Properties

Label 900.2.k.a
Level $900$
Weight $2$
Character orbit 900.k
Analytic conductor $7.187$
Analytic rank $1$
Dimension $2$
CM discriminant -20
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(307,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.307");
 
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.k (of order \(4\), degree \(2\), 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: \(1\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 100)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + (i - 1) q^{2} - 2 i q^{4} + (3 i - 3) q^{7} + (2 i + 2) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (i - 1) q^{2} - 2 i q^{4} + (3 i - 3) q^{7} + (2 i + 2) q^{8} - 6 i q^{14} - 4 q^{16} + ( - i - 1) q^{23} + (6 i + 6) q^{28} - 6 i q^{29} + ( - 4 i + 4) q^{32} - 12 q^{41} + ( - 9 i - 9) q^{43} + 2 q^{46} + (7 i - 7) q^{47} - 11 i q^{49} - 12 q^{56} + (6 i + 6) q^{58} - 8 q^{61} + 8 i q^{64} + (3 i - 3) q^{67} + ( - 12 i + 12) q^{82} + ( - 11 i - 11) q^{83} + 18 q^{86} - 6 i q^{89} + (2 i - 2) q^{92} - 14 i q^{94} + (11 i + 11) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 6 q^{7} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 6 q^{7} + 4 q^{8} - 8 q^{16} - 2 q^{23} + 12 q^{28} + 8 q^{32} - 24 q^{41} - 18 q^{43} + 4 q^{46} - 14 q^{47} - 24 q^{56} + 12 q^{58} - 16 q^{61} - 6 q^{67} + 24 q^{82} - 22 q^{83} + 36 q^{86} - 4 q^{92} + 22 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
1.00000i
1.00000i
−1.00000 + 1.00000i 0 2.00000i 0 0 −3.00000 + 3.00000i 2.00000 + 2.00000i 0 0
343.1 −1.00000 1.00000i 0 2.00000i 0 0 −3.00000 3.00000i 2.00000 2.00000i 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.k.a 2
3.b odd 2 1 100.2.e.c yes 2
4.b odd 2 1 900.2.k.e 2
5.b even 2 1 900.2.k.e 2
5.c odd 4 1 inner 900.2.k.a 2
5.c odd 4 1 900.2.k.e 2
12.b even 2 1 100.2.e.a 2
15.d odd 2 1 100.2.e.a 2
15.e even 4 1 100.2.e.a 2
15.e even 4 1 100.2.e.c yes 2
20.d odd 2 1 CM 900.2.k.a 2
20.e even 4 1 inner 900.2.k.a 2
20.e even 4 1 900.2.k.e 2
24.f even 2 1 1600.2.n.l 2
24.h odd 2 1 1600.2.n.b 2
60.h even 2 1 100.2.e.c yes 2
60.l odd 4 1 100.2.e.a 2
60.l odd 4 1 100.2.e.c yes 2
120.i odd 2 1 1600.2.n.l 2
120.m even 2 1 1600.2.n.b 2
120.q odd 4 1 1600.2.n.b 2
120.q odd 4 1 1600.2.n.l 2
120.w even 4 1 1600.2.n.b 2
120.w even 4 1 1600.2.n.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.2.e.a 2 12.b even 2 1
100.2.e.a 2 15.d odd 2 1
100.2.e.a 2 15.e even 4 1
100.2.e.a 2 60.l odd 4 1
100.2.e.c yes 2 3.b odd 2 1
100.2.e.c yes 2 15.e even 4 1
100.2.e.c yes 2 60.h even 2 1
100.2.e.c yes 2 60.l odd 4 1
900.2.k.a 2 1.a even 1 1 trivial
900.2.k.a 2 5.c odd 4 1 inner
900.2.k.a 2 20.d odd 2 1 CM
900.2.k.a 2 20.e even 4 1 inner
900.2.k.e 2 4.b odd 2 1
900.2.k.e 2 5.b even 2 1
900.2.k.e 2 5.c odd 4 1
900.2.k.e 2 20.e even 4 1
1600.2.n.b 2 24.h odd 2 1
1600.2.n.b 2 120.m even 2 1
1600.2.n.b 2 120.q odd 4 1
1600.2.n.b 2 120.w even 4 1
1600.2.n.l 2 24.f even 2 1
1600.2.n.l 2 120.i odd 2 1
1600.2.n.l 2 120.q odd 4 1
1600.2.n.l 2 120.w even 4 1

Hecke kernels

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

\( T_{7}^{2} + 6T_{7} + 18 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display
\( T_{23}^{2} + 2T_{23} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$29$ \( T^{2} + 36 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( (T + 12)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 18T + 162 \) Copy content Toggle raw display
$47$ \( T^{2} + 14T + 98 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 6T + 18 \) 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
$83$ \( T^{2} + 22T + 242 \) Copy content Toggle raw display
$89$ \( T^{2} + 36 \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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