# 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 $2$

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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: $$\Q(\sqrt{22})$$ 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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