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Properties

Label 399.2.w.c
Level $399$
Weight $2$
Character orbit 399.w
Analytic conductor $3.186$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [399,2,Mod(170,399)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(399, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("399.170");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 399 = 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 399.w (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: \(3.18603104065\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.3317760000.8
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + 2 \beta_{4} q^{4} + \beta_{6} q^{5} + ( - 2 \beta_{4} - 1) q^{7} + (\beta_{6} + 2 \beta_{4} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + 2 \beta_{4} q^{4} + \beta_{6} q^{5} + ( - 2 \beta_{4} - 1) q^{7} + (\beta_{6} + 2 \beta_{4} - 2) q^{9} - 2 \beta_{2} q^{11} + 2 \beta_{5} q^{12} + ( - \beta_{7} - \beta_{5} + \beta_1) q^{13} + \beta_{7} q^{15} + (4 \beta_{4} - 4) q^{16} + \beta_{2} q^{17} + ( - \beta_{5} + \beta_{4} + \beta_{3} - 1) q^{19} + (2 \beta_{6} + 2 \beta_{2}) q^{20} + ( - 2 \beta_{5} - \beta_1) q^{21} + \beta_{6} q^{23} + (\beta_{7} + 2 \beta_{5} - 2 \beta_1) q^{27} + ( - 6 \beta_{4} + 4) q^{28} + ( - \beta_{7} + 5 \beta_{5} - 5 \beta_1) q^{29} - 2 \beta_{3} q^{33} + ( - 3 \beta_{6} - 2 \beta_{2}) q^{35} + (2 \beta_{6} + 2 \beta_{2} - 4) q^{36} + ( - 2 \beta_{5} + 2 \beta_{3}) q^{37} + ( - 3 \beta_{4} - 3 \beta_{2}) q^{39} + (\beta_{7} - 5 \beta_{5} + 5 \beta_1) q^{41} + 5 q^{43} + 4 \beta_{6} q^{44} + (5 \beta_{4} + 2 \beta_{2}) q^{45} - 2 \beta_{6} q^{47} + (4 \beta_{5} - 4 \beta_1) q^{48} + (8 \beta_{4} - 3) q^{49} + \beta_{3} q^{51} + ( - 2 \beta_{7} - 2 \beta_{3} + 2 \beta_1) q^{52} + (\beta_{7} + \beta_{3} + 5 \beta_1) q^{53} + 10 q^{55} + ( - 3 \beta_{6} + \beta_{5} - 3 \beta_{2} + \cdots - 3) q^{57}+ \cdots + (4 \beta_{6} + 4 \beta_{2} + 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} - 16 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{4} - 16 q^{7} - 8 q^{9} - 16 q^{16} - 4 q^{19} + 8 q^{28} - 32 q^{36} - 12 q^{39} + 40 q^{43} + 20 q^{45} + 8 q^{49} + 80 q^{55} - 24 q^{57} + 16 q^{61} + 40 q^{63} - 64 q^{64} - 32 q^{73} - 16 q^{76} + 4 q^{81} - 40 q^{85} - 60 q^{87} + 80 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} + 14\nu^{4} - 7\nu^{2} - 36 ) / 63 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 14\nu^{5} - 7\nu^{3} - 36\nu ) / 63 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4\nu^{6} + 7\nu^{4} + 28\nu^{2} + 144 ) / 63 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{7} + 7\nu^{5} + 28\nu^{3} + 144\nu ) / 63 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -8\nu^{6} - 14\nu^{4} + 7\nu^{2} - 162 ) / 63 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -8\nu^{7} - 14\nu^{5} + 7\nu^{3} - 162\nu ) / 63 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + 2\beta_{4} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 2\beta_{5} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 4\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 4\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{6} - 7\beta_{2} - 22 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -7\beta_{7} - 7\beta_{3} - 22\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/399\mathbb{Z}\right)^\times\).

\(n\) \(115\) \(134\) \(211\)
\(\chi(n)\) \(-1 + \beta_{4}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
170.1
−1.40294 − 1.01575i
−0.178197 − 1.72286i
0.178197 + 1.72286i
1.40294 + 1.01575i
−1.40294 + 1.01575i
−0.178197 + 1.72286i
0.178197 − 1.72286i
1.40294 − 1.01575i
0 −1.40294 − 1.01575i 1.00000 + 1.73205i 1.93649 + 1.11803i 0 −2.00000 − 1.73205i 0 0.936492 + 2.85008i 0
170.2 0 −0.178197 − 1.72286i 1.00000 + 1.73205i −1.93649 − 1.11803i 0 −2.00000 − 1.73205i 0 −2.93649 + 0.614017i 0
170.3 0 0.178197 + 1.72286i 1.00000 + 1.73205i −1.93649 − 1.11803i 0 −2.00000 − 1.73205i 0 −2.93649 + 0.614017i 0
170.4 0 1.40294 + 1.01575i 1.00000 + 1.73205i 1.93649 + 1.11803i 0 −2.00000 − 1.73205i 0 0.936492 + 2.85008i 0
284.1 0 −1.40294 + 1.01575i 1.00000 − 1.73205i 1.93649 − 1.11803i 0 −2.00000 + 1.73205i 0 0.936492 − 2.85008i 0
284.2 0 −0.178197 + 1.72286i 1.00000 − 1.73205i −1.93649 + 1.11803i 0 −2.00000 + 1.73205i 0 −2.93649 − 0.614017i 0
284.3 0 0.178197 − 1.72286i 1.00000 − 1.73205i −1.93649 + 1.11803i 0 −2.00000 + 1.73205i 0 −2.93649 − 0.614017i 0
284.4 0 1.40294 − 1.01575i 1.00000 − 1.73205i 1.93649 − 1.11803i 0 −2.00000 + 1.73205i 0 0.936492 − 2.85008i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 170.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
19.b odd 2 1 inner
21.h odd 6 1 inner
57.d even 2 1 inner
133.r odd 6 1 inner
399.w even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 399.2.w.c ✓ 8
3.b odd 2 1 inner 399.2.w.c ✓ 8
7.c even 3 1 inner 399.2.w.c ✓ 8
19.b odd 2 1 inner 399.2.w.c ✓ 8
21.h odd 6 1 inner 399.2.w.c ✓ 8
57.d even 2 1 inner 399.2.w.c ✓ 8
133.r odd 6 1 inner 399.2.w.c ✓ 8
399.w even 6 1 inner 399.2.w.c ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
399.2.w.c ✓ 8 1.a even 1 1 trivial
399.2.w.c ✓ 8 3.b odd 2 1 inner
399.2.w.c ✓ 8 7.c even 3 1 inner
399.2.w.c ✓ 8 19.b odd 2 1 inner
399.2.w.c ✓ 8 21.h odd 6 1 inner
399.2.w.c ✓ 8 57.d even 2 1 inner
399.2.w.c ✓ 8 133.r odd 6 1 inner
399.2.w.c ✓ 8 399.w even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(399, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{5}^{4} - 5T_{5}^{2} + 25 \) Copy content Toggle raw display
\( T_{31} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 4 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{4} - 5 T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4 T + 7)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 20 T^{2} + 400)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 18)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 5 T^{2} + 25)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 2 T^{3} + \cdots + 361)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 5 T^{2} + 25)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 90)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} - 72 T^{2} + 5184)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 90)^{4} \) Copy content Toggle raw display
$43$ \( (T - 5)^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} - 20 T^{2} + 400)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 90 T^{2} + 8100)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T + 16)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 162 T^{2} + 26244)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{2} + 8 T + 64)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 162 T^{2} + 26244)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 245)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 90 T^{2} + 8100)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 72)^{4} \) Copy content Toggle raw display
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