Properties

Label 2450.2.a.bj
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 98)
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.

\(f(q)\) \(=\) \( q - q^{2} + \beta q^{3} + q^{4} -\beta q^{6} - q^{8} - q^{9} +O(q^{10})\) \( q - q^{2} + \beta q^{3} + q^{4} -\beta q^{6} - q^{8} - q^{9} -2 q^{11} + \beta q^{12} + q^{16} + \beta q^{17} + q^{18} -5 \beta q^{19} + 2 q^{22} + 4 q^{23} -\beta q^{24} -4 \beta q^{27} + 2 q^{29} + 6 \beta q^{31} - q^{32} -2 \beta q^{33} -\beta q^{34} - q^{36} -10 q^{37} + 5 \beta q^{38} -7 \beta q^{41} -2 q^{43} -2 q^{44} -4 q^{46} -2 \beta q^{47} + \beta q^{48} + 2 q^{51} + 2 q^{53} + 4 \beta q^{54} -10 q^{57} -2 q^{58} -\beta q^{59} + 2 \beta q^{61} -6 \beta q^{62} + q^{64} + 2 \beta q^{66} -12 q^{67} + \beta q^{68} + 4 \beta q^{69} -12 q^{71} + q^{72} + \beta q^{73} + 10 q^{74} -5 \beta q^{76} -4 q^{79} -5 q^{81} + 7 \beta q^{82} -7 \beta q^{83} + 2 q^{86} + 2 \beta q^{87} + 2 q^{88} -5 \beta q^{89} + 4 q^{92} + 12 q^{93} + 2 \beta q^{94} -\beta q^{96} -7 \beta q^{97} + 2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} - 2q^{8} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} - 2q^{8} - 2q^{9} - 4q^{11} + 2q^{16} + 2q^{18} + 4q^{22} + 8q^{23} + 4q^{29} - 2q^{32} - 2q^{36} - 20q^{37} - 4q^{43} - 4q^{44} - 8q^{46} + 4q^{51} + 4q^{53} - 20q^{57} - 4q^{58} + 2q^{64} - 24q^{67} - 24q^{71} + 2q^{72} + 20q^{74} - 8q^{79} - 10q^{81} + 4q^{86} + 4q^{88} + 8q^{92} + 24q^{93} + 4q^{99} + O(q^{100}) \)

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 −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.bj 2
5.b even 2 1 98.2.a.b 2
5.c odd 4 2 2450.2.c.v 4
7.b odd 2 1 inner 2450.2.a.bj 2
15.d odd 2 1 882.2.a.n 2
20.d odd 2 1 784.2.a.l 2
35.c odd 2 1 98.2.a.b 2
35.f even 4 2 2450.2.c.v 4
35.i odd 6 2 98.2.c.c 4
35.j even 6 2 98.2.c.c 4
40.e odd 2 1 3136.2.a.bm 2
40.f even 2 1 3136.2.a.bn 2
60.h even 2 1 7056.2.a.cl 2
105.g even 2 1 882.2.a.n 2
105.o odd 6 2 882.2.g.l 4
105.p even 6 2 882.2.g.l 4
140.c even 2 1 784.2.a.l 2
140.p odd 6 2 784.2.i.m 4
140.s even 6 2 784.2.i.m 4
280.c odd 2 1 3136.2.a.bn 2
280.n even 2 1 3136.2.a.bm 2
420.o odd 2 1 7056.2.a.cl 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.2.a.b 2 5.b even 2 1
98.2.a.b 2 35.c odd 2 1
98.2.c.c 4 35.i odd 6 2
98.2.c.c 4 35.j even 6 2
784.2.a.l 2 20.d odd 2 1
784.2.a.l 2 140.c even 2 1
784.2.i.m 4 140.p odd 6 2
784.2.i.m 4 140.s even 6 2
882.2.a.n 2 15.d odd 2 1
882.2.a.n 2 105.g even 2 1
882.2.g.l 4 105.o odd 6 2
882.2.g.l 4 105.p even 6 2
2450.2.a.bj 2 1.a even 1 1 trivial
2450.2.a.bj 2 7.b odd 2 1 inner
2450.2.c.v 4 5.c odd 4 2
2450.2.c.v 4 35.f even 4 2
3136.2.a.bm 2 40.e odd 2 1
3136.2.a.bm 2 280.n even 2 1
3136.2.a.bn 2 40.f even 2 1
3136.2.a.bn 2 280.c odd 2 1
7056.2.a.cl 2 60.h even 2 1
7056.2.a.cl 2 420.o odd 2 1

Hecke kernels

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

\( T_{3}^{2} - 2 \)
\( T_{11} + 2 \)
\( T_{13} \)
\( T_{17}^{2} - 2 \)
\( T_{19}^{2} - 50 \)
\( T_{23} - 4 \)
\( T_{37} + 10 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( -2 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 2 + T )^{2} \)
$13$ \( T^{2} \)
$17$ \( -2 + T^{2} \)
$19$ \( -50 + T^{2} \)
$23$ \( ( -4 + T )^{2} \)
$29$ \( ( -2 + T )^{2} \)
$31$ \( -72 + T^{2} \)
$37$ \( ( 10 + T )^{2} \)
$41$ \( -98 + T^{2} \)
$43$ \( ( 2 + T )^{2} \)
$47$ \( -8 + T^{2} \)
$53$ \( ( -2 + T )^{2} \)
$59$ \( -2 + T^{2} \)
$61$ \( -8 + T^{2} \)
$67$ \( ( 12 + T )^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( -2 + T^{2} \)
$79$ \( ( 4 + T )^{2} \)
$83$ \( -98 + T^{2} \)
$89$ \( -50 + T^{2} \)
$97$ \( -98 + T^{2} \)
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