Properties

Label 220.2.l.a
Level $220$
Weight $2$
Character orbit 220.l
Analytic conductor $1.757$
Analytic rank $1$
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] = [220,2,Mod(23,220)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(220, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("220.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 220.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: \(1.75670884447\)
Analytic rank: \(1\)
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.

\(f(q)\) \(=\) \( q + (i - 1) q^{2} + ( - 2 i - 2) q^{3} - 2 i q^{4} + ( - 2 i - 1) q^{5} + 4 q^{6} + (i - 1) q^{7} + (2 i + 2) q^{8} + 5 i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (i - 1) q^{2} + ( - 2 i - 2) q^{3} - 2 i q^{4} + ( - 2 i - 1) q^{5} + 4 q^{6} + (i - 1) q^{7} + (2 i + 2) q^{8} + 5 i q^{9} + (i + 3) q^{10} - i q^{11} + (4 i - 4) q^{12} + (4 i - 4) q^{13} - 2 i q^{14} + (6 i - 2) q^{15} - 4 q^{16} + (2 i + 2) q^{17} + ( - 5 i - 5) q^{18} + 2 q^{19} + (2 i - 4) q^{20} + 4 q^{21} + (i + 1) q^{22} + ( - 4 i - 4) q^{23} - 8 i q^{24} + (4 i - 3) q^{25} - 8 i q^{26} + ( - 4 i + 4) q^{27} + (2 i + 2) q^{28} - 2 i q^{29} + ( - 8 i - 4) q^{30} + 8 i q^{31} + ( - 4 i + 4) q^{32} + (2 i - 2) q^{33} - 4 q^{34} + (i + 3) q^{35} + 10 q^{36} + ( - 5 i - 5) q^{37} + (2 i - 2) q^{38} + 16 q^{39} + ( - 6 i + 2) q^{40} - 6 q^{41} + (4 i - 4) q^{42} + (i + 1) q^{43} - 2 q^{44} + ( - 5 i + 10) q^{45} + 8 q^{46} + (4 i - 4) q^{47} + (8 i + 8) q^{48} + 5 i q^{49} + ( - 7 i - 1) q^{50} - 8 i q^{51} + (8 i + 8) q^{52} + ( - 9 i + 9) q^{53} + 8 i q^{54} + (i - 2) q^{55} - 4 q^{56} + ( - 4 i - 4) q^{57} + (2 i + 2) q^{58} - 8 q^{59} + (4 i + 12) q^{60} - 10 q^{61} + ( - 8 i - 8) q^{62} + ( - 5 i - 5) q^{63} + 8 i q^{64} + (4 i + 12) q^{65} - 4 i q^{66} + ( - 4 i + 4) q^{68} + 16 i q^{69} + (2 i - 4) q^{70} - 12 i q^{71} + (10 i - 10) q^{72} + ( - 2 i + 2) q^{73} + 10 q^{74} + ( - 2 i + 14) q^{75} - 4 i q^{76} + (i + 1) q^{77} + (16 i - 16) q^{78} - 10 q^{79} + (8 i + 4) q^{80} - q^{81} + ( - 6 i + 6) q^{82} + (i + 1) q^{83} - 8 i q^{84} + ( - 6 i + 2) q^{85} - 2 q^{86} + (4 i - 4) q^{87} + ( - 2 i + 2) q^{88} - 6 i q^{89} + (15 i - 5) q^{90} - 8 i q^{91} + (8 i - 8) q^{92} + ( - 16 i + 16) q^{93} - 8 i q^{94} + ( - 4 i - 2) q^{95} - 16 q^{96} + ( - 3 i - 3) q^{97} + ( - 5 i - 5) q^{98} + 5 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{3} - 2 q^{5} + 8 q^{6} - 2 q^{7} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 4 q^{3} - 2 q^{5} + 8 q^{6} - 2 q^{7} + 4 q^{8} + 6 q^{10} - 8 q^{12} - 8 q^{13} - 4 q^{15} - 8 q^{16} + 4 q^{17} - 10 q^{18} + 4 q^{19} - 8 q^{20} + 8 q^{21} + 2 q^{22} - 8 q^{23} - 6 q^{25} + 8 q^{27} + 4 q^{28} - 8 q^{30} + 8 q^{32} - 4 q^{33} - 8 q^{34} + 6 q^{35} + 20 q^{36} - 10 q^{37} - 4 q^{38} + 32 q^{39} + 4 q^{40} - 12 q^{41} - 8 q^{42} + 2 q^{43} - 4 q^{44} + 20 q^{45} + 16 q^{46} - 8 q^{47} + 16 q^{48} - 2 q^{50} + 16 q^{52} + 18 q^{53} - 4 q^{55} - 8 q^{56} - 8 q^{57} + 4 q^{58} - 16 q^{59} + 24 q^{60} - 20 q^{61} - 16 q^{62} - 10 q^{63} + 24 q^{65} + 8 q^{68} - 8 q^{70} - 20 q^{72} + 4 q^{73} + 20 q^{74} + 28 q^{75} + 2 q^{77} - 32 q^{78} - 20 q^{79} + 8 q^{80} - 2 q^{81} + 12 q^{82} + 2 q^{83} + 4 q^{85} - 4 q^{86} - 8 q^{87} + 4 q^{88} - 10 q^{90} - 16 q^{92} + 32 q^{93} - 4 q^{95} - 32 q^{96} - 6 q^{97} - 10 q^{98} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/220\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(111\) \(177\)
\(\chi(n)\) \(1\) \(-1\) \(i\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
1.00000i
1.00000i
−1.00000 1.00000i −2.00000 + 2.00000i 2.00000i −1.00000 + 2.00000i 4.00000 −1.00000 1.00000i 2.00000 2.00000i 5.00000i 3.00000 1.00000i
67.1 −1.00000 + 1.00000i −2.00000 2.00000i 2.00000i −1.00000 2.00000i 4.00000 −1.00000 + 1.00000i 2.00000 + 2.00000i 5.00000i 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
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 220.2.l.a 2
4.b odd 2 1 220.2.l.b yes 2
5.c odd 4 1 220.2.l.b yes 2
20.e even 4 1 inner 220.2.l.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.2.l.a 2 1.a even 1 1 trivial
220.2.l.a 2 20.e even 4 1 inner
220.2.l.b yes 2 4.b odd 2 1
220.2.l.b yes 2 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 4T_{3} + 8 \) acting on \(S_{2}^{\mathrm{new}}(220, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$3$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$11$ \( T^{2} + 1 \) Copy content Toggle raw display
$13$ \( T^{2} + 8T + 32 \) Copy content Toggle raw display
$17$ \( T^{2} - 4T + 8 \) Copy content Toggle raw display
$19$ \( (T - 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 8T + 32 \) Copy content Toggle raw display
$29$ \( T^{2} + 4 \) Copy content Toggle raw display
$31$ \( T^{2} + 64 \) Copy content Toggle raw display
$37$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$47$ \( T^{2} + 8T + 32 \) Copy content Toggle raw display
$53$ \( T^{2} - 18T + 162 \) Copy content Toggle raw display
$59$ \( (T + 8)^{2} \) Copy content Toggle raw display
$61$ \( (T + 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 144 \) Copy content Toggle raw display
$73$ \( T^{2} - 4T + 8 \) Copy content Toggle raw display
$79$ \( (T + 10)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$89$ \( T^{2} + 36 \) Copy content Toggle raw display
$97$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
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