Properties

Label 2450.2.a.bk
Level $2450$
Weight $2$
Character orbit 2450.a
Self dual yes
Analytic conductor $19.563$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.5633484952\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 490)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q - q^{2} + \beta q^{3} + q^{4} -\beta q^{6} - q^{8} - q^{9} +O(q^{10})\) \( q - q^{2} + \beta q^{3} + q^{4} -\beta q^{6} - q^{8} - q^{9} + \beta q^{12} + 3 \beta q^{13} + q^{16} -4 \beta q^{17} + q^{18} -3 \beta q^{19} -6 q^{23} -\beta q^{24} -3 \beta q^{26} -4 \beta q^{27} + 6 q^{29} -6 \beta q^{31} - q^{32} + 4 \beta q^{34} - q^{36} -6 q^{37} + 3 \beta q^{38} + 6 q^{39} + 6 \beta q^{41} -12 q^{43} + 6 q^{46} + 2 \beta q^{47} + \beta q^{48} -8 q^{51} + 3 \beta q^{52} + 6 q^{53} + 4 \beta q^{54} -6 q^{57} -6 q^{58} + 3 \beta q^{59} -3 \beta q^{61} + 6 \beta q^{62} + q^{64} -4 \beta q^{68} -6 \beta q^{69} + 6 q^{71} + q^{72} -6 \beta q^{73} + 6 q^{74} -3 \beta q^{76} -6 q^{78} -10 q^{79} -5 q^{81} -6 \beta q^{82} -11 \beta q^{83} + 12 q^{86} + 6 \beta q^{87} -6 q^{92} -12 q^{93} -2 \beta q^{94} -\beta q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} - 2q^{8} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} - 2q^{8} - 2q^{9} + 2q^{16} + 2q^{18} - 12q^{23} + 12q^{29} - 2q^{32} - 2q^{36} - 12q^{37} + 12q^{39} - 24q^{43} + 12q^{46} - 16q^{51} + 12q^{53} - 12q^{57} - 12q^{58} + 2q^{64} + 12q^{71} + 2q^{72} + 12q^{74} - 12q^{78} - 20q^{79} - 10q^{81} + 24q^{86} - 12q^{92} - 24q^{93} + O(q^{100}) \)

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
−1.41421
1.41421
−1.00000 −1.41421 1.00000 0 1.41421 0 −1.00000 −1.00000 0
1.2 −1.00000 1.41421 1.00000 0 −1.41421 0 −1.00000 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(7\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2450.2.a.bk 2
5.b even 2 1 2450.2.a.bp 2
5.c odd 4 2 490.2.c.f 4
7.b odd 2 1 inner 2450.2.a.bk 2
35.c odd 2 1 2450.2.a.bp 2
35.f even 4 2 490.2.c.f 4
35.k even 12 4 490.2.i.e 8
35.l odd 12 4 490.2.i.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.c.f 4 5.c odd 4 2
490.2.c.f 4 35.f even 4 2
490.2.i.e 8 35.k even 12 4
490.2.i.e 8 35.l odd 12 4
2450.2.a.bk 2 1.a even 1 1 trivial
2450.2.a.bk 2 7.b odd 2 1 inner
2450.2.a.bp 2 5.b even 2 1
2450.2.a.bp 2 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2450))\):

\( T_{3}^{2} - 2 \)
\( T_{11} \)
\( T_{13}^{2} - 18 \)
\( T_{17}^{2} - 32 \)
\( T_{19}^{2} - 18 \)
\( T_{23} + 6 \)
\( T_{37} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( -2 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( -18 + T^{2} \)
$17$ \( -32 + T^{2} \)
$19$ \( -18 + T^{2} \)
$23$ \( ( 6 + T )^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( -72 + T^{2} \)
$37$ \( ( 6 + T )^{2} \)
$41$ \( -72 + T^{2} \)
$43$ \( ( 12 + T )^{2} \)
$47$ \( -8 + T^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( -18 + T^{2} \)
$61$ \( -18 + T^{2} \)
$67$ \( T^{2} \)
$71$ \( ( -6 + T )^{2} \)
$73$ \( -72 + T^{2} \)
$79$ \( ( 10 + T )^{2} \)
$83$ \( -242 + T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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