Properties

Label 80.6.c.c
Level $80$
Weight $6$
Character orbit 80.c
Analytic conductor $12.831$
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] = [80,6,Mod(49,80)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(80, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("80.49");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 80.c (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: \(12.8307055850\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 10)
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 7 \beta q^{3} + ( - 5 \beta + 55) q^{5} - 79 \beta q^{7} + 47 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 7 \beta q^{3} + ( - 5 \beta + 55) q^{5} - 79 \beta q^{7} + 47 q^{9} + 148 q^{11} - 342 \beta q^{13} + (385 \beta + 140) q^{15} + 1024 \beta q^{17} + 2220 q^{19} + 2212 q^{21} - 623 \beta q^{23} + ( - 550 \beta + 2925) q^{25} + 2030 \beta q^{27} + 270 q^{29} + 2048 q^{31} + 1036 \beta q^{33} + ( - 4345 \beta - 1580) q^{35} - 2186 \beta q^{37} + 9576 q^{39} - 2398 q^{41} + 1147 \beta q^{43} + ( - 235 \beta + 2585) q^{45} + 5341 \beta q^{47} - 8157 q^{49} - 28672 q^{51} - 1482 \beta q^{53} + ( - 740 \beta + 8140) q^{55} + 15540 \beta q^{57} - 39740 q^{59} - 42298 q^{61} - 3713 \beta q^{63} + ( - 18810 \beta - 6840) q^{65} - 16049 \beta q^{67} + 17444 q^{69} + 4248 q^{71} - 15052 \beta q^{73} + (20475 \beta + 15400) q^{75} - 11692 \beta q^{77} + 35280 q^{79} - 45419 q^{81} - 13913 \beta q^{83} + (56320 \beta + 20480) q^{85} + 1890 \beta q^{87} + 85210 q^{89} - 108072 q^{91} + 14336 \beta q^{93} + ( - 11100 \beta + 122100) q^{95} - 48616 \beta q^{97} + 6956 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 110 q^{5} + 94 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 110 q^{5} + 94 q^{9} + 296 q^{11} + 280 q^{15} + 4440 q^{19} + 4424 q^{21} + 5850 q^{25} + 540 q^{29} + 4096 q^{31} - 3160 q^{35} + 19152 q^{39} - 4796 q^{41} + 5170 q^{45} - 16314 q^{49} - 57344 q^{51} + 16280 q^{55} - 79480 q^{59} - 84596 q^{61} - 13680 q^{65} + 34888 q^{69} + 8496 q^{71} + 30800 q^{75} + 70560 q^{79} - 90838 q^{81} + 40960 q^{85} + 170420 q^{89} - 216144 q^{91} + 244200 q^{95} + 13912 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.00000i
1.00000i
0 14.0000i 0 55.0000 + 10.0000i 0 158.000i 0 47.0000 0
49.2 0 14.0000i 0 55.0000 10.0000i 0 158.000i 0 47.0000 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 80.6.c.c 2
3.b odd 2 1 720.6.f.a 2
4.b odd 2 1 10.6.b.a 2
5.b even 2 1 inner 80.6.c.c 2
5.c odd 4 1 400.6.a.c 1
5.c odd 4 1 400.6.a.k 1
8.b even 2 1 320.6.c.a 2
8.d odd 2 1 320.6.c.b 2
12.b even 2 1 90.6.c.a 2
15.d odd 2 1 720.6.f.a 2
20.d odd 2 1 10.6.b.a 2
20.e even 4 1 50.6.a.c 1
20.e even 4 1 50.6.a.e 1
40.e odd 2 1 320.6.c.b 2
40.f even 2 1 320.6.c.a 2
60.h even 2 1 90.6.c.a 2
60.l odd 4 1 450.6.a.c 1
60.l odd 4 1 450.6.a.w 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.b.a 2 4.b odd 2 1
10.6.b.a 2 20.d odd 2 1
50.6.a.c 1 20.e even 4 1
50.6.a.e 1 20.e even 4 1
80.6.c.c 2 1.a even 1 1 trivial
80.6.c.c 2 5.b even 2 1 inner
90.6.c.a 2 12.b even 2 1
90.6.c.a 2 60.h even 2 1
320.6.c.a 2 8.b even 2 1
320.6.c.a 2 40.f even 2 1
320.6.c.b 2 8.d odd 2 1
320.6.c.b 2 40.e odd 2 1
400.6.a.c 1 5.c odd 4 1
400.6.a.k 1 5.c odd 4 1
450.6.a.c 1 60.l odd 4 1
450.6.a.w 1 60.l odd 4 1
720.6.f.a 2 3.b odd 2 1
720.6.f.a 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 196 \) acting on \(S_{6}^{\mathrm{new}}(80, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 196 \) Copy content Toggle raw display
$5$ \( T^{2} - 110T + 3125 \) Copy content Toggle raw display
$7$ \( T^{2} + 24964 \) Copy content Toggle raw display
$11$ \( (T - 148)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 467856 \) Copy content Toggle raw display
$17$ \( T^{2} + 4194304 \) Copy content Toggle raw display
$19$ \( (T - 2220)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 1552516 \) Copy content Toggle raw display
$29$ \( (T - 270)^{2} \) Copy content Toggle raw display
$31$ \( (T - 2048)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 19114384 \) Copy content Toggle raw display
$41$ \( (T + 2398)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 5262436 \) Copy content Toggle raw display
$47$ \( T^{2} + 114105124 \) Copy content Toggle raw display
$53$ \( T^{2} + 8785296 \) Copy content Toggle raw display
$59$ \( (T + 39740)^{2} \) Copy content Toggle raw display
$61$ \( (T + 42298)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1030281604 \) Copy content Toggle raw display
$71$ \( (T - 4248)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 906250816 \) Copy content Toggle raw display
$79$ \( (T - 35280)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 774286276 \) Copy content Toggle raw display
$89$ \( (T - 85210)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 9454061824 \) Copy content Toggle raw display
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