Properties

Label 700.2.i.a
Level $700$
Weight $2$
Character orbit 700.i
Analytic conductor $5.590$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [700,2,Mod(401,700)]
 
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([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("700.401");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.i (of order \(3\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (3 \zeta_{6} - 3) q^{3} + ( - 3 \zeta_{6} + 1) q^{7} - 6 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (3 \zeta_{6} - 3) q^{3} + ( - 3 \zeta_{6} + 1) q^{7} - 6 \zeta_{6} q^{9} + ( - 2 \zeta_{6} + 2) q^{11} + 6 q^{13} + ( - 2 \zeta_{6} + 2) q^{17} + (3 \zeta_{6} + 6) q^{21} - 9 \zeta_{6} q^{23} + 9 q^{27} + 3 q^{29} + (2 \zeta_{6} - 2) q^{31} + 6 \zeta_{6} q^{33} + 8 \zeta_{6} q^{37} + (18 \zeta_{6} - 18) q^{39} + 5 q^{41} - q^{43} + 8 \zeta_{6} q^{47} + (3 \zeta_{6} - 8) q^{49} + 6 \zeta_{6} q^{51} + ( - 4 \zeta_{6} + 4) q^{53} + ( - 8 \zeta_{6} + 8) q^{59} - 7 \zeta_{6} q^{61} + (12 \zeta_{6} - 18) q^{63} + (3 \zeta_{6} - 3) q^{67} + 27 q^{69} + 8 q^{71} + ( - 14 \zeta_{6} + 14) q^{73} + ( - 2 \zeta_{6} - 4) q^{77} - 4 \zeta_{6} q^{79} + (9 \zeta_{6} - 9) q^{81} + q^{83} + (9 \zeta_{6} - 9) q^{87} - 13 \zeta_{6} q^{89} + ( - 18 \zeta_{6} + 6) q^{91} - 6 \zeta_{6} q^{93} + 10 q^{97} - 12 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} - q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} - q^{7} - 6 q^{9} + 2 q^{11} + 12 q^{13} + 2 q^{17} + 15 q^{21} - 9 q^{23} + 18 q^{27} + 6 q^{29} - 2 q^{31} + 6 q^{33} + 8 q^{37} - 18 q^{39} + 10 q^{41} - 2 q^{43} + 8 q^{47} - 13 q^{49} + 6 q^{51} + 4 q^{53} + 8 q^{59} - 7 q^{61} - 24 q^{63} - 3 q^{67} + 54 q^{69} + 16 q^{71} + 14 q^{73} - 10 q^{77} - 4 q^{79} - 9 q^{81} + 2 q^{83} - 9 q^{87} - 13 q^{89} - 6 q^{91} - 6 q^{93} + 20 q^{97} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −1.50000 2.59808i 0 0 0 −0.500000 + 2.59808i 0 −3.00000 + 5.19615i 0
501.1 0 −1.50000 + 2.59808i 0 0 0 −0.500000 2.59808i 0 −3.00000 5.19615i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.2.i.a 2
5.b even 2 1 140.2.i.b 2
5.c odd 4 2 700.2.r.c 4
7.c even 3 1 inner 700.2.i.a 2
7.c even 3 1 4900.2.a.v 1
7.d odd 6 1 4900.2.a.a 1
15.d odd 2 1 1260.2.s.b 2
20.d odd 2 1 560.2.q.a 2
35.c odd 2 1 980.2.i.a 2
35.i odd 6 1 980.2.a.i 1
35.i odd 6 1 980.2.i.a 2
35.j even 6 1 140.2.i.b 2
35.j even 6 1 980.2.a.a 1
35.k even 12 2 4900.2.e.b 2
35.l odd 12 2 700.2.r.c 4
35.l odd 12 2 4900.2.e.c 2
105.o odd 6 1 1260.2.s.b 2
105.o odd 6 1 8820.2.a.w 1
105.p even 6 1 8820.2.a.k 1
140.p odd 6 1 560.2.q.a 2
140.p odd 6 1 3920.2.a.bi 1
140.s even 6 1 3920.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.i.b 2 5.b even 2 1
140.2.i.b 2 35.j even 6 1
560.2.q.a 2 20.d odd 2 1
560.2.q.a 2 140.p odd 6 1
700.2.i.a 2 1.a even 1 1 trivial
700.2.i.a 2 7.c even 3 1 inner
700.2.r.c 4 5.c odd 4 2
700.2.r.c 4 35.l odd 12 2
980.2.a.a 1 35.j even 6 1
980.2.a.i 1 35.i odd 6 1
980.2.i.a 2 35.c odd 2 1
980.2.i.a 2 35.i odd 6 1
1260.2.s.b 2 15.d odd 2 1
1260.2.s.b 2 105.o odd 6 1
3920.2.a.d 1 140.s even 6 1
3920.2.a.bi 1 140.p odd 6 1
4900.2.a.a 1 7.d odd 6 1
4900.2.a.v 1 7.c even 3 1
4900.2.e.b 2 35.k even 12 2
4900.2.e.c 2 35.l odd 12 2
8820.2.a.k 1 105.p even 6 1
8820.2.a.w 1 105.o odd 6 1

Hecke kernels

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

\( T_{3}^{2} + 3T_{3} + 9 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$13$ \( (T - 6)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$29$ \( (T - 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$37$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$41$ \( (T - 5)^{2} \) Copy content Toggle raw display
$43$ \( (T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$53$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$59$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$61$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
$79$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$83$ \( (T - 1)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$97$ \( (T - 10)^{2} \) Copy content Toggle raw display
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