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Properties

Label 338.4.e.g
Level $338$
Weight $4$
Character orbit 338.e
Analytic conductor $19.943$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [338,4,Mod(23,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.23");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 338.e (of order \(6\), degree \(2\), 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: \(19.9426455819\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.45979465625856.49
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 109x^{6} + 8965x^{4} - 317844x^{2} + 8503056 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} + ( - \beta_{6} - \beta_{4} - 2 \beta_{2} + 1) q^{3} + 4 \beta_{2} q^{4} + ( - 5 \beta_{7} + \beta_{5} + \cdots - \beta_1) q^{5} + ( - 2 \beta_{3} - 2 \beta_1) q^{6} + ( - 2 \beta_{3} - \beta_1) q^{7}+ \cdots + ( - 393 \beta_{7} + 150 \beta_{5} + \cdots - 150 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{3} + 16 q^{4} - 118 q^{9} + 76 q^{10} + 48 q^{12} + 56 q^{14} - 64 q^{16} - 26 q^{17} - 72 q^{22} + 196 q^{23} - 156 q^{25} - 1332 q^{27} - 748 q^{29} - 548 q^{30} - 350 q^{35} + 472 q^{36} - 960 q^{38}+ \cdots - 324 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 109x^{6} + 8965x^{4} - 317844x^{2} + 8503056 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -109\nu^{6} + 8965\nu^{4} - 977185\nu^{2} + 34644996 ) / 26141940 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 175231\nu ) / 242055 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} - 175231 ) / 8965 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -109\nu^{7} + 8965\nu^{5} - 977185\nu^{3} + 34644996\nu ) / 26141940 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} + 163\nu^{4} - 8965\nu^{2} + 317844 ) / 8802 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3079\nu^{7} - 493075\nu^{5} + 27603235\nu^{3} - 978641676\nu ) / 705832380 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{4} - 54\beta_{2} + 55 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -27\beta_{7} - 55\beta_{5} - 27\beta_{3} + 55\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 109\beta_{6} - 2970\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2943\beta_{7} - 3079\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -8965\beta_{4} - 175231 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 242055\beta_{3} - 175231\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(171\)
\(\chi(n)\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
−6.81169 − 3.93273i
5.94566 + 3.43273i
6.81169 + 3.93273i
−5.94566 − 3.43273i
−6.81169 + 3.93273i
5.94566 − 3.43273i
6.81169 − 3.93273i
−5.94566 + 3.43273i
−1.73205 + 1.00000i −2.93273 − 5.07964i 2.00000 − 3.46410i − 2.13454i 10.1593 + 5.86546i 3.34759 + 1.93273i 8.00000i −3.70181 + 6.41172i 2.13454 + 3.69713i
23.2 −1.73205 + 1.00000i 4.43273 + 7.67771i 2.00000 − 3.46410i − 16.8655i −15.3554 − 8.86546i −9.40976 − 5.43273i 8.00000i −25.7982 + 44.6838i 16.8655 + 29.2118i
23.3 1.73205 − 1.00000i −2.93273 − 5.07964i 2.00000 − 3.46410i 2.13454i −10.1593 − 5.86546i −3.34759 − 1.93273i − 8.00000i −3.70181 + 6.41172i 2.13454 + 3.69713i
23.4 1.73205 − 1.00000i 4.43273 + 7.67771i 2.00000 − 3.46410i 16.8655i 15.3554 + 8.86546i 9.40976 + 5.43273i − 8.00000i −25.7982 + 44.6838i 16.8655 + 29.2118i
147.1 −1.73205 − 1.00000i −2.93273 + 5.07964i 2.00000 + 3.46410i 2.13454i 10.1593 − 5.86546i 3.34759 − 1.93273i − 8.00000i −3.70181 − 6.41172i 2.13454 − 3.69713i
147.2 −1.73205 − 1.00000i 4.43273 − 7.67771i 2.00000 + 3.46410i 16.8655i −15.3554 + 8.86546i −9.40976 + 5.43273i − 8.00000i −25.7982 − 44.6838i 16.8655 − 29.2118i
147.3 1.73205 + 1.00000i −2.93273 + 5.07964i 2.00000 + 3.46410i − 2.13454i −10.1593 + 5.86546i −3.34759 + 1.93273i 8.00000i −3.70181 − 6.41172i 2.13454 − 3.69713i
147.4 1.73205 + 1.00000i 4.43273 − 7.67771i 2.00000 + 3.46410i − 16.8655i 15.3554 − 8.86546i 9.40976 − 5.43273i 8.00000i −25.7982 − 44.6838i 16.8655 − 29.2118i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.4.e.g 8
13.b even 2 1 inner 338.4.e.g 8
13.c even 3 1 26.4.b.a ✓ 4
13.c even 3 1 inner 338.4.e.g 8
13.d odd 4 1 338.4.c.h 4
13.d odd 4 1 338.4.c.i 4
13.e even 6 1 26.4.b.a ✓ 4
13.e even 6 1 inner 338.4.e.g 8
13.f odd 12 1 338.4.a.f 2
13.f odd 12 1 338.4.a.i 2
13.f odd 12 1 338.4.c.h 4
13.f odd 12 1 338.4.c.i 4
39.h odd 6 1 234.4.b.b 4
39.i odd 6 1 234.4.b.b 4
52.i odd 6 1 208.4.f.d 4
52.j odd 6 1 208.4.f.d 4
65.l even 6 1 650.4.d.d 4
65.n even 6 1 650.4.d.d 4
65.q odd 12 1 650.4.c.e 4
65.q odd 12 1 650.4.c.f 4
65.r odd 12 1 650.4.c.e 4
65.r odd 12 1 650.4.c.f 4
104.n odd 6 1 832.4.f.h 4
104.p odd 6 1 832.4.f.h 4
104.r even 6 1 832.4.f.j 4
104.s even 6 1 832.4.f.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.b.a ✓ 4 13.c even 3 1
26.4.b.a ✓ 4 13.e even 6 1
208.4.f.d 4 52.i odd 6 1
208.4.f.d 4 52.j odd 6 1
234.4.b.b 4 39.h odd 6 1
234.4.b.b 4 39.i odd 6 1
338.4.a.f 2 13.f odd 12 1
338.4.a.i 2 13.f odd 12 1
338.4.c.h 4 13.d odd 4 1
338.4.c.h 4 13.f odd 12 1
338.4.c.i 4 13.d odd 4 1
338.4.c.i 4 13.f odd 12 1
338.4.e.g 8 1.a even 1 1 trivial
338.4.e.g 8 13.b even 2 1 inner
338.4.e.g 8 13.c even 3 1 inner
338.4.e.g 8 13.e even 6 1 inner
650.4.c.e 4 65.q odd 12 1
650.4.c.e 4 65.r odd 12 1
650.4.c.f 4 65.q odd 12 1
650.4.c.f 4 65.r odd 12 1
650.4.d.d 4 65.l even 6 1
650.4.d.d 4 65.n even 6 1
832.4.f.h 4 104.n odd 6 1
832.4.f.h 4 104.p odd 6 1
832.4.f.j 4 104.r even 6 1
832.4.f.j 4 104.s even 6 1

Hecke kernels

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

\( T_{3}^{4} - 3T_{3}^{3} + 61T_{3}^{2} + 156T_{3} + 2704 \) Copy content Toggle raw display
\( T_{5}^{4} + 289T_{5}^{2} + 1296 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - 3 T^{3} + \cdots + 2704)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 289 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 133 T^{6} + \cdots + 3111696 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 12280707219456 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + 13 T^{3} + \cdots + 1726596)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 314741094011136 \) Copy content Toggle raw display
$23$ \( (T^{4} - 98 T^{3} + \cdots + 9144576)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 374 T^{3} + \cdots + 592240896)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 84484 T^{2} + 1747908864)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 59\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( (T^{4} - 219 T^{3} + \cdots + 480311056)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 116901 T^{2} + 2466314244)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 24 T - 7668)^{4} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 61\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{4} - 282 T^{3} + \cdots + 9008287744)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 31\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{4} + 722500 T^{2} + 8100000000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 722 T - 166752)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 1623976 T^{2} + 797271696)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 68\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 56\!\cdots\!56 \) Copy content Toggle raw display
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