Properties

Label 1800.2.d.f
Level $1800$
Weight $2$
Character orbit 1800.d
Analytic conductor $14.373$
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] = [1800,2,Mod(1549,1800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("1800.1549");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1800.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: \(14.3730723638\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 200)
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.

\(f(q)\) \(=\) \( q + ( - i + 1) q^{2} - 2 i q^{4} + 2 i q^{7} + ( - 2 i - 2) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - i + 1) q^{2} - 2 i q^{4} + 2 i q^{7} + ( - 2 i - 2) q^{8} - 5 i q^{11} - 6 q^{13} + (2 i + 2) q^{14} - 4 q^{16} + 3 i q^{17} + i q^{19} + ( - 5 i - 5) q^{22} - 4 i q^{23} + (6 i - 6) q^{26} + 4 q^{28} - 6 i q^{29} - 8 q^{31} + (4 i - 4) q^{32} + (3 i + 3) q^{34} - 2 q^{37} + (i + 1) q^{38} - 7 q^{41} + 4 q^{43} - 10 q^{44} + ( - 4 i - 4) q^{46} - 2 i q^{47} + 3 q^{49} + 12 i q^{52} - 4 q^{53} + ( - 4 i + 4) q^{56} + ( - 6 i - 6) q^{58} + 4 i q^{59} - 10 i q^{61} + (8 i - 8) q^{62} + 8 i q^{64} + 3 q^{67} + 6 q^{68} - 2 q^{71} - i q^{73} + (2 i - 2) q^{74} + 2 q^{76} + 10 q^{77} - 10 q^{79} + (7 i - 7) q^{82} - 9 q^{83} + ( - 4 i + 4) q^{86} + (10 i - 10) q^{88} + 5 q^{89} - 12 i q^{91} - 8 q^{92} + ( - 2 i - 2) q^{94} + 2 i q^{97} + ( - 3 i + 3) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 4 q^{8} - 12 q^{13} + 4 q^{14} - 8 q^{16} - 10 q^{22} - 12 q^{26} + 8 q^{28} - 16 q^{31} - 8 q^{32} + 6 q^{34} - 4 q^{37} + 2 q^{38} - 14 q^{41} + 8 q^{43} - 20 q^{44} - 8 q^{46} + 6 q^{49} - 8 q^{53} + 8 q^{56} - 12 q^{58} - 16 q^{62} + 6 q^{67} + 12 q^{68} - 4 q^{71} - 4 q^{74} + 4 q^{76} + 20 q^{77} - 20 q^{79} - 14 q^{82} - 18 q^{83} + 8 q^{86} - 20 q^{88} + 10 q^{89} - 16 q^{92} - 4 q^{94} + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1549.1
1.00000i
1.00000i
1.00000 1.00000i 0 2.00000i 0 0 2.00000i −2.00000 2.00000i 0 0
1549.2 1.00000 + 1.00000i 0 2.00000i 0 0 2.00000i −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
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1800.2.d.f 2
3.b odd 2 1 200.2.f.a 2
4.b odd 2 1 7200.2.d.c 2
5.b even 2 1 1800.2.d.d 2
5.c odd 4 1 1800.2.k.b 2
5.c odd 4 1 1800.2.k.h 2
8.b even 2 1 1800.2.d.d 2
8.d odd 2 1 7200.2.d.j 2
12.b even 2 1 800.2.f.b 2
15.d odd 2 1 200.2.f.b 2
15.e even 4 1 200.2.d.a 2
15.e even 4 1 200.2.d.d yes 2
20.d odd 2 1 7200.2.d.j 2
20.e even 4 1 7200.2.k.e 2
20.e even 4 1 7200.2.k.g 2
24.f even 2 1 800.2.f.a 2
24.h odd 2 1 200.2.f.b 2
40.e odd 2 1 7200.2.d.c 2
40.f even 2 1 inner 1800.2.d.f 2
40.i odd 4 1 1800.2.k.b 2
40.i odd 4 1 1800.2.k.h 2
40.k even 4 1 7200.2.k.e 2
40.k even 4 1 7200.2.k.g 2
60.h even 2 1 800.2.f.a 2
60.l odd 4 1 800.2.d.b 2
60.l odd 4 1 800.2.d.c 2
120.i odd 2 1 200.2.f.a 2
120.m even 2 1 800.2.f.b 2
120.q odd 4 1 800.2.d.b 2
120.q odd 4 1 800.2.d.c 2
120.w even 4 1 200.2.d.a 2
120.w even 4 1 200.2.d.d yes 2
240.z odd 4 1 6400.2.a.e 1
240.z odd 4 1 6400.2.a.f 1
240.bb even 4 1 6400.2.a.s 1
240.bb even 4 1 6400.2.a.t 1
240.bd odd 4 1 6400.2.a.r 1
240.bd odd 4 1 6400.2.a.u 1
240.bf even 4 1 6400.2.a.d 1
240.bf even 4 1 6400.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.d.a 2 15.e even 4 1
200.2.d.a 2 120.w even 4 1
200.2.d.d yes 2 15.e even 4 1
200.2.d.d yes 2 120.w even 4 1
200.2.f.a 2 3.b odd 2 1
200.2.f.a 2 120.i odd 2 1
200.2.f.b 2 15.d odd 2 1
200.2.f.b 2 24.h odd 2 1
800.2.d.b 2 60.l odd 4 1
800.2.d.b 2 120.q odd 4 1
800.2.d.c 2 60.l odd 4 1
800.2.d.c 2 120.q odd 4 1
800.2.f.a 2 24.f even 2 1
800.2.f.a 2 60.h even 2 1
800.2.f.b 2 12.b even 2 1
800.2.f.b 2 120.m even 2 1
1800.2.d.d 2 5.b even 2 1
1800.2.d.d 2 8.b even 2 1
1800.2.d.f 2 1.a even 1 1 trivial
1800.2.d.f 2 40.f even 2 1 inner
1800.2.k.b 2 5.c odd 4 1
1800.2.k.b 2 40.i odd 4 1
1800.2.k.h 2 5.c odd 4 1
1800.2.k.h 2 40.i odd 4 1
6400.2.a.d 1 240.bf even 4 1
6400.2.a.e 1 240.z odd 4 1
6400.2.a.f 1 240.z odd 4 1
6400.2.a.g 1 240.bf even 4 1
6400.2.a.r 1 240.bd odd 4 1
6400.2.a.s 1 240.bb even 4 1
6400.2.a.t 1 240.bb even 4 1
6400.2.a.u 1 240.bd odd 4 1
7200.2.d.c 2 4.b odd 2 1
7200.2.d.c 2 40.e odd 2 1
7200.2.d.j 2 8.d odd 2 1
7200.2.d.j 2 20.d odd 2 1
7200.2.k.e 2 20.e even 4 1
7200.2.k.e 2 40.k even 4 1
7200.2.k.g 2 20.e even 4 1
7200.2.k.g 2 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1800, [\chi])\):

\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 25 \) Copy content Toggle raw display
\( T_{13} + 6 \) Copy content Toggle raw display
\( T_{37} + 2 \) Copy content Toggle raw display
\( T_{41} + 7 \) Copy content Toggle raw display
\( T_{53} + 4 \) 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
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( T^{2} + 25 \) Copy content Toggle raw display
$13$ \( (T + 6)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 9 \) Copy content Toggle raw display
$19$ \( T^{2} + 1 \) Copy content Toggle raw display
$23$ \( T^{2} + 16 \) Copy content Toggle raw display
$29$ \( T^{2} + 36 \) Copy content Toggle raw display
$31$ \( (T + 8)^{2} \) Copy content Toggle raw display
$37$ \( (T + 2)^{2} \) Copy content Toggle raw display
$41$ \( (T + 7)^{2} \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 4 \) Copy content Toggle raw display
$53$ \( (T + 4)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 16 \) Copy content Toggle raw display
$61$ \( T^{2} + 100 \) Copy content Toggle raw display
$67$ \( (T - 3)^{2} \) Copy content Toggle raw display
$71$ \( (T + 2)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 1 \) Copy content Toggle raw display
$79$ \( (T + 10)^{2} \) Copy content Toggle raw display
$83$ \( (T + 9)^{2} \) Copy content Toggle raw display
$89$ \( (T - 5)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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