Properties

Label 2400.2.k.a
Level $2400$
Weight $2$
Character orbit 2400.k
Analytic conductor $19.164$
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] = [2400,2,Mod(1201,2400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2400.1201");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2400.k (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: \(19.1640964851\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
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.

\(f(q)\) \(=\) \( q + i q^{3} - 2 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + i q^{3} - 2 q^{7} - q^{9} - 4 i q^{13} + 2 q^{17} + 4 i q^{19} - 2 i q^{21} + 4 q^{23} - i q^{27} + 6 i q^{29} - 2 q^{31} + 8 i q^{37} + 4 q^{39} + 2 q^{41} + 4 i q^{43} - 12 q^{47} - 3 q^{49} + 2 i q^{51} + 6 i q^{53} - 4 q^{57} + 4 i q^{59} + 2 q^{63} + 12 i q^{67} + 4 i q^{69} - 12 q^{71} + 6 q^{73} - 10 q^{79} + q^{81} - 16 i q^{83} - 6 q^{87} - 10 q^{89} + 8 i q^{91} - 2 i q^{93} + 2 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{7} - 2 q^{9} + 4 q^{17} + 8 q^{23} - 4 q^{31} + 8 q^{39} + 4 q^{41} - 24 q^{47} - 6 q^{49} - 8 q^{57} + 4 q^{63} - 24 q^{71} + 12 q^{73} - 20 q^{79} + 2 q^{81} - 12 q^{87} - 20 q^{89} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1601\) \(1951\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1201.1
1.00000i
1.00000i
0 1.00000i 0 0 0 −2.00000 0 −1.00000 0
1201.2 0 1.00000i 0 0 0 −2.00000 0 −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
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2400.2.k.a 2
3.b odd 2 1 7200.2.k.d 2
4.b odd 2 1 600.2.k.b 2
5.b even 2 1 96.2.d.a 2
5.c odd 4 1 2400.2.d.b 2
5.c odd 4 1 2400.2.d.c 2
8.b even 2 1 inner 2400.2.k.a 2
8.d odd 2 1 600.2.k.b 2
12.b even 2 1 1800.2.k.a 2
15.d odd 2 1 288.2.d.b 2
15.e even 4 1 7200.2.d.d 2
15.e even 4 1 7200.2.d.g 2
20.d odd 2 1 24.2.d.a 2
20.e even 4 1 600.2.d.b 2
20.e even 4 1 600.2.d.c 2
24.f even 2 1 1800.2.k.a 2
24.h odd 2 1 7200.2.k.d 2
35.c odd 2 1 4704.2.c.a 2
40.e odd 2 1 24.2.d.a 2
40.f even 2 1 96.2.d.a 2
40.i odd 4 1 2400.2.d.b 2
40.i odd 4 1 2400.2.d.c 2
40.k even 4 1 600.2.d.b 2
40.k even 4 1 600.2.d.c 2
45.h odd 6 2 2592.2.r.g 4
45.j even 6 2 2592.2.r.f 4
60.h even 2 1 72.2.d.b 2
60.l odd 4 1 1800.2.d.b 2
60.l odd 4 1 1800.2.d.i 2
80.k odd 4 1 768.2.a.a 1
80.k odd 4 1 768.2.a.h 1
80.q even 4 1 768.2.a.d 1
80.q even 4 1 768.2.a.e 1
120.i odd 2 1 288.2.d.b 2
120.m even 2 1 72.2.d.b 2
120.q odd 4 1 1800.2.d.b 2
120.q odd 4 1 1800.2.d.i 2
120.w even 4 1 7200.2.d.d 2
120.w even 4 1 7200.2.d.g 2
140.c even 2 1 1176.2.c.a 2
180.n even 6 2 648.2.n.c 4
180.p odd 6 2 648.2.n.k 4
240.t even 4 1 2304.2.a.e 1
240.t even 4 1 2304.2.a.o 1
240.bm odd 4 1 2304.2.a.b 1
240.bm odd 4 1 2304.2.a.l 1
280.c odd 2 1 4704.2.c.a 2
280.n even 2 1 1176.2.c.a 2
360.z odd 6 2 648.2.n.k 4
360.bd even 6 2 648.2.n.c 4
360.bh odd 6 2 2592.2.r.g 4
360.bk even 6 2 2592.2.r.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.d.a 2 20.d odd 2 1
24.2.d.a 2 40.e odd 2 1
72.2.d.b 2 60.h even 2 1
72.2.d.b 2 120.m even 2 1
96.2.d.a 2 5.b even 2 1
96.2.d.a 2 40.f even 2 1
288.2.d.b 2 15.d odd 2 1
288.2.d.b 2 120.i odd 2 1
600.2.d.b 2 20.e even 4 1
600.2.d.b 2 40.k even 4 1
600.2.d.c 2 20.e even 4 1
600.2.d.c 2 40.k even 4 1
600.2.k.b 2 4.b odd 2 1
600.2.k.b 2 8.d odd 2 1
648.2.n.c 4 180.n even 6 2
648.2.n.c 4 360.bd even 6 2
648.2.n.k 4 180.p odd 6 2
648.2.n.k 4 360.z odd 6 2
768.2.a.a 1 80.k odd 4 1
768.2.a.d 1 80.q even 4 1
768.2.a.e 1 80.q even 4 1
768.2.a.h 1 80.k odd 4 1
1176.2.c.a 2 140.c even 2 1
1176.2.c.a 2 280.n even 2 1
1800.2.d.b 2 60.l odd 4 1
1800.2.d.b 2 120.q odd 4 1
1800.2.d.i 2 60.l odd 4 1
1800.2.d.i 2 120.q odd 4 1
1800.2.k.a 2 12.b even 2 1
1800.2.k.a 2 24.f even 2 1
2304.2.a.b 1 240.bm odd 4 1
2304.2.a.e 1 240.t even 4 1
2304.2.a.l 1 240.bm odd 4 1
2304.2.a.o 1 240.t even 4 1
2400.2.d.b 2 5.c odd 4 1
2400.2.d.b 2 40.i odd 4 1
2400.2.d.c 2 5.c odd 4 1
2400.2.d.c 2 40.i odd 4 1
2400.2.k.a 2 1.a even 1 1 trivial
2400.2.k.a 2 8.b even 2 1 inner
2592.2.r.f 4 45.j even 6 2
2592.2.r.f 4 360.bk even 6 2
2592.2.r.g 4 45.h odd 6 2
2592.2.r.g 4 360.bh odd 6 2
4704.2.c.a 2 35.c odd 2 1
4704.2.c.a 2 280.c odd 2 1
7200.2.d.d 2 15.e even 4 1
7200.2.d.d 2 120.w even 4 1
7200.2.d.g 2 15.e even 4 1
7200.2.d.g 2 120.w even 4 1
7200.2.k.d 2 3.b odd 2 1
7200.2.k.d 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 2 \) acting on \(S_{2}^{\mathrm{new}}(2400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 16 \) Copy content Toggle raw display
$17$ \( (T - 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 16 \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 36 \) Copy content Toggle raw display
$31$ \( (T + 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 64 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( (T + 12)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 16 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 144 \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( (T - 6)^{2} \) Copy content Toggle raw display
$79$ \( (T + 10)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 256 \) Copy content Toggle raw display
$89$ \( (T + 10)^{2} \) Copy content Toggle raw display
$97$ \( (T - 2)^{2} \) Copy content Toggle raw display
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