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Properties

Label 336.6.q.f
Level $336$
Weight $6$
Character orbit 336.q
Analytic conductor $53.889$
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] = [336,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.193");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(53.8889634572\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{9601})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2401x^{2} + 2400x + 5760000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
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.

\(f(q)\) \(=\) \( q + (9 \beta_{2} - 9) q^{3} + (26 \beta_{2} + \beta_1) q^{5} + (\beta_{3} + 3 \beta_{2} - 2 \beta_1 - 3) q^{7} - 81 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (9 \beta_{2} - 9) q^{3} + (26 \beta_{2} + \beta_1) q^{5} + (\beta_{3} + 3 \beta_{2} - 2 \beta_1 - 3) q^{7} - 81 \beta_{2} q^{9} + (5 \beta_{3} - 98 \beta_{2} + \cdots + 93) q^{11}+ \cdots + ( - 405 \beta_{3} - 7533) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 18 q^{3} + 53 q^{5} - 6 q^{7} - 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 18 q^{3} + 53 q^{5} - 6 q^{7} - 162 q^{9} + 191 q^{11} - 758 q^{13} - 954 q^{15} + 340 q^{17} - 1769 q^{19} - 27 q^{21} + 3236 q^{23} + 45 q^{25} + 2916 q^{27} + 8918 q^{29} + 1994 q^{31} + 1719 q^{33} + 4562 q^{35} - 20587 q^{37} + 3411 q^{39} + 17628 q^{41} - 31706 q^{43} + 4293 q^{45} + 33912 q^{47} + 9598 q^{49} + 3060 q^{51} - 49239 q^{53} - 37882 q^{55} + 31842 q^{57} - 56735 q^{59} - 67508 q^{61} + 729 q^{63} + 42762 q^{65} + 75723 q^{67} - 58248 q^{69} + 17984 q^{71} + 3201 q^{73} + 405 q^{75} + 120299 q^{77} + 26612 q^{79} - 13122 q^{81} + 1898 q^{83} + 210040 q^{85} - 40131 q^{87} - 176562 q^{89} - 210085 q^{91} + 17946 q^{93} + 234098 q^{95} - 258846 q^{97} - 30942 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 2401x^{2} + 2400x + 5760000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 2401\nu^{2} - 2401\nu + 5760000 ) / 5762400 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 4801 ) / 2401 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 2400\beta_{2} + \beta _1 - 2401 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2401\beta_{3} - 4801 \) Copy content Toggle raw display

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

Character values

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

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

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
−24.2462 − 41.9956i
24.7462 + 42.8616i
−24.2462 + 41.9956i
24.7462 − 42.8616i
0 −4.50000 + 7.79423i 0 −11.2462 − 19.4789i 0 96.4847 + 86.5893i 0 −40.5000 − 70.1481i 0
193.2 0 −4.50000 + 7.79423i 0 37.7462 + 65.3783i 0 −99.4847 − 83.1252i 0 −40.5000 − 70.1481i 0
289.1 0 −4.50000 − 7.79423i 0 −11.2462 + 19.4789i 0 96.4847 − 86.5893i 0 −40.5000 + 70.1481i 0
289.2 0 −4.50000 − 7.79423i 0 37.7462 − 65.3783i 0 −99.4847 + 83.1252i 0 −40.5000 + 70.1481i 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 336.6.q.f 4
4.b odd 2 1 42.6.e.c ✓ 4
7.c even 3 1 inner 336.6.q.f 4
12.b even 2 1 126.6.g.h 4
28.d even 2 1 294.6.e.s 4
28.f even 6 1 294.6.a.w 2
28.f even 6 1 294.6.e.s 4
28.g odd 6 1 42.6.e.c ✓ 4
28.g odd 6 1 294.6.a.r 2
84.j odd 6 1 882.6.a.bb 2
84.n even 6 1 126.6.g.h 4
84.n even 6 1 882.6.a.bh 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.e.c ✓ 4 4.b odd 2 1
42.6.e.c ✓ 4 28.g odd 6 1
126.6.g.h 4 12.b even 2 1
126.6.g.h 4 84.n even 6 1
294.6.a.r 2 28.g odd 6 1
294.6.a.w 2 28.f even 6 1
294.6.e.s 4 28.d even 2 1
294.6.e.s 4 28.f even 6 1
336.6.q.f 4 1.a even 1 1 trivial
336.6.q.f 4 7.c even 3 1 inner
882.6.a.bb 2 84.j odd 6 1
882.6.a.bh 2 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 53T_{5}^{3} + 4507T_{5}^{2} + 89994T_{5} + 2883204 \) acting on \(S_{6}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9 T + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 53 T^{3} + \cdots + 2883204 \) Copy content Toggle raw display
$7$ \( T^{4} + 6 T^{3} + \cdots + 282475249 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 2589384996 \) Copy content Toggle raw display
$13$ \( (T^{2} + 379 T - 254520)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 867133440000 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 8227948033600 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 6653923430400 \) Copy content Toggle raw display
$29$ \( (T^{2} - 4459 T - 3960660)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 40\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 99\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( (T^{2} - 8814 T + 8966160)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 15853 T + 44661910)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 52\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 23\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 64\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{2} - 8992 T - 4289674884)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 33\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 46\!\cdots\!69 \) Copy content Toggle raw display
$83$ \( (T^{2} - 949 T - 7511023590)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 59\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( (T^{2} + 129423 T - 5231869258)^{2} \) Copy content Toggle raw display
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