Properties

Label 4400.2.a.bh
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: \(\Q(\sqrt{13}) \)
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.

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

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