Properties

Label 5700.2.f.n
Level $5700$
Weight $2$
Character orbit 5700.f
Analytic conductor $45.515$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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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.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(45.5147291521\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1140)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{2} - \beta_1) q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + (\beta_{2} - \beta_1) q^{7} - q^{9} + ( - \beta_{3} - 1) q^{11} + (\beta_{2} + \beta_1) q^{13} - 2 \beta_1 q^{17} + q^{19} + (\beta_{3} - 1) q^{21} - 2 \beta_1 q^{23} + \beta_1 q^{27} + ( - \beta_{3} + 1) q^{29} + 4 q^{31} + (\beta_{2} + \beta_1) q^{33} + (\beta_{2} - 7 \beta_1) q^{37} + (\beta_{3} + 1) q^{39} + (\beta_{3} + 3) q^{41} + (\beta_{2} + 7 \beta_1) q^{43} - 6 \beta_1 q^{47} + (2 \beta_{3} - 7) q^{49} - 2 q^{51} - \beta_1 q^{57} + ( - 2 \beta_{3} + 2) q^{59} - 2 \beta_{3} q^{61} + ( - \beta_{2} + \beta_1) q^{63} - 4 \beta_1 q^{67} - 2 q^{69} + ( - 2 \beta_{3} - 2) q^{71} + 6 \beta_1 q^{73} - 12 \beta_1 q^{77} - 8 q^{79} + q^{81} + ( - 2 \beta_{2} - 4 \beta_1) q^{83} + (\beta_{2} - \beta_1) q^{87} + ( - \beta_{3} - 3) q^{89} - 12 q^{91} - 4 \beta_1 q^{93} + ( - \beta_{2} - 13 \beta_1) q^{97} + (\beta_{3} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} - 4 q^{11} + 4 q^{19} - 4 q^{21} + 4 q^{29} + 16 q^{31} + 4 q^{39} + 12 q^{41} - 28 q^{49} - 8 q^{51} + 8 q^{59} - 8 q^{69} - 8 q^{71} - 32 q^{79} + 4 q^{81} - 12 q^{89} - 48 q^{91} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 4\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 10\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{2} + 5\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5700\mathbb{Z}\right)^\times\).

\(n\) \(1901\) \(2851\) \(3877\) \(4201\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3649.1
2.30278i
1.30278i
1.30278i
2.30278i
0 1.00000i 0 0 0 4.60555i 0 −1.00000 0
3649.2 0 1.00000i 0 0 0 2.60555i 0 −1.00000 0
3649.3 0 1.00000i 0 0 0 2.60555i 0 −1.00000 0
3649.4 0 1.00000i 0 0 0 4.60555i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5700.2.f.n 4
5.b even 2 1 inner 5700.2.f.n 4
5.c odd 4 1 1140.2.a.e 2
5.c odd 4 1 5700.2.a.u 2
15.e even 4 1 3420.2.a.i 2
20.e even 4 1 4560.2.a.bl 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1140.2.a.e 2 5.c odd 4 1
3420.2.a.i 2 15.e even 4 1
4560.2.a.bl 2 20.e even 4 1
5700.2.a.u 2 5.c odd 4 1
5700.2.f.n 4 1.a even 1 1 trivial
5700.2.f.n 4 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5700, [\chi])\):

\( T_{7}^{4} + 28T_{7}^{2} + 144 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} - 12 \) Copy content Toggle raw display
\( T_{13}^{4} + 28T_{13}^{2} + 144 \) Copy content Toggle raw display
\( T_{17}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 28T^{2} + 144 \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T - 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 28T^{2} + 144 \) Copy content Toggle raw display
$17$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$19$ \( (T - 1)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 2 T - 12)^{2} \) Copy content Toggle raw display
$31$ \( (T - 4)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 124T^{2} + 1296 \) Copy content Toggle raw display
$41$ \( (T^{2} - 6 T - 4)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 124T^{2} + 1296 \) Copy content Toggle raw display
$47$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 4 T - 48)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 52)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 4 T - 48)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$79$ \( (T + 8)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 136T^{2} + 1296 \) Copy content Toggle raw display
$89$ \( (T^{2} + 6 T - 4)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 364 T^{2} + 24336 \) Copy content Toggle raw display
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