Properties

Label 4400.2.a.bu
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: \(\Q(\sqrt{21}) \)
Defining polynomial: \(x^{2} - x - 5\)
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.

\(f(q)\) \(=\) \( q + \beta q^{3} + ( 3 - \beta ) q^{7} + ( 2 + \beta ) q^{9} +O(q^{10})\) \( q + \beta q^{3} + ( 3 - \beta ) q^{7} + ( 2 + \beta ) q^{9} + q^{11} - q^{13} + ( -2 + \beta ) q^{17} + ( -1 - 2 \beta ) q^{19} + ( -5 + 2 \beta ) q^{21} + ( 1 + \beta ) q^{23} + 5 q^{27} + ( 4 + \beta ) q^{29} + ( 3 + 2 \beta ) q^{31} + \beta q^{33} + ( 3 - 2 \beta ) q^{37} -\beta q^{39} + ( -7 + 2 \beta ) q^{41} + 10 q^{43} + ( 7 - 2 \beta ) q^{47} + ( 7 - 5 \beta ) q^{49} + ( 5 - \beta ) q^{51} + ( 3 + 3 \beta ) q^{53} + ( -10 - 3 \beta ) q^{57} + ( 5 + 2 \beta ) q^{59} + ( 7 - \beta ) q^{61} + q^{63} + 4 q^{67} + ( 5 + 2 \beta ) q^{69} + ( -6 + 6 \beta ) q^{71} + ( -5 - \beta ) q^{73} + ( 3 - \beta ) q^{77} + ( -4 + 7 \beta ) q^{79} + ( -6 + 2 \beta ) q^{81} + ( -4 + 5 \beta ) q^{83} + ( 5 + 5 \beta ) q^{87} + ( 2 - \beta ) q^{89} + ( -3 + \beta ) q^{91} + ( 10 + 5 \beta ) q^{93} + ( -9 + \beta ) q^{97} + ( 2 + \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 5 q^{7} + 5 q^{9} + O(q^{10}) \) \( 2 q + q^{3} + 5 q^{7} + 5 q^{9} + 2 q^{11} - 2 q^{13} - 3 q^{17} - 4 q^{19} - 8 q^{21} + 3 q^{23} + 10 q^{27} + 9 q^{29} + 8 q^{31} + q^{33} + 4 q^{37} - q^{39} - 12 q^{41} + 20 q^{43} + 12 q^{47} + 9 q^{49} + 9 q^{51} + 9 q^{53} - 23 q^{57} + 12 q^{59} + 13 q^{61} + 2 q^{63} + 8 q^{67} + 12 q^{69} - 6 q^{71} - 11 q^{73} + 5 q^{77} - q^{79} - 10 q^{81} - 3 q^{83} + 15 q^{87} + 3 q^{89} - 5 q^{91} + 25 q^{93} - 17 q^{97} + 5 q^{99} + O(q^{100}) \)

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
−1.79129
2.79129
0 −1.79129 0 0 0 4.79129 0 0.208712 0
1.2 0 2.79129 0 0 0 0.208712 0 4.79129 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.bu 2
4.b odd 2 1 1100.2.a.g 2
5.b even 2 1 4400.2.a.bi 2
5.c odd 4 2 4400.2.b.s 4
12.b even 2 1 9900.2.a.bh 2
20.d odd 2 1 1100.2.a.h yes 2
20.e even 4 2 1100.2.b.d 4
60.h even 2 1 9900.2.a.bz 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 4.b odd 2 1
1100.2.a.h yes 2 20.d odd 2 1
1100.2.b.d 4 20.e even 4 2
4400.2.a.bi 2 5.b even 2 1
4400.2.a.bu 2 1.a even 1 1 trivial
4400.2.b.s 4 5.c odd 4 2
9900.2.a.bh 2 12.b even 2 1
9900.2.a.bz 2 60.h 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4400))\):

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

Hecke characteristic polynomials

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