} - 3 \beta_1 + 7) q^{67} + ( - \beta_{2} + 4 \beta_1) q^{68} + (6 \beta_{2} - 8 \beta_1) q^{69} - 4 q^{70} + (4 \beta_{2} - 2 \beta_1 - 2) q^{71} + ( - \beta_1 + 3) q^{72} + ( - 7 \beta_{2} + 3 \beta_1 - 3) q^{73} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{74} + (\beta_{2} - 7 \beta_1 - 5) q^{75} + (2 \beta_{2} - 3 \beta_1 + 2) q^{76} + (4 \beta_{2} - 2 \beta_1 - 4) q^{77} + ( - 6 \beta_{2} - 8) q^{79} + 2 \beta_1 q^{80} + (\beta_{2} - 3 \beta_1 - 7) q^{81} + (4 \beta_{2} - \beta_1 + 4) q^{82} + ( - 2 \beta_{2} + 3 \beta_1 - 2) q^{83} + (6 \beta_{2} - 4 \beta_1 + 2) q^{84} + (6 \beta_{2} + 14) q^{85} + (2 \beta_{2} - 5 \beta_1 + 6) q^{86} + ( - 6 \beta_{2} + 14 \beta_1 - 10) q^{87} + ( - \beta_{2} + 2 \beta_1) q^{88} + ( - 3 \beta_{2} - 5 \beta_1 + 9) q^{89} + ( - 2 \beta_{2} + 6 \beta_1 - 4) q^{90} + (4 \beta_{2} - 2 \beta_1 + 2) q^{92} + (4 \beta_{2} + 8 \beta_1 - 2) q^{93} + ( - 4 \beta_{2} - 2) q^{94} + ( - 2 \beta_{2} + 4 \beta_1 - 8) q^{95} + ( - \beta_{2} - \beta_1 + 1) q^{96} + ( - 3 \beta_{2} + 8 \beta_1 + 4) q^{97} + ( - 8 \beta_{2} + 4 \beta_1 - 3) q^{98} + ( - 4 \beta_{2} + 6 \beta_1 - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 2 q^{5} + 3 q^{6} - 4 q^{7} + 3 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 2 q^{5} + 3 q^{6} - 4 q^{7} + 3 q^{8} + 8 q^{9} + 2 q^{10} + 3 q^{11} + 3 q^{12} - 4 q^{14} - 12 q^{15} + 3 q^{16} + 5 q^{17} + 8 q^{18} + q^{19} + 2 q^{20} - 4 q^{21} + 3 q^{22} + 3 q^{24} + 5 q^{25} + 6 q^{27} - 4 q^{28} - 10 q^{29} - 12 q^{30} - 16 q^{31} + 3 q^{32} - 4 q^{33} + 5 q^{34} - 12 q^{35} + 8 q^{36} - 14 q^{37} + q^{38} + 2 q^{40} + 7 q^{41} - 4 q^{42} + 11 q^{43} + 3 q^{44} - 4 q^{45} - 2 q^{47} + 3 q^{48} + 3 q^{49} + 5 q^{50} - 16 q^{51} + 6 q^{54} + 16 q^{55} - 4 q^{56} + 8 q^{57} - 10 q^{58} - 3 q^{59} - 12 q^{60} - 4 q^{61} - 16 q^{62} - 6 q^{63} + 3 q^{64} - 4 q^{66} + 21 q^{67} + 5 q^{68} - 14 q^{69} - 12 q^{70} - 12 q^{71} + 8 q^{72} + q^{73} - 14 q^{74} - 23 q^{75} + q^{76} - 18 q^{77} - 18 q^{79} + 2 q^{80} - 25 q^{81} + 7 q^{82} - q^{83} - 4 q^{84} + 36 q^{85} + 11 q^{86} - 10 q^{87} + 3 q^{88} + 25 q^{89} - 4 q^{90} - 2 q^{93} - 2 q^{94} - 18 q^{95} + 3 q^{96} + 23 q^{97} + 3 q^{98} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) 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
1.80194
0.445042
−1.24698
1.00000 −2.04892 1.00000 3.60388 −2.04892 −1.10992 1.00000 1.19806 3.60388
1.2 1.00000 2.35690 1.00000 0.890084 2.35690 −4.49396 1.00000 2.55496 0.890084
1.3 1.00000 2.69202 1.00000 −2.49396 2.69202 1.60388 1.00000 4.24698 −2.49396
\(n\): 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 338.2.a.h yes 3
3.b odd 2 1 3042.2.a.z 3
4.b odd 2 1 2704.2.a.w 3
5.b even 2 1 8450.2.a.bn 3
13.b even 2 1 338.2.a.g 3
13.c even 3 2 338.2.c.h 6
13.d odd 4 2 338.2.b.d 6
13.e even 6 2 338.2.c.i 6
13.f odd 12 4 338.2.e.e 12
39.d odd 2 1 3042.2.a.bi 3
39.f even 4 2 3042.2.b.n 6
52.b odd 2 1 2704.2.a.v 3
52.f even 4 2 2704.2.f.m 6
65.d even 2 1 8450.2.a.bx 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
338.2.a.g 3 13.b even 2 1
338.2.a.h yes 3 1.a even 1 1 trivial
338.2.b.d 6 13.d odd 4 2
338.2.c.h 6 13.c even 3 2
338.2.c.i 6 13.e even 6 2
338.2.e.e 12 13.f odd 12 4
2704.2.a.v 3 52.b odd 2 1
2704.2.a.w 3 4.b odd 2 1
2704.2.f.m 6 52.f even 4 2
3042.2.a.z 3 3.b odd 2 1
3042.2.a.bi 3 39.d odd 2 1
3042.2.b.n 6 39.f even 4 2
8450.2.a.bn 3 5.b even 2 1
8450.2.a.bx 3 65.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(338))\):

\( T_{3}^{3} - 3T_{3}^{2} - 4T_{3} + 13 \) Copy content Toggle raw display
\( T_{5}^{3} - 2T_{5}^{2} - 8T_{5} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 3 T^{2} - 4 T + 13 \) Copy content Toggle raw display
$5$ \( T^{3} - 2 T^{2} - 8 T + 8 \) Copy content Toggle raw display
$7$ \( T^{3} + 4 T^{2} - 4 T - 8 \) Copy content Toggle raw display
$11$ \( T^{3} - 3 T^{2} - 4 T + 13 \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - 5 T^{2} - 22 T + 97 \) Copy content Toggle raw display
$19$ \( T^{3} - T^{2} - 16 T - 13 \) Copy content Toggle raw display
$23$ \( T^{3} - 28T + 56 \) Copy content Toggle raw display
$29$ \( T^{3} + 10 T^{2} - 4 T - 104 \) Copy content Toggle raw display
$31$ \( T^{3} + 16 T^{2} + 76 T + 104 \) Copy content Toggle raw display
$37$ \( T^{3} + 14 T^{2} + 56 T + 56 \) Copy content Toggle raw display
$41$ \( T^{3} - 7 T^{2} - 14 T + 91 \) Copy content Toggle raw display
$43$ \( T^{3} - 11 T^{2} - 4 T + 1 \) Copy content Toggle raw display
$47$ \( T^{3} + 2 T^{2} - 36 T - 8 \) Copy content Toggle raw display
$53$ \( T^{3} - 28T - 56 \) Copy content Toggle raw display
$59$ \( T^{3} + 3 T^{2} - 60 T + 127 \) Copy content Toggle raw display
$61$ \( T^{3} + 4 T^{2} - 144 T - 64 \) Copy content Toggle raw display
$67$ \( T^{3} - 21 T^{2} + 84 T + 287 \) Copy content Toggle raw display
$71$ \( T^{3} + 12 T^{2} + 20 T + 8 \) Copy content Toggle raw display
$73$ \( T^{3} - T^{2} - 86 T - 251 \) Copy content Toggle raw display
$79$ \( T^{3} + 18 T^{2} + 24 T - 232 \) Copy content Toggle raw display
$83$ \( T^{3} + T^{2} - 16 T + 13 \) Copy content Toggle raw display
$89$ \( T^{3} - 25 T^{2} + 94 T + 757 \) Copy content Toggle raw display
$97$ \( T^{3} - 23 T^{2} + 62 T + 883 \) Copy content Toggle raw display
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