Properties

Label 2450.2.a.bo
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: \(\Z[a_1, \ldots, a_{19}]\)
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{8} -3 q^{9} +O(q^{10})\) \( q + q^{2} + q^{4} + q^{8} -3 q^{9} -4 q^{11} + 3 \beta q^{13} + q^{16} -3 \beta q^{17} -3 q^{18} -4 \beta q^{19} -4 q^{22} + 3 \beta q^{26} -4 q^{29} + 4 \beta q^{31} + q^{32} -3 \beta q^{34} -3 q^{36} -6 q^{37} -4 \beta q^{38} + \beta q^{41} -12 q^{43} -4 q^{44} + 3 \beta q^{52} -12 q^{53} -4 q^{58} + 8 \beta q^{59} -5 \beta q^{61} + 4 \beta q^{62} + q^{64} -12 q^{67} -3 \beta q^{68} + 8 q^{71} -3 q^{72} -3 \beta q^{73} -6 q^{74} -4 \beta q^{76} + 9 q^{81} + \beta q^{82} + 12 \beta q^{83} -12 q^{86} -4 q^{88} + 3 \beta q^{89} + 3 \beta q^{97} + 12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 2q^{8} - 6q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 2q^{8} - 6q^{9} - 8q^{11} + 2q^{16} - 6q^{18} - 8q^{22} - 8q^{29} + 2q^{32} - 6q^{36} - 12q^{37} - 24q^{43} - 8q^{44} - 24q^{53} - 8q^{58} + 2q^{64} - 24q^{67} + 16q^{71} - 6q^{72} - 12q^{74} + 18q^{81} - 24q^{86} - 8q^{88} + 24q^{99} + O(q^{100}) \)

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
−1.41421
1.41421
1.00000 0 1.00000 0 0 0 1.00000 −3.00000 0
1.2 1.00000 0 1.00000 0 0 0 1.00000 −3.00000 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.bo 2
5.b even 2 1 2450.2.a.bi 2
5.c odd 4 2 490.2.c.g 4
7.b odd 2 1 inner 2450.2.a.bo 2
35.c odd 2 1 2450.2.a.bi 2
35.f even 4 2 490.2.c.g 4
35.k even 12 4 490.2.i.d 8
35.l odd 12 4 490.2.i.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.c.g 4 5.c odd 4 2
490.2.c.g 4 35.f even 4 2
490.2.i.d 8 35.k even 12 4
490.2.i.d 8 35.l odd 12 4
2450.2.a.bi 2 5.b even 2 1
2450.2.a.bi 2 35.c odd 2 1
2450.2.a.bo 2 1.a even 1 1 trivial
2450.2.a.bo 2 7.b odd 2 1 inner

Hecke kernels

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

\( T_{3} \)
\( T_{11} + 4 \)
\( T_{13}^{2} - 18 \)
\( T_{17}^{2} - 18 \)
\( T_{19}^{2} - 32 \)
\( T_{23} \)
\( T_{37} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( -18 + T^{2} \)
$17$ \( -18 + T^{2} \)
$19$ \( -32 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( 4 + T )^{2} \)
$31$ \( -32 + T^{2} \)
$37$ \( ( 6 + T )^{2} \)
$41$ \( -2 + T^{2} \)
$43$ \( ( 12 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( ( 12 + T )^{2} \)
$59$ \( -128 + T^{2} \)
$61$ \( -50 + T^{2} \)
$67$ \( ( 12 + T )^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( -18 + T^{2} \)
$79$ \( T^{2} \)
$83$ \( -288 + T^{2} \)
$89$ \( -18 + T^{2} \)
$97$ \( -18 + T^{2} \)
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