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Properties

Label 338.3.f.b
Level $338$
Weight $3$
Character orbit 338.f
Analytic conductor $9.210$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [338,3,Mod(19,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 338.f (of order \(12\), degree \(4\), not 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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{2} + 2 \zeta_{12} q^{4} + (3 \zeta_{12}^{3} - 3) q^{5} + (2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{7} + ( - 2 \zeta_{12}^{3} - 2) q^{8} + ( - 9 \zeta_{12}^{2} + 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{2} + 2 \zeta_{12} q^{4} + (3 \zeta_{12}^{3} - 3) q^{5} + (2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{7} + ( - 2 \zeta_{12}^{3} - 2) q^{8} + ( - 9 \zeta_{12}^{2} + 9) q^{9} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{10} + ( - 6 \zeta_{12}^{3} + \cdots + 6 \zeta_{12}) q^{11}+ \cdots + ( - 54 \zeta_{12}^{3} - 54) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 12 q^{5} - 4 q^{7} - 8 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 12 q^{5} - 4 q^{7} - 8 q^{8} + 18 q^{9} - 12 q^{11} + 16 q^{14} + 8 q^{16} - 36 q^{18} - 52 q^{19} - 12 q^{20} - 24 q^{22} - 8 q^{28} + 96 q^{29} - 56 q^{31} + 8 q^{32} - 24 q^{34} + 24 q^{35} - 74 q^{37} + 48 q^{40} + 18 q^{41} + 48 q^{44} - 54 q^{45} - 48 q^{46} + 168 q^{47} - 14 q^{50} + 120 q^{53} + 72 q^{55} + 96 q^{58} + 108 q^{59} + 36 q^{61} + 36 q^{63} + 44 q^{67} + 24 q^{68} - 48 q^{70} - 12 q^{71} - 36 q^{72} + 68 q^{73} - 148 q^{74} + 104 q^{76} - 432 q^{79} - 24 q^{80} - 162 q^{81} + 312 q^{83} - 36 q^{85} - 144 q^{86} + 18 q^{89} + 192 q^{92} - 168 q^{94} + 94 q^{97} + 82 q^{98} - 216 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(171\)
\(\chi(n)\) \(\zeta_{12}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 + 0.500000i
0.866025 − 0.500000i
0.366025 + 1.36603i 0 −1.73205 + 1.00000i −3.00000 + 3.00000i 0 0.732051 − 2.73205i −2.00000 − 2.00000i 4.50000 + 7.79423i −5.19615 − 3.00000i
89.1 0.366025 − 1.36603i 0 −1.73205 − 1.00000i −3.00000 − 3.00000i 0 0.732051 + 2.73205i −2.00000 + 2.00000i 4.50000 − 7.79423i −5.19615 + 3.00000i
249.1 −1.36603 − 0.366025i 0 1.73205 + 1.00000i −3.00000 + 3.00000i 0 −2.73205 + 0.732051i −2.00000 − 2.00000i 4.50000 − 7.79423i 5.19615 − 3.00000i
319.1 −1.36603 + 0.366025i 0 1.73205 − 1.00000i −3.00000 − 3.00000i 0 −2.73205 − 0.732051i −2.00000 + 2.00000i 4.50000 + 7.79423i 5.19615 + 3.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner
13.d odd 4 1 inner
13.f odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.3.f.b 4
13.b even 2 1 338.3.f.g 4
13.c even 3 1 26.3.d.a ✓ 2
13.c even 3 1 inner 338.3.f.b 4
13.d odd 4 1 inner 338.3.f.b 4
13.d odd 4 1 338.3.f.g 4
13.e even 6 1 338.3.d.a 2
13.e even 6 1 338.3.f.g 4
13.f odd 12 1 26.3.d.a ✓ 2
13.f odd 12 1 338.3.d.a 2
13.f odd 12 1 inner 338.3.f.b 4
13.f odd 12 1 338.3.f.g 4
39.i odd 6 1 234.3.i.a 2
39.k even 12 1 234.3.i.a 2
52.j odd 6 1 208.3.t.b 2
52.l even 12 1 208.3.t.b 2
65.n even 6 1 650.3.k.b 2
65.o even 12 1 650.3.f.e 2
65.q odd 12 1 650.3.f.b 2
65.q odd 12 1 650.3.f.e 2
65.s odd 12 1 650.3.k.b 2
65.t even 12 1 650.3.f.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.3.d.a ✓ 2 13.c even 3 1
26.3.d.a ✓ 2 13.f odd 12 1
208.3.t.b 2 52.j odd 6 1
208.3.t.b 2 52.l even 12 1
234.3.i.a 2 39.i odd 6 1
234.3.i.a 2 39.k even 12 1
338.3.d.a 2 13.e even 6 1
338.3.d.a 2 13.f odd 12 1
338.3.f.b 4 1.a even 1 1 trivial
338.3.f.b 4 13.c even 3 1 inner
338.3.f.b 4 13.d odd 4 1 inner
338.3.f.b 4 13.f odd 12 1 inner
338.3.f.g 4 13.b even 2 1
338.3.f.g 4 13.d odd 4 1
338.3.f.g 4 13.e even 6 1
338.3.f.g 4 13.f odd 12 1
650.3.f.b 2 65.q odd 12 1
650.3.f.b 2 65.t even 12 1
650.3.f.e 2 65.o even 12 1
650.3.f.e 2 65.q odd 12 1
650.3.k.b 2 65.n even 6 1
650.3.k.b 2 65.s odd 12 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
\( T_{7}^{4} + 4T_{7}^{3} + 8T_{7}^{2} + 32T_{7} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 6 T + 18)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 4 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{4} + 12 T^{3} + \cdots + 5184 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$19$ \( T^{4} + 52 T^{3} + \cdots + 1827904 \) Copy content Toggle raw display
$23$ \( T^{4} - 576 T^{2} + 331776 \) Copy content Toggle raw display
$29$ \( (T^{2} - 48 T + 2304)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 28 T + 392)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 74 T^{3} + \cdots + 7496644 \) Copy content Toggle raw display
$41$ \( T^{4} - 18 T^{3} + \cdots + 26244 \) Copy content Toggle raw display
$43$ \( T^{4} - 1296 T^{2} + 1679616 \) Copy content Toggle raw display
$47$ \( (T^{2} - 84 T + 3528)^{2} \) Copy content Toggle raw display
$53$ \( (T - 30)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 108 T^{3} + \cdots + 34012224 \) Copy content Toggle raw display
$61$ \( (T^{2} - 18 T + 324)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 44 T^{3} + \cdots + 937024 \) Copy content Toggle raw display
$71$ \( T^{4} + 12 T^{3} + \cdots + 5184 \) Copy content Toggle raw display
$73$ \( (T^{2} - 34 T + 578)^{2} \) Copy content Toggle raw display
$79$ \( (T + 108)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 156 T + 12168)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 18 T^{3} + \cdots + 26244 \) Copy content Toggle raw display
$97$ \( T^{4} - 94 T^{3} + \cdots + 19518724 \) Copy content Toggle raw display
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