Properties

Label 2450.2.a.bn
Level $2450$
Weight $2$
Character orbit 2450.a
Self dual yes
Analytic conductor $19.563$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

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: \(\Q(\sqrt{2}) \)
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.

\(f(q)\) \(=\) \( q + q^{2} + ( -2 + \beta ) q^{3} + q^{4} + ( -2 + \beta ) q^{6} + q^{8} + ( 3 - 4 \beta ) q^{9} +O(q^{10})\) \( q + q^{2} + ( -2 + \beta ) q^{3} + q^{4} + ( -2 + \beta ) q^{6} + q^{8} + ( 3 - 4 \beta ) q^{9} + ( 2 + 2 \beta ) q^{11} + ( -2 + \beta ) q^{12} + ( 2 - 2 \beta ) q^{13} + q^{16} + ( -4 - \beta ) q^{17} + ( 3 - 4 \beta ) q^{18} + ( 2 + \beta ) q^{19} + ( 2 + 2 \beta ) q^{22} + ( 4 + 2 \beta ) q^{23} + ( -2 + \beta ) q^{24} + ( 2 - 2 \beta ) q^{26} + ( -8 + 8 \beta ) q^{27} + ( -2 + 2 \beta ) q^{29} + 2 \beta q^{31} + q^{32} -2 \beta q^{33} + ( -4 - \beta ) q^{34} + ( 3 - 4 \beta ) q^{36} + ( 2 - 4 \beta ) q^{37} + ( 2 + \beta ) q^{38} + ( -8 + 6 \beta ) q^{39} + ( 4 + 5 \beta ) q^{41} + ( 6 - 2 \beta ) q^{43} + ( 2 + 2 \beta ) q^{44} + ( 4 + 2 \beta ) q^{46} + ( -8 - 2 \beta ) q^{47} + ( -2 + \beta ) q^{48} + ( 6 - 2 \beta ) q^{51} + ( 2 - 2 \beta ) q^{52} + ( 2 + 6 \beta ) q^{53} + ( -8 + 8 \beta ) q^{54} -2 q^{57} + ( -2 + 2 \beta ) q^{58} + ( 10 + \beta ) q^{59} + ( -2 - 8 \beta ) q^{61} + 2 \beta q^{62} + q^{64} -2 \beta q^{66} + ( 4 + 4 \beta ) q^{67} + ( -4 - \beta ) q^{68} -4 q^{69} + ( 4 + 6 \beta ) q^{71} + ( 3 - 4 \beta ) q^{72} + ( 8 - \beta ) q^{73} + ( 2 - 4 \beta ) q^{74} + ( 2 + \beta ) q^{76} + ( -8 + 6 \beta ) q^{78} + ( -4 + 2 \beta ) q^{79} + ( 23 - 12 \beta ) q^{81} + ( 4 + 5 \beta ) q^{82} + ( -2 - 3 \beta ) q^{83} + ( 6 - 2 \beta ) q^{86} + ( 8 - 6 \beta ) q^{87} + ( 2 + 2 \beta ) q^{88} -9 \beta q^{89} + ( 4 + 2 \beta ) q^{92} + ( 4 - 4 \beta ) q^{93} + ( -8 - 2 \beta ) q^{94} + ( -2 + \beta ) q^{96} + ( 12 + 3 \beta ) q^{97} + ( -10 - 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 4q^{3} + 2q^{4} - 4q^{6} + 2q^{8} + 6q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 4q^{3} + 2q^{4} - 4q^{6} + 2q^{8} + 6q^{9} + 4q^{11} - 4q^{12} + 4q^{13} + 2q^{16} - 8q^{17} + 6q^{18} + 4q^{19} + 4q^{22} + 8q^{23} - 4q^{24} + 4q^{26} - 16q^{27} - 4q^{29} + 2q^{32} - 8q^{34} + 6q^{36} + 4q^{37} + 4q^{38} - 16q^{39} + 8q^{41} + 12q^{43} + 4q^{44} + 8q^{46} - 16q^{47} - 4q^{48} + 12q^{51} + 4q^{52} + 4q^{53} - 16q^{54} - 4q^{57} - 4q^{58} + 20q^{59} - 4q^{61} + 2q^{64} + 8q^{67} - 8q^{68} - 8q^{69} + 8q^{71} + 6q^{72} + 16q^{73} + 4q^{74} + 4q^{76} - 16q^{78} - 8q^{79} + 46q^{81} + 8q^{82} - 4q^{83} + 12q^{86} + 16q^{87} + 4q^{88} + 8q^{92} + 8q^{93} - 16q^{94} - 4q^{96} + 24q^{97} - 20q^{99} + O(q^{100}) \)

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 −3.41421 1.00000 0 −3.41421 0 1.00000 8.65685 0
1.2 1.00000 −0.585786 1.00000 0 −0.585786 0 1.00000 −2.65685 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

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 2450.2.a.bn 2
5.b even 2 1 490.2.a.m yes 2
5.c odd 4 2 2450.2.c.t 4
7.b odd 2 1 2450.2.a.bs 2
15.d odd 2 1 4410.2.a.bt 2
20.d odd 2 1 3920.2.a.bm 2
35.c odd 2 1 490.2.a.l 2
35.f even 4 2 2450.2.c.w 4
35.i odd 6 2 490.2.e.j 4
35.j even 6 2 490.2.e.i 4
105.g even 2 1 4410.2.a.by 2
140.c even 2 1 3920.2.a.ca 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.a.l 2 35.c odd 2 1
490.2.a.m yes 2 5.b even 2 1
490.2.e.i 4 35.j even 6 2
490.2.e.j 4 35.i odd 6 2
2450.2.a.bn 2 1.a even 1 1 trivial
2450.2.a.bs 2 7.b odd 2 1
2450.2.c.t 4 5.c odd 4 2
2450.2.c.w 4 35.f even 4 2
3920.2.a.bm 2 20.d odd 2 1
3920.2.a.ca 2 140.c even 2 1
4410.2.a.bt 2 15.d odd 2 1
4410.2.a.by 2 105.g even 2 1

Hecke kernels

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

\( T_{3}^{2} + 4 T_{3} + 2 \)
\( T_{11}^{2} - 4 T_{11} - 4 \)
\( T_{13}^{2} - 4 T_{13} - 4 \)
\( T_{17}^{2} + 8 T_{17} + 14 \)
\( T_{19}^{2} - 4 T_{19} + 2 \)
\( T_{23}^{2} - 8 T_{23} + 8 \)
\( T_{37}^{2} - 4 T_{37} - 28 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( 2 + 4 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( -4 - 4 T + T^{2} \)
$13$ \( -4 - 4 T + T^{2} \)
$17$ \( 14 + 8 T + T^{2} \)
$19$ \( 2 - 4 T + T^{2} \)
$23$ \( 8 - 8 T + T^{2} \)
$29$ \( -4 + 4 T + T^{2} \)
$31$ \( -8 + T^{2} \)
$37$ \( -28 - 4 T + T^{2} \)
$41$ \( -34 - 8 T + T^{2} \)
$43$ \( 28 - 12 T + T^{2} \)
$47$ \( 56 + 16 T + T^{2} \)
$53$ \( -68 - 4 T + T^{2} \)
$59$ \( 98 - 20 T + T^{2} \)
$61$ \( -124 + 4 T + T^{2} \)
$67$ \( -16 - 8 T + T^{2} \)
$71$ \( -56 - 8 T + T^{2} \)
$73$ \( 62 - 16 T + T^{2} \)
$79$ \( 8 + 8 T + T^{2} \)
$83$ \( -14 + 4 T + T^{2} \)
$89$ \( -162 + T^{2} \)
$97$ \( 126 - 24 T + T^{2} \)
show more
show less