↑

Properties

Label 6336.2.f.n
Level $6336$
Weight $2$
Character orbit 6336.f
Analytic conductor $50.593$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6336,2,Mod(3169,6336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6336, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6336.3169");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6336 = 2^{6} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6336.f (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: \(50.5932147207\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 15x^{10} + 84x^{8} - 187x^{6} + 141x^{4} + 108x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 704)
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 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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{5} - \beta_{9} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{5} - \beta_{9} q^{7} - \beta_{3} q^{11} + (\beta_{8} + \beta_{6}) q^{13} - \beta_{2} q^{17} + (\beta_{5} + 2 \beta_{3}) q^{19} + (\beta_{11} + \beta_{10} + \beta_{9}) q^{23} + (\beta_{2} - \beta_1 - 5) q^{25} + ( - \beta_{8} - 2 \beta_{7}) q^{29} + (\beta_{11} + \beta_{10} - \beta_{9}) q^{31} + (3 \beta_{5} - 2 \beta_{4} - 2 \beta_{3}) q^{35} + ( - 2 \beta_{8} - \beta_{7} - 2 \beta_{6}) q^{37} + ( - \beta_{2} - 2 \beta_1 + 4) q^{41} + ( - 2 \beta_{4} - 4 \beta_{3}) q^{43} - \beta_{11} q^{47} + ( - 2 \beta_{2} + 5) q^{49} - 2 \beta_{8} q^{53} - \beta_{10} q^{55} + (\beta_{4} - 8 \beta_{3}) q^{59} + ( - \beta_{8} - 2 \beta_{7} + 2 \beta_{6}) q^{61} + (\beta_{2} - 4 \beta_1 + 2) q^{65} + (2 \beta_{5} + \beta_{4} - 4 \beta_{3}) q^{67} + (\beta_{11} + \beta_{10} - \beta_{9}) q^{71} + (\beta_{2} - 2 \beta_1) q^{73} + \beta_{6} q^{77} - 2 \beta_{10} q^{79} + ( - 2 \beta_{5} + 2 \beta_{4} - 4 \beta_{3}) q^{83} + ( - 2 \beta_{8} + 3 \beta_{6}) q^{85} + (\beta_{2} + \beta_1 + 4) q^{89} + (4 \beta_{4} - 8 \beta_{3}) q^{91} + (2 \beta_{11} + 2 \beta_{10} + 3 \beta_{9}) q^{95} + (\beta_{2} + \beta_1 + 8) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 60 q^{25} + 48 q^{41} + 60 q^{49} + 24 q^{65} + 48 q^{89} + 96 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 15x^{10} + 84x^{8} - 187x^{6} + 141x^{4} + 108x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 67\nu^{10} - 870\nu^{8} + 4056\nu^{6} - 5925\nu^{4} - 3698\nu^{2} + 21684 ) / 8084 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 199\nu^{10} - 2946\nu^{8} + 15184\nu^{6} - 23269\nu^{4} - 14362\nu^{2} + 32672 ) / 8084 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 321\nu^{11} - 4681\nu^{9} + 25224\nu^{7} - 51915\nu^{5} + 33411\nu^{3} + 43440\nu ) / 16168 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 829\nu^{11} - 12303\nu^{9} + 67560\nu^{7} - 143895\nu^{5} + 91461\nu^{3} + 119288\nu ) / 16168 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 562\nu^{11} - 8655\nu^{9} + 49828\nu^{7} - 113406\nu^{5} + 71509\nu^{3} + 93204\nu ) / 8084 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 951\nu^{10} - 14038\nu^{8} + 77600\nu^{6} - 172541\nu^{4} + 155402\nu^{2} + 49216 ) / 8084 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -1167\nu^{10} + 18170\nu^{8} - 107568\nu^{6} + 266329\nu^{4} - 249658\nu^{2} - 78220 ) / 8084 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -19\nu^{10} + 286\nu^{8} - 1616\nu^{6} + 3737\nu^{4} - 3410\nu^{2} - 1076 ) / 94 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -1688\nu^{11} + 25629\nu^{9} - 146468\nu^{7} + 342808\nu^{5} - 306719\nu^{3} - 93724\nu ) / 8084 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -4907\nu^{11} + 74969\nu^{9} - 433640\nu^{7} + 1043377\nu^{5} - 986979\nu^{3} - 297928\nu ) / 16168 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -65\nu^{11} + 991\nu^{9} - 5696\nu^{7} + 13483\nu^{5} - 12341\nu^{3} - 3752\nu ) / 172 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} + 2\beta_{9} + 2\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} + 2\beta_{6} - 4\beta _1 + 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{11} + \beta_{10} + 4\beta_{9} - 3\beta_{4} + 8\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 13\beta_{8} - 4\beta_{7} + 18\beta_{6} + 4\beta_{2} - 20\beta _1 + 38 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -25\beta_{11} + 12\beta_{10} + 28\beta_{9} + 10\beta_{5} - 60\beta_{4} + 122\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 54\beta_{8} - 23\beta_{7} + 65\beta_{6} + 9\beta_{2} - 33\beta _1 + 52 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -47\beta_{11} + 28\beta_{10} + 44\beta_{9} + 98\beta_{5} - 448\beta_{4} + 810\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 723\beta_{8} - 336\beta_{7} + 826\beta_{6} + 4\beta_{2} + 44\beta _1 - 138 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 219\beta_{11} - 93\beta_{10} - 264\beta_{9} + 324\beta_{5} - 1377\beta_{4} + 2392\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 4055\beta_{8} - 1932\beta_{7} + 4558\beta_{6} - 684\beta_{2} + 3060\beta _1 - 5470 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 6797\beta_{11} - 3180\beta_{10} - 7732\beta_{9} + 3366\beta_{5} - 14036\beta_{4} + 24110\beta_{3} ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6336\mathbb{Z}\right)^\times\).

\(n\) \(1729\) \(3521\) \(4159\) \(4357\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3169.1
1.51496 − 0.500000i
−1.51496 + 0.500000i
0.0585812 + 0.500000i
−0.0585812 − 0.500000i
2.43956 + 0.500000i
−2.43956 − 0.500000i
2.43956 − 0.500000i
−2.43956 + 0.500000i
0.0585812 − 0.500000i
−0.0585812 + 0.500000i
1.51496 + 0.500000i
−1.51496 − 0.500000i
0 0 0 − 3.97405i 0 −4.76197 0 0 0
3169.2 0 0 0 − 3.97405i 0 4.76197 0 0 0
3169.3 0 0 0 − 2.90798i 0 −1.84921 0 0 0
3169.4 0 0 0 − 2.90798i 0 1.84921 0 0 0
3169.5 0 0 0 − 2.39804i 0 −3.14708 0 0 0
3169.6 0 0 0 − 2.39804i 0 3.14708 0 0 0
3169.7 0 0 0 2.39804i 0 −3.14708 0 0 0
3169.8 0 0 0 2.39804i 0 3.14708 0 0 0
3169.9 0 0 0 2.90798i 0 −1.84921 0 0 0
3169.10 0 0 0 2.90798i 0 1.84921 0 0 0
3169.11 0 0 0 3.97405i 0 −4.76197 0 0 0
3169.12 0 0 0 3.97405i 0 4.76197 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3169.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6336.2.f.n 12
3.b odd 2 1 704.2.c.c ✓ 12
4.b odd 2 1 inner 6336.2.f.n 12
8.b even 2 1 inner 6336.2.f.n 12
8.d odd 2 1 inner 6336.2.f.n 12
12.b even 2 1 704.2.c.c ✓ 12
24.f even 2 1 704.2.c.c ✓ 12
24.h odd 2 1 704.2.c.c ✓ 12
48.i odd 4 1 2816.2.a.t 6
48.i odd 4 1 2816.2.a.u 6
48.k even 4 1 2816.2.a.t 6
48.k even 4 1 2816.2.a.u 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
704.2.c.c ✓ 12 3.b odd 2 1
704.2.c.c ✓ 12 12.b even 2 1
704.2.c.c ✓ 12 24.f even 2 1
704.2.c.c ✓ 12 24.h odd 2 1
2816.2.a.t 6 48.i odd 4 1
2816.2.a.t 6 48.k even 4 1
2816.2.a.u 6 48.i odd 4 1
2816.2.a.u 6 48.k even 4 1
6336.2.f.n 12 1.a even 1 1 trivial
6336.2.f.n 12 4.b odd 2 1 inner
6336.2.f.n 12 8.b even 2 1 inner
6336.2.f.n 12 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(6336, [\chi])\):

\( T_{5}^{6} + 30T_{5}^{4} + 273T_{5}^{2} + 768 \) Copy content Toggle raw display
\( T_{7}^{6} - 36T_{7}^{4} + 336T_{7}^{2} - 768 \) Copy content Toggle raw display
\( T_{17}^{3} - 24T_{17} - 24 \) Copy content Toggle raw display
\( T_{19}^{6} + 60T_{19}^{4} + 912T_{19}^{2} + 4096 \) Copy content Toggle raw display
\( T_{41}^{3} - 12T_{41}^{2} - 24T_{41} + 456 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + 30 T^{4} + \cdots + 768)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} - 36 T^{4} + \cdots - 768)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$13$ \( (T^{6} + 48 T^{4} + \cdots + 192)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} - 24 T - 24)^{4} \) Copy content Toggle raw display
$19$ \( (T^{6} + 60 T^{4} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 90 T^{4} + \cdots - 5292)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 108 T^{4} + \cdots + 3072)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 66 T^{4} + \cdots - 10092)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 198 T^{4} + \cdots + 190512)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} - 12 T^{2} + \cdots + 456)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + 120 T^{4} + \cdots + 12544)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 12)^{6} \) Copy content Toggle raw display
$53$ \( (T^{2} + 48)^{6} \) Copy content Toggle raw display
$59$ \( (T^{6} + 210 T^{4} + \cdots + 197136)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 252 T^{4} + \cdots + 248832)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 282 T^{4} + \cdots + 283024)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 66 T^{4} + \cdots - 10092)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 48 T - 56)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} - 120 T^{4} + \cdots - 49152)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 264 T^{4} + \cdots + 451584)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} - 12 T^{2} + \cdots + 18)^{4} \) Copy content Toggle raw display
$97$ \( (T^{3} - 24 T^{2} + \cdots - 274)^{4} \) Copy content Toggle raw display
show more
show less