Properties

Label 440.2.b.a
Level $440$
Weight $2$
Character orbit 440.b
Analytic conductor $3.513$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 440.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.51341768894\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + (\beta - 1) q^{5} + \beta q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + (\beta - 1) q^{5} + \beta q^{7} - q^{9} - q^{11} + ( - \beta - 4) q^{15} - 2 \beta q^{17} - 4 q^{19} - 4 q^{21} + 3 \beta q^{23} + ( - 2 \beta - 3) q^{25} + 2 \beta q^{27} - 2 q^{29} + 8 q^{31} - \beta q^{33} + ( - \beta - 4) q^{35} - 2 \beta q^{37} - 6 q^{41} + 3 \beta q^{43} + ( - \beta + 1) q^{45} - \beta q^{47} + 3 q^{49} + 8 q^{51} + 6 \beta q^{53} + ( - \beta + 1) q^{55} - 4 \beta q^{57} - 4 q^{59} + 14 q^{61} - \beta q^{63} + 5 \beta q^{67} - 12 q^{69} + 8 q^{71} - 2 \beta q^{73} + ( - 3 \beta + 8) q^{75} - \beta q^{77} + 8 q^{79} - 11 q^{81} - \beta q^{83} + (2 \beta + 8) q^{85} - 2 \beta q^{87} + 10 q^{89} + 8 \beta q^{93} + ( - 4 \beta + 4) q^{95} - 4 \beta q^{97} + q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} - 2 q^{9} - 2 q^{11} - 8 q^{15} - 8 q^{19} - 8 q^{21} - 6 q^{25} - 4 q^{29} + 16 q^{31} - 8 q^{35} - 12 q^{41} + 2 q^{45} + 6 q^{49} + 16 q^{51} + 2 q^{55} - 8 q^{59} + 28 q^{61} - 24 q^{69} + 16 q^{71} + 16 q^{75} + 16 q^{79} - 22 q^{81} + 16 q^{85} + 20 q^{89} + 8 q^{95} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1
1.00000i
1.00000i
0 2.00000i 0 −1.00000 2.00000i 0 2.00000i 0 −1.00000 0
89.2 0 2.00000i 0 −1.00000 + 2.00000i 0 2.00000i 0 −1.00000 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 440.2.b.a 2
3.b odd 2 1 3960.2.d.c 2
4.b odd 2 1 880.2.b.b 2
5.b even 2 1 inner 440.2.b.a 2
5.c odd 4 1 2200.2.a.c 1
5.c odd 4 1 2200.2.a.i 1
15.d odd 2 1 3960.2.d.c 2
20.d odd 2 1 880.2.b.b 2
20.e even 4 1 4400.2.a.h 1
20.e even 4 1 4400.2.a.x 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.b.a 2 1.a even 1 1 trivial
440.2.b.a 2 5.b even 2 1 inner
880.2.b.b 2 4.b odd 2 1
880.2.b.b 2 20.d odd 2 1
2200.2.a.c 1 5.c odd 4 1
2200.2.a.i 1 5.c odd 4 1
3960.2.d.c 2 3.b odd 2 1
3960.2.d.c 2 15.d odd 2 1
4400.2.a.h 1 20.e even 4 1
4400.2.a.x 1 20.e even 4 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 36 \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( (T - 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 16 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 36 \) Copy content Toggle raw display
$47$ \( T^{2} + 4 \) Copy content Toggle raw display
$53$ \( T^{2} + 144 \) Copy content Toggle raw display
$59$ \( (T + 4)^{2} \) Copy content Toggle raw display
$61$ \( (T - 14)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 100 \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 16 \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 4 \) Copy content Toggle raw display
$89$ \( (T - 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 64 \) Copy content Toggle raw display
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