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Properties

Label 336.6.q.n
Level $336$
Weight $6$
Character orbit 336.q
Analytic conductor $53.889$
Analytic rank $0$
Dimension $12$
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] = [336,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.193");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(53.8889634572\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 2041 x^{10} - 63452 x^{9} + 3932036 x^{8} - 70117724 x^{7} + 1560078988 x^{6} + \cdots + 472919482810944 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{18}\cdot 3\cdot 7^{4} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (9 \beta_1 - 9) q^{3} + ( - \beta_{4} + \beta_{2} + 3 \beta_1) q^{5} + (\beta_{6} - 9 \beta_1 - 8) q^{7} - 81 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (9 \beta_1 - 9) q^{3} + ( - \beta_{4} + \beta_{2} + 3 \beta_1) q^{5} + (\beta_{6} - 9 \beta_1 - 8) q^{7} - 81 \beta_1 q^{9} + ( - \beta_{11} + \beta_{6} - \beta_{5} + \cdots + 94) q^{11}+ \cdots + (81 \beta_{10} + 162 \beta_{5} + \cdots - 7776) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 54 q^{3} + 17 q^{5} - 144 q^{7} - 486 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 54 q^{3} + 17 q^{5} - 144 q^{7} - 486 q^{9} + 565 q^{11} - 842 q^{13} - 306 q^{15} + 52 q^{17} - 107 q^{19} + 1377 q^{21} - 700 q^{23} - 4881 q^{25} + 8748 q^{27} + 12958 q^{29} + 3552 q^{31} + 5085 q^{33} + 938 q^{35} - 3453 q^{37} + 3789 q^{39} - 15148 q^{41} - 38278 q^{43} + 1377 q^{45} - 12136 q^{47} - 39926 q^{49} + 468 q^{51} - 37591 q^{53} + 68670 q^{55} + 1926 q^{57} - 3905 q^{59} + 13296 q^{61} - 729 q^{63} + 45638 q^{65} + 2889 q^{67} + 12600 q^{69} - 132992 q^{71} + 127931 q^{73} - 43929 q^{75} - 306817 q^{77} - 100226 q^{79} - 39366 q^{81} + 146246 q^{83} - 181328 q^{85} - 58311 q^{87} - 3418 q^{89} + 275865 q^{91} + 31968 q^{93} - 108810 q^{95} + 28934 q^{97} - 91530 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 2041 x^{10} - 63452 x^{9} + 3932036 x^{8} - 70117724 x^{7} + 1560078988 x^{6} + \cdots + 472919482810944 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 84\!\cdots\!10 \nu^{11} + \cdots + 67\!\cdots\!64 ) / 88\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 54\!\cdots\!41 \nu^{11} + \cdots + 13\!\cdots\!16 ) / 92\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10\!\cdots\!85 \nu^{11} + \cdots + 27\!\cdots\!88 ) / 13\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 23\!\cdots\!15 \nu^{11} + \cdots + 93\!\cdots\!40 ) / 22\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 21\!\cdots\!01 \nu^{11} + \cdots - 15\!\cdots\!96 ) / 13\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10\!\cdots\!98 \nu^{11} + \cdots - 14\!\cdots\!96 ) / 68\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 86\!\cdots\!95 \nu^{11} + \cdots - 24\!\cdots\!52 ) / 45\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 33\!\cdots\!43 \nu^{11} + \cdots + 58\!\cdots\!68 ) / 13\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 20\!\cdots\!93 \nu^{11} + \cdots - 28\!\cdots\!48 ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 24\!\cdots\!87 \nu^{11} + \cdots - 25\!\cdots\!92 ) / 34\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 15\!\cdots\!31 \nu^{11} + \cdots - 90\!\cdots\!56 ) / 19\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} + \beta_{10} + \beta_{7} - 3\beta_{6} - 3\beta_{5} - 11\beta_{4} + \beta_{3} + 2\beta_{2} - 9\beta _1 + 10 ) / 56 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 12 \beta_{11} - 7 \beta_{10} + 16 \beta_{9} + 27 \beta_{8} - 16 \beta_{7} - 248 \beta_{6} + 107 \beta_{5} + \cdots + 88 ) / 56 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 881 \beta_{11} - 80 \beta_{10} - 1767 \beta_{9} - 2949 \beta_{8} + 11189 \beta_{6} + 447 \beta_{5} + \cdots + 854552 ) / 56 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 28507 \beta_{11} + 13579 \beta_{10} + 68330 \beta_{8} + 60421 \beta_{7} + 200669 \beta_{6} + \cdots - 66605268 ) / 56 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 214072 \beta_{11} - 1639401 \beta_{10} + 3846645 \beta_{9} + 4442791 \beta_{8} - 3846645 \beta_{7} + \cdots + 6463068 ) / 56 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 52596103 \beta_{11} + 24471648 \beta_{10} - 169236913 \beta_{9} - 376278467 \beta_{8} + \cdots + 152076711848 ) / 56 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 3985553385 \beta_{11} + 3109587113 \beta_{10} + 7837190810 \beta_{8} + 9153261831 \beta_{7} + \cdots - 7267209008612 ) / 56 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 52708634544 \beta_{11} - 196514795911 \beta_{10} + 439765022173 \beta_{9} + 543735963837 \beta_{8} + \cdots + 905276874028 ) / 56 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 7515194137457 \beta_{11} + 2420387138752 \beta_{10} - 22572086964135 \beta_{9} - 47227483738197 \beta_{8} + \cdots + 18\!\cdots\!36 ) / 56 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 495573699183511 \beta_{11} + 368700747752023 \beta_{10} + \cdots - 93\!\cdots\!92 ) / 56 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 62\!\cdots\!68 \beta_{11} + \cdots + 11\!\cdots\!44 ) / 56 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
7.97360 − 13.8107i
10.9316 − 18.9342i
−4.17642 + 7.23378i
13.5741 − 23.5110i
−25.0588 + 43.4031i
−2.74409 + 4.75291i
7.97360 + 13.8107i
10.9316 + 18.9342i
−4.17642 − 7.23378i
13.5741 + 23.5110i
−25.0588 − 43.4031i
−2.74409 − 4.75291i
0 −4.50000 + 7.79423i 0 −45.7916 − 79.3134i 0 −58.7448 − 115.568i 0 −40.5000 − 70.1481i 0
193.2 0 −4.50000 + 7.79423i 0 −30.2849 − 52.4550i 0 −76.1691 + 104.906i 0 −40.5000 − 70.1481i 0
193.3 0 −4.50000 + 7.79423i 0 −0.634624 − 1.09920i 0 62.9648 − 113.324i 0 −40.5000 − 70.1481i 0
193.4 0 −4.50000 + 7.79423i 0 16.3419 + 28.3050i 0 −23.3641 + 127.519i 0 −40.5000 − 70.1481i 0
193.5 0 −4.50000 + 7.79423i 0 23.1486 + 40.0946i 0 126.558 + 28.1086i 0 −40.5000 − 70.1481i 0
193.6 0 −4.50000 + 7.79423i 0 45.7207 + 79.1905i 0 −103.245 − 78.4062i 0 −40.5000 − 70.1481i 0
289.1 0 −4.50000 − 7.79423i 0 −45.7916 + 79.3134i 0 −58.7448 + 115.568i 0 −40.5000 + 70.1481i 0
289.2 0 −4.50000 − 7.79423i 0 −30.2849 + 52.4550i 0 −76.1691 − 104.906i 0 −40.5000 + 70.1481i 0
289.3 0 −4.50000 − 7.79423i 0 −0.634624 + 1.09920i 0 62.9648 + 113.324i 0 −40.5000 + 70.1481i 0
289.4 0 −4.50000 − 7.79423i 0 16.3419 − 28.3050i 0 −23.3641 − 127.519i 0 −40.5000 + 70.1481i 0
289.5 0 −4.50000 − 7.79423i 0 23.1486 − 40.0946i 0 126.558 − 28.1086i 0 −40.5000 + 70.1481i 0
289.6 0 −4.50000 − 7.79423i 0 45.7207 − 79.1905i 0 −103.245 + 78.4062i 0 −40.5000 + 70.1481i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.6
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 336.6.q.n 12
4.b odd 2 1 168.6.q.d ✓ 12
7.c even 3 1 inner 336.6.q.n 12
12.b even 2 1 504.6.s.f 12
28.g odd 6 1 168.6.q.d ✓ 12
84.n even 6 1 504.6.s.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.6.q.d ✓ 12 4.b odd 2 1
168.6.q.d ✓ 12 28.g odd 6 1
336.6.q.n 12 1.a even 1 1 trivial
336.6.q.n 12 7.c even 3 1 inner
504.6.s.f 12 12.b even 2 1
504.6.s.f 12 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 17 T_{5}^{11} + 11960 T_{5}^{10} - 262771 T_{5}^{9} + 112417320 T_{5}^{8} + \cdots + 94\!\cdots\!96 \) acting on \(S_{6}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9 T + 81)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 94\!\cdots\!96 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 22\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 43\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots - 59595303023616)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 28\!\cdots\!44)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 47\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 17\!\cdots\!52)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 10\!\cdots\!16)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 39\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 24\!\cdots\!80)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 13\!\cdots\!81 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 70\!\cdots\!36)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 65\!\cdots\!68)^{2} \) Copy content Toggle raw display
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