Properties

Label 1575.2.b.d
Level $1575$
Weight $2$
Character orbit 1575.b
Analytic conductor $12.576$
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] = [1575,2,Mod(251,1575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1575, 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("1575.251");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.b (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: \(12.5764383184\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{2} q^{7} - 2 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{2} q^{7} - 2 \beta_1 q^{8} - \beta_1 q^{11} + ( - 2 \beta_{2} + 1) q^{13} + ( - \beta_{3} + \beta_1) q^{14} - 4 q^{16} + ( - 2 \beta_{3} + \beta_1) q^{17} + (2 \beta_{2} - 1) q^{19} - 2 q^{22} + 5 \beta_1 q^{23} + ( - 2 \beta_{3} + \beta_1) q^{26} - \beta_1 q^{29} + ( - 2 \beta_{2} + 1) q^{31} + (4 \beta_{2} - 2) q^{34} + 8 q^{37} + (2 \beta_{3} - \beta_1) q^{38} + 5 q^{43} + 10 q^{46} + ( - 2 \beta_{3} + \beta_1) q^{47} + (\beta_{2} - 7) q^{49} - 4 \beta_1 q^{53} + ( - 2 \beta_{3} + 2 \beta_1) q^{56} - 2 q^{58} + ( - 2 \beta_{3} + \beta_1) q^{59} + (2 \beta_{2} - 1) q^{61} + ( - 2 \beta_{3} + \beta_1) q^{62} - 8 q^{64} - 7 q^{67} + 2 \beta_1 q^{71} + ( - 4 \beta_{2} + 2) q^{73} - 8 \beta_1 q^{74} + ( - \beta_{3} + \beta_1) q^{77} + 14 q^{79} + (2 \beta_{3} - \beta_1) q^{83} - 5 \beta_1 q^{86} - 4 q^{88} + ( - 4 \beta_{3} + 2 \beta_1) q^{89} + (\beta_{2} - 14) q^{91} + (4 \beta_{2} - 2) q^{94} + (2 \beta_{2} - 1) q^{97} + (\beta_{3} + 6 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{7} - 16 q^{16} - 8 q^{22} + 32 q^{37} + 20 q^{43} + 40 q^{46} - 26 q^{49} - 8 q^{58} - 32 q^{64} - 28 q^{67} + 56 q^{79} - 16 q^{88} - 54 q^{91}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{2} - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 6\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{2} + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 + 0.707107i
−1.22474 + 0.707107i
−1.22474 0.707107i
1.22474 0.707107i
1.41421i 0 0 0 0 −0.500000 2.59808i 2.82843i 0 0
251.2 1.41421i 0 0 0 0 −0.500000 + 2.59808i 2.82843i 0 0
251.3 1.41421i 0 0 0 0 −0.500000 2.59808i 2.82843i 0 0
251.4 1.41421i 0 0 0 0 −0.500000 + 2.59808i 2.82843i 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
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.2.b.d 4
3.b odd 2 1 inner 1575.2.b.d 4
5.b even 2 1 1575.2.b.e yes 4
5.c odd 4 2 1575.2.g.b 8
7.b odd 2 1 inner 1575.2.b.d 4
15.d odd 2 1 1575.2.b.e yes 4
15.e even 4 2 1575.2.g.b 8
21.c even 2 1 inner 1575.2.b.d 4
35.c odd 2 1 1575.2.b.e yes 4
35.f even 4 2 1575.2.g.b 8
105.g even 2 1 1575.2.b.e yes 4
105.k odd 4 2 1575.2.g.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1575.2.b.d 4 1.a even 1 1 trivial
1575.2.b.d 4 3.b odd 2 1 inner
1575.2.b.d 4 7.b odd 2 1 inner
1575.2.b.d 4 21.c even 2 1 inner
1575.2.b.e yes 4 5.b even 2 1
1575.2.b.e yes 4 15.d odd 2 1
1575.2.b.e yes 4 35.c odd 2 1
1575.2.b.e yes 4 105.g even 2 1
1575.2.g.b 8 5.c odd 4 2
1575.2.g.b 8 15.e even 4 2
1575.2.g.b 8 35.f even 4 2
1575.2.g.b 8 105.k odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1575, [\chi])\):

\( T_{2}^{2} + 2 \) Copy content Toggle raw display
\( T_{37} - 8 \) Copy content Toggle raw display
\( T_{47}^{2} - 54 \) Copy content Toggle raw display
\( T_{67} + 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 54)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 50)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$37$ \( (T - 8)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T - 5)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 54)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 54)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$67$ \( (T + 7)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$79$ \( (T - 14)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 54)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 216)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
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