Properties

Label 2450.2.a.br
Level $2450$
Weight $2$
Character orbit 2450.a
Self dual yes
Analytic conductor $19.563$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \(x^{2} - 7\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} + q^{8} + 4 q^{9} +O(q^{10})\) \( q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} + q^{8} + 4 q^{9} + 5 q^{11} + \beta q^{12} -2 \beta q^{13} + q^{16} + \beta q^{17} + 4 q^{18} + 3 \beta q^{19} + 5 q^{22} -4 q^{23} + \beta q^{24} -2 \beta q^{26} + \beta q^{27} + 6 q^{29} -4 \beta q^{31} + q^{32} + 5 \beta q^{33} + \beta q^{34} + 4 q^{36} -4 q^{37} + 3 \beta q^{38} -14 q^{39} -\beta q^{41} -8 q^{43} + 5 q^{44} -4 q^{46} + 2 \beta q^{47} + \beta q^{48} + 7 q^{51} -2 \beta q^{52} + 4 q^{53} + \beta q^{54} + 21 q^{57} + 6 q^{58} + 2 \beta q^{59} -2 \beta q^{61} -4 \beta q^{62} + q^{64} + 5 \beta q^{66} + 5 q^{67} + \beta q^{68} -4 \beta q^{69} + 6 q^{71} + 4 q^{72} -\beta q^{73} -4 q^{74} + 3 \beta q^{76} -14 q^{78} + 10 q^{79} -5 q^{81} -\beta q^{82} -\beta q^{83} -8 q^{86} + 6 \beta q^{87} + 5 q^{88} -5 \beta q^{89} -4 q^{92} -28 q^{93} + 2 \beta q^{94} + \beta q^{96} + 20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 2q^{8} + 8q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 2q^{8} + 8q^{9} + 10q^{11} + 2q^{16} + 8q^{18} + 10q^{22} - 8q^{23} + 12q^{29} + 2q^{32} + 8q^{36} - 8q^{37} - 28q^{39} - 16q^{43} + 10q^{44} - 8q^{46} + 14q^{51} + 8q^{53} + 42q^{57} + 12q^{58} + 2q^{64} + 10q^{67} + 12q^{71} + 8q^{72} - 8q^{74} - 28q^{78} + 20q^{79} - 10q^{81} - 16q^{86} + 10q^{88} - 8q^{92} - 56q^{93} + 40q^{99} + O(q^{100}) \)

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
−2.64575
2.64575
1.00000 −2.64575 1.00000 0 −2.64575 0 1.00000 4.00000 0
1.2 1.00000 2.64575 1.00000 0 2.64575 0 1.00000 4.00000 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.br yes 2
5.b even 2 1 2450.2.a.bm 2
5.c odd 4 2 2450.2.c.u 4
7.b odd 2 1 inner 2450.2.a.br yes 2
35.c odd 2 1 2450.2.a.bm 2
35.f even 4 2 2450.2.c.u 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2450.2.a.bm 2 5.b even 2 1
2450.2.a.bm 2 35.c odd 2 1
2450.2.a.br yes 2 1.a even 1 1 trivial
2450.2.a.br yes 2 7.b odd 2 1 inner
2450.2.c.u 4 5.c odd 4 2
2450.2.c.u 4 35.f even 4 2

Hecke kernels

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

\( T_{3}^{2} - 7 \)
\( T_{11} - 5 \)
\( T_{13}^{2} - 28 \)
\( T_{17}^{2} - 7 \)
\( T_{19}^{2} - 63 \)
\( T_{23} + 4 \)
\( T_{37} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( -7 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -5 + T )^{2} \)
$13$ \( -28 + T^{2} \)
$17$ \( -7 + T^{2} \)
$19$ \( -63 + T^{2} \)
$23$ \( ( 4 + T )^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( -112 + T^{2} \)
$37$ \( ( 4 + T )^{2} \)
$41$ \( -7 + T^{2} \)
$43$ \( ( 8 + T )^{2} \)
$47$ \( -28 + T^{2} \)
$53$ \( ( -4 + T )^{2} \)
$59$ \( -28 + T^{2} \)
$61$ \( -28 + T^{2} \)
$67$ \( ( -5 + T )^{2} \)
$71$ \( ( -6 + T )^{2} \)
$73$ \( -7 + T^{2} \)
$79$ \( ( -10 + T )^{2} \)
$83$ \( -7 + T^{2} \)
$89$ \( -175 + T^{2} \)
$97$ \( T^{2} \)
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