Properties

Label 2175.1.h.b
Level $2175$
Weight $1$
Character orbit 2175.h
Self dual yes
Analytic conductor $1.085$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -87
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2175,1,Mod(1826,2175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2175.1826");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2175.h (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: yes
Analytic conductor: \(1.08546640248\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 87)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.87.1
Artin image: $D_6$
Artin field: Galois closure of 6.2.946125.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + q^{2} - q^{3} - q^{6} + q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} - q^{6} + q^{7} - q^{8} + q^{9} - q^{11} + q^{13} + q^{14} - q^{16} + q^{17} + q^{18} - q^{21} - q^{22} + q^{24} + q^{26} - q^{27} + q^{29} + q^{33} + q^{34} - q^{39} + 2 q^{41} - q^{42} + q^{47} + q^{48} - q^{51} - q^{54} - q^{56} + q^{58} + q^{63} + q^{64} + q^{66} + q^{67} - q^{72} - q^{77} - q^{78} + q^{81} + 2 q^{82} - q^{87} + q^{88} - q^{89} + q^{91} + q^{94} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1451\) \(2002\)
\(\chi(n)\) \(1\) \(1\) \(0\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1826.1
0
1.00000 −1.00000 0 0 −1.00000 1.00000 −1.00000 1.00000 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
87.d odd 2 1 CM by \(\Q(\sqrt{-87}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2175.1.h.b 1
3.b odd 2 1 2175.1.h.a 1
5.b even 2 1 87.1.d.a 1
5.c odd 4 2 2175.1.b.a 2
15.d odd 2 1 87.1.d.b yes 1
15.e even 4 2 2175.1.b.b 2
20.d odd 2 1 1392.1.i.a 1
29.b even 2 1 2175.1.h.a 1
45.h odd 6 2 2349.1.h.a 2
45.j even 6 2 2349.1.h.b 2
60.h even 2 1 1392.1.i.b 1
87.d odd 2 1 CM 2175.1.h.b 1
145.d even 2 1 87.1.d.b yes 1
145.f odd 4 2 2523.1.b.b 2
145.h odd 4 2 2175.1.b.b 2
145.l even 14 6 2523.1.h.a 6
145.n even 14 6 2523.1.h.b 6
145.s odd 28 12 2523.1.j.b 12
435.b odd 2 1 87.1.d.a 1
435.l even 4 2 2523.1.b.b 2
435.p even 4 2 2175.1.b.a 2
435.w odd 14 6 2523.1.h.a 6
435.bb odd 14 6 2523.1.h.b 6
435.bk even 28 12 2523.1.j.b 12
580.e odd 2 1 1392.1.i.b 1
1305.w even 6 2 2349.1.h.a 2
1305.ba odd 6 2 2349.1.h.b 2
1740.k even 2 1 1392.1.i.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.1.d.a 1 5.b even 2 1
87.1.d.a 1 435.b odd 2 1
87.1.d.b yes 1 15.d odd 2 1
87.1.d.b yes 1 145.d even 2 1
1392.1.i.a 1 20.d odd 2 1
1392.1.i.a 1 1740.k even 2 1
1392.1.i.b 1 60.h even 2 1
1392.1.i.b 1 580.e odd 2 1
2175.1.b.a 2 5.c odd 4 2
2175.1.b.a 2 435.p even 4 2
2175.1.b.b 2 15.e even 4 2
2175.1.b.b 2 145.h odd 4 2
2175.1.h.a 1 3.b odd 2 1
2175.1.h.a 1 29.b even 2 1
2175.1.h.b 1 1.a even 1 1 trivial
2175.1.h.b 1 87.d odd 2 1 CM
2349.1.h.a 2 45.h odd 6 2
2349.1.h.a 2 1305.w even 6 2
2349.1.h.b 2 45.j even 6 2
2349.1.h.b 2 1305.ba odd 6 2
2523.1.b.b 2 145.f odd 4 2
2523.1.b.b 2 435.l even 4 2
2523.1.h.a 6 145.l even 14 6
2523.1.h.a 6 435.w odd 14 6
2523.1.h.b 6 145.n even 14 6
2523.1.h.b 6 435.bb odd 14 6
2523.1.j.b 12 145.s odd 28 12
2523.1.j.b 12 435.bk even 28 12

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2175, [\chi])\):

\( T_{2} - 1 \) Copy content Toggle raw display
\( T_{7} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T - 1 \) Copy content Toggle raw display
$17$ \( T - 1 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 1 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T - 2 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T - 1 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T - 1 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T + 1 \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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