Properties

Label 338.6.b.a
Level $338$
Weight $6$
Character orbit 338.b
Analytic conductor $54.210$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [338,6,Mod(337,338)]
 
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([1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.337");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 338.b (of order \(2\), degree \(1\), 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: \(54.2097310968\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta q^{2} - 16 q^{4} - 7 \beta q^{5} + 85 \beta q^{7} + 32 \beta q^{8} - 243 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta q^{2} - 16 q^{4} - 7 \beta q^{5} + 85 \beta q^{7} + 32 \beta q^{8} - 243 q^{9} - 56 q^{10} + 125 \beta q^{11} + 680 q^{14} + 256 q^{16} - 1062 q^{17} + 486 \beta q^{18} - 39 \beta q^{19} + 112 \beta q^{20} + 1000 q^{22} - 1576 q^{23} + 2929 q^{25} - 1360 \beta q^{28} + 2578 q^{29} - 4327 \beta q^{31} - 512 \beta q^{32} + 2124 \beta q^{34} + 2380 q^{35} + 3888 q^{36} - 5493 \beta q^{37} - 312 q^{38} + 896 q^{40} + 525 \beta q^{41} + 5900 q^{43} - 2000 \beta q^{44} + 1701 \beta q^{45} + 3152 \beta q^{46} + 2981 \beta q^{47} - 12093 q^{49} - 5858 \beta q^{50} + 29046 q^{53} + 3500 q^{55} - 10880 q^{56} - 5156 \beta q^{58} + 6961 \beta q^{59} - 32882 q^{61} - 34616 q^{62} - 20655 \beta q^{63} - 4096 q^{64} - 34783 \beta q^{67} + 16992 q^{68} - 4760 \beta q^{70} - 25271 \beta q^{71} - 7776 \beta q^{72} + 23375 \beta q^{73} - 43944 q^{74} + 624 \beta q^{76} - 42500 q^{77} - 19348 q^{79} - 1792 \beta q^{80} + 59049 q^{81} + 4200 q^{82} - 43719 \beta q^{83} + 7434 \beta q^{85} - 11800 \beta q^{86} - 16000 q^{88} - 47085 \beta q^{89} + 13608 q^{90} + 25216 q^{92} + 23848 q^{94} - 1092 q^{95} + 91393 \beta q^{97} + 24186 \beta q^{98} - 30375 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{4} - 486 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 32 q^{4} - 486 q^{9} - 112 q^{10} + 1360 q^{14} + 512 q^{16} - 2124 q^{17} + 2000 q^{22} - 3152 q^{23} + 5858 q^{25} + 5156 q^{29} + 4760 q^{35} + 7776 q^{36} - 624 q^{38} + 1792 q^{40} + 11800 q^{43} - 24186 q^{49} + 58092 q^{53} + 7000 q^{55} - 21760 q^{56} - 65764 q^{61} - 69232 q^{62} - 8192 q^{64} + 33984 q^{68} - 87888 q^{74} - 85000 q^{77} - 38696 q^{79} + 118098 q^{81} + 8400 q^{82} - 32000 q^{88} + 27216 q^{90} + 50432 q^{92} + 47696 q^{94} - 2184 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(171\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
1.00000i
1.00000i
4.00000i 0 −16.0000 14.0000i 0 170.000i 64.0000i −243.000 −56.0000
337.2 4.00000i 0 −16.0000 14.0000i 0 170.000i 64.0000i −243.000 −56.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.6.b.a 2
13.b even 2 1 inner 338.6.b.a 2
13.d odd 4 1 26.6.a.a 1
13.d odd 4 1 338.6.a.d 1
39.f even 4 1 234.6.a.g 1
52.f even 4 1 208.6.a.b 1
65.f even 4 1 650.6.b.a 2
65.g odd 4 1 650.6.a.a 1
65.k even 4 1 650.6.b.a 2
104.j odd 4 1 832.6.a.d 1
104.m even 4 1 832.6.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.6.a.a 1 13.d odd 4 1
208.6.a.b 1 52.f even 4 1
234.6.a.g 1 39.f even 4 1
338.6.a.d 1 13.d odd 4 1
338.6.b.a 2 1.a even 1 1 trivial
338.6.b.a 2 13.b even 2 1 inner
650.6.a.a 1 65.g odd 4 1
650.6.b.a 2 65.f even 4 1
650.6.b.a 2 65.k even 4 1
832.6.a.d 1 104.j odd 4 1
832.6.a.e 1 104.m even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{6}^{\mathrm{new}}(338, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 196 \) Copy content Toggle raw display
$7$ \( T^{2} + 28900 \) Copy content Toggle raw display
$11$ \( T^{2} + 62500 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 1062)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 6084 \) Copy content Toggle raw display
$23$ \( (T + 1576)^{2} \) Copy content Toggle raw display
$29$ \( (T - 2578)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 74891716 \) Copy content Toggle raw display
$37$ \( T^{2} + 120692196 \) Copy content Toggle raw display
$41$ \( T^{2} + 1102500 \) Copy content Toggle raw display
$43$ \( (T - 5900)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 35545444 \) Copy content Toggle raw display
$53$ \( (T - 29046)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 193822084 \) Copy content Toggle raw display
$61$ \( (T + 32882)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 4839428356 \) Copy content Toggle raw display
$71$ \( T^{2} + 2554493764 \) Copy content Toggle raw display
$73$ \( T^{2} + 2185562500 \) Copy content Toggle raw display
$79$ \( (T + 19348)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 7645403844 \) Copy content Toggle raw display
$89$ \( T^{2} + 8867988900 \) Copy content Toggle raw display
$97$ \( T^{2} + 33410721796 \) Copy content Toggle raw display
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