Properties

Label 700.2.g.d
Level $700$
Weight $2$
Character orbit 700.g
Analytic conductor $5.590$
Analytic rank $0$
Dimension $4$
CM discriminant -20
Inner twists $8$

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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: \(\Q(\sqrt{-2}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_{2} + 2 \beta_1) q^{3} - 2 q^{4} + 2 \beta_{3} q^{6} + ( - 2 \beta_{2} + \beta_1) q^{7} - 2 \beta_{2} q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + ( - \beta_{2} + 2 \beta_1) q^{3} - 2 q^{4} + 2 \beta_{3} q^{6} + ( - 2 \beta_{2} + \beta_1) q^{7} - 2 \beta_{2} q^{8} + 7 q^{9} + (2 \beta_{2} - 4 \beta_1) q^{12} + (\beta_{3} + 3) q^{14} + 4 q^{16} + 7 \beta_{2} q^{18} + ( - 3 \beta_{3} + 5) q^{21} + \beta_{2} q^{23} - 4 \beta_{3} q^{24} + ( - 4 \beta_{2} + 8 \beta_1) q^{27} + (4 \beta_{2} - 2 \beta_1) q^{28} - 6 q^{29} + 4 \beta_{2} q^{32} - 14 q^{36} + 2 \beta_{3} q^{41} + (2 \beta_{2} + 6 \beta_1) q^{42} + 9 \beta_{2} q^{43} - 2 q^{46} + (3 \beta_{2} - 6 \beta_1) q^{47} + ( - 4 \beta_{2} + 8 \beta_1) q^{48} + ( - 3 \beta_{3} - 2) q^{49} + 8 \beta_{3} q^{54} + ( - 2 \beta_{3} - 6) q^{56} - 6 \beta_{2} q^{58} - 6 \beta_{3} q^{61} + ( - 14 \beta_{2} + 7 \beta_1) q^{63} - 8 q^{64} + 3 \beta_{2} q^{67} + 2 \beta_{3} q^{69} - 14 \beta_{2} q^{72} + 19 q^{81} + (2 \beta_{2} - 4 \beta_1) q^{82} + (3 \beta_{2} - 6 \beta_1) q^{83} + (6 \beta_{3} - 10) q^{84} - 18 q^{86} + (6 \beta_{2} - 12 \beta_1) q^{87} - 8 \beta_{3} q^{89} - 2 \beta_{2} q^{92} - 6 \beta_{3} q^{94} + 8 \beta_{3} q^{96} + ( - 5 \beta_{2} + 6 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} + 28 q^{9} + 12 q^{14} + 16 q^{16} + 20 q^{21} - 24 q^{29} - 56 q^{36} - 8 q^{46} - 8 q^{49} - 24 q^{56} - 32 q^{64} + 76 q^{81} - 40 q^{84} - 72 q^{86}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - \nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{2} + \beta_1 \) Copy content Toggle raw display

Character values

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

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

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
−1.58114 0.707107i
1.58114 0.707107i
−1.58114 + 0.707107i
1.58114 + 0.707107i
1.41421i −3.16228 −2.00000 0 4.47214i −1.58114 + 2.12132i 2.82843i 7.00000 0
251.2 1.41421i 3.16228 −2.00000 0 4.47214i 1.58114 + 2.12132i 2.82843i 7.00000 0
251.3 1.41421i −3.16228 −2.00000 0 4.47214i −1.58114 2.12132i 2.82843i 7.00000 0
251.4 1.41421i 3.16228 −2.00000 0 4.47214i 1.58114 2.12132i 2.82843i 7.00000 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
28.d even 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.g.d 4
4.b odd 2 1 inner 700.2.g.d 4
5.b even 2 1 inner 700.2.g.d 4
5.c odd 4 2 140.2.c.a 4
7.b odd 2 1 inner 700.2.g.d 4
20.d odd 2 1 CM 700.2.g.d 4
20.e even 4 2 140.2.c.a 4
28.d even 2 1 inner 700.2.g.d 4
35.c odd 2 1 inner 700.2.g.d 4
35.f even 4 2 140.2.c.a 4
35.k even 12 4 980.2.s.b 8
35.l odd 12 4 980.2.s.b 8
40.i odd 4 2 2240.2.e.a 4
40.k even 4 2 2240.2.e.a 4
140.c even 2 1 inner 700.2.g.d 4
140.j odd 4 2 140.2.c.a 4
140.w even 12 4 980.2.s.b 8
140.x odd 12 4 980.2.s.b 8
280.s even 4 2 2240.2.e.a 4
280.y odd 4 2 2240.2.e.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.c.a 4 5.c odd 4 2
140.2.c.a 4 20.e even 4 2
140.2.c.a 4 35.f even 4 2
140.2.c.a 4 140.j odd 4 2
700.2.g.d 4 1.a even 1 1 trivial
700.2.g.d 4 4.b odd 2 1 inner
700.2.g.d 4 5.b even 2 1 inner
700.2.g.d 4 7.b odd 2 1 inner
700.2.g.d 4 20.d odd 2 1 CM
700.2.g.d 4 28.d even 2 1 inner
700.2.g.d 4 35.c odd 2 1 inner
700.2.g.d 4 140.c even 2 1 inner
980.2.s.b 8 35.k even 12 4
980.2.s.b 8 35.l odd 12 4
980.2.s.b 8 140.w even 12 4
980.2.s.b 8 140.x odd 12 4
2240.2.e.a 4 40.i odd 4 2
2240.2.e.a 4 40.k even 4 2
2240.2.e.a 4 280.s even 4 2
2240.2.e.a 4 280.y odd 4 2

Hecke kernels

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

\( T_{3}^{2} - 10 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display
\( T_{37} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - 10)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 4T^{2} + 49 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$29$ \( (T + 6)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 162)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 90)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 180)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 90)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 320)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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