Properties

Label 336.4.b.a
Level $336$
Weight $4$
Character orbit 336.b
Analytic conductor $19.825$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [336,4,Mod(223,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.223");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 336.b (of order \(2\), degree \(1\), 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: \(19.8246417619\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = \sqrt{-6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 q^{3} + \beta q^{5} + ( - 3 \beta - 17) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + \beta q^{5} + ( - 3 \beta - 17) q^{7} + 9 q^{9} + 23 \beta q^{11} - 30 \beta q^{13} - 3 \beta q^{15} + 21 \beta q^{17} - 10 q^{19} + (9 \beta + 51) q^{21} - 47 \beta q^{23} + 119 q^{25} - 27 q^{27} + 126 q^{29} - 8 q^{31} - 69 \beta q^{33} + ( - 17 \beta + 18) q^{35} + 244 q^{37} + 90 \beta q^{39} - 151 \beta q^{41} + 66 \beta q^{43} + 9 \beta q^{45} + 180 q^{47} + (102 \beta + 235) q^{49} - 63 \beta q^{51} + 594 q^{53} - 138 q^{55} + 30 q^{57} + 540 q^{59} - 144 \beta q^{61} + ( - 27 \beta - 153) q^{63} + 180 q^{65} - 432 \beta q^{67} + 141 \beta q^{69} - 73 \beta q^{71} - 270 \beta q^{73} - 357 q^{75} + ( - 391 \beta + 414) q^{77} + 420 \beta q^{79} + 81 q^{81} + 864 q^{83} - 126 q^{85} - 378 q^{87} + 527 \beta q^{89} + (510 \beta - 540) q^{91} + 24 q^{93} - 10 \beta q^{95} + 330 \beta q^{97} + 207 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 34 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 34 q^{7} + 18 q^{9} - 20 q^{19} + 102 q^{21} + 238 q^{25} - 54 q^{27} + 252 q^{29} - 16 q^{31} + 36 q^{35} + 488 q^{37} + 360 q^{47} + 470 q^{49} + 1188 q^{53} - 276 q^{55} + 60 q^{57} + 1080 q^{59} - 306 q^{63} + 360 q^{65} - 714 q^{75} + 828 q^{77} + 162 q^{81} + 1728 q^{83} - 252 q^{85} - 756 q^{87} - 1080 q^{91} + 48 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
223.1
2.44949i
2.44949i
0 −3.00000 0 2.44949i 0 −17.0000 + 7.34847i 0 9.00000 0
223.2 0 −3.00000 0 2.44949i 0 −17.0000 7.34847i 0 9.00000 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
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.4.b.a 2
3.b odd 2 1 1008.4.b.a 2
4.b odd 2 1 336.4.b.d yes 2
7.b odd 2 1 336.4.b.d yes 2
8.b even 2 1 1344.4.b.c 2
8.d odd 2 1 1344.4.b.b 2
12.b even 2 1 1008.4.b.f 2
21.c even 2 1 1008.4.b.f 2
28.d even 2 1 inner 336.4.b.a 2
56.e even 2 1 1344.4.b.c 2
56.h odd 2 1 1344.4.b.b 2
84.h odd 2 1 1008.4.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.4.b.a 2 1.a even 1 1 trivial
336.4.b.a 2 28.d even 2 1 inner
336.4.b.d yes 2 4.b odd 2 1
336.4.b.d yes 2 7.b odd 2 1
1008.4.b.a 2 3.b odd 2 1
1008.4.b.a 2 84.h odd 2 1
1008.4.b.f 2 12.b even 2 1
1008.4.b.f 2 21.c even 2 1
1344.4.b.b 2 8.d odd 2 1
1344.4.b.b 2 56.h odd 2 1
1344.4.b.c 2 8.b even 2 1
1344.4.b.c 2 56.e even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(336, [\chi])\):

\( T_{5}^{2} + 6 \) Copy content Toggle raw display
\( T_{19} + 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 6 \) Copy content Toggle raw display
$7$ \( T^{2} + 34T + 343 \) Copy content Toggle raw display
$11$ \( T^{2} + 3174 \) Copy content Toggle raw display
$13$ \( T^{2} + 5400 \) Copy content Toggle raw display
$17$ \( T^{2} + 2646 \) Copy content Toggle raw display
$19$ \( (T + 10)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 13254 \) Copy content Toggle raw display
$29$ \( (T - 126)^{2} \) Copy content Toggle raw display
$31$ \( (T + 8)^{2} \) Copy content Toggle raw display
$37$ \( (T - 244)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 136806 \) Copy content Toggle raw display
$43$ \( T^{2} + 26136 \) Copy content Toggle raw display
$47$ \( (T - 180)^{2} \) Copy content Toggle raw display
$53$ \( (T - 594)^{2} \) Copy content Toggle raw display
$59$ \( (T - 540)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 124416 \) Copy content Toggle raw display
$67$ \( T^{2} + 1119744 \) Copy content Toggle raw display
$71$ \( T^{2} + 31974 \) Copy content Toggle raw display
$73$ \( T^{2} + 437400 \) Copy content Toggle raw display
$79$ \( T^{2} + 1058400 \) Copy content Toggle raw display
$83$ \( (T - 864)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 1666374 \) Copy content Toggle raw display
$97$ \( T^{2} + 653400 \) Copy content Toggle raw display
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