Properties

Label 2175.1.b.b
Level $2175$
Weight $1$
Character orbit 2175.b
Analytic conductor $1.085$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -87
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2175,1,Mod(2174,2175)]
 
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, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2175.2174");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2175.b (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: \(1.08546640248\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
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: \(\Z[a_1, a_2]\)
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: $C_4\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - i q^{2} - i q^{3} - q^{6} + i q^{7} - i q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{2} - i q^{3} - q^{6} + i q^{7} - i q^{8} - q^{9} + q^{11} - i q^{13} + q^{14} - q^{16} - i q^{17} + i q^{18} + q^{21} - i q^{22} - q^{24} - q^{26} + i q^{27} + q^{29} - i q^{33} - q^{34} - q^{39} - 2 q^{41} - i q^{42} - i q^{47} + i q^{48} - q^{51} + q^{54} + q^{56} - i q^{58} - i q^{63} - q^{64} - q^{66} + i q^{67} + i q^{72} + i q^{77} + i q^{78} + q^{81} + 2 i q^{82} - i q^{87} - i q^{88} - q^{89} + q^{91} - q^{94} - q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{6} - 2 q^{9} + 2 q^{11} + 2 q^{14} - 2 q^{16} + 2 q^{21} - 2 q^{24} - 2 q^{26} + 2 q^{29} - 2 q^{34} - 2 q^{39} - 4 q^{41} - 2 q^{51} + 2 q^{54} + 2 q^{56} - 2 q^{64} - 2 q^{66} + 2 q^{81} - 2 q^{89} + 2 q^{91} - 2 q^{94} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2174.1
1.00000i
1.00000i
1.00000i 1.00000i 0 0 −1.00000 1.00000i 1.00000i −1.00000 0
2174.2 1.00000i 1.00000i 0 0 −1.00000 1.00000i 1.00000i −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 \(\Q(\sqrt{-87}) \)
5.b even 2 1 inner
435.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2175.1.b.b 2
3.b odd 2 1 2175.1.b.a 2
5.b even 2 1 inner 2175.1.b.b 2
5.c odd 4 1 87.1.d.b yes 1
5.c odd 4 1 2175.1.h.a 1
15.d odd 2 1 2175.1.b.a 2
15.e even 4 1 87.1.d.a 1
15.e even 4 1 2175.1.h.b 1
20.e even 4 1 1392.1.i.b 1
29.b even 2 1 2175.1.b.a 2
45.k odd 12 2 2349.1.h.a 2
45.l even 12 2 2349.1.h.b 2
60.l odd 4 1 1392.1.i.a 1
87.d odd 2 1 CM 2175.1.b.b 2
145.d even 2 1 2175.1.b.a 2
145.e even 4 1 2523.1.b.b 2
145.h odd 4 1 87.1.d.a 1
145.h odd 4 1 2175.1.h.b 1
145.j even 4 1 2523.1.b.b 2
145.o even 28 6 2523.1.j.b 12
145.p odd 28 6 2523.1.h.a 6
145.q odd 28 6 2523.1.h.b 6
145.t even 28 6 2523.1.j.b 12
435.b odd 2 1 inner 2175.1.b.b 2
435.i odd 4 1 2523.1.b.b 2
435.p even 4 1 87.1.d.b yes 1
435.p even 4 1 2175.1.h.a 1
435.t odd 4 1 2523.1.b.b 2
435.bc odd 28 6 2523.1.j.b 12
435.bg even 28 6 2523.1.h.a 6
435.bj even 28 6 2523.1.h.b 6
435.bn odd 28 6 2523.1.j.b 12
580.o even 4 1 1392.1.i.a 1
1305.bi even 12 2 2349.1.h.a 2
1305.bk odd 12 2 2349.1.h.b 2
1740.v odd 4 1 1392.1.i.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.1.d.a 1 15.e even 4 1
87.1.d.a 1 145.h odd 4 1
87.1.d.b yes 1 5.c odd 4 1
87.1.d.b yes 1 435.p even 4 1
1392.1.i.a 1 60.l odd 4 1
1392.1.i.a 1 580.o even 4 1
1392.1.i.b 1 20.e even 4 1
1392.1.i.b 1 1740.v odd 4 1
2175.1.b.a 2 3.b odd 2 1
2175.1.b.a 2 15.d odd 2 1
2175.1.b.a 2 29.b even 2 1
2175.1.b.a 2 145.d even 2 1
2175.1.b.b 2 1.a even 1 1 trivial
2175.1.b.b 2 5.b even 2 1 inner
2175.1.b.b 2 87.d odd 2 1 CM
2175.1.b.b 2 435.b odd 2 1 inner
2175.1.h.a 1 5.c odd 4 1
2175.1.h.a 1 435.p even 4 1
2175.1.h.b 1 15.e even 4 1
2175.1.h.b 1 145.h odd 4 1
2349.1.h.a 2 45.k odd 12 2
2349.1.h.a 2 1305.bi even 12 2
2349.1.h.b 2 45.l even 12 2
2349.1.h.b 2 1305.bk odd 12 2
2523.1.b.b 2 145.e even 4 1
2523.1.b.b 2 145.j even 4 1
2523.1.b.b 2 435.i odd 4 1
2523.1.b.b 2 435.t odd 4 1
2523.1.h.a 6 145.p odd 28 6
2523.1.h.a 6 435.bg even 28 6
2523.1.h.b 6 145.q odd 28 6
2523.1.h.b 6 435.bj even 28 6
2523.1.j.b 12 145.o even 28 6
2523.1.j.b 12 145.t even 28 6
2523.1.j.b 12 435.bc odd 28 6
2523.1.j.b 12 435.bn odd 28 6

Hecke kernels

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

\( T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{11} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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