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Properties

Label 700.2.g.b
Level $700$
Weight $2$
Character orbit 700.g
Analytic conductor $5.590$
Analytic rank $0$
Dimension $4$
CM no
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(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: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} - \beta_{3} q^{3} - 2 \beta_1 q^{4} + (\beta_{3} - \beta_{2}) q^{6} + ( - \beta_{3} + \beta_1) q^{7} + (2 \beta_1 + 2) q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} - \beta_{3} q^{3} - 2 \beta_1 q^{4} + (\beta_{3} - \beta_{2}) q^{6} + ( - \beta_{3} + \beta_1) q^{7} + (2 \beta_1 + 2) q^{8} + 3 q^{9} + 5 \beta_1 q^{11} + 2 \beta_{2} q^{12} - \beta_{2} q^{13} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{14} - 4 q^{16} + 2 \beta_{2} q^{17} + (3 \beta_1 - 3) q^{18} + ( - \beta_{2} + 6) q^{21} + ( - 5 \beta_1 - 5) q^{22} - \beta_1 q^{23} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{24} + (\beta_{3} + \beta_{2}) q^{26} + (2 \beta_{2} + 2) q^{28} - 5 q^{29} - 3 \beta_{3} q^{31} + ( - 4 \beta_1 + 4) q^{32} - 5 \beta_{2} q^{33} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{34} - 6 \beta_1 q^{36} + 3 q^{37} + 6 \beta_1 q^{39} - 5 \beta_{2} q^{41} + (\beta_{3} + \beta_{2} + 6 \beta_1 - 6) q^{42} - 11 \beta_1 q^{43} + 10 q^{44} + (\beta_1 + 1) q^{46} - 2 \beta_{3} q^{47} + 4 \beta_{3} q^{48} + ( - 2 \beta_{2} + 5) q^{49} - 12 \beta_1 q^{51} - 2 \beta_{3} q^{52} + 4 q^{53} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots - 2) q^{56}+ \cdots + 15 \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 8 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 8 q^{8} + 12 q^{9} - 4 q^{14} - 16 q^{16} - 12 q^{18} + 24 q^{21} - 20 q^{22} + 8 q^{28} - 20 q^{29} + 16 q^{32} + 12 q^{37} - 24 q^{42} + 40 q^{44} + 4 q^{46} + 20 q^{49} + 16 q^{53} - 8 q^{56} + 20 q^{58} + 24 q^{72} - 12 q^{74} - 20 q^{77} - 24 q^{78} - 36 q^{81} + 44 q^{86} - 40 q^{88} - 8 q^{92} + 72 q^{93} - 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

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

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

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

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

Character values

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

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

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
1.22474 − 1.22474i
−1.22474 + 1.22474i
1.22474 + 1.22474i
−1.22474 − 1.22474i
−1.00000 − 1.00000i −2.44949 2.00000i 0 2.44949 + 2.44949i −2.44949 − 1.00000i 2.00000 − 2.00000i 3.00000 0
251.2 −1.00000 − 1.00000i 2.44949 2.00000i 0 −2.44949 − 2.44949i 2.44949 − 1.00000i 2.00000 − 2.00000i 3.00000 0
251.3 −1.00000 + 1.00000i −2.44949 − 2.00000i 0 2.44949 − 2.44949i −2.44949 + 1.00000i 2.00000 + 2.00000i 3.00000 0
251.4 −1.00000 + 1.00000i 2.44949 − 2.00000i 0 −2.44949 + 2.44949i 2.44949 + 1.00000i 2.00000 + 2.00000i 3.00000 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
4.b odd 2 1 inner
7.b odd 2 1 inner
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.b ✓ 4
4.b odd 2 1 inner 700.2.g.b ✓ 4
5.b even 2 1 700.2.g.h yes 4
5.c odd 4 1 700.2.c.a 4
5.c odd 4 1 700.2.c.g 4
7.b odd 2 1 inner 700.2.g.b ✓ 4
20.d odd 2 1 700.2.g.h yes 4
20.e even 4 1 700.2.c.a 4
20.e even 4 1 700.2.c.g 4
28.d even 2 1 inner 700.2.g.b ✓ 4
35.c odd 2 1 700.2.g.h yes 4
35.f even 4 1 700.2.c.a 4
35.f even 4 1 700.2.c.g 4
140.c even 2 1 700.2.g.h yes 4
140.j odd 4 1 700.2.c.a 4
140.j odd 4 1 700.2.c.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.c.a 4 5.c odd 4 1
700.2.c.a 4 20.e even 4 1
700.2.c.a 4 35.f even 4 1
700.2.c.a 4 140.j odd 4 1
700.2.c.g 4 5.c odd 4 1
700.2.c.g 4 20.e even 4 1
700.2.c.g 4 35.f even 4 1
700.2.c.g 4 140.j odd 4 1
700.2.g.b ✓ 4 1.a even 1 1 trivial
700.2.g.b ✓ 4 4.b odd 2 1 inner
700.2.g.b ✓ 4 7.b odd 2 1 inner
700.2.g.b ✓ 4 28.d even 2 1 inner
700.2.g.h yes 4 5.b even 2 1
700.2.g.h yes 4 20.d odd 2 1
700.2.g.h yes 4 35.c odd 2 1
700.2.g.h yes 4 140.c even 2 1

Hecke kernels

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

\( T_{3}^{2} - 6 \) Copy content Toggle raw display
\( T_{11}^{2} + 25 \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display
\( T_{37} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 10T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T + 5)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 54)^{2} \) Copy content Toggle raw display
$37$ \( (T - 3)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 150)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 121)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$53$ \( (T - 4)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 150)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 150)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 54)^{2} \) Copy content Toggle raw display
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