Properties

Label 80.2.j.a
Level $80$
Weight $2$
Character orbit 80.j
Analytic conductor $0.639$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [80,2,Mod(43,80)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(80, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("80.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 80.j (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: \(0.638803216170\)
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: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - i + 1) q^{2} + 2 i q^{3} - 2 i q^{4} + (2 i + 1) q^{5} + (2 i + 2) q^{6} + ( - 3 i - 3) q^{7} + ( - 2 i - 2) q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - i + 1) q^{2} + 2 i q^{3} - 2 i q^{4} + (2 i + 1) q^{5} + (2 i + 2) q^{6} + ( - 3 i - 3) q^{7} + ( - 2 i - 2) q^{8} - q^{9} + (i + 3) q^{10} + ( - i - 1) q^{11} + 4 q^{12} - 2 q^{13} - 6 q^{14} + (2 i - 4) q^{15} - 4 q^{16} + (i + 1) q^{17} + (i - 1) q^{18} + (3 i + 3) q^{19} + ( - 2 i + 4) q^{20} + ( - 6 i + 6) q^{21} - 2 q^{22} + (i - 1) q^{23} + ( - 4 i + 4) q^{24} + (4 i - 3) q^{25} + (2 i - 2) q^{26} + 4 i q^{27} + (6 i - 6) q^{28} + ( - 7 i + 7) q^{29} + (6 i - 2) q^{30} + 2 i q^{31} + (4 i - 4) q^{32} + ( - 2 i + 2) q^{33} + 2 q^{34} + ( - 9 i + 3) q^{35} + 2 i q^{36} - 6 q^{37} + 6 q^{38} - 4 i q^{39} + ( - 6 i + 2) q^{40} - 4 i q^{41} - 12 i q^{42} + 4 q^{43} + (2 i - 2) q^{44} + ( - 2 i - 1) q^{45} + 2 i q^{46} + ( - 7 i + 7) q^{47} - 8 i q^{48} + 11 i q^{49} + (7 i + 1) q^{50} + (2 i - 2) q^{51} + 4 i q^{52} + 8 i q^{53} + (4 i + 4) q^{54} + ( - 3 i + 1) q^{55} + 12 i q^{56} + (6 i - 6) q^{57} - 14 i q^{58} + (3 i - 3) q^{59} + (8 i + 4) q^{60} + ( - i - 1) q^{61} + (2 i + 2) q^{62} + (3 i + 3) q^{63} + 8 i q^{64} + ( - 4 i - 2) q^{65} - 4 i q^{66} + 4 q^{67} + ( - 2 i + 2) q^{68} + ( - 2 i - 2) q^{69} + ( - 12 i - 6) q^{70} + (2 i + 2) q^{72} + ( - 3 i - 3) q^{73} + (6 i - 6) q^{74} + ( - 6 i - 8) q^{75} + ( - 6 i + 6) q^{76} + 6 i q^{77} + ( - 4 i - 4) q^{78} - 8 q^{79} + ( - 8 i - 4) q^{80} - 11 q^{81} + ( - 4 i - 4) q^{82} + 2 i q^{83} + ( - 12 i - 12) q^{84} + (3 i - 1) q^{85} + ( - 4 i + 4) q^{86} + (14 i + 14) q^{87} + 4 i q^{88} + 6 q^{89} + ( - i - 3) q^{90} + (6 i + 6) q^{91} + (2 i + 2) q^{92} - 4 q^{93} - 14 i q^{94} + (9 i - 3) q^{95} + ( - 8 i - 8) q^{96} + ( - 11 i - 11) q^{97} + (11 i + 11) q^{98} + (i + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{5} + 4 q^{6} - 6 q^{7} - 4 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{5} + 4 q^{6} - 6 q^{7} - 4 q^{8} - 2 q^{9} + 6 q^{10} - 2 q^{11} + 8 q^{12} - 4 q^{13} - 12 q^{14} - 8 q^{15} - 8 q^{16} + 2 q^{17} - 2 q^{18} + 6 q^{19} + 8 q^{20} + 12 q^{21} - 4 q^{22} - 2 q^{23} + 8 q^{24} - 6 q^{25} - 4 q^{26} - 12 q^{28} + 14 q^{29} - 4 q^{30} - 8 q^{32} + 4 q^{33} + 4 q^{34} + 6 q^{35} - 12 q^{37} + 12 q^{38} + 4 q^{40} + 8 q^{43} - 4 q^{44} - 2 q^{45} + 14 q^{47} + 2 q^{50} - 4 q^{51} + 8 q^{54} + 2 q^{55} - 12 q^{57} - 6 q^{59} + 8 q^{60} - 2 q^{61} + 4 q^{62} + 6 q^{63} - 4 q^{65} + 8 q^{67} + 4 q^{68} - 4 q^{69} - 12 q^{70} + 4 q^{72} - 6 q^{73} - 12 q^{74} - 16 q^{75} + 12 q^{76} - 8 q^{78} - 16 q^{79} - 8 q^{80} - 22 q^{81} - 8 q^{82} - 24 q^{84} - 2 q^{85} + 8 q^{86} + 28 q^{87} + 12 q^{89} - 6 q^{90} + 12 q^{91} + 4 q^{92} - 8 q^{93} - 6 q^{95} - 16 q^{96} - 22 q^{97} + 22 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(i\) \(-i\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
1.00000i
1.00000i
1.00000 + 1.00000i 2.00000i 2.00000i 1.00000 2.00000i 2.00000 2.00000i −3.00000 + 3.00000i −2.00000 + 2.00000i −1.00000 3.00000 1.00000i
67.1 1.00000 1.00000i 2.00000i 2.00000i 1.00000 + 2.00000i 2.00000 + 2.00000i −3.00000 3.00000i −2.00000 2.00000i −1.00000 3.00000 + 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
80.j even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.2.j.a 2
3.b odd 2 1 720.2.bd.a 2
4.b odd 2 1 320.2.j.a 2
5.b even 2 1 400.2.j.a 2
5.c odd 4 1 80.2.s.a yes 2
5.c odd 4 1 400.2.s.a 2
8.b even 2 1 640.2.j.a 2
8.d odd 2 1 640.2.j.b 2
15.e even 4 1 720.2.z.d 2
16.e even 4 1 320.2.s.a 2
16.e even 4 1 640.2.s.a 2
16.f odd 4 1 80.2.s.a yes 2
16.f odd 4 1 640.2.s.b 2
20.d odd 2 1 1600.2.j.a 2
20.e even 4 1 320.2.s.a 2
20.e even 4 1 1600.2.s.a 2
40.i odd 4 1 640.2.s.b 2
40.k even 4 1 640.2.s.a 2
48.k even 4 1 720.2.z.d 2
80.i odd 4 1 640.2.j.b 2
80.i odd 4 1 1600.2.j.a 2
80.j even 4 1 inner 80.2.j.a 2
80.k odd 4 1 400.2.s.a 2
80.q even 4 1 1600.2.s.a 2
80.s even 4 1 400.2.j.a 2
80.s even 4 1 640.2.j.a 2
80.t odd 4 1 320.2.j.a 2
240.bd odd 4 1 720.2.bd.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.j.a 2 1.a even 1 1 trivial
80.2.j.a 2 80.j even 4 1 inner
80.2.s.a yes 2 5.c odd 4 1
80.2.s.a yes 2 16.f odd 4 1
320.2.j.a 2 4.b odd 2 1
320.2.j.a 2 80.t odd 4 1
320.2.s.a 2 16.e even 4 1
320.2.s.a 2 20.e even 4 1
400.2.j.a 2 5.b even 2 1
400.2.j.a 2 80.s even 4 1
400.2.s.a 2 5.c odd 4 1
400.2.s.a 2 80.k odd 4 1
640.2.j.a 2 8.b even 2 1
640.2.j.a 2 80.s even 4 1
640.2.j.b 2 8.d odd 2 1
640.2.j.b 2 80.i odd 4 1
640.2.s.a 2 16.e even 4 1
640.2.s.a 2 40.k even 4 1
640.2.s.b 2 16.f odd 4 1
640.2.s.b 2 40.i odd 4 1
720.2.z.d 2 15.e even 4 1
720.2.z.d 2 48.k even 4 1
720.2.bd.a 2 3.b odd 2 1
720.2.bd.a 2 240.bd odd 4 1
1600.2.j.a 2 20.d odd 2 1
1600.2.j.a 2 80.i odd 4 1
1600.2.s.a 2 20.e even 4 1
1600.2.s.a 2 80.q even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(80, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$3$ \( T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{2} - 2T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$11$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$19$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$23$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$29$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$31$ \( T^{2} + 4 \) Copy content Toggle raw display
$37$ \( (T + 6)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 16 \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$53$ \( T^{2} + 64 \) Copy content Toggle raw display
$59$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$61$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$67$ \( (T - 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 4 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 22T + 242 \) Copy content Toggle raw display
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