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Properties

Label 336.6.q.b
Level $336$
Weight $6$
Character orbit 336.q
Analytic conductor $53.889$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (9 \zeta_{6} - 9) q^{3} - 11 \zeta_{6} q^{5} + (7 \zeta_{6} - 133) q^{7} - 81 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (9 \zeta_{6} - 9) q^{3} - 11 \zeta_{6} q^{5} + (7 \zeta_{6} - 133) q^{7} - 81 \zeta_{6} q^{9} + ( - 269 \zeta_{6} + 269) q^{11} - 308 q^{13} + 99 q^{15} + (1896 \zeta_{6} - 1896) q^{17} - 164 \zeta_{6} q^{19} + ( - 1197 \zeta_{6} + 1134) q^{21} - 3264 \zeta_{6} q^{23} + ( - 3004 \zeta_{6} + 3004) q^{25} + 729 q^{27} + 2417 q^{29} + ( - 2841 \zeta_{6} + 2841) q^{31} + 2421 \zeta_{6} q^{33} + (1386 \zeta_{6} + 77) q^{35} + 11328 \zeta_{6} q^{37} + ( - 2772 \zeta_{6} + 2772) q^{39} - 16856 q^{41} + 7894 q^{43} + (891 \zeta_{6} - 891) q^{45} + 21102 \zeta_{6} q^{47} + ( - 1813 \zeta_{6} + 17640) q^{49} - 17064 \zeta_{6} q^{51} + ( - 29691 \zeta_{6} + 29691) q^{53} - 2959 q^{55} + 1476 q^{57} + (8163 \zeta_{6} - 8163) q^{59} - 15166 \zeta_{6} q^{61} + (10206 \zeta_{6} + 567) q^{63} + 3388 \zeta_{6} q^{65} + (32078 \zeta_{6} - 32078) q^{67} + 29376 q^{69} + 38274 q^{71} + (34866 \zeta_{6} - 34866) q^{73} + 27036 \zeta_{6} q^{75} + (35777 \zeta_{6} - 33894) q^{77} + 13529 \zeta_{6} q^{79} + (6561 \zeta_{6} - 6561) q^{81} + 68103 q^{83} + 20856 q^{85} + (21753 \zeta_{6} - 21753) q^{87} + 114922 \zeta_{6} q^{89} + ( - 2156 \zeta_{6} + 40964) q^{91} + 25569 \zeta_{6} q^{93} + (1804 \zeta_{6} - 1804) q^{95} + 154959 q^{97} - 21789 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 9 q^{3} - 11 q^{5} - 259 q^{7} - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 9 q^{3} - 11 q^{5} - 259 q^{7} - 81 q^{9} + 269 q^{11} - 616 q^{13} + 198 q^{15} - 1896 q^{17} - 164 q^{19} + 1071 q^{21} - 3264 q^{23} + 3004 q^{25} + 1458 q^{27} + 4834 q^{29} + 2841 q^{31} + 2421 q^{33} + 1540 q^{35} + 11328 q^{37} + 2772 q^{39} - 33712 q^{41} + 15788 q^{43} - 891 q^{45} + 21102 q^{47} + 33467 q^{49} - 17064 q^{51} + 29691 q^{53} - 5918 q^{55} + 2952 q^{57} - 8163 q^{59} - 15166 q^{61} + 11340 q^{63} + 3388 q^{65} - 32078 q^{67} + 58752 q^{69} + 76548 q^{71} - 34866 q^{73} + 27036 q^{75} - 32011 q^{77} + 13529 q^{79} - 6561 q^{81} + 136206 q^{83} + 41712 q^{85} - 21753 q^{87} + 114922 q^{89} + 79772 q^{91} + 25569 q^{93} - 1804 q^{95} + 309918 q^{97} - 43578 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\zeta_{6}\)

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
0.500000 + 0.866025i
0.500000 − 0.866025i
0 −4.50000 + 7.79423i 0 −5.50000 − 9.52628i 0 −129.500 + 6.06218i 0 −40.5000 − 70.1481i 0
289.1 0 −4.50000 − 7.79423i 0 −5.50000 + 9.52628i 0 −129.500 − 6.06218i 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.b 2
4.b odd 2 1 21.6.e.a ✓ 2
7.c even 3 1 inner 336.6.q.b 2
12.b even 2 1 63.6.e.a 2
28.d even 2 1 147.6.e.g 2
28.f even 6 1 147.6.a.d 1
28.f even 6 1 147.6.e.g 2
28.g odd 6 1 21.6.e.a ✓ 2
28.g odd 6 1 147.6.a.c 1
84.j odd 6 1 441.6.a.h 1
84.n even 6 1 63.6.e.a 2
84.n even 6 1 441.6.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.a ✓ 2 4.b odd 2 1
21.6.e.a ✓ 2 28.g odd 6 1
63.6.e.a 2 12.b even 2 1
63.6.e.a 2 84.n even 6 1
147.6.a.c 1 28.g odd 6 1
147.6.a.d 1 28.f even 6 1
147.6.e.g 2 28.d even 2 1
147.6.e.g 2 28.f even 6 1
336.6.q.b 2 1.a even 1 1 trivial
336.6.q.b 2 7.c even 3 1 inner
441.6.a.g 1 84.n even 6 1
441.6.a.h 1 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 11T_{5} + 121 \) acting on \(S_{6}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$5$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$7$ \( T^{2} + 259T + 16807 \) Copy content Toggle raw display
$11$ \( T^{2} - 269T + 72361 \) Copy content Toggle raw display
$13$ \( (T + 308)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 1896 T + 3594816 \) Copy content Toggle raw display
$19$ \( T^{2} + 164T + 26896 \) Copy content Toggle raw display
$23$ \( T^{2} + 3264 T + 10653696 \) Copy content Toggle raw display
$29$ \( (T - 2417)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 2841 T + 8071281 \) Copy content Toggle raw display
$37$ \( T^{2} - 11328 T + 128323584 \) Copy content Toggle raw display
$41$ \( (T + 16856)^{2} \) Copy content Toggle raw display
$43$ \( (T - 7894)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 21102 T + 445294404 \) Copy content Toggle raw display
$53$ \( T^{2} - 29691 T + 881555481 \) Copy content Toggle raw display
$59$ \( T^{2} + 8163 T + 66634569 \) Copy content Toggle raw display
$61$ \( T^{2} + 15166 T + 230007556 \) Copy content Toggle raw display
$67$ \( T^{2} + 32078 T + 1028998084 \) Copy content Toggle raw display
$71$ \( (T - 38274)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 34866 T + 1215637956 \) Copy content Toggle raw display
$79$ \( T^{2} - 13529 T + 183033841 \) Copy content Toggle raw display
$83$ \( (T - 68103)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 114922 T + 13207066084 \) Copy content Toggle raw display
$97$ \( (T - 154959)^{2} \) Copy content Toggle raw display
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