Properties

Label 4400.2.a.bs
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: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + ( 3 - \beta ) q^{7} + \beta q^{9} +O(q^{10})\) \( q + \beta q^{3} + ( 3 - \beta ) q^{7} + \beta q^{9} + q^{11} -5 q^{13} -3 \beta q^{17} + q^{19} + ( -3 + 2 \beta ) q^{21} + ( -5 - \beta ) q^{23} + ( 3 - 2 \beta ) q^{27} + ( -6 + 3 \beta ) q^{29} + ( -1 - 4 \beta ) q^{31} + \beta q^{33} + ( -7 + 2 \beta ) q^{37} -5 \beta q^{39} + ( -1 - 2 \beta ) q^{41} + ( -2 + 4 \beta ) q^{43} -3 q^{47} + ( 5 - 5 \beta ) q^{49} + ( -9 - 3 \beta ) q^{51} + ( -1 + \beta ) q^{53} + \beta q^{57} + ( 5 + 4 \beta ) q^{59} + ( -1 - 3 \beta ) q^{61} + ( -3 + 2 \beta ) q^{63} -4 q^{67} + ( -3 - 6 \beta ) q^{69} + ( -2 + 2 \beta ) q^{71} + ( 1 + 3 \beta ) q^{73} + ( 3 - \beta ) q^{77} + ( 4 + 3 \beta ) q^{79} + ( -6 - 2 \beta ) q^{81} + ( 8 - 5 \beta ) q^{83} + ( 9 - 3 \beta ) q^{87} + ( 4 - \beta ) q^{89} + ( -15 + 5 \beta ) q^{91} + ( -12 - 5 \beta ) q^{93} + ( -13 - \beta ) q^{97} + \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 5 q^{7} + q^{9} + O(q^{10}) \) \( 2 q + q^{3} + 5 q^{7} + q^{9} + 2 q^{11} - 10 q^{13} - 3 q^{17} + 2 q^{19} - 4 q^{21} - 11 q^{23} + 4 q^{27} - 9 q^{29} - 6 q^{31} + q^{33} - 12 q^{37} - 5 q^{39} - 4 q^{41} - 6 q^{47} + 5 q^{49} - 21 q^{51} - q^{53} + q^{57} + 14 q^{59} - 5 q^{61} - 4 q^{63} - 8 q^{67} - 12 q^{69} - 2 q^{71} + 5 q^{73} + 5 q^{77} + 11 q^{79} - 14 q^{81} + 11 q^{83} + 15 q^{87} + 7 q^{89} - 25 q^{91} - 29 q^{93} - 27 q^{97} + q^{99} + O(q^{100}) \)

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
−1.30278
2.30278
0 −1.30278 0 0 0 4.30278 0 −1.30278 0
1.2 0 2.30278 0 0 0 0.697224 0 2.30278 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.bs 2
4.b odd 2 1 275.2.a.e 2
5.b even 2 1 4400.2.a.bh 2
5.c odd 4 2 4400.2.b.y 4
12.b even 2 1 2475.2.a.t 2
20.d odd 2 1 275.2.a.f yes 2
20.e even 4 2 275.2.b.c 4
44.c even 2 1 3025.2.a.n 2
60.h even 2 1 2475.2.a.o 2
60.l odd 4 2 2475.2.c.k 4
220.g even 2 1 3025.2.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.a.e 2 4.b odd 2 1
275.2.a.f yes 2 20.d odd 2 1
275.2.b.c 4 20.e even 4 2
2475.2.a.o 2 60.h even 2 1
2475.2.a.t 2 12.b even 2 1
2475.2.c.k 4 60.l odd 4 2
3025.2.a.h 2 220.g even 2 1
3025.2.a.n 2 44.c even 2 1
4400.2.a.bh 2 5.b even 2 1
4400.2.a.bs 2 1.a even 1 1 trivial
4400.2.b.y 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4400))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -3 - T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 3 - 5 T + T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( ( 5 + T )^{2} \)
$17$ \( -27 + 3 T + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( 27 + 11 T + T^{2} \)
$29$ \( -9 + 9 T + T^{2} \)
$31$ \( -43 + 6 T + T^{2} \)
$37$ \( 23 + 12 T + T^{2} \)
$41$ \( -9 + 4 T + T^{2} \)
$43$ \( -52 + T^{2} \)
$47$ \( ( 3 + T )^{2} \)
$53$ \( -3 + T + T^{2} \)
$59$ \( -3 - 14 T + T^{2} \)
$61$ \( -23 + 5 T + T^{2} \)
$67$ \( ( 4 + T )^{2} \)
$71$ \( -12 + 2 T + T^{2} \)
$73$ \( -23 - 5 T + T^{2} \)
$79$ \( 1 - 11 T + T^{2} \)
$83$ \( -51 - 11 T + T^{2} \)
$89$ \( 9 - 7 T + T^{2} \)
$97$ \( 179 + 27 T + T^{2} \)
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