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Properties

Label 336.4.q.h
Level $336$
Weight $4$
Character orbit 336.q
Analytic conductor $19.825$
Analytic rank $0$
Dimension $4$
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] = [336,4,Mod(193,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.193");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 336.q (of order \(3\), 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: \(19.8246417619\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \beta_{2} q^{3} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{5} + (\beta_{3} - 6 \beta_{2} - 3 \beta_1 + 3) q^{7} + ( - 9 \beta_{2} - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 \beta_{2} q^{3} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{5} + (\beta_{3} - 6 \beta_{2} - 3 \beta_1 + 3) q^{7} + ( - 9 \beta_{2} - 9) q^{9} + ( - 2 \beta_{3} + 25 \beta_{2} + \cdots - 1) q^{11}+ \cdots + (9 \beta_{3} + 9 \beta_{2} + \cdots + 234) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{3} + 3 q^{5} + 20 q^{7} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{3} + 3 q^{5} + 20 q^{7} - 18 q^{9} - 51 q^{11} + 122 q^{13} - 18 q^{15} - 24 q^{17} + 169 q^{19} + 15 q^{21} - 192 q^{23} - 11 q^{25} + 108 q^{27} - 78 q^{29} - 92 q^{31} - 153 q^{33} + 294 q^{35} + 173 q^{37} - 183 q^{39} + 348 q^{41} + 994 q^{43} + 27 q^{45} + 180 q^{47} - 146 q^{49} - 72 q^{51} - 285 q^{53} - 666 q^{55} - 1014 q^{57} - 1269 q^{59} - 328 q^{61} - 225 q^{63} + 1374 q^{65} - 875 q^{67} + 1152 q^{69} + 2808 q^{71} + 1361 q^{73} - 33 q^{75} + 897 q^{77} - 182 q^{79} - 162 q^{81} + 798 q^{83} - 4176 q^{85} + 117 q^{87} - 822 q^{89} - 1955 q^{91} - 276 q^{93} - 510 q^{95} + 1682 q^{97} + 918 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 4\nu^{2} + 56\nu - 25 ) / 20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 4\nu^{2} - 4\nu - 25 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{3} + 2\nu^{2} + 28\nu + 35 ) / 10 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 14\beta_{2} + 14 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{3} + 4\beta _1 + 19 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
−1.63746 − 1.52274i
2.13746 + 0.656712i
−1.63746 + 1.52274i
2.13746 − 0.656712i
0 −1.50000 + 2.59808i 0 −4.91238 − 8.50848i 0 16.3248 + 8.74657i 0 −4.50000 − 7.79423i 0
193.2 0 −1.50000 + 2.59808i 0 6.41238 + 11.1066i 0 −6.32475 − 17.4068i 0 −4.50000 − 7.79423i 0
289.1 0 −1.50000 − 2.59808i 0 −4.91238 + 8.50848i 0 16.3248 − 8.74657i 0 −4.50000 + 7.79423i 0
289.2 0 −1.50000 − 2.59808i 0 6.41238 − 11.1066i 0 −6.32475 + 17.4068i 0 −4.50000 + 7.79423i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.4.q.h 4
4.b odd 2 1 84.4.i.b ✓ 4
7.c even 3 1 inner 336.4.q.h 4
7.c even 3 1 2352.4.a.cb 2
7.d odd 6 1 2352.4.a.bp 2
12.b even 2 1 252.4.k.d 4
28.d even 2 1 588.4.i.i 4
28.f even 6 1 588.4.a.h 2
28.f even 6 1 588.4.i.i 4
28.g odd 6 1 84.4.i.b ✓ 4
28.g odd 6 1 588.4.a.g 2
84.h odd 2 1 1764.4.k.z 4
84.j odd 6 1 1764.4.a.p 2
84.j odd 6 1 1764.4.k.z 4
84.n even 6 1 252.4.k.d 4
84.n even 6 1 1764.4.a.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.i.b ✓ 4 4.b odd 2 1
84.4.i.b ✓ 4 28.g odd 6 1
252.4.k.d 4 12.b even 2 1
252.4.k.d 4 84.n even 6 1
336.4.q.h 4 1.a even 1 1 trivial
336.4.q.h 4 7.c even 3 1 inner
588.4.a.g 2 28.g odd 6 1
588.4.a.h 2 28.f even 6 1
588.4.i.i 4 28.d even 2 1
588.4.i.i 4 28.f even 6 1
1764.4.a.p 2 84.j odd 6 1
1764.4.a.x 2 84.n even 6 1
1764.4.k.z 4 84.h odd 2 1
1764.4.k.z 4 84.j odd 6 1
2352.4.a.bp 2 7.d odd 6 1
2352.4.a.cb 2 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 3T_{5}^{3} + 135T_{5}^{2} + 378T_{5} + 15876 \) acting on \(S_{4}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 3 T^{3} + \cdots + 15876 \) Copy content Toggle raw display
$7$ \( T^{4} - 20 T^{3} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{4} + 51 T^{3} + \cdots + 272484 \) Copy content Toggle raw display
$13$ \( (T^{2} - 61 T - 2276)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 24 T^{3} + \cdots + 65028096 \) Copy content Toggle raw display
$19$ \( T^{4} - 169 T^{3} + \cdots + 49168144 \) Copy content Toggle raw display
$23$ \( (T^{2} + 96 T + 9216)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 39 T - 36684)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 92 T^{3} + \cdots + 114682681 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 1506681856 \) Copy content Toggle raw display
$41$ \( (T^{2} - 174 T - 140688)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 497 T + 15454)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 180 T^{3} + \cdots + 611671824 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 5356483344 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 161150862096 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 19408947856 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 32767516324 \) Copy content Toggle raw display
$71$ \( (T^{2} - 1404 T + 361476)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 165260136484 \) Copy content Toggle raw display
$79$ \( T^{4} + 182 T^{3} + \cdots + 38800441 \) Copy content Toggle raw display
$83$ \( (T^{2} - 399 T - 197334)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 11416495104 \) Copy content Toggle raw display
$97$ \( (T^{2} - 841 T + 120262)^{2} \) Copy content Toggle raw display
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