Properties

Label 336.4.bl.f
Level $336$
Weight $4$
Character orbit 336.bl
Analytic conductor $19.825$
Analytic rank $0$
Dimension $6$
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(31,336)]
 
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([3, 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("336.31");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 336.bl (of order \(6\), degree \(2\), 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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.633537072.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 13x^{4} - 16x^{3} + 158x^{2} - 168x + 196 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \beta_{2} q^{3} + (\beta_{4} + \beta_{3} - \beta_{2} + 1) q^{5} + ( - 2 \beta_{5} + \beta_{3} - 2 \beta_{2} - 3) q^{7} + ( - 9 \beta_{2} - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 \beta_{2} q^{3} + (\beta_{4} + \beta_{3} - \beta_{2} + 1) q^{5} + ( - 2 \beta_{5} + \beta_{3} - 2 \beta_{2} - 3) q^{7} + ( - 9 \beta_{2} - 9) q^{9} + (4 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - 4 \beta_{2} + \beta_1 - 8) q^{11} + (\beta_{5} + \beta_{4} + 10 \beta_{2} + 5) q^{13} + ( - 3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + 6 \beta_{2} - 3 \beta_1 + 3) q^{15} + ( - 2 \beta_{5} - 10 \beta_{2} - 2 \beta_1 - 20) q^{17} + (\beta_{5} - 6 \beta_{4} + 7 \beta_{3} - 23 \beta_{2} + \beta_1 - 23) q^{19} + ( - 3 \beta_{5} - 9 \beta_{3} - 3 \beta_{2} + 6) q^{21} + (6 \beta_{5} - 18 \beta_{4} - 12 \beta_{3} - 18 \beta_{2} - 6 \beta_1 + 18) q^{23} + ( - 2 \beta_{5} - \beta_{4} - \beta_{3} + 47 \beta_{2} + \beta_1) q^{25} + 27 q^{27} + ( - \beta_{5} + \beta_{4} - 5 \beta_{3} + 5 \beta_1 + 63) q^{29} + ( - 11 \beta_{5} - 21 \beta_{4} - 21 \beta_{3} - 14 \beta_{2} - 10 \beta_1) q^{31} + ( - 9 \beta_{5} + 12 \beta_{4} + 3 \beta_{3} - 12 \beta_{2} + 9 \beta_1 + 12) q^{33} + (5 \beta_{5} - 7 \beta_{4} + 8 \beta_{3} - 16 \beta_{2} + 14 \beta_1 + 88) q^{35} + (31 \beta_{5} + 8 \beta_{4} + 23 \beta_{3} + 71 \beta_{2} + 31 \beta_1 + 71) q^{37} + ( - 3 \beta_{4} + 3 \beta_{3} - 15 \beta_{2} - 3 \beta_1 - 30) q^{39} + (12 \beta_{5} + 12 \beta_{4} + 24 \beta_{3} - 36 \beta_{2} + 24 \beta_1 - 18) q^{41} + ( - 13 \beta_{5} - 13 \beta_{4} + 6 \beta_{3} + 54 \beta_{2} + 6 \beta_1 + 27) q^{43} + (9 \beta_{5} - 9 \beta_{2} + 9 \beta_1 - 18) q^{45} + (2 \beta_{5} - 32 \beta_{4} + 34 \beta_{3} - 114 \beta_{2} + 2 \beta_1 - 114) q^{47} + (21 \beta_{5} + 21 \beta_{4} + 14 \beta_{3} + 35 \beta_{2} + 56 \beta_1 - 70) q^{49} + ( - 6 \beta_{4} - 6 \beta_{3} - 30 \beta_{2} + 30) q^{51} + ( - 35 \beta_{5} - 46 \beta_{4} - 46 \beta_{3} - 45 \beta_{2} - 11 \beta_1) q^{53} + (31 \beta_{5} - 31 \beta_{4} + 17 \beta_{3} - 17 \beta_1 - 159) q^{55} + ( - 21 \beta_{5} + 21 \beta_{4} - 18 \beta_{3} + 18 \beta_1 + 69) q^{57} + ( - 52 \beta_{5} - 47 \beta_{4} - 47 \beta_{3} + 36 \beta_{2} + 5 \beta_1) q^{59} + ( - 52 \beta_{5} + 32 \beta_{4} - 20 \beta_{3} - 168 \beta_{2} + \cdots + 168) q^{61}+ \cdots + ( - 9 \beta_{5} - 9 \beta_{4} - 36 \beta_{3} + 72 \beta_{2} - 36 \beta_1 + 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 9 q^{3} + 12 q^{5} - 7 q^{7} - 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 9 q^{3} + 12 q^{5} - 7 q^{7} - 27 q^{9} - 48 q^{11} - 84 q^{17} - 77 q^{19} + 42 q^{21} + 108 q^{23} - 141 q^{25} + 162 q^{27} + 372 q^{29} + 11 q^{31} + 144 q^{33} + 546 q^{35} + 159 q^{37} - 135 q^{39} - 108 q^{45} - 378 q^{47} - 567 q^{49} + 252 q^{51} + 78 q^{53} - 1044 q^{55} + 462 q^{57} - 150 q^{59} + 1608 q^{61} - 63 q^{63} - 78 q^{65} - 339 q^{67} - 273 q^{73} - 423 q^{75} - 2394 q^{77} + 3429 q^{79} - 243 q^{81} - 1464 q^{83} + 648 q^{85} - 558 q^{87} + 768 q^{89} + 1029 q^{91} + 33 q^{93} - 258 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 13x^{4} - 16x^{3} + 158x^{2} - 168x + 196 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -6\nu^{5} + 1021\nu^{4} - 71\nu^{3} + 13207\nu^{2} - 29298\nu + 105518 ) / 13202 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -78\nu^{5} + 71\nu^{4} - 923\nu^{3} + 65\nu^{2} - 11218\nu - 1274 ) / 13202 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 148\nu^{5} - 981\nu^{4} - 449\nu^{3} - 11125\nu^{2} - 3426\nu - 85596 ) / 13202 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -604\nu^{5} - 635\nu^{4} - 4947\nu^{3} - 1697\nu^{2} - 58094\nu - 27468 ) / 13202 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -702\nu^{5} + 639\nu^{4} - 8307\nu^{3} + 13787\nu^{2} - 74558\nu + 94150 ) / 13202 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} - \beta_{3} - 3\beta_{2} - 2\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} + \beta_{4} + \beta_{3} - 24\beta_{2} + 2\beta _1 - 24 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -11\beta_{5} + 11\beta_{4} - 7\beta_{3} + 7\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -4\beta_{5} - 9\beta_{4} - 9\beta_{3} + 89\beta_{2} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 49\beta_{5} - 82\beta_{4} + 131\beta_{3} - 69\beta_{2} + 49\beta _1 - 69 ) / 3 \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.594460 1.02964i
1.66918 2.89111i
−1.76364 + 3.05472i
0.594460 + 1.02964i
1.66918 + 2.89111i
−1.76364 3.05472i
0 −1.50000 + 2.59808i 0 −8.37970 + 4.83802i 0 −15.8619 9.56041i 0 −4.50000 7.79423i 0
31.2 0 −1.50000 + 2.59808i 0 6.21705 3.58942i 0 1.45104 + 18.4633i 0 −4.50000 7.79423i 0
31.3 0 −1.50000 + 2.59808i 0 8.16265 4.71271i 0 10.9108 14.9651i 0 −4.50000 7.79423i 0
271.1 0 −1.50000 2.59808i 0 −8.37970 4.83802i 0 −15.8619 + 9.56041i 0 −4.50000 + 7.79423i 0
271.2 0 −1.50000 2.59808i 0 6.21705 + 3.58942i 0 1.45104 18.4633i 0 −4.50000 + 7.79423i 0
271.3 0 −1.50000 2.59808i 0 8.16265 + 4.71271i 0 10.9108 + 14.9651i 0 −4.50000 + 7.79423i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.4.bl.f 6
4.b odd 2 1 336.4.bl.h yes 6
7.d odd 6 1 336.4.bl.h yes 6
28.f even 6 1 inner 336.4.bl.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.4.bl.f 6 1.a even 1 1 trivial
336.4.bl.f 6 28.f even 6 1 inner
336.4.bl.h yes 6 4.b odd 2 1
336.4.bl.h yes 6 7.d odd 6 1

Hecke kernels

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

\( T_{5}^{6} - 12T_{5}^{5} - 45T_{5}^{4} + 1116T_{5}^{3} + 4113T_{5}^{2} - 105462T_{5} + 428652 \) Copy content Toggle raw display
\( T_{11}^{6} + 48T_{11}^{5} - 657T_{11}^{4} - 68400T_{11}^{3} + 2342529T_{11}^{2} - 27778950T_{11} + 126672012 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 9)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} - 12 T^{5} - 45 T^{4} + \cdots + 428652 \) Copy content Toggle raw display
$7$ \( T^{6} + 7 T^{5} + 308 T^{4} + \cdots + 40353607 \) Copy content Toggle raw display
$11$ \( T^{6} + 48 T^{5} + \cdots + 126672012 \) Copy content Toggle raw display
$13$ \( T^{6} + 579 T^{4} + 93024 T^{2} + \cdots + 3048192 \) Copy content Toggle raw display
$17$ \( T^{6} + 84 T^{5} + 2700 T^{4} + \cdots + 248832 \) Copy content Toggle raw display
$19$ \( T^{6} + 77 T^{5} + \cdots + 71197181584 \) Copy content Toggle raw display
$23$ \( T^{6} - 108 T^{5} + \cdots + 429632335872 \) Copy content Toggle raw display
$29$ \( (T^{3} - 186 T^{2} + 8253 T - 5184)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} - 11 T^{5} + \cdots + 6131274251881 \) Copy content Toggle raw display
$37$ \( T^{6} - 159 T^{5} + \cdots + 6785587367056 \) Copy content Toggle raw display
$41$ \( T^{6} + 103140 T^{4} + \cdots + 702930633408 \) Copy content Toggle raw display
$43$ \( T^{6} + 109803 T^{4} + \cdots + 4094157830832 \) Copy content Toggle raw display
$47$ \( T^{6} + 378 T^{5} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 525419367632784 \) Copy content Toggle raw display
$59$ \( T^{6} + 150 T^{5} + \cdots + 35939689280784 \) Copy content Toggle raw display
$61$ \( T^{6} - 1608 T^{5} + \cdots + 12\!\cdots\!12 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 145376921152512 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 831854523313152 \) Copy content Toggle raw display
$73$ \( T^{6} + 273 T^{5} + \cdots + 29\!\cdots\!48 \) Copy content Toggle raw display
$79$ \( T^{6} - 3429 T^{5} + \cdots + 64\!\cdots\!43 \) Copy content Toggle raw display
$83$ \( (T^{3} + 732 T^{2} + 109989 T - 5856786)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 768 T^{5} + \cdots + 14\!\cdots\!72 \) Copy content Toggle raw display
$97$ \( T^{6} + 2015814 T^{4} + \cdots + 19\!\cdots\!32 \) Copy content Toggle raw display
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