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Properties

Label 336.3.bh.d
Level $336$
Weight $3$
Character orbit 336.bh
Analytic conductor $9.155$
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] = [336,3,Mod(145,336)]
 
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, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.145");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 336.bh (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: \(9.15533688251\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} + 1) q^{3} + ( - 3 \zeta_{6} + 6) q^{5} + ( - 3 \zeta_{6} - 5) q^{7} + 3 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} + 1) q^{3} + ( - 3 \zeta_{6} + 6) q^{5} + ( - 3 \zeta_{6} - 5) q^{7} + 3 \zeta_{6} q^{9} + ( - 15 \zeta_{6} + 15) q^{11} + ( - 16 \zeta_{6} + 8) q^{13} + 9 q^{15} + (6 \zeta_{6} + 6) q^{17} + ( - 6 \zeta_{6} + 12) q^{19} + ( - 11 \zeta_{6} - 2) q^{21} + ( - 2 \zeta_{6} + 2) q^{25} + (6 \zeta_{6} - 3) q^{27} - 9 q^{29} + (7 \zeta_{6} + 7) q^{31} + ( - 15 \zeta_{6} + 30) q^{33} + (6 \zeta_{6} - 39) q^{35} - 10 \zeta_{6} q^{37} + ( - 24 \zeta_{6} + 24) q^{39} + ( - 12 \zeta_{6} + 6) q^{41} + 74 q^{43} + (9 \zeta_{6} + 9) q^{45} + (39 \zeta_{6} + 16) q^{49} + 18 \zeta_{6} q^{51} + (33 \zeta_{6} - 33) q^{53} + ( - 90 \zeta_{6} + 45) q^{55} + 18 q^{57} + ( - 9 \zeta_{6} - 9) q^{59} + ( - 52 \zeta_{6} + 104) q^{61} + ( - 24 \zeta_{6} + 9) q^{63} - 72 \zeta_{6} q^{65} + (76 \zeta_{6} - 76) q^{67} - 84 q^{71} + ( - 36 \zeta_{6} - 36) q^{73} + ( - 2 \zeta_{6} + 4) q^{75} + (75 \zeta_{6} - 120) q^{77} - 43 \zeta_{6} q^{79} + (9 \zeta_{6} - 9) q^{81} + (138 \zeta_{6} - 69) q^{83} + 54 q^{85} + ( - 9 \zeta_{6} - 9) q^{87} + (42 \zeta_{6} - 84) q^{89} + (104 \zeta_{6} - 88) q^{91} + 21 \zeta_{6} q^{93} + ( - 54 \zeta_{6} + 54) q^{95} + (214 \zeta_{6} - 107) q^{97} + 45 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} + 9 q^{5} - 13 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{3} + 9 q^{5} - 13 q^{7} + 3 q^{9} + 15 q^{11} + 18 q^{15} + 18 q^{17} + 18 q^{19} - 15 q^{21} + 2 q^{25} - 18 q^{29} + 21 q^{31} + 45 q^{33} - 72 q^{35} - 10 q^{37} + 24 q^{39} + 148 q^{43} + 27 q^{45} + 71 q^{49} + 18 q^{51} - 33 q^{53} + 36 q^{57} - 27 q^{59} + 156 q^{61} - 6 q^{63} - 72 q^{65} - 76 q^{67} - 168 q^{71} - 108 q^{73} + 6 q^{75} - 165 q^{77} - 43 q^{79} - 9 q^{81} + 108 q^{85} - 27 q^{87} - 126 q^{89} - 72 q^{91} + 21 q^{93} + 54 q^{95} + 90 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 1.50000 − 0.866025i 0 4.50000 + 2.59808i 0 −6.50000 + 2.59808i 0 1.50000 − 2.59808i 0
241.1 0 1.50000 + 0.866025i 0 4.50000 − 2.59808i 0 −6.50000 − 2.59808i 0 1.50000 + 2.59808i 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.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.3.bh.d 2
3.b odd 2 1 1008.3.cg.a 2
4.b odd 2 1 21.3.f.a ✓ 2
7.c even 3 1 2352.3.f.a 2
7.d odd 6 1 inner 336.3.bh.d 2
7.d odd 6 1 2352.3.f.a 2
12.b even 2 1 63.3.m.d 2
20.d odd 2 1 525.3.o.h 2
20.e even 4 2 525.3.s.e 4
21.g even 6 1 1008.3.cg.a 2
28.d even 2 1 147.3.f.a 2
28.f even 6 1 21.3.f.a ✓ 2
28.f even 6 1 147.3.d.c 2
28.g odd 6 1 147.3.d.c 2
28.g odd 6 1 147.3.f.a 2
84.h odd 2 1 441.3.m.g 2
84.j odd 6 1 63.3.m.d 2
84.j odd 6 1 441.3.d.a 2
84.n even 6 1 441.3.d.a 2
84.n even 6 1 441.3.m.g 2
140.s even 6 1 525.3.o.h 2
140.x odd 12 2 525.3.s.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.f.a ✓ 2 4.b odd 2 1
21.3.f.a ✓ 2 28.f even 6 1
63.3.m.d 2 12.b even 2 1
63.3.m.d 2 84.j odd 6 1
147.3.d.c 2 28.f even 6 1
147.3.d.c 2 28.g odd 6 1
147.3.f.a 2 28.d even 2 1
147.3.f.a 2 28.g odd 6 1
336.3.bh.d 2 1.a even 1 1 trivial
336.3.bh.d 2 7.d odd 6 1 inner
441.3.d.a 2 84.j odd 6 1
441.3.d.a 2 84.n even 6 1
441.3.m.g 2 84.h odd 2 1
441.3.m.g 2 84.n even 6 1
525.3.o.h 2 20.d odd 2 1
525.3.o.h 2 140.s even 6 1
525.3.s.e 4 20.e even 4 2
525.3.s.e 4 140.x odd 12 2
1008.3.cg.a 2 3.b odd 2 1
1008.3.cg.a 2 21.g even 6 1
2352.3.f.a 2 7.c even 3 1
2352.3.f.a 2 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 9T_{5} + 27 \) acting on \(S_{3}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$5$ \( T^{2} - 9T + 27 \) Copy content Toggle raw display
$7$ \( T^{2} + 13T + 49 \) Copy content Toggle raw display
$11$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$13$ \( T^{2} + 192 \) Copy content Toggle raw display
$17$ \( T^{2} - 18T + 108 \) Copy content Toggle raw display
$19$ \( T^{2} - 18T + 108 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 9)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 21T + 147 \) Copy content Toggle raw display
$37$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$41$ \( T^{2} + 108 \) Copy content Toggle raw display
$43$ \( (T - 74)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 33T + 1089 \) Copy content Toggle raw display
$59$ \( T^{2} + 27T + 243 \) Copy content Toggle raw display
$61$ \( T^{2} - 156T + 8112 \) Copy content Toggle raw display
$67$ \( T^{2} + 76T + 5776 \) Copy content Toggle raw display
$71$ \( (T + 84)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 108T + 3888 \) Copy content Toggle raw display
$79$ \( T^{2} + 43T + 1849 \) Copy content Toggle raw display
$83$ \( T^{2} + 14283 \) Copy content Toggle raw display
$89$ \( T^{2} + 126T + 5292 \) Copy content Toggle raw display
$97$ \( T^{2} + 34347 \) Copy content Toggle raw display
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