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Properties

Label 336.3.bh.a
Level $336$
Weight $3$
Character orbit 336.bh
Analytic conductor $9.155$
Analytic rank $0$
Dimension $2$
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} - 1) q^{3} + (3 \zeta_{6} - 6) q^{5} + ( - 7 \zeta_{6} + 7) q^{7} + 3 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} - 1) q^{3} + (3 \zeta_{6} - 6) q^{5} + ( - 7 \zeta_{6} + 7) q^{7} + 3 \zeta_{6} q^{9} + (11 \zeta_{6} - 11) q^{11} + ( - 8 \zeta_{6} + 4) q^{13} + 9 q^{15} + (14 \zeta_{6} + 14) q^{17} + ( - 2 \zeta_{6} + 4) q^{19} + (7 \zeta_{6} - 14) q^{21} + 28 \zeta_{6} q^{23} + ( - 2 \zeta_{6} + 2) q^{25} + ( - 6 \zeta_{6} + 3) q^{27} + 25 q^{29} + (19 \zeta_{6} + 19) q^{31} + ( - 11 \zeta_{6} + 22) q^{33} + (42 \zeta_{6} - 21) q^{35} + 58 \zeta_{6} q^{37} + (12 \zeta_{6} - 12) q^{39} + ( - 4 \zeta_{6} + 2) q^{41} - 26 q^{43} + ( - 9 \zeta_{6} - 9) q^{45} + ( - 44 \zeta_{6} + 88) q^{47} - 49 \zeta_{6} q^{49} - 42 \zeta_{6} q^{51} + (31 \zeta_{6} - 31) q^{53} + ( - 66 \zeta_{6} + 33) q^{55} - 6 q^{57} + (5 \zeta_{6} + 5) q^{59} + ( - 8 \zeta_{6} + 16) q^{61} + 21 q^{63} + 36 \zeta_{6} q^{65} + (52 \zeta_{6} - 52) q^{67} + ( - 56 \zeta_{6} + 28) q^{69} - 64 q^{71} + (4 \zeta_{6} + 4) q^{73} + (2 \zeta_{6} - 4) q^{75} + 77 \zeta_{6} q^{77} + 17 \zeta_{6} q^{79} + (9 \zeta_{6} - 9) q^{81} + (62 \zeta_{6} - 31) q^{83} - 126 q^{85} + ( - 25 \zeta_{6} - 25) q^{87} + (46 \zeta_{6} - 92) q^{89} + ( - 28 \zeta_{6} - 28) q^{91} - 57 \zeta_{6} q^{93} + (18 \zeta_{6} - 18) q^{95} + ( - 106 \zeta_{6} + 53) q^{97} - 33 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} - 9 q^{5} + 7 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} - 9 q^{5} + 7 q^{7} + 3 q^{9} - 11 q^{11} + 18 q^{15} + 42 q^{17} + 6 q^{19} - 21 q^{21} + 28 q^{23} + 2 q^{25} + 50 q^{29} + 57 q^{31} + 33 q^{33} + 58 q^{37} - 12 q^{39} - 52 q^{43} - 27 q^{45} + 132 q^{47} - 49 q^{49} - 42 q^{51} - 31 q^{53} - 12 q^{57} + 15 q^{59} + 24 q^{61} + 42 q^{63} + 36 q^{65} - 52 q^{67} - 128 q^{71} + 12 q^{73} - 6 q^{75} + 77 q^{77} + 17 q^{79} - 9 q^{81} - 252 q^{85} - 75 q^{87} - 138 q^{89} - 84 q^{91} - 57 q^{93} - 18 q^{95} - 66 q^{99}+O(q^{100}) \) 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 3.50000 + 6.06218i 0 1.50000 − 2.59808i 0
241.1 0 −1.50000 − 0.866025i 0 −4.50000 + 2.59808i 0 3.50000 − 6.06218i 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.a 2
3.b odd 2 1 1008.3.cg.g 2
4.b odd 2 1 21.3.f.b ✓ 2
7.c even 3 1 2352.3.f.d 2
7.d odd 6 1 inner 336.3.bh.a 2
7.d odd 6 1 2352.3.f.d 2
12.b even 2 1 63.3.m.c 2
20.d odd 2 1 525.3.o.g 2
20.e even 4 2 525.3.s.c 4
21.g even 6 1 1008.3.cg.g 2
28.d even 2 1 147.3.f.c 2
28.f even 6 1 21.3.f.b ✓ 2
28.f even 6 1 147.3.d.b 2
28.g odd 6 1 147.3.d.b 2
28.g odd 6 1 147.3.f.c 2
84.h odd 2 1 441.3.m.e 2
84.j odd 6 1 63.3.m.c 2
84.j odd 6 1 441.3.d.b 2
84.n even 6 1 441.3.d.b 2
84.n even 6 1 441.3.m.e 2
140.s even 6 1 525.3.o.g 2
140.x odd 12 2 525.3.s.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.f.b ✓ 2 4.b odd 2 1
21.3.f.b ✓ 2 28.f even 6 1
63.3.m.c 2 12.b even 2 1
63.3.m.c 2 84.j odd 6 1
147.3.d.b 2 28.f even 6 1
147.3.d.b 2 28.g odd 6 1
147.3.f.c 2 28.d even 2 1
147.3.f.c 2 28.g odd 6 1
336.3.bh.a 2 1.a even 1 1 trivial
336.3.bh.a 2 7.d odd 6 1 inner
441.3.d.b 2 84.j odd 6 1
441.3.d.b 2 84.n even 6 1
441.3.m.e 2 84.h odd 2 1
441.3.m.e 2 84.n even 6 1
525.3.o.g 2 20.d odd 2 1
525.3.o.g 2 140.s even 6 1
525.3.s.c 4 20.e even 4 2
525.3.s.c 4 140.x odd 12 2
1008.3.cg.g 2 3.b odd 2 1
1008.3.cg.g 2 21.g even 6 1
2352.3.f.d 2 7.c even 3 1
2352.3.f.d 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
$7$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$11$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$13$ \( T^{2} + 48 \) Copy content Toggle raw display
$17$ \( T^{2} - 42T + 588 \) Copy content Toggle raw display
$19$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$23$ \( T^{2} - 28T + 784 \) Copy content Toggle raw display
$29$ \( (T - 25)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 57T + 1083 \) Copy content Toggle raw display
$37$ \( T^{2} - 58T + 3364 \) Copy content Toggle raw display
$41$ \( T^{2} + 12 \) Copy content Toggle raw display
$43$ \( (T + 26)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 132T + 5808 \) Copy content Toggle raw display
$53$ \( T^{2} + 31T + 961 \) Copy content Toggle raw display
$59$ \( T^{2} - 15T + 75 \) Copy content Toggle raw display
$61$ \( T^{2} - 24T + 192 \) Copy content Toggle raw display
$67$ \( T^{2} + 52T + 2704 \) Copy content Toggle raw display
$71$ \( (T + 64)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 12T + 48 \) Copy content Toggle raw display
$79$ \( T^{2} - 17T + 289 \) Copy content Toggle raw display
$83$ \( T^{2} + 2883 \) Copy content Toggle raw display
$89$ \( T^{2} + 138T + 6348 \) Copy content Toggle raw display
$97$ \( T^{2} + 8427 \) Copy content Toggle raw display
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