Properties

Label 700.2.m.b
Level $700$
Weight $2$
Character orbit 700.m
Analytic conductor $5.590$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [700,2,Mod(293,700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(700, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("700.293");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.m (of order \(4\), 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: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{6} + \beta_{2}) q^{7} + \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{6} + \beta_{2}) q^{7} + \beta_{3} q^{9} + 3 q^{11} + \beta_1 q^{13} - 2 \beta_{7} q^{19} + (\beta_{5} + 4) q^{21} - 3 \beta_{4} q^{23} + 2 \beta_{6} q^{27} + 3 \beta_{3} q^{29} + \beta_{5} q^{31} + 3 \beta_1 q^{33} - 3 \beta_{2} q^{37} + 4 \beta_{3} q^{39} + 3 \beta_{5} q^{41} - 5 \beta_{4} q^{43} - 6 \beta_{6} q^{47} + (2 \beta_{7} - \beta_{3}) q^{49} - 8 \beta_{2} q^{57} - 3 \beta_{7} q^{59} - \beta_{5} q^{61} + (\beta_{4} + \beta_1) q^{63} + 5 \beta_{2} q^{67} + 3 \beta_{7} q^{69} - 15 q^{71} - \beta_1 q^{73} + (3 \beta_{6} + 3 \beta_{2}) q^{77} + \beta_{3} q^{79} + 11 q^{81} - 3 \beta_1 q^{83} - 3 \beta_{6} q^{87} - 3 \beta_{7} q^{89} + (\beta_{5} + 4) q^{91} + 4 \beta_{4} q^{93} + 7 \beta_{6} q^{97} + 3 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 24 q^{11} + 32 q^{21} - 120 q^{71} + 88 q^{81} + 32 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{5} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2\zeta_{24}^{7} - \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 4\zeta_{24}^{4} - 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\zeta_{24}^{5} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -2\zeta_{24}^{6} + 4\zeta_{24}^{2} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{6} + 2\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{7} + 2\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{5} + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{6} + 2\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( 2\beta_{4} + \beta_1 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(-1\) \(1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
293.1
−0.965926 + 0.258819i
0.258819 0.965926i
−0.258819 + 0.965926i
0.965926 0.258819i
−0.965926 0.258819i
0.258819 + 0.965926i
−0.258819 0.965926i
0.965926 + 0.258819i
0 −1.41421 + 1.41421i 0 0 0 −2.63896 0.189469i 0 1.00000i 0
293.2 0 −1.41421 + 1.41421i 0 0 0 −0.189469 2.63896i 0 1.00000i 0
293.3 0 1.41421 1.41421i 0 0 0 0.189469 + 2.63896i 0 1.00000i 0
293.4 0 1.41421 1.41421i 0 0 0 2.63896 + 0.189469i 0 1.00000i 0
657.1 0 −1.41421 1.41421i 0 0 0 −2.63896 + 0.189469i 0 1.00000i 0
657.2 0 −1.41421 1.41421i 0 0 0 −0.189469 + 2.63896i 0 1.00000i 0
657.3 0 1.41421 + 1.41421i 0 0 0 0.189469 2.63896i 0 1.00000i 0
657.4 0 1.41421 + 1.41421i 0 0 0 2.63896 0.189469i 0 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 293.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
7.b odd 2 1 inner
35.c odd 2 1 inner
35.f even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.2.m.b 8
5.b even 2 1 inner 700.2.m.b 8
5.c odd 4 2 inner 700.2.m.b 8
7.b odd 2 1 inner 700.2.m.b 8
35.c odd 2 1 inner 700.2.m.b 8
35.f even 4 2 inner 700.2.m.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.m.b 8 1.a even 1 1 trivial
700.2.m.b 8 5.b even 2 1 inner
700.2.m.b 8 5.c odd 4 2 inner
700.2.m.b 8 7.b odd 2 1 inner
700.2.m.b 8 35.c odd 2 1 inner
700.2.m.b 8 35.f even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 16 \) acting on \(S_{2}^{\mathrm{new}}(700, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 94T^{4} + 2401 \) Copy content Toggle raw display
$11$ \( (T - 3)^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{2} - 48)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 729)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 9)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 12)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 729)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 108)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 5625)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 20736)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{2} - 108)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 12)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 5625)^{2} \) Copy content Toggle raw display
$71$ \( (T + 15)^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 1296)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 108)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 38416)^{2} \) Copy content Toggle raw display
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