Properties

Label 700.2.p.a
Level $700$
Weight $2$
Character orbit 700.p
Analytic conductor $5.590$
Analytic rank $0$
Dimension $4$
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(451,700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(700, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 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("700.451");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.p (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: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
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: \( 1 \)
Twist minimal: no (minimal twist has level 28)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{2} + (2 \zeta_{12}^{3} - \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} + \cdots - 1) q^{6}+ \cdots + (2 \zeta_{12}^{3} + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{2} + (2 \zeta_{12}^{3} - \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} + \cdots - 1) q^{6}+ \cdots + ( - 3 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + \cdots + 8) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 8 q^{8} - 12 q^{12} + 2 q^{14} + 8 q^{16} + 6 q^{17} + 6 q^{21} + 4 q^{22} - 12 q^{24} - 12 q^{26} + 20 q^{28} + 16 q^{29} - 8 q^{32} - 6 q^{33} + 6 q^{37} - 18 q^{38} - 12 q^{42} + 4 q^{44} - 2 q^{46} - 4 q^{49} - 2 q^{53} - 18 q^{54} + 16 q^{56} + 36 q^{57} + 8 q^{58} - 18 q^{61} - 6 q^{66} - 30 q^{73} - 6 q^{74} + 10 q^{77} - 24 q^{78} + 18 q^{81} - 12 q^{82} - 4 q^{86} - 4 q^{88} + 54 q^{89} - 8 q^{92} - 6 q^{93} - 30 q^{94} - 24 q^{96} + 22 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(\zeta_{12}^{2}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.366025 + 1.36603i 0.866025 1.50000i −1.73205 1.00000i 0 1.73205 + 1.73205i −1.73205 + 2.00000i 2.00000 2.00000i 0 0
451.2 1.36603 + 0.366025i −0.866025 + 1.50000i 1.73205 + 1.00000i 0 −1.73205 + 1.73205i 1.73205 2.00000i 2.00000 + 2.00000i 0 0
551.1 −0.366025 1.36603i 0.866025 + 1.50000i −1.73205 + 1.00000i 0 1.73205 1.73205i −1.73205 2.00000i 2.00000 + 2.00000i 0 0
551.2 1.36603 0.366025i −0.866025 1.50000i 1.73205 1.00000i 0 −1.73205 1.73205i 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
4.b odd 2 1 inner
7.d odd 6 1 inner
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.2.p.a 4
4.b odd 2 1 inner 700.2.p.a 4
5.b even 2 1 28.2.f.a 4
5.c odd 4 1 700.2.t.a 4
5.c odd 4 1 700.2.t.b 4
7.d odd 6 1 inner 700.2.p.a 4
15.d odd 2 1 252.2.bf.e 4
20.d odd 2 1 28.2.f.a 4
20.e even 4 1 700.2.t.a 4
20.e even 4 1 700.2.t.b 4
28.f even 6 1 inner 700.2.p.a 4
35.c odd 2 1 196.2.f.a 4
35.i odd 6 1 28.2.f.a 4
35.i odd 6 1 196.2.d.b 4
35.j even 6 1 196.2.d.b 4
35.j even 6 1 196.2.f.a 4
35.k even 12 1 700.2.t.a 4
35.k even 12 1 700.2.t.b 4
40.e odd 2 1 448.2.p.d 4
40.f even 2 1 448.2.p.d 4
60.h even 2 1 252.2.bf.e 4
105.o odd 6 1 1764.2.b.a 4
105.p even 6 1 252.2.bf.e 4
105.p even 6 1 1764.2.b.a 4
140.c even 2 1 196.2.f.a 4
140.p odd 6 1 196.2.d.b 4
140.p odd 6 1 196.2.f.a 4
140.s even 6 1 28.2.f.a 4
140.s even 6 1 196.2.d.b 4
140.x odd 12 1 700.2.t.a 4
140.x odd 12 1 700.2.t.b 4
280.ba even 6 1 448.2.p.d 4
280.ba even 6 1 3136.2.f.e 4
280.bf even 6 1 3136.2.f.e 4
280.bi odd 6 1 3136.2.f.e 4
280.bk odd 6 1 448.2.p.d 4
280.bk odd 6 1 3136.2.f.e 4
420.ba even 6 1 1764.2.b.a 4
420.be odd 6 1 252.2.bf.e 4
420.be odd 6 1 1764.2.b.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.f.a 4 5.b even 2 1
28.2.f.a 4 20.d odd 2 1
28.2.f.a 4 35.i odd 6 1
28.2.f.a 4 140.s even 6 1
196.2.d.b 4 35.i odd 6 1
196.2.d.b 4 35.j even 6 1
196.2.d.b 4 140.p odd 6 1
196.2.d.b 4 140.s even 6 1
196.2.f.a 4 35.c odd 2 1
196.2.f.a 4 35.j even 6 1
196.2.f.a 4 140.c even 2 1
196.2.f.a 4 140.p odd 6 1
252.2.bf.e 4 15.d odd 2 1
252.2.bf.e 4 60.h even 2 1
252.2.bf.e 4 105.p even 6 1
252.2.bf.e 4 420.be odd 6 1
448.2.p.d 4 40.e odd 2 1
448.2.p.d 4 40.f even 2 1
448.2.p.d 4 280.ba even 6 1
448.2.p.d 4 280.bk odd 6 1
700.2.p.a 4 1.a even 1 1 trivial
700.2.p.a 4 4.b odd 2 1 inner
700.2.p.a 4 7.d odd 6 1 inner
700.2.p.a 4 28.f even 6 1 inner
700.2.t.a 4 5.c odd 4 1
700.2.t.a 4 20.e even 4 1
700.2.t.a 4 35.k even 12 1
700.2.t.a 4 140.x odd 12 1
700.2.t.b 4 5.c odd 4 1
700.2.t.b 4 20.e even 4 1
700.2.t.b 4 35.k even 12 1
700.2.t.b 4 140.x odd 12 1
1764.2.b.a 4 105.o odd 6 1
1764.2.b.a 4 105.p even 6 1
1764.2.b.a 4 420.ba even 6 1
1764.2.b.a 4 420.be odd 6 1
3136.2.f.e 4 280.ba even 6 1
3136.2.f.e 4 280.bf even 6 1
3136.2.f.e 4 280.bi odd 6 1
3136.2.f.e 4 280.bk odd 6 1

Hecke kernels

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

\( T_{3}^{4} + 3T_{3}^{2} + 9 \) Copy content Toggle raw display
\( T_{17}^{2} - 3T_{17} + 3 \) 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
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 2T^{2} + 49 \) Copy content Toggle raw display
$11$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$13$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$23$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$29$ \( (T - 4)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$37$ \( (T^{2} - 3 T + 9)^{2} \) 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
$47$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$53$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$61$ \( (T^{2} + 9 T + 27)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$71$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 15 T + 75)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 81T^{2} + 6561 \) Copy content Toggle raw display
$83$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 27 T + 243)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 300)^{2} \) Copy content Toggle raw display
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