Properties

Label 5700.2.a.u
Level $5700$
Weight $2$
Character orbit 5700.a
Self dual yes
Analytic conductor $45.515$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5700.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(45.5147291521\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1140)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{13}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( - \beta - 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + ( - \beta - 1) q^{7} + q^{9} + (\beta - 1) q^{11} + (\beta - 1) q^{13} - 2 q^{17} - q^{19} + ( - \beta - 1) q^{21} + 2 q^{23} + q^{27} + ( - \beta - 1) q^{29} + 4 q^{31} + (\beta - 1) q^{33} + ( - \beta - 7) q^{37} + (\beta - 1) q^{39} + ( - \beta + 3) q^{41} + (\beta - 7) q^{43} - 6 q^{47} + (2 \beta + 7) q^{49} - 2 q^{51} - q^{57} + ( - 2 \beta - 2) q^{59} + 2 \beta q^{61} + ( - \beta - 1) q^{63} - 4 q^{67} + 2 q^{69} + (2 \beta - 2) q^{71} - 6 q^{73} - 12 q^{77} + 8 q^{79} + q^{81} + ( - 2 \beta + 4) q^{83} + ( - \beta - 1) q^{87} + ( - \beta + 3) q^{89} - 12 q^{91} + 4 q^{93} + (\beta - 13) q^{97} + (\beta - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} - 2 q^{11} - 2 q^{13} - 4 q^{17} - 2 q^{19} - 2 q^{21} + 4 q^{23} + 2 q^{27} - 2 q^{29} + 8 q^{31} - 2 q^{33} - 14 q^{37} - 2 q^{39} + 6 q^{41} - 14 q^{43} - 12 q^{47} + 14 q^{49} - 4 q^{51} - 2 q^{57} - 4 q^{59} - 2 q^{63} - 8 q^{67} + 4 q^{69} - 4 q^{71} - 12 q^{73} - 24 q^{77} + 16 q^{79} + 2 q^{81} + 8 q^{83} - 2 q^{87} + 6 q^{89} - 24 q^{91} + 8 q^{93} - 26 q^{97} - 2 q^{99}+O(q^{100}) \) 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
2.30278
−1.30278
0 1.00000 0 0 0 −4.60555 0 1.00000 0
1.2 0 1.00000 0 0 0 2.60555 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(19\) \(1\)

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 5700.2.a.u 2
5.b even 2 1 1140.2.a.e 2
5.c odd 4 2 5700.2.f.n 4
15.d odd 2 1 3420.2.a.i 2
20.d odd 2 1 4560.2.a.bl 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1140.2.a.e 2 5.b even 2 1
3420.2.a.i 2 15.d odd 2 1
4560.2.a.bl 2 20.d odd 2 1
5700.2.a.u 2 1.a even 1 1 trivial
5700.2.f.n 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5700))\):

\( T_{7}^{2} + 2T_{7} - 12 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} - 12 \) Copy content Toggle raw display
\( T_{13}^{2} + 2T_{13} - 12 \) Copy content Toggle raw display
\( T_{17} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2T - 12 \) Copy content Toggle raw display
$11$ \( T^{2} + 2T - 12 \) Copy content Toggle raw display
$13$ \( T^{2} + 2T - 12 \) Copy content Toggle raw display
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( (T - 2)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 2T - 12 \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 14T + 36 \) Copy content Toggle raw display
$41$ \( T^{2} - 6T - 4 \) Copy content Toggle raw display
$43$ \( T^{2} + 14T + 36 \) Copy content Toggle raw display
$47$ \( (T + 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 4T - 48 \) Copy content Toggle raw display
$61$ \( T^{2} - 52 \) Copy content Toggle raw display
$67$ \( (T + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 4T - 48 \) Copy content Toggle raw display
$73$ \( (T + 6)^{2} \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 8T - 36 \) Copy content Toggle raw display
$89$ \( T^{2} - 6T - 4 \) Copy content Toggle raw display
$97$ \( T^{2} + 26T + 156 \) Copy content Toggle raw display
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