Properties

Label 980.3.l.a
Level $980$
Weight $3$
Character orbit 980.l
Analytic conductor $26.703$
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] = [980,3,Mod(197,980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("980.197");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 980.l (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: \(26.7030659073\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 20)
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.

\(f(q)\) \(=\) \( q + (i - 1) q^{3} + ( - 4 i + 3) q^{5} + 7 i q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (i - 1) q^{3} + ( - 4 i + 3) q^{5} + 7 i q^{9} + 10 q^{11} + (9 i - 9) q^{13} + (7 i + 1) q^{15} + ( - i - 1) q^{17} + 8 i q^{19} + (23 i - 23) q^{23} + ( - 24 i - 7) q^{25} + ( - 16 i - 16) q^{27} - 8 i q^{29} + 14 q^{31} + (10 i - 10) q^{33} + (33 i + 33) q^{37} - 18 i q^{39} + 14 q^{41} + (15 i - 15) q^{43} + (21 i + 28) q^{45} + (39 i + 39) q^{47} + 2 q^{51} + (7 i - 7) q^{53} + ( - 40 i + 30) q^{55} + ( - 8 i - 8) q^{57} + 56 i q^{59} - 42 q^{61} + (63 i + 9) q^{65} + ( - 7 i - 7) q^{67} - 46 i q^{69} + 98 q^{71} + (49 i - 49) q^{73} + (17 i + 31) q^{75} + 96 i q^{79} - 31 q^{81} + ( - 63 i + 63) q^{83} + (i - 7) q^{85} + (8 i + 8) q^{87} + 112 i q^{89} + (14 i - 14) q^{93} + (24 i + 32) q^{95} + ( - 33 i - 33) q^{97} + 70 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 6 q^{5} + 20 q^{11} - 18 q^{13} + 2 q^{15} - 2 q^{17} - 46 q^{23} - 14 q^{25} - 32 q^{27} + 28 q^{31} - 20 q^{33} + 66 q^{37} + 28 q^{41} - 30 q^{43} + 56 q^{45} + 78 q^{47} + 4 q^{51} - 14 q^{53} + 60 q^{55} - 16 q^{57} - 84 q^{61} + 18 q^{65} - 14 q^{67} + 196 q^{71} - 98 q^{73} + 62 q^{75} - 62 q^{81} + 126 q^{83} - 14 q^{85} + 16 q^{87} - 28 q^{93} + 64 q^{95} - 66 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(1\) \(i\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
1.00000i
1.00000i
0 −1.00000 + 1.00000i 0 3.00000 4.00000i 0 0 0 7.00000i 0
393.1 0 −1.00000 1.00000i 0 3.00000 + 4.00000i 0 0 0 7.00000i 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.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.3.l.a 2
5.c odd 4 1 inner 980.3.l.a 2
7.b odd 2 1 20.3.f.a 2
21.c even 2 1 180.3.l.a 2
28.d even 2 1 80.3.p.a 2
35.c odd 2 1 100.3.f.a 2
35.f even 4 1 20.3.f.a 2
35.f even 4 1 100.3.f.a 2
56.e even 2 1 320.3.p.g 2
56.h odd 2 1 320.3.p.c 2
84.h odd 2 1 720.3.bh.e 2
105.g even 2 1 900.3.l.a 2
105.k odd 4 1 180.3.l.a 2
105.k odd 4 1 900.3.l.a 2
140.c even 2 1 400.3.p.d 2
140.j odd 4 1 80.3.p.a 2
140.j odd 4 1 400.3.p.d 2
280.s even 4 1 320.3.p.c 2
280.y odd 4 1 320.3.p.g 2
420.w even 4 1 720.3.bh.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.f.a 2 7.b odd 2 1
20.3.f.a 2 35.f even 4 1
80.3.p.a 2 28.d even 2 1
80.3.p.a 2 140.j odd 4 1
100.3.f.a 2 35.c odd 2 1
100.3.f.a 2 35.f even 4 1
180.3.l.a 2 21.c even 2 1
180.3.l.a 2 105.k odd 4 1
320.3.p.c 2 56.h odd 2 1
320.3.p.c 2 280.s even 4 1
320.3.p.g 2 56.e even 2 1
320.3.p.g 2 280.y odd 4 1
400.3.p.d 2 140.c even 2 1
400.3.p.d 2 140.j odd 4 1
720.3.bh.e 2 84.h odd 2 1
720.3.bh.e 2 420.w even 4 1
900.3.l.a 2 105.g even 2 1
900.3.l.a 2 105.k odd 4 1
980.3.l.a 2 1.a even 1 1 trivial
980.3.l.a 2 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 2T_{3} + 2 \) acting on \(S_{3}^{\mathrm{new}}(980, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$5$ \( T^{2} - 6T + 25 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 10)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 18T + 162 \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$19$ \( T^{2} + 64 \) Copy content Toggle raw display
$23$ \( T^{2} + 46T + 1058 \) Copy content Toggle raw display
$29$ \( T^{2} + 64 \) Copy content Toggle raw display
$31$ \( (T - 14)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 66T + 2178 \) Copy content Toggle raw display
$41$ \( (T - 14)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 30T + 450 \) Copy content Toggle raw display
$47$ \( T^{2} - 78T + 3042 \) Copy content Toggle raw display
$53$ \( T^{2} + 14T + 98 \) Copy content Toggle raw display
$59$ \( T^{2} + 3136 \) Copy content Toggle raw display
$61$ \( (T + 42)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 14T + 98 \) Copy content Toggle raw display
$71$ \( (T - 98)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 98T + 4802 \) Copy content Toggle raw display
$79$ \( T^{2} + 9216 \) Copy content Toggle raw display
$83$ \( T^{2} - 126T + 7938 \) Copy content Toggle raw display
$89$ \( T^{2} + 12544 \) Copy content Toggle raw display
$97$ \( T^{2} + 66T + 2178 \) Copy content Toggle raw display
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