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Properties

Label 700.2.c.i
Level $700$
Weight $2$
Character orbit 700.c
Analytic conductor $5.590$
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] = [700,2,Mod(699,700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(700, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("700.699");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.c (of order \(2\), degree \(1\), not 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: \(8\)
Coefficient field: 8.0.342102016.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 140)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + (\beta_{4} + \beta_{3}) q^{3} + (\beta_{7} + \beta_{4} + \beta_{3}) q^{4} + ( - \beta_{6} + \beta_{4} + \cdots + \beta_{2}) q^{6}+ \cdots + ( - 2 \beta_{7} - \beta_{6} - \beta_{4} + \cdots - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + (\beta_{4} + \beta_{3}) q^{3} + (\beta_{7} + \beta_{4} + \beta_{3}) q^{4} + ( - \beta_{6} + \beta_{4} + \cdots + \beta_{2}) q^{6}+ \cdots + ( - 2 \beta_{7} + \beta_{6} + \cdots + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{4} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{4} - 16 q^{9} - 28 q^{12} - 16 q^{13} + 6 q^{14} - 14 q^{16} - 24 q^{17} - 36 q^{21} - 32 q^{28} + 16 q^{29} + 48 q^{33} - 30 q^{36} + 28 q^{38} - 12 q^{42} + 20 q^{44} + 44 q^{46} - 20 q^{48} - 36 q^{49} + 4 q^{52} + 2 q^{56} - 32 q^{62} - 2 q^{64} + 40 q^{68} - 24 q^{73} - 4 q^{74} + 16 q^{77} + 48 q^{81} + 40 q^{82} - 8 q^{84} - 20 q^{86} + 24 q^{97} - 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} - 3\nu^{4} + 8\nu^{3} - 10\nu^{2} - 8\nu - 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - \nu^{6} + 3\nu^{5} - 3\nu^{4} + 2\nu^{3} + 6\nu^{2} - 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 4\nu^{5} - 3\nu^{4} + 4\nu^{3} - 10\nu^{2} + 16\nu - 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} - 4\nu^{5} - 3\nu^{4} - 4\nu^{3} - 10\nu^{2} - 16\nu - 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} - \nu^{5} - 4\nu^{3} - 4\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} - \nu^{6} - 3\nu^{5} - 3\nu^{4} - 2\nu^{3} + 6\nu^{2} - 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{6} + \nu^{4} + 2\nu^{2} ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{6} - \beta_{5} - 2\beta_{4} - \beta_{3} - \beta_{2} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{7} - 3\beta_{6} - \beta_{4} - \beta_{3} - 3\beta_{2} - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{6} + 5\beta_{5} - 2\beta_{4} + \beta_{3} + 5\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -12\beta_{7} - \beta_{6} - 3\beta_{4} - 3\beta_{3} - \beta_{2} - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -3\beta_{6} - 13\beta_{5} + 10\beta_{4} - \beta_{3} + 3\beta_{2} - 9\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
699.1
−1.17915 + 0.780776i
−1.17915 − 0.780776i
−0.599676 + 1.28078i
−0.599676 − 1.28078i
0.599676 + 1.28078i
0.599676 − 1.28078i
1.17915 + 0.780776i
1.17915 − 0.780776i
−1.17915 − 0.780776i 3.02045i 0.780776 + 1.84130i 0 2.35829 − 3.56155i −1.51022 + 2.17238i 0.516994 − 2.78078i −6.12311 0
699.2 −1.17915 + 0.780776i − 3.02045i 0.780776 − 1.84130i 0 2.35829 + 3.56155i −1.51022 − 2.17238i 0.516994 + 2.78078i −6.12311 0
699.3 −0.599676 − 1.28078i 0.936426i −1.28078 + 1.53610i 0 1.19935 − 0.561553i 0.468213 + 2.60399i 2.73546 + 0.719224i 2.12311 0
699.4 −0.599676 + 1.28078i − 0.936426i −1.28078 − 1.53610i 0 1.19935 + 0.561553i 0.468213 − 2.60399i 2.73546 − 0.719224i 2.12311 0
699.5 0.599676 − 1.28078i − 0.936426i −1.28078 − 1.53610i 0 −1.19935 − 0.561553i −0.468213 − 2.60399i −2.73546 + 0.719224i 2.12311 0
699.6 0.599676 + 1.28078i 0.936426i −1.28078 + 1.53610i 0 −1.19935 + 0.561553i −0.468213 + 2.60399i −2.73546 − 0.719224i 2.12311 0
699.7 1.17915 − 0.780776i − 3.02045i 0.780776 − 1.84130i 0 −2.35829 − 3.56155i 1.51022 − 2.17238i −0.516994 − 2.78078i −6.12311 0
699.8 1.17915 + 0.780776i 3.02045i 0.780776 + 1.84130i 0 −2.35829 + 3.56155i 1.51022 + 2.17238i −0.516994 + 2.78078i −6.12311 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 699.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.2.c.i 8
4.b odd 2 1 inner 700.2.c.i 8
5.b even 2 1 700.2.c.j 8
5.c odd 4 1 140.2.g.c ✓ 8
5.c odd 4 1 700.2.g.j 8
7.b odd 2 1 700.2.c.j 8
15.e even 4 1 1260.2.c.c 8
20.d odd 2 1 700.2.c.j 8
20.e even 4 1 140.2.g.c ✓ 8
20.e even 4 1 700.2.g.j 8
28.d even 2 1 700.2.c.j 8
35.c odd 2 1 inner 700.2.c.i 8
35.f even 4 1 140.2.g.c ✓ 8
35.f even 4 1 700.2.g.j 8
35.k even 12 2 980.2.o.e 16
35.l odd 12 2 980.2.o.e 16
40.i odd 4 1 2240.2.k.e 8
40.k even 4 1 2240.2.k.e 8
60.l odd 4 1 1260.2.c.c 8
105.k odd 4 1 1260.2.c.c 8
140.c even 2 1 inner 700.2.c.i 8
140.j odd 4 1 140.2.g.c ✓ 8
140.j odd 4 1 700.2.g.j 8
140.w even 12 2 980.2.o.e 16
140.x odd 12 2 980.2.o.e 16
280.s even 4 1 2240.2.k.e 8
280.y odd 4 1 2240.2.k.e 8
420.w even 4 1 1260.2.c.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.g.c ✓ 8 5.c odd 4 1
140.2.g.c ✓ 8 20.e even 4 1
140.2.g.c ✓ 8 35.f even 4 1
140.2.g.c ✓ 8 140.j odd 4 1
700.2.c.i 8 1.a even 1 1 trivial
700.2.c.i 8 4.b odd 2 1 inner
700.2.c.i 8 35.c odd 2 1 inner
700.2.c.i 8 140.c even 2 1 inner
700.2.c.j 8 5.b even 2 1
700.2.c.j 8 7.b odd 2 1
700.2.c.j 8 20.d odd 2 1
700.2.c.j 8 28.d even 2 1
700.2.g.j 8 5.c odd 4 1
700.2.g.j 8 20.e even 4 1
700.2.g.j 8 35.f even 4 1
700.2.g.j 8 140.j odd 4 1
980.2.o.e 16 35.k even 12 2
980.2.o.e 16 35.l odd 12 2
980.2.o.e 16 140.w even 12 2
980.2.o.e 16 140.x odd 12 2
1260.2.c.c 8 15.e even 4 1
1260.2.c.c 8 60.l odd 4 1
1260.2.c.c 8 105.k odd 4 1
1260.2.c.c 8 420.w even 4 1
2240.2.k.e 8 40.i odd 4 1
2240.2.k.e 8 40.k even 4 1
2240.2.k.e 8 280.s even 4 1
2240.2.k.e 8 280.y odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(700, [\chi])\):

\( T_{3}^{4} + 10T_{3}^{2} + 8 \) Copy content Toggle raw display
\( T_{11}^{4} + 28T_{11}^{2} + 128 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display
\( T_{19}^{4} - 28T_{19}^{2} + 128 \) Copy content Toggle raw display
\( T_{23}^{4} - 74T_{23}^{2} + 1352 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{6} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T^{4} + 10 T^{2} + 8)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 18 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{4} + 28 T^{2} + 128)^{2} \) Copy content Toggle raw display
$13$ \( (T + 2)^{8} \) Copy content Toggle raw display
$17$ \( (T^{2} + 6 T - 8)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 28 T^{2} + 128)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 74 T^{2} + 1352)^{2} \) Copy content Toggle raw display
$29$ \( (T - 2)^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} - 56 T^{2} + 512)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 52 T^{2} + 64)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 58 T^{2} + 8)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 126 T^{2} + 2592)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 124 T^{2} + 512)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 218 T^{2} + 2888)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 92 T^{2} + 2048)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 6 T - 144)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 20 T^{2} + 32)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 10 T^{2} + 8)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 144)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 6 T - 8)^{4} \) Copy content Toggle raw display
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