Properties

Label 4400.2.a.bq
Level $4400$
Weight $2$
Character orbit 4400.a
Self dual yes
Analytic conductor $35.134$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(35.1341768894\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2200)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + ( -1 + \beta ) q^{7} + ( -2 + \beta ) q^{9} +O(q^{10})\) \( q + \beta q^{3} + ( -1 + \beta ) q^{7} + ( -2 + \beta ) q^{9} + q^{11} + ( -1 + 2 \beta ) q^{13} + 3 \beta q^{17} + ( -1 + 4 \beta ) q^{19} + q^{21} + ( 5 - 7 \beta ) q^{23} + ( 1 - 4 \beta ) q^{27} + ( -6 + \beta ) q^{29} + ( 1 + 2 \beta ) q^{31} + \beta q^{33} + ( 5 - 2 \beta ) q^{37} + ( 2 + \beta ) q^{39} + ( -5 + 8 \beta ) q^{41} + ( 2 + 4 \beta ) q^{43} + ( -5 + 6 \beta ) q^{47} + ( -5 - \beta ) q^{49} + ( 3 + 3 \beta ) q^{51} + ( 1 - 3 \beta ) q^{53} + ( 4 + 3 \beta ) q^{57} + ( 7 - 2 \beta ) q^{59} + ( -1 + 5 \beta ) q^{61} + ( 3 - 2 \beta ) q^{63} + ( -7 - 2 \beta ) q^{69} + ( 10 - 6 \beta ) q^{71} + ( 1 - 5 \beta ) q^{73} + ( -1 + \beta ) q^{77} + ( 2 - 3 \beta ) q^{79} + ( 2 - 6 \beta ) q^{81} + ( 6 - 3 \beta ) q^{83} + ( 1 - 5 \beta ) q^{87} + ( 2 - 9 \beta ) q^{89} + ( 3 - \beta ) q^{91} + ( 2 + 3 \beta ) q^{93} + ( 13 - 5 \beta ) q^{97} + ( -2 + \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - q^{7} - 3 q^{9} + O(q^{10}) \) \( 2 q + q^{3} - q^{7} - 3 q^{9} + 2 q^{11} + 3 q^{17} + 2 q^{19} + 2 q^{21} + 3 q^{23} - 2 q^{27} - 11 q^{29} + 4 q^{31} + q^{33} + 8 q^{37} + 5 q^{39} - 2 q^{41} + 8 q^{43} - 4 q^{47} - 11 q^{49} + 9 q^{51} - q^{53} + 11 q^{57} + 12 q^{59} + 3 q^{61} + 4 q^{63} - 16 q^{69} + 14 q^{71} - 3 q^{73} - q^{77} + q^{79} - 2 q^{81} + 9 q^{83} - 3 q^{87} - 5 q^{89} + 5 q^{91} + 7 q^{93} + 21 q^{97} - 3 q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.618034
1.61803
0 −0.618034 0 0 0 −1.61803 0 −2.61803 0
1.2 0 1.61803 0 0 0 0.618034 0 −0.381966 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4400.2.a.bq 2
4.b odd 2 1 2200.2.a.n 2
5.b even 2 1 4400.2.a.bk 2
5.c odd 4 2 4400.2.b.ba 4
20.d odd 2 1 2200.2.a.r yes 2
20.e even 4 2 2200.2.b.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2200.2.a.n 2 4.b odd 2 1
2200.2.a.r yes 2 20.d odd 2 1
2200.2.b.j 4 20.e even 4 2
4400.2.a.bk 2 5.b even 2 1
4400.2.a.bq 2 1.a even 1 1 trivial
4400.2.b.ba 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4400))\):

\( T_{3}^{2} - T_{3} - 1 \)
\( T_{7}^{2} + T_{7} - 1 \)
\( T_{13}^{2} - 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -1 - T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( -1 + T + T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( -5 + T^{2} \)
$17$ \( -9 - 3 T + T^{2} \)
$19$ \( -19 - 2 T + T^{2} \)
$23$ \( -59 - 3 T + T^{2} \)
$29$ \( 29 + 11 T + T^{2} \)
$31$ \( -1 - 4 T + T^{2} \)
$37$ \( 11 - 8 T + T^{2} \)
$41$ \( -79 + 2 T + T^{2} \)
$43$ \( -4 - 8 T + T^{2} \)
$47$ \( -41 + 4 T + T^{2} \)
$53$ \( -11 + T + T^{2} \)
$59$ \( 31 - 12 T + T^{2} \)
$61$ \( -29 - 3 T + T^{2} \)
$67$ \( T^{2} \)
$71$ \( 4 - 14 T + T^{2} \)
$73$ \( -29 + 3 T + T^{2} \)
$79$ \( -11 - T + T^{2} \)
$83$ \( 9 - 9 T + T^{2} \)
$89$ \( -95 + 5 T + T^{2} \)
$97$ \( 79 - 21 T + T^{2} \)
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