↑

Properties

Label 700.2.g.g
Level $700$
Weight $2$
Character orbit 700.g
Analytic conductor $5.590$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [700,2,Mod(251,700)]
 
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, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("700.251");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.g (of order \(2\), degree \(1\), 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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{2} + (\beta_{3} - \beta_{2} + \beta_1) q^{3} + (\beta_{3} + \beta_1) q^{4} + (\beta_{3} + \beta_{2} + 2) q^{6} + (\beta_{3} - 3 \beta_{2} + \beta_1) q^{7} + (2 \beta_{2} + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{2} + (\beta_{3} - \beta_{2} + \beta_1) q^{3} + (\beta_{3} + \beta_1) q^{4} + (\beta_{3} + \beta_{2} + 2) q^{6} + (\beta_{3} - 3 \beta_{2} + \beta_1) q^{7} + (2 \beta_{2} + 2) q^{8} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{11} + ( - \beta_{3} + \beta_1 + 2) q^{12} + (2 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 2) q^{13} + (3 \beta_{3} - \beta_{2} + 2) q^{14} + ( - 2 \beta_{3} + 2 \beta_1) q^{16} + (2 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 2) q^{17} - 6 q^{19} + (2 \beta_{3} - 2 \beta_1 + 5) q^{21} + ( - 2 \beta_{3} + 3 \beta_{2} + \cdots - 2) q^{22}+ \cdots + ( - 3 \beta_{2} - 5 \beta_1 + 8) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 6 q^{6} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 6 q^{6} + 8 q^{8} + 12 q^{12} + 2 q^{14} + 8 q^{16} - 24 q^{19} + 12 q^{21} - 2 q^{22} + 6 q^{26} + 20 q^{28} + 4 q^{29} + 24 q^{31} - 8 q^{32} + 18 q^{34} + 24 q^{37} - 12 q^{38} + 18 q^{42} - 8 q^{44} - 8 q^{46} - 4 q^{49} + 12 q^{52} - 8 q^{53} - 18 q^{54} + 16 q^{56} - 22 q^{58} + 12 q^{62} - 6 q^{66} - 12 q^{68} + 16 q^{77} + 6 q^{78} - 36 q^{81} - 12 q^{82} + 48 q^{83} - 4 q^{86} - 48 q^{87} - 16 q^{88} - 24 q^{91} - 8 q^{92} - 6 q^{94} + 24 q^{96} + 22 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{2} + \zeta_{12} \) Copy content Toggle raw display

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( -\beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
−0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 − 0.500000i
0.866025 + 0.500000i
−0.366025 − 1.36603i −1.73205 −1.73205 + 1.00000i 0 0.633975 + 2.36603i −1.73205 − 2.00000i 2.00000 + 2.00000i 0 0
251.2 −0.366025 + 1.36603i −1.73205 −1.73205 − 1.00000i 0 0.633975 − 2.36603i −1.73205 + 2.00000i 2.00000 − 2.00000i 0 0
251.3 1.36603 − 0.366025i 1.73205 1.73205 − 1.00000i 0 2.36603 − 0.633975i 1.73205 + 2.00000i 2.00000 − 2.00000i 0 0
251.4 1.36603 + 0.366025i 1.73205 1.73205 + 1.00000i 0 2.36603 + 0.633975i 1.73205 − 2.00000i 2.00000 + 2.00000i 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d 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.g.g 4
4.b odd 2 1 700.2.g.f 4
5.b even 2 1 140.2.g.b yes 4
5.c odd 4 1 700.2.c.c 4
5.c odd 4 1 700.2.c.f 4
7.b odd 2 1 700.2.g.f 4
15.d odd 2 1 1260.2.c.b 4
20.d odd 2 1 140.2.g.a ✓ 4
20.e even 4 1 700.2.c.b 4
20.e even 4 1 700.2.c.e 4
28.d even 2 1 inner 700.2.g.g 4
35.c odd 2 1 140.2.g.a ✓ 4
35.f even 4 1 700.2.c.b 4
35.f even 4 1 700.2.c.e 4
35.i odd 6 1 980.2.o.a 4
35.i odd 6 1 980.2.o.c 4
35.j even 6 1 980.2.o.b 4
35.j even 6 1 980.2.o.d 4
40.e odd 2 1 2240.2.k.a 4
40.f even 2 1 2240.2.k.b 4
60.h even 2 1 1260.2.c.a 4
105.g even 2 1 1260.2.c.a 4
140.c even 2 1 140.2.g.b yes 4
140.j odd 4 1 700.2.c.c 4
140.j odd 4 1 700.2.c.f 4
140.p odd 6 1 980.2.o.a 4
140.p odd 6 1 980.2.o.c 4
140.s even 6 1 980.2.o.b 4
140.s even 6 1 980.2.o.d 4
280.c odd 2 1 2240.2.k.a 4
280.n even 2 1 2240.2.k.b 4
420.o odd 2 1 1260.2.c.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.g.a ✓ 4 20.d odd 2 1
140.2.g.a ✓ 4 35.c odd 2 1
140.2.g.b yes 4 5.b even 2 1
140.2.g.b yes 4 140.c even 2 1
700.2.c.b 4 20.e even 4 1
700.2.c.b 4 35.f even 4 1
700.2.c.c 4 5.c odd 4 1
700.2.c.c 4 140.j odd 4 1
700.2.c.e 4 20.e even 4 1
700.2.c.e 4 35.f even 4 1
700.2.c.f 4 5.c odd 4 1
700.2.c.f 4 140.j odd 4 1
700.2.g.f 4 4.b odd 2 1
700.2.g.f 4 7.b odd 2 1
700.2.g.g 4 1.a even 1 1 trivial
700.2.g.g 4 28.d even 2 1 inner
980.2.o.a 4 35.i odd 6 1
980.2.o.a 4 140.p odd 6 1
980.2.o.b 4 35.j even 6 1
980.2.o.b 4 140.s even 6 1
980.2.o.c 4 35.i odd 6 1
980.2.o.c 4 140.p odd 6 1
980.2.o.d 4 35.j even 6 1
980.2.o.d 4 140.s even 6 1
1260.2.c.a 4 60.h even 2 1
1260.2.c.a 4 105.g even 2 1
1260.2.c.b 4 15.d odd 2 1
1260.2.c.b 4 420.o odd 2 1
2240.2.k.a 4 40.e odd 2 1
2240.2.k.a 4 280.c odd 2 1
2240.2.k.b 4 40.f even 2 1
2240.2.k.b 4 280.n even 2 1

Hecke kernels

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

\( T_{3}^{2} - 3 \) Copy content Toggle raw display
\( T_{11}^{4} + 14T_{11}^{2} + 1 \) Copy content Toggle raw display
\( T_{19} + 6 \) Copy content Toggle raw display
\( T_{37}^{2} - 12T_{37} + 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 2T^{2} + 49 \) Copy content Toggle raw display
$11$ \( T^{4} + 14T^{2} + 1 \) Copy content Toggle raw display
$13$ \( T^{4} + 42T^{2} + 9 \) Copy content Toggle raw display
$17$ \( T^{4} + 42T^{2} + 9 \) Copy content Toggle raw display
$19$ \( (T + 6)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 32T^{2} + 64 \) Copy content Toggle raw display
$29$ \( (T^{2} - 2 T - 47)^{2} \) Copy content Toggle raw display
$31$ \( (T - 6)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 12 T + 24)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$53$ \( (T + 2)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 96T^{2} + 576 \) Copy content Toggle raw display
$67$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 56T^{2} + 16 \) Copy content Toggle raw display
$73$ \( T^{4} + 168T^{2} + 144 \) Copy content Toggle raw display
$79$ \( T^{4} + 222T^{2} + 1521 \) Copy content Toggle raw display
$83$ \( (T^{2} - 24 T + 132)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 96T^{2} + 576 \) Copy content Toggle raw display
$97$ \( T^{4} + 234T^{2} + 9801 \) Copy content Toggle raw display
show more
show less