↑

Properties

Label 336.3.bn.f
Level $336$
Weight $3$
Character orbit 336.bn
Analytic conductor $9.155$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [336,3,Mod(65,336)]
 
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, 3, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.65");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 336.bn (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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 25 \) 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: $\mathrm{SU}(2)[C_{6}]$

$q$-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 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.

\(f(q)\) \(=\) \( q + (\beta_{3} + 2 \beta_{2} - \beta_1) q^{3} + \beta_1 q^{5} - 7 \beta_{2} q^{7} + ( - \beta_{2} - 4 \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 2 \beta_{2} - \beta_1) q^{3} + \beta_1 q^{5} - 7 \beta_{2} q^{7} + ( - \beta_{2} - 4 \beta_1 + 1) q^{9} + (5 \beta_{3} - 5 \beta_1) q^{11} - 2 q^{13} + (2 \beta_{3} - 5) q^{15} + (12 \beta_{3} - 12 \beta_1) q^{17} + ( - 16 \beta_{2} + 16) q^{19} + ( - 14 \beta_{2} + 7 \beta_1 + 14) q^{21} - 6 \beta_1 q^{23} - 20 \beta_{2} q^{25} + ( - 7 \beta_{3} + 22) q^{27} - 7 \beta_{3} q^{29} - 3 \beta_{2} q^{31} + ( - 25 \beta_{2} - 10 \beta_1 + 25) q^{33} - 7 \beta_{3} q^{35} + (12 \beta_{2} - 12) q^{37} + ( - 2 \beta_{3} - 4 \beta_{2} + 2 \beta_1) q^{39} - 14 \beta_{3} q^{41} - 44 q^{43} + ( - \beta_{3} - 20 \beta_{2} + \beta_1) q^{45} + 6 \beta_1 q^{47} + (49 \beta_{2} - 49) q^{49} + ( - 60 \beta_{2} - 24 \beta_1 + 60) q^{51} + (9 \beta_{3} - 9 \beta_1) q^{53} - 25 q^{55} + (16 \beta_{3} + 32) q^{57} + (9 \beta_{3} - 9 \beta_1) q^{59} + ( - 26 \beta_{2} + 26) q^{61} + (28 \beta_{3} - 7) q^{63} - 2 \beta_1 q^{65} + 52 \beta_{2} q^{67} + ( - 12 \beta_{3} + 30) q^{69} + 42 \beta_{3} q^{71} - 18 \beta_{2} q^{73} + ( - 40 \beta_{2} + 20 \beta_1 + 40) q^{75} + 35 \beta_1 q^{77} + (79 \beta_{2} - 79) q^{79} + (8 \beta_{3} + 79 \beta_{2} - 8 \beta_1) q^{81} - 63 \beta_{3} q^{83} - 60 q^{85} + ( - 14 \beta_{3} + 35 \beta_{2} + 14 \beta_1) q^{87} + 22 \beta_1 q^{89} + 14 \beta_{2} q^{91} + ( - 6 \beta_{2} + 3 \beta_1 + 6) q^{93} + ( - 16 \beta_{3} + 16 \beta_1) q^{95} - 93 q^{97} + (5 \beta_{3} + 100) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 14 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 14 q^{7} + 2 q^{9} - 8 q^{13} - 20 q^{15} + 32 q^{19} + 28 q^{21} - 40 q^{25} + 88 q^{27} - 6 q^{31} + 50 q^{33} - 24 q^{37} - 8 q^{39} - 176 q^{43} - 40 q^{45} - 98 q^{49} + 120 q^{51} - 100 q^{55} + 128 q^{57} + 52 q^{61} - 28 q^{63} + 104 q^{67} + 120 q^{69} - 36 q^{73} + 80 q^{75} - 158 q^{79} + 158 q^{81} - 240 q^{85} + 70 q^{87} + 28 q^{91} + 12 q^{93} - 372 q^{97} + 400 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
1.93649 + 1.11803i
−1.93649 − 1.11803i
1.93649 − 1.11803i
−1.93649 + 1.11803i
0 −0.936492 + 2.85008i 0 1.93649 + 1.11803i 0 −3.50000 − 6.06218i 0 −7.24597 − 5.33816i 0
65.2 0 2.93649 + 0.614017i 0 −1.93649 − 1.11803i 0 −3.50000 − 6.06218i 0 8.24597 + 3.60611i 0
305.1 0 −0.936492 − 2.85008i 0 1.93649 − 1.11803i 0 −3.50000 + 6.06218i 0 −7.24597 + 5.33816i 0
305.2 0 2.93649 − 0.614017i 0 −1.93649 + 1.11803i 0 −3.50000 + 6.06218i 0 8.24597 − 3.60611i 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
3.b odd 2 1 inner
7.c even 3 1 inner
21.h 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.bn.f 4
3.b odd 2 1 inner 336.3.bn.f 4
4.b odd 2 1 21.3.h.b ✓ 4
7.c even 3 1 inner 336.3.bn.f 4
12.b even 2 1 21.3.h.b ✓ 4
21.h odd 6 1 inner 336.3.bn.f 4
28.d even 2 1 147.3.h.b 4
28.f even 6 1 147.3.b.c 2
28.f even 6 1 147.3.h.b 4
28.g odd 6 1 21.3.h.b ✓ 4
28.g odd 6 1 147.3.b.d 2
84.h odd 2 1 147.3.h.b 4
84.j odd 6 1 147.3.b.c 2
84.j odd 6 1 147.3.h.b 4
84.n even 6 1 21.3.h.b ✓ 4
84.n even 6 1 147.3.b.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.h.b ✓ 4 4.b odd 2 1
21.3.h.b ✓ 4 12.b even 2 1
21.3.h.b ✓ 4 28.g odd 6 1
21.3.h.b ✓ 4 84.n even 6 1
147.3.b.c 2 28.f even 6 1
147.3.b.c 2 84.j odd 6 1
147.3.b.d 2 28.g odd 6 1
147.3.b.d 2 84.n even 6 1
147.3.h.b 4 28.d even 2 1
147.3.h.b 4 28.f even 6 1
147.3.h.b 4 84.h odd 2 1
147.3.h.b 4 84.j odd 6 1
336.3.bn.f 4 1.a even 1 1 trivial
336.3.bn.f 4 3.b odd 2 1 inner
336.3.bn.f 4 7.c even 3 1 inner
336.3.bn.f 4 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(336, [\chi])\):

\( T_{5}^{4} - 5T_{5}^{2} + 25 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 4 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{4} - 5T^{2} + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 125 T^{2} + 15625 \) Copy content Toggle raw display
$13$ \( (T + 2)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 720 T^{2} + 518400 \) Copy content Toggle raw display
$19$ \( (T^{2} - 16 T + 256)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 180 T^{2} + 32400 \) Copy content Toggle raw display
$29$ \( (T^{2} + 245)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 12 T + 144)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 980)^{2} \) Copy content Toggle raw display
$43$ \( (T + 44)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 180 T^{2} + 32400 \) Copy content Toggle raw display
$53$ \( T^{4} - 405 T^{2} + 164025 \) Copy content Toggle raw display
$59$ \( T^{4} - 405 T^{2} + 164025 \) Copy content Toggle raw display
$61$ \( (T^{2} - 26 T + 676)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 52 T + 2704)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 8820)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 18 T + 324)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 79 T + 6241)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 19845)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 2420 T^{2} + 5856400 \) Copy content Toggle raw display
$97$ \( (T + 93)^{4} \) Copy content Toggle raw display
show more
show less