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Properties

Label 336.5.bh.d
Level $336$
Weight $5$
Character orbit 336.bh
Analytic conductor $34.732$
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] = [336,5,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, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.145");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
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: \(34.7323075962\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 42)
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.

\(f(q)\) \(=\) \( q + (3 \beta_1 + 6) q^{3} + ( - \beta_{3} + \beta_{2} + 11 \beta_1 - 11) q^{5} + (7 \beta_{3} - 35 \beta_1) q^{7} + (27 \beta_1 + 27) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (3 \beta_1 + 6) q^{3} + ( - \beta_{3} + \beta_{2} + 11 \beta_1 - 11) q^{5} + (7 \beta_{3} - 35 \beta_1) q^{7} + (27 \beta_1 + 27) q^{9} + (2 \beta_{3} - 4 \beta_{2} - 81 \beta_1) q^{11} + ( - 34 \beta_{2} + 72 \beta_1 + 36) q^{13} + ( - 6 \beta_{3} + 3 \beta_{2} - 99) q^{15} + (34 \beta_{3} - 34 \beta_1 - 68) q^{17} + ( - 34 \beta_{3} + 34 \beta_{2} + \cdots + 74) q^{19}+ \cdots + (108 \beta_{3} - 54 \beta_{2} + 2187) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 18 q^{3} - 66 q^{5} + 70 q^{7} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 18 q^{3} - 66 q^{5} + 70 q^{7} + 54 q^{9} + 162 q^{11} - 396 q^{15} - 204 q^{17} + 444 q^{19} + 630 q^{21} - 312 q^{23} - 476 q^{25} + 2724 q^{29} + 3786 q^{31} + 1458 q^{33} - 672 q^{35} + 1396 q^{37} - 648 q^{39} + 632 q^{43} - 1782 q^{45} + 7896 q^{47} - 98 q^{49} - 612 q^{51} - 1038 q^{53} + 2664 q^{57} + 966 q^{59} + 5088 q^{61} + 3780 q^{63} - 744 q^{65} - 14600 q^{67} + 9696 q^{71} + 22584 q^{73} - 4284 q^{75} - 3654 q^{77} - 3974 q^{79} - 1458 q^{81} + 1224 q^{85} + 12258 q^{87} - 33156 q^{89} + 18984 q^{91} + 11358 q^{93} - 3252 q^{95} + 8748 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + 2\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display

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

Character values

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

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

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
0.707107 − 1.22474i
−0.707107 + 1.22474i
0.707107 + 1.22474i
−0.707107 − 1.22474i
0 4.50000 − 2.59808i 0 −20.7426 − 11.9758i 0 47.1985 + 13.1645i 0 13.5000 − 23.3827i 0
145.2 0 4.50000 − 2.59808i 0 −12.2574 − 7.07679i 0 −12.1985 + 47.4573i 0 13.5000 − 23.3827i 0
241.1 0 4.50000 + 2.59808i 0 −20.7426 + 11.9758i 0 47.1985 − 13.1645i 0 13.5000 + 23.3827i 0
241.2 0 4.50000 + 2.59808i 0 −12.2574 + 7.07679i 0 −12.1985 − 47.4573i 0 13.5000 + 23.3827i 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 336.5.bh.d 4
4.b odd 2 1 42.5.g.a ✓ 4
7.d odd 6 1 inner 336.5.bh.d 4
12.b even 2 1 126.5.n.b 4
28.d even 2 1 294.5.g.c 4
28.f even 6 1 42.5.g.a ✓ 4
28.f even 6 1 294.5.c.a 4
28.g odd 6 1 294.5.c.a 4
28.g odd 6 1 294.5.g.c 4
84.j odd 6 1 126.5.n.b 4
84.j odd 6 1 882.5.c.a 4
84.n even 6 1 882.5.c.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.5.g.a ✓ 4 4.b odd 2 1
42.5.g.a ✓ 4 28.f even 6 1
126.5.n.b 4 12.b even 2 1
126.5.n.b 4 84.j odd 6 1
294.5.c.a 4 28.f even 6 1
294.5.c.a 4 28.g odd 6 1
294.5.g.c 4 28.d even 2 1
294.5.g.c 4 28.g odd 6 1
336.5.bh.d 4 1.a even 1 1 trivial
336.5.bh.d 4 7.d odd 6 1 inner
882.5.c.a 4 84.j odd 6 1
882.5.c.a 4 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 66T_{5}^{3} + 1791T_{5}^{2} + 22374T_{5} + 114921 \) acting on \(S_{5}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 9 T + 27)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 66 T^{3} + \cdots + 114921 \) Copy content Toggle raw display
$7$ \( T^{4} - 70 T^{3} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{4} - 162 T^{3} + \cdots + 39350529 \) Copy content Toggle raw display
$13$ \( T^{4} + 63264 T^{2} + 569108736 \) Copy content Toggle raw display
$17$ \( T^{4} + 204 T^{3} + \cdots + 589324176 \) Copy content Toggle raw display
$19$ \( T^{4} - 444 T^{3} + \cdots + 128051856 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 525265461504 \) Copy content Toggle raw display
$29$ \( (T^{2} - 1362 T + 460233)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 1025818531929 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 219747187984 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 8311481425296 \) Copy content Toggle raw display
$43$ \( (T^{2} - 316 T - 10371836)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 5862054484224 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 6375054362769 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 105068670590601 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 2285563428864 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 26\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{2} - 4848 T + 1589184)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 45\!\cdots\!41 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 109011202550649 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 31\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 9242250571449 \) Copy content Toggle raw display
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