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Properties

Label 336.4.q.g
Level $336$
Weight $4$
Character orbit 336.q
Analytic conductor $19.825$
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] = [336,4,Mod(193,336)]
 
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, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.193");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 336.q (of order \(3\), 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: \(19.8246417619\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{505})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 127x^{2} + 126x + 15876 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (3 \beta_{2} - 3) q^{3} + ( - 4 \beta_{2} - \beta_1) q^{5} + (\beta_{3} + 17 \beta_{2} - 9) q^{7} - 9 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (3 \beta_{2} - 3) q^{3} + ( - 4 \beta_{2} - \beta_1) q^{5} + (\beta_{3} + 17 \beta_{2} - 9) q^{7} - 9 \beta_{2} q^{9} + ( - 5 \beta_{3} - 5 \beta_1 + 5) q^{11} + (\beta_{3} - 10) q^{13} + ( - 3 \beta_{3} + 15) q^{15} + (4 \beta_{3} - 4 \beta_{2} + 4 \beta_1) q^{17} + (55 \beta_{2} + 7 \beta_1) q^{19} + ( - 3 \beta_{3} - 24 \beta_{2} + \cdots - 24) q^{21}+ \cdots + (45 \beta_{3} - 45) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{3} - 9 q^{5} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{3} - 9 q^{5} - 18 q^{9} + 5 q^{11} - 38 q^{13} + 54 q^{15} + 4 q^{17} + 117 q^{19} - 153 q^{21} + 148 q^{23} - 43 q^{25} + 108 q^{27} - 190 q^{29} - 360 q^{31} + 15 q^{33} + 482 q^{35} + 53 q^{37} + 57 q^{39} - 340 q^{41} - 806 q^{43} - 81 q^{45} + 368 q^{47} - 362 q^{49} + 12 q^{51} - 697 q^{53} - 2570 q^{55} - 702 q^{57} - 585 q^{59} + 1160 q^{61} + 459 q^{63} + 338 q^{65} + 233 q^{67} - 888 q^{69} - 1232 q^{71} + 817 q^{73} - 129 q^{75} - 1135 q^{77} - 802 q^{79} - 162 q^{81} + 566 q^{83} + 1984 q^{85} + 285 q^{87} + 1858 q^{89} + 505 q^{91} - 1080 q^{93} + 2294 q^{95} + 3458 q^{97} - 90 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 127x^{2} + 126x + 15876 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 127\nu^{2} - 127\nu + 15876 ) / 16002 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 253 ) / 127 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 126\beta_{2} + \beta _1 - 127 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 127\beta_{3} - 253 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
5.86805 + 10.1638i
−5.36805 − 9.29774i
5.86805 − 10.1638i
−5.36805 + 9.29774i
0 −1.50000 + 2.59808i 0 −7.86805 − 13.6279i 0 −11.2361 + 14.7224i 0 −4.50000 − 7.79423i 0
193.2 0 −1.50000 + 2.59808i 0 3.36805 + 5.83364i 0 11.2361 + 14.7224i 0 −4.50000 − 7.79423i 0
289.1 0 −1.50000 − 2.59808i 0 −7.86805 + 13.6279i 0 −11.2361 − 14.7224i 0 −4.50000 + 7.79423i 0
289.2 0 −1.50000 − 2.59808i 0 3.36805 − 5.83364i 0 11.2361 − 14.7224i 0 −4.50000 + 7.79423i 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.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.4.q.g 4
4.b odd 2 1 168.4.q.d ✓ 4
7.c even 3 1 inner 336.4.q.g 4
7.c even 3 1 2352.4.a.cc 2
7.d odd 6 1 2352.4.a.bo 2
12.b even 2 1 504.4.s.f 4
28.f even 6 1 1176.4.a.u 2
28.g odd 6 1 168.4.q.d ✓ 4
28.g odd 6 1 1176.4.a.r 2
84.n even 6 1 504.4.s.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.q.d ✓ 4 4.b odd 2 1
168.4.q.d ✓ 4 28.g odd 6 1
336.4.q.g 4 1.a even 1 1 trivial
336.4.q.g 4 7.c even 3 1 inner
504.4.s.f 4 12.b even 2 1
504.4.s.f 4 84.n even 6 1
1176.4.a.r 2 28.g odd 6 1
1176.4.a.u 2 28.f even 6 1
2352.4.a.bo 2 7.d odd 6 1
2352.4.a.cc 2 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 9T_{5}^{3} + 187T_{5}^{2} - 954T_{5} + 11236 \) acting on \(S_{4}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 9 T^{3} + \cdots + 11236 \) Copy content Toggle raw display
$7$ \( T^{4} + 181 T^{2} + 117649 \) Copy content Toggle raw display
$11$ \( T^{4} - 5 T^{3} + \cdots + 9922500 \) Copy content Toggle raw display
$13$ \( (T^{2} + 19 T - 36)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 4 T^{3} + \cdots + 4064256 \) Copy content Toggle raw display
$19$ \( T^{4} - 117 T^{3} + \cdots + 7639696 \) Copy content Toggle raw display
$23$ \( T^{4} - 148 T^{3} + \cdots + 11943936 \) Copy content Toggle raw display
$29$ \( (T^{2} + 95 T - 900)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 1017291025 \) Copy content Toggle raw display
$37$ \( T^{4} - 53 T^{3} + \cdots + 331776 \) Copy content Toggle raw display
$41$ \( (T^{2} + 170 T - 33680)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 403 T - 51434)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 16838695696 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 13993943616 \) Copy content Toggle raw display
$59$ \( T^{4} + 585 T^{3} + \cdots + 41990400 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 56368256400 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 434563097796 \) Copy content Toggle raw display
$71$ \( (T^{2} + 616 T - 634356)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 111698997796 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 16920546241 \) Copy content Toggle raw display
$83$ \( (T^{2} - 283 T - 546714)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 643101348096 \) Copy content Toggle raw display
$97$ \( (T^{2} - 1729 T + 365454)^{2} \) Copy content Toggle raw display
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