Properties

Label 1575.2.d.c
Level $1575$
Weight $2$
Character orbit 1575.d
Analytic conductor $12.576$
Analytic rank $0$
Dimension $2$
CM no
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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) 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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{4} - i q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{4} - i q^{7} + 3 q^{11} + 5 i q^{13} + 4 q^{16} + 3 i q^{17} - 2 q^{19} + 6 i q^{23} - 2 i q^{28} + 3 q^{29} - 4 q^{31} - 2 i q^{37} + 12 q^{41} - 10 i q^{43} + 6 q^{44} + 9 i q^{47} - q^{49} + 10 i q^{52} - 12 i q^{53} + 8 q^{61} + 8 q^{64} + 4 i q^{67} + 6 i q^{68} + 2 i q^{73} - 4 q^{76} - 3 i q^{77} + q^{79} - 12 i q^{83} - 12 q^{89} + 5 q^{91} + 12 i q^{92} + i q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{4} + 6 q^{11} + 8 q^{16} - 4 q^{19} + 6 q^{29} - 8 q^{31} + 24 q^{41} + 12 q^{44} - 2 q^{49} + 16 q^{61} + 16 q^{64} - 8 q^{76} + 2 q^{79} - 24 q^{89} + 10 q^{91}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1324.1
1.00000i
1.00000i
0 0 2.00000 0 0 1.00000i 0 0 0
1324.2 0 0 2.00000 0 0 1.00000i 0 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.c 2
3.b odd 2 1 175.2.b.a 2
5.b even 2 1 inner 1575.2.d.c 2
5.c odd 4 1 315.2.a.b 1
5.c odd 4 1 1575.2.a.f 1
12.b even 2 1 2800.2.g.l 2
15.d odd 2 1 175.2.b.a 2
15.e even 4 1 35.2.a.a 1
15.e even 4 1 175.2.a.b 1
20.e even 4 1 5040.2.a.v 1
21.c even 2 1 1225.2.b.d 2
35.f even 4 1 2205.2.a.e 1
60.h even 2 1 2800.2.g.l 2
60.l odd 4 1 560.2.a.b 1
60.l odd 4 1 2800.2.a.z 1
105.g even 2 1 1225.2.b.d 2
105.k odd 4 1 245.2.a.c 1
105.k odd 4 1 1225.2.a.e 1
105.w odd 12 2 245.2.e.b 2
105.x even 12 2 245.2.e.a 2
120.q odd 4 1 2240.2.a.u 1
120.w even 4 1 2240.2.a.k 1
165.l odd 4 1 4235.2.a.c 1
195.s even 4 1 5915.2.a.f 1
420.w even 4 1 3920.2.a.ba 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.a 1 15.e even 4 1
175.2.a.b 1 15.e even 4 1
175.2.b.a 2 3.b odd 2 1
175.2.b.a 2 15.d odd 2 1
245.2.a.c 1 105.k odd 4 1
245.2.e.a 2 105.x even 12 2
245.2.e.b 2 105.w odd 12 2
315.2.a.b 1 5.c odd 4 1
560.2.a.b 1 60.l odd 4 1
1225.2.a.e 1 105.k odd 4 1
1225.2.b.d 2 21.c even 2 1
1225.2.b.d 2 105.g even 2 1
1575.2.a.f 1 5.c odd 4 1
1575.2.d.c 2 1.a even 1 1 trivial
1575.2.d.c 2 5.b even 2 1 inner
2205.2.a.e 1 35.f even 4 1
2240.2.a.k 1 120.w even 4 1
2240.2.a.u 1 120.q odd 4 1
2800.2.a.z 1 60.l odd 4 1
2800.2.g.l 2 12.b even 2 1
2800.2.g.l 2 60.h even 2 1
3920.2.a.ba 1 420.w even 4 1
4235.2.a.c 1 165.l odd 4 1
5040.2.a.v 1 20.e even 4 1
5915.2.a.f 1 195.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 \(S_{2}^{\mathrm{new}}(1575, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{11} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 25 \) Copy content Toggle raw display
$17$ \( T^{2} + 9 \) Copy content Toggle raw display
$19$ \( (T + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 36 \) Copy content Toggle raw display
$29$ \( (T - 3)^{2} \) Copy content Toggle raw display
$31$ \( (T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T - 12)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 100 \) Copy content Toggle raw display
$47$ \( T^{2} + 81 \) Copy content Toggle raw display
$53$ \( T^{2} + 144 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4 \) Copy content Toggle raw display
$79$ \( (T - 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( (T + 12)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 1 \) Copy content Toggle raw display
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