Properties

Label 4400.2.a.bv
Level $4400$
Weight $2$
Character orbit 4400.a
Self dual yes
Analytic conductor $35.134$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 275)
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.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{3} + ( - 3 \beta + 2) q^{7} + (3 \beta - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{3} + ( - 3 \beta + 2) q^{7} + (3 \beta - 1) q^{9} - q^{11} + ( - 2 \beta - 3) q^{13} + (\beta - 1) q^{17} + (6 \beta - 3) q^{19} + ( - 4 \beta - 1) q^{21} + ( - 5 \beta + 4) q^{23} + (2 \beta - 1) q^{27} + (\beta - 3) q^{29} + 3 q^{31} + ( - \beta - 1) q^{33} + ( - 2 \beta - 7) q^{37} + ( - 7 \beta - 5) q^{39} - 3 q^{41} - 6 q^{43} + (8 \beta - 1) q^{47} + ( - 3 \beta + 6) q^{49} + \beta q^{51} + ( - 7 \beta + 2) q^{53} + (9 \beta + 3) q^{57} + (4 \beta - 7) q^{59} + (5 \beta - 8) q^{61} - 11 q^{63} + 8 q^{67} + ( - 6 \beta - 1) q^{69} + ( - 10 \beta + 8) q^{71} + (\beta - 12) q^{73} + (3 \beta - 2) q^{77} + ( - 3 \beta - 1) q^{79} + ( - 6 \beta + 4) q^{81} + (3 \beta - 15) q^{83} + ( - \beta - 2) q^{87} + (5 \beta - 15) q^{89} + 11 \beta q^{91} + (3 \beta + 3) q^{93} - \beta q^{97} + ( - 3 \beta + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} + q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{3} + q^{7} + q^{9} - 2 q^{11} - 8 q^{13} - q^{17} - 6 q^{21} + 3 q^{23} - 5 q^{29} + 6 q^{31} - 3 q^{33} - 16 q^{37} - 17 q^{39} - 6 q^{41} - 12 q^{43} + 6 q^{47} + 9 q^{49} + q^{51} - 3 q^{53} + 15 q^{57} - 10 q^{59} - 11 q^{61} - 22 q^{63} + 16 q^{67} - 8 q^{69} + 6 q^{71} - 23 q^{73} - q^{77} - 5 q^{79} + 2 q^{81} - 27 q^{83} - 5 q^{87} - 25 q^{89} + 11 q^{91} + 9 q^{93} - q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 0.381966 0 0 0 3.85410 0 −2.85410 0
1.2 0 2.61803 0 0 0 −2.85410 0 3.85410 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.bv 2
4.b odd 2 1 275.2.a.d 2
5.b even 2 1 4400.2.a.bg 2
5.c odd 4 2 4400.2.b.x 4
12.b even 2 1 2475.2.a.s 2
20.d odd 2 1 275.2.a.g yes 2
20.e even 4 2 275.2.b.e 4
44.c even 2 1 3025.2.a.m 2
60.h even 2 1 2475.2.a.n 2
60.l odd 4 2 2475.2.c.p 4
220.g even 2 1 3025.2.a.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.a.d 2 4.b odd 2 1
275.2.a.g yes 2 20.d odd 2 1
275.2.b.e 4 20.e even 4 2
2475.2.a.n 2 60.h even 2 1
2475.2.a.s 2 12.b even 2 1
2475.2.c.p 4 60.l odd 4 2
3025.2.a.i 2 220.g even 2 1
3025.2.a.m 2 44.c even 2 1
4400.2.a.bg 2 5.b even 2 1
4400.2.a.bv 2 1.a even 1 1 trivial
4400.2.b.x 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4400))\):

\( T_{3}^{2} - 3T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} - T_{7} - 11 \) Copy content Toggle raw display
\( T_{13}^{2} + 8T_{13} + 11 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - T - 11 \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 8T + 11 \) Copy content Toggle raw display
$17$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$19$ \( T^{2} - 45 \) Copy content Toggle raw display
$23$ \( T^{2} - 3T - 29 \) Copy content Toggle raw display
$29$ \( T^{2} + 5T + 5 \) Copy content Toggle raw display
$31$ \( (T - 3)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 16T + 59 \) Copy content Toggle raw display
$41$ \( (T + 3)^{2} \) Copy content Toggle raw display
$43$ \( (T + 6)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 6T - 71 \) Copy content Toggle raw display
$53$ \( T^{2} + 3T - 59 \) Copy content Toggle raw display
$59$ \( T^{2} + 10T + 5 \) Copy content Toggle raw display
$61$ \( T^{2} + 11T - 1 \) Copy content Toggle raw display
$67$ \( (T - 8)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 6T - 116 \) Copy content Toggle raw display
$73$ \( T^{2} + 23T + 131 \) Copy content Toggle raw display
$79$ \( T^{2} + 5T - 5 \) Copy content Toggle raw display
$83$ \( T^{2} + 27T + 171 \) Copy content Toggle raw display
$89$ \( T^{2} + 25T + 125 \) Copy content Toggle raw display
$97$ \( T^{2} + T - 1 \) Copy content Toggle raw display
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