Properties

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

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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: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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.

\(f(q)\) \(=\) \( q + 2 q^{2} + 5 \beta q^{3} + 4 q^{4} + 10 \beta q^{6} + 8 q^{8} + 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 5 \beta q^{3} + 4 q^{4} + 10 \beta q^{6} + 8 q^{8} + 23 q^{9} - 14 q^{11} + 20 \beta q^{12} - 36 \beta q^{13} + 16 q^{16} - \beta q^{17} + 46 q^{18} - \beta q^{19} - 28 q^{22} - 140 q^{23} + 40 \beta q^{24} - 72 \beta q^{26} - 20 \beta q^{27} - 286 q^{29} - 66 \beta q^{31} + 32 q^{32} - 70 \beta q^{33} - 2 \beta q^{34} + 92 q^{36} + 38 q^{37} - 2 \beta q^{38} - 360 q^{39} - 89 \beta q^{41} + 34 q^{43} - 56 q^{44} - 280 q^{46} - 370 \beta q^{47} + 80 \beta q^{48} - 10 q^{51} - 144 \beta q^{52} + 74 q^{53} - 40 \beta q^{54} - 10 q^{57} - 572 q^{58} + 307 \beta q^{59} + 10 \beta q^{61} - 132 \beta q^{62} + 64 q^{64} - 140 \beta q^{66} - 684 q^{67} - 4 \beta q^{68} - 700 \beta q^{69} + 588 q^{71} + 184 q^{72} + 191 \beta q^{73} + 76 q^{74} - 4 \beta q^{76} - 720 q^{78} + 1220 q^{79} - 821 q^{81} - 178 \beta q^{82} - 299 \beta q^{83} + 68 q^{86} - 1430 \beta q^{87} - 112 q^{88} + 437 \beta q^{89} - 560 q^{92} - 660 q^{93} - 740 \beta q^{94} + 160 \beta q^{96} - 1049 \beta q^{97} - 322 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 8 q^{4} + 16 q^{8} + 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} + 8 q^{4} + 16 q^{8} + 46 q^{9} - 28 q^{11} + 32 q^{16} + 92 q^{18} - 56 q^{22} - 280 q^{23} - 572 q^{29} + 64 q^{32} + 184 q^{36} + 76 q^{37} - 720 q^{39} + 68 q^{43} - 112 q^{44} - 560 q^{46} - 20 q^{51} + 148 q^{53} - 20 q^{57} - 1144 q^{58} + 128 q^{64} - 1368 q^{67} + 1176 q^{71} + 368 q^{72} + 152 q^{74} - 1440 q^{78} + 2440 q^{79} - 1642 q^{81} + 136 q^{86} - 224 q^{88} - 1120 q^{92} - 1320 q^{93} - 644 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
2.00000 −7.07107 4.00000 0 −14.1421 0 8.00000 23.0000 0
1.2 2.00000 7.07107 4.00000 0 14.1421 0 8.00000 23.0000 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.bx 2
5.b even 2 1 98.4.a.g 2
7.b odd 2 1 inner 2450.4.a.bx 2
15.d odd 2 1 882.4.a.bg 2
20.d odd 2 1 784.4.a.y 2
35.c odd 2 1 98.4.a.g 2
35.i odd 6 2 98.4.c.h 4
35.j even 6 2 98.4.c.h 4
105.g even 2 1 882.4.a.bg 2
105.o odd 6 2 882.4.g.ba 4
105.p even 6 2 882.4.g.ba 4
140.c even 2 1 784.4.a.y 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.4.a.g 2 5.b even 2 1
98.4.a.g 2 35.c odd 2 1
98.4.c.h 4 35.i odd 6 2
98.4.c.h 4 35.j even 6 2
784.4.a.y 2 20.d odd 2 1
784.4.a.y 2 140.c even 2 1
882.4.a.bg 2 15.d odd 2 1
882.4.a.bg 2 105.g even 2 1
882.4.g.ba 4 105.o odd 6 2
882.4.g.ba 4 105.p even 6 2
2450.4.a.bx 2 1.a even 1 1 trivial
2450.4.a.bx 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2450))\):

\( T_{3}^{2} - 50 \) Copy content Toggle raw display
\( T_{11} + 14 \) Copy content Toggle raw display
\( T_{19}^{2} - 2 \) Copy content Toggle raw display
\( T_{23} + 140 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 50 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 14)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2592 \) Copy content Toggle raw display
$17$ \( T^{2} - 2 \) Copy content Toggle raw display
$19$ \( T^{2} - 2 \) Copy content Toggle raw display
$23$ \( (T + 140)^{2} \) Copy content Toggle raw display
$29$ \( (T + 286)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 8712 \) Copy content Toggle raw display
$37$ \( (T - 38)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 15842 \) Copy content Toggle raw display
$43$ \( (T - 34)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 273800 \) Copy content Toggle raw display
$53$ \( (T - 74)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 188498 \) Copy content Toggle raw display
$61$ \( T^{2} - 200 \) Copy content Toggle raw display
$67$ \( (T + 684)^{2} \) Copy content Toggle raw display
$71$ \( (T - 588)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 72962 \) Copy content Toggle raw display
$79$ \( (T - 1220)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 178802 \) Copy content Toggle raw display
$89$ \( T^{2} - 381938 \) Copy content Toggle raw display
$97$ \( T^{2} - 2200802 \) Copy content Toggle raw display
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