Properties

Label 1122.2.c.a
Level $1122$
Weight $2$
Character orbit 1122.c
Analytic conductor $8.959$
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] = [1122,2,Mod(67,1122)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1122, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1122.67");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1122 = 2 \cdot 3 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1122.c (of order \(2\), degree \(1\), 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: \(8.95921510679\)
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: yes
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - i q^{3} + q^{4} + 4 i q^{5} + i q^{6} - 4 i q^{7} - q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - i q^{3} + q^{4} + 4 i q^{5} + i q^{6} - 4 i q^{7} - q^{8} - q^{9} - 4 i q^{10} + i q^{11} - i q^{12} - 6 q^{13} + 4 i q^{14} + 4 q^{15} + q^{16} + (4 i + 1) q^{17} + q^{18} + 8 q^{19} + 4 i q^{20} - 4 q^{21} - i q^{22} - 4 i q^{23} + i q^{24} - 11 q^{25} + 6 q^{26} + i q^{27} - 4 i q^{28} + 8 i q^{29} - 4 q^{30} + 4 i q^{31} - q^{32} + q^{33} + ( - 4 i - 1) q^{34} + 16 q^{35} - q^{36} + 8 i q^{37} - 8 q^{38} + 6 i q^{39} - 4 i q^{40} + 4 q^{42} + i q^{44} - 4 i q^{45} + 4 i q^{46} - 12 q^{47} - i q^{48} - 9 q^{49} + 11 q^{50} + ( - i + 4) q^{51} - 6 q^{52} - 2 q^{53} - i q^{54} - 4 q^{55} + 4 i q^{56} - 8 i q^{57} - 8 i q^{58} - 8 q^{59} + 4 q^{60} + 4 i q^{61} - 4 i q^{62} + 4 i q^{63} + q^{64} - 24 i q^{65} - q^{66} - 4 q^{67} + (4 i + 1) q^{68} - 4 q^{69} - 16 q^{70} + 12 i q^{71} + q^{72} + 4 i q^{73} - 8 i q^{74} + 11 i q^{75} + 8 q^{76} + 4 q^{77} - 6 i q^{78} - 4 i q^{79} + 4 i q^{80} + q^{81} + 4 q^{83} - 4 q^{84} + (4 i - 16) q^{85} + 8 q^{87} - i q^{88} + 6 q^{89} + 4 i q^{90} + 24 i q^{91} - 4 i q^{92} + 4 q^{93} + 12 q^{94} + 32 i q^{95} + i q^{96} + 16 i q^{97} + 9 q^{98} - i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9} - 12 q^{13} + 8 q^{15} + 2 q^{16} + 2 q^{17} + 2 q^{18} + 16 q^{19} - 8 q^{21} - 22 q^{25} + 12 q^{26} - 8 q^{30} - 2 q^{32} + 2 q^{33} - 2 q^{34} + 32 q^{35} - 2 q^{36} - 16 q^{38} + 8 q^{42} - 24 q^{47} - 18 q^{49} + 22 q^{50} + 8 q^{51} - 12 q^{52} - 4 q^{53} - 8 q^{55} - 16 q^{59} + 8 q^{60} + 2 q^{64} - 2 q^{66} - 8 q^{67} + 2 q^{68} - 8 q^{69} - 32 q^{70} + 2 q^{72} + 16 q^{76} + 8 q^{77} + 2 q^{81} + 8 q^{83} - 8 q^{84} - 32 q^{85} + 16 q^{87} + 12 q^{89} + 8 q^{93} + 24 q^{94} + 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1122\mathbb{Z}\right)^\times\).

\(n\) \(409\) \(749\) \(1057\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
1.00000i
1.00000i
−1.00000 1.00000i 1.00000 4.00000i 1.00000i 4.00000i −1.00000 −1.00000 4.00000i
67.2 −1.00000 1.00000i 1.00000 4.00000i 1.00000i 4.00000i −1.00000 −1.00000 4.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1122.2.c.a 2
3.b odd 2 1 3366.2.c.c 2
17.b even 2 1 inner 1122.2.c.a 2
51.c odd 2 1 3366.2.c.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1122.2.c.a 2 1.a even 1 1 trivial
1122.2.c.a 2 17.b even 2 1 inner
3366.2.c.c 2 3.b odd 2 1
3366.2.c.c 2 51.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1122, [\chi])\):

\( T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{19} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + 16 \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{2} + 1 \) Copy content Toggle raw display
$13$ \( (T + 6)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 2T + 17 \) Copy content Toggle raw display
$19$ \( (T - 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 16 \) Copy content Toggle raw display
$29$ \( T^{2} + 64 \) Copy content Toggle raw display
$31$ \( T^{2} + 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 64 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( (T + 12)^{2} \) Copy content Toggle raw display
$53$ \( (T + 2)^{2} \) Copy content Toggle raw display
$59$ \( (T + 8)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 16 \) Copy content Toggle raw display
$67$ \( (T + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 144 \) Copy content Toggle raw display
$73$ \( T^{2} + 16 \) Copy content Toggle raw display
$79$ \( T^{2} + 16 \) Copy content Toggle raw display
$83$ \( (T - 4)^{2} \) 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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