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Properties

Label 336.6.q.g
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: \(\Q(\sqrt{-3}, \sqrt{7081})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 1771x^{2} + 1770x + 3132900 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + ( - 9 \beta_{2} + 9) q^{3} + ( - 24 \beta_{2} + \beta_1) q^{5} + ( - \beta_{3} - 59 \beta_{2} + \cdots + 73) q^{7}+ \cdots - 81 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 9 \beta_{2} + 9) q^{3} + ( - 24 \beta_{2} + \beta_1) q^{5} + ( - \beta_{3} - 59 \beta_{2} + \cdots + 73) q^{7}+ \cdots + (81 \beta_{3} + 16443) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 18 q^{3} - 47 q^{5} + 174 q^{7} - 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 18 q^{3} - 47 q^{5} + 174 q^{7} - 162 q^{9} - 407 q^{11} + 898 q^{13} - 846 q^{15} + 1868 q^{17} - 1463 q^{19} - 783 q^{21} - 44 q^{23} + 1605 q^{25} - 2916 q^{27} + 1534 q^{29} - 11170 q^{31} + 3663 q^{33} - 9674 q^{35} + 3113 q^{37} + 4041 q^{39} - 15684 q^{41} + 25258 q^{43} - 3807 q^{45} + 9576 q^{47} + 4558 q^{49} - 16812 q^{51} - 13395 q^{53} + 26210 q^{55} - 26334 q^{57} - 47521 q^{59} + 63652 q^{61} - 21141 q^{63} - 28254 q^{65} + 44541 q^{67} - 792 q^{69} + 251680 q^{71} - 6039 q^{73} - 14445 q^{75} + 35407 q^{77} + 17588 q^{79} - 13122 q^{81} - 78650 q^{83} - 116120 q^{85} + 6903 q^{87} - 83082 q^{89} - 31747 q^{91} + 100530 q^{93} - 214946 q^{95} + 369570 q^{97} + 65934 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 1771x^{2} + 1770x + 3132900 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 1771\nu^{2} - 1771\nu + 3132900 ) / 3134670 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 3541 ) / 1771 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 1770\beta_{2} + \beta _1 - 1771 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 1771\beta_{3} - 3541 \) 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
−20.7872 − 36.0044i
21.2872 + 36.8705i
−20.7872 + 36.0044i
21.2872 − 36.8705i
0 4.50000 − 7.79423i 0 −32.7872 − 56.7890i 0 −40.6487 − 123.104i 0 −40.5000 − 70.1481i 0
193.2 0 4.50000 − 7.79423i 0 9.28717 + 16.0858i 0 127.649 + 22.6454i 0 −40.5000 − 70.1481i 0
289.1 0 4.50000 + 7.79423i 0 −32.7872 + 56.7890i 0 −40.6487 + 123.104i 0 −40.5000 + 70.1481i 0
289.2 0 4.50000 + 7.79423i 0 9.28717 − 16.0858i 0 127.649 − 22.6454i 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.g 4
4.b odd 2 1 84.6.i.b ✓ 4
7.c even 3 1 inner 336.6.q.g 4
12.b even 2 1 252.6.k.e 4
28.d even 2 1 588.6.i.m 4
28.f even 6 1 588.6.a.h 2
28.f even 6 1 588.6.i.m 4
28.g odd 6 1 84.6.i.b ✓ 4
28.g odd 6 1 588.6.a.l 2
84.n even 6 1 252.6.k.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.6.i.b ✓ 4 4.b odd 2 1
84.6.i.b ✓ 4 28.g odd 6 1
252.6.k.e 4 12.b even 2 1
252.6.k.e 4 84.n even 6 1
336.6.q.g 4 1.a even 1 1 trivial
336.6.q.g 4 7.c even 3 1 inner
588.6.a.h 2 28.f even 6 1
588.6.a.l 2 28.g odd 6 1
588.6.i.m 4 28.d even 2 1
588.6.i.m 4 28.f even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 47T_{5}^{3} + 3427T_{5}^{2} - 57246T_{5} + 1483524 \) 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
$5$ \( T^{4} + 47 T^{3} + \cdots + 1483524 \) Copy content Toggle raw display
$7$ \( T^{4} - 174 T^{3} + \cdots + 282475249 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 1571488164 \) Copy content Toggle raw display
$13$ \( (T^{2} - 449 T + 6144)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 712390017024 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 16559430371584 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 313361370464256 \) Copy content Toggle raw display
$29$ \( (T^{2} - 767 T - 20885268)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 439626033842121 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 2696203408144 \) Copy content Toggle raw display
$41$ \( (T^{2} + 7842 T + 15310512)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 12629 T + 39092230)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 425625782490000 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 44\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 17\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{2} - 125840 T + 3516363900)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 97\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 21\!\cdots\!29 \) Copy content Toggle raw display
$83$ \( (T^{2} + 39325 T - 6214225254)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{2} - 184785 T + 6255893750)^{2} \) Copy content Toggle raw display
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