Properties

Label 9200.2.a.bs
Level $9200$
Weight $2$
Character orbit 9200.a
Self dual yes
Analytic conductor $73.462$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9200.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(73.4623698596\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
Defining polynomial: \(x^{2} - x - 5\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{21})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{3} + ( 1 - \beta ) q^{7} + ( 2 + \beta ) q^{9} +O(q^{10})\) \( q -\beta q^{3} + ( 1 - \beta ) q^{7} + ( 2 + \beta ) q^{9} + ( -2 + \beta ) q^{11} + ( -3 - \beta ) q^{13} + ( 2 - \beta ) q^{17} + ( -3 - \beta ) q^{19} + 5 q^{21} + q^{23} -5 q^{27} + ( 2 + 2 \beta ) q^{29} + ( -5 + 3 \beta ) q^{31} + ( -5 + \beta ) q^{33} + 4 q^{37} + ( 5 + 4 \beta ) q^{39} + ( -4 - \beta ) q^{41} + 4 \beta q^{43} + ( -10 + 2 \beta ) q^{47} + ( -1 - \beta ) q^{49} + ( 5 - \beta ) q^{51} -6 q^{53} + ( 5 + 4 \beta ) q^{57} + ( 8 + 2 \beta ) q^{59} + ( 2 + 3 \beta ) q^{61} + ( -3 - 2 \beta ) q^{63} + 4 \beta q^{67} -\beta q^{69} -3 \beta q^{71} + ( 4 - 6 \beta ) q^{73} + ( -7 + 2 \beta ) q^{77} -8 q^{79} + ( -6 + 2 \beta ) q^{81} -6 q^{83} + ( -10 - 4 \beta ) q^{87} + ( 4 + 4 \beta ) q^{89} + ( 2 + 3 \beta ) q^{91} + ( -15 + 2 \beta ) q^{93} + ( -6 + 5 \beta ) q^{97} + ( 1 + \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} + q^{7} + 5q^{9} + O(q^{10}) \) \( 2q - q^{3} + q^{7} + 5q^{9} - 3q^{11} - 7q^{13} + 3q^{17} - 7q^{19} + 10q^{21} + 2q^{23} - 10q^{27} + 6q^{29} - 7q^{31} - 9q^{33} + 8q^{37} + 14q^{39} - 9q^{41} + 4q^{43} - 18q^{47} - 3q^{49} + 9q^{51} - 12q^{53} + 14q^{57} + 18q^{59} + 7q^{61} - 8q^{63} + 4q^{67} - q^{69} - 3q^{71} + 2q^{73} - 12q^{77} - 16q^{79} - 10q^{81} - 12q^{83} - 24q^{87} + 12q^{89} + 7q^{91} - 28q^{93} - 7q^{97} + 3q^{99} + O(q^{100}) \)

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
2.79129
−1.79129
0 −2.79129 0 0 0 −1.79129 0 4.79129 0
1.2 0 1.79129 0 0 0 2.79129 0 0.208712 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(23\) \(-1\)

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 9200.2.a.bs 2
4.b odd 2 1 1150.2.a.o 2
5.b even 2 1 1840.2.a.n 2
20.d odd 2 1 230.2.a.a 2
20.e even 4 2 1150.2.b.g 4
40.e odd 2 1 7360.2.a.bq 2
40.f even 2 1 7360.2.a.bk 2
60.h even 2 1 2070.2.a.x 2
460.g even 2 1 5290.2.a.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.a 2 20.d odd 2 1
1150.2.a.o 2 4.b odd 2 1
1150.2.b.g 4 20.e even 4 2
1840.2.a.n 2 5.b even 2 1
2070.2.a.x 2 60.h even 2 1
5290.2.a.e 2 460.g even 2 1
7360.2.a.bk 2 40.f even 2 1
7360.2.a.bq 2 40.e odd 2 1
9200.2.a.bs 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9200))\):

\( T_{3}^{2} + T_{3} - 5 \)
\( T_{7}^{2} - T_{7} - 5 \)
\( T_{11}^{2} + 3 T_{11} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -5 + T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( -5 - T + T^{2} \)
$11$ \( -3 + 3 T + T^{2} \)
$13$ \( 7 + 7 T + T^{2} \)
$17$ \( -3 - 3 T + T^{2} \)
$19$ \( 7 + 7 T + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( -12 - 6 T + T^{2} \)
$31$ \( -35 + 7 T + T^{2} \)
$37$ \( ( -4 + T )^{2} \)
$41$ \( 15 + 9 T + T^{2} \)
$43$ \( -80 - 4 T + T^{2} \)
$47$ \( 60 + 18 T + T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( 60 - 18 T + T^{2} \)
$61$ \( -35 - 7 T + T^{2} \)
$67$ \( -80 - 4 T + T^{2} \)
$71$ \( -45 + 3 T + T^{2} \)
$73$ \( -188 - 2 T + T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( ( 6 + T )^{2} \)
$89$ \( -48 - 12 T + T^{2} \)
$97$ \( -119 + 7 T + T^{2} \)
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