↑

Properties

Label 700.2.i.f
Level $700$
Weight $2$
Character orbit 700.i
Analytic conductor $5.590$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.2702336256.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{6} + \beta_{4}) q^{3} + ( - \beta_{4} + \beta_{3}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} + \beta_{4}) q^{3} + ( - \beta_{4} + \beta_{3}) q^{7} + ( - \beta_{2} + \beta_1 + 1) q^{11} + ( - 2 \beta_{6} - 2 \beta_{5} + \cdots + 2 \beta_{3}) q^{13}+ \cdots + 4 \beta_{4} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{11} + 2 q^{19} + 4 q^{29} + 2 q^{31} + 12 q^{39} - 60 q^{41} - 10 q^{49} - 18 q^{51} - 2 q^{59} + 24 q^{61} - 48 q^{69} + 24 q^{71} - 14 q^{79} + 36 q^{81} + 28 q^{89} + 20 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} - 56\nu^{4} - 224\nu^{2} - 895 ) / 280 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9\nu^{6} + 56\nu^{4} + 504\nu^{2} + 2025 ) / 1400 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\nu^{7} + 224\nu^{5} + 616\nu^{3} + 9475\nu ) / 7000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 16\nu^{5} + 44\nu^{3} + 175\nu ) / 500 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 14\nu^{5} + 126\nu^{3} + 505\nu ) / 350 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 47\nu^{7} + 448\nu^{5} + 1232\nu^{3} + 4975\nu ) / 7000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{6} - 9\nu^{4} - 31\nu^{2} - 100 ) / 25 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} - \beta_{4} + 3\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} + 13\beta_{2} - \beta _1 - 14 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{6} + 12\beta_{5} + \beta_{4} - 12\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{7} - 17\beta_{2} - 3\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 34\beta_{6} - 33\beta_{5} + 67\beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -56\beta_{7} + 56\beta_{2} + 112\beta _1 + 53 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 281\beta_{6} - 559\beta_{4} - 3\beta_{3} ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(-\beta_{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
−0.656712 + 2.13746i
1.52274 − 1.63746i
−1.52274 + 1.63746i
0.656712 − 2.13746i
−0.656712 − 2.13746i
1.52274 + 1.63746i
−1.52274 − 1.63746i
0.656712 + 2.13746i
0 −0.866025 − 1.50000i 0 0 0 0.209313 + 2.63746i 0 0 0
401.2 0 −0.866025 − 1.50000i 0 0 0 2.38876 − 1.13746i 0 0 0
401.3 0 0.866025 + 1.50000i 0 0 0 −2.38876 + 1.13746i 0 0 0
401.4 0 0.866025 + 1.50000i 0 0 0 −0.209313 − 2.63746i 0 0 0
501.1 0 −0.866025 + 1.50000i 0 0 0 0.209313 − 2.63746i 0 0 0
501.2 0 −0.866025 + 1.50000i 0 0 0 2.38876 + 1.13746i 0 0 0
501.3 0 0.866025 − 1.50000i 0 0 0 −2.38876 − 1.13746i 0 0 0
501.4 0 0.866025 − 1.50000i 0 0 0 −0.209313 + 2.63746i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 401.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j 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.i.f 8
5.b even 2 1 inner 700.2.i.f 8
5.c odd 4 1 140.2.q.a ✓ 4
5.c odd 4 1 140.2.q.b yes 4
7.c even 3 1 inner 700.2.i.f 8
7.c even 3 1 4900.2.a.be 4
7.d odd 6 1 4900.2.a.bf 4
15.e even 4 1 1260.2.bm.a 4
15.e even 4 1 1260.2.bm.b 4
20.e even 4 1 560.2.bw.a 4
20.e even 4 1 560.2.bw.e 4
35.f even 4 1 980.2.q.b 4
35.f even 4 1 980.2.q.g 4
35.i odd 6 1 4900.2.a.bf 4
35.j even 6 1 inner 700.2.i.f 8
35.j even 6 1 4900.2.a.be 4
35.k even 12 2 980.2.e.c 4
35.k even 12 1 980.2.q.b 4
35.k even 12 1 980.2.q.g 4
35.l odd 12 1 140.2.q.a ✓ 4
35.l odd 12 1 140.2.q.b yes 4
35.l odd 12 2 980.2.e.f 4
105.x even 12 1 1260.2.bm.a 4
105.x even 12 1 1260.2.bm.b 4
140.w even 12 1 560.2.bw.a 4
140.w even 12 1 560.2.bw.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.q.a ✓ 4 5.c odd 4 1
140.2.q.a ✓ 4 35.l odd 12 1
140.2.q.b yes 4 5.c odd 4 1
140.2.q.b yes 4 35.l odd 12 1
560.2.bw.a 4 20.e even 4 1
560.2.bw.a 4 140.w even 12 1
560.2.bw.e 4 20.e even 4 1
560.2.bw.e 4 140.w even 12 1
700.2.i.f 8 1.a even 1 1 trivial
700.2.i.f 8 5.b even 2 1 inner
700.2.i.f 8 7.c even 3 1 inner
700.2.i.f 8 35.j even 6 1 inner
980.2.e.c 4 35.k even 12 2
980.2.e.f 4 35.l odd 12 2
980.2.q.b 4 35.f even 4 1
980.2.q.b 4 35.k even 12 1
980.2.q.g 4 35.f even 4 1
980.2.q.g 4 35.k even 12 1
1260.2.bm.a 4 15.e even 4 1
1260.2.bm.a 4 105.x even 12 1
1260.2.bm.b 4 15.e even 4 1
1260.2.bm.b 4 105.x even 12 1
4900.2.a.be 4 7.c even 3 1
4900.2.a.be 4 35.j even 6 1
4900.2.a.bf 4 7.d odd 6 1
4900.2.a.bf 4 35.i 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
\( T_{11}^{4} - 3T_{11}^{3} + 21T_{11}^{2} + 36T_{11} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 3 T^{2} + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 5 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{4} - 3 T^{3} + \cdots + 144)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 44 T^{2} + 256)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 23 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( (T^{4} - T^{3} + 15 T^{2} + \cdots + 196)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 62 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$29$ \( (T^{2} - T - 14)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - T^{3} + 15 T^{2} + \cdots + 196)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 131 T^{6} + \cdots + 9834496 \) Copy content Toggle raw display
$41$ \( (T^{2} + 15 T + 42)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 47 T^{2} + 196)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 47 T^{6} + \cdots + 38416 \) Copy content Toggle raw display
$53$ \( T^{8} + 87 T^{6} + \cdots + 3111696 \) Copy content Toggle raw display
$59$ \( (T^{4} + T^{3} + 15 T^{2} + \cdots + 196)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 12 T^{3} + \cdots + 441)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 206 T^{6} + \cdots + 5764801 \) Copy content Toggle raw display
$71$ \( (T^{2} - 6 T - 48)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + 47 T^{6} + \cdots + 38416 \) Copy content Toggle raw display
$79$ \( (T^{4} + 7 T^{3} + 51 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 87 T^{2} + 1764)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 7 T + 49)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 48)^{4} \) Copy content Toggle raw display
show more
show less