Properties

Label 4400.2.a.bi
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: \(\Q(\sqrt{21}) \)
Defining polynomial: \( x^{2} - x - 5 \) 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 1100)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{21})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{3} + (\beta - 3) q^{7} + (\beta + 2) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} + (\beta - 3) q^{7} + (\beta + 2) q^{9} + q^{11} + q^{13} + ( - \beta + 2) q^{17} + ( - 2 \beta - 1) q^{19} + (2 \beta - 5) q^{21} + ( - \beta - 1) q^{23} - 5 q^{27} + (\beta + 4) q^{29} + (2 \beta + 3) q^{31} - \beta q^{33} + (2 \beta - 3) q^{37} - \beta q^{39} + (2 \beta - 7) q^{41} - 10 q^{43} + (2 \beta - 7) q^{47} + ( - 5 \beta + 7) q^{49} + ( - \beta + 5) q^{51} + ( - 3 \beta - 3) q^{53} + (3 \beta + 10) q^{57} + (2 \beta + 5) q^{59} + ( - \beta + 7) q^{61} - q^{63} - 4 q^{67} + (2 \beta + 5) q^{69} + (6 \beta - 6) q^{71} + (\beta + 5) q^{73} + (\beta - 3) q^{77} + (7 \beta - 4) q^{79} + (2 \beta - 6) q^{81} + ( - 5 \beta + 4) q^{83} + ( - 5 \beta - 5) q^{87} + ( - \beta + 2) q^{89} + (\beta - 3) q^{91} + ( - 5 \beta - 10) q^{93} + ( - \beta + 9) q^{97} + (\beta + 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 5 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - 5 q^{7} + 5 q^{9} + 2 q^{11} + 2 q^{13} + 3 q^{17} - 4 q^{19} - 8 q^{21} - 3 q^{23} - 10 q^{27} + 9 q^{29} + 8 q^{31} - q^{33} - 4 q^{37} - q^{39} - 12 q^{41} - 20 q^{43} - 12 q^{47} + 9 q^{49} + 9 q^{51} - 9 q^{53} + 23 q^{57} + 12 q^{59} + 13 q^{61} - 2 q^{63} - 8 q^{67} + 12 q^{69} - 6 q^{71} + 11 q^{73} - 5 q^{77} - q^{79} - 10 q^{81} + 3 q^{83} - 15 q^{87} + 3 q^{89} - 5 q^{91} - 25 q^{93} + 17 q^{97} + 5 q^{99}+O(q^{100}) \) 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
2.79129
−1.79129
0 −2.79129 0 0 0 −0.208712 0 4.79129 0
1.2 0 1.79129 0 0 0 −4.79129 0 0.208712 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.bi 2
4.b odd 2 1 1100.2.a.h yes 2
5.b even 2 1 4400.2.a.bu 2
5.c odd 4 2 4400.2.b.s 4
12.b even 2 1 9900.2.a.bz 2
20.d odd 2 1 1100.2.a.g 2
20.e even 4 2 1100.2.b.d 4
60.h even 2 1 9900.2.a.bh 2
60.l odd 4 2 9900.2.c.x 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1100.2.a.g 2 20.d odd 2 1
1100.2.a.h yes 2 4.b odd 2 1
1100.2.b.d 4 20.e even 4 2
4400.2.a.bi 2 1.a even 1 1 trivial
4400.2.a.bu 2 5.b even 2 1
4400.2.b.s 4 5.c odd 4 2
9900.2.a.bh 2 60.h even 2 1
9900.2.a.bz 2 12.b even 2 1
9900.2.c.x 4 60.l odd 4 2

Hecke kernels

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

\( T_{3}^{2} + T_{3} - 5 \) Copy content Toggle raw display
\( T_{7}^{2} + 5T_{7} + 1 \) Copy content Toggle raw display
\( T_{13} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T - 5 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 5T + 1 \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 3T - 3 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T - 17 \) Copy content Toggle raw display
$23$ \( T^{2} + 3T - 3 \) Copy content Toggle raw display
$29$ \( T^{2} - 9T + 15 \) Copy content Toggle raw display
$31$ \( T^{2} - 8T - 5 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 17 \) Copy content Toggle raw display
$41$ \( T^{2} + 12T + 15 \) Copy content Toggle raw display
$43$ \( (T + 10)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 12T + 15 \) Copy content Toggle raw display
$53$ \( T^{2} + 9T - 27 \) Copy content Toggle raw display
$59$ \( T^{2} - 12T + 15 \) Copy content Toggle raw display
$61$ \( T^{2} - 13T + 37 \) Copy content Toggle raw display
$67$ \( (T + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 6T - 180 \) Copy content Toggle raw display
$73$ \( T^{2} - 11T + 25 \) Copy content Toggle raw display
$79$ \( T^{2} + T - 257 \) Copy content Toggle raw display
$83$ \( T^{2} - 3T - 129 \) Copy content Toggle raw display
$89$ \( T^{2} - 3T - 3 \) Copy content Toggle raw display
$97$ \( T^{2} - 17T + 67 \) Copy content Toggle raw display
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