Properties

Label 440.2.r.d
Level $440$
Weight $2$
Character orbit 440.r
Analytic conductor $3.513$
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] = [440,2,Mod(67,440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(440, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("440.67");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 440.r (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: \(3.51341768894\)
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: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + ( - i + 1) q^{2} + (i - 1) q^{3} - 2 i q^{4} + ( - i + 2) q^{5} + 2 i q^{6} + ( - 2 i + 2) q^{7} + ( - 2 i - 2) q^{8} + i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - i + 1) q^{2} + (i - 1) q^{3} - 2 i q^{4} + ( - i + 2) q^{5} + 2 i q^{6} + ( - 2 i + 2) q^{7} + ( - 2 i - 2) q^{8} + i q^{9} + ( - 3 i + 1) q^{10} - q^{11} + (2 i + 2) q^{12} - 4 i q^{14} + (3 i - 1) q^{15} - 4 q^{16} + (2 i + 2) q^{17} + (i + 1) q^{18} - 8 i q^{19} + ( - 4 i - 2) q^{20} + 4 i q^{21} + (i - 1) q^{22} + (i + 1) q^{23} + 4 q^{24} + ( - 4 i + 3) q^{25} + ( - 4 i - 4) q^{27} + ( - 4 i - 4) q^{28} + 6 q^{29} + (4 i + 2) q^{30} + 2 i q^{31} + (4 i - 4) q^{32} + ( - i + 1) q^{33} + 4 q^{34} + ( - 6 i + 2) q^{35} + 2 q^{36} + (3 i - 3) q^{37} + ( - 8 i - 8) q^{38} + ( - 2 i - 6) q^{40} + 6 q^{41} + (4 i + 4) q^{42} + (6 i - 6) q^{43} + 2 i q^{44} + (2 i + 1) q^{45} + 2 q^{46} + (i - 1) q^{47} + ( - 4 i + 4) q^{48} - i q^{49} + ( - 7 i - 1) q^{50} - 4 q^{51} + ( - 3 i - 3) q^{53} - 8 q^{54} + (i - 2) q^{55} - 8 q^{56} + (8 i + 8) q^{57} + ( - 6 i + 6) q^{58} + 4 i q^{59} + (2 i + 6) q^{60} + 14 i q^{61} + (2 i + 2) q^{62} + (2 i + 2) q^{63} + 8 i q^{64} - 2 i q^{66} + ( - 3 i - 3) q^{67} + ( - 4 i + 4) q^{68} - 2 q^{69} + ( - 8 i - 4) q^{70} + 8 i q^{71} + ( - 2 i + 2) q^{72} + (6 i - 6) q^{73} + 6 i q^{74} + (7 i + 1) q^{75} - 16 q^{76} + (2 i - 2) q^{77} + 12 q^{79} + (4 i - 8) q^{80} + 5 q^{81} + ( - 6 i + 6) q^{82} + (10 i - 10) q^{83} + 8 q^{84} + (2 i + 6) q^{85} + 12 i q^{86} + (6 i - 6) q^{87} + (2 i + 2) q^{88} + 10 i q^{89} + (i + 3) q^{90} + ( - 2 i + 2) q^{92} + ( - 2 i - 2) q^{93} + 2 i q^{94} + ( - 16 i - 8) q^{95} - 8 i q^{96} + ( - 5 i - 5) q^{97} + ( - i - 1) q^{98} - i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 4 q^{5} + 4 q^{7} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{3} + 4 q^{5} + 4 q^{7} - 4 q^{8} + 2 q^{10} - 2 q^{11} + 4 q^{12} - 2 q^{15} - 8 q^{16} + 4 q^{17} + 2 q^{18} - 4 q^{20} - 2 q^{22} + 2 q^{23} + 8 q^{24} + 6 q^{25} - 8 q^{27} - 8 q^{28} + 12 q^{29} + 4 q^{30} - 8 q^{32} + 2 q^{33} + 8 q^{34} + 4 q^{35} + 4 q^{36} - 6 q^{37} - 16 q^{38} - 12 q^{40} + 12 q^{41} + 8 q^{42} - 12 q^{43} + 2 q^{45} + 4 q^{46} - 2 q^{47} + 8 q^{48} - 2 q^{50} - 8 q^{51} - 6 q^{53} - 16 q^{54} - 4 q^{55} - 16 q^{56} + 16 q^{57} + 12 q^{58} + 12 q^{60} + 4 q^{62} + 4 q^{63} - 6 q^{67} + 8 q^{68} - 4 q^{69} - 8 q^{70} + 4 q^{72} - 12 q^{73} + 2 q^{75} - 32 q^{76} - 4 q^{77} + 24 q^{79} - 16 q^{80} + 10 q^{81} + 12 q^{82} - 20 q^{83} + 16 q^{84} + 12 q^{85} - 12 q^{87} + 4 q^{88} + 6 q^{90} + 4 q^{92} - 4 q^{93} - 16 q^{95} - 10 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/440\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(177\) \(221\) \(321\)
\(\chi(n)\) \(-1\) \(i\) \(-1\) \(1\)

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 + 1.00000i 2.00000i 2.00000 1.00000i 2.00000i 2.00000 2.00000i −2.00000 2.00000i 1.00000i 1.00000 3.00000i
243.1 1.00000 + 1.00000i −1.00000 1.00000i 2.00000i 2.00000 + 1.00000i 2.00000i 2.00000 + 2.00000i −2.00000 + 2.00000i 1.00000i 1.00000 + 3.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 440.2.r.d yes 2
5.c odd 4 1 440.2.r.a 2
8.d odd 2 1 440.2.r.a 2
40.k even 4 1 inner 440.2.r.d yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.r.a 2 5.c odd 4 1
440.2.r.a 2 8.d odd 2 1
440.2.r.d yes 2 1.a even 1 1 trivial
440.2.r.d yes 2 40.k even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(440, [\chi])\):

\( T_{3}^{2} + 2T_{3} + 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 4T_{7} + 8 \) Copy content Toggle raw display
\( T_{23}^{2} - 2T_{23} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$5$ \( T^{2} - 4T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 8 \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 4T + 8 \) Copy content Toggle raw display
$19$ \( T^{2} + 64 \) Copy content Toggle raw display
$23$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 4 \) Copy content Toggle raw display
$37$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 12T + 72 \) Copy content Toggle raw display
$47$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$53$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$59$ \( T^{2} + 16 \) Copy content Toggle raw display
$61$ \( T^{2} + 196 \) Copy content Toggle raw display
$67$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$71$ \( T^{2} + 64 \) Copy content Toggle raw display
$73$ \( T^{2} + 12T + 72 \) Copy content Toggle raw display
$79$ \( (T - 12)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 20T + 200 \) Copy content Toggle raw display
$89$ \( T^{2} + 100 \) Copy content Toggle raw display
$97$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
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