Properties

Label 120.2.k.a
Level $120$
Weight $2$
Character orbit 120.k
Analytic conductor $0.958$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [120,2,Mod(61,120)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(120, 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("120.61");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 120.k (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: \(0.958204824255\)
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: yes
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.

\(f(q)\) \(=\) \( q + (i + 1) q^{2} - i q^{3} + 2 i q^{4} + i q^{5} + ( - i + 1) q^{6} + 2 q^{7} + (2 i - 2) q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (i + 1) q^{2} - i q^{3} + 2 i q^{4} + i q^{5} + ( - i + 1) q^{6} + 2 q^{7} + (2 i - 2) q^{8} - q^{9} + (i - 1) q^{10} - 4 i q^{11} + 2 q^{12} + (2 i + 2) q^{14} + q^{15} - 4 q^{16} - 6 q^{17} + ( - i - 1) q^{18} - 4 i q^{19} - 2 q^{20} - 2 i q^{21} + ( - 4 i + 4) q^{22} - 4 q^{23} + (2 i + 2) q^{24} - q^{25} + i q^{27} + 4 i q^{28} + 6 i q^{29} + (i + 1) q^{30} + 10 q^{31} + ( - 4 i - 4) q^{32} - 4 q^{33} + ( - 6 i - 6) q^{34} + 2 i q^{35} - 2 i q^{36} - 4 i q^{37} + ( - 4 i + 4) q^{38} + ( - 2 i - 2) q^{40} + 10 q^{41} + ( - 2 i + 2) q^{42} + 4 i q^{43} + 8 q^{44} - i q^{45} + ( - 4 i - 4) q^{46} - 4 q^{47} + 4 i q^{48} - 3 q^{49} + ( - i - 1) q^{50} + 6 i q^{51} + 10 i q^{53} + (i - 1) q^{54} + 4 q^{55} + (4 i - 4) q^{56} - 4 q^{57} + (6 i - 6) q^{58} - 8 i q^{59} + 2 i q^{60} + 8 i q^{61} + (10 i + 10) q^{62} - 2 q^{63} - 8 i q^{64} + ( - 4 i - 4) q^{66} + 12 i q^{67} - 12 i q^{68} + 4 i q^{69} + (2 i - 2) q^{70} - 4 q^{71} + ( - 2 i + 2) q^{72} + 10 q^{73} + ( - 4 i + 4) q^{74} + i q^{75} + 8 q^{76} - 8 i q^{77} - 14 q^{79} - 4 i q^{80} + q^{81} + (10 i + 10) q^{82} + 4 q^{84} - 6 i q^{85} + (4 i - 4) q^{86} + 6 q^{87} + (8 i + 8) q^{88} + 14 q^{89} + ( - i + 1) q^{90} - 8 i q^{92} - 10 i q^{93} + ( - 4 i - 4) q^{94} + 4 q^{95} + (4 i - 4) q^{96} - 10 q^{97} + ( - 3 i - 3) q^{98} + 4 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{6} + 4 q^{7} - 4 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{6} + 4 q^{7} - 4 q^{8} - 2 q^{9} - 2 q^{10} + 4 q^{12} + 4 q^{14} + 2 q^{15} - 8 q^{16} - 12 q^{17} - 2 q^{18} - 4 q^{20} + 8 q^{22} - 8 q^{23} + 4 q^{24} - 2 q^{25} + 2 q^{30} + 20 q^{31} - 8 q^{32} - 8 q^{33} - 12 q^{34} + 8 q^{38} - 4 q^{40} + 20 q^{41} + 4 q^{42} + 16 q^{44} - 8 q^{46} - 8 q^{47} - 6 q^{49} - 2 q^{50} - 2 q^{54} + 8 q^{55} - 8 q^{56} - 8 q^{57} - 12 q^{58} + 20 q^{62} - 4 q^{63} - 8 q^{66} - 4 q^{70} - 8 q^{71} + 4 q^{72} + 20 q^{73} + 8 q^{74} + 16 q^{76} - 28 q^{79} + 2 q^{81} + 20 q^{82} + 8 q^{84} - 8 q^{86} + 12 q^{87} + 16 q^{88} + 28 q^{89} + 2 q^{90} - 8 q^{94} + 8 q^{95} - 8 q^{96} - 20 q^{97} - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/120\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(41\) \(61\) \(97\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
1.00000i
1.00000i
1.00000 1.00000i 1.00000i 2.00000i 1.00000i 1.00000 + 1.00000i 2.00000 −2.00000 2.00000i −1.00000 −1.00000 1.00000i
61.2 1.00000 + 1.00000i 1.00000i 2.00000i 1.00000i 1.00000 1.00000i 2.00000 −2.00000 + 2.00000i −1.00000 −1.00000 + 1.00000i
\(n\): 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 120.2.k.a 2
3.b odd 2 1 360.2.k.b 2
4.b odd 2 1 480.2.k.a 2
5.b even 2 1 600.2.k.a 2
5.c odd 4 1 600.2.d.a 2
5.c odd 4 1 600.2.d.d 2
8.b even 2 1 inner 120.2.k.a 2
8.d odd 2 1 480.2.k.a 2
12.b even 2 1 1440.2.k.a 2
15.d odd 2 1 1800.2.k.g 2
15.e even 4 1 1800.2.d.c 2
15.e even 4 1 1800.2.d.h 2
16.e even 4 1 3840.2.a.d 1
16.e even 4 1 3840.2.a.w 1
16.f odd 4 1 3840.2.a.m 1
16.f odd 4 1 3840.2.a.r 1
20.d odd 2 1 2400.2.k.b 2
20.e even 4 1 2400.2.d.a 2
20.e even 4 1 2400.2.d.d 2
24.f even 2 1 1440.2.k.a 2
24.h odd 2 1 360.2.k.b 2
40.e odd 2 1 2400.2.k.b 2
40.f even 2 1 600.2.k.a 2
40.i odd 4 1 600.2.d.a 2
40.i odd 4 1 600.2.d.d 2
40.k even 4 1 2400.2.d.a 2
40.k even 4 1 2400.2.d.d 2
60.h even 2 1 7200.2.k.f 2
60.l odd 4 1 7200.2.d.e 2
60.l odd 4 1 7200.2.d.f 2
120.i odd 2 1 1800.2.k.g 2
120.m even 2 1 7200.2.k.f 2
120.q odd 4 1 7200.2.d.e 2
120.q odd 4 1 7200.2.d.f 2
120.w even 4 1 1800.2.d.c 2
120.w even 4 1 1800.2.d.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.k.a 2 1.a even 1 1 trivial
120.2.k.a 2 8.b even 2 1 inner
360.2.k.b 2 3.b odd 2 1
360.2.k.b 2 24.h odd 2 1
480.2.k.a 2 4.b odd 2 1
480.2.k.a 2 8.d odd 2 1
600.2.d.a 2 5.c odd 4 1
600.2.d.a 2 40.i odd 4 1
600.2.d.d 2 5.c odd 4 1
600.2.d.d 2 40.i odd 4 1
600.2.k.a 2 5.b even 2 1
600.2.k.a 2 40.f even 2 1
1440.2.k.a 2 12.b even 2 1
1440.2.k.a 2 24.f even 2 1
1800.2.d.c 2 15.e even 4 1
1800.2.d.c 2 120.w even 4 1
1800.2.d.h 2 15.e even 4 1
1800.2.d.h 2 120.w even 4 1
1800.2.k.g 2 15.d odd 2 1
1800.2.k.g 2 120.i odd 2 1
2400.2.d.a 2 20.e even 4 1
2400.2.d.a 2 40.k even 4 1
2400.2.d.d 2 20.e even 4 1
2400.2.d.d 2 40.k even 4 1
2400.2.k.b 2 20.d odd 2 1
2400.2.k.b 2 40.e odd 2 1
3840.2.a.d 1 16.e even 4 1
3840.2.a.m 1 16.f odd 4 1
3840.2.a.r 1 16.f odd 4 1
3840.2.a.w 1 16.e even 4 1
7200.2.d.e 2 60.l odd 4 1
7200.2.d.e 2 120.q odd 4 1
7200.2.d.f 2 60.l odd 4 1
7200.2.d.f 2 120.q odd 4 1
7200.2.k.f 2 60.h even 2 1
7200.2.k.f 2 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 2 \) acting on \(S_{2}^{\mathrm{new}}(120, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + 1 \) Copy content Toggle raw display
$7$ \( (T - 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 6)^{2} \) 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
$31$ \( (T - 10)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 16 \) Copy content Toggle raw display
$41$ \( (T - 10)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( (T + 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 100 \) Copy content Toggle raw display
$59$ \( T^{2} + 64 \) Copy content Toggle raw display
$61$ \( T^{2} + 64 \) Copy content Toggle raw display
$67$ \( T^{2} + 144 \) Copy content Toggle raw display
$71$ \( (T + 4)^{2} \) Copy content Toggle raw display
$73$ \( (T - 10)^{2} \) Copy content Toggle raw display
$79$ \( (T + 14)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T - 14)^{2} \) Copy content Toggle raw display
$97$ \( (T + 10)^{2} \) Copy content Toggle raw display
show more
show less