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Properties

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

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [80,2,Mod(49,80)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("80.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 80.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.638803216170\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Copy content comment:defining polynomial
 
Copy content 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: \( 2 \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + (\beta - 1) q^{5} - \beta q^{7} - q^{9} + 4 q^{11} - 2 \beta q^{13} + ( - \beta - 4) q^{15} - 4 q^{19} + 4 q^{21} - \beta q^{23} + ( - 2 \beta - 3) q^{25} + 2 \beta q^{27} - 2 q^{29} + \cdots - 4 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 2 q^{9} + 8 q^{11} - 8 q^{15} - 8 q^{19} + 8 q^{21} - 6 q^{25} - 4 q^{29} + 8 q^{35} + 16 q^{39} + 4 q^{41} + 2 q^{45} + 6 q^{49} - 8 q^{55} - 24 q^{59} - 20 q^{61} + 16 q^{65} + 8 q^{69}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
− 1.00000i
1.00000i
0 − 2.00000i 0 −1.00000 − 2.00000i 0 2.00000i 0 −1.00000 0
49.2 0 2.00000i 0 −1.00000 + 2.00000i 0 − 2.00000i 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.2.c.a 2
3.b odd 2 1 720.2.f.e 2
4.b odd 2 1 40.2.c.a ✓ 2
5.b even 2 1 inner 80.2.c.a 2
5.c odd 4 1 400.2.a.b 1
5.c odd 4 1 400.2.a.g 1
8.b even 2 1 320.2.c.b 2
8.d odd 2 1 320.2.c.c 2
12.b even 2 1 360.2.f.c 2
15.d odd 2 1 720.2.f.e 2
15.e even 4 1 3600.2.a.k 1
15.e even 4 1 3600.2.a.bb 1
16.e even 4 1 1280.2.f.b 2
16.e even 4 1 1280.2.f.e 2
16.f odd 4 1 1280.2.f.a 2
16.f odd 4 1 1280.2.f.f 2
20.d odd 2 1 40.2.c.a ✓ 2
20.e even 4 1 200.2.a.b 1
20.e even 4 1 200.2.a.d 1
24.f even 2 1 2880.2.f.h 2
24.h odd 2 1 2880.2.f.i 2
28.d even 2 1 1960.2.g.b 2
40.e odd 2 1 320.2.c.c 2
40.f even 2 1 320.2.c.b 2
40.i odd 4 1 1600.2.a.d 1
40.i odd 4 1 1600.2.a.u 1
40.k even 4 1 1600.2.a.f 1
40.k even 4 1 1600.2.a.v 1
60.h even 2 1 360.2.f.c 2
60.l odd 4 1 1800.2.a.j 1
60.l odd 4 1 1800.2.a.s 1
80.k odd 4 1 1280.2.f.a 2
80.k odd 4 1 1280.2.f.f 2
80.q even 4 1 1280.2.f.b 2
80.q even 4 1 1280.2.f.e 2
120.i odd 2 1 2880.2.f.i 2
120.m even 2 1 2880.2.f.h 2
140.c even 2 1 1960.2.g.b 2
140.j odd 4 1 9800.2.a.d 1
140.j odd 4 1 9800.2.a.bf 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.c.a ✓ 2 4.b odd 2 1
40.2.c.a ✓ 2 20.d odd 2 1
80.2.c.a 2 1.a even 1 1 trivial
80.2.c.a 2 5.b even 2 1 inner
200.2.a.b 1 20.e even 4 1
200.2.a.d 1 20.e even 4 1
320.2.c.b 2 8.b even 2 1
320.2.c.b 2 40.f even 2 1
320.2.c.c 2 8.d odd 2 1
320.2.c.c 2 40.e odd 2 1
360.2.f.c 2 12.b even 2 1
360.2.f.c 2 60.h even 2 1
400.2.a.b 1 5.c odd 4 1
400.2.a.g 1 5.c odd 4 1
720.2.f.e 2 3.b odd 2 1
720.2.f.e 2 15.d odd 2 1
1280.2.f.a 2 16.f odd 4 1
1280.2.f.a 2 80.k odd 4 1
1280.2.f.b 2 16.e even 4 1
1280.2.f.b 2 80.q even 4 1
1280.2.f.e 2 16.e even 4 1
1280.2.f.e 2 80.q even 4 1
1280.2.f.f 2 16.f odd 4 1
1280.2.f.f 2 80.k odd 4 1
1600.2.a.d 1 40.i odd 4 1
1600.2.a.f 1 40.k even 4 1
1600.2.a.u 1 40.i odd 4 1
1600.2.a.v 1 40.k even 4 1
1800.2.a.j 1 60.l odd 4 1
1800.2.a.s 1 60.l odd 4 1
1960.2.g.b 2 28.d even 2 1
1960.2.g.b 2 140.c even 2 1
2880.2.f.h 2 24.f even 2 1
2880.2.f.h 2 120.m even 2 1
2880.2.f.i 2 24.h odd 2 1
2880.2.f.i 2 120.i odd 2 1
3600.2.a.k 1 15.e even 4 1
3600.2.a.bb 1 15.e even 4 1
9800.2.a.d 1 140.j odd 4 1
9800.2.a.bf 1 140.j odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(80, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 4 \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 16 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 36 \) Copy content Toggle raw display
$47$ \( T^{2} + 36 \) Copy content Toggle raw display
$53$ \( T^{2} + 16 \) Copy content Toggle raw display
$59$ \( (T + 12)^{2} \) Copy content Toggle raw display
$61$ \( (T + 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 196 \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 64 \) Copy content Toggle raw display
$79$ \( (T - 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 4 \) Copy content Toggle raw display
$89$ \( (T + 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 256 \) Copy content Toggle raw display
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