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Properties

Label 336.2.bj.e
Level $336$
Weight $2$
Character orbit 336.bj
Analytic conductor $2.683$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [336,2,Mod(95,336)]
 
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, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.95");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 336.bj (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: \(2.68297350792\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.8275904784.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 4x^{4} - 18x^{3} + 45x^{2} - 81x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} - \beta_{5} - \beta_1 + 1) q^{3} + (\beta_{6} - \beta_{5} - \beta_{3} + \cdots + 1) q^{5}+ \cdots + (\beta_{7} + \beta_{5} - \beta_{3} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} - \beta_{5} - \beta_1 + 1) q^{3} + (\beta_{6} - \beta_{5} - \beta_{3} + \cdots + 1) q^{5}+ \cdots + ( - 3 \beta_{7} + 8 \beta_{6} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 3 q^{3} - 6 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 3 q^{3} - 6 q^{7} - q^{9} + 16 q^{13} + 6 q^{19} - 19 q^{21} + 12 q^{25} - 12 q^{31} - 11 q^{33} + 10 q^{37} - 6 q^{39} - 17 q^{45} + 10 q^{49} + 9 q^{51} - 22 q^{57} + 30 q^{61} + 27 q^{63} + 66 q^{67} - 26 q^{69} + 14 q^{73} - 66 q^{75} - 36 q^{79} + 7 q^{81} - 68 q^{85} + 54 q^{87} - 12 q^{91} + 3 q^{93} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 4x^{4} - 18x^{3} + 45x^{2} - 81x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} - 14\nu^{5} + 6\nu^{4} - 22\nu^{3} + 18\nu^{2} - 27\nu + 189 ) / 54 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 3\nu^{6} + 5\nu^{5} - 6\nu^{4} + 4\nu^{3} - 18\nu^{2} + 45\nu - 81 ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{7} + 6\nu^{6} - 16\nu^{5} + 12\nu^{4} - 2\nu^{3} + 108\nu^{2} - 135\nu + 270 ) / 54 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{7} + 9\nu^{6} - 16\nu^{5} - 2\nu^{3} + 66\nu^{2} - 153\nu + 297 ) / 54 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{7} - 9\nu^{6} + 17\nu^{5} - 9\nu^{4} + 19\nu^{3} - 105\nu^{2} + 144\nu - 324 ) / 54 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -10\nu^{7} + 9\nu^{6} - 23\nu^{5} + 9\nu^{4} - 13\nu^{3} + 123\nu^{2} - 135\nu + 378 ) / 54 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 3\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{7} + 2\beta_{6} - 3\beta_{5} - 5\beta_{3} - \beta_{2} - \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{7} - 6\beta_{6} - 2\beta_{5} - 2\beta_{4} - \beta_{3} - 4\beta_{2} + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -6\beta_{7} - 6\beta_{6} + 6\beta_{5} - 4\beta_{4} - 6\beta_{3} - 4\beta_{2} + 16\beta _1 - 11 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -18\beta_{7} - 6\beta_{6} + 6\beta_{5} + 12\beta_{4} + 6\beta_{3} + 6\beta_{2} + 11\beta _1 - 6 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
95.1
0.906034 − 1.47618i
1.73142 − 0.0465589i
−1.37009 − 1.05965i
0.232633 + 1.71636i
0.906034 + 1.47618i
1.73142 + 0.0465589i
−1.37009 + 1.05965i
0.232633 − 1.71636i
0 −1.73142 − 0.0465589i 0 −3.41502 + 1.97166i 0 1.13746 − 2.38876i 0 2.99566 + 0.161227i 0
95.2 0 −0.906034 − 1.47618i 0 3.41502 − 1.97166i 0 1.13746 − 2.38876i 0 −1.35821 + 2.67493i 0
95.3 0 −0.232633 + 1.71636i 0 −0.581054 + 0.335472i 0 −2.63746 − 0.209313i 0 −2.89176 − 0.798564i 0
95.4 0 1.37009 − 1.05965i 0 0.581054 − 0.335472i 0 −2.63746 − 0.209313i 0 0.754305 − 2.90362i 0
191.1 0 −1.73142 + 0.0465589i 0 −3.41502 − 1.97166i 0 1.13746 + 2.38876i 0 2.99566 − 0.161227i 0
191.2 0 −0.906034 + 1.47618i 0 3.41502 + 1.97166i 0 1.13746 + 2.38876i 0 −1.35821 − 2.67493i 0
191.3 0 −0.232633 − 1.71636i 0 −0.581054 − 0.335472i 0 −2.63746 + 0.209313i 0 −2.89176 + 0.798564i 0
191.4 0 1.37009 + 1.05965i 0 0.581054 + 0.335472i 0 −2.63746 + 0.209313i 0 0.754305 + 2.90362i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 95.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
28.g odd 6 1 inner
84.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.2.bj.e ✓ 8
3.b odd 2 1 inner 336.2.bj.e ✓ 8
4.b odd 2 1 336.2.bj.g yes 8
7.c even 3 1 336.2.bj.g yes 8
7.c even 3 1 2352.2.h.n 8
7.d odd 6 1 2352.2.h.m 8
12.b even 2 1 336.2.bj.g yes 8
21.g even 6 1 2352.2.h.m 8
21.h odd 6 1 336.2.bj.g yes 8
21.h odd 6 1 2352.2.h.n 8
28.f even 6 1 2352.2.h.m 8
28.g odd 6 1 inner 336.2.bj.e ✓ 8
28.g odd 6 1 2352.2.h.n 8
84.j odd 6 1 2352.2.h.m 8
84.n even 6 1 inner 336.2.bj.e ✓ 8
84.n even 6 1 2352.2.h.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.bj.e ✓ 8 1.a even 1 1 trivial
336.2.bj.e ✓ 8 3.b odd 2 1 inner
336.2.bj.e ✓ 8 28.g odd 6 1 inner
336.2.bj.e ✓ 8 84.n even 6 1 inner
336.2.bj.g yes 8 4.b odd 2 1
336.2.bj.g yes 8 7.c even 3 1
336.2.bj.g yes 8 12.b even 2 1
336.2.bj.g yes 8 21.h odd 6 1
2352.2.h.m 8 7.d odd 6 1
2352.2.h.m 8 21.g even 6 1
2352.2.h.m 8 28.f even 6 1
2352.2.h.m 8 84.j odd 6 1
2352.2.h.n 8 7.c even 3 1
2352.2.h.n 8 21.h odd 6 1
2352.2.h.n 8 28.g odd 6 1
2352.2.h.n 8 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(336, [\chi])\):

\( T_{5}^{8} - 16T_{5}^{6} + 249T_{5}^{4} - 112T_{5}^{2} + 49 \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display
\( T_{19}^{4} - 3T_{19}^{3} - T_{19}^{2} + 12T_{19} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 3 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} - 16 T^{6} + \cdots + 49 \) Copy content Toggle raw display
$7$ \( (T^{4} + 3 T^{3} + 2 T^{2} + \cdots + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 40 T^{6} + \cdots + 117649 \) Copy content Toggle raw display
$13$ \( (T - 2)^{8} \) Copy content Toggle raw display
$17$ \( T^{8} - 43 T^{6} + \cdots + 200704 \) Copy content Toggle raw display
$19$ \( (T^{4} - 3 T^{3} - T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 13 T^{6} + \cdots + 784 \) Copy content Toggle raw display
$29$ \( (T^{4} + 85 T^{2} + 1792)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 3 T + 3)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 5 T^{3} + 33 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 100 T^{2} + 448)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 92 T^{2} + 64)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 117 T^{6} + \cdots + 5143824 \) Copy content Toggle raw display
$53$ \( T^{8} - 232 T^{6} + \cdots + 167521249 \) Copy content Toggle raw display
$59$ \( T^{8} + 160 T^{6} + \cdots + 34656769 \) Copy content Toggle raw display
$61$ \( (T^{4} - 15 T^{3} + \cdots + 1764)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 33 T^{3} + \cdots + 2304)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 124 T^{2} + 1792)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 7 T^{3} + 51 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 9 T + 27)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 31 T^{2} + 112)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 247 T^{6} + \cdots + 102171664 \) Copy content Toggle raw display
$97$ \( (T^{2} - T - 14)^{4} \) Copy content Toggle raw display
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