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Properties

Label 1575.2.d.e
Level $1575$
Weight $2$
Character orbit 1575.d
Analytic conductor $12.576$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1575,2,Mod(1324,1575)]
 
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([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1575.1324");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.d (of order \(2\), degree \(1\), not 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: \(\Q(i, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} - 3) q^{4} + \beta_{2} q^{7} + (4 \beta_{2} - \beta_1) q^{8} - \beta_{3} q^{11} + (3 \beta_{2} + \beta_1) q^{13} + ( - \beta_{3} + 1) q^{14} + ( - 3 \beta_{3} + 3) q^{16}+ \cdots - \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{4} - 2 q^{11} + 2 q^{14} + 6 q^{16} + 12 q^{19} - 12 q^{26} + 2 q^{29} + 12 q^{34} - 4 q^{41} - 12 q^{44} - 32 q^{46} - 4 q^{49} - 18 q^{56} - 16 q^{59} + 12 q^{61} - 14 q^{64} - 32 q^{71} - 12 q^{74}+ \cdots - 56 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 5\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{2} - 5\beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1324.1
− 2.56155i
− 1.56155i
1.56155i
2.56155i
− 2.56155i 0 −4.56155 0 0 1.00000i 6.56155i 0 0
1324.2 − 1.56155i 0 −0.438447 0 0 − 1.00000i − 2.43845i 0 0
1324.3 1.56155i 0 −0.438447 0 0 1.00000i 2.43845i 0 0
1324.4 2.56155i 0 −4.56155 0 0 − 1.00000i − 6.56155i 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
5.b 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.d.e 4
3.b odd 2 1 175.2.b.b 4
5.b even 2 1 inner 1575.2.d.e 4
5.c odd 4 1 315.2.a.e 2
5.c odd 4 1 1575.2.a.p 2
12.b even 2 1 2800.2.g.t 4
15.d odd 2 1 175.2.b.b 4
15.e even 4 1 35.2.a.b ✓ 2
15.e even 4 1 175.2.a.f 2
20.e even 4 1 5040.2.a.bt 2
21.c even 2 1 1225.2.b.f 4
35.f even 4 1 2205.2.a.x 2
60.h even 2 1 2800.2.g.t 4
60.l odd 4 1 560.2.a.i 2
60.l odd 4 1 2800.2.a.bi 2
105.g even 2 1 1225.2.b.f 4
105.k odd 4 1 245.2.a.d 2
105.k odd 4 1 1225.2.a.s 2
105.w odd 12 2 245.2.e.h 4
105.x even 12 2 245.2.e.i 4
120.q odd 4 1 2240.2.a.bd 2
120.w even 4 1 2240.2.a.bh 2
165.l odd 4 1 4235.2.a.m 2
195.s even 4 1 5915.2.a.l 2
420.w even 4 1 3920.2.a.bs 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.b ✓ 2 15.e even 4 1
175.2.a.f 2 15.e even 4 1
175.2.b.b 4 3.b odd 2 1
175.2.b.b 4 15.d odd 2 1
245.2.a.d 2 105.k odd 4 1
245.2.e.h 4 105.w odd 12 2
245.2.e.i 4 105.x even 12 2
315.2.a.e 2 5.c odd 4 1
560.2.a.i 2 60.l odd 4 1
1225.2.a.s 2 105.k odd 4 1
1225.2.b.f 4 21.c even 2 1
1225.2.b.f 4 105.g even 2 1
1575.2.a.p 2 5.c odd 4 1
1575.2.d.e 4 1.a even 1 1 trivial
1575.2.d.e 4 5.b even 2 1 inner
2205.2.a.x 2 35.f even 4 1
2240.2.a.bd 2 120.q odd 4 1
2240.2.a.bh 2 120.w even 4 1
2800.2.a.bi 2 60.l odd 4 1
2800.2.g.t 4 12.b even 2 1
2800.2.g.t 4 60.h even 2 1
3920.2.a.bs 2 420.w even 4 1
4235.2.a.m 2 165.l odd 4 1
5040.2.a.bt 2 20.e even 4 1
5915.2.a.l 2 195.s even 4 1

Hecke kernels

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

\( T_{2}^{4} + 9T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} + T_{11} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 9T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 21T^{2} + 4 \) Copy content Toggle raw display
$17$ \( T^{4} + 21T^{2} + 4 \) Copy content Toggle raw display
$19$ \( (T^{2} - 6 T - 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 36T^{2} + 256 \) Copy content Toggle raw display
$29$ \( (T^{2} - T - 38)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 2 T - 16)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 84T^{2} + 64 \) Copy content Toggle raw display
$47$ \( T^{4} + 89T^{2} + 1024 \) Copy content Toggle raw display
$53$ \( T^{4} + 36T^{2} + 256 \) Copy content Toggle raw display
$59$ \( (T + 4)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 6 T - 144)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 144T^{2} + 4096 \) Copy content Toggle raw display
$71$ \( (T + 8)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 168T^{2} + 2704 \) Copy content Toggle raw display
$79$ \( (T^{2} - 9 T + 16)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 6 T - 8)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 253T^{2} + 7396 \) Copy content Toggle raw display
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