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Properties

Label 840.2.dc.b
Level $840$
Weight $2$
Character orbit 840.dc
Analytic conductor $6.707$
Analytic rank $0$
Dimension $8$
CM discriminant -24
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [840,2,Mod(53,840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(840, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 6, 6, 9, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("840.53");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.dc (of order \(12\), degree \(4\), 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: \(6.70743376979\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1 - 1) q^{2} + ( - \beta_{6} + \beta_{3}) q^{3} + (2 \beta_{4} - 2 \beta_1) q^{4} + (\beta_{7} + \beta_{4} + \beta_{2} - \beta_1) q^{5} + ( - \beta_{7} + \beta_{5} - \beta_{3}) q^{6} + ( - \beta_{7} - \beta_{6} + \cdots + \beta_{2}) q^{7}+ \cdots - 3 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1 - 1) q^{2} + ( - \beta_{6} + \beta_{3}) q^{3} + (2 \beta_{4} - 2 \beta_1) q^{4} + (\beta_{7} + \beta_{4} + \beta_{2} - \beta_1) q^{5} + ( - \beta_{7} + \beta_{5} - \beta_{3}) q^{6} + ( - \beta_{7} - \beta_{6} + \cdots + \beta_{2}) q^{7}+ \cdots + (3 \beta_{7} - 3 \beta_{6} - 6 \beta_{4} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} + 4 q^{5} + 4 q^{7} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} + 4 q^{5} + 4 q^{7} - 16 q^{8} - 16 q^{10} + 8 q^{11} - 8 q^{14} - 12 q^{15} + 16 q^{16} - 12 q^{18} + 8 q^{20} + 12 q^{21} - 16 q^{22} - 12 q^{30} + 16 q^{32} + 12 q^{33} - 16 q^{35} + 48 q^{36} - 48 q^{44} + 24 q^{45} + 20 q^{49} + 4 q^{50} - 60 q^{53} - 56 q^{55} - 8 q^{56} - 8 q^{58} + 48 q^{59} - 12 q^{66} - 32 q^{70} - 24 q^{72} - 28 q^{73} - 48 q^{75} + 28 q^{77} - 60 q^{79} + 32 q^{80} + 36 q^{81} + 16 q^{83} + 48 q^{84} + 72 q^{87} + 32 q^{88} + 12 q^{93} + 4 q^{97} + 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{5} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{24}^{7} + \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{5} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{7} + 2\zeta_{24}^{3} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{6} + \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{5} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{6} + 2\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{4} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{7} + 2\beta_{5} ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/840\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(281\) \(337\) \(421\) \(631\)
\(\chi(n)\) \(-1 + \beta_{2}\) \(-1\) \(-\beta_{4}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−0.258819 − 0.965926i
0.258819 + 0.965926i
−0.258819 + 0.965926i
0.258819 − 0.965926i
0.965926 − 0.258819i
−0.965926 + 0.258819i
0.965926 + 0.258819i
−0.965926 − 0.258819i
−1.36603 − 0.366025i −1.67303 + 0.448288i 1.73205 + 1.00000i 1.81431 + 1.30701i 2.44949 −1.62132 − 2.09077i −2.00000 − 2.00000i 2.59808 − 1.50000i −2.00000 − 2.44949i
53.2 −1.36603 − 0.366025i 1.67303 − 0.448288i 1.73205 + 1.00000i 0.917738 − 2.03906i −2.44949 2.62132 + 0.358719i −2.00000 − 2.00000i 2.59808 − 1.50000i −2.00000 + 2.44949i
317.1 −1.36603 + 0.366025i −1.67303 − 0.448288i 1.73205 − 1.00000i 1.81431 − 1.30701i 2.44949 −1.62132 + 2.09077i −2.00000 + 2.00000i 2.59808 + 1.50000i −2.00000 + 2.44949i
317.2 −1.36603 + 0.366025i 1.67303 + 0.448288i 1.73205 − 1.00000i 0.917738 + 2.03906i −2.44949 2.62132 − 0.358719i −2.00000 + 2.00000i 2.59808 + 1.50000i −2.00000 − 2.44949i
557.1 0.366025 − 1.36603i −0.448288 − 1.67303i −1.73205 − 1.00000i 1.30701 − 1.81431i −2.44949 −1.62132 − 2.09077i −2.00000 + 2.00000i −2.59808 + 1.50000i −2.00000 − 2.44949i
557.2 0.366025 − 1.36603i 0.448288 + 1.67303i −1.73205 − 1.00000i −2.03906 − 0.917738i 2.44949 2.62132 + 0.358719i −2.00000 + 2.00000i −2.59808 + 1.50000i −2.00000 + 2.44949i
653.1 0.366025 + 1.36603i −0.448288 + 1.67303i −1.73205 + 1.00000i 1.30701 + 1.81431i −2.44949 −1.62132 + 2.09077i −2.00000 − 2.00000i −2.59808 − 1.50000i −2.00000 + 2.44949i
653.2 0.366025 + 1.36603i 0.448288 − 1.67303i −1.73205 + 1.00000i −2.03906 + 0.917738i 2.44949 2.62132 − 0.358719i −2.00000 − 2.00000i −2.59808 − 1.50000i −2.00000 − 2.44949i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
35.l odd 12 1 inner
840.dc even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 840.2.dc.b yes 8
3.b odd 2 1 840.2.dc.c yes 8
5.c odd 4 1 840.2.dc.a ✓ 8
7.c even 3 1 840.2.dc.a ✓ 8
8.b even 2 1 840.2.dc.c yes 8
15.e even 4 1 840.2.dc.d yes 8
21.h odd 6 1 840.2.dc.d yes 8
24.h odd 2 1 CM 840.2.dc.b yes 8
35.l odd 12 1 inner 840.2.dc.b yes 8
40.i odd 4 1 840.2.dc.d yes 8
56.p even 6 1 840.2.dc.d yes 8
105.x even 12 1 840.2.dc.c yes 8
120.w even 4 1 840.2.dc.a ✓ 8
168.s odd 6 1 840.2.dc.a ✓ 8
280.bt odd 12 1 840.2.dc.c yes 8
840.dc even 12 1 inner 840.2.dc.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.dc.a ✓ 8 5.c odd 4 1
840.2.dc.a ✓ 8 7.c even 3 1
840.2.dc.a ✓ 8 120.w even 4 1
840.2.dc.a ✓ 8 168.s odd 6 1
840.2.dc.b yes 8 1.a even 1 1 trivial
840.2.dc.b yes 8 24.h odd 2 1 CM
840.2.dc.b yes 8 35.l odd 12 1 inner
840.2.dc.b yes 8 840.dc even 12 1 inner
840.2.dc.c yes 8 3.b odd 2 1
840.2.dc.c yes 8 8.b even 2 1
840.2.dc.c yes 8 105.x even 12 1
840.2.dc.c yes 8 280.bt odd 12 1
840.2.dc.d yes 8 15.e even 4 1
840.2.dc.d yes 8 21.h odd 6 1
840.2.dc.d yes 8 40.i odd 4 1
840.2.dc.d yes 8 56.p even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\):

\( T_{11}^{8} - 8T_{11}^{7} + 76T_{11}^{6} - 272T_{11}^{5} + 1879T_{11}^{4} - 6416T_{11}^{3} + 30700T_{11}^{2} - 48392T_{11} + 69169 \) Copy content Toggle raw display
\( T_{53}^{8} + 60 T_{53}^{7} + 1800 T_{53}^{6} + 36000 T_{53}^{5} + 514791 T_{53}^{4} + 5292000 T_{53}^{3} + \cdots + 466948881 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 9T^{4} + 81 \) Copy content Toggle raw display
$5$ \( T^{8} - 4 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( (T^{4} - 2 T^{3} - 3 T^{2} + \cdots + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 8 T^{7} + \cdots + 69169 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} + 232 T^{6} + \cdots + 529 \) Copy content Toggle raw display
$31$ \( T^{8} + 162 T^{6} + \cdots + 22667121 \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} + 60 T^{7} + \cdots + 466948881 \) Copy content Toggle raw display
$59$ \( T^{8} - 48 T^{7} + \cdots + 106977649 \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} + 28 T^{7} + \cdots + 6250000 \) Copy content Toggle raw display
$79$ \( (T^{4} + 30 T^{3} + \cdots + 441)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 8 T^{3} + \cdots + 47089)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} - 4 T^{7} + \cdots + 61984129 \) Copy content Toggle raw display
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