Properties

Label 2450.4.a.bs
Level $2450$
Weight $4$
Character orbit 2450.a
Self dual yes
Analytic conductor $144.555$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(144.554679514\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{22}) \)
Defining polynomial: \(x^{2} - 22\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 \(\beta = 2\sqrt{22}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 q^{2} + \beta q^{3} + 4 q^{4} -2 \beta q^{6} -8 q^{8} + 61 q^{9} +O(q^{10})\) \( q -2 q^{2} + \beta q^{3} + 4 q^{4} -2 \beta q^{6} -8 q^{8} + 61 q^{9} + 20 q^{11} + 4 \beta q^{12} -7 \beta q^{13} + 16 q^{16} -6 \beta q^{17} -122 q^{18} + \beta q^{19} -40 q^{22} -48 q^{23} -8 \beta q^{24} + 14 \beta q^{26} + 34 \beta q^{27} -166 q^{29} -22 \beta q^{31} -32 q^{32} + 20 \beta q^{33} + 12 \beta q^{34} + 244 q^{36} + 78 q^{37} -2 \beta q^{38} -616 q^{39} + 42 \beta q^{41} -436 q^{43} + 80 q^{44} + 96 q^{46} -22 \beta q^{47} + 16 \beta q^{48} -528 q^{51} -28 \beta q^{52} -62 q^{53} -68 \beta q^{54} + 88 q^{57} + 332 q^{58} -71 \beta q^{59} + 29 \beta q^{61} + 44 \beta q^{62} + 64 q^{64} -40 \beta q^{66} -580 q^{67} -24 \beta q^{68} -48 \beta q^{69} -544 q^{71} -488 q^{72} + 64 \beta q^{73} -156 q^{74} + 4 \beta q^{76} + 1232 q^{78} -680 q^{79} + 1345 q^{81} -84 \beta q^{82} -21 \beta q^{83} + 872 q^{86} -166 \beta q^{87} -160 q^{88} -160 \beta q^{89} -192 q^{92} -1936 q^{93} + 44 \beta q^{94} -32 \beta q^{96} + 70 \beta q^{97} + 1220 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{2} + 8q^{4} - 16q^{8} + 122q^{9} + O(q^{10}) \) \( 2q - 4q^{2} + 8q^{4} - 16q^{8} + 122q^{9} + 40q^{11} + 32q^{16} - 244q^{18} - 80q^{22} - 96q^{23} - 332q^{29} - 64q^{32} + 488q^{36} + 156q^{37} - 1232q^{39} - 872q^{43} + 160q^{44} + 192q^{46} - 1056q^{51} - 124q^{53} + 176q^{57} + 664q^{58} + 128q^{64} - 1160q^{67} - 1088q^{71} - 976q^{72} - 312q^{74} + 2464q^{78} - 1360q^{79} + 2690q^{81} + 1744q^{86} - 320q^{88} - 384q^{92} - 3872q^{93} + 2440q^{99} + O(q^{100}) \)

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
−4.69042
4.69042
−2.00000 −9.38083 4.00000 0 18.7617 0 −8.00000 61.0000 0
1.2 −2.00000 9.38083 4.00000 0 −18.7617 0 −8.00000 61.0000 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 2450.4.a.bs 2
5.b even 2 1 98.4.a.h 2
7.b odd 2 1 inner 2450.4.a.bs 2
15.d odd 2 1 882.4.a.w 2
20.d odd 2 1 784.4.a.z 2
35.c odd 2 1 98.4.a.h 2
35.i odd 6 2 98.4.c.g 4
35.j even 6 2 98.4.c.g 4
105.g even 2 1 882.4.a.w 2
105.o odd 6 2 882.4.g.bi 4
105.p even 6 2 882.4.g.bi 4
140.c even 2 1 784.4.a.z 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.4.a.h 2 5.b even 2 1
98.4.a.h 2 35.c odd 2 1
98.4.c.g 4 35.i odd 6 2
98.4.c.g 4 35.j even 6 2
784.4.a.z 2 20.d odd 2 1
784.4.a.z 2 140.c even 2 1
882.4.a.w 2 15.d odd 2 1
882.4.a.w 2 105.g even 2 1
882.4.g.bi 4 105.o odd 6 2
882.4.g.bi 4 105.p even 6 2
2450.4.a.bs 2 1.a even 1 1 trivial
2450.4.a.bs 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2450))\):

\( T_{3}^{2} - 88 \)
\( T_{11} - 20 \)
\( T_{19}^{2} - 88 \)
\( T_{23} + 48 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + T )^{2} \)
$3$ \( -88 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -20 + T )^{2} \)
$13$ \( -4312 + T^{2} \)
$17$ \( -3168 + T^{2} \)
$19$ \( -88 + T^{2} \)
$23$ \( ( 48 + T )^{2} \)
$29$ \( ( 166 + T )^{2} \)
$31$ \( -42592 + T^{2} \)
$37$ \( ( -78 + T )^{2} \)
$41$ \( -155232 + T^{2} \)
$43$ \( ( 436 + T )^{2} \)
$47$ \( -42592 + T^{2} \)
$53$ \( ( 62 + T )^{2} \)
$59$ \( -443608 + T^{2} \)
$61$ \( -74008 + T^{2} \)
$67$ \( ( 580 + T )^{2} \)
$71$ \( ( 544 + T )^{2} \)
$73$ \( -360448 + T^{2} \)
$79$ \( ( 680 + T )^{2} \)
$83$ \( -38808 + T^{2} \)
$89$ \( -2252800 + T^{2} \)
$97$ \( -431200 + T^{2} \)
show more
show less