Properties

Label 338.3.d.a
Level $338$
Weight $3$
Character orbit 338.d
Analytic conductor $9.210$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [338,3,Mod(99,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.99");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 338.d (of order \(4\), degree \(2\), minimal)

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: no
Analytic conductor: \(9.20983293538\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - i - 1) q^{2} + 2 i q^{4} + (3 i + 3) q^{5} + (2 i - 2) q^{7} + ( - 2 i + 2) q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - i - 1) q^{2} + 2 i q^{4} + (3 i + 3) q^{5} + (2 i - 2) q^{7} + ( - 2 i + 2) q^{8} - 9 q^{9} - 6 i q^{10} + (6 i - 6) q^{11} + 4 q^{14} - 4 q^{16} + 6 i q^{17} + (9 i + 9) q^{18} + ( - 26 i - 26) q^{19} + (6 i - 6) q^{20} + 12 q^{22} - 24 i q^{23} - 7 i q^{25} + ( - 4 i - 4) q^{28} - 48 q^{29} + (14 i + 14) q^{31} + (4 i + 4) q^{32} + ( - 6 i + 6) q^{34} - 12 q^{35} - 18 i q^{36} + (37 i - 37) q^{37} + 52 i q^{38} + 12 q^{40} + (9 i + 9) q^{41} + 36 i q^{43} + ( - 12 i - 12) q^{44} + ( - 27 i - 27) q^{45} + (24 i - 24) q^{46} + (42 i - 42) q^{47} + 41 i q^{49} + (7 i - 7) q^{50} + 30 q^{53} - 36 q^{55} + 8 i q^{56} + (48 i + 48) q^{58} + ( - 54 i + 54) q^{59} - 18 q^{61} - 28 i q^{62} + ( - 18 i + 18) q^{63} - 8 i q^{64} + (22 i + 22) q^{67} - 12 q^{68} + (12 i + 12) q^{70} + ( - 6 i - 6) q^{71} + (18 i - 18) q^{72} + (17 i - 17) q^{73} + 74 q^{74} + ( - 52 i + 52) q^{76} - 24 i q^{77} - 108 q^{79} + ( - 12 i - 12) q^{80} + 81 q^{81} - 18 i q^{82} + ( - 78 i - 78) q^{83} + (18 i - 18) q^{85} + ( - 36 i + 36) q^{86} + 24 i q^{88} + ( - 9 i + 9) q^{89} + 54 i q^{90} + 48 q^{92} + 84 q^{94} - 156 i q^{95} + (47 i + 47) q^{97} + ( - 41 i + 41) q^{98} + ( - 54 i + 54) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 6 q^{5} - 4 q^{7} + 4 q^{8} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 6 q^{5} - 4 q^{7} + 4 q^{8} - 18 q^{9} - 12 q^{11} + 8 q^{14} - 8 q^{16} + 18 q^{18} - 52 q^{19} - 12 q^{20} + 24 q^{22} - 8 q^{28} - 96 q^{29} + 28 q^{31} + 8 q^{32} + 12 q^{34} - 24 q^{35} - 74 q^{37} + 24 q^{40} + 18 q^{41} - 24 q^{44} - 54 q^{45} - 48 q^{46} - 84 q^{47} - 14 q^{50} + 60 q^{53} - 72 q^{55} + 96 q^{58} + 108 q^{59} - 36 q^{61} + 36 q^{63} + 44 q^{67} - 24 q^{68} + 24 q^{70} - 12 q^{71} - 36 q^{72} - 34 q^{73} + 148 q^{74} + 104 q^{76} - 216 q^{79} - 24 q^{80} + 162 q^{81} - 156 q^{83} - 36 q^{85} + 72 q^{86} + 18 q^{89} + 96 q^{92} + 168 q^{94} + 94 q^{97} + 82 q^{98} + 108 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/338\mathbb{Z}\right)^\times\).

\(n\) \(171\)
\(\chi(n)\) \(i\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
1.00000i
1.00000i
−1.00000 1.00000i 0 2.00000i 3.00000 + 3.00000i 0 −2.00000 + 2.00000i 2.00000 2.00000i −9.00000 6.00000i
239.1 −1.00000 + 1.00000i 0 2.00000i 3.00000 3.00000i 0 −2.00000 2.00000i 2.00000 + 2.00000i −9.00000 6.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.3.d.a 2
13.b even 2 1 26.3.d.a 2
13.c even 3 2 338.3.f.g 4
13.d odd 4 1 26.3.d.a 2
13.d odd 4 1 inner 338.3.d.a 2
13.e even 6 2 338.3.f.b 4
13.f odd 12 2 338.3.f.b 4
13.f odd 12 2 338.3.f.g 4
39.d odd 2 1 234.3.i.a 2
39.f even 4 1 234.3.i.a 2
52.b odd 2 1 208.3.t.b 2
52.f even 4 1 208.3.t.b 2
65.d even 2 1 650.3.k.b 2
65.f even 4 1 650.3.f.e 2
65.g odd 4 1 650.3.k.b 2
65.h odd 4 1 650.3.f.b 2
65.h odd 4 1 650.3.f.e 2
65.k even 4 1 650.3.f.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.3.d.a 2 13.b even 2 1
26.3.d.a 2 13.d odd 4 1
208.3.t.b 2 52.b odd 2 1
208.3.t.b 2 52.f even 4 1
234.3.i.a 2 39.d odd 2 1
234.3.i.a 2 39.f even 4 1
338.3.d.a 2 1.a even 1 1 trivial
338.3.d.a 2 13.d odd 4 1 inner
338.3.f.b 4 13.e even 6 2
338.3.f.b 4 13.f odd 12 2
338.3.f.g 4 13.c even 3 2
338.3.f.g 4 13.f odd 12 2
650.3.f.b 2 65.h odd 4 1
650.3.f.b 2 65.k even 4 1
650.3.f.e 2 65.f even 4 1
650.3.f.e 2 65.h odd 4 1
650.3.k.b 2 65.d even 2 1
650.3.k.b 2 65.g odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(338, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5}^{2} - 6T_{5} + 18 \) Copy content Toggle raw display
\( T_{7}^{2} + 4T_{7} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$11$ \( T^{2} + 12T + 72 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 36 \) Copy content Toggle raw display
$19$ \( T^{2} + 52T + 1352 \) Copy content Toggle raw display
$23$ \( T^{2} + 576 \) Copy content Toggle raw display
$29$ \( (T + 48)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 28T + 392 \) Copy content Toggle raw display
$37$ \( T^{2} + 74T + 2738 \) Copy content Toggle raw display
$41$ \( T^{2} - 18T + 162 \) Copy content Toggle raw display
$43$ \( T^{2} + 1296 \) Copy content Toggle raw display
$47$ \( T^{2} + 84T + 3528 \) Copy content Toggle raw display
$53$ \( (T - 30)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 108T + 5832 \) Copy content Toggle raw display
$61$ \( (T + 18)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 44T + 968 \) Copy content Toggle raw display
$71$ \( T^{2} + 12T + 72 \) Copy content Toggle raw display
$73$ \( T^{2} + 34T + 578 \) Copy content Toggle raw display
$79$ \( (T + 108)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 156T + 12168 \) Copy content Toggle raw display
$89$ \( T^{2} - 18T + 162 \) Copy content Toggle raw display
$97$ \( T^{2} - 94T + 4418 \) Copy content Toggle raw display
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