Properties

Label 80.2.q.b
Level $80$
Weight $2$
Character orbit 80.q
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(29,80)]
 
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([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("80.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 80.q (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.

\(f(q)\) \(=\) \( q + (i + 1) q^{2} + ( - i - 1) q^{3} + 2 i q^{4} + (i + 2) q^{5} - 2 i q^{6} + (2 i - 2) q^{8} - i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (i + 1) q^{2} + ( - i - 1) q^{3} + 2 i q^{4} + (i + 2) q^{5} - 2 i q^{6} + (2 i - 2) q^{8} - i q^{9} + (3 i + 1) q^{10} + ( - 3 i - 3) q^{11} + ( - 2 i + 2) q^{12} + ( - 3 i - 3) q^{13} + ( - 3 i - 1) q^{15} - 4 q^{16} + 4 i q^{17} + ( - i + 1) q^{18} + (i - 1) q^{19} + (4 i - 2) q^{20} - 6 i q^{22} + 8 q^{23} + 4 q^{24} + (4 i + 3) q^{25} - 6 i q^{26} + (4 i - 4) q^{27} + ( - 3 i + 3) q^{29} + ( - 4 i + 2) q^{30} + ( - 4 i - 4) q^{32} + 6 i q^{33} + (4 i - 4) q^{34} + 2 q^{36} + (3 i - 3) q^{37} - 2 q^{38} + 6 i q^{39} + (2 i - 6) q^{40} + (3 i - 3) q^{43} + ( - 6 i + 6) q^{44} + ( - 2 i + 1) q^{45} + (8 i + 8) q^{46} + 2 i q^{47} + (4 i + 4) q^{48} - 7 q^{49} + (7 i - 1) q^{50} + ( - 4 i + 4) q^{51} + ( - 6 i + 6) q^{52} + ( - 9 i + 9) q^{53} - 8 q^{54} + ( - 9 i - 3) q^{55} + 2 q^{57} + 6 q^{58} + (9 i + 9) q^{59} + ( - 2 i + 6) q^{60} + (5 i - 5) q^{61} - 8 i q^{64} + ( - 9 i - 3) q^{65} + (6 i - 6) q^{66} + (3 i + 3) q^{67} - 8 q^{68} + ( - 8 i - 8) q^{69} + 6 i q^{71} + (2 i + 2) q^{72} - 6 q^{73} - 6 q^{74} + ( - 7 i + 1) q^{75} + ( - 2 i - 2) q^{76} + (6 i - 6) q^{78} + 8 q^{79} + ( - 4 i - 8) q^{80} + 5 q^{81} + ( - 9 i - 9) q^{83} + (8 i - 4) q^{85} - 6 q^{86} - 6 q^{87} + 12 q^{88} - 12 i q^{89} + ( - i + 3) q^{90} + 16 i q^{92} + (2 i - 2) q^{94} + (i - 3) q^{95} + 8 i q^{96} - 12 i q^{97} + ( - 7 i - 7) q^{98} + (3 i - 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 4 q^{5} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{3} + 4 q^{5} - 4 q^{8} + 2 q^{10} - 6 q^{11} + 4 q^{12} - 6 q^{13} - 2 q^{15} - 8 q^{16} + 2 q^{18} - 2 q^{19} - 4 q^{20} + 16 q^{23} + 8 q^{24} + 6 q^{25} - 8 q^{27} + 6 q^{29} + 4 q^{30} - 8 q^{32} - 8 q^{34} + 4 q^{36} - 6 q^{37} - 4 q^{38} - 12 q^{40} - 6 q^{43} + 12 q^{44} + 2 q^{45} + 16 q^{46} + 8 q^{48} - 14 q^{49} - 2 q^{50} + 8 q^{51} + 12 q^{52} + 18 q^{53} - 16 q^{54} - 6 q^{55} + 4 q^{57} + 12 q^{58} + 18 q^{59} + 12 q^{60} - 10 q^{61} - 6 q^{65} - 12 q^{66} + 6 q^{67} - 16 q^{68} - 16 q^{69} + 4 q^{72} - 12 q^{73} - 12 q^{74} + 2 q^{75} - 4 q^{76} - 12 q^{78} + 16 q^{79} - 16 q^{80} + 10 q^{81} - 18 q^{83} - 8 q^{85} - 12 q^{86} - 12 q^{87} + 24 q^{88} + 6 q^{90} - 4 q^{94} - 6 q^{95} - 14 q^{98} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
1.00000i
1.00000i
1.00000 1.00000i −1.00000 + 1.00000i 2.00000i 2.00000 1.00000i 2.00000i 0 −2.00000 2.00000i 1.00000i 1.00000 3.00000i
69.1 1.00000 + 1.00000i −1.00000 1.00000i 2.00000i 2.00000 + 1.00000i 2.00000i 0 −2.00000 + 2.00000i 1.00000i 1.00000 + 3.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 2T_{3} + 2 \) 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
$3$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$5$ \( T^{2} - 4T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$13$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$17$ \( T^{2} + 16 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$23$ \( (T - 8)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$47$ \( T^{2} + 4 \) Copy content Toggle raw display
$53$ \( T^{2} - 18T + 162 \) Copy content Toggle raw display
$59$ \( T^{2} - 18T + 162 \) Copy content Toggle raw display
$61$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$67$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$71$ \( T^{2} + 36 \) Copy content Toggle raw display
$73$ \( (T + 6)^{2} \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 18T + 162 \) Copy content Toggle raw display
$89$ \( T^{2} + 144 \) Copy content Toggle raw display
$97$ \( T^{2} + 144 \) Copy content Toggle raw display
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