Properties

Label 5700.2.f.q
Level $5700$
Weight $2$
Character orbit 5700.f
Analytic conductor $45.515$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.5089536.1
Defining polynomial: \(x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 16 x^{2} - 24 x + 18\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + ( \beta_{1} - \beta_{4} ) q^{7} - q^{9} +O(q^{10})\) \( q + \beta_{4} q^{3} + ( \beta_{1} - \beta_{4} ) q^{7} - q^{9} + ( 2 - \beta_{2} - \beta_{3} ) q^{11} + ( \beta_{1} + \beta_{4} ) q^{13} + ( -2 \beta_{1} - \beta_{4} + \beta_{5} ) q^{17} + q^{19} + ( 1 - \beta_{3} ) q^{21} + ( -2 \beta_{1} + \beta_{4} - \beta_{5} ) q^{23} -\beta_{4} q^{27} + ( -\beta_{2} - \beta_{3} ) q^{29} + ( 1 - \beta_{2} + 2 \beta_{3} ) q^{31} + ( -\beta_{1} + 2 \beta_{4} + \beta_{5} ) q^{33} + ( -3 \beta_{1} + \beta_{4} ) q^{37} + ( -1 - \beta_{3} ) q^{39} + ( 4 + \beta_{2} + \beta_{3} ) q^{41} + ( \beta_{1} - 3 \beta_{4} - 2 \beta_{5} ) q^{43} + ( -2 \beta_{1} - 3 \beta_{4} - \beta_{5} ) q^{47} + ( \beta_{2} + 4 \beta_{3} ) q^{49} + ( 1 + \beta_{2} + 2 \beta_{3} ) q^{51} + ( -2 \beta_{1} + 3 \beta_{4} + 3 \beta_{5} ) q^{53} + \beta_{4} q^{57} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{59} + ( 5 - \beta_{2} ) q^{61} + ( -\beta_{1} + \beta_{4} ) q^{63} + ( 4 \beta_{1} + 4 \beta_{4} ) q^{67} + ( -1 - \beta_{2} + 2 \beta_{3} ) q^{69} + ( 4 + 2 \beta_{2} - 2 \beta_{3} ) q^{71} + ( -4 \beta_{4} - 2 \beta_{5} ) q^{73} + ( 6 \beta_{1} - 5 \beta_{4} - \beta_{5} ) q^{77} + ( 6 - 2 \beta_{2} ) q^{79} + q^{81} + ( \beta_{4} + \beta_{5} ) q^{83} + ( -\beta_{1} + \beta_{5} ) q^{87} + ( -\beta_{2} - \beta_{3} ) q^{89} + ( -5 + \beta_{2} + 2 \beta_{3} ) q^{91} + ( 2 \beta_{1} + \beta_{4} + \beta_{5} ) q^{93} + ( -\beta_{1} + 3 \beta_{4} ) q^{97} + ( -2 + \beta_{2} + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{9} + O(q^{10}) \) \( 6 q - 6 q^{9} + 12 q^{11} + 6 q^{19} + 8 q^{21} - 4 q^{39} + 24 q^{41} - 6 q^{49} + 4 q^{51} + 8 q^{59} + 28 q^{61} - 12 q^{69} + 32 q^{71} + 32 q^{79} + 6 q^{81} - 32 q^{91} - 12 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 16 x^{2} - 24 x + 18\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -5 \nu^{5} - 11 \nu^{4} + 101 \nu^{3} - 136 \nu^{2} + 292 \nu - 147 \)\()/393\)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{5} + 48 \nu^{4} - 12 \nu^{3} - 2 \nu^{2} + 12 \nu + 701 \)\()/131\)
\(\beta_{3}\)\(=\)\((\)\( -6 \nu^{5} + 13 \nu^{4} - 36 \nu^{3} - 6 \nu^{2} + 36 \nu + 7 \)\()/131\)
\(\beta_{4}\)\(=\)\((\)\( 23 \nu^{5} - 28 \nu^{4} + 7 \nu^{3} + 154 \nu^{2} + 386 \nu - 267 \)\()/393\)
\(\beta_{5}\)\(=\)\((\)\( -161 \nu^{5} + 196 \nu^{4} - 49 \nu^{3} - 292 \nu^{2} - 2702 \nu + 1869 \)\()/393\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + \beta_{3} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} + 7 \beta_{4}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{5} + 5 \beta_{4} - 4 \beta_{3} + \beta_{2} + 4 \beta_{1} - 5\)\()/2\)
\(\nu^{4}\)\(=\)\(-\beta_{3} + 3 \beta_{2} - 16\)
\(\nu^{5}\)\(=\)\((\)\(-7 \beta_{5} - 31 \beta_{4} - 18 \beta_{3} + 7 \beta_{2} - 18 \beta_{1} - 31\)\()/2\)

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
1.66044 1.66044i
0.675970 0.675970i
−1.33641 + 1.33641i
−1.33641 1.33641i
0.675970 + 0.675970i
1.66044 + 1.66044i
0 1.00000i 0 0 0 1.32088i 0 −1.00000 0
3649.2 0 1.00000i 0 0 0 0.648061i 0 −1.00000 0
3649.3 0 1.00000i 0 0 0 4.67282i 0 −1.00000 0
3649.4 0 1.00000i 0 0 0 4.67282i 0 −1.00000 0
3649.5 0 1.00000i 0 0 0 0.648061i 0 −1.00000 0
3649.6 0 1.00000i 0 0 0 1.32088i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3649.6
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.q 6
5.b even 2 1 inner 5700.2.f.q 6
5.c odd 4 1 1140.2.a.g 3
5.c odd 4 1 5700.2.a.w 3
15.e even 4 1 3420.2.a.m 3
20.e even 4 1 4560.2.a.br 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1140.2.a.g 3 5.c odd 4 1
3420.2.a.m 3 15.e even 4 1
4560.2.a.br 3 20.e even 4 1
5700.2.a.w 3 5.c odd 4 1
5700.2.f.q 6 1.a even 1 1 trivial
5700.2.f.q 6 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 \(S_{2}^{\mathrm{new}}(5700, [\chi])\):

\( T_{7}^{6} + 24 T_{7}^{4} + 48 T_{7}^{2} + 16 \)
\( T_{11}^{3} - 6 T_{11}^{2} - 12 T_{11} + 76 \)
\( T_{13}^{6} + 20 T_{13}^{4} + 112 T_{13}^{2} + 144 \)
\( T_{17}^{6} + 92 T_{17}^{4} + 2224 T_{17}^{2} + 5184 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( ( 1 + T^{2} )^{3} \)
$5$ \( T^{6} \)
$7$ \( 16 + 48 T^{2} + 24 T^{4} + T^{6} \)
$11$ \( ( 76 - 12 T - 6 T^{2} + T^{3} )^{2} \)
$13$ \( 144 + 112 T^{2} + 20 T^{4} + T^{6} \)
$17$ \( 5184 + 2224 T^{2} + 92 T^{4} + T^{6} \)
$19$ \( ( -1 + T )^{6} \)
$23$ \( 118336 + 7728 T^{2} + 156 T^{4} + T^{6} \)
$29$ \( ( 36 - 24 T + T^{3} )^{2} \)
$31$ \( ( -208 - 72 T + T^{3} )^{2} \)
$37$ \( 16 + 5136 T^{2} + 180 T^{4} + T^{6} \)
$41$ \( ( -4 + 24 T - 12 T^{2} + T^{3} )^{2} \)
$43$ \( 219024 + 11088 T^{2} + 184 T^{4} + T^{6} \)
$47$ \( 5184 + 2736 T^{2} + 156 T^{4} + T^{6} \)
$53$ \( 2214144 + 52032 T^{2} + 400 T^{4} + T^{6} \)
$59$ \( ( 864 - 144 T - 4 T^{2} + T^{3} )^{2} \)
$61$ \( ( 8 + 44 T - 14 T^{2} + T^{3} )^{2} \)
$67$ \( 589824 + 28672 T^{2} + 320 T^{4} + T^{6} \)
$71$ \( ( 1312 - 64 T - 16 T^{2} + T^{3} )^{2} \)
$73$ \( 107584 + 9264 T^{2} + 204 T^{4} + T^{6} \)
$79$ \( ( 384 - 16 T^{2} + T^{3} )^{2} \)
$83$ \( 576 + 496 T^{2} + 44 T^{4} + T^{6} \)
$89$ \( ( 36 - 24 T + T^{3} )^{2} \)
$97$ \( 144 + 336 T^{2} + 52 T^{4} + T^{6} \)
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