Properties

Label 900.2.e.b
Level $900$
Weight $2$
Character orbit 900.e
Analytic conductor $7.187$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,2,Mod(251,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("900.251");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.e (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: \(7.18653618192\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - 2 q^{4} - 2 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - 2 q^{4} - 2 \beta q^{8} + 4 q^{13} + 4 q^{16} + 5 \beta q^{17} + 4 \beta q^{26} + 7 \beta q^{29} + 4 \beta q^{32} - 10 q^{34} - 2 q^{37} + \beta q^{41} + 7 q^{49} - 8 q^{52} + 5 \beta q^{53} - 14 q^{58} - 10 q^{61} - 8 q^{64} - 10 \beta q^{68} + 16 q^{73} - 2 \beta q^{74} - 2 q^{82} + 13 \beta q^{89} - 8 q^{97} + 7 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{4} + 8 q^{13} + 8 q^{16} - 20 q^{34} - 4 q^{37} + 14 q^{49} - 16 q^{52} - 28 q^{58} - 20 q^{61} - 16 q^{64} + 32 q^{73} - 4 q^{82} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
1.41421i
1.41421i
1.41421i 0 −2.00000 0 0 0 2.82843i 0 0
251.2 1.41421i 0 −2.00000 0 0 0 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
3.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.e.b 2
3.b odd 2 1 inner 900.2.e.b 2
4.b odd 2 1 CM 900.2.e.b 2
5.b even 2 1 36.2.b.a 2
5.c odd 4 2 900.2.h.a 4
12.b even 2 1 inner 900.2.e.b 2
15.d odd 2 1 36.2.b.a 2
15.e even 4 2 900.2.h.a 4
20.d odd 2 1 36.2.b.a 2
20.e even 4 2 900.2.h.a 4
35.c odd 2 1 1764.2.e.b 2
40.e odd 2 1 576.2.c.b 2
40.f even 2 1 576.2.c.b 2
45.h odd 6 2 324.2.h.c 4
45.j even 6 2 324.2.h.c 4
60.h even 2 1 36.2.b.a 2
60.l odd 4 2 900.2.h.a 4
80.k odd 4 2 2304.2.f.d 4
80.q even 4 2 2304.2.f.d 4
105.g even 2 1 1764.2.e.b 2
120.i odd 2 1 576.2.c.b 2
120.m even 2 1 576.2.c.b 2
140.c even 2 1 1764.2.e.b 2
180.n even 6 2 324.2.h.c 4
180.p odd 6 2 324.2.h.c 4
240.t even 4 2 2304.2.f.d 4
240.bm odd 4 2 2304.2.f.d 4
420.o odd 2 1 1764.2.e.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.b.a 2 5.b even 2 1
36.2.b.a 2 15.d odd 2 1
36.2.b.a 2 20.d odd 2 1
36.2.b.a 2 60.h even 2 1
324.2.h.c 4 45.h odd 6 2
324.2.h.c 4 45.j even 6 2
324.2.h.c 4 180.n even 6 2
324.2.h.c 4 180.p odd 6 2
576.2.c.b 2 40.e odd 2 1
576.2.c.b 2 40.f even 2 1
576.2.c.b 2 120.i odd 2 1
576.2.c.b 2 120.m even 2 1
900.2.e.b 2 1.a even 1 1 trivial
900.2.e.b 2 3.b odd 2 1 inner
900.2.e.b 2 4.b odd 2 1 CM
900.2.e.b 2 12.b even 2 1 inner
900.2.h.a 4 5.c odd 4 2
900.2.h.a 4 15.e even 4 2
900.2.h.a 4 20.e even 4 2
900.2.h.a 4 60.l odd 4 2
1764.2.e.b 2 35.c odd 2 1
1764.2.e.b 2 105.g even 2 1
1764.2.e.b 2 140.c even 2 1
1764.2.e.b 2 420.o odd 2 1
2304.2.f.d 4 80.k odd 4 2
2304.2.f.d 4 80.q even 4 2
2304.2.f.d 4 240.t even 4 2
2304.2.f.d 4 240.bm odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(900, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{13} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T - 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 50 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 98 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 2 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 50 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 16)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 338 \) Copy content Toggle raw display
$97$ \( (T + 8)^{2} \) Copy content Toggle raw display
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